Communications and Control Engineering Series
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CCES published titles include;
Sampled-Data Control Systems
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Interactive System Identification
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Alberto Isidori
Nonlinear Control
Systems
Third Edition
With 47 Figures
Springer
Professor Alberto Isidori
Dipartimento di Informatica e Sistemistica,
Universita di Roma “La Sapienza”, Via Eudossiana 18, 00184 Roma, Italy
Department of Systems Science and Mathematics,
Washington University, 1 Brookings Drive, St Louis, MO 63130, USA
ISBN 3-540-19916-0 3rd edition Springer-Verlag Berlin Heidelberg New York
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Isidori, Alberto
Nonlinear Control Systems. - 3Rev.ed. -
(Communications & Control Engineering
Series)
I. Title II. Series
629.836
ISBN 3-540-19916-0
Library of Congress Cataloging-in-Publication Data
Isidori, Alberto
Nonlinear control systems/Alberto Isidori. - 3rd ed.
p. cm. - (Communications and control engineering series)
Includes bibliographical references and index.
ISBN 3-540-19916-0 (hard cover)
1. Feedback control systems. 2. Nonlinear control theory.
3. Geometry, Differential. I. Title. II. Series.
QA402.3.I74 1995	95-14976
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preface to the second edition
The purpose of this book i> to present a self-contained description of the fun-
damentals of the theory of nonlinear control systems, with special emphasis
on the differential geometric approach. The book is intended as a graduate
text as well as a reference to scientists and engineers involved in the analysis
and design of feedback systems.
The first version of this book was written in 1983. while I was teach-
ing at the Department, of Systems Science and Mathematics at Washington
University in St. Louis. This new edition integrates my subsequent teaching
experience gained at the University of Illinois in Urbana-Champaign in 1987.
at the Carl-Cranz Gesellschaft in Oberpfaffenhofen in 1987. at the University
of California in Berkeley in 1988. In addition to a major rearrangement of
the last two Chapters of the first version, this new edition incorporates two
additional Chapters at a more elementary level and an exposition of some
relevant research findings which have occurred since 1985.
In the past few years differential geometry lias proved to be an effective
means of analysis and design of nonlinear control systems as it was in the
past for the Laplace transform, complex variable theory and linear algebra
in relation to linear systems. Synthesis problems of longstanding interest like
disturbance decoupling, noninteracting control, output regulation, and the
shaping of the input-output response, can be dealt, with relative ease, on
the basis of mathematical concepts that can be easily acquired by a control
scientist. The objective of this text is to render the reader familiar with
major methods and results, and enable him to follow the new significant
developments in the constantly expanding literature.
The book is organized as follows. Chapter 1 introduces invariant, dis-
tributions. a fundamental tool in the analysis of the internal structure of
nonlinear systems. With the aid of this concept, it is shown that a non-
linear system locally exhibits decompositions into "reachable/unreachable”
parts and/or "observable/unobservable" parts, similar to those introduced
by Kalman for linear systems. Chapter 2 explains to what extent global
decompositions may exist, corresponding to a partition of the whole state
space into "lower dimensional"' reachability and/or indistinguishability sub-
sets. Chapter 3 describes various "formats" in which the in put-output map
of a nonlinear system may be represented, and provides a short description
viii
of the fundamentals of realization theory. Chapter 4 illustrates how a series
of relevant design problems can be solved for a single-input single-output
nonlinear system. It explains how a svstem can be transformed into a linear
and controllable one by means of feedback and coordinates transformation,
discusses how the nonlinear analogue of the notion of ‘‘zero' plays an im-
portant role in the problem of achieving local asymptotic stability, describes
the problems of asymptotic tracking, model matching and disturbance decou-
pling. The approach is somehow -‘elementary", in that requires only standard
mathematical tools. Chapter 5 covers similar subjects for a special class of
multivariable nonlinear systems, namely those systems which can be rendered
non interactive by means of static state feedback. For this class of systems,
the analysis is a rather straightforward extension of the one illustrated in
Chapter 4. Finally, the last two Chapters are devoted to the solution of the
problems of output regulation, disturbance decoupling, noninteracting con-
trol with stability via static state feedback, and noninteracting control via
dynamic feedback, for a broader class of multivariable nonlinear systems. The
analysis in these Chapters is mostly based on a number of key differential
geometric concepts that, for convenience, are treated separately in Chapter
6.
It has not been possible to include all the most recent developments in this
area. Significant omissions are, for instance: the theory of global linearization
and global controlled invariance, the notions of left- and r ight-in vert i bilit y
and their control theoretic consequences. The bibliography, which is by no
means complete, includes those publications which were actually used and
several -works of interest for additional investigation.
The reader should be familiar with the basic concepts of linear system
theory. Although the emphasis of the book is on the application of differential
geometric concepts to control theory, most of Chapters 1. 4 and 5 do not
require a specific background in this field- The reader who is not familiar with
the fundamentals of differential geometry may skip Chapters 2 and 3 in a first
approach to the book, and then come back to these after having acquired the
necessary skill. In order to make the volume as self-contained as possible,
the most important concepts of differential geometry used throughout the
book are described without proof—in Appendix A. In the exposition of each
design problem, the issue of local asymptotic stability is also discussed. This
also presupposes a basic knowledge of stability theory, for which the reader is
referred to well-known standard reference books. Some specific results which
are not frequently found in these references are included in Appendix B.
I wish to express my sincerest gratitude to Professor A, Ruberti, for his
constant encouragement, to Professors J. Zaborszky, P. Kokotovic. J. Acker-
mann, C-A. Desoer who offered me the opportunity to teach the subject of
this book in their academic institutions, and to Professor M. Thoma for his
continuing interest in the preparation of this book. I am indebted to Pro-
fessor A.J. Kroner from whom—in the course of a joint research venture I
IX
learned many of the methodologies which have been applied in the book. I
wish to thank Professor C.I. Byrnes, with whom I recently shared intensive
research activity and Professors T.J. Tarn, J.W. Grizzle and S.S. Sastry with
whom I had the opportunity to cooperate on a number of relevant research
issues. I also would like to thank Professors M. Fliess. S- Monaco and M.D.
Di Benedetto for their valuable advice.
Rome. March 1989
Alberto Isidori

Preface to the third edition
In the last six years, feedback design for nonlinear systems has experienced
a growing popularity and many issues of major interest, which at the time
of the preparation of the second edition of this book were still open, have
been successfully addressed. The purpose of this third edition is to describe
a few significant new findings as well as to streamline and improve some of
the earlier passages.
Chapters from 1 to 4 art1 unchanged. Chapter 5 now includes also the
discussion of the problem of achieving relative degree via dynamic extension,
which in the second edition was presented in Chapter 7 (former sections
7.5 and 7.6). The presentation is now based on a new "canonical" dynamic
extension algorithm, which proves itself very convenient from a number of
different viewpoints. Chapter 6 is also unchanged, with the only exception of
the proof of the main result of section 6.2. namely the construction of feedback
laws rendering invariant a given distribution, which has been substantially
simplified due to a valuable suggestion of C.Scherer. Chapter 7 no longer
includes the subject of tracking and regulation (former section 7.2) which
has been expanded and moved to a separate new Chapter and. as explained
before, the discussion of how to obtain relative degree via dynamic extension.
It includes, on the other hand, a rather detailed exposition of the subject of
noninteracting control with stability via dynamic feedback, which was not
covered in the second edition.
Chapters 8 and 9 are new. The first one of these covers the subject of
tracking and regulation, in a improved exposition which very easily leads to
the solution of the problem of how to obtain a "structurally stable"1 design.
The last Chapter deals with the design of feedback laws to the purpose of
achieving global or "scmiglobal" stability as well as global disturbance atten-
uation. This particular area has been the subject of major research efforts in
the last years. Among the several and indeed outstanding progresses in this
domain, Chapter 9 concentrates only on those contributions whose develop-
ment seems to have been particularly influenced by concepts and methods
presented in the earlier Chapters of the book. The bibliography of the sec-
ond edition has been updated only with those references which were actually
used in the preparation of the new materiah namely sections 5.4. 7.4; 7.5 and
Chapters 8 and 9.
xii
I wish to express my sincere gratitude to all colleagues who have kindly
expressed comments and advice on the earlier versions of the book. In par-
ticular. I wish to thank Prof. Ying-Keh Wu, Prof. M.Zeitz and Dr. U. Knopp
for their valuable suggestions and careful help.
St.Louis. December 1994
Alberto Isidori
Table of Contents
1.	Local Decompositions of Control Systems........................ 1
1.1	Introduction.............................................. 1
1.2	Notations ................................................ 5
1.3	Distributions............................................ 13
1.4	Frobenius Theorem........................................ 22
1.5	The Differential Geometric Point of View................. 33
1.6	Invariant Distributions.................................. 41
1.7	Local Decompositions of Control Systems.................. 49
1.8	Local Reachability....................................... 53
1.9	Local Observability...................................... 69
2.	Global Decompositions of Control Systems...................... 77
2.1	Sussmann's Theorem and Global Decompositions............. 77
2.2	The Control Lie Algebra ................................. 83
2.3	The Observation Space.................................... 87
2.4	Linear Systems and Bilinear Systems ..................... 91
2.5	Examples................................................. 99
3.	Input-Output Maps and Realization Theory......................105
3.1	Fliess Functional Expansions.............................105
3.2	Volterra Series Expansions...............................112
3.3	Output Invariance........................................116
3.4	Realization Theory.......................................121
3.5	Uniqueness of Minimal Realizations.......................132
4.	Elementary Theory of Nonlinear Feedback for Single-Input
Single-Output Systems .............................................137
4.1	Local Coordinates Transformations........................137
4.2	Exact Linearization Via Feedback ........................147
4.3	The Zero Dynamics........................................162
4.4	Local Asymptotic Stabilization...........................172
4.5	Asymptotic Output Tracking ..............................178
4.6	Disturbance Decoupling...................................184
4.7	High Gain Feedback.......................................189
Table of Contents
4.8	Additional Results on Exact Linearization.................194
4.9	Observers with Linear Error Dynamics......................203
4.10	Examples.................................................211
5.	Elementary Theory of Nonlinear Feedback for Multi-Input
Multi-Output Systems................................................219
5.1	Local Coordinates Transformations...............................219
5.2	Exact Linearization via Feedback................................227
5.3	Noninteracting Control..........................................241
5.4	Achieving Relative Degree via Dynamic Extension.................249
5.5	Examples........................................................263
5.6	Exact Linearization of the Input-Output Response................277
6.	Geometric Theory of State Feedback: Tools............................293
6.1	The Zero Dynamics...............................................293
6.2	Controlled Invariant Distributions..............................312
6.3	The Maximal Controlled Invariant Distribution in ker(cM) ... 317
6.4	Controllability Distributions.............................333
7.	Geometric Theory of Nonlinear Systems: Applications .... 339
7.1	Asymptotic Stabilization via State Feedback...............339
7.2	Disturbance Decoupling....................................342
7.3	Noninteracting Control with Stability via Static Feedback	..	.	344
7.4	Non interacting Control with Stability. Necessary Conditions	.	364
7.5	Noninteracting Control with Stability. Sufficient Conditions.	.	373
8.	Tracking and Regulation..............................................387
8.1	The Steady State Response in a Nonlinear System...........387
8.2	The Problem of Output Regulation..........................391
8.3	Output Regulation in the Case of Full Information.........396
8.4	Output Regulation in the Caj/e of Error Feedback ...............403
8.5	Structurally Stable Regulation .................................416
9.	Global Feedback Design for Single-Input Single-Output Sys-
tems ............................................................427
9.1	Global Normal Forms .........................................427
9.2	Examples of Global Asymptotic	Stabilization .................432
9.3	Examples of Semiglobal Stabilization.........................439
9.4	Artstem-Sontag;s Theorem ....................................448
9.5	Examples of Global Disturbance Attenuation ..................450
9.6	Semiglobal Stabilization by Output Feedback..................460
Table of Contents xv
A. Appendix A..................................................471
A.l	Some Facts from Advanced	Calculus.............471
A.2	Some Elementary Notions; of Topology..................473
A.3	Smooth Manifolds......................................474
A.4	Submanifolds .........................................479
A.5	Tangent Vectors.......................................483
A.6	Vector Fields ........................................493
B.	Appendix В ................................................503
B.l	Center Manifold Theory................................503
B.2	Some Useful Properties................................511
B.3	Local Geometric Theory	of	Singular Perturbations......517
Bibliographical Notes............................,.............529
References.................................................... 535
Index......................................................... 545

1.	Local Decompositions of Control Systems
1.1 Introduction
The subject of this Chapter is the analysis of a nonlinear control system,
from the point of view of the interaction between input and state and - re-
spectively between state and output, with the aim of establishing a number
of interesting analogies with some fundamental features of linear control sys-
tems. For convenience, and in order to set up an appropriate basis for the
discussion of these analogies, we begin by reviewing -- perhaps in a slightly
unusual perspective - a few basic facts about the theory of linear systems.
Recall that a linear multivariable control system with m inputs and p
outputs is usually described, in state space form, by means of a set of first
order linear differential equations
in which .r denotes the state vector (an element of HF1), и the input vector
(an element of B"') and у the output vector (an element of 3J'). The matrices
A,B.C are matrices of real numbers, of proper dimensions.
The analysis of the interaction between input and state, on one hand,
and between state and output, on the other hand, has proved of fundamen-
tal importance in understanding the possibility of solving a large number
of relevant control problems, including eigenvalues assignment via feedback,
minimization of quadratic cost criteria, disturbance rejection, asymptotic out-
put regulation, etc. Key tools for the analysis of such interactions - intro-
duced by Kalman around the 1960 are the notions of reachability and ob-
servability and the corresponding decompositions of the control system into
•lreachable/unreachable” and. respectively, ’‘observable/unobservable” parts.
We review in this section some relevant aspects of these decompositions.
Consider the linear system (1.1), and suppose that there exists a d-
subspace V of HF1 having the following property:
(i)	V is invariant under .4. i. e. is such that .4.r £ V for all т £ V.
Without loss of generality, after possibly a change of coordinates, we
can assume that the subspace V is the set of vectors having the form
г = соЦс]...., t\i. 0.... .0). i.e. of all vectors whose last n - d components
2
1. Local Decompositions of Control Systems
are zero. If this is the case, then, because of the invariance of V under A, this
matrix assumes necessarily a block triangular structure
. _ / -4n ^12 A
4 “	0	A -2 -2 J
with zero entries on the lower-left block of n — d rows and d columns.
Moreover, if the subspace I' is such that:
(ii)	Г contains the image (i.e. the range-space) of the matrix B. i.e. is such
that Ви G V for all «6
then, after the same change of coordinates, the matrix В assumes the form
i.e. has zero entries on the last n — d rows.
Thus, if there exists a subspace V which satisfies (i) and (ii). after a
change of coordinates in the state space, the first equation of (1.1) can be
decomposed in the form
Tl = AnTi + .412^2 + B]_U
. j2 — -422*1’2 
By Tj and z-2 we denote here the vectors formed by taking the first d and.
respectively, the last n — d new coordinates of a point j.
The representation thus obtained is particularly interesting when studying
the behavior of the system under the action of the control u. At any time T.
the coordinates of t(T) are
jq(T)= exp(.4j i T)zi (0) + / exp(.4n (T - т))А12ехр(.422т) dTT2(0)+
./o f
fT
+ exp{Au(T - r))Blu(T)dT
x2(T) - exp(-4*22Т)т2(0) .
From this, we see that the set of coordinates denoted by does not
depend on the input и but only on the time T. In particular, if we denote by
tc(T) the point of Ж" reached at time t — T when u(i) = 0 for all t € [0. Т].
i.e. the point
t°(T) = ехр(АТ)т(0)
we observe that any state which can be reached at time T. starting from z(0)
at time t — 0. has necessarily the form т°(Т) + г. where v is an element of
V.
This argument identifies only a necessary condition for a state x to be
reachable at time T, i.e. that of being of the form x — x°(T) + v, with с € V.
However, under the additional assumption that:
1.1 Introduction
3
(iii)	V is the smallest subspace which satisfies (i) and (ii) (i.e. is contained
in any other subspace of which satisfies both (i) and (ii)),
then this condition is also sufficient. As a matter of fact, it is known from the
theory of linear systems that (iii) occurs if and only if
U = Im(B AB ... An~lB)
(where Imb) denotes the image of a matrix) and, moreover, that under this
assumption the pair (Au.BJ is a reachable pair. i.e. satisfies the condition
rankfBi AuBi ... А*~1Ву) =d
or. what, is the same, has the property that for each zj e there exists an
input u. defined on [0. Т]. satisfying
zi = / ехр(Ац(Т - т))Вуи(т) dr .
Then, if V is such that the condition (iii) is also satisfied, starting from
z(0) it is possible to reach at time T every state of the form z°(T) + v. with
г € I .
This analysis suggests the following considerations. Given the linear con-
trol system (1.1), let Г be the smallest subspace of satisfying (i) and (ii).
Associated with I' there is a partition of Ж” into subsets of the form
Sp =- {.r e Rr' : z — p + r. c G I'}
characterized by the following property: the set of points reachable at time
T starting from z(0) coincides exactly with the element - of the partition
- which contains the point exp(.4T)z(0). i.e. with the subset 5exp, дплчО)-
Note also that these sets. i.e. the elements of this partition, are d-dimensional
planes parallel to V (see Fig.1.1).
Fig. 1.1.
4
1. Local Decompositions of Control Systems
An analysis similar to the one developed so far can be carried out by
examining the interaction between state and output. In this case one considers
instead a d-d intension al subspace IT of R'1 characterized by the following
properties:
(i)	П' is invariant under A
(ii)	IГ is contained in the kernel (the null-space) of the matrix C (i.e. is such
that Cx = 0 for all iGlf)
(iii)	И' is the largest subspace which satisfies (i) and (ii) (i.e. contains any
other subspace of Ж" which satisfies both (i) and (ii)).
Properties (i) and (ii) imply the existence of a change of coordinates in
the state space which induces on the control system (1.1) a decomposition of
the form
Z1 =	,4ц Г j + Al? J>2 + Bill
if 2	— *4224’2 H- B'zU
У = C.X?
(in the new coordinates, the elements of IT are the points having .r2 — 0).
This decomposition shows that the set of coordinates denoted by jq has no
influence on the output y. Thus, any two initial states whose coordinates
are equal, produce identical outputs under any input, i.e. are indistinguish-
able. Actually, since two states whose .r2 coordinates that are equal are such
that their difference is an element of IT. we deduce that any two states whose
difference is an element of IT are indeed indistinguishable.
Condition (iii). in turns, guarantees that only the pairs of states charac-
terized in this way (i.e. having a difference in IT) are indistinguishable from
each other. As a matter of fact, it is known from the linear theory that the
condition (iii) is satisfied if and only if
C
! C -
П’= ker
\C.4'; 1 /
(where ker(-) denotes the kernel of a matrix) and, if this is the case, the pair
(C2.A22) is an observable pair, i.e. satisfies the condition
or, what is the same, has the property that
С? схр(А22^)а,2 — 0 for all t > 0 => _r2 = 0 .
1.2 Notations
5
As a consequence, any two initial states whose difference does not belong
to IV are distinguishable from each other, in particular by means of the
output produced under zero input.
Again, we may synthesize the above discussion with the following con-
siderations. Given a linear control system, let IV be the largest subspace of
satisfying (i) and (ii). Associated with IV there is a partition of R'! into
subsets of the form
Sp = {r € Rri : x = p + w, w G IV}
characterized by the following property: the set of points indistinguishable
from a point p coincides exactly with the element - of the partition - which
contains p, i.e. with the set Sp itself. Note that again these sets - as in the
previous analysis - are planes parallel to IV.
In the following sections of this Chapter and in the following Chapter we
shall deduce similar decompositions for nonlinear control systems.
1.2 Notations
Throughout these notes we shall study multivariable nonlinear control sys-
tems with m inputs ui,.... um and p outputs t/i. - - -, t/p described, in state
space form, by means of a set of equations of the following type
x
Hi
m
j-i
hf(j)	1 < i < p .
(1-2)
The state
•r = (^i....
is assumed to belong to an open set U of Rn.
The mappings	which characterize the equation (1.2) are Re-
valued mappings defined on the open set U; as usual, /(t), ^(z),... ,gm(x)
denote the values they assume at a specific point x of L~. Whenever con-
venient, these mappings may be represented in the form of n-dimensional
vectors of real-valued functions of the real variables Tj...., zn, namely
The functions ,.... hp which characterize the equation (1.2) are real-valued
functions also defined on U, and hi(-r),.... hp(x) denote the values taken at
a specific point x. Consistently with the notation (1.3), these functions may
be represented in the form
6
1. Local Decompositions of Control Systems
= ЛДт]......тг!)	(1.4}
In what follows, we assume that the mappings f.g^...........gm and the func-
tions h[....................................................hp are smooth in their arguments, i.e. that all entries of (1.3) and
(1.4) are real-valued functions of .......rt1 with continuous partial deriva-
tives of any order. Occasionally, this assumption may be replaced by the
stronger assumption that the functions in question are analytic on their do-
main of definition.
The class (1.2) describes a large number of physical systems of interest in
many engineering applications, including of course linear systems. The latter
have exactly the form (1.2). provided that /(u-) is a linear function of x. i.e.
f(r) = Ar
for some n x n matrix .4 of real numbers. gY(t)..... gm(x) are constant func-
tions of .r. i.e.
p,(.r) = b,
where bi.....btn are n x 1 vectors of real numbers, and /qf-r)......hp[x) are
again linear in .r. i.e.
hfix) = Cj2:
where cq.....cP are 1 x n (i.e. row) vectors of real numbers.
We shall encounter in the sequel many examples of physical control sys-
tems that can be modeled by equations of the form (1.2). Note that, as a
state space for (1.2). we consider a subset Г of E” rather than itself. This
limitation may correspond either to a constraint established by the equations
themselves (whose solutions may nor be free ro evolve on the whole of Ж’1) or
to a constraint specifically imposed on the input, for instance to avoid points
in the state space where some kind of "singularity" may occur. We shall be
more specific later on. Of course, in many cases one is allowed to set Г =	.
The mappings f.gi.......gm are smooth mappings assigning to each point
x of U a vector of E" . namely /(.r). (.r)......For this reason, they
are frequently referred to as smooth rector fields defined on U. In many
instances, it will be convenient to manipulate together with vector fields -
also dual objects called covector fields, which are smooth mappings assigning
to each point J’ (of a subset. U) an element of the dual space (IP/1 A.
As we will see in a moment, it is quite natural to identify smooth covector
fields (defined on a subset U of 3") with 1 x n (i.e. row) vectors of smooth
functions of .r. For. recall that the dual space Г* of a vector space V is
the set of all linear real-valued functions defined on V. The dual space of
an л-dimensional vector space is itself an 77-dimensional vector space, whose
elements are called corectors. Of course, like any linear mapping, an element
tr* of V* can be represented by means of a matrix. In particular, since it*
is a mapping from tin1 77-dimensional space I' to the 1-dimensional space IP.
this representation is a matrix consisting of one row only, i.e- a row vector.
On these grounds, one can assimilate (R?1 )* with the set of all /(-dimensional
1.2 Notations
row vectors, and describe any subspace of (л" )* as the collection of all linear
combinations of some set of n-dimensional row vectors (for instance, the rows
of some matrix having n columns). Note also that if
l'i
is the column vector representing an element of V. and
1Г — iU.’] U>2  
is the row vector representing an element of V*. the "value’' of ic* at r is
given by the product
iv v
Most of the times, as often occurring in the literature, the value of u1* at v
will be represented in the form of an inner product, writing instead
of simply ie*r.
Suppose now that ^]...., are smooth real-valued functions of the real
variables .......defined on an open subset. U of 3?’. and consider the row
vector
.г(т) = (u-’iCj'r, .... lc2(xi.............r„) ... ^„(3'1-------J-n)) 
On the grounds of the previous discussion, it is natural to interpret the latter
as a mapping (a smooth one. because the ^fs are smooth functions) assigning
to each point j of a subset U an element of the dual space (Ж1)*, i-e.
exactly the object that was identified as a covector field.
A covector field of special importance is the so-called differential, or gra-
dient. of a real-valued function Л defined on an open subset U of . This
covector field, denoted dX. is defined as the 1 x n row vector whose г-th ele-
ment is the partial derivative of Л with respect to Xi. Its value at a point x
is thus
(1-5)
Note that the right-hand side of this expression is exactly the jacobian matrix
of Л, and that the more condensed notation
is sometimes preferable. Any covector field having the form (1.5)-(1.6), i.e.
the form of the differential of some real-valued function Л. is called an exact
differential.
8
1. Local Decompositions of Control Systems
Wc describe now three types of differential operation, involving vector
fields and covector fields, that are frequently used in the analysis of nonlinear
control systems. The first type of operation involves a real-valued function Л
and a vector field /, both defined on a subset U of Ж”. From these, a new
smooth real-valued function is defined, whose value -- at each т in U is
equal to the inner product
1=1 c 1
This function is sometimes called the derivative of X along f and is often
written as L/X. In other words, by definition
?—1
at each z of U.
Of course, repeated use of this operation is possible. Thus, for instance,
by taking the derivative of Л first along a vector field f and then along a
vector field g one defines the new function
LsLfX(x) =	.
Ox
If Л is being differentiated к times along /, the notation L^X is used: in other
words, the function satisfies the recursion
ipw = -XX—
}	ox
with LQjX(x) ~ A(z).
The second type of operation involves two vector fields f and g, both
defined on an open subset L7 of 3". From these a new smooth vector field is
constructed, noted [/.<?] and defined as
at each .r in I-. In this expression
	/ Offi dgi dxY dxX	ад \	( ад ад oxn	OXi дх2			dx„
dg = dx	dg2 dg2 cDq dxf	dg2 Bx^	df = dx	9R OXi OX-2	 ££
	\ 0X1 OX-2	' J OXrt /		. dfn Ofn \ dx\ dx2	 dh J
denote the Jacobian matrices of the mappings g and f. respectively.
1.2 Notations
9
The vector field thus defined is called the Lie product (or bracket) of f
and g. Of course, repeated bracketing of a vector field g with the same vector
field f is possible. Whenever this is needed, in order to avoid a notation of
the form	<?]]]• that could generate confusion, it is preferable to
define such an operation recursively, as
adKfg(r) = [fiad^gfix)
for any A* > 1, setting ad^glx) = g(x).
The Lie product between vector fields is characterized by three basic
properties, that, are summarized in the following statement. The proof of
them is extremely easy and is left as an exercise to the reader.
Proposition 1.2.1. The Lie product of vector fields has the following prop-
erties:
(i)	ii bilinear over Л. i.e. if f\. fz-Pi- g-z are vector fields and i^.r-z real num-
bers, then
[rifi +r->f-2.gi] = r^fi.g^ + rfffz-gi]
[fi^igi + r-zg-fi = r^fi.gi] + r2[/i.t?2] ,
(ii)	is skew commutative, i.e.
lf-.9} = -[?•/] -
(iii)	satisfies the Jacobi identity, i.e.. if f.g.p are vector fields, then
If- [g-P\] + [fL [P’ /]] + [P: If$]] = 0.
The third type of operation of frequent use involves a covector field л
and a vector field f. both defined on an open subset U of Ж". This operation
produces a new covector field, noted Lfw and defined as
at each x of th where the superscript ;T“ denotes transposition. This covector
field is called the derivative of w along f.
The operations thus defined are used very frequently in the sequel. For
convenience we list in the following statement, a series of '‘rules" of major
interest, involving these operations either separately or jointly. Again, proofs
are very elementary and left to the reader.
Proposition 1.2.2. The three types of differential operations introduced so
far are such that
(i)	if a is a real-valued function, f a vector field and A a real-valued function,
then
L,yfX(x) = (£;A(j-))o(j-) .
(1.7)
10
1. Local Decompositions of Control Systems
(ii)	ifa.3 are real-valued functions and. fig vector fields, then
[(if.3g]lx) = o(.r)J(z)[/..g](j-) + (£/- (Lga{x)}3(x)f(x) .
(1.8)
(iii)	if fig are vector fields and A a real-valued function, then
LggX(x] = LfLgX(x) - LgLfX(x) .	(1.9)
(iv)	if a.3 are real-valued functions, f a vector field and a co vector field,
then
Laf3^{fi = ci(x}3(x){Lf^(x)} + 3{x){^(x}. f{x})da(x)
+ (Lf3(t))o(./%•(.r) .
(v)	if f is a vector field and Л a real-valued function, then
LfdX{x) = dL/Л(т) .	(LH)
(vi)	if fig are vector fields and л a covector field. then
Lfix.gfix) = (Lf^(x).g(x)) + {fig](x)) .	(1.Г2)
Example 1.2.1. As an exercise one can check, for instance, (1-7). By defini-
tion. one has
Л>/А(.г) =	r)) =	= (^/АИ)ПИ-
Or. as far as (1.10) is concerned,
г г ji V~' t 03^,'i	j
A	, ,d^	fi	, dJ	O,	, dfj C-	, , On
dx,	dx,	dxj	J J dx,
j=l	J	j=l	J	j=L	j=l	'
- [od(L/u,j]; Hr [odLfdfifi +	/)do],.<
To conclude the section, we illustrate another procedure of frequent use
in the analysis of nonlinear control systems, the change of coordinates in the
state space. As is well known, transforming the coordinates in the state space
is often very useful in order to highlight sonic properties of interest, like e.g.
reachability and observability, or to show how certain control problems, like
e.g. stabilization or decoupling, can be solved.
In the case of a linear system, only linear changes of coordinates are
usually considered. This corresponds to the substitution of the original state
vector r with a new vector z related to .r by a transformation of the form
c = T.r
1.2 Notations
11
where T is a nonsingular n x 11 matrix. Accordingly, the original description
of the system
T	—	Ат	+	В и
у	=	Cx
is replaced by a new description
z	=	Ac	+ Bu
у	=	Cz
in which
A = TAT-1 B = TB C = CT~'.
If the system is nonlinear, it is more meaningful to consider nonlinear
changes of coordinates. A nonlinear change of coordinates can be described
in the form
c = Ф(х)
where Ф[x) represents a IP?1-valued function of n variables, i.e.
/ Pi (z) \	/01(^1.......
ф(2.) = I	= °2(‘Г1......J”'1
\b71{x)/	\on(.ri------j,,)/
with the following properties
(1) Ф(х) is invertible, i.e. there exists a function Ф~1 (z) such that
Ф~1 (Ф(х)) = x
for all x in ??'.
(ii') Ф(х) and Ф-1(с) are both smooth mappings, i.e. have continuous partial
derivatives of any order.
A transformation of this type is called a global diffeomorphism on A". The
first of the two properties is clearly needed in order to have the possibility of
reversing the transformation and recovering the original state vector as
т = ф-’(г)
while the second one guarantees that the description of the system in the
new coordinates is still a smooth one.
Sometimes, a transformation possessing both these properties and defined
for all т is difficult to find and the properties in question are difficult to be
checked. Thus, in most cases one rather looks at transformations defined only
in a neighborhood of a given point. A transformation of this type is called a
local diffeomorphism. In order to check whether or not a given transformation
is a local diffeomorphism. the following result is very useful.
12
1. Local Decompositions of Control Systems
Proposition 1.2.3. Suppose Ф(х) is a smooth function defined on some sub-
set U of R". Suppose the jacobian matrix of Ф is nonsingular at a point
j- = x°. Then, on a suitable open subset U° of U, containing x°, ф(т) defines
a local diffeomorphism.
Example 1.2.2. Consider the function
which is defined for all	in R*. Its jacobian matrix
ЭФ _ fl 1 \
dx \0 cos z9 J
has rank 2 at x° = (0.0). On the subset
: Ы < (tr/2)}
this function defines a diffeomorphism. Note that on a larger set the function
does not anymore define a diffeomorphism because the invertibility property
is lost. For. observe that for each number x2 such that l-r^l > (тг/2) . there
exists x2 such that. |j.j| < (тг/2) and sinj2 = sinx^. Any pair t-j), (zj.
such that Ji +j>2 = .t'j + J? yields Ф(.Т1,х2) = Ф(х\. x'2) and thus the function
is not injective. <
Example 1.2.3. Consider the function
I \
( ) = = r 1
\z? f	\ - —------TN /
x 7	X Jj + 1 /
defined on the set	{
U° ~ {(ti . z2) : -fi > -1} •
This function is a diffeomorphism (onto its image), because Ф(х^,х-2) =
Ф(х\, x'2) implies necessarily = x^ and x2 = x2. However, this function is
not defined on all R2. <
The effect of a change of coordinates on the description of a nonlinear
system can be analyzed in this way. Set.
z(t) = Ф(х(1))
and differentiate both sides with respect to time. This yields
d ~ с)Ф dx с)Ф
Then, expressing as т(/) = Ф~1 (c(t)), one obtains
1,3 Distributions
13
where
i(#) =
y(t) ~
/И*)) + ё(г(0)и(0
^(2(0)
» =	o.r	
=	дФ '	
h(z) = [Ш)]^		“ЧП 
The latter are the expressions relating the new description of the system
to the original one. Note that if the system is linear, and if Ф(т) is linear as
well. i.e. if Ф(х) = Tx, then these formulas reduce to ones recalled before.
1.3 Distributions
We have observed in the previous section that a smooth vector field f. defined
on an open set U of can be intuitively interpreted as a smooth mapping
assigning the 77-dimensional vector f(x) to each point т of U. Suppose now
that (1 smooth vector fields fi,.... fd are given, all defined on the same open
set U and note that, at any fixed point x in U. the vectors /1 (.r)..Л(т)
span a vector space (a subspace of the vector space in which all the /;(x):s
are defined, i.e. a subspace of R”). Let this vector space, which depends on
j, be denoted by i-e. set
J(.r) = span{/i(.r)....
and note that, in doing this, we have essentially assigned a vector space to each
point x of the set U. Motivated by the fact that the vector fields fY..... fd
are smooth vector fields, we can regard this assignment as a smooth one.
The object thus characterized, namely the assignment - to each point x
of an open set U of IR." of the subspace spanned by the values at r of some
smooth vector fields defined on U, is called a smooth distribution. We shall
now illustrate a series of properties, concerning the notion of smooth distri-
bution. that are of fundamental importance in all the subsequent analysis.
According to the characterization just given, a distribution is identified
by a set of vector fields, say {/1..... fd}- we will use the notation
A = span{/]..... fd}
to denote the assignment as a whole, and. as before. A(t) to denote the
“value” of A at a point j.
Pointwise, a distribution is a vector space, a subspace of HP’. Based on
this fact, it is possible to extend to the notion thus introduced a number of
elementary concepts related to the notion of vector space. Thus, if -b and
14
1. Local Decompositions of Control Systems
Jo are distributions, their sum Jj + Jo is defined by taking pointwise the
sum of the subspaces Jjz) and J2(z), namely
(Jl + Jzjfz) = Jl(-T) + JiGr) .
The intersection Ji Cl Jo is defined as
(Ji П J2)(z) = JJz) П Ji>(z) •
A distribution Jj contains a distribution Jo. and is written Ji □ J2. if
Ji(z) 3 J2(z) for all .r. A vector field f belongs to a distribution J. and is
written f € J. if /(z) € J(z) for all .r. The dimension of a distribution at a
point z of f? is the dimension of the subspace J(z).
If F is a matrix having n rows and whose entries are smooth functions of
.r. its columns can be considered as smooth vector fields. Thus any matrix of
this kind identifies a smooth distribution, the one spanned by its columns.
The value of such a distribution at each z is equal to the image of the matrix
Ffz) at this point
J(z) = Im(Ffz)) .
Clearly, if a distribution J is spanned by the columns of a matrix F. the
dimension of J at a point z° is equal to the rank of F(zc).
Example 1.3.1. Let U = л?, and consider the matrix
(Zi	Z[Z2	Zi \
1 + z3 fl + z3)z2 Zi
1	^2	0 /
Note that the second column is proportional to the first one. via the coefficient
z2. Thus this matrix has at most rank 2. The first and third columns are
independent (and, accordingly, the matrix F has rank exactly equal to 2) if
Zi is nonzero. Thus, we conclude that tyhe columns of F span the distribution
characterized as follows
/	°	\
J(z) — span{ 1 + z3 I} if zi = 0
\	1	J
/	zi	\	Zl\
J(z) - span{ I 1 + z3 I . 1 j } if / 0 .
\	1	/	\	° /
The distribution has dimension 2 everywhere except on the plane Z] = 0. <
Note that, by construction, the sum of two smooth distributions is
a smooth distribution. In fact, if Ji is spanned by smooth vector fields
f i....fh and J2 is spanned by smooth vector fields gY.....gk. then Ji + J2
is spanned by /i..... fa. gi,.... Qk- However, the intersection of two smooth
distributions may fail to be smooth. This may be seen in the following exam-
ple.
1.3 Distributions
15
Example 1.3.2. Consider the two distributions defined on F?
л = span{ Q)} A2 = span^1 4‘1'Г1 )}
We have
(Ji n _*L>)0) = {()}	if Z1 0
(Ji P A2)(.r) = Aj (x) = А2(т) if = 0 .
This distribution is not smooth because it is not possible to find a smooth
vector field on E’2 which is zero everywhere but on the line Ji = 0. <
Remark 1.3.3. The previous example shows that sometimes one may en-
counter an assignment A. of a vector space А(т) to each point j of a set U,
which is not smooth, in the sense that it is not possible to find a set of smooth
vector fields {/,:/€!}. defined on U. such that A(.r) — span{/,(z) : i 6 1}
for all т in C. If this is the case, it is convenient to replace A by an appropri-
ate smooth distribution, defined on the basis of the following considerations.
Suppose J] and Д are two smooth distributions, both contained in A. The
distribution Д + A2, is still smooth and contained in A. by construction.
From this one concludes that the family of all smooth distributions contained
in -A has a unique maximal element (with respect to distributions addition),
namely the sum of all members of the family. This distribution, which is the*
largest smooth distribution contained in A. will be denoted by smt(A) and
sometimes used as substitute for the original J whenever convenient. <J
Other important concepts associated with the notion of distribution are
the ones related to the "behavior" of this object as a ‘"function" of x. We have
already seen how it is possible to characterize the quality of being smooth,
but there are other properties to be considered. A distribution A. defined on
a open set U. is nonsingitlar if there exists an integer d such that
dim( A(z)) = d
for all x in U. A singular distribution, i.e. a distribution for which the above
condition is not satisfied, is sometimes called a distribution of variable di-
mension. A point. .r° of U is said to be a regular point of a distribution A. if
there exists a neighborhood Iго of with the property that A is nonsingular
on . Each point of U which is not a regular point is said to be? a point of
singularity.
Example 1.3.4- Consider again the distribution defined in the Example 1.3.1.
The distribution in question has dimension 2 at each x such that .ri 0 and
dimension 1 at each u such that, up = 0. The plane {r 6 E3 : iq = 0} is the
set of points of singularity of A. <i
In what follows we list some properties related to these notions, whose
proofs are rather simple, and either omitted or just sketched.
16
1. Local Decompositions of Control Systems
Lemma 1.3.1. Let A be a smooth distribution and xQ a regular point of A.
Suppose dim( A(.r°)) = d. Then, there exist an open neighborhood UQ of xQ
and a set {fi....fa} of smooth vector fields defined on Uz’ with the property
that
(i) the vectors fi(x')..... fifix} are linearly independent at each x in Lro,
(ii) —= span{/i(.r).........fAT)} M each f *n t"-
Moreover, every smooth rector field т belonging to A can be expressed, on
U3. as
d
= ^2 сил(<)
;= I
where t‘i(x).....mfix) are smooth real-valued function of x. defined on L’°.
Proof. The existence of exactly d smooth vector fields spanning Л around A
is a trivial consequence of the assumptions. If r is a vector field in -1 , then
for each .r near A. the n x (d + 1) matrix
(/10) /2 CH ... fifix} t(x))
has rank d. Thus, from elementary linear algebra we deduce the representa-
tion above, and the smoothness of the entries of this matrix implies that of
the сг-(т) s. <
Lemma 1.3.2. The set of all regular points of a distribution A, defined on
U, is an open and dense subset of U.
Lemma 1.3.3. Let Ai and A? be two smooth distributions, defined on IT.
with the. property that A? is nonsingular and Afix) C A (A M each point x
of a dense subset of L:. Then -b(x) С A2(t) at each x in U. i.e. А] С
Lemma 1.3.4. Let A and A> be twb smooth distributions, defined on U.
with the property that A is nonsingular, A C A? and Afix) = at
each point x of a dense subset of . Then A = -b.
As we have seen before, the intersection of two smooth distributions may
fail to be smooth. However, around a regular point this cannot happen, as
we see from the following statement.
Lemma 1.3.5. Let be a regular point of AL. A2, and Ai Й A2. Then,
there exists a neighborhood Uc of A such that the restriction of А й A to
P is smooth.
Proof. Let di and d> denote the dimensions of -1] and A2. By Lemma 1.3.1.
A and A can be described - around ,rc as
A = span{/, : 1 < i < d[}, A2 — span{^ : 1 < i < d2} .
1.3 Distributions
17
At a given point .r. the intersection Aj ( .r) A A2(.r) is found by solving the
homogeneous equation
di	rfj
52	- 52 ь>дм = °
in the unknowns u((j-). 1 < i < dY. and 6;(.г). 1 < i. < d-2. If -I] A A2 has
constant dimension d. the coefficient matrix
(Mr) 	-<7а2И)
of this equation has constant rank r = d\ + d-2 — d: the space of solutions of
this equation has dimension d and is spanned by d vectors of the form
colffitU-)....a,/, (j-)Ai (.r).
which are smooth functions of .r. As a consequence, it is easy to conclude
that At A A; is spanned around J’0 - by d smooth vector fields. <
A distribution A is involutiee if the Lie bracket [fi, r2] of any pair of
vector fields tj and r? belonging to A is a vector field which belongs to A.
i.e. if
Л € A. 72 € A => [t"i . Tj] c A .
Remark l.d.ij. Consider a nonsingular distribution A, and recall that, using
Lemma 1.3.1. it is possible to express any two vector fields tj and r-2 of A in
the form
d	d
= E а-или r2(.r) = 52
mi	i=i
where f\...../j art1 smooth vector fields locally spanning A. It is easy to see
that A is invohnive if and only if
M € A for all 1 < I.j<d.	(1.13)
The necessity of this follows trivially from the fact that f\..... /rf are vector
fields of A. For the sufficiency, consider the expansion (see (1.8))
d	d	d d
lE Cl  E J = E E(dj \f> ‘ fjj + A (кл dj )fj -dj(Lfj e,)/,)
( = 1	J = 1	( = 1 ( = 1
and note that all vector fields on the right-hand side are vector fields of A.
Because of (1.13). checking whether or not a nonsingular distribution is
invohnive amounts to check that
rankf/j» ...	= rank(/](j) ... fd(a*)
for all i- and all 1 < ?.J < d. <
18
1. Local Decompositions of Control Systems
Example 1.3.6. Consider, on 3/ . a distribution
J = span{/!./2}
with
This distribution has dimension 2 for each x e 3?. Since
/0	0	(A	/2jo\ /0	2	0\ / 1 X /0\
L/W-K-C = о	о	o|	i |-o	о	oo=o
\0	1	Oj	\ 0 J \0	0	0/ \x2J \1 J
we see that the matrix
/ 2x2	1	0	\
(/i h [U»=	1	0	0
\ 0	X-2	1	J
has rank 3 (for all j-). and therefore the distribution is not involutive. <
Example 1.3.7. Consider, on the set U = {.r G 3? : x? +	0}. a distribu-
tion
J = span{/i. f>}
with
This distribution has dimension 2 for each x G I-. Since
has rank 2 (for all t). and therefore the distribution is involutive, <
Remark 1.3.8. Any 1-dimensional distribution is involutive. As a matter of
fact, such a distribution is locally spanned by a nonzero vector field f and.
since
If- f](x) = ~f(E) - -J-f(x) = 0
ox ox
the condition of involutivity indicated in Remark 1.3.5 is indeed satisfied. <
1.3 Distributions
19
The intersection of two involutive distributions -C and -F is again an
involutive distribution, by construction. However, the sum of two involutive
distributions in general is not involutive1. This is shown, for instance1, in the
Example 1.3.6. if one interprets Jas-h т -12 with
Л = span{/i }	_V. - span{/2}
zA1 and -1-J are involutive (because both 1-dimensional), but -C + _12 is not.
Remark 1.3.9. Sometimes, starting from a distribution which is not invo-
lutive, it is useful to construct an appropriate involutive distribution, defined
on the basis of the following considerations. Suppose _li and tire two in-
volutive distributions, both containing _1. The distribution _\] И -12 i-s still
involutive and containing _h by construction. From this, one concludes that
the family of all involutive distributions containing has a unique minimal
element (with respect to distributions inclusion), namely the intersection of
all members of the family. This distribution, which is the smallest involutive
distribution containing A it is called the involutive closure of -1 and will be
denoted by inv(_\). <
In many instances, calculations are easier if. instead of distributions, one
considers dual objects, called codistributions. that are defined in the following
way. Recall that a smooth covector field л. defined on an open set U of
, can be interpreted as the smooth assignment - to each point j? of U
- of an element of the dual space (Ж")*- With a set ..........ла of smooth
covector fields, all defined on the same subset U of . one can associate
the assignment - to each point x of U - of a subspace of (Rfi)*. the one
spanned by the covectors -Ji..... Motivated by the fact that the covector
fields ......^'a are smooth covector fields, one may regard this assignment
as a smooth one. The object characterized in this way is called a .smooth
codistribution.
Coherently with the notations introduced for distributions, we use
Q = span^!.......
to denote the assignment as a whole, ami
<?(т) = span^Hj?)......^d(J-)}
to denote the "value1" of 1? at a point x of t/. Since, pointwise, codistributions
are vector spaces (subspaces of (Rn )*), one can easily extend the notion of
addition, intersection, inclusion. Similarly, one can define the dimension of a
codistribution at each point x of U, and distinguish between regular points
and points of singularity. If IF is a matrix having n columns and whose entries
are smooth functions of x„ its rows can be regarded as smooth covector fields.
Thus, any matrix of this kind identifies a codistribution, the one spanned by
its rows.
20
1. Local Decompositions of Control Systems
Sometimes, it is possible to construct codistributions starting from given
distributions, and conversely. The natural way to do this is the following one:
given a distribution A. for each .r in Г consider the annihilator of A(j-). that
is the set of all covectors which annihilates all vectors in A(.r’)
A~(,r) = {tr* € (E" )* : (ic*. r) = 0 for all г G A(.r)} .
Since is a subspace1 of (R")* , this construction identifies exactly a
codistribution, in that assigns to each r of U a subspace of (HE1 )*. This
codistribution, noted A*. is calk'd the annihilator of A.
Conversely, given a codistribution P. one can construct a distribution,
noted P- and called the annihilator of P. setting at each .rin Г
= {r e R” :	r> = 0 for all <r* G Pfir)}
Some care is required, for distributions/codistributious constructed in this
way. about the quality of being smooth. As a matter of fact, the annihilator of
a smooth distribution may fail to be smooth, as the following simple example
shows.
Example 1.3.10. Consider the following distribution defined on R1
A = span{.r} .
Then
A2(,r) = {0}	if ;r 0
A~(.r) = (R1 )*	if x = 0
and we see that A- is not smooth because it is not possible to find a smooth
covector field on A1 which is zero everywhere but on the point z = 0. <
Or, else, the annihilator of a non/smooth distribution can be a smooth
codistribution, as in the following example.
Example 1.3.11. Consider again the two distributions Ai and A2 described
in the Example 1.3.2. Their intersection is not smooth. The annihilator of
1AX П A2j is
[Aj П A2]^ (j') = (K2)*	if а-! ф 0
[Aj П Ao]^ (j) = span{( 1 -1)} if jq = 0 .
The codistribution thus defined is smooth because1 is spanned, for instance,
by the smooth covector fields
-'i	=	(1	-1)
= (1 -Ц-jq)) .<
1.3 Distributions
21
Distributions and codistributions related in this way possess a number of
interesting properties. In particular, the suni of the dimensions of -i and -V
is equal to n. The inclusion Э _F is satisfied if and only if the inclusion
c Ay is satisfied. Finally, the annihilator [—Xj П of an intersection
of distributions is equal to the sum -if -t- If a distribution -i is spanned
bv the columns of a matrix F, whose entries are smooth functions of j-, its
annihilator i> identified, at each т in by the set of row vectors tr* satisfying
the condition
m*F(.r) ~ 0.
Conversely, if a redistribution Q is spanned by the rows of a matrix IF. whose
entries are smooth functions of J. its annihilator is identified, at each .r. by
the set of vectors г satisfying
IF(z)c ~ 0.
Thus, in this case f?_L(.r) is the kernel of the matrix IF at the point т
*?-(/) = ker(IF(j-)) .
One can easily extend Lemmas 1.3.1 to 1.3.5. In particular, if is a
regular point of a smooth codistribution <2. and dim(f?(.F)) — d. it is possible
to find an open neighborhood F° of .F and a set of smooth covector fields
{uji ,...: .Ct/} defined on Cc, such that the covectors _y ..... ug/ are linearly
independent at each J1 in Iго and
= spanpH.r)....
at each J in U~. Moreover, every smooth covector field jj belonging to J2 can
be expressed, on Fc. as
-'(•H =
;=i
where cy.....rj are smooth real-valued functions of j’, defined on L °.
In addition one can easily prove the following result.
Lemma 1.3.6. Let ,rc be a regular point of a smooth, distribution A. Then
x° is a regular point of A^ and there exists a neighborhood U=' of xc such
that the restriction of A^ to U is a smooth codistribution.
Example 1.3.12. Let -1 be a distribution spanned by the columns of a matrix
F and *2 a codistribution spanned by the rows of a matrix IF. and suppose
the intersection 12 П -Iх is to be calculated. By definition, a covector in
12 A is an element of which annihilates all the elements of
A generic element in has the form yIF(.r). where is a row vector of
suitable dimension, and this (covector) annihilates all vectors of A(z) if and
only if
22
1. Local Decompositions of Control Systems
= 0 .	(1.14)
Thus, in order to evaluate f? П -Iх (г) at a point .r. one can proceed in the
following way: first find a basis (say *]..of the1 space of the solutions
of the linear homogeneous equation (1.14). and then express D Г -I^(.r) in
the form
.0 n -A“(J’) = £рап{~ДГ(.г) : 1 < i < d} .
Note that the -./s depend on the point x. If IГ(.г)Т'(.г) has constant rank for
all ,r in a neighborhood U. then the space of solutions of (1.14) has constant
dimension and the Vj's depend smoothly on x. As a consequence, tin1 row
vectors ~ j W(j*)...4f/H’(.r) are smooth covector fields spanning	<
1.4 Frobenius Theorem
In this section we shall investigate the sob-ability of a special system of partial
differential equations of the first order, which is of paramount importance in
the analysis and design of nonlinear control systems. Later on. in the same
Chapter, we will use the results of this investigation in order to establish
a fundamental correspondence between the notion of involutive distribution
and the existence of local partitions of S" into ’lower dimensional” smooth
surfaces. Such a correspondence is instrumental in the investigation of the
existence of decompositions of the system into "reachable"’ and "unreachable”
parts, as well as "observable’’ and "unobservable” parts, which very naturally
extends to the nonlinear setting the analysis anticipated in section 1.1. In the
subsequent Chapters, we shall encounter again the same system of partial
differential equations in several problems related to the synthesis of nonlinear
feedback control laws.
Consider a nonsingular distribution J. defined on an open set T of a" .
and let d denote its dimension. We know from the analysis developed in the
previous section that, in a neighborhood U* of each point fo of U. there exist
d smooth vector fields fl....fa. all defined on U°. which span 3, i.e. arc
such that
T(,r) = span{/i(.r)----fa(x)}
at each j: in Cc, We know also that the codistribution <? = _lx is again
smooth and nonsingular, has dimension n — d and. locally around each ,r°. is
spanned by ?i — d covector fields wi....By construction, the covector
field is such that
(z). ft4кг)} = 0 for all 1 < i < d, 1 < j < n — d
for all x in t’°. i.e. solves the equation
^’j(.r)F(j-) = 0	(1.15)
where F(x) is the n x d matrix
1.4 Frobenius Theorem
23
F(jj = (/i(.r)  f,ffr)).
At any fixed r in F. (1-15) can be simply regarded as a linear homogeneous
equation in tlie unknown (.r). The rank of the coefficient matrix F(j-) is d by
assumption and the space of solutions is spanned by n —d linearly independent
row vectors. In fact, the row vectors ~i(.r).....^n..rfO) are exactly a basis
of this space. Suppose now that, instead of accepting any solution of (1.15).
one seeks only solutions having the form
(9 A j
dr
for suitable real-valued smooth functions Л].....An. In other words, sup-
pose one is interested in solving the differential equation
^(filx)    fffiff = ^-F(x) = 0	(1.16)
dr	dr
and finding n — d independent solutions. By “independent", we mean that
the row vectors
(9Ai	dX^—rf
dr....... dr
are independent at each r. Observing that these row vectors (more precisely,
these covector fields) have the form of differentials of real-valued functions,
i.e. exact differentials, the problem of establishing the existence of n — d in-
dependent solutions of the equation (1.16) can be rephrased in the following
terms: when a nonsingular distribution A has an annihilator A- which is
spanned by exact differentials? This problem will be discussed in the1 present
section. Wc begin with some terminology. A nonsingular d-dimensional distri-
bution A. defined on an open set F of . is said to be completely integrable
if. for each point of F there exist a neighborhood F° of rc, and n — d
real-valued smooth functions A]......A,,^. all defined on F°. such that
span{dAj......dAn_rf} = A*	(1.17)
on F'J (recall the notation (1.6)). Thus, "complete integrability of the distri-
bution spanned by the columns of the matrix F(rf is essentially a synony-
mous for “existence of n -d independent solutions of the differential equation
(1.16)” . The following result illustrates necessary and sufficient conditions
for complete integrability.
Theorem 1.4.1. (Frobenius) .4 nonsingular distribution is completely inte-
grable if and only if it. is involutive.
Proof. We shall show first that the property of being involutive is a necessary
condition, for a distribution to be completely integrable. By assumption, there
exist functions AT.....A„_f/ such that (1.17). or. what is the same. (1.16) is
satisfied. Now. observe that the equation (1.16) can also be rewritten as
24
1. Local Decomposition^ of Control Systems
—i/Jz) = Ш.(z). /,Cr)> = 0 for all 1 < i < d. all z G (1-18)
dx
and that the latter, using a notation established in section 1.2, can in turn
be rewritten as
(dXfax). fdx)) = L/,Aj(z) = 0 for all 1 < i < d. all z G lT=.	(1.19)
Differentiating the function A; along the vector field [/,. Д]. and using (1.19)
and (1.9). one obtains
” LfhLf;Xj(x) = 0 .
Suppose now the same operation is repeated for all the functions AL,..., An
We conclude that
/	\	/ t/Ai(z) \
I    I = I   [fi. A](j) = 0 for all z G c3 -
\	, Д. ] Ar; - [f ( J ) J	у dA/t —4 (z) J
Since by assumption the differentials {dAi,..., cZAn_(/} span the distribution
_W. we deduce from this that the vector field [/(. Л] is itself a vector field
in Л. Thus, in view of the condition established in the Remark 1.3.5. we
conclude that the distribution -1 is involutive.
The proof of the sufficiency is constructive. Namely, it is shown how a
set of n — d functions satisfying (1.17) can be found. Recall, that, since _i is
nonsingular and has dimension d, in a neighborhood U~ of each point z° of
L: there exist d smooth vector fields f}...fa. all defined on L'°. which span
Л. i.e. are such that
J(z) = spanj/Jz).......fa(x)}
at each z in U°. Let fa+i....be a complementary set of vector fields, still
defined on Lr=. with the property thdt
span{/i(z)-----/rf(-r)Jd+i(-r)----/n(.r)} =
at each z in .
Let ф/(х) denote the flow of the vector field fa i.e. the smooth function of
t and z with the property that z(f) = ф{(х°) solves the ordinary differential
equation
> = №)
with initial condition z(0) = zc. In other words. ф{(z) is a smooth function
of t and z satisfying
J^(z) =/(ф'(.г))	ф£(т)=.г.
Recall also that, for any fixed z° there is a (sufficiently small) t such that the
mapping
1,4 Frobenius Theorem
25
гт^Ф^х)
is defined for all ,r in a neighborhood of r°, is a local diffeoinorphisni (onto
its image), and = Ф?_е Moreover, for any (sufficiently small) t..s
Ф^3И = (^{0)) •
We show now that a solution of the partial differential equation (1.16) can
be constructed by taking an appropriate composition of the flows associated
with the vector fields /]..fn. i.e. of
Ф^ И......U) •
To this end. consider the mapping
ф 	[	3»
f	(1-20)
(-1....гп) Ф{] о  - - о Ф^(,г )
where U.- = {г € E" : |c;-| < s} and "o" denotes composition with respect
to the argument j. If s is sufficiently small, this mapping has the following
properties:
(i) is defined for all z —	..., гп) € U.: and is a diffeoinorphisni onto its
image.
(ii) is such that, for all z (E L\. the first d columns of the Jacobian matrix
'ЭФ'
are linearly independent vectors in _1(Ф(с)).
Before proceeding with the proof of these two properties, it is important
to observe that they are sufficient to construct a solution of the partial differ-
ential equation (1.16). To this end. let denote1 the image of the mapping
Ф, and observe that U~ is indeed an open neighborhood of j-3. because is
exactly the value of Ф at the point г = 0. Since this mapping is a diffeoinor-
phisni onto its image (property (i)), the inverse Ф^1 exists and is a smooth
mapping, defined on . Set
/ Ф1И \
	= ^"V)
\Ofi(j-) J
where, di....d>n arc real-valued functions, defined for all j in U°. We claim
that the last n—d of these functions are independent solutions of the equation
(1.16). For. observe that, by definition
‘<ЭФ-11	Г<ЭФ1 _ J
dx ]	L J
26
1. Local Decompositions of Control Systems
where I is the identity matrix, for all z 6 (Л (i.e. for all .r 6	). By property
(ii). the first d columns of the second factor on the left-hand side form a basis
of _1 at any point .r = Ф(с) of As a consequence, the differentials
,	0o(^
dOrf-C.r) = —-
or
.	, dori
don{r) = -7—
Or
are annihilated by the vectors of A at each j- in Uz. These differentials (which
are independent by construction) are therefore a solution of (1.16). At this
point, to complete the proof of the sufficiency we only have to show that (i)
and (ii) hold.
Proof of (I). It is known that, for all r 6 and sufficiently small |t|. the
flow (,r) of a vector field f is defined and this renders the mapping Ф well
defined for all (cb .... 2,J with |cj sufficiently small. Moreover, since a flow
is smooth, so is Ф. Wo prove that Ф is a local diffeomorphisins by showing
that the rank of Ф at 0 is equal to n. To this purpose, let for simplicity (AZ)*
denote the jacobian matrix of a mapping d/(.r). i.e.
Ш),
OM
dr
and note that, by the chain rule
0Ф

о   - о фР (.г°))
 • • О фр (,гс ))
(С;).  - -	о  - - о «У ,(SW)i) .
In particular, at 2 = 0. since Ф(0) = rz
РФ
тНО'МН-
ozt (
The tangent vectors f\{r'z).... ,/Г|(.гс) are by assumption linearly indepen-
dent. and this proves that the n columns of (Ф)„ are linearly independent at
2 = 0. Thus, the mapping Ф has rank n at 2=0.
Proof of (ii). From the previous computations, we deduce also that, at
any 2 6 L\.
...{<₽{;:;) j,о  • -o^U)) =
where r = Ф(2). If we are able to prove that for all .r in a neighborhood of
.г0, for small jt| and for any two vector fields r and d belonging to A.
(Ф^)*7" о Ф''^(г} € A(z)
i.e. that (Ф/) тоФ^Дх) is a (locally defined) vector field of A. then we easily
see that (ii) is true. To prove this, one proceeds as follows. Let d be a vector
field of A and set
1.4 Frobenius Theorem
27
г,ш = (<?",) J, °Ф
for / = 1...d- Since
(differentiate the identity (Ф/) = I with respect to t and interchange
d/dt with d/d.r} and
4(Л°^'(С) = ^Д>°ф?'и
df	U.r
the functions Vf(t) just defined satisfy
о <₽;’.
Since both d and /, belong to A and -A is involutive, there exist functions
Xij defined locally around J’’ such that.
d
[od;\ = Y.x'i/j
and. therefore.
Wj/d о (ф;'(-'-0 = ^а,,(ф;'(п)со) .
\=1 7
The functions l}(f) are seen as solutions of a linear differential equation and.
therefore1, it is possible1 to set
! (t).,. L/ft = (l1 (0).,. A (t)
where1 A'(f) is a d x d fundamental matrix of solutions. By multiplying on the
left both sides of this equality by fTf), wc get
Л(ф;'и)) = ((ф/) jiM...W'),Mr))v(f)
and also, by replacing x by Ф^Дг)
(ли... /an) = ((ф;').л °фУ(.г)... (ф;'|,лофУ(^)).\'(/) .
Since1 A'tf) is nonsingular for all t. we have that, for i = 1.d
оФ-Д-Н e Span{/1 И.......fd(x)}
i.e.
(ф;')у, офУ(.г> e Jut
28
1. Local Decompositions of Control Systems
This result, bearing in mind the possibility of expressing any vector field т
of Л in the form
d
i=l
completes the proof of (ii). <
The proof of this theorem, in the part concerning the sufficiency, is quite
interesting, because it shows that the solution of the partial differential equa-
tion (1.16) (or. what is the same. (1.17)) can be reduced to the solution of n
ordinary differential equations of the form
i = fi (x)	1 < i < n
where /]......f,t are linearly independent vector fields, with /i. -  -, /</ span-
ning the distribution -A. As a matter of fact, if the solutions of these equations
are composed to build the mapping E defined by (1.20). a solution of (1-16)
can be found by taking the last n - d components of the inverse mapping
Ф'-1. This procedure is applied in the following examples.
Example l.J.l. Consider the distribution, defined on P?
This distribution has dimension 1 for each .r E R2. Thus, _1 is nonsingular
and. being 1-dimensional, is also involutive. Set
ah=(J).
The calculation of the flows of /1 ^iid f> is rather easy. As far as /] is
concerned, since
= exp .c->
= 1
is solved by
J'lU) = exp(./A)(exp(f) - 1) +
= t + .rf,
we have
фЛ = ( ехр(т2)(ехр(з1) - 1) + jq \
:i J	t]+;r-2	J'
About /2. since
>1	= 1
T2 = 0
is solved by
1.4 Frobenius Theorem
29
Г 1 ( И — t T .Г'j
Г_>С) = J 2
we have
\ ,r- /
The mapping Ф. choosing .г, = .ri = 0. has the form
and its inverse is given by
U'l .J-2) = f ‘Г2	. Y
\ - j J	\ J' 1 ~ expt J 2) + 1 /
The function C2i.r[..r2| is a solution of the partial differential equation
as a straightforward check also confirms. Note that this function is defined
in all ??. <]
Example 1.4-2. Consider the following distribution, defined on E2
Again, this distribution is 1-dimensional, and therefore completely integrable.
In order to integrate it wo set
The1 calculation of the flow of /, is not difficult. Since
•C = -П
j'2	=	-1
is solved by
•r l , ‘?1 Э. -^(0 =
1 - .1-11
we have
Note that the flow is not defined for jpci > 1 (i.e. the vector field /; is not
complete). The flow of f > is identical to the one calculated in the previous
example. The mapping Ф has the form
30
1. Local Decompositions of Control Systems
/	-2 +
^(<) =	1 - (32 +
\	-31 + J’2
and its inverse is given by
= f:1 "j = (	j -
Note that this mapping is not defined on all B2. However, provided |.r2 — ./Aj
is sufficiently small. the mapping is well-defined for any j:°. The function
c2(J'i. r-j) is then defined in a neighborhood of any .rc and solves the partial
differential equation
= 0 .<
ox
Example 1,4-3. Consider the distribution, defined on B3
J = spa„{^ij.
This distribution has dimension 2 at each point of the set
U = {.r e B3 : xj + x'l 0}.
The distribution is also involutive on C. as shown in the Example 1.3.7. Thus,
the distribution is completely integrable on U. Set
Therefore, the mapping Ф has the form
/ 2zi ехр(г2)хз + ехр(-с2)(~з + Л) \
^(з1.г2,-з)=	+ ехр(-2.г2).Г2
\ ехр(г2)-Гз	/
Consider for instance the point .C = (0.0.1). At. this point the mapping ’E-1
is given by
1.4 Frobenius Theorem
31
Thus, the partial differential equation
is solved by
A(.r i. .r-j. J’3) = C3(.r 1. .i’2. T3) = (т] + 2а>2.Гз) 7*3 .<
One of the most useful consequences of the notion of complete integrability
is related to the possibility of using the functions Ai.......Afi_rf. which solve
the partial differential equation (1.16). in order to define (locally around .rc) a
coordinates transformation entailing a particularly simple representation for
the vector Hehls of Л. For. observe that, by construction, the n — d differentials
dAi......dA„_(/	(1.21)
are linearly independent at the* point r°. Then, it is always possible to choose,
in the set of functions
jq (x) = uq. j--j(j-) =	------rZi(r) = 1‘n
a subset of d functions whose differentials at a12, together with those of the set
(1.21). form a set of exactly n linearly independent row vectors. Let tJq..... d»(f
denote the functions thus chosen and set
Orf+i(-Z-) = Ai(t)----
By construction, the Jacobian matrix of the mapping
z = Ф(т) = col(Q1(.r)....................---------------------©J:r))
has rank n at .r° arid, therefore, the mapping Ф qualifies as a local diffeo-
morphism (i.e. a local smooth coordinates transformation) around the point
r°. Now. suppose т is a vector field of _A. In the new coordinates, this vector
field is represented in the form
Since, by construction, the last n — d rows of the Jacobian matrix of Ф span
Л-. it is immediately deduced that the last n — d entries of the vector on the
right-hand side are zero, for all .r in the set where the coordinates transfor-
mation is defined. We conclude from this that any vector field of Л. in the
new coordinates, has a representation of the form
7(3) =со1(й(Л.....yffc),0....0) .	(1.22)
32
1. Local Decompositions of Control Systems
We end this section with an additional result that shows how the notion
of integrability can be extended to a collection of distributions -dj......_ДА..
all defined on an open set Suppose each distribution of this collection has
constant dimension, say di.......Suppose also that the distributions form
a nested sequence, i.e. that
Ji D _12 D  D _1a-
(so that, in particular. > d2 > -	If the distribution -li is com-
pletely integrable, by Frobenius Theorem, in a neighborhood of each point
.m there exist functions A,. 1 < i < n — di. such that
spanjdAi......dXn_dl } = -If.
Suppose now also is completely integrable. Then, again. -1-Г is locally
spanned by differentials of suitable functions p;.l < i < n - d2. However,
since
-1^ C -1^
it is immediate to conclude that one can choose
l-ir — A; for all 1 < i < n — di
thus obtaining
span{dA1......dAfi_rf] } + span {dp	. ,dpn_d2} =	.
Note also that the sum on the left-hand side of this relation is direct; i.e. the
two summands have zero intersection. The construction can be repeated for
all other distributions of the sequence, provided they are involutive. Thus,
one arrives at the following result.
Corollary 1.4.2. Let _li D -\2 D • -O -1^ be a collection of nested nonsin-
gular distributions. If and only if each distribution of the collection is involu-
tive then, for each point of U. there exists a neighborhood La of x°. and
real-valued smooth functions
\1 \1
A]....,An_til.A1......Adl _d2 ..... Aj....Adk-_1-iik
all defined on L'°. such that
-V = Span{dA{...........
-Af = L- spaii{dA’j..............dA^_t_rft}
for 2 < i < k.
1.5 The Differential Geometric Point of View
33
Remark 1,4-4- hi order to avoid the problem of using double subscripts, it is
sometimes convenient to state1 the previous, and similar, results by means of
a more1 condensed notation, defined in the following way. Given a set of pf
real-valued functions
o'] (.r)..Op, (t)
set
t/o' = (dpj..................................) -
In this notation, the last expressions of the previous statement can be clearly
rewritten in a form like
_1р = span{dAL}
Лу = t -у span{dA!} = span{dA!...........dA! }.<
1.5 The Differential Geometric Point of View
We present in this section some additional material related to the notion
of distribution and to the property, for a distribution, of being completely
integrable. The analysis requires some familiarity with a few basic concepts of
differential geometry, like the ones that - for convenience of the reader are
summarized in the Appendix A. This background, as well as the knowledge of
the material developed in this section, is indeed helpful in the understanding
the proofs of some later results and is essential in any non-local analysis (like
the one presented in Chapter 2). but. can be dispensed of in a first reading.
Throughout the whole section, we consider objects defined on an arbitrary
/г-di men si on al smooth manifold V. This point of view is interesting, for in-
stance. when the natural stare spare on which a control system is defined is
not jT nor a set diffeoniorphie to FT, but a more abstract set.
If this is the case, one can still describe the control system in a form like
i> = №) + 52	(1'23)
<=1
lh — h( (p)	1 < ? < I	(1-241
where f.gY......gni are smooth vector fields defined on a smooth manifold
iV. and	are smooth real-valued functions defined on .V. The first
relation represents a differential equation on A’, and p stands for the tangent
vector, at the point p of A', to the smooth curve which characterizes the
solution for some fixed initial condition. For the sake of clearness, we have1
used here p in order to denote a point in a manifold A\ leaving the symbol z
to denote the n-vector formed by the local coordinates of the point p in some
coordinate chart.
34
1. Local Decompositions of Control Systems
Example 1,5.1. The most common example in which such a situation occurs
is the one describing the control of the orientation of a rigid body around its
center of mass, for instance the attitude of a spacecraft. Let e = (e1.e-j.f3)
denote an inertially fixed triplet of orthonormal vectors (the reference frame)
and let a = (t/i.oj.03) denote a triplet of orthonormal vectors fixed in the
body (the body fra?ne). as depicted in Fig.1.2.
Fig. 1.2.
A possible way of defining the attitude of the rigid body in space is to
consider the angles between the vectors of a and the vectors of e. Let R he a
3x3 matrix whose element is the cosine of the angle between the vectors
n; and e.j. By definition, then, the elements on the? ?-th row of R are exactly
the coordinates of the vector a, with respect to the reference frame identified
by the triplet e. Since the two triplets are both orthonormal, the matrix R is
such that
Ш?7 = I
or. what is the same. J?-1 = RT (thaX is. R is an orthogonal matrix); in
particular, det(7?) = 1. The matrix R completely identifies the orientation of
the body frame with respect to the fixed reference frame, and therefore it is
possible and convenient - to use R in order to describe the attitude of the
body in space. We shall illustrate now how the equations of the motion of
the rigid body and its control can be derived accordingly.
First of all. note that if xe and x denote the coordinates of an arbitrary
vector with respect to e and, respectively, to n, these two sets of coordinates
arc related by the linear transformation
./ = R.r. .
Moreover, note that if one associates with a vector
w = col(uq, U’2, u-3)
the 3 x 3 matrix
1.5 The Differential Geometric Point of View
35
(О гг'з	\
— »'з	0	u'i |
UO	— Up	0 /
the usual ’‘vector" product between te and о can be written in the form
W x r = — S(U.')t’.
Suppose the body is rotating with respect to the inertial frame. Let 7?(f)
denote the value at time t of the matrix* R describing its attitude, and let w(t)
(respectively ujc(t)) denote its angular velocity in the a frame (respectively
in the e frame). Consider a point, fixed in the body and let x denote its
coordinates with respect to the body frame n. Since this frame is fixed with
the body, then .r is a constant with respect to the time and d.r/dt = 0. On
the other hand the coordinates zf(t) of the same point with respect to the
reference frame c: satisfy
C(0 = -S(wF(0)-MH 
Differentiating .r(/) = R[t)i‘({t}. and using the identity 7?S(w(-)j'c =
vields
0 = 7?.rt- + Ri'. = RRrx - RS(^( К = RRr;r - S(~).r
and. because of the arbitrariness of
W) = SMOW) -	(1.25)
This equation, which expresses the relation between the attitude R of
the body and its angular velocity (the latter being expressed with respect to
a coordinate frame fixed with the body), is commonly known as kinematic
equation.
Suppose now the body is subject to external torques. If In denotes the
coordinates of the angular momentum and T. those of the external torque
with respect to the reference frame e. the momentum balance equation yields
in(t) = Tt{t) .
On the other hand, in the body frame a. the angular momentum can be
expressed as
h[t) = J^'(t)
where J is a matrix of constants, called the inertia matrix. Combining these
relations one obtains
Jib = h = Rh, + Rh. = S(~)Rh. + RR =	-т- T
where T = RT. is the expression of the external torque in the body frame a.
The equation thus obtained, namely
Jw(C = S(w(0)dw(t) +T(t)	(1-2G)
is commonly known as dynamic, equation.
36
1. Local Decompositions of Control Systems
The equations (1.25) and (1.26). describing the control of the attitude of
the rigid body, are exactly of the forni (1.23). with
p = (Я.^) .
In particular, note that J? is not any 3x3 matrix, but is an orthogonal matrix,
namely a matrix satisfying RRr = I (and det(f?) = 1). Thus, the natural
state space for the system defined by (1.25) and (1.26) is not as one might
think just counting the number of equations R12. hut a more abstract set.
namely the set of all pairs (R.aA where R belongs to the set of all orthogonal
3x3 matrices (with determinant equal to 1) and w belongs to E3.
The subset of R3x3 in which R ranges, namely the set of all 3x3 matrices
satisfying RRr = / and det (R) = 1. is an embedded submanifold of E3'3. of
dimension 3. In fact, the orthogonality condition RR1 ~ I can be expressed
in the form of 6 equalities
- do = °	1 < ' < J < 3
*-!
and it is possible to show that the 6 functions on tin1 l< ft hand side of this
equality have linearly independent differentials for each nonsingular R (thus,
in particular, for any R such that RR1 = I). Thus, the1 set of matrices
satisfying these ('qualities is an embedded 3-dimensional submanifold of З3х3.
called the orthogonal group and noted 0(3). Any matrix such that RR1 = I
has a determinant which is equal either to 1 or to —1. and therefore 0(3)
consists of two connected components. Tin1 connected component of 0(3)
in which det(/?) = 1 is called rhe special orthogonal group (in R3*3) and is
denoted by 50(3).
We can conclude that the natural state space of (1.25) and (1.26) is the
6-dimensional smooth manifold
V = 50(3) x PA
This is a 6-dimensional smooth manifold, which however is not. diffeo-
morphic to R6 (because5O(3) is not diffeoinorphic to E3). <
We begin by showing how the notion of smooth distribution can be rig-
orously defined in a coordinate-free setting. For. recall that the set of all
smooth vector fields defined on .V. noted V(.V). can be given different alge-
braic structures. It can be given the structure of a vector space over the set R
of real numbers, the structure of a Lie Algebra (the product of vector Helds
fi and /2 being defined by their Lie bracket [/1./2]) and. also, of a module
over (_V). the ring of all smooth real-valued functions deHned on Л’. In
the latter structure, the addition fi -*- f•_> of vector fields fL and /2 is deHned
pointwisc. i.e. as
(fi + /-’)(/>) = fi (pl + /j(p)
1.5 The Differential Geometric Point of View
37
at each point p of A'. and so is tlie product cf of a vector field f by an clement
cofCx(A'). i.e.
(cf) ip) = c(p)f(p}
Suppose Л is a mapping which assigns to each point p of -V a subspace,
noted Др), of rhe tangent space Tj,A' to A’ at p. With A it is possible to
associate a submodule of V(Anoted ,4_j. defined as the set of all vector
fields in C(A') that pointwise take values in Д/Д i.e.
Ab - {f € V (_V) : f(p) e Др) for all p e A'} .
This set by construction is a submodule of V(Ar)- Note, however, that then*
may be many submodules of V(Ar) whose vector fields span _i(p) at each p:
the submodule .Via thus defined is the largest of them, in the sense that it
contains any submodule of I (4V) consisting of vector fields which span -Mp)
at each p.
Example 1.5.2. Suppose Л = 1Я, and let -A be defined in the following way
Д.г) = 0	at x = 0
_A(;r) — TrR	at x 0 .
The submodule .M_i is clearly the set of all vector fit’Ids of the form
f(x} = c(.r)^-
dx
where c is any element of (Ж.) such that c(0) = 0. The set Vf of all vector
fields of the form
.. ..	d
f(j-) - r(.r);r‘ —
dx
where c is any element of С'ДкД is by construction a submodule of
ami its vector fields span -1 at each x. However. ,Vt' docs not coincide with
because for instance
In fact, the smooth function т cannot be represented in the form .r = r(x)z2
with smooth c(j-). <
Conversely, with any submodule .M of V(A’). one can associate an assign-
ment. noted Доо of a subspace of rhe tangent space TPN with each point p
of A’, defining the value of Дц at p as the set of all the values assumed at p
by the* vector fields of i.e. setting
(Д\,1 )(p) = {r e TPN : V - f(p) with f e .
This argument shows how two objects of interest: a mapping which assigns
to each point p of A' a subspace of TpA’ and a submodule of V(A'). can be
related. For consistency reasons, it is desirable that the submodule associated
38
1, Local Decompositions of Control Systems
with the mapping A.m be the module ,M itself. For this to be true, it is
necessary and sufficient that, the submodule .VI has the property that, if f is
any smooth vector field of I '(Д) which is pointwise in d u • then f is a vector
field of .Vf. If this is the case, the submodule .VI is said to be complete.
A complete submodule of V(A’) is the object that, in a global and
coordinate-free setting, replaces the intuitive notion of a smooth distribu-
tion introduced in section 1.2. Of course, the mapping A>t associated with
.M has. locally, smoothness properties which agree with the ones considered
so far. i. e. can be (locally) described as the span of a finite set of smooth
vector fields.
A similar point of view leads to a coordinate-free notion of codistribution.
The latter can be defined, in fact, as a submodule of the module Г*(Л )
of all smooth covcctor fields of >V. satisfying a completeness requirement
corresponding to the one just discussed.
To the objects thus defined it is possible to extend, quite easily, all the
properties discussed in section 1.3. There is. however, a specific point that re-
quires a little extra attention: the difference between an involutive distribution
and a Lie subalgebra of the Lie algebra Г(ЛТ- We recall that an involutive
distribution is a distribution A having the property that the Lie bracket of
any two vector fields of A is again a vector field of A. hi the present setting,
we shall say that an involutive distribution is a complete submodule .Vf hav-
ing the property that the Lie bracket of any two vector fields of .M is again
in .M. Since a Lie subalgebra of V(A’) is a collection of vector fields having
the same property of closure under Lie bracket, one could be led to assimilate
the two objects. However, this is not possible, as for instance the following
simple example shows.
Example 1.5.3. Consider the two vector fields of®2
where r(j‘i) is a but not analytic - function vanishing at 0 with all its
derivatives and nonzero for jq 0.
It is easy to check that the Lie algebra L{fy,h} generated by j\ and /-j
consists of all vector fields having the form
when* F is any non-negative integer and а.Ь^ lg; real numbers. Tlnm. the
subspace A(.r) of TrK2 spanned by tin1 vectors of	at a point, x has
the following description
АД) = ТД'	if j 4~. 0
d
A(j-) — span{ -—} if .ri = 0 .
1.5 The Differential Geometric Point of View
39
However, the submodule .Mj consisting of all vector fields of Л is not
involutive. because the Lie bracket of the vector fields /1 and
r	0
/зИ =
UJ'2
(both are pointwise in _i, but /3 is not in £{/1-/2} I is the vector field
[/1J3] = TT-
(J Л ч
which does not belong to -A at jq = 0. <
We discuss now an important interpretation of the notion of complete in-
tegrability of a distribution. In the previous section, we defined a nonsingular
distribution _A. of dimension d. to be completely integrable if its annihilator
is locally spanned by n — d covector fields which are differentials of func-
tions. This definition is still meaningful in a coordinate-free setting, where
it requires, for each p° of X. the existence of a neighborhood b'° of p° and
n — d real-valued smooth functions Ai..... An (/ defined on , such that
spamfdA^p)......dArj_d(p)} = J-f/d	(1.27)
for all p in C'~. Note that the definition thus given although in coordinate-
free terms specifies only local properties of a distribution. We shall see in
the next Chapter a global version of the notion of complete integrability.
By definition, the и - d differentials of the functions Ai...., A;i_fi are
linearly independent at each point p of the set l'c' where they are defined.
Thus, there exist a neighborhood [г C C0 of p°. and functions Ol..........&d
defined on U. that, together with
Cd-t-i =	.....on — Ajj j .
define a coordinate chart at pz. Without loss of generality, we may suppose
that this is a cubic coordinate chart centered at p°. i.e. that cy(pc) = 0 for
all 1 < ? < n and is an open interval of the form {.rGR: |.r; < К].
Let Ci ~ Oj(p), 1 < i < m denote the /-th coordinate of the point p and
recall that, at each p € U. the choice of these coordinates induces the choice
of a basis in the tangent space1 TPX. namely
(A)........(A)	.
\Эс\ l>'	'd$n p
and that of a basis in the cotangent space T*X. namely
[d&\ ......(de,,)p.
These two bases arc dual. i.e. satisfy
40
1. Local Decompositions of Control Systems
«ЛОР-(Т-У = y.
The property (1.27) says that the last n — d covcctors of the basis of
T*jV are a basis of the codistribution A~(p). at each p e LT Thus, from
the relation of duality, we can conclude that the first d vectors of the basis
of TPN are a basis of -A(p). at each p € L~. From this argument, one can
deduce an alternative characterization of the property, for a distribution, of
being completely integrable. A nonsingidar distribution A, of dimension d.
is completely integrable if. at each p3 € Ah there exists a cubic coordinate
chart (('. o)- with coordinate functions' <?i..... <Лг such that
A(p) = spanL (-у—	?
for all p € Lb
This characterization lends itself to an interesting interpretation. Let p
be any point of the cubic coordinate neighborhood U and consider the slice
of U passing through p consisting of all points whose last n — d coordinates
are held constant; i.e. the subset of U
Sp = {qeU: <pd-i(9) = Od+i(p),  • •	= On(p)} 	(1-28)
This subset, which is a smooth submanifold of U, of dimension d, has the
property of having - at each point q - a tangent space that, by construction,
is exactly* the subspace _\(q) of TqX (Fig.1.3).
Fig. 1.3,
Note that the coordinate neighborhood V is partitioned into slices of the
form (1.28). Thus, a nonsingular and completely integrable distribution A
induces, at each point ph a local partition of A’ into submanifolds, each one
having, at any point, a tangent space which - viewed as a subspace of the
tangent space to A' coincides with the value of A at this point.
1.6 Invariant Distributions
41
1.6 Invariant Distributions
The notion of a distribution invariant under a vector field plays, in the theory
of nonlinear control systems, a role similar to the one played in rhe theory of
linear systems by the notion of subspace invariant under a linear mapping. A
distribution A is said to be invariant under a vector field f if the Lie bracket
[f. r] of f with every vector field г of A is again a vector field of J. i.e. if
T e A [/. Г] G J .
In order to represent this condition in a more condensed form, it is conve-
nient to introduce the following notation. We let [/.Al denote the distribution
spanned by all the л-ect or fields of the form [/, r], with r € A . i.e. we set
[/. A] = span{[/, r\.r e A} .
Using this notation, it is possible to say that a distribution A is invariant
under a vector field f if
[/• J] С V
Remark 1.6.1. Suppose the distribution A is nonsingular (and has dimension
d). Then, using Lemma 1.3.1. it is possible to express at. least locally every
vector field r of A in the form
d
т\.г) = ^2 Ur)т-гt.-r)
г=1
where ч.......r,; are vector fields locally spanning A. It is easy to see that A
is invariant under f if and only if
[/. rj G A for all 1 < i < d .
The necessity follows trivially from the fact that ........у/ are vector fields
of A. For the sufficiency, consider the expansion (see (1.8)}
d	a
1 = 1	I — l
and note that all the vector fields on the right-hand side are vector fields of
A.
The previous expression in particular shows that
[/. А] э span{[/. и]..... [f. y/j}
but note that the distribution on the left-hand side тал-, in general, be un-
equal to the one on the right-hand side. However, by adding to both sides
the distribution A. it is easy to deduce again from the previous expression
that
42
1. Local Decomposition^ of Control Systems
A + [/. A] = A + span{r/. n]......If.rfi}
i.e. that
J -r [f. J] = span{n......Г;}...........[f.	-/]} .
This property will be utilized in sonic later developments. <
Remark 1.6.2. The notion of invariance of a distribution under a vector field
incorporates, in sonic sense, the notion of invariance of a subspace under a
linear mapping. In order to see this, consider a subspace V of invariant
under a linear mapping A. i.e. such that AV С V. Define a distribution A^
as
Ai (.г) - V
at each .r € K". and a (linear) vector field Ja as
= Ar
at each z £ 3d1. It is easy to prove that the distribution Av is invariant
under the vector field /д. in the sense of the previous definition. On the basis
of the previous Remark, all we have to show is that, if ri.........tj is a set
of vector fields locally spanning Av- then [f. ’i].....[/, ~/] are again vector
fields of Av- To this end. note that if ?y..... rd is a basis of V. tin* vector
fields defined as
т,(.г) = 74	1 < 1< (I
ar each .r € K". 1 < i < d. locally span A) . The Lie bracket [/д. r,-j has the
expression
rr _1( t	Of A____1
at each r £	. Since, by assumption. Ac( is a vector of V, we conclude that
[/д.т,] is a vector field of Av. <	/
The notion of invariance under a vector field is particularly useful when
referred to completely integrable distributions, because it provides a way of
simplifying the local representation of the given vector field.
Lemma 1.6.1. Let A be a nonsingular involutive distribution of dimension
d and suppose that A is invariant under the vector field, f. Then at. each
point f there erist a neighborhood Гс' of .m and a coordinates transformation
z = Ф(х) defined on UT in which the vector field f is represented by a vector
of the form
/ /1 (~i...4,Mi.........\
/<i(m >   • Tn :,ui.zn 1
f d У1 (~ t +1.)
\	Li.’^_____)
(1.29)
L6 Invariant Distributions
43
Proof. The distribution -A, being nonsingular and involutive, is also inte-
grable. Therefore, ar each point ,r: there exists a neighborhood and a
coordinates transformation z = ф(х) defined on C'c with the property that
span{dorf).1..don} = J-.
Let f(z) denote the representation of the vector Held f in the new coordinates.
Consider now a vector field
7(c) = coif 7lU)....ojc))
and suppose
~c(t) = (J for A 7^ i
Tf,(z) = 1	for A- — i.
Then
rf =	=_2L
dz' dz,'
Recall that (see (1.22)). in the coordinates just chosen, every vector field of -A
is characterized by the property that the last n—d components are vanishing.
Thus, if 1 < i < d. the vector field г belongs to -A. Since -A is invariant under
f. \f. t] also belongs to -A. i.e. its last n — d components must vanish. This
yields
^=0
dzt
for all d + 1 < A < o. 1 < t < d. and proves the assertion. <
The representation (1.29) is particularly useful in interpreting the notion
of invariance of a distribution from a system-theoretic point of view. For.
suppose a dynamical system of the form
.r = /(.г)	(1-30)
is given and let _i be a nonsingular and involutive distribution, invariant
under the vector field f. Choose the coordinates as described in the Lemma
1.6-1 and set
<i ~	(~1 •    
Qi — (^d r 1.......)	•
Then, the system in question is represented by equations of the form
~ )
G = 7(G)
that is exhibits, in the new coordinates, an internal triangular decomposition.
The block diagram of Fig. 1.4 illustrates this decomposition.
44
1. Local Decomposition:? of Control Systems
Fig. 1.4.
Remark 1.6.3. Note that, if the vector field f is a linear vector field, i.e. if
/(.?•) = tire special form on the right-hand side of (1.31) reduces to
Thus, we may interpret Lemina 1.6.1 as an extension of the well known al-
ready recalled in section 1.1 - result according to which if a subspace V of R"
is invariant under a matrix Д. then choosing an appropriate (linear) change
of coordinates the matrix itself can be put into a block upper-triangulai'forni.
Geometrically, the decomposition described by (1.31) can be interpreted
in the following way (see also the end of section 1.5). Suppose, without loss of
generality, that Ф(.г°) = 0 (for. if Ф(.гс) is nonzero, consider the "translated"
transformation z = Ф!(.г) = Ф(.г) — Ф!^21) which still satisfies rhe requirements
of Lemina 1.6.1 and is such that Ф'(.гс) = 0). Suppose also, again without
loss of generality, that the neighborhood L;° on which the transformation is
defined is a neighborhood of the form
= {.г С : |yfo); < 5}
where г is a suitable small number. Such a neighborhood [’° is called a cubic
neighborhood centered at fo (Fig. 1.5a). Let ,r be a point of foc, and consider
the subset of f’° consisting of all points whose last n —d coordinates (namely
the fo coordinates) coincide with those of jx i.e. the set
5. = {/ e L’°: Gkh = G(fo} .	(1.32)
This set is called a slice of the neighborhood L'° (Fig. 1.5b). Note that any set
of this type, being rhe locus (of points of Uc") where the smooth coordinates
functions (r)............(z)	assume fixed values, can be regarded as a smooth
surface, of dimension d. Note also that the collection of all subsets of
having this form defines a partition of (Fig. 1.5c).
Suppose now that fo and are two points of L’21 satisfying the condition
1.6 Invariant Distributions
45
Fig. 1,5.
GlX) - G(/)	(1.33)
i.e. having the1 same G coordinates but possibly different G coordinates. Let
j-°(n and хь(!) denote the integral curves of the equation (1.30) starting
respectively from x'1 and jif> at time t — 0. Recalling that in the new coor-
dinates the equation (1.30) exhibits the decomposition (1.31). it is easy to
conclude that, so long as .r"(H and .rb(t) art' contained in the domain Гэ of
the coordinates transformation z = Ф(.г).
G(^(H) =	(1.34)
at any time t. As a matter of fact. C-?((^)) and G(-G'(t)) arc both solutions
of the saint1 differential equation - namely, second equation of (1.31) - and
both satisfy the same initial condition, because
6 (-гп (0)) = G G’11) = G (1 = G Ь'л(011 
Two initial conditions .rri and xf> satisfying (1.33) belong, by definition,
to a slice of the form (1.32). As we have just seen, the two corresponding
trajectories T'T) and xb(t} of (1.30) necessarily satisfy (1.34). i.e. at any
time t belong necessarily to a slice of the form (1.32). Thus, we can conclude
that the flow of (1.30) carries slices (of the form (1.32)) into slices (Fig. 1.6).
Example 1.6.4- Consider the 2-dimensional distribution
A — spanj Cj, <<> }
with
and the vector field
46
1. Local Decompositions of Control Systems
Fig. 1.6.
.Г2
•r;i
T3J 'i —
X sin .Г3 -I- .rry + T1T3 J
A simple calculation shows that
[ci. r2] = 0
and therefore (Remark 1.3.5) the distribution A is involutive. Moreover, since
[/• *’1] = 0 If-t’2j = - t’i
this distribution is also invariant under the vector field f (Remark l.G.l).
By Frobenius’ Theorem, in a neighborhood of any point .r° there exist
functions A1 (J’). A2 ) such that
span{dAi. dA^} = A*.
One can easily verify that, for instance. the functions
Aj(j-) =
A2CO ~ -Tj U‘2 + T.J
whose differentials have the form
dAj - (0 0 1 0)
dX‘2 = ( — JO — J?1	0 1 )
satisfy this condition.
As described in the proof of Lemma l.G.l. define new (local) coordinates
Zi — p,(.r). 1 < i < 4. choosing
O3(z) = АДт) o.i(t) = A2(z)
and completing, e.g.. the set of new coordinates functions with
1.6 Invariant Distributions
47
01 (.г) = .Г]	0-2 (.г) = .Г-2 -
In the new coordinates, the vector field f assumes the form
\ sin z3J
i.e. the form indicated by (1.31). with <1 — (-1 -~-j)-Qj ~ (-з-'д)- <
We discuss now some additional properties related to the notion of invari-
ant distribution, that shall be sometimes used in the sequel.
Lemma 1.6.2. Let Д be a distribution invariant under the rector fields fa
and fa. Then Л is also invariant under the vector field [/1./2]-
Proof Suppose т is a vector field in J. From the1 Jacobi identity, we get
\[fa-Hr} = [fa.[fa.r]\-[fa.[fa,fa\.
By assumption. [/?,?"] 6 Л and so is [/1. [fa. r]]. For the very same rea-
son [fa. [/i- r]] 6 A. and thus, from the above cqualitv we conclude that
Remark 1.6.5. Note that the notion of invariance under a given vector field
f can be also extended to a (possibly) nonsniootli distribution A. by simply
requiring that the Lie bracket [f.r] of f with every smooth vector field т of
A be a vector field in J. i.e. that
[fa Sinti J)] C J .
Since [/.smt(d)] is a smooth distribution, this is clearly equivalent to
[/. suit ( —1)] C SIIlt(A)
i.e. to the invariance under f of suit(A). <
When dealing with codistributions, one can as well introduce the notion
of invariance under a vector field in the following way. A codistribution fl
is said to be invariant under the vector field f if the derivative of any
cove ctor field lc of Q is again a co vector field of f?. i.e. if
£ Г2 => e <?.
Using the notation
Lfll — span{£/и.' : л e .0}
this condition can be rewritten in the form
Lf.Q C f? .
It is easy to prove that the notion thus introduced is the dual version
of the notion of invariance of a distribution, as expressed by the following
statement.
-18
1. Local Decompositions of Control Systems
Lemma 1.6.3. If a smooth distribution Д is invariant under the vector field
f. then the codistribution <? = Л1 is also invariant under f. If a smooth
codistribution .0 is invariant under the rector field f. then the distribution
Д = f?1 is idso invariant under f.
Proof. Suppose Л is invariant under f and let r be any vector field of Л.
Then [f. r] 6 Л. Let be any covoctor field in IP Then, by definition
(- -) = 0
and also
{-If--f) = 0-
The identity
= Lffj.r) -	[f. r])
yields
{L^.r}=0.
Since A is a smooth distribution, given any point ,rc and any vector r in
A(.rc) we may find a smooth vector field r in A with the property that
r(z°j = r. Thus, the previous equality shows that
{£z^(.rc).c) = 0
for all r 6 Л(г°), i.e. that £Zu.'(,r°) £ f?(.rc). This proves that L f~ is a
covector field in IP i.e. that <? is invariant under /- The second part of the
statement is proved in the same way. <
Note that, in rhe previous Lemma, first part, we don't need to assume
that the annihilator Лг of Л is smooth nor. in the second part, that the
annihilator <P of It is smooth. However, if both A - and A are smooth, we
conclude from the Lemma that the/iiivariance of Л under f implies and is
implied by that of A1 under the same vector field f. In view of Lemma 1.3.G
this is true, in particular, whenever A is nonsingular.
Bcm ark 1.6.6. As an exercise of application of the notion of invariance of a
codistribution, and of the previous Lemina, we suggest an alternative proof of
Lemina 1.6.1, First of all, note that if new coordinates are chosen as indicated
at the beginning of the proof of Lemma l.G.l, any covoctor field of J- has
a representation of the form
-’(-) = (f) ... 0 ~d_i(c l ...	(1.35)
(this is simply because, in these coordinates, A is spanned by vectors of
the form (1.22)). Observe now that, by construction. the expression of the
functions Oi..... p,i in the new coordinates is just
1.7 Loc al Decompositions of Control Systems
49
for all 1 < / < m This implies
аг, -
and- therefore, in the new coordinates all the entries of the differential do?
are zero but the t-th one. which is equal to к As a consequence
= (do,(ch ft.z)) = ftiz)
and
Lfdofz} = (IL= dfRz) 
Since A is invariant under f and nonsingular. by Lemma 1.6.3 we have
that Lf_\~ C _1~ and tlm. since do; £ A - for d -u 1 < i < n. yields
L;dot = d/; £ dw
The differential dfj. like any covector field of _b. must have the form (1.35)
and this proves that
if 1 < j d. d+1 < i < о. <
Remark 1.6. 7. L’sing 11.10) one can easily prove the dual version of Remark
1.6.1. namely the fact that if 12 is a nonsingular eodistribution of dimension
d. spanned by cuvcetor fields .....-,/ . then 1? is invariant under f if and
only if Lf^'t € f2 for all 1 < i < d. One also finds that
12 + LfO = span^m-. ..-'mTv-h.............<
1.7 Local Decompositions of Control Systems
In this section rhe notion of invariant distribution, and in particular Lemma
1.6.1. are used in order to obtain, for a nonlinear control system of the form
i'L2). namely
.f =	f\j-) -г у g	_
j .	,	(r3G)
!h = M-H 1 <'<P-
decomposition^ similar to those described at the beginning of the Chapter.
50
1. Local Decompositions of Control Systems
Proposition 1.7.1. Let Л be a nonsingular involutive. distribution of dimen-
sion d and assume that Л is invariant under the vector fields f.Si..-..!/»,-
Moreover, suppose, that the distribution span{f/].gm} is contained, in A.
Then, for each point J'" it is possible to find a neighborhood Uc of .m and
a local coordinates transformation с = Ф(а') defined on l'a such that, in the
new coordinates, the control system (1-36) is represented by equations of the
form (see Fig. 1.7 a)
U = /i Uh • <21 +
!=1	11.37)
G = Л(О)
lh = HGUh)
inhere <i = (ft..~f) and G = Ud-i.......-J-
Proof. From Lennna l.G.l it is known that there exists, around each ,r°. a
local coordinates transformation yielding a representation of the form (1.29,1
for the vector fields f.g\,....gm. In the now Coordinates the vector fields
gi....g„t. that by assumption belong to d. are represented by vectors whose
last n—(Icomponents are vanishing (set1 (1.22')). This proves the Propositions
Fig. 1.7.
Proposition 1.7.2. Let Д be a nonsingular involutive distribution of dimen-
sion d and assume that Д is invariant under the vector fields f\ gi.... .g)n.
Moreover, suppose that the codistribution span{d/?i.... , d/p,} is contained in
the codistribution ‘ . Then, for each point ,r° it is possible to find a neigh-
borhood t’° of ;rrj and a local coordinates transformation z = ф(х') defined on
Cc such that, in the new coordinates, the control system (1.36) is represented
by equations of the form (see Fig. 1.7b)
1.7 Local Decompositions of Control Systems
51
m
G — /1 (Ci - G) + AG•G1 fo
(1.38)
G — A ( G ) +	p21 (G )111
!=1
y> = fo(G)
where G = (~i....~A «nd G =	• •  • -n)-
Proof. As before, we know that there exists, around each .d. a coordinates
transformation yielding a representation of the form (1.29) for the vector
fields f. gY..... gnr. In the new coordinates, the covector fields (11ц ..... dhp.
that by assumption belong to must have the form (1.35). Therefore
for all 1 < j < d. 1 < ? < p. and this completes the proof. <
The two local decompositions thus obtained are very useful in understand-
ing the input-state and state-output behavior of the control system (1.36).
Suppose that the inputs u; are piecewise constant, functions of time. i.e. that
there exist real numbers To = 0 < Ti < T>... such that
»,(/) = for Tk < t <	
Then, on the time interval [Д-.Д.-ы) the state of the system evolves along
the integral curve of the vector field
+    +
passing through the point г(Т^). For small values of t. the state u(t) evolves
in a neighborhood of the initial point т(0).
Suppose now that the assumptions of the Proposition 1.7.1 are satisfied,
choose a point zc. and set u(0) = ,rc. For small values of t the state evolves
on U° and we may use the equations (1.37) to interpret the behavior of the
system. From these, we see that the G coordinates of z(t) are not affected
by the input. In particular, if we denote by jt°(T) the point of U° reached at
time t — T when no input is imposed (i.e. when u(f) = 0 for all t e [0. T]).
namely the point
хЦТ) = Ф-fdA)
(ф{(z) being the flow of the vector field /). we deduce from the structure
of (1-37) that the set of points that can be reached at time T, starting from
A is a set of points whose (-j coordinates are necessarily equal to the G
coordinates of J’°(T). In other words, the set of points reachable at time T
is necessarily a subset of a slice of the form (1.32), exactly the one passing
through the point zc(T) (see Fig.1.8).
52
1. Local Decompositions of Control Systems
Fig. 1.8.
Tints, we conclude that locally the system displays a behavior strictly
analogous to the one described in section 1.1. The1 state space can be parti-
tioned into d-dimensional smooth surfaces (the slices of ) and the states
reachable at time1 T. along trajectories that stay in for all t 6 [0.Т]. lit1
inside the slice passing through the point reached under zero input.
The Proposition 1.7.2 is useful in studying state-output interactions.
Choose a point j-“ and take two initial states .r" and .r' belonging to I J
with local coordinates (у'Лу?) and (у^,^) Slirh ^iat
C = У
i.e. two initial states belonging to the same slice of Cc. Let <(0 and .r^ff)
denote the values of the states readied at time t. starting from .r‘J and x1'.
under the action of the same input u. From the second equation of (1.38) wo
sec immediately that, if the input и is such that both J^jf) anti .r^(C evolve
on the C’ coordinates of J’“(d and J^(f) are rhe same, no matter what
input и we take. As a matter of fact. sTf-r^ft)) and <_>(;r,’z(t)) art1 solutions of
the same differential equation (the jiecond equation of (1.381) with the same
initial condition. If we take into account also the third equation of (1.38). we
see that
л,еа')) = л,(4('))
for every inpur u. We may thus conclude that the two states z" and
produce the same output under any input, i.e. are indistinguishable.
Again, we find that locally the state space may be partitioned into d-
dimensional smooth surfaces (the slices of L’°) and that all. the initial states
on the same slice are indistinguishable, i.e. produce the saint1 output under
any input which keeps the state trajectory evolving on F°.
In the next stayions we shall reach stronger conclusions. showing that
if we add to the hypotheses contained in the Propositions 1.7.1 and 1.7.2
the further assumption that the distribution A is ‘’minimal" (in the case of
Proposition 1.7.1) or "maximal” (in the case of Proposition 1.7.2). then from
the decompositions (1.37) and (1.38) we may obtain more precise information
about the states reachable from and. respectively, indistinguishable from
1.8 Local Rpachabilitv
53
1.8 Local Reachability
In the previous section we have seen That if there is a nonsingular distribution
Д of dimension d with the properties that
0) J is involutive1
(ii) J contains the distribution spanjtq....g„,}
(iii) Л is invariant under the vector fields /. c/i.gt„
then at each point T' 6 U it is possible to find a coordinates transformation
defined on a neighborhood Uc of .rc and a partition of Uc into slices of
dimension d. such that the points reachable at some rime T. starting from
some initial state j-3 g along trajectories that stay in L"° for all t g [0. T\
lie inside a slice of t’0. Now we want to investigate the actual "thickness” of
the subset of points of a slice reached at time T.
The obvious suggestion that comes from the decomposition (1.37) is to
look at the1 "minimal" distribution, if any. that satisfies (ii). (iii) and. then,
to examine what can be said about rhe properties of points which belong to
the same slice in the corresponding local decomposition of IT It turns out
that this program can be carried our in a rather satisfactory way.
We need first some additional results on invariant distributions. If V is a
family of distributions on 17. we define the smallest or minimal element as
the member of P (when it exists) which is contained in every other element
of V.
Lemma 1.8.1. Let Д be a given smooth distribution and	a given
set of vector fields. The family of all distributions which, are invariant under
rY.....r(j and contain Д has a minimal element, which is a smooth distribu-
tion.
Proof. The family in question is clearly nonempty. If and -A2 are two
elements of this family, then it is easily seen that their intersection А П A
contains Л and. being invariant under tj......rq. is an element of the same
family. This argument shows that the intersection of all elements in the
family contains J. is invariant under n.....r(/ and is contained in any other
element of the family. Thus is its minimal element. Л must be smooth because
otherwise smt(d) would be a smooth distribution containing Л (because Л
is smooth by assumption), invariant under q..... rq (see Remark 1.6.5) and
possibly contained in A <
In what follows, the smallest distribution which contains and is invari-
ant under the vector fields n.....тч will be denoted by the symbol
(n-----rJJ) .
While rhe existence of a minimal element in the family of distributions
which satisfy (ii) and (iii) is always guaranteed, the nonsingularity requires
54
1. Local Decompositions of Control Systems
some additional assumptions. We deal with the prob han in the following
way. Given a distribution Л and a set rL.....t.j of vector fields we define the
nondecreasing sequence of distributions
-do = Л
Л =	(1'39)
i=l
The sequence of distributions thus defined has the following property.
Lemma 1.8.2. The distributions Д^ generated with the. algorithm (1.39) are
such that
Дк C (n.......
for all k. If there exists an integer k* such that Лд.- = Лд• _ид. then
dt-* = (d....ч|Л> 
Proof. If Л' is any distribution which contains Л and is invariant under o-
1 < i< q. then it is easy to sec that Л' D Лд implies Л' Э Лд^ д. For . we
have (recall Remark 1.6.1)
q	Q
Лдч-i = Лд. +	 Лд] = Лд + ^2sPh11{ltc " ; т G Лд }
; = 1	г-1
<7
С Лд. -I- £span{lr(.T] : т е Л'} С Л\
i=i
Since Л' D Ло. by induction we see that Л' D Лд. for all Ar. If Лд- — Лд-^i
for some /г*, we easily see that Лд-- D Л (by definition) and Лд- is invariant
under ri r(/ (because [ту. Лд.-] С. Лд.^д = Лд- for all 1 < i < q). Thus
Лд- must coincide with (ту......тд]-Д). <
Remark 1.8.1. Before proceeding further with rhe analysis, we want to stress
that the recursive construction indicated by (1.39) can be interpreted as a
nonlinear analogue of the construction that, in a linear system
T = Л.г + Bn
!1 = Cx
ends up with the subspace
R = Im(B AB ... An~l B)
namely, the smallest subspace of R'1 invariant under Д and containing Im(£?).
For. suppose the set ту...., rQ consists of only one vector field, namely r. and
set
Im(B)
Д.г
-W)
r(.r)
1.8 Local Reachability 55
at each z £ ??. Observe that any vector field 8 of Jo can be locally expressed
in the form (see Lemma 1.3.Г)
0(.r) =
( = 1
where b\......6„t are the columns of B. Thus, in view of a property illustrated
in the Remark l.G.l.
Ji = Jo + - Jo] = span{bi........6r/J.[x. 61]....[r.6/f!]} .
Since
',r. C](.r) = [.l.r.6,J =	= -ЛЬ>
u.r
we obtain
J! - spanfbi.......bm. Jdi,.......46,r(}
i.e.
Ji (u) = Irn( В AB)
at each .r £ J,!. Continuing; in the saint* way. we easily deduce that, for any
k > 1.
JC.-c) = Im(£? AB ... .4a'£?J .
Each distribution of the sequence thus constructed is a constant distribution.
Since Ja-i D Jfr. a dimensionality argument proves that there exists an
integer < n with the property that Ja--i — Jjy. Thus Jfi_i. which
is indeed the largest distribution of the sequence, by Lemma 1.8.2 is the
smallest distribution invariant under the vector field 4z which contains the
distribution span{6i......b;!i}. At each j* € E'!. this distribution assumes the
value
J„-ib) = InifT? AB ...An~lB)
i.e. that of the smallest subspace of Rn invariant under A which contains
Ini(B) — span{6i.......6,rt}. <
Remark 1.8.2. Note that, in general, the actual calculation of the distribu-
tion JA. generated by the algorithm (1.39) can still take advantage of the
expression illustrated in rhe Remark 1.6.1. For. if J^-—i is nonsingular and
spanned by a set of vector fields 8[......8^. then
-V-1 + [’, Ф-i] = span{0i........8j. [c,. 0j],.... iTj.
and therefore
JA. — span{0s. [n. 8 A : 1 < s < d. 1 < i < </} .<
об
1. Local Decompositions of Control Systems
We return now to the analysis of the properties of the sequence of dis-
tributions generated by (1.39) which, in the nonlinear setting, is quite more
elaborate than the one illustrated in the Remark 1.8.1. Tin1 increase of dif-
ficulty depends, among the other things, on the fact that. in view of the
interest of using Proposition 1.7.1 for the purpose of obtaining a decompo-
sition of the system a nonsingular and involutive Д.........урД is sought.
First of all. we examine when the stopping condition identified in Lemma
1.8.2 can be met and then we discuss nonsiiigularity and involutivity. The
simplest practical situation in which the algorithm (1.39) converges in a finite
number of steps is when all the distributions of the sequence are nonsingular.
In this case, in fact, since by construction
dim Д < dim Д+i < n
it is easily seen that there exists an integer k* < n such that J*-- = Д--1.
If the distributions Д are singular, one has the following weaker result.
Lemma 1.8.3. There exists an open and dense subset U* of U with the
property that at each point т E L *
(d.....ДИ = X-iU) 
Proof. Suppose V is an open set with the property that, for some к*.
ДДт) =	for all r E I'. Then, it is possible to show that
Д......Tqi-1)(т) = Д. (j’) for all a- E V. For. we already know from Lemma
1.8.2 that (ту....й;|Д D Д-. Suppose the inclusion is proper at some
V and define a new distribution Л by setting
ДД = Д.Д) if re V
J(t) = Д,... ,r(/| J>(j:) if ,r £ V .
This distribution contains J and i£ invariant under	For. if г is a
vector field in -3. then [у-.т] E (ч...У/1Д (because J C (y..........rq'<A')}
and, moreover. [y.r](.r) e Д-Д) for all ;r € V (because, in a neighbor-
hood of J’. r E Лк- and [у. Д-] С Д-). Since J is properly contained in
(y.....y,| Д. this would contradict the minimal it у of (y..| Д. ^ow let
Uk be the set of regular points of Д. This set is an open and dense subset
of U (see Lemma 1.3.2) and so is the set L;* = Lo П [rt П - -  П Un_i. In a
neighborhood of every point ,r E Г* the distributions ......Д3-1 are non-
singular. This, together with the previous discussion and a dimensionality
argument, shows that Д^ = (у...........yj Д on [/* and completes the proof.
<3
We stress that the equality between Д,-! and (y,.... yj Д is true only
at points of an open and dense subset of U. and not everywhere on L'. A
simple example in which there are points at which the equality is not true is
the following one.
1.8 Local Reachability 57
Example 1.8.3. Let (7 = 1- , q = 2. and set
J = spanf-j} Ti(;r) = f 1 )	~
Then,
J„_i = Ji = span{n} + span{["i. ri], [ti,72D = span{n, [ri, t2]}.
Since
we see that the distribution -b lias dimension 2 at each point of the dense
set
[/* = {./ G I2 : 1 — j>2 —	0} .
Л1 is equal to (ri.TOiJ) for all л* 6 L~*. However	(n.	at
some j- £ ['*. For. note that
Thus. Ji is not invariant under n, because this vector is not in -b(.r) if r
is such that .rj =1 and x2 = 0. As a matter of fact, in this case, since
we have that (ту. т2Ц)(-с) — Ж2 at each т E й.2. <з
We illustrate now a property of (ту.......which is instrumental in
achieving involutivit.y.
Lemma 1.8.4. Suppose Л is spanned by some of the vector fields of the set
{ri,.... тч}. Then there exists an open and dense submanifold Г* of U with
the following property. For each xc E U* there exist a neighborhood V of .F
and d vector fields (with d = diin(~i.....rfiSfix0)} .........dj of the form
= [cr. ['(>_!........[pb t'o]]]
where r < n — 1 is an integer (which mag depend on i) and и0....,сг are
vector fields in the set {n...тч}, such that
(n......nd J) И = span{0H'.r).......0d(.r)}
for all x E I ’.
58
1. Local Decompositions of Control Systems
Proof. By induction, using as P* the subset of Г defined in the proof of
Lemma 1.8.3. Let d^ denote the dimension of Jf) ( which may depend on ,r
but is constant locally around .r if the latter is a point of P*}. Since, by
assumption. _i0 is the span of some vector fields in the set {ri..... tq}. there
exists exactly d0 vector fields in this set that span Jo locally around .r. Let
now (p denote the dimension of JA. (constant around .C ) and suppose JA. is
spanned locally around x by vector fields ..... 8^ of the form
= [е,.,	...[(у . ro]]J
where ?\)....rr (with г < к and possibly depending on i] are vector fields
in the set {n......r7}. Then, a similar result holds for JA.+ f. For. let т be
any vector field in JA-. From Lemma 1.3.1 it is known that there exist real-
valued smooth functions n q,.defined locally around .r such that r may
be expressed, locally around J*, as т = и#!	+ c(qt?(/k. If ту is any vector
in the set -(ту..... ту} we have
[Tj. c^i-F-------------------------+cdk [ту.	] + (LT] Cl ----ЦТ-
As a consequence
Ja-i = JA + [ri •	+----479--V-l - -span{0(. [ti . 0,]..[v : 1 < i < (Л }.
Since Ja-+j is nonsingular around .c. then it is possible to find exactly
vector fields of the form
......[c  C>]]]
where го....rr (with r < к q- 1 and possibly depending on i) are vector
fields in the set {щ..... which span JA-h locally around x. <з
On the basis of the previous Leiyina it is possible to find conditions under
which the distribution (ту...., ту |A} is also involutive.
Lemma 1.8.5. Suppose J us spanned by some of the vector fields ту.......TtJ
and that (ту..... ту| J) is nonsingular. Then (-|..rQ\S) is involutive.
Proof. We use first the conclusion of Lemma 1.8.4 to prove that if tri and
ст-2 are two vector fields in J(l_i. then their Lie bracket [tri.oy] is such that
[tri .cr-j](.r) 6	for all x e P*. Using again Lemma 1.3.1 and the
previous result we deduce, in fact, that in a neighborhood U of ,r
H = 1	J=1
e span{0o 8j. [вг, 8j] . 1 < i.j < d}
where 8j.0j are vector fields of the form described before.
In order to prove the claim, we have only to show that [0,. 0,](;r) is a tan-
gent vector in __x (.r). For this purpose, we recall that on P* the distribution
1,8 Local Reachability
59
Jn_i is invariant under the vector fields n...тг/ (see Lemma 1,8,3) and that
any distribution invariant under vector fields r; and т3 is also invariant under
their Lie bracket (see Lemma 1.6.2), Since each в; is a repeated Lie
bracket of the vector fields и...t,j. then [0;.	C	for all
1 < i < d and. thus, in particular :0(-. 0j'(J*) is a tangent, vector which belongs
to
Thus the Lie bracket of two vector fields cq. tr2 in -—i C such that
.rrj](j™) E -bi-jfir). Moreover, it has already been observed that =
(-x....Лл-1) in a neighborhood of xJ and, therefore, we conclude that at any
point u* of P the Lie bracket of any two vector fields .(7_> in (ti...
is such that [cq. oMfir) 6 (л....
Consider now the distribution
-i = (n.....o|J) + span{[0(-.0jj : 0,.0j G (n.....т,/J)}
which, by construction, is such that
....n/J) .
Krom the previous result it is seen that _i(r) = (iq,.,.. 7-QI —1И-r) at each
point j: of C'\ which is a dense set in By assumption, (n,.... r9|-l) is
nonsingular. So. by Lemina 1.3.4 we deduce that -1 = (г]...., and.
therefore, that [0,.07] E (n.....79|J) for all pairs 0,.0j E (п.......’J J).
This concludes the proof. <
Lemma 1.8.6. Suppose is spanned by some of the. vector fields 7i..... 7V
and that AT)_l is nonsingular. Then (71..... v9|-l) is involutive and
{ И....Uy I “d ) = —Ci - 1 .
Proof. An immediate consequence of Lemmas 1,8.3, 1.8.5 and 1.3.4. <
Wo now come back to the original problem of the study of the small-
est distribution which contains span{ffi,.... gm } and is invariant under the
vector fields f. g^....g7Jl. From the previous Lemmas it is seen that, if
{f.g{...., |span{t?!......9>n})	is nonsingular. then it is also involutive and.
therefore, the decomposition (1.37) may be performed. We will see later that
the minimality of (/. gi,..., gni |span{j7i...tfm}) niakes it possible to de-
duce an interesting topological property of the set of points reached at some
fixed time T starting from a given point However, before doing this, it
is convenient to illustrate the results obtained so far with the aid of a sim-
ple example and to analyze some other characteristics of the decomposition
(1.37).
Example 1.8.J. Consider the system
? = M +
with
60
1. Local Decompositions of Control Systems
Computing the sequence of distributions
Ji = span{.9. [/.9]}. with
(1.39b we find Ло = span{</}.
Note that the distribution has dimension 2 for all x. Proceeding further.
we have clearly
J2 =	- [/. Jj] + [g. Ji] = Ji +spaii{7;[/,t/]j. [g. [f. 9]]} .
However, in this specific cast' we have [f-[f-g]] = [.9, [/-</]] = 0. Therefore,
the construction terminates, and
(/. 9|span{,9}) = Ji = span{p. [f.g\]
is the smallest distribution invariant, under f.g and containing the vector
field g. Since this distribution is nonsingular anti involutive (Lemma 1.8.3).
we may use it in order to find a decomposition of the form indicated in section
1.7. To this end. we have first to integrate1 this distribution, that is to find
2 real-valued functions AL.A2 such that span{dA]. dX-2 } = [(/. #|span{f/})]-.
This amounts to solve the partial differential equation
\
dr
dX-2
dr /
1	0 I _
Of 0	\ 0 0J
r-3	0 /
i.e. to find 2 independent functions satisfying
OX	OX	OX
xj + —--h ——xj — 0 and 7?—f	— 0
OX?	OX 1	OX
Since the latter implies (dX/dxi) = 0. the former reduces to
OX OX
dr-2 dr
= 0 .
Two independent solutions of this equation are the functions
At
A2
J’3
x4
1.8 Local Reachability
61
We may now use these functions in order to construct a change of coordinate*
in the state space, as explained in section 1.7. setting
=	Ai(j‘)=.r{
CJ =	z\2( jj = ,F 4 — JOJtn
The change of coordinate* can be completed by choosing
~i —
~2	=	J’2
In the new coordinates, the system becomes
i.e. exactly in the form (1.37). <
We introduce now another distribution, which plays an important role
in the study of local decompositions of the form (1.37) and is related to
{f.gi....gl)t |span{pi......The distribution in question is
U-fP.....fli................g,„}) .
i.e. the smallest distribution invariant under f.gi......gm. which contains
span-f^!,... .(/,„} and. also, the vector field f. If this distribution is nonsin-
gular. and therefore involutive by Lemma 1.8.5. it may indeed be used in
defining a local decomposition of the control system (1.36) similar to the de-
composition (1.37). We are going to see in which way this im'w decomposition
is related to the decomposition (1.37) and why it may be of interest. In order
to simplify the notation, we set
F = (/.t/i... ..gm[spanl.Q!....
R = {fPh...........Уе-Ьрап{/..71.... .д,ц}) .
The relation between F and F is described in tht1 following statement.
Lemma 1.8.7. The distributions F and П are such that
(a)	F 4- span{/} C F
(b)	is a regular point of P 4- span{/}. then (P 4- span{/})(x) = F(,r).
Proof. By definition. F C F and / e F. so (a) is true. It is known from the
proof of Lemina 1.8.4 that, around each point of an open dense submanifold
F* of F. F is spanned by vector fields of the form
0, = h......[?-i. rol]
62
1. Local Decompositions of Control Systems
where r < zz — 1 is an integer which may depend on ?. and rr........zy are
vector fields in the set {/. ...., gm }.
It is easy to see that all such vector fields belong to P -f- span{/}. For. if
(R is just one of the vector fields in the set {/. g\.grri} it either belongs
to P (which contains gi,~-^gm) or to span{/}. If 0, has the general form
shown above we may. without loss of generality, assume that zy is in the set
{171....gm}. For. if r0 = rt = f. then вг = 0. Otherwise, if r0 = f and,
t‘i — 9j- then 6, = py......[/-.hj]] has the desired form. Any vector of the
form
3i = [D.....[O  9j]]
with zy.....t’i in the set [f,g\..gm} is in P because P contains g} and is
invariant under f.g\....,gm and so the claim is proved.
From this fact we deduce that on an open and dense submanifold of
U.
R G P + span{/}
and therefore, since П G) P + span{/} on U. that on [’*
R = P + span{/} .
Suppose that P + span{/} has constant dimension on some neighborhood
I'. Then, from Lemma 1.3.4 we conclude that the two distributions R and
P + span{/} coincide on V. <1
Corollary 1.8.8. If P mid P + span{/} are nonaingular. then
dim(F) - dim(F) < 1 .
If P and P 4-spanj f} are both nonsingular, so is R and. by Lemina 1.8.5,
both P and R are involutive. Suppose that P is properly contained in R.
Then, using Corollary 1.4.2. one can find for each j-° 6 th a neighborhood
L’° of jP and a coordinates transformation z — Ф(х) defined on such that
span{d6>r_!,. ,л . do,;} = /?-
span{dor,.do,,} = P~
on L:c>, where r — 1 = dim(F).
In the new coordinates the control system (1.36) is represented by equa-
tions of the form
~1	=	/1(~1...Cl) +	, . - - •
~r-i -	....Pt) + ^2gr-] .(("I....P>)ui
i-l
C. =	/r(?r.....2„)
- 0
0 .
1.8 Local Reachability
63
Note that this differs from the form (1.37). only in that the last n - r
components of the vector field f are vanishing (because, by construction.
f £ R). If. in particular. R = P then also the г-th component of / vanishes
and the corresponding equation for is
zr = 0 .
The decomposition (1.40) lends itself to considerations which, to some
extent, refine the analysis presented in sections 1.6 and 1.7. We knew from
the previous analysis that, in suitable coordinates, a set of components of the
state (namely, the last n — r + 1 ones) was not affected by the input; we see
now that in fact all these coordinates but. (at most) one are even constant
with the time.
If L'c is partitioned into г-dimensional slices of the form
S-r = {.r e pc : Or+iU) = or+i(.r).....o„(x) = о„(т))
then any trajectory -r(f) of the system evolving in Cc actually evolves on the
slice passing through the initial point ,гэ. This slice, in turn, is partitioned
into (r — l)-dimensional slices, each one corresponding to a fixed value of the
r-th coordinate function, which include the set of points reached at a specific
time T (see Fig. 1.9).
Fig. 1.9.
Remark 1.8.5. A further change of local coordinates makes it possible to
better understand the role of the time in the behavior of the control system
(1.40). We may assume, without loss of generality, that the initial point z" is
such that Ф(тс) = 0. Therefore we have Zj(t) = 0 for all f = r + 1,.... n and
zr = /г(зг,0.....0) .
Moreover, if we assume that f £ P, then the function fr is nonzero ev-
erywhere on the neighborhood U. Now, let cr(f) denote the solution of this
differential equation, which passes through 0 at t = 0. Clearly, the mapping
64
1. Local Decompositions of Control Systems
/m / Zr(t)
is a diffeoinorphisni from an open interval (—f.c) of the time axis into the
open interval of the axis (cr.(—y). zr(r)). If its inverse /t-1 is used as a local
coordinates transformation on the z,- axis, one easily sees that, since
P-1(cr) = t
the time t can be taken as new r-th coordinate.
In this way. points on the slice Sro passing through the initial state art*
parametrized by .........rr_tA). In particular, the points reached at time T
belong to the (r - 1 )-dimensional slice
S',, = {j? e l'3 : or(.r) = T. =0............= 0} .<
Remark 1.8.6. Note also that, if / is a vector field of F then the local rep-
resentation (1-40) is such that fr vanishes on Гс. Therefore, starting from a
point ,r° such that c(.rc) = 0 we shall have zpt) = 0 for all ? = r...n. <
By definition the distribution R is the smallest distribution which contains
f.gi.....gnt and is invariant under f,g{.....gm. Thus, we may say that in the
associated decomposition (1.40) rhe dimension r is "пшптаГ. in the sense
that it is not possible to find another set of local coordinates cj..... z,s..zt)
with s strictly less than r. with the property that rhe last n — я coordinates
remain constant with the time. We shall now show that, from the point of
view of the interaction between input and state, the decomposition (1.40)
has even stronger properties. Actually, we are going to prove that the states
reachable from rhe initial state .r° fill up at least an open subset of the r-
dimensional slice in which they are contained.
Theorem 1.8.9. Suppose the distribution R (i.e. the smallest distribution
invariant under f.gi,--..g7Il which {Contains f „ g i....g„J is nonsingular.
Let r denote the dimension of R. Then, for each F El' it is possible, to find
a neighborhood U° of ,rc and a coordinates transformation z = Ф(т) defined
on Uc with the. following properties
(a)	the set R(j‘°) of states reachable starting from j:c along trajectories en-
tirely contained in U'3 and under the action of piecewise, constant input func-
tions is a subset of the slice
Sr= =	£r+i(.r) = c>,.+ i(.rc)..0п(.г) = 6>,1(.rc')}
(b)	the set TZ(.r") contains an open subset ofS_r-.
Proof. The proof of the statement (a) follows from the previous discussion.
We proceed directly to the proof of (b). assuming throughout the proof to
operate on the neighborhood U~ on which the coordinates transformation
Ф(л-) is defined. For convenience, we break up the proof in several steps.
1.8 Local Reachability
03
(i)	Let в\....(R be a set of vector fields, with k < r. and let Ф}..
denote the corresponding flows.
Consider the mapping
F :	-> I-
(?1....tk) >> о -оФ'Дт3)
where .ri is a point of l’c and suppose that its differential has rank k at some -
., -Sa-, with 0 < s, < г for 1 < j < k (we shall show later that this is
true). For s sufficiently small the mapping
F : (sp.s) x -  - x (,sA..f)	-> Cc
(1.41)
(fi.... .tk ) FCi........F}
is an embedding.
Let .W denote tin* image of the mapping (1.41). Consider the slice of
STc = {,r e L’c : cy(.r) =	-r- 1 < i < n} .
If the vector fields ....have the form
ej=f+
with uJ( € E. for 1 < i < m and 1 < j< An then .V in view of statement (a)
- for small £ is an embedded submanifold of ,Sr=. This implies, in particular,
that for each т E M
TrMGW)	(1.42)
where /?, as before, is the smallest distribution invariant under f.g^.g,n
which contains f.gx.....^„dreeall that R[.r} is the tangent space to SiT= at
t).
(ii)	Suppose that the vector fields f. gY..... gin are such that
/(.r) e T,.v
(1.43)
th(.r) £ Tj..W 1 < i < m
for all .r G 4/. We shall show that this contradicts the assumption k < r. For.
consider the distribution Л defined by setting
J(j-) = rr.W for all .c G 4/
J(a-) = /?(.r) for all re (l.: \ 41).
This distribution is contained in R (because of (1.42)) and contains the
vector fields f.gi,.... grn (because these vector fields are in R and. moreover,
it is assumed that (1.43) are true). Let r be any vector field of A Then т E R
and since R is invariant under f, r/i...., д1л. then for all ,r e (Г \ M)
66
1. Local Decompositions of Control Systems
"](/) e J(j-)	1 < i < m.
Moreover since r. f. g^..... gm are vector fields which are tangent to Af at
each .r e .11. we have also that (1.44) hold for all л- g .V. and therefore all j- g
[L Having shown is invariant under f. g},.... g,„ and contains f. g}..... gm.
we deduce that Л must coincide with R. But this is a contradiction since at
all .r G .W
dim -A(.r) = A-
dim R(.r) = r > A- .
(iii)	If (1.43) are not true, then it is possible1 to find rn real numbers
Uj'11	1 and a point ,r G -V such that, the vector field
Vfi
0^1 = f +
? = 1
satisfies the condition ^+1(.r) $ T^M.
Let J = F(s\........s'k) be this point (.s' > .s;.l < i < A-) and let
denote the flow of  Then the mapping
Г :	(-5. s)^1	-> U
(Ai.....tkRk+x)	°.........
at the point. (.Sj...has rank A- + 1, For. note that
Г p1 (JL) 1	= ГгЛ —)1 ,
L *'dti .......С-0’	k(9L- J(si...SC
for ? — 1.....A- ami that
The first k tangent vectors at n are linearly independent, because F has
rank A- at all points of (,S| .s) x    x (s*..c). The (A- + l)-th one is indepen-
dent from the first A- by construction and therefore F' has rank A' + 1 at
(s'..... <4.0). We may thus conclude that the mapping F’ has rank A- + 1 at
a point, (s'j..... <s'A,, s-J, j). with s, < -s' < e for 1 < i < A- and 0 < .sA.+l < e.
Note that given any real number T > 0 it is always possible to choose the
point i in such a way that
(•4 - si) +    +	“ 's>) < T .
For. otherwise, we had that any vector field of the form
m
0 = f + ^д^1,
1.8 Local Reachability
67
would be tangent to the image under F of the open set
{ U1.....t к1 E (•‘’'1 • -) X ' ' * X f’s‘k • 7) : (f i — Si)	(f r ~ -rt-) < T}
and this, as in (ii), would be a contradiction.
(iv)	We can now construct a sequence of mappings of the form (1-41). Let
= f + ^9^}
; = 1
be a vector field which is not zero at. .r: (such a vector field can always be
found because, otherwise, we would have R(xc) = {0}) and let Л/j denote
the image of the mapping
A:(0,f)	U
fl	Ф\(.г°) .
Let .r = Fi(s*) be a point of ЛД in which a vector field of the form
a = f + JZ
; = 1
is such that A(T) $ Then we may define the mapping (see Fig. 1.10)
A : (.4.7) x (0,;) -о U
Iterating this procedure, at stage k we start with a mapping
Fk : (A“*-7) x  •  x (a{T}-7) x (0. s) C
Ui.....Д) ьо Ф^ о ... о Ф^ (,rc)
and we find a point .7 = А-(л'1.,,.. a£) of its image ЛД- and a vector field
1 = f +	fha-;+1 !5UC^ that	£ TsMk. This makes it possible
to define the next mapping A-i- Note that a* > -s*-1 for i — 1.........k — 1
and shk > 0.
The procedure clearly stops at the stage r. when a mapping Fr is defined
A : (.<-*.£) x x	x (0:s) -> U
(fi...../r) *-> ФД о ... о Ф^ (.rc) .
(v)	Observe that a point r = Fr(ti...,, tr) in the image Mr of the em-
bedding Fr can be reached, starting from the state .rc at time t = 0, under
the action of the piecewise constant control defined by
U((t) = и  for t E [0, fi)
U;(f) =	for f e [f] + • - - + f*_i, fi + f2 +  -  + Д) 
68
1. Local Decompositions of Control Svstems
Fig. 1.10.
W<‘ know from our previous discussions that *Vr must be contained in the
slice of Cc
= j.r e I'3 : рД.г) = oj.r i. /• + 1 < i < n} .
The images ihkIit F, of the open sets of
Fr =	x  x (xjTpf) x (0.5i
arc open in the topology of .V,. as a subset of U~ (because Fr is an embedding)
and therefore they are also open in the topology of d/r. as a subset of ST =
(because Sr* is an embedded submanifold of Fc). Thus tec have that. .Vr is
an embedded submanifold of 5.^ and a dimensionality argument tells us that
_Wr is actually an open submanifold of 5,r=- <
Theorem 1.8.10. Suppose the distributions P (i.e. the smallest distribution
invariant under f. fp....gtl) which contains cp....gni ) and P +span{/} are
nonsingular. Let p denote the dimension of P. Then, for each .F E U it is
possible to find a. neighborhood U~ of .F and a coordinates transformation
г = Ф(х) defined on l'Q with the. following properties
(a)	the set 'R(x'".T) of states reachable at time t ~ T starting from Xй at
t = 0. along trajectories entirely contained in Ur' and under the action of
piecewise constant input functions, is a subset of the. slice.
Я.гГ! ~{.reU:  Of^A-Г) = Op+| (${(.<)).........Qn[.v] = 0n($fr(F))}
where Ф^х?) denotes the state reached at tune t = T when n{t) = 0 for all
t E [0. Г
(b)	the set	contains an open subset of Sj°,t 
Proof. \Xc know from Lemina 1.8.7 that R is nonsingular. Therofort1 one
can repeat the construction list'd to provc the part (b) of Theorem 1.8.9.
Moreover, from Corollary 1.8.8 it follows that r, the dimension of R. is equal
1.9 Local Observability
69
either to p + 1 or to p. Suppose the first situation occurs. Given any real
number T G (0- s). consider the set
f-7 - {IG......G) e L? : 0 +- + fr = T}
where Ur is as defined at step (v) in the proof of Theorem 1,8.9. From the
last remark at the step (iii) we know that there exists always a suitable choice
of S'i“*....s'?’} after which this set is not empty. Clearly the image F, (L'f) ".
consists of {joints reachable at time T and therefore is contained in F(.rc. T).
Moreover, using the same arguments as in (v), we deduce that the set Fr(U,!. )
is an open subset of STe .7If p = r, i.e. if P = /?. the proof can be carried
out by simply adding an extra state variable satisfying the equation
~л —1 = 1
and showing that this reduces the problem to the previous one. The details
are left to the reader. <
1.9 Local Observability
We have seen in section 1.7 that if there is a nonsingular distribution of
dimension d with the properties that
(i)	-A is involutive
(ii)	-A is contained in the distribution (span{d/?i,... .dhr})J
(iii)	-A is invariant under the vector fields /,pi.g,ri
then, at each point ;r° G L~ it is possible to find a coordinates transforma-
tion defined in a neighborhood of .r3 and a partition of U" into slices of
dimension d. such that points on each slice produce the same output under
any input и which keeps the state trajectory evolving on
We want, now to find conditions under which points belonging to different
slices of fr3 produce different outputs, i.e. are distinguishable. In this case
we see from the decomposition (1.38) that the right object to look for is now
the "largest7' distribution which satisfies (ii), (iii). Since the existence of a
nonsingular distribution -1 which satisfies (i), (ii). (iii) implies and is implied
by the existence of a codistribution 1? (namely 3-) with the properties that
(i? *? is spanned, locally around each point .r G I’, by n — d exact covcctor
fields
(ii? f? contains the codistribution span {dh 1..... d/q,}
(iii’) P is invariant under the vector fields f„(p.g,n
we may as well look for the "smallest” codistribution which satisfies (ii?.
(iii?.
Like in the previous section, we need some background material. However,
most of the results stated below require proofs which are similar to those of
the corresponding results stated before and. for this reason, will be omitted.
70
1. Local Decompositions of Control Systems
Lemma 1.9.1. Let <? be a given smooth codistribution and tj...........tq a given
set of vector fields. The family of all redistributions which are invariant under
И......and contain C has a minimal clement, which is a smooth codistri-
bution.
Wo shall use the symbol (n........to denote the smallest codistribu-
tion which contains P and is invariant under ~i......t,}. Given a codistribu-
tion P and a set of vector fields 7j.................t,4 oik1 can consider the following dual
version of the algorithm (1.39)
= (>
fh = f4-i + Z^-i f-4-i
(1.45)
and have the following result.
Lemma 1.9.2. The codis t ri but ions Lf.... де n c z a t ed with the algor ith m
( Lf5) are such that
P* C (n.......7,IP)
for- all k. If there exists an integer k* such that	then
<4- = (n......
Remark. 1.9.1. In the case of the linear system
.r — .4.г
у = Cx
the sequence (1.45) can be interpreted as a nonlinear analogue of a sequence
leading to the largest subspace of invariant under -A and contained in
ker(C). Suppose as in Remark 1.8.1 that the set Т[...........r(l consists only
of the vector field т and set	f
-Oo(.r’) - >pan{ri.......r₽}
r(.r) = .A.r
for each j- G R'!. where ri.....cfl denote the rows of C, Since any covector
field u/ in Pa can be locally expressed as
V С.-.И
i=l
where -q......y;, are smooth real-valued functions, it is easy to deduce (see
Remark 1.G.7) that
Pi = P(> + £rf2() = span{f'i....cp. Lrcx......LT(‘p} .
Therefore, since
1.9 Local Observability
71
1	I	.	)	- J
L T? 1 — L \ т<? I — f	( .4
Ox
we have
= span{ci................................c}>. ci .4.гд,Л}
at each j- G j£'! . Continuing in the same way. we have, for any к > 1.
-r4LH = span{ci......rp.0^4------criA..............cP-4A’} .
Each codistribution of the sequence thus constructed is a constant redistri-
bution. Since f-Cri Э -C\. a dimensionality argument proves that there1 exists
an integer к* < n with the jjrojjcrty that = f?;-. Thus f?n_]. which
is indeed the largest codistribution of the sequence, by Lemina 1.9.2 is the
smallest redistribution invariant under the vector field Ar which contains the
codistribution f?0 = span{ri.....rp}.
By duality 1 is the largest distribution invariant under the vector field
Ar and contained in the1 distribution .Qq-. Observe now that. by construction,
at each x G K" .
= ker(C)
( c \
C 4
= ker _
XCA1-1 /
Thus, we can conclude that the value of Q-__ j at each x coincides with
that, of the largest subspace of Л" which is invariant under .4 and is rout aim'd
in the subspace ker(C). <
Returning to the case of nonlinear systems, one may obtain the following
dual versions of Lemmas 1.8.3. 1.8.4. 1.8.5. 1.8.G.
Lemma 1.9.3. There exists an open and dense subset A of Г with the
property that at each point x G L'*
(n.....-,|.Q)U) = W-i(r)
Lemma 1.9.4. Suppose J? z,s spanned by a set dX\.......dX^ of exact corer tor
fields. Then, there exists an open and dense submanifold T'* of U with- the
following property. For each xc G f’* there exist a neighborhood U'~J of x°
and d exact covector fields fwith d = dini^Tj...., r</|f?)(.r°)) wi which
hare the form
wi = dXj or = dL Vt ... Aj
where r < n - 1 is an integer (which may depend on i). i\........cr are vector
fields in the set {tj. ..., and Xj is a function in the set {Ai......As}, such
that
(n.....т,Д(г) = span{^i(.rl........~Д.г)}
for all x G C
72
1, Local Decompositions of Control Systems
Lemma 1.9.5. Suppose Q is spanned by a set dX\..... dXs of exact covector
fields and that {ty....Tq\fT) is nonsingular. Then (n ....,is involu-
tire.
Proof. From the previous Lemma, it is seen that in a neighborhood of
each point x in an open ami dense submanifold £’*. the codistribution
(ri,... .rffl) is spanned by exact covector fields. Therefore, the Lie bracket
of any two vector fields 0i.0-2 in (~i.....rjf?)- is such that	€
(see section 1.4) for each x e Lr+ - From this result, us-
ing again Lemma 1.3.4 as in the proof of Lemma 1.8.5. one deduces that
(n,.... vq : <7) - is involutive. <
Lemma 1.9.6. Suppose .0 is spanned by a set dX\..........dXs of exact covector
fields and that CfJ_i is nonsingular. Then (п ...., rq| <T)~ is involutive. and
(D.......o; = Cri_i .
In the study of the state-output interactions for a control system of the
form (1.36). we consider the distribution
Q = (f-9i......^Jspanjd/!!........dhp})-
From Lemma 1.6.3 we deduce that this distribution is invariant under
f,gi.....gr„ and we also see that, by definition, it is contained in (span{d/?i.
- - . dhp})1  If nonsingular. then, according to Lemma 1.9.5.is also involutive.
Invoking Proposition 1.7.2. this distribution may be used in order to find
locally around each x° E P an open neighborhood Pc of .r0 and a coordinates
transformation yielding a decomposition of the form (1.38). Let s denote
the dimension of Q. Since Q~ is the smallest codistribution invariant under
f,gi.....gU] which contains dhi......dhp. then in this case the decomposition
we find is maximal, in the sense that.it is not possible to find another set
of local coordinates C!..... Zf. Zf~ \ zn with r strictly larger than with
the property that only the last n — r coordinates influence the output. We
show now that this corresponds to the fact that points belonging to different
slices of the neighborhood P° art1 distinguishable.
Theorem 1.9.7. Suppose the distribution Q (i.e. the annihilator of the
smallest codistribution invariant under f, уi.....gm and which contains dhi-
....dhp) is nonsingular. Let s denote the dimension of Q, Then, for each
x:' G U it is possible to find a neighborhood Pc' of ,r3 and a coordinates
transformation z = Ф(х) defined PQ with the following properties
(a) Any two initial states ,r” and of Pc such that
Oi(xa ) = (pd-*’6)	' = a + 1....ri
produce identical output functions under any input which keeps the state tra-
jectories evolving on P"
1.9 Local Observability
73
(b) Any initial state x of [r= which cannot be distinguished from .r~ under
piecewise constant input functions belongs to the slice
S.r< = {.r e C° : Cgix} = cy (,r”).-s + 1 </</?} •
Proof- We need only to prove (I))- For simplicity, we break up the proof in
various steps.
(il Consider a piecewise-constant input function
upt) — nJ for t E [O.fi)
ut(t) =	for t E [h H----Fh-idJ----------Ha).
Define the vector field
= f + ^2
i=i
and let Фк denote the corresponding flow. Then, the1 state reached at time p
starting from j,c at time t = 0 under this input may be expressed as
.r(C-) = Фр о   • о (,Р )
and the corresponding output у as
yAh} = hi{.idtk)) 
Note that this output may be regarded as the value of a mapping
Ff‘ :	(-sp)*	-> К
(h-----tk) h, о Ф*. O’-’O^ („p ) .
If two initial states Fl and .r1’ ata1 such that they produce two identical outputs
for any' possible piecewise const ant input, we must have
Ff 71......7-) = Ff 71.......tk )
for all possible (C.....7)- with 0 < tt < x for 1 < i< p. From this we
deduce that
dkFr3	Эк Fr
^dt. ...OtJ^ = -=t^{) = 'ЙС ...с?/аЛ’=-=^-0 '
An easy calculation shows that
FJ'"
...'atA.= £": Fp.h'(.F)
and. therefore, we must have
ip ... ipA j-r'C = ip ... L^ hp-w) .
74
1. Local Decompositions of Control Svstems
(ii)	Xow. remember that f)r j = 1........k. depends on ............) anti
that the above equality must hold for all possible choices of (u-'.....u-^) G
F."’. By appropriately selecting these (tr^.......one easily arrives at an
('quality of the form
£t., ... L,..	) = L,! ...	/р-|.гА)	(1.461
where f’i....are vector fields belonging to the1 set {/. r/i......p,f(}. For. set
-,> = Lti.z ... L,ikh, . From the equality L„, ~2{.ra 1 = L^,.'>(.r1') wo obtain
= L/--'(.г1'} + ^2	
(=i	?-i
This, due to the arbitrariness of the (»}....implies that
L, y>(/) =
where r is any vector in the set {f.gi.......9m}- This procedure can be iter-
ated. by setting •'j = L,i?t   .Lt,khj. From the above equality one gets
ЕгЬ/~-3(.г") + LrL7l hr = LvLf';3(xh) + L'j> ':r(.c')u;
and, therefore.
Tri	} = £t.'3(.r J
for all ci. r-2 belonging to the' sea {f.g\..gni}. Finally, one arrives at (1.46).
(iii)	Let C:; be a neighborhood of the point J'" on which a coordinates
transformation 0(.r) is defined which makes the condition
r. o .	. d . s
Q(.r) =s>an{ ( -+ ..........O— ,}	11.47)
Cd I J ()q^ ~r
satisfied for all .r £ 1“. From Lemma 1.9.4. we know that there exists an open
subset F‘ of CJ. dense in £r;. with the property that, around each .r G C’
it is possible to find a set of n — -s real-valued functions A!...Ап_^ which
have the form
A, = £u. .. .L^. hj	(1.48)
with rj......г,- vector fields in {f.Q\..g>n} and 1 < j < p. such that
Q~ — span {//A]...., dXn-s} .
Suppose .m £ [’*. Since	has dimension n - s. it follows that
the tangent covectors dAi(J'J).......dA^-sij-::) are linearly independent. In
the local coordinates which satisfy (1.47). Ai...AfJ_.s arc functions only of
v>4-i...zri (see (1.35)). Therefore, we may deduce that the mapping
-I ;	1....~„) i—f Ai ..........zn).......A„_.Jc.s._i,.... c,;))
1,9 Local Observability
has a jacobian matrix which is square and nonsingular at (1 (.r3)............
г71(.г°)). bi particular, this mapping is locally injective. We may use this
property to deduce that, for some suitable neighborhood Г' of ,r=. any other
point т of I-' such that
A,(.rj = АДZ I
for 1 < i' < n - s. must be such that
= p.s. ,(.r )
for 1 < i < я - s. i.e. must belong to the slice of passing through ,r3. This,
in view of the results proved in (ii) completes the proof in the case where1
/ e Г*.
(iv) Suppose .r=	f’’. Let .r(u2. T. и) denote the state reached at time
t = T under the action of the piecewise constant input function a. If Г is
sufficiently small. л-{хл/Г.и} is still in I е. Suppose ,г(.г’.Г. и) 6 Г*. Then,
using the conclusions of (iii). we deduce that in some neighborhood U( of
x! = x(.r= .T. n}. the states indistinguishable from У lie on the slice of U'~
passing through .rh Xow. recall that the mapping
Ф :	-> г(.г2. T. u")
is a local diffeoinorphisni. Thus, there exists a neighborhood Г of .rc whose1
(diffeomorphic) image under Ф is a neighborhood I'" С I f of .rb Let ,r denote
a point of Г indistinguishable from under piecewise constant inputs. Then,
clearly, also .г" = .r(j\T.u) is indistinguishable from x1 — .r(,r“.T, it). From
the previous discussion we know that r" and x1 belong to the same slice of
t'°. But this implies also that and ,r belong to the same slice of t’°. Thus
the proof is completed, provided that
j-lX.T. a) G Г* .	(1.49)
(v) All we have to show now is that (1.49) can be satisfied. For. suppose
'/?(.r3). the set of states reachable from under piecewise constant control
along trajectories entirely contained in U°. is such that
) О F‘ = 0 .	(1.5U)
If this is true, we know from Theorem 1.8.9 that it is possible to find an
r-dimensional embedded submanifold I’ of Гг~ entirely contained in 'R\.r31
and therefore such that I ’ П I'* =0. For any choice of functions Ai,.... An_.s
of the form (1.48). at any point ,r G V the covectors с/АД.г)......dA,;_ ,#(^)
an1 linearly dependent. Thus, without loss of generality, we may assume that
there exist d < n — ,s functions m.....~a still of the form (1.48) such that,
for some open subset I"' of U.
- span{d/ti (u),.... (Mp(j-)} C span{(/"i (.r)..d~ d(z)} for all ,r G I '
- d'dr)......d~tj(T) are linearly independent covectors at all .r G V'.
ifj 1. Local Decompositions of Control Systems
- dLrj C span{d"i(r).......d~rf(r’)} for all .r € U' and г € {f.gi..gm}.
Now. we define a codistribution on U" as follows:	for
x £ Uh and P(.r) = ?pan{d-T| Ud.   ,d~,t[r)}. for x ё V'. Using the fact
that f.gi gnt art1 tangent to V'. it is not difficult to verify that this
codistribution is invariant under f.g\.....gm. contains span{d/ti....dh}>\
and is smaller than (f.gi gm |span{ dh-\ This is a contradiction
and therefore (1.50) must be false. <
2. Global Decompositions of Control Systems
2.1 Sussmann’s Theorem and Global Decompositions
In the previous Chapter. we have shown that a uonsingular and involutive
distribution A induces a local partition of the state space into lower dimen-
sional submanifolds and we have used this result to obtain local decomposi-
tions of control systems. The decompositions thus obtained arc very useful to
understand the behavior of control systems from the point of view of input-
state and. respectively. state-output interaction. However, it must be stressed
that the existence of decompositions of this type is strictly related to the as-
sumption that the dimension of the distribution is constant at least over a
neighborhood of rhe point around which we want to investigate1 the behavior
of our control system.
In this section we shall see that the assumption that A is uonsingular can
be removed and that global partitions of the state space can be obtained.
Since we are interested in establishing results which have a global validity,
it is convenient for more generality to consider, as anticipated in section
1.5. the case of control systems whose state space is a manifold N. Of course,
this more general analysis will cover in particular the case in which A' = U.
To begin with, we need to introduce a few more* concepts. Let A be a
distribution defined on the manifold _V. A submanifold S of Л' is said to be
an integral submanifold of the distribution A if. for every p € S, the tangent
space TPS to S at. p coincides with the subspace A(p) of Tp.\. A maumtil
integral submanifold of A is a connected integral submanifold S of A with
the property that every other connected integral submanifold of A which
contains S coincides with S'. We see immediately from this definition that
any two maximal integral submanifolds of A passing through a point p G -V
must coincide. Motivated by this, it is said that a distribution A on A’ has
the maximal integral manifolds property if through every point p € N passes
a maximal integral submanifold of A or. in other words, if there exists a
partition of A’ into maximal integral submanifolds of A.
It is easily seen that this is a global version of the notion of complete inte-
grability for a distribution. As a matter of fact, a uonsingular and completely
integrable distribution is such that for each p € A’ there exists a neighborhood
U of p with the property that A restricted to I' has the maximal integral
manifolds property.
78
2. Global Decompositions of Control System:
A simple con sequence of the previous definitions is the fol lowing one.
Lemma 2.1.1. A distribution Л which has the maximal integral manifolds
property is intolutiue.
Proof. If т is a vector field which belongs to a distribution A with the max-
imal integral manifolds property, then r must be tangent to every maximal
integral submanifold S of A. As a consequence, the Lie bracket irq.-ra] of two
vector fields и and т-2 both belonging to A must be tangent to every maximal
integral submanifold S of A. Thus. ‘Т1.Г2] belongs to A. <
Thus, involutivity is a necessary condition for A to have the maximal
integral manifolds property but. if A has points of singularity, this condition
may fail to be sufficient.
Example 2.1	.1. Let Ar — 3c and let A be a distribution defined by
J(.rl =spaii{(A-)j.A(.r1)(A-)i}
where A(.zq) is a C'x function such that A(jq) = 0 for ./q < 0 and A(.iq) > 0
for jq > 0. This distribution is involutive and
dim A(jc) = 1	if .r is such that iq < 0
dim A(jc) = 2	if x is such that jq > 0 .
Clearly, the open subset of .V
{(.tq.T^e^ :.zq >0}
is an integral submanifold of A (actually a maximal integral submanifold)
and so is anv subset of the form
i
{(aq.T-j) G 3? : jq < 0,x-> = c} .
However, it is not possible to find integral submanifolds of A passing through
a point (0, c). <
Another important point to be stressed, which emphasizes the difference
between the general problem considered here and its local version described
in .section 1.4. is that the elements of a global partition of Л' induced by
a distribution which has the integral manifolds property are immersed sub-
manifolds. On the contrary, local partitions induced by a nonsingular and
completely integrable distribution are always made of slices of a coordinate
neighborhood, i.e. of embedded submanifolds.
2,1 Sussmaim's Theorem and Global Decompositions
79
Example 2.1	.2. Consider a torus T> — S\ x Sj. We define a vector field on
the torus in the following way- Let т be a vector field on R2 defined by setting
r,.,-,..,, S	p -	; .J
At each point (.ri.-m) G Si this mapping defines a tangent vector in
T[X1 ,.r.2 )Si. and therefore a vector field on Sj whose flow is given by
(j*i, .r5) = cos f — r-] sin f..r\ sin t + j'.“> cos t) .
In order to simplify the notation we may represent a point Lr j, -tg) of Si with
the complex number z = л -1- jx >, |c, = 1. and have1 $f(z) = ejtz. Similarly,
bv setting
.	cl
= -rml-—) +
dxY r dx2 ’
we define a not tier vector field on Si. who^e flow is now given by (c) = e,,>fz.
From т and в we may define a vector field f on T> by setting
/(Ci. co) = (r(h), 9<z2 1)
and we readily see that the flow of f is given by
Ф^.гА = (e^.e^z-,} .
If a is a rational number, then there exists a T such that ф{ = Ф^к.т
for all t € R and all k € Z. Otherwise . if n is irrational, for each fixed
p = Oi- G T? the mapping Fp : t н-> ф{^. z2) is an injective immersion
of R into T2. and J7},(JR) is an immersed submanifold of T2.
Front the vector field f we can define the one-dimensional distribution
A = span{/} and see that, if о is irrational, the maximal integral submanifold
of A passing through a point p G T> is exactly FP(R) and J has the maximal
integral manifold property.
Fp(R) is an immersed but. not an embedded submanifold of T2. For. it
is easily seen that given any point p € T2 and any open (in rhe topology of
T2) neighborhood f ’ of p. the intersection FP(E) П f.’ is dense1 in E and this
excludes the possibility of finding a coordinate cube (Г. o) around p with the
property that FP(R) О U is a slice of <
The following Theorem establishes the desired necessary and sufficient
condition for a distribution to have the maximal integral manifolds property.
Theorem 2.1.2. (Sussmann) A distribution A has the. maximal integral,
manifolds property if and only if. for every rector field т 0 A and for every
pair (t.p) t R x .V such that the flow Ф) (p) of т is defined, the differential
(Ф[)+ at p maps the subspace A(p) into the subspace А(ФЦр)'),
We are not going to give the proof of this theorem, that can be found in
the literature. Nevertheless, some remarks are in order.
80
2. Global Decompositions of Control Svstems
Remark 2.1.3. An intuitive understanding' of the constructions that are be*
hind the statement of Sussmann’s theorem may be obtained in this wav.
Let "i.....Tf,- he a collection of vector fields of A and let Фр....фр'
denote the corresponding flows. It is clear that if p is a point of -V. and S is
an integral manifold of A passing through p. then Фр (p) should be a point
of S for all values of t, for which $'• (p) is defined. Thus. 5 should include all
points of A’ that can be expressed in the form
°	° оф[’ (pl .	(2.1 j
In particular, if т and в arc vector fields of A. the smooth curve
cr : ( — s) -> A’
t н-> Фр о Ф“ о <T_fl (р)
passing through p at t = 0. should be contained in S and its tangent vector
at p should be contained in A(p). Computing this tangent vector, we obtain
U(£Tn(pi) e A(pi
i.e. setting q = Фз^р)
{ФР')^({) e Л(фр{<Р)
and this motivates the necessity of Sussinann’s condition. <
According to the statement of Theorem 2.1.2. in order to "test" whether
or not a given distribution A is integrable, one should check that (Фр)* maps
A(p) into А(Ф^(р)) for all vector fields - in A. Actually one could limit
oneself to make this test only on some suitable subset of vector fields in A
because the statement of the Theorem 2.1.2 can be given the following weaker
version, also due to Sussmann.
Theorem 2.1.3. A distribution A has the maximal integral manifolds prop-
erty if and only if there exists a set of rector fields T. which spans A. with
the property that for every r^T and every pair (t.pl €?. x A" such that the
flow Фр (pi is defined, the differential \ Фр )A at p maps the sab space _\{p) into
the яиЬ$расе_\(Фр (pf).
Remark 2.1.4. It is clear that the proof of rhe "if" part of Theoremi 2.1.2 is
implied by the "if’" part of Theorem 2.1.3 because the set of all vector fields
in A is indeed a set of vector fields which spans A. Conversely, the "only if"
part of Theorem 2.1.3 is implied by the "only if" part of Theorem 2.1.2. <
We have semi that involutivity is a necessary but not sufficient condition
for a distribution A to have the maximal integral manifolds property. How-
ever. the involutivity is something easier to test in principle because it
involves only the coinput at ion of the Lie bracket of vector fields in A whereas
the test of the condition stated in the Theorem 2.1.3 requires the knowledge
2.1 Sitssniaim's Theorem and Global Decompositions
81
of the flows associated with all vector fields г of a subset T which sp<ui<
J. Therefore, one might wish to identify some special classes of distributions
for which the invoiutivity becomes a sufficient condition for them to have
the maximal integral manifolds property. Actually, this is possible with a
relatively little effort.
A set T of vector fields is locally finitely generated if. for every p E _V
there exists a neighborhood P of p and a finite set {ту..........} of vector
fields of T with the property that every other vector field belonging to T can
be represented on I' in the form
r = £e,r,	(2.2)
where each a is a real-valued smooth function defined on L'.
The class of the distributions which are spanned by locally finitely gener-
ated sets of vector fields is actually one of the classes we were looking for. as
it will bo shown hereafter.
We prove first a slightly different insult which will also be used indepen-
dently.
Lemma 2.1.4. Let T be a locally finitely generated set of vector fields which
spans Д and. в another vector field such that [6hr] € T for all т E T. Then,
for every pair (t.p) E ?,x.V such that the flout Ф'] (p] is defined, the differential
(Ф'( p at p maps the subspace -A[p) into the subspace A(Tf (pi).
Proof. The reader will have no difficulty in finding that t he same arguments
used for the statement (ii) in the proof of Theorem 1.4.1 can be used. <
Note that in the above statement the vector field в may not belong to T-
If the set T is involutive. i.e. if the Lie bracket . m] of any two vector fields
"i E T- m E ‘T is again a vector field in T. from the previous Lemma and
from Sussmanns Theorem wo derive immediately the1 following result.
Theorem 2.1.5. .4 distribution Л spanned by an inrolntire and locally
finitely generated set. of vector fields T has the maximal integral manifolds
property.
The exist('iice of an involutive and locally finitely generated set of vector
fields appears to be something easier to prove, at least in principle. In partic-
ular. there1 art1 some classes of distributions in which the existence of a locally
finitely generated set of vector fields is automatically guaranteed. This yields
the following corollaries of Theorem 2.1,5.
Corollary 2.1.6. .4 nonsingular distribution lias the maximal integral man-
ifold property if and only if it is involutive.
Proof. In this ('asm the set of all vector fields which belong to the distribution
is involutive and. because of Lemina 1.3.1. locally finitely generated. <
82
2. Global Decompositions of Control Systems
Corollary 2.1.7. An analytic distribution on a real analytic man/fold has
the maximal integral manifolds property if and only if it is involutive.
Proof. It depends on the fact that any set of analytic vector fields defined on
a real analytic manifold is locally finitely generated. <
Wo conclude this section with another interesting consequence of the pre-
vious results, which will be used later on.
Lemma 2.1.8. Let Л be a distribution with the maximal integral manifolds
property and let S be a maximal integral submanifold of Л. Then, given any
two points p and q in S. there exist rector fields ту ту m _1 and real
numbers t\ p such that q = о    о Ф^(р),
Theorem 2.1.9. Let. _1 be an inrolutu-e distribution invariant under a com-
plete rector field в. Suppose the set of all vector fields in. Л is locally finitely
generated. Let pi and p-z be two points belonging to the same maximal integral
submanifold of Л. Them for all T. Ф^ (pC and Фт(рР belong to the same
maximal integral submanifold of Л.
Proof. Observe, first of all. that _1 has the maximal integral manifolds prop-
erty (set1 Theorem 2.1.5). Let г be a vector field in J. Them, for s sufficiently
small the mapping
cr:(.-s.5) -> Л'
t ^ оф' o^_fip)
defines a smooth curve on Д’ which passes through p at t = 0. Computing
the tangent vector to this curve at t we get
But since г € _1. we know from Lemma 2.1.4 that, for all q.
(Ф°[Рт(Фв_т(д}') C -1(g)
and therefore we get
e -Ист((>)
for all t 6 (-£.£). This shows that the smooth curve <7 lies on an integral
.submanifold of _L Now. lot pT = Фе_т(р) and p2 — ФЦру). Then pi and p>
are two points belonging to a maximal integral submanifold of _1. and the
previous result shows that Ф^{р1) and Ф& (p->) again are two points belonging
to a maximal integral submanifold of _1. Thus the Theorem is proved for
points pi. р-z such that p-_> — Ф^ (pfi. If this is not the case, using Lemma 2.1.8
we can always find vector fields ту...т> of _1 such that p2 = Ф7{) °- • -°Ф^ (Pi)
and use the above result in order to prove the Theorem. <i
2.2 The Control Lie Algebra 83
2.2 The Control Lie Algebra
The notions developed in the previous section are useful in dealing with the
study of input-state interaction properties from a global point of view. As in
section 1.5. we consider here control systems described by equations of the
form
J71
V = ftp) + ^2 9itp)lh 	(2-3)
i = i
Recall that the local analysis of these properties was based upon the con-
sideration of the smallest distribution, denoted R. invariant under the vector
fields /.	.....g)ri and which contains f.g^.....g,n. It was also shown that
if this distribution is nonsingular. then it is involutive (Lemma 1.8.5). This
property makes it possible to use immediately one of the results discussed in
the previous section and to find a global decomposition of the state space Ah
Lemma 2.2.1. Suppose R is nonsingular. then R has the maximal integral
manifolds property.
Proof Just use Corollary 2.1.6. <
The decomposition of A' into maximal integral submanifolds of R has
the following interpretation from the point of view of the study of interac-
tions between inputs and states. It is known that each of the vector fields
f.g\.....gm is in R. and therefore tangent to each maximal integral subman-
ifold of R. Lot Sp-- be the maximal integral submanifold of R passing through
pz. From what we have said before, we know that any vector field of the for in
т - f w ^2’11 gpti- where iq.......u?f, are real numbers, will be tangent to
Sr° and. therefore, that the integral curve1 of г passing through p~ at time
t ~ 0 will belong to Sp-. We conclude that any state trajectory emanating
from the point pc, under the action of a piecewise constant control, will stay
in Spc.
Putting together this observation with the part (b) of the1 statement of
Theorem 1.8.9. one obtains the following result.
Theorem 2.2.2. Suppose R is nonsingular. Then there exists a partition of
N into maximal integral submanifolds of R. all with the same dimension. Let
Spc denote the maximal integral submanifold of R passing through pa. The set
RijP) of states reachable from pa under piecewise constant input functions
(a)	is a subset of
(b)	contains an open subset of Sps.
This result might be interpreted as a global version of Theorem 1.8.9.
However, then1 are more general versions, which do not require the assump-
tion that R is nonsingular. Of course, since one is interested in having global
84
2. Global Decompositions of Control Systems
decompositions, it is necessary to work with distributions having the maxi-
mal integral manifolds property. From tiie discussions of the previous section,
we see that a reasonable situation is the one in which the distributions are
spanned by a set of vector fields which is involutive and locally finitely gen-
erated. This motivates the interest in the following considerations.
Let {o : 1 < i < q] be a finite set of vector fields and £\.£> two subal-
gebras of V(A') which both contain the vector fields .......r(/. Clearly, tin
intersection £} A ZA is again a subalgebra of V(A') and contains tj.....t,v
Thus we conclude that there exists an unitpie subalgebra £ of Vi.V) which
contains rm...,-/ and has the property of being contained in all the subal-
gebras of V(A’) which contain the vector fields 7j....We refer to this as
the smallest subalgebra of l’( AT which contains the vector fields n....rr
Remark 2.2.1. Ont1 may give1 a description of the subalgebra £ also in the
following terms. Consider the sen
Lc = {7 £ V(A') : t =	....[г,.,. t(!]]]: 1 < d < q. 1 < k < ?c }
and let LC(L-) denote the set of all finite' Пс-linear combinations of elements
of L.-. Them it is possible to see that £ = LC(LA). For. by construction,
every element of Lz is an element of £ because £. being a subalgebra of
V(A') which contains t\......74. must contain (’very vector field of the form
LGy. .........[г,,. tix ]];. Therefore LC(Lc) G £ and also r, G L(?(LC) for
1 < z< q. To prove that £ = LC(L.Z) we only need to show that £C(Lc) is
a subalgebra of V(Aj. This follows from the fact that the Lie bracket of any
two vector fields in Lc is a S-lincar combination of elements of Lc. <
With the subalgebra £ we may associate a distribution -W in a natural
way, by setting
Ac ~ span{r : г e £} .
Clearly, A/; needs not to be noibingidar. Thus, in order to be able to operate
with A^. we have to set explicitly some suitable assumptions. In view of the
results discussed at the end of the previous section we shall assume that £ is
locally finitely generated.
An immediate consequence of this assumption is the following one.
Lemma 2.2.3. If the subalgebra £ is locally finitely generated. the distribu-
tion Ay has the maximal integral manifolds property.
Proof. The set £ is involutive by construction (because it is a subalgebra of
V(Ar))> Then, using Theorem 2,1.5 we see that Ac has the maximal integral
manifolds property. <
When dealing with control systems of the form (2.3). we take into con-
sideration the smallest subalgebra of V(A') which contains the vector fields
f,gx.....y„t. This subalgebra will be denoted by C and called Control Lie
Algebra. With C we associate the distribution
Ac = span{r ; т € C} .
2.2 Tilt1 Control Lie Algebra
85
Remark 2.2.2. It is not difficult to prove that the codistribution is in-
variant under the vector fields /, g} ..... g,ti. For. let т be any vector field in
C and w a covector field in _ic4. Then А т) ~ 0 and A J. г]) — 0 because
[f. r] is again a vector field in C. Therefore, from the equality
7~]) ~ 0
we deduce that LjjO annihilates all vector fields in C. Since _ic is spanned by
vector fields in C. it follows that Le^- is a covector field in Ay. i.e. that _1(4
is invariant under f. In the same war' it is proved that Ay is invariant under
9i.....9>n-
If the codistribution At4 is smooth (e.g. when the distribution -V is noti-
singular). then using Lemina 1.6.3. one concludes that Ac itself is invariant
under /, ,9i....g1u. <
Remark 2.2.2. Tin1 distribution Ac and the distributions P and R introduced
in the previous Chapter arc related in the following wax'.
(а) Л’ С P - span{/} C R
(bi if p is a regular point of A- then A (pi = (P + span{ f}) (p) = R(p}.
We leave to the reader the proof of this statement. <
The role of the control Lie algebra C in the study of interactions between
input and state depends on the following considerations. Suppose A’ 1ms
the maximal integral manifolds property and let Sp° be the maximal integral
submanifold of A’ passing through p°. Since the vector fields f.tp....grn. as
well as any vector field т of the form r - f + j gpij with wi.......um real
numbers, are in Af (mid therefore tangent to Sp-), then any state trajectory
of the control system (2.3) passing through pc at t = 0, due to the action of
a piecewise constant control, will stay in Sf)<>.
As a consequence of this we set1 that, when studying the behavior of a
control system initialized at p3 G AL we may regard as a natural state space
the submanifold Sp= of A’ instead of the whole AL Since for all p e S^. the
tangent vectors f(p).g\ (p).....9m(p)	are elements of the tangent space to 5;,c
at p. by taking tin1 restriction to Sp- of the original vector fields f.g{.... .д)Г1
one may define a set of vector fields f.tp,... .gni on Sp^ and a control system
evolving on Sp-
tn
P =	+	(2A
7=1
which behaves exactly as the original one.
By construction, the smallest subalgebra C of U(.SpO) which contains
f.gi.....glfl spans, at each p € Sr°. the whole tangent space TPSP=. This
may easily be seen using for C and C the description illustrated in the Re-
mark 2.2.1.
86
2. Global Decompositions of Control Systems
Therefore, one may conclude that foi the control system (2.4) (which
evolves on S},-') the dimension of Д. is equal to that of Sp- at each point
or, also, that rhe smallest distribution /? invariant under f.ip.   .i/m «'hich
contains /.t/i.....uonsingular (see Remark 2.2.3). with a dimension
equal to that of Sp=.
The control system (2.4 I is such that the assumptions of Theorem 2,2.2
are satisfied and this makes it possible* to state the following result.
Theorem 2.2.4. Suppose the distribution has the ma.rimal integral man-
ifolds property. Let Sj1C denote the inarimal integral submanifold of Ac pass-
ing through p'. The set R[p:) of states reachable from p~ under piecewise
constant input functions
(a)	is a subset of SP-
(b)	contains an open subset ofSp>.
Remark 2.2.4. Note that, if Д- has rhe maximal integral manifolds property
but is singular, then the dimensions of different maximal integral subman-
ifolds of Д* may be different. Thus, it may happen that at two different
initial states p1 and p~ one obtains two control systems of the form (2.4)
which evolve on two manifolds Sp. and Sp^ of different dimensions. We will
see examples of this in section 2.4. <
Remark 2.2.5. Note that the assumption ’'the distribution Д’ has the max-
imal integral manifolds property" is implied by the assumption "the distri-
bution Д* is nonsingular". In this case, in fact. Д* = R (see Remark 2.2,3)
and R has the maximal integral manifolds property (Lemma 2.2.1), <
We conclude this section by the illustration of some terminology which is
frequently used. The1 control system 1’2,3) is said to satisfy the controllability
rank condition at if	!
dim Д [pc) = n .	(2.5,1
Clearly, if this is the case, and Д has the maximal integral manifolds
property, then the maximal integral submanifold of Д- passing through p'~
has dimension n and. according to Theorem 2.2.4. the set of states reachable
from pc fill up at least an open set of rhe state space A'.
The following Corollary of Theorem 2.2.4 describes rhe situation which
holds when one is free to choose arbitrarily the initial state pT A control
system of the form (2.3) is said to be weakly controllable on A' if for every
initial state /Т 6 A’ the set of states reachable under piecewise constant input
functions contains at least an open subset of A'.
Corollary 2.2.5. A sufficient condition for a control system of the form
(2.3) to be weakly controllable on X is that
dim ДТр) = n
2.3 The Observation Space
87
for all P t AIf the distribution Ac the maximal integral manifolds
property then this condition is also necessary.
Proof. If thF condition is satisfied, Ac is nonsingular. involutive and there-
fore. from the previous discussion, wo conclude that the system is weakly
controllable. Conversely, if rhe distribution _!<. has the maximal integral man-
ifolds property and dim	< n at some p' £ -V then the set of states
reachable from p:' belongs to a submanifold of V whose dimension is strictly
less than n (Theorem 2.2.4). Therefore, this >et cannot contain an open subset
of X. <
2.3 The Observation Space
In this section we study state-output interaction properti--- from a global
point of view, for a system described by equations of the form (2,3). together
with an output map
у = h[p) .	(2.6)
The presentation will be closely analogue to the one given in the previous
section. First of all. recall that the local analysis carried out in section 1.9 was
based upon the consideration of the smallest codistribution invariant under
the vector fields f. <j\.g„,	and containing the covector fields dh\...dig.
If the annihilator Q of this codistribution is nonsingular. then it is also invo-
hitive fLemma 1.9.5) and max' bo used to perform a global decomposition of
the state space. Parallel to Lemma 2.2.1 we hare the following result.
Lemma 2.3.1. Suppose Q is nonsingular. Then Q has the maximal integral
manifolds property.
The role of this decomposition in describing the stare-output interaction
may be explained as follows. Observe that Q. being nonsingular and involu-
tive. satisfies t he assumptions of Theorem 2.1.9 (because the set of all vector
fields in a nonsingular distribution is locally finitely generated). Let S be any
maximal integral submanifold of Q. Since Q is invariant under f.gy,... .g}ll
and also under any vector field of the form т = f	дрд. where
tri....U),, arc real numbers, using Theorem 2.1.9 we deduce that given any
two points //' and // in 5 and any vector field of rhe form т = f	t/qq,
the points Ф^[рл ] and Ф’(pb i for all t belong to the same maximal integral
submanifold of Q. In other words, we set1 that from any two initial states
on some maximal integral submanifold of Q. under the action of the same
piecewise constant control one obtains two trajectories which, at any time,
pass through the1 same maximal integral submanifold of Q.
Moreover, it is easily seen that the functions /q...hj are constant on
each maximal integral submanifold of Q. For. let S be any of these subman-
ifolds and let denote the restriction of h, to S. Ar each point p of 5 the
88
2. Global Decompositions of Control Systems
derivative of /p along any vector r of TPS is zero because Q C (span{dh,})_.
and therefore the function /p is a constant.
As a conclusion, we immediately see that if //’ and pb art1 two initial
states belonging to the same integral manifold of Q then under the action of
the same piecewise constant control one obtains two trajectories which, at
any time, product1 identical values on each component of the output, tug. art1
indistinguishable.
These considerations enable us to state the following global version of
Theorem 1.9.7.
Theorem 2.3.2. Suppose Q is nonsingular. Then there exists a partition of
.V into maximal integral submanifolds ofQ. all with the same dimension. Let
denote the maximal integral submanifold of Q passing through p~. Then
fa) no other point of Sp~ can be distinguished from p" under piecewise con-
stant input functions
fb) there exists an open neighborhood L of jf in .V with the property that, any
point p € P which cannot be distinguished from /Т under piecewise, constant
input functions necessarily belongs to U Я Sp=.
Proof. The statement (a) has already been proved. The statement (b) re-
quires some remark. Since Q is nonsingular. we know that around any point
we can find a neighborhood P and a partition of P into slices each of
which is clearly an integral submanifold of Q. But also the intersection of S},~
with th which is a nonempty open subset of Sfl?, is an integral submanifold
of Q. Therefore, since Spc is maximal, we deduce that the slice of P passing
through pr~ is contained into I' Я Sp=. From the statement fb) of Theorem
1.9.7 we deduce that any other state of P which cannot be distinguished from
p° under piecewise constant inputs belongs to the slice of P passing through
p° and therefore to P Я S})°. <
If the distribution Q is singular, one may approach the problem on the
basis of the following considerations. Let {A; : 1 < i < 1} be a finite set of
real-valued functions and {r( : 1 < i < q} be a finite set of vector fields.
Let and S> be two subspaces of Cx (A’) which both contain the functions
AL А/ and have the property that, for all A £ 5, and for all 1 < j <
q.LTjX E Sj.i — 1.2. Clearly the intersection Si n<S-_> is again a subspace of
CX(.V) which contains Ai...., А/ and is such that, for all A e Si П& and for
all 1 < j < q,LTjX E 5] P S>. Thus we conclude that there exists a unitine
minimal subspace 5 of Cx (Aj which contains At..... А/ and is such that, for
all A e S and for all 1 < j < q. L-.X E S. This is the smallest subspace of Cx
which contains At.....А/ and is closed under differentiation along ..r7.
Remark d.'d. 1. The subspace 5 may be described as follows. Consider rhe set
So = {A e C^fA7) : A= Aj or
A = LTi[ .. .L^XjA <j <1.1 < iK. <q.l<k< oc}
2.3 The Observation Space
89
and lot LC(S-) denote the set of all 2-linear combinations of elements of
So. Then. LC(SZ) — 5- As a matter of fact, it is easily checked that every
element of LC(SZ) is an element of 5. so LClSOj G <S. that A; £ LC(So) for
1 < j <1 and that LC(SZ) is closed tinder differentiation along и.....тч. <
it li the subspace 5 we may associate a codistribution in a natural
way. by setting
= span{dX : A £ 5} .
The codistribution is smooth by construction, but as we know the
distribution may fail to be so. Since we are interested in smooth distribu-
tions because we use them to partition the state space into maximal integral
submanifolds, we should rather be looking at the distribution smt(J?^) (sec
Remark 1.3.3).
The following result is important when looking at smtflTy-) for the purpose
of finding global decompositions of ;V.
Lemina 2.3.3. Suppose the set of all vector fields in smt(-r7$) is locally
finitely generated. Then suit(1?^) has the maximal integral manifolds prop-
erty.
Proof, In view of Theorem 2.1.5. we have only to show that smt(f?^) is
involutive. Lot Ti and m be two vector fields in smt(.Chf) and A any function
in 5. Since (dX. tj) ~ 0 and fiiX.m} = 0 we have
(t/A. [ту . t2]) = LTl (dX. T-,} - LrfidX, n) = 0 .
The vector field [л.ту] is thus in .C?^. But	being smooth, is also in
smt(Lffr). <
In order to study the observability we consider the smallest subspace of
C^foA ) which contains the functions /ij.....hi and is closed under differen-
tiation along the vector fields f. g\,..,. g„t. This subspace will be denoted
by О and called the Observation Space. Moreover, with О we associate the
codistribution
Oo = span{dA : A e 0} .
Remark 2.3.2. It is possible to prove that the distribution is invariant
under the vector fields figx....gm. For. let A be any function in О and r a
vector field in O&. Then (dX. т) = 0 and (dLjX.r) = 0 because L/X is again
a function in O. Therefore, from the equality
(dA,[/.r]) = Lf(dX.r} - (dLfX.r) = 0
we deduce that [/, r] annihilates the differentials of all functions in O. Since
is spanned by differentials of functions in 0. it follows that [f. r] is a vector
field in Ofi. In the same way one proves the invariance under gi......gtn.
If the distribution is smooth (e.g. when the codistribution Qo is
nonsingular) the using Lemma 1.6.3 one concludes that Оо itself is invariant
under f. ep...., gm. <
90
2. Global Decompositions of Control Systems
Remark 2.3.3. The distribution and the distribution Q introduced in the
previous Chapter are related in rhe following way
(a)	C Q
(b)	if p is a regular point of f?c7. then f2^(p'j = Q(pL
We leave to the reader the proof of this statement, <
From the previous Remark 2.3.2 and from Remark 1.6.5 it is deduced that
the distribm ion siut() is invariant under the vector fields f. <p......g,u and
so under any vector field г of the form г = f + PP6- where щ................a,,,
are real numbers. Now suppose that the s<4 of all vector fields in smt(f?cl I is
locally finitely generated, so that suit(Qp,’) has the maximal integral manifolds
property. Using Theorem 2.1.9. as we did before in the case of nonsingular ().
wo may conclude that from any two start's on the same integral submanifold
of suit I X?—). under the action of rhe same piecewise constant control one
obtains two trajectories that at any time lie on the sann1 maximal integral
submanifold of sintff?^). Observe now that smtlf?^) is also contained in
(span{(///;-}U  1 < i <1- because every tangent vector in smt I )(pi is also
in f^(p) and every tangent vector г in ddp(.r) is such that (dh,(p).i'') = 0.
Therefore one may deduce that the functions /у are constant on each maximal
integral submanifold of smt(f?^).
This, together with the previous observations, diows that any two initial
states pa and pb on the same maximal integral submanifold of smt( <2^ 1 arc* in-
distinguishable under piecewise constant inputs. This extends the statement
(a) of Theorem 2.3.2. As for the statement (b). some regularity is required,
as it is seen hereafter.
Theorem 2.3.4. Suppose the set of all vector fields contained in smtlP^l
is locally finitely generated. Let Sp^ denote the maximal integral submanifold
o/smt(G(u) passing through p~. Thefi
(a)	no other point of Sj,-- can be distinguished from p~ under piecewise con-
stant inputs
(b)	if p° is a regular point of then there exists an open neighborhood U of
p~ in Л with the property that any point p G U which cannot be distinguished
from pc under piecewise constant inputs necessarily belongs to Г П S/t-.
Proof, The statement (a) has already been proved- The statement (b) is
proved essentially in the same way as the statement (b) of Theorem 2.3.2. <
The following example illustrates the need for the “regularity" assumption
in the statement (b) of the previous Theorem.
Example 2.3.4- Consider the following system with V = R and
j- == 0
У = h{.r)
2.4 Linear Systems and Bilinear Systems
91
where h(4’l is defined as
hi г i = exp( — —) sin (—) for .r 0
.r- .r
/HO'l = (J.
For this system. two start's .r,! and ff are indistinguishable if and only
if h(.r") = hl./1). In particular, the set of states which are indistinguishable
front the start1 .r = 0 coincides with the set of the roots of the equation
/г(.г) = (). Each point in this set is isolated but tin* point .r = 0. Thus, no
matter how small we choose an open neighborhood L of ,r — 0, U contains
points indistinguishable front .r = U.
It is also set'll that the codistribution = span{t//?} has dimension 1
everywhere but at the points ,r in which dh/d.r = 0 where its dimension is 0,
Thus, any smooth vector field belonging to must vanish identically on IE
and suit() = {()}. The maximal integral submanifold of smr(f2^j’) passing
through ,r is the point .r itself.
At tht' point ,r — (1. which is nor a regular point of bff- we have that
U Г S; = {0} for all Г. whereas we know there are other points of Г indis-
tinguishable from .?• = 0. <
We conclude this section with some global considerations. The control
system (2.3)-(2.6) is said to satisfy the observabddy rank condition ar pz if
dim -C\e(7>Jj = n .	(2.7)
Clearly, if this is the case then p~J is a regular point of and from the
previous discussion it is seen that any point p in a suitable neighborhood I
of pr can be distinguished under piecewise constant inputs. A control system
of the form (2.31-12.6) is said to be locally observable on A’ if for every state
p: there is a neighborhood fd of/г in which every point can be distinguished
from p1' under piece wise constant inputs.
Corollary 2.3.5. .4 sufficient condition for a control system of the form
(2.2 )-t 2.6) to be locally observable on N is that
dim = n
for all p £ A".
2.4 Linear Systems and Bilinear Systems
In this section we describe some elementary examples, in order to make the
reader more familiar with the ideas introduced so far.
As a first application, we shall compute the Lie algebra C and the distri-
bution Д/ for a linear system
92
2. Global Decompositions of Control Systems
j' — .4,г + Bu
у = C.r ,
Recall (see also section 1.2) that, this is indeed a system of the form (2.3)-
(2,6). with Л’ = FC anti
/(.r) = Л. г
yi(x) = 6,	1 < i < m
where C is the /-th column of the the matrix В and
/p(j’) = (‘).r	1 < i. <1
where c( is the /-th row of the matrix C.
We want to prove first that the control Lie algebra. C is the subspace of
VRV) consisting of all vector fields which are IR-linear combinations of the
vector fields in the set
{.4.r} U {.4A’C : 1 < ? < m. 0 < k < n - 1} .	(2.8)
For. observe that this set contains the vector fields Aj- and ...b,}) (i.e. the
vector fields f.gi....g,ri) and also that this set is contained in C. because
any of its elements is a repeated Lie bracket of f and g\...grn. As a matter
of fact.
до, = о1>Мл 
Moreover, it is easy to see that the set
LC({.-Lr} (J {Aa’6, : 1 < i < m.O < k < n - 1})	(2.9)
of all S-linear combinations of vector fields in the set (2.8) is already a Liz1
subalgebra, i.e. is closer] under Lie bracketing.
For. one easily sees that if 7i(t] and 72(*r') are vector fields of tin1 form
7i M = Akb,
7-2(т)	=	AKbj
then [7i.72l(.r) = 0. On the other hand, if
И (,r) = Aa'6,
7-j(z)	=	A.r
then
[n. 7->] = -4A‘^ 1 bt ,
If A < n — 1. this vector field is in the set (2.8) and, if A: = n — 1. this
vector field is an IR-linear combination of vector fields in the set (2.8) (by
Cayley-Hamilton Theorem).
2.4 Linear Svstern> and Bilinear Systems
93
If ту and ~2 are E-linear combinatioib of vector Helds of (2.8) then their
Lie bracket is still an Et-linear combination of vector fields of (2.8,]. and this
proves that the set (2.9) is a Lie subalgebra.
The set (2.9) is a Lie algebra which contains /. tp...g,u and is contained
in C. rhe smallest Lie subalgebra which contains .........gn,. Then, the sot
(2.9) coincides with C.
Evaluating the distribution _1L’ we get. at a point ,r £ T" .
_ltd.r) — span{.4.r} -i-span{.46, : 1 < t < m.O < A' < n — 1}
,!-1	(2.10)
= span{.4j-} - Iin(_4A’B) .
A-=0
We are now interested in the distribution F. the smallest distribution
which contains щ.......g,-,, and is invariant under f. g{.g,tl. By means of
calculations similar to the ones in Remark 1.8.1. it is easy to check that at
any point x G TT
P(.r) ~ span{.4A'6, : 1 < i < m.O < A‘ < r? - 1} .	(2.11)
Thus, we see that
-V = span{/} + F .
The distribution _1L’ is spanned by a set of vector fields which is locally
finitely generated (because any vector field in C i.s analytic on SL ), and there-
fore by Lemma 2,2.3 - the distribution _1L’ has the maximal integral man-
ifolds property. The distribution F is nonsingular and involutive and thus
by Corollary 2.1.6 it also has t he maximal integral manifolds property.
The maximal integral submanifolds of P. all of the same dimension, have
the form r + V. where
V = Im(B AB ... _4"-lB) .
The maximal integral submanifolds of Д; may have different, dimensions,
because _lc may have singularities,
If. at some point r E EC, /(.r) e F(j-). then the maximal integral sub-
manifold of passing through ,r coincides with the one of rhe distribution
F. i.e. is a subset of the form ,r V. Otherwise, if such a condition is not
verified, the maximal integral submanifold of Д’ is a submanifold whose1 di-
mension exceeds by 1 that of F and this submanifold, in turn, is partitioned
into subset,s of the form F + IL
Example 2-4. E The following simple example illustrates the case of a singular
Д’. Let the system be described by
/1	0	(A	/1\
,r -	I 0	-1	0	j +	0	u .
\o	0	1/	Vv
94
2. Global Decompositions of Control Systems
Then we easily sec that
V = {.r e ?? : ./-2 - ,r3 = 0}
and that
P = span{-—} .
()d 1
The tangent vector /(y) belongs to P only at those ,r in which = ,r3 =
{). i.e. only on I . Thus, the maximal integral submanifolds of -V will have
dimension 2 everywhere but on U. A direct computation shows that these
submanifolds may be described in the following way (Fig.2.1)
Fig. 2.1.
(i) if r' is such that jg - 0 (resp. .r3 = 0) then the maximal submanifold
passing through .P is the half open plane
{.r e : T-2 = 0 hud sgn(j’:>) = sgn(j-^)}
(resp. {j- e S3 : t3 = 0 and sgn(.r2) = sgn(.rj)})
(ii) if ,rc is such that both 0 and .r3	0. then the maximal submanifold
passing through ,r= is the surface
{.r E R3 : т2.г3 — .rTr|} .<
We turn now to the computation of the subspace О and the codistribution
It is easy to prove that (9 is the subspace of CY(.V) consisting of all
IE-linear combinations of functions of the form or CjAK'bj. namely that
О = LC{X e (W) : A(j') = c, Akj- or
(2 12
Air) = CjAkbj: 1 < ? < /.1 < j < m.O < k < n - 1} .
For, note that functions of the form c( AAhr or ctAkbj are such that
2.4 Linear Systems and Bilinear Systems
95
ctAkx - 7,£/q(.r)
У-4А7у = L^Lj/qU)
ami this implies that the right-hand side of (2.12) is contained in O. Moreover,
the functions hi....h; are elements of the right-hand side of (2.12). Then,
the proof of (2.12) is completed as soon as we show that its right-hand side
is closed under differentiation along f.gi..g,u.
If A(.r) = then LfX = c!.4A"'“I.r and £^A(.r) = г,Аа’67. If A(j-) =
сг.4А7у, then ZjA(.r) = Ly3X(x'} ~ 0. Thus, using again Cayley-Hamilton
Theorem, it is easily seen that the right-hand side of (2.12) is closed under
differentiation along f.gi...gm.
At each point .r. the codistribution _QC? is given by Co(t) = span {r, .4* :
1 < i < /. 0 < A' < n — 1}. Thus. is nonsingular and, in view of Remark
2.3.3 (part, (bl)
n-i
УУ.г) = П k('r(C.4‘') = Q(J-)
k=0
(see also Remark 1.9.1)- Tht1 maximal integral submanifolds of Q have now
the form j‘ + IT where
и — i
1Г = P| кег(С.4г) .
г=0
As a second application we consider a bilinear system. i.e. a system de-
scribed by equations of the form
i = A.r+ £Xt(A»tq
!J = Cf
Hen1 also the manifold on which the system evolves is the whole of R” . / and
h\.....hi are the same as before, and
= Xj- 1 < i < n? 
In order to compute the subalgebra C we note first that any vector field
r in the set {/.  has the form r(jj = Tx. where T is an n x n
matrix. If we want to take the Lie bracket of two vector fields iq. r2 of the
form
T[(.r) - T\x T?(.r} = T2x
we have
[n.njM - (T2Ti -T}T2)x = [Л. T2]x
where [Ti-Tb] = {T2T[ - is the commutator of Th and T2,
On the basis of this observation, it is easy to set up a recursive1 procedure
yielding the smallest Lie subalgebra which contains a set of vector Helds of
the form ri (.r) = T\ x...., тг(т) = T,x.
96
2. Global Decompositions of Control Systems
Lemma 2.4.1. Consider the nondccieasing segue nee of subspaces of
the space of all n x n matrices of real numbers, defined by setting
-M() = span{Ti....T,\
M, = .XI,+span{ir1.r].......[ТГ.Т] : T G Л/д-i } -
Then, there exists an integer k* such that
-Xh = M,
for all k > k*. The set of vector fields
C = {n G V(l" ) : r(.r) = Tj-.T G Л/д.}
is the smallest Lie subalgebra of vector fields uchich contains ^(.r) = Ti.r
7>(.r) = Tr,r.
Proof. The proof is rather simple and consists in the following steps. A dimen-
sionality argument proves the existence of the integer k* such that Л/д = M,--
for all к > к*. Then. one checks that the subspace Л/д- contains T\..T>.
and any repeated commutator of the form [T,.............	T(f]]] and is such
that P. 62] -Xlk- for all P G Л/д.. and Q G Л/д... From these properties, it i>
straightforward to deduce that £ is the desired Lie algebra. <
Based on this result, it is easy to construct the Lie algebra C by simply
initializing rite algorithm described in the above1 Lemina with the matrices
A.Vi.......V„(.
In this case, unlike the previous one. we cannot anymore give1 a simple
expression of -V’(.r) and/or its maximal integral submanifolds. In some spe-
cial situations, however, like rhe one illustrated in the following example, a
rather satisfactory analysis is possible,.
Example Consider tin* system '
.r = A.r + .Vzu.
where .г G IF3 and
/0	1	0\	/ 0	(J	1 \
A =	-1 0 0 .V = (J 0 0 .
\ 0	0	0/	\-1	0	0/
An easy calculation shows that
/ 0 0 0 X
[A. .V] = 0 0	1
\0 -1 0/
[A7. [A.A”-i - A [A, [A. .V]] = -Ah
2.1 Linear Systems and Bilinear Systems
97
Therefore, we have
C = {rG Vi E3 i : 7(.r) = Ti-.T G span{.4.V [A. V}} .
To compute the dimension of Ac we evaluate the rank of the matrix
/ .r-j	J’a	0 \
(Ar. _V.r. A.A'h) = I —J’!	О	,Г3
\	0 —.Cl — J‘2 /
and we find the following result
dim Ac fir) = i)	if .r = 0
dim Ac Lr) = 2	if .r (J .
A direct computation shows that the maximal integral submanifold of Ac
passing through .m i> the set
{z G E3 : r’2 -r J’2 + 4 = (.r{ )’2 - (,r; )2 - (,r:; )2}
i.e. the sphere centered at the origin passing through .r=.
Therefore, we can say that the stale of the system is not free to evolve on
the whole of E?. but rather on the sphere centered at the origin which passes
through the initial state.
Around any point .r 0 the distribution Ac is nomingular. so we can
obtain locally a decomposition of the form (1.40). by means of a suitable
coordinates transformation. To this end, we may make use of the construction
introduced in the proof of Theorem 1.4.1 and find a set of three vector fields
T\-T->.T2 with tlit1 property that Tj and r-i belong to Ac and rj'.r0). таЦ’0)
and тз(.r°) are linearly independent. If we consider an initial point on the line
p G : 71 = -r2 = 0}
we may take1 the vector fields
ЛИ = (-VjA
r, (r) = ([А.Лф-)
71 - 9
Accordingly we get
/ (cos thy + (sin t);r3 \
Ф* (,r) = I	JO
\ — (sinh-7] + (cos/):r3 У
I J1	\
Ф2(г) = (cost),r_> 4- (bint)jy,
(sinh-7 2 + (COS h-7.3 /
98
2. Global Decompositions of Control Systems
=	.r>
\f + 'Гз /
The local coordinate chart around the point .rJ is given by the inverse of
the function
/' : (. c-j. '3) ^ $!. о Ф:„ о Ф?’ (,r'?) .
For ,r\ = ./'2 = I) and = a we have
((sin cj )(cos i(-~3 + «} \
(sin 1-2 )(<j + (1)
(cos г1 Jfcos z-2 )Сз + n) J
The local representations of the vector fields f and g in the new coordinate
chart are given by
Gl = U.r'/UC)) = (ПГ ’.4C(-')
<)(Cl = icrWOfaWi:).
A simple but tedious computation yields
We conclude that around .rc the system, in the z coordinates, is described
by the equations
Cj = cos c1 ran z-> + и
z> = - sin .11
= 0 .<
The study of the observability of a bilinear system is much simpler. By
means of arguments similar to t.hose/used in the case of linear systems it is
easy to provc that О is given by
О = ЛС{А e CX(A) : A(.r> = c(-.r or
A(a-) = с,Л\ ... Ay, .г: 1 <?</. 1 < к < n - 1.0 < jx....jk < m}
(with .Vo = .4). Therefore
11 — l	in
'? P	А	кег(СЛу....Ул).
A —-E) ji-Jk=O
The distribution f?p = Q is nonsingular and its maximal integral sub-
manifolds have1 the form т + TV. where now
Ii — 1 in
1Г=П A k«r(C.y,...;VA).
A’=0
2.5 Examples
99
It may be worth observing that the subspace IE thus defined is invariant
under .4.Л\......._V(71. is contained in ker(C) and is the largest subspace of
]Rn having these properties. From linear algebra we know that by taking
a suitable change of coordinates in (see e.g. section 1.1) the matrices
Д,.........-Vt?) become block triangular and. therefore, the dynamics of the
system become described by equations of the form
tn
j‘i = .411 .r i + .412 .r-j ~	J A t, 11 .r i + A ,. i _> .r-j) a,
,= i
in
.r> =	Лг.22>Г2'О 
i = i
Moreover, the output // depends only on the .r-j coordinates, у = C_>.r_>.
The above equations are exactly of the form (1.38). this time obtained by
means of standard linear algebra arguments.
2.5 Examples
In this section we discuss an example of application of the theories illustrated
in the Chapter to a control system whose state space is a manifold A not dif-
feomorphic to ?/. More precisely, we study the system already introduced
in section 1.5 which describes the control of the attitude of a spacecraft by
means of equat ions of the form

(2-13)
В
SU}B
(2.1 1)
with state (-<;./?) G ?? x S(9(3) and input 1 g -M  The orthogonal matrix
В represents the orientation of the spacecraft with respect to au martially
fixed reference frame, the vector its angular velocity, and the vector T
represents the external torque. The matrix J is the so-called inertia matrix
of the spacecraft, and S(w) is tin1 skow-syinmetric matrix
If we suppose the external torque T generated by a set of r independent
pairs of gas jets (thi u.stcr.s). it is possible to set
T = biui + brur
when1 ........b, G F? represent rhe vectors of direction cosines with respect
to the body frame of the axes about which the control torque?, are applied
100
2. Global Decompositions; of Control Systems
and ?/1....ur the corresponding magnitudes. Of course, we assume the set
{61.....br} is a linearly independent sei (and thus r < 3).
We want, to analyze1 the partition induced by the distribution _iy. in the
two cases r = 3 and r = 2. For convenience, we begin by discussing the
dynamic equation (2.13) only. Note that, stating
,r = ./w
and using the property
Sim);' = -.Sir)»'
(which holds for any pair of vectors c. ir G B3 ) the equation in question can
be rewritten in rhe form
.r = -S(.r>.7-l.r + Bn
where В = (6L ... 6/1. i.e.
c = /(z) + fli I .r) tt i — •  > + gr (.r) и,
with
/uT = —	=6,	1 < i < г .
The cast1 in which r = 3 is rather trivial. In fact, since1 the control Lie
algebra C contains, by definition, the three vector fields t/j (.r). r/jl-r). t/j(.<).
and these vector fields which are constant art1 by assumption linearly
independent at each r g B3, we have immediately
A’(.r) = T/Z’ for all .r G B3.
In other' words, the controllability rank condition (2.5) is satisfied at each .r.
and the partition of T? induced by degenerates into one single element,
namely F? itself-
The cast1 г = 2 is more interesting (at least from the point of view of the
analysis). In this case, to obtain meaningful information, out1 has to compute
a few Lit1 brackets between f Lr) and the //, (-c)'s. Let n and c? be real numbers
and consider the (constant I vector field
t/U’,1 = CO <71 (./) + cm/jLr) -
Since, as a straightforward calculation shows.
+S(J~\r}
ch-
then. setting 6 = C]6] +c-2b-_>. it is immediate to see that
[/,,y](.r) =	6 - 5(	1 .r)6
2.5 Examples
101
By definition, the control Lie algebra contains the three* (constant) vector
fields .Qi(J-<?]("»’). Thus, if
гаик(Ъ| b2 S(b)J~4j) — 3	(2.15)
we again obtain that has dimension 3 at each z as before, and the
associated partition of R3 degenerates into oik* single element.
Note that the vector b is an arbitrary vector in the image* if the matrix
В = (bY b2)
and. therefore, the possibility of having the condition (2.15) fulfilled can be
restated in the following terms
S(b).J~xb Ini(B) for some 6 G Ini(f?) -	(2.16)
We show now that. If the condition (2.16) is not satisfied, then Ap lias
dimension 2 at all points of a certain plane in R3 . For. let denote a (nonzero)
row vector satisfying
а В = 0
and suppose* a linear (thus globally defined) coordinates transformation is
performed, changing z into z = T.r. with
А И = ~z .
By definition of z, we have
= -,z =	+ Ba) = -mSVV’-i- 	(2Л7)
If the condition (2.16) does not hold, at each point z of Im(B). S(.r).7~lz
is a vector in Im(B). Since
z € Im(£?) «* o.r = 0 <=> ~i (z) — 0
we see from (2.17) that, if the condition (2.16) docs not hold, at each point
where ct = 0. then i1 = 0 also. This means that any trajectory of the system
(2.13) starting in tin* plane
.U = {z G R3 : yz = 0}
remains in this plane for all times. As a consequence, in view of the results
established in section 2.2. we deduct* that necessarily Ac has at most dimen-
sion 2 at each point of A 7 (in fact, it has dimension 2 because and b2 are
independent).
At every other point, z .W. we assume that Ac has dimension 3. To have
this hypothesis fulfilled, it suffices to observe that the control Lie algebra
contains the three vector fields (}i (z).	[/. and to assume that at
least for one value of z and one value of i these three vectors are linearly
independent, i.e.
102
2. Global Decompositions of Control Systems
det(6] b-2	J’))	0 .
In fact, this determinant is linear in .r and. if not identically zero, can only
vanish at points of a plane. which necessarily are the points of .V.
Summarizing, if the condition (2.16) does not hold and tin* determinant
dct(&i hj [/. y,\(.J‘)) is not identically zero (for one value of /). the distribution
Д’ has dimension 2 at all points of Л/ and dimension 3 everywhere else. A<
a result, the state space of (2.13) is partitioned by Ae into three maximal
integral manifolds: the plane _W and the two (open) half-spaces separated by
.W.
We study now the same kind of problems for the full system (2.131-( 2.14).
whose state space is the* manifold
X = 5? x 50(31 .
To this end. a few preliminary remarks about the structure of rhe tangent
space to SO{3) are in order.
Recall that 5(1(3) is an embedded 3-dimensional submanifold of rhe man-
ifold A3> 3. As a consequence, the tangent space to 5(1(3) at /? can be viewed
as a(3-dimensional) subspace of rhe1 tangent space Тд23’3. Let .уу denotes
the (l.j) clement of a matrix X 6 T3>3. and choose1 the natural (globally
defined on 1R3<3) coordinate functions
{<afj(A ) — .I'tj : 1 < i.j < 3}
This choice induces, at each X. a choice of a basis for
set of tangent vectors
3. namely the
i.2. IS)
Using this basis, any tangent vector r ar a point X of.*3
in the form
{ will be represented
denotes the (?.j) element of a 3 x 3 matrix V.
where co
Now. consider the three matrices
/ ()	1 0\
At = I -1 0 0 A2 =
\ 0	0 f) /
()	0	1 \	/о	о	0
0 0 l) Аз = I о () 1
-1	0	0 /	\0	-1	(J
and the corresponding exponentials expl A[ f). <‘xp( A>t). exp(A:!/1. with f G X.
An easy calculation shows that . for each 1 < k < 3, exp( Art) is an orthogonal
matrix, with determinant equal to 1. Thus. exp(AC) € 5(1(3). Consider now
rhe mapping
П : й -о 5(1(3)
t i-> (expt AjA))/?
2.5 Examples
103
where R is an element of 569(3). By construction. is a smooth curve on
569(3). passing through /? at t = 0. Its tangent vector at f = 0. in the basis
(2-18). is represented by the matrix
[A-(/)y_o = R
i.e. has the form
Since the throe matrices . _42- Ai are linearly independent, so are the three
corresponding tangent vectors {»’i- tg- 1'з}- Moreover, each сд- is an element of
TrS()(3) by construction, and ТцЗСРЗ] is 3*dimensional. As a consequence',
we can conclude that rhe set {iq. n>. tg} is actually a basis of Tj{SO('3}.
In particular we see that, in the basis E2.1S). any vector of T^SO(3) can
be represented by means of a matrix of the form
ci .4! I? — c_> .4-jR + c3.43R
where tq.c2.c3 are real numbers, i.e. in view of the special structure of
Ai. .4-2. A3 in tin1 form
(0 ct ro \
-ci 0 C3 I R = Su')R
—ry, — C3 0 /
when' r = col(c3. -r-j,ci).
We return now to the problem of discussing the partition of the state
space of system (2.13'1-1'2.14) induct'd by the distribution Ay. The system in
question has the form
P = f(p') + .91 (../Ал 4-H 9r(PiMr
with
P = (.r.R) € -V = ?? x 569(3)
p e ТРУ = ТГА’ xTi{SO(3)
Цр] =	S(J- l.HR)
<p(p) = (Ml) 1 < ? < r
(recall that we set Лс = .r).
Suppost' r — 3 and note that, by definition, the control Lie algebra C
contains the six vector fields y;(p). [f. g,]pp}. 1 < i < 3. An easy calculation
shows that
[/.У(.г.Л) = (l.aS(g^ l,r)l>,.-.S(.7 4)7?) .
Note that the fe/s are linearly independent vectors, and so are the1 vectors
J }lp and the matrices 5(J-16;)R. 1 < i < 3- Thus, in view of the previous
104	2. Global Decompositions of Control Systems
discussion, we deduce that the1 matrices S(.7-16()/?. 1 < i < 3 represent, in
the basis (2.18). three independent tangent vectors that span ThS()(3). for
each J? E SO(3). On the other hand, the vectors 1 < i < 3, span Гг??.
Therefore, we can conclude that the set of six vectors gt(p).[f. <7,’]lpl. 1 <
i < 3 span the tangent space x T^SO(3) at each (.r./?) € A’. The
controllability rank condition (2.5) is satisfied at each point, and the partition
of .V induced by -V degenerates into one single ('lenient, namely .V itself.
The study of rhe case in which r = 2 can be carried out in the same way.
using the condition 12.16) to show that rhe mar rices S( J-1 bY)/?, S(J~} b_> )R.
and
(with b = cjn + ejb?) are linearly independent, and proving that TpA' is
spanned by p, (p), [/-y,] (- 1 < i < 2.[[f. g]. g](p). and [f. [[f. g\. g]]{p). with
g(p) = <3gi(p) +c>g2(p)-
3	. Input-Output Maps and Realization Theory
3.1 Fliess Functional Expansions
The purpose of this section and of the following one is to describe represen-
tations of the input-output behavior of a nonlinear system. We consider, as
usual, systems described by differential equations of the form
г =	(31j
!Jj = hjfo 1 < j < p .
This system, as in Chapter 1. is assumed to be defined on an open set
U of . Moreover, throughout the present Chapter, we constantly suppost1
also that the vector fields f. r/t.are analytic vector fields defined on
Ch Likewise, the output functions fo....hp are analytic functions defined
on Г.
For the sake of notational convenience most of the times we represent the
output of the system as a veer or-valued function
у - /for) - colf/q (.r).hfo J’) 1 .
We require first some combinatorial notations. Consider the set of m — 1
indices I = {(). 1........tn} (we represent here1, as Ui-aial. indices with integer
numbers, but we could as well represent the m + 1 indices with elements of any
set Z with card! Z) = in I). Let 7}. be the set of all sequences (q.    >i I of k
elements ......h of I. An element of this set Д. will be called a multiindex
of length k. For consistency we define also a set fo whose unique element is
the empty sequence (i.e. a multiindex of length U). denoted 0. Finally, let
/• = u
k>l)
It is easily seen that the set can be given a structure of free monoid
with composition rule
bк • • • i i)[jh   71)	(С-   b fo  J i)
with neutral element 0.
1()6	3. Input-Output Maps and Realization Theory
A formal poirtr series in m + 1 noncominutative indeterminate* and co-
efficients in E is a mapping
r : Г И .
In what follows we represent the value of c at some <’1< inf'iir b, ... m of I'
with the symbol c{i^. ... i0).
The second relevant object we have to introduce is called an iterated n.~
tec/ral of a given set of functions and is defined in the following way. Let
T be a fixed value of the time and suppose ... a,,, are real-valued piece-
wise continuous functions defined on 0.7:. For each nniltiindex I ц. ... fid the
corresponding iterated integral is a real-valued function of t
defined for (I < t < T by recurrence on the length, setting
ч	.НТ) = J() uj.rhfi for 1 < i < m
and
/ (/£,, . , . dft = / dfn. It] / r/^(. _ ... df hi 
m	Jo Jo
The iterated integral corresponding to rhe multiindex 0 is the real number
E.rample J.1.1. .Just for convenience. 1(4 us compute the first few iterated
integrals, in a case where m = 1.
Given a formal power scries in m — 1 non-commutative indcterniinates. it
fi possible to associate with thi> scries a functional of ay.........u!lt by taking
the sum over 7* of all the products of the form
z-f
(‘(h    h)l /	   dG.- 
Jo
The convergence of a sum of this kind is guaranteed by some growth
condition on the "coefficients'' c(b-    as stated below.
3.1 Fliess Functional Expansion1
107
Lemma 3.1.1. Suppose there exist real numbers К > 0. _U >0 such that
\(Vk...i[i} < K(A’+	(3.2)
for all h > (1 and all multiindices if,-... io.
Then, there exists a real number T > 0 such that, for each 0 < t < T ami
each set. of piecetrise. continuous functions u^....uni defined on 0. T] and
subject to the constraint
max //,( — )’ < 1.	(3.3i
o<-</
the series
f) = r(0) -t- У H if, ... toi d^.K...d^j	(3.4'.i
k-(i i .ч -U
is absolute Ip and uniformly converge nt.
Proof. It is easy to see, from the definition of iterated integral that, if the
functions щ ... um satisfy rhe constraint (3.3). then
rt
If the growth condition is satisfied, then
г1	!
c(if. ... io} / t/fy,    dh,i.,: < A~[M\ ш + 141 .
fr..
As a consequence. if T is sufficiently small, the series (3-4) converges
absolutely and uniformly on (). Т]. <
The expression (3.4) clearly defines a functional of m ... ut)l. This func-
tional is causal, in the sense that g[f) depends only on the restrictions of
U;.....uni to the time interval [Iff].
A representation of the form (3.4) is unique.
Lemma 3.1.2. Let c" and cb be tiro formal power series in in + 1 non-
conimutatire tn determinates and let the associated functionals of the form
(3.4/ be- defined on the same interred [0. Т]. Then the two functionals coincides
if and only if ca = r’1.
Proof. We provide only a siniplc sketch of rhe first few passages involved. Let
r". oh be two formal power series and y,![t). yf’it} the associated functionals
of the form (3.4). Note that
t/iT) = tfiit) - y1'(t)
108
3. Input-Output Maps and Realization Theory
is still a functional of tin* form (3.4) associated with a formal power series
c whose coefficients are defined as differences between the corresponding co-
efficients of <" and ch. To prove the Icmiua. all we need is to show that if
y(t] = 0 for all t £ h). T] and for all input functions, all the coefficients of the
series c vanish.
If. in particular, //1 =    = unl = (J on [0. T] then t/(f) = I) for all t E [0. 7”
implies
r(0l + ciOtf + r( 00)L_ + ...=()
for ah t e [0. Т]. i.e.
c(0) = 0
c((hMI) = (J 1 < A- < dc .
/ times
Taking tin* derivative of (3.4) with respect to time and evaluating it ar
/ = 0. one obtains
= У r(/)ip(0) .
t=o J=1
Therefore. (dt//dO/=o = 0 for all «i (0)...(()) implies
< m .
the second derivative of y[t") at
-г У^(с(0П +	",(0) 
(di) = 0	1 < i
Continuing this way. one may compute
t = 0 and get
/ j-2 \	m
( "TT )	=52 CiimhoJdimJU)
X dr- / ,_n	*—'
x	/	0.01 = 1
If this is zero for all iл (()).ur?J (0). .then
1 < Л.m < ni
1 < i < m .
In the third derivative, the contribution of terms
is
f yo/) + 2r(«)][^^.
If this is zero for all ldu,/dt)t-o- then c((h') = -2r(/0) which, together with
the previous equality r(0Fl ~ ~c(/0) implies
c(0i 1=0	1 < ? < m .
For a complete version of the proof, the reader is referred to the literatures
3.1 Fliers. Functional Expansions
109
We an' now going to show that rhe output yif) of rhe nonlinear system
(3.1) can bo represented as a functional of the inputs ....nm in the form
(3.4). To this end wo need some preliminary results.
Lemma 3.1.3. Let g(t gn, be a set of analytic vector fields and A a real-
valued analytic function defined on I'. Given a point rz € L~. consider the
formal power series defined by

. L:l A(F) .
Then, there e.rist real numbers Is > () and- 4/ > 0 such that the growth
condition /3.2) is satisfied.
Proof. The reader is referred to the literature. <
In view of this result and of Lemma 3.1.1. oik1 may associate with
gQ.....gni and A the functional
Oi = A(C> - £ £ О, O
I =[)	... . о =(!
(3.6)
Lemma 3.1.4. Let g{,.......y)n be us in the pre riotis Lemma and let Xt....X/
be real-valued analytic functions defined on I  Moreover, let' be a re al-valued
analytic function defined on ifi . Let r\(t),... . r/(t) denote the functionals
defined by setting, in a functional of the. form (3.6). X = Al......A = A?.
The coni posit ion "dm1?)....r/dl) is again a fimctional of the form /3.6).
corresponding to the setting X = MAl......A/).
Proof. Wo will only give* a trace to the reader for the proof. Let n.m denote
the formal power series defined by setting, in (3.5). A = Ai ami respectively
A = Aj. and let Ci (t).m(11 denote the associated functionals (3.6). Them
it is iinmediately seen that with the formal power series defined by setting
A = oi Al +сьА2- where Oi and o2 are real numbers, there is associated the
functional oi m (t) — n2r2(t}.
With a little work, it is also seen that with rhe formal power series defined
by setting A = AjA?. then* i> associated the functional m (t)c2(t). We show
only the verv first computations needed for that. For. consider the product
r1(f)c2(t)=(Ai
— £4] A] /	L
f1
i). A> /	+ L u-, Lfh, X-2 I <
4(|Us() -
where, for simplicity, we have omitted specifying that the value* of all the
functions of .r are to be taken at w = F. Multiplying term-by-term we have
110	3. Input-Output Maps and Realization Theory


The factors that multiply /J dfp and are clearly Lg.,X] X> and re-
spectively £У1А1А2. For the other three, we have
£yi	— -^r Fi;,,Fj,. A_> + AjZ7. £y,, Ai + 2( Lg. Ai)(£7, A-j)
but also
/	/ dfQ = 2 /
Jo	-/u	Jo
so that the three terms in question give exactly
h'ji.^-y.XiX> I	.
Jo
It is not difficult to set up a recursive formalism which makes it possible to
completely verify the claim.
If now у is any real-valued analytic function defined on Л1, we may fake its
Taylor series expansion at the origin and use recursively the previous result*
in order to show that the composition у(rj (t)...i'/(t),) may be represented
as a series like the (3.6) with A replaced by the Taylor series expansion of
w (Л]....A/T <
At this point, it is easy to obtain the desired representation of ?/(f) as a
functional of the form (3.6).	/
Theorem 3.1.5. Suppose the inputs tq..........m„ of the control system (3.1)
satisfy the constraint (3.3). If T is sufficiently small, then for all 0 < t < T
the j-th output уj (t) of th e s у stem (3.1) та у hr ewpa mied i n th c following
way
ci	t
yfft) = £/>‘Т +	52	(3.7)
r=(i C-..d--l)	,/u
where g(i = f.
Proof. We fii>t show that the j-th component of the solution of the differen-
tial equation in (3.1) may be expressed as
X tn	t
rjn = TWO + 52 5Z	(3.8)
A--0 oj.u. =-0
3.1 Fliess Functional Expansions
111
where the function .гд.г) stands for
J’j : U’i...........J’n i 
Note that, by definition of iterated integral

and	rf /'	f'
- /	- di,. = 4,4) / rff„, - 
do	/0
for 1 < I< in. Then, taking the derivative of the right-hand side of (3.8'i
with respect to the time and rearranging the terms we have
... Il7i. If.r ,i.r=)
  	'Г.1 'l 
»;T) .
.a =o
Now. let fj and denote the j-th components of f and (/j. 1 < j <
ml < i < m and observe that
-	+ У2
Lf-r.i =	....rn] 
Therefore, on the basis of Lemma 3.1.4. we may write
LyrJ.rT -	£-W. •МтгЛг') I
A—Din n=0
L9...   	/Л'""1 / f/sn.
—17 .'I. 4'
=	.....r„(H) .
A similar substitution can be performed on the other terms thus yielding
t/o = /At ...........-v	.....ejniupn.
Moreover, rhe .rj(t) satisfy the condition
r7 (0) = .Й
ami therefore an1 the components of the solution of rhe differential equa-
tion in (3.1).
A further application of Lemma 3.1.4 shows that the output of (3.1) can
be expressed in the form (3.7). <
112
3. Input-Output Maps and Realization Theory
The expansion (3.7) will be from now on referred to as the fundame ntal
formula, or Floss' fuwtiorial expansion of уj(t]. Obviously, one may deal
directly with the case of a vector-valued output with the same formalism, bv
just replacing the real-valued function hj{,r} with the vector-valued function
h{.r]. We stress that, from Lemma 3.1.1. it is known that the series (3.7«
converges absolutely and uniformly on [li.T].
Remark 3.1.2. The reader will immediately observe that the functions h}
and T7i ... I,j,, hj(a-). with 1 < j < p and (ik •   ri ) G (Z*\To), whose values
at .r characterize the functional (3.7). span the so called observation spurt
CL The latter, in fact, was characterized - in section 2.3 as the space 1'1 of all
w-linear combinations of functions of the form h,\.r] and L;ii. ... h , :> 
with 1 < j < p. 0 < E ri "i. 1 < k < oc. <
E.ramplr In the cast1 of a linear system, the formal power series which
characterizes the functional (3.7) is such that c(И) = Cj.t .
c( ri-  ml
= J *1-r"
" V-14
if hj =    = if. = H
if ri -  -  = ri- = H
and (‘{if- ... ri) = U elsewhere.
In the cast1 of a bilinear system, the formal power series which character-
izes the functional (3,7) rakes the form
ri-'"
r,.Vu
where* = .1. <
3.2 Volterra Series Expansions
Tin* input-output behavior of a nonlinear system of rhe form (3.1) mav also
be represented by means of a series of fpmeralizcd convolution inteyrals. A
generalized eoiivohnion integral of order k is defined as follows. Let (E <   ri ;
he a niultiindex of length k. with ri.......ri elements of the set {1......m}.
With this niiilriindcx then* is associated a real-valued continuous function
tc,. . defined on the sublet of 1
,5\. =	.. .7, ) e 1 : T > t > Tk >    > 7i > 0}
where1 Г is a fixed number. If ip...u,;i are real-valued piecewise continuous
functions defined on .0, Т]. the generalized convolution integral of order k of
ui.....to- with kernel tc)( (1 R defined as
Tj )U,. (TA. ) . . . (i)1 (7| ) d~i . . . d7A-
г
3,2 Volterra Series Expansions
113
for 0 < t < T.
For consistency, if A‘ = 0. rather than a generalized convolution integral,
one considers simply a continuous real-valued function ir(J defined on the set
So = {t e К : T > t > 0} .
The sum of a series of generalized convolution integrals may describe a
functional of ui....um. under the conditions stated below.
Lemma 3.2.1. Suppose there exist real numbers I\ > 0. M > 0 such that
|'G,...H(Ln......nil <	13.9)
for all k > 0. for all multi in dices (ik . .. ?h ), and all It. . 7]) 6 S^.
Then, there e.rists a real number T > 0 such that, for each 0 < t < T and
each set of piecewise continuous functions a^.......u,„ defined on [O.T] and
subject to the constraint
max I iti(т} I < 1 .	(3.10)
U<r<T' 1
the series
y(t) = wu(t)
ОС	W	л/ fTk f-r.-,
+ У2 / / "	....T KJn-)... Ult (T] )drfr.,. d-x
...= 7()
(3.11)
is absolutely and uniformly convergent.
Proof. It is similar to that of Lemma 3.1.1. <
The expression (3.11) clearly defines a functional of . itm. which is
causal, and is called a Voltcrra series expansion.
As in the previous section, we are interested in the possibility of using
an expansion of the form (3.11) for the output of the nonlinear system (3.1).
The existence of such an expansion and the expressions of the kernels may
be described in the following way.
Lemma 3.2.2. Let f.g\..........gm be a set of analytic vector fields and X a
real-valued analytic function defined on th Let denote the flow of f. For
each pair (t..r) ейх[' for' which the flow ф{ (.r) is defined, let Qf(r) denote
the function
(?,(*) =	(3.12)
and Pf(.r)....Pfn(x) the vector fields
=	(3.i3)
1 < i < m. Moreover, let
11-1	3. Input-Output Maps and Realization Theory
«М = <Ж)
(f.	.......7”i)
Lp’i ... Lp>k Qt(,xc) •
i
(3.14)
Then, there exist real numbers К > 0 and УI > 0 such that the condition
(3.9) is satisfied.
From this result it is easy to obtain the desired representation of y(t} in
the form of a Volterra series expansion.
Theorem 3.2.3. Suppose the inputs ux....um of the control system (3.1)
satisfy the constraint (3.10). If T is sufficiently small, then for all 0 < t < T
the output уj(t) of the system (3.1) may he expanded in the form of a Volterra
series, with kernels (3.1 J). where Qt(r) and P)(x) are as in (3.12)-(3.U3) and
X = hj 
This result may be proved either directly, by showing that the Volterra
series in question satisfies the equations (3.1). or indirectly, after establishing
a correspondence between the functional expansion described at the begin-
ning of the previous section and the Volterra series expansion. We take the
second way.
For, observe that for all (/*.... t’j) rhe kernel (t. ту......Tj) is ana-
lytic in a neighborhood of the origin, and consider the Taylor series expansion
of this kernel as a function of the variables t — ту, ту - ть_ i.т--> —	. и .
This expansion has clearly the form
(t. Tk
(t - Tk)”k  
nyl.
. n i !n0!
where
рПй-..Tlfe
1' к	 г 1

If we substitute this expression in rhe convolution integral associated with
obtain an integral of the form
(I - n-P
——;—
Ы -
——;---------uq (n) —L-- drk • dn.
nd.	nol
The integral which appears in this expression is actually an iterated inte-
gral of «[.... ,u.m, and precisely the integral
I W‘4,
0

(3.15)
(where (df0)n stands for n-times t/£0).
Thus, the expansion (3.11) may be replaced with the expansion
3.2 Volterra Series Expansions
115
rt=o "'°
+E E E C-"7	...hw4>«.f
fc = l it.. .,i\ = l rto in— 0	’
(3.16)
which is clearly an expansion of the form (3.4). Of course, one could rearrange
the terms and establish a correspondence between the coefficients Cg. г”;'
(i.e. the values of the derivatives of tro and at t —	=  • • = т-2 — tl =
Г1 = 0) and the coefficients r(0), r(/\ ... ?0) of the expansion (3-4), but this
is not needed at this point.
On the basis of these considerations it is very easy to find Taylor series
expansions of the kernels which characterize the Volterra series expansion
of	\\e see from (3.16) that the coefficient PP"-** of the Taylor series
expansion of	coincides with the coefficient of the iterated integral
(3.15) in the expansion (3.4). but we know also from (3.7) that the coefficient
of the iterated integral (3.15) has the form
Ln/LsllL’}'	.
This makes it possible to write down immediately the expression of the Taylor
series expansions of all tire kernels which characterize the Volterra series
expansion of
u!o(t)
)
n=0
(f~Tjril 7Г
nJ no!
(3.17)
«2—0 n i —0 nr- =0
п2^1!мо'-
and so on.
The last, step needed in order to prove Theorem 3.2.3 is to show that the
Taylor series expansions of the kernels (3.14). with Qt(x) and P/(.r) defined
as in (3.12). (3.13) for Л = h;(x) coincide with the expansions (3.17).
This is only a routine computation, which may be carried out with a little
effort by keeping in mind the well-known Campbell-Baker-Hausdorff formula,
which provides a Taylor series expansion of Р](Р). According to this formula
it is possible to expand Ptl(x) in the following way
ВД = офЦт) = ^2
n=0
where, as usual, ad’jg = [f.ad^~lg] and ad^g — g.
116
3. Input-Output Maps and Realization Theory
Example 3.2.1. In the case of bilinear systems, the flow ф] may be clearly
given the following closed form expression
ф{ (,r) = (rxp.4f).r .
From this it is easy to find the expressions of the kernels of the Vol terra st'ties
expansion of ydt). In this cast1
Qtl'x) = Cj(expAt)x
Pt4.r) = (exp(-AZ)RV,(cxp Af)z
and. therefore.
it'o(f) = cjexp Аф-С
i/tR/.T-i) = сДехрАр - -i))jV((expАр)?
u f2?i (^. 72 - m) = сДехрАД - r2)hV;2(exp.4(r> - л ))ЛД (exp An ).rc
and so on. <
3.3 Output Invariance
In this section we want to find the conditions under which the output of a
system is not affected by the input. These conditions will be used later on hi
the next Chapters when dealing with the disturbance decoupling or with the
noninteracting control.
Consider again a system of the form
Hi
3 = f(.r) + ^ДтДо
yj =
and let
ui,.... и)
denote the value at time t of the j-th output, corresponding to an initial
state r and to a set of input functions ip..... un>. We say that the output
yj is unaffected by (or invariant under) the input at. if for every initial state
€ th for every set of input functions tp.....u;-i.	..... um and for all
f
t/j(pzo:ui....tp-i, c“, ui+l....um)
,	(d-loj
- yj(t\tp.......u,_i.	....um)
for every pair of functions C1 and vb.
There is a simple test that identifies the system having the output y}
unaffected by the input tp.
3.3 Output Invariance 117
I^mma 3.3.1. The output y3 is unaffected by the input U; if and only if. for
gllr > 1 and for any choice of vector fields л ...., г,- m the set {f.gi.gm }
Ly'hjfr) = 0
LgLT1 ...L^hjfr) = 0	(3.19)
for all x € t' 
Proof. Suppose the above condition is satisfied. Then, one easily sees that
the function
£71 .. .L^hfr]	(3.20)
is identically zero whenever at least one of the vector fields л.coincides
with gn If we now look, for instance, at the Flicss expansion of i/j(t). we
observe that under these circumstances
c(ik    ?o) = 0
whenever one of the indexes г0,.... is equal to i. and this, in turn, implies
that any iterated integral which involves the input function u, is multiplied
by a zero factor. Thus the condition (3.18) is satisfied and the output yj is
decoupled from the input ut.
Conversely, suppose the condition (3.18) is satisfied, for ('very g U. for
every set of inputs itj..... iq_i.	,.... u,n and every pair of functions va
and vb. Take in particular va(t) = 0 for all t. Then in the Fliess expansion of
yj(t; xa: Uj..... u;-i. va. ....... u,n) an iterated integral of the form
Jo
will be zero whenever one of the indexes	is equal to i. All other
iterated integrals of this expansion (i.e. the ones in which none of the indexes
i0,...,ik is equal to i) will be equal to the corresponding iterated integrals
in the expansion of T3:tq..........иг-ь г\ и|+1........и„г) because the inputs
Ui,.... u;_i. щ-1,.... uTH are the same. Therefore , we deduce that the dif-
ference between the right-hand side and the left-hand side of (3.18) is a series
of the form
k=0 tl).....ik=0	do
in which the only nonzero coefficients are those with at least one of the in-
dexes	equal to i. The sum of this series is zero for every input
иi,.... и,-1. cb. j.......Ujn- Therefore, according to Lemma 3.1.2 all its co-
efficients must vanish, for all € U. <
118
3. Input-Output Maps and Realization Theory
The condition (3.19) can be given other formulations, in geometric terms.
Remember (Remark 3.1.2) that we have already observed that the coefficients
of the Fliess expansion of y{t) coincide with tin1 values at J'c of the functions
that span the observation space Cl. The differentials of these functions span.
by definition, the codistribution
Pp = span{dA : A G 0} .
If we fix our attention only on the j-th output, we may in particular define an
observation space Oj as the space of all jL-liuear combinations of functions
of the form hj and L4i_ ...Lg., hj. 0 < ц. < m. 0 < k < ос. The set of
differentials d/ij.dL,^ ... Ly with o-........70 € I and j fixed spans the
codistribution
Pcy = span{dA : A G C\} .
Now. observe that the condition (3,19) can be written as
(dh j.:gt )(.r)	- 0
(dLg,,,	=	0
for all k > 0 and for all ц...io G I. From the above discussion we conclude
that the condition stated in Lemma 3.3.1 is equivalent to the condition
g, G P^_ .	(3.211
Other formulations are possible. For. remember that we have shown in
section 2.3 that the distribution P^ is invariant under the vector fields
/.	....gm- For the same reasons, also the distribution P^ is invariant
under /..9i.....gm.
Now. let (f.g\.........9m|span{<9(})	denote, as usual, the smallest distribu-
tion invariant under /,91..............gm which contains span}#;}. If (3.21) is true,
then, since P^j is invariant under fdgi,.... gm, we must have
(/-Si.....3,»	I span {9,}) C PCo. 	(3.22)
Moreover, since
Pcy C (span{d/?j })"
we see also that if (3.22) is true, we must have
</-3i.....3njspan{3(}) C (spanjd/ij})^.	(3.23)
Thus, we have seen that (3.21) implies (3.22) and this, in turn, implies
(3.23). We will show now that (3.23) implies (3.21) thus proving that the
three conditions are in fact equivalent.
For. observe that any vector field of the form [r. 9;] with т G {/. 91...., 3,71}
is by definition in the left-hand side of (3.23). Therefore, if (3.23) is true.
0 = (dhj, [r. gt]) = LTLyi hj - L^L-hj .
3.3 Output Invariance"
119
But, again from (3.23). gt G (span{rfftj}) so we can conclude
L y;~hj — 0
i.e.
9i G (span{d£7M)“.
By iterating this argument it is easily seen that if Tk....n is any set of
к vector fields belonging to the set {f.gi......</„;). then
g, £ (span{t/Ln. ... Lr./tj})1 .	(3.24)
We know that consists of Z-lincar combinations of functions of the*
form hj or LTk ...LT,hj. with r, G {f-(Ji.......1 < ?' < An 1 < k < oc.
Thus, from (3.24) we deduce that g, annihilates the differential of any function
in Oj. i.e. that (3.21) is satisfied.
Summing up we may state the following result.
Theorem 3.3.2. The output ijj is unaffected by the input ut if and only if
any one of the following (equivalent) conditions is satisfied
(i) gt G
(ii) U-tT.....<7,n|span{<?,}) C (span{M,})_
(hi) (f-gi....I span {гл}) С .
Remark 3.3.1. It is clear that the statement of Lemma 3.3.1 can be slightly
modified (and weakened) by asking that
hj (т)	= 0
L,hL^ .. .Lrffij(j-} = 0
for all r > 1 and any choice of vector fields ri....rr in the set {/. <ji, gi-\.
gi+\. д,,,}. Consistently, instead of Oj. one should consider the subspace of
all Fi-lincar combinations of hj and LT1   - LTrhj, with ту..Tk vector fields
in the set {f. g\ 9i—i .щ... b .... gm }. <i
Remark 3.3.2. Suppose (f.gi span{<?;}) and *2^ are nonsingular.
Then, these distributions are also involutive (see Lemmas 1.8.5. 1.9.5 and
Remark 2.3.3). If the condition (iii) of Theorem 3.3.2 is satisfied, then around
each point .r G C it is possible to find a coordinate neighborhood on which
the nonlinear system is locally represented by equations of the form
.Г1	= /1(.Г1..г2) + ^2 3iC.r1.z2)uA. + J7b(j’t. J-2)ui
= 1, A ?
tn
= /2(^2)+ 52
У j = hffx-,)
from which one sees that the input tq has no influence on the output pj. <
120
3. Input-Output Maps and Realization Theory
Suppose there is a distribution A which is invariant under the vector fields
/. <71,.... 9m • contains the vector field and is contained in the distribution
(span{dfij}) ~. Then
(f-9\.....5Mspan{9;}} CdC (spanfc/hj})1.
We conclude from the above inequality that the condition (ii) of Theo-
rem 3.3.2 is satisfied. Conversely, if condition (i) of Theorem 3.3.2 is satis-
fied. we have a distribution, , which is invariant under the vector fields
/. r/i..... gm, contains gt and is contained in (span{t/hj}) Therefore we may
give another different and useful formulation to the invariance condition.
Theorem 3.3.3. The output yj is unaffected by the input щ if and only if
there exists a distribution _1 with the following properties
(i) _1 is invariant under .......gm
(ii) gt e -A C (spanjdftj})1.
Remark 3.3.3. Again the condition (i) may be weakened by simply asking
that
(i‘) _1 is invariant under f.gx..gi_\.g^\.... ,gm.
Note that this implies that if there exists a distribution A with the prop-
erties (i'J and (ii) then there exists another distribution A with the properties
(i) and (ii). <
We leave to the reader the task of extending the previous result to the
situation in which it is required that a specified set. of outputs ур.yj,. has
to be unaffected by a given set of inputs u(],..., uu. The conditions stated
in Lemma 3.3.1 remain formally the same, while the ones stated in Theorems
3.3.2 and 3.3.3 require appropriate modifications.
Example 3.3.j. In concluding this section it may be worth observing that in
case the system in question reduces to a linear system of the form
m
x = Ar + y^b,ul
j=i
!Jj = Q-7'	1 < J < P
then the condition (3.19) becomes
CjAkbt = 0 for all к > 0 .
The conditions (i). (ii), (iii) of Theorem 3.3.2 become respectively
n-1
bi e П ker(cJ-4*)
F
3.4 Realization Theory 121
я—1
^ImfA*’^) C ker(cj)
k=o
Tl!	n-i
У^1т(Ал~Ь;) с П ker (cjAfr) .
A-=(J	fr=0
These clearly imply and are implied by the existence of a subspace I' invariant
under A and such that
6, G Г C ker(cj) .<
3.4 Realization Theory
The problem of "realizing" a given input-output behavior is generally known
as the problem of finding a dynamical system with inputs and outputs able
to reproducet when initialized in a suitable state, the given input-output
behavior. The dynamical system is thus said to “realize", from the chosen
initial state, the prescribed input-output map.
Usually, the search for dynamical systems which realize the input-output
map is restricted to special classes in the universe of all dynamical systems,
depending on the structure and/or properties of the given map. For example,
when this map may be represented as a convolution integral of the form
t/(t) — / u>(t - r)u(r) dr
Jo
where w is a prescribed function of t defined for t > 0. then one usually looks
for a linear dynamical system
т ~ .4 j- + В и
у = Ст
able to reproduce, when initialized in = 0. the given behavior. For this to
be true, the matrices А, В. C must be such that
C exp(At)В = tc(7) .
We will now describe the fundamentals of the realization theory for the
(rather general) class of input-output maps which can be represented like
functionals of the form (3.4). In view of the results of the previous sections,
the search for “realizations" of this kind of maps will be restricted to the
class of dynamical system of the form (3.1).
From a formal point of view, the problem is stated in the following way.
Given a formal power series in tri + 1 noncommutative indeterminates with
coefficients in Kp. find an integer n. an element .r° of Kri. m + 1 analytic
vector fields go,..., gm and an analytic p-vector valued function h defined on
a neighborhood U of rc such that
122
3. Input-Output Maps and Realization Theory
/>(r) = r(0)
...	~ c{ik ... t0) 
If these conditions are satisfied, then it is clear that the dynamical system
lit
,i- = g0(Z) + У/М.Г) Uj
У = h(.r)
initialized in ;r" € produces an input-output behavior of the form
3C tn	-t
y(O=c(0) + ^2 Ц  — >0) /	.
A-Oj....д.=0	-
In vieyv of this, the set {yo....9m  h- } wih be called a realization of the
formal power series c.
In order to present the basic results of the realization theory, we need first
to develop some notations and describe some simple algebraic concepts re-
lated to the formal power series. In view of the need of dealing with sens of se-
ries and defining certain operations on these sets, it is useful to represent each
series as a formal infinite sum of "monomials". Let го.г,„ denote a set
of m + 1 abstract noncommutative indeterminates and let Z = {^o.г„(}.
With each multiindex (ik    fol we associate the monomial ... г;Г|) and
we represent the series in the form
m
c = c(0) + ^2 c(ik  • -Ь))гп •  •	•	(3.25)
A=0 ...u-=0
The set of all power senes in m +.1 noncommutative indeterminates (or.
in other words, in the noncom unit at iye indeterminates гп.zm) and coeffi-
cients in is denoted with the symbol K1P((Z)). A special subset of is
the set of all those series in which the number of nonzero coefficients (i.e. the
number of nonzero terms in the sum (3.25)) is finite. A series of this type is a
polynomial in m + 1 noncommutative indeterminates and the set of all such
polynomials is denoted with the symbol H£P(Z). In particular 3(Z) is the set
of all polynomials in the m + 1 noncommutative indeterminates гр..........z,1t
and coefficients in 31.
An element of R(Z) may be represented in the form
rf 7X1
p = p(0) + ^2 ZL pOo • -  ud-n.	(3.26)
A*=0 in.ifc =o
where d is an integer which depends on p and p(0).p(io ... ik) are real num-
bers.
3.4 Realization Theory
123
The sets R(Z) and ({Z}) may be given differ fait algebraic structures.
They can clearly be regarded as R-vector spaces, by hating R-linear com-
binations of polynomials and/or series be defined coefficient-wise. The set
R(Z) may also be given a ring structure, by letting the operation of sum of
polynomials be defined coefficient-wise (with the neutral element given by
the polynomial whose coefficients are all zero) and the operation of product
of polynomials defined through the customary product of the corresponding
representations (3.26) (in winch case the neutral element is the polynomial
whose coefficients are all zero but p(0) is equal to 1), Later on. in the proof
of Theorem 3.4.3 we shall also endow R(Z) and R((Z)) with structures of
modules over the ring E(Z). but, for the moment, those additional structures
are not required.
What is important at this point is to know that the set R(Z) can also
be given a structure of Lie algebra. by taking the above-mentioned R-vector
space structure and defining a Lie bracket of two polynomials pi, p2 by setting
[pi-p-j] — R?Pi — PiP'2- The smallest sub-algebra of R(Z) which contains the
monomials c().....will be denoted by C(Z). Clearly. C(Z) may be viewed
as a subspace of the R-vector space R(Z). which contains c(j........and is
closed under Lie bracketing with Ci>.....zlu. Actually, it is not difficult to see
that £(Z] is the smallest subspace of R(Z) which has these properties.
Now we return to the problem of realizing an in put-out put map repre-
sented by a functional of the form (3.4). As expected, the existence of real-
izations will be characterized as a property of the formal power series which
specifies the functional. We associate with the formal power series c two inte-
gers which will be called, following Fliess. the Hanke I rank and the Lie rank
of c. This is done in the following manner. We use the given formal power
series c to define a mapping
F(. : R(Z) -o F((Z))
in the following way:
(a)	The image under F,. of any polynomial in the set Z* = {zjk ...zJi?: €
R(Z) : (д-...jo) £ /*} (by definition, the polynomial associated with the
multiindex 0 £ /* will be the polynomial in which all coefficients are zero
but p(0) which is equal to 1. i.e. the unit of R(Z)) is a formal power series
defined by setting
. 37il)](?,.... /()) - c(ir... iojk ... jo)
for all Jk-- jo e I*.
(b)	The map Fc is an R-vector space morphism of R(Z) into RJJ((Z)).
Note that any polynomial in R(Z) may be expressed as an R-linear com-
bination of elements of Z* and. therefore, the prescriptions (a) and (b) com-
pletely specify the mapping Fe.
Looking at Fr as a morphism of R-vector spaces, we define the Hankel
rank рн{с) of c as the rank of Fc. i.e. the dimension of the subspace
124
3. Input-Output Maps and Realization Theory
МВД) c ЗВД»-
Moreover, we define the Lie rank piJs) of c as the dimension of tin* sub-
space
Fr(£(Z)) C??((Z}
i.e. the rank of the mapping F<-\c,z\-
It is easy to get a matrix representation of the mapping Fr. For. suppose
we represent an element p of R(Z) with an infinite column vector of real
numbers whose entries are indexed by the elements of /* and the entry in-
dexed by jh- -   Jo i-s exactly p(jk ... Jo). Of course, p being a polynomial, only
finitely many elements of this vector are nonzero. In the same way. we may
represent an element c of Rp ((Z)) with an infinite column vector whose entries
are p-vectors of real numbers, indexed by the elements of /* and such that
the entry indexed ir ... m is c[tr... /0). Then, any R-vector space morphism
defined on R(Z) with values in RP((Z)) will be represented by an infinite
matrix Hc. whose columns are indexed by elements of /* and in which each
block of p-rows of index (ir ... t0) on the column of index (д.- - - Jo) i* exactly
the coefficient
c(fr ...i0 jk . ..jo)
of c. We leave to the reader the elementary check of this statement.
The matrix Hc is called the Hankel matrix of the series c. It is clear from
the above definition that the rank of the matrix Hc coincides with the Hankel
rank of c.
Example ,'L4-1- If the set I consists of only one clement, then it is easily
seen that I* can be identified with the set Z+ of the non-negative integer
numbers. A formal power series in one indeterminate with coefficients in R.
i.e. a mapping
c : Z/“ —> R
may be represented, like in (3.25). as an infinite sum
f' = Zc‘?
A-=0
and the Hankel matrix associated with the mapping Fc coincides with the
classical Hankel matrix associated with the sequence r0.n....
/ c0 Ci c->	•• • \
H(. — Cl C2 C3 .<
C-2 Г3 C4 - - - I
The importance of the Hankel and Lie ranks of the mapping Fe depends
on the following basic results.
3.4 Realization Theory
125
Lemma 3.4.1. Let f.tp........<hlt-h and a point xs £ be given. Let Лс be
tfie distribution associated with the control Lie algebra C and the codistri-
bution associated, with the observation space O. Let K{xc) denote the subset
of vectors of Лс[.г-) which annihilate <2ti(.rc) i.e. the subspace of Tr=5? de-
fined by
K(xs) = A(.r) г = {? e Mr) - <dA(Z'). <.•> = i) VA e C?}.
Finally, led. c be the formal power series defined by
r(0) =
r ,	(3.27)
...qj = Llj;i: ... Lgjhx-')
with, go = f. Then the Lie rank of c has the value
...
pi (r) = dim A’l М -diinA(.r“) = dim —-------------- •••. ,- .
P	>U-C) П
Proof. Define a morphism of Lie algebras
piC(Z)^VOfi)
by setting
/H-q.) — g, 0 < i < ni .
Then, it is easy to check that if p is a polynomial in Cl Z) the (d ... z’ol'th
coefficient of Ffip) is Lfl^Lgii ... L:/i h(xz). Thus, the series Ffip) has the
expression
<	lh
T'c[p) — Lfl\{lih(x ) -r	* L/l{pjL,hi ... Lg^ h(x )~lk ....
A’ —U л. !i-—0
If wo lot г denote the value of the vector field p(p) at .rc. the above can be
rewritten as
Ffi.p) = {dh(.r~).r} +	$2	 LyiJt(xrfi. r)zt, .
k=() Ci.... a. =o
When p ranges over C(Z). the tangent vector r takes any value in
Moreover, the covectors dh(xc).... ,dLih. ... Lg h(rz ),... span	This
implies that the number of a-linearly in de] > endent power series in Fc{C(Z))
is exactly equal to
dim Ad-r0j - dim A(-rc) n L?fi(xs)
and this, in view of the definition of the Lie rank of c. proves the claim. <i
126
3- Input-Output Maps and Realization Theory
We immediately see from this that if an input-output functional of the
form (3.4) is realized by a dynamical system of dimension n, then necessar-
ily the Lie rank of the formal power series which specifies the functional is
bounded by n. In other words, the finiteness of the Lie rank pific) is a nec-
essary condition for the existence of finite-dimensional realizations. We shall
see later on that this condition is also sufficient. For the moment, wo wish to
investigate the role of the finiteness of the other rank associated with Fr i.e.
the Hankel rank. It comes from the definition that
Pl(c) < рн(с)
so that the Hankel rank may be infinite when the Lie rank is finite. However,
there are special cases in which pjfic) is finite.
Lemma 3.4.2. Suppose f.gi...........gmSi are linear in x. i.e. that
fix) = Ar. <7i MJ = -Vi-r.......(;r) = h(x) = Cx
for suitable matrices A A'i......V,P.C. Let f be a point of TV’. Let V de-
note the smallest subspace of №n which, contains A and is invariant under
.4. >Vi..... .V,h  Let IL denote the largest subspace of IK" which is contained in
ker(C) and is invariant under .4. Afi.....Arm. The Hankel rank of the formal
power series (3.27) has the value
p}} fc) = dim 1' — dim IF P 1 ’ = dim Tr? _ .
'	n m
Proof. We have already seen, in section 2.4. that the subspace IF may be
expressed in the following way
?c m
IF = (ker С) П [p| Q	kerfCAj. ... А\; )j
r=0 t,;,=0
with An = .4. With the same kind of arguments one proves that the subspace
Ir may be expressed as
>_ m
F = span{Z} +	^2	sPan(AA  	•
A-=0 jr,.jk=O
In rhe present case the Hankel matrix of Fc is such that the block of p
rows of index (?r... J) on the column of index (д. .. .jh), he- the coefficient
c(i,-... ?од.... Jo) of c has the expression
CA (r. ... A j,, A jk ... A x .
By factoring out this expression in the form
(CAJ ...A;,)(Ajfc ...A’j.j0)
3.4 Realization Theory 127
it is seen that the Hankel matrix can be factored as the product of two
matrices, of which the one on the left-hand side has a kernel equal to the
subspace 1Г. while the one on the right-hand side has an image equal to the
subspace U. From this the claimed result follows immediately. <
Thus, it is seen from this Lemma that if an input-output functional of
the form (3.4) is realized by a dynamical system of dimension n described by
equations of the form
7П
J‘ =	.4Т + У AjJ’U;
1 = 1
у = Cx
i.e. by a bilinear dynamical system of dimension n. then the Hankel rank of
the formal power series which specifies the functional is bounded by n. The
finiteness of the Hankel rank рц(с) is a necessary condition for the existence
of bilinear realizations.
We turn now to the problem of showing the sufficiency of the above two
conditions. We treat first the case of bilinear realizations, which is simpler.
In analogy with the definition given at the beginning of the section, we say
that the set { A'o..... Am. C. F}, where x° E H". Ah £ for 0 < i < m
and C 6 Bpx" is a bilinear realization of the formal power series c if the set
{ffen • • • • ,9m Jh } defined by
9o(.r) = A’oJ’. f/i (^) = Ah-r. .... gm(x) = Armj
h(x) = Cx
is a realization of c.
Theorem 3.4.3. Let e be a formal power series in m + 1 noncornmutative
indeterminates and coefficients in . There exists a bilinear realization ofc
if and only if the Hankel rank of c is finite.
Proof. We need only to prove the "if” part. For. consider again the mapping
Fc. The sets KP(Z) and № ({Z')') will now be endowed with structures of
modules. The ring IR(Z) is regarded as a module over itself. 1RP((Z)) is given
an 1R(Z)-module structure by letting the operation of sum of power series be
defined coefficient-wise and the product p  s of a polynomial p 6 IR(Z) by a
series s 6 W {{Z}) be defined in the following way
(a) 1 • s = s
(b) for all 0 < i < m the series - s is given by
(a ‘ s)(?r .. .io) = s(?r ... io?)
(c) for all/q./o € 'ZfZ') and Q1.Q2 6 K.
128	3. Input-Output Maps and Realization Theory
(«1P1 + П2Р2) ’ * =	''’)+ ('ztP? - Л’) •
Note that from (a) and (b) we have that for all д •-jo € Г
(zjk ... Zjn  s)(ir  •  OJ = *(F. - - ioJk • • .Jo) -
Note also that since the ring R(Z) is not commutative, the order in which
the products arc performed is essential.
We leave to the reader the simple proof that the map Fc previously defined
becomes an R(Z)-module morphism when RP({Z)} is endowed with this kind
of R(Z)-rnodule structure. As a matter of fact, it is trivial to check that
fc(p) = p-c.
Now consider the canonical factorization of Fe
Fc
R(Z) ---------------► KP«Z))
ker(Fc)
in which, as usual. Pc denotes the canonical projection p *-> (p + ker Fc) and
Qc the injection (p + ker FJ Fe{p}. Pc and Qc are R-vector space mor-
phisms. but. there is also a canonical R(Z)-module structure on R(Z)/ ker F,.
which makes Pc and Qc R(Z)-module morphisms.
Since, by definition. R(Z)/ker(FJ is isomorphic to the image of Fr. we
have that the dimension of R(Z)/ ker(FJ as an R-vector space is equal to the
Hankel rank рн(Р) of the formal power series 0. Let. for simplicity, denote
?;=
ker (Fr)
But X is also an R{Z)-module. so to (each of the indeterminates co,.... zm
we may associate mappings
ЛЛ- : A -> A
X I—> Zi  X .
The mappings AJ are clearly R-vector space morphisms. We also define
an R-vector space morphism
H : A -o R₽
bv taking
Нх= Д(т)](в) .
With the notation on the right-hand side we mean the coefficient with empty
index in the series Qc(x).
Finally, let zs be the element of A”
F
3.4 Realization Theory 129
= Pr(l)
where 1 is the unit polynomial in IR(Z).
We claim that
c(0) = H.r
с(ц. ...i'o)	= HM,t	.	(3'28)
For. it is seen immediately that
c=FJl) = Q(.oP(.(l) = Q,.(P).	(3.29)
Moreover, suppose that
...zJ = QeMik ...AIir,P	(3.30)
then we have
Fc(Vh -pJ = - -Fc(c,fc ...;(Q) = zfiQMk
= Qfiz,  Mlk .   -V11?P) = QcMMk ... ЛДХ
for 0 < f < m. Thus (3.30) is true for all (z\... .io) e P.
Now. keeping in mind the definition of Fc, one has
Tee;. c)](») = c(<;....iu)
and, therefore, in view of the definition of the mapping H. (3.28) are proved.
Take now a basis in the pH(c)-dimensional vector space AT The mappings
Mo..... Afm and Я will be represented by matrices Ao.........Ar,n and C; r
will be represented by a vector r°. These quantities are such that
t‘(u- - - • io) = CA^ ... A73xc
for all (ц ... Zq) e P. This shows that the set. {C, A'o...., A\rl.	} is a bilinear
realization for our series. <
The result which follows presents a necessary and sufficient condition
for the existence of realizations of an input-output functional of the form
(3.4). provided that the coefficients of the power series which characterize
the functional are suitably bounded.
Theorem 3.4.4. Let c be a formal power series whose coefficients satisfy the
condition
l|c(u...<o)|!<C(t+l)0^1>	(3.31)
for all . 7’0) E I*  for some pair of real numbers C > 0 and r > 0. Then
there exists a realization of c if and only if the Lie rank of c is finite.
130	3. Input-Output Maps and Realization Theory
Proof. Some more machinery is required. For each polynomial p E 3t(Z) we
define a mapping Sp : Rp ((Z)} —> PT ((Z)) in the following way
(a)	if p e Z* — {zjk .. .Zj,. G R(Z) : (д, ...j0) G /*} then Sp(c) is a formal
power series defined by setting
• --to) = <-(Л ... jQp ---io)
(b)	if oi.o2 G К and p\.p-> G R(Z) then
Sajpj_Q2p2 (c) — оiSPl (c) a- <~i-iSp2 {e) .
Moreover, suppose1 that, given a formal power series sq G ^f(Z')} and a
formal power series sq G R((Z)). the sum of the numerical series
•si (0)^2(0) + 52 52 51  -  *o)-'-(^ • • - hi) (3.32)
A=0	...ц.=0
exists. If this is the case, the sum of this series will be denoted by (si-.s-;).
We now turn our attention to the problem of finding a realization of o.
In order to simplify the notation, we assume p = 1 (i.e. we consider the
case of a single-output system). By assumption, there exist n polynomials
in £(Z). denoted pi......p,t. with the property that the formal power series
Fe{pi),.... tire Ж-linearly independent.
With the polynomials pi......pn we associate a formal power series
ir = exp\JTxtPi \ = 1 + 52 П ( 52 XiPi }	(3.33)
'	A=i K' M=i '
where iq , - -  • Jn are real variables.
The series c which is to be realized And the series tc thus defined are used
in order to construct a set of analytii? functions of	defined in a
neighborhood of 0 and indexed by the elements of /*, in the following way
= (c, te)
/гп....<оИ =	
The grow’th condition (3.31) guarantees the convergence of the series on
the right-hand side for all -r in a neighborhood of x = 0.
We will give now a sketch of the proof that there exist m + 1 vector fields,
defined in a neighborhood of 0, with the property
hJj;. ...i0 (t) — hjj.	(3.34)
for all (u- ---io) G I*- This is be actually enough to prove the Theorem
because1, at .r = 0. the functions hu...r) by construction are such that
3.4 Realization Theory
131
Л(0)
(0)
с(0)
c(ik --  ?о)
and this shows that the set {h.go.....gm] together with the initial state
x — 0 is a realization of c.
To find the vector fields p().g,„ one proceeds as follows. Since the n
series Fc(pi)...Fc(p,J are IR-linear independent, it is easily seen that there
exists n monomials rri]....rn„ in the set Z* with the property that the
(n x n) matrix of real numbers

(3.35)
has rank ii (where s, denotes the multiindex associated with the monomial
nij 'l. It is easy to see that
(c?	\
— (sm, (c), tc) )
^5	/ j=0
For. if Pi G Z* • then by definition
/ Q	\
[Fr(pj](.s/J = r(s/;) = [S,„/r)](tj =
(where t( denotes the multiindex associated with the monomial p,). From
this, using linearity, one concludes that the above expression is true also in
the (general) case where p, is an R-line ar combination of elements of Z*.
Using this property, we conclude that the j-th row of the matrix (3.35)
coincides with the value at 0 of the differential of one of the functions
the one whose multiindex corresponds to the monomial rtij.
Consider now the system of linear equations
:	9k(r) =	:
in the unknown vector pxTC- The coefficient matrix is nonsingular for all .r
in a neighborhood of 0 (because at т = 0 it coincides as we have seen
with the matrix (3.35)). Thus, in a neighborhood of 0 it is possible to find a
vector field pA-(-f) such that
L<7k{Sirit(c)- in) = (Snu;fc(c). w)
and this proves that (3.34) can be satisfied, at least for those whose
multiindices correspond to the monomials ...............nt„.
132	3. Input-Output Maps and Realization Theory
The proof that (3.34) holds for all other functions (.r) depends on
the fact that every formal power series in F,.(C\ Z)) is an ^-linear combination
of F, (y?i)...Fc(p„}. and is is not included here. The reader is referred to
the literature for complete version of it. <
It is seen from the above Theorem that if a formal power series c has
finite Lit1 rank, and its coefficients satisfy the growth condition (3.31). then
it is possible to find a dynamical system of dimension pn(c) which realizes
the series.
This fact, together with the result stated before in Lemma 3.4.1. induces
to some further remarks. A realization {/. g^.....g,l(.h.xz} of a formal power
series r is minimal if its dimension, i.e. the dimension of the underlying man-
ifold on which f.gi.....an1 defined, is less than or equal to rhe dimension
of any other realization of c. Thus, from Lemina 3.4.1 we immediately deduce
the following corollaries.
Corollary 3,4.5. A realization {f.(p........ty^.F.m} of a formal power se-
nes c i s m i n in i al if and only if its dimension is eq a al t о the L i e ra nk pcFn.
Corollary 3.4.6. A realization {f.(p........gm.h..F} of a formal power se-
nes c is minimal if and only if
dim Л-T') = dim ) — n
or. which is the. same, the realization satisfies the controllability rank, condi-
tion and the observability rank condition at jF .
3.5 Uniqueness of Minimal Realizations
In this station we prove an interesting ipiiqueiiess result, by showing that any
two minimal realizations of a formal power series are locally "diffcomorphi<.’\
Theorem 3.5.1. Let r be a formal power series and let n denote its Lie
rank. Let {t/q ..... g'^. /?". F} and {g^... /3', xh} be. two minimal, i.e.
a-dimensional, realizations ofc. Let iff. 0 < i < m. and lrl be defined on
a neighborhood I " of x'1 in R'! and g*fi 0 < ;’• < ni. and hfl be defined on a
neighborhood Fb of .rh in . Then, there exist open subsets I'" C L'a and
U1 C Uh and a diffeomorphism F : V" T6 such that
gb(.r) = F,.(ff о F-1(.r)	0 < i < in	(3.36)
h!Zx) = h“oF Ч.г)	(3.37)
for all .г E V1'.
3.5 Uniqueness of Minimal Realizations
133
Proof. We break up the proof in several steps.
(i)	Recall that a minimal realization {/. g\..yw, h, .r°} of c satisfies the
observability rank condition at (Corollary 3.4.6). From the definitions of
О and f?cj. one deduces that there exist n real-valued functions A;......A„.
defined in a neighborhood t of .r°. having the form
At(j-) = Lv,
with t?i___rr vector fields in the set {f.gi....д„г}. r (possibly) depending
on t and 1 < j < p such that the covectors dX\ (xc).....dX„ (,rc) arc1 linearly
independent (i.e. span the cotangent space Tf^U). From this property, using
the inverse function theorem, it is deduced that there exists a neighborhood
Uh C t of -r° such that the mapping
Я :.r^ (Ai (r)..... AJj))
is a diffeoinorphisni of Uh onto its image H(Uh)-
From any two minimal realizations, labeled “a1' and "b”. we will construct
twro of such mappings, denoted Ha and respectively Hb.
(ii)	Let.	be a set of vector fields, defined in a neighborhood U
of xc. having the form
J=1
with ti' 6 Ж for 1 < j' < m. Lot Ф) tienote the flow of fR and G denote the
mapping
G : (С-----fn) -о о -  - о Ф}{ (Я)
defined on a neighborhood (-5. с)'1 of 0.
From any two minimal realizations, labeled "a"’ and ,kb". we will construct
two of such mappings, denoted Ga and Gf> (rhe same set of iz'-'s being used
in both Ga and CC).
Recall that a minimal realization {/l3, <7", ..., g“t. ha, Я’} satisfies the con-
trollability rank condition at. .r'1 (Corollary 3.4.6). From the properties of
and R (see Remark 2.2.3). one deduces that the distribution R is nonsingular
and mdimensional around Яг. Then, using rhe1 same arguments as the ones
used in the proof of Theorem 1.8.9. it. is possible to see that there exists a
choice of a('s and an open subset IF of (0.e)!1 such that the restriction of Gt!
to IF is a diffeoniorphism of IF onto its image СЯ(И').
(iii)	It is easily proved that if
{/M........and {/‘.4......................к4.Л»./}
are two realizations of the same formal power series c, then, for all 0 < t, < 5.
1 < / < m with sufficiently small s.
Ha o6’a(h......Я) = Hh oGb(t\........(3.38)
134
3. Input-Output Марк and Realization Theory
As a matter of fact, if s is small then G(fi...f„) is a point of Гц reached
from F tinder the piecewise constant control defined by
ft j 10 — n j for t E [ti -t-'-' + b- iAi +  -  — b)
Moreover the values of the components of H (i.e. the values of the functions
A!.....A„) at a point were shown to coincide with the value's of certain
derivatives, at time t = 0. of some components of an output function y(/i
obtained under suitable piecewise constant controls (see proof of Theorem
1.9.7). So. oik' may interpret the components of H о G(t\.........tri) as the
values at time t — ti + — - + t„ of certain derivatives of an output function i/(t)
obtained under suitable piecewise constant controls. Two minimal realizations
of the same power series c characterize two systems which by definition display
the same input-output behavior. These two systems, initialized respectively
in .ra and .rb. under any piecewise constant control produce two identical
output, functions. Thus the two sides of (3.38) must coincide.
(iv)	Recall that, if the realization "a"' is minimal, if .t;1) G IT and s
is sufficiently small, the mapping H" oG" is composition of diffeomorphi>ms.
If also the realization “b" is minimal. Hb is indeed a diffeomorphism. but also
Gb must be a diffeomorphism of IT onto its image, because of the equality
(3.38) and of the fact that the left-hand-side is itself a diffeomorphism. The
following diagram
where Г1 =	Vb = GbiW). C Uf}. Vb C and IT = Я'1 о
Сг"'(И') — Hb о Gh(\V). is a commutative diagram of di ffeoi norph isms. Thus,
we may define a diffeomorphism
F : V'1 Vb
as
F = (Hbrl oHa	(3.39)
whose inverse may also be expressed as
F-1 = G11 о (GV1 .	(3.40)
(v)	By means of the same arguments as the ones already used in (iii) one
may easily prove a more general version of (3.38). More precisely, setting
Pl	Pl
= /" + £9;y C = fb + £.F,
mi	mi
3.5 Uniqueness of Minimal Realizations 135
one may deduce that, for sufficiently small f
Ha оФ*’ oGa(tl.......= НьоФ*;Ь oGb{tl............tn) .
Differentiating this one with respect to t and setting t — 0 one obtains
°<у'(с...........................м =	..../„>.
Because of the arbitrariness of tq...cl7! one has then
(//").</> G'J(F......G) =	.....
for all 0 < i < m. But these ones, in view of the definitions (3.39) (3.40) may
be rewritten as
pG.r) =	° F-1 (.r)	0 < / < rn
for all ,r G Vf‘. thus proving (3-36).
(vi)	Again, using the same arguments already used in (ii) one may easily
see that
ha oG'l(ti----b,) = h6 о G1’(h.......t„)
i.e. that
hb(,r) = о
for all j- e V6. thus proving also (3.37). <

4. Elementary Theory of Nonlinear Feedback
for Single-Input Single-Output Systems
4.1 Local Coordinates Transformations
Beginning with this Chapter, we will study - in order of increasing complexity
- a series of problems concerned with the synthesis of feedback control laws
for nonlinear systems of the form (1.2). We will discuss first the case of
single-input single-output systems, whose simple structure lends itself to a
rather elementary analysis, and then in the next Chapter - a special class
of multivariable systems, in which a straightforward extension of most of the
theory developed for single-input single-output systems is possible. Finally
in the last four Chapters - wo will present a set of more powerful tools for the
analysis and the design of more general classes of nonlinear control systems,
The purpose of this introductory section is to show how single-input
single-output nonlinear systems can be locally given, by means of a suit-
able change of coordinates in the state space, a "normal form'' of special
interest, on which several important properties can be elucidated.
The point of departure1 of the whole analysis is the notion of relative
degree of the system, which is formally doserilied in the following way. The
single-input single-output nonlinear system
j- = /(.r)+.y(.r)n
У = h(x}
is said to have relative degree r at a point j,: if
(i)	= 0 for all z in a neighborhood of j,a and all k < r — 1
(ii) L3L^4i(x^0.
Note that there may be points where a relative degree cannot be defined.
This occurs, in fact, when the first function of the sequence
Lgh(x). LgLffi{x)...LglJjh^r)....
which is not identically zero (in a neighborhood of z°) has a zero exactly at
the point x = za. However, the set of points where a relative degree can be
defined is clearly an open and dense subset of the set l: where the system
(4.1) is defined.
138
4. Nonlinear Feedback for Single-Input Single-Out put Systems
Example J.1.1. Consider the equation* describing a controlled Van dec Pol
oscillator in statt' space form
.Г = /И+дМи =	_Д,,	) + (1) " •
Suppose the output function is chosen as
</ = h(.r) = jq .
In this case we have
B;h(.r) =	“ (1 ° ( 1) = °
and
t)/? ,	.	(	ju	h
Lfh(r] = — /[t) = ( 1 0)	.	'	.>	= .to .
7	дх	\	1 -	- -е-j] /
Moreover
L„LfhM = ^^<lM = 1'0 1 ) Q’) = 1
and thus we sec that, rhe system in question ha> relative degree 2 ar any point
However, if the output function is. for instance
у = h(x) = sin.7‘2
then L9h(x) = coszj. The system has relative degree 1 at any point ,r':.
provided that (,rc )•_>	(2k + l)~/2. If the point ,rc is such that this condition
is violated, no relative degree can be defined. <
/
Remark 4.1.2. In order to compare the notion thus introduced with a familiar
concept, let us calculate the relative degree of a linear system
т = .4 j- + Bu
у = Cx .
In this case, since f(x) = .-hr. y(x} = B. h(.r} = Cx. it easily' seen that
L^h(x) = C.4A’.r
and therefore
LgLkfh(x] = CAkB .
Thus, the integer r is characterized by the conditions
САк В = 0 for all A- < r - 1
СА’-ЧЗ ± 0.
4.1 Local Coordinates Transformations
139
It is well-known that the integer satisfying these conditions is exactly equal
to the difference between the degree of the denominator polynomial and the
degree of the numerator polynomial of the transfer function
H(,s) = C(sI-Д)~’В
of the system. <
We illustrate now a simple interpretation of the notion of relative degree,
which is not restrict tai to the assumption of linearity considered in the pre-
vious Remark. Assume the system at some1 time t" is in the state zfB) = ,r:
and suppose we wish to calculate the value of the output y(t) and of its
derivatives with respect to time y[k"' (f). for k = 1.2.at t —	. We obtain
y(fc ) = h(x\ff)] = h(x=)
,,,, Oh dx Oh ,,
.</' (0	=	W--7T =
dx at dx
= Lfbixlt)'} -	
If the relative degree r is larger than 1. for all t such that x[t] is near .r':, i.e.
for all t near C. we have Lffi(x(t}') = 0 and therefore
y' 1110 = Lf/dx(t)) .
This yields
, OLrh dx Wff1 r< , >	, . ,. ,
tf-dt) = -^—-7.	=	+ </ .r f) u(t))
dx dtdx
= Ljh(x(ty} - LgLfh(x(t}')u(t) .
Again, if the relative degree is larger than 2. for all t near В we have
LgLyk(x{t)) = 0 and
= Ljh(x(t)'] .
Continuing in this way, we get
AJ(M =	for all k < r and all t near C
yirl(O - Ljhix^^LgLyhdx^idd) .
Thus, the relative degree r is exactly equal to the number of times one has
to differentiate the output y(t) ar time t — in order to have the value u(tc)
of the input explicitly appearing.
Note also that if
LgLkjh(x) = 0 for all x in a neighborhood of j-c and all k >Q
(in which case no relative degree can be defined at any point around .rc) then
the output of the system is not affected by the input, for all t near tJ As
140	4. Xoulinear Feedback for Single-Input Single-Output. Systems
a matter of fact, if this is the case, the previous calculations show that the
Taylor series expansion of y(t) at the point t = t° has the form
A=0
i.e. that y(t) is a function depending only on the initial state and not on the
input.
These calculations suggest that the functions h(x). Lfh(x)......
must have a special importance. As a matter of fact. it. is possible to show
that they can be used in order to define, at least partially, a local coordinates
transformation around .r3 (recall that is a point where LgLj~} h(.P)	0).
This fact is based on the following property.
Lemma 4.1.1. The. row vectors
dh(x°). dLfh(xQ)......dLj~} h(xc)
are linearly independent.
In order to prove this Lemma., we illustrate first another property, which
will also be used several other times in the sequel.
Lemma 4.1.2. Let о be a real-valued function and f.g vector fields, all de-
fined tn an open set t ' of TA . Then, for any choice, of s. k. r > 0.
(dL'j(?(x), adk+ry(x)') - ^(-1 V f \ 'J Lr~l(dL J~’O(t), adjy(x)) .	(4.2)
i=o
A.s a consequence., the two sets of conditions
(i) LgCfix) = LyLfo(x) - .. /= LgLkd>(x) = 0	for all x € Lr (4.3)
(ii) Lg(p{x) - Ladfgo(x) - - -  = Ladk-9<p{x} - 0	for all x € U (4.4)
are equivalent.
Proof. The proof of (4.2) is easily obtained by induction on r. in view of the
fact that
{</LyO(j-).a</J+' + l = (dL}<Sr). [/. u</pr</(z)]}
=	- <dL,f+lO(T).adt/+rg(O) .
The equivalence of (4.3) and (4.4) is a straightforward consequence of (4.2).
We can proceed now with the proof of Lemina 4.1.1.
г
4.1 Local Coordinates Transformations 141
Proof. Observe that by definition of relative- degree, using (4.2,1 we obtain for
all Л j such that i -+- j < r - 2
{dLJfh{.r).	x')) = 0 for all .r around ,rc
and
( — L(/Lrf~[ h[js ) ^0
for all i. j such that i + j = г - 1.
The above conditions, all together, show that the matrix
/ dh(x~)	\
dLfh(.r )	adf(j[.r°} ... adrf~{g{.r~) i =
\dL'r^[h(.rQ) /	_
/	0	(dh(.r').(idrf 'д(хсУ)\
_	0	...	*
\ (dLy~1 h(g{.r°))	*	*	/
has rank г and. thus, that the row vectors dh(,rc ). dLfh(x°)..(m l
are linearly independent, <
Lemina 4.1.1 shows that necessarily r < n and that the г functions
й(х). Ljh(x) qualify as a partial set of new coordinate func-
tions around the point x^ (recall Proposition 1.2.3). As we shall see in a
moment, the choice of these1 new coordinates entails a particularly simple
structure for the equations describing the system. However, before doing this,
it is convenient to summarize the results discussed so far in a formal state-
ment. that also illustrates a way in which the set of new coordinate’s can be
completed in case the relative degree r is strictly less than n.
Proposition 4.1.3. Suppose the system has relative degree r at J’3. Thea
r <n. Set
Oi(x) -- h(x)
сгф‘) — Lfh(gr)
or.(.r) = Lrf ^(.r) .
If r is strictly less than n. it ts always possible to find n — r more functions
Фг-v (u’)...., <i>,i (.r) such that the mapping
Ф(х) =
has a jacobian matrix which is nonsingular at x:' and therefore, qualifies as
a local coordinates transformation in a neighborhood of .T~. The value at xQ
142
4. Nonlinear Feedback for Single-Input Single-Output Systems
of these additional functions can be fired arbitrarily. Moreover, it is always
possible to choose G>r+1(j-).......On(.r) in such a way that
LyO^x) — 0 for all r 4- 1 < i < n and all. x around x?,
Proof, By definition of relative degree, the vector g(r~ ) is nonzero, and. thus,
the distribution G = span{g} is nonsingular around ,r°. Being 1-dimensional,
this distribution is also involutive. Therefore, by Frobenius’ Theorem, we
deduce the existence of n-1 real-valued functions. AJ.r).An-i (t)- defined
in a neighborhood of .rc, such that
spanjdAi...dA,;_] у — G .	|4.G)
It is easy to show that
dim(G- + span{dh, dLfh.....= n	14.7)
at z“. For. suppose this is false. Then
G(xc J П (span{/M .dLfh.dLrjT1 h }M.r= 1	{0}
i. c. the vector g(xJ) annihilates all the covectors in
span{c7h. dL fh...., dL'j~{ /?}(./ °) .
But this is a contradiction, because by definition {dL^-1 h(xc ).g{xz)) is
nonzero.
From (4.6). (4.7) and from the fact that spanjrf/ndLfh.......dL’f~xh} has
dimension r. by Lemma 4.1.1. we conclude that in the set {Ai,.... An_, } it is
possible to find n - r functions, without loss of generality Ai....An , with
the property that the n differentials dh.dLfh........dL^h.dXi..........dXn_r.
are linearly independent at x°. Since by construction the functions A!.......
A„_r are such that	f
{dXi (z). g(j')) — LgXtlx) = 0 for all .r near x" and all 1 < i < n — r
this establishes the required result. Note that any other set of functions of the
form A((£) = A;(z) +c;. where c, is a constant, satisfies the same conditions,
thus showing that the value of these functions at the point d can be chosen
arbitrarily. <
The description of the system in the1 new coordinates z; = d>i(x). 1 < i <
n. is found very easily. Looking at the calculations already carried out at the
beginning, we obtain for zi,....zr
dz\
dt
doi dx
dx dt
dzr-i
dt
t?or i dx
Ox dt
= Lfh(x(t)) =	= ~?(0
ax dt
OlLh ~h) dx i
—x-------r = L^GdxG)) = oMl) = -r(t) 
ax dt J
4.1 Local Coordinates Transformations
143
For zr we obtain
dt J	J
On the right*hand side of this equation we must now replace x(f’) with its
expression as a function of z(f). i-e- -r(t} — Ф~х lz(t))_ Thus, setting
a(z) = LgLrf~1h^~'l(zY)
b[z) =
the equation in question can be rewritten as
= b|T(7')) +	))u(f) .
dt
Note that at. the point z" = Ф(.гс). n(:c 1	() by definition. Thus, the
coefficient a(z) is nonzero for all c in a neighborhood of z~.
As far as the other new coordinates are concerned, we cannot expect any
special structure for the corresponding equations, if nothing else has been
specified. However, if дг-ч(-r)...have been chosen in such a way
that Lg&d.r) = 0. then
= 1/О;(.г(С)+£йог(.с(пМп =
Setting
qi(z) = L) for all r + 1 < i < n
the latter can he rewritten as
dzt
~r =	•
dt
Thus, in summary, the state-space description of the system in the new
coordinates will be as follows
*	-1 ~ ~_
-	2	=	-3
~r_1 ”	(4.8)
-	r = 6(c) +
zr+1 = q>’~ i (~ 1
zn = q,A.z) .
In addition to these equations one has to specify how the output of the
system is related to the new state variables. But. being у = /ifir). it is imme-
diately seen that
144
4, Nonlinear Feedback for Single-Input. Single-Output Systems
i) = Ci .	(4.9i
The structure of these equations is best illustrated in the block diagram
depicted in Fig. 4.1. The equations thus defined are said to be in normal form.
We will find them useful in understanding how certain control problems can
be solved.
Fig- 4.1.
Remark Note that sometimes it is not easy to construct n —r functions
G>r+l (jt)...such that LgO^r) = (J. because1 this, as shown in the proof
of Proposition 4.1.3. amounts to solve a system of n — r partial differential
equations. Usually, it is much simpler to find functions +	.....<pri(z)
with the only property that the jaeobian matrix of Ф(л') is nonsingular at ,r~.
and this is sufficient to define a coordinates transformation. Using a trans-
formation constructed in this way. one gets the same struct uro for the first г
equations, i. e.	•
C i =	Z_)
~ r — bl z) + d ( Z ) ti
but it is not possible to obtain anything special for the last и — г ones, that
therefore1 will appear in a form like
~r--i	— (b -U c) + Pr -1 (c )'U
7л(~) + Pnl du
with the input и explicitly present. <
4.1 Local Coordinates Transformations 145
Example 4-1-4- Consider the system
For this system we have
~ =	(0	0	1).	L,,h(x') = 0. L fh(x) — x>
ox	J
=	(0	1	0),	£,£/Л(.г) =	1 .
OX
In order to find tin1 normal fornn we set
-1	= Pl (<) = ll(x) = J‘3
-2 = po(.r) = Lfh(x) = x?
and we seek for a function p3(т) such that.
6>p3
Ox
t?p3 . дс>ч
yC’l = w— exp(.z2) + — = 0 -
Ox I	ox-.
It is easily seen tliat the function
Оз(х’) = 1 + JJ[ - СХр(-Г2)
satisfies this condition, This and the precious two functions define a trans-
formation ~ = Ф(х) whose jacobian matrix
0Ф
dx
(J
0
1
° 1\
1 °
- ехр(л'з) 0 /
is nonsingular for all z. The inverse transformation is given by
= -1 + c3 + exp(c2f
j-2	=	<>
T? = ~i -
Note also that Ф(0) = 0. In the new coordinates the system is described by
S = (-1 + 23 + ехр(г2))с2 + и
c3 = (1 - c3 - ехр(з2))(1 + ^ехр(г2)) .
These equations are globally valid because the transformation we considered
was a global coordinates transformation. <
У
146
4. Nonlinear Feedback for Single-In put Single-Output Systems
Example J. 1.5. Consider the system
For this system we have
dh
dr
d(Lfh)
dr
(0 0 0 1),
( 2xl 1 0 0).
Lgh(r) = 0. Lfh[.r) = x] 4- .m
LgLfh(x) = 2(1 4-	.
Note that LyLfh(r)	0 if r:i —1. This means that we shall he able to find
a normal form only locally, away from any point such that .r:1 = -1.
In order to find this normal form, we have to set first of all
-i = <?iИ =	- j-j
-2 = (M-r) = Lfh(r} = r2 + r'l
and then find <?3(r). which complete the transformation and are such
that Lgcxdr) = Lgdi(x) =0.
Suppose we do not want to search for these particular functions and we
just take any choice of 03(r), ddx) which completes the transformation. This
can be done. e.g. by taking
г:1 = <Рз(-г) = л-з
~1 =	= -Г1 •
The jacobian matrix of the transformation thus defined
	/ °	0	0	1\
ЭФ _	2ti	1	(J	0 ’
dr	0	0	1	0
	\ 1	0	0	0/
is nonsingular for all r. and the inverse transformation is given by
Г2
Cl
Note also that Ф(0) = 0. In these new coordinates the system is described by
4.2 Exact Linearization Via Feedback 147
~2 = -i + 2c 1(44(73 - 24) - C4 ) 4- (2 4- Zz^hi
Z3 =	-~3 + U
-4	=	—2?4 + 42Сд .
These equations are valid globally (because rhe transformation we considered
was a global coordinates transformation), but they are not in normal form
because of the presence of the input u in the equation for 23.
If one wants to get rid of и in this equation, it is necessary to use a
different function ©з(.г). making sure that
доз . , дфз ,9 , 9 . ддз
-x— ff(-c) = w—(2 + 2jV) + —- = 0 .
dr dx-i	дгз
An easy calculation shows that the function
Оз (z) = r2 - 2.Г3 -
satisfies this equation. Lsing this new function and still taking 04(4*) = ri
one finds a transformation (whose domain of definition does not include the
points at which = —1) yielding the required normal form. <1
4.2 Exact Linearization Via Feedback
As we anticipated at the beginning of the previous section, one of the main
purposes of those notes is the analysis and the design of feedback control
laws for nonlinear systems. In almost all situations, we assume the state z
of the system being available for measurements, and we let the input of the
system to depend on this state and. possibly, on external reference signals.
If the value of the control at time t depends only on the values, at the same
instant of time, of the state r and of the external reference input, the control
is said to be a Static (or Memoryless) State Feedback Control. Otherwise,
if the control depends also on a set of additional state variables, i.e. if this
control is itself the output of an appropriate dynamical system having its
own internal state, driven by ,r and by the external reference input, we say
that a Dynamic State Feedback Control is implemented.
In a single-input single-output system, the most convenient structure for
a Static State Feedback Control is the one in which the input variable u is
set equal to
u = o(z) 4- J(z)r	(4.10)
where г is the external reference input (see Fig. 4.2). In fact, the composition
of this control with a system of the form
r = /Gr) + s(.r)u
= h(r)
У
148	4. Nonlinear Feed back for Single-Input Single-Output Systems
yields a closed loop characterized by the similar structure
r = f(.r) 4- (jhrjnf.r) + gir)J(.rlr .
у = IM
The functions a(j-) and J(z) that characterize the control (4.10) art1 de-
fined on a suitable open set of . For obvious reasons, Jbr) is assumed to
be nonzero for all ,r in this set.
Fig. 4.2.
The first application that will be discussed is the use of state feedback
(and change of coordinates in the st ate-space) to the purpose of transforming
a given nonlinear system into a linear and controllable one. The point of
departure of this study will be the normal form developed and illustrated in
the previous section.
Consider a nonlinear system having relative degree r = n. he. exactly
equal to the dimension of the state space, at some point z = j-°. In this
case the change of coordinates required to construct the normal form is given
exactly by
/ h(z) \
=	= I Lfh(T)
\0гМ/
i.e. by the function h(x) and its first n — 1 derivatives along f(d-). No extra
functions are needed in order to complete the transformation. In the new
coordinates
1 < i < it
the system will appear described by equations of the form
4.2 Exact Linearization Via Feedback
149
where z = (z\....c„).	Recall also that at the point з° = Ф(х3). and thus at
all г in a neighborhood of z°. the function n(^) is nonzero.
Suppose now the following state feedback control law is chosen
a =	(-b(z) + r)
ti(c)
(4.11)
which indeed exists and is well-defined in a neighborhood of z~'. The resulting
closed loop system is governed by the equations (Fig. 4.3)
'ii-1 — 'ii
= r
i.e. is linear and controllable. Thus we conclude that any nonlinear system
with relative degree n at some point j?c can be transformed into a system
which, in a neighborhood of the point. zc = Ф(х°), is linear and controllable.
It is important to stress that the transformation in question consists of two
basic ingredients
(i) a change of coordinates, defined locally around the point. ,r°
(ii) a state feedback, also defined locally around the point .c°.
Fig. 4.3.
Remark 4-2.1. It is easily checked that the two transformations used in order
to obtain the linear form can be interchanged. One can first use a feedback
and then change the coordinates in the state space, and the result is the
same. The feedback needed to this purpose is exactly the same feedback just
used, but now expressed in the т coordinates, i.e. as
u = ran и	+ г) .
а(Ф(т))
150	4. Aonlinear Feedback for Single-Input Single-Output Systems
Comparing this with the expressions for </(;) and b(z) given in the previous
section, one realizes that this feedback - expressed in terms of the functions
/(t), g(r). h(s) which characterize1 the original system - has the form
An immediate? calculation shows that this feedback, together with the1 same1
change of coordinates used so far. exactly yields the same linear and control-
lable system already obtained.
Remark 4-2.2. Note that if J'3 is an equilibrium point for the original non-
linear system, i.e. if /(z3) = 0. and if also — 0. then г3 = Ф(.гс) — ().
As a matter of fact
cM-H = Л(т3,)-0
0,(^1 = -------------f(.rA = 0 for all 2 < i < n .
о/
Note also that a condition like /г(т3) = 0 can always be satisfied, by means
of a suitable translation of the Origin of the1 output space.
Thus, we conclude that if J’c is an equilibrium point for the original sys-
tem. and this system has relative degree n at rc. there is a feedback control
law (defined in a neighborhood of j,c) and a coordinates transformation (also
defined in a neighborhood of .r°) changing the system into a linear and con-
trollable one. defined in a neighborhood of 0. <
Remark 4-2.3. On the linear system thus obtained one can impose new feed-
back controls, like for instance
>7= K-
with
К = (c0 .. .cn-0
chosen e.g. in order to assign a specific set of eigenvalues, or to satisfy an
optimality criterion. Recalling the expression of the c/s as functions of z. the
feedback in question can be rewritten as
c = coh(r) + ciLfh^x) +------r cFJ_i£y-1/?(.r)	(4.13)
i.e. in the form of a nonlinear feedback from the state г of the original de-
scription of the system. Note that the composition of (4.12) and (4.13) is
again a state feedback, having the form
-L'jhM + Y.-3 CiL'.hW
U = --------------y-—--------—
L,,L'j-4i(r)
4.2 Exact Linearization Via Feedback
151
Example 4^-4- Consider the system
0	\	/exp(j-2) \
Л H + exp(z2) <i
j-i - z-] /	\	0	/
У = J‘3 •
For this system we have
Lgh(.r J
LyLfb[j-]
L^Lyh^r}
L3fh(x)
1). Ljhix) = .Г1 - ,r2.
0T L’yh(x) = —aq — л-3.
-(1 + 2,r2) oxp(j‘2),
— 2.1'2 (V ^'5 ) '
Thus; we see that the system has relative degree 3 {i.e. equal to n) at each
point such that 1	2.r2	0. Around any of such points, for instance around
x — 0. the system can be transformed into a linear and controllable system
by means of the feedback control
-2.r2(.ri +
11	2j'-_>) exp(.r2)
1
(1 + 2./'2) exp(>2)
and the change of coordinates
d =	=	,r3
c2 = Ljh{x) = jq — ,r2
-3 = L'yh(x) = — .л — j-i] .
Noto that both the feedback and the change of coordinates are defined only
locally around .r — 0. In particular, the feedback u is not defined at points x
such that 1 + 2j'2 = 0 and the jacobian matrix of the coordinates transfor-
mation is singular at these points.
In the new coordinates, the system appears as
/0 1 0\	/0\
: = 0 0 1	: •	0 r
poo/	\1/
which is linear and controllable. <
Of conisc. the basic feature of the system that made it possible to change
it into a linear and controllable one was the existence of an "output” function
h(r) for which the system had relative degree exactly n (at rc). We shall see
now that rhe existence of such a function is not only a sufficient - as the
previous discussion shows - but also a necessary condition for the existence
152
4. Nonlinear Feedback for Single-Input Single-Output Systems
of a state feedback and a change of coordinates transforming a given system
into a linear and controllable one.
More precisely, consider a system (without output)
* = ffir) + g(x)u
and suppose the following problem is set: given a point xc find (if possible),
a neighborhood U of .r°. a feedback
и = n(r) +
defined on and a coordinates transformation z = Ф(х) also defined on l\
such that the corresponding closed loop system
r = M + g( -r) + g(x)3(x)v
in the coordinates z = Ф(х). is linear and controllable, i.e. such that

j-ф-1 (г)
'ЭФ
— (g(x)3(x))
их
(4.14)
(4.15,1
for some suitable matrix A e and vector В G IR” satisfying the condi-
tion
rank(B AB ... An-1B) = n .
This problem is the "single-input” version of the so-called State Space
Exact Linearization Problem. The previous analysis has already established
a sufficient condition for the existence of a solution: we show now that this
condition is also necessary. •
Lemma 4.2.1. The State Space Exact Linearization Problem is solvable if
and only if there exist a neighborhood U of x° and a real-valued function A(z).
defined on L-, such that the system
x = f(x) + g(x)u
У = А(т)
has relative degree n at xc.
Proof. Clearly, we only have to show that the condition is necessary. We begin
by showing an interesting feature of the notion of relative degree, namely that
the latter is invariant tinder coordinates transformations and feedback. For.
let z = Ф(л*) be a coordinates transformation, and set
/N =
гаФт/ ,
dx
Т=Ф“ 1 ( zI
' дФ
dx
Л(г) =Л(ф-‘(г)).
g{-) =

4.2 Exact Linearization Via Feedback 153
| Then
Lfh(z}
=
'Oh '
aT(j,)
dh'	'ЭФ 11 'дФ
Or.	. dz J .dr J
— [Lf /ЦХ)]	-
х=Ф-Чс)
Iterated calculations of this kind show that
from which it is easily concluded that the relative degree is invariant under
coordinates transformation. As far as the feedback is concerned, note that
Lj^gah(x) = Lkjh.(r) for all 0 < к < r - 1 .	(4.16)
As a matter of fact, this equality is trivially true for к = 0. By induct ion,
suppose is true for some 0 < к < г — 1. Then
Ty^Q/?(.r) = L f~gC1L jhi,x) = Lj h(x) + LgLfh(x]a(x) = Lj+ h(x)
thus showing that the equality in question holds for к + 1. From (4.16). one
deduces that
L3JLhJ>(A = 0
for all 0 < к < r - 1
and that, if 3(r°)	0
LgiiLj+gnh(r ) 7^ 0 
This shows that r is invariant under feedback.
Now, let (A, B) be a reachable pair. Then, it is well-known from the theory
of linear systems that there exists a nonsingular n x n matrix T and a 1 x n
vector k such that
	/0 1 0
	0 0 1
T(A + Bk)T~l =	
	0 0 0
\0 0 0	
(4.17)
Suppose (4.14) and (4.15) hold and set
Ф(г) = ТФ(х)
a(x) + 3{х)кФ(х).
Then, it is easily seen that
154	4. Nonlinear Feedback for Single-lupin Single-О input Systems
O.r
^-(<Л.г).зИ)
/Jr'
/0 I 0
0 0 1
0 0 0
\o о 0
n
о/
1
0/
From this, it is deduced that there is no loss of generality in assuming that the
pair (.4. B) which renders the (4.14)-(4.15) satisfied has the form indicated
in the right-hand-sides of (4.17).
Define now the "output" function
У = (1 0	• 0)5 .
A straightforward calculation shows that the linear system with A and В in
the form of the right-hand-sides of (4.17) and with this output function has
exactly relative degree n. Thus, since the relative degree is invariant under
feedback and coordinates transformation, the proof is complete. <3
The problem of finding a function A(u™) such that the relative degree of
the system at .rc is exactly n. namely a function such that
LgX(x] = LgLfX(x) = ... = £y£r/-JA(.r) = 0 for all .r (4.18)
(4.19)
is apparently a problem involving the solution of a system of partial differen-
tial eq nations (namely the equations (4.18)). in which the unknown function
A(j-) is differentiated up to n — 1 tidies, together with a condition (namely the
condition (4.19)) which singles-out trivial solutions like e.g. A(.r) = 0. How-
ever. thanks to Lemma 4.1.2. this system is in fact equivalent to a system of
first order partial differential equations, of a rather simple form. As a matter
of fact, this Lemma exactly shows that the equations (4.18) are equivalent to
£уА(т) = LadfgX(r} = ... = Lad„-2gX{;r) = 0	(4.20)
and that the nontriviality condition (4.19) is equixalent to
(4.21)
The existence of a function satisfying these relations is an easy conse-
quence of Frobenius’ Theorem, as it can be seen in the proof of the following
result.
г
4.2 Exact Linearization Via Feedback 155
Lemma 4.2.2. There elists a real-valued function A(.r) defined in a neigh-
borhood U of xz solving the partial differential equations (4-20). and satisfying
Uie nontriviality condition (4-21). if and only if
(i) the matrix. adjglx'-} ... ad’ff~gUz) ad^~} g{ xz )) has rank n.
(ii) the distribution D — span-fry. adjg..adj~~g} is involutive in a neigh-
borhood of x°.
Proof. Suppose a function A(.r) satisfying (4.201 and (4.21) exists. Then, from
the proof of Lemma 4.1.1, in particular from the nonsingularity of the matrix
(4.5), we deduct1 that the n vectors
adfg(x=).....adj~2g(xz }.ady} g( ff)
are linearly independent. This proves the necessity of (i). If (i) holds then
the distribution D is nonsingular and (n — 1.)-dimensional around J‘°. The
equations (4.20). that can be rewritten as
dX{x) (jffr) adfg(.r') ... ad’^~2g{x)^ = 0 ,	(4-22)
tell us that the differential dX(x) is a basis of the 1-dimensional codistribution
around .rc. So. by Frobenius' Theorem, the distribution D is involutive.
and this proves the necessity of (ii). Conversely, suppose (i) holds. Then the
distribution D is nonsingular and (n — 1)-dimensional around T. If also (ii)
holds, by Frobenius’ Theorem we know there exists a real-valued function
A(t). defined in a neighborhood C of .r° whose differential tfA(j-) spans Dff
i.e. solves the partial differential equation (4.20). Moreover. (YA(j-) also satis-
fies (4.21). because otherwise dA(z) would be annihilated by a set of n linearly
independent vectors, i.e. a contradiction. <3
We can at this point summarize the results established so far in the fol-
lowing formal statement
Theorem 4.2.3. Suppose a. system
.r = /(J') -e g[x)u
is given. The State Space. Exact Linearization Problem is solvable near a
point jS (i.e. there exists an "output" function A(x) for which the system has
relative degree n at. x'z) if and only if the following conditions are satisfied
(i) the matrix ) adfg(x°) ... ad’l'~2g(xz) ad’}~1 д(т~)^ has rank n.
(ii) the distribution D = span{(/. adyg,... .adj~2g} is involutive near r':.
On the basis of the previous discussion, it is now clear that the procedure
leading to the construction of a feedback и — o(z) + 3(x)v and of a coordi-
nates transformation z = Ф(.г) solving the State Space Exact Linearization
problem consists of the following steps
lob
4. Aonlmear Feedback for Single-Input Single-Output Systems
-	from f(x) and g(x), construct the vector fields
дЛ)- ndfg(x).....adnf~2g(x). ad”~lg(x)
and check the conditions (i) and (ii).
-	if both are satisfied, solve for A(j’) the partial differential equation id.20).
- set
-LnfX(x)	1
°C) = r r"-'M t J(z) = T	l4 231
- set
Ф(х) = со1(Х(х)Л/Х(х)......L}-’A(jr)) .	(4.24)
The feedback defined by the functions (4.23) is called the linearizing feed-
back and the new coordinates defined by (4.24) are called the linearizing
coordinates. We illustrate now the whole Exact Linearization procedure in a
simple example.
Example 4-2.-5. Consider the system
/ j:3(l + x2) \	/ 0 \
x — jq 1 +	1 + x-2 u .
\.r2(l + Zi) /	\ -z3 /
In order to check whether or not this system can be transformed into a
linear and controllable system via state feedback and coordinates transfor-
mation. we have to compute the functions adfg(x) and ndjg(x) and test the
conditions of Theorem 4.2.3.
Appropriate calculations show that adfgfx) —
0 0 0 \ /.r3(l + j2) \	/ 0 x3
0 1	0	.Г] I 	1	0
0 0-1/ \т2(1 + xi)/ ! \r-3 1+J-i
1 +	\ / 0
0 I I 1 + T2
0 J \ -j3
0
jq
-(l+-r1)(l + 2j2)
and that
/ (1 + ,r2)(l + 2.Г2)(1 + Jq) - J3.C1
arf/ffU) =	т3(1 + ,r2)
\ -z3(l + jq>)(l + 2.r3) - 3zi (1 + 4q)
At x = 0. the matrix
/0 0	1\
(s(t) adfg{x) adjgtx))^ =	1	0	0 I
\0 -1 0/
4.2 Exact Linearization Via Feedback
157
у has rank 3 and therefore the condition (i) is satisfied. It is also easily checked
P that the product [g. adfg][x] has a form
;	[g.adfg\(x) = 1*1
' \ * /
and therefore also rhe condition (ii) is satisfied, because the matrix
I g(j-) adfg(x) [g.adfg](x))
has rank 2 for all ,r near x = 0.
In the present case, it is easily seen that a function A(j) that solves the
equation
~(g{j‘) ad/glx] ) - 0
их
is given by
A(jj = jq .
From our previous discussion, we know that considering this as "output"
will yield a system having relative1 degree 3 (i.e. equal to ?i) at the point
x = 0. We double-check and observe that
LgX(x) = 0. LgLjX(x) = 0. £a£yA(.r) = fl+ti)(1+-r-2)(l+2.r>) - .ri.r:{
and LgL‘jX((.X] = 1. Locally around x = 0, tire system will be transformed
into a linear and controllable one by means of rhe state feedback
— L J A( j) + г
“ = LgL2fX(x) ~
_ ~J’s(1 + .Г-j) - .Г2.Г3Ц + J’U2 - T] (1 + J] )(1 + 2z2) - Ja-r-jfl -Г1) + r
(1 zjfl + J’2)(l + 2х-г) ~ J’1J*3
and the coordinates transformation
= A(.r) =
Z-2 = LfX(r) = ,гз(1 -+- ду)
<1 - LjX(x) = X3XY 4- ( 1 + .Г!,)(1 -+ X-2)x2  <
Remark J. 2.6. Using the above result, it is easily seen that any nonlinear
system whose state space has dimension ti — 2 can be transformed into a
linear system, via srate feedback and change of coordinates, around a point
r°, if and only if the matrix
(flU’3) adfg(xQ))
has rank 2. As a matter of fact, this is exactly the condition (i) of the previous
Theorem, and condition (ii) is always satisfied, because D ~ spau{§} is 1-
dimensional. In this case it is always possible to find a function A(z) =
АСп.ь), defined locally around x°. such that
158	4. Nonlinear Feedback for Single-Input Single-Output Systems
dX t к ЭХ	,	OX
<T1' Tjj) +	<-’) ^°'<]
OX	01'1	0X2
Remark Jf.2.rl. The condition (i) of Theorem 4.2.3 has the following interest-
ing interpretation. Suppose the vector field fix} has an equilibrium at x~ = (j,
i.e. /(O’) = 0. and consider for /(x) an expansion of the form
/(x) = Ат + /2М
which separates the linear approximation At from the higher-order term
/2(т). Consider also for g(x) an expansion of the form
g{x\ = В + gi (.r)
with В — fl(O). These expansions characterize the linear approximation of
the system at x — 0. which is defined as
i = A.r 4- Bn .
An easy calculation shows that the vector fields adjg(x) can be expanded
in rhe following way
adjg(x) ~ (-1)a‘.4a’B + Pk(x)
where pfr(-r) is a function such that р^.(О) — T a matter of fact, the
expansion in question is trivially true for к ~ 0. By induction, suppose is
true for some An Then, by definition
ad^lg(x) =	f(x) i ^-adfax)
J	q ^x	ox J
=	+ Л(.г|) - (.4 + Sp)((-!)' AkB + PkM)
OX ,	OX
=	+ P1.+1(j-)
where pt-i (x). by construction, is zero at x = 0.
From this, we see that the condition (i) of Theorem 4.2.3 (written at
,r2 — 0) is equivalent to the condition
rank(Z? AB ... _4'i-1B) = n
i.	e. to the condition that the linear approximation of the system at x=0 is
controllable.
In other words, we conclude that the controllability of rhe linear approx-
imation of the system at т — .rc is a necessary condition for the solvability
of the State Space Exact Linearization Problem. <
4.2 Exact Linearization Via Feedback
159
Remark 4.2.8. It is interesting to observe that the conditions (i) and (ii ) of
Theorem 4.2.3 imply the invohit.ivity of the distribution
Dk- ~ spanfy. adjg.....adjg]
for any 1 < A: < ii - 3. As a matter of fact, since (i) and (ii) imply the
existence of a A(.r) such that (4.18) and (4.19) hold, from Lemma 4.1.2 it
follows that
dX(x) (p(.r) adfg(x)
dLfX(j-) (g(x) adjg(x)
adkg(x)) = 0
adjg(x)) = 0
dLj k 2X{xHg(.r] adfg(x)
adkfg(x))
0 .
These equalities show that
span{dA.d£/A....dL’^k-'2X} C D£
Moreover, since (Lemma 4.1.1) the differentials dX.dLfX.... .dL1} k 2X
are linearly independent around j-° and Djr has dimension n - £• — 1 around
x° (as a consequence of assumption (i)). it is concluded that Dj? is spanned
by exact differentials. Then, by Frobenius' Theorem. Dr is involutive.
We see from this property that the in vol utility of all the distributions
Dk- 1 < A' < n — 2, is a necessary condition for the solvability of the Exact
State Space Linearization Problem. <
Remark 4.2.9. Note that if the State Space Exact Linearization Problem is
solved by means of a feedback and a coordinates transformation z = Ф(т)
defined in a neighborhood E of the corresponding linear system is defined
on the open set Ф(1')- For obvious reasons, it is interesting to have
containing the origin of , and in particular to have Ф[х°) ~ 0. In this case,
in fact, one could for instance use linear systems theory concepts in order to
asymptotically stabilize at z = 0 the transformed system and then use the
stabilizer thus found in a composite loop to the purpose of stabilizing the
nonlinear system at x - x° (see Remark 4.2.3).
This is indeed the case when x° is an equilibrium of the vector field f(x).
In this case, in fact, choosing the solution A(r) of the differential equation
wfith the additional constraint A(tc) = 0, as is always possible, one gets
Ф(т°) = 0. as already shown at the beginning of the section (sec Remark
4.2.2).
If is not an equilibrium of the vector field f(x), one can manage to
have this occurring by means of feedback. As a matter of fact, the condition
Ф(х°) = 0. replaced into (4.14). necessarily yields
ЭФ
w-(/(j) + 5(j?)o(j))
dx
=-- 0
160
4. Nonlinear Feedback for Single-Input Single-Output Systems
i.e.
/(Z) c/(Z = 0 .
This clearly expresses the fact that the point Z is an equilibrium of tin*
rector field /(t) + g(j)o(.r). and can be obtained if and only if the vectors
/(/) and g(jc) arc such that
/(Z) = cg[.ra)
where c is a real number. If this is the case, an easy calculation shows that
the linearizing coordinates are still zero at .e':: (if A(Z is such), because, for
all 2 < ? < n
L^’A(Z) = cLyL’ff2\(xc) = 0 .
Moreover, the linearizing feedback a(Z is such that
as expected. <
Remark 4-2.10. Note that a nonlinear system
c = /(Z + д(х}и
У = h(r)
having relative degree strictly less than n could as well meet the requirements
(i) and (ii) of Theorem 4.2.3. If this is the case, there w ill be a different "out-
put" function, say A(j). with respect to which the system will have relative
degree exactly n. Starting from this new function it will be possible to con-
struct a feedback u = a(j-) + 3(x)v and a change of coordinates z ~ £(Z-
that will transform the state space equation
i- = /(/) + f/(j-)u
into a linear and controllable one. However, the real output of the system, in
the new coordinates
у = /г(ф-](с))
will in general continue to be a nonlinear function of the state z. u
If the system has relative degree r < n. for some given output h(x). and
either the conditions of Lemma 4.2.2 for the existence of another output
for which the relative degree is equal to n - are not satisfied, or more simply
one doesn't like to embark oneself in the solution of the partial differential
equation (4.20) yielding such an output, it is still possible to obtain - by
means of state feedback - a system which is partially linear. As a matter of
fact, setting again
a = -y— (~b(z) + r)	(4.25)
n(z)
4.2 Exact. Linearization Via Feedback 161
on the normal form of the equations, one obtains, if r < n, a system like
(4.26)

Cl —	Qh(~)
У = ~i •
This system clearly appears decomposed into a linea?- subsystem, of di-
mension r, which is the only one responsible for the input-output behavior,
and a possibly nonlinear subsystem, of dimension n -r. whose behavior how-
ever does not affect the output (Fig.4.4).
Fig. 4.4.
We summarize this result for convenience in a formal statement where, for
more generality, the linearizing feedback is specified in terms of the functions
/(«r). g(x) and /i(r) characterizing the original description of the system.
Proposition 4.2.4. Consider a nonlinear system having relative degree r at
a point .rc. The state feedback
transforms this system into a system whose input-output behavior is identical
to that of a linear system having a transfer function
WC = 4 
S’’
162	4. Nonlinear Feedback for Single-Input Single-Output Systems
4.3 The Zero Dynamics
In this section we introduce and discuss an important concept, that in many
instances plays a role exactly similar to that of the "zeros" of the transfer
function in a linear system. We have already seen that the relative degree r
of a linear system can be interpreted as the difference between the number
of poles and the number of zeros in the transfer function. In particular, anv
linear system in which r is strictly less than n has zeros in its transfer function.
On the contrary, if r = n the transfer function has no zeros: thus, the systems
considered at the beginning of the previous section are in some sense analogue
to linear systems without zeros. We shall see in this section that this kind of
analogy can be pushed much further.
Consider a nonlinear system with r strictly loss than n and look at its
normal form. In order to write the eq nations in a slightly more compact
maimer, we introduce a suitable vector notation. In particular, since there
is no specific need to keep track individually of each one of the last n — /
components of the state vector, we shall represent all of them together as
/ a-+i \
V = I “	
\ Zr> /
Sometimes, whenever convenient arid not otherwise required, we shall repre-
sent also the first r components together, as
(-1 1
I .
With the help of these notations, the normal form of a. single-input single-
output nonlinear system having r <, n (at some point of interest .rc. e.g. an
equilibrium point) can be rewritteidas

Recall that, if x° is such that /(тс) = 0 and h(xc) ~ 0, then necessarily
the first r new coordinates zt arc 0 at r°. Note also that it is always possible
to choose arbitrarily the value at x° of the last n — r new coordinates, thus
in particular being 0 at xG. Therefore, without loss of generality, one can
assume that £ = 0 and i] ~ 0 at xc'. Thus, if was an equilibrium for the
system in the original coordinates, its corresponding point (£. r/) = (0. 0) is an
4.3 The Zero Dynamics 163
equilibrium for rhe system in the new coordinates and from this we deduce
that	T,	.
b(Oh) = 0	at (£. ri) = (0.0)
ijfo'l) = 0	at «fo = (0.0) .
Suppose now we want to analyze the following problem, called rhe Problem
of Zeroing the Output. Find, if any. pairs consisting of an initial state ,rc and ,
of an input function tr(-), defined for all t in a neighborhood of t = 0. such
that the corresponding output y(t) of the system is identically zero for all t in
a neighborhood of t — 0. Of course, we are interested in finding all such pairs
(яо,ио) and not simply in the trivial pair r° = 0. tP = 0 (corresponding to
the situation in which the system is initially at rest and no input is applied).
We perform this analysis on the normal form of rhe system.
Recalling that in rhe normal form
we see that the constraint у It.) = 0 for all t implies
ifof) =i2(t) = ... = fo(t') = 0.
that is £(t) — 0 for all t.
Thus. we see that when the output of the system is identically zero its
state is constrained to evolve in such a way that also £(£) Is identically zero.
In addition, the input u(t') must necessarily be the unique solution of the
equation
0 = 6(0. ?/(0) + n(0, foOfo(t)
(recall that f/(O.t/(Z))	0 if iff) is close to 0). As far as the variable ?/(f) is
concerned, it is clear that, being Of) identically zero, its behavior is governed
by the differential equation
foO = qfbift}) .	(4.28)
From this analysis we deduce the following facts. If rhe output y{t) has to
be zero, then necessarily the initial state of the system must be set to a value
such that £(0) = 0. whereas 7?(0) — rf can be chosen arbitrarily. According
to the value of if. rhe input must be set as
,...	И0. iff)
lit =-------———-
fotkfot))
where r/(t) denotes the solution of the differential equation
rft) =q(O.q(t)) with initial condition//(()) - if.
Note also that for each set of initial data £ = 0 and 7/ = if the input thus
defined is the unique input capable to keep y(t) identically zero for all times.
164
4. Nonlinear Feedback for Single-Input Single-Output Systems
The dynamics of (4.28) correspond to the dynamics describing the “m-
гегпаГ behavior of the system when input and initial conditions have been
chosen in such a way as to constrain the output to remain identically zero.
These dynamics, which are rather important in many of our developments.
art> called the zero dynamics of the system.
Remark 4^.1. bi order to understand why we used the terminology ‘'zero"
dynamics in dealing with the dynamical system (4.28). it is convenient to
examine how these dynamics are related to the zeros of the transfer function
in a linear system. Let
тл \ гЛ) + Мс-----------Vbn r-1 m Cl~r
«о ~ m-s +    +
denote the transfer function of a linear system (where r characterizes, as ex-
pected. the relative degree). Suppose rhe numerator and denominator poly-
nomials are relatively prime and consider a minimal realization of H(t>)
.r = ,4.r + В a
У
C.r
with
0	1	0
0	0	1
C
0	I)	()	
\ —По	—Hi	—«2
( b[} 6]   • Ъп_г-\
1
Let us now calculate its normal form. For the first r new coordinates we
know we have ro take
/
г I — С.Г = bf)X J + 6] .Г'2 A • ‘ ‘ + ЬЦ_ r - I T;; _r + .1‘; j _ ,~ + 1
~2	=	С-1.Г = b^.T'2 + bi.r-2 -1- -   + bn ] .rtl -r-r-1 + -Г n-r-b‘2
= C.T l.r = bciiy + b[.rf.4 i a-	+brJ-r-+ .rr;.
For the other n — r new coordinates we have some freedom of choice (provided
that the conditions stated in Proposition 4.1.3 are satisfied), but the simplest
one is
= .ci
~ n — n - r 
This is indeed an admissible choice because the corresponding coordinates
transformation z = Ф(.г) has a jarobian matrix with the following structure
4.3 The Zero Dynamics
165
and therefore nonsingular.
In the new coordinates we obtain equations in normal form, which, be-
cause of the linearity of the system, have the following structure
zr =	+ St/ 4- Ku
j] =	+ Qi]
where R and S are row vectors and P and Q matrices, of suitable dimensions.
The zero dynamics of this system, according to our previous definition, are
those of
0 = Qv •
The particidar choice of the last и - г new coordinates (i.e. of the elements
oft?) entails a particularly simple structure for the matrix Q. As a matter of
fact, is easily checked that
dzr+i	ч
dt dt
=	—r— — .r,7-r+i (£) — -боЛ (?) - •   - 6,J_r_1.rrt_,(f) + Ci (t)
at	dt
~ — &o~r+l(O — ' — kfi-r -1 -ri(0 + -1(f)
from which we deduce	that				
	( °	1	0	 	0	\
	0	0	1	0	
Q =	0	0	0	• 	1	
	\ -6o	-61	-1)2  	 -b„_r.	1 /
From the particular form of this matrix, it is clear that rhe eigenvalues of
Q coincide with the zeros of the numerator polynomial of H(s). i.e. with the
166
4. X on linear Feedback for Single-In put Single-Output Systems
zeros of the transfer function. Thus it is concluded that in a linear system
the zero dynamics art* linear dynamics with eigenvalues coinciding with the
zeros of the transfer function of the system. <
Remark 4-3.2. The calculations carried out in rhe previous Remark are also
useful in showing that the linear approximation, at q — 0. of the zero dynam-
ics of a system coincide with the zero dynamics of the linear approximation
of the system at ,r = 0. i. e. that the operations of taking the linear approxi-
mation and calculating the zero dynamics commute.
In order to check this, all we have to show is that the linear approximation
of equations in normal form coincides with the* normal form of the3 linear
approximation of the original description of the system and this amount>
only to show that rhe relative degree of rhe system and that of its linear
approximation are rhe same. To this end, suppose that the system has relative*
degree r at .r — 0. Consider the expansions already introduced in Remark
4.2.7
/(.r) = Ar + Adz)
= В + ед (z)
and. in addition, expand h(.r) (which is 0 at ,r = 0) as
h(.r] = C.r + hits)
An easy calculation shows, by induction, that
LKfh(x} = СА*.г +
where is a function such that
From this one deduces that
CAkB = LQLkfh(0) = 0 for all к < r - 1
CA'-'B =
i.e. that the relative degree of the linear approximation of the system at ,r — (J
is exactly r.
From this fact, it is concluded that taking the linear approximation of
equations in normal form, based on expansions of t he form
6(C ?/) = /?£ + Si] + b-2(£. i])
= A' + «1 (^-z?)
?((•'/) = P£ + Qii + q2(£.ri)
4.3 The Zero Dynamics 167
yields a linear system in normal form. Thus, the jacobian matrix
which describes the linear approximation at 7/ — 0 of the zero dynamics of
the original nonlinear system has eigenvalues which coincide with the zeros
of the transfer function of the linear approximation of the system ar т = I). <
Example J.3.3. Suppose we want to calculate the zero dynamics of the system
already analyzed in the Example 4.1.4. The only thing we have to do is to
set zi = -2 = 0 in the last equation of the normal form of the equations and
get
'-3 =	
These are the zero dynamics of the system. <
Example J.3.j. Suppose we want to analyze the zero dynamics of the system
/ - 4 \	/ 0 \
X =	— .Г-j 1 4- I — 1 1 U
\ x 2} - .r3 /	\ 1 /
У = A 
For this system we have
= 0 Ljhlx) ~ хз — .4 LyL/hlx) = 1 + 34 .
We can calculate a normal form by taking
21	=	J1!
- 4
- x2 + .7’3
which is a globally defined coordinates transformation. Using these new co-
ordinates we obtain equations of the following form
2’2	=	6(21 . 29, 33) +	, 2-j.
2з = Ci — 23 .
The constraint y(t) = 0 for all t imposes cjt) = ;-2(t) = 0 for all t. and
this shows that when the output is identically zero the state must necessarily
evolve on the curve (see Fig 4.5)
M = {.r G ?? : .r j - 0 and = 4}
and be governed by its zero dynamics
168
4. Nonlinear Feedback for Single-Input Single-Output Systems
Fig. 4.5.
Although all the properties illustrated so far were discovered and discussed
using the normal form, it is not difficult to arrive at similar conclusions
starting from equations in different forms. If, for instance, one has not. been
able to find exactly the normal form because of the difficulty in constructing
functions (p7.+i (j),.... with the property that Lgdf(x) — 0 (see Remark
4.1.3). one can still identify the zero dynamics of the system working on
equations of the form
fy	-	Z2
=	*3
С г — 1	=	2 r
fy	'	77)u
f/	=	Cffy- 'h)+p(fyr/)u .
As a matter of fact, having seen that the zero dynamics of the system
describe its behavior when the output is forced to be zero, we impose this
condition on the equations above. We obtain, as before, fyf) =0 and
0 = f>(O, t/(f)) +«(0. q(t))u(t) .
Replacing u(f) from this equation into the last one, yields a differential equa-
tion for 77(f)
h = <7(0. ?/)
«(0.7/)
which describes the zero dynamics in the new coordinates chosen.
Example 4-3.5. Suppose we want to calculate the zero dynamics of the system
already analyzed in the Example 4.1.5. In this case we don't have the normal
form, but the calculation of the zero dynamics is still very easy. Setting
гх =z-2—0 in the second equation yields
4,3 The Zero Dynamics
169
- 4U
u = — ~;—
Replacing this, ami zj = z? ~ 0. in the third ami fourth equation yields

23	2 + 2z3
which describes the zero dynamics of the system. <
The problem of zeroing the output could also have been analyzed di-
rectly on rhe original form of the equations. Keeping in mind the calcula-
tions already done at the beginning of section 4.1, it is easy to deduce that
= Q implies h(,r(t)) = 0. for all 1 < t < r. Thus, as expected,
the system has to evolve on the subset
Z* = {.г G Ж'
which, locally around x3. is exactly the set of points whose new coordinates
Z],..., zr are 0 (see Fig. 4.61. If one writes the additional constraint
0 =	= Lrfh(r(t)) + LaLrs '
this turns out to be exactly the same constraint previously obtained for u(f’).
but now expressed in terms of the functions which characterize the original
equations.
Fig. 4.6.
Note that, since the differentials dLljh(x). 0 < i. < r — 1. arc linearly inde-
pendent at .r: (Lemma 4.1.1), the set Z* is a smooth manifold of dimension
n — r. near The state feedback
170	4. Nonlinear Feedback for Single-Input Single-Out put Systems
—Ljh(.r)
LgLrf~4iix)
by construction is such that
dh(x)
dL/Ь(х)
(/(j-) -f- p(.r)u’(.r) ]
\dL'f
/ Lfh(.r) + Lgh(x)u*(r) \
L'jh(x) + LgLfh(r)u*{x) '
\ LTjh(x) + LgL’^~1h(x)n*(x) /
Thus
/ Mi(x) \
dLjh(.i)	= о
\dLj~1 b(x') /
for all x e Z* (because h[xI = Lfh(.r] =   • = L1'^1 h(x] = 0 if x 6 Z+) and
therefore the vector field
Г И = f(x) + д(х'цГ(х)
is tangent to Z*. As a consequence. any trajectory of the1 closed loop system
x =
starting ar a point of Z* remains in Z* (for small values of t). The restriction
of /*(>) to Z* is a well-denned vector field of Z*. which exactly
describes in a coordinate-free setting - rhe zero dynamics of the system.
We will illustrate in the sequel a series of relevant issues in which rhe
notion of zero dynamics, and in particular its asymptotic properties, plays
an important role. For the time being we can show, for instance, how the
zero dynamics are naturally imposed as internal dynamics of a closed loop
system whose input-output behavior has been rendered linear by means of
state feedback. For. consider again a system in normal form and suppose rhe
feedback control law (4.25) is imposed, under which the input-output behav-
ior becomes identical with that of a linear system consisting of a string of r
integrators between input and output (see Fig. 4.4). The closed loop system
thus obtained is described by rhe equations (4.26). that can bo rewritten in
rhe form
sc = A£ + Br
= qM
П
У
г
4.3 The Zero Dynamics 171
with
.4
C
/0 1 0
0 0 1
0 0 0
\ о о 0
(.1 0 
If the linear subsystem is initially at rest and no input is applied, then y(t) = 0
for all values of t, and the corresponding internal dynamics of the whole
(closed loop) system are exactly those of (4.28). namely the zero dynamics
of the open loop system.
We conclude the section by showing that rhe interpretation of
z)(0 = t/(0.r/(O) •
as of the dynamics describing the internal behavior of the system when the
output is forced to track exactly the output y(t) = 0. can easily be extended
to the case in which the output to be tracked is any arbitrary function.
Consider rhe following problem, which is called the Problem of Reproducing
the Reference Output yn[t). Find, if any, pairs consisting of an initial stare
x° and of an input function u=(-). defined for all t in a neighborhood of t = 0.
such that the corresponding output y(t) of the system coincides exactly with
for all t in a neighborhood of t — 0. Again, we arc interested in finding
all such pairs (.r°,ufo. Proceeding as before, we deduce that y(t) = yn(t)
necessarily implies
for all t and all 1 < i < r .
Setting
frfo - colft/fof).........(4.29)
we then see that the input u(f) must necessarily satisfy
4t) = b(£ipt).T](t)) +	foOfoD)
where r/(t) is a solution of the differential equation
MWfyWOb	(4-30)
Thus, if the output y(f) has to track exactly yx(t), then necessarily the
initial state of the system must be set to a value such that £(0) = £r(0),
whereas tj(O) = rf can be chosen arbitrarily. According to the value of ?f,
the input must be set as

(4.31)
172
4. Nonlinear Feedback for Single-Input Single-Output Systems
where z/(7) denotes the solution of the differential equation (4.30) with initial
condition r/(0) = if. Note also that for each set of initial data £(0) = ^(0)
and r/(0) = if the input thus defined is the unique, input capable of keeping
y(t) = gift) foi' all times.
The (forced) dynamics (4.30) clearly correspond to rhe dynamics describ-
ing the "internal'' behavior of the system when input and initial conditions
have been chosen in such a way as to constrain the output to track exactly
Note that the relations (4.30) and (4.31) describe a system with input
^ri(f). output u(t) and state q(t) that can be interpreted as a realization of
the inverse of the original system.
4.4 Local Asymptotic Stabilization
In this section we illustrate how the notion of zero dynamics can be helpful
in dealing with the problem of asymptotically stabilizing a nonlinear system
at a given equilibrium point. Suppose, as usual, a nonlinear system of the
form
.r = /(x) + g(x)u
is given, with fix) having an equilibrium point at fo that, without loss <jf
generality, we assume to be ,rc = 0. The problem we want to discuss is the
one of finding a smooth state feedback
и ~ a(x)
defined locally around the point .rc = 0 and preserving the equilibrium, i.e.
such that q(0) = 0. with the property that the corresponding closed loop
system
j- = f(x) + pfofofo)
has a locally asymptotically stable equilibrium at x = 0. We shall refer to it
as to the Local Asymptotic Stabilization Problem.
First of all. we review a rather well-known property, by discussing to
what extent the possibility of solving the problem in question depends on
rhe properties of the linear approximation of the system near .rc = 0. To this
end, recall that the linear approximation of a system having an equilibrium
at r = 0 is defined by expanding f(x) and g(x) as (see Remark 4.2.7)
/И = .4z + /2(x)
g(x) = B + gfx)
with
df
4 = —-	and В = g(0) .
J j.=0
From the point of view of the stability properties of the dosed loop sys-
tem, the importance of the linear approximation is essentially related to the
following result.
IF
4.4 Local Asymptotic' Stabilization
173
Proposition 4.4.1. Suppose the linear approximation is asymptotically .sta-
bilizable, i.e. either the pair (A.B) is controllable or in case the pair [A. Bi
is not controllable - the uncontrollable modes correspond to eigenvalues with
negative real part. Then, any linear feedback which asymptotically stabilizes
the linear approximation is also able to asymptotically stabilize the original
nonlinear system, at least locally. If the pair (A. Bi is not controllable, and.
there exist uncontrollable modes associated with eigenvalues with positive real
part, the original nonlinear system cannot be stabilized at all.
Proof. Suppose the linear approximation is asymptotically stabilizable. Let
P be any matrix such that (.4 + BF] has all eigenvalues with negative real
part, and set
и — Fa-
on the nonlinear system. Tin1 resulting closed loop system
j- ~ f(x] + g[x}Fx - (A + BF']x + /o(J,) + yiU)Fx
has a linear approximation having all the eigenvalues in the left-half complex
plane. Thus, the Principle of St ability* in the First Approximation proves that
the nonlinear closed loop system is locally asymptotically stable at x = 0.
Conversely, suppose rhe linear approximation has uncontrollable modes
associated with eigenvalues having positive real part. Lot и = o(r) be any
smooth state feedback. The corresponding closed loop system has a linear
approximation of the form (recall that o(0) = 0)
d[f(Fl + g(j‘)n(u-)]
A + B
da
dx
which has eigenvalues with positive real part, irrespectively of what о is.
Thus, again by the Principle of Stability in the First Approximation, the
nonlinear closed-loop system is unstable at .r = 0. <i
Note that the previous result does not cover the whole spectrum of cases.
As a matter of fact, if the pair (A. B) is not controllable and there are uncon-
trollable inodes associated with eigenvalues with zero real part (but none of
them has positive real part), nothing can be said from the linear approxima-
tion. in the sense that the nonlinear system might be locally asymptotically
stabilizable by means of a nonlinear feedback even though its linear ap-
proximation is not. Problems in which this situation occurs are said to be
critical problems of local asymptotic stabilization.
We show now in which way the notion of zero dynamics is useful when
dealing with critical problems of local asymptotic stabilization. Consider
again a system in normal form
174
4. Nonlinear Feedback for Single-Input Single-Output Systems
b(h-V)
q(^l)
where
s = colfsi,.... cr)
and. without loss of generality, assume that (s-d) = (0.0) is an equilibrium
point. Impose a feedback of the form
и = "	- c03i - ci г-2 -------fr-ib )	(4.32)
ti(C p)
where r()...cr_i art1 real numbers.
This choice of feedback yields a closed loop system
.4£

(4.33)
with
	/ °	1	0 	' 0 \
	0	0	1	0
4 =	t °	0	0 •		1	j
	X -Co	“Cl	— C>	 ~er-i /
In particular, the matrix .4 has a characteristic polynomial
P(s') - c0 + cPs +/-------------------------к c,._iSr-1 + sr.
From this form of the equations describing the closet! loop system we
deduce the following interesting property.
Proposition 4.4.2. Suppose the equilibrium p = 0 of the zero dynamics of
the system is locally asymptotically stable and all the roots of the polynomial
pis') have negative real part. Then the feedback law Ц.32) locally asymptoti-
cally stabilizes the equilibrium (thy) = (0.0).
Proof We only need to use the first Lemma of section B.2. In fact, the closed
loop system has the form (B.8) and that, by assumption, the subsystem
// = <7(0. t/)
is locally asymptotically stable at i] = 0. <
F
4.4 Local Asymptotic Stabilization
175
Note that the matrix
0(1^-J])
Q =
Ajl J i^.rp-m.oi
characterizes the linear approximation of the zero dynamics at 7 = 0 (see Re-
mark 4.3.2). If this matrix had had all the eigenvalues in the left-half complex
plane, then the result stated in the previous Proposition would have been a
trivial consequence of the Principle of Stability in the First Approximation,
because the linear approximation of (4.33) has rhe form
7 /	\ * Q ) \JU
However. Proposition 4.4.2 establishes a stronger result, because it only
relies upon the assumption that 7 = 0 is simply an asymptotically stable
equilibrium of the zero dynamics of the system, and this as is well known
- does not necessarily require, for a nonlinear dynamics, asymptotic stability
of the linear approximation (i.e.. all eigenvalues of Q having negative real
part). In other words, the result in question may also hold in the presence* of
some eigenvalue of Q with zero real part.
In order to design rhe stabilizing control law there is no need to know
explicitly the expression of the system in normal form, bur only to know
the fact that the system has a zero dynamics with a locally asymptotical!)'
stable equilibrium at 7 = 0. Recalling how the coordinates ci........c, and
the functions and 6(£.p) are related to the original description of
the system, it is easily seen that, in rhe original coordinates, the stabilizing
control law assumes the form
и —
1
LgLf~lh(r}
(~Lrfh(x) - eoh(.r) -
с,- iLrf~lh{.r))
which is particularly interesting, because expressed in terms of quantities
that can be immediately calculated from the original data.
By this method we can asymptotically stabilize also systems whose linear
approximation has uncontrollable modes corresponding to eigenvalues on rhe
imaginary axis. i.e. we can solve critical problems of local asymptotic stabi-
lization. provided we know that for some choice of an "output" the system
has an asymptotically stable zero dynamics.
Example Consider the system already discussed in the Example 4.1.5.
Its linear approximation at r
form
/0
.4=^1	=	1
I. th- j т=о	0
\o
= 0 is described by matrices A and В of the
0	0	0\
0	0	0'
0 -1 0
1	0	0/
В = f/(0) =
176
4. Nonlinear Feedback for Single-Input Single-Output Systems
and has exactly one uncontrollable inode corresponding to the eigenvalue
A = 0. However, its zero dynamics (see Example 4.3.5)
-coh(x) - ciLfh(x))
have an asymptotically stable equilibrium at '.3 — cj — (J. Thus, from our
previous discussion, we conclude that a control law of the form
1
.r
locally stabilizes the equilibrium т = 0. <
If an output, function is not defined, the zero dynamics are not defined a>
well. However, it may happen that one is able to design a suitable dummy
output whose associated zero dynamics have an asymptotically stable equi-
librium. In this case a control law of the form discussed before will guarantee
asymptotic stability. This procedure is illustrated in the following simple ex-
ample.
Example 4'4-2- Consider the system
?	__	r2 „3
•П - .r [ X.,
1'2	=	^2 + it
whose linear approximation ar .r = 0 has an uncontrollable mode correspond-
ing to the eigenvalue A = (J. Suppose (Hie is able to find a function -'-(.ri) such
that
•h =
is asymptotically stable at jq ~ 0. /Then, setting
у - f/[X) ~ У(Х!) - x->
a system with an asymptotically stable zero dynamics is obtained. As a mat-
ter of fact, we know that the zero dynamics are those induced by the con-
straint y(t) = 0 for all t. This constraint, in the present case, implies
Thus, the zero dynamics evolve exactly according to
ci -	)J3
and the system can be locally stabilized by means of the procedure discussed
above. A suitable choice of will be. e.g.
‘ (>i) - -J’i
4.4 Local Asymptotic Stabilization
177
Accordingly, a locally stabilizing feedback is the one given by

with c > 0. <
Remark 4-4-3- It i-s not difficult to see that the eigenvalues associated with
uncontrollable modes of the linear approximation of the system, if any. cor-
respond necessarily to eigenvalues of the jacobian matrix Q. that is of the
linear approximation of the zero dynamics. For. observe that the linear ap-
proximation of the equations in normal form has this structure
51/ + Ku
- Qv
where
П
P
'Pb
~9<f
'Ob'
~dq'
^3.
and К = (z(O.O). Suppose the linear approximation is uncontrollable. Then
for some complex number A. the matrix
has rank less than r?. More specifically, the values of A such that this matrix
has rank loss than n are exactly the eigenvalues associated with the uncon-
trollable modes. From the structure of the matrix in question it is easily seen
that, since К is nonzero, its rank can be less than л only if A annihilates the
determinant of (AI — Q ], i.e. if A is an eigenvalue of the linear approximation
of the zero dynamics.
Thus, if the system has a linear approximation with uncontrollable modes
corresponding to eigenvalues on the imaginary axis and an output is defined
178	4. Nonlinear Feedback for Single-Input Single-Output Systems
such that the zero dynamics (on the nonlinear system) are locally asymp-
totically stable at >) = 0. the latter cannot asymptotically stable in the first
approximation. However, the system can still be stabilized by the method
described before because, as observed, the asymptotic stability in the first
approximation of the zero dynamics is not an issue. <
Remark 4-4-4- H- instead of the feedback (4.32) one imposes a control
" =	~ co^i - ... - cr_. iCr + c)
«(Ch)
in whit'h r is an additional reference input, a closed loop system of the form
£ = ,4£ + Br
(4.34)
h = q(f.q}
is obtained, with
В = col(0.....0.1) .
This, of course, for c = 0 reduces to the system (4.33). If the latter is lo-
cally asymptotically stable at (Ch) = (0.0). then for sufficiently small г the
trajectories of (4.34) are bounded. More precisely, using the results of section
B.2. it is possible to conclude that for each s > 0 there exist <5 > 0 and К > 0
such that
|| J'(O) P< 6 and |r(f)| < IT for all t > 0
imply || x(t) ||< 5 for all t > 0. <
4.5 Asymptotic Output Tracking
In section 4.3. we have established conditions under which a prescribed ref-
erence output function yfi(t) can be exactly reproduced. As we have seen,
for this to be possible, certain components of the state of the system at time
t = 0 must fit with the values at this time of the desired output в(/д(Н and of
its first г — 1 derivatives. However, the possibility of presetting the initial state
to a prescribed value is quite unusual in practice and, in addition, one cannot
neglect the event of unexpected perturbations causing the initial state to be
different from the desired one. More realistically, one is then led to investigate
the problem of producing an output that, irrespectively of what the initial
state of the system is. converges asymptotically to the prescribed reference
function yn(t). This problem is called the Problem of Tracking the Output
Again, an elementary analysis of the problem in question (although
not the most, general one. as we will observe later in Remark 4.5.2) is made
possible by an appropriate use of the results developed in sections 4.1 and
4.3.
4,5 Asymptotic Output Trackin:
179
Consider again a system in normal form
O- =	+^Ю)"
У = О
and choose
н = -77— (-Ж,??) + Ун - ^c,_i(c, - .(//Г1’))	(4-35)
where eg,.... cr_j. are real numbers.
Define an "error” e(t). the difference between the real output y(t) and the
reference output y^ft), as
t'U) = y(t) - yM .
Then, since by construct ion zt = y'f-1*(t) for 1 < i < r. it is immediately
seen that the input (4.35) has the expression
u=ttibn (~bicr>)+s«rl -
Note that the input in question, if e(t) = 0 for all t. reduces exactly to
the input needed in order to have precisely yn(t) as an output (see end of
section 4.3). For the sake of completeness, note also that the input (4.35), in
the original coordinates has the expression
u = т г'-,,,	* № - Z e, _! (zy-1 ’Л (J-) -	11)). (4.36)
Imposing the input (4.35) yields
ir = у{г] = Ун ’ “ r-r-ie(r-1>-----ср?1’ - roe
i.e.
e(r| + cr„ie(r-1) 4--+ С]в( ° + rue = 0 .	(4.37)
The error function e(t) satisfies a linear differential equation, of order r.
whose coefficients can be arbitrarily preset. The roots of the characteristic
equation associated with (4.37) can be arbitrarily assigned, and one can con-
clude that under the effect, of an input of the form (4.35) the output of the
180	4. Nonlinear Feedback for Single-Input Single-Output Systems
system "tracks’" the desired output ijrF)- with an error which can be made
to converge to zero, as t —> oc. with arbitrarily fast exponential decay.
Of course, an ever present concern in the design of control laws is that
the variables representing the internal behavior of the system remain bounded
when a specific control law is imposed. In the present situation, the asymp-
totic- analysis of the internal behavior of the system obtained by imposing
the control law (4.36) on (4.1) can be carried out in the following way.
First of all. note that if we consider, as we implicitly did. the reference
output ijrF) to be a filed function of rime, then the system (4.1) driven
by the input (4.36) can be interpreted as a time-varying nonlinear system.
In particular, looking at the behavior of the state variables in the coordi-
nates used for the normal form, it is easy to check that Ci......satisfy the
identities
=у'гГ1'1
whereas ?/ satisfies a differential equation of the form
b = <?(£«(*)	(4.38)
where, as in (4.29).
and
\(t) = col(r(^).e<il(t)...cir'H(t)) .
Equation (4.38). in view of the remarks at the1 end of section 4.3. can be seen
as an equation describing the -‘response’1 of the inverse system "driven” by
the function у/fit) + e(f).
Sufficient conditions for the boundedness of the z, (t)’s and g(f) arc ex-
pressed in the following statement.
Proposition 4.5.1. Suppose yR(t). 0v (t),.... y^ ^(t) are defined for all
t > 0 and bounded. Let v/rF) denote 'the solution of
V = q(fifit)-V)	(4.39)
satisfying i}r(0) = 0. Suppose this solution is defined for all t > 0, bounded
and uniformly asymptotically stable. Finally, suppose the roots of the polyno-
mial
S + Cr _ i Л 1 +    + Cl S -e Co = 0
all have negative real part. Then, for sufficiently small a > 0. if
~ yR~1}(t°)\ < a. 1 < г < r.	||t/(r) - rtfi(F) jI < a
the. corresponding response zfit), ij{t). t >F > 0- of the closed loop system
(J.l)-(J.36) is bounded. More precisely, for all - > П there exists <5 > 0 such
that
|г((Г) - Уд’1!(Л| < => i-JO “ y'R~1}(F\ < =	f°r (ll} f > 0
1ЫЛ -	<b => ||7?U) - 7/д(Г)|| < г	for all t > F > 0 .
4.5 Asymptotic Output Tracking
181
Proof. Observe that the system (4.1)-(4.36) can be rewritten in the form
\ = A \	(4.40)
Й =	+ X-t/)	(4.41)
where К is a matrix in companion form, whose characteristic equation is
that of (4.37). Set ir = - /рИС and XT) = <z(£;t(C + X- Unit) + И -
t//f(O)- Them the system
ti- = F(u’,\.f)
X = A'\
is in the form (B.13) and (w.y) = (0.0) is an equilibrium. Note that,
since is smooth and £д(7), i)x(t) are bounded. F(a\\.t) is locally
Lipschitzian in (?r.\) uniformly with respect to t. The solution tc = 0 of
w = F(tm0Л) is uniformly asymptotically stable and К has eigenvalues with
negative real part. Thus, from section B.2. we deduce that (О.г?д(П) is an
uniformly stable solution of (4.40). and the indicated estimates follow. <
Remark ^.5.7. Note that the solution of (4.39) needs not to be a con-
stant solution (even in a linear system this is not necessarily the case, as the
reader may easily check). The assumption that the solution qri(t) of (4.39) is
uniformly asymptotically stable can be interpret tai as the rather natural
requirement that, in the conditions in which yn(t) is exactly reproduced (in
which case T]{t) cm is exactly a solution of (4.39)). the internal behavior of
the system is that of an uniformly asymptotically stable system. <
Remark 4.5.2. It is important to observe that, the approach presented so far
is not the unique possible, and the assumption for r//?(f) of a being uniformly
asymptotically stable solution of (4.39) is a not necessary condition for hav-
ing bounded response in the state variables when a tracking problem is ad-
dressed. As a matter of fact, one might have the feeling that this assumption
is somewhat necessary because it naturally comes together with the impo-
sition of the control law (4.35) which, in turn, was naturally suggested as
an adaptation - to the case of mismatched initial state - of the control law
(4.31) that was proved to be necessary for exact reproduction of yR(t). The
approach followed here, i.e. the choice of the control law (4.35). incorporates
the property that, in the closed loop system, if e = 0 at time t = 0. then
e(f) = 0 for all t. However, as we shall see in more detail in Chapter 8 (where
we will present a more general approach to the problem), there is no need
in principle for such a requirement if one wants a closed loop system whose
output tracks a reference output, with the internal variables being bounded.
<
Sometimes the reference output is not just a fixed function of time, but
the output of a reference model, which in turn is subject to some input tc.
for instance a linear model, described by equations of the form
182	4. Nonlinear Feedback for Single-Input Single-Output Systems
< = A^Bir	(-1.42)
ул = C< .	(4.43)
If this is the case, one may pose the problem of finding a feedback control
which, irrespectively of what the initial states of the system and of tin1 model
are. causes - for every input w(t) to the model an output y(t) asyniptotieallv
coiiverging to the corresponding output y^(f) produced by the model under
the effect of w(t). This problem is commonly known as asymptotic model
matching.
In order to solve this problem one could, in principle, think to use the
same input. (4.36) considered at the beginning, with and its first r
derivatives replaced by the ones calculated from the reference model (4.42)-
(4.43). However, since
y^Jt} = C.4\’(h +C.4'
the control law thus obtained would depend on the first г — 1 derivatives of
the input vjt) to the reference model, a situation whic h is not desirable if the
control law in question has to be realized by a device which receives ic(t) as
an input and produces u(t) as an output. In fact, the differentiation of u’(t)
would indeed boost the effect of unavoidable additive noise.
If we .suppose that
CB = CAB = ... = CAr~2B = 0	(4.44)
i.e.. that the model has a relative degree equal to or possibly larger than the
relative degree r of the system. we have that
yjP’(t) = C.4'C(t) for all 0 < i < r — 1
УяР) = C.W(c4c.4r-'B«’(O .
The first г — 1 derivatives of уя(Н do not depend explicitly on and the1
r-th one depends explicitly on ic(f) but not on its derivatives. Replacing this
in the expression (4.36) of и yields
“ =	+CVW) + СЛг~1Ви-
LgL. 7/(j) J
1	r	(4.45)
up1 !o) -
f=l
By construction the system (4.1). subject to an input of this form - if the
coefficients cq..c,-_] are appropriately chosen - will produce an output
asymptotically converging to the output уиС) of the model. Since the latter
has the form
УнО = CcAtQ(0) + [ Се1|М1Вф) ds
Jo
4.5 Asymptotic Output Tracking 183
we can conclude that the output of the closed loop system (4.T)-(4.45) (sec
fig. 4.7) will be of the form
= c (t) — Cf 4'<i0) + [ Cc ' Bir(s) (is
Ja
with e(t) solution of the differential equation (4.37).
Fig. 4.7.
Note that the input (4.45) depends explicitly at each time t ~ on the
state .c(t) of the system, on the input w(t) of the model, and on the state
((t) of the model, which in turn obeys the differential equation (4.42). Thus.
u(t) can be regarded as the "output'" of a dynamical system of the form
и
~(G л -t- <)(;.*)«
a(G t) + T(G -c)«’
(4.46)
with internal state G driven by th*' "inputs" tc and r. As a matter of fact,
the first one of these two equations can be identified with (4.42) and the
second one with (4.45). We see then that the solution of the problem of
asymptotically tracking the output of a reference model entails the use of a
more general type of state feedback than the one considered so far. in that
it includes also an internal dynamics. A feedback of this form is called a
dynamic state feedback.
Summarizing the whole discussion, we can conclude that, if the relative
degree of the model (4.42)-(4.43) is larger than or equal to the relative degree
of the system, there exists a dynamic feedback of the form (4.46) yielding an
output y(t) that con verge's asymptotically to the output of the mode1!,
for every possible input u'(t). and for every possible initial state .r(0). <(())
184	4. Nonlinear Feedback for Single-Input Single-Output Systems
The analysis of the internal asymptotic properties of the system thus
obtained is quite similar to the one developed earlier for the system (4.1)-
(4.36). As a matter of fact, it is immediate to check that, in the present case
the closed loop system can be described in proper coordinates by equations
of the form
< = -V + Bic
\ M
i) = 'jiR + x.t?)
with Г ~ col(C. CA.....CAr-1). The first one of these equations describes
the dynamics of the model (driven by its own input), the second one the
dynamics of the error (which is an autonomous equation), the latter the
dynamics of the inverse system driven by the function CC — \i-
4.6 Disturbance Decoupling
The normal form introduced in section 4.1 is also useful in understanding how
the output response of a given system can be protected from disturbances
affecting the state. Consider a system of the form
r = f(T) + g(x)u + p(x)ir
у = h(x)
in which ir represents an undesired input, or disturbance. We want to examine
under what conditions there exists a static state feedback control
U = o(x) + J(.Z')p
yielding a closed loop system in which the output у is completely independent
of. or decoupled from, the disturbance ir- This problem is commonly known
as Disturbance Decoupling Problem.
As usual, we discuss first the solution looking at the normal form of the
equations. Let the system have relative degree r at. ;r=. and suppose the vector
field p(x) which multiplies the disturbance in the state equation being such
that
LpL^hlr) = 0 for all () < I < r - 1 and all ,r near .rG.
If we write the state space equations choosing the same coordinates used
before to describe the normal form, we obtain
dci _ дг-L dx _ Oh dr
dt Ox dt dx dt
= Lfh(x(t)) + I,/i(.r(t)|u(t) + Lph(x(t)}ir(t)
= Lfh(x(t)j = j2(0
4.6 Disturbance Decoupling
185
I because, by assumption, Lph(x) — 0 for all t such that i-(t) is near У. A
I- similar situation happens for all other subsequent equations, and thus we get.
dzr_
dt
? For zr we still obtain
because, again. Up£y-1/?(j-) = 0. Thus, the first r equations are exactly the
same as those of a normal form of a system without disturbance. This is not
anymore the case for the remaining ones, that will now appear depending
also on the disturbance w.
Using, as in the previous sections, a vector notation, we can rewrite the
system in the form
У = 6(£.t/)+n(£,r/)u
rj =	+ Ш-h)»’ 
In addition we have, as usual
У = -i 
Suppose now the following state feedback is chosen
K «(sDi) «(Ch)
This feedback yields a system which is described by the equations
*-1	=	~2
=	~3
V
from which it is easily seen that the output, i.e. the state variable ci. has
been completely decoupled from the disturbance ;c.
186
4. Nonlinear Feedback for Single-Input Single-Output Systems
The block-diagram interpretation of the closed loop system thus obtained,
described in Fig. 4.8. clearly explains what happened. The effect of the input
has been that of isolating from the output that part of the system on which
the disturbance has effect.
Fig. 4.8.
We have thus found a sufficient condition for the existence of solutions
to the problem of decoupling the output of a system from a disturbance, and
explicitly constructed a decoupling feedback. It is not difficult to prove that
the condition in question is also necessary, as we shall see in a moment. For
convenience, we summarize all the results of interest in a formal statement
in which, for more generality, we specify the decoupling feedback in terms of
the functions /(>). g(.r) and h(x) characterizing the original description of
rhe system.
Proposition 4.6.1. Suppose the system has relative degree r at . The
problem of finding a feedback и =	defined locally around .m.
such that the output of the system is decoupled from the disturbance can be
solved if and only if
= 0 for all 0 < i < r — 1 and all .r near ,r“,	(4.47)
If this is the case, then a solution is given by
и
L.ffi’ffifiiU) LgL'ffi'hix)
Proof We have only to prove the necessity. Let и = o(j') 4- denote
any feedback decoupling the output from the disturbance, and consider the
corresponding closed loop system
4.6 Disturbance Decoupling
187
By assumption, the output у has to be independent of ir. and this has to
be true also when r(t) = 0 for all t. i.e. for the system
.r = /(r) + y(.Hu(.r) + p(s)tr
у = /((.г) .
By repeating, in the present case calculations similar to the ones done in
section 4.1. we obtain
y11 '(f) = Lf^^h(.r(t)") +
from which we see that y(H can bo independent from tc(f) only if Lph[.r) = 0
for all t such that r(t] is near ,r:. Assuming this condition satisfied, we
calculate	and get
У:2|(Н =	.
Again, we conclude that	h(.r) must be 0. The same arguments can
be repeated for all the higher derivatives of y(t), until we get
and see that also	Ь(х') must be 0. We conclude that if the feedback
decouples у from u\ necessarily
LpLj, /((/) = 0 for all 0 < i < r — 1 and all ,r near xz.
This is indeed the condition we wanted to prove because, as seen in the
proof of Lemma 4.2.1.	= L^hix) for all 0 < i < r - 1. <
Remark 4-6.1. Note that on the system thus decoupled one can further
choose the new control r in order to achieve additional performances, like
e.g. asymptotic stability. If the original system had an asymptotically stable
zero dynamics, in view of the properties illustrated in section 4.4. we see that
this goal can be accomplished by means of a feedback of the* form
r = — (cq/R j'J + ... + cr_i L’f 1 It(j*)) + г .
Suppose, without loss of generality. .m =0. i. e. that /(0) = 0 and Л(0) = 0.
Let the coefficients
Co.....c,.
be such that the vector field f(r) + yfj')o(.r) has an asymptotically stable
equilibrium at ,r = 0. Then, using the results illustrated in section B.2. it is
possible to conclude - as in Remark 4.4.4 -- that for each s > 0 there exist
5 > 0 and A' > 0 such that
|| z(0) ||< 6 and ffi'(f)| < Ab |b(t)| < A', for all t > 0
imply || .r(f) ||< s for all />().<
188
4. Nonlinear Feedback for Single-Input Single-Output Systems
Sometimes, it is useful to write the condition of Proposition 4.6.1 in a
slightly different manner. Recall that
LpLkfh(r) ~ (dLkf(x).p{r)')
and consider the codistribution
17 = spanjdh. dL,-h......
I’hen. it is immediate to realize that the condition (4.47) is equivalent to the
following one
p(j-) € -(’2±(х) for all z near j’“.	14.481
Sometimes it is possible to obtain "measurements" of the disturbance and
use them in the design of the control law. If the disturbance w is available
for measurements one can think to use a control
u — u(.r) 4- J(.r)r + v (j-)ic
which, besides to a feedback on t. includes a feedforward on rhe disturbance
If this is the case, then decoupling the output from the disturbance' is
possible under conditions that are obviously weaker than those established
before. Looking at the form of the closed loop system
z = /(n) +	+ (д(т)у(т) + p(.r))w
У = fi(-r)
it is immediately understood that what is needed is the possibility of finding
a function y(.r) such that
(д(.г)~(.г)+р(т))еО~(х) for all i'near .rc. (4491
This condition is equivalent to .
f
0 -	= LgL'fh{jr)y(x) + LpL'fh(x)
for all 0 < i < r — 1 and all z near ,r°. and this, recalling the1 definition of
relative degree, is in turn equivalent to
LpL^h(.r) = 0 for all 0 < i. < г - 2
LpLyAh{x} - -LaL,f"1h(x)y(x)
for all ,r near nc. The second one of these can always be satisfied, by choosing
L,,Lrf-lhU) '
Thus, the necessary and sufficient condition for solving the problem of decou-
pling the disturbance from the output by means of a feedback that, incorpo-
rates measurements of the1 disturbance is simply the first о tie. Note that this
4.7 High Gain Feedback 189
condition weaken? the condition [4.47) of Proposition 4.6.1. in the sense that
now LpL’jhi.r) = (> must be zero for all the values of i up to r - 2 included,
but not necessarily for i — г - 1.
If this is the c ase, a control law solving the problem of decoupling у from
w is clearly given by
Lyhl.r)	r	LpLj 4i(x)
L:!L!f~4i(x) L(lL!'f~x h\x\	T jC^' h(x)
Note that, geometrically, rhe condition (4.49) has the following interpre-
tation: at each .r. the vector p(x) can be decomposed in the form
pl.Z‘) = Cl (— P] (,r)
where Ci(,r) is a real-valued function and pi(.r) is a vector in G- (r). This
can be expressed in the form
p(z) G .0*(.r) — span{<7(r)} for all .r near .rc ( 1.50)
thus showing to what extent the condition (4.48) can be weakened by allowing
a feedback which incorporates measurements of the disturbance.
4.7 High Gain Feedback
In this section we consider again the problem of the design of a locally sta-
bilizing feedback and we show that under the stronger assumption that
the zero dynamics are asymptotically stable in the first approximation a
nonlinear system can be locally stabilized by means of output feedback. First
of all. we consider the case of a system having relative degree 1 at the point
— 0. and we show that asymptotic stabilization can be achieved by means
of a memory less linear feedback.
Proposition 4.7.1. Consider a system of the. form (4-1), with /[()) = 0 and
h(0) = 0. Suppose this system has relative degree 1 at x — 0. and suppose the
zero dynamics arc asymptotically stable in the first approximation, i.e. that
all the eigenvalues of the matrix

d)
0.0 j
have negative real part. Consider the closed loop system
x = /(.r) + g(x}u
a = — Ixh(x)
(4.51)
where
190	4. Nonlinear Feedback for Single-Input Single-Output Systems
f A' > 0 if L9h(D) > ()
[ К < 0 if Lgh(0) < 0.
Then, there exists a positive nurnbei' Ac such that, for all IT satisfying :A’| >
A‘o the equilibrium x — 0 of (J.51) is asymptotically stable.
Proof. Au elegant proof of this result is provided by the Singular Perturba-
tions Theory (see Appendix B). Suppose Lgh(0) < 0 (the other case can bo
dealt with in the same manner), and set
Note that the closed loop system (4,51), rewritten in the form
£.r = ef(x) + g(x)h(x) = F(x. e)	(4.52)
can he interpreted as a system of the form (B.22). Since A(z.O) — g(x)h(x]
and p(0) ф- 0 (because Lgh(0) 0). in a neighborhood U of the point ,r = 0
the set of equilibrium points of x' ~ F(x.O) coincides with the set
E = {т e Г : h(z) = 0} .
Moreover, since dh(x) is nonzero at x = 0. one can always choose V so that
the set E is a smooth (n — 1 )-dimensional submanifold.
We apply the main Theorem of section B.3 to examine the stability prop-
erties of this system. To this end. we need to check the corresponding as-
sumptions on the two "limit" subsystems (B.23) and (B.24). Note that, at
each x E E,
Tj.E = ker(c//t(;r)) .
Moreover, it is easy to check that
W = sp^'n{f/(jr)} .
In fact, at each r € E.
d(g(x)h(x))	dh(x)
= —сУ— =
(because h(x) = 0) and therefore.
Л,9(.г) = g{x}Lgh{x) .
Thus, the vector g(x) is an eigenvector of >IT. with eigenvalue A(.r) = Lgh(x).
At each x E E. the system x' = E(x. 0) has (n - 1) trivial eigenvalues and one
nontrivial eigenvalue. Since by assumption A(0) < 0. we see that at each point
x in a neighborhood of 0, the nontrivial eigenvalue A(j:) of is negative.
We will show now that the reduced vector field associated with the system
(4.52) coincides with the zero dynamics vector field. The easiest way to see
this is to express the first equation of (4.51) in normal form, that is
4.7 High Gain Feedback
191
z = b(z. q) + a(z, q)u
f) = q(z.q)
in which z — h(.r) € й and q G Accordingly, system (4.52) becomes
f 7 \	/eb(z.q) - a(z,q)I\z\
\*1) V -q^-П) Л
In the coordinates of the normal form, the set E is rhe set of pairs (z.q) such
that z=0. Thus
from which we deduce that the reduced system is given by
- q{().q)
i.e. by the zero dynamics of (4.1). Since, by assumption, the latter is asymp-
totically stable in the first approximation at q = 0. it is possible to conclude
that there exists > 0 such that, for each c G (0. sc) the system (4.52) has
an isolated equilibrium point ;r=- near 0 which is asymptotically stable. Since
F(0,c) = 0 for all s G (0.io). wo have necessarily z.- = 0. and this completes
the proof. <
Remark 4-7-E Note that this result, is the nonlinear version of the well known
fact that the root locus of a transfer function having relative degree 1 and all
zeros in the left-half complex plane has all branches contained in the left-half
complex plane for sufficiently large values of the loop gain. <
We turn now our attention to the case of systems having higher relative
degree, and we will show that the problem can be solved by reduction to the
case of a system with relative degree 1. For. suppose we can replace the real
output у by a “dummy" output of the form
a- = k(.r)
with A'(.r) defined as
к(Е] = Lrj + cr.,2^/ 2h(;r) + ... + (qLfh(x) + сиЬ(х)
where cq. .... cr_-„> are real numbers.
We obtain in this way a new system
F = /И+зОф
w = k(x)
having relative degree 1 at 0, because
£^(0) =£,LJ’1//(O) #0
192	4. Nonlinear Feedback for Single-Input Single-Output Systems
In order to decide whether or not this new system can be stabilized hy
an output feedback of the form considered before, i.e1. of the form
и = -I\ ir .
in view of Proposition 4.7.1. we need to examine the asymptotic behavior of
its zero dynamics. To this end. recall that the zero dynamics describe^ the
internal behavior of a system constrained in such a way as to produce zero
output. Observe also that, in the coordinates used to represent the normal
form of the original system, the "dummy” output tc is described by
H' =	+ C,._2Cr_j + ... +	+ e'u~i .
The constraint ir = 0 implies
= —{Cr- 2zr— 1 + - . . + cyc-j +	.
Substituting this into the normal form of rhe original system and choosing
the input in order to impose ir(t) = 0. one obtains the (z? — 1)-dimensional
dynamics
Ci —
-2	=	-3
-r-1	—	-(f,r-2rr-l + • • • + r1 + COC1 )

~(cr_93r_i
+ . . . + C] Z-2 + Ci)2] ). q)
which therefore describes the zero dynamics associated with the new output.
These equations have a "block triangular” form and from this it is easy to
conclude, looking for instance at the corresponding jacobian matrix, that, if
the zero dynamics of the original system is asymptotically stahle in the first
approximation, and if all the roots of the polynomial
n(s1 = Sf 1 -r- Cr--2^r 2 + •   + Cl S’1 + Cq
have negative real part, the dynamics in question is also asymptotically stable
in the first approximation. Thus, from Proposition 4.7.1. we can conclude that
if all the roots of the polynomial have negative real part, and К has the
same sign as that of L9L^~l the feedback
u -	+ ... +dLfh(x) +c0/;Gr))	(4.53)
asymptotically stabilizes the equilibrium z = 0 of the system (4.-51)-(4-53).
From the feedback (4.53). which actually is a state feedback, it is possible
to deduct1 an output feedback in the following way. Observe that the function
for 0 < t < r — 1. coincides- with the i-th derivative of the function
y(t) with respect to time. Thus the function m(0 is related to y(t) by
4.7 High Gain Feedback 193
w(f) = jj(r 1 ' it) + C>_2 t/‘'	1 (f ) +   - -r	+ C07/(t)
and can therefore be interpreted as the output of a system obtained by
cascade-connecting the original system with a linear filter having transfer
function exactly equal to the polynomial zi(.s). Clearly such a filter is not
physically realizable, but it is not difficult to replace it by a suitable physi-
cally realizable approximation, without impairing the stability properties of
the corresponding closed loop system.
To this end. all we need is the following simple result.
Proposition 4.7.2. Suppose the system
x = fix) - g(x')k(x)K
is asymptotically stable in the first approximation (at the equilibrium x — ()).
Then. if T is a sufficiently small positive number, a Is о th e s ys t e m
-i' = f(-r}-g(x)f
C - (l/T)(-£ + AV)A')
is asymptotically stable in the first approximation fat (x.f) = (().())/
Proof. The proof of this result is another simple application of Singular Per-
turbations Theory. For, change the variable £ into a new variable z defined
by
~ = — £ 4- k (.r) A
and note that the system in question becomes
F = f(x) - ,g(.r)(-c + k(x)K)
О у
Tz = -z + TK~[f(x)-g(x)(-z + k(x)K}] = -z+Tb{z,x} .
This system has exactly the structure of the system (B.1G), with f = T.
There is only one nontrivial eigenvalue, which is equal to —1. and the reduced
system, which is given by
j’ = /(.r) - g{x)k(x'}K
is by assumption asymptotically stable in the first approximation. There-
fore, for sufficiently small positive T the equilibrium (.r.£) = (0.0) is indeed
asymptotically stable1 in the first approximation. <
Note that the system discussed in this Proposition is nothing (‘Ise than
the system
F ~ /(•**) “ 9(-r)n
у = k(x)
in closed loop with a linear system having transfer function (Fig. 4.9)
19-1
4. Nonlinear Feedback for Single-Input Single-Output Systems
Thus we may interpret this result as the fact that the introduction of a "small
time constant" in a stable control loop does not impair (at least locally) its
asymptotic stability, Using this property r - 1 times, we can iimncdiately
deduce the result indicated in the* following statement.
Fig. 4.9.
Proposition 4.7.3. Suppose a system has relative degree r at ,rc — 0 and its
zero dynamics are asymptotically stable in the. first approximation. Suppose
also that all the roots of the polynomial
n(s) = yr-1 у Cr-2Sr~2 + . . . + Cj.S'1 + G)
have negative real part. Л linear dynamic output feedback with transfer func-
tion
stabilizes the system, provided К is a suitably large constant with the same
sign as that of LyL^-1 h(0) and T is a sufficiently small positive constant.
4.8 Additional Results on Exact Linearization
We have illustrated in section 4.2 a sc*t of necessary and sufficient conditions
for tin' existence of a (locally defined) state feedback and change of coordi-
nates transforming the system described by the first equation of (4.1) into a
linear and controllable system. Of course, if the conditions specified in The-
orem 4.2.3 are not satisfied, there is no way to obtain a linear controllable
system via feedback and change of coordinates. However, taking advantage
of the construction indicated at the end of the section, i.e. of the possibility
4.8 Additional Results on Exact. Linearization
195
of achieving always a decomposed system in which one of the two compo-
nent subsystems is linear, one may wish at least to search for a feedback and
a coordinates transformation which {if possible) maximize the dimension of
the linear subsystem. In view of other results established in section 4.2. the
problem is clearly equivalent to that of finding a suitable "output" шар A(.r)
for which the relative degree of the system at a point is the highest possible.
As a matter of fact, the solution of such a problem is not much difficult, as
the result hereafter discussed shows,
In the following statement, we make use of the notion of involutive clo-
sure of a distribution _1. that has been introduced in Remark 1.3.9 and. in
particular, of tin1 following property.
Lemma 4.8.1. Consider a distribution A and suppose Л is a real-valued
function such that dA{x0} 0 and dA € -I-. Then, in a neighborhood of .
dA 6 (invfA))1, where. inv(A) denotes the involutive closure of
Proof. Consider the distribution
Г = {span{(/A})-.
This distribution is (r? - 1 )-dirnensional in a neighborhood of /. and invo-
lutive, by Frobenius theorem. Moreover, by construction, А С Г. Since, by
definition. inv(A) is the smallest involutive distribution containing A. then
inv(_X) С Г, that is
spanjdA} C (inv(A) I1 .<
Theorem 4.8.2. Consider a pair of vector fields f(.r) and (fix). Suppose,
for some integer и
dim(inv(span{t/.adfp.....ad'f '1 g})) ~ k < n	(4,54)
for all x around .C and
dim (i nv (span { (/.adj t/.«dy-1/?})) = n	(4.55)
at x = .ro Then, in a neighborhood I‘ = of .r . there exists a function A(x)
such that
LgA(x) = L,LfA{x) =  = £vL^-~A(.r) = 0 for all x E t’c
and A('.r) is not identically zero on . Moreover, ifA(x) is any func-
tion, defined in a neighborhood L~° of x '. such that
L,jA(x) = L3LfA(.r) — ... = L3L’f~~ ~A(.C = 0 for all x E I
and LgLj~1 A(.r=)	(). then necessarily r < n.
196
4. Nonlinear Feedback for Single-In put Single-Output Systems
Proof. The distribution
inv(span{g. adjg.....ad‘j ’#})	(4.56,i
is involutive by construction and A‘-dimensional by assumption, with A‘ < n.
Thus, by Frobenius’ Theorem, there exist n — A’functions A, (jj......
whose differentials locally span the annihilator of (4.56)- If we set. e.g.. A(.rl =
Ai (j‘) we have, by construction
L.yX[.r) = L,lti,gX[.r'} = •   = L^-^Xfr) = 0
for all ,r near Moreover.
L r-1 A(.r)
is not identically zero near .P. For. if this were not true, the1 nonzero covector
</A(.r) would be an element of (span{fl. ad/g....ad‘j V*/})- and. by Lemma
4.8.1. also an element of (inv(span{p. odgg......ad^	. that is a contra-
diction. because the latter has dimension (I by assumption. Thus, by Lemma
4.1.2, wo can conclude that the function Ayr) has the required properties.
Now. consider any other function A(j’) having the properties indicated in
the Theorem. By Lemma 4.1.2 we have
dA E (span{/7<cidfg.....ad'j
and therefore, by Lemma 4.8.1. also
dX € (inv(span{f/. adjg..... od^~~g} })~ .
Since dXf.P) 0. we deduce that
dini(inv(span{<y.(idjg.....ad'j~~g])) < и
for all ,r near ,r'. Thus, from the assumptions (4.54)-(4-55). we conclude1 that
г < n. <	<
Note that the result of the previous Theorem incorporates that of Theo-
rem 4.2.3. As a matter of fact, if the conditions (i) and (ii) of Theorem 4.2.3
are satisfied, the integer о defined in the previous statement is exactly equal
to n. If the condition^ in question are not satisfied, in order to find an output
map A(.r) which "maximizes" the relative degree of the system, one has to
solve a partial differential equation of the form
dX(.r) ( Г]	1 = 0 .	(4.5/)
where T].....are such that
spaii{T"i....rA.J - (inv(span{p.adfg.......a<Pf~2g}))
and о defined as before. Once this solution has been constructed, then rhe
feedback (4.27) will transform the system into one which, in suitable coordi-
nates. contains a linear subsystem of maximal dimension.
4.8 Additional Results on Exac t Linearization
197
Example 4-8.1. Consider the system
In order to cheek whether or not this system can be transformed into a linear
and controllable system via state feedback and coordinates transformation,
we have to compute the vector fields adfg.adyg. adjg and test the condition^
of Theorem 4.2.3. Appropriate calculations show that
Since

one observes that
£ span{j7,ad/.9}
and therefore the distribution span{#. ad/g] is not involutive1. Thus, the con-
ditions of Theorem 4.2.3 are violated (see Remark 4.2.8). However, in this
case
inv(span{,9. adjg}) = span{</. adjg. [g.ad/g]} = span]
W
and
inv(span{9. adjg. adjg}) = inv(span{9,adjg. [g.(idjg].adjg}}
0	0
0	0
1/ \о/
so that the conditions of Theorem 4.8.2 are1 satisfied, with v = 3 and к = 3.
Then, the maximal relative degree one can obtain for this system is r = v = 3.
In order to find an output for which the relative1 degree is 3. one has to solve
the differential equation (4.57), which yields, in this case
A(.r) = *r i
198
4. Unilinear Feedback for Single-Input Single-Output Systems
From the previous discussion, it is clear that choosing a feedback
— LT X(x) + г
" = ~LgL}X(xf = “J’? + "
and new coordinates
Ci =	A(\r) = J"!
c2 = LfX(.r] ~ .r2 - J’5
z-3 = L’jX{x) = .r3
one obtains a. system which contains a linear subsystem of dimension 3. Com-
pleting the choice of coordinates with
Ц = 9(.r’J = -fj
yields
— c2
-2	=	-3
<5	= C
й =	+ 4 <
We conclude the section by discussing an additional problem. We have
already observed in Remark 4.2.10 that if an Exact State Space Linearization
Problem has been solved and the system has an output, the output, map in
the linearizing coordinates is not necessarily a linear map. Thus, ош1 might
pose the question of when there exist a feedback and a change of coordinates
transforming the entire description of the system, output function included,
into a linear and controllable one. An answer to this question is described in
the following statement.
Theorem 4.8.3. Consider a system idith relative degree. r at x = .C. Sup-
pose also f{xJ) = 0 and h(x") = 0. There exists a feedback of the form
(4.10) and a coordinates transformation z = Ф(х), defined locally around xz.
changing the system (4-1) into a lincxrr and controllable system
z = A; - Be
P = C:
if an d о n I у if t h e following con d it io ns are .s atisfi cd
Ii) the matrix ^у(.гэ) adf(fix~) ad'} '2(fixc) g(xz}^ has rankn,
(ii) the vector fields f(x) — f(x") + (x) and (fix) ~ ffix)J(x). irith u(.r)
and fix) defined by
-Lrfh[x)	1
o(j-) = -----^-j----J(.r) = -----------7-----
L„Lrf ^fix)	L’lL}
4.8 Additional Results on Exact Linearization
199
are such that
\а^д.ш1^д](х) - 0	(4.38)
for all 0 < i. j < n. and all .r near .r~.
Remark 4-8.4. Note that the system
j- = /(.;) — y(.r)o(.r(	p(j-)4(z)г ~ /(,r) — fd-r)c (4.59)
у = h(.r)	(4.69)
with o(.r) and i(J-) chosen in the way indicated in (ii). has already a linear
input-output response*, by Proposition 4.2.4. Then, this Theorem shows that
under the additional condition (4.58) it is possible to achieve linearity - via
feedback and coordinates transformation - also in the state space equations.
On the other hand, since the conditions (i) and (ii) imply the conditions (i)
and (ii) of Theorem 4.2.3 (as we shall see in a moment), this Theorem also
describes to what extent the conditions of Theorem 4.2.3. necessary and suf-
ficient to achieve linearity of the state space equations, must be strengthened
in order to achieve linearity also in the input-output response. In fact, as the
reader may easily verify, since 3Lr=)	9- condition (i) implies
rank(y(.r“) ttdyy/(.r) ... adj-1 у(.г")) = n
and condition (ii) implies (see Remark 1.3.5) that the distribution
spanfy. adjg..............................ad'~~[g}
is involutive. Thus system (4.59). by Theorem 4-2.3. can be transformed into
a linear and controllable system by means of state feedback and coordinates
transformation. But since this system has been obtained from [4.11 by means
of a state feedback, namely a = <i(z) -t- 3(.r}r. then also (4.1) can be trans-
formed into a linear and controllable system by means of state feedback and
coordinates transformation, i.e. (4.1) must satisfy the conditions of Theorem
4.2.3. <
We proceed now with the proof of Theorem 4.8.3.
Proof. Sufficiency. For convenience, we separate rhe proof in several steps.
(i) Observe that, by construction, the system (4.59)-(4.60) satisfies
L^LkJi(.r] = 0 for all 0 < k < r - 2	(4.61)
(because the relative degree is invariant under feedback).
LgL'P1 h(.r) = (L,?£'f-1 /((.r))3(j‘) = 1	(4.62)
(because Lk/?pr) = Lkjh(x} for all к < г - 1) and
200
4. Nonlinear Feedback for Single-Input Single-Output Systems
L^h(.r] - 0 for all A- > r	(4,63)
(because Ljh{.r} = L	h(x) - L'jh(x') + LyLrf~1h(x}a(x) = 0). Using
the formula (4.2). we deduce from these
(dLyi(x). adt~g{x)') is independent of т for all ,s. A > 0 .	(4.64)
(ii) As observed in the previous Remark, the system (4.59) satisfies con-
ditions (i) ami (ii) of Theorem 4.2.3. Therefore, by Lemma 4.2.1. there exists
a real-valued function A(.r). defined in a neighborhood U of .rc, satisfying
{dX(x). ad^g(x)) = 0 for all ,r near xc\ 0 < k < n - 2
and the function
r(.zj - (dAf.r). mfU1 g(f))
is nonzero at the point xc. We show now that, because of the assumption
(4.58), the function A(.r) can always be chosen in such a way that c(j-) = 1.
For. recall that by construction the functions
Zj = L^Xix) 1 < i < и
have linearly independent differentials, so that they can be chosen as new
coordinate functions near zc. As a consequence, there exists a function
- (ci....с,,) such that
5 (A(jr). LjXix]...Lj-'Xtx)) = c(j‘)
(7(3) is exactly the function r(.r) expressed in the z coordinates).
Observe also that, because of (4.58). for all 0 < A- < 7? — 2
C) = <<7A(jt), [ad^g(x). adj-1 g^x)]}
~~	-X(.L ) —	- Lac{kg A(j ) Lajk^(\x) .
Using this with k — 0 - in the previous expression for c(z) we obtain
" d. dL^Xix}^
0 -	- £ gzi Ox S(U _ ( 1)" g^ c(x)
and we deduce
21=0.
Recursively, it is possible to show that
i.e. that '(c) depends only on 34. In other words
4.8 Additional Results on Exact Linearization
201
с(т) = (А(л-))
where -<(£) is a real-valued function of the real variable £. defined in a neigh-
borhood of A(r=). Let be such that
du _ 1
~ -(C) ‘
Then, the function A(.r) ~ с:(А(т)) satisfies
(dX(r). (idk~g(.r)}
(dX(r.).ad^~lg(,rC
0 for all 0 < к < n - 2
’<M	.< \ _ 1
for all ,r near J‘°.
(iii) The function A(.r) considered in the previous step is such that
(dLyX(;r).ad^g(.r')} is independent of т	U-65)
for all .s. к such that 0 < s + к < n — 1. We show now that, because of (4.58).
this property holds for any value of .s. k. For. observe that, if t’i..<v-i is
a collection of vector fields satisfying
rank[ri(,r) ... rn(z)l = n
[с; (j-J, гД.г)] = 0 for all 1 < i, j < n + 1
then
ег^1(т) = У CA'd-r)
i-i
where ry.....c„ are real numbers. To see this, express r,( 4 ((.r) as


anti note that
0 =	-Г). сДл’)*',(j‘): = JjLr/G-Hkif.r) .
1=1 1=1
Thus
(U/iU) ...	= dc^J-) (o pr) ... r,t(.r)) — П.
i.e. <Zcj(.r) = 0. and r,(.r) is independent of ,r for all 1 < i < n. Using this
property we deduce that
ud^(jr) = ^cj'ndh-1 g{r)
; = i
202	4. Nonlinear Feedback for Single-Input Single-Output Systems
and. with a simple induction, also that
ad^gU} =	pi.r)
t=i
for all k > 11. where the rf's art1 real numbers. From this we have
(dX(.r).adk:g(x)) is independent of j.' for all k > 0
and. using again the formula (4.2). it is easy to conclude that 14.6-5) holds
for any value of x. k.
(iv) Arrange (4.64) and (4.6-5) in the matrix relation
/ dX(j-) \
dL f-A(.zd
(j7(.r) adjijtd') ... m/'l lg(-r) ) = constant.
\ dh(j‘) /
The last row of the matrix on the left-hand side i_s linearly dependent from
the first n ones through constant coefficients (because of the constancy of
the right-hand side). Then, since the right factor of the left-hand side i»
nonsingular for all r near .r3. we deduce that
n -1
dhdr) = b,dZ/f-A(.r)
;=u
where b0......are real numbers. This implies
h{r) = btL^Xir') + c	(4.66)
(==o
where r is a constant. Moreover, this constant is zero if A(.r3i = О. becau»e
of the assumptions h{.r~) = 0 and /(.rc) = 0.
tv) We know for the theory developed in section 4-2 (see in particular
(4.23)-( 1.24)) that rhe system (4.59). after the feedback
-T^Aud	!
ZJrpIfT) ~ L^-lXdr)L
in the new coordinates
cf = £M1A(.r)	0 < i < n - 1
is a controllable linear system. But in these new coordinates the output map
(4.60) also is linear, as (4-66) shows. Thus, the proof of the sufficiency is
completed.
The proof of the necessity requires only straightforward calculation». and
is left to the reader. <
4.9 Observers with Linear Error Dynamics
203
4.9 Observers with Linear Error Dynamics
We consider in this section a problem which is in some sense dual of that
considered in section 4.2. We have seen that the soivabilty of the Exact State
Space Linearization Problem enables us to design a feedback under which the
system in suitable local coordinates becomes linear with prescribed eigen-
values. In the case of a linear system, the dual notion of spectral assignability
via static state feedback is the existence of observers with prescribed eigen-
values. Moreover, it is known that the dynamics of an observer and that of
the observation error (i.e. the difference between the unknown state and the
estimated state) are the >ame. In view of this, if we wish to dualize the results
developed so far. we are led to consider the problem of the synthesis of (non-
linear) observers yielding an error dynamics that, possibly after some suitable
coordinates transformation, becomes linear and spectrally assignable.
For the sake of simplicity, we restrict ourselves to the consideration of sys-
tems without input and with scalar output, i.e. systems described by equa-
tions of the form
with у e IE.
Suppose there exists a coordinates transformation г = Ф(.г) under which
the vector field f and rhe output map h become respectively
[&Ф	1
.	J j —Ф - ’ ' Z I
/цф-ЧМ) = Cz
where (J.C) is an observable pair and k is a п-vpctor valued function of a
real variable.
If this is the case, then an observer of the form
s = U GCtf ~Gy + k(y)
yields an observation error fin the z coordinates)
f = £ - c = < - Ф(г)
governed by the differential equation
e = (.4 GC]e
which is linear and spectrally assignable (via the n-vector G of real numbers).
Motivated by these considerations, we examine the following problem,
called the Observer Linearization Problem. Given a system without input
(4.67). and an initial state rs. find (if possible) a neighborhood I ° of .r:. a
204	4. Nonlinear Feedback for Single-Input Single-Output Systems
coordinates transformation с = Ф(х) defined on T’°. a mapping A- : h(l'z}
T". such that
Гс*М,
dx
AzPk{Cz}
(4.681

=	- Cz
(4.69)
for all z E Ф(1'А- for some suitable matrix .4 and row vector C. satisfying
the condition
n .
rank
(4.70)
W.4"-1/
The conditions for the solvability of this problem can be described as
follows.
Lemma 4.9.1. The Observer Linearization Problem is solvable only if
dim(span{d/t(.rc). dLfh\xz).....dL’j 1Л(.г3)}) = m (4.71)
Proof. The observability condition (4.70) implies the existence of a nonsin-
gtdar 7i x n matrix T and a n x 1 vector G such that
/0	0	•••	0	0\
T(A + GC')T~l =	1 °	0 0
\()	0		1	0/	(4'/2>
CT~l = (0 0	0 1).
Suppost1 (4.68) and (4.69) hold, and se(
7 = Ф(т) = ТФ(х)
k(y) = T(kfy)-Gy).
Them it is easily seen that
/ЬФ-1^)) = (0 o o i);
/0 0	0 0\
0Ф f 1	1 0  о 0 ] .
dr T -	~~	..................z
L~ j т=ф-] (S)	,
\0 0	1 0/
+ A-((C) 0 ••• 0 IM).
4.9 Observers with Linear Error Dynamics
205
Front this, we deduce that there is no loss of generality in assuming that
the pair (.4. C) that makes (4.68) and (4.69) satisfied has directly the form
specified by the right-hand sides of (4.72). Now. set
Z = Ф(Т) ~ (4)1(3) (y)...........cniz)) .
If (4.68) and (4.69j hold, we have, for all ,r € t"

dr
od-d
ЫоД.Н)
~t(.r) + Ao (z„ (r))
dr
when1 k\....kr, denote the n components of the vector Ac
Observe that
Ljh(r') =	= згг_) (r) + kn
dr
T > f i	dzn_} L-fh(r) =	f(r] - 7	dr		dk^'		dz!t °-r	
— -il-ld-H +	'dkn~ . д.У .			^f(x) - A:;j_i	
=	3,,-2(.r) + A‘r; _ I ( 3rl (,r). z,;-! (J-))
where
A'n-iI зГ|,	з(!_1 + ~~~A’„(zn) + A‘TJ_i(c,f) .
д-r.
Proceeding in this way one obtains for each L^h\r). for 1 < i < n — 1. an
expression of the form
L	—- зГ} _ ; ( t) + kn _ [ I (1 (,r).........z r! _ ) (.r)) .
Differentiating with respect to ,r and arranging all these expressions to-
gether. one obtains
dh
. Or
OL f h
dr
Or
/ (d \
dz
$Lf}i dz
^.z
dur}h
\ dz ~ f
0
1
о 0
о 0
1 *
206	4. Nonlinear Feedback for Single-Input Single-Output Systems
This, because of the nonsingularity of the matrix on the right-hand side,
proves the claim. <j
If the condition (4. < 1) is satisfied, then it is possible to define, in a neigh-
borhood of j'°. a unique vector field т which satisfies the conditions
LfiFr)
LrLfh(j-) = ... — LrLj 2/t(.r) = 0
1
for all .r € 1°. As a matter of fact, one only needs to solve the set of equations
/ dh(T) \
dL fh{y}
\dL"-' ЛИ У
The construction of this vector field r is useful in order to find necessary
and sufficient conditions for the solution of our problem.
Lemma 4.9.2. The Observer Linearization Problem is solvable if and only
if
(i) dim(span{d/j(r°). dLfh(F).   , dL’j~[h(F)}) ~n
(iii there exists a mapping F of some open set V of IP1 onto a neighborhood
Uz of F that satisfies the equation
dF
= -adfT(j-) ... (-1)"“	b=jF{;}	(4.74)
for all z G V. where т is the unique vector field solution of (4-73).
Proof. Necessity. We already know th^t (i) is necessary. Suppose (4.68) and
(4.69) are satisfied and set F(~) = Ф-1(с) for all z G IF. Set also
dF
~ —r-pj'i 	(4.(5)
We claim that
dF
ad^eo, =	(4.76)
for all 0 < k < n - 1. This equality, which is true by definition for к — 0.
will be proved by induction, using the fact that (4.68) and (4.69) imply (see
proof of Lemma 4.9.1)
with
4.9 Observers with Linear Error Dynamics
207
/(:) =
In fact, suppose (4.76) is true for some 0 < k < n — 1. and let f.j denote the
i-th column of the и x n identity matrix. Then
adf
OF
=
= i-1 )A’-1 f—e )
1	' 0z = = F-1	'
Collecting, all (4.76) together, we obtain
c) F
- (0(aj -adf6(.Fi ... (- 1 C ' lnd'j " l6*(jj ),r^/--( - j 
If we show that в necessarily coincides with the unique solution of (4.73). the
proof is completed, because the p.d.e. 14.74) will coincide with the one just
found.
To this end, observe that
(-1)A'Z
Oh dF
dx dzk^
.dh(F(z))
but. since Н(Ф l(z)} = zn. we have
= 0
for all 0 < < n — 2 and
Using Lemina 4.1.1. we deduce that
L^Lkfh[j'} = 0
for all 0 < A- < n — 2 and
LffU‘-4dx) = 1 .
Thus, the vector field 0 necessarily coincides with the unique solution of
(4.73).
Sufficiency. Suppose (i) holds and lot. т denote the solution of (4.73). Using
Lemma 4.1.1 one may immediately note (see (4.5)) that the matrix
208	4. Nonlinear Feedback for Single-In put Single-Output Sy sterna
/ dh(j') \
dLjh(x) (-(.г) adfrijr} асГ^~}т(т))
YdLf-'hU)/
has rank n for all .r near ,r°. Thus, the vector fields {-, ad/r.ad1} are
linearly independent at all J’ near F.
Let F denote a solution of the p.d.e. (4.74) and let zrj be a point such
that F(zF = F. From the linear independence of the vector fields on the
right-hand side of (4.74) we deduce that the differential of F has rank n at c:'.
i.e. that F is a diffeoinorphisni of a neighborhood of onto a neighborhood
of F. Set Ф = F 1 and
= (—= l 	(4.77>
By definition, the mapping F is such that
LUz^i F-f 1! ,r j
so that
[a7fZ^7t‘rl] -Ф >	(4.78)
for all 0 < A- < n - 1.
Using (4.77) and (4.78). one obtains, for all 0 < k < ii — 2
|-l|bV> =
L U.f J	J -Г=Ф~ 1(7!
гдФ	i
1р.Г }	1.г = Ф-1|:!
= 7(c). (-1 )A’r^l]
= F^/F
that is
2	.	Of,	.
77----- - 1.	7~---= 0 for I F k + 2 .
(7^+1	ОЗД + 1
From these, one deduces that A depends only on cn. and that for
2 < i < n. is such that f, - depends only on z„. This proves that (4.68)
holds. Moreover, since1
for 0 < k < n — 1.
we deduce that
0h(F(z)) _
0zk
thus proving (4.69). <
for 1 < A- < n.
= (-1 )"-*
7"
dh[F(z]} _
Oz,,
Т()(Л/J?(.ri = 0
4.9 Observers with Linear Error Dynamics
209
The integrability of the p.d.e. (4.74) may be expressed in terms of a prop-
erty of the vector fields r.adfT.... ,ad'j~lT. To this end. one may use the
following consequence of Frobenius Theorem.
Theorem 4.9.3. Let n..........r!t be vector fields of 33. Consider the partial
differ?.ntial equatiоns
= D(.r(c))	(4.79)
(JZ;
where, i- denotes a mapping from an open set of T" to an open set of 33 .
Let	be a point in R'! x R" and suppose Ty\.r:')....r„(z^)	are linearly
independent. There e.nsts ne.ighborhoods Uz of .C and V? of ~c and a diffeo-
morphism .r : I ° —> I ; solving the equation and such that ,r(z~} — ,rc
if and only if
[Зф]=0	IT80)
for all 1 < i.j < n.
Proof. We limit ourselves to give a sketch of the proof of the sufficiency. To
this end, set
-F - spaiifn.......rr_i. .........
This distribution is involutive (because of (4.80)) and has constant dimension
n — 1 in a neighborhood of .r’3. Therefore, by Frobenius Theorem, there exists
a function 0, whose' differential spans Ay. i.e. such that
(dd,, Tj) = 0
for all j i. We claim that the differentials .......dori are linearly indepen-
dent for all r in a neighborhood of .rc. For. let c,(j‘) denote the real-valued
function
C,(.r) = {dodo-}. -фТ)
and note that effr0) 0 because dofr2} / 0 and spat^n..................rFi} has
dimension n at :C. If the differentials doilr0}.....dQnprc) were linearly de-
pendent. then for some nonzero row vector v such that
j = ()
t = l
we would have	,
0 = У2	PCA rd-H) = фдФЗ
i = i
and this would imply = 0 for some J, i.e. a contradiction.
As a consequence, the mapping f = Ф[а:) = colfoAx).............on(j-)) F a
local diffeomorphism at .r3. By construction
ЭФ
~ ( и (,r) ... гп(^) ) = diagfri (.r)......c.n(j•)) .
210	4. (Nonlinear Feedback for Single-Input Single-Output Systems
Moreover, using again (4.80), it is easy to see that с,(Ф *(£)) depends only
on U- Thus, there exist functions zt = such that
1
00 ~ с^Ф-Ч^У)
(recall that сДгс) 0). The composed function
z = T(r) = (col(pi(M);  
is such that
anti therefore ;r = T-1(c) solves the p.d.e. (4.79). <
Merging Lemma 4.9.2 with Theorem 4.9.3 yields the desired result.
Theorem 4.9.4. The Observer Linearization Problem is solvable if and only
if
(i) dim(span{d/?(.r°). dL fh(r3),.... dL’j~lh(j-°)}) = n
(ii) the unique vector field т solution of bf.73) is such that
[adfT.ad^r] = 0	(4.81)
for all 0 < i < j < n — 1.
Remark f.9.1. Using the Jacobi identity repeatedly, one can easily show that
the condition (4.81) can be replaced by the condition
[r. tidy"] = 0
for all к = 1.3....2n - 1. <
In summary, one may proceed as' follows in order to obtain an observer
with linear (and spectrally assignable) error dynamics. If condition (i) holds,
one finds first a vector field r solving the equation (4.73). If also condition
(ii) holds, one solves the p.d.e. (4.74) and finds a function F, defined in a
neighborhood V° of 2°. such that F(z°) = rc. Then one sets Ф = F-1.
Eventually, one computes the mapping к as
At this point, the observer
£ — (A + GC)f — Gy + k(y)
with (A.C) in the form of the right-hand side of (4.72) yields the desired
result.
4.10 Examples
211
4.10 Examples
We discuss in this section two simple examples of physical control systems
that can be modeled by equations of the form (4.1) and to which the design
methodologies illustrated in the Chapter can be applied.
The first example is the one of a d.c. motor in which the rotor voltage is
kept constant, while the stator voltage is used as a control variable (sec fig.
4.10).
Fig. 4.10.
The system in question is characterized by a set of three first order differential
equations. The first one describes the electrical balance in the stator winding,
and has the form
+ RSI, = U
dt
in which Is represents the stator current. Vs the stator voltage. Rs and Ls
the resistance and. respectively, inductance of the stator winding. The second
equation describes the electrical balance in the rotor winding, and has the
form
Lr(^ + RrIr=V,.-E
dt
in which Ir represents the rotor current. Vr the rotor voltage (constant by
assumption). Rr and Lr the resistance and. respectively, the inductance of
the rotor winding, and E is the so-called “back e.m.f.''. The third equation
describes the mechanical balance of the load that, in the hypothesis of vis-
cous friction only (namely friction torque purely proportional to the angular
velocity) has the form
J— + FQ--T
dt
in which Q denotes the angular velocity of the motor shaft. J the inertia of
the load. F the viscous friction constant, and T the torque developed at the
shaft. The coupling between the three equations is expressed by the relations
212
4. Nonlinear Feedback for Single-Input Single-Output Systems
E = К.ФЕ
T = К,„Ф1,.
Ф = LJ,
in which Ф represents the flux associated with the stator winding, and A’, and
A\t are constants. In the ideal hypothesis of 11Ю1/ efficiency in the energy
conversion. E\ = А\, — A'.
Choosing, as state variables.
.ci = A .r2 = Ir j’3 = f?
and considering the voltage Cs as the input variable, the equations in question
can be rewritten in the form
r = /(A +
with
The first thing we want to check is whether or not this system is fully
linearizable by state feedback and coordinates changes. To this end, we have
to test the conditions (i) and (ii) of Theorem 4.2.3. Since
= -
(Tri Ls
we see that the distribution
D = span{(/. [/.<?]}
has dimension 2 at each point of the dense set
: J"2 ф 0 or .r3 ф 0}
and is involutive on U. Thus, around any point of A the condition (ii) is
fulfilled. In order to check the condition (i) we calculate also the vector field
[/.[/. </]](т) and we find that the condition in question is satisfied for all A
in an open and dense subset Vе of U.
In order to transform the system into a linear and controllable one. it is
necessary first to solve the partial differential equation
4,10 Examples 213
( 9^)	) = 0
L, Ц
f dX	OX	dX \	K'	, n.
N—	0	—z:i = (0 0).
\ dx\	d.r >	oj’3 /	Lr
К
\°	-j'2/
An easy calculation shows that a possible solution is
A(;r) = Lrx2, + Jx%
From this, the linearizing feedback and the linearizing coordinates are
calculated by means of (4.23) and (4.24).
Next, we illustrate on this system the notion of zero dynamics and. to this
end. an output map /i(j’) has to be defined first. A natural output variable to
look at in a motor is indeed the angular velocity of the shaft. More precisely,
since the mode of control we are considering in this ease (namely, holding Vr
constant, and using Vs as an input) is particularly suited to the problem of
controlling the velocity around a nominal nonzero value, we can choose as an
output the quantity
у =	= Q-	- Qz
i.e. the deviation of the angular velocity Q from a fixed reference value _QC.
The problem of zeroing the output, for this system, clearly corresponds to
the search of all initial states and inputs which produce an angular velocity
constantly equal to .Qc.
For the system thus defined we have
Is/i(.r)=0 isL/A(T) = j^
and we see that the relative degree is r = 2 at each point in which x-i / 0.
Imposing zero output implies having the state evolving on the set
Z' = (.r£l3: h(j-) = £;h(T) =0}
that is on the manifold (see Fig. 4.11)
FPC
Z’ = {> e IR3 : .r3 = IF.j-jTs = —yr}
i s A
and this can be accomplished, as shown in section 4.3. by means of the input.
-L2Ji.(t)
u’w = глад'
214
4. Nonlinear Feedbark for Single-Input Single-Output Systems
The zero dynamics of a system describe its internal behavior when the
input is set equal to u*(.r) and the initial conditions have1 been chosen on the
manifold Z*. In the present example, the zero dynamics are l~dimensional
and can be easily obtained, for instance, by replacing the constraints
(which define the manifold Z*) in the system equations. Imposing these con-
straints one obtains
/?,.	FfF'~	1
г? =	~----+ y- -
L,-	Lr.r->	Lr
The differential equation thus found describes the projection, on the jw-axis.
of the motion of the system on the manifold Z* on which the zero dynamics
are defined. Note that x2 = 0 is a singular value of the right-hand side, as
expected from the shape of the manifold Z* itself.
Suppose, for instance. > 0- The differential equation characterizing the
zero dynamics has two equilibria, which correspond to the roots of the second
order equation
R,.jr.2 — 1 ’Г,Г9 + FRF'2 = 0 .
These roots, on the	plane, span an ellipse (Fig. 4.11). This, in par-
ticular. shows that only angular velocities satisfying- the condition
ARrFfF- < V;
can be imposed, and that, if 4RrFQ°2 < Vr2. the same fixed angular velocity
<2° can be obtained from two different steady-state values of the rotor current.
Accordingly, on Z* we find two equilibria /' and x- for the zero dynamics,
with	f
17 - Jv’2 -4FRr.Qc~	I 7 + JV2 -+FRFF2
-_______У_________________ -_____________ A-———------------
2	2Rr	2	2Rr
The sign of the right-hand side of the differential equation defining the
zero dynamics is negative for 0 < x-> < positive for .r" < ^2 < and
again negative for	< oc. Thus it is easy to conclude that the point
.rfl is an asymptotically stable equilibrium for the zero dynamics, whereas the
point .ra is an unstable equilibrium for the zero dynamics.
The second example we want to consider is the one of a simple one-link
robot arm. whose rotary motion about one end is controlled by means of an
elastically coupled actuator. Elastic coupling between actuators and links is a
phenomenon that cannot be neglected in many practical situations, and the
experience has shown that robot, arms in which the motion is transmitted
hy means of long shafts or transmission belts, or in which the actuator is an
4.10 Examples 215
Fig, 4.11.
harmonic drive, show a resonant behavior in the same range of frequencies
used for control.
The effect of elastic coupling between actuators and links, that is com-
monly referred to as joint elasticity, can be modeled by inserting a linear
torsional spring at each joint, between the shaft of the actuator and the
end about which the link is rotating. In the case of a simple one-link arm.
the model thus obtained is like the one illustrated in Fig 4.12. The system in
question is described by means of two second order differential equations, one
characterizing the mechanical balance of the actuator shaft, and the other
one characterizing the mechanical balance of the link. Using qi and q> to
denote the angular positions, with respect to a fixed reference frame, of the
actuator shaft and respectively - of the link, the actuator equation can he
written in the form
ЛУ1 + F\<ii +	= T
Л	Л
in which ,7i and F\ represent inertia and viscous friction constants, К the
elasticity constant of the spring which represents the clastic coupling with
the joint, and .V the1 transmission gear ratio. T is the torque produced at the
actuator axis. On the other hand, the link equation can be written in the
similar form
Fzq-2 + К-t- mgdeosq-i = 0
in which m and d represent the mass and. respectively, the position of the
center of gravity of the link.
Choosing the state vector
•r = col(qi.g>,qi.q2)
216
4. Nonlinear Feedback for Single-Input Single-Output Systems
Fig. 4,12.
the system can be represented in the form (4.1), with input u = T. and
As output of this system, it is natural to choose the angular position
of the link with respect to the fixed reference frame, i.e.
У — h(x) = X-2 
An easy calculation shows that
Lfh(x) L‘2fh(.r) L3fh(x)	= = fiO) 0f\	. Of a	0f< —		^3 + Zt	~ uX\	UJr-2	иТд
and. therefore, since Л(т) does not depend on j-3.
Lgh(x)	= LgLfM.x) = LgL'jh(x) =0
LgL3h(.r)	OL3fh 1 _	1 _ к Эхз	dxi Jl J1J2X
The system in question has relative degree r = 4 = n at each point z0 of
the state space. Thus, on the basis of the results established at the beginning
of section 4.2, we conclude that this system can be exactly linearized via
state feedback and coordinates transformation around any point xc of the
state space. The linearizing feedback has the expression
4.10 Examples
217
) 4- г
LgL3fh(j-)
and the linearizing coordinates are
£1 = h(j-). z2 = Lfh(x)
г4 = L’jh(x) .
Note that, since by definition of relative degree,
h(x) = y, Lfh(x) =
r>h/ > _ (}2У	гм( \ - c^y
it is possible to identify the linearizing coordinates with the output and its
first three derivatives with respect to time: these variables are in fact the
angular position, velocity, acceleration and jerk of the link with respect to
the fixed reference frame.
It may be of interest to linearize the system around a state .?,э having
x| = 0 (which corresponds to an horizontal position of the link). However,
it is immediately seen that a state of this type cannot be an equilibrium for
the vector field because the constraints = 0 and — 0 are not
compatible, and therefore the corresponding linearized system will not be
necessarily defined in a neighborhood of the point c — 0 (see Remark 4.2.9).
In this case, one may try to render a point of this type an equilibrium by
means of feedback, as described in Remark 4.2.9. The condition for this to
be possible is that, at z°, the vector /(/) is in the span of g(-r5) or. in other
words, that
/Н + <7(.r)c = 0
for some real number c. In the present situation, this condition is satisfied
for a state ,r: having = 0. because it reduces to
— j-j — mgd = 0
that can indeed be (uniquely) solved for c and .rj. Thus, if instead of the
former linearizing feedback one considers
— L'jh(x') + u
ZgLpi(j-)
J J
with c satisfying the previous equation, the corresponding linearized system
(in the same linearizing coordinates) will be defined around the point c = 0.

5. Elementary Theory of Nonlinear Feedback
for Multi-Input Multi-Output Systems
5.1 Local Coordinates Transformations
In this Chapter we shall see how the theory developed for single-input single
output systems can be extended to nonlinear systems having many inputs
and many outputs. In particular, in the first three sections we shall consider
a special class of multivariable nonlinear systems, those for which there is a
meaningful multivariable analogue of the notion of relative degree. For these
systems it is an easy matter to extend in a straightforward way - most
of the design procedures illustrated in Chapter 4. Then, in section 5.4. we
shall proceed to the study of more general classes of multivariable systems.
In order to avoid unnecessary complications, we shall restrict our analysis to
the consideration of systems having the same number in of input and output
channels. Occasionally, we shall specify how the results should be adapted in
order to include systems having a different number of inputs and outputs.
The multivariable nonlinear systems we consider are described in state space
form by equations of the following kind
i = f(x) 4-
г-1
t/i ” Ы-Н	(5.1)
in which /(z). (?i (.r) gm (z) are smooth vector fields, and ..................hnt (x)
smooth functions, defined on an open set of R'1. Whenever possible and con-
venient. these equations will be rewritten in the more condensed form
x
f(x) + g(x)u
having set
U = colfui..........Hrn)
У = col(t/i.... ,ym )
and where
220
5, Nonlinear Feedback for Multi-Input Multi-Output Systems
g№ = (gi(^)    gmi-r))
hU) = соЦМт)..........
are respectively an n x m-matrix and an z/i-vector-
The point of departure of the analysis is an appropriate multivariable
version of the notion of relative degree, which, as a matter of fact, identifies
the class of multivariable nonlinear systems which will be studied in the first
three sections of this Chapter. A multivariable nonlinear system of the form
(5.1) has a (vector) relative degree {zq.rm } at a point .r° if
in
L91L = 0
for all 1 < j < m. for all k < r, — 1. for all 1 < i < m. and for all ,c in a
neighborhood of .C.
(ii) the m x m matrix
4(r)= onr’oe) 	(5,2)
is nonsingular at.?' = .rc.
Remark 5.1.1. It is immediately seen that this definition incorporates the
one given at the beginning of the previous Chapter, for a single-input single-
output nonlinear system. As far as the numbers r-[,....rm are concerned,
note that each integer rt is associated with the z-th output channel of the
system, By definition, for all k < r( — 1, the row vector
Lg.2L^hi{x) ••• La,nLkfhi(x))
is zero for all j* in a neighborhood of .R and. for k = r1 — 1. this row vector
is nonzero (i.e. has at least a nonzero element, at T“), because the matrix
A(r°) is assumed to be nonsingular, As a consequence, in view also of the
condition (i). we see that for each i there is at least one choice of j such
that the (single-input single-output) system having output yt and input
has exactly relative degree r, at x° and, for any other possible choice of j
(i.e. of input channel), the corresponding relative degree at x~ - if any -
is necessarily higher than or equal to r}. It is important to stress that the
characterization of r, as the integer such that
L^hiW =0	(5,3)
for all 1 < j < m. for all 1 < i < m. for all к < r, — 1. and for all z in a
neighborhood of .C. and
L9] Lrf -1 ЬДт3) 0 for at least one 1 < j < m
(5-4)
51 Local Coordinates Transformations
221
is only implied by, but not equivalent to. (i) and (ii). As a matter of fact,
(ii) also incorporates the assumption that the matrix .4(.r) is nonsingular.
This assumption - although quite restrictive plays a crucial role in mak-
ing possible a straightforward extension of most of the results developed for
single-input single-output systems.
Note, finally, that r; is exactly the number of times one has to differentiate
the z-th output у pt} at t = C in order to have at least one component of the
input vector n(t ’~} explicitly appearing (see section 4.1). <
The nonsingularity of .4(r") may be interpreted as the appropriate mul-
tivariable version of the assumption that the coefficient
«(/) = )
is nonzero in a single-input single-output system. As we shall see. this greatly
simplifies the problem of calculating normal forms and reduces it and several
related issues to an essentially trivial extension of the theory illustrated so
far. The deduction of normal forms is based on a proper choice of new local
coordinates, specified in the following statements, which are multivariable
versions of Lemma 4.1.1 and Proposition 4.1.3.
Lemma 5.1.1. Suppose the system has a (vector) relative degree {n..г,л}
at J?a. Then, the roir vectors
dhi(r=). dLfhl(rc]....dL’j-1-1 hi(j-")
dhiQ’.'Q.dLdLr^~] /n>(.ro)
(,r3L dLfhm(x^)----
are linearly independent.
Proof. It. is very similar of the proof of Lemma 4.1.1. Suppose, without loss
of generality, that 7'] > n, 2 < i < m. Consider the matrices
Q - col(d/7i (;r).dLy- ............dh„,(;r)...... dU'f1 -1 hf(J Cr))
and
P = col(f;](;r)...g,nU)....ad'f' ^g^-r)....ad^ g,„(,?}') .
Using Lemma 4.1.2 and the definition of relative degree, it is easy to see
that the matrix QP. after possibly a reordering of the rows, exhibits a block
triangular structure in which the diagonal blocks consist of rows of the matrix
(5.2). This shows the linear independence of the rows of the matrix QP. i.e.
that of the rows of the matrix Q and this completes the proof. <
222
5. Nonlinear Feedback for Multi-Input Multi-Output Systems
Proposition 5.1.2. Suppose a system has a (vector) relative degree {ry.....
rm } at r. Then
П + ... + rtu < n .
Set, for 1 < i < in.
M-t)
£ /hfix)
W
<ArA-G =	%(-*•)
If r = П + ... + rm is strictly less than n. it is always possible to find n — r
more functions	....0л(.г) such that the mapping
Ф(х) = col(<5} (t).....0^	(x)....; <?"'(*)-(^), Or—1 (-Г)........«М-И)
has a jacobian matrix which is nonsingular at x° and therefore qualifies as a
local coordinates transformation in a neighborhood of .r°. The value at tc of
these additional functions can be chosen arbitrarily. Moreover, if the distri-
bution
G = spaii^i......c/,„}
is involutive near .r°. it is always possible to choose &r+fi.r)..in
such a way that
~ 0	(5.5)
for all r -+- 1 < i < n, for all 1 < j < m. and all x around x°.
Proof. We only have to prove the second part of the statement, namely the
possibility of choosing the remaining n — r new coordinates in such a way
that (5.5) holds. Since the matrix Л(л,с) can be written as
A(Z) =
/ dLrfi	\
dLrf--4ifixA
(Ы-Л

)

from the nonsingularity of this matrix we deduce that the m vectors
..., gmAA) are linearly independent. Thus, the distribution G has dimen-
sion m near r. Since the distribution is also involutive by assumption, by
Frobenius" Theorem we deduce the existence of n — m real-valued functions
Ai(j-). .... A„_m(j). defined in a neighborhood of such that
span{dAi........t£Xrt_rn} = G
Consider now the codistribution
Q — spaii{tZ£*/g : 0 < k < r,- 1.1 < i < m}
which has dimension r. and note that
5.1 Local Coordinates Transformations 223
G(t3) П (2^{тс ) = 0 .
(5.6)
For, if this were not true, there would exist a nonzero vector in G(.r°). i.e. a
vector of the form
£ = 52 Off,(-r0)
;=i
that, would annihilate all vectors of Pf.r0). but this is a contradiction, because
implies ci = c2 = cm ~ 0. by the nonsingularity of .4(tc). Since (5.6)
implies
dim(G±(j‘°) +	= n
the proof can be continued exactly as in the proof of the corresponding Propo-
sition 4,1.3. <
Remark 5.1.2. Note that this result incorporates that of Proposition 4.1.3.
An important point to be stressed is that the choice of the additional functions
satisfying the condition (5.5) is possible if and only if the distribution
G spanned by the vector fields gY (t), .... gm(x) is involutive. Such a condition
is always satisfied when the system has only one input, because this set
consists of only one vector, and this is why a similar assumption was not
mentioned earlier, <
Remark 5.1.3. The reader will have no difficulties in proving that most of
the results established so far can be extended to a system having a different
number of inputs and outputs, provided that the condition (ii). namely the
nonsingularity of the matrix A(j-). is replaced by the assumption that this
matrix has rank equal to the number of its rows (i.e, to the number of output
channels). Note that this implies dealing with a system having a number
of inputs larger than or equal to the number of outputs. As a matter of
fact, under this assumption Lemma 5.1.1 is still true, and from this one
deduces that the same type of coordinates transformation introduced in the
Proposition 5.1.2 can be considered. <
The calculations leading to the description of the system in the new co-
ordinates are exactly the same done earlier for single-input single-output
nonlinear systems. As a matter of fact, differentiating with respect to time,
one obtains, e.g.. for the first set of new coordinates
224
5. Nonlinear Feedback for Multi-Input Multi-Output Systems
do]
lit
dt
dolr
dt
У')
j=i

Note that the coefficient that multiply iij(t) in the latter equation is exactly
equal to the (1. j) entry of the matrix ,4(.r).
Consistently with the notations already introduced in the previous Chap-
ter. set now
for 1 < ? < m.
and
MCh)
^Ц~Чь(Ф-\^
Lf h rf)}
for 1 < i.j < m
for 1 < j < m .
Then, the equations in question can be rewritten as

m
C, - MP1) +
4=1
di - £1
for 1 £ ? < m. As far аь the remaining set of new coordinates is concerned,
we cannot expect any special form for the corresponding equations. If the
distribution spanned by the vector fields gi (<)..<5?i(ir)	i^ not involutive
(which is likely to be the most general case), we can only write generically,
with a vector notation.
TH
n =	+p{^r))u .	(5.8)
i-i
5.1 Local Coordinate? Transformations
225
Otherwise, if the distribution in question is involutive. it is always possible
to choose the remaining set of coordinates or+l	in such a way as to
obtain an equation of the type
n = я(£,-п) 
However, as we observed earlier, this is not always very easy because it
involves, in general, solving partial differential equations for G>r+1....o7i.
The equations (5.7) and (5.8) characterize the normal form of the equa-
tions describing (locally around a point j-=) a nonlinear system, with m
inputs and m outputs, having a (vector) relative degree {rq...........at.
x°. Observe, in particular, that if ;r; is an equilibrium point of /(.r). if
/ii(j"°) =  ~ /?7J1(.rc) — 0. and if d,— l(.r=') — ... = o77 (.rc) = 0. the normal
form thus found is defined in a neighborhood of the point (£. z/1 = (0.0). Note
also that in the equation (5.7) - the coefficients «^(C’t?) are exactly the
entries of the matrix (5.2), with ,r replaced by Ф-1 (^,7)1. and the coefficients
ar(1 the entries of a vector
again with .r replaced by Ф~1(^1?/).
We conclude the section by discussing the interpretation of the equations
(5.8), thus illustrating the multivariable version of the analysis developed
in section 4.3. This will provide an appropriate extension of the notion of
zero dynamics to a system having relative degree {п..........The idea is
always that of solving first the Problem of Zeroing the Output, i.e. to find
initial conditions and inputs consistent with the constraint that the output,
function y(t) is identically zero for all times in a neighborhood of t =0. and
then to analyze the corresponding internal dynamics. Calculations similar to
those carried out at the beginning of section 4.3 show that, if y(t) = () for all
t, then
M-r(0) = i/hi (.*(<)) =  •  = -1hi (t(H) = 0
Ы-ф)) - LfhOMt}) =	= L?-4r2(x(t}) = 0
i.e. £(t) = 0 for all t near 0.
Imposing the derivative of order r; of y;(/) to be zero, for all 1 < г <
m. constrains the inputs ui(/).......u7ll(f) to be solutions of the system of
equations
0 = y^’U) = 6Д0.;;(/)) +	niO)uj(t) for 1 < ? < rn
which, using a vector notation, can be rewritten as
226
5. Nonlinear Feedback for Multi-Input Multi-Output Systems
«надпн) = 0.
Recall now that the* matrix (5.2) is nonsingular at ,r = xc by definition. Thus
the matrix A(£. !j) is nonsingular at (7.7/) = (0,0). and the above equation
can bt1 solved for u(t) if r/(t) is close to 0.
From these considerations we deduce, in close analogy with the results
established in section 4.3. that if the output y(t) has to be 0 for all Л then
necessarily the initial state of the system must be set to a value such that
7(0) = 0, whereas 7/(0) = r/° can be chosen arbitrarily. According to the value
of if. the input must be set as
= -1.4(0, t/unrW/dtn
with 7/(0 solution of a differential equation of the form
r/(7) = Qo(0.r/(t))	(5.10)
where <Zo(7.;/) is defined as
<Zo(Ah) =7(A-7/) - P(A~ 7) И 7- z/)]"15(7- ?/)
with initial condition r/(0) = if. Note also that for each set of initial data
7 = 0 and i] = if the input thus defined is the unique input capable to keep
y(t] identically zero for all times. The dynamics of (5.10) characterize the
internal dynamics consistent with the constraint y(t) = 0. and are called the
zero dynamics of the system. Moving from these calculations to a coordinate-
free setting, the reader will have no difficulties in realizing that, in order to
yield y(t) = 0 for all times, the system must evolve on the subset
Z* = {x £ E” : Тро(т) = 0.0 < A- < 7-; - 1.1 < i < m}
under the effect of an input u(t) solution of the equation
5(.r(A)) +A(/(i))u(t) = 0.
Moreover, an easy calculation (similar to the corresponding one presented
towards the end of section 4.3). shows that, the state feedback thus obtained,
namely
tt*(r) = -A-1(j-)5(t)
is such that the vector field
ГИ = f(x) + g(x)u*(x)
is tangent to Z*. As a consequence, any trajectory of the closed loop system
starting at a point of Z* remains in Z’ (for small values of t) and the re-
striction /’'(j')lz’ of /*(x) to Z*. which is a well-defined vector field of Z*.
describes in a coordinate-free setting - the zero dynamics of the system.
5,2 Exact Linearization via Feedback
227
The Problem of Reproducing a Refewnce Output function
Ы0 = a)l(y1/?U),	....ymn{t}')
is dealt with in a similar manner. Setting
ыо =
for 1 < i < m
we find that the problem is solved if and only if
(i) the initial state of the system is such that £(0) = £/?(()), whereas r/( O') = if
can be chosen arbitrarily.
(ii) the input u(t) is set as
)	(5.id
where q(t) denotes the solution of the differential equation
.	(y[^\
q =	+ p^R{t),jf]A	+	...	)
\1/шЯ W /
(5.12)
with initial condition r/(0) = if'. For each set of initial data £(0) = ^(0)
and r/(0) = if the input thus defined is the unique input capable of keeping
y(t) = yrt(t) for all times. The (forced) dynamics of (5.12) correspond to
the dynamics describing the internal behavior of the system when input and
initial conditions have been chosen in such a way as to constrain the output
to track exactly yR(t). Thus, the relations (5.11)-(5.12) describe a system
with input yR(t}. output. u(t) and state /ft) that can be interpreted as a
realization of the inverse of the original system.
5.2 Exact Linearization via Feedback
The purpose of this section is to illustrate how a system having m inputs can
be transformed into a linear and controllable system by means of feedback
and change of coordinates in the state space, thus extending to multi-input
systems the results already discussed in section 4,2.
The appropriate multivariable version of the state feedback considered
in the corresponding single-input single-output problem is the one in which
each input u, depends on the state .r of the system and on the new reference
inputs ci, с|Л as
228
Nonlinear Feedback for Multi-Input Multi-Output Systems
a,	=	'iA-rii'j	Io. 131
j-i
when1 n,(z) and f°r 1 < '• J' < tn. are smooth functions defined on an
open subset of K" . Nott1 that the number of components of the1 new reference
input
r = colG‘[...., r,„ )
has been chosen for simplicity exactly equal to the1 number of components
of tht1 original input it.
The composition of (5.13) with the system (5.1) yields a closed loop sys-
tem having the same structure and described by equations of the form
i=]	i^i j=i
.Vi = hi(J’)	(5.14)

Using for (5.13) tilt1 more condensed expression
и = o(j-) + J(j-)c
(5.15)
in which
are an m-vector and. respectively, an in. x m matrix, the closed loop (5.14)
can be rewritten in more convenient w^y as
t 1
'r = f(J’) +9(у)н(.г) + f/(.r)3(.r)r
We also systematically assume that the matrix J(z) is nonsingular for all
.r. Accordingly, the feedback (5.13) is called a regular static state feedback.
As anticipated, the main problem dealt with in this section is that, of
using feedback and coordinates transformat ion to the purpose of changing a
nonlinear system into a linear atid controllable one. Formally, the problem in
question can be stated in the following way.
State-Space Exact Linearization Problem. Given a sc*t of vector
fields /(t) and g{ (.r)..... g,n (j-) and an initial state .rc, find (if possible), a
neighborhood U of .r°. a pair of feedback functions n(z) and 3(j-) defined
on U, a coordinates transformation z — Ф(т) also defined on U. a matrix
.4 e R'!>and a matrix В E R"x”‘. such that
5.2 Exact Linearization via Feedback
229
ЭФ
+ 9( J’lfi (x))	= A;	(a. 17)
Ъ-=Ф-Чс|
~дФ	1
—	I	=£	(o-18)
and
rank ( В AB  -  Л"-1 В ) = n .
The point of departure of our discussion will be the normal form developed
and illustrated in the previous section. Consider a nonlinear system having
(vector) relative degree {тр...г„;} at .rc and suppose that the sum r =
П + r-y + ... + rul is exactly equal to the dimension it of the state space. If
this is the case, the set of functions
Qfc(r) = Lk~lht(x) for 1 < к < r,. 1 < i < m
completely defines a local coordinates transformation at. ,r=. In the new co-
ordinates the system is described by m sets of equations of the form
G =
bite +
for 1 < f < m. and no extra, equations are involved.
Now. recall that, in a neighborhood of the point = Ф-1(.гс) the matrix
A(£) is nonsingular and therefore the equations
= ед + .-lie»
can be solved for u. As a matter of fact, the input u solving these equations
has the form of a state feedback
11 = .r’lOHfHr] .
Imposing this feedback yields a system characterized by the nt sets of
equations
230
Nonlinear Feedback for Multi-Input Multi-Output Systems
for 1 < i < m. which is clearly linear and controllable.
From these calculations, which extend in a trivial way the ones performed
at the beginning of section 4.2. we see that the conditions that the system
for some choice of output functions /у (>)..hin(x) has a (vector) relative
degree {у.....} at .r=. and that tq + r> + ...4- rni = n. imply the existence
of a coordinates transformation and a state feedback, defined locally around
A which solve the State Space Exact Linearization Problem. Nott1 that, in
terms of the original description of the system, the linearizing feedback, has
the form
и = n(r) +
with n(j-) and 3(,r) given by
o(t) = - .4-1(.r)6(.r)
with .4(.c) and 6(.r) as in (5.2) and (5.91, whereas the linearizing coordinates
are defined as
= £у-1/г((т) for 1 < A- < r(. 1 < i < hi .
We show now that the conditions in question are also necessary.
Lemma 5.2.1. Suppose the. matrix g(x~ ) has rank in. Then, the State Space
Exact Linearization Problem is solvable if and only if there exist a neighbor-
hood U of xc and m real-valued functions /у (j),....(т), defined on I',
such that, the system
j = JW + yU)u
У = h(x')
has some (vector) relative degree {zy...r„f} at x° and ix+r-^-^.  - + rni ~ n.
Proof. We need only to show the necessity. We follow very closely the proof
of Lemma 4.2.1. First of all. it is shown that the integers r,. 1 < i < m. are
invariant under a regular feedback. Recall that, for any a(>)
L^^-Jidx} — L^hdx] for all 0 < A- < rt - 1. 1 < i < m .
From this, one deduces that
Li7,?);LyhSx} =	L9;!Ll)hl(x}3Sj(x') = 0
«-1
for all 0 < k < r, — 1, for all 1 < i.j <m, and all x near A Moreover
b„J)inL-+-;(1A(A)) =
(L^L^^h^) ---
and thus, if the matrix J(tc) is nonsingular,
5.2 Exact Linearization via Feedback
231
(	I/-go (-i" ) ) / (0 • 0).
This completes the proof of the fact that the integers r,, 1 < i < m. are
invariant under regular feedback.
We return now to the proof of the necessity. Since, by assumption, the
matrix </(jc) has rank m, from (5.18) wo deduct1 that any В satisfying this
relation has also rank m. Therefore, without loss of generality, as in the proof
of Lemma 4.2.1. we ina.v assume that the matrices .4 and В considered in the
statement of the Problem have the form (Brunowsky canonical form)
.4 = diag(.4i.......4)fi) В — diagf^...........b,„ 1
where .4; is the k(- x k, matrix
0	1	0	0\
0	0	1	  •	o'
о	0	0	1
о	0	0	—	0 /
and b, is the к, x 1 vector
б,	= col(0...0.1} .
Now. decompose the vector 2 = Ф(и-) in the form
= = col(?...............................Д")
and set
1Л = ( 1 0	0)c'	(5.19)
with dini(c’) = кр. for 1 < 1 < m. A straightforward calculation shows that
the linear system
2 = .4; + Be
with output functions defined as in (5.19) has vector relative degree {k1t ....
Km} and Ki -+• h--2 + ... + кт = n. Thus, since a vector relative degree is
invariant under regular feedback and coordinates transformation, the proof
is completed. <
Remark 5,2.1, Note that the condition that the matrix gi-B) has rank rn is
indeed necessary for the existence of any sec of m "output'’ functions such
that the system has some relative degree at j‘° because, as already observed
in the proof of Proposition 5Д.2. this matrix is a factor of the* matrix (5.2).
If the matrix g(r} has a rank p < m, but this rank is constant for all j- near
r°. then the State Space Exact Linearization Problem is solvable1 if and only
if there exist p functions /11 (j-).Лр(-Н defined in a neighborhood f ’ of .
such that the system has some (vector) relative degree {ty ,.... zy;} at j-° (see
Remark 5.1.3) and ty -r m + • -  +— n. As a matter of fact, if the matrix
232
5. Nonlinear Feedback for Multi-Input Multi-Output Systems
д(т) has constant rank p < m. it is possible to find a nonsingular matrix J{ j-)
such that
= (У(.г) 0)
with g'(x) consisting of p columns and having rank p. Thus, a preliminary
feedback of the form
i ( u' \
U =	( n" J
changes the original system into the system
j = /И + g’Uh'
which satisfies the assumptions of Lemina 5.2.1. for m = p. <
On tin1 basis of this result, we proceed now to describe how, under suitable
conditions on the vector fields /(.ij.pj (j-)..... gm (j-), it is possible to find m
functions (t). hz(jr). .., h,„ (.r) satisfying the requirements of the previous
Lemma. Extending the solution of the corresponding problem dealt with in
the previous Chapter (Lemma 4.2.2). the conditions in question will be stated
in terms of properties of suitable distributions spanned by vector fields of the
form
(/i(j').  •. ,£рф').т//,91И..adfgm(x).......adr}~1 gfix).... g,n(x} ,
More precisely, having set
Go = span{^!.........
Gj = span{tyi--------gm-ddfgl.......adf9m}
Gr = span{«dpQj : 0 < k < i, 1 < j < m]
for i — 0.1....ii — 1. the following result will be proved.
Lemma 5.2.2. Suppose the matrix g(x°) has rank in. Then, there exist a
neighborhood Lr of ,r° and m real-valued functions Ai(x). A^x)....,
defined on U. such that the system
j’ = /W + g(*)u
У = A(j-)
has some (vector) relative degree {zq....rm} at t°. with
T'l + 1'2 + -  . + f'rn = П
if and only if:
(i)	for each 0 < i < n — 1, the distribution Gt has constant dimension near
(ii)	the distribution Gn_i has dimension n:
(iii)	for each 0 < i < n — 2. the distribution Gj is involutive.
5.2 Exact Linearization via Feedback 233
Note that, in view of this result and of the previous discussion, we can
state the conditions for rhe solvability of the State Space Exact Linearization
Problem in the following way.
Theorem 5.2.3. Suppose the matrix g(xc) has rank nt. Then, the State
Space Exact Linearization Problem is solvable if and only if
(i)	for each 0 < i < n — 1, the distribution Gi has constant dimension near
„С -
£ r
(ii)	the distribution Glt-y has dimension n:
(iii)	for each 0 < i < n — 2. the distribution Gi is involutive.
We provide now a proof of this result, and in doing this - we also indicate
how the functions Ai(r).., A?„(z) can be constructed.
Proof. The proof that the conditions in question are sufficient is conceptually
similar to that of the corresponding result of Chapter 1 (Lemma 4.2.2). but
- unfortunately not as straightforward as that one. The main issue is to
find solutions Ai(j-). .... A,„(,r) of equations of the form
LPj L*A, (.г) = 0 for all 0 < к < г, - 2.1 < j < in (5.20)
and to impose, as a nontriviality condition, the nonsingularity of the matrix
(5.2). In addition, one has also to make sure that zu — m + ... + r)l( = n.
The equations (5.20) are clearly equivalent (by Lemma 4.1.2) to equations
of the form
(dA|.(.G.ad^gj;(z)) = 0 for all ,r near .r:. all () < Ar < r, - 2. all 1 < j < in
and this suggests that, for each value of z, the differential dA;(j") must be a
covector belonging to the codistribution
(span{«d*.gj : 0 < к < r, — 2. 1 < j < m})-*- = G~_2 .
On the basis of this observation, it is convenient to proceed in this way.
Recall that the distributions Go.....Gn.-i all have constant dimension near
(assumption (i)) and that in particular Gn-i has dimension n (assumption
(ii)). Thus, there exists an integer, that we shall denote by к anti which is
less than or equal to n. such that
diin(G\-..-?) < n
dim(Gh _i) = n .
Set
UZ[ = n - dim(G\_.2)
and note that, since GK_-> is involutive (assumption (iii)). by Frobenius' The-
orem there exist mL functions, that we shah denote by Aj(j-)- 1 < t < mL.
such that
234
5. Xonlinear Feedback for Multi-Input Multi-Output. System^
span^t/A, : 1 < i < он } =	
By construction, those functions arc such that
(dXM.adKfg, (.г]') = 0
for all .r around .r: and 0 <k < к - 2. 1 < j < m. 1 < i < z/q. and thi>. bv
Lemma 4.1.2 implies
LfjjLjX^x} = 0	15.21)
for all ,r around .r°. and (1 < k < к — 2, 1 < j < in. 1 < i' < zzzi. Moreover,
we claim that the пц x zzz matrix
ML(.r) = {<U)} = {L9iL^~1 X,[x)}
has rank rri i at .r5. For. suppose1 this is false. Then, using (5-21J and again
Lemma 4.1.2. we would have that
z;Ly, Lf 1Af(.r°') = УУ--1Г lCj{dXi[j-c').cidhf'
,= i
for all 1 < j < in. for some set of real numbers с;. 1 < z < ezt j - But this.
together with (5.21) implie
c;{dX,l.xc).ad^gj^r-)) = 0 for all 0 < £ < к - 1, 1 < j < m
?=i
This shows that
У2 <v/Afbr=) e (?;.
;=i
Since Gh_i has dimension n. the vector on the left-hand side must be zero,
and this in turn implies all r(‘s are zero, because rhe row vectors dXi (x'j.
dXl!n(xc) are by construction linearly independent.
The properties thus established, namely rhe equalities (5.21) and the fact
that _4’(.rc) has frill rank, show that the functions A(p’). 1 < i < zzq. are
good candidates to the solution of the problem. As a matter of fact, if the
integer zzq is exactly equal to m (note that zz/i < zzz. always, because A1 (.rc)
has m columns and rank z?z i) then these functions indeed solve rhe problem.
For. if this is the case. (5.21) imply that the matrix .4l (x) is exactly equal to
the matrix (5.2) with
Cl = /у = •   = Z‘,;; = K
Thus. the system with outputs A,(.r). 1 < i < rm has (vector) relative degree
{к. ь-,.... k}. Moreover, zq 4- zo -+-... + ?qf7 = n. because
1ПК < n
(see Proposition 5.1.2). and
5.2 Exact Linearization via Feedback
235
n —	) < nm
by construction. This shows that the functions thus found satisfy the required
conditions.
If the integer mi is strictly less than m. the set {АДг) : 1 < i < /rq]
provides only a partial solution of the problem, and then om1 has to continue
the search for an additional set of m — mi new functions. The idea is to
move one step backward, look at and try to find the new functions
among those whose differentials span G~_. 3. In order to show how these new
functions must be constructed, we need first to show a preliminary property.
We claim that
(a)	the codistribution
(?! = span{dA}.....dAni. .dLfX,......1
has dimension 2tni around
(b)	c Ge-3 
The proof of (b) is immediate. As a matter of fact, the differentials dXj(x').
1 < i G mi- which are in G~_,2 by construction, are also in G’E_;j because1
Ga--3 C Gk-2- The differentials с/ТуАД;г). 1 < i < zui. by (5,21) and Lemma
4.1.2, are such that
{dLfXdr].(idKfgj(G)} = 0
for all j around .r::. all 0 < k < к - 3. 1 < j < m. 1 < i < rri]. Therefore
these differentials are in G’y_3. To prove (a), suppose is false at ,r = u:' and
there exist numbers ct and d,. 1 < i < ng. such that
У^(с,с/А, Lrs) + dtdL jXd.x31) = 0 -
? = i
This would imply
(^(cp/AdZ) + didLf АД.г0))• adKf~’gj(xc"]) = 0
i=i
for all 1 < j < m. This, in turn, implies
m i
У с/ДЛАД.гс).аДу“1 (}j(.rc 1) = 0 .
i = l
By the linear independence of the dAd.rfs and the linear independence of
the rows of the matrix A1 G). we conclude that all d/s and r('s must be zero.
From (a) and (b). we deduce that the dimension of G’^^ is larger than or
equal to 2m i. Suppose is larger, and set
tji-2 = dim(GE_3) - 2mi
236
5. Nonlinear Feedback for Multi-Input Multi-Output Systems
Since is involutive (assumption (iii)). invoking Frobenius' Theorem we
know that is spanned by 2m1 -e exact differentials. Properties (a)
and (b) already identify 2rzq of such differentials (namely, those spanning
Г21). Thus, we can conclude that there exist, m> additional functions, that we
shall denote by A((.r). nq + 1 < i < up + i?n. such that
= f?i + span{tZA((.r).mi + 1 < i < mi + /т>} .	(5.22)
Observe that, by construction, these new functions are such that
LajLkfX,(x) = 0	(5.23)
for all x near r\ and 0 < k < к — 3. 1 < j < in, nt\ + 1 < 1 < in 1 + ni2.
Moreover, we claim that
(c)	the (ttii + пи) x m matrix
M-(.r) = {uf/z)}
wit h
f a2j(x} = (cZA, (j-). ndj-1 g}(x')) if 1 < i < niY
(z) = (dXj(x), ad'f-’gj (x)) if nil + 1 < i < mj + m2
has rank equal to nu + n?2 at x = x'~J.
For. suppose there exist real numbers сг, 1 < ? < mi and d;. inY + 1 <
/ < mi + in2, such that
m ;	m 1 4- JT12
-	c;(dA;(T°). ady~lgj(x°)') +	(z°). adhf~~gj(x=)) = 0 .
1 = 1	i = ,n 1 — 1
Then, using Lemma 4.1.2, it follows /that
mi	Oil—r^2
{^('dLfXdx0) + У2 dtdXl(x°),adKf~2gj(xo)) = 0
(=1	( = UI1+1
i.e.
m !	m 1
^CjdLfXdx3) + У2 dtdXi(xQ) e (span {ad J-2 <7//°) : 1 < j < m})^.
i-l	r=Ult + l
By construction, the vector on the left-hand side of this relation belongs also
to G^-3- and therefore we have
О?1	ГО1— P? 2
CjdL fXt (x°) +	djdXj(x ) 6 G^_2 
; = 1	г=(7ц-|-1
Г).2 Exact Linearization via Feedback
237
Since the codistribution G~_2 is spanned by dX].........dXmi. the previous
relation shows that
cp/L^A,(j'°) 4	d,dX;[J'z") G span{dA,(.r=”) : 1 < z < nii}
but. this is in contradiction with the property expressed by (3.22). unless all
Ci’s and d,‘s are zero.
The properties thus illustrated enable us to prove that, if пц + m2 b equal
to m (note that uq 4 ni2 < zn. always, because .42(.rc ) has in columns and
rank rz!] 4 m?). the set of functions {A, : 1 < i < ???} is a solution of the
problem. As a matter of fact, from (5.21). (5.23) and the nonsingularity of
the matrix A’2^). we immediately deduct' that the system has a (vector)
relative degree {rt....rm }. with
7'2 =
t’rn, = «
. = Cf?! = Л' ~ 1
Moreover. /4 4 /g 4 - •  + r,„ — n. because
71 = dini(G'A.-2) 4 nil < 70 (S' — 1) + 77? 1 = 77/1 К + 7/12 (ft “ 1) < П 
If г/н 4 n?2 is strictly less than m (and note that this includes also the
case m2 — 0). one has to continue, searching for an additional set of functions
among those whose differentials span G~_4.
After к — 1 iterations of this procedure, one lias found rn^-i functions
with the property that the differentials
' dXi(jr).dLfXj(jr). ...dLhj~2Xi(r) for 1 < i < m t
dApjq, dL rAj.r).... dLf~ri A((z) for m i — 1 < ? < r/q 4
dX, (t). dL fXt (j-j
c/A;(t)
for 777 ]-r. . 777 ^-3 4 1 < 7 < 77? 14- - .4/77 K-2
for in 14.  .4?7?л--2 41 < i < 7)114.. 4mh -]
are a basis of Gq . Since G» has dimension in by assumption, the total number
of differentials in this table is equal to
n - m = dim(G(y) = (к - l)mi 4 (к - 2}m2 4  -  4 niK-i 	(5.24)
With arguments similar to those used to prove the property (a) above, it
is possible to prove that also the кпц 4 (a- - l)m2 4 . . 4 2mK_i differentials
' dAt(T). dLfX, (j:). ... dL^~x A, ( t) for 1 < i < in j
dA;(z). dLjXjG).... dL^^XjiJ’) for 41 < i < //?] 4 rn2
dA;(.r). dL fXjlj-j.dLyX^x)
dXi(x).dL fXf(-G
for /П 14- . 47»к-з41< i < 77?. 14. - -4777k-2
fol' 777 ]4.  .4771 к - 2 4 1 < ? < 7П1 — • -4/Пк-1
238
5. Хеш linear Feedback for Multi'Input Multi-Output Systems
are independent in a neighborhood of Thus, we may deduce that zz -
(ктi + (к — l)m.9 4- ... 4- 2n?K_i) >0. If this inequality is strict, set
7HK = n — (K7??1 + (h‘ — 1 )zzt2 + 2z7th-_i )
and note that, by (5.24).
zzii + ... + 7ZiK = zzz .
Clearly, there exist functions A,(.r). 7/11 + ... 4- 7nK-i +1 < i < m. such
that the differentials
dA;f.r) for 7Pi 4- • •. 4- m^-i 4- 1 < i < m
together with those of the peer-ions table form a set of exactly n independent
differentials (in a neighborhood of r). With arguments similar to those list'd
to prove property (c) above, it is possible to prove that the system, with
outputs АД.г), 1 < i < m. has relative degree {zq..........r„(} at ,r°. with
rt = a- for 1 < i < zzii
r( — K‘ — 1 for 772 ] + 1 < i < 77? 1 -+- ?Zi2
(5.23)
= ‘2	for 77?! + . . . 4- 77!k_2 + 1 < ? <	772[ 4- ... 4- ШН-1
< Tj = 1	for 7П1 +  -  4- I??*-] + 1 < } <	772 .
Moreover. rY	4-	r2 4- ... 4- rm — n. and thus the	proof of the sufficiency	is
complete. The	proof of the necessity is quite straightforward and is	left,	as
an exercise, to the reader. <
Remark 5.2.2, It may be interesting to observe that the conditions stated
in Theorem 5.2.3. in the case of a single-input system, reduce exactly to
those described in Theorem 4.2.3. For, if this is the case, i.e. if m = 1. the
distribution Gt reduces to
Gt = span{5.	..., adjg] .
The condition (ii) above, i.e. dim(Gri-i) = n, implies that diin(G() — i 4- 1.
i.e. the condition (i). This being the case, the involutivity of Gn_2 implies
(see Remark 4.2.8) also that of Go, .... Gn_3- <
Remark 5.2.3. Note that if 7??2 = 0. no useful function can be found at the
second iteration of the procedure and one has to proceed directly with the
third iteration. If this is the case, then it is clear that the condition '"GK-3
is involutive" (which is part of the conditions (iii)) is superfluous because,
as shown in the proof, it is in fact implied by the involutivity of GK-i. The
same consideration is of course true for any GK-i such that nii-i = 0. Thus,
strictly speaking, the requirement (iii) is in some sense redundant, because the
involutivity of some distributions of the sequence Go. .... G,(-2 might imply
that of the others. On the other hand, the way the condition was presented is
much simpler, in that it does not require identifying what distributions must
necessarily be involutive in order to let the procedure go. <
5.2 Exact Linearization via Feeriback
239
Remark 5.2-4. The arguments illustrated in the proof enable us to identify
the numbers rq.....rfJi directly in terms of the dimensions of the distributions
{70t ..., (well-defined by assumption). For. it suffices to use (5.2-5) and
to keep in mind that
n?;-] ~ n — dim(GK-,) — (i — 1 )/ni - (J — 2)т? —  •  — 2//i,<
Remark 5-2.5. Note that if the system were linear, conditions (i) and (iii)
of Lemma 5.2.2 would be automatically satisfied and condition (ii) would
reduce to the condition that the system is controllable. In this case the previ-
ous construction will end up with a set of linear functions Af (.r). 1 < i< m.
Using these functions in the expressions of the linearizing feedback and of
the linearizing coordinates would produce a linear feedback and a linear co-
ordinates change that brings the system to its Brunowsky canonical form.
<i
We illustrate now in a simple example how a system satisfying the con-
ditions of Theorem 5.2.3 can be transformed into a linear and controllable
system via feedback and coordinates change.
Example 5.2.6. Consider the system
In this system the distribution Go = span{r/i, cy-j} has dimension 2 = m in a
neighborhood of .rc = 0. Moreover, since
[Я\>.Я2}^2 = 0
using Remark 1.3.5 we see that the distribution in question is also involutive.
Consider now
G! = spaiij^i.c/j.rtdyt/brtd/^}
where
ad/t?i(.H ~
m//.9-,(.r) =
This distribution has maximal dimension 4 at = 0. Therefore its dimension
is constant near j3. Moreover, since
[gi.adj(}A]{.r) = [gi.adf(}-2](.r} ~ [g^ad/g^r) - [g-> < ad f g-2](.r) = 0
240
5, Nonlinear Feedback for Multi-Input Multi-Output Systems
and
[adfgl.adfg-2](.r) = tanfzx - z-,ff/i(z)
this distribution is also involutive.
Finally, similar calculations show that the distribution
G> = span{ffi. g-2.adjgi, adjg2.adjgi. adjg->}
has maximal dimension 5 at zc = Г). and therefore at each z in a neighborhood
of zc = 0.
Since by definition G,_i C G, for any i > 1. and G2 has a dimension which
is equal to the dimension n of the state space, we see that G2 = G2 = GY
and G>.G;i are (trivially) involutive. The system satisfies the hypotheses of
Theorem 5.2.3.
In order to solve the State Space Exact Linearization Problem, we lune io
construct two functions A] (z| and A2(.r) according to the procedure indicated
in the proof of Lemma 5.2.2. Since in this case к = 3. one has to consider
first the1 codistribution Gy. This codistribution has dimension 1. Therefore,
there exists a real-valued function A^z) such that
span{dA!} = Gy .
As a matter of fact, it is not difficult to check that the function
Ai(.r) = Zi -z5
does the job. Then, we know from the proof of Lemma 5.2.2 that the function
LfX\(z) has a differential which is linearly independent from that of Aj(z)
and that
spau{dAi (z). Ai(.r)} G Gy .
The left-hand side of this relation lyis dimension 2. whereas the right-hand
side has dimension 3. Therefore, there exists another real-valued function
A-j(z) such that
span{dAi (.r j.dLfXi (z). dX2 (z)} = Gy .
Since
dA, (z)	=	(1 () 0	0	-1 )
d£j-A](z)	=	d.i’2 — I 0	1	0 0	0)
a function Aj(z) whose differential is linearly independent of dAjf.r) and
JT/A^z) and is annihilated by tin* vectors of Go is indeed the function
AMz) - Z] .
At this point, the procedure illustrated in the proof of Lemina 5.2.2 is
terminated. By construction, the two functions Ai(z) and A2(.r) are such
that
5.3 bioninreracting Control
241
L tjl A i (л* 1 — £^2 A i (z) — L Lf Ai (z) —	— d
£P1A^(j-) = £92A2(z) = U
and, moreover, the matrix
^si (x)
L f A? (x)
£уг£уА1(т) \
£31£/>2W/
is nonsingular at .r = 0. Thus, the system in question, with dummy outputs
yx = Ai(j^) and у2 = Al>(j‘) will have relative degree {tq. r2} = {3,2}. with
П + r2 = 5 = n. <
5.3 Noninteracting Control
In a multivariable system, in addition to standard synthesis problems like
exact linearization (already examined in the previous section), asymptotic
stabilization, disturbance decoupling, output tracking, one may wish to use
feedback in order to reduce the system, at least from an input-output point of
view, to an aggregate of independent single-input single-output channels. This
problem, known as the problem of noninteracting control, will be discussed
in the present section. For convenience, we start from a formal definition.
We suppose that the point .C about which the problem is to be solved is an
equilibrium point of the vector field /(z) (i.e. /(z-c) = 0), that Ь,(т°) = 0
for all 1 < i < m. and that the feedback (5.13) preserves this equilibrium.
Moreover, without loss of generality, we assume .rc = 0.
Noninteracting Control Problem. Given a nonlinear system of the
form	, i
-r =
г=1
У1 = filCr)
Ут =
find a regular static state feedback control law
u, = аг(-г) +
5=1
defined in a neighborhood U of .r = 0, with a;(0) — 0. such that the closed
loop system
242
5, Nonlinear Feedback for Multi-Input Multi-Output Systems
m	ni	?n
i = f(-r) + 52<7(Иа,(.г) +
i=i	j=i ;=i
У1 - /*1И
У in	— hm(j')
has a vector relative degree at the equilibrium point j: = 0 and. for each
1 < i < m, the output t/( is affected only by the corresponding input c( and
not by Cj. if j / о
Remark 5.3.1. The property that, the output yr is not affected by the input
i’j if i 7^ j- can be characterized in any one of the alternative ways illustrated
in section 3.3. Thus, in particular, the output y, of system
j- = fU) + fofoofo) 4-
У =
is not affected by the input ip if and only if. for all r > 0 and for any choice
of the vector fields Ti,.... тг in the set
{f +	....
the identities
^(5Jjj Cri ' ' '	) — 0
hold for all .c. or what is the same - the identities
= 0
(dh,. [тг. [.... [p.fo]]]>(T) = 0
hold for all .c.	
The property that the closed loop system has some vector relative degree
at the equilibrium point r = 0 takes care of avoiding tritual solutions, namely
solutions in which in the closed loop system some output is not affected
by any input at all. <i
The main result about the Noninteracting Control Problem is that this
problem is solvable if and only if the system has some vector relative degree,
i.e. belongs to the special class of multivariable systems introduced in section
5.1. The sufficiency is discussed first.
Suppose that the system has been given the normal form illustrated in
section 5.1 and suppose the following feedback law is imposed
=	+ .-r1(CO
(5.26)
5.3 2soninteracting Control
243
An immediate calculation shows that the imposition of this feedback
yields a system characterized by the m sets of equations
C,
C'
£
for 1 < i < m. together with an additional set of the form
У =	- Р(^У)А~1 (<. y)b(^.y) + p(Ch)-4-1	.
The structure of these equations (which correspond to the block diagram of
Fig.5.1) shows that the noninteraction requirement has been achieved. As a
matter of fact, the input n controls only the output yA, throughout, a chain
of ri integrators, the input to controls only rhe output y-2. throughout a chain
of Г2 integrators, etc. If r = r\ + r2 + ... 4- r,n is not equal to zn in the closed
loop system an unobservable part is present, which behaves like a "sink"',
namely is affected by all inputs and all the states, but has no effect on the
outputs. If. on the other hand, r ~ n. no "sink"' is present and the closed
loop system consists - as already shown in the previous section only of m
chains of rt integrators each. We observe also that in either cases the input-
output behavior of the closed loop thus obtained is that of a linear system,
characterized by a transfer function matrix of the form
Although the use of the normal form is very helpful in understanding
how the noninteracting control problem can be solved, it is clear that the
achievement of an input-output non inter active behavior is independent of
the coordinates used in the state space description. Thus, we deduce that a
feedback of the form
it — o(t) + J(t)c	(5.27)
with ft(x) and 3(t) given by
o(z) = -.4-1(t)5(z) J(t) = .4-I(j?)	(5.28)
with A(z) and b(x) as in (5.2) and (5.9) (which is the expression of (5.26)
in the original state space coordinates) solves the Noninteracting Control
Problem. We shall refer to this as to the standard iioninteractwe feedback.
It is clear from the previous discussion that for any system in which the
matrix .4(x) is nonsingular at .r =- 0, i.e. any system which has a (vector)
244
5. Nonlinear Feedback for Multi-Input Multi-Output Systems
Fig. 5.1.
relative degree at this point, the noninteracting control problem can be solved,
by means of a static state feedback which is defined for all .r in a neighborhood
of the point .r — 0. It will be shown now that the existence of a (vector)
relative degree it is also a necessary condition for the existence of solutions
of the problem in question.
Proposition 5.3.1. Consider a multivariable nonlinear system with m in-
puts and m outputs
? = /U) +
i=i
У1 =
•  f
Ут =	.
The Noninteracting Control Problem is solvable if and only if the matrix -4(0)
is nonsingular, i.e. if the system has a some vector relative degree {гх.rw}
at r = 0.
Proof. Suppose, for some integer fj.
= 0
for all 1 < j < rn. for all A* <	- 1, and for all .r in some neighborhood of
x = 0, and
(	•   LgmLrf~lh1(x))
is not identically zero in some neighborhood of z = 0. Then (see Lemma
•5.2.1) also
5.3 Noninferacting Control 245
for all 1 < j < m, for all k < rt - 1, and for all r in a neighborhood of
x = 0. Thus, if the Problem of Noninteracting Control has been solved by
some feedback и = а(т) + 3(z)i’ and the corresponding closed loop system
has (vector) relative degree {fi,.... rm }. necessarily r, > rt (which, by the
way, shows that each of the r/s is necessarily finite).
Suppose fi is strictly larger than zy. Then,
0 = (Lf+gah((x) -  • L(g3) n Lf‘+gaht(x))
-	L^L^hM^x).
It is easy to see that this implies rank(3(0)) < m. If fact, if rank(.3(0)) = zn.
then rank(,3(z)) = zn for all z in a neighborhood of z = 0 and this contradicts
the hypothesis that
(LgiL^hM ••• L^L^hds))
is not identically zero in some neighborhood of .r = 0. Thus rank)3(0)) < m
and, therefore, also rank(^(0),3(0)) < m.
Now (recall the proof of Proposition 5.1.2) that if the closed loop system
has vector relative degree {г1я..., fm }, the matrix
-4(z) =	 - • I <7(z)3(z)
f+gcM1) /
is nonsingular at z - 0. But this contradicts the fact that гапк(</(0)3(0)) < zn
and therefore it is concluded that ri = rt.
This being the case, we obtain
-4(z) = .4(z).3(z)
from which it is deduced that -4(z) and 3(z) are nonsingular at z = 0. <
In view of its importance in connection with the solution of the noninter-
acting control problem, the matrix _4(z) is sometimes called the decoupling
matrix of the system (in this case "decoupling1' means "'separation of the in-
dividual input-output channels1'). From the previous Proposition we see that
the class of systems having a vector relative degree at the point x = 0 and the
class of systems in which the nonintcracting control problem can be solved,
locally around x = 0, by means of static state feedback actually coincide.
In other words, we may say that the special class of multivariable nonlinear
system considered so far in this Chapter is exactly the class of those systems
that can be made noninteractive via static state feedback.
Remark 5.3.2. The previous analysis can easily be extended to deal with
systems having a number m of inputs which is larger than the number p
of outputs. In this case, the Noninteracting Control Problem is the one of
246	5. Nonlinear Feedback for Multi-Input Multi-Output Systems
finding a regular static state feedback and a partition of the input vector г
into p disjoint sets
c = col(t’i. Г-2..Cp)
such that, in the corresponding closed loop, each output channel ?/,. 1 < / <
p, is affected only by the corresponding set of inputs r, (and not by c;. if
j i). A rather straightforward extension of Proposition 5.3.1 shows that
the problem in question is solvable if and only if the matrix .4(.r:) has rank
p (i.e. equal to the number of output channels).
The proof of the necessity is almost identical to that of Proposition 5.3.1.
As far as the sufficiency is concerned, the proof is based on an appropriate
extension of the normal form. The reader will have no difficulty in under-
standing that, in the case of systems with m > p inputs and p outputs, a
normal form similar to the one utilized so far can be developed under the
assumption that the matrix .4(7*) has rank p. because under this assumption
the choice of local coordinates indicated in the Proposition 5.1.2 is still valid
(see Remark 5.1.3). The normal form thus deduced has a structure which is
identical to that of the one discussed in section 5.1. with the only formal dif-
ference that m > p input components are present in the appropriate places.
If .4(j-) has rank p. the equations
\£^h/((.r) -rp/
can be solved for u, for any p-tuplet n...cp. The imposition of the cor-
responding feedback yields a closed loop system in which, for 1 < i < p.
affects only yi. <
We conclude this section with some considerations about the stability of a
system which has been made noninter active by means of static state feedback.
From the block diagram of Fig.5.1. we see that the internal structure of the
non hit er active closed loop obtained using the feedback (5.28) consists of m
chains of r, integrators each, all feeding the? (unobservable) subsystem
0 = <?(£•'/) - p(^.q)A~]	+p(^r/)A^1(Cb);' 
Imposing on this system an additional feedback of the form

that, in the original coordinates, reads as
G = -c^h^x) - c\Lfhi(x) - ... - c(.._1T^“1h;(.r) + c; (5.29)
for 1 < i< m, yields an overall closed loop which is still noninteractive. and
characterized by equations of the form
5.3 Xoiiinteracting Control 247
for 1 < i < m. and

in which p) and /)(£. r?) are suitable functions.
In particular, the system thus obtained has a linear input-output behavior.
characterized by the diagonal transfer functions matrix
with
d((.s) = Cq + ('is' 4-... + _j-s*‘ 1 + .C- .	(5.30)
As far as the internal asymptotic stability is concerned, we see from the
previous equations that the system has essentially the same structure as the
one we obtained, via a similar feedback, in section 4.4. Thus, using the results
of section B.2 we ('an conclude that if the zero dynamics of the system are
asymptotically stable, and the polynomials (5.30) have all roots in the left-
half complex plane, the system in question is locally asymptotically stable at
(^.7/) = (0.0).
Remark 5.3.3. It is apparent from the previous discussion that the asymp-
totic stability of the zero dynamics is a sufficient condition to achieve non-
interacting control with internal asymptotic stability. However, it must be
stressed that such a condition is not in general a necessary one. As a matter
of fact, there may exist systems whose zero dynamics are not asymptotically
stable (or even unstable) in which the achievement of noninteract ive control
with internal asymptotic stability is still possible. The precise characteri-
zation of those nonlinear systems that (‘an be rendered nonint er active and
simultaneously internally stable by means of static state feedback requires a
rather more sophisticated analysis, that will be pursued in the next Chapters,
•a
Remark 5.3.4- The previous analysis considers only the property of (internal)
asymptotic stability, i.e. the asymptotic behavior (of the closed loop system)
in the particular situation in which all the reference inputs t’i,.... are set
to zero. In general, the equations describing the closed loop system have the
form
248
5. Nonlinear Feedback for Multi-Input Multi-Output Systems
rn
ЭД = /(z) +
i=l
Recall that, by hypothesis, ,r° = 0 is an equilibrium of the system, i.e. that
/(0) = 0. and that. ft(0) = 0. If the zero dynamics of the system are asymp-
totically stable, and the feedback has been chosen as the composition of the
standard noninteractive feedback (5.28) with the stabilizing feedback (5.29)
(of course, with the polynomials (5.30) having all roots in the left-half com-
plex plane), the vector field f(x) has an asymptotically stable equilibrium at.
.r = 0. Thus, using the results illustrated in section B.2 it is possible, as in
section 4.4, to conclude that for each - there exist, d and К such that
|| < <5. |t’,(t)| < К for all t > 0.1 < i < m
imply ||r(t)|| < s for all t > 0. <
In concluding this section, we observe that for a multivariable system hav-
ing a (vector) relative degree at a point xc of equilibrium, it is possible to ad-
dress problems like asymptotic output reproduction, disturbance decoupling,
and model matching, in much the same way as in the case of single-input
single-output systems. The corresponding procedures are straightforward ex-
tensions of those already illustrated, and their derivation can be left, as an
exercise, to the reader. The following statement shows, for instance, how the
problem of disturbance decoupling can be addressed.
Proposition 5.3.2. Consider the system
T = f(x) + ^g^xjui + p(x)u:
i=l
.У1 - /h(-f)
Ут = hTtl(x) .
Suppose this system (mewed as a system with input	and output
У1- • • • чУт) has a (vector) relative degree {rq...., rm } (say, at x = 0j. There
exists a feedback of the form a — a(r) + 3(т)г which renders the output у
independent of the disturbance w if and only if
LpL^hi(x) — 0 for all 0 < A: < r, — 1, 1 < i < m .
There exists a feedback of the form и = o(z) + 5(x)r + q(z)u> which renders
the output у independent of the disturbance w if and only if
LpLfhi(x) = 0 for all 0 < к < zq — 2. 1 < i < m .
5.4 Achieving Relative Degree via Dynamic Extension
249
5.4 Achieving Relative Degree via Dynamic Extension
The analysis developed in the previous sections has shown that a nonlinear
system of the form (5.1) which has a (vector) relative degree at the point
xQ lends itself to the implementation of some relevant control strategies. For
instance; this system can be rendered noninteractive (from an input-output
point of view) via state feedback. If, in addition, the equality rL +.. . + rm = n
is satisfied, this system can be changed into a fully linear and controllable
system by means of feedback and coordinates transformation. Note that the
latter condition, in view of a property illustrated in section 5.1, is exactly the
condition under which the manifold Z\ on which the zero dynamics of the
system is defined, degenerates to the single point r°. In this case, the system
is said to have a trivial zero dynamics.
The purpose of this section is to show that, under certain assumptions, it
is possible to modify - by means of control laws which are more general than
those considered so far - a system which does not have a vector relative degree
into a new system which does have a relative degree. Of course, this cannot
be achieved by means of static state feedback of the form (5.15) because, as
shown for instance in the proof of Lemma 5.2.1. the property for a system
- of having relative degree is invariant under this type of feedback. We will
rather use a feedback structure which incorporates an additional set of state
variables, namely a dynamic state feedback. As anticipated in section 4.5
(see in particular (4.46)). this type of feedback is modeled by equations of
the form
и = q(x,<) +J(t.C)v
/ I г/ m	(5.31)
г) —	() + 5(z, Qu .
The reason why the addition of auxiliary state variables majr be helpful
in achieving relative degree can be easily motivated with the aid of a simple
example.
Example 5.^.1. Consider a system of the form (5.1). with 2 inputs and 2
outputs, defined on R4, with
hi(z) =
Zto(x) — j*9 .
This system has no relative degree, because the matrix (5.2), which in
this case has the form
has rank 1 for all x.
250	5. Nonlinear Feedback for NIulti-Input Multi-Out put Svstems
The reason why this system has no relative degree is that the lowest
derivatives of t/i and y-2 which are affected by the input (in this case
and y^1). are affected both by щ and none by Thus, in order to obtain
a relative degree, one could try to render y^11 and yV' independent of ?q.
that is to "delay" the appearance of Ui to higher order derivatives of yi and
y2 . and hope that when this happens also shows up. In order to render
y^! and уV 1 independent of the input, in particular of its first component
tzi. it suffices to set tp equal to the output of another (auxiliary) dynamical
system, with some internal state and driven by a new reference input cj.
The simplest way in which this result can be achieved is to set ui equal to
the output of an "integrator" driven by ty. i.e. to set
ui = C
(see Fig. 5.2).
< - “i
Fig. 5.2,
For consistency of notation it is also set. for the second input channel
which has been left unchanged. ’
u--> = IV.
The composed system thus obtained is described by equations of the form
.r - ft?) + yi(.r)t’i +y2Mi'2
у = h(x)
with I* = and

/(/.() =
•t‘4 +
ф .Гд
=

Straightforward calculations show that now
5.4 Achieving Relative Degree via Dynamic Extension
251
LyLjh(j\ <)
0 Oh
0 0 J
1
•га 1)
i.e. that the system in question hus (vector) relative degree {'2.2}. <
Having explained why the addition of auxiliary state variables, in par-
ticular the addition of integintions on certain input channels, is helpful in
obtaining a relative degree, we describe now a recursive procedure which es-
sentially identifies the channels on which the integrations must be added and
the number of integrators needed in order to achieve the desired goal, that is
some vector relative degree. As we shall see. the procedure in question incor-
porates also a feedback-type modification of the original system and thus, the
entire control structure that will be determined is that of a dynamic feedback
of the form (5.31). In what follows, we consider, as done throughout most of
these notes, a multivariable system with the same number m of inputs and
outputs channels. Tin1 symbol r, is still used to denote the largest integer
such that
£Pi fjh,(.r) = 0
for all k < fi - 1. all 1 < j' < in, all ,r near .r: but. of course, it is not
necessarily assumed that the system has relative degree {ri rm} (i.e.
that the matrix (5.2) is nonsingular).
Dynamic extension algorithm. Consider the matrix ,4(.r) defined by
(5.2) and suppose the rank of -4(>) is constant on a neighborhood of .r0. If
the rank in question is equal to m. the system has a (vector) relative degree
at js. Suppose this is not the case, and let m(J1), 1 <i < in. denote the /-th
row of .4(.r). Without loss of generality (after possibly having rearranged the
order of the output channels), it is possible to find an integer 1 < p < m. a
set of p - 1 smooth functions cjj1)......rp_i(.r) (defined in a neighborhood
of and two integers such that Gju) is not identically zero.
p-1
UyJ.r) =	(> (.Z‘)u( (j )
(=1
and
u;ilJ,,(-C 1 = Lgj. L’j" lhh (.rc) ^ 0 .
Define the dynamic feedback
и j = i j for j
1
“D =	+	V Г^,:;(-Г)г7)	(5.32)
ucj. m j	J = 1
i ~	
252	5. Nonlinear Feedback for Multi-In put Multi-Output Systems
in which p(x) and q(x) are arbitrary functions satisfying p(xrj) = 0 and
7(^°) - 1-
The composition of (5.1) and (5.32) defines a new system
Ш	7 V	m
= f(x) + 9j(-r)vj + '9j0 \ (рИ +	- 52 a‘oj(z)rj)
утт	°Г0Лэ1-Г1	“7
J#JQ
£	= rju
Ul = М-Г)
IJm — hTTl{x) .
(5.33)
Replace system (5.1) by system (5.33) and iterate the procedure.
Remark 5-4-2. Note that, since pU’c) = 0, the point (r.£) = (.r°,0) is an
equilibrium of (5.33). <
Remark 5-4-3. Note that the state £ of the dynamic extension (5.33) satisfies
£ = ^7) (Xa’;o>uJ =	~рИ) 	(5.34)
This property will be exploited later in the proof of Proposition 5.4.1. a
Remark 5-4-4- The two functions p(x) and q(x) considered in the definition
of uJO may sometimes help to obtain simpler expressions in the composite
system (5.33). In particular, observe that, by definition, the r,0-th derivative
of Угг. (0 can be expressed in the form
У^’ =	.
J = 1
Thus, choosing
p(r) = -Lrf'Qhio{x)
and q(x) = 1 in the law (5.32) yields, for the rl0'th derivative of t/i0(t), the
simple expression
The latter in turn yields
and this shows that, in the composed system (5.33). the lowest derivative
of y(0(t) which explicitly depends on the input is precisely the (rIn 4- l)-t.h
derivative. Accordingly, in. the z’o-th row of the matrix (5.2) of (5.33), all
entries are zero but the jo-th one. which is equal to 1. <
г
5.4 Achieving Relative Degree via Dynamic Extension
253
The purpose of the Dynamic Extension Algorithm is to construct, start-
ing with a system in which the rank of the matrix (5.2) is not equal to m.
an extended (and feedback-modified) system in which the rank of the corre-
sponding matrix is possibly larger, and therefore - possibly after a number
of iterations equal to m. In order to figure out under what conditions this
will be the case, a detailed discussion of some interesting features of this
algorithm is necessary.
First of all. it will be shown that the dynamic extension constructed by
means of this algorithm is. in some sense, always "intrinsically built-in” in any
dynamic extension yielding a system having vector relative degree. In order
to explain this important property, suppose without loss of generality -
that = 0. consider a dynamic feedback law of the form (5.31). let n denote
the dimension of its state vector ( and suppose n(0, 0) = 0 and ".(0.0) = 0.
In this case, the point (т.у) = (0,0) is an equilibrium point of the closed
loop system
•	r = /(-r) + g(x)a (jt-O + з(т):3(т.<)г
<	= " (.r.0 + <5(.r.<)t‘	(5.35)
<	/ = h(.r) .
The dynamic feedback (5.31) is said to be a regularizing dynamic extension
for (5.1) if the composite svsteni (5.35) has a vector relative degree at (x. () =
(0,0).
Remark 5-4-5. Note that, if dim(() = 0. a feedback of the form (5.31) reduces
to a static state feedback
и = a(x) + 3(x)v .
If the corresponding closed loop system has a vector relative degree at x = 0.
then 5(t) is necessarily nonsingular at т = 0 (see proof of Proposition 5.3.1).
Thus a regularizing dynamic extension of trivial dimension is necessarily a
regular static feedback. <a
Proposition 5.4.1. Suppose the Dynamic Extension Algorithm has been it-
erated к times. Let
и = H(r,& + K(x.&v
a.36
e = г(л£) + с(т.£Н’;
with £ E . denote the composition of the к feedback laws of the form. (5.32)
constructed at each stage of the algorithm. If there exists any regularizing
dynamic extension
и = a(x. C) + 3{x. C)u
< = 7(t,() +d(z,<)e
254
5, Nonlinear Feedback for Multi-Input Multi-Output Systems
for (5.1). then necessarily к < a and there is a local coordinates transforma-
tion in the state space of the composed system (5.35), defined in a neighbor-
hood of(x.(() = [0.0). in which the .r coordinates are left unchanged and the
< coordinates are replaced by a set of coordinates
= Ф(х. <). with $ e	.
changing (5.35) into a system of the form
r = f(-r)	+ .»»»'])
£ = P(x.£) + G(T.£)[a(z.£. c) + 3(.r.£»r]
z = S(z,c) -r dfr. G г)с
У = h(x) .
In other words, the feedback (5.31) can be seen as the composition of the
feedback (5.36). constructed by means of the Dynamic Extension Algorithm,
and of an additional regularizing dynamic extension of the. form
г = a|.r.£,c) + .J(.r.f.:)i’
6 =	c) + z)r .
Proof. Consider the composite system (5.35). Define the function
= -Ц (rt,J.r)(a(j-,0 +	C)<f) -Pl»') 
r/(.r) V	/
In justification of the notation used on the left-hand side, we prove first that
the right-hand side of this expression is independent of the variable r. To this
end. recall that, by construction,
y\'"'} = Lrfht{x) + a/» (o(z. <) + 3(x. <);)
and that, by hypothesis, the composite system (5.35) has some vector relative
degree ».......at »<) = (0; 0). Thus
r; = r,
if and only if aAx)3(x. () is not identically zero.
Suppose, by contradiction, that tq»^) depends on r. Then the function
m,,(•*')'»» is not identically zero and. accordingly. ri:, = rl:y. Next, recall
that
p
Civ{x)ai.A.r} = - 22 G»n,(.r)
(with cp(x) = —1) and multiply - on the right - both sides of this inequality
by o(t.C) + 3(.r.0c. Equating the coefficient matrices of c. we obtain
5.4 Achieving Relative Degree via Dynamic Extension
255
p
= - 52 r'(() .	(5.37)
i=i
Since riD(r) is not identically zero nor is u(fl(z)J(t.0. we deduce that
the right-hand side of (5.37) is not identically zero and this, in turn, implies
that rtj(.r)d(.r. 0 is not identically zero for every i in some nonempty subset
I of {1..... to ~ 1. hi + 1..p}- Thus, for each (6 I. we have that r, — г(.
From this, we conclude that for i = t'o and I E /. пг(т).'Л(х. 0 is a row of the
decoupling matrix of the composite system (5.35). If (5.37) holds, this matrix
cannot have rank zn. i.e. a contradiction, and this proves that i'i(j‘. 0 cannot
depend on e.
We will prove now that

(0f 0) ^0.
(5.38)
Consider again the composite system (5.35) and note that, by definition
of th (x. 0.
“jC:
(p(.r) +y(T)t'i(J--<) - 52 n‘ojWwj)
j
pi
(p(t) +y(z)v1(j';0) - 52 a'oj(-r)«j) + °(^0
where
о(т,<) =	- C'i(t,0))
If (5.38) is not true, the function d(.r:0 thus defined is such that
0(0,0) = 0,
(0.0) =0.
OX
Hj(°.0) = 0.
(5.39)
Consider now the new dynamic feedback law
«j
«AO + J/T.Qf for j ± j0
(5.40)
a irijtj
v0,0+d(^.0c .
and note that, because of (5.39). the latter and the feedback law (5.31) have
the same linear approximation about (-т.(.т) = (0,0.0).
This implies that also the composite system (5.1)~(5.31) and the compos-
ite system (5.1)-(5.40) have the same linear approximation about (r.(,tj =
(0,0.0). The former, which is system (5.35). has by hypothesis a decoupling
256	5. Nonlinear Feedback for Multi-Input Multi-Output Systems
matrix which is nonsingular at (j,<) = (0.0). Thus, also the latter has a
decoupling matrix which is nonsingular at (j.<) = (0.0), because the value
at an equilibrium point of the decoupling matrix of a system depends
exclusively on the linear approximation of the system at this point. In other
words, we have shown that, if (5.38) is not true, the linear approximation
about (.r.C, r) = (0.0.0) of the composite system (5.1)-(5.40) has a nonsin-
gular decoupling matrix.
Observe now that in the feedback law (5.40). the jo-th component of a
does not depend explicitly on £. but only on j and on the other components
of u. Thus. (5.1)-(5.40) can be viewed as a system of the form
m
x = f(j') + 52
j
(resulting from the substitution of into (5.1)). which has only m — 1 input
channels, composed with a dynamic feedback of the form
Uj = tij(j-.() + 3/т.<)и for j =4 j0
C = y(T.<) +d(j,<)t- .
Taking the linear approximations of both these systems, we see that their
composition cannot have a nonsingular decoupling matrix, because the former
has only m — 1 inputs. Thus, it is concluded that. (5.38) must be true.
Since (5.38) holds, the new variable
£1 = t'l (z, c)
can replace one of the n components of Q (namely any component Q* for
which
)
In fact, the mapping
Q = Q for i / Г
has a Jacobian matrix which is nonsingular at (т. 0 = (0.0).
Set
г = col(C1;.. ..C-bG-n..........................
We will express now the closed loop system (5.35) in the new coordinates
, z). To this end, observe that, since t?i(лг. C) does not depend on o. we
have
(j = -vp-(/И +.?(^)а(лС) +5(j?),3(j.C)e)
+	+ <5(a<»)
= dJo(j7.^. г) + 3j0(t;81.3)l’ ,
5.4 Achieving Relative Degree via Dynamic Extension
257
and
1 ,r'
Ujc =-------'(p(i) +	- V afoj(jr)uj) .
«/ojot jtj	7Z7
On the other hand, as far as z and the remaining components of a are
concerned, generic expressions of the form
A	forj/j0
- =	+ (Ц.г.£1,г)г
hold- This proves the Proposition for A = 1, A simple iteration completes the
proof for arbitrary k. <
Remark 5.4.6. The hypothesis, indicated in the description of the algorithm,
required to perform one iteration of the Dynamic Extension Algorithm is
that the matrix A(t) has constant rank in a neighborhood of ,r — 0. Thus,
to assume that the algorithm can be iterated k times is to assume that the
hypothesis in question is satisfied for the original system and for all composite
systems which are subsequently built at the end of each iteration. Actually,
it may be worth remarking that, in the proof of the above Proposition, a
weaker hypothesis was requested, namely just the possibility of having
P-i
«pH =
i=i
satisfied, with c;0(j) is not identically zero and	0 for some hn.7o-<
The previous result says that, if the Dynamic Extension Algorithm can be
iterated k times and if there exists any dynamic feedback yielding a composite
system having some vector relative degree at (л, £) — (0.0), then this feedback
necessarily contains, as a subsystem, the A-dimensional dynamic feedback
constructed by means of the Dynamic Extension Algorithm (see Fig. 5.3). In
this sense, the Algorithm in question can be viewed as a sort of ‘"canonical"
way to attack the problem of constructing a regularizing dynamic extension,
if such an extension exists at all.
Proposition 5.4.1 also shows that. if the Dynamic Extension Algorithm
Succeeds, in a finite number of steps, in producing an extended system hav-
ing a vector relative degree, then the dynamic feedback generated by this
algorithm (i.e., the composition of the elementary one-dimensional dynamic
extensions of the form (5.32) defined at each step) has necessarily the least
possible dimension (compared with that of any other regularizing dynamic
extension). Typically an elementary dynamical extension of the form (5.32)
includes a number of arbitrary selections (the integers io. jo, and the functions
p(z), д(т)). These selections may indeed affect the possibility of continuing
the algorithm, in the sense that they may have an influence, at some later
258
5. Nonlinear Feedback for Multi-Input Multi-О input Systems
Fig. 5.3.
stage, on the standing hypothesis that the rank of the matrix .4(j) is con-
stant in a neighborhood of the equilibrium. However, if for different selections
the algorithm can be continued up to a final successful stage, the different
regularizing dynamic extensions thus generated have always the same dimen-
sion (which means, in particular, that for different successful selections the
algorithm always consists of the same number of iterations). In fact, using
Proposition 5.4.1. on can say than any regularizing dynamic extension gen-
erated by the algorithm is a subsystem of any other one. Thus, any two
regularizing dynamic extensions generated by the algorithm have necessar-
ily the same dimension and only differ by change of coordinates and regular
static feedback.
It is useful, in preparation to a future use of these properties in the solution
of the problem of noninteracting control with stability, to express the result
of Proposition 5.4.1 in the following way. Let S be a dynamical system of the
form (5.1) and let R be a regularizing dynamical extension for S. Let S о R
denote the composition of S and R. i.e. the system defined by
j = /(j) +
C = о(т.() + d(j.()i-
y = ад 
If Ri, a regularizing dynamical extension for S. is described by equations
of the form
и = aiOG) +
6 — G) + f’h Ci )*'i 
and if R-j. a regularizing dynamical extension for SoRb is described by
equations of the form
t’l = гпадСьС-з) + ‘32(x,Ci. C'2h'
Сз = ад-.Ci-Cd + (bU-Ci-Gk •
5.4 Achieving Relative Degree via Dynamic Extension
259
the composition R_> ° Ri of and Ri. which is described by equations of
the form
it. = oi (j-.0) + (t'-G )(cu>(.r.G-G>) + -Mr. G- ц-?)'1)
G> - ".>(>.G-G) + G(.r.G-Gb-,
is indeed a regularizing dynamical extension for S. Note also that, if S is a
system having a vector relative degree at x ~ 0. any regular static feedback
F is a regularizing dynamic extension (of trivial dimension) for S.
Let R denote the set of all regularizing dynamic extensions for S and let S
denote the subset of R. consisting of all regularizing dynamic extensions gen-
erated by the Dynamic Extension Algorithm. Then, the result of Proposition
5.4.1 can be re-expressed in the following way.
Proposition 5-4.2. Suppose £ is nonempty. Then also R is nonempty and
for each R E R there exists E G £ and a (possibly dynamic) feedback R'.
which is a regularizing dynamic extension for S о E. such that
S о R and S о E ° R' are locally diffeoniorphic.
In particular, for each pair E[ € £. E? € £, there exists a regular static
feedback F fur S о E? such that
S о Ei and S о E2 ° F are locally diffeoniorphic.
Of course, in this setup, the obvious question arises of how many times the
algorithm should be iterated before reaching a positive or negative conclusion
about the possibility of achieving relative degree via dynamic feedback. An
answer to this question ib provided by the following result.
Proposition 5.4.3. Consider a system of the form (5.1). Suppose the matrix
(5.2) has constant rank q < m. for all x in a neighborhood of x = 0. Without
loss of generality (after a change in the order of the outputs, if necessary)
suppose the first q rows of the matrix (5.2) are linearly independent at each
x in a neighborhood of x — 0. Let rx = min{rj : q -1- 1 < j < m}. If the. set
£ is nonempty, then after at most (n - m - ... - r7 — C)q itemtions of the
Dynamic Extension Algorithm a system is obtained in which the rank of the
matrix (5.2) is larger than or equal to q+\ at some point of any neighborhood
U of the origin.
Proof. A possible way to implement the first iteration of the algorithm is
indeed the following one: take, if necessary after a change in the order of
the inputs and outputs, L'(). j0) — (1.1) and choose, as suggested in Remark
5.4.4, p(x) = -£ylhi(j*) and (fix) = 1. Then, it is easy to realize that this
yields an extended system in which
Oa-i:
У i	= !'1
260
5. Nonlinear Feedback for Multi-Input Multi-Output Systems
while
iAr2' \
• • • =
where 6(t. £1) is a (in — l)-vector. -4(j™) a (m — 1) x (m — l)-niatrix and
г = col(r>,.... crrj. In particular the matrix А(т). which by construction is
such that
<r)
/ 1/rtn (!)
0
\	0
— «12(-Г)/«11 И
1
0
- П1Ш(т)/а11 (j •)
0
1
1
*
0
Л(х)
has constant rank q — 1 in a neighborhood of j = 0. If q
iteration one can choose, similarly. (?□-Jo)
system in which
1. in the second
= (2.2) and obtain an extended
Ут
У-2
while, for i > 2, y\r'] does not depend on t'i-i’2 but may depend, if q > 2.
on из.....vrn. Then, after q iterations of this kind, one obtains an extended
system in which
•У1
гч
v;(t.£1.....q + 1 < i < m .
From the inspection of the last equation it is deduced that, for each 7+1 <
1 < in. the least integer rt such that, yff’1 explicitly depends on r is strictly
larger that r(. For notational convenience, set
Г( — Г,- + 1 + .5;i .
where ,s‘(1 > 0. If. for some q + 1 < i < m. y\r,] depends explicitly on any one
of the inputs rg+i. •••.	. then the rank of the matrix (5.2) has increased, at
least by one unit, at some point of any neighborhood of the origin. Otherwise,
the extended system thus obtained (which is a system of dimension n + q) is
a system in which the matrix (5.2) has the following form

0\
0J ‘
On such a system it is possible to iterate q more times the Dynamic
Extension Algorithm, so as to obtain an extended system (which now has
dimension n + 2q) in which
5.4 Achieving Relative Degree via Dynamic Extension
261
and in which, for each 9+1 < i < m. the least integer fi such that y^
explicitly depends on v can be expressed in the form
f, = Fi + 2 + Sj-2 ,
with 8,2 > 0- Again, if y\r'} depends explicitly on any one of the inputs
Vq+i.  - -; then the rank of the matrix (5.2) has increased. Otherwise, one
can iterate the algorithm q more times.
Since, by hypothesis, the set 8 is not empty and. by Proposition 5.4.1, all
elements of £ are equivalent (up to a regular state feedback and a change of
coordinates), after a finite number of iterations this procedure must produce
an extended system in which the rank of the matrix (5.2) is at least q + 1 at
some point of any neighborhood of the origin.
Suppose that kq iterations are needed in order to obtain such a system,
which for convenience is denoted as
* =	+
У =	.
For some q + 1 < i < m and some > 0 the matrix
has rank q + 1 at some point ic. As a consequence (see Lemma 5.1.1 and
Remark 5.1.3) at each point of a neighborhood L*° of L°. the differentials of
the functions
{LJ/i/i-) : 0 < s < Tji + k. 1 < j < qj = ?}
are linearly independent. The dimension of the state variable r is equal to
n + kq, whereas the number of these functions is equal to (7*1 + ...+ rg + rj +
k(q +1). The linear independence of the differentials of these functions implies
(t‘l + ... + г+ гt) + k(q + 1) + 71 + kq
i.e.
fc < n. - (и + ... + rq + rj
and this completes the proof. <
262
5. Nonlinear Feedback for Multi-Input Multi-Output Systems
Of course, if there exists a feedback of the form 15.31) which changes the
original system (5.1) into a new system (of the form (5.351) having some
vector relative degree at (.r.<) — (0.0) of the extended state space, then an
additional static feedback (determined on the basis of the1 results illustrated
in section 5.3. e.g. a standard noninteract ive feedback) of the form
r = n(,r. C) 3(.r. <)r
can make each output y, depending only on the /-th component of the new
reference input г and not on the other ones. In other words, the original
system (5.1) can be rendered nonmtemetive. by means of dynamic state
feedback.
Another property of the systems having a vector relative degree is that,
if
/у + ... + r m = fi	(5.411
where n is the dimension of the state space, there exist a feedback and a
coordinates transformation that can change the system into a fully linear
and controllable one. Thus, if relative degree can be achieved via dynamic
feedback and the condition (5.41) is satisfied in the extended system, then
the original system can be changed into a fully linear ami controllable one
via dynamic feedback and coordinates transformations.
If the relative degree has been achieved via dynamic feedback, then the
data included in the previous condition, namely the integer n and the r/s.
are not known until the dynamic feedback has been constructed, i.e*. for in-
stance until all the iterations of the Dynamic Extension Algorithm have been
successfully completed. However, it is possible to prove, under rather mild as-
sumptions, that the fulfillment of a condition like (5.41) depends directly on a
simple property of the original system, namely the absence of zero dynamics.
To this end, consider a system of,the form (5.1). with /(0) = 0 and
h(0) = 0 and suppose that if y(t) = 0 for all t then necessarily .r(0) — 0 and
n(t) = 0 for all t (i.e. suppose that the trivial pair consisting of initial state
x3 — () and input u2(t) = 0 is the only solution of the Problem of Zeroing the
Output). If this is the case, the system is said to have a trivial zero dynamics.
Note also that the definition of this property does not require the system to
have any (vector) relative degree.
Now, it is immediate to realize that the composition of a system having a
trivial zero dynamics with a dynamic feedback of the form (5.32) is again a
system having a trivial zero dynamics. In fact, recall that the state f of this
dynamic feedback satisfies tin* identity (5.34). Since, by hypothesis, system
(5.1) is such y{t") = 0 implies u(t) — () and also .r(t) = 0 and the function
p(.r) vanishes at x — 0. it is concluded that, in the composite system (5.33).
the constraint y(f) = 0 implies ,r(t) = 0. f(t) =0 and c(0 ~ 0.
Suppose now that the Dynamic Extension Algorithm has been iterated,
say n times, to yield an extended n — //-dimensional system
5.5 Examples
263
i - f(.r) 4- g{.г)a(.r. ;) -+- g(j’)3(.r. <)r
< =	(5.42)
g — h(x) .
having some vector relative degree {z’i.... rm } at (.r. (,') = (0.0). If the original
system (5.1) had a trivial zero dynamics, then also (5.42) has a trivial zero
dynamics and this, since the system in question has a vector relative degree
at (j'.Q = (0.0). implies (actually, is equivalent to) the property that
П +------h rm = n + n .
Therefore, the extended system (5.42) can be rendered linear and controllable
via feedback and coordinates transformation.
We summarize this interesting property as follows.
Proposition 5.4.4. Consider a system of the form (5.1). Suppose this sys-
tem has a trivial zero dynamics. Suppose the Dynamic Extension. Algorithm
can be iterated, say и times, to yield a regularizing dynamic extension. Then
(5.1) can be changed into a linear and controllable system ria (locally defined)
dynamic feedback and coordinates transformation.
Applications of this property will be illustrated in the next section.
5.5 Examples
We begin by discussing an elementary application of the design methodologies
illustrated in this Chapter to the system which describes the control of the
rotation of a rigid spacecraft around its center of mass. Recall (see section
1.5) that the system in question can be modeled by captations of the form
R
SCS)R
in which R is a 3 x 3 orthogonal matrix (with det(R) = 1). which describes
the rotation of the spacecraft with respect to an inert.ially fixed reference
frame, and w is a 3-dimensional vector which expresses its angular velocity
(with respect to a reference frame fixed to the spacecraft). In what follows,
we assume as in section 2.5 that the external control force is exerted by
a set of gas jets. Accordingly, we set
T = Du
where a is a vector which represents the magnitudes of the control torques,
and В is a constant, matrix. In particular, we assume that 3 independent
control torques are available, so that the matrix В is nonsingular.
264	5. Nonlinear Feedback for Multi-Input Multi-Output Systems
Our purpose is to obtain, by means of a feedback of the form (5.13). a
system in which each component of the new reference input controls, inde-
pendently of the other ones, the rotation of the spacecraft around one of its
reference axis. As customary in aircraft and space mechanics, the maneuver
needed to rotate the spacecraft - from an initial position in which its refer-
ence axes are aligned with the ones of the fixed reference frame to a generic
attitude R. can be executed in the following way. A rotation (yaw) of an an-
gle t’ around the axis «3. followed by a rotation (pitch) of an angle в around
the resulting axis cw, followed by a rotation (roll ) of an angle ф around the
resulting axis (see Fig.5.4).
Fig. 5.4.
The three elementary rotations thus described can be represented, as
any rotation, by means of an orthogonal matrix wdiose entries are direction
cosines. An immediate calculation shows that the matrices corresponding to
the three elementary rotations in question are. respectively
(cosv/ sinip 0
— sint? cosv 0
0	0	1
(cos в 0 — sin# \
0	10
sin# 0 cos# /
/1	0	0
R($) — 0	cos ф	sin d
\ 0 - sin d> cos ф
Note that
R(c) =	R(0) =	R(d>) = с(ЛзО)
where the matrices Ai- A2, A3 are the three matrices, already introduced in
section 2.5,
5.5 Examples
265
/ °	1	0\
Ai =	-1	0	0 |
\ 0 0 0/
/ 0 0 1\	/00 0\
A-> =	0 0 0] A3 = 0 0	1
\-l 00/	\0 -1 О/
Thus, the maneuver previously described brings the attitude of the space-
craft ' from an initial value R = I in which its reference axes are aligned
with the ones of the fixed reference frame - to a final value R given by
ft —	-43ol	A20	.4) c)
This expression, can be interpreted as a smooth mapping
F : R3 SO(3)
which assigns to each triplet [u.6. d) an element
R = Fiy.e.o} =^^-^^(.4^4	(5,43)
of the set 50(3) of orthogonal 3x3 matrices (whose determinant is equal to
1). It is easy to show that the mapping F is locally invertible, in a neighbor-
hood of the value R = I (this is. in fact, a consequence of the property that
the mapping in question has rank 3 at the point	— 0, because
rar"
[<9d _
z Аз
OF
_~дё
= -A->
'OF]
-d1-' J (v.0.m=o

and the three matrices Aj. A2. A3 are linearly independent). In other words,
there exists a neighborhood U of the point R = I in 50(3) with the property
that, for each R G F. the relation (5.43) can be satisfied by one and only one
triplet (t’,0.o). Moreover, the mapping
F~-} : F R3
which assigns to each R G L* the (unique) triplet (y.O.Q) = F~l(R) which
satisfies (5.43) is a smooth mapping.
We see from these arguments that the three angles (t".0,o) can be used
to parametrize, locally around the point R = I. the set of rotation matrices
which define rhe attitude of the spacecraft. Considering these three quantities
as outputs of the control system, one can pose the problem of finding, if there
exists, a static state feedback of the form (5.13), namely
3
it — a(R.^) + У/ 3I[R.^)rt	(5.44)
(=i
which renders the angle w dependent only on the input tq, the angle в de-
pendent only on the input c2. and the angle 0 dependent only on the input
1'3, that is to solve for the system in question - the noninteract ing control
problem.
266
5. Nonlinear Feedback for Multi-Input Multi-Output Systems
Note that the functions which characterize the feedback 15,44) are for-
mally expressed as functions of the state (/?._e) of rhe system. However, if
the value of the attitude 1? belongs to rhe set Г in which the mapping (5.43)
is invertible, we can replace В by F(и. 0. c>). and therefore rewrite the right-
hand side of (5.44) as a function of the six variables (с. в. о. -Ci .	).
In order to check whether or not the1 noninteracting control problem is
solvable, one has to calculate the integers rq. r2- Fi and check whether or not
the matrix (5.2) is invertible. However, the calculation of quantities of the
form
cannot bo directly pursued in this case, because an explicit expression of t ho
function Лда’). which is the i-rh component of mapping
is not available. Instead, we calculate r}. r2. and the matrix (5.2) indirectly,
by appealing to the interpretation of rt as the least integer for which the rv-th
derivative1 of yt with respect to time depends explicitly on rhe input.
The problem is to differentiate with respect to t the functions c(f). #(0.
d(t). To this end, it is to convenient compare the expression of
. 1 j о (t: ।	( U 2 f? i f j i : .41 <.  i O J
” dt
with
B = S^(t))B(t) .
Since
— i(.' -Fwco y.-i Ь'ш: if! Ui l it; ।	/
t/f
= (o.4:i - 0f' Ьт_42С-;.1зо: + Ало if-< .4,w	। .4^ 7.-1 4m
and В is an invertible matrix, we obtain from these expressions a relation
С1.4:з -ве[ be,^2f-i.43Ol	f -1.42 H) J .4^) f, -1,43<U = s^}
which must be solved for с. 0. о.
Observe that all matrices in this expression art1 skew-symmetric and that,
in particular
(0 sin о coso\
— sin о	0	0	j
— cos о	0	0 /
/	0
c( 1 c~1	1.41 el 1 - ЬоI _ cos у ct>s &
\ cos 0 sin о
ct)>e cos 0
0
sin 0
- cos 0 sin о
— sin 0
0
m5 Examples
267
Solving the previous relation for 0. о yields, after some simple calculations
col(c. 0. 0’) = 4/(г.0.О);л?
where
(0 sin 6sei-0 cos q sec0 \
0 COS O - sin о I
1 sin о tan 6 cos о tan 0/
is a matrix which, as shown, depends only on (t'.0,o). which is invertible for
all (l\0.q) in a neighborhood of the origin.
Since no component of the first derivative of y(t) depends explicitly on
the input u. we go to the second derivative. Clearly.
,, _	+ .v^ =	- .1/ J-1	+ .MJ-'Bu .
dy dt dt dy
The second derivative of y(t) has a form of the type
f/2) = b{ l.\0. O. vv'j . -с-?-~'з) + -4(c". 0. o}u .
From this we deduce that щ = m =	= 2. Moreover, since the matrix
А(у- 0- d) = .ЩкЯ
is invertible at {v.0.o] — (0.0.Oh we conclude that the system has rela-
tive degree {2,2.2} at this point and the noninteracting control problem is
solvable. A static state feedback which solves this problem is given by
u = A"1 (c. 0. ©)( г - b( t\0. o. uj;i)’) .	(5.45)
Note also that, since1 the state space of the system has dimension 6 (see
section 1.5). the condition
Л = Г1 + I\> + П
is also satisfied and the system is exactly linearizable. In fact, in the coordi-
nates
,rj = col(r. 0. o)
_r2 = J/U\0.dU’
the closed loop system obtained by means of the feedback (5.45) becomes
In the next, two examples, we show the application of some of the results
developed in section 5.4 to the control of a general aviation aircraft and to
the control of a two-link robot arm with nonnegligible joint elasticity.
268
5. Nonlinear Feed b ark for Multi-In put Multi-Out put Systems
The dynamical model of an aircraft can be described by means of three
sets of first order differential equations, involving the following sets of state
variables:
-	the angles (w. d. q) which characterize the attitude of the aircraft with
respect to the so-called wind axes (these three angles are respectively called
yaw angle, pitch angle and roll angle),
-	the components, denoted (p.q. r). of the angular velocity vector with
respect to a reference frame fixed with the aircraft (these three quantities are
respectively called roll rate, pitch rate and yaw rate).
-	the amplitude V of the velocity along the flying path, and two angles о
and ,3 which identify the direction of the tangent vector to the flying path
with respect to the main symmetry axis of the aircraft (which are respectively
called angle of attack and sideslip angle): о is the angle between the tangent
to the flying path and the longitudinal axis in the pitch direction (Fig. 5.5).
and 3 is the angle between the tangent to the flying path and the longitudinal
axis in the yaw direction).
Fig. 5.5.
The derivatives with respect to time of the angles (c’.d. d>) can be ex-
pressed in the form
col(y\ d. q) = AI(<’. d. o)/
where w* — col(p*. q*. r*) is the angular velocity vector expressed with re-
spect to the wind axes and AI(u.d, ф) is the matrix already introduced in
the previous example.
The derivative with respect to time of the angular velocity vector =
col(p. q, r) can be expressed in the form
w =	+ J~YT
in which S(w) is the matrix already introduced in section 1.5. J is the inertia
matrix, which in this case has the form
/ ir	о
j = о	i у	о
\-Tr_-	о	I2	/
5.0 Examples
269
and T represents the vector of external torques.
Finally, the derivatives of V. a. J with respect to rime have the form
1' = — (D/m) — g sin t)
a = q - q* see J — (pcos о + r sin a) tan 3
j = r* + psin a - r cos о
in which D is a scalar quantity called the drag force, in is the mass of the
aircraft, and is g the gravity acceleration.
In order to complete the model it is necessary to specify how rhe three
rates (p*,g*.r*). which appear in the first and third set of equations, are
related to the other state variables. The relations in question have the form
p* = pcosa cos 3 + (q - 6) sin 3 + r sin a cos 3
q* — -—-(L — mg cost? cost»)
"i1
rr — —— (— S + mg cos d sin o)
ml
in which S and L are two scalar quantities called the side and lift, forces.
Replacing (p* . g*. r*) in the previous equations and solving for a. one obtains
a system of nine first order differential equations in the state variables t". d.
0, p. q. г. V. a. 3. which describes the dynamics of the aircraft.
The vector T of the external torques and the vector col(£>, L. S'] of the
external forces contain the input variables, The first one of these two vectors
can be approximately expressed in the form
«12 г + «13p
a-23 q
«32 f + «ззР
«и sin 3
«21 + «22 sin О
«31 sin 3
61 j cos 3
0
0
0
62? cos a
0
613 COS ,3
0
633 COS .3
in which the «,/s and the 6,/s are fixed aerodynamic parameters (dependent
on the geometry of the aircraft, the air density, etc.) and dQ. 6e. denote
the deflections of the aileron, of the elevator and of the rudder. The vector
col(£>.£,5) can be given an approximate expression of the form
D \	/ «и + cj 2 cos a \	/ — cosacos.3\
L I — V2 C21 + «22 sin 2a | + P I sin a | dp
S I	\	C31 sin 23	/	\ cos a cos ,3 /
in which the c?J:s are again fixed parameters, P indicates the maximal thrust
and 6p the setting of the throttle. Note that, in the previous description, the
effect of the thrust on the vector T of external torques is neglected, and so
270
5. Nonlinear Feedback for Multi-Input Multi-Output Systems
are the effects of the deflections (d*. de, d’r) on the vector col(D. Z..S) of the
external forces.
The equations thus illustrated describe a system whose state is defined in a
certain open neighborhood F of R9, subject to the action of the 4-dimensional
input vector
и — col(dp. . 6,. <5,J .
Our purpose is to show that this system can be locally modified, via dy-
namic feedback and coordinates transformation, into a fully linear and con-
trollable system. To this end. we first observe that, if the nine state1 variables
arc rearranged into the following three subsets
.Г] - (V. 0. c)
.Го — (<?> O. d)
,r:s = (p.q.r)
and the input variables into the following two subsets
a i = hp
= (дп.6,.дг)
then the previous equations exhibit a structure of the form
•П = FJj-i. jo) + Gi(a. -Г'>)иг
jo zr F-2(.r[.	+ Gafj-bJ-oJu!	(5.46 j
= F'2(.Ci. т2. Тз) + Сз(г1,Т2Тз)гЬ
in which the Fj’s are 3x1 vectors, Gi, G2 are 3x1 vectors and G2 is a
3x3 matrix (for reasons of space, the explicit expressions of the functions
involved in these relations whose determination involves no difficulty are
omitted).
It will be shown that, for a suitable choice of output functions, the sys-
tem in question is such that the design procedure illustrated in the previous
section can be successfully applied. To begin with, consider the output
у = j’i = col(U, d. с)
(note that this output is 3-di mens ion al, whereas the input to the system is
4-dimensional) and observe that, by definition
j/11 = colb^1’.	= F1(j’1,t2) +Gi(x1.t2)u1.
Since none of the three entries of the 3x1 vector G\(jp.t2) is identically
zero, it is concluded that п = r2 = r3 = 1, but the system cannot have
relative degree {1,1. 1} because	the matrix	(5.2) has the following form
/ (Gib	0	0	0\
А(т) =	(Gi)2	0	0	0
\ (G’i)з	0	0	0/
5.5 Examples
271
where (СД- denotes the /-th entry of G'i. We apply once the Dynamic' Ex-
tension .Algorithm. In the present case, we can set io = 1 and jo = 1 and. in
order to obtain a simplified expression for t//1. we choose
p(r) = -(Д )j ( jq . jq). qU) = 1 .
This yields
ui = 77—7--------------(-(Fi)i(.ri ..m) + £i)
— IS
si = <T -
The composition of (5.46) with the feedback thus defined is a system of
equations which has the form
.Fj = Hi(jq..r2) + Ah(jq..m)6
G = n
>2	= Я2(,Г1 ..г2..сз) +	j-jX,
= F3 (jq . .r2. .r31 + Ch Ln , ^2  J’3) t’2 -
(recall that 7 is a scalar, r2 is 3-dimensional vector, and the first equation
of the first set is simply = G ), In particular.
A\(jq.j-2) =	1 GU-iq.-m) -
(Ст! I !
We now recalculate the n's, in order to check whether or not the extended
system has some relative degree. By construction.
y( 11 = HY (jq . .m) + Ah (jq. jq)G
does not depend on the input, In order to obtain a shortened expression for
yl'2) W(1
Н1(л,г>) + A| (.Fi .t2)G - Bl (.Fl ,.m, G)
thus obtaining
y'2} =	+ A’jG) + 77-^ (#2 AEG) w A'in-
03'1	UX-)
Since all the three entries of the 3x1 vector A\ (jq . ,r2) which is propor-
tional to Gi(j‘i.j’2) - arc1 nonzero, we set' that all the entries of y[~' depend
on t'i but not on c2. Thus zq = z2 = zq = 2, but the system cannot have1 rel-
ative degree {2, 2, 2} because the matrix (5.2) has only one nonzero column.
We proceed with another cycle of dynamic extension. In this case, the first
row of the matrix (5.2) is equal to
(1 0 0 0)
272
5. Nonlinear Feedback for Multi-Input Multi-Output Systems
and therefore, choosing i0 — 1. j0 = 1. p(x) = 0 and q(x) = 1. one obtains
the dynamic feedback
l'i = 6
1’2	—	W2
i'2 = H’j.
This yields an extended system of the form
zi =	+ A'i(j-1,jt2)Ci
6 = &
G> = «’i
X2 = H2(Xi . X-2. .Т3) +	T2)C1
x3 = FAx^x-2. T3) + G'sUl-.^.Ts)^ -
In this new system, t/11 and yl,'2> do not the depend on the input, by
construction. In order to obtain a shortened expression for y'3' we set
(Hi -r A'^i) +	(H2 + A’^Ci) + Ah£> = B-2(j-i.io. J"3. Ci -C?)
OX]	О X-2
thus obtaining
0B2 IT r. 5B2/	. 4	, dB2
У' = —(Hi +A1Ci)+ —(H> + A2Ci) + w— (F3+G3ir>) +	i»’i •
СЛГ1	OX2	ox3	oG
Now Ci = r> — 7-:j = 3, and the input ix2 appears explicitly. The matrix (5.2)
has the form
(/ v f ЭВ2„\	( г. ,дВ2^дН2^. \
-^(-oCi) — ( M ——G3 1 — ( Ai (——)(-=—G3) , 1
\	/	\ dj'y	J
A simple (but tedious) calculation shdws that this matrix has rank 3 at any
point (in the extended state space) characterized by C = ct = 3 = г = d —
o = 0 and V 0. Thus the extended system has relative degree {3,3,3} at
any point, of an open and dense subset of the state space.
We introduce now a fourth output function
У4 = о
(which is the first component of the vector лч) and we note that, in the1
extended system, y^' does not depend on the input, whereas y\* does. More
specifically, setting
(H2)i + (Ab)iCi - B2(ti, т2;.г3.С1)
we find that
Q 7T	О F"'b	5-1 TT
t/.}’ — w— (Hi + A]Ci) + "x— (H2 + A2C1) + w— (At -+-	+ (A2)iC2 -
ch"i	ox2	ax3
5.5 Examples
273
The extended system with the four outputs
У1 = 1'- /ri = (Л Уз = <', y4 = о
(5.47)
has ri = /'j — i'3 = 3 and r.i = 2. The assoeiated matrix [5.2) has the form
G3
V
d№h
сЭ.Гз
and. as an appropriate calculation shows, is nonsingular at any point (of the
extended state space) at which < = n = 3 = (.• = 1) = q = 0 and ri ф 0. The
system thus defined has relative1 degree {3.3. 3,2} at each of these points.
Fig. 5.6.
In summary, we can conclude the following. The system coni posed (see
Fig, 5.6) by the original equations describing the dynamics of the aircraft
with a dynamic state1 feedback, of the form
has relative degree {t’i. гз, Г3. r.}} = {3.3. 3.2} with respect to the choice of
outputs (5.47). Since the extended system thus defined has dimension n — 11.
the condition
zy -e r-2 + Г3 + П = »
is fulfilled and. therefore, by means of an additional static state feedback
(see section 5.2) the system in question can be transformed (Fig. 5.7) into a
system which, in suitable coordinates, is linear and controllable (Fig. 5.8).
274
о. Nonlinear Feedback for Multi-Input Multi-Output Systems
Fig. 5.7.
The second example of a system which can be rendered both noninterac-
tive (from the input-output point of view) and linear (in suitable coordinates)
by means of dynamic state feedback is that of a multi-link robot arm with
nonnegligible elasticity between actuators and links. We have already briefly
illustrated the phenomenon of elastic coupling between actuators and links
of a robot, arm in section 4.10. where we showed that the elementary model
of a single-link arm can be exactly linearized via static feedback and change
of coordinates. This is not anymore the case in general when the arm
consists of two or more links. As we shall see in a moment on a specific case,
even in very simple configurations the features of the models are such that
not only the exact linearization problem but even the less demanding nonin-
teracting control problem is not solvable via static state feedback. However,
by means of the design methodologies described in the previous section, these
two goals can still be achieved via dynamic feedback.
The simplest model on which tMse features can be illustrated is the one
of an arm consisting of two links moving on an horizontal plane. The first
link is rotating (about a fixed point) of the bast1 frame, and the second link
is rotating about the end point of the first link. For simplicity it is assumed
that only the coupling between the first and second link (i.e. the second joint)
exhibits significant elasticity.
The description of this system requires three angular coordinates, that
can be chosen in the following way: the rotation qY of the first link with
respect to the base frame, the rotation q2 of the axis of the actuator which
moves the second link (with respect to its own base fixed to the first link),
the rotation q?> of the second link with respect to the first link. The equations
describing the motion of the arm. whose derivation is not within the scope
of these notes and can be found in the appropriate literature, have the form
B(q)q + C(q,q) + r(q) = T
5.5 Examples
275
Fig. 5.8,
in which B(q) (the so-called inertia matrix) is a 3 x 3 symmetric and positive
definite matrix of the form
(Л1 + 2.4з cos q3 Д4 Л2 + Лз cos t/з \
.44	,44	0 j
-4-j + .43 eosQ3 0	.4o	/
C(qfq) (the Coriolis and centrifugal forces) is a 3 x 1 vector of the form
/ - .4;1 sin q3 (2<h q3 4- q3) \
C(q.q)=	0
\ A3 snig3qf /
and, finally. r(q) is a 3 x 1 vector of the form
r(?) =
у у 1
Лг
The coefficients .4, , 1 < i < 4, which appear in these expressions are
parameters related to the mass distribution in the arm. К is an elasticity
constant and Л' represents the gear ratio of the coupling between the second
actuator and the second link. The 3x1 vector T on the right-hand side
includes the two control forces ift and im imposed by the two actuators, and
has the form
T = col( U ! . ll>. 0).
Note that the third entry of tins vector is 0 because there is no independent
input, available for the coordinate q3.
276	5. Nonlinear Feedback for Multi-Input Multi-Output Systems
Choosing the state variables .r, , 1 < i < 6 as
jq	= q,	for 1 < i < 3
.r,	— qt_3 for 4 < i < 6
rhe system of equations can be put in the customary form
•r = /(.r) + pi(.fhti + gitJ'ju-i.
More precisely, it is easy to check that
A very natural choice of outputs in this system (as in any robot arm) A
the set of the angular coordinates which define the relative positions of the
links, namely
,9i - qi = -ci
02 = q-s = -г-л .
With respect to these outputs the system does not have a relative degree
because, as immediate calculations show.
Lgh(.r} = 0
and
AU) = LgLfh\r) =
y.W-Ct)
has rank 1 for all r.
-041 (ТЙ
-,9giUs)
However, relative degree can be achieved after two iterations of the1 Dy-
namic Extension Algorithm. Standard calculations, that are k'ft as an exercise
to the reader, show that cascading the system in question with a compensator
of the form
«1 = ------( -AU) + £11 + Г-2
911И
и'J =	1-2
£1 = £2
6 = 1’1
yields a system having relative degree {zq. r->} = {4. 4}. The composite system
has dimension n = 8 and therefore the condition zq -4- r2 ~ n is also fulfilled.
It is concluded that by means of an additional static state1 feedback (sec
section 5.2) the system in question can be transformed into a system which,
in suitable coordinates, is linear and controllable.
5.6 Exact Linearization of the Input-Output Response
277
5.6 Exact Linearization of the Input-Output Response
In section 5.2. we have shown that if a system has relative degree {o..... rffl}
at a point and
Г! + Г2 + . . . + Гт = П
then this system can be rendered linear by пк'ans of feedback and change of
coordinates. If the1 latter condition is not satisfied (but the system continues
to have relative degree {гр..rm } at a certain point ), one can at least obtain
a system whose input-output behavior is linear. As a matter of fact, we have
already shown in section 5.3 that a result of this kind can always be achieved
by means of the so-called Standard Non interactive Feedback
u(.r) = —	1 (j ) - A-1 (j:)c .
The possibility of using feedback in order to achieve linearity in the input-
output response is not restricted to systems having a certain (vector) relative
degree at a point of interest, but holds for a broader class of systems: we
shall see in this section how this broader class can be characterized and how
a feedback producing a linear input-output behavior can be designed. To
this end. we need to start with a precise formulation of what we mean by
achieving "linear input-output behavior’" via feedback. Looking again at a
nonlinear system having relative degree {ry...on which the Standard
Noninteractive Feedback had been imposed, we find that its outputs
for 1 < i < m. are related to the input by expressions of the form
Г1
yi(t) = C,(f)£'(0) + /	- s)vi(s) .ds
Jo
where
/	fr.-l \
= ( 1 f 7 " ' ------------FA )	= Г-—Tv
\	2 (n - 1)! /	(r> -1):
and £'(0) represents the value at time t = 0 of certain components of the
state vector, in the coordinates associated with the normal form.
The latter is (obviously) linear in the input ami in the initial state. How-
ever. the linearity in the initial state is due only to the fact that special co-
ordinates have been chosen, and does not hold anymore if ц' (0) is expressed
as a (generally nonlinear) function of the initial value of the state in the
original coordinates. Nevertheless, in any case the response is always given
by the sum of the response under zero input, which is a function of the time
and of the initial condition only, and of a response depending on the input
and not on the initial state, which is linear in the input itself. In other words,
the response has a structure of the following kind
m -f
HO = Q(L-r°) + У /	- rOt’dnldri. (5.48)
278	5. \onlinear Feedback for Multi-Input Multi-Output Systems
Comparing this with the general expression of the Volterra series expan-
sion of the input-output response of a nonlinear system (see (3.11)). we may
thus conclude that a nonlinear system haying relative degree {n.......}
subject to the Standard Noninteractive Feedback is characterized by an out-
put response in which the first order kernels <r((f.ri) depend only on the
difference t - и and not on A and all kernels of order higher than ош1 are
vanishing.
Note also that if the first order kernels of a Volterra series expansion
depend only on the difference t - rY and not on A. then necessarily all the
kernels of higher order are vanishing, and therefore the condition that the
U’((L depend only on the1 difference t - n and not on .rc is necessary and
sufficient for the response of a nothinear system to be of the form (5.48).
On the basis of these observations, our goal is now to try to tise feedback
in order to achieve (on a class of systems possibly broader than the one of
systems having some vector relative degree) a response in which all the first
order kernels of a Volterra series expansion depend only on the difference
t — и and not on A. In order to simplify the formulation of the problem,
note that if one considers the Taylor scries expansion (3.17) of а\(Лп)- it
is easily found that a necessary and sufficient condition for this kernel to be
independent of .rc and dependent only on t - ту. or - in other words for a
response of the form (5.48) to hold, is that
Lyi Ljhffir) = independent of .c	(-5.49)
for all A1 > 0 and all 1 < i. j < m.
In general, the conditions (5.49) will not be satisfied for a specific non-
linear system. If this is the case, we may wish to have them satisfied via
feedback, as expressed in the following statement.
Input-Output Exact Linearization Problem. Given a set of m +
1 vector fields /(.r).	.....дгг1(-г)-. a set of m real-valued functions
hi (j*)..... hj7, (j-) and an initial state .r°, find (if possible), a neighborhood
U of A and a pair of feedback functions o(z) and J(u’) defined on I/. such
that for all A- > 0 and all 1 < i. j < m
= independent of .r on U.	(5.50)
First, of all. we show that because of finite dimensionality of rhe un-
derlying system - the apparently infinite set of conditions (5.50) is actually
completely determined by a finite subset of them. It is possible to prove, in
fact, the following result.
Lemma 5.6.1. Suppose (5.50) holds for all Q < k < 2n — 1 and all 1 < i.j <
di. Then (5.50) holds for all k > 0 and all 1 < i.j < m.
Proof. We can indeed prove the result for a (infinite1) set of functions of the
form
3.6 Exact Linearization of the Input-Output Response
279
(5.51)
(k > 0 and 1 < i.j < m) thus simplifying the notation. First of all, recall
(see Lemma 1.9.4) that, given any neighborhood £' of .r2. on an open and
dense subset I'1 of [' the largest codistribiition Q invariant, under the vector
fields f.ij]...у,,, which contains span{d/q.....dhm } is locally spanned by
exact differentials of the type
=	:	hj
with г < n — 1. 0 < < on and yo = /. Since, by assumption, the functions
(5.51) are constant on l~' for all 0 < k < 2n — 1 and all 1 < i.j < m. we
deduct' that
d£M    L.h ht = 0
whenever if 0. 1 < / < r. so that Q is necessarily spanned by differentials
of the type dLjhj. with 1 < j< m and 0 < /г < n - 1. Let q denote the
dimension of Q at a point of I ' and define, in a neighborhood 1 of this point,
new local coordinates (сд. gj) = Ф(.г). where the q elements of art1 chosen
in the set {£*/;Д.с) : 1 < j < m. 0 < k < n — 1}. Them, by Proposition 1.7.2.
the1 vector fields /,	....дП) and the* functions /q..../;п( are transformed
into
/«1.0) =	s.U'iWi =	= л,«2) -
Replacing the expressions thus found into (5.51). we obtain
so that the constancy of the (5.51) with respect to r (on the* neighborhood
V) is equivalent to the constancy of the functions on the right.'hand side with
respect to <>.
We use now again the assumption that the functions in question are con-
stant for all 0 < k < 2n - 1 and all 1 < i.j < m. and we note1 that this
implies (see the formula (4.2))
{dLjJi7(<2).adj.^dQ}') = (-iV'L^L^'/bK-d - independent of
for all r. s such that 0 < c + -s < 2n — 1. Recall that, by construction, for each
value of 1 < k < q. there exist some 1 < j < m. 0 < s < n - 1, such that
(<2)A. =	= £}2/(ДО)
where ((,’_>)* denotes the £-th component of £>. Replacing this into the previous
expression yields
(t£)a-.adj.лу->1 (<•_>)) = Аг-th component of ud;.,//2i(0) ~ independent of G
280	5. Nonlinear Feedback for Multi-Input Multi-Output Systems
for all 1 < i< in and all 0 < r < n. In other words, the vector fields
adrj2g-2t (<2) are constant vector fields for all 1 < i < m and all 0 < r <
n. Let P denote the smallest distribution invariant under the vector fields
/2^21, •   • 92m and containing the vector fields	Recalling the
algorithm described in section 1.8. it is easy to realize that this distribution
can be expressed (because of the constancy of the vector fields ndy <72((G))
as
P = span{nt7^,pL>, : 1 < i < m. 0 < A' < tt - 1}
and that, for any 1 < f < m.
ad^g-ji E P .
Since ad’^g-^i also is a constant vector field, we conclude that the latter can be
expressed" as a linear combination, with constant coefficients, of vector fields
of the set {adj^g?} *. 1 < i < m.O < A' < n — 1}. and the same property holds
for any vector field of the form ad'^sg-2h with s > 0 (as a simple induction
argument shows).
Exactly as in the step (iii) of Theorem 4.8.3. this fact can be used to show
that
L92! L^Jij (<2) — independent of <2
for all A' > 0 and all 1 < i.j < rn. Thus, the functions (5.51) are constant on
a neighborhood V of every point т of a dense subset C’ of U. Being smooth,
they are constant on all P and this completes the proof. <
We come back now to the Input-Output Exact Linearization Problem.
Our goal is to find necessary and sufficient conditions under which this prob-
lem is solvable, and to show how a pair of feedback functions o(.r) anti J(j‘)
which actually solves the problem can be constructed. First of all. from the
data f(x).gj(r), р(х). 1 < i.j < тп. tfe construct the set of real-valued func-
tions L<hLfhd-rb 0 < A’ <	— 1. and we arrange all these functions into a
set of m x rfi matrices of the form
/ LgiLkfhPx) 	\
Tpjj = I -	 - -		1	0 < A- < 2n - 1.
\ Д91LLjhm(x) J
As a matter of fact, the possibility of solving the problem in question
depends 011 a property of the set of matrices thus constructed. This property
can be expressed in different forms, depending on how the data /^(.r). 0 <
к < 2n — 1. arc arranged.
One way of arranging these data is to consider a formal power series
T(s,x) in the indeterminate я, defined as
T(.s. x) =	(5.52)
5.6 Exact Linearization of the Input-Output Response
281
\Ve will see below that the problem in question may be solved if and only if
Tt#. t) satisfies a suitable separation condition. Another equivalent condition
for the existence1 of solutions is based on the construction of a sequence of
Toeplitz matrices, denoted ЛЛ-(.г). 0 < fc < 2n. — 1. and defined as
\ 0	0	Г0(г) /
In this case one is interested in the special situation in which linear de-
pendence between rows may be tested by taking linear combinations with
constant co efficients only.
In view of the relevance of this particular property throughout, all the
subsequent analysis, we discuss the point with a little more detail. Let M(x)
be a p x m matrix whose entries are smooth real-valued functions. We say
that .r12 is a regular point of M if there exists a neighborhood L" of .r° with
the property that
rank(Af(.rl) — rank(Af(.r°))	(5.54)
for all r e L". In this case, the integer rank(AL(x°)) is denoted (AT): clearly
гк(М) depends on the point .rc, because on a neighborhood V of another
point z1. rankQWfx1)) may be different.
With the matrix M we will associate another notion of •Tank’’. in the
following way. Let x° be a regular point of _U. Г an open set on which
(5.54) holds, and .V a matrix whose entries are the restrictions to U of the
corresponding entries of Л/. We consider the vector space defined by taking
linear combinations of rows of Mover the field E.. the set of real numbers,
and denote гц[М) its dimension (note that again гк(М') may depend on z°).
Clearly, the two integers гн(М) and j’k(.M) are such that
rnfM) > rK(M).	(5.55)
The equality of these two integers may easily be tested in the following
way. Note that both remain unchanged if M is multiplied on the1 left by a
nonsingular matrix of real numbers. Let us call a row-reduction of M the
process of multiplying M on the left by a nonsingular matrix I" of real num-
bers with the purpose of annihilating the maximum number of rows in W
(here also the row-reduction process may depend on the point. ). Then, it is
trivially seen that the two sides of (5.55) are equal if and only if any process
of row-reduction of M leaves a number of nonzero rows in I’ M which is equal
to гл'(-Н).
We may now return to the original synthesis problem and prove the main
result.
Theorem 5.6.2. There exists a solution at ,r° to the Input-Output Exact
Linearization Problem if and only if either one of the following equivalent
conditions is satisfied
282	5. Nonlinear Feedback for Multi-Input Multi-Output Systems
(a)	there, exists a forinal power series
A-~0
whose coefficients are m x m matrices of real numbers, and a formal pawn
senes
R(s.x) = Я-Н.Г1 +	1
A=0
whose coefficients are m x m matrices of smooth functions defined on. a neigh-
borhood I of rQ. with invertible 7?_i(.r). which factorize the formal power
series T(rrx) as follows
T(s..r) = A'(s) • /?(.$.t)	(5.56)
(b)	for all 0 < i < 2n — 1. the. point xc is a regular point of the Toeplitz
matrix and
rpfiMfi -	-	(5.57,1
The proof of this Theorem consists in the following steps. First of all we
introduce a recursive algorithm, known as the Structure Algorithm, which
operates on the sequence of matrices Тд.(.г). Then, we prove the sufficiency of
(b). essentially by showing that this assumption makes it possible to continue
the Structure Algorithm at each stage and that from the data thus extracted
one may construct a feedback solving the problem. Then, we complete the
proof that (a) is necessary and that (a) implies (b).
Remark 5.6.1. For the sake of notational compactness, from this point on we
make systematic use of the following notation. Let be a s x 1 vector of
smooth functions and {r/i...., gm } a/set of vector fields. We let L;/z denote
the ,s x m matrix whose г-th column is the vector re.
v • -  £Уп1 v ) .<
Structure Algorithm. Step 1. Let .rc be a regular point of To and
suppose rn(To) = rp(Tofi Then, there exists a nonsingular matrix of real
numbers, denoted by
where P^ performs row permutations, such that
where Si (a*) is an r0 x m matrix and rank (Sj (P)) = r0. Set
5.6 Exact Linearization of the Input-Output Response
283
rt'i	—	f0
"'1W	=	IWr)
= A\4!(.rJ
and note that	Lg-itr'} = Si(.r) Lg^it-r) = 0.
If T(}(\r) = 0. then Pi must be considered as a matrix with no rows and A'/
is the identity matrix.
Step i. Consider rhe matrix
/ £C'i(-d	\
\ LyLf--,_} (.r) /
/ Sf_i(.r) \
and let ,r° be a regular pen nt of this matrix. Suppose
r ( V • ( S'-*
(5.58)
Then, there exists a nonsingular matrix of real numbers, denoted by
()
0
n
0
P, ,
a; /
where Pr performs row permutations, such that
where S,dr) is an r,_i x m matrix and rank(S,(J’0)) = 0-1- Set
f'i- i
Рг£7^_1О)
A i~ i (.r) + -   + A  _ j i_ i (z) + A  Lf - i _! (r)
and note that
£s7i(-r) \
Lyy,(r) /
£35г (z)
5г(.г)
0 .
Lj/Ci (U
\ 0 /
d;
284	5. Nonlinear Feedback for Multi-Input Multi-Output Systems
If the condition (5.58) is satisfied but the last in — г;_-> rows of the matrix
depend on the first r^-j. then the step degenerates, P, must be considered as
a matrix with no rows, A  is the1 identity matrix. d; = 0 and St (jt) =	(j-).
As we said before, this algorithm may be continued at each stage if and
only if the assumption (b) is satisfied, because of the following fact.
Lemma 5.6.3. Let be a regular point ofT^ and suppose. tr(Tq) = rkITo)-
Then r zs a regular point of
( S’~l
mid the condition (5.58) holds for all 2 < i < k if and only if .P is a regular
point of T, and the condition (5.57) holds for all 1 < i < к - 1.
Proof. We sketch the proof for the case k = 2. Recall that
-Vi
( L(Jh
\ 0
LgL/h
L9h

Moreover, let V]. M and yj be defined as in the first step of the algorithm.
Multiply ЛА on the left by
0
Vi
As a result, one obtains
0
ГЛЛ
/LgP^h
LgV}Lfh\ _ .	0
\\L„h /	0
.	\ 0
/ Si LgLfyi \
0 LgLfy}
LgLfP^h \
LgLfI\ (h
LgPl h
0	/
Note that i’r(Si) =	Thus, because of the special structure of
Idfi. ,rc is a regular point of A/j and the condition пг(АД) = cr(AR) is
satisfied if and only if is a regular point of
к )
and
ro ( LgLf' i \	, ( LgLj-^ \
f n c J — A. | r* ]
X *^1	/	X	/
i.e. the condition (5.58) holds for i = 2. For higher values of к one may
proceed by induction. <
5.6 Exact Linearization of the Input-Output Response
285
Front this, we see that the Structure Algorithm may be continued up to
the A1-th step if and only if the condition f5.57) holds for all i up to A: — 1.
The Structure Algorithm may he continued up to the 2n-th step if and only
if the assumption (b) is satisfied. We can therefore proceed with the proof of
Theorem 5.6.2.
Proof. Sufficiency of (b): construction of the lint1 arizing feedback. If the Struc-
ture Algorithm can be continued up to the 2mth iteration, two possibilities
may occur. Either there is a step q < 2n such that the matrix
/ \
Е((Г ig— i (j I ,
' 9 ' f Q ~ 1 1 ‘ffi
has rank m at .r:. Then the algorithm terminates. Formally, one can still set
P7 = identity, Vfl = identity
" 7 (-r) — ^7 /7 7 1 ()
and
and consider Kf......Kf as matrices with no rows. Or. else, from a certain
step on all further steps are degenerate. In this case, let q denote the index
of the last nondegenerate step. Then, for all q < j < 2m Pj will be a matrix
with ni) rows. Kj the identity and 6j = 0.
From the functions m ..... -,r/ generated by tin1 Structure Algorithm, one
may construct a linearizing feedback in the following way. Ser
and recall that Sl} = LgT is an rg_l x m matrix, of rank r7_i at r. Then
the equations
0)	(o.o.)
on a suitable neighborhood I of t0 are solved by a pair of smooth functions
о and 3.
Sufficiency of (b): proof that the feedback defined by (5.59) solves the
problem. Set f(j-) = f(.r) + г/(.г)(к(;г) and = g(.r)3(r). We show first
that, for all 0 < k < 2n — 1
PiLgL^h(.r) - independent of z	(5.60)
286	5. Nonlinear Feedback for Multi-Input Multi-Output Systems
Р,К'_у    K[LgLkjh(x) = independent of .r
for all 2 < i < q and that
  - A"i' LgL^h(x) = independent of z.
To this end. note that (5.59) imply
Lp( = 0
Lg'-'n = independent of .r
for all 1 < i < q. Moreover, since Lg^i = 0 for all i > 1, also
T y“ ।	— Lf^j
L^!t = 0
for all 1 < Л Using (5.63) and (5.64) repeatedly, it is easy to see that, if k > i
'	3 f
(5.65)
(5.61)
(5.62)
(5.63)
(5.64)
and. if к < i
k;  . I<; Ць = К'--- K^Lpt .	(5.GG)
These expressions hold for every i > 1 (recall that, if i > q, K- is an identity
matrix).
Thus, if i < q and k > i — 1 we get from (5.65)
PtK‘:;  • • K^L-gL^h - LgP'L^-1--^ = LgL-yi+\
which is either independent of x (if Z — i — 1) or zero, while for i < q and
k < i — 1 we get from (5.66)
k
PpJ  -  KlL}Lkfh = P,- • • K^LS (5t+I - У А-‘-у;) .
J = 1
The right-hand side of this expression is again independent of z and this
completes the proof of (5.61).
Moreover, if k > q. (5.65) yields
   K^LgL^h = Kjj  • • A\lLgL^h
= T^Lyyr-
= LgK^L^
q

5.6 Exact Linearization of the Input-Output Response
287
anti tiiis, together with (5.66) written for z = q. which holds for к < q. shows
that also (5.62) is true. Finally, (5.60) is also true because Pi L^L^-h —
f a f '
anti the latter is either independent of .r (if к = 0) or zero.
Suppose now that the matrix
p4k^--k;
H =
(5.67)
is square and nonsingular. This, together with the (5.60)-(5.61)-(5.62) already
proved, shows in fact that
LgLkjh{.r) = independent of j
for all 0 < Ar < 2n — 1 and. in view of Lenin la 5.6.1, proves the sufficiency of
(b). But the non singularity of (5.67) is a straightforward consequence of the
fact that this matrix may be deduced from the matrix (I ’9... 11) by means
of elementary row operations.
Necessity of (a). Let
J(t) = 3 for)
a(z) = -,J-1 (х)п(т)
and let
= L-gLkh(x) .
If the feedback pair о and ,3 is such as to make Tk(r) independent of .r
for all к (i.e. to solve the problem), then
Lkfh — L^h + Tk_iq + _>Ljet + ,.. + T()L-k for. (5.68)
This expression may be easily proved by induction. In fact, one has
Lp'/i = 1^5бур + 0(П-16 + ... + Го1/“'<1)
= Г О	LgL^hA + Tk _ । L + ... + 7., /-С1 .
From (5.68) one then deduces
LgLjh = { LgL^h}3 +	+ Tk^2LgLfa(:r) + ... + T0LgLkf~la
TkH = tk3(j:) + Тк^Ьда(х) + ... +	.	(5.69)
Now. consider the formal power series
288	5. Nonlinear Feedback for Multi-Input Multi-Output Systems
АЪ)
A(.s. .r)
t=0
А-г)+E(£t/£/<^;r^s~A*'
A=0
and note that the latter is invertible (i.e. the coefficient of the О-th power of
s is an invertible matrix). At this point, the expression (5.69) tells us exactlv
that the Cauchy product of the two series thus defined is equal to the serie>
(5.52). thus proving the necessity of (a).
(a)=>(b). If (5.56) is true, we may write
A'o Ah
0 Ар
0	0
Ay-i(.r) \
TKk )
M-i
Ao /
An(.r>
A-i(.r)
0
The factor on the left of this matrix is a matrix of real numbers, whereas the
factor on the right is nonsingular at ,E. as a consequence of tin1 nousingularity
of А-Д.Г). Thus is a regular point of and condition (5.57) holds. <
Remark 5.6.2. We stress again the importance of the Structure Algorithm as
a test for the fulfillment of the conditions (a) (or (b)) as well as a procedure
for the construction of the linearizing feedback. <i
Remark 5.6.3. An obvious sufficient condition for the solvability of the Input-
Output Exact Linearization Problem is that the system has relative degree
{ri.....rm} at ,rc. The reader may easily verify that, if this is the case, the
Structure Algorithm can be continued up to a step q = max{ri............
yielding S7(z) = A(.r). <
Example 5.6.4. Consider the system .
On this system, the Structure Algorithm proceeds as follows. Construct
the matrix
T„(.r) = Lsh(x) = (j “
This matrix satisfies the condition (5.57). and one can set
/ 1
U
°
-1 J
5.6 Exart Linearization of the Input-Output Response 289
that yields
Si LH = (1 0)
= .r3
Consider now the matrix
which still satisfies the condition (5.58). Thus tee can proceed with the algo-
rithm. and set
that yields
S2(.r) = (1 01
~2(.r)	=	= T1
(no *2(.r) exists, because n = ?-0 = 1). At the third step, we consider the
matrix
у Ly Lj~i2(,r) J у 0	1 J
which now has r2 = - Thus, the algorithm terminates, with q — 3. and
~3(.r) = P^Lj%(.H =	.
The system can be rendered linear from an input-output point of view, by
means of the feedback u = ri(.r) + J(r)r. with a(.r) and J(x) solutions of (see
(5.59))
f U’) \	(.r) \
\Т97з(х) /	‘	“	y£/v3(.r) J
and J(z) the identity matrix.
Note that this system does not have any relative degree, because the
matrix (5.2). which in this case coincides with 7b(.r). is singular, nor the
state-input equations can be exactly linearized by means of feedback and
coordinates transformations, because the distribution G = span{(?i. (?•.>} is
not involutive. <
290
5. Nonlinear Feedback for Multi-Input Multi-Out put Systems
We conclude the section with an application of the Structure Algorithm
to the solution of the problem of matching the input-output behavior of a
linear model. Consider a system of the form
= Л-H + У/лЬНт	,__n.
'	(o.dl)
; = 1
У = h(j‘)
anti a linear reference model
< = +
(5-i И
Ун = Ц.
Suppose that the system (5.70) satisfies the conditions of Theorem 5.6.2
for solvability of the Input-Output Exact Linearization Problem. Let F;. A'{.
.... I\’ be the set of matrices determined at the i-rh step of the Structure
Algorithm (performed on the set of data /(z). сд (j-),    • f/nd-r), M.r)). Set
and, for i. > 2
С = ЛС-1-4
С = A{C’t + ... + A  . ^,-1 + А'С, i A .
Set also
D = col(Ci.....C4) .
Then the following result, (whose proof is left to the reader) holds.
Proposition 5.6.4. If and only if /
CtB = 0 for all i > 1	(5.72)
there exists a feedback of the form
< = o«.t) -U(O>’
и = o(<,z) +
yielding, for the corresponding closed loop, an input-output response of the
form
y(t) = Q(L C°, -r< ) + / CeA[t~a} Bir(cr) da .	(5.73)
Jo
In particular, if (5.72) holds, then one. can obtain a lesponse of the form
(5.75) by choosing
5.6 Exact Linearization of the Input-Output Response
291
— -'i
d«.jJ = В
= n(.r) - 3(.r)DAQ
.)((./) = — 3(j-)DB
where o(r) and 3(x) are solutions of (5.59).
Hint. Construct an "error" system
j = /U) +
г = 1
< = .4< + Bit
e = h(j-) - C<
and solve for this one the Input-Output Exact Linearization Problem. <

6. Geometric Theory of State Feedback: Tools
6.1 The Zero Dynamics
The purpose of the next two Chapters is to analyze in a more general
differential-geometric (and coordinate-free) setup some of the most important
concepts and design methodologies which have been introduced in Chapters
4 and 5. For convenience, we present in this Chapter the fundamental geo-
metric tools, zero dynamics and controlled invariant distributions, on which
the analysis is based, and we defer to Chapter 7 the illustration of how these
tools can be used in the solution of specific control problems.
We begin by discussing - form a rather general viewpoint the problem
of how the output of a nonlinear system of the form
r = /И +
(b.l)
У =
with the same number m of input, and output components, and state r defined
on an open subset L of . can be set to zero by means of a proper choice
of initial state and input.
Consider a point J10 in the state space of (6.1) and suppose that /(.r°) = 0
and Л(х°) = 0. Thus, if the initial state of (6.1) at time t — 0 is equal to j-'0
and the input u(t) is zero for all t > 0. then also the output y(t) is zero for
all t > 0. Our purpose is to identify, if possible, the set of all pairs consisting
of an initial state and an input function which produce an identically zero
output. To this end, it is convenient to introduce first some terminology.
Let SI be a smooth connected submanifold of I which contains the point
C. The manifold SI is said to be locally controlled invariant at if there
exists a smooth mapping и : SI -л P?71. and a neighborhood of .rc, such
that the vector field ) = /(.r)+,9(.r)u(.r) is tangent, to SI for all .r € _VcC;
or. what, is the same, SI is locally invariant under the vector field /(a1)-
An output zeroing submanifold of (6.1) is a smooth connected submanifold
SI of U which contains the point .r° and satisfies
(i) for each x £ SI. h(x) = 0;
(ii) SI is locally controlled invariant at .rc.
294	6. Geometric Theory of State Feedback: Tools
In other words, an output zeroing submanifold is a submanifold M of the
state space with the property that for some choice of feedback control t/(x)
- the trajectories of the closed loop system
,r = f(x) + g(x)u(.r'}
У = /Нт)
which start in 3/ stay in 3f for all times in a neighborhood of the time t = 0.
and the corresponding output is identically zero in the meanwhile.
If M and 31' are two connected smooth submanifolds of U which both
contain the point .rQ. it is said that .M locally contains .M1 (or, .XI coincitbs
with 3/') if, for some neighborhood Cc of ,r°. 3/PCc D 3/'PCD (or 3/PC" =
3C PCC). An output zeroing submanifold 3/ is locally maximal if. for some
neighborhood I “ of ,r~. any other output zeroing submanifold .XI' satisfies
XI P C° D ЗГ P Uz.
In general, it is not clear whether or not a locally maximal output zeroing
submanifold might exist at all. However, under some mild regularity assunip-
tions. in a neighborhood of j-° a manifold Z* satisfying the said requirements
can be found rather easily, as the recursive construction that we will describe
in a moment shows. Note that requirement (i) implies h(j-c) = 0. i.e. that
the point belongs to the inverse image, noted h~1 (0). of the point у — I)
with respect to th*1 output mapping Л. This motivates the consideration of
the sequence of nested submanifolds M() □ ЗД D ... э ЛД- Э ... defined in
the following way.
Zero Dynamics Algorithm. Step 0: set .Mo = h "1 (0). Step k: suppost1
that, for some neighborhood Ca-_i of /, ЗЛ-i P is a smooth sub-
manifold. let 3/^_T denote the connected component of 3A-_i PtT-i which
contains the point (IMf is nonempty because = 0) and define 3//.
as	/
3C = {.r G ЗЯ., : f(x) € span^Cr)----------+ TJ-.M'k_] } .	(6.2)
The following statements describe conditions under which the sequence
thus defined converges to a locally maximal output zeroing submanifold.
Proposition 6.1.1. Suppose that, for each k > 0. there exists a neighbor-
hood Uk of .C such that Л/д. P L\. is a smooth submanifold. Then, for some
k* < ri and some neighborhood l'k- ofxz. 3C-+1 = Mf.. Suppose also that
dini(span{<7i(.r“)....g,Zi(.r=')}) = m .	(6.3)
and that the subspace spanf^f.r).....glf/[т)}РТтЛ/д,'. has constant dimension
for all x £ 3/£.. Then, the manifold Z* = Mf. is a locally maximal output
zeroing submanifold for (6.1).
6.1 The Zero Dynamics
295
proposition 6.1.2. If, in addition
spaii{W1 (j'25).....	)} n W = 0 .	(6.41
then there exists a unique smooth mapping u* : Z' -> 3’" sach that, the vector
field
f'fr) = f(.r) - g(.r)u*(;r}
is tangent to Z'.
Proof. Proposition 6.1.1. Since all ЛЛ-’s are locally smooth submanifolds and
Mk A a dimensionality argument shows that for some integer k* < n
and some neighborhood of .r:.	= ^I£-- Set Z* = .Mf.. Since
Л/л-. + 1 = Z*. then by construction, at each point ,r of Z*. there exists a
vector и € 3”' such that
f{,r) + g(.r)u G Tj-Z*.	(6.5)
Since Z* is a smooth submanifold, in a neighborhood U' of .rc it is possible
to define a submersion H : l:f —) 3J (where q — n — dim(Z*)l. such that
Z’ ПГ = {.r G Гг : H(.r) = 0} .
As a consequence. since TXZ* = ker(dH(.r)) at each .r G (Z* А Г!). the
condition (6.5) can be reexpressed in the equivalent form
(dHfr). f(x} +g(.r)u} = 0 .
From the fact that this equation can be solved for a we deduce that
f(j-f) G 1т((г/Я(.т).^(.г)))	(6.6)
at each point .r of Z* p U'.
If (6.3) holds and the subspace span{f/i (J-),.... ym (т)} A T.rZ* has con-
stant dimension for all j- G Z* near /. the matrix {dH {r},g(x)} has constant
rank at each ,r G Z* near j,= . Therefore from (6.6) we deduce the existence
on some neighborhood I’" C U' of .r° of a smooth mapping u* : Z* —> 3,ri
such that
/(J') + J’)u*(j’) £ TrZ*
for all .r G Z’ А Thus. Z* is locally controlled invariant at zc'.
Z* is also such that h(x) = 0 for all J’ G Z*. by construction. Observe
now that any other output zeroing submanifold Z' is necessarily such that
Z’ C .Wa near тг. for all k > 0. This is proved, by induction, showing that
Z' C Mf_i implies Z' c Mt-. In fact
r £ Z' => f(j-) G span{r/i(j)........g!:M}	+ Tj.Z'
=> /(t) g spanf^br)........+ Tj-J/'.'
4 ,r 6 ЛД. .
296	6. Geometric Theory of State Feedback: Tools
From this we deduce that Z' is locally contained into Z*. i.e. that Z* is
locally maximal.This concludes the proof of Proposition G.1.1.
Proposition 6.1.2. Note that, if (G.4) holds, the matrix	g(.r)} has
rank m for all z near z:. For. the identity (dH(.r).^(z))y — 0 would imply
either g{j~)^: — 0. which is contradicted by (G.3) or p(.r)" € ker (dH j which
is contradicted by (6.4). As a consequence, in this case the mapping tz’fzi
found in the proof of Proposition G.1.1 is unique. <
Suppose the hypotheses listed in the precious Propositions are satisfied.
Since the vector field /*(z) is tangent to Z*. the restriction /*(z)|z- of /*(z)
to Z* is a well defined vector field of Z* (in what follows, whenever there is no
danger of confusion, in order to simplify the notation we will often use f^t.r)
instead of /*(z)|z- )• The submanifold Z* is called the (local) zero dynamics
submanifold and the vector field /x(z) of Z* is called the zero dynamics vectoi
field. The pair (Z*./*) is called the zero dynamics of the system (6.1).
By construction, the dynamical system
i- = Ш	e Z*	(6.7)
identifies the internal dynamical behavior induced on rhe system when the
<jutput. has been forced, by proper choice of initial state and input, to remain
zero for some interval of time. In fact, setting z(0) = .r° € Z* and
u(i) = u*(z(f))
when1 z(f) is the solution of (6.7) initialized at z5 € Z*. an output y(t') is
obtained which is identically zero as long as z(f) remains in Z* (i.e. for some
open interval of the time axis).
Remark 6.1.1. The approach followed here arrives at a local characterization
of the notion of zero dynamics. Of course, a global characterization would
also in principle - be possible, identifying Z* as the largest (with respect to
inclusion) smooth submanifold of h~1 (0) having the property that, for some
smooth input u* : Z* 3"'. the vector field /* — f + gu* is tangent to Z*.
<
Remark 6.1.2. The hypotheses (6.3) and (G.4) of Propositions G.1.1 and 6.1.2
can be interpreted as a special property of invertibility for the system. As a
matter of fact . from the proofs of these Propositions, it is easy to see that if
(and only if) these two conditions are satisfied then, for each initial state z'
(in a neighborhood of z°). any two pairs (z'. u'1) and (z'. ufj) producing zero
output are necessarily equal, i.e. have ua = id'. <
Remark 6.1.6. It is useful to show what the arguments illustrated reduces
to in the case of a linear system. As far as the Zero Dynamics Algorithm is
concerned, the reader should not haw1 much difficulty in realizing that.
- ker(C)
6.1 The Zero Dynamics
297
and
ЛД. = {;r e : Аг e Im(B) — Mk-1} 
Thus, all ЛЛ-’s are subspaces of the state space. The smoothness assumptions
are indeed satisfied, and there is an integer A-* < n such that 3F- + i =	
The subspace Г* = Mk. is by construction the largest subspace of ker(C)
satisfying
.4Г* с V* +Im(B) .
The hypotheses (6.31 and (6.4) reduce to the following ones
dim(Im( £?)) = m. V* Q Im(B) = 0
and these, as is known from the theory of linear systems, are exactly con-
ditions under which rhe transfer function matrix C(sl — .4) }B (which is
square because by assumption the system has the same number m of input
and output components) is invertible (see also Remark 6.1.2).
These two conditions imply the existence1 of a unique state feedback
which now is a linear function of t. namely
(Г i t) = Ft
such that /*(т) = Ar + Bu*(.r) is tangent to 1*. i. e. such that (.4 + BF)x
is In VT for all те I*, namely.
(-4 + BF)V* Q V’.
The subspace I '* is invariant under the linear mapping (.4 + BF) and the
restriction (.4 + BF)\v- identifies a linear dynamical system, defined on U*.
whose dynamics are by definition the zero dynamics of the original system.
We will show now that the eigenvalues of (.4 + BF)|y- (more precisely, its
invariant polynomials) coincide with the so-called trans mission zeros (more
precisely, with the transmission polynomials) of the transfer function matrix
of the system. This property enables us to extend to the case of multivariable
systems the interpretation, already given in the Remark 4.3.1. of the zero
dynamics as a nonlinear analogue of the notion of "zeros" of a linear system.
For. recall that the transmission polynomials of a multivariable minimal
linear system are defined as the invariant factors of the so-called system
matrix.
(si - A B\	._
C 0 ) '	(6 1
We will prove that the invariant factors of this matrix coincide with those of
the linear mapping (A + BF)\\ -. To this end. choose suitable new coordinates
т = col(J‘i. Tj )
such that the subspace V* assumes the form
1	— {(т [. t -_> ) G л : т i — 0 }
298
G. Geometric Theory of State Feedback: Tools
and such that
Im(B) C	G !> =0} .
This is indeed possible because U*Dlm(B) = 0. Accordingly. the inatrices
.4. B. f1 will be represented in a form of the type
 1ц
A-2l
-412
-4 22
B(
0
(c. 0)
(the special structure of C is due to the fact that l’’C ker(C)).
Observe that, since AU* C U* + Im( В). the matrices Ai2 and B( satisfy
the condition
Im(Ai21 C Im(Bl) .
Therefore there exists a (unique, because B( has rank m) matrix Fi such
that
BlF2 = -A12 .
Setting
F = ( 0 F, j
it follows that
A + BF =
/Ац
уА-л
0 \
A22 J '
We see from this that U* is invariant under (A — BF] and. in particular,
that Ao? is a representation of the linear mapping (A + BFl!p. Moreover,
it is easy to see that the maximality of F* implies the nonsingularity of the
matrix
/ -s / — A [ 1 В । \
\	° J
(6.9)
for all 5 e C. In fact, suppose that,’for some srj the matrix in question i>
singular. Them there exists vectors jq and ti such that
F.rj - A [ [ jq — Bi и = 0. Ci .r ( = 0 .
If this is the case, then the subspace
V = U* + span {roll ,r I . (1)}
satisfies
AU C U+ Irn(B). U c kerfCl
i.e. a contradiction, because U properly contains U* and U* is rhe largest
subspace satisfying these conditions.
Observe now that
si -A - BF B\ _( si - А В \ / I 0 \
C	0 J ~ \ C 0 ) \ -F I)
(6.10)
G.l The Zero Dynamics
299
and therefore that the invariant factors of the matrix (6.8) and those of
the left-hand side of (6.10) coincide. Replacing in the latter the expressions
previously established for .4 + BF. В. C and taking a permutation of rows
and columns, one obtains a matrix -- whose invariant factors still coincide
with those of (6.8) of the form
(6.11)
Since the subinatrix (6.9) is nonsingular for all s E C. the invariant factors
of (6.11) coincide with those of and this shows that the invariant factors
of (6.8) arc exactly those of (--1 H- BF}\\--.
Before closing this Remark, note also that the nonsingularity of the matrix
(6.9) for all s e C implies, in view of a well known controllability condition,
that the pair (.411. B/) is a controllable pair. From this, it can be immediately
deduced that, if all the eigenvalues of the matrix A22 have negative real part,
the pair
-In 0 \	( ВЛ
A22J	/
is stabilizable. Since the latter has been deduced from the original pair (.4. B)
by means of a regular feedback, which preserves stabilizability, we can con-
clude that a sufficient condition for a linear system to be stabilizable by
means of static state feedback is that its zero dynamics art1 asymptotically
stable. We will find in section 7.1 a nonlinear version of this condition. <
We proceed now to illustrate how the zero dynamics algorithm can be
implemented in practice on a given system of the form (6.1) and. in doing
so. we also show how rhe various regularity assumptions indicated in the
description of the algorithm (namely, the smoothness of г F\. for all
k > 1) as well as the hypothesis (6.4) can be tested.
At the beginning. 4/q is defined as the set of points where the mapping h
is zero; if the differential dh of this mapping has constant rank, say <?(). near
z°, for some neighborhood th- then the set 4/q П I о is a smooth in — so)-
dimensional manifold. If so is strictly less than the number m of components
of h. without loss of generality it is assumed that exactly the first s-0 rows of
dh are independent (otherwise, rhe order of the rows of h is changed). Thus,
if So denotes rhe matrix which selects the first rows of an .s-dimensional
vector, namely the (.s(j x ,sj matrix
So - (I 0 )
the mapping
= Soh{.r)
is clearly such that
300	6. Geometric Theory of State Feedback: Tools
3/q P ho — {T € f n : H^o(-c) — 0} .
Let Л/q denote the connected component of Mo П I о which contains the
point M. At rhe first step of the Zero Dynamics Algorithm, can be cal-
culated (see also the proof of Proposition G. 1.1) as the set of all т G M^ such
that
+ /?(-г)п> = LjHidjr} +	= 0	(6.12 j
is solvable in a. Suppose the matrix LgHoM) has constant rank, say rH. for
all .r e M$ (note that the constancy of rhe rank is required only on this
submanifold and nor on the whole f'o). Then, the space of solutions v of the
linear equation
= 0
has constant dimension (nanrely, - r0) for all z e M£. Since L,?Hn(j’) is
smooth, for some neighborhood [у C fh of ;rc it is possible to define an
(a-0 — co) x .s() matrix Н(|(т) of smooth functions of z. such that at each
J~ G (3/q PI7,) the rows of 2?0(.r) span the1 space of solutions of this equation.
In particular
= 0
for all ,r g [M(} P t'o), and it is immediately seen that at each point r g
(My A I'q) the equation (6.12) is solvable in и if and only if .r is such that
Bo(MLfHo{.r) =0 .
Setting
then for some neighborhood th of J‘° the set ЛЛ PLh fan be expressed in the
form
JV/i П I. i = {.c G C [ :	= (J and ФоМ) — 0} .
!
If the smooth mapping col(H0(z). Фо(л*)) has constant, rank, say s(J + ,-q.
near .P. then the previous construction can be iterated once more. Note that
since the vector Фо(л’) has (s0 — r0) rows, then
si < ’S'o — t’o .	(6.13)
At step A’ + 1 > 2. the iteration is started with mappings	and
(j?), where Hk-i is Such that the rank of dHk-i is exactly equal to the
number .s() + ... .sq._] of irs rows. and. for some neighborhood l\- of jt°.
П UK. = {r G Гк ‘ Нк„М') = 0 and Фа-i (т) = 0} .
Suppose the mapping coif Ha— i (т). Фа—i (J*)) has constant rank (,s0 + ... + sk)
near M (thus, for a suitable choice of L\-. the set Мк P is a smooth
manifold). Without loss of generality, assume that the first (.?[) + ...+
rows of the differential of this mapping are linearly independent (otherwise,
6.1 The Zero Dynamics 301
change the order in the last set of rows, namely in Фа-i)- let Sa-i demote
the matrix which selects the' first	rows of Фа-i	and set
Яа(.с) = соКЯа._1(.г).5д._1Фа._1(^)1 .
Obviously
ЛЛ- nli = {.r e L\ : HfcU) - 0} .
One has to look now at the equation
Т/Яа(.г) — LgHk(.r}u = 0 .
If the matrix LgHiRj-) has constant rank rA. for all j- e 5/j.' (the connected
component of 3/A. П through nfo it is possible to find a matrix R^-lr) of
smooth functions whose (.sq -r ... -t- $/, — rows span (locally around .r~)
the space of solutions of
~£,;Яа(.г) = 0 .
Clearly it is possible to choose
о /..o _ ( KA--i for) 0	\
JWj)- U-W)
where Pk -i for) anti Qa-H-H are suitable matrices, because by construction
Rk-\ {.r}LfHk- \ (.r) = 0 for all ji near .r° in Moreover, since Яа-i (-И ha>
(s0 + ... + .s-A._l - rA._i) rows. Pa-iU) and Qa--i(-H have - rk - rfr-i)
rows each. From this choice of Rk (>). one can define a new mapping Фк (.r)
as
Фк(.г] = Pk-PrjLfH),-^) 4- Qk_ i f.r)Lf.Si -u Фа-i (j’1
and continue the construction. (Note that, since Фк has (sa — га га-j) tows,
then
s’a—i < -4 — rk + rA-l 	(6.14)
Remark 6.1-4- Note that the integers sA. and r\ introduced in the previous
calculations can be characterized also as
sa =dim(-VA-i) - dim(.MA )
rfr = dim(span{<7i(.r).ford-H}) - dim(span{(п).......fo„(.r)}	nTJh.j.
Thus, in particular, the invertibility hypotheses (6.3) and (6.4) can be ex-
pressed in the form
га* = гапк(Т(/ЯА* (,.rc)) = m .<
Before proceeding further with the analysis of some properties of rhe con-
struction thus described, we illustrate it with the aid of two simple examples.
302
6. Geometric Theory of State Feedback: Tools
Example 6.1.5. Consider a system of the form (6.1). with 2 inputs and 2
outputs, defined on T4. with
h j (J')	= .r ।
h-Hx) = jy>.
First of all, note that this system has no relative degree, because the
matrix (5.2). which in this case has the form
T .. .	/ 1 0\
£3Ш) -	0 J
lias rank 1 for all z. Proceeding with the zero dynamics algorithm, we sec
that dh has rank 2 for all x. Thus = 2. Ho = h and
.V0 = {r G ?? ; n = ,r, = 0} .
We construct the matrix LgHo(x). whose rank r0 as already observed is
1 for all .r and we set
^o(-f) = ( --Gi 1 ) 
Tims
Since the rank of the mapping col(/f0(j*). Фо(.г)) is 3 for all we have .*0 = 1.
Ffj (.r) = col(.-Г2..Г4) and
AFj = T f IR4 : .ip ~ x > — x^ :: 0} 
The matrix	/
/ i 0\
LyHA-H) = t3 0 I
\ 0 1/
has rank ri = 2 ar all x e ЛЛ- and the algorithm terminates. We have
Z* = A/i. and the unique that keeps the state of the system evolving
on Z' must solve, at each x E Z*. the equation
/°\	/1 0\
LfHY(x} -+- Lf,H1(.r)i/*(.r’) = I .r4 j + l -r3 0 I u*(x) = 0 .
V/ Vi/
Thus
u*(x} = 0 .
Accordingly, the1 zero dynamics of the system those of	- are de-
scribed by
= Aj-3 .<3
6.1 The Zero Dvnaniics
303
Example 6.1.6- Consider a system of the form (6.1). with 2 inputs and 2
outputs, defined on R’. with
		/ ~ X		(c			
		fl		,r3			
/И	—	Z1 z4	9\f-r) =	0	(12 [Л	) =	1
		ft		r-,			X-2
		\ J.3		\ J			\1J
hi (z) = др
/ь(.г)
This system has no relative degree at z~ = 0. because the matrix (5.2).
which in this case has rhe form
Lgh(x) = ( 1
is singular at x" = 0. Proceeding with the zero dynamics algorithm, we find
So = 2. //q = h and
.V() = {.r 6 R’ : др = ,r-2 = 0} .
The matrix LgH(}(x) has rank r{) = 1 for all ,r E Mq and the algorithm can
be continued. Choosing
— -Pt 1,)
the product
= (I) .m )
vanishes at each x E _U(). Then
Фо(.г) = др - .map 
The rank of rhe mapping соЦЯ0(.г).Фо(.г)) is 3 for all .r. we have .sp = 1.
H\ (x) = collzi. Ху. x.i —X2X3). and
ЛЛ = {.г e R’1 :лр = до = др =0} .
The matrix
/ f	t) \
=	jp	др j
\ J’5 - др	-.Г2.Г,} /
has still rank rp = 1 at all z e *W(. We choose now
thus obtaining
Ф; (z) = -z2z5 + т-2.Гз - z3z4 - n z2z4 + x-y .
304	6. Geometric Theory of State Feedback: Tools
The mapping col (Яг (т). Ф] (г)) has rank 4. and we may set
H-lix) = colfj’i . .r?. J- J — .r2Z3. ——.ГеЛ'з “ Г3.Г4 — АЛЛ + -Я )
and
AT = {.;• e ?? : n = jo = zi = j-5 = 0} .
The matrix LyH->(.r) has rank 2 at Я = 0. and therefore the algorithm
terminates. We have Z* = Af>. and the unique W(.r) that keeps the state of
the system evolving on Z* is a solution of
(LfH2[.r) — L9H2(.r)iZ ('./))L'GZ- = 0
that is
t/(.r) = col(U. -r:i) .
Accordingly, the zero dynamics of the system those of f(.r) z- are de-
scribed by
h = -r3 .<
The constructions indicated above, and consequently the' various regu-
larity assumptions about the ranks of the mappings со1(7Д.(.г). Фу(.г)) and
about the ranks of the matrices ЬуН^(.г). are apparently depending on the
choice that, ar each iteration, is made of the' matrices 7?д-(./') which annihi-
late LyHfffr). However, we shall see in a moment that this is not the' case
if the invertibility hypotheses (6.3) and (6.4) are assumed. To this end. rhe
following result is helpful.
Proposition 6.1.3. Assume the following
(i)	dh(.r) has constant rank for all jf around .rc and. for some. choice of matri-
ces Rf_d.r)...Tf• _i (r). the differentials of the mappings \.r). Фд. (,r i).
i.e. the matrices со1рШд( .г). с/Фу (jj). flarc constant rank for all .r around Я.
0 < k < k* - 1.
(ii)	the matrices	hare constant rank for all .r g Mf. around Я.
()<£<£•* —1.
(iii)	the matrix LgHк- (.r~} has rank m.
Then .s() = in. s-] — sf] — r0. and for all k > 1. .sp.-j =	- rk + />_,. As-
a consequence
Я0(.г) = Mr)
and
fee. there is no need to discard rows of Ф^.(г) in order to define
Moieoiw.r. any other choice of matrices Ro(r),.... Rr--i U') is such that
the conditions (i). (ii). (iii) are still satisficd.
G.l The Zero Dynamics 305
Proof. Recall that
•so < m .	(6.16J
By definition. = 0 and. by assumption, — m. L’sing this and all
(6-13). (6.14). (6.16) together, we obtain
m = ry. < ,4. + rk>-} < sk-+ rk. -2 <.-<*! + r0 < .s‘o < m
thus concluding that the sign of equality must hold in all of them. This
proves the first parr of the statement because, since sq--] = sk ~ rk + />._], all
the rows of col(d#r (т). (1Фк (jj) are linearly independent, and the selection
matrix Sk reduces to the identity matrix.
In order to prove the second part, we will show first, by induction, that a
different choice of 7?o(<).^-(Jj (i-f. a different choice of 7?<j(r) and Pk(F).
Qk(F). for к > 0) yields a sequence of mappings H0(F). Ф0(х).......Фк(-г)
related to rhe former by
Я0(.г) = Я0(.г)
ФоИ = Тп(;г)Ф0(.г) +
(6.1J
Фк\.г} = Fk{x)Hk(.r} + Тк[г)Фк(.г) + \‘к1х}
where ГДт) is a matrix which is nonsingular at each .r E arid !'Дт)
vanishes on Л/г . for all 0 < i < k.
To this end. we show - again by induction that these relations imply,
for each ,r G Mk.
Lg,Hk{.r) = Sk(x)LyiHk[.r}
where Sk 1 .r) is a nonsingular matrix, for all 0 < i < m. In fact, since
г f, = (= (	1,.вд	A
‘‘”l 1 ( УВД J	+
and since, at each r G Mk + i
(L;/iFk(r))Hk(r) +	= 0
UUd = Ga.(t)£5,^,(z)
for some suitable matrix СЛ-(.г) (the latter because the differentials of the
entries of li(^) are linear combinations of the differentials of the entries of
Hy(j’) at each .r G i). we have
Lg,Hk+i(-i-)=	.{^’^.(z) rX) ) Ly’Hk~df) = sk~iFFLaiHk+i(F)-
Recall now that U’) is a matrix whose rows, at each ir G Mk^i near
.rc. are a basis of the space of solutions of the homogeneous linear equation
306	6. Geometric Theory of State Feedback: Tools
''!LgHk+i(^) = 0- Thus, in view of the expression established for L3i (x).
it is concluded that tffc+i(.r) has necessarily a form of the type
= M{x)Rk+i
in which Af(r) is nonsingular for all j e and Lk~i(x) vanishes on
ЛД-ri- Moreover, since it is requested that the upper-right block of (<i
be zero (and so is the corresponding block of Л\.+] (j)). the .V(j?) must have,
for each x e АЛ+ь the form
Using these expressions in the construction of Фк~] (-r)- a simple calculation
shows that the latter can be given the form
Фк—Лх) = Fk^i(x)Hk+l(x) + ГА.+ 1(.г)ФА—1Сг) + 1'ж(.г)
thus proving the correctness of (6.17).
From the expressions thus proved, using again the fact that Hq(x) and
the Ф,0 < i < к - 1. are vanishing for all x € near xs. it follows
that
dHk(x) = Sk(x)dHk(x)
LgHk(x) = Sk(x)LgHk(x)
for each x e Af*: near xc. where S^(j) is a nonsingular matrix, and this
completes the proof of the second part of the statement. <з
This result essentially shows that the regularity assumptions (i) and (ii).
if the invertibility hypothesis (iii) is satisfied, do not depend on the particular
choice of matrices introduced at each/it erat ion of the algorithm. In view of this
property, we will say that a point is a regular point of the zero dynamics
algorithm if the conditions (i), (ii). (iii) of Proposition 6.1.3 are satisfied.
We show now that h(x) and the mappings Ф^.(т) constructed at each
step of the zero dynamics algorithm are helpful in defining a new set of local
coordinates around z°, which induces on the equations describing the system
a structure of special interest (although not as simple as the normal form
analyzed in the previous Chapter). The point of departure is the following
result.
Proposition 6.1.4. If x° is a regular point of the zero dynamics algorithm,
the differentials of the entries of
Ф(х) = со1(Л(т),Ф0(-г)-•  ,Фк'-1(х))	(6.18)
are linearly independent at x*.
Proof. It is immediate from Proposition 6.1.3. <
6.1 The Zero Dynamics 307
The next step of our program is that of using the components of the
mapping (6.18) in order to define a new set of local coordinates in the state
space. However, before proceeding with this, it is convenient to explain the
forthcoming constructions with the aid of a simple example.
Example 6.1.1. Consider a system with m = 3 and suppose the zero dynamics
algorithm proceeds in the following way.
Step 1. Let (dh.g} = Lgh have the form
(L gh 1 \
0 /
and rank 1 at each .r 6 Afo = {-r : ^i(t) = h2(x) — йз(^) = 0}, locally
around xc. Then, there exists a smooth function ?. defined locally around
x°, such that
Lgh2(x) = --(^Lgh^x) + сг2(х)
with (J^x) = 0 for all x € Mo (note that 02 (-r) is not necessarily zero if x
is not in 2W0. because the rank of Lgh(x) is not necessarily 1 at one of these
points). We can set
0 0 2 J
and therefore
Step 2. Consider the matrix
which is 5 x 3, and suppose it has rank 2 at each x € ЛЛ = {y € Mo : <p2 (-r) —
фз(х) = 0}, locally around x°, with the first and fourth rows being linearly
independent (note that the second one is already dependent on the first one
and the third one is vanishing). If and 62 are smooth functions, defined
locally around such that
Ьдф3(х) - ~6}(x)Lgh1(x') - 62(x)Lg®2(x) +<хз(х)
with 03 (t) ~ 0 for all .r E АЛ, we can use
p f\ _ (	^o(-r)
о о)
(о о) A
(ВД Ш
and then set
Ф1(т) = S^Lfh^x) + 52(t)L/c>2(z) + £/дз(х) = v3(^) 
Step 3. Suppose now that the matrix
308
6. Geometric Theorv of State Feedback: Tools
Г LT _ I LyH} \
" “ " l Ь,,Ф1 J
which is 6 x 3 lias rank 3 at ,r: (thus, in particular, its first, fourth and sixth
rows will be linearly independent). If this is the cast1, then the algorithm
terminates, and Z’ can be locally described, in a neighborhood U of r3. as
Z* - {.r € U : hi(j-) = /qU’) =	= Gq(J’) = Оз(-1‘) = GiU’) = 0} 
The input u + (.r) which renders the vector field f*(s) = /(.r) + y(.r)u*orl
tangent to Z* must solve, at each z G Z*. rhe equation
and is therefore given by
LfH-ilr) — L,.}H-2(d-')u* (.r) = ()

(note that the equation for u*(.r) apparently consists of 6 scalar equations,
in which however - the second, third and fifth one an1 automatically solved
at each x G ZZ).
By the Proposition 6.1.4. the functions /q. /q. h3, c>2, cq. tq have linearly
independent differentials at .r:. so that they can be chosen as a partial set of
new local coordinates. Denoting by r/ the set of complementary coordinates
(with rj(.r°) = 0), it is easy to check that in the new coordinates the system
is described by
.У1	— Lfh\ + Lgh\ti
У2 — Lfh-2 + Lyhin = L — 'tLghiu — <t2u
— 02 — ''•(Lfhi -r Lgh^u) + oou
Уз = Lffi3 + Lgh-зи = Lfh‘3 =
0’2 — LfQ-2 +
03	— Lfd-з + LgO-.^i = LfO-з - (diLyhi 4- dzLgOi'lu + <73;/
= t"3 — <h (Lfhi + Lghiu) — 6-2(LfO2 +	+ <73 и
t-3	= L/tq +
У1- У2^Уз-О2.о-з, 1’з) + gtAy- У\  Уз. 1’3)u .
Note that, setting u — u*. one obtains
6.1 The Zero Dynamics
309
У i	=	o
y-2 =	+ cr-2 ie
Уз	=	Оз
&>	=	о
O.i =	- o3u*
03	=	(J
0 = fo(n- У\ У2-Уз-&2-ОЗ-Ы * до(Ч-У1 -У2‘Уз-О2. Оз- tql’U
and from this, since both ou and гт3 are 0 on Z*. we see that the zero dynamics
- in the new coordinates is described by
У = /*(h) = /o(r/.0....0) +5o(h-0......O)u*(r/.O.....0)	.<
To extend the constructions described in this example is not too much
difficult. What is needed is essentially to give an appropriate notation to the
entries of h(x) and of Фо(.г)...Ф*-_1 (j). To this end. we suppose first of
all without loss of generality - that the outputs of the system have been
rearranged in such a way that in the matrix
\ (1Фк... Jt) J '
the last (5a- —	+ D.-i 1 rows art1 dependent on the previous ones (at each x
on ЛД-. near .U ). If this is the case, then the matrix Qk-i(x) may be chosen
of the form
Qk-i(yr) = (Qk-M I).
Now. set Zq(t) — h(x) and. recalling that Фд.-Мг) has by construction
entries, let T\(x) be a vector consisting of exactly m elements in which the
first (m — <57.) ones are zero, while the last ones coincide; with those of (z).
1 < A’ < A’*. Then, in the rectangular m x (k* + 1) matrix
T(3-j = (r0(j) T.ix)  7VUD
each row consists of a sequence of nonzero entries, say in number of n( (where
i is the index of the row), followed by a sequence of zero entries. Moreover,
by construction.
Hi <n2 < ..
Now set. for 1 < A- < rq, 1 < i < m. Q(x) equal to the entry of T(x) on the
t’-th row and A-- th column, and
e* =coi(^'.............	(6.i9)
Using these functions as new coordinates, together with an extra set у
consisting of (n. — Sq — ... — .57 J components, the equations of the system
can be put into a form that, to some extent, generalizes the normal form
310	6. Geometric Theory of State Feedback: Tools
introduced in the previous Chapter. Note that, by construction, the new
coordinates ^.(x) are such that ^(z°) — 0. for all for 1 < к < n,. 1 < i < ni.
and one can always choose the complementary set of coordinates r/(z) such
that i](xc) — 0.
Proposition 6.1.5. In the local coordinates z = Ф(х) —	.. f"’, r/) de-
fined by (6.19fi the system (6.1) assumes the form (in which .r stands for
Ci = Ci
Cii;~i = C^
Cr = C? + И +	-1- (rj (x)u
CiL-l = Cn2 +С-1.1И(Ь1И +0.‘(t)u) +tT“2_1(.r)u
Й2 = Ь2(т) + O2(;r)u
? —1
Cl = Ci +	+ aJ(T)u) + a{(x)u
j=i
Сл.-i = Cn, +	+«JUM + a^_1(x)u
J-1
С», = b'(.r)+alGr)u
n = /o(C].......ci-rA + gvUl......
and
yt = Ci
for i = 1.....m.
In particular
(6.20)
а'И = Lg^nfix)
bl(x) = Lf&ffir) .
The. coordinate functions ^.(x). the coefficients and сг[(т) are. such
that
i-i
CI’+iU) z	=	+ Lf^) 1 < к < Hf-1.2 < i < m j=i
z	i-i -	j	(t) + tTj.(.r) 1 < A’ <	2 < i < m . j=i
(6.21)
6.1 The Zero Dynamics
311
In the ne.tr coordinates, the submanifold Z* is described as
Z* — {z e : £'(*') = 0.1 < i < m}
and the functions n!k(xi vanish on Z*.
The matrix
Л(т) = ci>l(a1(.r)......а"гИ)	(6.22)
is nonsingular at .г", and the unique и*\х) which solves the equation
bl(x) + u'(.r)u‘(j-) = 0	1 < i < m	(6,23)
is such that f*(,r) = fh‘) + glx)iT(x) is tangent tn Z*. Thus, the zero dy-
namics of the system, in the new coordinates, are described by
V =	= /o(O...., 0.>}) + ,t?o(0.  • • t0- z?)u*(0.().	t/) .	(6.24)
Proof. It is quite simple, although a little tedious, and is loft as an exercise
to the reader. We suggest to check first the (6.21) that, on the basis of the
definitions (6.20). descend directly from the properties of the zero dynamics
algorithm: then the special form of the system equations follows trivially. <
Remark 6.1.8. It is important to note that the results illustrated in this sec-
tion. in particular the generalized normal form and the corresponding charac-
terization of the zero dynamics, incorporate completely the results discussed
in section 3.1,
In fact, suppose the integers zq......rtrt are such that the vector
(L^I^/рИ	... £^niLj7z,(j') )
is zero for all ,r near .rc and for all к < zq — 1. and nonzero for к = tq — 1 at
j- = j-c. Without much effort, it is possible to realize that, after possibly a
reordering of the outputs, the integers zq..... zqn thus defined are related to
the integers n (...., n!n associated with the generalized normal form in the
following way
zq = tii, zq < щ for 2 < i < m.
and also that
д[; (л-)	=	0	for all 1 < к <	rt; — 1.1 < j < i -	1. 2 < i < m
<Tk(x}	=	0	for all 1 < к <	r}; — 1,2 < i < m .
If. in addition , the matrix (5.2) is nonsingular, i.e. the system has vector
relative degree {zq.......rm } at f. then zq = nt for all 1 < i < m. and the
previous normal form reduces exactly to the one introduced in section 5.1.<
312
6. Geometric Theorv of State Feedback; Tools
6.2 Controlled Invariant Distributions
In the next sections of this Chapter we develop a series of results that are
very helpful in studying the effect of a static state feedback on a nonlinear
system of the form (6.1). In accordance with the set-up already established
in Chapter 5. we consider a feedback control law of the form (5.15). namely
a = n(.r) +	(6.25)
with о and J defined on a neighborhood U° of the point of interest (which
sometimes could be the state space C on which the system (6.1) is defined),
and T(j-) nonsingular for all x.
The effect of this feedback is that of changing the original system (6.1)
into one of the same structure, noted
m
± = /(j-) + ^.(ф
i=l
in which we have set
f(x) = /(x) + J2^(.r)a((T)
j = i
In more condensed form, the latter will be almost always rewritten as
f(x') = f(x) + g(x)a(x)
g(x) = g(x)3(x) .
The purpose for which feedback (s introduced is to obtain a dynamics
with some nice properties that the original dynamics does not have. As we
shall see later on. a typical situation is the one in which a modification is
required in order to obtain the invariance of a given distribution J under the
vector fields which characterize the new dynamics. This kind of problem is
usually dealt with in the following way.
A distribution Л is said to be controlled invariant on U if there exists a
feedback pair (a. J) defined on U with the property that -A is invariant under
the vector fields f.g}... ..gm. i.e. if
!/» c _1(X)
,	6.2b
[(/,. -A](r) C J(-r) for 1 < i < m
for all r G U.
A distribution J is said to be locally controlled invariant if for each x G U
there exists a neighborhood L'0 of x with the property that -A is controlled
6.2 Controlled Invariant Distributions
313
invariant on In view of the previous definition, this requires the existence
of a feedback pair (o. 3) defined on P such that (6.26) are true for all j- e
The notion of local controlled invariance lends itself to a simple geometric
test. If wo set
G = span]#!.......gjn}
we may express the test in question in the following terms.
Lemma 6.2.1. Let Д be an involutive distribution. Suppose Л. and Д + G
are nonsingular on U, Then. is locally controlled invariant if and only if
c J + G	(6.27)
[(p. С Д + G for 1 < i. < m .	(6.28)
Proof. Necessity. Suppose J is locally controlled invariant. Let tro be a neigh-
borhood of .r° and (a. J) a feedback pair defined on L’° which makes (6.26)
satisfied on L'°. Let т be a vector field of J. Then we have
Hi	ГЛ
[LT = If +	= [M1 +
j=i	j=i
ш	m
[&-r] =	_
j=i	j=i	i=i
for 1 < i < ni.
Since 3 is invertible, one may solve the last m equalities for [<7j.r]. ob-
taining
e	+ G
j=t
for 1 < j < m. Therefore, from the second equation of (6.26) we deduce
(6.28). Moreover, since
Ут]е[/.-Ч + £;л,-1’+с
J=1
again from (6.26) and (6.28) we deduce (6.27). <
Remark 6.2.1. Note that in proving the necessity of conditions (6.27) and
(6.28) we have not yet used the hypothesis that _i and J+G are nonsingular.<
In order to prove the sufficiency, we need to explain first the properties
of a certain construction. Let 1 < d < n be a fixed integer and consider the
d-dimensional nonsingular distribution A' defined as follows
A'(j-) = Im
at each ;r G R”. where I is the d x d identity matrix.
314	6. Geometric Theorv of State Feedback: Tools
Suppose A +G has constant dimension, say d + q (with q > 0). Then, it
is easily seen that in some neighborhood f’c of each G S!i there exists a
nonshigular rn x m-matrix of smooth functions d(x) such that such that
	/ .9i i	и	91	91	q-l(x]	91 m fo’)
g(j-) = g(x)dU) =	9di	(.г) у	\	9dq(G	9d		9dltd-G
	9d^i	l(fo 			0	0
					0	0
	V 9m.l	(х)	•••			0	0 /
for all .r € f.’c. In particular, it is possible to choose 5(.r) in such a way that.
for some set of indices A. i->. -  , i4. with d + 1 < if,- < n for 1 < k < q. the
submatrix of g(x) formed by the rows A- ?->.......and by the first q columns
coincides, for all rgfin with the q x q identity matrix.
This being the case, it is always possible to find an m-veetor of smooth
functions such that in /(t) = /(•**)+ fofoo(t) the entries of index
A. i->..... i4 are zero for all .r G l'°.
The vectors /. .......gm constructed in this way enjoy the following prop-
erty.
Proposition 6.2.2. If
f/.A’] C
lgt-K]
for 1 < i < m .
(6.29)
(6.3(1)
the vector fields f.g}....g,n defined above are such that
[/.A’] C A	(6.31)
[gt. A'] С K, for 1 < i < in .	(6.32)
/
Proof. Observe that, after having reordered the last n — d rows, the matrix
g(x) and the vector /(j) can be given the following form
/ g(,U) giM\	/ A(-r)\
дИ - J 0	.	/(t) =	0
o /	\/cCr)/
As a consequence, for all 1 < i < d.
/ 0fa \
Arguments identical to those presented above in the proof of the necessity
of Lemma 6.2.1 show that the hypotheses (6.29) and (6.30) imply
6.2 Controlled Invariant Distributions 315
[f.K] C K + G
1 < i < m .
for
a К
Thus, in particular.
dxi
0
Ofc
e im о
\o
d
' Ox.
g<M дь(х)\
I 0
о /
and this implies

This identity shows that all the last n — cl entries of f(x) are independent
of -Ti...jrf and thus, since
..	I 0	° I
A=spail{sy.........a^1,
it is concluded (see e.g. section 1.6) that (6.31) holds. Similar arguments
easily prove that also (6.32) holds. <j
At this point, it is easy to complete the proof of Lemma 6.2.1. In fact, using
the property that A is nonsingular and involutive. it is possible to define, in a
neighborhood of each point j-c of U. a new set of local coordinates r = Ф(х)
such that
0 d
J = sl>an<ay.......эУ}'
Using the hypothesis that A -y G is non singular, it is possible to construct
a nonsingular m x zn-matrix of smooth functions 3(x) and an m-vector of
smooth functions t>(i) with the properties indicated above and for which
the result described in Proposition 6.2.2 holds. Then, since the property of
invariance is independent of the choice of coordinates, it is concluded that
the functions
3(j) = 3(Ф(т)). o(z) = ,5(т)о(Ф(т))
are such that A is invariant under f + да and any column of g.3.
We see from Lemma 6.2.1 that, under reasonable assumptions (namely,
the nonsingularity of A and A y- G) an involutive distribution is locally con-
trolled invariant if and only if the conditions (6.27) and (6.28) are satisfied.
These conditions are of special interest because they don’t invoke the exis-
tence of feedback functions о and 3. as the definition does, but are expressed
only in terms of the vector fields f-gi,-- -  gin which characterize the given
control system and of the distribution itself. The fulfillment of conditions
(6.27) and (6.28) implies the existence of a pair of feedback functions which
316	6. Geometric Theory of State Feedback: Tools
make J invariant under the new dynamics. As we have seen, the actual con-
struction of such a feedback pair involves the determination of a change of
coordinates to the purpose of transforming -A into a distribution spanned bv
constant vector fields (which generally requires the solution of an appropri-
ate system of partial differential equations) and the solution of certain linear
(r-dependent) algebraic equations.
We conclude this section with an interesting result which describes a
uniqueness property of any feedback which renders invariant a given distri-
bution.
Lemma 6.2.3. Let .U be an equilibrium point of the vector field f(x'). Sup-
pose Л is a nonsingular and involutive controlled invariant distribution and
suppose also
diin(G) = m
Anb = {0} -
Let o1 and or be any two feedback functions such that [/ + .yrh.-A] С -A
for i - 1,2 and o1(j'°) =	= 0. Let АЦ.О be the maximal integral
submanifold of Л which contains the point xQ. Then
a1 (jr) = a2(.r)	(6.33)
for each x e ЛЦ.°.
Proof. Let J(r) be a nonsingular matrix such that [giL -A] C -A. Proving
(6.33) is equivalent to prove that Af.zjo1 (r) ~ J(lr)o2(.r). Using the fact that
[/ + c one deduces that
[f + ОДО"1»1 - f - (9.J),r ‘a2, _1] с -1
that is	/
- o2), r] G -A
for all vector fields r of -A, which yields
i(gS)S~l (a1 - cr).r]
m
=	- LJ3-l(cP - o2)h(<?cOi) 6 -A .
f=i
Using the fact that [(ff3)j.r] € -A for all 1 < j < m, -А (T G = {0} and the
fact that the {дЗУ'ъ are linearly independent for all j*. one deduces that
LT(3~x(c3 — q2))7 = 0 for all re A.
This implies that .PHq1 -o2)(r) is constant on AU= and therefore, since
о1 (г0) = ci2(.Uh (6.33) must follow. <
6.3 The Maximal Controlled Invariant Distribution in ker(rM)
317
6.3 The Maximal Controlled Invariant Distribution in
ker(dh)
The notion of controlled invariant distribution is of particular interest in the
problem of using feedback to the purpose of rendering some outputs of a
system independent of certain inputs. In fact, suppose a control system of
the form
+ p(.r)ic
У = h(.r)
is given, in which the additional input ir represents an undesired perturbation
that affects the behavior of the system through the vector Held p, and consider
the following problem: find, if possible, a static state feedback (of the form
(6.25)), with the property that, in the corresponding closed loop system
r = /(-r) +.9U‘)f‘V) + y^(ff(.r)d(j‘));r, +p(-c>
У = h(-c)
the perturbation ?r has no influence on the output y.
In view of some results established in Chapter 3 (see Theorem 3.3.3 and
Remark 3.3.3). this problem has a solution if and only if there is a distribution
Л which is
(i)	invariant under the vector fields f = f + ya. = (^d);. 1 < i < m, and
p which characterize the closed-loop system.
(ii)	contains the vector field p.
(iii)	is contained in the distribution
ker(d/;) — Q ker(dhj) = (span{d/ii.......dhfri})~ 
j=i
According to the terminology established in the previous section, we see
from (i) that J is a controlled invariant distribution, invariant under the
vector field p. which as (ii) and (iii) specify - satisfies
p e J C ker(dh).	(6.34)
On the basis of this simple observation, we can conclude that the problem
of using feedback in order to make the output of a given system independent
of a certain input implies the problem of finding a distribution M which is
controlled invariant for the system (6.1) and satisfies the constraint (6.34).
Among the conditions which this distribution must satisfy there is also the
invariance under the vector field p but. as it is immediate to check, this is not
really an additional constraint if the distribution itself is involutive. In fact,
318	6- Geometric Theory of State Feedback: Tools
if (6.34) is satisfied, p is a vector field of -Л and. if the latter is involutive. the
invariance under p is achieved by definition.
Note also that, if the distribution in question is involutive and nonsingu-
lar, then in a neighborhood of each point x in the state space it is possible to
change the coordinates (see e.g. Remark 3.3.2) in such a way that the closed
loop system
j = /fy) +	+ pfy)w
У = ЛО)
is locally represented by equations of the form
ii = /1 (j~l , z2) + z2)v +p(zi,x2)u’
j-2 = /2(^2) + 92^2)v
у = ЬЫ 
We see from this that the disturbance w has no effect on the output y.
just because the feedback has rendered unobservable the closed loop system.
In fact, all pairs of states whose j2 components are equal produce identical
outputs under any input. We observe then that seeking a pair of functions
a and 3 which makes (i) and (iii) satisfied for some distribution J essen-
tially corresponds to search for a feedback that induces a certain amount of
unobservability into the system.
The problem of finding, for the system (6.1). a (possibly involutive) con-
trolled invariant distribution which satisfies the constraint (6.34) can be dealt
with in the following way. First of all, it is examined whether or not the fam-
ily of all controlled invariant distributions of (6.1) which are contained in
ker(dh) has a maximal element (in the sense of distributions inclusion, i.e. an
element which contains all other members of the family). Then, it is checked
whether or not the maximal element thus defined is involutive and contains
the vector field p. We shall see in thfysection how a program of this kind can
be accomplished.
As explained in the previous section, a necessary condition for a distri-
bution to be controlled invariant is that the conditions (6.27) and (6.28) are
satisfied, and these conditions - which turn out to be also sufficient, at least
for local controlled invariance, under some mild regularity assumptions are
particularly interesting because they do not involve explicitly the feedback
functions a and 3. Motivated by this, we are naturally led to consider the
family, noted J7(/, g, ker(dh)), of all smooth distributions which satisfy the
conditions (6.27) and (6.28) and are contained in ker(dh). Since this family
is closed under distribution addition (in fact, a trivial calculation shows that
if Al and -12 satisfy (6.27) and (6.28) then also -b + J2 satisfies these con-
ditions). then this family has a well defined maximal element, namely the
sum of all the members of the family. In view of Lemma 6,2.1. the maxi-
mal element of j7(/.y, ker(dft)) is the natural candidate in the search for the
maximal locally controlled invariant distribution contained in ker(dfi).
6.3 The Maximal Controlled Invariant Distribution in ker(dh)
319
The calculation of the maximal element of J(f.g. ker(cZh)) is made pos-
sible by the following recursivc construction.
Controlled invariant distribution algorithm. Step 0: set f?0 —
span{d/?}. Step k: set
«I. = -Qj.-! + £/№-, n GO + jci,, (/?*._! nG1) .	(6.35)
? —1
Remark 6.3.1. Note that the codistribution .C^-i ПСл being defined as an
intersection of codistributions, may fail to be smooth. However, it is still
possible to define the codistribution Ly(f2^_] nG1), as the one which is
spanned by all covector fields of the form Lfu,'. with smooth covector field
in -Qjt-i П Gx. <
Lemma 6.3.1. Suppose there exists an integer k* such that -W-i = f?*-.
Then l?k = for all к > к*. If П G~ and f/jf- are smooth, then 12^
is the maximal element of J(f. g,kex(dh)).
Proof. The first part of the statement is a trivial consequence of the defini-
tions. As for the other, note first that from the equality -Qjt'+i — -C4* we
deduce
MfMG1)) c Qk.
for 1 < i < m and also for i = 0 if we set f = go. as sometimes we did before.
Let ix1 be a covector field in П G1. and т a vector field in • Fi the
expression
we have
<0.-0 = 0
because Lae 14* and
o, r} = 0
because т E I?k,. Thus
ЫМ = 0 .
Since .r4- nGx is smooth by assumption, [<л,т] annihilates every covector in
П G1, i.e,
[50 d £ if +
for 0 < 1 < m. Thus. f2p is a member of J(f, g, ker(d/i)). Let Л be any
other element of this collection. We will prove that A c 12^.  First of all,
note that if a? is a covcctor field in А1 П G± and r a vector field in A we
have
t) = 0
so that (recall that A is a smooth distribution)
M^nG1) cd1 .
320	6. Geometric Theorv of Staff; Feedback: Tools
Suppose
-V D P,
for some k > 0. Then
<4 + 1 C <4- + L f (J- П G-) + L,)t О- П G-) c
; = L
Thus, since f?0 = span-{c//i} С Л-, we deduce that
Л c
and <?p is the maximal element of J(f. g. ker(dh')). <
It is important to observe that the algorithm (6.35) is invariant under
feedback transformation.
Lemma 6.3.2. Let f .g\.......gm be any set of vector fields deduced from
fi-gi....g,,, by settmg f = f + get. gt = {g3)j. 1 < f < rn. Then each
codistribution ffi of the sequence (6.35) is such that
Gk = <4'-i + Lj(G~~ n <?k-i) ^^Тдг (G1 n <4-i) 
Proof. Recall that, given a covector field a vector field т and a scalar
function n.
iL4-.T4.d(h -
If is a covector field in G~ П <4-i, then
Lf^ = Lf^ +
i = /	,=!
^'3.**'' =	"b ^2(-*'’• gj)33j, .
J-I	3=1
But {tv.gj} — 0 because a; G G1 and therefore
Lj(G~ n <4-i) + (G^nf4-i) c L/(G~ n <-4-_!) +	(G1 n f4-i).
!-l	i=l
Since 3 is invertible, one may also write f = f — g3~lci and g^ = (g3 "1)(
and. using the same arguments, prove the reverse inclusion. The two sides of
the inclusion are thus equal and the Lemma is proved. <
6.3 The Maximal Controlled Invariant Distribution in ker(dh)
321
For convenience, we introduce a terminology which is useful to indicate
the convergence of the sequence (6.35) in a finite number of stages. We set
-Г = (.Co + -Ci + ... + Ca- + ...)	(6.36)
and we say that -V is finitely computable if there exists an integer /Т such
that, in the sequence (6.35). .Сд- — fik' + i- If this is the case, then obviouslv
A* = Cf..
In the Lemma 6.3.1 we have seen that if d* is finitely computable and
if (-И - И G~ and J* arc smooth, then J* is the maximal element, of
J(f.y. ker(dh)). In order to let this distribution be locally controlled in-
variant all we need are the assumptions of Lemina 6.2.1. as stated below.
Lemma 6.3.3. Suppose 3* is finitely computable. Suppose and -+- G
are nonsingular. Then A* is involutive and is the largest locally controlled
invariant distribution contained in ker(dh).
Proof. First, observe that, the assumption of nonsingularity of J* and -Д* + С/
indeed implies the smoothness of (-A*)- П G~ and J*. So. we need only to
show that -Л* is involutive.
For. let d denote the dimension of J*. At any point .m' one may find a
neighborhood of .rc and vector fields n, - - -. тр such that
J* = span{-!......Td}
on L'°. Consider the distribution
D = span{r; : 1 < i < d} + span-dr,. Tj j : 1 < i.j < d}
and suppose, for the moment, that D is nonsingular on Cc. Then, every
vector field т in D can be expressed as the sum of a vector field r' in J* and
a vector field r" of the form
d d
T" =
(=i J=i
where c,j. 1 < i.j < d. are smooth real-valued functions defined on L‘°.
We want to show that
[gt.D] CD+G
for all 0 < к < m. In view of the above decomposition of any vector field r
in D. this amounts to show that
l9k-lT-rj]] C D +G .
The expression of the vector field on the left-hand side via Jacobi identity
yields
\9k- MJ] = [ту. [да- Tj]] - [rj.^A-D-]] 
322
6. Geometric Theorv of State Feedback: Tools
The vector field [<7/г.т,] is in J* 4- G and therefore, because of the nonsingu-
larity of Л* and -A* + G, it can be written as the sum of a vector field г in
and a vector field g in G. Since [г;.g] C J* + G for any g E G. we have
to, [SA- - "j]] ~ [ту. т -4- gi C D -r -A* + G = D T G
and we conclude that D is such that
[gk .D'iCD + G
for all 0 < A- < m.
Now. recall that ker{d/i) is involutive by definition, and therefore that
D G ker(dh) .
From this and from the previous inclusions we deduce that D is an element
of J(f. <?. ker(dh)). Since D D Л* by construction and -A* is the maximal
element of J(f.g, ker(d/t)) we see that
D = J*.
Thus, any Lie bracket of vector fields of d*. which is in D by construction,
is still in -Л* and the latter is an involutive distribution.
If we drop the assumption that D has constant dimension on U°. we can
still conclude that D coincides with _1* on the subset U C L ° consisting of
all regular points of D. Then, using Lemma 1.3.4, we can as well prove that
D = Д* on the whole of U°. <i
In practice, the largest locally controlled invariant distribution contained
in ker(d/i) can be calculated, in a neighborhood of a fixed point in the
following manner. Suppose has constant dimension, say &k-\, near .r=
and let this redistribution be spanned.by the rows of a (о>_] xn) matrix И\._1
of smooth functions. In order to calculate a basis of using (6.35). one has
to determine first the intersection —i nG1. A covector uj in f4-_i nG_L(z).
being a linear combination of the rows of ИУ-i (.r) which annihilates the
vectors of G(a-), has the form = уП\._i(x), with 7 solution of
= 0 .	(6.37)
If the matrix
= П;._] (j-)g(z)
has constant rank, say pk-i, near r°. the space of solutions of (6.37) has
constant dimension (сц._1 - p^-i) near F and there exists a ~ Pk-i ) x
(Тк-i matrix of smooth functions, noted 5jt_i(a’), whose rows span, at each
u. this space. As a consequence, Gk-i П G1^) is spanned by the rows of the
matrix (-r)H	(z). From this, using (6.35) and also recalling Remark
1.6.7, it is concluded that LG can be described in the form
6.3 The Maximal Controlled Invariant Distribution in ker(dft) 323
= Г4-1 + span{L/(5A,._1n fc-1), : 1 < t < crfc-i - pt-i}
+ span{L^(5fc_iH\._i)i : 1 < i < cta-i - pk-i. 1 < j < m}
(where (Sk-1HA-1)( denotes the z-t.h row of Sa--iIFa:-i )• From the covector
fields indicated on the right-hand side, one can easily find a basis for Qk. if
the latter has constant dimension <jk near z°.
Of course, the recursive construction is initialized by setting h'o(z) =
dhfir). If сгд._1 = ста- for some k. then by definition
C G^) c <4-1
Lg, (<4-i П G1) C <4-1; 1 < j < m
and the construction terminates. In other words, if appropriate regularity
conditions are satisfied (namely, the constancy of the dimensions of Qk and
of <4 F1G1). after a finite number k* of iterations the condition -*4’^i — f4*
of Lemma 6.3.1 is achieved.
Remark 6.3.2. Note that the integer pk. the rank of -4-. can be characterized
as
pk = dim(<4) - dim(<4 bl G1) .<
For convenience, we incorporate into a suitable definition all the regularity
conditions introduced in the previous discussion and we say that the point
x° is a regular point of the controlled invariant distribution algorithm if, in a
neighborhood of i°. the codistributions Qk and <4 bl G,J-. for all к > 0. are
nonsingular.
Proposition 6.3.4. Suppose P is a regular point of the controlled invari-
ant distribution algorithm. Then the hypotheses of Lemma 6.3.3 are locally
satisfied, i.e.. in a neighborhood Uc' of xc. A* is finitely computable and A*
and A* + G are nonsingular.
Proof. It is an immediate consequence of the previous discussion. <i
Example 6.3.3. Consider again the system already discussed in the Example
6.1.5. In this case,
Tr f V _ ( 1 о 0 I
|^o i о ।
and
J t \	( 1
•To CH = I	n
\	t) /
Thus, ctq = 2 and po = 1. We can choose
So И = ( -J-3 1)
and we find
f?oblG1(Jr) = span{S0(z)H o(z)} = span{^-}
324
6. Geometric Theory of State Feedback: Tools
with = (. — тз 1 (J 0). Now, observe that
Lfx =	+^’(.r) (= ( -(Ax3 4*t) 0 0 1)
\ U.l- /	\UJ-)
and
£j;i~ = (0 0 10)
L.^ = (0 0 0 0 ) .
From these, since
— f?o 4- span{iy^. Ly^'.
we conclude that A'* = 1. <?k. (.r) — (F? )* for all ,r. As a consequence. A* = 0
for all r. <
Example 6.3. J. Consider a system of the form
outputs, defined on Ei;>. with
(6.1). with 2 inputs and 2
In this case, again
ir«o = C J
() 0 (A
0 0 0 J
and
Ao
(A
0 )
Thus, сто = 2 and p0 = 1. Wc can use the same S'oA) as in the previous
example, hating
A) П G~(x) = span{.%(.r)H'o(.r)} = span A}
with = ( —./'3	1 0 0 O'). Since
Lj~
Lg^
( --ri-Tj -j:3 0 0 0)
(0 0 1 () 0)
(-1 0 0 0 0)
we can choose, as a basis of f?[. the rows of the matrix
/1 0 0 0 0\
H’j(.r) =	0 1 0 0 0 .
\0 0 1 0 0/
1
6-3 Th? Maximal Controlled Invariant Distribution in ker(dh)
325
We calculate now
/ 1 0\
Л (r) - H i (z)y(r) =	,r3 0
\ 0 1 /
whose rank is 2 for all Therefore, the construction terminates. In fact,
we can set
SJr) - ( -r3 1 0)
and find that Sl (r)H*i (z) has its (single) row coincudent with the1 one already
found for 5()(т)И’0(г). This clearly implies
Lf(^h nG1) G f?i
GJfGnG1) C Pi	l<J<m
i.e., k* — 1. Thus. is spanned by the rows of П, (./) and
J* = kerf И'i) = span{
We have seen before that, if the hypotheses of Lemina G.3.3 hold, the dis-
tribution d* is the largest locally controlled invariant distribution contained
in ker(dh). This means that there exist feedback functions о and .3, defined
locally in a neighborhood of each given point, such that this distribution is
left invariant by the vector fields f — f + ga. and g, = (g3)t. 1 < i < m. How-
ever, for the actual construction of these feedback functions the only result
available so far is the one described in the proof of Lemma 6.2.1. which - in
general requires the solution of a set of partial differential equations in or-
der to find the change of coordinates which transforms _1* into a distribution
spanned by constant vector fields. If a slightly stronger set of hypotheses is
assumed, it is possible to avoid the solution of partial differential equations,
and to find n and 3 at the end of a recursive procedure which involves only
solving linear (.r-dependent) algebraic equations. This result, is summarized,
for convenience, in the next statement.
Proposition 6.3.5. Suppose .c° is a regular point foi‘ the controlled invariant
distribution algorithm. Then, in a neighborhood l’° of xz . the following prop-
erties hold. For each к > 0, there exists a. og.-dimensional vector of smooth
functions
-Ц- = col( Aj....A^.. Ap^i......Auk)
such that
Qk = span{dAj : 1 < i< ст*}
(G.38)
and
326	6. Geometric Theory of State Feedback: Tools
spanfdA, : 1 < i < pk} D Gr - {0} .	(6.39)
Moreover. J?a-h can be expressed tn the form
= f2jt + span{t/Lz^aA( : pk + 1 < i < cq-}
+	: Pa- + 1 < / <	1 < j < m}
where a and 3 are solutions of
(dX^x), f(x') + p(.r)a(j-)) 3= 0	1 < i < pa-
{dXl{x].g(x)3j(x)') = d,j 1 < i < pk
and dj(x) denotes the j-th column of 3(x)-
.4 s a consequence A*, the largest locally controlled invariant distribution
contained in kcr(tM), can be expressed in the form
_V* = P| ker(JAi) ,
;=i
Л pair of feedback functions that solve (6.J1) for к — A1* such that
[f + pn;a*] c a*
i(pj);.a*] c a* i < i < m.
Remark 6-3.5. Note that, obviously.
,l0 = со1(Лi....hm) .
Because of (6.39), the row vectors {dX}.g(xf), 1 < i < pk. arc linearly inde-
pendent for all j’ near xc. Thus, the equations (6.41) can always be solved.
In particular, because of the special form of the right-hand side of the second
equation of (6,41). the latter can always be solved by a matrix 3(x) which is
nonsingular in a neighborhood of x°: <
Remark 6.3.6. Note also that the involutivity of the distribution a*, that was
proved in Lemma 6.3.3 under some weaker hypotheses, now follows trivially
from the fact that (a*)1 is spanned by exact differentials. <
We give now the proof of Proposition 6.3.5.
Proof. We proceed by induction, since an expression of the form (6.38) cer-
tainly holds for k = 0. because J?o = span{d/q..,.. dhm }. Suppose (6.38)
holds. Since by assumption the intersection 4?*. П G~ has constant dimen-
sion сгк — рк- then it is always possible to reorder the entries of Ta- in such
a way that also (6.39) holds. Because of (6.39) no linear combination of
dXi.....dXpl; can be in G^- and we deduce that fh n G~ is spanned by
vectors of the form
— dX, T ci\d\^ h~ ... +- dX^
6.3 The Maximal Controlled Invariant Distribution in ker(d/<)
327
where c,!: •  • • (-ipk are suitable functions and (pk + 1) < i < ak. Recall now
that the controlled invariant distribution algorithm is invariant under feed-
back (Lemma 6.3.2). so that we can calculate C\+1 as
m
-C4-! = <4 + £С1МС~|
1=0
assuming that, the feedback functions о and d are exactly those given by
(6.41). The derivative of u,- along pj. 0 < j < m, has the form
АЧ-	Pl-
= dLy j A; +	(‘is) d As -4-	f/Z.y i Ag 
Since, by construction (dAs..g;) is either 0 or 1. the third term of this sum
is zero. On the other hand, the second term is already in because it is a
combination of covectors of /2*. We sec from this that
L'g]^ = dL^Xi + У with и/ G *?a-	(6.42)
and therefore
.Ch—1 -	+ span{d£ ^ A; : pk + 1 < i < (Jk-0 < j < ш} .
This proves (6.40). At this point it is clearly possible to choose, in the set of
functions whose differentials span the second term of this sunn an additional
set of (cFfc^i - (Jk) new functions, that will be denoted by Aajl._i.A^+1.
such that
<?A-i = span{dA, : 1 < i < cr^i}
thus proving the validity of (6.38) for к + 1.
The last part of the statement is a trivial consequence of the previous
construction. As a matter of fact, consider again the expression (6.42) of the
derivative along gj. 0 < j < m. of a covector field in	If the algorithm
terminates at к - к*. then
£^.(f4‘ nc1) c .Qk.
and therefore we see from (6.42) that
£y dA; G fh-*
for (pk- +1) < ? < (7k- - On the other hand, this relation is valid also for
1 < г < />*•. because, in this case, by construction
Tg^dA; --- (ILg^Xi — 0 .
Therefore, since the dXi ‘s. 1 < ? <	, span P*., we obtain that
Lg, -W C !?f
i.e. that is invariant under pj. 0 < j < m. Since <4- is nonsingular.
and therefore smooth, in view of Lemma 1.6.3 we conclude that A* = Qk. is
invariant under the new dynamics. <
328
6. Geometric Theory of State Feedback: Tools
It is quite interesting to establish a relationship between the controlled
invariant distribution algorithm and some concepts introduced earlier, like
the zero dynamics algorithm. To this end. observe that if is a regular
point of the controlled invariant distribution algorithm, the distribution _1*
is nonsingular and involutive in a neighborhood L'° of j‘°. Thus, by Corol-
lary 2.1.6. has the maximal integral manifolds property on CD. i.e. I"'
is partitioned into maximal integral submanifolds of Let denote the
integral submanifold of which contains the point ,rc. In what follows, we
will characterize the relation existing between and the zero dynamics
manifold Z*.
Proposition 6.3.6. Suppose x° is a regular point for the controlled invariant
distribution algorithm, and dim(G(;rc)) =- m. Suppose also that
PG*) C Qk	(6.43)
(=i
for all k > 0. Then the assumptions of Proposition 6.1.1 hold, and for all
x E Z* in a neighborhood of r',
T(r) = TtZ*.
.4.$ a consequence. Z* locally coincides with the integral submanifold Lj.^ of
_T.
Proof. We prove, by induction, that if the assumption (6.43) holds, then in
a neighborhood L’c' of xc,
_Mk n Гс = {r e	= 0} .	(6.44)
This is true, by definition, for k — 0/Suppose is true for some A* > 0. Since
the differentials dX,(x) are by assumption linearly independent at ;rc. and
the matrix
col((dAi(r).g(j-))..... {dXffk(xTg(x})) = LgAk(r.)
has constant rank pk near x~. then, according to the zero dynamics algorithm.
ЛД.^1 is obtained in the following way. Let Rk(x) be a matrix whose rows
at each ;r - form a basis in the space of all vectors у such that у£3Лд-(.г) = 0.
Then
n L- = U e 17° : ли = 0,	= 0} .
On the other hand, if the assumption (6.43) is satisfied. -Од-^i is given (see
the proof of Proposition 6.3.5) by
ttk~-i = span{dA; : 1 < i < nk} + spari{d£/+S(1A, : pk + 1 < i < uk } .
Observe that, by definition of o(z) and of Rk(x)
6.3 The Maximal Controlled Invariant Distribution in ker(dh)
329
Lf+g(iAkLr) = 0 « (dA,. f + go) = 0. 1 < i < pk-
and RGx'jLj-^AiAx) - 0
» 0 = Rk(x)Lf~goAk{x}
= Rk{x}LfAk{x) -r R/AxjLyЛд.(r)o(.r)
О 0 = Rk(x)LfAklx) .
Thus.
,r G P Г° О -U(.r) = 0 and Lf ^tj(lAk{x) = 0 о .l^i(.r) = 0
and this proves the assertion (6.44). <
Remark 6.3.7. Note that, in case the condition (6.43) holds, then
so ^    +	= (7k and rk - pk. <
There are two special classes of systems which satisfy the assumption
(6.43): the linear systems, and the nonlinear systems having a relative degree
at the point xA We discuss first the case of a linear system.
Corollary 6.3.7. In a linear system, the zero dynamics algorithm and the
controlled invariant distribution algorithm produce the same result. More pre-
cisely, let I* denote the largest subspace ofker(C) satisfying
.41'* CT’- Im(B) .
Then.
Z* = V*
_l*(x) = V‘ at each x E xT .
Proof. In this case, the controlled invariant distribution algorithm proceeds as
follows. Note that the codistribution Pq = span{d/;} is spanned by constant
covector fields, namely the rows cL.	. c„, of the matrix C. Suppose1 also Рд
is spanned by constant covector fields, the ak rows of a matrix П).. Then the
intersection Рд. A GL is also spanned by constant vector fields, the (n> — pk)
rows of the matrix 5д.П in which is a matrix whose rows span the space
of solutions x of the equation
7-4д = x ll kВ — 0 .
Since gj is a constant vector field, the j-th column of the matrix R. it is
immediately deduced that
L^SiAVG, =0
and this implies (see also Remark 1.6.7) that
330	6. Geometric Theory of State Feedback: Tools
i.e. that the condition (6.43) holds. Moreover
and this shows that also f?A.+1 is spanned by constant covector fields. For
each .r. the codistributions and <?*. C'G'1 have indeed constant dimension
and any point is a regular point for the1 controlled invariant distribution
algorithm. The hypotheses of Proposition 6.3.6 are satisfied, and the result
follows. <i
We consider now the case of nonlinear systems having a vector relative
degree at a point P. In order to prove that for these systems the hypotheses
of Proposition 6.3.6 are satisfied, we prove first a property related to the
notion of controlled invariant distribution.
Lemma 6.3.8. Suppose the integers Г].........r„7 are such that the. rector
(Lg.Ly/btr) L^L^hS.-r)  L^L^hjaj )
is zero for all ,r near :r~ and for all к < r, — 1. and nonzero for к = ig — 1 at
.r = ,r°. Then, in a neighborhood Ua of the point ,r°. every controlled invari-
ant distribution contained in ker(dh) is also contained in the distribution D
defined by
D = Q p| ker(t/£pX) 	(6.451
г-1Г=1
Suppose D is a smooth distribution. 4 pair of feedback functions (n.J)
defined on 1‘ is such that
[f + gn.D] c D
,	,	(6.46j
С/ D 1 < i < in
!
if and only if
d({dLrf~lht. f(x) +	G	for all 1 < I < m
.	1	(6.47)
dlfidLf 1 h,.	E	for all 1 < i.j < in .
In particular, if the system has relative degree {n..rm} at xc. i.e. if
the matrix .4(j‘) defined by (5.2) is nonsingular. then D satisfies (6.46). with
afir) and 3{x] solutions of
_4(n)o(j-) -+- b(.r) = 0
.4(z).3(.r) = I
(6.48)
(where b(x) is the vector defined by (5.9)).
6.3 The Maximal Controlled Invariant Distribution in ker(dh) 331
Proof. Let be a locally controlled invariant distribution contained in
ker(dfi). Then, by definition, Л с (spanjd/q})1- for all 1 < i < m.
Moreover, for some locally defined feedback a. [/.-1] C _L Suppose _1 C
(span{dL^hj})- for some fc < r( - 1: then, using the property
L/^L^h, =1^'11,
we have, for any vector field r of J.
0 = (dLKfhh[f.r]} = Lj(dLkfhi.T) - (dLjUfhi.r) = -(dL^h^T)
i.e. _i c (span{d£Jr4-1 ht] . This proves that _i C D.
Now, suppose there exists a pair of feedback functions that makes (6.46)
satisfied. Let ~ be a vector field in D. Then
(dLfhj. т) = 0
= 0
{dldfhi^gj.r]) = 0
for all 1 < i.j < m. 0 < к < Fj — 1. From the second one, written for
к = r, — 1. we deduce
0 = Lj{dLy-1hi^T') - (dLfLy-'hi.T) = -W(dLrf--'h,.f + ga}'l.T)
i.e. the first condition in (6.47). Similarly, from the third one we obtain the
second condition in (6.47). Conversely, if the conditions (6.47) hold, then the
previous equalities are true for к = rt — 1. For other values of к < гг — 1.
these equalities hold for any feedback (o.5) by definition of r;. Thus, we
deduce that D is invariant under f and pit 1 < г < m. if and only if (o.T)
are solutions of (6.47).
The third part of the statement is a trivial consequence of the second
one. In fact, if the matrix .4(j) is nonsingular for all z in a neighborhood of
r°, the equations (6.48) have a (unique) solution, and this solution trivially
satisfies (6.47), because in this case
(dL^1 hi(x), f(.T) + 5(z)o(.r));	(dLrf’~lhd^^9^)d(x)}j)
are constant. <i
Using this Lemma, it is not difficult to see that, in the case of a system
having relative degree {tq....rm} at a point rc. the condition (6.43) is
satisfied. This and other properties of interest are collected in the following
statement.
332	6. Geometric Theory of State Feedback: Tools
Corollary 6.3.9. Suppose the system has relative degree {/‘i,....	} at a
point A . Then this point is a regular point of the controlled invariant distribu-
tion algorithm and the condition (6.43) holds. In particular the largest lo-
cally controlled invariant distribution, contained in kor(dh). can be. expressed,
in a neighborhood L J of x'~ . as
m r,
J- = П П kcrtrfi1/1/!,)
and is rendered invariant by the standard nonintemotive feedback (5.28). The
results of Proposition 6.3.6 apply and,
Z’ = {.r G C= : L^h^x} = 0. 1 < к < r(. 1 < i < zn} .	(6.191
Proof. It is left, as an exercise, to the reader. <
Remark 6.3.8. It is immediate to check that, the results stated in the last parr
of Lemma 6.3.8 an1 also valid in case the system has a number p of output-
which is less than the number zn of inputs, provided that the matrix (5.2)
has rank p at the point .rc (see also Remark 5.1.3). Thus, in particular, they
are valid for a system with only one output y, ~ ht(x}. because in this case,
by definition, the matrix in question reduces to a single nonzero row. As a
byproduct, it is found that the largest locally controlled invariant distribution
contained in ker(d/q). noted _J*. has the form
J* = P| ker(dLp'fo) .<	(6.50)
A.-1
In general, if the condition (6.43) is not satisfied, it is not possible to
identify the zero dynamics manifold Z* with an integral submanifold of d*.
In other words, the problems of using feedback in order to constrain the
output of a system to be zero for a certain time and the problem of using
feedback in order to induce a certain amount of unobservability, in a general
nonlinear setting, are not equivalent (although in a linear system they tire, as
the statement of Corollary 6.3.7 shows). There is however always a relation
between Z* and the integral submanifolds of Д*. which is expressed in the
following statement.
Proposition 6.3.10. Suppose .r= is both a regular point for the controlled
invariant distribution algorithm, and for the zero dynamics algorithm. Let
Lj° denote the integral submanifold of A* which contains the point xc. Then
is a locally controlled invariant submanifold and h(x) = 0 at each point
j’ G . i.e. Lx-> is an output zeroing manifold for (6.1). A.s a consequence.
Lj.-> is locally contained in Z*.
6.4 Controllability Distributions 333
Proof. Recall that, if the assumptions are satisfied. (A+)^ is spanned by the
differentials of certain functions A,. 1 < / <	. Thus, for some neighborhood
U° of
= {.r e C'z : A((.r) - AfU': ). 1 < i < tn- }
Suppose o(.r) is a function which solves the first equation in (6.41) for A' = A*.
Since f{rc) = 0 by assumption, we can always suppose that a(.r:”| = 0.
and therefore that the point .rc is an equilibrium of the vector field J(.r) =
/(r) + <7lA‘)a(.rl. The statement of Proposition 6.3.5 says that the distribution
J* is invariant under the vector field f(.r). and therefore, according to the
interpretation of invariance given in section 1.6. the flow of ft ri locally carries
into another integral submanifold of A*. But the point rc' is fixed under
the flow of /(j‘) and we conclude that the flow of f[.r) carries £.(>o into itself:
in other words. f(r) is tangent to .
We haw1 found in this way a smooth mapping, namely а (.г). defined at
each point of near .r=. with the property that /(.r) -n ,q(j‘)o(~c) is tangent
to L;°. Thus £.r= is locally controlled invariant. The other statements are
immediate consequences. <
Note that the previous analysis also clarifies, to some extent, the differ-
ence between the notion of a controlled invariant submanifold and that of a
controlled invariant distribution.
6.4 Controllability Distributions
A distribution A is said to be a controllability distribution on Г if it is invo-
lutive and there exist, a feedback pair (a. 3) defined on fc and a subset I of
the index set {1.....m} with the property that dnG = span{tfl : i G /}.
and A is the smallest distribution which is invariant under rhe vector fields
f. (fl ,.... gm and contains g; for all i e I.
A distribution A is said to be a local controllability distribution if for each
.rc e L' there exists a neighborhood of .rc with the property that A is a
controllability distribution on L’°.
It is clear that, by definition, a (local) controllability distribution is (lo-
cally) controlled invariant. Therefore, according to the result of Lemma 6.2.1.
such a distribution must satisfy (6.27) and (6.28) (recall that the necessity of
these conditions is not dependent on the assumptions made in Lemma 6.2.1
but only on the property of controlled invariance and on the nonsingularity
of 3). The main purpose of this section is to study under what additional
conditions a given distribution satisfying (6.27) and (6.28)) turns out to be
a local controllability distribution. To this purpose, it is useful to introduce
the following algorithm.
Controllability Distribution Algorithm. Let A be a fixed distribu-
tion. Step 0: sot So ~ А П G. Step A: set
334	6. Geometric Theory of State Feedback: Tools
m
Sk = Л П ([/. Sfc_i] +	S’*—i] + G).	(6.51)
j=i
Lemma 6.4.1. The sequence (6.51) is nondecreasing. If there exists an in-
teger k* such that Sk- = then Sk = for all k > k*.
Proof. We need only to prove that Sk Э Sk-i- This is clearly true for к = 1.
If true for some Av then
m	m
([/ 5t] + £[.9,. S,.]) э ([/. Sa-j ] + £[9,, Sa-J)
>1	J=1
and, therefore,
Э .<i
Remark 6-4 T Note that we may as well represent Sk as
ГП
Sk = An ([/,St_i] + Y. [Sj' S*-i ] + G) +
j=l
or as
m
Sk = A P ([/,Sa--i] +	[g j • Nt-]] + Sa-_i + G).
1=1
The last one of these formulas derives from the first one and from the modular
distributive rule, which holds because Sk-i C A. <
As we did for the algorithm (6.35) we introduce now a terminology which
will be used in order to express both the convergence of the sequence (6.51)
in a finite number of stages and the .dependence of its final element on the
distribution A. We set.	(
5(A) = (So + Si + ... + Sk + - ..)	(6.52)
and we say that 5(A) is finitely computable if there exists an integer k* such
that, in the sequence (6.51), Sk- — 5д--+]. If this is the case, then obviously
S(A)=Sk-.
An interesting property of the algorithm (6.51) is the following one.
Lemma 6.4.2. Let f. ifi ..... gm be any set of vector fields deduced from
f-gi- ’-iCJm by setting f = f + get and g; = (g3fi, 1 < i < m. with .3
invertible. Then each distribution Sk of the sequence (6.51) is such that
m
Sk = j n ([/. SV-!] +	St-,] + G) .
J=l
6.4 Controllability Distributions 333
Proof. Let. т be a vector field of Sk-\- Then, we have
[M] = If + П = lf~r] +	- [LrCijjfjj)
J=l
m
= [(.y,3);.r] =	- (Lr,3ji')gj) .
Therefore
Hi	ni
[f - Sk-i] +	([,9j  Sk-1 ] + G C [/, Sr._i I -e ([<jj •	-1 ] + G .
j=i	j=i
But. since 3 is invertible, then f — f — g.3~Gi and gi = (g3~l)i so that,
by doing the same computations, it is found that the reverse inclusion holds.
The two sides are thus equal and the Lemma is proved. <
From this it is now possible to deduce the desired 'intrinsic" characteri-
zation of a local controllability distribution.
Lemma 6.4.3. Let. Л be an involutive distribution. Suppose _J, G. _i + G
are nonsingular and that S(_\) is finitely computable. Then 3 is a local con-
trollability distribution if and only if
[f.A] c J + G
[g;. J] C -d + G 1 < i < m	(6.53)
SM) = -3.
Proof. Necessity. Suppose _i is a local controllability distribution. Then it
is locally controlled invariant, and (6.27) and (6.28) are satisfied. Moreover,
locally around each ./ there exists a feedback pair (ci, 3) with the property
that _i П G = span{g;.z e I}, where I is a subset of {1..... tn}, and _i is
the smallest distribution which is invariant under /. <h..  , gm and contains
ep for all i € I. Consider the sequence of distributions defined by
Л = JPG
. r ? . -i	V^r-	i • i	(6.54)
A	= [f • A-l] +	/	-V--C	+ A-I	-
!=1
It is easily seen, by induction, that
A c J
for all k. This is true for к = 0 and, if true for some A- > 0. the invariance of
J under f,ch.......gm shows that _lfe+i C A Therefore, one has
A = -^ П ([/. A-ij +	-ifr-1] + -V’-i + G)
;'=1
336
6. Geometric Theory of State Feedback: Tools
i.e.. from Lemma 6.4.2 (see also Remark 6.4.1)
Ла- = Sf,..	(6.55 j
Note also that, by definition. Л(> = span{<fi : i G I}. Thus, the sequence of
distributions generated by the algorithm (6.54) is exactly the same as the one
yielding (f.g\.......|span{,g, : I G I}). the smallest distribution invariant
under f.g}......gm and containing span{g, : i G I}. From (6.55) and from rhe
assumption that 5(Л) is finitely computable we know that there L an integer
k* such that Ла-- = Лд.^]. Therefore, in view of Lemma 1.8.2. the largest
distribution in the sequence (6.54) is exactly (f,gi......£m|span{(fi : i G /}).
From this, one concludes that the largest distribution in the sequence (6.54)
must coincide with Л. i.e.. again from (6.55). that the last condition of (6.53)
is satisfied.
Sufficiency. We know from Lemma 6.2.1 that if Л is involutive. if Л and
G + Л are nonsingular and if the conditions (6.27) and (6.28) are satisfied,
then locally around each .r there exists a pair of feedback functions (a. i)
with the property that Л is invariant under f.g^........g)7l. From this fact one
may deduce that
Л (~) ([f. St. i: +	[g>. Sa-—i j + G*) + Sk-1
= [/•	 Sr_i] + Л P G т S>-_i
j = L
(=1
In view of Lemma 6.4.2 and Remark 6.4.1. this shows that
-$7- — If • i +	[g,, 5a -i j + Si- - i .
;=1
Without loss of generality, we may assume that g\.........g,„ are such that
Л Pl G = span{.9( : i G /} for some index set I. In fact. Л P G is nonzero
because otherwise 5( Л) would be zero, thus contradicting the last of (6.53).
Since Л n G is nonsingular. one may find a new feedback function 3 and
construct new vector fields	1 < <’ < hi. such that, for some index
set I. span-ft/; : i 6 J} = Л P G and gt = gt for i I. This new set of vector
fields still keeps Л invariant, because g; G Л for i G I and Л is involutive.
So. .Sb = G Г) Л = span{(b : i G I}, and the sequence of distributions Si
coincides with the sequence of distributions yielding (J.g]....,c/T,Jspan{t), :
i G /}). Since, by assumption, for some A’*. Si- = we deduce from
Lemma 1.8.2 that Sa- is the smallest distribution which is invariant under
f. gx....g,r, and contains span{</j : i G I}. But the last condition of (6.53)
says that Si- coincides with Л and this completes the proof. <
6.4 Controllability Distributions
337
In view of rhe use of the notion of local controllability distribution in
problems of decoupling or noninteracting control, it is useful to bo able to
construct a •‘maximal" local controllability distribution contained in a given
distribution. To this end one may use the1 following result.
Lemma 6.4.4. Let Д be an involutive distribution. Suppose G. A G + Д
are nonsingular and
[/.J] c A-G
[у,. J] c J + G 1 < i< m .
Moreover, suppose S(Al is finitely computable and non.si.ngu.lar. Then S(A)
is the largest local controllability distribution contained in Д.
Proof. As in the proof of Lemma 6.4.3 (sufficiency) it is easily seen that the
assumptions imply that locally around each .r there exists a pair of feedback
functions with the property that A Cl G = span{^; : i 6 /} and 5(A) is
the smallest distribution which is invariant under f.g^.....gin and contains
span{.g( : i 6 /}. Moreover, since
span{t9;  i e 1} Q 5(A) C A
and AnG — span{.g, : i 6 I}, it is seen that
5(A) л G = span{ch  i & 1} 
Thus 5(A) is a local controllability distribution.
Let A be another local controllability distribution contained in A. Then,
by definition, in a neighborhood of each ./ there exists a feedback pair
(a, 3) with the property that А П G = span{(h : i E 1} for some subset I
of {1 m}. and A is invariant under f.fp.....gtn. where f = f + go and
9i —	for 1 < i < m. Consider the sequence of distributions
A() = span]/), : i e /}
Ад = [/• Ад.-цAfc-j] + A/._! .
?=i
Note that Ад C A G A. Thus
m
Ад С A л ([f. Ад_] 1 +	[ед. Ад_t] + Ад_i + G) .
i=i
Since Ao = A Л G G А Г G = Sq, it is easy to show, by induction, by means
of Lemma 6.4.2 and Remark 6.4.1. that Ад С 5д for all к > 0, i.e.
Ад сад •
338	6. Geometric Theory of State Feedback: Tools
Xow recall (see Lemma 1.8,3) that there exists a dense subset of Гс with the
property that at each j. 3(x) = -ViW- Thus, we have that
J(j') CS(»
for all r iii a dense subset. Since 3 is smooth and 5(3) is nonsingular, this
implies J C 5(J). <
Using the same arguments it is also possible to prove the following char-
acterization of the maximal controllability distribution contained in a given
distribution 3.
Lemma 6.4.5. Let Д be an involutive controlled invariant distribution. Let
(a, 3) be a pair of feedback functions such that
[/. j] c j
C 3 for 1 < i < m
Д П G = span{th : z G /}
for some suitable subset I o/{l........m}. Consider the sequence
Jo = span{(/i : i G /}
Jfc — Jfc-i + [f, Jfr-i] +	•
f=i
Then 5(3) is finitely computable if and only if Зд. - = Jf + i for some k*. If
this is the case. 5(3) = Л-. Moreover, if Дк- k> nonsingular, then Дк- is
the largest controllability distribution contained in Д,
The previous results can be used, for instance, in order to find the max-
imal controllability distribution in ker(dh). if so is requested. To this end,
using the results of section 6.3 one first finds - provided the assumptions of
Lemma 6.3.3 are satisfied -- the distribution 3*, which is the largest locally
controlled invariant distribution contained in ker(dh). Then, using Lemina
6.4.4. it is possible to conclude that if 5(3*) satisfies the assumptions of this
Lemma, then 5(3*) is exactly the largest local controllability distribution
contained in ker(t//i). In fact. 5(3*) is not only the largest controllability
distribution in 3* but also the largest controllability distribution in ker(dh)
because any controllability distribution contained in ker(dh). being locally
controlled invariant, must be contained in 3*. Alternatively, using Lemma
6.4.5. one can compute a feedback (a. 3) which renders 3* invariant, and
then find 5(3*) by means of the algorithm (1.39). In this case, the finite
computability of 5(3*) is implied by the existence of an integer к* such that
3e = 3p + 1-
7. Geometric Theory of Nonlinear Systems:
Applications
7.1 Asymptotic Stabilization via State Feedback
In this Chapter we show how the concepts introduced and developed in Chap-
ter 6 can be effectively used in rhe solution of a number of important syn-
thesis problems. We begin by considering the problem of local asymptotic
stabilization at a certain equilibrium point. Our purpose is to extend the
results developed in section 4.4. by showing that if the zero dynamics of a
system are asymptotically stable at this point, the system itself can be lo-
cally asymptotically stabilized via state feedback. Of course, as stressed at
the beginning of that section, our results are of special relevance only in case
the linear approximation of the system is not stabilizable.
To this end. suppose that the system satisfies the regularity assumptions
described in section 6.1. so that the functions (6.18) can be taken as a (partial)
set of local coordinates around the point Л and the generalized normal
form illustrated in Proposition 6.1.5 can be defined. Suppose, without loss of
generality, that — 0 and choose the input и which satisfies the equations
6'(j*) + a'(j)u = c; ,	1 < i < m ,	(7-1)
where the b1 (.r)'s and o'(t)’s are defined as in (6.20). Note (see (6.23)) that
the input thus defined is related to the (unique) input u*(j). which imposes
the vector field /*(z) = f (т) + д(х)и*(х) to be tangent to the zero dynamics
manifold Z*. by the following relation
и = u*(r) + .4“* (jt)t’	(7.2)
where .4(t) is the matrix (6.22). The effect of this feedback is to modify
the normal form of the equations describing the system into one having a
structure of the following type (recall that, on the right-hand sides. ./ stands
for	and г = (£*....,	77))
340
7. Geometric Theory of Nonlinear Systems: Applications
ё =
ё	+ Ь-21‘2 - ZAi(-F)Ci + 5\> (.г) f U * (.Г) + А-1(т)г)
о
ш — 1
&т"4 Ьт (х) (и (.Г) -4- .4 ‘(.Г)С)
2 = 1
/о(А + <7о(.г)0*И + А"1 Ис)
in which
/ 0 1 0    0 \
0 0 1    О
0 0 0	-1
\о о о  о/
and
/ \

for 2 < i < т. 1 < j < /77 — 1. Since the coefficients are vanishing at each
z 6 Z+. so are the matrices S((t). In view of the fact that, by construction.
u*(0) = 0 and 5/(0) = 0 for 2 < i < m. it is immediate to observe that the
linear approximation at ~ — 0 of the first m sets of the equations thus found
is controllable. As a matter of fact, the equations in question have the form
ё = Л1ё-^1А
ё = Азэё b2v2 + BL>1 (0)t'l + h(z) + g2(z)u
C" — m £’rl + by„ l'„i a- Dni 1 (0) t'l + ... + 7dm.m- 1 (0)l’n< -1
+ fm (r) + Sm (г)г
with g,(z) vanishing at 2 = 0, and /;(z) vanishing at z = 0 together with its
first order derivatives, and the pair
Mu 0
0 A2
\ о 0
0 \		/ b,	0	 °\
0	в =	0-21(0)	b-2	 	0
Л...J	\O,„i(0) D,„,(0)  Ь,„/
is indeed a controllable pair.
Set now
е = сокё.--..г) -
rewrite the equations in question in the more condensed form
7.1 Asymptotic Stabilization via State Feedback 341
£ = A£ + Bv + /(£. t/) + g(^.7])v
0 =	+p(^T])v
and note that, by construction.
7/ = <7(0.7?)
characterizes the zero dynamics of the system. Moreover /(0,?/) — 0.
We easily deduce from this that, if the zero dynamics of the system are
asymptotically stable, any linear feedback
c = F£
which stabilizes (.4 + BF) will also asymptotically stabilize the equilibrium
(C^) = (0,0) of (7.3). In fact, the corresponding closed loop system will have
the form of the equations (B.8). and the hypotheses of rhe corresponding
Lemina are satisfied.
For convenience. the result thus established is summarized in a formal
statement.
Proposition 7.1.1» Consider a nonlinear system of the form (6.1). Sup-
pose /' 75 a regular point for the zero dynamics algorithm. Suppose zs an
asymptotically stable equilibrium of the zero dynamics- Then, there exists a
matrix F such that the feedback
и = u*(x) + A-1 (ir)F£(.r)
asymptotically stabilizes the corresponding closed loop system at the equilib-
rium point x = j°.
We stress again that this result - as the corresponding result presented in
section 4.4 - does not require asymptotic stability in the first approximation
for the zero dynamics, so that it may be useful in order to solve critical
problems of local asymptotic stabilization.
Remark 7,1.1. The result we have established, namely the fact that the linear
approximation ar x = 0 of the first equation of (7.3) is controllable, can be
interpreted as a nonlinear version of a property of linear systems that was
already observed at the end of Remark 6.1.3. namely the controllability of
the pair (.4ц. BY) in (6.9). <
Remark 7.1.2. Observe that, by means of essentially the same arguments
as the ones used above, it is possible to prove the following result. Let j10
be a regular point for the zero dynamics algorithm for the system (6.1).
and suppose this system has an asymptotically stable zero dynamics (at the
equilibrium point j ~ xc). Then, there exists a smooth mapping k :	*->
Ж". where is a neighborhood of x°. such that the dynamics of (6.1). with
output
342
7. Geometric Theorv of Nonlinear Systems: Applications
у = A’(.r)
has relative degree {1..... 1} at and a zero dynamics which is still asymp-
totically stable fat the equilibrium point ,c = .rc). The proof of this is left as
an exercise to the reader. <
7.2 Disturbance Decoupling
A major outcome of theory of controlled invariant distributions, developed in
section 6.3. is the synthesis of feedback control laws which render the output
of a system independent of certain disturbances. Given, as in sections 4.6 and
6.3. a system of the form
t =	+p(r)ir
У = М-И
the matter is to solve following problem.
Disturbance Decoupling Problem. Consider a system of the form
(7.4) and a point ,rc. Find, if possible, a regular feedback of the form u —
o(t) + defined in a neighborhood Г of which renders the output
у independent of the disturbance ir.
The discussions at the beginning of section 6.3. together with the prop*
erties established in Lemma 6.4.2. already provide the desired answer which,
for convenience, is summarized in the following statement.
Proposition 7.2.1. Suppose A* is finitely computable and A* and A* + G
are nonsingular in a neighborhood offjr0. Them the disturbance decoupling
problem is solvable if and only if
p € A*	(7.3)
in a neighborhood of G .
Note that, under the slightly stronger assumption that the point G is a
regular point of the controlled invariant distribution algorithm one can easily
construct, by means of the procedure described in the Proposition 6.3.5. a
state feedback which solves this problem. Note also that the solvability of the
problem does not require at all that the system has some relative degree at
the point .r3. Of course, if the system has a relative degree at. .r°. then the
distribution J* is rendered invariant by the standard noninteracting feed-
back (see Corollary 6.3.9). and if the condition (7.5) is satisfied this feedback
provides also a solution to the disturbance decoupling problem. We recover
in this way the preliminary results already established in Chapters 4 and 5.
7.2 Disturbance Decouplin;
343
Example 7.2.1, Consider again the system described in the Example 6.3.4
and note that the system in question does not have a relative degree at x°.
The disturbance decoupling problem will be solvable if and only if the vector
field p(x) has the form
pi.r) = соЦО. И. 0 ,pi(x).R>( -c))-
Suppose this is the case. In order to solve the1 problem, one has to find a
feedback which renders Л* invariant. To this end. note that, performing the
controlled invariant distribution algorithm, we already obtained
h (j’) —	(x) = span {t/A}. r/A'j. dX^}
with
Al — X [ . Аз = J';j . A3 = J’-J
and
span {dX ]. dX2} T G * - 0 .
Thus, according to rhe results illustrated in the last parr of Proposition 6.3.5.
a feedback which renders J* invariant is a solution of
dAi
dX2
dX]
dX2
(f(x) +	=
p(x).J(x) = Q
° A
0 /
and d(x) rhe identity matrix. The corresponding closed loop system will then
be
-z3r3 ЛРЧ
i'3	=	1'2
i‘4	= J'i -грз --rpr-i +^51’1 +	+p.i(.r)u'
x-y = Xi ~ x-j — x^x-fx^x.x +	+ x2X'pv2 + po(x)m
and its decomposed structure shows that the output depends on a set of
state variables (./]. .r2. <3) which are independent of the others [./'.(,./'5) ami
not affected by the disturbance. <
344
7. Geometric Theory of Nonlinear Systems: Applications
7.3 Noninteracting Control with Stability via Static
State Feedback
In section 5.3. we have addressed the problem of finding a feedback law which
renders the input-output behavior of a nonlinear system with m inputs and
tn outputs equivalent to that of an aggregate of m independent single-input
single-output subsystems. In particular, we have seen that this problem is
solvable (locally around a point in the state space) by means of static
feedback, i.e. hy means of a feedback of the form
u = a(jj +	(7.6)
if and only if the matrix (5.2) is invertible at j-°. i.e. if and only if the system
has some vector relative degree {rlt... .rm} at this point. A solution of this
problem is provided by the standard noninter active feedback
u - А-1(;г)(-Ь(т) + r)	(7.7)
in which A(j) and &(т) are given by (5,2) and (5.9).
At the end of the same section, we have also pointed out that, if rhe zero
dynamics of the nonlinear system are asymptotically stable, it is easy to find
a feedback which, simultaneously, renders the system noninteractive from the
input-output point of view, and also internally asymptotically stable. In fact,
it suffices to add to the standard noninteractive feedback law a control of the
form (see (5.29))
c = col( t’l..cm)
with
G = -cloh,(x} - c\Lfh,(x) - ... - c^Ly -1Н;(т) + c; .
However, as stressed in the Remark 5.3.3. the hypothesis that the zero dy-
namics are asymptotically stable njtay not be necessary in order to obtain
nonintcracring control with stability and. in fact, there may be cases of sys-
tems having unstable zero dynamics, in which the simultaneous achievement
of these two goals is still possible.
We wish now to discuss this problem in more detail. For convenience, we
start with a formal definition. As in section 5.3. we suppose that the point
at which the problem is to be solved is an equilibrium point of the vector field
/(j). that h(xc) = 0 and that the feedback (7,6) preserves this equilibrium,
i.e. a(r°) = 0. Moreover, without loss of generality, we assume .r° = 0.
Problem of Noninteracting Control with Stability (via Static
Feedback). Consider a nonlinear system of the form (5.1). Find a regular
feedback of the form (7.6), defined in a neighborhood of j = 0. with n(0) = 0,
such that
(i) the equilibrium point .r = 0 of
j- - f(x) + ^(j)q(z)
7.3 Noninteracting Control with Stability via Static Feedback 345
is asymptotically stable in the first approximation.
(ii) the closed loop system
i = /(.r) + ^(j)o(t) -+- д(г)Л(т)г
У =
has a vector relative degree at the equilibrium point т = 0 and. for each
1 < i < hi, the output yt is affected only by the corresponding input r(- and
not by Vj, for any j i.
Remark 7.3.1. Note that, in view of the results illustrated in section 13.2
(and already utilized in a similar way in Remark 5.3.4). the fulfillment of (i)
guarantees that for each s there exist d and К such that
||t(0)|| < <5. |i’i(t)| < /С for all t > 0.1 < i < m
implies |b(/)|| < г. for all f > 0. in the corresponding noninteractive closed
loop system. <
We begin by identifying a necessary condition for the solution of such
a problem. The idea is to establish some common features of all feedback
laws which solve the noninteracting control problem, i.e. satisfy requirement
(ii). and then to check whether or not the fulfillment of requirement (i) is
compatible with the features thus found.
First of all. it will be shown that, with any system of rhe form (5.1) which
is noninteractive and has a vector relative degree at j = 0. it is possible
to associate certain objects (more precisely, certain distributions), which are
left unchanged by any regular static feedback which preserves the property of
noninteraction.
Consider a system of the form
m
j- = fO) + У !)j(Ouj
J-l	<1 -S>
yi = hj(j').	1 < i < ni .
and suppose this system is noninteractive and has vector relative degree
.......at i = 0. In view of the results described in section 5.3, this
is equivalent to assume that, for any J Л L^h^x) = 0 and
L4jLTr . ..LT{hi(r) = 0
for all r > 1 and any choice of the vector fields гг.....Г! in the set {/. g\.
  and - in addition that
L9t L^hi(x) =0 for all < r, - 1, and L3t Z^-1/?,(0)	0 .
Suppose this system has been composed with a regular static feedback
346
7. Geometric Theory of Nonlinear Systems: Applications
u = a(;r) 4- ,3(.r)r	(7.9)
to yield a noninteractive closed loop system (which necessarily has vector
relative degree {n.....r,n} at .r = 0) and let the latter be denoted by
уг = ht(x), 1 < i < m
with
/hr) - f(:r] -у ^2 Ы’ФнИ. с)/,т) =	.
A--1
Set.
= <f,9i......^Ispanf?; : j ;})	1 < ? < ш.
?71
1 = 1
and, respectively.
Ff = (f-ffi........<hn|span{pj : j # 0).	1 < i < m.
m
= n /?
Then, the following result holds.
Proposition 7.3.1. Suppose (7.8) is noninteractive and has vector relative
degree {g r,„} at x = 0 and suppose also (7.10) is noninteractive. For
each 1 < i < m. P* = Pf. As a consequence. P* = F* .
In order to prove this result, we need a preliminary lemma, which estab-
lishes some interesting features of any regular static feedback which preserves
the property of noninteraction.
Lemma 7.3.2. Suppose (7.8) is noninteractive and has vector relative de-
gree ]ri...., rm} at x = 0 and suppose also (7.10) is noninteractive. Then.
.ЗгД0) () for all 1 < i <m. Moreover, for any I j
;313(х) = 0. Lg3 3lt(x) = 0. o,U) = 0
and
Тд:ТГг .. .Ьт,3ц{х) = 0. Lg,L-T .. .L-.afix) = 0
for all г > 1 and any choice of the vector fields r7...... ту in the set {f.gi,
  , 9 m}-
7.3 Noiiinreracting Control with Stability via Static Feedback 347
Proof. Observe that by definition since (7.8) is noninteractive.
= bjfr) + пг;(.г)о,(.г) + alt(x) ^2	•
where M-r) = L’f hji.r) and = L^iL!f~}hl(,r). Moreover. an;(0) 0.
Since (7.10) is noninteractive as well, necessarily 3,j(.r) = 0 for i j. As a
consequence, since 3(0) is nonsingular. 3„(0)	0,
To prove all other identities, one should proceed as follows. First of all,
using tin1 property that (7.8) is noninteractive. observe that
L j	h, = a;; 3;;
Ту f	tqj (3i;) + (Lg ci!; ) 3113(i
L f Lgt L jr	hi — a it[ L ^Зц ] + (L )3ц + (L ci; j )o f 3t,
from which, by induction, it is not difficult to realize that, for any choice of
vector fields rr......rL in the set {f~gt}.
L-,.--- L71 L,h L’f 1 ht = alt (LTr. • L713lt) + dr
r
whore or is a product of the form
dr = (T• Lf/j о,;)    {LTp   • LTi 3tl)   	,, •   Lиj cit)
with 0s......0! in the set {f.fl,}- with tp.........n in the set {f.g,}. with (t7.
.... fT! in the set {f. cp}, Moreover, p < r and q < r.
In a similar way. starting from
Td/q = a,,<->, + b,
L^, L f h i = it a g г} + (L4; a,i Jo ,3,, + (L3i b,) 3 ц
LfLrfhi = (ijPLfoP + (I/tzu)a, - (I^.n/JOjO/ + {Pfb,) + (Lr/,b; mt
it is not difficult to deduce that, for any choice of vector fields tt....... н in
the set
L-.    LTxLrj'h' = a„(L-r   • T-._o,) +
r r
where Ok is a product of the form described before and 17. is a product of
the form
t’A- = (to   ‘ н 1 )    (TTp • ‘  Lт-j 3ti) -   (L/jf ‘ • - Lff,оj) .
From these, using the fact that (7.10) i_s noninteractive. and therefore for
any j i
= 0. L-4jLT„ L71L’)hi =0.
• j	J	j
one can inductively prove that all the identities indicated in the Lemma hold.si
348	7, Geometric Theory of Nonlinear Systems: Applications
We ('an proceed now with the proof of the Proposition.
Proof, Observe that gt = д^ц. Thus,
span{Sj	span{£j : j i} .
Now, take any vector field 0 e P,*. By Lemma 1.8.4, on some open and
dense subset P* of the set L’ on which the vector fields of (7.8) are defined, r
can be expressed in the form 0(x) = Y^k=i ck(P$k(Pp where 0k are vectors
of the form
0k = [ty.. L .., [ту, ,9j]]]
with ~r.....Ti in the set {f,g\......g,n} and j P i<
Note (see e.g. section 3.3). the identities established in Lemma 7.3.2 imply
(doy. [тг. [,... [71..gj]]]>(j-) = 0
for i j. Thus, Lf}Cii(r) is identic'all у zero on P* and therefore on P. because
is a smooth function.
Therefore
m
LM = [f.0]	ё P* 
J=1
In a similar way. one can show that
[ук-в]ер;
for every 1 < k < m. Thus, Pf is invariant under f,gi........gtn and contains
span{()j ; j ф i}, It follows that
Ff C P* .
Since the feedback which relates (7.8) and (7.10) is invertible, one can
reverse the roles of (7.8) and (7.10). to show that
F( * C P *
and this completes the proof. <
Consider now a system of the form (5.1) for which the problem of non in-
teracting control is solvable by means of static feedback (i.e. a system having
some vector relative degree at r = 0). and let и = o(j-) + be any
feedback which solves the problem in question. Set
f(p = /(.r) + ^^(j')oa.(t).	д3(;г) = ^2 ЫЛЛ-Д.Г) .	(7.11)
fc-l	A=1
The result expressed by Proposition 7.3.1 enables us to claim that the distri-
butions
7.3 Noninteraciing Control with Stability via Static Feedback
349
....j'4- ?}).	1 < i < m .
art1 independent of the choice of ci(.r) and J(.r) (so long as the feedback they
define is a solution of the problem of noninteracting control). In fact, any
other feedback law u — a'(-c) -1- .T(.r)r' which solves the problem in question
t'an always be viewed as a composition
{[ ~ n(.r) -+- >)(a(.r) + J(.r)r')
of the law n = <a(.r) -r J(.r)r used to define the vector fields f(j').	......
gfll (r). with the law
г - d(j‘) + J(j’)c' = J-1 O)(-n(.r) + o'(j') + J(.r)r')
and the latter, by definition, is a feedback law which preserves the property
of noninteraction.
In other words, we can conclude that for any system the form (5.1) for
which the problem of notiinteracting control is solvable by means of static
feedback, the distributions
F,* = (f-di.........|ьрап{Д, : j # i}).	1 < ? < zz?.
(with /(j‘). zb (t).defined as in (7.11), for some choice of a feedback
law u. = n(r) + J(.r)c which renders the system noninteractive) are well-
defined objects, independent of the choice of n(r) and T(t), The next step
is to show that the distribution F* is helpful in identifying an obstruction to
the solution of the problem of noninteractive control with stability via static
feedback.
To this end. we need to show some additional properties of the distribution
F*.
Lemma 7.3.3. Consider a system of the form (5.1) and suppose it has rel-
ative degree {/q....rni} at r ~ 0. For each 1 < i < m. define a distribution
J’ as follows
j;(r) - P| kerfdLp^Gr)) .	(7.13)
A’ = l
Then P* Cd- and

i = i
Proof. On some open and dense subset U* of the set U on which the vector
fields (7.11) are defined. F* is spanned by vector fields of the form
9 = [ТГ,[.. . .[Л.р;]]]
350
7. Geometric Theory of Aonlinear Systems; Applications
with тг....,т\ in the set {/. ifi...., g,n} and j i. Since the system char-
acterized by the vector fields (7.11) is noninteractive. any vector field of this
form is such that
0 = (dL^p.Vfix) = (dLkf-lhh0)(T)
for all 1 < A < r(. Thus. 9 € -1*. i,e. Pt*(x) C -A’(j-) for all r G C*. But
is nonsingular (as a consequence of the property that the matrix (5.2) is
nonsingular) and therefore it is concluded that P* C -1*. The other property
follows immediately. <J
Lemma 7.3.4. Consider a system of the form (5,1) and suppose it has rel-
ative degree {/q...., rm } at x = 0. Let a = q(t) + T(ir)r be any regular state
feedback which solves the problem, of noninteracting control. Let f(x), ,71 (-r)-
.... ginfir) be the vector fields defined by (7.11) and let P* be the distribu-
tion defined by (7.12). Suppose x = (J is a regular point for Pf....Pf.P*.
Then, in a neighborhood U° of the point x = 0. P* is involutive. and U° can
be partitioned into maximal integral submanifolds of P*. Let S* denote the
integral submanifold of P* which contains the point x = 0. The submanifold
S* is locally invariant under the vector field f(x). and the restriction of f fir)
to S* does not depend upon the particular choice of
Proof. Each Pf, being nonsingular, is an involutive distribution (see Lemma
1.8.5). Thus, also F* is involutive. Moreover, since each P* is invariant under
f. so is also P*, Therefore, according to the interpretation of invariance given
in Section 1.6. the flow of ffir) locally carries the integral submanifold 5* into
another integral submanifold of F*. But the point x = 0 is fixed under the
flow of f(x) and therefore it is concluded that the flow of ffir) carries S*
into itself; in other words. S* is locally invariant under ffir). The last part
of the statement follows from Lemma 6,2.3 because the nonsingularity of the
matrix (5.2) implies	/
dirn(G) = m
F* П G C П G = {()} .u
The property thus found immediately implies a necessary condition for the
existence of solutions of the problem of noninteracting control with stability
(via static feedback). In fact, this property essentially establishes that for
each system in which the Noiiinteracting Control Problem is solvable it is
possible to identify a submanifold 5*. the integral submanifold of F* through
j = 0, with the property that, in any closed loop system which has been
rendered noninteractive via static state feedback, the vector field f(x) leaves
S* invariant and the restriction /(.r)|s* of f(x) to S* is always a fixed vector
field, independent of what feedback law is chosen to obtain noniteraction.
As a consequence, the requirement (i) of asymptotic' stability in the first
approximation can only be achieved if the equilibrium .r = 0 of
7.3 Xoninteracting Control with Stability via Static Feedback 351
= /(-r)ls*	(7.14)
is asymptotically stable in the first approximation.
In other words, we have proved the following result.
Theorem 7.3.5. Suppose the system (5.1) has relative degree {ri...........rm}
at x = 0, and x = 0 is a regular point of Pf........P*rP*. Then the Prob-
lem of Noninteracting Control with Stability via Static Feedback is solvable
only if the restriction of the vector field f(x) to its invariant manifold S' is
asymptotically stable in the first approximation, at the equilibrium x ~ 0.
Remark 7.3,2, In the statement of Lemma 7.3.3. we have observed that the
distribution P* is contained in the distribution 3*. the largest locally con-
trolled invariant distribution contained in ker(dh). If both these distributions
are nonsingular and the inclusion _1* D P* is proper, i.e. the dimension of
_i* exceeds that of P*. the integral submanifolds of P* are proper submani-
folds of the integral submanifolds of 3*. More precisely, each of the integral
submanifolds of _V is partitioned into integral submanifolds of P*. Since, in
the case of systems having a relative degree at the point x — 0, the integral
submanifold of through x = 0 locally coincides with the zero dynamics
submanifold Z' (see Corollary 6.3.9), we conclude that S* is a proper sub-
manifold of Z*. Moreover, it is also known that the standard noninteractive
feedback (5.28) renders the submanifold Z* invariant under the vector field
f + да. and therefore the restriction of f + да to Z* coincides with the zero
dynamics vector field (see again Corollary 6.3.9) of the system in question.
Thus, the vector field (7.14) is nothing else than the vector field which de-
scribes the restriction of the zero dynamics of the system (5.1) to its invariant
manifold S*. In other words, the dynamical system (7.14) is a subsystem of
the system
L =	(7.15)
which describes the zero dynamics of (5.1). Of course, if the zero dynamics
of (5,1) are asymptotically stable in the first approximation, so are those of
any subsystem of (7.15) and the necessary condition established in Theorem
7.3.5 is automatically satisfied. <
In order to test whether or not the condition expressed by the previous
Theorem is satisfied, it is convenient to introduce suitable local coordinates.
To this end. consider, in addition to the distributions Pf.....P^.P* defined
before, also the distribution
P = {f,Vi......ffm|span{ffj : 1 < j < m})
introduced in Section 1.8, All these distributions, if nonsingular, are also
involutive. by Lemma 1.8.5. Therefore they are completely integrable by
Frobenius’ Theorem, and it is possible to find suitable sets of real-valued
functions whose differentials span (Pf)*.......(P*;	)x, (P*)1. P^. The follow-
ing Lemma illustrates that using these functions it is possible to construct a
352	7. Geometric Theory of Aonlinear Systems: Applications
local coordinates transformation which induces special forms for the vectors
of Pf......P*.P*,P.
Lemma 7.3.6. Suppose the distributions P*. Q Pf. foi' all 1 < i < in. P*
and P are nonsingular in a neighborhood of the point .r3. Then there, enst a
neighborhood P~ of .K and a coordinates transformation defined on l:c.
2 = Col(?.........= Ф(-Г)
= COKZ1!»........3'”-’’(.r))
such that
P~ = span{rf:',!*2}
(Pf}~ = spanK'.d;m+2}	(7.16)
(P*)- = span{(P1.,... dzm. dzm+~} .
In particular, it is possible to choose, for each 1 < i < ni. the coordinate
functions z’ip in the. form
Zl[r) col^'t-rb G'(.r))	(7.17)
= со1(/р(т\ Lfh{(P),....
as in the local normal form (5.1).
Proof. Recall that the distribution P does not change if the vector fields f
and gt. 1 < i < m. are modified by means of a regular feedback, i.e. that
= {/(]}---9™\sp*n{gj : 1 < j < m}) 
Consider the distribution
'< = g + П c* •
J#!
Since, by definition. P* с P for all 1 < i < m. then also Kj с P. Observe
also that is a nonsingular distribution (because so are P;*. Q Pf and
their intersection P*).
Using Lemma 1.8.4. it is not difficult to realize that, on some open and
dense subset C* of the set. U on which these distributions are defined, the
distribution Ki is invariant under f.g1:.,.. gm and contains all gfs. Thus, on
U*, Ki necessarily coincides with P. Since both A; and P are nonsingular,
they are equal, i.e.
р’ + Г|р* = р.
p
By duality
m 	(т-is)
t .3 Xoninteracting Control with Stability via Static Feedback 353
Let	be a collection of functions whose differentials span P~. For
each 1 < i < hi. since Pf and P are simultaneously integrable, it is possible
(see Corollary 1.4.2) to find a collection of functions z‘(.r) such that dz’ and
dz"l~2 span (P*)1. i.e. satisfy the second one of (7.16) (and therefore also
the third one). The property (7.18) guarantees that the differentials of all
the functions thus defined are linearly independent at zc, so that they can
be considered as a partial set of local coordinates in a neighborhood of r°.
For. suppose they were linearly dependent at Then there would exist, row
vectors C[...с7Г).cm+2. with cf 0 for some 1 < i < m. such that
ctd=' + cm^dzm"2 = ^CjcH .
The vector on the left-hand side belongs to (Pf}~. by construction, and that
on the right-hand side belongs to
E'cc-
Thus, by (7.18), rhe vector on the left-hand side is a vector in P~. and ct is
necessarily 0. i.e. a contradiction.
If the number of functions thus construe:!ed is not exactly equal to n. one
can find an additional collection гп,^1(г) of functions which completes the
coordinates transformation near j3.
To prove the last part of the statement, observe that
UXnr- = {0}
(because L^LL-1 /q(.r°)	0). Moreover.
u;cc(p,v.
Since by definition is spanned by the differentials of the elements of
^(z). it is indeed possible (using again Corollary 1.4.2) to find d»i(-r) >n order
to have (7.17) satisfied. <
Consider now a system which satisfies the assumptions of Theorem 7.3.5.
and suppose that a feedback which solves the noninteracting control prob-
lem has been implemented, Using the coordinates c1.in-
troduced in the previous Lemma, it is possible to represent the equations
describing the corresponding closed loop system in a particularly interesting
form.
Proposition 7.3.7. Suppose the system (5.1) has relative degree {rq.
r„t} at r = 0. and x = 0 a regular point of Pf. Q P*. for all 1 < ? < m.
P* and P, Let и = n(.r) + ;3(t)c be. any regular feedback law which solves the.
354	7. Geometric Theory of Nonlinear Systems: Applications
non-interacting control problem at .r = 0, In the coordinates z ~ Ф(.г) defined
by Lemma 7.3,6, the closed loop system
F = /(.r) + ,y(r)o(.r) + g(,r}3(r)v
У = h(x)
is represented by equations of the form
< 1 _ f / l „ m — 2 \ ।	„ I _ nt+2
-	- jiG  ~	) + <7ii G  ~	)<'i
**	— J fA b / U m щ - ' -*	) t 7ti
- w + l	=	/?fi_ 1 (г) + л (с)ri	+ ... + (рп^1,и (г)1’гП	(7-19)
F”~2	-	/^(F'"2)
У1	=	Ьф'.г"^2)
уИ1	=	h,n(z!\z’^2) .
In these coordinates, the submanifold S* is the set
S* = ]-r e U° : ?(.r) = 0....F"(.r) = O.F7”2(.r) =0}
mid the system (7.14) is represented by the differential equation
= m-‘ = /„,-1(0....0, z"'+1.0) .	(7.20)
Proof. The proof is based on arguments essentially identical to those used
in the Remark 1.6.6, Let f(z) = co^/ifc)	frJl+-i(z)) denote the represen-
tation of the vector held f[x) +	in the new coordinates, and note
that
df,(z) = Lfdz^z) .
Since (F,*)1 is invariant under /(c) by construction, for 1 < i < m. then
Lfdz‘(z) € йрап{с/г'. dzm+2}
and, thus, fi(z) depends only on zl and clr!+~. For the same reason, the
invariance of P~ under f(z) proves that fm+ffz) depends only on cm+'2,
(F*)1, 1 < i < m, and F~ are also invariant under all vector fields
(f/(z),i(z))j, and therefore the representation col(<n j (c).	g„l+2.ffz\) of the
latter in the new coordinates has similar properties, i.e. giffz) depends only
on F and F,)+“. Moreover, since gffz) in contained in F/. for all 1 < i < in
with i j, and also in F, we have
{dC.gjtz')') = 0
for all 1 < ? < m with i j. and for i = m + 2. This proves that gffz) has
nonzero entries only on the j-th and (m + l)-th block.
Finally. dht belongs to (P*ff by construction. Thus, hfz) depends only
on z! and c”i-k2. The last part of the statement is an immediate consequence
of the choice of the new coordinates. <
7.3 .\oiiiiiteracting Control with Stability via Static Feedback 35-5
Remark 7.3.3. It. may be interesting to compare the form of the equations
described in the previous Proposition with the local normal form introduced
in Section 5.1. To this end. observe that the equations (5,7) and (5.8) are
simply a local description of the system (5.1) in suitable coordinates, while the
ones introduced above describe a closed loop system which has been rendered
noninteractive by means of state feedback. Thus, in order to compare the two
sets of equations, it is necessary to impose on (5.7) and (5.8) a feedback (any
one can be used to this purpose) which solves the noiiinteracting control
problem. Suppose rhe standard non interacting control feedback is imposed
on (5.7) and (5,8). Then, one obtains a system of equations of the form
= -4u£J+Wi
’j	=	4(Ch) +Zbm(^-h)''i + •• 
У i	=	c^1
f !Jn>	—	Члч
in which, for all 1	< i	< m.
On the other hand, if the functions c'Uj. 1 < i < ni. arc chosen in the
way specified by (7.17). i.e. with the first r, components exactly equal to
those utilized to derive the normal form (5.7). it is immediate to realize that
each of the first m sets of equations introduced in the Proposition 7.3.7 can
be decomposed as

In other words, the functions ft( z'. z"‘+'2), gn 2). h,(г', c”’4 J) can be
expressed in the form
356	7. Geometric Theory of Nonlinear Systems: Applications
We deduce from the comparison of the two forms thus obtained that the
set of equations
Q1 = /1	. o1. z m+'2) + g} 1 (^ . o1, :	)r!
om = W-o'". c"“2) - д^дС1-O”1. z’n^)rtn
Z’ 1	— fm + 1 (- ) T j j (s.) i’i + . . - — (Jj,ltl ( C ) l'm
л	_ f t _ iri - 2 \
- Jm + L’t-	)
is nothing else than a decomposition of the equation
П = Qis- h) + lWv -   + Pounfs- 
In particular, setting r, = (I and C = 0. for all 1 < i < tn. one finds a
decomposed description of the zero dynamics of the system in the form
су1 = /] (0. o’.	)

л ai—2
;ii4((W.......О.о"!.с"! + 1.с”^2)

The set 5* corresponds to the subset of points having o' = 0 and г'7'+~’ —
(). This is clearly an invariant set of the zero dynamics manifold Z*. and
the restriction of the zero dynamics to this sot is a description in local
coordinates - of system (7.14). <
We show now that the necessary condition indicated in Theorem 7.3.5 is
essentially also sufficient for the solvability of the problem under considera-
tion. This fact is based on the following property.
Lemma 7.3.8. Suppose the assumptions of Proposition 7.9.7 are satisfied.
Suppose the linear approximation of (5.11 at the equilibrium point r = 0 is
stabilizable. Then, for each 1 < i < m. the linear approximation, at z‘ = 0.
of the subsystem
P =/;(Z,()) + 9i,(-.0)ri	(7.21)
of (7.19) is stabilizable.
Proof Clearly, if the linear approximation of (5.1) at r = () is stabilizable, so
is that, of the system (7.19). which has been obtained from (5.1) via regular
feedback and coordinates transformations, at the point Ф(0). Without loss of
generality, we may suppose Ф(0) = 0. The linear approximation of (7.19} at
z — 0 has a form
7.3 Non interacting Control with Stability via Static Feedback
357
If the latter is stabilizable, then the matrix .4ГЛ-1-2.m-2 has all eigenvalues
with negative real part and the pairs (,4/, bf) are stabilizable. Since the latter
define the linear approximations at c, = 0 of (7.21). the result follows. <
From this Lemma, we deduce that, if the linear approximation of (5.1) at
,r = ,rc is stabilizable, it is possible to find matrices A\.Km such that
the linear feedback
rf = A ' + c,	(t .22)
stabilizes in the first approximation the subsystem (7.21). This feedback pre-
serves the noninteractive structure of (7.19) (because depends only on C
and A and y, is affected only by C).
The linear approximation of
V"12 =
at Cn""2(0) is already asymptotically stable in the first approximation, as
a consequence of the stabilizability of the linear approximation of (5.1) at
x = 0 (see proof of Lemma 7.3.8). Thus, because of the special structure of
the equations (7.19). we can conclude that, if also the system (7.20) is asymp-
totically stable in the first approximation at cr'i+1(0), imposing the feedback
(7.22) on (7.19) yields a closed loop system which satisfies both the require-
ments (i) and (ii) of the Problem of Xoninteracting Control with Stability. In
other words, the composition of the standard noninteractive control feedback,
which induces the structure described by (7.19). with the additional feedback
(7.22) solves the problem under consideration. We formalize this result in the
following statement.
Theorem 7.3.9. Suppose the system (5.1) has relative degree
at x = 0. and x = 0 is a regular point of P*. Q Pf, for all 1 < i < m. P*
and P. Then, the Problem of Noninterac.ting Control with Stability via Static
Feedback is solvable if and only if
(i) the linear approximation of (5.1) at x: = 0 is stabilizable.
(ii) the linear approximation of (7.Ц) dt -C = 0 is asymptotically stable.
Proof. The necessity of (i) follows immediately by the requirement of achiev-
ing asymptotic stability in the first approximation for f(x) 4- y(.r)a(.r). The
necessity of (ii) and the sufficiency of both (i) and (ii) have already been
proved. <
358	7, Geometric Theory of Nonlinear Systems: Applications
Remark 7.3-4- Уохе that the standard noninteractive feedback renders the
distribution _1* invariant under the vector fields of the corresponding closed
loop system. However, the composition of this feedback with the law (7.22)
which, as we have shown, solves the Problem Noninteracting Control with
Stability via Static Feedback - does not anymore leave _i* invariant. <
The following example illustrates an application of the results discussed
so far.
Example 7.3.5. Consider the system
У] = Л
tj> — J-’l 
A simple calculation shows that this system has relative degree {1.1} at
,r = 0. In fact.
is/1(J-) = .4(,r)= {7 A.
1 U J
The zero dynamics of the1 system are defined on the submanifold
Z* = {.r £	: jq ~ г-, - 0}
and the zero dynamics vector field is given by tin1 restriction of rhe
vector field
f(x) -г t/(.r)(-A-1 (.r)6(z))
to Z*. Since
the representation of tin1 zero dynamics, in the (j-3. ,rt) coordinates of Z*. is
following one
j--> =
j’4	=	“^4 •
Note that the point .r = 0 is an unstable equilibrium of these equations, and
therefore the approach to noninteracting control used in Section j.3 would
yield an unstable closed loop system.
In order to check whether or not the Problem of Noninteracting Control
with Stability is solvable, we have to calculate tin1 vector field (7.1 1). This
requires first the calculation of the distributions P*. P-> and P*. We have
= (7-/71
= (Л <J\  <7-2Тра11{</1 })
7.3 Xoninteracting Control with Stability via Static Feedback 339
where f{x) = f(x) + ,д(г)о(.г). <)i(.r) = (c/(.r) J(r)h • <hU') = (g(-r).1(xp2.
and a(z).3(.r) is any feedback solving the Xoninteracting Control Problem.
Choosing the standard noninteracting feedback, one obtains
The calculation of Pf and Pf can be carried out by means of the1 algorithm
(1.39). In order to obtain P}* . we set
J() = span{p-2(jj}
and then we iterate, using
-1/. =	-V--i] + [ffi • -V--i] +• [#2- -V--i] -
Standard calculations show that
This distribution, which is nonsingular in a neighborhood of .r = 0 and in*
variant under the vector fields /(j). g\(„r).	J*), X the required distribution
Pf. Note that, it is possible to simplify the expression of the vectors which
span this distribution and obtain for instance
Pf = span{
Proceeding in a similar way, one obtains
and concludes that
P4 = Pf П pf = span{ ( 0
0 0 I)7’}.
The integral submanifold of P* which contains the point x = 0 is clearly
the set
5* = {.r e 3? :	= x-j = ur3 — 0} .
360
Geometric Theory of Nonlinear Systems: Applications
Tliis is an invariant manifold for f(z) = f(z) + g(x)ct(.r). and the restriction
of this vector field to S* is by the definition the vector field (7.14) whose
properties determine the solvability of the Problem of Xoninteracting Control
with Stability. Note that S* is also an invariant manifold of the zero dynamics
vector field (see Remark 7.3.2). and therefore we can immediately obtain a
representation of (7.14) by setting z3 = 0 in the representation of the vector
field f* (.?’)- This yields
z4 = -z4 .
This system has an asymptotically stable equilibrium at the origin and
therefore, by Theorem 7.3.9. the problem in question is solvable.
In order to find a solution, it is convenient to put the closed loop system
i = f(z) 4- fo(z)n + fo(z)m
(obtained by means of the standard noninteractive feedback) in the form
(7.19). To this end. note that
(P*)~ = span{dz! } = span{dh i }
= .span{dz2. dz3] = span{d/i2} + span{dz3} .
Thus. one can set
?	=	zt
z~	=	col(z2. z3)
гл	=	z4
(and no variable г-’ exists, because (P)~ — (Pf j-1-О (P_*)x = 0). Accordingly,
one obtains a system in the form (7.19)
Z1 = l’i
fo = t’a
z3 = z2 +
Z4 = Z2 - Z-jC*3 - Z4 + t'l + V-j
(see Fig. 7.1).
At this point, it suffices to stabilize by means of linear feedback - the
two subsystems with state variables г1 and c2. One can set. for instance,
О =	-Z! + Ci
to =	-z2 — Z3 + i~2 •
This additional feedback preserves the noninteractive structure and stabilizes
the system. In summary, a feedback law which solves the Problem of Xon-
intcracting Control with Stability, obtained by composition of the feedback
just determined with the standard noninteractive feedback, has the form
Ui ~ —.г2е'Гл — z2 - z3 + 1’2
ii2 = — Z] — Zjz4 + г,/^3 - zj + t“i + z3z2 + Z3 - z3r2 <
7.3 Noninteracting Control with Stability via Static Feedback 361
Fig. 7.1.
Before2 concluding the section, it is useful to discuss an alternative inter-
pretation of the distributions P*, on which the main results presented so far
were based and. also, to introduce an additional set of distributions that will
be used later in section 7.5 to solve the problem of non in teracting control
with stability via dynamic feedback.
The alternative interpretation of the distributions P*, which is contained
in the following statement, consists in showing that these distributions -
under appropriate hypotheses - can be directly characterized as special con-
trollability distributions of system (5.1).
Proposition 7.3.10. Suppose the system (5.1/ has relative degree. {> i........
r?r,} at x°. For each 1 < i < rn. consider the. distribution A* defined by (7.13/.
LetS(A*) be. the. distribution associated to A* by means of the controllability
distribution algorithm (see (6.53)). Suppose S(A() is finitely computable and
j'Q' is a regular point ofS(A*). Then, in a neighborhood of .r'~. Si-If) the
largest local controllability distribution contained in kor(dh,).
Let и =	4- d(.r)c be any regular feedback which, solves the noninter-
acting control problem at .4 . Set
/(J-) = f[.r] + (/(.r)n(r)
gfx) = {g(x]5(.r})l	1 < i < m .
Then, in a neighborhood of F .
= (Ltr......gm\sphn{(j;: j ?:}) = p*.	(7.23)
Proof As observed in the Remark 6.3.8. the distribution (7.13) is the largest
locally controlled invariant distribution contained in kerfdh,). Note that,
around jF. the distribution G has constant dimension ni (because the matrix
362
7. Geometric Theory of Nonlinear Systems: Applications
(5.2) is iiotisingular). the distribution (7.13} has constant dimension n - r,
(see Lemma 5.1.1 ). and the distribution 3* PG has constant dimension hi - 1
(in fact, the latter is spanned by vectors of the form <y(j*)p - with ' such that
(d£y-1 ht. (;(./))'- = 0 for all 1 < k < r/t and the set of all Vs which satisfy
this condition is an (m — l)-dimensional subspace of й"'). By Lemma 6.4.4.
5(3;) is the largest local controllability distribution containc*d in J*, and
then in ker(d/?().
To prove (7.23). recall that, by definition of relative degree (see proof of
Lemma 5.2.1)
L^l) ( = Lkfh, for all 0 < A’ < r( — 1.
J	J
If (a. 3) is any regular feedback which solves the non inter acting control prob-
lem at V. then (see Theorem 3.3.2, condition (iii)}
(/. f/i...(hi, >pan{.9j : j ф C (span{d£^?; : 0 < A- < r, - l}m = 3; .
Consider now the sequence of distributions Зд. generated by means of the1
following algorithm
30 = span{^ : j /}
ni
3A. = 3;._! - gs, Зд._ J
s=0
(here <7o = /) and note that (recall Lemma 1.8.2)
3a’ G \f. gi....gm Ispan{	. J jA j}) C -1( 
We show now that the distributions generated by means of the controlla-
bility distribution algorithm starting from 3* satisfy
= 3a .
This is certainly true for A1 = 0 because
So = 3* П G - spanf?, j i] .
Suppose is true for some A- and note that [<).,. Зд.] C 3*. because the distri-
bution (f.(h....(/„Jspan-f^ : j £ /}) is invariant under gs. Then
m
Sa-i = 3( P ([<?.,. 3a  + 3/. + G) — Зд.. i + 3; П G = Зд.^ j .
We obtain in this way
5(3;) c {f.yi......yfJspan{y7 : J ф с з;.
However, since {f.gi.....|span{c)j : j /}) is by construction a control-
lability distribution contained in 3* and 5(3;) is the largest local control-
lability distribution contained in 3;, necessarily
5(3;) = {f.gY......gm\wAn{gj :j ?})
i.e. (7.23) holds. <
7.3 Noniliteracting Control with Stability via Static Feedback
363
In section 7.5 we will find it convenient to consider, in the study of the
problem of noninteracting control with stability via dynamic feedback, also
another set of m distributions, which are denoted by Я’........./?*, and are
defined as follows
R* = (f-fh......j7,„|span{O
for 1 < I< Hl.
By definition, R* C /’* for all j i. Thus.
E л; c p; 
i 7е j
Suppose now that the distribution
(7.24)
is nonsingular. By definition, this distribution contains span{^, : i J}.
Moreover (see proof of Lemma 7.3.6 for a similar argument), on some1 open
and dense subset C’ of the set U on which the P*'s are defined, this distri-
bution is invariant under f.gy.....gm. Thus this distribution must coincide
with Pj1 on ['* and since both distributions are assumed to be nonsingular,
they coincide on i.e.
E"' m	u.25)
Note also that
P; G P|P; .	(7.26)
J^i
The distributions thus introduced lend themselves to an interpretation
similar to that illustrated in Proposition 7.3.10. In particular, it is not difficult
to set1 that, under appropriate "regularity"’ hypotheses, /?* can be interpreted
as the largest, local controllability distribution contained in
U, = P| ker(dhj) .
j*;
The details are left to the reader.
Using (7.26). we deduce that, in the local coordinates introduced in
Lemina 7.3.7.
Ri + P C span{ —.	} 
Moreover, if (7.24) is nonsingular. (7.25) yields
52 P* - span{ Д :i ф j.i ф m + 2} .
364
7. Geometric Theory of Nonlinear Systems: Applications
If also /?* is nonsingular. then arguments identical to the ones indicated
above show that
nt	„
Z = P = sPail< : 'V )}l + 2} •
t = i
and therefore
span{o?'a^'1 cR< +P' 
As a consequence, in the local coordinates introduced in Lemma 7.3.7,
_.	f d 9
Н, +P =SI>an{^7'5WTT} •
Finally, it may be worth observing that, if m = 2 and i j.
r- = p;
and therefore, since P* D P*.
However, this may not be the case if m > 2.
7.4 Necessary Conditions for Noninteracting Control
with Stability
As we have seen in section 5.4. a system which does not have a vector relative
degree at a some equilibrium point. ;rc may still be rendered noninteractive by
means of dynamic feedback. The existence of a dynamic feedback which does
this job can be checked, for instance, by iterating a certain number of times
the Dynamic Extension Algorithm. However, on the basis of the existence
of a dynamic feedback which merely renders a system non interactive, it is
not possible to deduce in general the existence of a dynamic feedback
rendering the system simultaneously noninteractive and stable (at least in
the first approximation). A further investigation is necessary, which is the
subject of this section.
As usual, we begin with a precise characterization of the problem in ques-
tion. Without loss of generality, we assume = 0.
Problem of Noninteracting Control with Stability (via Dynamic
Feedback). Consider a nonlinear system of the form (5.1). Find a dynamic
extension of the form (5.31), defined in a neighborhood of (j?. Q = (0, 0). with
a(0.(J) — 0 and 7(0.0) = 0. such that
(i) the equilibrium point (J-O = (0,0) of
7.-1 Noninteracting Control with Stability: Necessary Conditions 365
.г = /(.г)+t/(z)n (.?,<)
o2()
< -
is asymptotically stable in the first approximation.
(ii) the closed loop system
r = JUr) + r/(.r|(a(.r.<) + J(r.<)r|
s =	4-hl'.r.<)e	(7.28)
'/ = A(;r)
has a vector relative degree at the equilibrium point (z.<) = (0-0) and. for
each 1 < i < m. the output j/, is affected only by the corresponding input r;
and not by i'j, for any j / i.
An obvious particular case in which the Problem of Xoninteracting Con-
trol with Stability is not solvable via static feedback but it is solvable via
dynamic feedback is the one in which the system (5.1) dot's not have a vector
relative degree at .r = 0. but there' exists a regularizing dynamic extension
(of the form (-5.31)) and, moreover, in the1 corresponding composed system
(which has the form (7.28)). the various conditions indicated in Theorem
7.3.9 regarded as a set of sufficient conditions for noiiinteracting control
with stability via static feedback are fulfilled.
In this section, however, we wish to push the analysis a little further
and discuss to what extent it is possible to take advantage of the dynamic
feedback not only to impose the existence of a vector relative degree but.
also to weaken, if possible, the conditions indicated in Theorem 7-3.9. More
specifically, we wish to investigate tlie possibility of using dynamic feedback
in order to weaken the condition that the autonomous system (7.14) be stable
in the first approximation. The latter in fact, under the hypothesis that the
point .r = 0 is a point of regularity of certain distributions, was found to be
the main condition requested to have noninteracting control with stability
via static feedback.
The point of departure (d the analysis is pretty similar to t lie one described
in the previous section. In particular, it will be shown that, with any system
of the form (5.1) which is noninteractive and has a vector relative degree at
z = 0. it is possible to associate a distribution, which is left unchanged by
any dynamic feedback which preserves the property of noninteraction.
More precisely, consider a system of the form
+ (7 29)
yl = hi(jr). 1 < i < m ,
and suppose this system has a vector relative at z = 0 and is noninteractive.
Consider also the set of vector fields
36G	7. Geometric Theory of Nonlinear Systems: Applications
L.n.x = {r 6 V(E” ) : t = [adkf‘(h [... Pad^gl2. adklg^]:
'	J J	(i.3u)
2 < q, 0 < kt. i,. £ G for some pair (r. .s)} .
in which every element is a repeated Lie bracket involving two or more vector
fields of the form adkjgi (where к is any integer number) and each one of these
repeated brackets involves at least two vectors of the form
adjg, and adjgj
with ? j. Note that the set ZS,,,ix of all ^-linear combinations of vectors in
Z.inix is an ideal of the Control Lie Algebra of the system.
Define now
—ijnix — span{r . г E £inix} 	(-.31)
Remark 7.4-1  Suppose system (7.29) satisfies the1 hypotheses of Proposition
7.3.7. Since system (7.29) is by hypothesis noninteractive, rhe distributions
P* have the following expressions
P* =	,0,n|span{£j : j ;}) .
It is easily seen that all vector fields of £mix are in the distribution P* and.
therefore,
Xix C F* .	(7.32)
Moreover, if Jn,ix is nonsingular, it is also involutive and is invariant under
/-.Qi....9™-
The inclusion (7.32) implies, in particular, that the local coordinates in-
troduced in Proposition 7.3.7 are such that
span{dzl......dz^.dz”^’2} C Atlix .
Since the coordinates cm+) have to satisfy the only requirement of completing
the set. z1...., z'ri. £"’+2 to a full coordinate system, it is always possible -- if
it Allix is nonsingular - to choose --'N*1 as
with z™~^] such that
span{d?.......
In the coordinates thus defined
Xux = span{
and the (m + l)-th set of equations in (7.19) splits as
^m+l _	/771—1/	-711 + 1	-771 — 1	\ I „771-1-1/	1П + 1 ~1tl — 1
- Ja {'-a	• J + 9a L ‘ • ~(J	• G
+ГН-1	Г m — 1 /	771 — ] \ , >П — 1 !	-.771 + 1 \ , ,
-b = Jb
because of the invariance of Дп;х under f.g^... ,grjl. <
7.4 Noninteracting Control with Stability: Necessary Conditions 367
Suppose system (7.29) has been composed with some //-dimensional dy-
namic feedback of the form
и - a('j\ <) -r ,3(.r. (,’)c
(7.33)
< = o(.r.0 + d(j-.0r ,
to yield an (extended) noninteractive closed loop system, and let the latter
be denoted by

y; -	1 < i < m .
Gj-r) = [	j . /i,(Z) = l-ф:} .
X	/
With this extended system it is possible to associate a set defined as
in (7.30). which will be denoted by ai1^ a distribution defined as in
(7.31). which will be denoted by The reason why the distribution (7.31)
is important in the analysis of the problem of noninteracting control with
stability is that no matter what dynamic extension is considered - the
distribution can be viewed as an “extension" of the distribution JmjX-
P гор os it ion 7.4.1. Suppose (7.29) is noninteractive and has a vector rel-
ative degree at r = 0 and suppose also (7.34) is noninteractive and has a
vector relative degree at (,r.y) = (0.0), Let ~ denote the canonical projection
- : R" x	R"
(jr.() 1—> .r .
Then J®nix and Jniix are ~-related. i.e.
'l*—Xnix	—^mix ° 11 -
In order to prove this result, we need a preliminary lemma, which is an
extension of the result presented in Lemma 7.3.2,
Lemma 7.4.2. Suppose (7.29) is noninteractive and has a vector relative
degree at .r ~ 0 and suppose also (7.34) 7s non interact i re and has a vector
relative degree at (x, (J) = (0.0). Then, for any i. j
,J?j(jt,<) = 0, LGj3iSx.C,} = 0. LG/O(.r-<)=0
368
7. Geometric Theory of Nonlinear Systems: Applications
and
L(;,LTi, ... LTl3t,(r.(,) =0. L(}. LTr ... Lr.(ii{jr.Q) = 0
for all г > 1 and any choice' of the rector fields rr....и in the set {F.Gl,
.... G„;}. If 77.29} has vector relative degree {n.......r,„} and (7.94)
re I at i v e d egree { r |.г f)n }. then
P) = ly if and only if (U. 0)	0.
7 = r, 4- зг if and only if З^г.С) = 0.
and
Lc.Lj.-cifi.r. 4) = d for all 0 < A’< .у - 1
L(;t Lf- lo/((). 0) 7 0 
The proof of this Lemma is essentially similar to the proof of Lemina 7.3.2
and will not be repeated here.
The proof of Proposition 7.4.1 consists in showing using inductively
the identities established in the previous Lemma that the vectors in the
set projects into vectors which span A simplified sketch of the
calculations involved goes as follows.
Proof Observing that 3(J — 0 for i J, express the vector fields F and G,.
1 < i < in. which characterize the extended system (7.34). in the form
F = / + £2	+ F, G,
j-i
where, with some abuse of notation, f and g, are written instead of
(o) 7‘ (o)
Consider, for instance, the case in which all r,'s are equal to the corre-
sponding r®’-s- Then, a standard calculation yields
[FG,] = Fg^.-Gf
~ (Lr.fPg, --AJF./yJ + [F G',]
Hl
=	- l.Af-gA +	+ FiF.,?,]
j=i
H>	Hi
+	~	+ F.GJ .
,?=i	J=i
From this, using some1 of the properties indicated in Lemma 7.4.2. one obtains
7.4 Noninteracting Control with Stability: Necessary Conditions 369
Д’. GJ -	+ JH[f. gt] +	+ A
j=i
where A is a vector in кег(~ж).
In a similar way. one obtains, for k i and j i.
[G’c.GJ = ДЛ/Дг-g>] + I !
[G’^AF.G’J] = (Lf3„ -
m
+	- Yz^^jAsj-[дь-дА • z t
[GJ. [G^.G’J] = LCij ^n3kk){gk.gi\ + 'U;,G> Ak-gt]] + П'.
where Y. Z. and П’ are vectors in kerf тгт).
F'roin this, using the fact that all 3,;(0)‘s are nonzero, it is deduced that
Mspan{[GJ.GJ, [Gj. [F.GJJ [GJ. [GYGJ] : i / j.k ф ?})
= span{>j.0j. Д • Д-.9 JL [gj. k-^J] : i j-k / ?} 
Appropriate induction arguments, based on calculations of this type,
prove the Proposition. <
Having proven that the distribution Апих is left unchanged by any dy-
namic extension which preserves the property of noninteraction, it will be
shown now that this distribution is helpful in identifying an obstruction to
the solution of the problem of noninteracting control with stability via dy-
namic feedback. For convenience, in view of the results described in section
5.4. we restrict our attention to those systems of the form (5.1) - for which
the set v. consisting of all regularizing dynamic extensions which are gener-
ated by the Dynamic Extension Algorithm, is nonempty. This is a reasonable
hypothesis which guarantees that we are dealing with a system for which the
problem of non interacting control via dynamic feedback is solvable.
Given a system S for which the set Z is nonempty, let. E be any element
of Z and let F be any regular state feedback which solves^the problem of
noninteracting control for S о E. Let rhe composed system S = SoEoF be
described by equations of the form
Di
i = /Д) + Ург(.1-)г;
—	(t -3o)
у = fi(Y) .
By hypothesis, this system is noninteractive and has a some vector relative
degree at ./• = 0.
The results expressed by Propositions 5.4.2 and 7.4.1 enable us to claim
that the distribution
Д„;х = span{7 : г 6
(7.36)
3i0	7. Geometric Theory of Nonlinear Systems: Applications
with
imix = {т 6 Г(?Л) : r = {ad^lp [... .
2 < c/,0 < k,. ir ig for some pair (r. si}
is independent of the choice of E and F (so long as the feedback they define
is a solution of the problem of noninteracting control).
In fact, observe that - by Proposition 5.4.2 * any other dynamic extension
E € 8 is necessarily such that
S о E and S о E'
have the same dimension and. possibly after a change of coordinates in the
state space, only differ by a regular static feedback. Therefore, if F' is any
other regular state feedback which solves the problem of noninteracting con-
trol for S о E'. the two systems
S о E о F and S о E' о F'
have the same dimension and. possibly after a change of coordinates in the
state space, only differ by a regular state feedback which preserves the prop-
erty of noninteraction.
In other jvords. possibly after a change of coordinates in the state space,
the system S' = S о E' о F' can described by equations of the form
m
i - m + p/'mc	(7.38)
У ~ h(j')
with
f'(t) = №) + д(т)й(т);	g'(i) =
and г = 5(т) + 3(т)г' is a regular state feedback which preserves the property
of noninteraction. By Proposition /.4.1. the distributions of (7.35) and
of (7.38) are equal.
In other words, we can conclude that, for any system the form (5.1) for
which the set 8 is nonempty, the distribution defined by (7.36) is a well-
defined object, independent of the choice of E and F.
The next result, is an extension of the result presented in Lemma 7.3.4.
Lemma 7.4.3. Consider a system S of the form (7.29) and suppose the set
8 of all regularizing dynamic extensions generated by the Dynamic Extension
Algorithm, is nonempty. Let E be any edement of 8 and let F be any regu-
lar state feedback which solves the problem of noninteracting control for the
composed system SoE, Let the composed system S = SoEcF be described
by equations of the form (7.35) and let ДП|Х be the distribution defined by
(7.36). Suppose x = 0 is a regular point for Let L* denote the integral
submanifold of ЛШ-1Х which contains the point x = 0. The subnianifold L* is
locally invariant under the vector field f(xfi and the restriction of f(x) to L*
does not depend upon the particular choice ofE and F.
r.4 Non interacting Control with Stability: Necessary Conditions 371
Proof. By construction, is invariant under /(.r). For any other choice
E and F. one obtains a system S' described by equations of the form ( 7.38)
in which
/'(.r) = /(}) + g(x)a(x) .
Moreover.
Allix П G С P* П G С -С П G = {0}
(where G = span{^; : 1 < i < m}), Thus, invoking again (as in Lemina 7.3.4)
a uniqueness property of all state feedback which leave invariant a certain
distribution, the result follows. <
Remark 7.4-2. Consider the local coordinates introduced in Proposition 7.3.7.
with C1'1 split, as indicated in Remark 7,4.1. The set of points for which all
coordinates are zero except r',i + l and identifies the invariant manifold
S* (the maximal integral manifold of Pr which contains z = 0). while the
set of points for which all coordinates are zero except Д1+1 identifies the in-
variant manifold £* (the maximal integral manifold of -imjX which contains
x = 0). System
=	/;,,+i(0....+
X"' =	.....0.г"'*‘.0)
is a description - in these local coordinates - of the restriction of f(x) to S*.
while the subsystem
= ,,»-h(0.....o..-r-‘.o.o)
is a description in the same local coordinates of the restriction of f(x) to
£*. <J
We are now in a position to formulate a necessary condition for the solu-
tion of a problem of noninteracting control with stability via dynamic feed-
back.
Theorem 7.4.4. Consider a system S of the form (7.29) and suppose the
set S of all regularizing dynamic extensions generated by the Dynamic Ex-
tension Algorithm is nonempty. Let E be any element of £ and let F be any
regular state feedback which solves the problem of noninteracting control for
the composed system SgE, Let the composed system S = SoEoF be described
by equations of the. form (7.35) and let Jmix be the distribution defined by
(7.36). Suppose x = 0 is a regular point for Дп1х. Let L* denote, the. inte-
gral submanifold of Ani\x which contains the point i - 0. Then the Problem
of Noninteracting Control with. Stability via Dynamic Feedback is solvable
only if the restriction of the vector field f (x) to its invariant manifold L* is
asymptotically stable, in the first approximation, at the equilibrium x = 0.
372 t. Geometric Theory of Nonlinear Systems: Applications
Proof. Suppose the problem of iionintcracting control with stability has beeti
solved by some (dynamic) feedback R Then, by Proposition 3,4.2. there
exists R' £ 7? such that
S о R and S о E oR1 are locally diffeoinorphic.
Since F. which is a regular state feedback, has a unique inverse, it is deduced
that also
S о R and SoEoFoF 1 oR' are locally diffeoinorphic.
This shows that S о R. which is noninteractive and has a vector relative
degree, can be obtained from S о E о F. which also is noninteractive and has
’vector relative degree, via dynamic feedback and change of coordinates. In
other words, possibly after a change of coordinates. So R can be viewed as
obtained form SoEoF via a (dynamic) feedback which preserves the* property
of noninteraction, Let these two systems be described the equations of the
form (7.34) and. respectively. (7.29).
Set now
'0£'
dx
.4°
i j-=oi
'OF
0.F
A =
and observe that, as a consequence of the mere definition. Amix(0) is an
invariant subspace of A (and. therefore, Aplix(0) is an invariant subspace of
Ap). For. observe that any vector field r £ Anix is such that [f, rj 6 Ani;x.
Thus, recalling that /(0) = 0. this yields

Atr(0) -
7(0) = [/. t](U) e A,nix(0)
(т = 01
By definition, for any e e ДП1;х(0) theye exists т £ £т;х such that t(0) = r
and therefore
AAm;x(0) C Amtx(0) .
Observe that
w_ f A + 0(O)^(O.O) y(0)^(U.0)\
— I	ox	oC,
у	*	★	/
If rc is any vector in	there exists a vector field re £ Дрп!х such that
ee = re(0). Since (7.34) is noninteractive. the vector field rp is such that (see
Lemma 7.4.2)
Lrf0;(Z) = 0
for every 1 < i < m. Thus, in particular
(^(0.0) ^(0,0)) V = 0
7.5 Xoninteractiiig Control with Stability: Sufficient Conditions
373
and therefore
W») = lJ X>x(0) -
From this, using the "invariance projection property’" proven in Propo-
sition 7.4.1 and some standard results in linear algebra, it is possible to
complete the proof of the Theorem. <
Remark 7.4-3- Xote that the condition identified in this Theorem is trivial
in the case of a linear system. In a linear system, in fact, all vector fields of
the form are constant vector fields. Thus, all vector fields of £Inix are
trivially zero and -ДП1;Х = 0. <J
7.5 Sufficient Conditions for Noninteracting Control
with Stability
We address now the problem of constructing a dynamic feedback law which
solves the Problem of Xonint er acting Control with Stability. To this end. we
need some appropriate hypotheses. First of allt in view of the results illus-
trated in section 5.4. we assume that the set £ of all regularizing dynamic ex-
tensions generated by the Dynamic Extension Algorithm is nonempty. Then,
keeping in mind the results illustrated at the end of the previous section, we
take any element E e £ and any regular state feedback F which solves the
problem of noninteracting control for S oE and we assume that the extended
system S = SoEoF. which is by construction noninteractive and has a vec-
tor relative degree at the equilibrium point x = 0. satisfies all the hypotheses
of Proposition 7.3.7. i.e. x = 0 is a regular point of the distributions P(\
Q Pj  for all 1 < i < m, P* and P. Under these hypotheses, the system S
can be locally transformed, by means of a change of coordinates defined in
neighborhood of T = 0. into a system represented by equations of the form
(7.19) (note that the hypotheses in question are independent of the particular
choice of E and F).
On the system S thus constructed we impose some additional restrictions.
The first one of these is the hypothesis that the distribution P coincides with
the entire tangent space. Under this hypothesis, the (m + 2)-th set of local
coordinates in the form (7.19) is empty and the latter reduces to a system of
the form
—	/1(^1 ) + 9i i (-О
X 1Г1
i* nt—i
Vi
(7.39)
374 t. Geometric Theory of Nonlinear Systems: Applications
(note that the notation ,r, has replaced the notation z1 of (7.19)).
Remark 7.5.1. The hypothesis that P coincides with the entire tangent space
docs not involve, actually, a loss of generality. For. it h clear that the suit*
system of (7.19) associated with the (m -r 2)*th set of local coordinates is not
influenced ar all by rhe inputs to the system. If this subsystem is stable in the
first approximation at = 0 (which is indeed a necessary condition for
stabilizability of the full system (7.19)). any feedback law which solves rhe
Problem of Xoninteracting Control with Stability for the system obtained be-
setting zm^'2 = 0 in (7.19) also solves the same problem for the full system
(7.19). <
To indicate system (7.39). which is a diffeormorphic copy of the system
S described at the beginning, we continue to use rhe notation chosen for S
in the previous section, i.e.
T = /(.r)
J=1
= /|ДТ)	1 < i < //? .
A second hypothesis on the system S is that .r = 0 is a regular point
of the distribution Дп;х of S. Then, as expected, we need to assume that
the necessary condition identified in Theorem 7.4.4. namely the condition
that the restriction of the vector field /(i) to its invariant manifold Z* is
asymptotically stable in the first approximation at the equilibrium ,r = 0. is
fulfilled. However, in order to streamline rhe presentation, we illustrate first
the case in which the stronger assumption
Дшх =ft
holds, deferring to the end of the section the discussion of the more general
case.
A third hypothesis on the system S is that, for each 1 < i < in. the
distribution
R* =	....g„i |5Pan{9;}>
in a neighborhood of the point T = 0 is nonsingular and is spanned by the
vector field g, together with a finite set of vector fields of the form
[9jp ’ i.hjp-; 	[,9л ‘ (JJjjJ
in which p > 1 and 0 < д < m (as usual, go = /). It is also assumed
that the distributions 52 "=i anf^’ ^or eac^ K 52j^( ^2 arc llonshtgular in
a neighborhood of the point F = 0.
Remark 7.5.2. The hypothesis in question indeed is satisfied about any .r::
in an open and dense subset C* of the state space. What is assumed here is
that the point J'° = 0 is a point of L *. <i
7.5 X on inter ас ting Control with Stability: Sufficient Conditions 375
A fourth hypothesis on the system S is a stabilizability hypothesis, which
consists in the following. Recall that, if R* is nonsingular. then it is also
involutive ami invariant under f and (the latter, in particular, is a vector
field of R*; Let S, denote the maximal integral manifold of /?* which contains
the point ,r = 0. Since both f and (R are tangent to S,. the -restriction of
i - f[,r} +	(7.401
to Si is a well defined i single-input j subsystem. In what follows, it will be as-
sumed that, for each 1 < i < m. the restriction of (7.401 to the- corresponding
manifold S(- fins a stabilizable linear approximation at ,r = 0.
Remark 7.5.S. The hypothesis in question is satisfied, for instance, if
/?; = span{,7.adp7......adj’-1 g,}
where -s, is the dimension of R*. In this case, in fact, the re-.riiction of (7.40)
to S, has a controllable linear approximation at x = 0. <
Under these hypotheses, it is possible to construct a dynamic feedback
which solves the Problem of Xoninteracting Control with Stability. In what
follows, in order to simplify the exposition, we describe the construction in
the particular case of a system in which m = 2. The reader should have no
difficulty in extending the construction to the general case.
In the case m = 2. system (7.39) is a system of the form
h ~ /(h) + ,9i(i)ui +<72(.r)m2
.91 = /?!(?)	(7.41)
?/_> — h-2(h)
with
and
hi(.fl = hpj’i).	h->(h) = h->(r2) .
Let /р.гм.пз denote the dimensions of .zp..r_>.,r;}, respectively. By con-
struction. the decomposition of the state vector j- into z^.i’2. J’s is such that
the distributions
pi =	^/span{p2})
P1 = </ .91	| Spall {(/j})
have the following expressions
376	7. Geometric Theory of Nonlinear Systems: Applications
P*	f 9	9 X
p' = spa,1WaA}
f 9	9 X
P.f = span{ —.	.
PJ’l OX-j
Consider now an extended system defined as follows
xe	=	Г(те) +	G] (,re)u!	+ GGxe)l/2
*/i	=	ней
y2	=	H2(xe)
(7.42)
with
.rp = col( j’!. ,r2. J’3. Ai, pi. X-2-P‘>)
where Ai G Ani. A? € 5'la • pi € K'’3. p? € Br’3 • and
/ /1U1) \
Ш)
/з(^1-;г2.Лз)
fiUi)
Л(^1.О,Р!)
Л(-)
\ ЖЗД) /
ММ =
j/31 (zi _т2. x3)
<731 Mi - 0. pi)
0
\	0	/
tfi(Z) = MM
Standard calculations show that the vector fields F. and G2 of the
extended system (7.42) have properties indicated below.
Lemma 7.5.1. Suppose -Аш1х = 0. Let Djpji^1...J1g\ denote the repeated
bracket
(7.43)
where p > 1 and 0 < jk <2 (as usual, go = f). Let	denote the
repeated bracket

where p > 1 and 0 < д- < 2 (and. Go = F). Then. DJpJp_l  has an
expression of the form
7.5 > on inter acting Control with Stability: Sufficient Conditions 377

(7.45)
and D^j	has an expression of the form
Dc,.	=
J p J p 1 J J
0
7-31(^1.Т2..Гз)
n(xi)
"31 (j*l
0
0
(7.46)
(for each fixed string of integers jpjp_ (    ji. the functions ti (') and r:n (-. )
in (7.4-5) and the functions n(-) and t31 (•. •. •) in (7.4G) are exactly the same).
Corresponding expressions hold for Djpjp_l...j1g-2 and D? jp i,..jxG2.
Proof. It goes by induction, and simply consists in using the definition of
Lie bracket, the property that /2 (0) = 0. and the fact that any repeated Lie
bracket in which jk = 2 vanishes if Jmix = 0. <
Using this property it is possible to prove the important result described
below. This result holds under the hypotheses indicated at the beginning of
the section and which, for obvious reasons, are not repeated here.
Lemma 7.5.2. Let .$1 and demote the dimensions of R) and RC respec-
tively. The distributions
= (F- 6’i.G’2|span{Gi})
= (KGlG2|span{G2})
(which by definition are invariant under F. Gj. G?) have constant dimension
.Si and. respectively. in a neighborhood of .re = 0. A,s a consequence, they
are involutive. Moreover, they are independent, i.e. 7?^ P 7?® = {0}. and
7?i	C spanjdTG}^
R% С нрап{^Я1}±.	)
Proof By hypothesis, in a neighborhood of x = 0. 7?is spanned by <fi and
by .si — 1 vectors of the form (7.43). Let these vector fields be denoted by
01(7’)...0,si(7). By construction. 7?[ is involutive. invariant under f. Ifi and
g-2. Thus, for any 1 < i < s\.
Si
[9,9,1 L
where в is either f, gY or g?.
378	7. Geometric Theory of Nonlinear Systems: Applications
It can be shown that the (uniquely defined) coefficients eik only depend on
j~i . In fact. let. tri (i).a(Jr) denote a set of vector fields which generate
F*. Using the hypothesis -l[niX = 0, we see that, for every pair nJ.
о = [[&. e.j.crj] =
k=\
which, in view of the linear independence of the ffi-'s. yields £^с,7(т) = 0.
Since the cy's span R*> and (see section 7.4).
R? = span{^-.
О Z 2 САГ3
it is concluded that cp(7) are functions of aq only.
Note that, by Lemma 7.5.1, the @t’s are vector fields of the form
7i ( J‘i)
0
and that, again hy Lemma 7.5.1. the vectors
7^(Х|.;Г>.Г3)
731 (71.0, Pl)
0
\	0
1 < i < si
(7.49)
1 < ? < sp
are in (F. Gt. G2 |span{Gi}). Using the property just proven and again
Lemma 7.5.1, it is easily deduced that the distribution spanned by the sq
vectors (7.49) is invariant under F, Gi, G-2- Thus, the latter coincides with
(F, Gj. G-2 jspan{Gi}). The .sy vectors are independent and therefore, the dis-
tribution R\ has constant dimension and is involutive. Identical arguments
prove the properties of FL The independence of F^ and F5 is an easy con-
sequence of the structure of the vectors in F^ and FL Property (7-48) is
obvious. <
In the defining system (7.42), we have added a set. of
p =	4- it-2 + 2«з
state variables. The system thus obtained is still noninteractive. but the sta-
bility properties of the original system (7.41) have not been improved. To
achieve stabilizability (in a way which, as it will be shown later, is compat-
ible with noninteraction) the next step consists in adding to system (7.42)
a set of new p input functions. More precisely, we consider a system of the
form
।.j Nouintoracting Control with Stability: Sufficient Conditions 379
.re — 75	) ~b G i (.C )и i + G3(C )tt'2 + £T
.Vi =	(7.50)
ij> =	#2 (•*’*')
in which F(.rp). G'i(-re). G-2(xe).	H2(E) are the same as in (7.42).
i: G R" and the matrix E is a matrix of the form
with I a i2 x у identity matrix.
Note that the system thus defined can still be viewed as a dynamical
feedback acting on the original system (7.41). because the new -auxiliary"
input vector c affects only the dynamics of the extra state variables added in
(7-41) and not the dynamics of .r.
Observe now that the distributions R\ and R2 are independent and in-
volutive. and moreover, since Дп;х = 0. also the distribution R\ + Z?.^ is
involutive. Thus, it is possible to choose new local coordinates
with
£i e P/:. G e Rs"2
so that.
(7?^	=	span{d£3}
(7?^)-	=	spanjd^tf^}
(7?3 )*	=	span{d£i. d£3}	.
In the new coordinates, the equations characterizing system (7.50) assume
a special form. In particular, since 7?| and 7?® are both invariant under F.
G\. G>. since Ci £ I?i and G2 G 7?5. and since properties (7.48) hold, it is
easy to conclude that, in the new coordinates, system (7.50) is described by
equations of the form
SI = Vl (si  6) + <'l (£l-£s)Ul +	(£)f
6 = лМ) +
6 = рды + адь-
У i = Xl(si.£:i)
y-i = v(G-sS)-
We choose now the additional input r as
r =
in such a way as to further simplify the equations (7.51).
380
7. Geometric Theory of Nonlinear Systems: Applications
Lemma 7.5.3. Let .S3 denote the dimension of £3 in (7.51 j. The S3 rows of
the matrix #з(£) are linearly independent at £ = 0 (and therefore at each £
n eai ‘ ц = 0). Th e n, t h e re axis ts a no n s i ng ular m a trix 3 (£) s и eh th a t
ww = (/ oi
with I an S3 x S3 identity matrix. The feedback r = d(£)r' changes system
(7.51) into a system of the form
£l = r'db-fr) +	+ A 1 (<1. Сз)
&	=	M;i)W	(7.52)
У1	=	ti(si.b)
У'2	=	\2^2-b) -
in. which К — (I 0), with I an s;} x S3 identity matrix.
Proof First of all. we establish that the S3 rows of the matrix 6b(£) are
linearly independent, at £ = 0. Recall that, in (7.51), has dimension sy and
<$> has dimension s2- Thus
S‘1 + S2 + s3 = n + V
where v is the number of state variables added in (7.42). which also is equal
to the dimension of c, To show that the matrix #з(0) has precisely S3 indepen-
dent rows, let V denote the subspace spanned by the columns of the matrix
E in (7.50) and observe that, from the construction of the distributions
and /?!>. it can be deduced that
dim((7?i (0) +$(0))П1') = Si + s-2 - n .
By definition, ker(d£:i(0)) = R\ (0) + J^(0). and therefore
rank(d£3(0)E) = dim(V) - dim((R?(0) +	) (T V)
= n — si — S‘> + n = S3 .
Since #з(0) — г/£з(0)Е this con (‘hides the proof that 03(O) has precisely S3
independent rows.
Now. let Cj denote the j-th column of the matrix E in (7,50) and observe
that the distributions R\ and Rf> by construction arc such that
[e.j,Ret] G Я • + span{eA.: 1 < к <
for all 1 < j < n and i — 1.2. Thus, the same arguments used to prove
Proposition 6.2.2 show that
жюад) =	<W£W) = n
дь	db
and this completes the proof. <3
, .5 XonintPrac ting Control with Stability: Sufficient Conditions
381
The next, and final, step of the construction will be to show that is possible
to choose, for system (7.52). inputs of the form
«1	=	T'ls!	+	i'l
t/2	=	F?Cj	+	<’l>	(7.53)
l'1	=	Ft S3
in such a way that the corresponding closed loop system is stable' in the first
approximation at the equilibrium (^ . <2-чз) = (0.0.0). Since, with this choice
of inputs, the1 closed loop system is still noninteractive. this will complete the
construction of a dynamic feedback solving the Problem of Noninteracting
Control with Stability.
Lemina 7.5.4. There exists a state feedback lair
u_> = T2C2
H = Ыз
which stabilizes, in the first approximation. the equilibrium xe = 0 of system
(7.52).
Proof. The third subsystem of (7.52) is trivially stabilizable in the first ap-
proximation. because of the special structure of the matrix К.
To show that
a =^i(Ti-0) + t’1(el.0)u1	(7.54)
is stabilizable in the first approximation, observe that
= F(Z) + Gh (Z )U]	(7.55)
and
G = Ci (£i • СзJ + t'i 1st  s-C“i
в = уЫв-Ы	(7.56)
Ct	= Сз(Сз) 
are by hypothesis diffeormorphic- Let L? denote the integral manifold of R*
which contains Z = 0. In the new coordinates. is precisely the (invariant)
manifold of (7.56) in which & — 0 and Ct = 0. Thus, it is concluded that
(7.54) is nothing else than a description in suitable local coordinates - of
the restriction of (7.55) to the invariant manifold L*.
Recall now that, by hypothesis. the restriction of
x = /(.r) -r gi (T‘)Ui	(7.57)
to its invariant manifold Si has a stabilizable linear approximation at x = 0.
It is easy to see that there is a natural diffeomorphism between the restriction
382
7. Geometric Theory of Nonlinear Systems: Applications
of this system to and the restriction of (7.55) to L\. In fact, consider the
submanifold M of defined by
M = {Z e	e 5i. A! = n. pi = .r3,A2 = O.p2 = 0}
Using the property that / and <p arctangent to Si. it is easy to see that F and
Gi are tangent to Л1. As a consequence, all vectors fields of /?( are tangent
to M. Moreover, this manifold has precisely dimension $i = dim(/?j). Thus,
this manifold is a maximal integral manifold of and necessarily coincides
with L\. The diffeoniorphism
Q:	5( -О L*
(Zb, л'з) H-> (J-J»jr3.лу. J"3.0,0)
carries trajectories of (7.57) into trajectories of (7.55). Thus since the restric-
tion of (7.57) to Sj is by hypothesis stabilizablc in the first approximation, so
is the restriction of (7.55) to L? and. therefore, its diffeoinorphic copy (7.54).
Identical arguments indeed work also for the second subsystem in (7.51)
and this completes the proof. <
In summary, we have proven that, if the various hypotheses indicated at
the beginning of the section (which, among others, included the hypothesis
-Amix — 0) are fulfilled, it is possible to find a dynamic feedback law which
solves the Problem of Non interacting Control with Stability for the system
S. The feedback in question, which is the composition of the various control
actions successively introduced in (7.42). (7.50) and (7.53). assumes the form
Fi<i(Z) + iy X
F2^(Z) + /
/ /iZi) +pn(^)Fi^i(Z)	\
fiU’t0.pi) + д.ц (A .0, pi)FiG(ZJ
kir-F + (^ZMFi^MZ)
\/з (-Г2, 0-/F>) + 532(-r2.0!P2)F2^(Z)/
+ j(e(z))Fz3(Z)
/	X
P3i (.Fi, 0. pi)
which is a standard form of a dynamic state feedback.
The analysis conducted so far can be extended, without much difficulty,
to cover the case in which Jril;x = 0. Considering again, for simplicity, a
system with m = 2. one has to replace (assuming that .r = 0 is a regular
point of J,nix) system (7.41) by a system of the form (see Remark 7-4.1)
7.5 Noninteracting Control with Stability: Sufficient Conditions
383
ri = fi(J'i) +5n(Ji)ui
Ь =
Cv, =	-Гз^-Гзь) +
(t -58)
л-зь = /:tb ( J1 • ?? • -Гзь) + 5? 93b J (-Г1  -Г2; J3t>) Uj
2=1
iq = Лг(л) l<t<2
in which
\	- -  j d 1
-^mix — 5pa.Il{ Q } -
In these coordinates, the hypothesis that the restriction of the vector field
f(x) to the integral manifold L* of -lmix is stable in the first approximation
is the hypothesis that
хза = Ле,(0,0. r3„.O)	(7.59)
is stable in the first approximation at the equilibrium т3й = 0.
Now. it is clear from the structure of (7.58) that, if (7.59) satisfies this
hypothesis, any dynamic feedback law which solves the problem of noninter-
acting control with stability for the subsystem
•ri = /1(л) + 9u (-ri)iC
j? = h(^) + gaijin
2
X‘3b = /зь(-Г1.-Г2,-Гзь) + 57 93bj^l-^2.^3b)«j
2=1
Mi — hit-Ti) 1 < i < 2
of (7.58) also solves this problem for the full system (7.58). To obtain such
a feedback, it suffices to assume that the subsystem in question, which has
precisely the same structure as (7.41), satisfies the hypotheses indicated at the
beginning of the section and workout the construction procedure discussed
above.
We conclude the section with a simple illustrative example.
Example 7.5.4- Consider the following system, which is a modification of the
system discussed in the example 7.3.5,
±i = Ui
J’2	—
Т’з = -r? + -r3
±4	=	J*2 - -Г2СГз + 2*4 + «1 + U-2
9\ = J-1
92	=	-
384	7. Geometric Theory of Nonlinear Systems: Applications
Calculations identical to those described in the example 7.3.5 show that
P* = P’ — span{
P* = P* n P7 — span{( 0 0 0 1 )I} .
and
In the present example, however, the vector field (7.14) has the following
form
.
from which we deduce that the necessary condition for non inter acting control
with stability via static feedback is violated.
Seeking a solution via dynamic feedback, we compute the distribution
As easy calculation shows that
[adkjg^adhfg2] = 0
for all к > 0 and all h > 0- Thus, it is concluded that Да;х = 0 and.
in particular, that the necessary condition for noninteracting control with
stability via dynamic feedback is fulfilled.
Note also that.
P* = span{^l. adfgi }. P^ = span{j72-
t
so that also the remaining (sufficient) conditions indicated at the beginning
of the section are fulfilled.
Following the construction indicated above, we set
For this extended svstem we find
7.5 Non interacting Control with Stability: Sufficient Conditions
385
	A	у			A	A			0
f?^' = span{	0 0 1		0 1 0	} , f?2 — sP<iI1{	0 0 0		1 0 0		0 1 0 }
	0 0 w		1 0 0 M		0 1 0 \o/		0 0 A		0 0 \1)
Adding the extra 5-diin oiision al input e yields system (7.50). which has
the following form
Zi	=	u,
±2	=	li‘j
±3	=	X'2	+	<3
;r 1	—	J'2	—	Jl-2 f J’4 4~ u j + a 2
T5 =	+ i'l
+ U1 + t'2
_f7	=	«о	+	t’3
Тй	=	‘1'2	+	^3 +
J'y	—	JW	—	jyt'*''3	+ J~ci + 119 + С.",
The change of coordinates leading to the form (7.51) is defined as follows
In the new coordinates, extended system (7.50) splits into
Gi
62
Щ + П
Gl + П] + i’2
G1 =	+ r3
G*2 = G’l + 6'2 + (G'2 +	+ ;,4
6з - Gs + (Gl + S32)(l - С£- + Ьз) + U-> + r->
and
386
7. Geometric Theory of Nonlinear Systems: Applications
£31 = “П
U> = -ГЗ
£зз = -i'i
£з 1 = £з I - i'2 - Gi •
Since the matrices which multiply г are constant matrices, there is no
need to manipulate this input further. In other words, this system has already
the form (7.52). At. this point, it is immediate to check that an additional
feedback law of the form
«1	= Fi^id-Ui
a-2	=	+ й-2
V - F-з^з
stabilizes the system in the first approximation, and preserves the property
of noninteraction. <3
8. Tracking and Regulation
8.1 The Steady State Response in a Nonlinear System
In this Chapter, we discuss the problem of how to control a nonlinear system
in order to have its output asymptotically converging towards a prescribed
steady state response. To this end. we begin by showing in what specific sense
the "intuitive" notion of steady state response must be understood, in the
general setup of nonlinear systems, and we identify appropriate conditions
under which such a response exists. Then, beginning with the next section,
we show how a prescribed steady state response can be achieved.
The intuitive notion of steady state response is that of a particular re-
sponse towards which any other response of a system converges, as time in-
creases. In order to characterize this concept in more rigorous terms, consider
a system
.r = /(.r. u)	(8.1)
with state .r defined in a neighborhood [ of the origin in and input
it. € 3"’, assume that /(0.0) = 0. and let j-5. u(-)) denote the value of
rhe state achieved at a time t > 0 under the effect of the input u(-). starting
from the initial stare .r': at time t = 0. Let »*(-) be a specific input function
and suppose there exists an initial state / with the property that
^liin ||z(Lzc', it*(•)) - r(t..r'. n"())|| = 0
for every J‘° in some neighborhood [7* of /. If this is the case, then the
response
Mb =/!//.«*(•))
is called the steady state response of (8.1) to the specific input »"().
The notion of a steady state response is particularly useful in tin1 analysis
of the response of a system to inputs which are '‘persistent" in time, as it is in
the case of any periodic (and - of course bounded) function. In these cases,
in fact, the steady state response is itself a persistent function of time whose
characteristics depend entirely on the specific input imposed on the system
and not on the state in which the system was at the initial time. Usually,
inputs of this kind can be thought of as "generated" by a suitable dynamical
system modeled by equations of the form
388
8. Tracking and Regulation
ti’ = s(u-)
(8.2i
a = p[n:)
whose state ir is defined in a neighborhood IT of the origin in 3.’’ and in
which s(0) = 0 and p(0) = 0. To impost' that the inputs generated by such
a system are bounded, it suffices to assume that the point <c _ 0 is a stable
equilibrium (in the ordinary sense of Lyapunov) of the vector field .s( tr) and to
choose tin1 initial condition at time t = 0 in some appropriate neighborhood
IVе С IV of the origin. To impose that the inputs are pemVtent in time (that
is. to exclude the possibility that some input decays to zero as time tends
to infinity), it is convenient to assume that every point ;c of IV’ is Poisson
stable.
We recall that a point irz is said to be Poisson stable if the flow Фруи }
of the vector field я(?г) is defined for all t € x and. for each neighborhood
of it'3 and for each real number T > 0. there exists a time p > T such
that Фр (irc) € Lc'. and a time P < -T such that ФррР) e ГТ In other
words, a point. w~J is Poisson stable if the trajectory u’(f) which originates in
?rc passes arbitrarily close to w3 for arbitrarily large times, in forward and
backward direction. Thus, it is clear that if every point of II c is Poisson
stable1, no trajectory of (8.2) can decay to zero as time tends to infinity.
In what follows, we will study the stead)' state1 response to inputs gen-
erated by systems of the form (8.2). in which we assume that the vector
field s(zr) has the two properties indicated above, namely that the point
a* = l) is a stable equilibrium (in the ordinary sense) and there exists an open
neighborhood of the point tc = 0 in which every point is Poisson stable. For
convenience, these two properties together will be referred to as property of
neutral stability.
Remark 8.1.1. The hypothesis of neutral stability implies that the matrix
which characterizes the linear approximation of the vector field .фг) at ir = 0.
has all its tigenralttes on the imaginary ans. In fact, no eigenvalue of S can
have positive real part, because otherwise the equilibrium it = 0 would be
unstable. Moreover, the assumed Poisson stability of each point in a neigh-
borhood of ir = I) implies that no trajectory of the exosystem can converge
to ir — 0 as time tends to infinity, and this, in turn, implies the absence of
eigenvalues of S with negative real part. In fact, if 5 had eigenvalues with
negative real parr, the exosystem would have a stable invariant manifold near
the equilibrium, and the trajectories originating on this manifold would con-
verge to w = Ct as time tends to infinity. Note that the hypothesis in question
includes for instance systems in which every trajectory is a periodic tra-
jectory (and. accordingly, the input generattai by (8.2) is a periodic function
of time). <
8.1 The Steady State Response in а Хеш lit) ear System 389
It is rather easy to show that, if the equilibrium j = 0 of z — /(.r.O) is
asymptotically stable in the first approximation. a steady state response can
be defined for any input generated by (8.2), so long as its initial condition
ir° ranges over a sufficiently small neighborhood of the origin.
Proposition 8.1.1. Assume (8.2) is neutrally stable. Assume the equilib-
rium x — 0 of x = /(.r.O) is asymptotically stable in the first approximation.
Then, there exists a mapping x = 77(tr) defined in a neighborhood IVе С IV
of the origin, with 7t(0) = 0. which satisfies
= f(~(w),p(w))	(8.3)
UW
for all w C IV°. Moreover, for each u~ G IVе. the input
1Т(1)=р(ФЦт^))
produces a well-defined steady state response, which is given by
x.fit) = z(t. -(»). (;“()) •
Proof. The Jacobian matrix of the composite system
.r = f(x,p(w)}
w = s(ir)
at the equilibrium (z. u.’) = (0.0) has the following form
f . 4 * \
\ ° S )
where, by hypothesis. .4 has all eigenvalues with negative real part, while
S has all eigenvalues on the imaginary axis. Thus, the system in question
(see section B.l) has a center manifold at (j.lc) = (0.0), the graph of a
mapping j* = ~(ш) satisfying (8.3). Moreover, the associated reduced system
is precisely given by w ~ sfw). Thus, the equilibrium point (z, tr) = (0,0)
is stable (in the ordinary sense). The center manifold is locally exponentially
attractive and. for all pairs (r°. to*) in some neighborhood of (0.0).
||.r(/) - 7г( w{t))|| < A'eЦ.А - 7t(?c* )||
for all t > 0 and suitable A" > 0. о > 0. Observe that, by definition,
z(t) - ,r(t, ,r°, u*(-))
and. since the graph of z = -(«) is an invariant manifold,
x(t. 7г(аЛ). (t*(J) = 77(ш( 0) 
As a consequence
;liiu ||z(t..rc.u*(-)) - x(t. 77{w’f u“(-))|| = 0
and the result follows. <
390
S, Tracking and Regulation
Remark 8.1.2. If f(.r. piir)) and .s(y) are C^. the composite system
> = fi;t\p(w))
th = .s-(if)
has a CK‘ center manifold for any к < эс (see section B.l). Thus. (8.3) is
satisfied by a Ck mapping r = ~(«’) for any к < ос. <
The following simple example shows how center manifold theory is helpful
in determining the steady state response’ of a nonlinear system.
Example 8.1.8. Consider the nonlinear system
J-i = -rj -f- и
j‘2	=	—Xj + J’] ll
with input a generated by a system of the form (8.21
fC i = (иг?
ir-> = —noy
a = it-] .
Since the hypotheses of Proposition 8.1.1 hold, there exists a mapping
.r = - (in) satisfying the identity (8.3). which in the present case reduces to
o»"i	<9tti
тг~(пг> ~ -7.— ait'] = -~t(uy. ir2) + it'i
CziTi	Ulf'-)
<?-,	d-k
-7r--aie-2 -	- — "„qny . ay) + ~i (uy ./mpty .
dirt	dn'2
The first one of these relations is an equation in "Jffy. uy). which is
solved by a linear function of tc /
"i (tri. tt'-j) = --y( tt'i - (tlt'l) .
1 + a-
Substitution of this function into the second relation yields an equation in
7t2(uy. uo). which can be solved by a polynomial of second degree in fry. uy.
Simple calculations yield in fact
-.J»’]. m) = -------—-----—7-((1 + a2 )ud — 3« tty tr> За2 ir2) .
(1 + am +4a4	1
Note that the solution thus found is defined for all ir c R~. For any
(гд'7 - r/'2) £ ^2  1 he input
a*(t) = uqeosat + tr*simd
produces a well-defined steady state response, which is given by
8.2 The Problem of Out pur Regulation 391
Fig. 8.1.
7Ti(tCi(O. \
7T2(U'i(f). U’2(f)) J
Note also that the convergence of any other response to the steady state
response occurs for every initial state r', In fact, the differences
Cl = Zi - 7ti(tl'i. tc2)
- 7Г2 («’I, W2 )
sat isfy
Cl =	-Cl
e-2 = — e-> + t?i и
form which it is easily concluded that Ci(t) and e2(f) both converge to zero,
as time tends to infinity, for every value of ci (0) and e2(0). <
8.2 The Problem of Output Regulation
A classical problem in control theory is the design of a feedback law for the
purpose of imposing a prescribed steady state response to every external
command in a prescribed family. This may include, for instance, the problem
of having the output y(-) of a controlled plant asymptotically tracking any
prescribed reference output t/ref(-) in a given family, as well as the problem
of having t/(-) asymptotically rejecting any undesired disturbance in a
certain class of disturbances. In both cases, the matter is to impose that the
so called tracking error, i.e. the difference between the reference output and
the actual output, be a function of time
392	8. Tracking and Regulation
= w(0 - y(t}
which decays to zero as time tends to infinity, for every reference output ami
every undesired disturbance ranging over prespecified families of functions.
In other words, the matter is to impose that the control system exhibits, to
each external command in a given family, a steady state response for which
the associated tracking error is identically zero.
From the point of view of having zero steady state error, there is no
need to keep separate the roles of the required output response and that of
the undos ire d perturbation, since both can be viewed as components of an
"augmented"' exogenous command, which is required to he asymptotically
rejected by the error. Motivated by these (standard) arguments we consider,
in what follows, nonlinear systems modeled by equations of the form
The first equation of (8.4) describes the dynamics of a plant, whose state x
is defined in a neighborhood U of the origin in Ik/1 . with control input a 6 R"1
and subject to a set of exogenous input variables w E лЬ which includes
disturbances (to be rejected) and/or references (to be tracked). The second
equation defines an error variable e €	, which is expressed a.s a function
of the state z and of the exogenous input w.
For the sake of mathematical simplicity, and also because in this way a
large number of relevant practical situations can be covered, it is assumed
that the family of the exogenous inputs w(-) which affect the plant, and for
which asymptotic decay of the error is to be achieved, is the family of all
functions of time which are solution of a (possibly nonlinear) homogeneous
differential equation
ib = sfuf	(8.5)
with initial condition u’(0) ranging on some neighborhood 1Г of the origin of
FT. This system, which is viewed as a mathematical model of a "generator"
of all possible exogenous input functions, is called the exosystem.
As usual, it is assumed that f(x, w, u), h(x, w), s(w) are smooth functions.
Moreover, it is also assumed that /(0.0.0) = 0. ,s(0) = 0, h(0.0) = 0. Thus,
for и = 0. the composite system (8.4)-(8.5) has an equilibrium state (r. tr) —
(0,0) yielding zero error.
The control action to (8.4) is to be provided by a feedback controller which
processes the information received from the plant in order to generate the
appropriate control input. The structure of the controller usually depends
on the amount of information available for feedback. The most favorable
situation, from the point of view of feedback design, occurs when the set of
measured variables includes all the components of the stare x of the plant
and of the exogenous input w. In this case, it is said that the controller is
provided with full information and the latter is a memoryless system, whose
8.2 The Problem of Output. Regulation
393
output u is a function of the states ./ and ir of the plant and. respectively, of
the exosystem
u = n(z. w) .	(8.6)
The interconnection of (8.4) and (8.6) yields a closed loop system described
by the equations
z = f(j'.tc.a(x. ?c))
(8.0
tc = s(tc) .
In particular, it is assumed that a(O.O) = 0. so that the closed loop system
(8.7) has an equilibrium at (r. ic) — (0.0).
A more realistic, and rather common, situation is the one in which only
the components of the error e are available for measurement. In this case, it
is said that the controller is provided with error feedback and the latter is a
dynamical nonlinear system, modeled by equations of the form
with internal state £ defined in a neighborhood E of the origin in . The
interconnection of (8.4) and (8.8) yields in this case a closed loop system
characterized by the equations
? - /(.Г. 1Г.Я(£Ш
ё =	(8.9)
ii- = s(m).
Again, it is assumed that ?/(0.0) = 0 and 0(0) = 0. so that the triplet
= (0.0.0) is an equilibrium of the closed loop system (8.9).
The purpose of the control is to obtain a closed loop system in which, for
every exogenous input ufi>) (in the prescribed family) and every initial state
(in some neighborhood of the origin), the output e(-) decays to zero as time
tends to infinity. When this is the case, the closed loop system is said to have
the property of output regulation. Note that, in view of the discussion held in
the previous section, the requirement in question is essentially the require-
ment that each exogenous input u'fi) induces, in the closed loop system, a
steady state response As(’ l ^uch that
hU^(t). ir(t)) = 0
for all t > 0. Since a basic requirement, in this setup, is the existence of a
well defined steady state response to each input generated by the exosystem
(8.5). we appeal to the sufficient conditions presented in the previous section
(see Proposition 8.1.1) for the existence of such a response. As far as the ex-
osystem is concerned, we assume throughout the entire Chapter the property
of neutral stability, while for the interconnection of controlled plant and feed-
back controller we seek stability in the first approximation. This yields the
following formal characterization of the two design problems outlined above.
394
8. Tracking and Regulation
Full Information Output Regulation Problem. Given a nonlinear
system of the form (8.4) and a neutrally stable exosy мет (8.5). find, if pos-
sible. a mapping а(.г. /г) such that
(S)fi the equilibrium .г = 0 of
,r = f(r. 0. o(.r, 0))	(8.101
is asymptotically stable in the first approximation.
(R)fi there exists a neighborhood Г C U x П’ of (0.0) such that, for each
initial condition (.r(0). tr(O)) € Г. the solution of (8.7) satisfies
lim ir(t)) = 0 .
Error Feedback Output Regulation Problem. Given a nonlinear
system of the form (8.4) and a neutrally stable exosystem (8.5). find, if pos-
sible. an integer n and two mappings 0(£) and such that
(S)ef the equilibrium (,r.£) = (0.0) of
i- = /(-г.о.бч-)))
(8.11!
£ = //(£. /((>’, 0j)
is asymptotically stable in the first approximation.
(R)ef there exists a neighborhood Г С I' x r x IF of (0.0.0) such that, for
each initial condition (,r(0). £(()). u'(0)) € V. the solution of (8.9) satisfies
lim h(.r(f), a'(l')) = 0 .
/ —* X
Remark 8.2.1. Note that the requirements (Shi and (S)ef зге rather strong,
in that they ask for stability in the fi/st approximation for the closed loop
system. A characterization of this kind guarantees - under the hypothesis
of neutral stability of the exosysteni the existence of a well defined steady
state response. However, it is rather demanding. in that it requires (sec1 section
4.4) asymptotic stabilizability of the linear approximation of the controlled
plant. The possibility of fulfilling (S)fi and (S)ef depends entirely on the
properties of the linear appro.rinuition of the controlled plant at j‘ = 0. and
the design of a feedback law providing either one of these two properties is
a problem whose solution requires only standard results from linear system
theory. However, as we shall see in a moment, the simultaneous fulfillment of
(S)i-1 and (R)fi (respectively (S)ef and (R)ef’) i* a problem whose solution
requires a specific nonlinear analysis. <
Since, as we have just remarked; the properties of the linear approximation
of rhe controlled plant play a determinant role in the solution of a regulation
problem, it is convenient, to set up an appropriate notation in which the
8.2 The Problem of Output Regulation
395
parameters of this approximation are explicitly shown. To this end. note that
the closed loop system (8.7) can be written in the form
.г — (.4 + Б A ).r + (P + BL')ir + 0(j". w)
ii' = Str + <( ie)
where o(z. uj and vanish at the origin with their first order derivatives,
and 4, B. P. Ah L. S are matrices defined by
.4 =	'£/1		В =		du iFA		Z (0.0.0)		 clo	'£/’ dir	;o,o.o;	(8.12)
	CJJf j du	; U,0.(i -									
A’ =			S =				L -				
	dr	(0.0 I		dw			0:	die	J :oo		
On the other hand, the closed loop system (8.9) can be written in the form
.r = T.r + BHf + Ptr + G»(.r. О tr)
- Ff + GC.r + GQir -e \(t.	и-)
ii — Str “b t'(и' 1
where o(.r. G if). \(т. G ?r) and I’ftr) vanish at the origin with t heir first order
derivatives, and C. Q. F, H. G are matrices defined by
Using this notation, it is immediately realized that the requirement (S)H
is rhe requirement that the Jacobian matrix of (8.10) at ,r = 0.
J = A + BI\
has all eigenvalues with negative real part, whereas (S)ef b the requirement
that the Jacobian matrix of (8.11) at О,£) ~ (0.0).
f A BH\
\ GC F )
has all eigenvalues with negative real part.
From the- theory of linear systems, it is then easy to conclude that (S)ft
can be achieved only if the pair of matrices (.4.B) is stabibzable (i.e. there
exists К such that all the eigenvalues of (44 + BK] have negative real part)
and (S)ei- can be achieved only if the pair of matrices (.4.B) is stabilizable
and the pair of matrices (С. .4) is detectable (i.e. there exists G such that
all the eigenvalues of (.4 + GC\ have negative real part). These properties of
the linear approximation of the plant (8.4) at (r, ir. u) = (0,0. 0) are indeed
necessary conditions for the solvability of a problem of output regulation.
396	8. Tracking and Regulation
8.3 Output Regulation in the Case of Full Information
In this section, we show how the problem of output regulation via full infornia-
tion can be solved. To this end. we present first a simple but very important
preliminary result which later on will provide the key to the solution of the
problem in question.
Lemma 8.3.1. Assume that, for some o(j, tc). the condition. (S)i i is satis-
fied. Then, the conditioTi (R)fi is also satisfied if and only if there exists a
mapping .r = тг(<с). with тг(О) = 0. defined in a neighborhood П” C IT of the
origin, satisfying the conditions
^-s(uA - f(~(w), w.	w))	(8 14)
0 = h(~(ir).ir)
for all icGllh
Proof. Note that the Jacobian matrix of the closed loop system (8.7) at the
equilibrium (x. w) = (0. 0) has the1 following form
f A + В К * \
\ ° S) '
By assumption, the eigenvalues of the matrix (.4 + BI\) have negative real
part, and those of the matrix 5 are on the imaginary axis. Thus, using the
results of section B.l. we deduce the existence, for the system (8.7). of a local
center manifold at (0,0). This manifold can be expressed as the graph of a
mapping
J’ — ”(tr)
with "(w) satisfying an equation of the form (B.6). In the present setup, the
equation in question reduces precisely to the first one of (8.14).
Choose a real number /? > 0. and let trG be a point ofITC. with [!w°|| < B.
Since, by the hypothesis of neutral stability, the equilibrium tr = 0 of the
exosystem is stable, it is possible to choose В so that the solution «’(f) of
(8.5) satisfying ;c(0) = trc remains in ITC for all t > 0. If x(0) = .r° = "(tC),
the corresponding solution ,r(T) of (8.7) will be such that x(t) = -(w(t)) for
all t > 0 because the manifold x — ~(ir) is by definition invariant under the
flow of (8.7). Note that the mapping
p : IT'3 -4 C x
W (7r(tc).U')
(whose rank is equal to the dimension r of IT ° at each point of И’°), defines a
diffeomorphism of a neighborhood of IT0 onto its image. Thus, the restriction
of the flow of (8.7) to its center manifold is a diffeomorphic copy of the flow
of the exosystem, and any point on the center manifold sufficiently close to
8.3 Output. Regulation in the Case of Full Information
397
the origin is Poisson stable by hypothesis. We will show that this and the
fulfillment of requirement <R)h imply the second equation of (8.14).
For. suppose (8.14) is not true at some (~(tc3). И) sufficiently dost1 to
(0.0). Then.
Al = ||h(~(tr“). m )|| > 0
and there exists a neighborhood U of (~( h,c ). ir~) such that
!|/?17C'ic). H|| > M/2
at each (~(tc).ic) 6 U. If ('R)fi holds for a Trajectory starting at f~(ir’’). tc3).
there exists T such that
||/d~( НС). НП )p < -U/2
for all t > T. But if (—(tr3). ic=) is Poisson stable, then for some /' > 7\
(7r(tt’(P) )• u'(f')) 6 U and this contradicts the previous inequality. As a con-
sequence. the second one of (8.1 1) must be true.
In order to prove tin1 sufficiency, observe that, if the first equation of
(8.14) is satisfied, the graph of the mapping .r = tr) is by construction a
center manifold for (8.7). Moreover, by the second equation of (8.14). the
error satisfies
c(n = h{x(t). tr(U) ~ /|(7Г((Г(И ). !(•(/)) .
Observe that, by assumption, the point (,r. m) = (0.0) is a stable equilibrium
of (8.7). Then, for sufficiently small (.r(O). tc(Oj). the solution (J‘(/). ir(t)) of
(8.7) remains in any arbitrarily small neighborhood of (0.0) for all t > 0.
Using a property of center manifolds illustrated in section B.l. it is deduced
that there exist real numbers M > 0 and a > 0 such that
l-UU - v(tc(U).| < Mt ~,‘l ||.r(0) - 7t(tr(0) 11)
for all t > 0. By continuity of h(x.u:}. lim?_+ e(t) = 0. i. e. the condition
(R)i i is satisfied. <
Using this result, it is very easy' to establish a necessary' and sufficient
condition for the solution of the Full Information Output Regulation Problem.
Theorem 8.3.2. The Full Information Output Regulation Probleni is solv-
able if and only if the pair (A.B) is stabilizable and there exist mappings
j- = -(ir) and и = c(ia). with y(0) = 0 and c(0) = 0. both defined in a
neighborhood И'° C IT of the origin, satisfying the conditions
= /(-(«). »,<•(«.)) fg45)
0 = h(~ (ic).ir)
for all ir G П
398	8. Tracking and Regulation
Proof. The necessity of the condition that (Л,В) is stabilizable has already
been discussed in the previous section. To deduce the necessity of (8.15). it
suffices to observe that, by Lemma 8.3.1. any feedback law which solves the
problem in question is necessarily such that the identities (8.14) hold for some
"(ir). Now. setting
c(nj = О ( 7Г( 1Г ). 1Г)
immediately yields (8.15).
In order to establish the sufficiency, observe that, by hypothesis, there
exists a matrix A' such that (.4 + BE) has eigenvalues with negative real
part. Suppose the conditions (8.15) are satisfied for some 7t(tc) and ('(«’)- and
define a feedback law in the following way
O(a*, U') = c(u') a- A'(.r “ "(?('))
It is immediate to check that this is a solution of the Full Information Out-
put Regulation Problem. In fact, this choice clearly satisfies the requirement
(S)ki- because o(.r.O) — A'.r. Moreover, by construction
о(тг(П’).»’) — c(ic)
and. therefore, the first equation of (8.15)) becomes identical to the first
equation of (8.14). On the other hand, the second equation of (8.15) is already
exactly equal to the second equation of (8.14)). Thus, again using Lemma
8.3.1, we conclude that also the requirement (R)n is satisfied. <J
lipmark 8.3.1. The first one of the two conditions (8.15) expresses the fact
that there is a submanifold in the state space of the composite system
.r	=	f(j\ tr. u)
d-	=	sfir)	(8.16)
c	=	/t(.i\ U’) ,
namely the graph of the mapping z = ~(ic). which is rendered locally invari-
ant by means of a suitable feedback law, namely a — c(a'). The second con-
dition expresses the fact that the error map, I.e. the output of the composite
system (8.16), is zero at each point of this manifold. Altogether, the condi-
tions (8.15) express the property that the graph of the mapping x = "(»’) is
an output zeroing submanifold of the system (8.16). <
liemark 8.3.2. Recall (see section B.l) that a Ck vector field has a
center manifold. If the problem of output regulation is solved by some
feedback law o(z. tr). (8.15) hold for a pair of CA’-1 mappings т = "(m) and
и = c(ir). Conversely, if (8.15) hold for a pair of Ck mappings ,r = тг(т) and
a = c(ir). the problem of output regulation is solved by a CA' feedback law
o(.r. ir). <3
8.3 Output Regulation in the Case of Full Information
399
Remark 8.3.3. If the system (8.16) is a linear system, the conditions (8.15J
reduce to linear matrix equations. In this case the system in question can be
written in the form
.r — Ar + Pir + Bit
it- = Sr
e - Cr + Qic .
and. if the mappings j’ — тг(<г) and a = c(tc) are put in the form
тг(<с) = Пи- + x(?c)
cfir) = Г it’ + c(<c) .
the equations (8.15) have a solution if and only if the linear matrix equations
П5 = АП + Р + ВГ
0 = cn-Q
are solved by some П and Г. Note that, if this is the case, the mappings v( ir|
and c(u’) which solve (8.15) are actually linear mappings (i.e. тг(?г) = Пи:
and c(и') ~ Га-). <
The proof of the sufficiency, in Theorem 8.3.2. shows in particular that,
once a solution 7t(?r). c(ir) of the equations (8.15) is known, a control law
which solves the problem of output regulation is provided by
a{r. i/') - c(ic) + f\(r - -(if))	(8.17)
where К is any matrix which places the eigenvalues of (.4+ BK) in the open
left-half complex plane. A block-diagram interpretation of the feedback law
thus found is described in Fig. 8.2.
Fig. 8.2.
400
8. Tracking and Regulation
Remark 8.3.4- It might be instructive to compare the results obtained hero
with those illustrated in section 4.5. In that case, convergence to zero of the
error was implied by the fact that c( t) was a solution of a certain homogeneous
linear differential equation. Here, in the proof of Theorem 8.3.2 (and Lemma
8.3T) the error e(t) has been shown to converge to zero as a consequence of
a general property of center manifolds. The approach followed here, which is
much less demanding, shows that there is no need to impose that (•(/) obeys
a homogeneous differential equation. In particular, c(f) may be I) at some t
and nonzero for larger values of t. <
We reserve the last part of this section to the illustration of how the
existence conditions (8.15) can be tested in the particular case in which m = 1
(one-dimensional control input and one-dimensional error) and the equations
(8.4) assume the form
j- =	f(J')	r/(.r)u
J	(8.18)
e = h(;r) + p(fc) .
which corresponds to the case of a single-input single-output system whose
output is required to track any reference trajectory produced by
1Г = A'(ff)
t/ref =	
We also assume that the triplet {f (.г). д{х).Ь(.гУ\ has relative degree r at
r = 0 so that coordinates transformation to a normal form is possible. In the
new coordinates, the system in question assumes the form
— i	-r ;
h	= <7l£- n)
e	= u + P( <e) -
In order to check whether or not the1 equations (8.15) can be solved, it is
convenient to set
-(tr) = col(A'(u'). A(u’))
with
£(<*') — col(Aq [if).....kr(tc)J .
In this case, the equations in question reduce to
8.3 Output Regulation in the Case of Full Information 401
du?

Okr-Ax)
Oir
dkr(x)
div
«(<<’)
5A(t)
—s(.UT)
dir
0
Ы»0
b(k(u'). A( it)) + a(k( ir), X(u'))c(w)
q(k(u'),X(wY)
A'i(if) + p(ir) .
•S(>')
The last one of these. together with the first r — 1. yields immediately
kM = -L^ptw}	(8.19)
for all 1 < i < r. The r-th equation can be solved by
Lskr(iv'] -b(k(w).X(w))
ФС = ---------и/ w и --------- (8'20)
a(k(w). X(u-))
and, therefore, we can conclude that the solvability of equations (8.15) is in
this case equivalent to the solvability of
л\
— s(w)=q(k(lr).X(u-))	(8.21)
dir
for some mapping 7/ = А(гс).
We formalize this in the following statement.
Corollary 8.3.3. Suppose (8.4) has the form (8.18) and the triplet {fix).
g(x). /?(j*)} has relative degree r at x = 0. Define kt(w). 1 < ?' < r. as in
(8.19). Then, the Full Information Output Regulation Problem is solvable if
and only if the pair (Д. B) As stabilizable and the equation (8.21) can be solved
by some mapping A(u’). with A(0) = 0.
Recall that the linear approximation at tv — 0 of the exosystem has by
assumption all eigenvalues on the imaginary axis. Thus, if the linear approx-
imation of
r) = q(O.q)
at 7/ = 0 has no eigenvalue on the imaginary axis, the equation (8.21) is
exactly the equation which must be satisfied by any center manifold (see
section B.l) of the system
9
d-
q(k(w).q]
a(tz-) .
Thus, we have
402
8. Tracking and Regulation
Corollary 8.3.4. Suppose (8.f) has the form (8.18) and the triplet {/(t),
g(x). h(x)} has relative degree r at x = 0. Suppose (A.B) is stabilizable. If the
linear approximation at x = 0 of the zero dynamics of {f(x). g(x), hix'j} has
no eigenvalue on the imaginary axis, the Full Information Output Regulation
Problem is solvable.
We conclude the section with a simple example of application.
Example 8.3.5. Consider the system already in normal form
±1	=	X~2
= a
g = g + xy + x?
У =
and suppose it is desired to asymptotical!)' track any reference output of the
form
= M sin(nt + &)
where a is a fixed (positive) number, and M, & arbitrary parameters.
Note that the zero dynamics of this system are unstable and. therefore,
the approach described in Section 4.5 cannot be pursued. Note also that
the system is not exactly linearizable via feedback, because the distribution
span{(?. adfg} is not. involutive, as a simple calculation shows. Thus, it is not
possible to solve the problem by reduction of the plant to a linear system.
In this case, any desired reference output can be imagined as the output
of an exosystem defined by
/ x ( au"i \
Sit’ =
\ —«?C'l J
p(w) =	“U.'i
f
and therefore we could try to solve the problem via the theory developed in
this section, i.e. posing a Full Information Output Regulation Problem.
Since the linear approximation of the system is controllable and the (sin-
gle) eigenvalue of the linear approximation of the zero dynamics of the plant
is not on the imaginary axis, the hypotheses of Corollary 8.3.4 are satisfied
and the problem in question is solvable.
Following the procedure illustrated above, one has to set
^(tc) = -p(u0 = Wi
k2(w) = Lgkilu') = aw2
and then search for a solution A(wi,wa) of the partial differential equation
(8.21). i.e.
5 A	dX	. . v
——aw? - --------au’t = X(wi,w2) + uq + (auoy -
OWi	dw-2
8.4 Output Regulation in the Case of Error Feedback 403
A tedious, but elementary, calculation shows that this equation can be
solved by a complete polynomial of second degree, i.e.
A(u'i. ir2) — Uj ici + a-2n'2 + a11 tt’f + ai2(ri ^'2 + «22»'2 
Once A(wi,U’2) has been calculated, from the previous theory it follows
that the mapping

~(w) = k2(ir)
\ A(uii. tc2)
in
air?
A(uq. ?r2)
and the function
c(tc) = Lsk?(u') = —a“ u’i
are solutions of the equations (8.15). In particular, a solution of the regulator
problem is provided by
o(t. U’) = c(tc) + К(j — 7t(u’))
in which К = (ki k? A’3) is any matrix which places the eigenvalues of
0 1
0 0
1 0
0 \	/0\
0 I + I 1 I Л’
1 / \o/
in the left-half complex plane.
As expected, the difference
&
\	/	.Г] - uq
2*2	— tr(tt’) =	J>2 — «U.’2
П J	\i)~ A(u:lfH;2)
is asymptotically decaying to zero, and so is the error e(t), which in this case
is exactly equal to jq. In fact, the variables £1. £>• £3 satisfy
(i\	/	°	1	° А /Ц /	0	\
(2 1	= 1	*2	*’з I I £2 I	+ I	0	I	•<
£3 /	\ 1	0	1/\£з/	\ sf + 2a£iU.l2(t) /
8.4 Output Regulation in the Case of Error Feedback
The first step towards the solution of the output regulation problem in the
case of error feedback consists of showing a result which precisely corresponds,
in the present setup, to the result expressed by Lemina 8.3.1.
404
8. Tracking and Regulation
Lemma 8.4.1. Assume that, for some	the condition (S)ef is
satisfied. Then, the condition (R)ef IS (dso satisfied if and only if there e.nst
mappings r = 7r(tc) and f = cr(ir). with tf(O) = 0 and <т(0) = I), defined in a
neighborhood IVе С IV of the. origin, satisfying the conditions
^-s{w) ~ f{A(w),w.e(fj(w))]
Bin =	(8-22)
Uw
0 =	w)
for all w G И °.
Proof By assumption, all the eigenvalues of the matrix
f A BH\
\GC F J
have negative real part and those of the matrix S are on the imaginary axis.
Thus, the closed loop system (8.9) has a center manifold at (0. 0.0). the graph
of a mapping
T =	"(if)
< = ^(»') ;
with “(tc) and rr(w) satisfying the first one of (8.22) and
(8.23)
As in the proof of Lemma 8.3.1. the hypothesis of neutral stability and the
fulfillment of (R)ef imply that the mapping ,r = ~(m) must satisfy the last
one of (8.22) and this, together with (8.23). implies the second one of (8.22).
The sufficiency can be proven exactly qfi in Lemma 8.3.1. <
t
From this result, it is immediate to deduce that the fulfillment of the
identities (8.15) which were established in the analysis of the problem of
output regulation in the case of full information - continues to be a necessary
condition for the existence for solutions of a problem of output regulation also
in the case of error feedback. In fact, it suffices to set
c(ic) = 0(cr(ir))
in the first one of (8.22) to conclude that the mappings z = тг(и’) and a =
c(w) necessarily fulfill the identities (8.15). However, while in the ease of full
information it was possible to prove that the condition in question and the
other (trivially necessary) condition that the pair (A.B) is stabilizable were
together sufficient for the existence of a solution for the output regulation
problem, the situation is slightly more complicated in the present setup. As a
matter of face in general, the condition in question (namely, the fulfillment of
(8.15)) together with the (trivially necessary) conditions that the pair (A.B)
8.4 Output Regulation in the Case of Error Feedback
405
is stabilizable and the pair (C, .4) is detectable do not provide yet a set of
sufficient conditions for the solution of the problem of output regulation in
the case of error feedback. There is an additional condition which needs to
be fulfilled, which - as we will see can be expressed as a special property
of the solution tt(u-), c(u’) of the equations (8.15).
In order to describe this new condition, some preliminary material is
needed. First of all. it is convenient to return for a moment to the problem
of output regulation in the case of full information and to observe that, if
(8.15) hold, the graph of the mapping z = тг(ц') is an invariant manifold for
the composite system
./ = f (t, w. r(ic)J
} '	(8.24)
w = s(w)
and the error map e — h(x, tc) is zero at each point of this manifold. From
this interpretation, it is easy to observe that, for any initial (namely, at time
t = 0) state ir* of the exosystem, i.e. for any exogenous input
w*(t) =	•
if the plant is in the initial state r’ = and the input is equal to
u*(t) =
then e(t) — 0 for all t > 0. In other words, the control input generated by the
autonomous system
w = s(w)
и — c(ic)
is precisely the control required to impose, for any exogenous input, a response
producing an identically zero error, provided that the initial condition of the
plant is appropriately set (namely, at x* = 7г(?г*)).
The question of whether or not such a response is actually the steady state
response (that is whether or not the error converges to zero, as time tends
to infinity, when the initial condition of the plant is other than — тг(т*))
depends indeed on the asymptotic properties of the equilibrium x = 0 of
/(.r,0, 0). If such equilibrium is not stable in the first, approximation, then
in order to achieve the required steady state response, the control law must
also include a stabilizing component, as it is in the case of the control law
(8.17) indicated in the previous section. Under this control law. the composite
system
x = /(j, imc(u’) + K{x — tt(u'))
th — s (it;)
still has an invariant manifold of the form т = тг( tc), but the latter is now lo-
cally exponentially attractive. In this configuration, from any initial condition
in a neighborhood of the origin, the response of the closed loop system
x - f(x. w. c(w) + К(x - 7r(w))
406
8. Tracking and Regulation
to any exogenous input tr-* () converges towards rhe1 response of the open loop
system
x — f(x, и:, и)
produced by the same exogenous input by the control input «*(•) =
c(tc“(-))- with initial condition / =
In what follows, it will be shown that the existence of a solution of the
problem of output regulation in the case of error feedback depends - among
other things - on a particular property of the autonomous system (8.25)
which, as we have seen, may be thought of as a generator of those input
functions which produce responses yielding zero error. The description of the
property in question requires however - a preliminary digression on the
notion of immersion of a system into another system.
Consider a pair of smooth autonomous systems with outputs
j- - /(z). у =
and
= /(f). у = ВД
defined on two different, state spaces. A’ and .V. but having the same output
space У = R7rt. Assume, as usual. /(0) = 0. h(0) — 0 and /(0) = 0, h(0) =
0 and let the two systems in question be denoted - for convenience by
{Ah/, h} and {Ah/./<}• respectively.
System {Ah/, h} is said to be immersed into system {Ah f, /?} if there
exists a Ck mapping т . X —> A. with k > 1. satisfying r(0) = 0 and
h(.r) ± h(z) h(r(x)) ± h(i-(<))-
such that
дт
~dx^^ 7	(8-26)
h(j-)	=
for all j- € Ah
It is easy to realize that the two conditions indicated in this definition
express nothing else than the property that any output response generated
by {Ah /. /1} is also an output response of {Ah /. h}. In fact, the first condition
implies that the flows Ф{ (x ) and ф/(т) of the two vector fields / and / (which
are т-related), satisfy
т(ф{ (x)) = Ф{(г(х))
for all x € X and all t > 0, from which the second condition yields
Ь(ф{(х)) = Ь(т(ф{ (т))) = Н(ф{(т(хУ)}.
for all x € A' and all t >0. thus showing that the output response produced
by {Ah/, h}. when its initial state is any x € AC is a response that can also
be produced by {Ah/.Л}, if the latter is set in the initial state т(х) E Ah
8-4 Output Regulation in the Case of Error Feedback
407
The reason why the notion of immersion is relevant is because, sometimes.
{A'. f. /?} may have some special property that {A'. /. h} doesn't have. For
example, any linear system can always be immersed into an observable linear
system, and a similar thing occurs - under appropriate hypotheses also
in the case of a nonlinear system. Or. for instance, one may wish to have
a nonlinear system immersed into a linear system, if possible. The reason
why the notion of immersion is important for the solution of the problem
of output regulation in the case of error feedback is that the possibility of
having the autonomous system (8.25) immersed into a system with special
properties is actually a necessary and sufficient condition for the existence of
such a solution. Before discussing this point, however, we wish to present a
pair of results related to the notion of immersion, which will be used in the
sequel.
Proposition 8.4.2. Suppose there exists integers pi.p>........Pm such that
171
dim( span{d/i;, d.L fht......dLpf~'hi}} — py 4- p-> +   • + p JU
1 = 1
at r — () and.
dLPjhi € span{dh,. dL/h,..... dLp‘^} ht } j
i=i
for all 1 < г' < p. Then, there exists a neighborhood .V° G A of the origin
such that {X°.f,h} is immersed into a (pi + j>2 + •  • + p,n)-dimensional
system {X.f.h} whose linear approximation at x = 0 is observable.
Proof. Consider, for the sake of simplicity, the case m — 1. set p = pi and
Viewing the p components of r as a partial set of new local coordinates in
A'. it is readily seen that, since by hypothesis
dLpjh (E span{dh.dLfh.....d.Lp~lh\ ,
there exists a function <g(x\ . x!>. - - , t'J such that
Lph = d(fi. Lfh. •  . h) .
This shows that, for some neighborhood A'c G A' of the origin. {A'°. f.h} is
immersed into a system h} in which
408
8. Tracking and Regulation
/W =
h(x) = x\ .
The latter has indeed a linear approximation, at x — 0. which is observable.
The extension of these arguments to the case in which rn > 1 is straightfor-
ward. <
The following statement provides conditions for immersion into a linear
observable system.
Proposition 8.4.3. The following are equivalent:
(i)	{X. f.h} is immersed into a finite dimensional and observable linear sys-
tem,
(ii)	the observation space О of {X, f.h} has finite dimension.
(iii)	there exist an integer q and a set of real numbers Qo-ai, - -   o9-i such
that
L^h{w) ~ aoh(w) + a\Lfh(w) +  • - +	1 h(tc) .
Proof. To prove that (ii) implies (i) consider, for the sake of simplicity, the
case in which m = 1 and suppose the observation space О of {Д', f.h} has
finite dimension r. Then, by definition.
h(x). Lfh(xLrj 1 h(x)
is a basis of O. In particular’ the function L^h{x), which is an element of O.
can be expressed in the form
£^/i,(t) = aoh(x) +aiLfh(ir} + +ar_i£y 1h(r)
for some set of real numbers ak, 0 < k < r - I . Thus, {A*./, h} is indeed
immersed into an observable linear system {Kr./.h} in which
/(*) =
h(i) = j-)
\ Go-r'l + «1^2 + ‘ '
• + ar-iT'r /
via
/ h(x) \
Lfh(x)
r[x) =
£y “h(x)
8.4 Output Regulation in the Case of Error Feedback
409
The extension of these arguments to the case in which m > 1 is straightfor-
ward.
To prove that (i) implies (iii). observe that by definition
(J.г
h(x) = ffr(r) .
where F and H are matrices of real numbers. From this it. is easy to deduce
that
Lkfh(x) = HFkr(x)
for any к > 0. Let
p( A) = p0 + pi A + •   + Pq-i A5 1 * * + A?
denote the minimal polynomial of F. Then
pohfir) + piLfh(x) +  + pq_iL4f~lh(x) + Lgfh(x) ~ Hp(F)r(x) = 0
from which the result, follows.
The proof that (iii) implies (ii) is immediate. <
We are now in a position to state an important result concerning the
solution of the Error Feedback Output. Regulation Problem.
Theorem 8.4.4. The Error Feedback Output Regulation Problem is solvable
if and only if there exist mappings x — тг(гь’) and и = c(ir). with тг(0) - 0 and
c(0) = 0, both defined in a neighborhood TF° С IF of the origin, satisfying
the conditions
= /(^(u.1), tc. c(u:))	27)
0 = 7?(tt(u.') . ir) .
for all w € IF0, and such that the autonomous system with output {lF°.s.c}
is immersed into a system
i =
и = -к).
defined on a neighborhood xF of the origin in Bp. in which ip(0) — 0 and
y(0) = 0. and the two matrices
.	5=0
5=0
(8.28)
are such that the pair
A 0 \	/ В X
лтс Ф)  о у
(8.29)
410	8. Tracking and Regulation
is stabilizable for some choice of the matrix A . and the pair
/ 4 BF\
(C 0).	ф j	(8.3(J)
rs detectable.
Proof. Necessity. Suppose a controller of the form (8.8) solves the problem
of output regulation. Then, by Lemma 8.4.1. there exist mappings r = "(ir)
and £ = with тг(О) = 0 and cr(0) = 0. such that (8.22) are satisfied. Set
e(u') =8^(0-)).	=
and observe that 7t(uj and r(w) satisfy the conditions (8.27), while w'} and
t(u') satisfy
c(;r) = y(cr(u.-)) .
die
thus showing that {H'°.,s. <?} is immersed into {.ЕЛ v}, where T = <т(1Г =).
Observe now that, by definition (recall (8.13) and (8.28)), the mappings
^(£) and y(£) introduced above are such that
F = Ф, H = Г
and therefore, since all the eigenvalues of the matrix
f A BH\
\GC F )
have negative real part, so have the eigenvalues of the matrix
(А ВГ\
\GC /Ф J '
This indeed implies that (8.29) is stabilizable, for AT = G. and that (8.30) is
detectable.
Sufficiency. Choose A7 so that (8.29) is stabilizable. Then, observe that,
as a consequence of the hypotheses on (8,29) and (8.30), the triplet
is stabilizable and detectable. Choose	L. M	so that
( ( А	ВГ\	(B\	\
И A7C	Ф J	)
\ L(C	0)	К	;
has all eigenvalues with negative real part.
Now. consider the controller
8.4 Output Regulation in the Case of Error Feedback
411
so
u
A'£() + Le
p(£i ) + A>
-V^o + - (£J .
(8.31)
It is easy to see that rite controller thus defined solves the problem of
output regulation. In fact, it is immediate to see that the .Jacobian matrix of .
the vector field
/ f[x. 0. 4/£0 + " (,£i)) \
FGr.^j) =	A£0 + Z/t(z.O)
\ y(si) + -Vh(z. 0) /
at (z.£o.£i) = (0.0.0). which has the form
.4
LC
A'C
BM
к
0
вг\
о
ф )
has all eigenvalues with negative real part. Moreover, by hypothesis, there
exist mappings .r = ~('tr). u = c(<r) and = t(u’) such that. (8.27) hold and
dr
.s(u') = ^(t(u')).	c(u') = (7(ud) •
uw
This shows that the sufficient conditions of Lemma 8.4.1 are satisfied by
and completes the proof of the sufficiency- <
The statement of Theorem 8.4.4 essentially says that the problem of out-
put regulation in the case of error feedback is solvable if and only if it is
possible to find a mapping c(u') which renders the identities (8,27) satis-
fied for some ,т(ш) and. moreover, is such that the autonomous system with
output
satisfies a certain number of special conditions, which are expressed as prop-
erties of the linear approximation of an "auxiliary" system in which the latter
is requested to be immersed. We will discuss now in some detail the role of
these additional conditions. First of all. we observe that the condition that
the pair (8.29) is stabilizable implies the condition that the pair (.4.B) is
stabilizable and. similarly, the condition that the pair (8.30) is detectable
implies the condition that the pair (C. .4) is detectable (a simple application
of a standard stabilizability/detectability test suffices to check this claim).
Thus, the conditions of Theorem 8.4.4 include as expected the trivial
necessary conditions requested for the fulfillment of (S)ff-
412	8. Tracking and Regulation
Second, we observe that the condition that the pair (8.30) is detectable
also implies the condition that, the pair (Г, Ф) is detectable. Therefore, it is
deduced that a necessary condition for the solution of the problem of output
regulation in the case of error feedback is that, for some r(u’) satisfying
(8,27). the autonomous system with outputs (8.32) is immersed into a system
whose linear approximation at the equilibrium £ — 0 is detectable. This can
always be achieved if (8.32) is a linear system, but it may not be possible in
general. To verify the property that (8.32) is immersed into a system having
a detectable linear approximation one may use the (sufficient) conditions
indicated in the previous Propositions 8.4,2 and 8.4.3. and this may yield a
number of alternative versions (actually. Corollaries) of Theorem 8,4.4, which
will be presented at the end of the section.
Finally, it may be worth finding out whether or not the special properties
requested on (8.29) and on (8.30) could be more directly formulated as prop*
erties of the triplet (C, .4. B) which characterizes the linear approximation of
the controlled plant. This is actually possible to some extent, in view of the
following result (and of its dual version), whose proof is left as an exercise to
the reader.
Lemma 8.4.5. Suppose. (C. .4) and (H.F) are. detectable pairs. .4 sufficient
condition for the pair
to be detectable is that the matrix
A- XI
(8.33)
has independent columns for every A which is an eigenvalue of F having
non-negative real part. If
BH(ker(F-A/)) = Im(B)
for each eigenvalue A of F having non-negative real part, then this condition
is also necessary. In particular this condition is necessary if m = 1.
The controller constructed in the proof of the sufficiency of Theorem
8.4.4 lends itself to an interesting interpretation. The controller in question,
in fact, consists of the parallel connection (see Fig. 8.3) of two subsystems:
the subcontroller	+AV 4	(8.34) и = 9(6).
and the subcontroller	6 = K(o+Le , rr	(8.3o) u = XI .
8.4 Output Regulation in the Case of Error Feedback
413
Fig. 8.3.
As made clear in the proof of the Theorem, the role of the second sub-
controller. which is a linear system, is nothing else than that of stabilizing in
the first approximation the interconnection
> = /(t.?c,;(Ci) - n)
Ci = у (Ci) + A h.(z. «)
e — h{j\»') .
that is the interconnection of controlled plant and first subcontrollcr. The
role of the first sub co nt roll er. on the other hand, is that of producing an
input which generates the desired steady state response. As a matter of fact,
the identities
-^.s(?c) =	W7. - (-( tr)))
W’
= y(r(«d)
(which hold by construction) render the submanifold
Mc = {(.r.CmCi. w) : .r = tt(u;).Co = 0, Ci = r(u.’)}
an invariant manifold of the composite system
.r = /(.r. tc;y(Ci) +-VCo)
Co = It Co + Lh(jr.w)
Ci = V?(C1) + A'h(.r. ?C)
zr = x(zc)
i.e. of the closed loop system driven by the exosystem, and on this manifold
the error map e = h(-.r. ir) is zero.
The role of the subcontroller (8.34) is that of producing, for each initial
condition in Mc. an input, which keeps the trajectory of this composite sys-
tem evolving on Mc (and thereby producing a response for which the error is
414	8. Tracking and Regulation
zero). For this reason, the subcontroller [8.34) is sometimes referred to as an
internal model of the generator of exogenous inputs. The role of the snbcon-
troller (8.35) is that of rendering locally exponentially attractive so that
every motion starting in a sufficiently small neighborhood of the equilibrium
u’) = (0-0.0.0) exponentially converges towards the desired steady
state response.
We conclude the section with two Corollaries of Theorem 8.4.4 which,
as it was anticipated, can be deduced as consequences of the conditions for
immersion given in Propositions 8.4.2 and 8.4.3 as well as of the test indicated
in Lemma 8.4.5.
Corollary 8.4.6. The Error Feedback Output Regulation Problem is solvable
if the pair (A.B) is stabilizable, the pair (C. A) is detectable, there exist map-
pings .r — д'(ie) and и = c(w}. with тг(0) = 0 and c(0) — 0. both defined in
a neighborhood И ° C IF of the origin, satisfying the conditions (8.27) and
such that, for some set of integers Pi,p2 - • • • --
dim(^2 span{dc,_ d£sc;;.... dLps' 1 с;}) = Pi + p2 + • • • + Pm
at x = 0 and
dLps'ct e (J2 span {de,,	.....dLp' 1ci})
where cRw) is the. i-th entry ofc(w). for all 1 < i < p. and - moreover - the
matrix (8.33) is nonsingular for every A which is an eigenvalue of S.
Proof. The conditions indicated imply (see Proposition 8.4.2) that. {IW. ,ч, r}
is immersed, via
£ = r(w) =

Cm(w)

into a system whose linear approximation at £ = 0 is observable. Observe
now that, equating the first order terms of both sides of the identity
vields
TS = ФТ
where T. the .Jacobian matrix of r(tr) at w = 0. is a matrix having p inde-
pendent rows, with p = p] +  - • + p„7. Suppose A is an eigenvalue of Ф. Then
8,4 Output Regulation in the Case of Error Feedback
415
there is a row vector г such that сФ = Xv and the previous identity yields
vTS = t'TX. where vT Ф 0, thus showing that A is necessarily an eigenvalue
of S.
Choose any p x m. matrix Ar such that the pair (Ф. A') is stabilizable
(this is always possible because by construction the pair (Г.Ф) is detectable
and Г is a m x p matrix). Then, using Lemma 8.4.5. one can conclude that
the remaining conditions of Theorem 8.4.4 hold since the matrix (8.33) is
nonsingular for every A which is an eigenvalue of S and thus, in particular,
for every A which is an eigenvalue of Ф. <
Corollary 8.4-7. The Error Feedback Output Regulation Problem is solvable
by means of a linear controller if the pair (A.B) is stabilizable, the pair
(C. A) is detectable, there exist mappings x = 7?(w) and и = with
tt(O) — 0 and c(0) = 0, both defined in a neighborhood IVе C IT of the origin,
satisfying the conditions (8.27) and such that, for some set of q real numbers
«0,01......«<?-] -
Lgsc(w) = a[yc(w) + rt]£sc(u!) 4-----H	.	(8-36)
and - moreover - the matrix (8.83) is nonsingular for every A which is a root
of the polynomial
p(X) — ciq + ti 1A + ... +	1 A9 1 — A9
having поп-negative real part.
Proof. The proof is essentially identical to that of the previous Corollary. In
the present, case, condition (8.36) implies that {lTc.s,c} is immersed into a
linear observable system. In particular, it is very easy to check that {H‘°. s.c}
is immersed into the linear system
in which
Ф
Г
and
ё =
и = Г£
diag(^....Ф)
diag(f....;f)
0
0
1
Oq-l /
Г = (1 0 0 ••• 0).
In this case, the minimal polynomial of Ф is equal to p(A). After having chosen
a matrix .V such that the pair (Ф, A’) is stabilizable. for instance
416	8. Tracking and Regulation
.V = diag(.V....A')
with	~
A’ = col(0.0....0.1) .
using Lemma 8.4.5 one can conclude that the remaining conditions of The-
orem 8.4.4 hold since the matrix (8.33) is nonsingular for every A which is
an eigenvalue of Ф. Note also that, unlike the case examined in the previous
Corollary, the eigenvalues of Ф are not necessarily eigenvalues of S and this
explain why. in the statement of the Corollary, explicit reference was made
to the polynomial p(A). <
Remark 8.4-1- Note that the condition (8.36) is indeed a necessary condition
for the existence of a linear controller
£ =	+ Ge
u =
solving the problem of output regulation via error feedback. In fact, if such
a controller exists, from the proof of necessity in Theorem 8.4.4 it is deduced
that
= F(r(ir), c(u') = Gcr(ir) .
du-
for some mapping £ — cr(ir)- Thus {IVе. s. c) is immersed into a linear system
and. by Proposition 8-4.7. condition (8.36) necessarily holds. <
8.5 Structurally Stable Regulation
In this section, we consider the case in which the mathematical model of the
controlled plant depends on a certain ^et of parameters, which are assumed
to be fixed, but whose actual values are not known. The purpose is to design
a control law capable to solve the problem of output regulation via error
feedback for each set of values of the unknown parameters, at least in some
neighborhood of their nominal values.
For convenience, we continue to consider the case of a family of plants
modeled by equations of the form (8-4), in which we now explicitly introduce
a vector ц E of unknown parameters, in the form
i = /(j,atu,g)
e = h(j\ w, p).
(8-37)
Without loss of generality, we suppose p = 0 to be the nominal value of
the parameter /i and, for consistency with the analysis developed earlier, we
assume /(.r. w, u, p) and h(x, u\ /.i) to be smooth functions of their arguments.
Moreover, we also assume /(0.0,0,p) = 0 and h(0,0.p) = 0 for each value
of p. Finally, we assume that the exosystem - which models the family of
8.5 Structurally Stable Regulation
417
external commands against which regulation is to be achieved is not affected
by parameter uncertainty of any kind, and we continue to use (8.5) to denote
it. In this setup, we address the following design problem.
Structurally Stable Output Regulation Problem. Given a nonlinear
system of the form (8.37) and a neutrally stable exosystem (8.5). find, if
possible, an integer n. two mappings and //(£.*9 and a neighborhood P
of p = 0 in IF? such that, for each p € P:
(S) the equilibrium (t.£) = (0.0) of
i = (8 38)
5 = ^.hU-O.O)
is asymptotically stable in the first approximation,
(R) there exists a neighborhood V c U x E x Iff of (0,0.0) such that, for
each initial condition (t(0), £(0). m(0)) G V. the solution of
j =
£ =	H'./x))	(8.39)
11'	=	.S'(tr)
is such that
lim e(t) = 0 .
The solution of the problem in question is easily provided by the results
illustrated in the previous section. In fact, it suffices to look at w and p as if
they were components of an "augmented” exogenous input
which is generated by the "augmented1' exosystein
and regard the family of plants (8.37) as a single plant of the form (8.4).
modeled by equations of the form
j /а(т. гса.?г)
e — /ia(r.wa) .
It is easy to realize that a controller which solves the problem of output
regulation for the plant thus defined also solves the problem of structurally
stable output regulation for the family (8.37). In fact, by construction, this
controller wall stabilize in the first approximation the equilibrium (t, £) =
(0, 0) of
418
8. Tracking and Regulation
j- = Г^.о.ад)
£ = ?/(£• lP(x, 0) .
that is the equilibrium (z.^) = (0-0) of
j- = /(.r.O.0(£),p)
£ = rfifi.hfir.O.pY) .
for p = 0. Since the property of stability in the first approximation in not
destroyed by small parameter variations, the controller in question stabilizes
any plant of the family (8.37). so long /t stays on some open neighborhood
P of the origin in the parameter space. Moreover, this controller will be such
that linp-»-^, e(t) = 0 for every (-r(O). £(0). w’a(O)j in a neighborhood of of the
origin. Since
tca(0) =
\ /
the controller in question trivially yields the required property of output
regulation for any plant of the family (8.37), so long as p stays on some open
neighborhood P of the origin in the parameter space.
The conditions provided in Theorem 8.4.4 can be easily translated into
necessary and sufficient conditions for the existence of solutions of the prob-
lem of structurally stable regulation.
In other to put these conditions in explicit form, set
=
W) =
<э/1
(О.О.О.дО
I0.0-0.fj)
~ dx
J (0.0.pj
Moreover, observe that, because of the special form of the vector field sa(u:a).
dtra(uyp)	= дтга(?лр)
dwA	f dw
Then, we have the following.
Theorem 8.5.1. The Structurally Stable Output Regulation Problem is solv-
able if and only if there exist mappings x = тга(ш.р) and и = cA(w.p), with
тга(0,р) = 0 and c^fO.p) = 0, both defined in a neighborhood И'° x P c
П' x Kp of the origin, satisfying the conditions
дтРОг н }
-—= f^^pfiw.c^w.pfip)	(840)
0	—	/t(7Ta(w, p), W. fl)
for all (ш, p) G И ° x P, and such that the autonomous system with output
{1Г° x P,5a.ca} is immersed into a system
ДО
?(e) -
8.5 Structurally Stable Regulation
-119
defined on a neighborhood — ° of the origin in R1', in which ^(0) = 0 and
y(0) — 0, and the two matrices
are such that the pair
( M(O) 0A /B(0)A
\ VC(0) Ф J ‘ V 0 J
ts stabilizable for some choice of the matrix Л’. and the pair
(C(0) 0).
Л(0) В[0)Г
0 Ф
is detectable.
Remark 8.5.1. The proof of the necessity of this Theorem is exactly rhe
same as the proof of Theorem 8.4.4. In particular, from the hypothesis that
f{x. w. u. p) and h(x, w, p) are smooth functions of their arguments, it is eas-
ily deduced that if the problem of structurally stable output regulation is
solved by a controller in which t/(^.</) and 0(£) are Ck functions, the map-
pings тга and ca are functions. Moreover, from the hypothesis that
/(0,0. 0,p) ~ 0 and h(0.0. p) — 0, using the property that a center manifold
contains all other equilibria which are sufficiently close to the one at which
this manifold is defined, it is also deduced that тга(0, p) = 0 and ca(0.p.) = 0
in a neighborhood of p = 0. <j
Remark 8.5.2. Note that the first condition indicated in this Theorem,
namely the existence of a solution тга( jc. p). ca(u’. p) of the equations (8.40)
for each p in a neighborhood of p — 0 is a trivial necessary condition for
the existence of a solution of the problem of structurally stable output regu-
lation. The condition in question, in fact, is one of the necessary conditions
(see Theorem 8.4.4) for the existence of a solution of the standard problem
of output regulation via error feedback for any fixed value of p. <J
Remark 8.5.3. Note that the linear approximation of {TV3 x Р..ча.са} at the
equilibrium (w,p) = (0.0) cannot be detectable. In fact, since ca(0, p) = 0
by hypothesis.
<Эеа ,. ч
^-(O.p) = 0 .
<Эр
and the linear approximation in question is characterized by a pair of matrices
of the form
0)
OA
0 )
s
0
which is indeed not. detectable. Thus, it is not possible to have the conditions
of the Theorem directly satisfied by the trivial immersion of {IV° x P, sa. ca}
420
8. Tracking and Regulation
into itself. However, as shown below, {И'° x P. .s-a.cA} may be immersed into
another .system {Sc, }, having a detectable linear approximation at C = I).
<j
Remark 8.5.4- The condition that system {H’° x P. sa.ca} is immersed into
a system {Scy?.".} is the existence of a mapping r'*(w. /j ) such that
дта
—	= р(та(ш;р)). ra(ir.p) = :.(ra(tc.p)).
aw
Choose A\ L, .V, A' as suggested in the proof of Theorem 8.4.4. Then, a simple
calculation (see also section 8.4) shows that.
Л4 = {(.r-Co.Ci- ш./t) : x = -al>.p).Co = 0,Ci = />//)}
is a center manifold for the system
i	~	f(x.	tr.	" (Cl ). p)
Co	=	КCo	+ Lh(x. ir.fi)
si -	) + A7;(a tmp)
ii,-	=	s(tr)
p	=	0.
at the equilibrium (r. Co-Ci  p) — (0,0. 0,0.0)- Since a center manifold
contains all other equilibria which an1 sufficiently close to this particular one,
it is deduced that any point (j*. Co- Ci  p) = (0,0.0.0. p) is a point of .Wc.
In particular. ra(0.p) = 0- <
Clearly, it is possible to establish results which are analogous to those
indicated in the statements of Corollaries 8.4.6 and 8.4.7. We present hereafter
the result which corresponds to Corollary 8-4.7- To this end. observe that,
because of the special form of the vector field -sa(u,a), the derivative of any
function A(m.p) along aa(u:a) reduces to
г X/ i c?A( u-.p.)
L^X(w.p) = ------------s'(ic) .
dw
For convenience, the latter will be simply indicated as
LsX(w.p) .
Corollary 8.5.2. The. Str net anally Stable Output Regulation Problem is solv-
able by means of a linear controller if the pair (-4(0). B(0)) is stabilizable,
the pair (C(0).-4(0)) is detectable, there exist mappings x = va(ic.p) and
и = ca(imp). with 7ra(0.p) = 0 and c^O.p) = 0. both defined in a neighbor-
hood IFC x P С (V x of the origin, satisfying the conditions (8.40) and
such that, for some set of q real numbers ар. щ ,. . . .	,
Z^ra(u\ p) = nora((c.p) + «! L.sca(m.p) +  •  + aq_iLl.~iea(ir.fi) ,
8.5 Structurally Stable Regulation
421
for all (u-,p) G H'° x Pf and - moreover - the matrix
/А(0) - XI 5(0) \
C(0)	0 )
(8.41)
is nonsingular for every X which is a root of the polynomial
p( A) — no 4- a i Л + ... + tiq-] X4 1 — A'7
having non-negative real part.
We conclude the section by discussing some applications of this Corollary.
The first and most simple application is indeed found in the problem of
structurally stable output regulation of a linear system, modeled by equations
of the form
> = A(p)x + P(p)w + B(p)u
e = C(p)x + Q(p)u'
In this case, the conditions of the Corollary assume the following form.
First of all. it is required that the pair (*4(0).5(0)) is stabilizable and the
pair (C(O).A(O)) is detectable. Then, it is requirt'd that for every p in a
neighborhood of p = 0 the pair of linear equations
П(р)3 = А(р)П(р) + P(p) 4- 5(p)/'(p)
ч	)
0 = С(р)П(р) + Q(p) .
has a solution П(р). Г(р).
As far as the remaining condition is concerned, observe that
L'etw.p) = r(ti)Stw
for any к > 0 and thus, if we let
p(A) = po 4- pi A 4- ... 4- 4У-1А'7 1 4- Xq
denote the minimal polynomial of S, we see that
LAPAimp) = -poc*(unp) - p}Lsca(w. p)-----------p<l_iLQs~\ii(w.p).
for every (imp). Thus, the remaining condition is simply that the matrix
(8.41) is nonsingular for every A which is an eigenvalue of S. We also observe
that the condition in question - according to a well-known result about linear
matrix equations guarantees that equations (8.43) have solutions for every
p in a neighborhood of p = 0. from which it can be concluded that, in the
case of linear systems, a set of sufficient conditions for the existence of a
solution of the problem of structurally stable regulation is simply that the
pair (.4(0), 5(0)) is stabilizable, the pair (C(O).A(O)) is detectable and the
matrix (8.41) is nonsingular for every A which is an eigenvalue of S.
422
8. Tracking and Regulation
A controller which solves the problem is question can be constructed along
the lines indicated in the proof of Corollary 8.4.7 and consists of the parallel
connection (see also (8.34) and (8.351) of a system of the form
6	— ^si+Af-
u = Г£] .
with a system of the form
— A 4- Lr
и =	,
(8.44)
(8.45)
in which (A\ L. M) are such as to place the eigenvalues of
// .4(0)	В(0)Г\
\Л'С(0) Ф J
\ цс(0) о
in the open left-half complex plane. The former subsystem, as seen in the
proof of Corollary 8.4.7. is simply the collection of m identical single-input
single-output subsystems of the form
/ 0	1	0
0	0	1
th
I о 0
X ~Po -Pl
( 1 о 0 
0
(with pQ.pi.... coefficients of the minimal polynomial of S).
In the case of nonlinear systems, a/first interesting although rather
elementary application of this result ris the one in which the exosystcm
only generates constant commands (which is the case in any set point control
problem). In this case, in fact, s(u') = 0 and. no matter what the solution
ca(ir.jz) of (8.40) is, the condition
= unra(h’./J)
is trivially satisfied by n0 = 0. If (8.41) is iionsingular at A = 0. the problem of
structurally stable regulation can be solved by a linear controller. The latter
consists of two parallel connected subsystems as (8.44) and (8.45) in which
the former is given by
- e
u =	.
that is the classical form of an integral controller.
Another and more interesting example of application, in the case of
nonlinear systems, of the result indicated in Corollary 8.5.2 is the one in which
8.5 Structurally Stable Regulation 423
the exosystcm is still a linear system and the mapping cA(ir.p) is. for each p
in a neighborhood of p = 0. a polynomial (with ^-dependent coefficients} in
the components u.q. ..., of ir. whose degree does not exceed a fixed integer
Ac In fact, observe that if r(ir) is a polynomial in ir of degree less than or
equal to k. and .s(w) = Six. then also
_. . . de
Lsc{u:) = w—
is a polynomial in te of degree less than or equal to k. In other words, the set
Pk of all polynomials in ir with real coefficients is a finite dimensional vector
space, which is closed under the action of the mapping
I, : Pk Pk
( .	de c	(8.46)
r tr)	— Sa- .
ow
Since Ls is a linear mapping of the finite dimensional vector space Pk into
itself, its minimal polynomial
p( A) = Po + P\ A + . . . pq-l X4 + A4
is such that
L^c(u’) = -pnc(w) - p}Lsc(u')-----------pq_}LQ~'c(in) .
Thus, in view of the result of Corollary 8.5.2. it can be concluded that if.
for each p in a neighborhood of p — 0. the mapping ra(u\p) is a polynomial
in the components u.q.......wr of u- of degree not exceeding a fixed integer
к. structurally stable regulation can be achieved, provided that the matrix
(841) is nonsingular for each A with nonnegative real part which is an eigen-
value of the linear mapping (8.46). The controller yielding structurally stable
regulation has the same structure as the one described before in the case of
linear systems, but now the parameters which appear in the matrix Ф are the
coefficients of the minimal polynomial of (846).
The following elementary example shows how such a result can be used.
Example 8.5.5. Consider the nonlinear system
Л = x-2 + (tix'l
= (1 +	+ (1 + Рз)-Т? + U
e = 7xi - u'i
in which p = col(pi. р-2-рз) is a vector of unknown parameters, and suppose
»'i is generated by the linear exosystem
trq = w?
lE> = -tt'i .
424
8. Tracking and Regulation
An immediate calculation shows that the equations (8.40) have solutions
for each p. namely
7Tj (if. fl) = U’l ,	7Г^ ( IC. fl) = 11'2 — p] tt'f
ca(tc. fl) = -(2 + fi-pu'i - (1 + рз)а-2 + (1 + /r3)piU’f - 2piWj u'2 
in which ca(iu,fi) is a polynomial of degree not exceeding 2. is a space of
dimension 5. and Ls maps a polynomial
c(u’) = Uif/’i -f-(12lt’2 + nillt'1 + «12?t’i U’2 + а22У'2
into the polynomial
Lse(ir) = —a2it’i + aiti’2 — (iv’it'l + (2ац — 2«22)'i'i it's +	
Choosing a basis in P-2 consisting of {u’i. u--2-ao, it is readily
seen that L# is represented by the matrix
	(°	-1 0	0 0	0 0	° 0
M =	0	0	0	-1	0
	0	0	2	0	—2
	\0	0	0	1	0 /
whose characteristic polynom	ial is				
pW	=	A(A2	h l)(Aa +4) .		
Since the matrix
/.4(0)-AC B(0)\ f ?	°
I C'(°)	° Я ( i “0 о
is nonsingular for every Л. it is concluded that a structurally stable regulator
exists. The latter has the structure indicated ahove. where in particular (8.44)
is given by
Ci
и
/0 1 0
0 0	1
0 0 0
0 0 0
\ 0 -4 0
(10 0 0
0R1 -
Note that the explicit knowledge of ca(w,/i) and 7ta(w.p) is not requested
for the construction of the controller, о
8.5 Structurally Stable Regulation
425
Example 8.5.6- The system examined in the previous example is a particular
case of a nonlinear system of the form
ii
j-2
«2(^)Т2 4- /Ji (Xi t p)
«зМ-Гз + P2(-гч:.р)
и» (/t)-Tri + pn — i (. X2.....Jrn — i. a', p)
Pr^T^X-2,. . ..xn,u\p) + b(p)u
c(p)*l + <l(u-,p)
in which рДт], x-_>,... , Xi.it'-p) are polynomials in (x. w) and q(u\ p) is poly-
nomial in и-, of degree not exceeding a fixed number k. and aj(0). пз(0).
an(0). ^(0), f(0) are all nonzero. If the exosysteni is a linear system, then the
equations (8.40) have a solution, for each p in a neighborhood of p — 0. in
which (?(u:.p) is a polynomial whose degree does not exceed a fixed number,
and the previous analysis easily applies.
Note also that, if aa(p). аз (/г). .... an(p). b(p). c(p) are nonzero for all
p € Kp. then equations (8.40) have a solution which is defined for all (w, p) E
T x Г <

9. Global Feedback Design for Single-Input
Single-Output Systems
9.1 Global Normal Forms
In Chapters 4 and 5. we have presented a number of important concepts
which lead to the design of feedback laws which solve the problems of trans-
forming a nonlinear system into an equivalent linear system (possibly after
a change of coordinates in the state space), locally asymptotically stabilizing
a given equilibrium point (for those nonlinear systems whose zero dynam-
ics has an asymptotically stable equilibrium at this point), and rendering
certain outputs independent of certain inputs (the problems of disturbance
decoupling and noninteracting control). As pointed out several tiqies. all the
procedures illustrated in these Chapters have a local character, in the sense
that they lead to the design of feedback laws which are defined only in a
neighborhood of a given (equilibrium) point. We want to discuss now under
what conditions and how these design methodologies can be extended so as to
yield globally defined solutions to the above mentioned design problems. For
the sake of simplicity, and also for reasons of space, we restrict our consid-
eration to the case of single-input single-output systems. As seen in Chapter
5, in most cases, the analysis of the more general situation of a multi-input
multi-output system is not conceptually harder and only notationally more
involved.
To this end we begin by addressing, in this section, the problem of deriving
the global version of the coordinates transformation and normal form intro-
duced in section 4.1. Consider a single-input single-output system described
by equations of the form
.r = f(j-)	g(x)u
J J	3	9.1)
У = h(j')
in which /(.r) and g(x) are smooth vector fields, and h(.r') is a smooth func-
tion. defined on S". Assume, as usual, that /(0) = 0 anti /1(0) = 0. This
system is said to have uniform relative degree r if it has relative degree r at
each ;rc e .
If system (9.1) has uniform relative degree r. the r differentials
(ih(i-).dL / h.(;r)......h(j-)
428
9. Global Feedbac k Design for Single-Input Single-Out put Systems
are linearly independent at each .r € K" and therefore the set
Z* = {z e ПГ : Ш) = Lfh(x) = ...= Lr~lh(x) = 0}
(which is nonempty in view of the hypothesis that /(0) =0 and h(0) =0) is a
smooth embedded submanifold of R'1 . of dimension n - r. In particular, each
connected component of Z* is a. maximal integral manifold of the (nonsingular
and involutive) distribution (see section 6.3)
.Г = (span{dh.dLjh......h})" .
The submanifold Z* is the point of departure for the construction a glob-
ally defined version of the coordinates transformat ion considered in section
4.1.
Proposition 9.1.1. Suppose (9.1) has uniform relative degree r. Set
-Lrfh(x)	1
abr) =------7 т	Jb) =--------__
LgLrf-'h(x) ' LgLrf~lh(x)
and consider the (globally defined) vector fields
f(-r) = /CH + ff(.r)ab). g(x) = g(x)3(x) .
Suppose the vector fields
T, = (-D'-'ady'sU), l<c<r	(9.2)
are complete.
Then Zr is connected. Moreover, the smooth mapping
Ф  Z* x Р/ ->  R"
(г.(Ci....Cr))	(M)
in which as usual ФЦх) denotes the flow of the vector field t. has a
globally defined smooth inverse
(:.(C1.....е))	= Ф’1О)	(9.4)
in which
z = Фт\< , о - -  о Фт-	. Ь)
— Л{л->	-LI-1 М-гЛ
= Llf-]h(x) l<Si<r.
The globally defined diffeomorphism (9.4) changes system (9.1) into a
system described by equations of the form
9.1 Global Normal Forms 429
г = Л0Ч1.....sS)
= £2
e-i = &
4r =	.....^r) + ci(z.^....£r)lt
9 = Ь
where
b(z.£i.................U = Ц1юФ(:.^1.............6J)
«(=41....£r)	=	°#( = -(6...£r)l 
If. and only if, the vector fields (9.2) are such that
[тг, 7j] =0 for all 1 < i, j < r .
then the globally defined diffeomorphism (9-4) changes system (9.1)
system described by equations of the form
= = /o(=-£i)
6 = &
£r-l = &
& =	....^-) + «(=^i------£r)u
У =	
Proof. Set. for convenience.
Afor) = LP]h(x}. 1 < i < r .
and observe that, by construction.
r	f 1 if j 4- к = r -t- 1
A,(.r) = <
I 0 otherwise
It is also easy to prove that, for each j G Kn. the point
q = dff (x)
is such that
f АДт) + .? if i + k = r + 1
Ai'(^ ” 1 \	\	-
I A|(.r) otherwise
This property derives from the equality
a,(O - a,(a = Г ^а,(ф;*(л)л = Г
19.5)
into a
(9.6)
(9.7)
(9-8)
430	9. Global Feedback Design for Single-Input Single-Output Systems
and from the property (9.7).
Now. set
^-(j) = ф?Л11г1оФ2г^10"-офГ-'л„(гУ) 
and observe that (9.8) recursively yields
А;^(т)) -0 .
Thus, the1 point yj(j-) is in Z*. As a consequence, the mapping
Ф : .r (;(r). (AH-r). A2(t)...Аг(т)))
maps Rrl into Z* xRr. Actually, the image of this mapping is precisely Z* xRr.
In fact, using again (9.8), it is easy to deduce that, for each (з,(fh.£r)) €
Z’ x Rr.
АДФ^ оФ^ о...<££(») = £
and therefore
.............................
where Ф is the mapping defined by (9.3). This relation shows also that Ф ~
Ф"1. and. since both Ф and Ф are smooth mappings and RrI is connected, it
can be concluded that Z* is connected and Ф is a diffeo morphism.
By definition
?, =	+ L^-'hlzyu = LfL^hM = £i+i
for all 1 < ? < r - 1. Moreover, it is easy to check that
Ф* {--J Q о Ф
a?/.-
Thus, the diffeomorphism (9.4) changes system (9.1) into a system described
by equations of the form (9.5).
To prove the last part of the proposition observe that, by means of ar-
guments identical to those used in the proof of Frobenius* Theorem, it is
possible to deduce that
°-"оф^ЛфИ))-
Moreover, invoking again an argument already used in the proof of Frobenius'
Theorem, observe that if the two vector fields d and т commute, the function
Vi(f) = (Ф^)+т оФ?(аг)
is independent of t. i.e.
(Ф^ )#т о Ф?(т) = t(j) .
9.1 Global Normal Forms
431
Using this property repeatedly one obtains, from the previous expression.
which holds for all 0 < i < г - 1.
In the equations (9.5). the vectors f and g have the form
/ = фУ/оф = /0(;,?1......£,.)#	+	+ • + «>
O~	oG'-I
and
-	a. °
9 = Ф/доФ = — .
Thus.
dfo(z.^............................£,.) д _ d
d^r dz
On the other hand, by (9.9).
, _ a
nrf/9 = ~^Z?-
and this proves that /о(^. 0- - - - Лг) is independent of <$.. A simple induction
argument completes the proof. <
An immediate corollary of this result is that if a system has uniform
relative degree r and the vector fields (9.2) are complete, the globally defined
feedback law
-£W)	1
W = ------------1----;— c
LgLrj } h(j-} LgL'f 1h(x)
and the globally defined diffeomorphism
change the system into a new system described by equations of the form
i = /o(~.£i........£r)
=	$2
£r-1
<$r
У
6-
V
0 .
(9.10)
432	9. Global Feedback Design for Single-Input Single-Out put Systems
If. in addition, the vector fields (9.2) commute, the equations (9.10) assume
the special form
z = /o(^-G)
6 = b
у =	
Of course, if r = /?, the system in question is linear, controllable and
observable.
Note also that, if r < n, the submanifold Z‘ is the largest (with respect
to inclusion) smooth submanifold of h-1(0) with the property that, at each
x G Z*. there is id(r) such that f*(x) = f(x) + g(x)u*(x,) is tangent to Zr.
Actually, for each ,r t Z+ there is only one u*(r) rendering this condition
satisfied, namely.
a (j- = ------.
LgLrf-'h(x)
In particular, the vector field f*(x)\z- which characterizes the zero dynamics
of the system can be identified with the vector field
/о(з,О.....0)^
of Z*.
Remark 9.1.1. In the previous analysis, the (n - r)-dimensional sumbanifold
Z* is not required to be diffeoniorphic to Kn-r. However, in all subsequent
sections, we will - almost always consider the case in which the vector field
/*(t)!z- has a gioball}' asymptotically stable equilibrium at .r = 0. If this the
case, then necessarily Z* is diffeoniorphic to (see section B.2). Thus,
for the sake of simplicity, we will assume throughout that, in the equations
(9.10) and (9.11).
(c,^) G H?T-r x .<
9.2 Examples of Global Asymptotic Stabilization
In this section we discuss a number of cases in which it is possible to design a
feedback law which globally asymptotically stabilizes the equilibrium x — 0
of system (9.1). We restrict our attention to those system which have (some)
uniform relative degree r, in which the submanifold Z* is diffeoniorphic to
and in which the vector fields (9.2) are complete. Thus, without loss
of generality, in view of the results established in the previous section, we
9.2 Examples of Global Asymptotic Stabilization 433
can assume that, the system in question is modeled by equations of the form
(9.10) or, more particularly, of the form (9.11) if the vector fields (9.2) also
commute.
The results which follows describe a simple ’‘modular1’ property which is
instrumental in proving an important stabilizability result about the system
in question. Recall that a smooth function V : TT —> 3 is said to be positive
definite if V(0) =0 and V(z) > 0 for .r 0. and proper if, for any a > 0, the
set V-1(;0. cq) = {т E K'1 : 0 < V(j) < «} is compact.
Lemma 9.2.1. Consider a system described by equations of the form
z = fC.Ci
f = и
(9.12)
in which (c.<f) € FT x 1, and /(0.0) = 0. Suppose there exists a smooth
real-valued function I "(г). which is positive definite and proper, such that
f^/UO) <0
for all nonzero z. Then, there exists a smooth static feedback law и =
with u(0,0) =0. and a smooth real-valued function H'(z,£), which is positive
definite and proper, such that
(w air\	. n
\ dz d^ J \ и(г.£) )
for all nonzero (c.fj).
Proof. Observe that the function f{z.£) can be put in the form
/(c-0 - /(-.0)
(9.13)
where />(-.£) is a smooth function. For. it suffices to observe that the differ-
ence
/(.-?)	0)
is a smooth function vanishing at £ = 0. and express f(z.£) as
,/0 os J(1 L c\ J<=.<
Now, consider the positive definite and proper function
=Г(г)+ I?2 .
(9-14)
and observe that
/ап’ елг \ ff(z.n\ ai' с?г , ч л
(ch )( и ) “ dz + dz	'
43-4	9. Global Feedback Design for Single-Input Single-Output Systems
Choosing
ar
u = u(z.£) =	—р(г.£)	(9.15)
yields the required result. <
In view of the converse Lyapunov Theorem (sec section B.2). the hy-
pothesis of this Lemma (namely the hypothesis of the existence of a smooth
positive definite and proper function V(c) such that ^-f(z. 0) is negative for
each nonzero з) is implied by the hypothesis that the subsystem
i = f(z.Q)
has a globally asymptotically stable equilibrium at, z = 0. On the other hand,
now by the direct Lyapunov Theorem, the conclusion of the Lemma implies
that system
MM
has a globally asymptotically stable equilibrium at (£.£) = (0.0). Thus, the
result indicated in this Lemma simply says that, if z — /(г.О) a globally
asymptotically stable equilibrium at z — 0. then the equilibrium (з,£) =
(0,0) of system (9.12) can be rendered globally asymptotically stable by
means of a smooth feedback law и = u(z.£).
In the next Lemma (which contains Lemma 9.2.1 as a particular case)
this result is extended, by showing that, to the purpose of stabilizing the
equilibrium (г.£) = (0.0) of system (9.12), it suffices to assume that the
equilibrium z = 0 of
is stabilizable, by means of smooth law £ = г,+ (з).
Lemma 9.2.2. Consider a system ^escribed by equations of the form (9.12).
Suppose there exists a smooth real-valued function
with t'*(0) = 0. and a smooth real-valued function 1(2), which is positive
definite and proper, such that
<0
oz
far all nonzero z. Then, there exists a smooth static feedback law и = «(*,£)
with u(0,0) = 0. and a smooth real-valued function IT(z, <$), which is positive
definite and proper, such that
(SW
\ дг d( /[«MV
for all nonzero (z.ff).
9.2 Examples of Global Asymptotic Stabilization
435
Proof. It suffices to consider the (globally defined) change of variables
y = £-v*(z) ;
which transforms (9.12) into
and observe that the feedback law
ch.'*	,
и = —/(«.с (г) + у) + и
changes the latter into a system satisfying the hypotheses of Lemma 9.2.l.<
Using repeatedly the property indicated in Lemma 9.2.2 it is straightfor-
ward to derive the following stabilization result about a system in the form
(9.11).
Theorem 9.2.3. Consider a system of the form
i = /o(z.£i)
£1 = в
(9.17)
= £r
U- = « .
Suppose there exists a smooth real-valued function
£1 = е*(з).
with r*(0) = 0. and a smooth real-valued function U(z). which is positive
definite and proper, such that
^foO.CO))<0
for all nonzero z. Then, there exists a smooth static feedback law
и = u(z.^---------------------------------C)
with u(0.0.... .0) = 0, which globally asymptotically stabilizes the equilibrium
(z, £i..... £r) — (0- 0, -  •  0) of the corresponding closed loop system.
Of course, a special case in which the result of Theorem 9.2.3 holds is
when е*(г) ~ 0 i.e. when z = /0(2.0) has a globally asymptotically stable
equilibrium at z = 0. This is the case of a system of the form (9.11) whose
zero dynamics have a globally asymptotically stable equilibrium at z = 0,
which, for the sake of completeness, is described separately in the following
(trivial) Corollary of Theorem 9.2.3.
436	9. Global Feedback Design for Single-Input Single-Output Systems
Corollary 9.2.4. Consider a system of the form (9.17). Suppose its zero
dynamics have a globally asymptotically stable equilibrium at z = 0. Then,
there exists a smooth static feedback law
a =	....Cd
with ?z(0.0,..., 0) = 0, which globally asymptotically stabilizes the equilibrium
.........C) — (0-0....0) of the corresponding closed loop system.
Remark 9.2.1. In analogy with the case of linear systems, which are tradi-
tionally said to be “minimum phase” when all their transmission zeros have
negative real part, nonlinear systems (of the form (9.10)) whose zero dynam-
ics have a globally asymptotically stable equilibrium at z = 0 are also called
minimum phase systems. <
We now present an extension of Lemma 9.2.1. in which the hypothesis
that |p/(c,0) is negative definite is replaced by the hypothesis that this
function is just negative semide finite, together with a “controllability”-like
assumption.
Lemma 9.2.5. Consider a system described by equations the form (9.12).
Suppose there exists a smooth real-valued function V(z). which is positive
definite and proper, such that
for all z. Set
rm = fo.o'i 9*(y =
and
s* = П П ee x"cm = o}.
! >0 4 >0
Suppose S* = {0}. Then, there exists a smooth static feedback law и = ?j(z.£)
with u(0.0) = 0. and a smooth real-valued function H’(z. £). which is positive
definite and proper, such that
(9W 0W\(f(.z.e\<{}
\ dz df J J
for all nonzero (z, £).
Proof. Consider again the expansion (9.13) and observe that, by definition
<?*(>) = р(г.0) .
Choosing the input (9.15). the positive definite function (9.14) satisfies
9.2 Examples of Global Asymptotic Stabilization 437
/ air
air \
ae J
/(мП
J
= Lf.v(z)-e
(9.18)
This function is nonpositive for each (z.£). Thus, 1Г(г.£) is nondecreasing
along any trajectory (z(t).	) of the closed loop system
£

(9.19)
Since IT(~-C is positive and proper, it is deduced that all trajectories are
bounded and that the equilibrium (г.£) — (0,0) is stable (in the sense of
Lyapunov).
Let (c(t).£(f)J be any fixed trajectory of this closed loop system and let
ac > 0 denote the limit
ac = hm 1Г(г(0.£(ф 	(9.20)
This trajectory, being bounded, has a nonempty -а-limit set f?’ (see section
B.2). By continuity of IT(z, £) and by definition of u;-liniit set,
1Г(т,£) = a °	for all (x. e .
Now. take any initial condition (z°,£c) €	• The corresponding trajectory
(2c(f).^°(f)) of (9.19) is in f2° for all t. because f?5 is invariant (see again
section B.2) under the flow of (9.19). Thus, lT(^c(f). £c(t)) = a3 for all t and,
bv (9.18).
L/.r(;"(t))-[f(f)F = 0.
This condition, since 1у*1'(г) is nonpositive, shows that along the trajec-
tory in question £c(t) is identically zero. Thus, £°(t) is necessarily a trajectory
of
i = ГМ
satisfying
1лГ(гс(0) = 0.	(9.21)
Moreover, since £°(f) = u(z°(t). £°(t)) is identically zero, the trajectory in
question also satisfies (see (9.15))
ТгГ(г°(0) = 0.	(9.22)
We show now that the two conditions (9.21) and (9.22) imply zQ(t) E 5*
for all t. To this end. observe that since Z,/*V(z) is nonpositive and (9.21)
holds. LfA'(z) is maximal at any point of the trajectory z = z°(t) and.
therefore.
for all t. losing this identity and the fact that is an integral curve of
/*(z). one obtains
438
9. Global Feedback Design for Single-Input Single-Output Systems
= L{.La.V(z‘[ty}- Ll,.Ll.V(--{t'l'l
Iterating this argument and using the property that, for any function t'(c).
the identity U (z°(t)) = 0 implies
it is deduced that
L^LQd^a.V(z4t})=0
for every i > 0 and к > 0.
Having proven that c°(t) G S* for all t, the hypothesis S* = {0} implies
that the trajectory (ic(t).£°(t)) coincides with the trivial equilibrium trajec-
tory (0.0) and. therefore, the limit no in (9.20) is equal to 0. Since H'(z.^)
is positive definite and continuous, it is concluded that
,lim г(t) — 0. liin £(f) - 0 .
Thus, the equilibrium (?.£) = (0.0) of (9.19) is globally asymptotically
stable. By the converse Lyapunov theorem, it is deduced that there exists a
function, possibly different from the function И'(з.£) considered so far in the
proof, with the properties indicated in the Lemma. <
Remark 9.2.2. Note that, in particular, at each z e S*
adf.g*(z) ...	) = ( 0 0 ... 0).
Thus, the condition S* = {0} is satisfied, for instance, if vanishes only at
z = 0 and the matrix
(g*(z) adf-g*(z) ... adnfrlg*i^B
has rank n for each z. <
From this result it is straightforward to derive the corresponding extended
versions of Lemma 9.2,2. Theorem 9.2,3 and Corollary 9.2.4. which are im-
mediate and therefore not included here.
9.3 Examples of Semiglobal Stabilization 439
9.3 Examples of Semiglobal Stabilization
The global stabilization results presented in the previous section are indeed
conceptually appealing but their actual implementation requires the explicit
knowledge of a Lyapunov function V(rr) which satisfies either the conditions
of Lemma 9.2.1. or those of Lemma 9.2.2. This function, in fact, explicitly
determines the structure of the feedback law which globally asymptotically
stabilizes the system. Moreover, in the case of system whose relative degree
in higher than 1, the computation of the feedback law is somewhat cumber-
some. in that requires to iterate a certain number of times the manipulations
described in the proof of Lemma 9.2.2. In this section we show how these
drawbacks can be overcome, in a certain sense, if a less ambit ious design goal
is pursued, namely if instead of seeking global stabilization one is interested
in a feedback law capable of asymptotically steering to the equilibrium point
all trajectories which have origin in a a priori fixed (thus arbitrarily large)
bounded set.
The intuitive concept of achieving asymptotic stability with arbitrary
large basin of attraction can be formulated in the following way. A system
x = f(x} + gix}u
is said to be semiglobally stabilizable if. for each compact subset К c 31”.
there exists a feedback law и = u(?r). which in general depends on K. such
that in the corresponding closed loop system
r = fUf 9^)u(x)
the equilibrium x — 0 is locally asymptotically stable and
j(0) e К => ^lim x(t) = 0
(i.e. the compact subset К is contained in the basin of attraction of the
equilibrium x = 0).
The concept of semiglobal stabilizability, as we will see in the sequel, has
relevant practical consequences. As a first example of application, we will
show that systems having the special form (9.11) and a globally asymptoti-
cally stable zero dynamics are semiglobally stabilizable (which of course is an
obvious consequence of the fact that they are globally stabilizable), by means
of a feedback which has a very simple structure and above all - does not
require the explicit knowledge of a Lyaponov function for the zero dynamics.
More specifically, for a system described by equations of the form
i = ЯЧ1
	_	(9.23)
fr = U
9 = fi ,
440
9. Global Feedback Design for Single-Input Single-Output Systems
it is possible to prove that the following semiglobal stabilization result holds.
Theorem 9.3.1. Consider a system described by equations the form (9.23f.
Suppose its zero dynamics have a globally asymptotically stable equilibrium
at z = 0. Let
p(A) = A' 4- ar~i Ar	4- ... 4- ui A 4- no
be an arbitrary polynomial having all roots with negative real part and set
и = ~(A*rflg£i 4- kr 4- • • * 4- kar_i£r) .	(9.24)
For each real number Л > 0 there exists a real number fc* > 0 such that, if
к > к’. in the closed loop system (9.23)-(9.24) the equilibrium (z.£) = (0,0)
is locally asymptotically stable and. moreover.
f lim z{t) = 0
nw)ii<ft.ii-(o)ii<R =. (;gw=o.
Proof. We break up the proof in three steps. In the first, step we show that
the equilibrium (z.£) = (0,0) is locally asymptotically stable. In the second
one we prove that, if к is sufficiently large, all trajectories satisfying
||^0)|| <H.||z(0)||< 7?
are bounded. Finally, in the third step we prove that all such trajectories
eventually tend to the equilibrium as t tends to эс.
(i)	For any к > 0. all the roots of the polynomial
Рк (A) = А ц-кдг_]A 4“ -   4“ к 1 u i A 4- кгад
have negative real part. Thus, as shown in section 4.4. the feedback law
(9,24) locally asymptotically stabilizes the equilibrium (z.£) = (0.0) of the
corresponding closed loop system. '
(ii)	Ser
G = HTtG. 1 < » < r
к
and observe that the closed loop system (9.23)-(9.24). after this transforma-
tion of coordinates, is described by equations of the form
in which
9-3 Examples of Semiglobal Stabilization
441
Let P be a positive definite solution of the Lyapunov equation
ATP+ PA = -I
and let Г(-) be a positive definite and proper function satisfying
^-/o(-.O) <0
dz
for all nonzero z. The existence of such a matrix P and such a function V(z)
is implied by the hypothesis on the polynomial p(A) and. respectively, on the
zero dynamics of (9-23), Set
1Г(-.<) = Г(;) + <ТР<
and observe (writing, as in the proof of Lemma 9.2.1, /o(--Ci) = /oU-0) 4-
p(z, (i )G) that
+ |bp(j.C1)C1 _ t.||c||2 . (9.-26)
Consider now the compact set
К = {(г.С) e Rn"r X Г : ||;|[ < В and ||C!| < /?}
and set
a = max П'(Т,Q .
(;,<)€ A'
Also, observe that, the set
ЛЛ = {(z,Q e IF-'r X Г : !Т(з.<) <a}
is a compact set (because IT(z,0 is a proper function) and. by definition.
К C Ma. Finally, let dMa denote the boundary of AIa.
It will be shown now that, if к is sufficiently large, the quantity (9.26) is
strictly negative at each point of dMa. To this end, observe that the quantity
in question is indeed negative at each point of the compact subset
d.Ma П {(z,0 e x Г : ( = 0} .
Thus, by continuity, it is negative on some open neighborhood U of this
subset. Observing that the set dMa \ U is a compact set. define
dV
= max	j—-р(г, Q )G	b2 = min
(i.<)ec>Afo \C dz
and note that b2 > 0 because (	0 on dMa \ U. Set
26i
к - -i— .
442	9. Global Feedback Design for Single-Input Single-Output Systems
(z(0);C0)) E A =>
Then, it is easily concluded that, if A’ > A*. (9.261 is strictly negative at each
point of dMa.
Suppose A’ > A*. choose any initial condition (z(0). £(0)) E A' and let
(). <(t)) denote the corresponding trajectory. The previous arguments show
that (c(Z).('(f)) cannot cross the boundary дУ1а of .V(1. a set. which contains
K. If fact, if this were the case, the derivative (with respect to time) of the
function IT(-U)-^U)) would be nonnegative at some point of boundary of
.W(i. i.e. a contradiction. Thus, the trajectory in question remains in ЗА for
all t > 0. In other words
(z(0). <(0)) e(z(t). <(f)) e ЛА for all t > 0.
Without loss of generality, one can assume A > 1. Thus, 1^(0)) < |C(0)|
for all 1 < j < r and
UfO). £(0)) e A => (z(O).(J(O)) E к => (y(f).<D)) E ДА for all t > 0.
Observing that ^(t) converges to 0 for every initial condition £(0), it is
possible to conclude that, if k > A*, in the closed loop system (9.23)-(9.24)
z(t) is bounded
is bounded and lim = 0 .	(9-27)
(iii)	Choose (c(0), ^(0)) £ A and let f?c denote the ---limit set of the cor-
responding trajectory, which is nonempty because the trajectory in question
is bounded. By definition, since £(t) tends to zero as t tends to x.
Ac e {(z.f) e W1"’’ xT:f^o).
Pick any point (zc,0) in f?c and let (z°(t).CU)) denote the trajectory of
(9.23)-(9.24) satisfying (t°(0). C(0)) =£ (zTO). Clearly. CO = 0 for all f > 0
and. therefore, since z°(0 is an integral curve of z = /o(g0).
lim C(f) = 0 .	(9.28)
Since the equilibrium (z.£) — (0.0) is locally asymptotically stable, there
exists an open neighborhood I j of (0. 0) with t he property that every trajec-
tory starting in Ci asymptotically converges to (0.0) as t tends to x. From
(9.28). we see that there is a real number A > 0 such that
(гс(А).0) E int(Vi) .
Let Ф{(г,£) denote the flow of (9.23)-(9.24). For each fixed t. AGgC defines
a diffeomorphism of a neighborhood of (z.£) onto its image. Thus, since
(zc, 0) e int(Vi). there exists a neighborhood of (zAO) such that
for all (z.<f) E 12 .
9.3 Examples of Semiglobal Stabilization 443
By definition of u>liniit set. there exists 74 > 0 such that the trajectory
(c(f),£(0) satisfies
ЫТ-,)МТ2)) e V>.
Thus, this trajectory satisfies also
(.-ffi + iWi + OMi.
that is the trajectory in question reaches, in finite time, a point from which
asymptotic convergence to the equilibrium point is guaranteed. <
Remark 9.3.1. Note that the feedback law (9.24) can be simply expressed, in
the original coordinates of (9.1). as
a = a(j-) - ,J(t)((krafth(T) + kr“l«iLfh(j-) +   + kar--[L^~xh(r))) 
The possibility of expressing the (semiglobally stabilizing) feedback law in the
original coordinates is indeed another advantage of the concept of semiglobal
stabilization. <
As a second application of the notion of semiglobal stabilizability, we
consider now the class of systems described by equations the form
- -
(9.29)
£r-l - £r
6 = и
У = 6
in which j is an integer larger than 1. still with the hypothesis that the
zero dynamics are globally asymptotically stable- For this class of systems,
despite of its apparent simplicity, there is no general global stabilization result
available. In fact, the results derived in the previous section heavily depend on
the hypothesis that the differential equation governing the flow of z depends
only on and not on any one of the other components • 6- of the vector
However, if the flow of z is affected by just one single component of as
in (9.29). the system in question proves to be semiglobally stabilizable.
The intuitive idea which makes this result possible is the following one.
Suppose the actual output of map у = of system (9.29) is replaced by a
new dummy output map defined as follows
= C + C’Wi + C-2C1{3 +  •  + гс;-2С-1 •	<9.30)
The dynamics of (9.29) together with the “new output" (9.30) characterize
a system having uniform relative degree r — j у- 1. In fact.
LgLjh(g) = 0 for all к < r - j and all £
444
9. Global Feedback Design for Single-Input Single-Output Systems
and
L9Lrf~Jh(t} = 1 .
For the system thus defined it is possible, after having changed the coor-
dinates and imposed an appropriate feedback, to obtain a normal form which
has the exactly the same structure as (9.11). This is accomplished by leaving
the z and ......iq-i coordinates unaltered, changing .......into
....Lr~Jh^) .
and using the feedback law
и =	4- u' .
It is easy to verify, also, that the new coordinates and the feedback law thus
defined are functions of .....only (actually, linear functions).
Having obtained a system with the same structure as system (9.11). one
might wish to try the semiglobal stabilizing feedback of Theorem 9.3.1, which
in the present, case would be a feedback of the form (see Remark 9.3.1)
u1 = -(kr~j+laoh(^) + kr~JaiLfh(£) +-------(- kar-.jLrf~jh^)) .
that is. for the original system (9.29), a feedback of the form
и = ^Lrf~j + lh(O -	+ IT^Ljhtf) +  + kar-jL’^h^))
(9.31)
in which uq.«i; • • • • ar-j are coefficients of a polynomial
p(A) = A •j-*’ + nr—jX 2 + ... + яi A + uq
having all roots with negative real part. Of course, for this to be successful, the
zero dynamics of the system in question must have appropriate asymptotic
properties.	!
It is easy to check that the zero dynamics under considerations (namely
those of (9.29) with the output map у = replaced by у = h(£)) are de-
scribed by' equations of the form
Z = fo(z.	-EJ~2C^2---------5Cj_2^_1)
£1 - b
(9.32)
Cj-1 =	------cCj-2^^] .
If г is positive and cp, ci, ,.., с;_2 are coefficients of a polynomial
?(A) =	4- Cj-2 AJ ~ 4- ... 4- C] A 4- cq
9.3 Examples of Semiglobal Stabilization 445
having all roots with negative real part, the dynamics in question have a
locally asymptotically equilibrium at (c,£i.£j-i) = (0.0...0) (see sec-
tion 4,4). However, to the present, purposes, a stronger property is required,
namely the property that the basin of attraction of this equilibrium con-
tains an arbitrarily large compact set, This can be achieved by appropriately
tuning the design parameter s. as shown in the proof of the following result..
Theorem 9.3.2. Consider a system described by equations the form (9.29).
Suppose its zero dynamics hare a globally asymptotically stable equilibrium
at z — 0. For each real number В > 0 there exist a real number A,+ > 0 and.
for each k > k*. a number > 0 such that, if k > к* and 0 < e < ef.. in
the closed loop system. (9.29)-f9.31) the equilibrium (c.£) = (0.0) is locally
asymptotically stable and. moreover.
f lim z(t') = 0
IlfObl </М--(0)Н < л =	ilin{(()=0.
Proof. Having already realized that the equilibrium (c.£) = (0.0) of the
closed loop system is locally asymptotically stable, the crucial part of the
proof as in Theorem 9,3.1 is to establish that c(t) is bounded. Set
ч< = e-’-m
C, = jXir'AU).	l<-<r-J + l.
A'
ami observe that the closed loop system (9.29)-(9.31). after this transforma-
tion of coordinates, is described by equations of the form
г	-	foG, (i + zHi])
h	=	sFp + GCt:	(9.33)
C	=	E4(.
446
9. Global Feedback Design for Single-Input Single-Output Systems
	( °	1	0	° \
	0	0	0	0
.4 -=				
	t o	0	0	1
	\ -a0	-Gi 	-Qr-j+1	U г — j
Note also that, if	0 < г	< 1,		
	(0)1	< l^-(0)|,	1 < I < J	-1,
and that there exist, a number .V > R such that, if к > 1.
< R => 1K(O)II<-V.
for all 0 < £ < 1. Thus, if к > 1 and 0 < s < 1.
U(0)||</?	|H0)!| < я. |K(O)H < a;
Choose an initial condition (z(0),/?(()). ((0)) = (z°, in the compact
set
К = {(z. I/. Q e Rn~r X В?-1 X r“j+1 : ||z|| < R. ||r/|| < R. ||(|| < V]
and let	C°(C) denote the corresponding trajectory. Clearly,
z?o(t) = exp(sFt)if + / exp(eF(t — s))GCe.xp(kAs)(>od.ti ,	(9.34)
Jo
while (zc(7), C(0) can be viewed as an integral curve of the time-varying
system
i =
(9.3a
C - k.4(..
Observe that, when £ = 0. system (9.35) reduces to the system
C = /oO:G)
C = kAQ ,
which has precisely the form (9.25). Thus (see the proof of Theorem 9.3.1).
there exists a positive definite and proper function 1Г(г.£) and a number
k* > 1 such that, if к > к*, the derivative (9.26) is negative at each point, of
the boundary dMa of the compact set
= {(z.Q 6 x Г : П'(г-0 < a}
where a is a number such that
Ma D {(г.0 e R"-r X r-J'+I . IHI < -Mell < -V} 
Fix к > к* and note that /о(". О +гЯг/°(0) can be expressed in the form
9.3 Examples of Semiglobal Stabilization 447
/[)(£, <1 +	= /o(~-<! ) +?(-<! +	.
Thus, the derivative of П’(г.О along the trajectories of (9.35) can be ex-
pressed in the form
(dW_ £IT\ Шг.С
\ dz / x	/	(9.36 r
= ,W(z.<) +	+ £Hrf(ty)EH7f(t) .
dz
in which ,U(z.() is a function which is negative at each point of 031 a.
Observe the function defined by (9.34). in which A- now is a fixed number,
and note that there exists a real number L > U such that
IIHOll <L
for every t > 0. for every 0 < £ < T and every (^°.(°) satisfying ||^°|| < /?.
||C|i < AI. As a consequence, there exists a real number Jj > 0 such that
< 3L
for every (A.	€ 031a, for every t > 0. for every 0 < s < 1, and every (77е. C)
satisfying ||rgj| < Я, ||(c|j < А/. Set
;3o = max M(z.Q
and note that 3? < 0. Set also
* Л
£ “ 23?
Thus, if s < c*. the quantity (9.36) is negative on dMa. As in the proof of
Theorem 9.3.1. this implies that any trajectory of (9.35) with initial condition
in 3Ia is bounded, in particular the trajectory (г°(/), £°(t)).
Having shown that z°(i) is bounded, and knowing that £°(t) asymptoti-
cally decays to 0 as t tends to oc. the proof can continue precisely as in part
(iii) of the proof of Theorem 9.3.1. <s
The previous result shows that semiglobal stabilization is possible, for a
system in normal form (9.10), if the flow of z is affected by only one compo-
nent of the vector £. It is important to stress - however that this limitation
cannot be further weakened, without extra hypotheses. In fact there are cases
in which, if two or more components of £ are affecting the flow of z. semiglobal
stabilization is not possible, as shown by the following example.
448	9. Global Feedback Design for Single-In put Single-Output Systems
Example 9.3.2. Consider the following system, defined on 3?,
;	= -г + ^G
- Cz
C'2 = «
in which i/(z. £]. £>) is a function to be determined. Observe that the variable
9 = -Ci
satisfies
V =	+ 4i = ~9 + 9~ + Ci • С2И1 + -6 •
Thus, if
^(>-Ci-C2ki+^2>0.	(9.37)
one has
9 > ~9 + E •
Condition (9.37) is satisfied; for instance, at. each point of the set
S = {CCfe)613
if 1

4:
Note also that -1] + if > 2 at each point of the set S. so that // > 0 at
each point of S. This shows that any trajectory of the system with initial
condition in the interior of S cannot enter the set
S = {(.-y1.?2)eK3:I)<2}.
no matter how the input is chosen, and this proves that semiglobal stabiliza-
tion is not possible. <	'
9.4 Artstein-Sontag’s Theorem
In this section, we describe another approach of major conceptual relevance
to the problem of globally asymptotically stabilizing a nonlinear system
b =	+ д(-Ф 	(9.38)
Recall that - according to the converse Lyapunov theorem if system
(9.38) is globally asymptotically stabilized by some smooth feedback law
a = q(j). there exists a positive definite and proper smooth function V(r)
such that
Qi "
(f(r) + g (-г)сл(л-)) = LfV(:r) + Q(r)LffV(jT) < 0
9.4 Artstcin-Sontag's Theorem 449
for each z 0. This requires, in particular, that rhe function L fV(x) is nega-
tive at each nonzero z such that LyVfid = 0- Thus, it can be deduced that a
necessary condition for the system to be globally asymptotically stabilizable
(via smooth feedback), is the existence a positive definite and proper smooth
function V(z) with the property that
(z) = 0	=> LfV(x) <0
for each z 0. Such a function is called a control Lyapunov function.
The importance of the concept of control Lyapunov function is that the
existence of one of such functions is also a sufficient condition for the existence
of a stabilizing feedback. More precisely, we will see below that, given a
control Lyapunov function V(z). it is possible to construct - by means of a
very simple formula a stabilizing feedback law и — o(z). which is defined
on E”. satisfies o(0) = 0, is smooth on the open (and dense) subset R'1 \ {0}
of Ж" and at least continuous at z = 0. A function with these properties is
sometimes called an almost smooth function.
In fact, the following result holds.
Theorem 9.4.1. Consider a system described by equations of the form
(9.38), tn which f(.r) and g(x) are smooth vector fields and /(0) = 0. There
exists an almost smooth feedback law и ~ o(z) which globally asymptotically
stabilizes the equilibrium z = 0 of (9.38) if and only if there exists a positive
definite and proper smooth function V(z) with the following properties:
(i) LgV(x) ~ 0 implies LfV(x) < 0 for all z 0,
(ii) for each s > 0 there exists d > 0 such that, if x 0 satisfies ||z|| < fi.
then there is some и with [u| < £ such that
L/V(z) + LgV(x}u < 0 .
Proof. The necessity of condition (i) derives - as shown above by the con-
verse Lyapunov theorem in section B.2. according to which if /(z) is a vector
field defined on E'!, satisfying /(0) = 0. smooth on the open (and dense)
subset IR" \ {0} of R” and continuous at z = 0. and the equilibrium z = 0
of z — /(z) is globally asymptotically stable, there exists a positive defi-
nite and proper function U(z) defined on IR”, and smooth on IR”. such that
L/U(z) < 0 for all nonzero z. The necessity of (ii) is a simple consequence
of the hypothesis that the feedback law which is supposed to stabilize (9.38)
is continuous at z = 0.
To prove the sufficiency, consider the following open subset of R2
S = {(a. b) e IR2 : b > 0 or a < 0} .
and define on S a function ф(а. b) as follows
{0. if b = 0 and a < 0
a + Vo2 + b2 ,	,
-----—:-—-— elsewhere .
b
450
9- Global Feedback Design for Single-Input Single-Output Svstems
It is possible to show that this function is real-analytic on S. In fact, set
F(a. b.p) = bp~ - 2ap - b
and observe that the equation F(a,6,p) = 0. for each (a.b) € S, is satisfic'd
by p = &(a,b). Now. the quantity
fc’Fi
—	= 2(bo(a. b) - a)
L dp J p=o{(i.bi
is never zero on S. Thus, by the implicit function theorem, since F(a.b.p) is
real-analytic, the solution p = &{a.b) of F(a,b.p) = 0 is real-analytic.
Suppose now V(j) is a function satisfying (i). Observe that for each .r the
pair (a. b) = (Lf V(.r). [LfJ V(z)]2) is in S. and set
f 0 if т = 0
) -Lgelsewhere1.
This feedback law. which is a composition of the real-analytic function
d(a.b) and of the smooth functions £/V(j?) and \Lg V(z)]2, is indeed smooth
on R'1 \ {0}. Moreover, it is possible to prove that property (ii) implies that
this function is continuous at r = 0. Thus. q(j) is almost smooth. By con-
struction, ci(t) is such that
(/(-г) +	V(;r)]- +	< 0
for all j- ф 0. Thus, this feedback law globally asymptotically stabilizes the
equilibrium z = 0 of the closed loop system (9.38)-(9.39). <
Remark 9.4-1. Note that the proof of Theorem 9.4.1 provides a simple ex-
plicit formula for an almost smooth ^Feedback law globally asymptotically
stabilizing system (9.38). which is
f 0 if LgV(r) = 0
«И = < LfV(F) + yU/Vtz)]’2 + [iJ'W]4 , ,
-----------------------------—---------------------— else w her e.<i
I	W
9.5 Examples of Global Disturbance Attenuation
In section 4.6. we have considered the problem of rendering the output у of
a nonlinear system
T = f (•**) + g(F)u + p(F)w
(9.40)
£/ - /Цт)
9.5 Examples of Global Disturbance Attenuation 451
completely independent of (i.e. decoupled from) the disturbance w. The local
analysis described in that section can easily be given a global version, by
simply asking that system
i = /(т) + g(x)u
У =
satisfies the conditions indicated in section 9.1 for the existence of a globally
defined normal form. If this is the case, in fact, there exists a feedback law
which decouples у from ir if and only if (see (4.47))
LpL'jhix) - 0 for all 1 < i < r and all j: G IF .	(9.41)
However, it must be observed that a condition of the form (9.41) is a rather
severe condition, which is likely to be respected only in very special cases. In
view of this observation, we address in this section a less demanding problem,
which consists in seeking a feedback law which does not really "decouples"
rhe output, у of a system from a given disturbance tc. but simply “keeps
small" the influence which w has on y. Of course, such a feedback law should
also guarantee (global) asymptotic stability of the corresponding closed loop
system.
In order to pursue this approach, it is necessary to establish first a pre-
cise criterion by which the “influence" of a given input (in this case, the
disturbance) on the output of the system can be measured. A notion which
is particularly suited to this purpose, especially in the case of a nonlinear
system, is that of the so-called L? gain, which is defined as follows.
Consider a single-input single-output system described by equations of
the form
j' = f(-r) + ,?(.r)u
у = h(.r)
(9-42)
in which /(z) and y(.r) arc smooth vector fields, and h(r) is a smooth func-
tion. defined on IF. Assume, as usual, that /(()) = 0 and h(0) = 0. Let
£> denote the set of all piecewise constant input functions n : [0. ос) -о T
satisfying
I u2(f>)ds < ос .
Jo
Let j-(L jt°. u(-)) denote the value of the state achieved at a time t > 0 under
the effect of the input u(-) e £3. starting from the initial state £ at time
t = 0. This system is said to have L> gain less than or equal to о if for every
u(-) G £-_> the response z(L0.u(-)) from the initial state ,r(0) = 0 exists for
all t > 0 and satisfies
||h(x(.s. 0. u(‘)))||2ds < y2 / ||u(,s)||2ds.
Jo
for all t > U.
452
9. Global Feedback Design for Single-Input Single-Output Systems
The following Proposition presents a useful sufficient condition, for a sys-
tem of the form (9.42). to have an L2 gain which is less than or equal to
Proposition 9.5.1. Consider a system of the form (9.42). Suppose there
exists a positive definite and proper smooth function V(t) satisfying
+ Л[L3C(.r)]2 + [Л(т)12 < 0	(9.43)
1'
for all nonzero t. Then, system (9.4'2) has a globally asymptotically stable,
equilibrium at .r — 0 and has an L2 gain which is less than or equal to y.
Proof. First of all. observe that (9.43) is equivalent to the condition that
+ gCCip + [h(,r)]2 - - 2ir < U	(9.44)
or
for all nonzero j: and all и G R. In fact, for each fixed j*. the left-hand side of
(9.44) is a polynomial of degree 2 in a. which has a maximum at
a = n*(j-) =	.
Thus, (9.44) holds for all и if and only if the value of its left-hand side at.
и = u*(j~) is negative, which as a simple calculation shows is precisely
the condition (9.43).
Choose an input u°(-) G C2. let .r°(-) denote the corresponding response
of (9.42) from the initial state .r(0) = 0 and suppose z“(t) exists for all
0 < t < Г. Since (9.44) implies
,n (д(0) s Gy(f)]-’ - |G(< •
integrating with respect to time over an interval [O.t). with t < T. yields
л/	G
V(.r(t)) < V / [nc'(.s)]2d-s - / [/i(.rc(s‘))]2d.s .	(9.45)
Jo	Jo
because V(0) — 0.
This inequality can be used to prove that .r°(t) exists for all t > 0. For.
suppose [0.Г) is the maximal interval on which j~°(t) is defined, and Г is
finite. Since V(.r) is a positive definite proper function, given any number
I\ > 0 there exists a time a < T such that Г(х°(а)) > К. Let t be such that
Г(гс(0) > Л. where
.4 = <2 Г[пс(<.
Jo
Then, from (9.45). we obtain
9.5 Examples of Global Disturbance Attenuation
453
< 7 2 / [u°(s)]'2d.s < .4
Jo
i.e. a contradiction. Thus. J?°(t) exists for all t > 0. At this point. (9.45) yields
/* [u°(.s)]2d$ > /" [/((.rc(.s))]2t/,5- 4- F(-Ht)) > f [h(xc[s)')]2ds
о	Jo	Jo
as required.
Finally, observe that V(.r) is a positive definite and proper function sat-
isfying LyV(z) < U for all nonzero t. Thus V(t) is a Lyapunov function for
the autonomous system
j = f(x) .
which therefore has a globally asymptotically stable equilibrium at x = 0.<
Remark 9.5.1. The result indicated in the previous Proposition has a partial
converse. In fact, suppose (9.42) has a globally asymptotically stable equilib-
rium at x = 0 and consider the function I * : К" —> К

(9.46)
which is defined for each x E Ж'1 which is reachable from x = 0 in finite time
by means of a control u(-) E £<> If (9.42) has an L-2 gain which is less than
<jr equal to 7. this function is necessarily nonnegative. Suppose that for each
x there exist u(-) E Cj and t > 0 such that the infirnum in (9.46) is attained.
Then, from the identity
it is found that, by definition, the inequality
r.t~h
\\(x(t + h)) -i;(z(f)) < у (72[u.(s)]‘2 - [h(jr(.s))]2)d.s	(9.47)
holds for each sufficiently small h > 0.
Finally, suppose that is a smooth function. Then, dividing both
sides of (9.47) by h and letting h 0. one obtains a inequality of the form
= L,v(e + Lgvn« < -v - [ад]2.
from which, proceeding as in the proof of the previous Proposition, one can
see that rhe left-hand side of (9.43) is non-positive. <
454	9. Global Feedback Design for Single-In put Single-Output Systems
Remark 9.5.2. Note that in the case of a linear system
.г = A j + В it
у — Cx
a positive definite quadratic form V(jt) = satisfies the inequality (9.44)
if and only if the positive definite (symmetric) matrix P satisfies
PA + 4TF+ CTC + ^PBBTP < 0 .
This is a well-known (necessary and sufficient) condition for a system to be
asymptotically stable and to have an gain which is strictly less than
known as rhe Bounded Real Lemma. <
The existence of a positive definite and proper function V(j‘) satisfying
the partial differential inequality (9.43). which is called a Hamilton-.Jacobi
inequality, is an expressive, and in many cases practically useful, way to
ascertain whether a given system is globally asymptotically stable and has
an L? gain which is less than or equal to 4. Taking the existence of such a
function as a criterion to establish an estimate for the influence of the input
on the output of a system, it is possible at this point to formulate the problem
of achieving a prescribed level of disturbance attenuation - in a system of the
form (9.40) - as the problem of finding a feedback и — о(т) law such that,
in the corresponding closed loop system
x = f(x) + g(x)ci(x) + p(x)w
у = /d-r) •
an inequality of the form (9.43) holds for some positive definite and proper
function V(r).
1
Problem of Disturbance Attenuation with Stability. Consider a
system of the form (9.40). Given a real number у > 0 find, if possible, a
feedback law и = a(x) with a(0) = 0 and a positive definite and proper
smooth function V(t) such that
LfV(x) + a(.r)LaV(.r) + —’(.c)]2 + [ft.(j-)]2 < 0	(9.48)
4* "
for all nonzero x.
We present now. about the Problem of Disturbance Attenuation with Sta-
bility. a series of results which arc totally analogous to the results described
in section 9.2 about the problem of global stabilization.
9.5 Examples of Global Disturbance Attenuation -155
Lemma 9.5.2. Consider a system described by equations of the form
i = /(;•£)
£ = ii+q(z,O“'	ОД9)
У = h(z.£)
in which (c.£) G R" x /(0.0) = 0 and Л(0.0) = 0. Suppose there exists d
smooth real-valued function V(c). which is positive definite and proper, such
that	1
+ (ft(s-O)f < 0
for all nonzero z. Then, there exists a smooth static feedback law a — u(z.C)
with u(L).0) =0. and a smooth real-valued function IE( z. £), which is positive
definite and proper, such that
+ A (Ipirc. {)]’ + [M-’.o!2 < 0
for all nonzero (z.f,}. where
Fl.F}~(fwtW\	п,Г}_(рМ}
(«(--.о)' Co.oJ '
Proof. Observe that the functions /(£,£), p(z.£). h(z,£,). can beqint in the
form
/(.-,{) = /(.-.0) +f,o,i)i
p(=,t) = р(г.0)+р,(г.^)^
h(:,0 = Л(2.0)+Л1(г.?)? .
Consider the positive definite and proper function
1Г(М) = Г(-) + Ip .	(9.50)
and observe that
iriru.t) = ^/(c.o) +	+ „(-.<))«
Irli(r.t) =	\ q. )pi(--£) +
Thus
+ jb|LpH-C-{)]-’ + !'>(г.е)]-’
= ^/(,0) + A Г A_p((J A + (h(.-.O)]-’ + (./(.-.{) +
C/—	4 / L (/4.	J
where M(z-C is a suitable smooth function of (c.£). Then, choosing
»(M) = -е-МС'С)
yields the required result. <
456	9. Global Feedback Design for Single-Input. Single-Output Systems
Using the property indicated in this Lemina, it is immediate to prove the
following result (which indeed contains Lemma 9.5,2 as a particular case).
Lemma 9.5.3. Consider a system described by equations of the form (9.49).
Suppose there exists a smooth real-valued function
with r*(0) = 0, and a smooth real-valued function Г(г), which is positive
definite and proper, such that
^-f(z.v*(z)) +	+ [Л(с.	< 0
(JZ	-40 L C/i	J
for all nonzero z. Then, there exists a smooth static feedback law и =
with n(0.0) = 0. and a smooth real-valued function lU(c.£). which is positive
definite and proper, such that
+ Х[£рц-(г.{)]2 + [Л(г.i)]2 < o
for all nonzero where
Proof. It goes exactly as the proof of Lemma 9,2.2. <
Then, using repeatedly the property indicated in Lemma 9.5.3 it is
straightforward to derive the following important conclusion.
Theorem 9.5.4. Consider a system of the form
i
£i =	& + PiU.£i)tc
& = £,3 + pfiz £,!&) U'
(9.51)
£r-l = C + Pr-l (-•&•	• • • :^r-1 )«;
= и + pr(z. <fj.	i *$r-i • C)w
у = Ыг.£1) .
Suppose there exists a smooth real-valued function
with c*(0) = 0, and a smooth real-valued function V(s), which is positive
definite and proper, such that
9.5 Examples of Global Disturbance Attenuation 457
v*(e))l + [Ло( = . r*(z))]2 < 0
oz	4" - L oz	J
for all nonzero z. Then, there exists a smooth static feedback law
U =	.....£r)
with u(0.0......0)	= 0. which solves the Problem of Disturbance Attenuation
with Stability.
It is interesting to relate the results thus established to the asymptotic
properties of the zero dynamics of the system. To this end. consider again
the case described in Lemma 9.5.2 and suppose h(z.f} = f (in which case
the system, with w = 0. viewed as a system with input и and output y. has
relative degree 1). If this is the case, the hypothesis of Lemma 9.5.2 reduces
to the hypothesis that
,, 1
“X- / (о. 0) + —
oz	4"
Since the second term on the left-hand side is nonnegative, it is deduced that
the positive definite and proper function Г(с) is such that ^/(c.0) < 0 at
each nonzero x. In other words, the hypothesis of Lemma 9.5.2 can only be
fulfilled if the zero dynamics of rhe system have a globally asymptotically
stable equilibrium at z = 0.
As we shall see in a moment, if an appropriate additional condition is sat-
isfied. the necessary condition thus identified (namely the global asymptotic
stability of the zero dynamics of the system) is also a sufficient condition
for the existence of a function Г(г) which satisfies the hypothesis of Lemma
9,5.2. for any choice of the level of attenuation y.
Lemma 9.5.5. Let f(z) andp(z) be smooth rector fields of R” , with /(0) =
0. The following two properties are equivalent.
(i) there exists a positive definite and proper smooth function L7t z} such that
LfL (z) + —y[LpL (s)]“ < 0
(9.52)
for all nonzero z.
(ii) there exists a positive definite and proper smooth function V(y) such that
LfV(z}<0
for all nonzero z and a smooth function p : К —> S. satisfying
/1(0) = 0,	/1(a) >0
for all a > 0,
(9.53)
such that, for all a > 0,
h(«)
\LfV(z)\
< mm TV ------------A
{;;l (: ,i=a} [Lpl (s)]"
(9.54)
458	9. Global Feedback Design for Single-Input Single-Gut put Systems
Proof. Observe that (9.52) implies LfU(z) < 0 for all nonzero z. so that
(9.52) itself is equivalent to
om<0. and	>^.
This indeed establishes that (i) => (ii), with p(a) a smooth function having
the properties (9.53) and such that
To prove that (ii) => (i), set
Then, it is easily seen that the various properties of the function p(u) indi-
cated in the Lemina imply that the smooth function
t;O) = ^V(O)
is positive definite and proper. Moreover.
and therefore
LfV{z) = ^p(y{z))LfV(z).
LpL\z) = y2yV(z))LpV(z) ,
By definition of p(u),
for all z 0. Thus
=	1 lA/VIc)	1
[У'(:)Р 4-,МГ(П) IVW С2
from which the result follows. <
From this, it is possible to deduce - as a corollary of Theorem 9.5.4 - an
interesting result about the solution of the Problem of Disturbance Attenu-
ation with Stability for nonlinear systems which are described by equations
of the form
9.5 Examples of Global Disturbance Attenuation
459
z ~ /o(z-Ci) + Po(-,6 )«’
Ci - C2 + pi ('-Cl hr
6 = Сз +P-2(>.C1-C2>
(9.55)
Cr —1 Cr T Pr —1 ( - ' SI  ^2 i . Cr — I ) Ш
Sr = u+Pr(3-Cl-C2---------Cr-i.CJtP
у = Ci •
and have an asymptotically stable zero dynamics. More precisely, it can be
shown that if a condition corresponding to condition (ii) of the previous
Lemma holds, the problem in question can be solved for any choice of y.
Corollary 9.5.6. Consider a system of the. form (9.55). Set
ГО) =/о(г,(1). p*(c) = poO.O) ,
Suppose there exists a smooth function V(y). which is positive definite and
proper, such that
LfA'C) < 0
for all nonzero z and such that, for some smooth function p(-] satisfying
(9.53).
for all a > 0. Then, for every у > 0 there exists a smooth static feedback law
и = "(-Ci.................................Cr)
with u(0.0....0) = 0. which solves the Problem of Disturbance Attenuation
with Stability.
Remark 9.5.3. Note that any linear system whose zero dynamics are asymp-
totically stable satisfies the conditions indicated in the Corollary, In fact, in
the case of a linear system. f*{z) is linear in 2 and p*(z) is independent of г.
i.e. there exists a matrix P and a vector P such that
fCz) = Fz. p*(z) = P.
If the zero dynamics are asymptotically stable there exists a positive definite
matrix .V such that
FTX + A'F = -I
and thus, taking V(z) = гтА>. it follows that
\LfVC)\ _ i|r|rj >	1
[Lp-V(3)P 4pYRr|P - 4[|.VF||-
for all nonzero z. Therefore a condition like (9.56) indeed holds. <
460
9- Global Feedback Design for Single-Input Single-Output Systems
Finally, to conclude this section, we wish to observe that the approach
of section 9.4. based on the concept of control Lyapunov function, to the
problem of globally asymptotically stabilizing a system can easily be adapted
to obtain a necessary and sufficient condition for the existence of an almost
smooth feedback law which solves the Problem of Disturbance Attenuation
with Stability. To this end, it suffices to observe that the concept of control
Lyapunov function can be replaced, in the problem under consideration, by
tliat of a positive definite and proper smooth function with the property that.
for each .c A 0 such that L91’(•*') = 9.
L/l (z) 4- —^[Lp\	+ [h(-r)]~ < 0 -
4^ “
LTsing precisely the same arguments described in Artstein-Sontag’s The-
orem. it is straightforward to conclude that if (and only if) such a function
exists and a property corresponding to the one indicated in condition (ii) of
Theorem 9.4.1 is fulfilled, there exists an almost smooth feedback law which
solves the Problem of Disturbance Attenuation with Stability. The feedback
in question, actually, can be expressed in the form
0 if = 0
elsewhere
where
OU) = (LfVU) +	+ [Л(О
4 •
9.6 Semiglobal Stabilization by Output Feedback
t
In sections 9,2. 9.3 and 9.4 we have studied problems of globally (or semiglob-
ally) asymptotically stabilizing the equilibrium x = 0 of a nonlinear system
described by equations of the form (9.1) by means of control laws of the
form и = a(-r). The actual implementation of laws of this type requires, in
general, on-line measurements of all components of the stare vector j, which
is certainly a rather imposing constraint from a practical viewpoint. In this
section, we address the more realistic and indeed more challenging situation
in which the control law used to stabilize the system depends only on a
measured variable, which we may assume coincides with the output у of the
system itself. Of course, the possibility of succeeding in this task depends on
the possibility of tracking the state of the system when the only measure-
ments available are its input and its output, and this indeed requires some
appropriate "observability” properties. Thus, we examine this problem first.
Consider a system of the form (9.1) and consider the sequence of n + 1
mappings
9-6 Semiglobal Stabilization by Output Feedback
461
. TDf? v пт
. J£ -4- Ut
,Г
and
:	FT x	-4 FT
pt(r0........O--1,))	(.Г. t'Q...1>-1)	1 < A- < П
defined in the following way
To(-r) = /?(т)
TiU'- r0) -	+ </(.r)ru)
их
(9-57)
A' —9
rk( Г- Vo.....(4’-1 ) = —x’"1 ( f f J)	J?(.r)L’o) + V ^-L Г(_]
UX	Or,
1=0
for 1 < к < n.
It is immediate to realize that these mappings, if the input u(t.) of (9.1)
is a C*’-1 function of t. are precisely the mappings which express - for each
к and any given time t - the dependence of the A'-tli derivative ylk'(t) of y(t}
on x(t) and u(t)........u1 A'-11 (f). As a matter of fact.
------u|A*’ 1!(fl) .
Suppose now that the first n of these mappings are put together, to define
a mapping
Ф
HtT x FT-1 -► FT
(.Г. r) i-4 W = Фи. r)
(9.58)
in which
Ф(х. r) =
rl l-f- to)
By construction,
Ф :	.....col(y(t).....................
Suppose now that, at some (jc.tc) G 5?" x ffif1 4
и .
(9,59)
Then, by the implicit function theorem, there exist a neighborhood U3 of jc in
11T!. a neighborhood И'с x Vе of (u,= . ес) in K'! xK”-1 (where ?r° =	t,c))
and a unique smooth mapping
462
9- Global Feedback Design for Single-Input Single-Output Systems
Ф ; n ° x V3 -> Uc
(W.V)	1-4 jr —	I')
such that
(9.60)
ic = Ф(Ф(и\ г). с)
for all (?/. r) G IVе x Vе.
In view of the previous interpretation of the mapping Ф. it is possible to
conclude that if. at some time t°.
,r° = j(t°) and i’° - col(u(t°).......u'n
are such that the rank condition (9.59) holds, the mapping Ф can be used to
reconstruct the value of x(t) as
c(t)) -
where
tr(f) = col(y(t),...	1((t))
v(f) = col(u(f)..........
(9.61)
for all values of t in a neighborhood of t = f'. In other words, if the rank
condition (9.59) holds, the value of x(t) - at any time t sufficiently close to
can he expressed as a function of the values of the first n — 1 derivatives
of y[t) and the first n — 2 derivatives of u(t) at this time.
Of course, this "observability" property only enables us to determine the
value of j(f) in a neighborhood of a rime tc at which the rank condition (9.59)
holds. If a global reconstruction is sought, a stronger hypothesis is required,
which can be characterized as follows. A system of rhe form (9.1) is said to
be uniformly observable if the following conditions are satisfied
(i) the mapping
H : Rn ПГ
j »-> co1(A(t). Lf h(a-),.... Zy-1h(j?))
is a global diffeomorphism,
(ii) the rank condition
,ЭФ\
rank —— (t. r) = n
\ O.T /
holds for each (т. u) E ЕГ x R"-1.
It turns out, in fact, that if a system is uniformly observable, the mapping
Ф is globally defined.
9-6 Semiglobal Stabilization by Output Feedback
463
Proposition 9.6.1. Suppose f9.1) is uniforinly observable. Then there exists
a unique smooth mapping
$  "ST x3,,_|	) PJ’
(m. r)
j- = 4Su\ v)
such that iv = Ф(Ф(и\ ij. c) for all (tc, r) G IR'1 x JV1"1 and x = Ф{Ф(х. г), г)
for all (j, г) G R" x R"-1.
Proof. Using property (i). define a change of variables
x7i(.r)	1 < i < ii .
Then, it is easily seen that the vector field f(x) + g(x)i\} is changed into a
vector field of the form
/ 6 + 9i (f, 1 t’o \
£з + 9'2 Ю «’u
\/J«l +0.,(«)ro/
We will show that, if condition (ii) holds, then depends only on
giif)) depends only on £].£> and so on. For. let Ф(£.г) denote the expression
assumed by Ф(х, e) in the new coordinates and observe that, by construction.
b
A-2 (£) + i/i (£)t:o
Аз(£- t'o) + ,9i (0t’i
Ф(^,с) =
' An(£. (;0, . . . . t’n — 3) + 91	-2 /
where1 Aj(£), Лз(^, co).... are suitable functions, Thus, for i > 2.
дФ^.г)
d\-2 dgi
yy +
If
^(C) CO
УС
at some £°, there exists r° such that
ЭФ(^. r)
(Г H = 0
and this indeed contradicts (ii). By repeating this argument, one arrives at
the conclusion that is independent of £г+[...........
464
9. Global Feedback Design for Single-Input Single-Output Systems
Using this property it is easily found that, in the new coordinates. r)
has an expression of the form
=
& + ~ t’o)
s3 + "з(С • С-’- Gn t‘1)
V Ci + 9P (C , -   • Ci - 1 d'O- •
which shows that for each (tn. r) G ПР‘ x K'1 1. the equation w = Ф(С r) has
a unique solution £ = Ф(и\ c). which is a smooth mapping. <
Remark 9,6.1. Suppose system (9.1) is uniformly observable and consider
also the function ^r,(z.z?o...c„_i) defined above, which by construction is
such that
y["1 (C = JdU..............
Using the function iPljr. r). whose existence has been shown in Proposition
9.6.1. define the system
ti.’o = uq
U’l =	U'-2
(9.62)
6’n -2	—	n'jj-i
U'fl_i =	Г ). P0: ... .U,,-,) .
Then, it is clear from the previous discussion that, if
v,(t) — ul!l(t) for all t > 0 and for all 0 < г < n - 1
and
гсДО) = y‘”(0) fqr all 0 < i < - 1 .
then
= .y,!'U) f°r all t > 0 and for all 0 < i < n - 1 .
In other words, if the initial state (at rime t = 0) and the inputs to system
(9.62) are appropriately set, the various components of state of this system
reproduce the output of (9.1) and its first n - 1 derivatives. <
We proceed now to the description of how the system (9.1) can be
semiglohally stabilized using (dynamic) output feedback. More precisely, we
will describe a recent important result obtained by Teel and Praly. who have
shown that if the equilibrium .г = 0 of
•r = f(-r) +
is globally asymptotically stabilizable by means of a state feedback law of the
form и — q(t) and if the system is uniformly observable, then the system is
semiglobally stabilizable by means of a dynamic output feedback of the form
9.6 Semiglobal Stabilization by Output Feedback 465
и = в(£) .
(9.63)
To this end, assume that a feedback law и = o(j) which globally stabilizes
the equilibrium .г = 0 of
•r = /И + j?(.r)o(j)
exists. Since the state x is not directly available for feedback, one may wish
to use the results illustrated above in order to obtain a on-line reproduction
of x(t) via
j(t) =
with u’(f) and u(t') as in (9.61). However, one cannot simply replace the
argument j-(t) of o(z(f)) by	r(t)) because this would yield a feedback
law requiring ‘'differentiators". Thus, a more elaborate stabilization scheme
must be developed.
In order to bypass the need of derivatives of u(t), it is possible to proceed
in the following way. Consider the extended 2n-dimensional system
•f = f(x) + flUH’o
t'o ~	?'i
(9.64)
f'n — 2 — t’n —1
c„_i = il .
This system satisfies the hypotheses of Theorem 9.2.3. Thus, there exists a
smooth feedback law of the form
й = 8(x. ........
(9.65)
which globally asymptotically stabilizes its equilibrium
(t. c0, -..	= (0.0.......0)	.
The interconnection of (9.64) and (9.65) can be viewed as the intercon-
nection of the original system (9.1) and the dynamic feedback
/ t’o \
(9.66)
Thus, if in the latter x is replaced by	one obtains a dynamical
feedback which, if u^(t') were equal to y[ ,J (t) for all 0 < i < n - 1 . would
466	9. Global Feedback Design for Single-Input Single-Output Systems
stabilize system (9.1). This dynamic feedback no longer requires derivatives
the input (which now arc part of the state) but still requires derivatives of
the output.
To avoid the explicit computation of derivatives of y(t) one may take
advantage of the properties illustrated in Remark 9.6.1 and reproduce these
derivatives by means of an ^auxiliary"’ system of the form (9.62). Of course,
system (9.62) produces the appropriate derivatives of y(t) only if its initial
condition is correctly set. which is unlikely to be the real case. Thus, in
order to overcome this drawback, one is lead -- as in the design of asymptotic
state observers for linear systems - to add to the right-hand side of (9.62)
a "compensation" term proportional to the mismatch between the measured
output, у and its estimate. This leads to the consideration of an "estimator’"
described by equations of the form
(9.67)
in which L > 0 is a constant to be determined later and co, ci,..., crn are
coefficients of a polynomial
p( A) — A'1 + Cn-i A'! 1 +  - • + Cj A + cq
having all roots with negative real.
These ideas lead to a candidate dynamic (actually 2n-dimensional) feed-
back law which consists of the interconnection of (9.66). with т replaced by
•?(r/, r); and (9.67). This feedback law does not contains differentiators and, as
we shall see in a moment, can locally asymptotically stabilize the equilibrium
point	।
j- = 0. c, = 0. tj, = 0	0 < i < n - 1
of the closed loop system. This law, however, only guarantees local stability
(even if L is chosen large enough so as to induce "fast" convergence to zero of
the estimation error of (9.67)) and therefore, if semiglobal stability is sought,
a further adjustment is necessary.
Before showing what this adjustment consists of. we first prove the claim
about the property of local asymptotic stability. To this end. it is convenient
to define a vector e of estimation errors
e = col(e0,ei.....e„_i)
with
e0 = Ln ЧаЛ*) ~
e, = Ln i ^(vo.
1 < i < n — 1
9.6 Semiglobal Stabilization by Output Feedback
467
and the vector
Z = СО1(т,С’0. O. • • • , l’n-1) 
Then, standard calculations show that the closed loop system consisting
of (9.1), of (9.66) with т replaced by •£(//, c) and of (9.67) is described by
equations of the form
di(z.e)
LAe + P2(z.e)
in which the following properties hold
(i)	if £ is sufficiently large, all the eigenvalues of the matrix
LA+ [^1(0.0)
L de J
have negative real part.
(ii)	the function P2(z.e) is such that p2(z.O) = 0.
(iii)	the function pi (z, e) is such that the equilibrium z = 0 of
z = Pi(z. 0)
(9.68)
(9.69)
is globally asymptotically stable.
Actually, property (i) is a direct consequence of the fact that all the
eigenvalues of A, which has the following form
1 0
0 1
0 0
о 0
o)
have negative real part. To check properties (ii) and (iii) it is useful to observe
that, by construction, if e = 0,
По = po(-r)
Hi = рДт, t’o,..., t'f—i) 1 < i < n - 1.
and therefore
[«^(77,t')]e=0 - Ф^(Ф(х, v). v) = X .
from which it is easy to deduce property (ii). This also proves that the sub-
system in (iii), by construction, is exactly
x	= /(ar) + g(x)v0
co	= t’l
t'n —2
t'n — 1
t’n-1
0(x. v0,
468	9. Global Feedback Design for Single-Input Single-Output Systems
which by hypothesis has a globally asymptotically stable equilibrium at
(.r.t'o.......Gi-i) = (0.0... ..0) .
and this gives property (iii).
If L is large enough, system (9.68) satisfies the hypotheses of the first
Lemma in section B.2. Thus, its equilibrium (c.e) = (0,0) is locally asymp-
totically stable.
The properties thus indicated also show that, in the closed loop system,
the subset e = 0 is an invariant manifold and that the restriction of the system
to this invariant manifold, which is precisely system (9.69). has a globally
asymptotically stable equilibrium at z = 0. Thus the basin of attraction of
the equilibrium (z.tf) = (0.0) of the closed loop system contains the subset
e = 0. However, as anticipated earlier, the dynamic feedback law constructed
above is not sufficient - in general - to secure that tire basin of attraction of
the equilibrium (г. rj) = (0,0) contains any arbitrary compact set. To obtain
this more demanding result one has to choose a sufficiently large value of the
parameter L (for reasons which are similar to the ones which motivate the
results described in section 9.3) and. simultaneously, also to prevent that the
choice of a large L induce unacceptably high values of the input и to (9.1).
This result can be achieved by replacing, in the dynamic feedback law
constructed above, the function r) by another function o) defined
as follows
if < .u
u) = <
||^Ньс)1Г
(9.70)
if t’)l > -V .
where M > 0 is a design parameter to be determined later. In other words.
!?*(//. u) is a function which coincides with ^(z/. c) for all (zj.v) such that the
norm of <?(г/, и) is less than a fixed number M, and bounded (in norm) by Л/
elsewhere.
This yields a control law, for (9.1) described by equations of the form
(
l:2
Oi-l
\	f). l’o. 
(9.71)
и = ro .
(9.73)
9.6 Semiglobal Stabilization by Output Feedback
469
Note that the system thus defined is a feedback law of the form (9.63).
with
£ = CO1( l'(j. L’i ,.... t’n-i-t/o ЛЛ.7/r(_l) .
This feedback law is such that the following result holds.
Theorem 9.6.2. For each 7? > 0 there exist numbers B1 >0. Af* >0 and.
for each M > .V*, a number L\f > 0 such that, if M > iF in (9.70) and
L > L\j tn (9.72), the equilibrium (c,£) = (0.0) of the closed loop system
(9.1)-(9.72)-(9.71)-(9.73) is locally asymptotically stable and. moreover.
( lim x(t') = 0
1И0)|[<Л,|к(0)||<й'	{шп>) = 0.
We are not giving the proof of this result, for which we address the reader
to the original source. We only stress that the main, and more challenging,
issue in this proof is to establish the existence of numbers Af and L\f such
that, for all L > L\}, any trajectory of the closed loop system, with initial
condition satisfying |jr(O)|f < В and ||£(0)|j < B1. is bounded and, moreover.
satisfies ||T(f)|| < _W for all t > 0. Having proven this, one can look at
the equivalent description (9.68) of the closed loop system and observe that,
since ||z|| < M implies
[^*(r/. r)]c=0 - Ф*(Ф(х, с), г?) = Ф(Ф(х. v).v) - x .
the function Ф-2(z.e) is such that (see condition (ii) above)
O2(z(t), 0) = 0 for all t>0.
Thus. e(t) is a bounded integral curve of a system of the form
ё — LAe + o2(x(f). e)
in which z(t) is bounded and <D?(c(t), e) vanishes at e = 0. This can be used
to show that, if L is large enough. e(f) converges to zero as t tends to oc.
Having proven this, the proof can continue exactly as in step (iii) of the proof
of Theorem 9.3.1.

A. Appendix A
A.l Some Facts from Advanced Calculus
Let A be an open sublet of and f : .4 —> R a function. The value of f
at. z =	....т„) is denoted /(r) — f(xi..... xn). The function f is said
to be a function of class (or. simply. or. also, a smooth function) if
its partial derivatives of any order with respect to j-j..r,, exist and are
continuous. A function f is said to be analytic (sometimes denoted as C~) if
it is C' and for each point .r:1 £ .4 there exists a neighborhood V of .r°, such
that the Taylor series expansion of f at .r2 converges to f(x) for all .r E U.
Example. A typical example of a function which is but not analytic is
the function f : R —> R defined by
f /(.г) - 0	if x < 0
/(r) = exp(- -) if x > 0. <
A mapping F : .4 —> R,ri is a collection (/i../,„) of functions ft : .4
R. The mapping F is CiyL if all /('s are .
Let Г E R’1 and V £ Rn be open sets. A mapping F : E V is a
diffeomorphism if it is bijective (i.e. one-to-One and onto) and both F and
F-1 are of class . The jacobian matrix of F at a point x is the matrix
/ 9 fi	ЭЛ	\
OF	dx-i	0xn
d1'	... £21
\ Oxi	0xn	/
The value of yy at a point z = xc is sometimes denoted [37]
Theorem. (Inverse function theorem) Let .4 be an open set of№n and F :
.4 -> R" a C* mapping. If [|y] is nonsingular at some xQ E .4, then there
exists an open neighborhood L: of xa in .4 such that V = F(E) is open in Rn
and the restriction of F to L' is a diffeomorphism onto Г.
472
A. Appendix A
Theorem. (Rank theorem) Let A C R" and В C Rm be open sets. F : A —>
В a Cx mapping. Suppose [^yj has rank к for all x G A. For each point
x° G A there exist a neighborhood Ao of xrj in A and a neighborhood Bo of
F(xc) in B. two open sets L C Rrl and V C Rm . and two diffeomorphisms
G : U —> A2 and H : B^ —> Г such that H о F о G(L') C Vr and such that for
all (ti...Tfl) G U
(H о F о G)(xy....xn) - (rb ...	, 0) .
Remark. Let P^ denote the mapping : R” —> P defined by
PkLri.....= (^i..........^-.0....0).
Then, since H and G are invertible, one may restate the previous expression
as
F = H~x о об'”1
which holds at all points of Ao, <
Theorem. (Implicit function theorem) Let A c Rm and В c RT1 be open
sets. Let F : A x В —> Rn be a Cx mapping. Let
(м) = (-Г1.....xm,yi,...	.yn)
denote a point of A x B. Suppose that for some (zc.t/c) € A x В
F(x*.y^ = 0
and the matrix	~ .
/	df\	ЭЛ	\
dF _	dyi	dyn
dy	df^
\	дух /	dyri	/
is nonsingular at (xQ.yc). Then, there exist open neighborhoods Ao of xc in
A and Bc of yQ in В and a unique Cx' mapping G : Ao -> Bc such that
F(x,G(x')) =0
for all x E Ac,
Remark. As an application of the implicit function theorem, consider the
following corollary. Let A be an open set in R,!. let Af be a fc x n matrix
whose entries are real-valued Cx functions defined on A and b a ^-vector
whose entries are also real-valued Cx functions defined on A. Suppose that
for some x° G A
rankA/(z°) = к .
Then, there exist an open neighborhood L" of x° and a C'x mapping G : L"
Rn such that
Af(x)G(z) = b(z)
A.2 Some Elementary Notions of Topology
473
for all x € U.
In other words, the equation
АЦх)у = b(x)
has at least a solution which is a function of x in a neighborhood of
If A’ = n this solution is unique. <
A.2 Some Elementary Notions of Topology
This section is a review of the most elementary topological concepts that will
be encountered later on.
Let S be a set. A topological structure, or a topology, on S is a collection
of subsets of S. called open sets, satisfying the axioms
(i)	the union of any number of open sets is open
(ii)	the intersection of any finite number of open sets is open
(iii)	the set S and the empty set 0 are open.
A set S with a topology is called a topological space.
A basis for a topology is a collection of open sets, called basic open sets.
with the following properties
(i) S is the union of basic open sets
(ii) a nonempty intersection of two basic open sets is an union of basic open
sets.
A neighborhood of a point p of a topological space is any open set which
contains p.
Let Si and So be topological spaces and F a mapping F : Si —> So. The
mapping F is continuous if the inverse image of every open set of S? is an
open set of Si. The mapping F is open if the image of an open set of Sl is
an open set of S-j. The mapping F Is an homeomorphism if it is a bijection
and both continuous and open.
If F is an homeomorphism, the inverse mapping F”"1 is also an homeo-
morphism.
Two topological spaces Si, S? such that there is an homeomorphism F :
Si —> So are said to be homeomorphic.
A subset L’ of a topological space is said to be closed if its complement U
in S is open. It is easy to see that the intersection of any number of closed
sets is closed, the union of any finite number of closed sets is closed, and both
S and 0 are closed.
If So is a subset, of a topological space S, there is a unique open set. noted
int(So) and called the interior of Sc. which is contained in So and contains
any other open set contained in Sc. As a matter of fact. int(Sc) is the union
of all open sets contained in So. Likewise, there is a unique closed set, noted
cl(S0) and called the closure of Sc. which contains So and is contained in any
474
A. Appendix A
other closed set which contains Sc. Actually. cl(S:.) is the intersection of all
closed sets which contain S'c.
A subset of S is said to be dense in S if its closure coincides with S.
If Si and S-> are topological spaces, then the cartesian product Si x S2
can be given a topology taking as a basis the collection of all subsets of the
form tri x with l\ a basic open set of St and th a basic open set of S2.
This topology on Si x S2 is sometimes called the product topology.
If S is a topological space and Si a subset of S. then Si can be given a
topology taking as open sets the subsets of the form Si П V with 17 any open
set in S. This topology on Si is sometimes called the subset topology.
Let F : S'i —> S-2 be a continuous mapping of topological spaces, and let
F(Si) denote the image of F. Clearly. F(SF with the subset topology is a
topological space. Since F is continuous, the inverse image of any open set of
F(SJ is an open set of Si. However, not all open sets of Si are taken onto
open sets of F(Si). In other words, the mapping F' : Si -> F(Si) defined by
F'(p) = F(p) is continuous but not necessarily open. The set F(SJ can be
given another topology, taking as open sets in F(Si) the images of open sets
in S]. It is easily seen that this new topology, sometimes called the induced
topology, contains the subset topology (i.e. any set which is open in the subset
topology is open also in the induced topology), and that the mapping F! is
now open. If F is an injection, then Si and F(S[) endowed with the induced
topology are homeomorphic.
A topological space S is said to satisfy the Hausdorff separation axiom
(or. briefly, to be an Hausdorff space) if any two different points p{ and p2
have disjoint neighborhoods.
A.3 Smooth Manifolds
Definition. Л locally Euclidean space .Y of dimension n is a topological
space such that, for each p G A', there exists a homeomorphism о mapping
some open neighborhood of p onto an open set in E/’.
Definition. A manifold .V of dimension n is a topological space which is
locally Euclidean of dimension n, is Hausdorff and has a countable basis.
It is not possible that an open subset U of Е7 be homeomorphic to an
open subset V of . if n / m (Brouwer's theorem on invariance of domain).
Therefore, the dimension of a locally Euclidean space is a well-defined object.
A coordinate chart on a manifold .V is a pair ([.'. o), where L’ is an open
set of .V and b a homeomorphism of U onto an open set of E" . Sometimes о is
represented as a set (<pi..... <9n) and bi : U —> Ct is called the i-th coordinate
function. If p e L:, the /(-tuple of real numbers (c>i(p) bn(p)) is called
the set of local coordinates of p in the coordinate chart (L\ <p). A coordinate
chart (17. 6) is called a cubic coordinate chart if o(L’) is an open cube about
А.З Smooth Manifolds 475
the origin in K”. If p G U and o(p) = 0. then the coordinate chart is said to
be centered at p.
Let (C,o) and (Г, L) be two coordinate charts on a manifold Л’. with
[rnV 0. Let (ta........t’ri) be the set. of coordinate functions associated
with the mapping c. The homeomorphism
с о 0-1 : o(U ПГ) -> i?(C nV)
taking, for each p G LT П Г. the set of local coordinates (oi (p),.... on(p))
into the set of local coordinates (bq (p),.... i.-fl(p)). is called a coordinates
transformation on f Ab. Clearly. о t--1 gives the inverse mapping, which
expresses (ojp).....<?;l(p)) in terms of (04 (p)...., C'n(p)).
Frequently, the set (<Mp)-• • • - 0n(p)) is represented as an n-vector z =
col(zi....). and the set (l’i (p).....c\(p))	as an n-vector у — col......
p,J. Consistently, the coordinates transformation	can be represented
in the form
У И
and the inverse transformation Qoy 1 in the form
Two coordinate charts ({/.<?) and (V, v) are C*-compatible if, whenever
F П V 0. the coordinates transformation с о о-1 is a diffeomorphism.
i.e. if y(r) and z(p) are both maps (see Fig. A.l).
Fig. A.l.
476 A. Appendix A
A Cx atlas on a manifold *V is a collection Д = {(СД,	: i G /} of
pairwise Cx-compatible coordinate charts, with the property that. (J =
f E I
N. An atlas is complete if not properly contained in any other atlas.
Definition. A smooth or Cx manifold is a manifold equipped with a com-
plete Cx atlas.
Remark. If Д is an}’ Cx atlas on a manifold A*, there exists a unique complete
Cx atlas Д* containing Д. The latter is defined as the set of all coordinate
charts {V. p) which are compatible with every coordinate chart (L). oj of Д.
This set contains Д, is a Cx atlas, and is complete by construction. <
Some elementary examples of smooth manifolds are the ones described
below.
Example. Any open set U of Rri is a smooth manifold, of dimension n. For.
consider the atlas Д consisting of the (single) coordinate chart (L\ identity
map on U) and let Д* denote the unique complete atlas containing Д. In
particular. R" is a smooth manifold. <1
Remark. One may define different complete Cx atlases on the same manifold,
as the following example shows. Let A" = R. and consider the coordinate
charts (R, й>) and (R. u). with
0(x) = z
Ф) = A
Since d"1 (t) = j- and (.r) = z1^3.
ф о v-1 (<) 2 т1/3
and the two charts are not compatible. Therefore the unique complete atlas
Д* which includes (R.d) and the unique1 complete atlas Л* which includes
(R, v) are different. This means that the same manifold Ar may be considered
as a substrate of two different objects (two smooth manifolds), one arising
with the atlas Д* and the other with the atlas Д*,.. <
Example. Let C be an open set of Rr,! and let Ai,... be real-valued
Cx functions defined on U. Let Ar denote the (closed) subset of U on which
all functions Ai..... Am_„ vanish, i.e. let
A* = {z £ U : A((j*) — 0
Suppose the rank of the jacobian matrix
/ dXi
Oxi
9Xm~n
\ ()xi
, 1 < i < m — n} .
^Ar \
dzm
dXni—fi
dxm )
А.З Smooth Manifolds 477
is m — n at all ,r G AL Then A' is a smooth manifold of dimension n.
The proof of this essentially depends on the Implicit Function Theorem,
and uses the following arguments. Let .rc = (ay, • -  r<r .r' + 1 -rfoJ be a
point of Ar and assume, without loss of generality, that the matrix
/ dAi	d*A]	\
0xn^	0xn>
д^т—11	^А,;?Г[
X (Э.Г,|+1	thr™	'
is nonsingular at r'. Then, there exist neighborhoods Ao of (.г’...in
3.'1 and of U’pj-! •.. , J'p;) in UL"“Ti and a Cx mapping G : Ao —> B5 such
that
A,(j’i....rn - ЙГ1 (Jt-1 -. ..J„)-9т-л(л-  -• -Лг)) = О
for all 1 < i < m—n. This makes it possible to describe points of Л around .rc
as m-tuples (.Ti ) such that ^п+; = дфх^............rn) for 1 < ? < m - zi.
In this way one can construct a coordinate chart around each point .r° of A
and the coordinate charts thus defined form a Cx atlas.
A manifold of this type is sometimes called a smooth hypersurface in .
An important example of hypersurface is the sphere Sm~l, defined by taking
n = m — 1 and
The set of points of IR'fI on which Aifo) — 0 consists of all the points on a
sphere of radius 1 centered at the origin. Since
/ЭАТ	dAi \
\ dxi	0x,„ J
never vanishes on this set. the required conditions are satisfied and the set is
a smooth manifold, of dimension m — 1. <
Example. An open subset A"' of a smooth manifold A’ is itself a smooth
manifold. The topology of _V' is the subset topology. If (LL d) is a coordinate
chart of a complete atlas of AL such that Lfo Ar' / fl. then the pair
(L’L <У) defined as
U' = U П Ar/
o' = restriction of о to U1
is a coordinate chart of Afo In this way, one may define a complete C^atlas
of A’L The dimension of A*' is the same as that of AL <
Example. Let M and A" be smooth manifolds, of dimension m and zi. Then
the cartesian product ;V x Ar is a smooth manifold. The topology of M x A’
is the product topology. If (U. ф) and (IL o) are coordinate charts of M and
A*, the pair (Lr x V. (<h;tfo) is coordinate chart of M x AL The dimension of
M x A' is clearly zzi + n.
478
A. Appendix A
An important example of this type of manifold is the torus T~ = S1 x S1.
the cartesian product of two circles. <i
Let Л be a real-valued function defined on a manifold AL If (L\ <p) is a
coordinate chart, on Ar. the composed function
A = A opr1 : o(L') -> к
which takes, for each p € the set of local coordinates (j?i ,..., zn) of p into
the real number A(p), is called an expression of X in local coordinates.
In practice, whenever no confusion arises, one often uses the same symbol
A to denote A op-1, and write A(ti ,.... xn) to denote the value of A at a
point p of local coordinates .......t„).
If Ar and M are manifolds, of dimension n and m, F : N —> If is a
mapping. (L. d>) a coordinate chart on A’ and (I’, o) a coordinate chart on
_W, the composed mapping
F = о о F о 0-1
is called an expression of F in local coordinates. Note that this definition
makes sense only if F(U) A V* 0. If this is the case, then F is well defined
for all n-tuples (ri,..., z„) whose image under F о dr1 is a point in V.
Here again, one often uses F to denote г? о F о <p-1, writes yt =
ft(xi,... ,xn) to denote the value of the г-th coordinate of F(p). p being
a point of local coordinates .......zn), and also
Definition. Let N and M be smooth manifolds. A mapping F : Лг —> Д/ is
a smooth mapping if for each p € A7 there exist coordinate charts (U. p) of
A’ and (V. p) of XI, with p € U and F(pf € V, such that the expression of F
in local cooidinates is C * .
Remark. Note that the property of being smooth is independent of the choice
of the coordinate charts on A7 and ЛЛ Different coordinate charts (Uf, o') and
(V'', o') are by definition C'x compatible with the former and
F' = boFo^1
= p* 0 t1’-1 0 l-’ 0 F О О-1 О О о р'^1
= (о' О О-1 ) О F О (р' о о-1)-1
being a composition of Сх functions is still <1
Definition. Let N and XI be smooth manifolds, both of dimension n. A
mapping F : N —> M is a diffeomorphism if F is bijective and both F and
F~l are smooth mappings. Two manifolds N and M are diffeoniorphic if
there exists a diffeomorphism F : Ar —> Af.
A.4 Submanifolds 479
The rank of a mapping F : А' —> Л/ at a point p € A7 is the rank of the
jaeobian matrix
/3/i	...
3л	dxn
dfm	dfm
\ 3л	dxn	/
at x = o(p). It must be stressed that, although apparently dependent on
the choice of local coordinates, the notion of rank thus defined is actually
coordinate-independent. The reader may easily verify that the ranks of the
jaeobian matrices of two different expressions of F in local coordinates are
equal.
Theorem. Let .V and Al be smooth manifolds both of dimension n. A map-
ping F : .V —> AL is a diffeo morphism if and only if F is bijective, F is smooth
and rank(F) — n at all points of X.
Remark. In some cases, the assumption that functions, mappings, etc. are
Cx, may be replaced by the stronger assumption that functions, mappings,
etc. are analytic. In this way one may define the notion of analytic manifold,
analytic mappings of manifolds, and so on. We shall make this assumption
explicitly whenever needed. <i
A.4 Submanifolds
Definition. Let F : A’ —> Al be a smooth mapping of manifolds.
(i)	F is an immersion z/rank(F) — dim(-V) for all p € A’
(ii)	F is an univalent immersion if F is an immersion and is injective
(iii)	F is an embedding if F is an univalent immersion and the topology
induced on F(X) by the one of X coincides with the topology of F(X) as a
subset of M.
Remark. The mapping F, being smooth, is in particular a continuous map-
ping of topological spaces. Therefore (sec section A.2) the topology induced
on F(X) by the one of A7 may properly contain the topology of F(A’) as a
subset of AL. This motivates the definition (iii). <
The difference between (i). (ii) and (iii) is clarified by the following ex-
amples.
Example. Let A’ = R and Al = R2. Let t denote a point in A7 and (xj.x-2) a
point in AL The mapping F is defined by (Fig. A.2)
xT(t) — at — sin t
XL>(f) = COSf
and. then.
480 A. Appendix A
rank(F) = rank [ a
\ — sin t
If a = 1 this mapping is not an immersion because rank(F) = 0 at t = ‘2k~
(for any integer A-)-
Fig. A.2.
If 0 < a < 1 the mapping is an immersion, because rank(F) = 1 for all
t. but not an univalent immersion, because F(tj) — F(M) for all t^t-y such
that t; = '2k:i — т. f-> — 2 кт + т and sinr — ат.
Fig. A.3.
As a second example we consider the so-called "figure-eight" (Fig. A.3).
Let TV be the open interval (0, 2тг) of the real line and Л/ = R2. Let t denote
a point in _V and (Xj. j-2) a point in M. The mapping F is defined by
a*i(t) = sin2t
^2(0 = sin t .
This mapping is an immersion because
rank(F) = rank
dt
%
dt /
= rank
2 cos 2t
cos t
= 1
A.4 Submanifolds 481
for all 0 < t < 2~. It is also univalent because
F(C) = F(t2) => h - t-2 •
However, the mapping is not an embedding. For. consider the image of F.
The mapping F takes the open set (тг - г. - + s) of A” onto a subset U' of
F(A') which is open by definition in the topology induced by the one of Ab
but is not an open set in the topology of F(A7) as a subset of M. This is
because Uf cannot be seen as the intersection of F(A’) with an open set of
IR2.
As a third example one may consider the mapping F : R K3 given by
j'j (t) = cos 2~t
x-At) = sin2Ttf
•гз(О = t
whose image is an "helix" winding on an infinite cylinder whose axis is the
,r3 axis. The reader may easily check that this is an embedding. <
The following theorem shows that every immersion locally is an embed-
ding.
Theorem. Let F : X —> M be an immersion. For each p € A' there exists
a neighborhood U of p with the property that the restriction of F to L’ is an
embedding.
Example. Consider again the "figure-eight'’ discussed above. If V is any inter-
val of the type (d. 2тг —d), then the critical situation we had before disappears
and the image V of (тг — s. ~ + c) is now open also in the topology of F(A')
as a subset of R2. <
The notions of univalent immersion and of embedding are used in the
following way.
Definition. The image F(X) of a univalent immersion is called an im-
mersed submanifold of M. The image F(X) of an embedding is called an
embedded submanifold of M.
Remark. Conversely, one may say that a subset M1 of M is an immersed
(respectively, embedded) submanifold of M if there is another manifold A'
and a univalent immersion (respectively, embedding) F : А7 -о M such that
F(N) = ML
The use of the word "submanifold” in the above definition clearly indicates
the possibility of giving F(N') the structure of a smooth manifold, and this
may actually be done in the following way. Let M' = F(A’) and F' : Д' M'
denote the mapping defined by
F'(p) = F(p)
482
A- Appendix A
for all p € Ar. Clearly. F' is a bijection. If the topology- of AF is the one
induced by the one of A’ (i.e. open sets of AF are the images under F! of
open sets of A7). F' is a homeomorphism. Consequently, any coordinate chart
(E.<?) of .V induces a coordinate chart (V. c‘) of AF. defined as
C = F'(U). r — фо {F'F1 -
C -compatible charts of A7 induce C^-compatible charts of AF and so com-
plete -atlases induce complete C0'--atlases. This gives AF the structure of
a smooth manifold.
The smooth manifold AF thus defined is diffeomorphic to the smooth
manifold A7. A diffeomorphism between Al' and A" is indeed F' itself, which
is bijective, smooth and has rank equal to the dimension of A7 at each p € A*.
Embedded submanifolds can also be characterized in a different way.
based on the following considerations.
Let .V be a smooth manifold of dimension in and (U.&) a cubic coordinate
chart. Let n be an integer. 0 < n < tv. and p a point of U. The subset of U
Sp - {q G E : .гг(</1 =	= n 4- 1....m}
is calk'd an n-dimensional slice of U passing through p. In other words, a slice
of U is the locus of all points of E for which some coordinates (e.g. the last
tv - n.) are constant.
Theorem. Let Al be a smooth manifold of dimension m. A subset AF of
Al is an embedded submanifold of dimension n < m if and only if for each
p 6 Al1 there exists a cubic coordinate chart (E. g») of Al. with p 6 E . such
that E П AF coincides with an n-dimensional slice ofU passing through p.
This theorem provides a more “intrinsic*’ characterization of the notion of
an embedded submanifold (of a manifold Al), directly related to the existence
of special coordinate charts (of Al). Note that, if (E. c) is a coordinate chart
of .V such that FfAAF is an n-dimensional/slice of U, the pair (EL E) defined
as
E' = E П AF
E(p) = foi(P)........jrn(p))
is a coordinate chart of AF. This is illustrated in Fig. A.4 (where Af = Ж3
and n = 2).
Remark. Note that an open subset AF of Al is indeed an embedded subman-
ifold of .V. of the same dimension m. Thus, a submanifold AF of Al may be
a proper subset of .V. although being a manifold of the same dimension. <
Remark. It can be proved that any smooth hyper surf ace in is an embed-
ded submanifold of K"1. Moreover, it has also been shown that if A’ is an
n-dimensional smooth manifold, there exist an integer m > n and a mapping
F : .V —> JE' which is an embedding (Whitney's embedding theorem). In
other words, any manifold is diffeoniorphic to an embedded submanifold of
R’7’. for a suitably large in. <
А.э Tangent Vectors 483
Fig. A.4.
Remark. Let I' be a «-dimensional subspace of . Any subset of of the
form
j° + r={zG Rm : .r = ./ + G V}
where r0 is some fixed point of R,7i, is indeed a smooth hypersurface and so
an embedded submanifold of Rm, of dimension n. This is sometimes called a
flat submanifold of Я'7’. <
A.5 Tangent Vectors
Let A7 be a smooth manifold of dimension n. A real-valued function A is said
to be smooth in a neighborhood of p. if the domain of A includes an open set.
U of AT containing p and the restriction of A to U is a smooth function. The
set of all smooth functions in a neighborhood of p is denoted C^(p). Note
that Cx{p) forms a vector space over the field R. For. if A. у are functions
in C,x"(p) and a. b are real numbers, the function aX + defined as
(nA + byflq) ~ aA(g) + by(q)
for all q in a neighborhood of p; is again a function in Cx(p). Note also that
two functions А. у G Cx{p) may be multiplied to give another element of
С,эс(р). written Ay and defined as
(Ay)(<?) = X(q)-y(q)
for all q in a neighborhood of p.
Definition. A tangent vector i: at p is a map v : Cx(p) —> R with the
following properties:
(i) (linearity): v(aX + by) — nr(A) + bc(y) for all Л. у G Cx(p) and a.b GR
(ii) (Leibniz rule): r(Ay) = y(p)v(X) + X(p)v(y) for all A, у G Cx(p) .
Definition. Let N be a smooth manifold. The tangent space to A' at p,
written TPN, is the set of all tangent vectors at p.
484 A. Appendix A
Rent ark. A map which satisfies the properties (i) and (ii) is also called a
derivation. <
Remark. The set TPX forms a vector space over the field EL under the rules of
scalar multiplication and addition defined in the following way. If C[- r2 are
tangent vectors and t’i, c2 real numbers, cici + c2r2 is a new tangent vector
which takes the function Л G Cx(p) into the real number
(qi’i + c2c2)(A) = CinfA) + r2r2(A) .
Remark. We shall see later on that, if the manifold A' is a smooth hypersur-
face in T"!. the object previously defined may be naturally identified with
the intuitive notion of "tangent hyperplane’" at a point. <
Let (th <p) be a (fixed) coordinate chart around p. With this coordinate
chart one may associate n tangent vectors at p. denoted
д A / d \
d<^/p......\d<t>,Jp
defined in the following way
for 1 < i < n. The right-hand side is the value taken at r =	.....тп) =
ф(р) of the partial derivative of the function Aoq-1 (j-j...., .rn) with respect to
jy (recall that the function Aod»-1 is an expression of A in local coordinates).
Theorem. Let N be a smooth manifold of dimension n. Let p be any point
of A . The tangent space TPAT to A’ at p is an n-dimensional vector space
over the field B.. If ([’, o) is a coordinate chart around p. then the tangent
vectors ..........form a basis!of TpN.
The basis •!	.....I of is sometimes called the natural
basis induced by the coordinate chart (U. (fi).
Let r be a tangent vector at p. From the above theorem it is seen that
v
where 1.4....are real numbers. One may compute the p/s explicitly in the
following way. Let. be the г-th coordinate function. Clearly ot €
and then
д(Ф1 о Ф 1)

A.5 Tangent Vectors
485
because оо-1(/],....т„) = тг. Thus the real number сг coincides with
the \-alue of r at the z-th coordinate function.
A change of coordinates around p clearly induces a change of basts in TPA7.
The computations involved are the following ones. Let (F.p) and (Г. r) be
coordinate charts around p. Let { (^-) . - - , () } denote the natural
basis of Tp.V induced by the coordinate chart (U. tA. Then
d(0j o v 1)
dy
In other words
Note that the quantity
d(pj о C"1)
dyt
is the element on the J-th row and г-th column of the jaeobian matrix of the
coordinates transformation
z - x(y) .
So the elements of the columns of the jaeobian matrix of j* — x(y) are the
coefficients which express the vectors of the “new" basis as linear combina-
tions of the vectors of the '"old’* basis.
If r is a tangent vector, and (14,....	, (114	the n-tuples of real
numbers which express 0 in the form
Definition. Let A7 and M be smooth manifolds. Let F : A7 —> Л/ be a smooth
mapping. The differential of F at p e A7 is the map
F* : TpA -> TF(p]M
defined as follows. For v G TPX and Л G Cx(F(p)).
(FMW -r(AoF) .
486 A. Appendix A
Remark. F* is a map of the tangent space of Л' at a point p into the tangent
space of M at the point F(p). If с E TPN. the value F*(t’l of F* at v is a
tangent vector in Tp^.M. So one has to express the way in which F*(r’) maps
the set CDC(F(p)), of all functions which are smooth in a neighborhood of
F(p). into R This is actually what the definition specifies. Note that there
is one of such maps for each point p of .V (see Fig. A.5). <
Fig. A.5.
Theorem. The differential F„ is a linear map.
Since F* is a linear map. given a basis for TP.X and a basis for Tp{p}M one
may wish to find its matrix representation. Let (F. <?) be a coordinate chart
around p. (V. <?) a coordinate chart around q = F(p). {(^7).......
the natural basis of TPN and { (Щ77) , .... (^j j the natural basis of TqM.
In order to find a matrix representation of F* one has simply to see how F*
maps (^7) for each 1 < 1 < n.
Observe that
А.З Tangent Vectors
487
Thus,
/ d \	_
( w— ) (A о F) =
<?(A о г-1 О ?/ O I
’д{Х о t — 1) 1
. dVj J и
nt
/fl
д(А о F о о 1)
dj't
 = ОI Р I
р О F О <
д.г}
d( i 'j о F о о-1)
’ d(i.
Ox,
L


G>(Vj ° F O o’ 1)
&N 7 p
dxt
dt'j
Now. recall that г о F о о 1 is an expression of F in local coordinates. Then,
the quantity
0{l’j о F ° q 1)
d.i‘i
is the element on the j-th row and i-th column of the jacobian matr}.r of rhe
mapping expressing F in local coordinates. Using again
F(:r} - F(xi...........rlt] =
\ Tf!J (j" i
to denote и о F о о 1. one has simply
—=vl^U—
Р	j V^ J / ч
If с 6 TpJV and tn = F*(r) G TrlfliAI arc expressed as
Remark. The matrix representation of F„ is exactly the jacobian of its ex-
pression in local coordinates. From this, it is seen that the rank of a mapping
coincides with the rank of the corresponding differential. <
488 A. Appendix A
Remark. (Chain rule). It is easily seen that, if F and G are smooth mappings,
then
(Go FA = G*F*.<
The following examples may clarify the notion of tangent space and the
one of differential.
Example. The tangent vectors on R” . Let E." be equipped with the "natural’"
complete atlas already considered in previous examples (i.e. the one including
the chart (3". identity map on Then, if v is a tangent vector at a point
x and A a smooth function
” / o \ Ami
i’(A 7Г- (л)=^2 и‘
\ их, /	Ox,
i-l x 17 t	t=l L J -f
So. r(A) is just the value of the derivative of A along the direction of the
vector
col(fi....l’„)
at the point x. <
Remark. Let F : .V -> .W be a univalent immersion. Let n = dim(jV) and
m = dim(-iV). By definition, Fx has rank n at each point. Therefore the image
Е*(ТрХ) of F„. at each point p. is a subspace of Tf[ppW isomorphic to TpN.
The subspace F*(TpAr) can actually be identified with the tangent space at
F(p) to the submanifold Al' = F(X). In order to understand this point, let
F' denote the function F' ; X —> AT defined as
F'(p) = F(p)
for all p G X. F' is a diffeomorphism and so F( is an isomorphism. Therefore
the image F'fiTpN) is exactly the tangent space at F'(p) to AF. Any tangent
vector in	is the image F'(v) of a (unique) vector r 6 TpX and can
be identified with the (unique) vector F*(u) of F^(TPX).
In other words, the tangent space at p to a submanifold Al' of Al can be
identified with a subspace of the tangent space at p to Al.
The same considerations can be repeated in local coordinates. It is known
that an immersion is locally an embedding. Therefore, around every point
p G Al' it is possible to find a coordinate chart (F. <>) of Al. with the property
that the pair (L'L o') defined by
U' = {q G C : Oi (<y) = ot (p). i — n + 1.....m}
d' = (oi........dp)
is a coordinate chart of Al'. According to this choice, the tangent space to
Al' at p is identified with the n-dirnensional subspace of TpA'l spanned by the
tangent vectors ..............(afdpb <
A.5 Tangent Vectors 489
Example. The tangent vector to a smooth curve in . We define first the
notion of a smooth curve1 in T’. Let E = (h-E) be an open interval on
the real line. A smooth curve in Rn is the image of a univalent immersion
<7 : E . Thus, a smooth curve is an immersed submanifold of R'!. In
E and Ж'1 one may choose natural local coordinates as usual and. letting
t denote an element of U. express a by means of an n-tuple of real-valued
functions oi...., of t.
A smooth curve is a 1-diinensional immersed submanifold of . At a
point cr(to). the tangent space to the curve is a 1-dimensional vector space
which, as we have seem may be identified with a subspace of the tangent
space to R" at this point. A basis of the tangent space to the curve at. cr[E)
is given by the image under of (^) . a tangent vector at tc to E. This
image is computed as follows
, d	, da, . ( d \
1 = 1	v	I г О 1
Thinking oft 6 E as time and a(t) as a point moving in УА . we may interpret
the vector
as the velocity along the curve, evaluated at the point a(tc). So. we have that
the velocity vector at a point of the curve spans the tangent space to the curve
at this point. From this point of view, we see that the notion of tangent space
to a 1-dimensional manifold may be identified with the geometric notion of
tangent line to a curve in a Euclidean space (Fig. A.6). <
Fig. A.6.
Example. Let h be a smooth function h : л2 —> R and F : R2 —> R3 a
mapping defined by
490
A. Appendix A
F(.ri.rL>) - (j'l. T	J<)).
This mapping is an embedding and therefore F(xh’), a surface in T3, is an
embedded submanifold of л?. At each point F(.r) of this surface, the tangent
space, identified as a subspace of the tangent space to B3 at this point, may
be computed as
spanjF* —	. F*	}.
ydan/j. \0x>JT
Now.
d
dx3
The tangent space to F(R2) at some point (zj .z.L h(zp jy)) is the set of
tangent vectors whose expressions in local coordinates art1 from the form
o, 3 being real numbers and 4^-. being evaluated at .ri = and x? = z.i.
From this point of view, we see that the notion of tangent space to a 2-
dimensional manifold may be identified with the geometric notion of tangent
plane to a surface in a Euclidean space. <
Example. Let Aj..... ArJi_n be real-valued functions defined on R!!i and
set
= col( A] 3 ... Arri_. J .
As explained before, if rank of the jacobian matrix is m - n at all x G E™ 
the set
;V = {z G Г" : .l(z) = 0}
is a smooth manifold of dimension n (a smooth hypersurface in S'” ). Consider
now a smooth mapping о : U —>	. where Г is an open interval in jL with
the property that a(t] E .V for all t E I (i.e. the smooth curve o(F) is a
subset of Д'). By definition, о satisfies
Л(<т(0) = 0 for all IGF.
Thus, by the chain rule, we obtain
.1*0* (-7-). = 0 for all t G E .
at f
In other words, the tangent vector cr*(^) is an element of kerf.i*) at o(t).
A.5 Tangent Vectors 491
Now. any vector v G FpJV can be expressed as
for some a and some t. Thus, we conclude that TPX can be expressed as
TP;V = kcrClJp .<
One may define objects dual to the ones considered so far.
Definition. Let -V be. a smooth manifold. The cotangent space to Лт at p.
written TpX. is the dual space of TpN. Elements of the. cotangent space are
called tangent covectors.
Remark. Recall that a dual space V* of a vector space V is the space of all
linear functions from V to IR. If v* eV*, then r* : V —> R and the value of
r* at г 6 V is written as (e*.c). V* forms a vector space over the field R.
with rules of scalar multiplication and addition which define cit’[ + c^t’* in
the following terms
(Cj + C21’2, 0 = Cj (0; 0 + C2(c;. 0 .
If €i....erj is a basis of V. the unique basis 0...c* of V‘ which satisfies
«^0 = dij
is called a dual basts.
If V and TV are vector spaces. F : V —> IV a linear mapping, v 6 Г and
in* G IV*. the mapping F* : IT* —> V* defined by
(F*(w*).0 = (u’*.F(0)
is called the dual mapping (of F). <
Let A be a smooth function A : Л' —> R. There is a natural way of identi-
fying the differential A+ of A at p with an element of T*X. For, observe that
A* is a linear mapping
A* : ТРЛ -> 7\(pylR
and that Ta(P)]R is isomorphic to R. The natural isomorphism between IR and
TA(pI1R is the one in which the element c of R corresponds to the tangent
vector c(4) . If c(4L is the value at г of the differential A* at p. then c
must depend linearly on c, i.e. there must exist a covector, denoted (dA)p.
such that
^*(0 — {(dA)p, 0( —)( •
Given a basis of TPX, the covector (dA)p (like any other covector). may be
represented in matrix form. Let { (5^7)^- • • — (a^“)p} be the natural basis of
TPX induced by the coordinate chart (U. 0). The image under A* of a vector
492 A. Appendix A
v
is the vector
and this shows that
<(rfA)p,r) =
Remark. Note1 also that the value at A of a tangent vector r is equal to the
value at r of the tangent covector (dA)p. i.e.
e(A) = ((dA)p; i’).<
The dual basis of {(^fp)p...(a3“)p} computed as follows. From the
equality r(A) = ((dX)p.e) we deduce that
( — ') («,) =
\eojjr \aOiJP
d(&t oo 1)
0t1
d^j
— $ij
so that the desired dual basis is exactly provided by the set of tangent cov-
ectors {(cfoi )p,.... (dQn)p}.
If r* is any tangent covector. expressed as
i = i
the real numbers r*.r* are such that
P
Note also that, if r is any tangent vector expressed as
the real numbers ci,..., i’n are such that
r, = {(do^p.v) .
A.6 Vector Field* 493
A.6 Vector Fields
Definition. Let -V be a smooth manifold, of dimension n. A vector field f
on. X is a mapping assigning to each point p G Ar a tangent vector f(p) in
TPN. A vector field f is smooth if for each p 6 A’ there exists a coordinate
chart (U.o) about p and n real-valued smooth functions /i..fi, defined on
U such that, for all q G Г
Л,	/ a \
/(<;) = 2_,Мч)\ ,	
“t	VoJ,
Remark. Because of C1OC-compatibility of coordinate charts, given any coor-
dinate chart (Ai l'd about p other than ([’. dh one may find a neighborhood
V' с V of p and n real-valued smooth functions f[......f!Tl defined on V'.
such that, for all q G V'
/(?> = /,'(?) f 
X
Thus, the notion of smooth vector field is independent of the coordinates
used. <i
Remark. If (Г. <p) is a coordinate chart of AT on the submanifold U of A’ one
may define a special set of smooth vector fields, denoted (^-)..(ao-)
the following way
f d \	/ Q \
‘ P ydd,J p
It must be stressed, however, that such a set of vector fields is an object
defined only in lT. <
For any fixed coordinate chart (C.o). the set of tangent vectors
{(A)............................(A) }
* ' A<?i 4	Ufin q
is a basis of TfiX at each q G U and, therefore, there is a unique set of smooth
functions {/]....fn} that makes it possible to express the value of a vector
field f at q in the form
t=l	7 q
Expressing each fi in local coordinates, as
provides an expression in local coordinates of the vector field f itself. So. if
p is a point of coordinates (j1!,.... r„) in the chart (U. d). ftp) is a tangent
494 A. Appendix A
vector of coefficients; (A ( X] ....,	......fn (xi ...., xn)) in the natural basis
{ (af?)p’   • ’ (afr)p} TPX induced by (A. o). Most of the times, whenever
possible, the symbol A replaces A 0 £>-1 and the expression of f in local
coordinates is given a form of an n-vector f = col(A...........fn)-
Remark. Let f be a smooth vector field. (L’.d) and (V.v) two coordinate
charts about p and /(x) = /(.rx..... xn), f'(y) = f'(yi, •   -Un) the corre-
sponding expressions of f in local coordinates. Then
Ш =
их

.<
The notion of vector field makes it possible to introduce the concept of
differential equation on a manifold X. For, let f be a smooth vector field. A
smooth curve a : (A-A) —> A is an integral curve of f if
for all t E (ti, A)- The left-hand side is a tangent vector to the submanifold
cr((h,A)) tit the point a(t)‘. the right-hand side is a tangent vector to Ar at
cr(t). As usual, we identify the tangent space to a submanifold of Ar at a point
with a subspace of the tangent space to A’ at this point.
In local coordinates. aft) is expressed as an n-tuple (<T] (t).crn(f)). and
/И0) as
№(A) = 52 A)...........^n(A)
Moreover
, d . AA d(7i / d \
dt '
Therefore, the expression of a in local coordinates is such that
da i . ,	, ..
= A(M0---^(A)
for all 1 < i < n. This shows that, the notion of integral curve of a vector
field corresponds to the notion of solution of a set of n ordinary differential
equations of the first order.
For this reason one often uses the notation
to indicate the image of under the differential cr* at t.
The following theorem contains all relevant information about the prop-
erties of integral curves of vector fields.
A.6 Vector Fields
495
Theorem. Let f be a smooth vector field on a manifold Д'. For each p G .V
there exists an open interval - depending on p and. written Ip of ® such
that 0 G Ip and a smooth mapping
Ф : ГГ -> .V
defined on the subset IF of Ж x .V
П’ = {(t.p) g R x x t g ip}
with the following properties
(i)	Ф(О.р) = p,
(ii)	for each p the mapping : Ip -л A' defined by
ap(t) = Ф{1.р)
is an integral curve of f.
(iii)	if p : (7i. t-fi) —> -V is another integral curve of f satisfying the condition
p(0) = p, then (ti.t-z) C Ip and the restriction of ap to (t-[,t-L) coincides with
/'•
(iv)	Ф($.Ф(кр\) = Ф(я + t.p) whenever both sides are defined.
(v)	whenever Ф(Ьр) is defined, there exists an open neighborhood U of p such
that the mapping Ф} : U —> Л' defined by
ф^д) = Ф(/,д)
is a diffeomorphism onto its image, and
ФГ1 =
Remark. Properties (i) and (ii) say that ap is an integral curve of f passing
through p at t = 0. Property (iii) says that this curve is unique and that the
domain Ip on which &p is defined is maximal. Properties (iv) and (v) say that
the family of mappings {$?} is a one-parameter (namely, the parameter t)
group of local diffeomorphisms. under the operation of composition. <
Example. Let Лг = Ж and use x to denote a point in EL Consider the vector
field
/(T) = m + i)(A)
\ Ox J x
A solution of this equation has the form
ct(C = t.an(t + arotan(j*c))
with r3 being indeed the value of u at t = 0. Clearly, for each .r° the solution
is defined for
496 A. Appendix А
2
< t + arctan(T°) <
2 '
Thus IT is the set
IT = {(t / ) : t e - arctan(.r°). - агегап(тс))}
which has the form indicated in Fig. A.7. <
Fig. A.7.
The mapping Ф is called the flow of f. Often, for practical purposes, the
notation $>t replaces Ф. with the understanding that t is a variable. To stress
the dependence on /. sometimes Фг is written as ф{.
Definition. A vector field f is complete if. for all p £ A’. the interval Ip
coincides with Ft, i.e. - in other words т if the flow Ф of f is defined on the
whole cartesian product К x Ah /
The integral curves of a complete vector field are thus defined, whatever
the initial point p is. for all t e K.
Definition. Let f be a smooth vector field on .V and A a smooth real-valued
function on Ah The derivative of A along f is a function N —> R. written
L/A and defined as
(LfX)(p) = (№))(>)
(i.e. (LjXflp) is the value at X of the tangent vector ftp) at p).
The function LfX is a smooth function. In local coordinates. LfX is rep-
resented by
А.6 Vector Fields 49“
If /1. f2 are vector fields and A a real-valued function, we denote
^/i	(Lf2X) .
The set of all smooth vector fields on a manifold Д’ is denoted by rhe
symbol I'(.V). This set. is a vector space over JR since if f. g are vector fields
and «, b are real numbers, their linear combination af + bg is a vector field
defined by
(«/ + bg}{p) = af(p) + bg{p) .
If a. b are smooth real-valued functions on Д'. one may still define a linear
combination af bg by
(af + bg)(p) = a(p)f(p) + b(p)g(p)
and this gives I'(Д') the structure of a module over the ring, denoted Cx (Лг)-
of all smooth real-valued functions defined on Д'. The set V(.V) can be given,
however, a more interesting algebraic structure in this way.
Definition. .4 vector space Г over 5 is a Lie algebra if in addition to its
vector space structure it is possible to define a binary operation ГхГ->1 .
called a product and written which has the following properties
(i) it is skew commutative, i.e.
[mtc] = -[tc.r]
Iii) it is bilinear over IR. i.e.
[(i|Ci + rejig, tc] = щ [;]. ?e] + O2[t’3, «’]
(where o1; o2 are real numbers)
(iii) it satisfies the so called Jacobi identity, i.e.
+ [w.[z. c]] + [3.[c.m]] = 0 .
The set I'(.V) forms a Lie algebra with the vector space, structure already
discussed and a product [-. -] defined in the following way. If f and g are. vector
fields, [f. g] is a new vector field whose value at p, a tangent vector in Tpi\ ,
maps C'xfp) into JR according to the rule
(lf-g]{p])(M = (LfL^XUp) - (L9LfX}(p).
In other words. [f.g](p) takes A into the real number (LfLaX)(p) — (LgLfX)(p).
Note that, one may write more simply
Lj i у ц A — L у L g X	L g L у A.
Theorem. I‘(Д’) with the product [f,g] thus defined is a Lie algebra.
498 Л- Appendix А
The product [/.(?] is called the Lie bracket of the two vector fields f and
9-
It is not difficult to check that the expression of [f,g] in local coordinates
is given by the n-vector
/
a.r1
In fact.
and
LyL f A
a2 a	a
di-jd-Ti } +	d?( vax^/ i
J	1 = 1
If. in particular. -V = A1 and
/И = Ar
= B.r
then
LMA) = (BA-AB)r.
The matrix [A. B] — (BA - AB) is called the commutator of .4. B.
The importance of the notion of Lie bracket of vector fields is very much
related to its applications in the study of nonlinear control systems. For the
moment, we give hereafter two interesting properties.
Theorem. Let N' be an embedded submanifold of A'. Let. U1 be an open set
of Л’* and f. g two smooth rector fields of -V such that for all p G L '
ftp) e TPA( and g(p) e TPX' .
Then also
[/gW e Tr,.\'
for all p e V.
In other words, the Lie bracket of two vector fields '‘tangent'1 to a fixed
submanifold is still tangent to that submanifold.
A.6 Vector Fields
499
Theorem. Let f. g be two smooth vector fields on .V, Let4>{ denote the flow
of f. For each p e .V
Jim i [&f_t)*g(p{(p)) - g(p)] = [flg](p] 
Remark. The expression under bracket, can be interpreted in the following
way. Take a point p. let q = ф{ (p) be the point uniquely associated with .
p by mapping Ф{ (always defined for sufficiently small t) and consider the
value of the vector field g at q. i.e. g(q). The idea is to compare g(q} with the
value g(p) of the vector field at p. This cannot be done directly, because g(p)
and g(q) belong to different tangent spaces TPN and T^X. Thus, the tangent
vector g(q) e TqX is first taken back to TPX via the differential )* (which
maps the tangent space at q onto the tangent space at p ~	Then the
difference on the left-hand side can be formed and the limit can be taken.
To see that the result indicated in the Theorem holds, observe - using for
instance expressions in local coordinates for all quantities involved - that
ф{(х) = x + f(x)t + P(x. t)
9^+y) = g(x) + ~y + Q(x.y]
= I -	+ R(x.t)
dx
where P(x.t). Q(x,y) and R(x,t) are residual terms satisfying
hm-------- = 0. Inn -—7—7—1 = 0.
Thus.
д(ф{(х)) = g(x) + ^-f(x}t + P’(x.tfi
lint J----------- = I) .
f—»0	f
_Q
t
(Ф^Хд(фЦр)) = g(x) +
Ox	Ox
lim№-j)V0.
(-X) t
As a consequence
Jim j[flPf_tflg(<p{(xfi) -p(x)] =	~
Let f be a smooth vector field on V. g a smooth vector field on .W and
F : X —> .W a smooth function. The vector fields f. g are said to be F-related
if
=goF.
Note that rhe vector field (Ф^_{Ад(ф{(p)) considered in the above Remark is
Ф^-related to g.
500 A. Appendix A
Remark. If f is F-related to f and g is F-related to g. then [/.9] is F-related
to If-9]- <
Remark. The Lie bracket of g and f may be interpreted as the value at t = 0
of the derivative with respect to t of a function defined as
(p)) .
Moreover, it is possible to prove that for any A- > 0
where adkg is the vector field recursively defined by
= 9- adkg = [f.ad^g] .
If IT(0 is analytic in a neighborhood of t = 0. then H'(0 can be expanded
in the form
и= ^2 adkfg(p) —
*=()
known as the Campbell-Baker-Hausdorff formula. <i
One may define an object which dualizes the notion of a vector field.
Definition. Let- X be a smooth manifold of dimension n. A covector field
(also called one-formj jj on X is a mapping assigning to each point p E X
a tangent eovector in TfX. A covector field f is smooth if for each
p E X there exists a coordinate chart (U.X) about p and n real-valued smooth
functions u,']..defined on U. such that, for all q E I
Tire notion of smooth covector field is clearly independent of the coor-
dinates used. The expression of a covector field in local coordinates is often
given the form of a row vector lc = roxxCp..... ~оц) in which the u.-('s are
real-valued functions of Ji..xn.
If is a covector field and f is a vector field. f) denotes the smooth
real-valued function defined by
Ы}[Р) = ЦрЦ(р)) .
With any smooth function A : A’ —> iR one may associate a eovector field
by taking at each p the cotangent vector (dX)p. The covector field thus defined
is usually still represented by the symbol dX. However, the converse is not
always true.
A.6 Vector Fields
501
Definition, A covector field u.’ is exact if there exists a smooth real-valued
function A : .V —> R such that
= dX .
The set of all smooth covector fields on a manifold _V is denoted by the
symbol
In a previous Theorem, the Lie bracket of the vector fields f and g was
interpreted, in a suitable sense, as "’derivative" of g along f. In a similar way.
it. is possible to define the concept of "derivative" a covector field гс along
a vector field f. In order to do this, it is convenient to introduce first, for
covector fields, a notion corresponding to that of F-relation between vector
fields. Let p be a point of the domain of ф{. Recall that (ф{)* : TP.X —>
Тфс1р}Х is a linear mapping and let (Ф^)* : T*f ( Tp.X denote the dual
mapping. With w and ф{ we associate a new covector field whose value at a
point p in the domain of ф{ is defined hy
(ф')МФ'(р))-
The covector field thus defined is said to be Ф^-related to w.
Lemma. Let f be a smooth vector field and a smooth covector field on X.
For each p e A the limit
Jim |	ip)) -
exists.
Definition. The derivative of along f is a covector field on A , written
Lf^j, whose value at p is set equal to the value of the limit
hni )Мф{(р)) - Лр)]-
The expression of Lf^ in local coordinates, which can be deduced by
means of arguments similar to the ones used above in the case of the Lie
bracket [/..g], is given hy the (row) n-vector
	/ dwj	dwn \		/£Л	£A\
	Эху	dx-t		dx\	dxn
/1 /«)		+ (Л’| “ ’	)	xl	ал
	\ cAr,!	dxn /		\ dxy	dxn /
1T df
Ox J dx
where the superscript “T’‘ denotes ‘'transpose".
502 A. Appendix A
Let. л be a smooth covector field and g a smooth vector field. Then, it
makes sense to consider the derivative of the smooth function (л.д) along a
new vector field /.It is possible to show (looking, for instance, at the expres-
sions in local coordinates described above) that the following "Leibniz”-type
rule holds
+ (ьс. [/.<?]) 
В. Appendix В
B.l Center Manifold Theory
Consider a nonlinear system
.r = f(x)
(B.l)
where f is a Cr vector field (r > 2) defined on an open subset I- of FT. and
let j-° e U be a point of equilibrium for f. i.e. a point such that = 0,
Without loss of generality we may assume x: — 0. It is well known that the
(local) asymptotic stability of this point can be determined, to some extent,
by the behavior of the linear approximation of f at x = 0. For. let
R1
F =
dx.
denote the jaeobian matrix of f at x = 0. Then
(i) if all the eigenvalues of F are in the (open) left-half complex plane,
then x = 0 is an asymptotically stable equilibrium of (B.l).
(ii) if one or more eigenvalues of F are in the right-half complex plane,
then .r = 0 is an unstable equilibrium of (B.l).
This important result, is commonly known as the Principle of Stability in
the First Approximation. It is also well understood that this principle does
not completely cover the analysis of the local stability of the equilibrium
x = 0, because nothing can be inferred - in general - about the asymptotic
properties of (B.l) when some eigenvalue of F has zero real part. The case of a
system whose matrix F has some eigenvalue with zero real part is commonly
referred to as a critical case of the asymptotic analysis.
In this section we describe an interesting set of results known as Center
Manifold Theory that in many instances is of great help in analyzing' critical
cases. We begin with some definitions.
Definition. .4 Cr submanifold S of U is said to be locally invariant for
(B.l). if for each 6 S. there exist C < 0 < t? with the property that the
integral curve x(t) of (B.l) satisfying r(0) = xQ is such that x(t) G S for all
t e (c.t2).
504
В. Appendix В
Suppose the matrix F has nc eigenvalues with zero real part. ns eigenval-
ues with negative real part and nu eigenvalues with positive real part. Then,
it is well known from linear algebra that the domain of the linear mapping
F can be decomposed into the direct sum of three invariant subspaces, noted
Ec. Es. Eu (whose dimensions are respectively nr. ns. nu), with the property
that Fl^- has all eigenvalues with zero real part, F|fS has all eigenvalues
with negative real part and F|f,- has all eigenvalues with positive real part.
If the linear mapping F is viewed as a representation of the differential (at
j- = 0) of the nonlinear mapping f :i £ U —> /(x) €	. its domain is the
tangent space to U at x = 0. and the three subspaces in question can
be viewed as subspaces of Tol: satisfying
T0U = E'-1C - E".
Definition. Let x = 0 be an equilibrium of (B.l). T manifold S. passing
through x = 0.. is said to be a center manifold for (B.l) at x = 0, if it is
locally invariant and the tangent space to S at 0 is exactly Ec.
In what follows, we will consider only cases in which the matrix F has
all eigenvalues with nonpositive real part, because these are the only cases
in which j’ = 0 can be a stable equilibrium. In any of these cases, one can
always choose coordinates in U such that the system (B.l) is represented in
the form
у = Ay + g(y,z)
z = Bz + f(y,z)
where .4 is an (ns x ns) matrix having all eigenvalues with negative real part,
В is an (nc x nc) matrix having all eigenvalues with zero real part, and the
functions g and f are Cr functions vanishing at (y.z) = (0,0) together with
all their first order derivatives. In fact, ft suffices to expand the right-hand
side of (B.l) in the form	(
J(t) = Fx + f(x)
where f(x) vanishes at x — 0 together with all its first order derivatives, and
then to reduce F to a block diagonal form
TFZ-‘ = (0 «)
by means of a linear change of coordinates
(0 = TX '
We shall henceforth consider only systems in the form (B.2). Existence of
center manifolds for (B.2) is illustrated in the following statement.
B.l Center Manifold Theory
505
Theorem. There exist a neighborhood V С UtM of z = 0 and a Cr 1 map-
ping a i V —> Bn such that
5 = {[у.г)еЯп1 хС:^7г(г)}
is a center manifold for (B.2).
By definition, a center manifold for the system (B.2) passes through (0,0)
and is tangent to the subset of points whose у coordinate is 0. Thus, the
mapping у; satisfies
дтг
тг(О)=О — (0) - 0 .	(В.З)
Moreover, this manifold is locally invariant for (B.2). and this imposes on
the mapping 7t a constraint that can be easily deduced in the following way.
Let (y(t).z(t)) be a solution curve of (B.2) and suppose this curve belongs
to the manifold S. i.e. is such that y(t) = 7r(z(t)). Differentiating this with
respect to time we obtain the relation
+ 9(-(z(f)). i(t)) =	.-(*)) 
dt	dz dt
Since a relation of this type must be satisfied for any solution curve of (B.2)
contained in S. we conclude that the mapping тг satisfies the partial differ-
ential equation
Q~
+ /(MM-M) = МММ +	•
Remark. Consider, instead of (B.2).a system of the form
У = Ay + Pz + g(y.z)
i = Bz + f(y.z)
where .4 is an (ns x ns) matrix having all eigenvalues with negative real part.
В is an (nc x nc) matrix having all eigenvalues with zero real part. Suppose
77 : Г —>	is a mapping satisfying 7t(0) — 0. The submanifold
S-{(.y,z)er? x I' : у = тг(с)}
is locally invariant for (B.5) if the mapping тг satisfies the partial differential
equation
^(Bz + /(-(г), г)) = -4тг(г) + Pz + 9(тг(г). г).	(В.6)
Comparing the first order terms on both sides, it is seen that the matrix
(Этт t 4
(B.4)
(B.5)
satisfies
506 В. Appendix В
А Р\ (П\
0 В J \1 ) ~ \1 )
from which it is deduced that
Im(
) = EC .
Thus, in view of the definition given above, it is concluded that S is a center
manifold for (B.5) if and only if (B.6) holds. <
The previous statement describes the existence, but not the uniqueness
of a center manifold for (B.2). As a matter of fact, a system may have many
center manifolds, as the following example shows.
Example. Consider the system
У = -у
z = — -3
The function у = tt(z) defined as
тг(г) = <?exp(-|z-2) if z 0
7r(z) =0	‘ if z = 0
is a center manifold for every value of R. <
Note also that if g and / are f’v functions, the system (B.2) has a Ck
center manifold for any k > 1. but not necessarily a center manifold.
Lemma. Suppose у = z(z) is a center manifold for (B.2) at (0.0). Let
be a solution of (B.2). There exist a neighborhood L:° of (0.0) and
real numbers M > 0, К > 0 such that, if (t/(0), z(0)) G Uc, then
II M - TrHt)) |l< Me-lKt || y(0) - 7T(z(0)) II
for all f > 0, so long as (y(t\z(t)) G L’°.
This Lemma shows that any trajectory of the system (B.2) starting at
a point sufficiently close to (0.0). i.e. close to the point at which the center
manifold has been defined, converges to the center manifold as t tends to oc,
with exponential decay (Fig. B.l). In particular, this shows that if (t/°.z°) is
an equilibrium point of (B.2) sufficiently close to (0.0). then this point must
belong to any center manifold for (B.2) passing through (0.0). In fact, in this
case the solution curve of (B.2) satisfying ((/(O).z(O)) = (ya-zQ) is such that
y(t) = У° z(f) = for all t > 0
and this is compatible with the estimate given by the Lemma only if =
7r(z°). For the same reasons, if Г is a periodic orbit of (B.2) all contained
in a sufficiently’ small neighborhood of (0.0). then Г must lie on any center
B.l Center Manifold Theory 507
Fig. B.l.
manifold for (B.2) at (0,0). Thus, despite of the non uniqueness of center
manifolds, there are points that must always belong to any center manifold.
The following theorem provides a more detailed picture about the role
of the center manifold in the analysis of the asymptotic properties of the
system (B.2) near (0,0). Recall that, by definition, if (,y(0).2(0)) is any initial
condition on the center manifold у = ~(c). then necessarily y(t) = ~(z(t)) for
all t in a neighborhood of t = 0. As a consequence, any trajectory of (B.2)
starting at a point yc = тг(г°) of this center manifold can be described in the
form
i/(t) = 7r(cu))	= at)
where ((t) is the solution of the differential equation
С = ВС + /(7Г(О,0	(B.7)
satisfying the initial condition £(0) = 2°. The essence of the following results
is that the asymptotic behavior of (B.2) - for small initial conditions - is
completely determined by its behavior for initial conditions on the center
manifold, i.e. by the asymptotic behavior of (B.7).
Theorem. (Reduction principle). Suppose £ =0 is a stable (resp. asymptot-
ically stable, unstable) equilibrium of (ВЦ)- Then (y.z) = (0.0) is a stable
(resp. asymptotically stable, unstable) equilibrium of (B.2).
Example. As an immediate application of the reduction principle to the anal-
ysis of critical cases, consider a system of the form (B.2). with g such that
3(0,2) -0.
In this case, the center manifold equation (B.4) is trivially solved by тг(г) = 0.
and the reduction principle establishes that the stability properties of (B.2)
at (0.0) can be completely determined from those of the reduced system
508 В. Appendix В
< = в< + /(0. <) .<
This Theorem is rather important, for it reduces the stability analysis
of an n-dimensional system to that of a lower dimensional (namely. n‘-
dimensional) one. but its practical application requires solving the center
manifold equation, and this in cases other than the one illustrated in the
previous example - is in general quite difficult. It is however always possible
to approximate the solution у = тг(з) of the equation (B.4) to any required
degree of accuracy and, then, to use the approximate solution thus found in
the reduced equation (B.7). In this way. one may still be able to determine
the asymptotic properties of the equilibrium £= 0 of (B.7).
Theorem. Let у = тгд-U) be a polynomial of degree k. 1 < k < r. satisfying
= o ~m = o
dz
and suppose
n
+ f(nk(z). z)) — .4тгд.(с) — g(~k(с), з) = Ht(c)
oz
where Rk(z) is some (possibly unknown) function vanishing atO together with
all partial derivatives of order less than or equal to k. Then, any solution тг(з)
of the center manifold equation (B.f) is such that the difference
Dk(z) = ^z)--k(z)
vanishes at 0 together with all partial derivatives of order less than or equal
to k.
The practical application of this result, is illustrated in the following ex-
amples. In all of them, the reduced equation (B.7) is 1-dimensional, and its
stability can be easily determined on the basis of the following property.
Proposition. Consider the one-dimensional system
j =	+ Qm(.r)
with m > 2. and Qm(x) a function vanishing at 0 together with all derivatives
of order less than or equal to m. The point of equilibrium x = 0 is asymptot-
ically stable if m is odd and a < 0. The equilibrium is unstable if m is odd
and a >0, or if m is even.
Example. Consider the system
У - -у + ~'2
z = azy .
The center manifold equation (B.4) is in this case
B.l Center Manifold Theory
509
|^-(агтг(г)) = -tt(z) + z2.
The simplest approximation we may try for тг(г) is a polynomial of the second
order, namely тггС-г) = <az2, where a must be such as to satisfy the center
manifold equation at least up to terms of order 2. This yields
-—(az-^z)') ~ (-7m(z) + zL) = (a - 1)з' + глсгг1.
Setting a = l we obtain, on the right-hand side of this expression, a remainder
R;ilz} that vanishes at 0 together with all derivatives of order less than or
equal to 3. We may thus set
= z2 + D3(z)
where D-^z) is some unspecified function of z (vanishing at 0 together with
all derivatives of order less than or equal to 3). Replacing tt(z) in (B.7), we
obtain
<, = «С3 + Q\ (0
where Q-i(O is an unknown remainder, vanishing at 0 together with all the
derivatives of order less than or equal to 4. On the basis of the previous
Proposition we deduce that the equilibrium £ = 0 of the reduced equation
(B.7) is asymptotically stable if and only if a < 0. At this point, on the
basis of the Reduction Principle, we can conclude that the full system is
asymptotically stable at the equilibrium (y. z) = (0, 0) if and only if a < O.<
Example. Consider the system
у = -y + y~~z3
z = az2 + zJT’y
where m is any positive integer. The center manifold equation (B.4) is in this
case
— (rtc3 4- Cm-(y)) = — 7t(z) + 772(z) -
Again, we start trying for тг(г) an approximation of the second order, namely
7Г2(г) = az2. with a such as to satisfy the center manifold equation at least
up to terms of order 2. However, since.
д_
- (-7T2(Z) + 772(Z) - Z3) = Q? + R^z)
we deduce that necessarily о = 0. Thus an approximation of the second order
is meaningless, and we have to try with a polynomial of the third order . We
set ттз(-г) = ,.3z3 because we already know that the coefficient of z2 must be
zero. In this case we have
^(аг3 + _-"V3(.-)) - (-7ГЗ(C +	- г3) = О + 1 )г3 + Я3( = )
510
В. Appendix В
and the center manifold equation (B.4) will be satisfied up to terms of order
3 if J = -1, Thus we may set
тг(.') = -C + D3(z) .
Replacing ~(^) in (B.7), we obtain
С = a? - C"-3 + em+;i(O
where Q»i+3(O is an unknown remainder, vanishing at 0 together with all
the derivatives of order less than or equal to m 4- 3. Since m > 1. we can
invoke the previous Proposition and conclude that, for any m, the equation
(B.7) is asymptotically stable at £ — 0 if and only if a < 0. As a consequence
of the Reduction Principle, this is true also for the equilibrium (y. z) — (0, 0)
of the full system. <
Example. Consider the system
у = -у + ayz + bz2
z = cyz - Л
Again, we try first an approximation for тг(г) of the form тг?(sA = Qc2. In
this case we find
^(сгтрДг) - г3) - (-тг2(-) А пгтг2(г) + bz2) = (a - b)z2 + R->{z)
Oz
and therefore a — b. Replacing
7T(O = b(2 + DM
in the equation (B.7), we obtain
< = (cb -1)<7+ QM
Again, on the basis of the previous Proposition, we can conclude that the
reduced equation - and so the full system - is asymptotically stable if
(cb — 1) < 0, and unstable if (eb — 1) > 0. If cb = 1. the right-hand side
of this equation is totally unspecified, and thus we have to find a better ap-
proximation for the center manifold. Choosing тгз(с) = bz2 + Jz3. we find
now
^-(cz^z) - г3) - (-тг3(г) + az-3(z) + bz2) = (J - ab)z3 + R3(z)
Oz
and so 3 = ab. Replacing
7Г( <) = Ц'2 + abf3 + D-M
in (B.7) we obtain (assuming cb = 1)
C = «C4pQ4(<)
B.2 Some Useful Properties 511
and we can conclude that the system is unstable if a 0. If a = 0 we don't
know yet, because the right-hand side of this equation is unspecified Thus,
the only case left is the one in which cb = 1 and a = 0. In this particular-
sit nation. however, the center manifold equation (B.4) is satisfied exactly by
the function тгС) = bz2 and the reduced system is then
< = 0 .
Its equilibrium < = 0 is stable (not asymptotically) and so is the equilibrium
of the full system. <
Example, Consider the system
у = az + u(y}
z = -z'2 +byzm
where m > 0. and u(y) represents a feedback, depending on the state variable
у only. Choose
u(y) = -Ky
and show that, the equilibrium (у, г) = (0,0) of the full system
-	if m = 0 and ab < 0. is asymptotically stable for all values of К > 0.
-	if m = 1. is always unstable.
-	if m = 2. is asymptotically stable for all values of К > max(0. ab).
-	if m > 3. is asymptotically stable for all values of К > 0.
Show that these conclusions remain unchanged if
’Hj/J = -Ky + f(y)
where f{y) is a function of у vanishing at 0 together with its first derivative.<
B.2 Some Useful Properties
We present in this section some interesting results about the asymptotic prop-
erties of certain nonlinear systems, that, arc used several times throughout
the text.
Lemma. Consider a system
(B.8)
у = Ay + p(z.y)
and suppose that p(z.Q} = 0 for all z near 0 and
y'(O.O) = 0.
Oy
If z — f(z.O) has an asymptotically stable equilibrium at z = 0 and the
eigenvalues of A all have negative real part, then the system (B.8) has an
asymptotically stable equilibrium at (z.y) = (0.0).
512 В, Appendix В
Proof. Expand f(z.y) as
f(z.y) = Fz + Gy + g(z.y).
Using a linear change of coordinates hi.z-j) = Tz + Fy it is possible to
rewrite the system (B.8) in the form
ii = Fi~i + <7i (-1, z-z-y)
Z-2 = F2Z2 + G>y +	y}
if = Ay +p(zi-Z2-y)
with F> having all the eigenvalues with negative real part and Fj having all
the eigenvalues with zero real part. Moreover, the functions g}. g2 vanish at
(0.0,0) together with their first-order partial derivatives.
By assumption, the equilibrium (0.0) of
= Fi;i+91(3,.i2,0)	(B9)
z2 — F2z2 + g-2(z^, z2.0)
is asymptotically stable. Let co = trohi) be a center manifold for (B.9) at
(0.0). By assumption, ~2 satisfies
(Fi 4- Qi hi, тг-2(<;]). 0)) = F2772(^1) + 92 hi • ~i )• 0)
and then - by the reduction principle the reduced dynamics
j- = FXX + y} (j, ТГ-2 (-r), 0)
has necessarily an asymptotically stable equilibrium at x = 0. Consider now
the full system (B.8). A center manifold for this system is a pair
Гдтг2
(hi
such that
Г 9k2 1 ,.
*	/ I
(hi (и + 9i (3i. Ay(~~i), Ay (zi)))	— -Hihi) + Phi-^shi)
A trivial calculation shows that these actuations are solved by
As a consequence, using again the reduction principle, we see that the dy-
namics (B.8) has an asymptotically stable equilibrium at (0,0) if the reduced
dynamics
has. But this reduced dynamics is exactly the reduced dynamics of (B.9) and
the claim follows. <
B.2 Some Useful Properties
513
Remark. We stress that the result of this Lemma requires, for the dynamics
of
z =
just asymptotic stability, anti not necessarily asymptotic stability in the first
approximation, i.e. a jaeobian matrix
'df(z.oy
. Lo
having all the eigenvalues in the open left-half plane. <
In a similar way. one can prove the following result.
Lemma. Consider a system
z ~
J	(B.10)
У = Pty)
and suppose that у = p(z) has an asymptotically stable equilibrium at у = 0.
If z = f(z.O) has an asymptotically stable equilibrium at. 2 =0. then the
system (B.10) has an asymptotically stable equilibrium at (z.y) — (0.0).
In the next Lemma the asymptotic properties of a time-varying system are
illustrated. To this end. recall that the equilibrium ,r = 0 of a time-varying
system
x=f(x.t]	(B.ll)
is said to be uniformly stable if. for all s > 0. there exist a <5 > 0 (possibly
dependent on г but independent of C) such that
j| ,r“' ||< <5 =>|| :r(f.	|| < e for all t > ta > 0
where /(M’.r5) denotes the solution of (B23) satisfying x(C. f~. x") = /.
The equilibrium т = 0 of (B.ll) is said to be uniformly asymptotically stable.
if it is uniformly stable and. in addition, there exist у > 0 and. for all M > 0.
а Г > 0 (possibly dependent on 31 but independent of .r3 and f°) such that
II xc ||< у =>|| j-(t.r.Z) j|< M for all t > С +T. C > 0 .
Lemma. Consider the system
x = f (x, t) 4- p(x. t) .	(В.Г2)
Suppose the equilibrium x ~ 0 of x = f(x.t) is uniformly asymptotically
stable. Suppose f(x.t) is locally Lipschitzian in x. uniformly with respect to
t. i.e. there exists L (independent oft) such that
514
В. Appendix В
for all r(. x,f in a neighborhood of x = 0 and all t > 0. Then, for all s > 0.
there exist <5i > 0 and dA > 0 (both dj and dA possibly depend on £ but are
independent of t~) such that, if || /’ ||< and |l p(r.f) ||< for all (x.t)
such that || r ||< s and t > tc', the solution x(t. tc’. x°) of (B.12) satisfies
|| x(t,tz .x°) || < £ for all t > t° > 0 .
The property expressed by this statement is sometimes referred to as total
stability, or stability under persistent disturbances. Note that the function
p(x. t) need not to be zero for x = 0, From this Lemma it is easy to deduce
some applications of interest for systems in triangular form.
Corollary. Consider the system
z - q(z.y.t)
(B.13)
у = gly) 
Suppose
(i) (r.y) = (0,0) is an equilibrium of (B.13). and the function q(z.y,t) is
locally Lipschitzian in (z.y). uniformly with respect to t. i.e. there exists L
(independent oft) such that
II q(C.yr.t)-q(z".yTt) ||<£(!|?-;" || -r || y’ - y" ||)
for all z1, z" in a neighborhood of z = 0, all y' ,y” in a neighborhood of у = 0.
and all t > 0.
(ii) the equilibrium z — 0 of z ~ q(z.O.t) is uniformly asymptotically stable,
(iii) the equilibrium у — 0 of у = g(y ) is stable.
Then the equilibrium (z,y) — (0.0) of (B.13) is uniformly stable.
Proof. It is a simple consequence of the previous Lemma. For. set
mw =
p(z.t) = q(z.y(t).t) - qCCht)
where y(t) is the solution of у = g(y) satisfying y(tQ) =	. Thus the first
equation of (B.13) has the form (В.Г2). Note that if || y(t) ||< zy for all t > C.
then, by assumption (i). p(z.t) satisfies
!| pC-C II = || g(z.y(t),t) - q(z.O.t) ||< Ley
for all z in a neighborhood of z — 0 and all t > t°. By assumption (ii) and
the previous Lemma, for all > 0. there exists <5i > 0 and dA > 0. such that
|| 2° ||< d'i and || p(z,t) ||< dA. for all (z.f) such that || z ||< = . and t > tc.
imply
|| s(t,fc.F) ||< for all t > tc > 0 ,
By assumption (iii) one can find 6y such that ||	||< dy implies || y(t) ||<
<P>jL for all t > tc. and this completes the proof. <
B.2 Some Useful Properties 515
Remark. This result has an obvious counterpart in the study of the stability
of (nonequilibrium) solutions of a differential equation. To this end. recall
that a solution T*(f) (defined for all t > (J) of a differential equation of the
form (B.ll) is said to be uniformly stable if. for all s > 0. there exists art>0
(possibly dependent on .5 but independent of U) such that
|| ;rc - r*(F) ||< 6 =^|| r(C t°, x°) - j-* (f) || < £ for all t > F > 0
where x(f.F,J,Q) denotes the solution of (B.ll) satisfying .r(t°. F. zc) = rU
The solution x*(t) of (B.ll) is said to be uniformly asymptotically stable, if
it is uniformly stable and. in addition, there exist у > 0 and. for all .V > 0.
a T > 0 (possihly dependent, on Af but independent of C and F) such that
|i тс -r(F) ||< у =>|| i-(t.t\rz) -T*(f) ||< -W for all t > F +F.F > 0.<
The study of the stability of a solution x*(t) of (B.ll) can be reduced to
the study of the stability of the equilibrium of a suitable differential equation.
For. it suffices to set
1Г = ,r - .7'*
and discuss the stability of the equilibrium w = 0 of
F = /(tr +	- /(T*(t). t) .
Thus, the previous Corollary is helpful also in determining the uniform
stability of some nonequilibrium solution of equations of the form (B.13). For
instance, suppose z = q(z. 0. t) has a uniformly asymptotically stable solution
z*(t). defined for all t. Set
F(u-.y.t) = g(w + £*(*). t/.f) - q(z*(f).O.f) .
Then w - 0 is a uniformly asymptotically stable equilibrium of tr =
F(u\0.f). If q(z,y,t) is locally Lipschitzian in {z.y). uniformly in t. so is
F(w.y.t), provided that s*(f) is sufficiently small for all t > 0. Assumptions
(i), (ii) and (iii) are satisfied, and it is possible to conclude that the solution
(z’(t).O) of (B.13) is uniformly stable.
If the system (B.13) is time invariant, then the result of the previous
Corollary can be expressed in a simpler form.
Corollary. Consider the. system
у = <M -
(B.14)
Suppose (z.y) = (0.0) is an equilibrium of (В.Ц), the equilibrium z — 0
of z = y(c,0) is asymptotically stable, the equilibrium у = 0 of у = g(y) is
stable. Then the equilibrium (z.y) = (0,0) of (В.Ц) is stable.
516
В. Appendix В
Another interesting application of the previous Lemma is the one de-
scribed in the following statement.
Corollary. Consider the system
x = f(x) +	•	(B.15)
Suppose .r = 0 is an asymptotically stable equilibrium of Jr = fix'). Then, for
all г > 0 there exist > 0 and К > 0 such that, if |l x° l|< di and p/.(t)| < К
for all t > t°. the solution x(t.tD.,rc) of (B.15) satisfies
|| ,r(t. t°. j’° ) || < 5 for all t > t° > 0
Proof. . Since g{x) is smooth, there exists a real number .M > 0 such that
J g{x) || < M for all j such that |[ x ||< s. Choosing К ~ 5-y/M yields
|| g(x)u(t) ]|< d'_> and the result follows from the Lemma.
In concluding the section, we summarize also a few important concepts
and results about some "global’’ asymptotic properties of the trajectories of
a system of the form (B.l). which we now assume defined for all ,r € R”.
Recall that a smooth function V : R” —> R is said to be positive definite if
V(0) = 0 and Г(т) > 0 for x 0. and proper if, for any а e К the set
- {.r e R*1 : 0 < V(z) < a} is compact.
The first two Theorems are the well known criterion of Lyapunov for
global asymptotic stability and its converse.
Theorem. (Direct Lyapunov theorem) Consider a. system of the form
x = fix)
in which x £ Rf<. and f(x) is a smooth function, with /(0) = 0. If there exists
a positive definite and proper smooth function V(r) such that
for all nonzero x. then the equilibrium .r = 0 of the system is globally asymp-
totically stable.
Theorem. (Converse Lyapunov theorem) Consider a system of the form
x = f(x)
in which x € R,!, and f(x) is a function which is smooth on . Rn \ {0} and
continuous at x = 0. with /(0) = 0. If the equilibrium x = 0 of the system is
globally asymptotically stable, then there exists a positive definite and proper
smooth function V(z) such that
for all nonzero x.
В.З Local Geometric Theory of Singular Perturbations
517
The next result illustrates an interesting global property of any smooth
manifold on which it is possible to define a globally asymptotically stable
vector field.
Theorem. (Milnor) Consider a system of the form
I = f(.r)
in which .r E -V. an 71-dimensional smooth manifold, and f(r') is a smooth
vector field. If the equilibrium .r = 0 of the system is globally asymptotically
stable, then M is globally diffeomorphic to ET .
Finally, we recall the dt'hi lit ion of w-limit set of a trajectory and some of
its properties. Let .гДС denote an integral curve of system (B.l). which is
assumed to be defined for all 0 < t < x. A point .r is said to be a ~dimit
point of rz(t) if there exists an increasing sequence of values of t
0 < fi < t-z < ... < tj,- < •   • ^lim t/,. = x ,
such that
lim = .f .
The set of all л-limit points of ^"(f) is the ai-limit set of ,rc(fb
Theorem. (G.D.Birkhoff) Suppose ft) is a bounded trajectory. Its л-limit
set IC is nonempty, closed and invariant unde?' the flow of (B.l).
B.3 Local Geometric Theory of Singular Perturbations
Consider a system of differential equations of the form
" =	(BIG,
i = f(y.z.s)
with ly. z) defined on an open subset of НТ x EL. and s a small positive real
parameter. A system of this type is called a singularly perturbed system.. In
fact, at s = 0. this system degenerates to a set of only p differential equations
z =f(y.z.O)	(B.17)
subject to a constraint of the form
0 = gig- z.0).	fB. 18)
Let. К denote the set of solution points of the equation (B. 18). and suppose
rank — I = n
\vyj
518 В. Appendix В
at some point (y^.z'^) of K. By tlic implicit function Theorem, there exist
neighborhoods A ° of and Bc of yc and a unique smooth mapping h :
A3 ->	. such that g(h(z). c.O) =0 for all z G A3. Therefore, locally around
(y~. :°) the degenerate system (B.17)-(B.18) is equivalent to a dimension al
differential system defined on the graph of the mapping h. i.e. on the set
S = {[y. z) E BQ x Ac : у = h(z)}
and thereby represented by the equation
z=f(h(z).z.O) .	(B.19)
This system is called the reduced system.
Note that, after a change of variables
?/ - у - h(z)
rhe set S can be identified as the set of pairs (ug г) such that tc — 0. In the
new variables (B.16) is represented by the system
sd- = g(w + h(z').z,3) -	+ h(z). z..s) = g^d'.z.e)
z = f(w /i(z),z.e) =	z.s) .
Since 9o(0. c,0) — 0 by construction, the reduced system is now described by
з -/с(0.г.0) .
The form of the singularity of (B.16) suggests also a change of variable
in the time axis, namely the replacement of t by a "rescaled" time variable
t defined as
7 = t/q.
Since Asa small number, the variables t and т are usually referred to as
the “slow" time and "fast" time. Moreover, to indicate differentiation with
respect to t, the superscript is used.
The substitution of t by t. together with that of у by w. yield a system
of the form
in which, since </o(0. c,0) = 0. any point (0. c) (i.e. each point of the set S) is
an equilibrium point at = ~ 0. Note that the behavior of the system (B.20) at
s = 0 is characterized by the family of //-dimensional differential equations
w’ = g.~J,w, z. 0)	(B.21)
(in which z can be regarded as a constant parameter).
В.З Local Geometric Theory of Singular Perturbations
519
The two equations (B.19) and (B.21) (which, it must be stressed, are
defined on two different time axis) represent in sonic sense two kinds of “ex-
treme"' behaviors associated with the original system (B.16). The purpose of
the singular perturbation theory is the study of the behavior of a singularly
perturbed system for small (nonzero) values of = and, if possible, to infer its
asymptotic properties from the knowledge of the asymptotic behavior of the
two “limit"1 systems (B.19) and (B.21).
Before proceeding further, it is convenient to observe that a system of the
form (B.16) is a particular case of a more general class of systems that, can
bo characterized in a coordinate-free manner, without explicitly asking for a
separation of the variables into groups z and t/. For. consider a system
У = F(.r.s)	(B.22)
with j defined on an open set 17 of Ji" . г a “parameter1, ranging on an interval
(— sy. ) of R and F : I' x (—sy. +sc) —> Rn a Cr mapping. Suppose also
there exists a p-dimensional (with p < n) submanifold E of Г consisting
entirely of equilibrium points of r' = F(r.O), i.e. such that
F(x. 0) = 0 for all x G E .
The class of systems thus defined contains as a special case the system
(B.16). with time rescaled; in fact, the set S is exactly a p-dimensional sub-
manifold of Rr x R/J consisting of equilibrium points of the rescaled system
at у = 0. In view of this fact, we shall henceforth proceed with the study of
the more general class of systems (B.22).
Let
_ raF(.r.of
J “ dx
denote the Jacobian matrix of F(z.O) at a point x of E. It is easy to verify
that the tangent space TTE to E at x is contained in the kernel of this matrix.
For. let (7 : Ж E be a smooth curve such that a(0) = x and note that,
since every point of E annihilates F(t.O). by definition F(a(f).O) — 0 for all
t. Differentiating this with respect to time yields
~<9F(.r,0)~
dx
d(t)
= 0 .
At t — 0. we have Уа(0) = 0. and this, in view of the arbitrariness of o.
proves that TXE С кег(Л)-
From this property, we deduce that 0 is an eigenvalue of with multi-
plicity at least, p. The p eigenvalues of Л associated with the eigenvectors
which span the subspace TXE are called the trivial eigenvalues of J£. whereas
the remaining n — p are called the nontrivial eigenvalues.
From now on, we assume that all the nontrivial eigenvalues of Jx have
negative real part. As a consequence, the two sets of trivial and nontrivial
520 В. Appendix В
eigenvalues аге disjoint sets and, from linear algebra, it. follows that, there
exists a unique subspace of TXU, noted Uj.. which is invariant under Jx and
complementary to TXE. i.e. such that.
Tcu = TTEpVT .
As a matter of fact, is exactly the subspace of Тх1: spanned by the eigen-
vectors associated with the nontrivial eigenvalues of Jx. Let Px denote the
projection of TXU onto TXE along Vr. i.e. the unique linear mapping satisfy-
ing
кег(Р^) = Vj. and Im(PT) = TXE .
We use Px to define a vector field on E. Namely, we set.
fR : z e E ->	= Л
de
This vector field is called the reduced vector field of the system (B.22). Note
that this definition agrees with the one given at the beginning. As a matter
of fact, if the system is in the special form (B.20). with gjO.c.O) = 0. the
jacobian matrix has the form
and its nontrivial eigenvalues arc those of G. The subspace Tj-E is the set of
all vectors whose first c coordinates arc zero, the subspace I j. is the set of
all vectors whose last ft coordinates are zero. and Px is described, in matrix
form, as
Л = ( 0 / ) 
Them it is clear that
The following statement describes conditions under which the local asymp-
totic behavior of the system (B.22) for small nonzero c, in particular its
asymptotic stability at some equilibrium point, can be described in terms of
properties of the two "limit" systems
F(.r.O)
(B.23)
fiM r G E.
(B.24)
В.З Local Geometric Theory of Singular Perturbations
521
Theorem. Let E3 be a subset of E suck that, for all j? € E°. the nontrivial
eigenvalues of have negative, real part.. Suppose x'3 & Ez is an equilibrium
point of the reduced system (B.2J) and suppose that all the eigenvalues of the
jacobian matrix
have negative real part. Then there exists f-, > 0 such that, for each 5 £ (0,c = )
the system (B.22) has an equilibrium point r.: near x°. with the following
properties
(i) ,r_* is the unique equilibrium of x1 = E(x.s) contained in a suitable neigh-
borhood of the point x°.
(ii) is an asymptotically stable equilibrium of xr = F(x.e).
Deferring the proof for a moment, we show first that by means of suitable
(local) changes of coordinates, a system of the form (B.22). satisfying the
assumptions of the previous Theorem, can be put into a form that closely
resembles the one considered at the beginning, namely the form (B.20). To
this end. begin by choosing local coordinates (£,//) on E in such a way that
E is represented, locally around J’1' , in the form E = {(£,;/) : £ = 0}. anti
x3' = (0.0). Accordingly, the system (B.22) is represented in the form
=	9^-9-G
n' = f(Efl^) 
By construction, since E consists of points of equilibrium for E(j-.O).
<7(0,77.0) = 0
/(O.ryO) = 0
for all ip Thus, it is possible to expand f and g near (0,0.0) in the form
= G^ + 9^ +
fiCk-G = F{ + fe+f>if.ihS)
where f2 and g2 vanish at (0.0. 0) together with their first order derivatives,
and /2(0. r/.O) = 0. </2(0. г/. 0) = 0 for all rp By construction, the eigenvalues
of G are the nontrivial eigenvalues of Л at x = xc.
The equations of this system can be simplified by means of the Center
Manifold Theory. For. consider the "extended" system
£'	=	Gt, + д~л	+	g-y(f. tpp}
9’	=	Ft + f^	+	f^-T^
F	=	0
and note that after a linear change of variables
522 В. Appendix В
,V = £ + As
z r} +
(with К = -FG~[ and A = G-1yo) the system in question can be rewritten
in the form
y' = Gy + q(y.z.G)
( z‘\ = /О /Л Zz\	/p(y.z.s)\
UJ \° ° / V /	\ 0 /
By construction, q and p vanish at {(), 0,0) together with their first derivatives,
and q(0. 2.0) — 0, p(0. c.O) = 0. Note that, in the new coordinates, points of
the set E correspond to points having у = 0.
Choose now a center manifold у — тг(с. =) for this system at (0.0,0).
and note that points of the form (O.c.O) - being equilibrium points of the
extended system - if z is sufficiently small belong to any center manifold
through (0.0.0). Therefore
0- 77(2,0)
for small 2. After a new change of variables
u- у - 77(2,5)
the extended system becomes
tF	—	a(u\ z. 5)
z'	=	b(u:.z,s)	(B.25)
s'	=	0
and by construction
a(0,z.s) = G7t(3,s) + 9(77(2.5). z. 5) - $f(b -г р(тг(2.е). >-£)) = 0
6(0.2.0) = р(0.г.0) =0	'
for small (2.5). Note that, in the new coordinates, the center manifold is
the set of points having it — 0, and that b(w.z.s) can be represented in the
following way
f d
b(io.z.s) =	/ —b(ciic. г. as) da
°ya	f1
= г I (aw. г, as) da + / bu (aw. z.ae) da  w
Jo	Jo
= sfa(u\z^) + Fi(w,z.e> .
We can therefore conclude that, choosing suitable local coordinates, the
system (B.22) can be put in the form
w' =z 2,5)
2'	= cfo(w.2,s) + Fi(tc.2.s)w
В.З Local Geometric Theory of Singular Perturbations 523
with a(0. :,£) = 0-
Note that we don't have used yet the assumption on the jacobian matrix
Af. We can now proceed with to the proof of the Theorem.
Proof. Suppose that /o(0.0.0) = 0 and that the jacobian matrix
(B.26)
‘dfc(O,z.O)
has all the eigenvalues in the left-half plane (we will show at the end that
this is implied by the corresponding assumption on the reduced vector field
ffi). Then, if s is sufficiently small, the equation /o(0,z.£) = 0 has a root
z, for each £ near 0 (with zc =0). The point (0. z5) is an equilibrium of the
system (B.25)
iF — a(w. z. c)
z' = zf0(u,\z.A + Fi(w,z.s)w .
Observe that (0,zs) is the point jy whose original coordinates are
= -(2,.c)-Ae
= z? - A'-(Zs.c) + KXz .
Clearly, .ty is the unique possible equilibrium point of F(z.f) for small
fixed £ in a neighborhood of the point j:3 (i.e. of the equilibrium point of the
reduced system). In fact, suppose jq is another equilibrium point of the sys-
tem (B.22) close to . Then the point (ti . г) (an equilibrium of the extended
system) must belong to any center manifold (for the extended system) pass-
ing through (t°,0). Since in the coordinates (m.z.c) the set of points having
ir — 0 describes exactly a center manifold of this type, we deduce that in
these coordinates the point (xi.s) must be represented in the form (0, zi.s).
Being two equilibria, the two points (O.zi.c) and (0, zs, s) must satisfy
6(0, 3i. e) — 6(0. z?.f)
i.e
/o(0. 3i.e) = /Д0.
but this, if £ is sufficiently small, implies zl = z.~, in view of the nonsingularity
of the matrix (B.26). This, in turn, implies
Since all the eigenvalues of the matrix (B.26) have negative real part, also
those of
6/o(u'.z,s)
dz
all have negative real part for small s. Moreover, since
z,0)1	= G
dw u-=o
L	J г=0
524 В. Appendix В
also the matrix
z. s’)
dir
lias all eigenvalues with negative real part for small s.
If s is positive, the equilibrium (0. cf) of (B.251 is asymptotically stable.
In fact, the jacobian matrix of the right-hand side, evaluated at this point,
has the form
and all the eigenvalues in the left-half plane.
In order to complete the proof we have to show that the matrix (B.26)
has all the eigenvalues in the left half plane. To this end. recall that in the
(io. z. sj coordinates, points of the set E correspond to points having w = 0
and s = 0. Note that
' д д'
F<f-;>57a?
~0F(x^Y
dz
5^0
and that the right-hand side of this expression, in the ( tc. c
г) coordinates,
becomes
0 д 1	d
6uc.c..-.)-.-	= -/o(0. z. 0)--
dz de ;-=o	dz
Thus, it is easily deduced that the tangent vector
Л(о.2.о)|
represents exactly the vector field jr at t/ie point (0. c.0) of E. From this,
the conclusion follows immediately. <
Remark. Note that, for a system given in the simplified form (B.16). the
previous Theorem establishes that if a point	satisfies
9<Mzy.	=	0
= 0.
the system
и/ = g(m + /ф°).2°.0)
is asymptotically stahle in the first approximation at tc = 0. and the reduced
svstem
~= f(h(z).z.(])
is asymptotically stable in the first approximation at c = z°. then for each
sufficiently small e > 0, there exists an equilibrium of (B.16) near (h(c=). г°)
which is asymptotically stable in the first approximation. <
В.З Local Geometric Theory of Singular Perturbations
525
Remark. Observe that, in a sufficiently small neighborhood of 1/. 0). the
equilibrium [joints of the extended system
x' = F(x.e)
,	(B.27)
s' = 0
are only those of the set E*. and those belonging to the graph of the function
f : £ —> x.-. <
Fig. B.2.
Fig B.2 illustrates some of the ingredients introduced in the previous
discussion, in the particular case of a system having n = 2. p = 1. In the
(у. г. s) coordinates, it shows the set E°. the center manifold S of the extended
system, and the location of the equilibrium points (xr.£). The trajectories
of the extended system are contained in planes parallel to the (y.z) plane
and. on each of these planes, obviously coincide with those of the system
(B.22) for a specific value of £. Note that, since the center manifold S is by
definition an invariant manifold, the intersections of S with planes parallel to
the (у, г) plane are invariant manifolds of (B.22) for the corresponding value
of г : E° is an invariant manifold consisting of equilibrium points, the other
ones contain only one equilibrium, which is asymptotically stable. Fig. B.3
shows a possible behavior of these trajectories for some ~ > 0. Clearly, for
c = 0 different trajectories converge to different equilibrium points on E3.
whereas, for e > 0 all trajectories locally converge to the equilibrium x.-.
Note that if the reduced system is asymptotically stable but not in the
first approximation (at the point x°), i.e. .4has not all the eigenvalues in
the left half plane, the results illustrated above are not anymore true. This
is illustrated for instance in the following simple example.
Example. Consider the system
526
В. Appendix В
Fig. В.З.
= У = -</ +
z = (у2 - г'5) .
In this case the set S is the set of points with у = 0. and the reduced system,
given by
is asymptotically stable at z = 0. In order to study the behavior of the entire
system, we rescale the time
y1 = -y + - z2
z' =
and we note that the rescaled system has twp equilibria, one at (у. з) = (0.0)
and the other at (y.z) = (dtr). The firyt one is a critical point; and its
stability may be analyzed by means of the Center Manifold Theory. A center
manifold at (0.0) is a function у = k(z) satisfying
and it is easily seen that
7ф)=5С2+-/ад
where is a remainder of order 5. The flow on the center manifold is
given bv
z'=£(fV4-.-5) + fl7(;)
(with /?7(г) a remainder of order 7) and is unstable at г = 0 for any e.
Thus, from Center Manifold Theory, we conclude that the point (0.0) is an
unstable equilibrium of the system. The analysis of the stability at (M.c2) is
simpler because, as the reader can easily verify, the linear approximation of
В.З Local Geometric Theory of Singular Perturbations 527
the system at this point is asymptotically stable. Thus the point in question
is an asymptotically stable equilibrium of the system.
We conclude that, for any arbitrarily small value of 5 > 0. there exist al-
ways two equilibria of the system near the equilibrium of the reduced system,
one being unstable and the other asymptotically stable. <
We conclude the Section by stating another interesting result, that pro-
vide> an additional ‘'geometric'' insight to the previous analysis.
Theorem. Suppose the assumptions of the previous Theorem are satisfied.
Then, in a neighborhood of the point (,r3.U) in L' x ( —4-so). there exists a
smooth integrable distribution _i with the following properties
(i) diin(J) = n — p
iii) if S is a center manifold for the extended system (Ii.d7) at	at
each point .r of S
TTS C _1( j) = 0
(iii) _i is invariant under the vector field
In other words, this Theorem says that, in a neighborhood of the point
there is a partition of U x (—£□-+=□) into submanifolds of dimension
11 - p (the integral submanifolds of _1) and that each of these manifolds
intersects S transversally, exactly in one point. Moreover, by property (iii).
each of these submanifolds is contained in a subset of the form U x {<}
and the How of F(x. carries submanifolds into submanifolds (the partition
being invariant under the flow of fix,;)). In particular, every submanifold
belonging to the set F x {0} is a locally invariant submanifold of F(j-.O).

Bibliographical Notes
Chapter 1. The definition of distribution used here is taken from Sussmann (1973):
in most of the references quoted in the Appendix A, the term "distribution17 without
any further specification is used to indicate what we mean here for 11 nonsingular
distribution". Different proofs of Frobenius' Theorem are available. The one used
here is adapted from Lobrv (1970) and Sussmann (1973). Additional results on
simultaneous integrability of distributions can be found in Respondek (1982).
The importance in control theory of the notion of invariance of a distribution
under a vector field was pointed out independently by Hirschorn (1981) and by
Isidori et al. (1981a). A more general notion of invariance, under a group of local
diffeomorphisms. was given earlier by Sussmann (1973). The local decompositions
described in section 1.7 are consequences of ideas of Keener (1977).
Theorems 1.8.9 and 1.8.10 were first proved by Sussmann-Jurdjevic (1972). The
proof described here is due to Keener (1974). An earlier version of Theorem 1.8.9.
dealing with trajectories traversed in either time direction, was given by Chow
(1939). Additional and more complete results on local controllability can be found
in the work of Sussmann (1983). (1987). Controllability of systems evolving on Lie
groups was studied by Brockett (1972a). Controllability of polynomial systems was
studied by Ballieul (1981) and Jurdjevic-Kupca (1985). Theorem 1.9.7. although
in a slightly different version, is due to Hermann-Krener (1977). Additional results
on observability, dealing with the problem of identifying the initial state from the
response under a fixed input function, can be found in Sussmann (1979b).
Chapter 2.	The proof of Theorems 2.1.2 and 2.1.3 may be found in Sussmann
(1973). An independent proof of Theorem 2.1.5. was given earlier by Hermann
(1962) and an independent proof of Corollary 2.1.7 by Xagano (1970). The rele-
vance of the control Lie algebra in the analysis of global reachability derives from the
work of Chow (1939) and was subsequently elucidated by Lobrv (1970). Haynes-
Hermes (1970). Elliott (1971) and Sussmann-Jurdjevic (1972). The properties of
the observation space were studied by Hennann-Krencr (1977). and. in the case
of discrete-time systems, by Sontag (1979). Reachability, observability and decom-
positions of bilinear systems were studied by Brockett. (1972b), Goka et al. (1973)
and d'Alessandro et al. (1974). The application to the study of attitude control of
spacecraft is adapted from Crouch (1984).
Chapter 3.	The functional expansions illustrated in section 3.1 were intro-
duced in a series of works by Fliess; a comprehensive exposition of the subject,
together with several additional results, can be found in Fliess (1981). A complete
proof of Lemina 3.1.2 can be found in Wang-Sontag (1992). The expressions of the
kernels of the Volterra series expansion were discovered by Lesjak-Krener (1978):
the expansions (3.17) are due to Fliess et al. (1983). The structure of the Volterra
kernels was earlier analyzed by Brockett (1976). who proved that, any individual
kernel can always be interpreted as a kernel of a suitable bilinear system, and re-
530 Bibliographical Notes
lated results may also be found in Gilbert (1977). The expressions of the kernels of
bilinear systems were first calculated by Bruni et al. (1970). Multivariable Laplace
transforms of Volterra kernels and their properties are extensively studied by Rugh
(1981). Functional expansions for nonlinear discrete-time systems have been studied
by Sontag (1979) and Monaco-Normand Cyrot (1986),
The conditions under which the output of a system is unaffected by some specific
input channel were studied by Isidori et al. (1981a) and Claude (1982): the former
contains, in particular, a different proof of Theorem 3.3.3.
Definitions and properties of generalized Hankel matrices were developed by
Fliess (1974). Theorem 3.4.3 was proved independently by Isidori (1973) and Fliess.
The notion of Lie rank and Theorem 3.4.4 are due to Fliess (1983). Equivalence of
minimal realizations was extensively studied by Sussmann (1977): the version given
here of the uniqueness Theorem essentially develops an idea of Hermann-Kroner
(1977); related results may also be found in Fliess (1983). An independent approach
to realization theory was followed by Jakubczyk (1980). (1986). A complete proof
of Theorem 3.4.4 can be found in Sussmann (1994). Additional results on this
subject can be found in Celle-Gauthier (1987). Realization of finite Volterra series
was studied by Crouch (1981). Constructive realization methods from the Laplace
transform of a Vol terra kernel may be found in the work of Rugh (1983). Realization
theory of discrete-time response maps was extensively studied by Sontag (19791.
Chapter 4.	The convenience of describing a system in the special local co-
ordinates considered in sections 4.1 and 5.1 was first explicitly suggested in the
work of Isidori et al. (1981a). Additional material on this and similar subjects can
be found in the work of Zeitz (1983). Bestle-Zeitz (1983) and of Kroner (1987).
The exact state-space linearization problem was proposed and solved. for single-
input systems, by Brockett (1978). A complete solution for multi-input systems was
found by Jakubczyk-Rcspondek (1981)). Independent work of Su (1982) and Hunt-
Su-Meyer (1983a) led to a slightly different formulation, together with a procedure
for the construction of the linearizing transformation. The possibility of using non-
interacting control techniques for the solution of such a problem was pointed out
in Isidori et al, (1981a), Additional results on this subject can be found in the
work of Sommer (1980) and Marino et al. (1985). The existence of globally defined
ti ansformatious was investigated by Dayawansa et al. (1985). Exact linearization
of discrete-time systems was studied by Lee e| al. (1986) and by Jakubczyk (1987).
The notion of zero dynamics was introduced by Byrnes-Isidori (1984). Its appli-
cation to the solution of critical problems of asymptotic stabilization was described
in Byrnes-Isidori (1988), Additional material on this subject can be found in the
work of Aevels (1985), where for the first time the usefulness of center manifold the-
ory for the solvability of critical problems of asymptotic stabilization was pointed
out. and Marino (1988). The concept of zero dynamics for discrete-time svsrems
and its properties are developed in the works of Glad (1987) and Monacn-Xormand
Cyrot (1988).
The subject of asymptotic stabilization via state feedback is only marginallv
touched m these notes, and there are several important, issues that have not been
covered here. These include, for instance, the problem of equivalence between stabi-
lizability and controllability (see Jurdjevic-Quinn (1979) and Brockett (1983)). the
smoothness properties of a stabilizing feedback (see Sussmann (1979a) and Sontag-
Sussmann (1980)). the input-output approach to stability of feedback systems (see
Hammer (1986). (1987) and Sontag (1989a)), and the recent work of Ceron on the
links between stabilizability and controllability (1992).
The singular perturbation analysis of high-gain feedback systems was indepen-
dently studied by Byrnes-Isidori (1984) and by Marino (1985). A link with the
so-called variable structure control theory developed by Utkin (1977) can be found
Bibliographical Notos 531
in the work of Marino (19851. An application of singular perturbation theory to
the design of adaptive control can be found in the work of Khalil-Saberi (1987): an
application to the so-called almost disturbance decoupling problem can be found
in the work of Marino et al. (1989).
The problem of finding the largest linearizable subsystem in a single-input non-
linear system was addressed and solved by Krener et al. (1983). The solution of the
corresponding problem for a multi-input system was found by Marino (1986). The
problem of exact linearization of a nonlinear system with outputs is extensively’
discussed by Cheng et al. (1988). The use of output, injection in order to obtain ob-
servers with linear error dynamics was independently suggested by Krener-Isidori
(1983) and Bestle-Zeitz (1983) for single-output systems. A complete analysis of
the corresponding problem for multi-output systems can be found in the work of
Krcner-Respondek (1985). Hammouri-Gauthier (1988) have suggested the use of
output injection in order to obtain bilinear error dynamics. Additional results on
the design of nonlinear observers can be found in the work of Zeitz (1985).
Chapter 5.	The solution of the problem of noninteracting control is due to
Porter (1970). Additional results can be found in Singh-Rugh (1972) and Freund
(1975). The work of Isidori et al. (1981a) showed how the solution of the non inter-
acting control problem can be analyzed from the differential-geometric viewpoint.
Related results can be found in Knobloch (1988). In the case of discrete-time non-
linear systems, the problem was studied by Grizzle (1985b) and Monaco-Normand
Cyrot (’1986).
The possibility of using dynamic feedback in order to achieve relative degreeh
was first shown by Singh (1980) ami subsequently elaborated by Descusse-Moog
(1985). (1987) and Nijmeijer-Respondek (1986). (1988). The approach presented
here is based on the "canonical’’ dynamic extension algorithm of Zhan et al. (1991).
This approach is particularly useful in understanding the relationships between
different dynamic extensions yielding relative degree and. as a consequence, in the
characterization of necessary conditions for noninteracting control with stability via
dynamic feedback presented in section 7.4.
The notions of left and right invertibility proposed by Fliess (1986). together
with the introduction of differential-algebraic methods in the analysis of control
systems, provide a precise conceptual framework in which the equivalence between
(right) invertibility and the possibility of achieving noninteracting control via dy-
namic feedback can be established. Additional results on the use of differential
algebra in control theory can be found in the work of Pommaret (1988). Additional
results on the subject system invertibility can be found in the works of Hirschorn
(1979a). (1979b). Singh (1981). Isidori-Moog (1988). Moog (1988) and Di Benedetto
et al. (1989).
As shown at the end section 5.4. the absence of zero dynamics, together with the
possibility of achieving relative degree via dynamic feedback; are properties which
imply the existence of a fet'd back and coordinates change which transform the sys-
tem into a fully linear and controllable one. This property, which was recognized by
Isidori et al. (1986) and since then "rediscovered’’ several other times in the litera-
ture. finds a very natural application to rhe control of the nonlinear dynamics of an
aircraft as well as to the control of a robot arm wit h joints elasticity. The first appli-
cation was pursued by Meyer-Cicolani (1980) and. more recently, by Lane-Stengel
(1988). The second application was developed by De Luca et al. (1985). Other rel-
evant applications of the theory discussed in this Chapter to process control are
those pursued by Hoo-Kantor (1986) and by Levine-Rouchon (1989).
The exact linearization of the input-output response was studied by Isidori-
Ruberti (1984). who proposed an approach (exposed in section 5.6) inspired by the
works of Silverman (1969) on the inversion of linear systems and Van Dooren et
532
Bibliographical Note?
al. (1979) on the calculation of the so-called zero structure at infinity. For discrete-
time systems, the corresponding problem was investigated by Monaco-Normand
Cyrot (1983) and Lee-Markus (1987). An interesting approach, alternative to exact
linearization, is the one based on approximate linearization around an operating
point, considered as a smoothly varying parameter. This approach, which is not
included here for reasons of space, was pursued by Baumann-Riigh (1986). Reboulet
et al. (1986). Wang-Rugh (1987). Sontag (1987a). (1987b), The matching of the
input-output, behavior of a prescribed system was studied by Isidori (1985) and Di
Benedetto-Isidori (1986).
Chapter 6,	Controlled invariant submanifold and controlled invariant distri-
bution are nonlinear versions of the notion of controlled invariant subspace, intro-
duced independently by Basile-Marro (1969) and by Wonham-Morse (1970). As
illustrated in section 6.3, the two notions are not equivalent in a nonlinear setting:
the former lends itself to the definition of the nonlinear analogue of the notion of
transmission zero, while the latter is particularly suited to the study of decoupling
and noninteracting control problems.
The properties of controlled invariant distribution were studied earlier. The no-
tion of controlled invariant, distribution was introduced by Isidori et al. (1981a) and
independently (although in a less general form) by Hirschorn (1981). The proof of
Lemma 6.2.1 presented here, which differs from rhe one contained in the second
edition, has been suggested by Scherer (personal communication). The calculation
of the largest, controllability distribution contained in ker(dh) by means of the con-
trolled invariant distribution algorithm was suggested by Isidori et al. (1981a). The
simpler procedure described in Proposition 6.3.3 is due to Kroner (1985). Lemma
6.3.8 is due to Claude (1981). The notion of controllability distribution, the nonlin-
ear version of the one of controllability subspace, and the corresponding properties
were studied by Isidori-Krener (1982) and by Nijmeijer (1982). The theory of glob-
ally controlled invariant distributions can be found in the work of Davawansa et al.
(1988).
The calculation of the largest output zeroing submanifold by means of the zero
dynamics algorithm was suggested by Isidori-Moog (1988). The work of Byrnes-
Isidori (1988) has shown how this algorithm is useful in order to derive the normal
forms illustrated in Proposition 6.1.5.
Controlled invariance for general nonlinear Systems (i.e. systems in which the
control does not enter linearly) has been studiqfl by Nijmeijer-Van dec Schaft (1983).
Controlled invariance for discrete-time nonlinear systems has been studied by Griz-
zle (1985a) and Monaco-Normand Cyrot (1985).
Chapter 7.	The results described in section 7.1. which are the multivariable
version of the results illustrated in section 4.4, have been adapted from Byrnes-
Isidori (1988). The usefulness of the differential geometric approach in the solution
of the nonlinear disturbance decoupling problem was pointed out by Hirschorn
(1981) and Isidori et al. (1981a).
The solution of the problem of noninteracting control with stability via static
state feedback is due to Isidori-Grizzle (1988). Leminas 7.3.1 and 7.3.1 incorporate
some earlier results by Nijmeijer-Schumacher (1986) and Ha-Gilbert (1986). An im-
portant property of linear systems is that the possibility of achieving noninteracting
control via dynamic feedback implies the possibility of achieving nonint.eracting con-
trol together with asymptotic stability (see Wonham (1979)). This property is not
anymore true in a nonlinear setting. In other words, there are nonlinear systems,
as shown in Isidori-Grizzle (1988). for which it is possible to obtain nonint er acting
control but no (either static or dynamic) feedback exists which yields a stable nonin-
teractive closed loop. The obstruction to the achievement of noninteracting control
Bibliographical Notes 533
with stability via dynamic state feedback, which depends on certain Lie brackets
of the vector fields which characterize rhe noninteractive system, has been studied
by Wagner (1989) who proved Lemma 7 4.2 and Proposition 7.4,1. The necessary
condition of Theorem 7-4.4 is a consequence of the results in Wagner (1989) and
Zhan et al. (1991). The results on noninteracting control with stability via dvnamic
feedback present ed in sect ion 7.5 are derived form the work of Batt dot t i (1991). For
a comprehensive ( overage of the subject of nonintrracting control with stability, the
reader is referred to Battilotti (1994).
Chapter 8-	The nonlinear regulator theory described in this Chapter is taken
from the work of Isidori- By rues (1990) and form the recent work of Byrnes et
aL (1994). The notion of immersion of a system into another system, which is
instrumental in this presentation, was developed by Fliess (1982). The special case
of constant reference signals was treated earlier by Hnang-Rugh (1999), Additional
results and an approximation method for output regulation can be found in Huaug-
Rugli (1992). Necessary conditions for the existence of error feedback nonlinear
regulators were investigated earlier by Hepburn and Wonham (1984). Sufficient
conditions for the solution of problem of structurally stable nonlinear regulation
were established by Huang-Lin (1991). (1993). A nonlocal analysis of the problem
of output regulation can be found in Knobloch et al. (1993).
Chapter 9- The proof, described in sta tion 9.1, of the existence of global nor-
mal forms for nonlinear systems was suggested by Sussiuaiin (personal comniunica-
tion). For additional related material, see Marino et al. (1985) and Byrnes-Isidori
(1991b). Lemmas 9.2.] and 9,2.2 were proven independently by Byrnes-Isidori
(1989) and Tsinias (1989). Corollary 9.2.4 was originally given in Byrnes-Isidori
(1991b). Lemma 9.2.5. which was proven in Byrnes et al. (1991c), contains as par-
ticular cases some earlier results of Jiirdjievic-Qninn (1979) and Lee-Araposthatis
(1988). Actually, a key idea in Lee-Araposthatis (1988) is the main ingredient of
the proof of Lemma 9.2.5.
The concept of semiglobal stabilizability, to the best of our knowledge, appears
to have been introduced by Bacciotti (1989). as property of "potentially global’1
stabilizability. However, the terminology which seems to be more frequently used
in the literature is that of "semiglobar” stabilizability. Theorem 9-3.1 was proven
in a earlier version of Byrnes-Isidori (1991b). However, the proof presented here
repeats an elegant and simpler proof suggested by Bacciotti (1992). The possibility
of extending Theorem 9.3.1 in the sense indicated by Theorem 9.3.2 was originallv
understood by Teel (1992). However, the control law indicated here is based on
a different construction suggested by Lin-Saberi (1992). The proof of Theorem
9.3.2 is somewhat, different from the one originally proposed in the literature. The
counterexample to semiglobal stabilizability in more general cases was suggested
by Sussmann (1990). Additional results and more general considerations about the
possible shrinking of the domain of asymptotic stability induced the use of "high-
gain” feedback can be found in Sussmann-К okot о vic (1991).
The concept of control Lyapunov function was introduced by Artstein (1983).
The constructive proof of Theorem 9.4,1 presented here is due to Sontag (1989b).
The problem of disturbance attenuation, or the equivalent so-called problem of
"almost disturbance decoupling”, was studied by Marino et al. (1989) and (1994).
under the slightly stronger hypotheses. The extension provided by Lemma 9.5.G is
due to Isidori (1994). As shown in Theorem 9.5.4. if the vector field which character-
izes the influence of an external perturbation can be given in suitable coordinates
the form of a vector field in purely triangular form, the output of this system
can be protected, to an arbitrary degree of accuracy, from the in fine nee of the per-
turbation in question. The exploitation of this property has recently load to the
534 Bibliographical Notes
development of some very successful robust stabilization and/or adaptive stabi-
lization schemes, for systems in which the unmodeled dynamics can be bounded
by vector fields in purely triangular form. The interested reader is referred to the
works of Kanellakopoulos et. al (1992). Kokotovic-Kristic (1993) and Marino-Tomei
(1993a). (1993b).
Section 9.G is entirely devoted to the exposition of a recent outstanding result
of Teel-Praly (1994). In this paper, rhe authors generously acknowledge that the
their construction uses a previous result of Gant, hi er-Born hard (1981). on the char-
acterization of those systems whose state is uniquely determined on-line by the
values of a finite number of derivatives of the input and the output, a suggestion of
Tornambe (1992) of incorporating integrators into a stabilizing feedback law to the
purpose of facilitating state estimation, and the idea of Khalil-Esfandjari (1992)
of using saturations to the purpose of securing boundedness of trajectories in the
presence of "high-gain" observers. The idea of using high output-injection gains to
the purpose of achieving asymptotic state estimation predates the later contribu-
tion and is due to Gauthier et al. (1992). For the proof of Theorem 9.G.2. the reader
is referred to the original source Teel-Praly (1994).
Appendix A. A comprehensive exposition of all the subjects summarized in
this Appendix can be found in the books of Boothby (1975). Brickell-Clark (1970).
Singer-Thorpe (1967), Warner (1979).
Appendix B. For a comprehensive introduction to the stability theory, the
reader is referred e.g. to the books of Hahn (1967). Vidyasagar (1978) and Khalil
(1992). The purpose of this Appendix is to cover some specific subjects, that are
frequently used throughout the text, which are not usually treated in standard
reference books on stability of control systems. The exposition of center manifold
theory follows closely the one of Carr (1981). The concepts of stability under per-
sistent disturbances and a proof of the third Lemma of section B.2 can be found in
Hahn (1967), pages 275-276 (see also Vidyasagar (1980)). A proof of the converse
Lyapunov theorem given in B.2 can be found in Kurzweil (1956). The proof of the
last Theorem of section B.2 can be found in Nemytskii-Stepanov (1960). pages 338-
343. Section B.3 is essentially a synthesis of some results taken from the work of
Fenichel (1979). Additional material on this subject can be found in the works of
Knobloch-Aulbach (1984) and Marino-Kokotoyic (1988). A comprehensive exposi-
tion of theory and applications of singular perturbation methods in control can be
found in the book of Kokotovic-Khalil-O’Reilly (1986).
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Index
Actuator 215
Aileron 269
Almost smooth feedback 449
Aualvtic
-	function 472
mapping 472
-	system 6
Angle of attack 268
Annihilator
-	- of a codistribution 20
of a distribution 20
Asymptotic
model matching 182
-	output tracking 178.391
stability of interconnected svstems
511
-	stabilization (local) 172.339
stabilization (global) 432
Artstein-Sontag Theorem 449
Atlas 476
Basic open sets. 473
Basis of a topology 473
Bilinear system 95
Birkhoff Theorem 517
Body frame 34
Bounded real Lemma 454
Brower s Theorem 474
Brunowsky canonical form 247
Campbell*Baker-Hausdorff formula
115.500
Cayley-Hamilton Theorem 92
Center manifold 504
equation 505
-	theory 434
Centrifugal forces 275
Change of basis in the tangent space
485
Change of coordinates sec Coordinates
C^-compatible coordinate charts 475
Closed set 473
Closure of a set 473
Codistribution 19
Compatible charts 475
Complete
-	atlas 476
-	submodule 38
vector field 496
Completely integrable distribution 23
Commutator 95. 498
Control Lie algebra 84
Continuous mapping 473
Controllability
distribution 333
*	- algorithm 333
-	rank condition 86
Controllable 86
Controlled invariant
distribution 312
-	- algorithm 319
submanifold 293
Control Lyapunov function 449
Convolution integral 112
Coordinate
-	chart 474
function 474
Coordinates 10.475
transformation 10.475
Coriolis forces 275
Cotangent space 491
Covector field 6. 500
Covectors 6.491
Critical
-	case of asymptotic analysis 503
-	problem of asymptotic stabilization
173
Cubic coordinate neighborhood 44,
474
DC Motor 211
Decompositions
546
Index
of bilinear systems 94
of linear systems 1.91
-	of nonlinear systems 49. 77
Decoupling matrix 245
Dense subset 474
Derivative
of a covector field 9. 488
of a real-valued function 8. 421
Detectable 395
Diffeomorphism 11
-	of manifolds 478
of sets 471
Differential
of a mapping 485
of a real-valued function 7
equation on a manifold 494
Distribution 13. 38
Disturbance 392
attenuation 450
decoupling 184, 248. 342
via disturbance measurements
188
-	- witlistability 187
Drag force 269
Dual
basis 491
-	space 6,491
Dynamic
equation 35
extension 249
algorithm 251
-	feedback see Feedback
Elevator 269
Embedded submanifold 78. 481
Embedding 479
Equilibrium point 503
Exact
eovector field 501
differential 7
linearization
- of multi-input systems 277, 262
-- of single-input systems 147
of the input-output response 160.
198. 277
Exogenous
input 392
-	disturbance 392
-	reference 392
Exosystem 392
Expression in local coordinates,
of a function 478
-	- of a mapping 478
Fast time 518
Feedback.
- dynamic
output 193
st ate 14i. 249
static.
output 189
stare 147.228
Finitely computable 321. 334
Fliess functional expansion 112
Flow 496
Flux 212
Formal power series 106. 282
Frobenius Theorem 23
Fundamental formula 112
Global
-	asymptotic stability 432
diffeomorphism see Diffeomorphism
normal form 427
Gradient 7
Growth condition 107. 113
Hankel
-	matrix 124
rank 123
Harmonic drive 205
Hamilt on-.Jacobi inequality 454
Hausdorff separation axiom 474
High gain 189
Homeomorphism 473
Hvpersurface 477
Image of a matrix 2
Immersed submanifold 78, 481
InAnersion
of a manifold into another manifold
479
univalent 479
-	of a system into
-	- another system	406
a linear system	408
Implicit function Theorem 472
Indistinguishability 4.52
Induced topology 474
Inertia matrix 35, 275
Inertial frame 38
Inner product 7
Integral
curve 494
- submanifold 78
controller 422
Integrator 250
Interior, of a set 473
Index
547
Internal model 414
Invariance, of the output 124
Invariant
codistribution 47
distribution 41
- submanifold 503
subspace 1
Inverse
- function Theorem 471
of a MI MO system 227
of a SISO system 172
Invertibility condition 290
Involutive
closure 19,195
- distribution 17
Iterated integral 106
Jacobi identity 9, 497
Jacobian matrix 8. 472, 485
Jerk
Joint elasticity 215. 274
Kernel
of a matrix 4
-	of a Volterra series 112
Kinematic equation 35
Leibniz rule 483. 502
Lie
-	algebra 497
of vector fields 36. 497
-	bracket 9, 498
-	rank 123
subalgebra 38. 84
Lift force 269
Linear
approximation 158. 172,503
-	system 1
Linearizing
coordinates 156, 230
- feedback 156, 230
Lipschitzian 514
Local coordinates 474
Locally
- controlled invariant see Controlled
invariant
euclidean space 474
-	finitely generated distribution 81
invariant manifold 503
-	lipschitzian see Lipschitzian
-	observable see Observable
Lyapunov
converse theorem 516
direct theorem 516
£_> gain 451
Manifold 474
Maximal
integral manifold property 77
linear subsystem 194
Memoryless feedback 147
Milnor Theorem 517
Model matching.
for a MIMO system 290
for a SISO system 182
Module 36.497
Multiindex 105
Natural basis of the tangent space
484
Neighborhood 473
Nested sequence of distributions 32
Neutral stability 388
Noninteracting control
- via static feedback 241
via dynamic feedback 262
with stability
via static feedback 344
—	via dynamic feedback 364. 373
Nonintcractive feedback 243
Nonsingular distribution 15
Nontrivial eigenvalues 450
Normal form
-	of a general nonlinear system 309
-	of a MIMO system 225
of a SISO system 144
Observability 1,69
rank condition 97
Observable 91
pair 4
Observation space 89
Observer
linearization 203
-	with linear error dynamics 20,3
One-form 500
Open
-	mapping 473
set 473
Orthogonal
-	group 36
-	matrix 34
Output
-	feedback see Feedback
- invariance 116
regulation
--in the case of error feedback 395.
403
548 Index
in the case of full information
395, 396
tracking
zeroing submanifold 293, 395
Parameter 272
Partition of state space
into integral submanifolds 77
into parallel planes 3, 5
into slices of a coordinates
neighborhood 44
Perturbations see Singular perturba-
tions
Pitch 264 268
Poisson stable 388
Positive definite function 516
Product topology 474
Proper function 516
Rank Theorem 472
Reachability 1, 53
Reachable pair 3
Realizability conditions,
-	via bilinear systems 127
via nonlinear systems 129
Realization. 121
diffeomorphism of 132
-	minimality of 132
-	uniqueness of 132
Reduced
system 507. 518
vector field 520
Reduction principle 507
Reference
frame 34
-	model 182
output 178
Regular
feedback 228
-	point
of a distribution 15
- of the controlled invariant
distribution algorithm 323
of the zero dynamics algorithm
306
Regularizing dynamic extension see,
dynamic extension
Regulation see Output regulation
Related
covector fields 501
vector fields 499
Relative degree
of a MIMO system 220
of a S1SO system 137
Reproducing a reference output
- for a MIMO system 227
for a SISO system 171
Rescaled time variable 518
Restriction of a system to a submanifold
85
Rigid body 34, 99
Robot arm 274
Roll 264, 268
Rotation matrix 34
Row reduction 281
Rudder 269
Semiglobal stabilizability 439, 461
Set point control 422
Side force 269
Sideslip angle 268
Singular perturbations theory 190.
517
Singularly perturbed system 517
Skew symmetric matrix 99
Slice of a neighborhood 44
Slow time 518
Small time constant 193
Smooth
curve 490
distribution see Distribution
function 471
-	manifold see Manifold
mapping 471
-	system 6
Smoothing of a distribution 15
Sphere 477
Stability in the First Approximation
(Principle of) 173, 503
Stabilizable 395
Standard noninteractive feedback лее
Noninteract ive feedback
State feedback лес Feedback
State space exact linearization see
Exact linearization
Static feedback see Feedback
Steady state response 387
Structure Algorithm 282
Structurally stable regulation 416
Submanifoids 479
Submodule 37
Subset topology 474
Sussmann Theorem 79
System
-	defined on a manifold 33
matrix 297
Index 519
Tangent
- space
— to a manifold 183
to SO(3) 102
vector 483
Time-varying system 313
Throttle 269
Thruster 99
Toeplitz matrix 281
Topological
space 473
structure 473
Topology 473
Total stability 514
Torus 478
Tracking
error 391
problem 391
Transmission
polynomials 297
- zeros 297
Triangular decompositions 43
Trivial eigenvalues 519
Uncontrollable modes 173
Uniform relative degree 428
Uniformly
-	asymptotically stable 513
observable 462
-	stable 513
Univalent immersion 479
Weakly controllable see Controllable
Vector
-	field 6. 493
relative degree 220
Volterra series 113
Whitney's theorem 482
Wind axes 268
Yaw 264. 268
Zero dynamics
-	algorithm 29 1
of a general nonlinear system 296
of a MIMO system 225
of a SISO system 164
-	submanifold 296
-	vector field 296
Zeroing the Output
for a general nonlinear system 293
for a MIMO system 225
for a SISO system 163
Zeros of a transfer function 164
u-'-limit
set 517
point 517