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<p>Communications and Control Engineering</p><p>Published titles include:</p><p>Stability and Stabilization of Infinite Dimensional Systems with Applications</p><p>Zheng-Hua Luo, Bao-Zhu Guo and Omer Morgul</p><p>Nonsmooth Mechanics (Second edition)</p><p>Bernard Brogliato</p><p>Nonlinear Control Systems II</p><p>Alberto Isidori</p><p>LrGain and Passivity Techniques in nonlinear Control</p><p>Arjan van der Schaft</p><p>Control ofLinear Systems with Regulation and Input Constraints</p><p>Ali Saberi, Anton A. Stoorvogel and PeddapuUaiab Sannuti</p><p>Robust and H,., Control</p><p>BenM.Chen</p><p>Computer Controlled Systems</p><p>Efim N. Rosenwasser and Bernhard P. Lampe</p><p>Dissipative Systems Analysis and Control</p><p>Rogeüo Lozano, Bemard Brogliato, Olav Egeland and Bernhard Maschke</p><p>Control ofComplex and Uncertain Systems</p><p>Stanislav V. Emelyanov and Sergey K. Korovin</p><p>Robust Control Design Using H,.,Methods</p><p>!an R. Petersen, Valery A. Ugrinovski and Andrey V. Savkin</p><p>Model Reduction for Control System Design</p><p>Goro Obinata and Brian D.O. Anderson</p><p>Control Theory for Linear Systems</p><p>Harry L. Trentelman, Anton Stoorvogel and Malo Hautus</p><p>Functional Adaptive Control</p><p>Sirnon G. Fabri and Visakan Kadirkamanathan</p><p>Positive lD and 2D Systems</p><p>Tadeusz Kaczorek</p><p>Identification and Control Using Volterra Models</p><p>F.J. Doyle III, R.K. Pearson and B.A. Ogunnaike</p><p>Non-linear Control for Underactuated Mechanical Systems</p><p>IsaheUe Fantoni and Rogelio Lozsno</p><p>Robust Control (Second edition)</p><p>Jürgen Ackermann</p><p>Flow Control by Feedback</p><p>Oie Morten Aamo and Miroslav Krstic</p><p>Leaming and Generalization (Second edition)</p><p>Mathukumalli Vidyasagar</p><p>Constrained Control and Estimation</p><p>Graham C. Goodwin, Marfa M. Seron and Josc! A. Oe Don&</p><p>Randomized Algorithms for Analysis and Control of Uncertain Systems</p><p>Roberto Tempo, Giuaeppe Calafiore and Fabrizio Dabbene</p><p>Switched Linear Systems</p><p>Zhendong Sun and Shuzhi S. Ge</p><p>Subspace Methods for System Identification</p><p>Tohru Katayama</p><p>Digital Control Systems</p><p>Ioan D. Landau and Gianluca Zito</p><p>Alberto Isidori</p><p>Nonlinear Control</p><p>Systems</p><p>Third Edition</p><p>With 47 Figures</p><p>~Springer</p><p>Professor Alberto Isidori</p><p>Dipartimento di Informatica e Sistemistica</p><p>"Antonio Ruberti"</p><p>Via Eudossiana 18</p><p>00184Roma</p><p>Italy</p><p>Series Editors</p><p>E.D. Sontag • M. Thoma • A. lsidori • J.H. van Schuppen</p><p>British Library Cataloguing in Publication Data</p><p>Isidori, Alberto</p><p>Nonlinear Control Systems. - 3Rev.ed -</p><p>(Communications & Control Engineering Series)</p><p>I. Title II. Series</p><p>629.836</p><p>ISBN 978-1-4471-3909-6</p><p>Library of Congress Cataloging-in-Publication Data</p><p>lsidori, Alberto</p><p>Nonlinear control systems/Alberto Isidori.- 3rd ed</p><p>p. cm. - ( Communications and control engineering series)</p><p>Includes bibliographical references and index.</p><p>ISBN 978-1-4471-3909-6 ISBN 978-1-84628-615-5 (eBook)</p><p>DOI 10.1007/978-1-84628-615-5</p><p>1. Feedback control systems. 2. Nonlinear control theory.</p><p>3. Geometry, Differential. I. Title II. Series</p><p>QA402.3.174 1995 95-14976</p><p>629.8'36-dc20</p><p>Communications and Control Engineering Series ISSN 0178-5354</p><p>ISBN 978-1-4471-3909-6 Printedon acid-free paper</p><p>@ Springer-Verlag London 1995</p><p>Originally published by Springer-Verlag London Limited in 2000</p><p>Softcover reprint of the bardeover 3rd edition 2000</p><p>Frrstpublished1985</p><p>Second edition 1989</p><p>Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted</p><p>under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or</p><p>transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case</p><p>of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing</p><p>Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.</p><p>The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a</p><p>specific Statement, that such names are exempt from the relevant laws and regulations and therefore free for</p><p>general use.</p><p>The publisher makes no representation, express or implied, with regard to the accuracy of the information</p><p>contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that</p><p>maybemade.</p><p>9876</p><p>For Maria Adelaide</p><p>Preface to the second edition</p><p>The purpose of this book is to present a self-contained description of the fun</p><p>damentals of the theory of nonlinear control systems, with special emphasis</p><p>on the differential geometric approach. The book is intended as a graduate</p><p>text as weil as a reference to scientists and engineers involved in the analysis</p><p>and design of feedback systems.</p><p>The first version of this book was written in 1983, while I was teach</p><p>ing at the Department of Systems Science and Mathematics at Washington</p><p>University in St. Louis. This new edition integrates my subsequent teaching</p><p>experience gained at the University of Illinois in Urbana-Champaign in 1987,</p><p>at the Carl-Cranz Gesellschaft in Oberpfaffenhofen in 1987, at the University</p><p>of California in Berkeley in 1988. In addition to a major rearrangement of</p><p>the last two Chapters of the first version, this new edition incorporates two</p><p>additional Chapters at a more elementary level and an exposition of some</p><p>relevant research findings which have occurred since 1985.</p><p>In the past few years differential geometry has proved to be an effective</p><p>means of analysis and design of nonlinear control systems as it was in the</p><p>past for the Laplace transform, complex variable theory and linear algebra</p><p>in relation to linear systems. Synthesis problems of longstanding interest like</p><p>disturbance decoupling, noninteracting control, output regulation, and the</p><p>shaping of the input-output response, can be dealt with relative ease, on</p><p>the basis of mathematical concepts that can be easily acquired by a control</p><p>scientist. The objective of this text is to render the reader familiar with</p><p>major methods and results, and enable him to follow the new significant</p><p>developments in the constantly expanding literature.</p><p>The book is organized as follows. Chapter 1 introduces invariant dis</p><p>tributions, a fundamental tool in the analysis of the internal structure of</p><p>nonlinear systems. With the aid of this concept, it is shown that a non</p><p>linear system locally exhibits decompositions into "reachable/unreachable"</p><p>parts and/or "observablejunobservable" parts, similar to those introduced</p><p>by Kaiman for linear systems. Chapter 2 explains to what extent global</p><p>decompositions may exist, corresponding to a partition of the whole state</p><p>space into "lower dimensional" reachability and/or indistinguishability sub</p><p>sets. Chapter 3 describes various "formats" in which the input-output map</p><p>of a nonlinear system may be represented, and provides a short description</p><p>viii</p><p>of the fundamentals of realization theory. Chapter 4 illustrates how a series</p><p>of relevant design problems can be solved for a single-input single-output</p><p>nonlinear system. It explains how a system can be transformed into a linear</p><p>and controllable one by means of feedback and coordinates transformation,</p><p>discusses how the nonlinear analogue of the notion of "zero" plays an im</p><p>portant role in the problern of achieving local asymptotic stability, describes</p><p>the problems of asymptotic tracking, model matehing and disturbance decou</p><p>pling. The approach is somehow "elementary", in that requires only standard</p><p>mathematical tools. Chapter 5 covers similar subjects for a special dass of</p><p>multivariable nonlinear systems, namely those systems which can be rendered</p><p>noninteractive by means of static state feedback. For this dass of systems,</p><p>the analysis is a rather Straightforward extension of the one illustrated in</p><p>Chapter 4. Finally, the last two Chapters are devoted to the solution of the</p><p>problems of output regulation, disturbance decoupling, noninteracting con</p><p>trol with stability via static state feedback, and noninteracting control via</p><p>dynamic feedback, for a broader dass of multivariable</p><p>what is the same, (1.17)) can be reduced to the solution of n</p><p>ordinary differential equations of the form</p><p>where h, ... , f n are linearly independent vector fields, with h, ... , /d span</p><p>ning the distribution .1. As a matter of fact, if the solutions of these equations</p><p>are composed to build the mapping tJ! defined by (1.20), a solution of (1.16)</p><p>can be found by taking the last n - d components of the inverse mapping</p><p>tP-1. This procedure is applied in the following examples.</p><p>Example 1..4.1. Consider the distribution, defined on JR2</p><p>A _ { (exp(x2))} .u- span 1 .</p><p>This distribution has dimension 1 for each x E IR2 . Thus, .1 is nonsingular</p><p>and, being I-dimensional, is also involutive. Set</p><p>h(x)=(~)·</p><p>The calculation of the ßows of h and h is rather easy. As far as h is</p><p>concerned, since</p><p>is solved by</p><p>we have</p><p>About /2, since</p><p>is solved by</p><p>±1 = expx2</p><p>x2 = 1</p><p>x1(t) = exp(x2)(exp(t) -1) + x]'</p><p>x2(t) = t + x2</p><p>~!t (x) = (exp(x2)(exp(zl)- 1) + x1) .</p><p>Zt Z1 + X2</p><p>1.4 Frobenius Theorem 29</p><p>Xt(t) t+x}'</p><p>x2(t) x2</p><p>we have</p><p>4>{~(x) = ( Z2 ~X1).</p><p>The mapping l]i, choosing x}' = x2 = 0, has the form</p><p>( :~) = lJi(zt, z2) = ( exp(zt) z7 Z2 - 1)</p><p>and its inverse is given by</p><p>(Zl) q;-1( ) ( X2 )</p><p>z2 = Xt' X2 = x1 - exp(x2) + 1 .</p><p>The function z2 ( x1, x2) is a solution of the partial differential equation</p><p>8z2 ft(x) = 0 ax</p><p>as a Straightforward check also confirms. Note that this function is defined</p><p>in all JR2 . {~ (x) = ( 1 -x~1z1 ) . -Zl + X2</p><p>Note that the flow is not defined for x1z1 ~ 1 (i.e. the vector field ft is not</p><p>complete). The flow of h is identical to the one calculated in the previous</p><p>example. The mapping l]i has the form</p><p>30 1. Local Decompositions of Control Systems</p><p>and its inverse is given by</p><p>Note that this mapping is not defined on all IR2. However, provided lx2- x21</p><p>is sufficiently small, the mapping is well-defined for any x 0 • The function</p><p>z2 (x1, x2 ) is then defined in a neighborhood of any X 0 and solves the partial</p><p>differential equation</p><p>OZ2</p><p>äx h(x) = 0 ... (~t</p><p>8x 0</p><p>= {0 0)</p><p>is solved by</p><p>A(XI,X2,X3) = Z3(X1,X2,X3) =(XI+ 2X2X3)X3 ...1, ... , An-d, which solve</p><p>the partial differential equation {1.16), in order to define (locally araund x 0 ) a</p><p>coordinates transformation entailing a particularly simple representation Jqr</p><p>the vector fields of Ll. For, observe that, by construction, the n-d differC:mtials</p><p>{1.21)</p><p>are linearly independent at the point x 0 • Then, it is always possible to choose,</p><p>in the set of functions</p><p>X! (x) = X1, X2(x) = X2, ... , Xn(X) = Xn</p><p>a subset of d functions whose differentials at x 0 , tagether with those of the set</p><p>{1.21), form a set of exactly n linearly independent row vectors. Let 1, ... , d</p><p>denote the functions thus chosen and set</p><p>4>d+l (x) = >..1 (x), ... , 4>n(x) = An-d(x) .</p><p>By construction, the Jacobian matrix of the mapping</p><p>z = ![j(x) = col{1 (x), ... , d(x), d+l (x), ... , 4>n(x))</p><p>has rank n at x 0 and, therefore, the mapping !fj qualifies as a local diffeo</p><p>morphism (i.e. a local smooth coordinates transformation) araund the point</p><p>x 0 • Now, suppose r is a vector field of Ll. In the new coordinates, this vector</p><p>field is represented in the form</p><p>r(z) = [88![j r(x)] .</p><p>X x= d2 > · · · > dk)· If the distribution Ll1 is com</p><p>pletely integrable, by Frobenius Theorem, in a neighborhood of each point</p><p>X 0 there exist functions >.i, 1 ::; i ::; n - db such that</p><p>span { d>.1, ... , d>.n-d1 } = Llf.</p><p>Suppose now also Ll2 is completely integrable. Then, again, Llf is locally</p><p>spanned by differentials of suitable functions J.Li, 1 ::; i ::; n - d2. However,</p><p>since</p><p>Llf c Llf</p><p>it is immediate to conclude that one can choose</p><p>J.Li = >.i for all 1 ::; i :$ n - d1</p><p>thus obtaining</p><p>span { d>.1, ... , d>.n-d1 } + span { dJ.Ln-d1 +1, ... , dJ.Ln-d2 } = Llf.</p><p>Note also that the sum on the left-hand side of this relation is direct, i.e. the</p><p>two summands have zero intersection. The construction can be repeated for</p><p>all other distributions of the sequence, provided they are involutive. Thus,</p><p>one arrives at the following result.</p><p>Corollary 1.4.2. Let Ll1 ~ Ll2 ~ · • · ~ Llk be a collection of nested nonsin</p><p>gular distributions. If and only if each distribution of the collection is involu</p><p>tive then, for each point X 0 of U, there exists a neighborhood U0 of X 0 , and</p><p>real-valued smooth functions</p><p>all defined on uo, such that</p><p>Llf = span { d>.L ... , d>.~-dJ</p><p>.L .L {i i} Lli Lli-1 EB span d>.l' ... 'd>.d,_l-d,</p><p>for 2 :$ i ::; k.</p><p>1.5 The Differential Geometrie Point of View 33</p><p>Remark 104040 In order to avoid the problern of using double subscripts, it is</p><p>sometimes convenient to state the previous, and similar, results by means of</p><p>a more condensed notation, defined in the following wayo Given a set of Pi</p><p>real-valued functions</p><p>a more abstract seto</p><p>If this is the case, one can still describe the control system in a form like</p><p>m</p><p>P = f(p) + L gi(p)ui (1.23)</p><p>i=l</p><p>(1.24)</p><p>where /, g1 , o 0 o , 9m are smooth vector fields defined on a smooth manifold</p><p>N, and h1 , 0 o o , h, are smooth real-valued functions defined on No The first</p><p>relation represents a differential equation Oll N' and p Stands for the tangent</p><p>vector, at the point p of N, to the smooth curve which characterizes the</p><p>solution for some fixed initial conditiono For the sake of clearness, we have</p><p>used here p in order to denote a point in a manifold N, leaving the symbol x</p><p>to denote the n-vector formed by the local coordinates of the point p in some</p><p>coordinate charto</p><p>34 1. Local Decompositions of Control Systems</p><p>Example 1.5.1. The most common example in which such a situation occurs</p><p>is the one describing the control of the orientation of a rigid body around its</p><p>center of mass, for instance the attitude of a spacecraft. Let e = (e1, e2, e3)</p><p>denote an inertially fixed triplet of orthonormal vectors (the reference frame)</p><p>and let a = (a1 , a2 , a3 ) denote a triplet of orthonormal vectors fixed in the</p><p>body (the body frame), as depicted in Fig.l.2.</p><p>Fig. 1.2.</p><p>A possible way of defining the attitude of the rigid body in space is to</p><p>consider the angles between the vectors of a and the vectors of e. Let R be a</p><p>3 x 3 matrix whose element rij is the cosine of the angle between the vectors</p><p>ai and ei. By definition, then, the elements on the i-th row of Rare exactly</p><p>the Coordinates of the vector a; with respect to the reference frame identified</p><p>by the triplet e. Since the two triplets are both orthonormal, the matrix R is</p><p>suchthat</p><p>or, what is the same, R- 1 RT (that is, R is an orthogonal matrix); in</p><p>particular, det(R) = 1. The matrix R completely identifies the orientation of</p><p>the body frame with respect to the fixed reference frame, and therefore it is</p><p>possible - and convenient - to use R in order to describe the attitude of the</p><p>body in space. We shall illustrate now how the equations of the motion of</p><p>the rigid body and its control can be derived accordingly.</p><p>First of all, note that if Xe and x denote the coordinates of an arbitrary</p><p>vector with respect to e and, respectively, to a, these two sets of coordinates</p><p>are related by the linear transformation</p><p>X= Rxe.</p><p>Moreover, note that if one associates with a vector</p><p>the 3 x 3 matrix</p><p>1.5 The Differential Geometrie Point of View 35</p><p>the usual "vector" product between w and v can be written in the form</p><p>w X V= -S(w)v.</p><p>Suppose the body is rotating with respect to the inertial frame. Let R(t)</p><p>denote the value at timet of the matrix R describing its attitude, and let w(t)</p><p>(respectively we(t)) denote its angular velocity in the a frame (respectively</p><p>in the e frame). Consider a point fixed in the body and let x denote its</p><p>coordinates with respect to the body frame a. Since this frame is fixed with</p><p>the bödy, then x is a constant with respect to the time and dx/dt = 0. On</p><p>the other hand the coordinates xe(t) of the same point with respect to the</p><p>reference frame e satisfy</p><p>Xe(t) = -S(we(t))xe(t).</p><p>Differentiating x(t) = R(t)xe(t), and using the identity RS(we)Xe = S(w)x,</p><p>yields '</p><p>. · . T . T</p><p>0 = Rxe + Rxe = RR X- RS(we)Xe = RR X- S(w)x</p><p>and, because of the arbitrariness of x,</p><p>R(t) = S(w(t))R(t). (1.25)</p><p>This equation, which expresses the relation between the attitude R of</p><p>the body and its angular velocity ( the latter being expressed with respect to</p><p>a coordinate frame fixed with the body), is commonly known as kinematic</p><p>equation.</p><p>Suppose now the body is subject to external torques. If he denotes the</p><p>coordinates of the angular momentum and Te those of the external torque</p><p>with respect to the reference frame e, the momentum balance equation yields</p><p>On the other hand, in the body frame a, the angular momentum can be</p><p>expressed as</p><p>h(t) = Jw(t)</p><p>where J is a matrix of constants, called the inertia matrix. Combining these</p><p>relations one obtains</p><p>Jw = h = Rhe + Rhe = S(w)Rhe + RTe = S(w)Jw + T</p><p>where T = RTe is the expression of the external torque in the body frame a.</p><p>The equation thus obtained, namely</p><p>Jw(t) = S(w(t))Jw(t) + T(t) (1.26)</p><p>is commonly known as dynamic equation.</p><p>36 1. Local Decompositions of Control Systems</p><p>The equations (1.25) and (1.26), describing the control of the attitude of</p><p>the rigid body, are exactly of the form (1.23), with</p><p>p = (R,w).</p><p>In particular, note that R is not any 3 x 3 matrix, but is an orthogonal matrix,</p><p>namely a matrix satisfying RRT = I (and det(R) = 1). Thus, the natural</p><p>state space for the system defined by (1.25) and (1.26) is not- as one might</p><p>think just counting the number of equations- IR.12 , but a more abstract set,</p><p>namely the set of all pairs (R,w) where R belongs to the set of allorthogonal</p><p>3 x 3 matrices (with determinant equal to 1) and w belongs to IR.3 •</p><p>The subset of JR.3 X 3 in which R ranges, namely the set of all3 x 3 matrices</p><p>satisfying RRT =I and det(R) = 1, is an embedded submanifold of IR.3 x 3 , of</p><p>dimension 3. In fact, the orthogonality condition RRT =I can be expressed</p><p>in the form of 6 equalities</p><p>3</p><p>L T;kTjk - 8;j = 0</p><p>k=l</p><p>and it is possible to show that the 6 functions on the left hand side of this</p><p>equality have linearly independent differentials for each nonsingular R (thus,</p><p>in particular, for any R such that RRT = I). Thus, the set of matrices</p><p>satisfying these equalities is an embedded 3-dimensional submanifold of JR.3 X 3 ,</p><p>called the orthogonal group and noted 0(3). Any matrixsuchthat RRT =I</p><p>has a determinant which is equal either to 1 or to -1, and therefore 0(3)</p><p>consists of two connected components. The connected component of 0(3)</p><p>in which det(R) = 1 is called the special orthogonal group (in JR.3 X 3 ) and is</p><p>denoted by 80(3).</p><p>We can conclude that the natural state space of (1.25) and (1.26) is the</p><p>6-dimensional smooth manifold</p><p>N = 80(3) x r.</p><p>This is a 6-dimensional smooth manifold, which - however - is not diffeo</p><p>morphic to JR.6 (because80(3) is not diffeomorphic to IR.3 ).</p><p>vector fields span .::1 at each x. However, M' does not coincide with</p><p>M .1, because for instance</p><p>x!rtM'.</p><p>In fact, the smooth function x cannot be represented in the form x = c(x)x2</p><p>with smooth c(x). ), with coordinate functions 4>1> ... , 4>n suchthat</p><p>for all p EU.</p><p>This characterization lends itself to an interesting interpretation. Let p</p><p>be any point of the cubic coordinate neighborhood U and consider the slice</p><p>of U passing through p consisting of all points whose last n - d coordinates</p><p>are held constant, i.e. the subset of U</p><p>(1.28)</p><p>This subset, which is a smooth submanifold of U, of dimension d, has the</p><p>property of having - at each point q - a tangent space that, by construction,</p><p>is exactly the subspace Ll(q) of TqN (Fig.l.3).</p><p>A(p) A(s)</p><p>00</p><p>~</p><p>A(q)/~</p><p>Fig. 1.3.</p><p>Note that the coordinate neighborhood U is partitioned into slices of the</p><p>form (1.28). Thus, a nonsingular and completely integrable distribution Ll</p><p>induces, at each point p0 , a local partition of N into submanifolds, each one</p><p>having, at any point, a tangent space which - viewed as a subspace of the</p><p>tangent space to N - coincides with the value of Ll at this point.</p><p>1.6 Invariant Distributions 41</p><p>1.6 Invariant Distributions</p><p>The notion of a distribution invariant under a vector field plays, in the theory</p><p>of nonlinear control systems, a role similar to the one played in the theory of</p><p>linear systems by the notion of subspace invariant under a linear mapping. A</p><p>distribution .:1 is said to be invariant under a vector field f if the Lie bracket</p><p>[/, Tj of f with every vector field T of .:1 is again a vector field of .:1, i.e. if</p><p>T E .:1 :::} [/, Tj E .:1 .</p><p>In order to represent this condition in a more condensed form, it is conve</p><p>nient to introduce the following notation. We let [/, .:1] denote the distribution</p><p>spanned by all the vector fields of the form [!, Tj, with TE .:1 , i.e. we set</p><p>[!, .:1] = span{[/, Tj, T E .:1} .</p><p>Using this notation, it is possible to say that a distribution .:1 is invariant</p><p>under a vector field f if</p><p>[!, .:1] c .:1.</p><p>Remark 1.6.1. Suppose the distribution .:1 is nonsingular (and has dimension</p><p>d). Then, using Lemma 1.3.1, it is possible to express-atleast locally- every</p><p>vector field T of .:1 in the form</p><p>d</p><p>T(x) = L ci(xh(x)</p><p>i=l</p><p>where T1, .•. , Td are vector fields locally spanning</p><p>.:1. It is easy to see that .:1</p><p>is invariant under f if and only if</p><p>for all 1 :5 i :5 d .</p><p>The necessity follows trivially from the fact that T1, .•. , Td are vector fields</p><p>of .:1. For the sufficiency, consider the expansion (see (1.8))</p><p>d d</p><p>[/,T] =:Lei[/, Ti]+ L(Ltci)Ti</p><p>i=l i=l</p><p>and note that all the vector fields on the right-hand side are vector fields of</p><p>.:1.</p><p>, The previous expression in particular shows that</p><p>[/, Llj ::> Span{[/, Tt], ... , [j, Td]}</p><p>but note that the distribution on the left-hand side may, in general, be un</p><p>equal to the one on the right-hand side. However, by adding to both sides</p><p>the distribution .:1, it is easy to deduce - again from the previous expression</p><p>- that</p><p>42 1. Local Decompositions of Control Systems</p><p>Ll + [/, Ll] = Ll + span{[/, TI], ... ,[/, Td]}</p><p>i.e. that</p><p>Ll + [/, Ll] = span{ T1, •.• , Td, [/, T1], ..• , [/, Td]} .</p><p>This property will be utilized in some later developments. (x) defined on U0 , in which the vector field f is represented by a vector</p><p>of the form</p><p>/(z) = /d(Zt, · ·., Zd, Zd+l>. ·.,Zn)</p><p>/d+l (zd+l, ··.,Zn)</p><p>(1.29)</p><p>1.6 Invariant Distributions 43</p><p>Proof. The distribution L1, being nonsingular and involutive, is also inte</p><p>grable. Therefore, at each point X 0 there exists a neighborhood uo and a</p><p>coordinates transformation z = .P(x) defined on U0 with the property that</p><p>span{ dcf>d+l, ... , dc/>n} = L1_1_.</p><p>Let f ( z) denote the representation of the vector field f in the new coordinates.</p><p>Consider now a vector field</p><p>and suppose</p><p>Then</p><p>7(z) = colh (z), ... , 7n(z))</p><p>7k(z) = 0</p><p>7k(z) = 1</p><p>for k =j:. i</p><p>for k = i.</p><p>[/,7] =-aJ 7 =- aJ.</p><p>az azi</p><p>Recall that (see (1.22)), in the coordinatesjust chosen, every vector'field of L1</p><p>is characterized by the property that the last n- d components are vanishing.</p><p>Thus, if 1 :::; i :::; d, the vector field 7 belongs to L1. Since L1 is invariant under</p><p>J, [/, 7] also belongs to L1, i.e. its last n- d components must vanish. This</p><p>yields</p><p>aA =O</p><p>azi</p><p>for all d + 1 :::; k :::; n, 1 :::; i :::; d, and proves the assertion. '(x0 ) = 0). Suppose also, again without</p><p>loss of generality, that the neighborhood U0 on which the transformation is</p><p>defined is a neighborhood of the form</p><p>uo = {x E JR» : jzi(x)j (x),</p><p>(1.34)</p><p>at any timet. As a matter offact, (2(xa(t)) and (2(xb(t)) are both solutions</p><p>of the same differential equation - namely, second equation of (1.31) - and</p><p>both satisfy the same initial condition, because</p><p>Two initial conditions xa and xb satisfying (1.33) belang, by definition,</p><p>to a slice of the form (1.32). As we have just seen, the two corresponding</p><p>trajectories xa(t) and xb(t) of (1.30) necessarily satisfy (1.34), i.e. at any</p><p>timet belang necessarily to a slice of the form (1.32). Thus, we can conclude</p><p>that the flow of (1.30) carries slices (ofthe form (1.32)) into slices (Fig. 1.6).</p><p>Example 1.6.4. Consider the 2-dimensional distribution</p><p>with</p><p>and the vector field</p><p>46 1. Local Decompositions of Control Systems</p><p>Fig. 1.6.</p><p>( X3X4 ~~1X2X3 ) .</p><p>sinx3 + x~ + x1x3</p><p>!=</p><p>A simple calculation shows that</p><p>and therefore (Remark 1.3.5) the distribution Ll is involutive. Moreover, since</p><p>this distribution is also invariant under the vector field f (Remark 1.6.1).</p><p>By Frobenius' Theorem, in a neighborhood of any point X 0 there exist</p><p>functions A1(x),A2(x) suchthat</p><p>span{ dA1, dA2} = Ll.L.</p><p>One can easily verify that, for instance, the functions</p><p>A1(x) = X3</p><p>A2(x) = -x1x2 + X4</p><p>whose differentials have the form</p><p>dA1 = (0 0 1 0)</p><p>dA2 = ( -X2 -Xl 0 1)</p><p>satisfy this condition.</p><p>As described in the proof of Lemma 1.6.1, define</p><p>new (local) coordinates</p><p>zi = 1/>i(x), 1 ~ i ~ 4, choosing</p><p>4>3(x) = A1(x)</p><p>and completing, e.g., the set of new coordinates functions with</p><p>1.6 Invariant Distributions 47</p><p>(h(x) = x2.</p><p>In the new coordinates, the vector field f assumes the form</p><p>i.e. the form indicated by (1.31), with (1 = (z1,z2),(2 = (z3,Z4). 1 , ... , 4>n in the new coordinates is just</p><p>1. 7 Local Decompositions of Control Systems 49</p><p>for all 1 :$ i :$ n. This implies</p><p>ßl/Ji _ ~ ..</p><p>- u,J</p><p>OZj</p><p>and, therefore, in the new coordinates all the entries of the differential dl/Ji</p><p>are zero but the i-th one, which is equal to 1. As a consequence</p><p>Ltl/Ji(z) = (dl/Ji(z), f(z)) = /i(z)</p><p>and</p><p>Ltdl/Ji(z) = dLtl/Ji(z) = dfi(z) .</p><p>Since L\ is invariant under f and nonsingular, by Lemma 1.6.3 we have</p><p>that LtLll. C L\1. and this, since dl/Ji E L\1. for d + 1 :$ i :$ n, yields</p><p>Ltdl/Ji = dfi E L\1..</p><p>The differential dfi, like any covector field of L\1., must have the form (1.35)</p><p>and this proves that</p><p>if 1 :$ j :$ d, d + 1 :$ i :$ n. (x) defined on</p><p>U0 such that, in the new coordinates, the control system {1.36) is represented</p><p>by equations of the form {see Fi9. 1. 7b)</p><p>1.7 Local Decompositions of Control Systems 51</p><p>m</p><p>'1 = !1((1,(2) + L9H((I.(2)Ui</p><p>i=l</p><p>m</p><p>'2 = /2((2) + L92i((2)Ui</p><p>(1.38)</p><p>i=l</p><p>Yi = hi((2)</p><p>where (1 = (z1, ... , zd) and (2 = (zd+l, ... , Zn).</p><p>Proof. As before, we know that there exists, around each X 0 , a Coordinates</p><p>transformation yielding a representation of the form (1.29) for the vector</p><p>fields J, 01, ... , Um. In the new Coordinates, the covector fields dh1. ... , dhp,</p><p>that by assumption belong to Ll.l., must have the form (1.35). Therefore</p><p>ahi = 0</p><p>azj</p><p>for all 1 :5 j :5 d, 1 :5 i :5 p, and this completes the proof.</p><p>passing through the point x(Tk)· For small values oft, the state x(t) evolves</p><p>in a neighborhood of the initial point x(O).</p><p>Suppose now that the assumptions of the Proposition 1.7.1 are satisfied,</p><p>choose a point X 0 , and set x(O) = X 0 • For small values of t the state evolves</p><p>on U 0 and we may use the equations (1.37) to interpret the behavior of the</p><p>system. From these, we see that the (2 coordinates of x(t) are not affected</p><p>by the input. In particular, if we denote by x 0 (T) the point of U 0 reached at</p><p>timet= T when no input is imposed (i.e. when u(t) = 0 for all t E [O,T]),</p><p>namely the point</p><p>c~t (x) being the flow of the vector field J), we deduce from the structure</p><p>of (1.37} that the set of points that can be reached at timeT, starting from</p><p>x 0 , is a set of points whose (2 coordinates are necessarily equal to the (2</p><p>coordinates of x 0 (T). In other words, the set of points reachable at time T</p><p>is necessarily a subset of a slice of the form (1.32), exactly the one passing</p><p>through the point x0 (T) (see Fig.1.8).</p><p>52 1. Local Decompositions of Control Systems</p><p>Fig. 1.8.</p><p>Thus, we conclude that locally the system displays a behavior strictly</p><p>analogaus to the one described in section 1.1. The state space can be parti</p><p>tioned into d-dimensional smooth surfaces ( the slices of U 0 ) and the states</p><p>reachable at time T, along trajectories that stay in U 0 for all t E [0, T], lie</p><p>inside the slice passing through the point X 0 (T) reached under zero input.</p><p>The Proposition 1.7.2 is useful in studying state-output interactions.</p><p>Choose a point X 0 and take two initial states xa and xb belanging to U0</p><p>with local Coordinates ((f, (~) and ((t, (~) suchthat</p><p>1a- 1b</p><p>'>2- '>2</p><p>i.e. two initial states belanging to the same slice of U 0 • Let x~ ( t) and xt ( t)</p><p>denote the values of the states reached at time t, starting from xa and xb,</p><p>under the action of the same input u. From the second equation of (1.38) we</p><p>see immediately that, ifthe input u is suchthat both x~(t) and xt(t) evolve</p><p>on U 0 , the (2 coordinates ofx~(t) and xt(t) are the same, no matter what</p><p>input u we take. As a matter offact, (2 (x~(t)) and (2 (xt(t)) are solutions of</p><p>the samedifferential equation (the second equation of (1.38)) with the same</p><p>initial condition. If we take into account also the third equation of (1.38), we</p><p>see that</p><p>hi(x: (t)) = hi(x~ (t))</p><p>for every input u. We may thus conclude that the two states xa and xb</p><p>produce the same output under any input, i.e. are indistinguishable.</p><p>Again, we find that locally the state space may be partitioned into d</p><p>dimensional smooth surfaces ( the slices of U0 ) and that all the initial states</p><p>on the same slice are indistinguishable, i.e. produce the same output under</p><p>any input which keeps the state trajectory evolving on U 0 •</p><p>In the next sections we shall reach stronger conclusions, showing that</p><p>if we add to the hypotheses contained in the Propositions 1.7.1 and 1.7.2</p><p>the further assumption that the distribution Ll is "minimal" (in the case of</p><p>Proposition 1.7.1) or "maximal" (in the case of Proposition 1.7.2), then from</p><p>the decompositions (1.37) and (1.38) we may obtain more precise information</p><p>about the states reachable from X 0 and, respectively, indistinguishable from</p><p>xo.</p><p>1.8 Local Reachability 53</p><p>1.8 Local Reachability</p><p>In the previous section we have seen that if there is a nonsingular distribution</p><p>Ll of dimension d with the properties that</p><p>(i) Ll is involutive</p><p>(ii) Ll contains the distribution span{g~, ... , Um}</p><p>(iii) Ll is invariant under the vector fields /, 91, ... , Ym</p><p>then at each point X 0 E U it is possible to find a coordinates transformation</p><p>defined on a neighborhood U 0 of X 0 and a partition of U 0 into slices of</p><p>dimension d, such that the points reachable at some time T, starting from</p><p>some initial state X 0 E U 0 , along trajectories that stay in U 0 for all t E (0, T],</p><p>lie inside a slice of U 0 • Now we want to investigate the actual "thickness" öf</p><p>the subset of points of a slice reached at time T.</p><p>The obvious suggestion that comes from the decomposition (1.37) is to</p><p>look at the "minimal" distribution, if any, that satisfies (ii), (iii) and, then,</p><p>to examine what can be said about the properties of points whidi belong to</p><p>the same slice in the corresponding local decomposition of U. It turns out</p><p>that this program can be carried out in a rather satisfactory way.</p><p>We need first some additional results on invariant distributions. If V is a</p><p>family of distributions on U, we define the smallest or minimal element as</p><p>the member of V (when it exists) which is contained in every other element</p><p>ofV.</p><p>Lemma 1.8.1. Let Ll be a given smooth distribution and r 1 , •.. , Tq a given</p><p>set of vector fields. The family of all distributions which are invariant under</p><p>r 1 , ... , Tq and contain Ll has a minimal element, which is a smooth distribu</p><p>tion.</p><p>Proof. The family in question is clearly nonempty. If ..:11 and ..:12 are two</p><p>elements of this family, then it is easily seen that their intersection ..:11 n ..:12</p><p>contains Ll and, being invariant under r 1 , ... , Tq, is an element of the same</p><p>family. This argument shows that the intersection Ll of all elements in the</p><p>family contains Ll, is invariant under r 1 , ... , Tq and is contained in any other</p><p>element of the family. Thus is its minimal element. Ll must be smooth because</p><p>otherwise smt(Ll) would be a smooth distribution containing Ll (because Ll</p><p>is smooth by assumption), invariant under 71, ... , Tq (see Remark 1.6.5) and</p><p>possibly contained in J. Llk implies Ll' ::> Llk+l· For , we</p><p>have (recall Remark 1.6.1)</p><p>q q</p><p>Llk+l = Llk + Lh, Llk] = Llk + L:span{h, Tj : T E Llk}</p><p>i=l i=l</p><p>q</p><p>C Llk + L:span{h, Tj: TELl'} C Ll'.</p><p>i=l</p><p>Since Ll' ::> Ll0 , by induction we see that Ll' ::> Llk for all k. If Llk· = Llk' +1</p><p>for some k*, we easily see that Llk· ::> Ll (by definition) and Llk· is invariant</p><p>under T1, ..• , Tq (because [Ti, Llk·] C Llk'+l = Llk· for all 1 ~ i ~ q). Thus</p><p>Llk· must coincide with (T1 , ... ,Tq1Ll).</p><p>we easily deduce that, for any</p><p>k ~ 1,</p><p>Llk(x) = Im(B AB ... Ak B) .</p><p>Each distribution of the sequence thus constructed is a constant distribution.</p><p>Since Llk+l ::) Llk, a dimensionality argument proves that there exists an</p><p>integer k* L1k·· Suppose the inclusion is proper at some</p><p>x E V and define a new distribution Ll by setting</p><p>Ll(x) = L1k• (x) if x E V</p><p>.d(x) = (Tl, ... , Tql.d)(x) if x ~V.</p><p>This distribution contains .::1 and is invariant under T1 , ••• , Tq. For, if T is a</p><p>vector field in Ll, then [Ti,T] E (Tl, ... ,Tql.d) (because Ll C (Tl, ... ,Tql.d))</p><p>and, moreover, [Ti, Tj(x) E .dk· (x) for all x E V (because, in a neighbor</p><p>hood of x, T E .dk. and [Ti, .dk·] C L1k· ). Since Ll is properly contained in</p><p>(Tl, ... , Tql.d), this would contradict the minimality of (Tl, ... , Tql.d). Now let</p><p>Uk be the set of regular points of .dk. This set is an open and dense subset</p><p>of U (see Lemma 1.3.2) and so is the set U* = Uo n U1 n · · · n Un-1· In a</p><p>neighborhood of every point x E U* the distributions .do, ... , L1n-l are non</p><p>singular. This, tagether with the previous discussion and a dimensionality</p><p>argument, shows that L1n-l = (Tl, ... , Tql.d) on U* and completes the proof.</p><p>vo]]]</p><p>where v0 , ••• , Vr (with r ::;; k and possibly depending on i) are vector fields</p><p>in the set { r1, ... , Tq }. Then, a similar result holds for ..:1k+l· For, let T be</p><p>any vector field in ..:1k. From Lemma 1.3.1 it is known that there exist real</p><p>valued smooth functions c1, ... , cdk defined locally araund x such that T may</p><p>be expressed, locally araund x, as T = c101 + · · · + cdk(}dk. If Tj is any vector</p><p>in the set {TI, ... , Tq} we have</p><p>As a consequence</p><p>..:1k+l =..:1k+h,..:1k]+ · +[rq,..:1k] = span{Oi, [rl>Oi], ... , [rq,Oi]: 1::;; i::;; dk}·</p><p>Since ..:1k+1 is nonsingular araund x, then it is possible to find exactly dk+l</p><p>vector fields of the form</p><p>(}i = [vn [vr-1, ... , [v1, vo]]]</p><p>where vo, ... , Vr (with r ::;; k + 1 and possibly depending an i) are vector</p><p>fields in the set {TI, ... , Tq}, which span ..:1k+l locally araund x. ... , Tql..:1) is also involutive.</p><p>Lemma 1.8.5. Suppose ..:1 is spanned by some of the vector fields r1, ... , Tq</p><p>and that (rl, ... ,rql..:1) is nonsingular. Then (rl>···,rql..:1) is involutive.</p><p>Proof. We use first the conclusion of Lemma 1.8.4 to prove that if u1 and</p><p>u2 are two vector fields in L1n-1, then their Lie bracket [u1, u2] is such that</p><p>[u1,u2](x) E L1n-l(x) for all x E U*. Using again Lemma 1.3.1 and the</p><p>previous result we deduce, in fact, that in a neighborhood V of x</p><p>d d</p><p>[u1,u2] = [~c}Oi,~c~(}i] E span{Oi,Oj,[Oi,Oj]: 1::;; i,j::;; d}</p><p>where (}i, (}i are vector fields of the form described before.</p><p>In order to prove the claim, we have only to show that [Oi, Oj](x) is a tan</p><p>gent vector in L1n-1 (x). Forthis purpose, we recall that an U* the distribution</p><p>1.8 Local Reachability 59</p><p>Lln_1 is invariant und er the vector fields r1, 0 0 0, rq (see Lemma 1.8o3) and that</p><p>any distribution invariant under vector fields Ti and Tj is also invariant under</p><p>their Lie bracket h, ri] (see Lemma 1.6o2)o Since each ()i is a repeated Lie</p><p>bracket of the vector fields r1,ooo,Tq, then [Oi,Lln-d(x) C Lln-1(x) for all</p><p>1 :::;; i:::;; d and, thus, in particular [Bi, Oj](x) is a tangent vector which belongs</p><p>to L1n-1(x)o</p><p>Thus the Lie bracket of two vector fields a 1, a 2 in Lln-1 is such that</p><p>[a1, a2 ](x) E Lln_1(x)o Moreover, it has already been observed that Lln-1 =</p><p>(r1, 0 0 0, rqiLl) in a neighborhood of X 0 and, therefore, we conclude that at any</p><p>point x of U* the Lie bracket of any two vector fields a 1, az in ( r1, 0</p><p>0 0 , Tq I Ll)</p><p>is suchthat [a1,az](x) E (r1, o 0 0, rqiLl)(x)o</p><p>Consider now the distribution</p><p>which, by construction, is such that</p><p>.1 :::> {rt, 0 0 0 ,rqiLl) o</p><p>From the previous result it is seen that Ll(x) = {rt, 0 0 0 ,rqiLl)(x) at each</p><p>point x of U*, which is a dense set in U o By assumption, ( r1, 0 0 0 , rq I Ll) is</p><p>nonsingularo So, by Lemma 1.3.4 we deduce that .1 = (r1,ooo,rq1Ll), and,</p><p>therefore, that [Oi,Oj] E (r1,ooo,rq1Ll) for all pairs ()i,()i E (r1,ooo,rq1Ll)o</p><p>This concludes the proofo then (}i = [vr, ... , [!, UilJ has the desired form. Any vector of the</p><p>form</p><p>(}i = [vr. .. ·, [v1, UilJ</p><p>with Vr, ..• , v1 in the set {f, U1, ... , Um} is in P because P contains Ui and is</p><p>invariant under I, Ul, . .. , Um and so the claim is proved.</p><p>From this fact we deduce that on an open and dense submanifold U* of</p><p>u,</p><p>R c P + span{f}</p><p>and therefore, since R :J P + span{f} on U, that on U*</p><p>R= P+span{f}.</p><p>Suppose that P+span{f} has constant dimension on some neighborhood</p><p>V. Then, from Lemma 1.3.4 we conclude that the two distributions R and</p><p>P + span{f} coincide on V. r+l, ... , dn} = Rl.</p><p>Span{ dr, .. ·, dn} = pl.</p><p>on U 0 , where r- 1 = dim(P).</p><p>In the new coordinates the control system (1.36) is represented by equa</p><p>tions of the form</p><p>m</p><p>z1 = ft(zl, ... ,zn) + LUH(Zl,···•Zn)Ui</p><p>i=l</p><p>m</p><p>Zr-l = lr-l (zl, ···,Zn) + LUr-l,i(Zl, ·. ·, Zn)Ui</p><p>i=l</p><p>(1.40)</p><p>Zr = lr(Zr, ... , Zn)</p><p>Zr+l = 0</p><p>Zn = 0.</p><p>1.8 Local Reachability 63</p><p>Note that this differs from the form (1.37), only in that the last n- r</p><p>components of the vector field f are vanishing . (because, by construction,</p><p>f ER). If, in particular, R = P then also the r-th component of f vanishes</p><p>and the corresponding equation for Zr is</p><p>Zr= 0.</p><p>The decomposition (1.40) lends itself to considerations which, to some</p><p>extent, refine the analysis presented in sections 1.6 and 1.7. We knew from</p><p>the previous analysis that, in suitable coordinates, a set of components of the</p><p>state (namely, the last n- r + 1 ones) was not affected by the input; we see</p><p>now that in fact all these coordinates but (at most) one are even constant</p><p>with the time.</p><p>If U 0 is partitioned into r-dimensional slices of the form</p><p>Sx = {x E uo: tPrH(x) = tPr+l(x), ... ,1/>n(x) = tl>n(x)}</p><p>then any trajectory x(t) of the system evolving in U0 actually evolves on the</p><p>slice passing through the initial point x 0 • This slice, in turn, is partitioned</p><p>into (r -1)-dimensional slices, each one corresponding to a fixed value of the</p><p>r-th coordinate function, which include the set of points reached at a specific</p><p>time T (see Fig. 1.9).</p><p>Fig. 1.9.</p><p>Remark 1. 8. 5. A further change of local coördinates makes it possible to</p><p>better understand the role of the time in the behavior of the control system</p><p>(1.40). We may assume, without loss of generality, that the initial point X 0 is</p><p>suchthat !l>(x0 ) = 0. Therefore we have zi(t) = 0 for all i = r + 1, ... , n and</p><p>Zr = fr(Zr, 0, ...</p><p>, 0) .</p><p>Moreover, if we assume that f ~ P, then the function fr is nonzero ev</p><p>erywhere on the neighborhood U. Now, Iet Zr(t) denote the solution of this</p><p>differential equation, which passes through 0 at t = 0. Clearly, the mapping</p><p>64 1. Local Decompositions of Control Systems</p><p>J.l: t ~ Zr(t)</p><p>is a diffeomorphism from an open interval ( -c, c) of the time axis into the</p><p>open interval ofthe Zr axis (zr( -c), Zr(c)). Ifits inverse J.t-1 is used as a local</p><p>coordinates transformation on the Zr axis, one easily sees that, since</p><p>the time t can be taken as new r-th coordinate.</p><p>In this way, points on the slice Sx• passing through the initial state are</p><p>parametrized by ( z1, ... , Zr-1 , t). In particular, the points reached at time T</p><p>belong to the (r- !)-dimensional slice</p><p>S~. = {x E U0 : ... ,9m· Thus, we may saythat in the</p><p>associated decomposition (1.40) the dimension r is "minimal", in the sense</p><p>that it is not possible to find another set of local coordinates z1, ... , z8 , ••• , Zn</p><p>with s strictly less than r, with the property that the last n - s coordinates</p><p>remain constant with the time. We shall now show that, from the point of</p><p>view of the interaction between input and state, the decomposition (1.40)</p><p>has even stronger properties. Actually, we are going to prove that the states</p><p>reachable from the initial state X 0 fill up at least an open subset of the r</p><p>dimensional slice in which they are contained.</p><p>Theorem 1.8.9. Suppose the distribution R (i.e. the smallest distribution</p><p>invariant under J, 91, ... , 9m which contains J, 91, . .. , 9m) is nonsingular.</p><p>Let r denote the dimension of R. Then, for each X 0 E U it is possible to find</p><p>a neighborhood U0 of X 0 and a coordinates transformation z = ~(x) defined</p><p>on U0 with the following properlies</p><p>(a) the set 'R(x0 ) of states reachable startin9 from X 0 along trajectories en</p><p>tirely contained in U0 and under the action of piecewise constant input func</p><p>tions is a subset of the slice</p><p>(b) the set 'R(x 0 ) contains an open subset of Bx•.</p><p>Proof. The proof of the statement (a) follows from the previous discussion.</p><p>We proceed directly to the proof of (b), assuming throughout the proof to</p><p>operate on the neighborhood U0 on which the coordinates transformation</p><p>~(x) is defined. For convenience, we break up the proof in several steps.</p><p>loS Local Reachability 65</p><p>(i) Let fh, 0 0 0 , Bk be a set of vector fields, with k ~, 0 0 0 , ~0</p><p>denote the corresponding flowso</p><p>Consider the mapping</p><p>F : ( -€, c)k</p><p>(h,ooo,tk)</p><p>--+ uo</p><p>H pk 0 0 0 0 0 pl (xo)</p><p>tk tt</p><p>where X 0 is a point of U0 and suppose that its differential has rank k at some</p><p>s1, 00 0, sk, with 0 00 o, 9m(recall that R(x) is the tangent space to Sxo at</p><p>x)o</p><p>(ii) Suppose that the vector fields /,91, 0 o o ,9m aresuchthat</p><p>f(x) E TxM</p><p>9i(x) E TxM</p><p>(1.43)</p><p>for all x E Mo Weshallshow that this contradicts the assumption k ... , 9m (because these vector fields are in R and, moreover,</p><p>it is assumed that (1.43) are true)o Let T be any vector field of .1. Then T E R</p><p>and since R is invariant under /, 91, ... , 9m, then for all x E (U \ M)</p><p>66 1. Local Decompositions of Control Systems</p><p>[f, T}(x)</p><p>[9;, T](x)</p><p>E Ll(x)</p><p>E Ll(x) 1 ::; i ::; m.</p><p>(1.44)</p><p>Moreover since T, f, 91 , ... , 9m are vector fields which are tangent to M at</p><p>each x E M, wehavealso that (1.44) hold for all x E M, and therefore all x E</p><p>U. Having shown .1 is invariant under J, 91, ... , 9m and contains J, 91, ... , 9m,</p><p>we deduce that .1 must coincide with R. But this is a contradiction since at</p><p>allxEM</p><p>dim Ll(x)</p><p>dimR(x)</p><p>k</p><p>r > k.</p><p>(iii) If (1.43) are not true, then it is possible to find m real numbers</p><p>u~+1 , ... , u~+l and a point x E M suchthat the vector field</p><p>m</p><p>fJk+1 = f + L 9;u~+ 1</p><p>i=1</p><p>satisfies the condition f)k+l (x) tf. T:rM.</p><p>Let x = F(s~, ... , sU be this point (s; > s;, 1 ::; i ~~1, oF(t1, ... ,tk)</p><p>at the point (s~, ... , s/.:, 0) has rank k + 1. For, note that</p><p>[F~(~i)](s~, ... ,s~,O) = [F*(~i)](s~, ... ,s~)</p><p>for i = 1, ... , k and that</p><p>[F~(ßt~+l )](s~, ... ,s~,o) = f)k+1(x).</p><p>The first k tangent vectors at x are linearly independent, because F has</p><p>rank k at all points of (s1 ,c) x · · · x (sk,E). The (k + 1)-th one is indepen</p><p>dent from the first k by construction and therefore F' has rank k + 1 at</p><p>( s~, ... , s/.:, 0). We may thus conclude that the mapping F' has rank k + 1 at</p><p>a point (s~, ... , s/.:, s/.:+1), with s; 0 it is always possible to choose the</p><p>point x in such a way that</p><p>(s~ - s1) + · · · + (s/.: - sk) s~-t for i = 1, ... , k - 1</p><p>and s~ > 0.</p><p>The procedure clearly stops at the stage r, when a mapping Fr is defined</p><p>Fr: (s~-t,c) x · · · x (s~=Lc) x (O,c) --+ U</p><p>(tt, ... ,tr) t-+ PLo ... ot.Pt(x0 ).</p><p>(v) Observe that a point x = Fr(tt, ... , tr) in the image Mr of the em</p><p>bedding Fr can be reached, starting from the state X 0 at time t = 0, under</p><p>the action of the piecewise constant control defined by</p><p>u}</p><p>= u~</p><p>for t E [0, tt)</p><p>for t E [tt + · · · + tk-t, tt + t2 + · · · + tk) .</p><p>68 1. Local Decompositions of Control Systems</p><p>s~</p><p>Fig. 1.10.</p><p>We know from our previous discussions that Afr must be contained in the</p><p>slice of uo</p><p>Sxo = {x E U0 : c/>;(x) = c/>i(X0 ),r+ 1 ~ i ~ n} o</p><p>The images under Fr of the open sets of</p><p>Ur = (s~- 1 , e) x o 0 0 x (s;=L e) x (0, e)</p><p>are open in the topology of Mr as a subset of U0 (because Fr is an embedding)</p><p>and therefore they are also open in the topology of Mr as a subset of</p><p>Sxo</p><p>(because Sxo is an embedded submanifold of U0 )o Thus we have that Mr is</p><p>an embedded submanifold of Sxo and a dimensionality argument teils us that</p><p>Mr is actually an open submanifold of Sxo 0 ... 'Cp, LTcl, ... 'LTcp}.</p><p>Therefore, since</p><p>1.9 Local Observability 71</p><p>we have</p><p>nl (x) = span{cb ... 'Cp,clA, ... 'CpA}</p><p>at each x E !Rn. Continuing in the same way, we have, for any k ~ 1,</p><p>nk(x) = span{c1, ... ,cp,c1A, ... ,cpA, ... ,c1Ak, ... ,cpAk}.</p><p>Each codistribution of the sequence thus constructed is a constant codistri</p><p>bution. Since nk+l ::> nk, a dimensionality argument proves that there exists</p><p>an integer k* .1, ... , d)..s of exact covector</p><p>fields and that Dn-1 is nonsin9ular. Then (TI. ... , Tqlfl}.L is involutive and</p><p>(Tl, ... , Tqlfl} = Dn-1 .</p><p>In the study of the state-output interactions for a control system of the</p><p>form (1.36), we</p><p>nonlinear systems. The</p><p>analysis in these Chapters is mostly based on a number of key differential</p><p>geometric concepts that, for convenience, are treated separately in Chapter</p><p>6.</p><p>It has not been possible to include all the most recent developments in this</p><p>area. Significant omissions are, for instance: the theory of globallinearization</p><p>and global controlled invariance, the notions of left- and right-invertibility</p><p>and their control theoretic consequences. The bibliography, which is by no</p><p>means complete, indudes those publications which were actually used and</p><p>several works of interest for additional investigation.</p><p>The reader should be familiar with the basic concepts of linear system</p><p>theory. Although the emphasis of the book is on the application of differential</p><p>geometric concepts to control theory, most of Chapters 1, 4 and 5 do not</p><p>require a specific background in this field. The reader who is not familiar with</p><p>the fundamentals of differential geometry may skip Chapters 2 and 3 in a first</p><p>approach to the book, and then come back to these after having acquired the</p><p>necessary skill. In order to make the volume as self-contained as possible,</p><p>the most important concepts of differential geometry used throughout the</p><p>book are described-without proof-in Appendix A. In the exposition of each</p><p>design problem, the issue of local asymptotic stability is also discussed. This</p><p>also presupposes a basic knowledge of stability theory, for which the reader is</p><p>referred to well-known standard reference books. Some specific results which</p><p>arenot frequently found in these references are induded in Appendix B.</p><p>I wish to express my sincerest gratitude to Professor A. Ruberti, for his</p><p>constant encouragement, to Professors J. Zaborszky, P. Kokotovic, J. Acker</p><p>mann, C.A. Desoer who offered me the opportunity to teach the subject of</p><p>this book in their academic institutions, and to Professor M. Thoma for his</p><p>continuing interest in the preparation of this book. I am indebted to Pro</p><p>fessor A.J. Krener from whom-in the course of a joint research venture-I</p><p>ix</p><p>learned many of the methodologies which have been applied in the book. I</p><p>wish to thank Professor C.l. Byrnes, with whom I recently shared intensive</p><p>research activity and Professors T.J. Tarn, J.W. Grizzle and S.S. Sastry with</p><p>whom I had the opportunity to cooperate on a number of relevant research</p><p>issues. I also would like to thank Professors M. Fliess, S. Monaco and M.D.</p><p>Di Benedetto for their valuable advice.</p><p>Rome, March 1989 Alberto Isidori</p><p>Preface to the third edition</p><p>In the last six years, feedback design for nonlinear systems has experienced</p><p>a growing popularity and many issues of major interest, which at the time</p><p>of the preparation of the second edition of this book were still open, have</p><p>been successfully addressed. The purpose of this third edition is to describe</p><p>a few significant new findings as well as to streamline and improve some of</p><p>the earlier passages.</p><p>Chapters from 1 to 4 are unchanged. Chapter 5 now includes also the</p><p>discussion of the problern of achieving relative degree via dynamic extension,</p><p>which in the second edition was presented in Chapter 7 (former sections</p><p>7.5 and 7.6). The presentation is now based on a new "canonical" dynamic</p><p>extension algorithm, which proves itself very convenient from a number of</p><p>different viewpoints. Chapter 6 is also unchanged, with the only exception of</p><p>the proof of the main result of section 6.2, namely the construction of feedback</p><p>laws rendering invariant a given distribution, which has been substantially</p><p>simplified due to a valuable suggestion of C.Scherer. Chapter 7 no Ionger</p><p>includes the subject of tracking and regulation (former section 7.2) which</p><p>has been expanded and moved to a separate new Chapter and, as explained</p><p>before, the discussion of how to obtain relative degree via dynamic extension.</p><p>It includes, on the other hand, a rather detailed exposition of the subject of</p><p>noninteracting control with stability via dynamic feedback, which was not</p><p>covered in the second edition.</p><p>Chapters 8 and 9 are new. The first one of these covers the subject of</p><p>tracking and regulation, in a improved exposition which very easily leads to</p><p>the solution of the problern of how to obtain a "structurally stable" design.</p><p>The last Chapter deals with the design of feedback laws to the purpose of</p><p>achieving global or "semiglobal" stability as well as global disturbance atten</p><p>uation. This particular area has been the subject of major research efforts in</p><p>the last years. Among the several and indeed outstanding progresses in this</p><p>domain, Chapter 9 concentrates only on those contributions whose develop</p><p>ment seems to have been particularly influenced by concepts and methods</p><p>presented in the earlier Chapters of the book. The bibliography of the sec</p><p>ond edition has been updated only with those references which were actually</p><p>used in the preparation of the new material, namely sections 5.4, 7.4, 7.5 and</p><p>Chapters 8 and 9.</p><p>xii</p><p>I wish to express my sincere gratitude to all colleagues who have kindly</p><p>expressed comments and advice on the earlier versions of the book. In par</p><p>ticular, I wish to thank Prof. Ying-Keh Wu, Prof. M.Zeitz and Dr. U. Knöpp</p><p>for their valuable suggestions and careful help.</p><p>St.Louis, December 1994 Alberto Isidori</p><p>Table of Contents</p><p>1. Local Decompositions of Control Systems . . . . . . . . . . . . . . . . . 1</p><p>1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . •. 1</p><p>1.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5</p><p>1.3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13</p><p>1.4 Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22</p><p>1.5 The Differential Geometrie Point of View . . . . . . . . . . . . . . . . . . 33</p><p>1.6 Invariant Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41</p><p>1. 7 Local Decompositions of Control Systems . . . . . . . . . . . . . . . . . . 49</p><p>1.8 Local Reachability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53</p><p>1.9 Local Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69</p><p>2. Global Decompositions of Control Systems . . . . . . . . . . . . . . . 77</p><p>2.1 Sussmann's Theorem and Global Decompositions........... 77</p><p>2.2 The Control Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83</p><p>2.3 The Observation Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87</p><p>2.4 Linear Systems and Bilinear Systems . . . . . . . . . . . . . . . . . . . . . 91</p><p>2.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99</p><p>3. Input-Output Maps and Realization Theory .............. 105</p><p>3.1 Fliess Functional Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105</p><p>3.2 Valterra Series Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112</p><p>3.3 Output Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116</p><p>3.4 Realization Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121</p><p>3.5 Uniqueness of Minimal Realizations ....................... 132</p><p>4. Elementary Theory of Nonlinear Feedback for Single-Input</p><p>Single-Output Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137</p><p>4.1 Local Coordinates Transformations ....................... 137</p><p>4.2 Exact Linearization Via Feedback ........................ 147</p><p>4.3 The Zero Dynamics ..................................... 162</p><p>4.4 Local Asymptotic Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . 172</p><p>4.5 Asymptotic Output Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178</p><p>4.6 Disturbance Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . .</p><p>consider the distribution</p><p>Q = (/,91, ... ,9mlspan{dh1, ... ,dhp}}.L</p><p>From Lemma 1.6.3 we deduce that this distribution is invariant under</p><p>/, 91, ... , 9m and we also see that, by definition, it is contained in (span{ dh1,</p><p>... , dhp} ).L. If nonsingular, then, according to Lemma 1.9.5, is also involutive.</p><p>Invoking Proposition 1.7.2, this distribution may be used in order to find</p><p>locally around each X 0 E U an open neighborhood U0 of X 0 and a coordinates</p><p>transformation yielding a decomposition of the form (1.38). Let s denote</p><p>the dimension of Q. Since QJ. is the smallest codistribution invariant under</p><p>/, 91, ... , Um which contains dh1, ... , dhp, then in this case the decomposition</p><p>we find is maximal, in the sense that it is not possible to find another set</p><p>of local Coordinates Zl' ... 'Zr, Zr+!' ... ' Zn with r strictly }arger than s, with</p><p>the property that only the last n - r Coordinates influence the output. We</p><p>show now that this corresponds to the fact that points belonging to different</p><p>slices of the neighborhood U 0 are distinguishable.</p><p>Theorem 1.9.7. Suppose the distribution Q (i.e. the annihilator of the</p><p>smallest codistribution invariant under J, 91, ... , Um and which contains dh1,</p><p>... , dhp} is nonsin9ular. Let s denote the dimension of Q. Then, for each</p><p>X 0 E U it is possible to find a nei9hborhood U 0 of X 0 and a coordinates</p><p>transformation z = 4>( x) defined U 0 with the jollowin9 properlies</p><p>(a) Any two initial states xa and xb of U 0 such that</p><p>i = s + 1, ... ,n</p><p>produce identical output functions under any input which keeps the state tra</p><p>jectories evolvin9 on U 0</p><p>1.9 Local Observability 73</p><p>(b) Any initial state x of uo which cannot be distinguished from X 0 under</p><p>piecewise constant input functions belongs to the slice</p><p>Proof. We need only to prove (b). For simplicity, we break up the proof in</p><p>various steps.</p><p>(i) Consider a piecewise-constant input function</p><p>u;(t)</p><p>u;(t)</p><p>for t E [0, tr)</p><p>for t E [t1 + · · · + tk-1, t1 + · · · + tk) .</p><p>Define the vector field</p><p>m</p><p>fh = !+ Lg;u7</p><p>i=1</p><p>and Iet P~ denote the corresponding flow. Then, the state reached at time tk</p><p>starting from X 0 at time t = 0 under this input may be expressed as</p><p>x(tk) = P~k o · · · o P~1 (x 0 )</p><p>and the corresponding output y as</p><p>Note that this output may be regarded as the value of a mapping</p><p>pxo</p><p>'</p><p>-+ IR</p><p>(t1, ... ,tk) 1-+ h;oPt O···oPt(x0 ).</p><p>lf two initial states xa and xb are such that they produce two identical outputs</p><p>for any possible piecewise constant input, we must have</p><p>F( (h, ... , tk) = Ft (t1, ... , tk)</p><p>for all possible (t1 , ... , tk), with 0 :::; t; 1 )x, ... ' (8)x} (1.47)</p><p>satisfied for all x E U 0 • From Lemma 1.9.4, we know that there exists an open</p><p>subset U* of uo, dense in U 0 , with the property that, around each x E U*</p><p>it is possible to find a set of n - s real-valued functions A1 , ... , An-s which</p><p>have the form</p><p>Ai = Lvr · · · Lv1 hj (1.48)</p><p>with v1 , ... , Vr vector fields in {!, u1 , ... , Um} and 1 ~ j ~ p, such that</p><p>Ql_ = span{ dA1, ... , dAn-s} .</p><p>Suppose X 0 E U*. Since Qj_(x0 ) has dimension n - s, it follows that</p><p>the tangent covectors dA1 (x 0 ), ••• , dAn-s(X0 ) are linearly independent. In</p><p>the local coordinates which satisfy (1.47), A1 , ... , An-s are functions only of</p><p>Zs+I, ... , Zn (see (1.35)). Therefore, we may deduce that the mapping</p><p>A : (zs+l, ... , Zn) f-t (AI (zs+l, .. . , Zn), ... , An-s(Zs+I, ... , Zn))</p><p>1.9 Local Observability 75</p><p>has a jacobian matrix which is square and nonsingular at (zs+l (x 0 ), ••• ,</p><p>zn(x0 )). In particular, this mapping is locally injective. We may use this</p><p>property to deduce that, for some suitable neighborhood U' of x 0 , any other</p><p>point x of U' such that</p><p>Ai(x) = Ai(X0 )</p><p>for 1 ~ i ~ n- s, must besuchthat</p><p>for 1 ~ i ~ n- s, i.e. must belang to the slice of U0 passing through x 0 • This,</p><p>in view of the results proved in (ii) completes the proof in the case where</p><p>X 0 EU*.</p><p>(iv) Suppose X 0 ~ U*. Let x(x0 ,T,u) denote the state reached at time</p><p>t = T under the action of the piecewise constant input function u. If T is</p><p>sufficiently small, x(x0 ,T,u) is still in U0 • Suppose x(x0 ,T,u) EU*. Then,</p><p>using the conclusions of (iii), we deduce that in some neighborhood U' of</p><p>x' = x(x0 , T, u), the states indistinguishable from x' lie on the slice of U0</p><p>passing through x'. Now, recall that the mapping</p><p>~: X 0 -+ x(x0 ,T,u)</p><p>is a local diffeomorphism. Thus, there exists a neighborhood Ü of X 0 whose</p><p>( diffeomorphic) image under ~ is a neighborhood U" C U' of x'. Let x denote</p><p>a point of Ü indistinguishable from X 0 under piecewise constant inputs. Then,</p><p>clearly, also x" = x(x, T, u) is indistinguishable from x' = x(x0 , T, u). From</p><p>the previous discussion we know that x" and x' belang to the same slice of</p><p>U0 • But this implies also that X 0 and x belang to the same slice of U0 • Thus</p><p>the proof is completed, provided that</p><p>x(x0 ,T,u) EU*. (1.49)</p><p>(v) Allwehave to show now isthat (1.49) can be satisfied. For, suppose</p><p>R(x0 ), the set of states reachable from x 0 under piecewise constant control</p><p>along trajectories entirely contained in U 0 , is such that</p><p>(1.50)</p><p>If this is true, we know from Theorem 1.8.9 that it is possible to find an</p><p>r-dimensional embedded submanifold V of U0 entirely contained in R(x0 )</p><p>and therefore such that V n U* = 0. For any choice of functions At, ... , An-s</p><p>of the form (1.48), at any point x E V the covectors dAt(x), ... ,dAn-s(x)</p><p>are linearly dependent. Thus, without lass of generality, we may assume that</p><p>there exist d</p><p>.:1 induces a local partition of the state space into lower dimen</p><p>sional submanifolds and we have used this result to obtain local decomposi</p><p>tions of control systemso The decompositions thus obtained are very useful to</p><p>understand the behavior of control systems from the point of view of input</p><p>state and, respectively, state-output interactiono However, it must be stressed</p><p>that the existence of decompositions of this type is strictly related to the as</p><p>sumption that the dimension of the distribution is constant at least over a</p><p>neighborhood of the point around which we want to investigate the behavior</p><p>of our control systemo</p><p>In this section we shall see that the assumption that .:1 is nonsingular can</p><p>be removed and that global partitions of the state space can be obtainedo</p><p>Since we are interested in establishing results which have a global validity,</p><p>it is convenient - for more generality - to consider, as anticipated in section</p><p>1.5, the case of control systems whose state space is a manifold N 0 Of course,</p><p>this more general analysis will cover in particular the case in which N = U o</p><p>To begin with, we need to introduce a few more conceptso Let .:1 be a</p><p>distribution defined on the manifold No A submanifold S of N is said to be</p><p>an integrol submanifold of the distribution .:1 if, for every p E S, the tangent</p><p>space TpS to S at p coincides with the subspace Ll(p) of TpNo A maximal</p><p>integral submanifold of .:1 is a connected integral submanifold S of .:1 with</p><p>the property that every other connected integral submanifold of .:1 which</p><p>contains S coincides with So We see immediately from this definition that</p><p>any two maximal integral submanifolds of .:1 passing through a point p E N</p><p>must coincideo Motivated by this, it is said that a distribution .:1 on N has</p><p>the maximal integral manifolds property if through every point p E N passes</p><p>a · maximal integral submanifold of .:1 or, in other words, if there exists a</p><p>partition of N into maximal integral submanifolds of .:10</p><p>It is easily seen that this is a global version of the notion of complete inte</p><p>grability for a distributiono As a matter of fact, a nonsingular and completely</p><p>integrable distribution is such that for each p E N there exists a neighborhood</p><p>U of p with the property that .:1 restricted to U has the maximal integral</p><p>manifolds propertyo</p><p>78 2. Global Decompositions of Control Systems</p><p>A simple consequence of the previous definitions is the following one.</p><p>Lemma 2.1.1. A distribution Ll which has the maximal integral manifolds</p><p>property is involutive.</p><p>Proof. If T is a vector field which belongs to a distribution Ll with the max</p><p>imal integral manifolds property, then T must be tangent to every maximal</p><p>integral submanifold S of Ll. As a consequence, the Lie bracket [rt, r2] of two</p><p>vector fields Tt and T2 both belonging to Ll must be tangent to every maximal</p><p>integral submanifold S of Ll. Thus, [r1 , r2) belongs to Ll. 0</p><p>for x1 > 0. This distribution is involutive and</p><p>dim Ll(x)</p><p>dim Ll(x)</p><p>1</p><p>2</p><p>Clearly, the open subset of N</p><p>if x is such that Xt :$ 0</p><p>if x is such that x1 > 0 .</p><p>{(xt,X2) E IR2 : Xl > o}</p><p>is an integral submanifold of Ll (actually a maximal integral submanifold)</p><p>and so is any subset of the form</p><p>{(x1,x2) E IR2 : Xt</p><p>also due to Sussmann.</p><p>Theorem 2.1.3. A distribution Ll has the maximal integral manifolds prop</p><p>erty if and only if there exists a set of vector fields T, which spans Ll, with</p><p>the property that for every r E T and every pair (t,p) E lR x N such that the</p><p>ftow 4't(p) is defined, the differential (4't)* at p maps the subspaceLl(p) into</p><p>the subspaceLl ( 4'[ (p)) .</p><p>Remark 2.1.4. lt is clear that the proof of the "if" part of Theorem 2.1.2 is</p><p>implied by the "if" part of Theorem 2.1.3 because the set of all vector fields</p><p>in Ll is indeed a set of vector fields which spans Ll. Conversely, the "only ir'</p><p>part of Theorem 2.1.3 is implied by the "only if" part of Theorem 2.1.2.</p><p>of states reachable from p0 under piecewise constant input functions</p><p>( a) is a subset of Spo</p><p>(b) contains an open subset of Spo .</p><p>This result might be interpreted as a global version of Theorem 1.8.9.</p><p>However, there are more general versions, which do not require the assump</p><p>tion that R is nonsingular. Of course, since one is interested in having global</p><p>84 2. Global Decompositions of Control Systems</p><p>decompositions, it is necessary to work with distributions having the maxi</p><p>mal integral manifolds property. From the discussions of the previous section,</p><p>we see that a reasonable situation is the one in which the distributions are</p><p>spanned by a set of vector fields which is involutive and locally finitely gen</p><p>erated. This motivates the interest in the following considerations.</p><p>Let {Ti : 1 ~ i ~ q} be a finite set of vector fields and C1, C2 two subal</p><p>gebras of V ( N) which both contain the vector fields T1, ... , Tq. Clearly, the</p><p>intersection C1 n C2 is again a subalgebra of V ( N) and contains T1, ... , Tq.</p><p>Thus we conclude that there exists an unique subalgebra C of V(N) which</p><p>contains T1, ... , Tq and has the property of being contained in all the subal</p><p>gebras of V ( N) which contain the vector fields T1, ... , Tq. We refer to this as</p><p>the smallest subalgebra of V ( N) which contains the vector fields T1, ... , Tq.</p><p>Remark 2.2.1. One may give a description of the subalgebra C also in the</p><p>following terms. Consider the set</p><p>Lo ={TE V(N): T = [Tik,[Tik-P"""'[Ti2 ,Ti1 ]]];1 ~ ik ~ q,1 ~ k</p><p>every</p><p>initial state p0 E N the set of states reachable under piecewise constant input</p><p>functions contains at least an open subset of N.</p><p>Corollary 2.2.5. A sufficient condition for a control system of the form</p><p>{2.3} to be weakly controllable on N is that</p><p>dim Llc(p) = n</p><p>2.3 The Observation Space 87</p><p>for alt p E N. lf the distribution Llc has the maximal integral manifolds</p><p>property then this condition is also necessary.</p><p>Proof. If this condition is satisfied, Llc is nonsingular, involutive and there</p><p>fore, from the previous discussion, we conclude that the system is weakly</p><p>controllable. Conversely, if the distribution Llc has the maximal integral man</p><p>ifolds property and dim Llc(p0 ) ..3 E LC(So) for</p><p>1 ~ j ~ l and that LC(So) is closed under differentiation along T1, ... , Tq. ..: >..ES}.</p><p>The codistribution ns is smooth by construction, but - as we know - the</p><p>distribution n.g may fail to be so. Since we are interested in smooth distribu</p><p>tions because we use them to partition the state space into maximal integral</p><p>submanifolds, we should ratherbe looking at the distribution smt(ng) (see</p><p>Remark 1.3.3).</p><p>The following result is important when looking at smt( n.g) for the purpose</p><p>of finding global decompositions of N.</p><p>Lemma 2.3.3. Suppose the set of all vector fields in smt( n_t) is locally</p><p>finitely generated. Then smt( n-}) has the maximal integral manifolds prop</p><p>erty.</p><p>Proof. In view of Theorem 2.1.5, we have only to show that smt(n-}) is</p><p>involutive. Let T1 and T2 be two vector fields in smt( n.g) and >.. any function</p><p>inS. Since (d>.., Tl}= 0 and (d>.., T2} = 0 we have</p><p>(d>.., [Tl, T21) = Lr1 (d>.., T2}- Lr2 (d>.., T1} = 0.</p><p>The vector field [T1,T2] is thus in n.g. But [T1,T2], being smooth, is also in</p><p>smt(n-} ). ... , Ym· This subspace will be denoted</p><p>by 0 and called the Observation Space. Moreover, with 0 we associate the</p><p>codistribution</p><p>no = span{d>..: >.. E 0}.</p><p>Remark 2.9.2. It is possible to prove that the distribution nfj is invariant</p><p>under the vector fields j,g1, ... ,Ym· For, let >.. be any function in 0 and Ta</p><p>vector field in nfj. Then (d>.., T} = 0 and (dLJ>.., T} = 0 because L1>.. is again</p><p>a function in 0. Therefore, from the equality</p><p>(d>.., [/,Tl) = Lj(d>..,</p><p>T} - (dLJ>.., T} = 0</p><p>we deduce that [/, Tj annihilates the differentials of all functions in 0. Since</p><p>[}0 is spanned by differentials offunctions in 0, it follows that [/, Tj is a vector</p><p>field in nfj. In the same way one proves the invariance under Yl, ... , Ym.</p><p>lf the distribution na is smooth ( e.g. when the codistribution no is</p><p>nonsingular) the using Lemma 1.6.3 one concludes that no itself is invariant</p><p>under /, Y1, ... , Ym·</p><p>may have singularities.</p><p>If, at some point x E !Rn, f(x) E P(x), then the maximal integral sub</p><p>manifold of Llc passing through x coincides with the one of the distribution</p><p>P, i.e. is a subset of the form x + V. Otherwise, if such a condition is not</p><p>verified, the maximal integral submanifold of Llc is a submanifold whose di</p><p>mension exceeds by 1 that of P and this submanifold, in turn, is partitioned</p><p>into subsets of the form x' + V.</p><p>Example 2.4 .1. The following simple example illustrates the case of a singular</p><p>Llc. Let the system be described by</p><p>94 2. Global Decompositions of Control Systems</p><p>Then we easily see that</p><p>V = {X E ~3 : X 2 = X3 = 0}</p><p>and that a</p><p>P = span { i:l} .</p><p>UXl</p><p>The tangent vector f(x) belongs toP only at those x in which X2 = X3 =</p><p>0, i.e. only on V . Thus, the maximal integral submanifolds of Llc will have</p><p>dimension 2 everywhere but on V. A direct computation shows that these</p><p>submanifolds may be described in the following way (Fig.2.1)</p><p>X3: Ü</p><p>/</p><p>Fig. 2.1.</p><p>(i) if x0 is such that x2 = 0 (resp. x3 = 0) then the maximal submanifold</p><p>passing through x 0 is the half open plane</p><p>{x E ~3 : x2 = 0 and sgn(x3)</p><p>(resp. { x E ~3 : X3 = 0 and sgn(x2)</p><p>sgn(x3)}</p><p>= sgn(x2)})</p><p>(ii) if X 0 is suchthat both x2 =j:. 0 and x3 =j:. 0, then the maximal submanifold</p><p>passing through X 0 is the surface</p><p>{x E ~3 : X2X3 = X~xn . k* o The set of vector fields</p><p>.C = {r E V(JR.n): r(x) = Tx,T E Mk·}</p><p>is the smallest Lie subalgebra of vector fields which contains r 1 (x) = T1x, 0 0 0,</p><p>Tr(x) = TrXo</p><p>Proof. The proof is rather simple and consists in the following stepso A dimen</p><p>sionality argument proves the existence ofthe integer k* suchthat Mk = Mk·</p><p>for all k > k*o Then, one checksthat the subspace Mk· contains T1, o o o, Tr</p><p>and any repeated commutator of the form [Ti1 , o 0 0, [Tih-l, Tih]] and is such</p><p>that [P, Q] E Mk· for all PE Mk· and Q E Mk· 0 From these properties, it is</p><p>Straightforward to deduce that .C is the desired Lie algebrao</p><p>contained in ker( C) and is the largest subspace of</p><p>JR.n having these properties. From linear algebra we know that by taking</p><p>a suitable change of coordinates in JRn (see e.g. section 1.1) the matrices</p><p>A, N1 , ... , Nm become block triangular and, therefore, the dynamics of the</p><p>system become described by equations of the form</p><p>m</p><p>±1 = Aux1 + A12X2 + L(Ni,HXl + Ni,12x2)ui</p><p>i=l</p><p>m</p><p>±2 = A22X2 + L:mNi,22x2ui .</p><p>i=l</p><p>Moreover, the output y depends only on the X2 Coordinates, y = C2x2.</p><p>The above equations are exactly ofthe form (1.38), this time obtained by</p><p>means of standard linear algebra arguments.</p><p>2.5 Examples</p><p>In this section we discuss an example of application of the theories illustrated</p><p>in the Chapter to a control system whose state space is a manifold N not dif</p><p>feomorphic to JR.n. More precisely, we study the system - already introduced</p><p>in section 1.5 - which describes the control of the attitude of a spacecraft by</p><p>means of equations of the form</p><p>Jw = S(w)Jw + T</p><p>R = S(w)R</p><p>(2.13)</p><p>(2.14)</p><p>with state (w, R) E JR3 x 80(3) and input T E JR3 . The orthogonal matrix</p><p>R represents the orientation of the spacecraft with respect to an inertially</p><p>fixed reference frame, the vector w its angular velocity, and the vector T</p><p>represents the external torque. The matrix J is the so-called inertia matrix</p><p>of the spacecraft, and S(w) is the skew-symmetric matrix</p><p>(</p><p>0 wa</p><p>S(w) = -wa 0</p><p>w2 -w1</p><p>If we suppose the external torque T generated by a set of r independent</p><p>pairs of gas jets ( thrusters), it is possible to set</p><p>T = b1u1 + · · · + brUr</p><p>where b1, ... , br E JR3 represent the vectors of direction cosines·- with respect</p><p>to the body frame - of the axes about which the control torques are applied</p><p>100 20 Global Decompositions of Control Systems</p><p>and u1, 0 0 0 , Ur the corresponding magnitudeso Of course, we assume the set</p><p>{b1, 0 0 0, br} is a linearly independent set (and thus r:::; 3)o</p><p>We want to analyze the partition induced by the distribution Llc, in the</p><p>two cases r = 3 and r = 20 For convenience, we begin by discussing the</p><p>dynamic equation (2013) onlyo Note that, setting</p><p>x= Jw</p><p>and using the property</p><p>S(w)v = -S(v)w</p><p>(which holds for any pair of vectors v, w E !R_3) the equation in question can</p><p>be rewritten in the form</p><p>;i; = -S(x)J-1x + Bu</p><p>where B = (b1 0 0 0 br), i.eo</p><p>;i; = f(x) + 91(x)u1 + 0 0 0 + 9r(x)ur</p><p>with</p><p>f(x) = -S(x)J-1x 9;(x) = b; 1:::; i:::; r o</p><p>The case in which r = 3 is rather trivial. In fact, since the control Lie</p><p>algebra C contains, by definition, the three vector fields 91 ( x), 92 ( x), 93 ( x),</p><p>and these vector fields - which are constant - are by assumption linearly</p><p>independent at each x E JR3 , we have immediately</p><p>for all x E JR3 0</p><p>In other words, the controllability rank condition (205) is satisfied at each x,</p><p>and the partition of JR3 induced by Llc degenerates into one single element,</p><p>namely JR3 itselfo</p><p>The case r = 2 is more interesting ( at least from the point of view of the</p><p>analysis) 0 In this case, to obtain meaningful information, one has to compute</p><p>a few Liebrackets between f(x) and the 9;(x)'so Let c1 and c2 be real numbers</p><p>and consider the ( constant) vector field</p><p>Since, as a Straightforward calculation shows,</p><p>äf = -S(x)J-1 + S(J-lx)</p><p>äx</p><p>then, setting b = c1b1 + c2b2 , it is immediate to see that</p><p>(f,9](x) = S(x)J- 1b- S(J-1x)b</p><p>[[J,9],9](x) = -2S(b)J-1b 0</p><p>2.5 Examples 101</p><p>By definition, the control Lie algebra contains the three ( constant) vector</p><p>fields 91(x),g2(x), [[f,g],g](x). Thus, if</p><p>(2.15)</p><p>we again obtain that Llc(x) has dimension 3 at each x as before, and the</p><p>associated partition of IR3 degenerates into one single element.</p><p>Note that the vector b is an arbitrary vector in the image if the matrix</p><p>and, therefore, the possibility of having the condition (2.15) fulfilled can be</p><p>restated in the following terms</p><p>for some b E Im(B) . (2.16)</p><p>We show now that, if the condition (2.16) is not satisfied, then Llc has</p><p>dimension 2 at all points of a certain plane in IR3 . For, Iet 'Y denote a ( nonzero)</p><p>row vector satisfying</p><p>'YB = 0</p><p>and suppose a linear (thus globally defined) coordinates transformation is</p><p>performed, changing x into z = Tx, with</p><p>By definition of "(, we have</p><p>(2.17)</p><p>If the condition (2.16) does not hold, at each point x oflm(B), S(x)J-1x</p><p>is a vector in Im(B). Since</p><p>x E Im(B) {::}"(X= 0 {::} z1(x) = 0</p><p>we see from (2.17) that, if the condition (2.16) does not hold, at each point</p><p>where z1 = 0, then i 1 = 0 also. This means that any trajectory of the system</p><p>(2.13) starting in the plane</p><p>M = {x E IR3 :"(X= 0}</p><p>remains in this plane for all times. As a consequence, in view of the results</p><p>established in section 2.2, we deduce that necessarily Llc has at most dimen</p><p>sion 2 at each point of M (in fact, it has dimension 2 because b1 and b2 are</p><p>independent).</p><p>At every other point x f/. M, we assume that Llc has dimension 3. To have</p><p>this hypothesis fulfilled, it suffices to observe that the control Lie algebra</p><p>contains the three vector fields g1(x),g2(x), [f,gi](x), and to assume that at</p><p>least for one value of x and one value of i these three vectors are linearly</p><p>independent, i.e.</p><p>102 2. Global Decompositions of Control Systems</p><p>det(b1 b2 [!, gi](x)) # 0.</p><p>In fact, this determinant is linear in x and, if not identically zero, can only</p><p>vanish at points of a plane, which necessarily are the points of M.</p><p>Summarizing, if the condition (2.16) does not hold and the determinant</p><p>det(b1 b2 [f,gi](x)) is not identically zero (for one value of i), the distribution</p><p>Llc has dimension 2 at all points of M and dimension 3 everywhere else. As</p><p>a result, the state space of (2.13) is partitioned by Llc into three maximal</p><p>integral manifolds: the planeM and the two (open) half-spaces separated by</p><p>M.</p><p>We study now the same kind ofproblems for the full system (2.13)-(2.14),</p><p>whose state space is the manifold</p><p>N = !Ra x 80(3) .</p><p>To this end, a few preliminary remarks about the structure of the tangent</p><p>space to 80(3) are in order.</p><p>Recall that 80(3) is an embedded 3-dimensional submanifold of the man</p><p>ifold JR.a x a . As a consequence, the tangent space to 8 0 ( 3) at R can be viewed</p><p>as a(3-dimensional) subspace of the tangent space TRJR.axa. Let Xij denotes</p><p>the (i,j) element of a matrix X E JR.axa, and choose the natural (globally</p><p>defined on JR.a x a ) coordinate functions</p><p>{c/>ij(X) = Xij: 1:5 i,j :53}</p><p>This choice induces, at each X, a choice of a basis for TxlR.axa, namely the</p><p>set of tangent vectors</p><p>{ 8</p><p>8 : 1 :5 i, i :5 3} .</p><p>Xij</p><p>(2.18)</p><p>Using this basis, any tangent vector v at a point X of JR.axa will be represented</p><p>in the form</p><p>a a</p><p>V= L Vij--</p><p>0 • l ßXij</p><p>t,J=</p><p>where Vij denotes the (i,j) element of a 3 x 3 matrix V.</p><p>Now, consider the three matrices</p><p>( 0 1 0)</p><p>A1 = -1 0 0</p><p>0 0 0</p><p>A2 = ( ~</p><p>-1</p><p>0 1) 0 0</p><p>0 0</p><p>and the corresponding exponentials exp(A1t),exp(A2t), exp(Aat), with t E lll</p><p>An easy calculation shows that, for each 1 :5 k :53, exp(Akt) is an orthogonal</p><p>matrix, with determinant equal to 1. Thus, exp(Akt) E 80(3). Consider now</p><p>the mapping</p><p>pk : lR. -t 80(3)</p><p>t t-t (exp(Akt))R</p><p>2.5 Examples 103</p><p>where R is an element of 80(3). By construction, Pk(t) is a smooth curve on</p><p>80(3), passing through R at t = 0. Its tangent vector at t = 0, in the basis</p><p>(2.18), is represented by the matrix</p><p>[A(t)]t=o = AkR</p><p>i.e. has the form</p><p>Since the three matrices A1 , A2 , A3 are linearly independent, so are the three</p><p>corresponding tangent vectors { v1 , v2 , v3 }. Moreover, each Vk is an element of</p><p>TR80(3) by construction, and TR80(3) is 3-dimensional. As a consequence,</p><p>we can conclude that the set { v1 , v2 , v3 } is actually a basis of TR80(3).</p><p>In particular we see that, in the basis (2.18), any vector of TR80(3) can</p><p>be represented by means of a matrix of the form</p><p>where c1 , c2, c3 are real numbers, i.e. - in view of the special structure of</p><p>A1,A2,A3- in the form</p><p>( -~1 ~ ~:) R = 8(c)R</p><p>-C2 -C3 0</p><p>where c = col(c3,</p><p>-c2, cl).</p><p>We return now to the problern of discussing the partition of the state</p><p>space of system (2.13)-(2.14) induced by the distribution Llc. The system in</p><p>question has the form</p><p>with</p><p>p</p><p>jJ E</p><p>f(p)</p><p>9i(p)</p><p>(recall that we set Jw = x).</p><p>(x, R) E N = JR3 x 80(3)</p><p>TpN = TxlR3 X TR80(3)</p><p>( -S(x)J-1x, 8(J- 1x)R)</p><p>(bi, 0) 1 :::; i :::; r</p><p>Suppose r = 3 and note that, by definition, the control Lie algebra C</p><p>contains the six vector fields gi(p), [/, Ui](p), 1 :::; i :::; 3. An easy calculation</p><p>shows that</p><p>88(x)J-1x _1</p><p>[/, 9i](x, R) = ( ( ox )bi, -8(J bi)R) .</p><p>Note that the b/s are linearly independent vectors, and so are the vectors</p><p>J- 1bi and the matrices 8(J-1bi)R, 1 :::; i :::; 3. Thus, in view of the previous</p><p>104 2. Global Decompositions of Control Systems</p><p>discussion, we deduce that the matrices S(J- 1bi)R, 1 S i S 3 represent, in</p><p>the basis (2.18), three independent tangent vectors that span TRS0(3), for</p><p>each R E S0(3). On the other hand, the vectors bi, 1 S i S 3, span TxlR3 •</p><p>Therefore, we can conclude that the set of six vectors 9i(p), [f,gi](p), 1 S</p><p>i S 3 span the tangent space TxlR3 x TRS0(3) at each (x, R) E N. The</p><p>controllability rank condition (2.5) is satisfied at each point, and the partition</p><p>of N induced by Llc degenerates into one single element, namely N itself.</p><p>The study of the case in which r = 2 can be carried out in the same way,</p><p>using the condition (2.16) to show that the matrices S(J- 1bi)R, S(J- 1b2)R,</p><p>and</p><p>S(J-1 S(b)J- 1b)R</p><p>(with b = c1b1 + c2b2) are linearly independent, and proving that TpN is</p><p>spanned by gi(p), [f,gi](p), 1 Si S 2, [[j,g],g](p), and [j, [[f,g],g]](p), with</p><p>g(p) = CI91 (p) + C292(p).</p><p>3. Input-Output Maps and Realization Theory</p><p>3.1 Fliess Functional Expansions</p><p>The purpose of this section and of the following one is to describe represen</p><p>tations of the input-output behavior of a nonlinear system. We consider, as</p><p>usual, systems described by differential equations of the form</p><p>m</p><p>f(x) + Lgi(x)ui</p><p>i=l</p><p>(3.1)</p><p>Yi hj(X) 1~j~p.</p><p>This system, as in Chapter 1, is assumed to be defined on an open set</p><p>U of ~n. Moreover, throughout the present Chapter, we constantly suppose</p><p>also that the vector fields J, g1 , ... , gm are analytic vector fields defined on</p><p>U. Likewise, the output functions h1 , ... , hp are analytic functions defined</p><p>Oll U.</p><p>For the sake of notational convenience most of the times we represent the</p><p>output of the system as a vector-valued function</p><p>y = h(x) = col(h1 (x), ... , hp(x)).</p><p>We require first some combinatorial notations. Consider the set of m + 1</p><p>indices I = {0, 1, ... , m} (we represent here, as usual, indices with integer</p><p>numbers, but we could as well represent the m+ 1 indices with elements of any</p><p>set Z with card(Z) = m + 1). Let Ik be the set of all sequences (ik ... il) of k</p><p>elements ik, ... , h of I. An element of this set Ik will be called a multiindex</p><p>of length k. For consistency we define also a set I0 whose unique element is</p><p>the empty sequence (i.e. a multiindex of length 0), denoted 0. Finally, let</p><p>It is easily seen that the set I* can be given a structure of free monoid</p><p>with composition rule</p><p>with neutral element 0.</p><p>106 3. Input-Output Maps and Realization Theory</p><p>A formal power series in m + 1 noncommutative indeterminates and co</p><p>efficients in IR is a mapping</p><p>c:I*t--t!R.</p><p>In what follows we represent the value of c at some element ik ... i 0 of I*</p><p>with the symbol c(ik ... io).</p><p>The second relevant object we have to introduce is called an iterated in</p><p>tegral of a given set of functions and is defined in the following way. Let</p><p>T be a fixed value of the time and suppose u1 ... Um are real-valued piece</p><p>wise continuous functions defined on (0, T]. Foreach multiindex (ik ... io) the</p><p>corresponding iterated integral is a real-valued function of t</p><p>Eik···itio(t) = 1td~ik ... d~itd~io</p><p>defined for 0 :::; t:::; T by recurrence on the length, setting</p><p>and</p><p>~o(t)</p><p>~i(t) for 1:::; i:::; m</p><p>1td~ik ... d~io = 1t d~ik(r) 1r d~ik-1 ... d~io.</p><p>The iterated integral corresponding to the multiindex 0 is the real number</p><p>1.</p><p>Example 3.1.1. Just for convenience, let us compute the first few iterated</p><p>integrals, in a case where m = 1.</p><p>1t d6 = 1t u1(r) dr</p><p>1t d~od6 = 1t 1r u1 ( 0) d(}dr</p><p>1t d6d6 = 1t u1(r) 1r u1(0) dOdr, · · · 0, M > 0 suchthat</p><p>lc(ik ... io)l 0 such that, for each 0 ~ t ~ T and</p><p>each set of piecewise continuous functions u 1 , ... , Um defined on [0, T] and</p><p>subject to the constraint</p><p>(3.3)</p><p>the series</p><p>y(t) = c(0) + f: . :t c(ik ... io) 1tdeik ... deio (3.4)</p><p>k=O to, ... ,tk=O</p><p>is absolutely and uniformly convergent.</p><p>Proof. It is easy to see, from the definition of iterated integral that, if the</p><p>functions u 1 ... Um satisfy the constraint (3.3), then</p><p>{t tk+l</p><p>lo deik ... deia ~ (k + l)! ·</p><p>If the growth condition is satisfied, then</p><p>f c(ik · ·. io) 1tdeik ... deio ~ K[M(m + l)t]k+1 .</p><p>io, ... ,ik=O 0</p><p>As a consequence, if T is sufficiently small, the series (3.4) converges</p><p>absolutely and uniformly on [0, T].</p><p>defined on U. Given a point X 0 E U, consider the</p><p>formal power series defined by</p><p>c(0) = A(x0 )</p><p>c(ik ... io) = L9,0 ••• L9,k A(x0 ) •</p><p>(3.5)</p><p>Then, there exist real numbers K > 0 and M > 0 such that the growth</p><p>condition (3.2} is satisfied.</p><p>Proof. The reader is referred to the literature. 0, M > 0 such that</p><p>(3.9)</p><p>for alt k > 0, for alt multiindices (ik ... il), and alt (t,rk ... r1) E Sk.</p><p>Then, there exists a real number T > 0 such that, for each 0 ::; t ::; T and</p><p>each set of piecewise continuous functions u1, ... , Um defined on [0, T] and</p><p>subject to the constraint</p><p>the series</p><p>y(t) = wo(t)</p><p>max lui(r)l</p><p>. . . . . 184</p><p>4. 7 High Gain Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189</p><p>xiv Table of Contents</p><p>4.8 Additional Results on Exact Linearization ................. 194</p><p>4.9 Observers with Linear Error Dynamics .................... 203</p><p>4.10 Examples .............................................. 211</p><p>5. Elementary Theory of Nonlinear Feedback for Multi-Input</p><p>Multi-Output Systems .................................... 219</p><p>5.1 Local Coordinates Transformations ....................... 219</p><p>5.2 Exact Linearization via Feedback ......................... 227</p><p>5.3 Noninteracting Control .................................. 241</p><p>5.4 Achieving Relative Degree via Dynamic Extension .......... 249</p><p>5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263</p><p>5.6 Exact Linearization of the Input-Output Response .......... 277</p><p>6. Geometrie Theory of State Feedback: Tools . . . . . . . . . . . . . . . 293</p><p>6.1 The Zero Dynamics ..................................... 293</p><p>6.2 Contralied Invariant Distributions ........................ 312</p><p>6.3 The Maximal Contralied Invariant Distribution in ker(dh) ... 317</p><p>6.4 Cantrollability Distributions ............................. 333</p><p>7. Geometrie Theory of Nonlinear Systems: Applications .... 339</p><p>7.1 Asymptotic Stabilization via State Feedback ............... 339</p><p>7.2 Disturbance Decoupling ................................. 342</p><p>7.3 Noninteracting Control with Stability via Static Feedback ... 344</p><p>7.4 Noninteracting Control with Stability: Necessary Conditions . 364</p><p>7.5 Noninteracting Control with Stability: Sufficient Conditions .. 373</p><p>8. Tracking and Regulation .................................. 387</p><p>8.1 The Steady State Response in a Nonlinear System .......... 387</p><p>8.2 The Problem of Output Regulation ....................... 391</p><p>8.3 Output Regulation in the Case of Full Information. . . . . . . . . . 396</p><p>8.4 Output Regulation in the Case of Error Feedback .......... 403</p><p>8.5 Structurally Stahle Regulation ........................... 416</p><p>9. Global Feedback Design for Single-Input Single-Output Sys-</p><p>tems ...................................................... 427</p><p>9.1 Global Normal Forms ................................... 427</p><p>9.2 Examples of Global Asymptotic Stabilization .............. 432</p><p>9.3 Examples of Semiglobal Stabilization ...................... 439</p><p>9.4 Artstein-Sontag's Theorem .............................. 448</p><p>9.5 Examples of Global Disturbance Attenuation .............. 450</p><p>9.6 Semiglobal Stabilization by Output Feedback .............. 460</p><p>Table of Contents xv</p><p>A. Appendix A .............................................. 471</p><p>A.1 Some Facts from Advanced Calculus ...................... 471</p><p>A.2 Some Elementary Notions of Topology .................... 473</p><p>A.3 Smooth Manifolds ...................................... 474</p><p>A.4 Submanifolds .......................................... 479</p><p>A.5 Tangent Vectors ........................................ 483</p><p>A.6 Vector Fields .......................................... 493</p><p>B. Appendix B .............................................. 503</p><p>B.1 Center Manifold Theory ................................. 503</p><p>B.2 Some Useful Properties .................................. 511</p><p>B.3 Local Geometrie Theory of Singular Perturbations .......... 517</p><p>Bibliographical N otes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529</p><p>References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535</p><p>Index .................................................. .' ...... 545</p><p>1. Local Decompositions of Control Systems</p><p>1.1 Introduction</p><p>The subject of this Chapter is the analysis of a nonlinear control system,</p><p>from the point of view of the interaction between input and state and - re</p><p>spectively - between state and output, with the aim of establishing a number</p><p>of interesting analogies with some fundamental features of linear control sys</p><p>tems. For convenience, and in order to set up an appropriate b'asis for the</p><p>discussion of these analogies, we begin by reviewing - perhaps in a slightly</p><p>unusual perspective - a few basic facts about the theory of linear systems.</p><p>Recall that a linear multivariable control system with m inputs and p</p><p>outputs is usually described, in state space form, by means of a set of first</p><p>order linear differential equations</p><p>x = Ax+Bu</p><p>{1.1}</p><p>y = Cx</p><p>in which x denotes the state vector (an element of llln }, u the input vector</p><p>(an element of lllm ) and y the output vector (an element of JllP). The matrices</p><p>A, B, C are matrices of real numbers, of proper dimensions.</p><p>The analysis of the interaction between input and state, on one hand,</p><p>and between state and output, on the other hand, has proved of fundamen</p><p>tal importance in understanding the possibility of solving a large number</p><p>of relevant control problems, including eigenvalues assignment via feedback,</p><p>minimization of quadratic cost criteria, disturbance rejection, asymptotic out</p><p>put regulation, etc. Key tools for the analysis of such interactions - intro</p><p>duced by Kaiman around the 1960 - are the notions of reachability and ob</p><p>servability and the corresponding decompositions of the control system into</p><p>"reachable/unreachable" and, respectively, "observablefunobservable" parts.</p><p>We review in this section some relevant aspects of these decompositions.</p><p>Consider the linear system {1.1}, and suppose that there exists a d</p><p>subspace V of llln having the following property:</p><p>{i) V is invariant under A, i. e. is such that Ax E V for all x E V.</p><p>Without loss of generality, after possibly a change of coordinates, we</p><p>can assume that the subspace V is · the set of vectors having the form</p><p>v = col( v1, ... , vd, 0, ... , 0}, i.e. of all vectors whose last n - d components</p><p>2 1. Local Decompositions of Control Systems</p><p>are zero. If this is the case, then, because of the invariance of V under A, this</p><p>matrix assumes necessarily a block triangular structure</p><p>with zero entries on the lower-left block of n - d rows and d columns.</p><p>Moreover, if the subspace V is such that:</p><p>(ii) V contains the image (i.e. the range-space) of the matrix B, i.e. is such</p><p>that Bu E V for all u E JRm ,</p><p>then, after the same change of coordinates, the matrix B assumes the form</p><p>i.e. has zero entries on the last n - d rows.</p><p>Thus, if there exists a subspace l" which satisfies (i) and (ii), after a</p><p>change of coordinates in the state space, the first equation of (1.1) can be</p><p>decomposed in the form</p><p>XI Anxi + A.IzXz +BI u</p><p>Xz AzzXz .</p><p>By XI and x2 we denote here the vectors formed by taking the first d and,</p><p>respectively, the last n - d new coordinates of a point x.</p><p>The representation thus obtained is particularly interesting when studying</p><p>the behavior of the system und er the action of the control u. At any time T,</p><p>the coordinates of x(T) are</p><p>XI (T) = exp(An T)xi (0) + 1:xp(An (T - T) )Aiz exp(AzzT) dTXz (0)+</p><p>T 0</p><p>+ 1 exp(An(T-T))Biu(T)dT</p><p>xz(T) = exp(A22T)xz(O) .</p><p>From this, we see that the set of coordinates denoted by x 2 does not</p><p>depend on the input u but only on the timeT. In particular, if we denote by</p><p>x 0 (T) the point of ]Rn reached at timet= T when u(t) = 0 for all t E [O,T],</p><p>i.e. the point</p><p>X0 (T) = exp(AT)x(O)</p><p>we observe that any state which can be reached at timeT, starting from x(O)</p><p>at timet= 0, has necessarily the form x 0 (T) + v, where v is an element of</p><p>V.</p><p>This argument identifies only a necessary condition for a state x to be</p><p>reachable at timeT, i.e. that of being of the form x = x 0 (T) + v, with v E V.</p><p>However, under the additional assumption that:</p><p>1.1 Introduction 3</p><p>(iii) V is the smallest subspace which satisfies (i) and (ii) (i.e. is contained</p><p>in any other subspace of !Rn which satisfies both (i) and (ii)),</p><p>then this condition is also sufficient. As a matter of fact, it is known from the</p><p>theory of linear systems that (iii)</p><p>set of analytic vector fields and A a</p><p>real-valued analytic function defined on U. Let 0 and M > 0 such that the condition</p><p>{9.9} is satisfied.</p><p>From this result it is easy to obtain the desired representation of y(t) in</p><p>the form of a Volterra series expansion.</p><p>Theorem 3.2.3. Suppose the inputs u1 , ... , Um of the control system {9.1)</p><p>satisfy the constraint {9.10}. If T is sufficiently small, then for all 0 ~ t ~ T</p><p>the output Yi(t) of the system {9.1} may be expanded in the form of a Volterra</p><p>series, with kemels {9.14}, where Qt(x) and Pf(x) are as in {9.12}-{9.19} and</p><p>>. = hi.</p><p>This result may be proved either directly, by showing that the Volterra</p><p>series in question satisfies the equations (3.1), or indirectly, after establishing</p><p>a correspondence between the functional expansion described at the begin</p><p>ning of the previous section and the Volterra series expansion. We take the</p><p>second way.</p><p>For, observe that for all (ik •.. it) the kernel wi •... i1 (t, Tk, ••• , Tt) is ana</p><p>lytic in a neighborhood of the origin, and consider the Taylor series expansion</p><p>of this kernel as a function of the variables t - Tk, Tk - Tk-1, .•• , T2 - T1, T1.</p><p>This expansion has clearly the form</p><p>oo (t )n• ( )n1 no """ cf!O···.nk - Tk • · · T2 -Tl T1</p><p>L.J '• ···'1 nk!. .. n1 !no!</p><p>no, ... ,n~e=O</p><p>where</p><p>If we substitute this expression in the convolution integral associated with</p><p>Wi• ... i 1 , we obtain an integral of the form</p><p>The integral which appears in this expression is actually an iterated inte</p><p>gral of u1, ... , Um, and precisely the integral</p><p>1t (deo)n•llei • ... (.deo)n1 deit (lleo)no (3.15)</p><p>(where (lleo)n stands for n-times lleo).</p><p>Thus, the expansion (3.11) may be replaced with the expansion</p><p>3.2 Valterra Series Expansions 115</p><p>y(t) = ~ c[; Iot (cl{o)n</p><p>+ f: 'f f: c~k0.".".i~k1t (cl{o)nkcl{ik ... (deo)n1cl{il (deo)no</p><p>k=l i1 , ... ,ik=l no, ... ,nk=O 0</p><p>(3.16)</p><p>which is clearly an expansion of the form (3.4). Of course, one could rearrange</p><p>the terms and establish a correspondence between the coefficients cl), c~0.".".i~k</p><p>(i.e. the values of the derivatives of Wo and wik ... i 1 at t- Tk = · · · = T2- T1 =</p><p>T1 = 0) and the coefficients c(0), c(ik ... io) of the expansion (3.4), but this</p><p>is not needed at this point.</p><p>On the basis of these considerations it is very easy to find Taylor series</p><p>expansions of the kernels which characterize the Volterra series expansion</p><p>of Yi(t). We see from (3.16) that the coefficient c~0.".".i~k of the Taylor series</p><p>expansion of Wik ... i 1 coincides with the coefficient of the iterated integral</p><p>(3.15) in the expansion (3.4), but we know also from (3.7) that the coefficient</p><p>of the iterated integral (3.15) has the form</p><p>LnoL Ln1 Lnk-lL Lnkh ( o) I g;l I . . . I g;k I j X •</p><p>This makes it possible to write down immediately the expression of the Taylor</p><p>series expansions of all the kernels which characterize the Volterra series</p><p>expansion of y i ( t)</p><p>(3.17)</p><p>n2=0 n1 =0 no=O</p><p>(t- T2)n2(T2- TI)nlTfo</p><p>n2!n1!no!</p><p>and so on.</p><p>The last step needed in order to prove Theorem 3.2.3 is to show that the</p><p>Taylor series expansions of the kernels (3.14), with Qt(x) and P/(x) defined</p><p>as in (3.12), (3.13) for .X= hj(x) coincide with the expansions (3.17).</p><p>This is only a routine computation, which may be carried out with a little</p><p>effort by keeping in mind the well-known Campbell-Baker-Hausdorffformula,</p><p>which provides a Taylor series expansion of Pl(x). According to this formula</p><p>it is possible to expand Pl(x) in the following way</p><p>. oo tn</p><p>Pt'(x} = (4>~t}•9i o 4>{ (x) = L adfgi(x}-,</p><p>n=O n.</p><p>where, as usual, adjg = [f,adj- 1g] and adjg = g.</p><p>116 3. Input-Output Maps and Realization Theory</p><p>Example 3.2.1. In the case of bilinear systems, the flow ~{ may be clearly</p><p>given the following closed form expression</p><p>~{ (x) = (expAt)x.</p><p>From this it is easy to find the expressions of the kernels of the Volterra series</p><p>expansion of Yi ( t). In this case</p><p>Qt(x) = ci(expAt)x</p><p>Pl(x) = (exp(-At))Ni(expAt)x</p><p>and, therefore,</p><p>Wo(t) = Cj(expAt)x0</p><p>wi(t,r1) = Cj(expA(t- r1))Ni(expAr1)x0</p><p>Wi2 h (t, r2, r1) = Cj(exp A(t- r2))Ni2 (exp A(r2 - rl))Ni1 (exp Arl)x0</p><p>and so on.</p><p>space Oj as the space of all ~-linear combinations of functions</p><p>of the form hj and L9;0 ••• L9;,. hj, 0 ~ ik ~ m, 0 ~ k ... , 9i-1, 9i+l, ... , 9m}·</p><p>of dealing with sets of se</p><p>ries and defining certain Operations on these sets, it is useful to represent each</p><p>series as a formal infinite sum of "monomials". Let zo, ... , Zm denote a set</p><p>of m + 1 abstract noncommutative indeterminates and letZ= {zo, ... , Zm}·</p><p>With each multiindex (ik ... i 0 ) we associate the monomial (zi~o ... Zi0 ) and</p><p>we represent the series in the form</p><p>00 m</p><p>c=c(0) + L L c(ik ... io)zi• ... Zi0 • (3.25)</p><p>The set of all power series in m + 1 noncommutative indeterminates ( or,</p><p>in other words, in the noncommutative indeterminates zo, ... , Zm) and coeffi</p><p>cients in JRP is denoted with the symbol JRP ((Z)). A special subset of JRP ((Z)) is</p><p>the set of all those series in which the number of nonzero coefficients (i.e. the</p><p>number of nonzero terms in the sum (3.25)) is finite. Aseries ofthistype is a</p><p>polynomial in m + 1 noncommutative indeterminates and the set of all such</p><p>polynomials is denoted with the symbol JRP (Z). In particular R(Z) is the set</p><p>of all polynomials in the m + 1 noncommutative indeterminates zo, ... , Zm</p><p>and coefficients in IR.</p><p>An element of R(Z) may be represented in the form</p><p>d m</p><p>p=p(0)+ I: L p(io ... ik)Zi~o ... Zi0 (3.26)</p><p>where d is an integer which depends on p and p(0),p(i0 ••• ik) arereal num</p><p>bers.</p><p>3.4 Realization Theory 123</p><p>The sets JR.(Z) and JR.P ((Z)) may be given different algebraic structureso</p><p>They can clearly be regarded as JR.-vector spaces, by letting JR.-linear com</p><p>binations of polynomials and/or series be defined coefficient-wiseo The set</p><p>JR.(Z) may also be given a ring structure, by letting the operation of sum of</p><p>polynomials be defined coefficient-wise (with the neutral element given by</p><p>the polynomial whose coefficients are all zero) and the operation of product</p><p>of polynomials defined through the customary product of the corresponding</p><p>representations (3026) (in which case the neutral element is the polynomial</p><p>whose coefficients are all zero but p(0) is equal to 1)0 Later on, in the proof</p><p>of Theorem 3.403 we shall also endow JR.(Z) and JR.((Z)) with structures of</p><p>modules over the ring JR.(Z), but, for the moment, those additional structures</p><p>are not requiredo</p><p>What is important at this point is to know that the set JR.(Z) can also</p><p>be given a structure of Lie algebra, by taking the above-mentioned JR.-vector</p><p>space structure and defining a Lie bracket of two polynomials P1, P2 by setting</p><p>(p1,P2] = P2P1 - P1P20 The smallest sub-algebra of JR.(Z) which contains the</p><p>monamials z0 , 0 0 0, Zm will be denoted by .C(Z)o Clearly, .C(Z) may be viewed</p><p>as a subspace of the JR.-vector space JR.(Z), which contains z0 , 0 0 0, Zm and is</p><p>closed under Lie bracketing with z0 , 0 0 0, Zmo Actually, it is not difficult to see</p><p>that .C(Z) is the smallest subspace of JR.(Z) which has these propertieso</p><p>Now we return to the problern of realizing an input-output map repre</p><p>sented by a functional of the form (3.4)0 As expected, the existence of real</p><p>izations will be characterized as a property of the formal power series which</p><p>specifies the functional. We associate with the formal power series c two inte</p><p>gers which will be called, following Fliess, the Hankel rank and the Lie rank</p><p>of Co This is clone in the following mannero We use the given formal power</p><p>series c to define a mapping</p><p>Fe : JR.( Z) --t JR.P (( Z))</p><p>in the following way:</p><p>(a) The image under Fe of any polynomial in the set Z* = {zik 0 0 0 Zj0 E</p><p>JR.( Z) : (j k 0 o 0 j 0 ) E I*} (by definition, the polynomial associated with the</p><p>multiindex 0 E I* will be the polynomial in which all coefficients are zero</p><p>but p(0) which is equal to 1, i.eo the unit of JR.(Z)) is a formal power series</p><p>defined by setting</p><p>[Fe(Zjk o o o Zj0 )] (ir o o o io) = c(ir o o o ioJk o o o jo)</p><p>for all Jk o 0 oJo E l*o</p><p>(b) The map Fe is an JR.-vector space morphism of JR.(Z) into JR.P((Z))o</p><p>Note that any polynomial in JR.(Z) may be expressedas an IR-linear com</p><p>bination of elements of Z* and, therefore, the prescriptions (a) and (b) com</p><p>pletely specify the mapping Fe o</p><p>Looking at Fe as a morphism of JR.-vector spaces, we define the Hankel</p><p>rank PH(c) of c as the rank of Fe, i.eo the dimension of the subspace</p><p>124 3. Input-Output Maps and Realization Theory</p><p>Fe{IR(Z}) C JRP((Z}}.</p><p>Moreover, we define the Lie rank p L ( c) of c as the dimension of the sub-</p><p>space</p><p>Fe { C( Z)) c JRP ((Z}}</p><p>i.e. the rank of the mapping Fel.c(Z)·</p><p>It is easy to get a matrix representation of the mapping Fe. For, suppose</p><p>we represent an element p of IR(Z} with an infinite column vector of real</p><p>numbers whose entries are indexed by the elements of I* and the entry in</p><p>dexed by ik .. . jo is exactly p(jk .. . jo). Of course, p being a polynomial, only</p><p>finitely many elements of this vector are nonzero. In the same way, we may</p><p>represent an element c of JRP ((Z}} with an infinite column vector whose entries</p><p>are p-vectors of real numbers, indexed by the elements of I* and suchthat</p><p>the entry indexedir ... io is c(ir ... io). Then, any IR-vector space morphism</p><p>defined on IR(Z} with values in JRP ((Z}} will be represented by an infinite</p><p>matrix He, whose columns are indexed by elements of I* andin which each</p><p>block of p-rows of index (ir ... io) on the column of index (ik ... io) is exactly</p><p>the coeflicient</p><p>c(ir ... io ik · · · io)</p><p>of c. We leave to the reader the elementary check of this statement.</p><p>The matrix He is called the Hankel matrix of the series c. It is clear from</p><p>the above definition that the rank of the matrix He coincides with the Hankel</p><p>rank of c.</p><p>Example 3.4.1. If the set I consists of only one element, then it is easily</p><p>seen that I* can be identified with the set z+ of the non-negative integer</p><p>numbers. A formal power series in one indeterminate with coeflicients in IR,</p><p>i.e. a mapping</p><p>c: z+ --t IR</p><p>may be represented, like in (3.25), as an infinite sum</p><p>and the Hanke} matrix associated with the mapping Fe coincides with the</p><p>classical Hankel matrix associated with the sequence eo, c1 , ...</p><p>C2 ···) C3 • • •</p><p>.</p><p>In other words, the finiteness of the Lie rank PL(c) is a nec</p><p>essary condition for the existence of finite-dimensional realizations. We shall</p><p>see later on that this condition is also sufficient. For the moment, we wish to</p><p>investigate the role of the finiteness of the other rank associated with Fe i.e.</p><p>the Hankel rank. It comes from the definition that</p><p>PL(c) ~ PH(c)</p><p>so that the Hankel rank may be infinite when the Lie rank is finite. However,</p><p>there are special cases in which PH(c) is finite.</p><p>Lemma 3.4.2. Suppose j,g1, ... ,gm, h arelinear in x, i.e. that</p><p>f(x) = Ax, 91(x) = N1x, ... ,gm(x) = Nmx, h(x) = Cx</p><p>for suitable matrices A, N1 , ... , Nm, C. Let X 0 be a point of Rn . Let V de</p><p>note the smallest subspace of Rn which contains X 0 and is invariant under</p><p>A, N1, ... , Nm. Let W denote the Zargest subspace ofRn which is contained in</p><p>ker(C) and is invariant under A, N1, ... , Nm. The Hankel rank of the formal</p><p>power series {3.27} has the value</p><p>PH(c) = dim V- dim W n V= dim W ~V .</p><p>Proof. We have already seen, in section 2.4, that the subspace W may be</p><p>expressed in the following way</p><p>00 m</p><p>w = (kerc) n [n n ker(CNi~ ... Ni0 )]</p><p>r=O io, ... ,i,.=O</p><p>with No = A. With the same kind of arguments one proves that the subspace</p><p>V may be expressed as</p><p>oo m</p><p>V= span{x0 } + E E span{Ni• ... Nj0 X 0 }.</p><p>k=O io, ... ,j"=O</p><p>In the present case the Hankel matrix of Fe is such that the block of p</p><p>rows of index (ir ... io) on the column of index (jk .. . j 0 ), i.e. the coefficient</p><p>c( ir ... ioj k ••• io) of c has the expression</p><p>By factoring out this expression in the form</p><p>3.4 Realization Theory 127</p><p>it is seen that the Hankel matrix can be factored as the product of two</p><p>matrices, of which the one on the left-hand side has a kernel equal to the</p><p>subspace W, while the one on the right-hand side has an image equal to the</p><p>subspace V. From this the claimed result follows immediately. 0 and r > 0. Then</p><p>there exists a realization of c if and only if the Lie rank of c is finite.</p><p>130 3. Input-Output Maps and Realization Theory</p><p>Proof. Some more machinery is required. Foreach polynomial p E IR(Z) we</p><p>define a mapping Sp : JRP ((Z)) -+ JRP ((Z)) in the following way</p><p>(a) if p E Z* = {zik ... Zj0 E IR(Z): (jk ... jo) E /*} then Sp(c) isaformal</p><p>power series defined by setting</p><p>[Sz;k ... z;0 (c)) (ir ... io) = c(jk ... )oir ... io)</p><p>(b) if a1, a2 E IR and P1,P2 E IR(Z) then</p><p>Sa: 1p 1 +a:2 p 2 (c) = a1 Sp1 (c) + a2Sp2 (c) .</p><p>Moreover, suppose that, given a formal power series s1 E IR((Z)) and a</p><p>formal power series s2 E IR((Z)), the sum of the numerical series</p><p>oo m</p><p>s1(0)s2(0) + L L s1(ik .. . io)s2(ik .. . io) (3.32)</p><p>k=O io, ... ,ik=O</p><p>exists. If this is the case, the sum of this series will be denoted by (s1, s2).</p><p>We now turn our attention to the problern of finding a realization of c.</p><p>In order to simplify the notation, we assume p = 1 (i.e. we consider the</p><p>case of a single-output system). By assumption, there exist n polynomials</p><p>in C(Z), denoted p1 , ... ,pn, with the property that the formal power series</p><p>Fc(p1), ... , Fc(pn) are IR-linearly independent.</p><p>With the polynomials p1 , ... , Pn we associate a formal power series</p><p>(</p><p>n ) oo 1 ( n )k</p><p>w = exp ~XiPi = 1 + L k! ~XiPi</p><p>~=1 k=1 ~=1</p><p>(3.33)</p><p>where x1 , ... , Xn are real variables.</p><p>The series c which is to be realized and the series w thus defined are used</p><p>in order to construct a set of analytic functions of x1 , ... , Xn, defined in a</p><p>neighborhood of 0 and indexed by the elements of I*, in the following way</p><p>h(x)</p><p>hik ... i0 (x)</p><p>(c,w)</p><p>(Sz,k ... z,0 (c),w).</p><p>The growth condition (3.31) guarantees the convergence of the series on</p><p>the right-hand side for all x in a neighborhood of x = 0.</p><p>We will give now a sketch of the proofthat there exist m + 1 vector fields,</p><p>go(x), ... , Ym(x), defined in a neighborhood of 0, with the property</p><p>L h· · (x)- h· · ·(x) 9i tJe .•. z.o - z.~e ... z.oz. (3.34)</p><p>for all (ik ... io) E /*. This is be actually enough to prove the Theorem</p><p>because, at x = 0, the functions hik ... i0 (x) by construction aresuchthat</p><p>3.4 Realization Theory 131</p><p>h(O) c(0)</p><p>hi •... i0 (0) = c(ik ... io)</p><p>and this shows that the set { h, go, ... , 9m} together with the initial state</p><p>x = 0 is a realization of c.</p><p>To find the vector fields go, ... , 9m one proceeds as follows. Since the n</p><p>series Fc(pi), ... , Fc(pn) are IR-linear independent, it is easily seen that there</p><p>exists n monomials m1, ... , mn in the set Z* with the property that the</p><p>( n x n) matrix of real numbers</p><p>(</p><p>[Fc(p~)](si) ::~</p><p>[Fc(pd] (sn)</p><p>[Fc(p~)] (sl))</p><p>[Fc(pn)] (sn)</p><p>(3.35)</p><p>has rank n (where Sj denotes the multiindex associated with the monomial</p><p>mj)· It is easy to see that</p><p>For, if PiE Z*, then by definition</p><p>(where ti denotes the multiindex associated with the monomial Pi)· From</p><p>this, using linearity, one concludes that the above expression is true also in</p><p>the (general) case where Pi is an IR-linear combination of elements of Z*.</p><p>Using this property, we conclude that the j-th row of the matrix (3.35)</p><p>coincides with the value at 0 of the differential of one of the functions hi• ... io,</p><p>the one whose multiindex corresponds to the monomial mi.</p><p>Consider now the system of linear equations</p><p>in the unknown vector 9k(x). The coefficient matrix is nonsingular for all x</p><p>in a neighborhood of 0 (because at x = 0 it coincides - as we have seen -</p><p>with the matrix (3.35)). Thus, in a neighborhood of 0 it is possible to find a</p><p>vector field 9k(x) suchthat</p><p>L9• (Sm, (c), w) = (Sm;z• (c), w)</p><p>and this proves that (3.34) can be satisfied, at least for those hi .... io whose</p><p>multiindices correspond to the monomials m1, ... , mn.</p><p>132 3. Input-Output Maps and Realization Theory</p><p>The proofthat (3.34) holds for all other functions hi •... io (x) depends on</p><p>the fact that every formal power series in Fc(C(Z)) is an IR-linear combination</p><p>of Fc(pt), ... , Fc(pn), and is is not included here. The reader is referred to</p><p>the Iiterature for complete version of it. ... , 9m, h, X 0 } of c satisfies the</p><p>observability rank condition at X 0 (Corollary 3.4.6). From the definitions of</p><p>0 and ilo, one deduces that there exist n real-valued functions At, ... , An,</p><p>defined in a neighborhood U of x 0 , having the form</p><p>Ai(x) = Lvr ... Lv1 hj(X)</p><p>with Vt, ... , Vr vector fields in the set {!, 91> ... , 9m}, r (possibly) depending</p><p>on i and 1 ~ j ~ p suchthat the covectors dA1 (x0 ), ••• ,dAn(x0 ) are linearly</p><p>independent (i.e. span the cotangent space T;oU). From this property, using</p><p>the inverse function theorem, it is deduced that there exists a neighborhood</p><p>U H C U of X 0 such that the mapping</p><p>H:xf-t {At(x), ... ,An(x)}</p><p>is a diffeomorphism of U H onto its image H (U H).</p><p>From any two minimal realizations, labeled "a" and "b", we will construct</p><p>two of such mappings, denoted na and respectively Hb.</p><p>(ii) Let 01 , ..• , On be a set of vector fields, defined in a neighborhood U</p><p>of X 0 , having the form</p><p>m</p><p>(}i = f + L9iü)</p><p>j=l</p><p>with ü~ E lR. for 1 ~ j ~ m. Let 4i~ denote the flow of (}i and G denote the</p><p>mapping</p><p>G: (h, ... , tn) f-t 4if,. o • • · o 4i:1 (x0 )</p><p>defined on a neighborhood (-c;,c;)n ofO.</p><p>From any two minimal realizations, labeled "a" and "b", we will construct</p><p>two of such mappings, denoted aa and Gb (the same set of ü~'s being used</p><p>in both aa and Gb).</p><p>Recall that a minimal realization ua, gf, ... , 9~, ha, xa} satisfies the con</p><p>trollability rank condition at xa (Corollary 3.4.6). From the properties of Llc</p><p>and R (see Remark 2.2.3), one deduces that the distribution R is nonsingular</p><p>and n-dimensional around xa. Then, using the same arguments as the ones</p><p>used in the proof of Theorem 1.8.9, it is possible to see that there exists a</p><p>choice of urs and an open subset W of (O,c;)n suchthat the restriction of aa</p><p>to W is a diffeomorphism of W onto its image aa(W).</p><p>(iii) It is easily proved that if</p><p>{Ja a a ha a} d {/b b b hb b} ,go, ... ,gm, ,x an ,go, ... ,gm, ,x</p><p>are two realizations of the same formal power series c, then, for all 0</p><p>is suffi.ciently small, the mapping na 0 aa is composition of diffeomorphisms.</p><p>If also the realization "b" is minimal, Hb is indeed a diffeomorphism, but also</p><p>Gb must be a diffeomorphism of W onto its image, because of the equality</p><p>(3.38) and of the fact that the left-hand-side is itself a diffeomorphism. The</p><p>following diagram</p><p>/ ~-w w</p><p>~h yb</p><p>where ya = aa(W), yb = Gb(W), ya c Ufi, yb c Uk and W = Ha o</p><p>aa(W) = Hb o Gb(W), is a commutative diagram of diffeomorphisms. Thus,</p><p>we may define a diffeomorphism</p><p>F: ya--+ yb</p><p>as</p><p>F = (Hb)- 1 o Ha</p><p>whose inverse may also be expressed as</p><p>p-1 = aa 0 (Gb)-1 .</p><p>(3.39)</p><p>(3.40)</p><p>(v) By means of the same arguments as the ones already used in (iii) one</p><p>may easily prove a more general version of (3.38). More precisely, setting</p><p>m m</p><p>(Ja= r + 2:urvi ()b = fb + 2:utvi</p><p>i=l i=l</p><p>3.5 Uniqueness of Minimal Realizations 135</p><p>one may deduce that, for sufficiently small t</p><p>a ea aa ( ) b ob Gb ( ) H 0 iPt 0 t1, ... ,tn = H 0 iPt 0 t1, ... , tn .</p><p>Differentiating this one with respect to t and setting t = 0 one obtains</p><p>(Ha)*ea o Ga(t1, ... , tn) = (Hb)*()b o Gb(t1, ... , tn).</p><p>Because of the arbitrariness of v1 , ... , Vm one has then</p><p>(Ha)*gf o Ga(t1, ... , tn) = (Hb)*g~ o Gb(t1, ... , tn)</p><p>for all 0 ~ i ~ m. But these ones, in view of the definitions (3.39)-(3.40) may</p><p>be rewritten as</p><p>g~(x) = F*gf o F-1 (x)</p><p>for all XE Vb, thus proving (3.36).</p><p>(vi) Again, using the same arguments already used in (ii) one may easily</p><p>see that</p><p>i.e. that</p><p>hb(x) = ha o F-1 (x)</p><p>for all X E Vb, thus proving also (3.37).</p><p>of Lemma 4.1.1.</p><p>4.1 Local Coordinates Transformations 141</p><p>Proof. Observe that by definition of relative degree, using ( 4.2) we obtain for</p><p>all i, j such that i + j :::; r - 2</p><p>(dL}h(x), ad}g(x)} = 0 for all x around X 0</p><p>and</p><p>(dL}h(x 0 ), ad}g(x0 )} = ( -lr-1-i L9 L'j-1 h(x0 ) =1- 0</p><p>for all i, j suchthat i + j = r- 1.</p><p>The above conditions, all tagether, show that the matrix</p><p>( df;~;o) ) (g(xo) adtg(xo) .. . ad'j-1g(xo)) =</p><p>dLr-1h(x0 )</p><p>(</p><p>0 .. . (dh(x 0 ), ad'j-1 g(x0 )))</p><p>0 ... *</p><p>. ... *</p><p>(dL'j- 1 h(x0 ), g(x0 )) * *</p><p>(4.5)</p><p>has rank r and, thus, that the row vectors dh(x0 ), dLth(x0 ), ••• , dL'j-1 h(x0 )</p><p>are linearly independent. r+l ( x), ... , cf>n ( x) in such a way that</p><p>for all r + 1 :5 i :5 n and all x around X 0 •</p><p>Proof. By definition of relative degree, the vector g(x0 ) is nonzero, and, thus,</p><p>the distribution G = span{g} is nonsingular around X 0 • Being I-dimensional,</p><p>this distribution is also involutive. Therefore, by Frobenius' Theorem, we</p><p>deduce the existence of n-1 real-valued functions, A1 (x), ... , An-l (x), defined</p><p>in a neighborhood of X 0 , such that</p><p>span{dAl, ... ,dAn-d = Gl..</p><p>It is easy to show that</p><p>dim(Gl. + span{ dh, dLfh, ... , dL'j- 1 h}) = n</p><p>at X 0 • For, suppose this is false. Then</p><p>G(x0 ) n (span{dh,dLth, . .. ,dL'j-1h})l.(x0 ) -::p {0}</p><p>i. e. the vector g(x0 ) annihilates all the covectors in</p><p>span{dh, dLth, ... ,dL'j-1h}(x0 ) •</p><p>(4.6)</p><p>(4.7)</p><p>But this is a contradiction, because by definition (dL'j- 1 h(x0 ), g(x0 )} is</p><p>nonzero.</p><p>From (4.6), (4.7) and from the fact that span{dh,dLfh, ... ,dL'j-1h} has</p><p>dimension r, by Lemma 4.1.1, we conclude that in the set {A1, ... , An-d it is</p><p>possible to find n- r functions, without loss of generality A1, ... , An-r, with</p><p>the property that the n differentials dh,dLth, ... ,dL'j-1h,dAl, ... ,dAn-r,</p><p>are linearly independent at x0 • Since by construction the functions A1, ... ,</p><p>An-r are such that</p><p>for all x near X 0 and all 1 :5 i :5 n - r</p><p>this establishes the required result. Note that any other set of functions of the</p><p>form AHx) = Ai(x) +Ci, where Ci is a constant, satisfies the same conditions,</p><p>thus showing that the value of these functions at the point X 0 can be chosen</p><p>arbitrarily. q</p><p>The description of the system in the new coordinates Zi = cf>i ( x), 1 :5 i :5</p><p>n, is found very easily. Looking at the calculations already carried out at the</p><p>beginning, we obtain for z1, ... , Zr</p><p>dzr-1</p><p>dt</p><p>ßc/>1 dx 8h dx</p><p>= 8x dt = ax dt = Lth(x(t)) = cf>2(x(t)) = Z2(t)</p><p>{} ß(Lr-2 h)</p><p>= cf>r-l dx = I dx = Lr1- 1h(x(t)) = cf>r(x(t)) = Zr(t).</p><p>8x dt 8x dt</p><p>4.1 Local Coordinates Transformations 143</p><p>For Zr we obtain</p><p>~; = L/h(x(t)) + L9L/-1h(x(t))u(t).</p><p>On the right-hand side of this equation we must now replace x(t) with its</p><p>expression as a function of z(t), i.e. x(t) = p-1(z(t)). Thus, setting</p><p>a(z) = L9L/-1h(4i-1(z))</p><p>b(z) = L/h(4i-1(z))</p><p>the equation in question can be rewritten as</p><p>~; = b(z(t)) + a(z(t))u(t).</p><p>Note that at the point Z0 = 4i(x0 ), a(z0 ) =/= 0 by definition. Thus, the</p><p>coefficient a( z) is nonzero for all z in a neighborhood of z0 •</p><p>As far as the other new coordinates are concerned, we cannot expect any</p><p>special structure for the corresponding equations, if nothing eise has been</p><p>specified. However, if 4>r+I ( x), ... , 4>n ( x) have been chosen in such a way</p><p>that L9 4>i(x) = 0, then</p><p>dz· 84>·</p><p>dt~ = ox~ (f(x(t))+g(x(t))u(t)) = Lf4>i(x(t))+L9 4>i(x(t))u(t) = L 14>i(x(t)).</p><p>Setting</p><p>for all r + 1 ::=; i ::=; n</p><p>the latter can be rewritten as</p><p>Thus, in summary, the state-space description of the system in the new</p><p>coordinates will be as follows</p><p>z1 Z2</p><p>z2 Z3</p><p>Zr-1 = Zr</p><p>Zr b(z) + a(z)u</p><p>(4.8)</p><p>Zr+l = qr+I(z)</p><p>Zn qn(z) .</p><p>In addition to these equations one has to specify how the output of the</p><p>system is related to the new state variables. But, being y = h(x), it is imme</p><p>diately seen that</p><p>144 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>Y = Z1 · (4.9)</p><p>The structure of these equations is best illustrated in the block diagram</p><p>depicted in Fig. 4.1. The equations thus defined are said tobe in normal form.</p><p>We will find them useful in understanding how certain control problems can</p><p>be solved.</p><p>u</p><p>b(z) + a(z)u</p><p>Z1 = Y</p><p>Fig. 4.1.</p><p>Remark 4.1.3. Note that sometimes it is not easy to construct n-r functions</p><p>rPr+l (x), ... , rPn(x) suchthat L9 r/Ji(x) = 0, because this, as shown in the proof</p><p>of Proposition 4.1.3, amounts to solve a system of n- r partial differential</p><p>equations. Usually, it is much simpler to find nmctions rPr+1 (x), ... , rPn(x)</p><p>with the only property that the jacobian matrix of 4i(x) is nonsingular at x0 ,</p><p>and this is sufficient to define a Coordinates transformation. Using a trans</p><p>formation constructed in this way, one gets the same structure for the first r</p><p>equations, i. e.</p><p>z1 = Z2</p><p>z2 = Z3</p><p>Zr-1 = Zr</p><p>Zr = b(z) + a(z)u</p><p>but it is not possible to obtain anything special for the last n - r ones, that</p><p>therefore will appear in a form like</p><p>Zr+l = qr+l(z) + Pr+l(z)u</p><p>Zn = qn(z) + Pn(z)u</p><p>with the input u explicitly present.</p><p>and we</p><p>just take any choice of cjJ3 (x), cjJ4 (x) which completes the transformation. This</p><p>can be done, e.g. by taking</p><p>Z3 cP3(x) = X3</p><p>Z4 cP4(x) = Xt ·</p><p>The jacobian matrix of the transformation thus defined</p><p>{)~ = 2Xt 1 0 0 (</p><p>0 0 0 1)</p><p>ox 0 0 1 0</p><p>1 0 0 0</p><p>is nonsingular for all x, and the inverse transformation is given by</p><p>Xt = Z4</p><p>X2 = z2- zl</p><p>X3 = Z3</p><p>X4 Zt ·</p><p>Note also that ~(0) = 0. In these new coordinates the system is described by</p><p>4.2 Exact Linearization Via Feedback 147</p><p>i1 Zz</p><p>iz Z4 + 2z4(z4(zz- zi)- zD + (2 + 2z3)u</p><p>Z3 -Z3 + u</p><p>.z4 -2zr + zzz4 .</p><p>These equations are valid globally (because the transformation we considered</p><p>was a global coordinates transformation), but they are not in normal form</p><p>because of the presence of the input u in the equation for z3 .</p><p>lf one wants to get rid of u in this equation, it is necessary to use a</p><p>different function</p><p>at these points.</p><p>In the new coordinates, the system appears as</p><p>(</p><p>0 1</p><p>i = ~ ~</p><p>which is linear and controllable. -1</p><p>] [8ci> f(x)]</p><p>8z 8x x=(x) = Tci>(x)</p><p>ä(x) = o:(x) + ß(x)kci>(x).</p><p>Then, it is easily seen that</p><p>( 4.17)</p><p>154 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>(0 I 0</p><p>... ~} [ ~! (f(x) + g(x)ä(x))] x=-'(z)</p><p>0 0 1</p><p>=</p><p>0 0 0 ... 1</p><p>0 0 0 ... 0</p><p>[ ~! (g(x)ß(x))] x=(x) solving the State Space Exact Linearization</p><p>problern consists of the following steps</p><p>156 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>- from f(x) and g(x), construct the vector fields</p><p>g(x), ad,g(x), ... , adj-2g(x), ad'j- 1g(x)</p><p>and check the conditions (i) and {ii),</p><p>- if both are satisfied, solve for .X(x) the partial differential equation (4.20),</p><p>- set</p><p>- set</p><p>-Ln1.X(x)</p><p>a(x) - -~;--</p><p>- L9L'j-1</p><p>.X(x)</p><p>1</p><p>ß(x) = LgL'j 1 .X(x)</p><p>!l>(x) = col{.X(x), LJ.X(x), ... , L'j-1 .X(x)) .</p><p>(4.23)</p><p>(4.24)</p><p>The feedback defined by the functions (4.23) is called the linearizing feed</p><p>back and the new coordinates defined by ( 4.24) are called the linearizing</p><p>coordinates. We illustrate now the whole Exact Linearization procedure in a</p><p>simple example.</p><p>Example 4.2.5. Consider the system</p><p>:i:= (xa{1x~x2)) + (1:x2) u.</p><p>x2{1 + xl) -xa</p><p>In order to check whether or not this system can be transformed into a</p><p>linear and controllable system via state feedback and coordinates transfor</p><p>mation, we have to compute the functions ad,g(x) and ad}g(x) and test the</p><p>conditions of Theorem 4.2.3.</p><p>Appropriate calculations show that ad,g(x) =</p><p>( 0 ) - Xl</p><p>-{1 + x1){1 + 2x2)</p><p>and that</p><p>At x = 0, the matrix</p><p>(g(x) ad,g(x) ad}g(x) l.~. = 0 ~1 n</p><p>4.2 Exact Linearization Via Feedback 157</p><p>has rank 3 and therefore the condition (i) is satisfied. lt is also easily checked</p><p>that the product [g, ad 1 g] ( x) has a form</p><p>[g,adty)(x) ~ m</p><p>and therefore also the condition (ii) is satisfied, because the matrix</p><p>(g(x) adtg(x) [g,adtg](x))</p><p>has rank 2 for all x near x = 0.</p><p>In the present case, it is easily seen that a function A(x) that solves the</p><p>equation</p><p>ÖA</p><p>äx(g(x) adtg(x))=O</p><p>is given by</p><p>A(x) =x1.</p><p>From our previous discussion, we know that considering this as "output"</p><p>will yield a system having relative degree 3 (i.e. equal to n) at the point</p><p>x = 0. We double-check and observe that</p><p>L9 A(x) = 0, L9 LJA(x) = 0, L9 LJA(x) = (1 + x1)(l + x2)(l + 2x2)- x1x3</p><p>and L9 L}A(0) = 1. Locally around x = 0, the system will be transformed</p><p>into a linear and controllable one by means of the state feedback</p><p>-L}A(x) + v</p><p>u = L9 L}A(x) =</p><p>-x~(l + X2)- X2X3(l + X2)2 - X1 (1 + xt)(l + 2x2)- X1X2(l + Xl) +V</p><p>(1 + xt)(l + x2)(l + 2x2)- X1X3</p><p>and the coordinates transformation</p><p>z1 = A(x) = x1</p><p>Z2 LtA(x) = X3(l + x2)</p><p>Z3 L}A(x) = X3X1 + (1 + xt)(l + X2)X2 .(x)</p><p>defined in a neighborhood U of X 0 , the corresponding linear system is defined</p><p>on the open set 4>(U). For obvious reasons, it is interesting to have 4>(U)</p><p>containing the origin of ]Rn, and in particular to have 4>(x 0 ) = 0. In this case,</p><p>in fact, one could for instance use linear systems theory concepts in order to</p><p>asymptotically stabilize at z = 0 the transformed system and then use the</p><p>stabilizer thus found in a composite loop to the purpose of stabilizing the</p><p>nonlinear system at x = X 0 (see Remark 4.2.3).</p><p>This is indeed the case when X 0 is an equilibrium of the vector field f ( x).</p><p>In this case, in fact, choosing the solution .A(x) of the differential equation</p><p>with the additional constraint .A(x0 ) = 0, as is always possible, one gets</p><p>4>(x0 ) = 0, as already shown at the beginning of the section (see Remark</p><p>4.2.2).</p><p>If X 0 is not an equilibrium of the vector field f(x), one can manage to</p><p>have this occurring by means of feedback. As a matter of fact, the condition</p><p>4>(x0 ) = 0, replaced into (4.14), necessarily yields</p><p>160 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>i.e.</p><p>f(x 0 ) + g(x0 )a(x0 ) = 0 .</p><p>This clearly expresses the fact that the point X 0 is an equilibrium of the</p><p>vector field f(x) + g(x)a(x), and can be obtained if and only if the vectors</p><p>f(x 0 ) and g(x0 ) are such that</p><p>where c is a real number. If this is the case, an easy calculation shows that</p><p>the linearizing coordinates are still zero at x0 (if A(x) is such), because, for</p><p>all2$;i$;n</p><p>L/-1 A(x0 ) = cL9 L/-2 A(X0 ) = 0 .</p><p>Moreover, the linearizing feedback o:(x) is suchthat</p><p>LnA(X0 )</p><p>a(x0 ) -- 1 - -c</p><p>- L9Lj 1 A(x0 ) -</p><p>as expected. (x),</p><p>that will transform the state space equation</p><p>x = f(x) + g(x)u</p><p>into a linear and controllable one. However, the real output of the system, in</p><p>the new coordinates</p><p>y = h(4>-1(z))</p><p>will in general continue to be a nonlinear function of the state z.</p><p>occurs if and only if</p><p>V = Im(B AB ... An-! B)</p><p>(where Im(·) denotes the image of a matrix) and, moreover, that under this</p><p>assumption the pair (Au, BI) is a reachable pair, i.e. satisfies the condition</p><p>rank( BI AuB1 ... At11 B1) = d</p><p>or, what is the same, has the property that for each x1 E JRd there exists an</p><p>input u, defined on [0, T], satisfying</p><p>XI= 1T exp(Au(T- r))B1u(r) dr.</p><p>Then, if V is such that the condition (iii) is also satisfied, starting from</p><p>x(O) it is possible to reach at timeT every state of the form x0 (T) + v, with</p><p>v E V.</p><p>This analysis suggests the following considerations. Given the linear con</p><p>trol system (1.1), Iet V be the smallest subspace of !Rn satisfying (i) and (ii).</p><p>Associated with V there is a partition of !Rn into subsets of the form</p><p>Sv={xEIRn :x=p+v,vEV}</p><p>characterized by the following property: the set of points reachable at time</p><p>T starting from x(O) coincides exactly with the element - of the partition</p><p>- which contains the point exp(AT)x(O), i.e. with the subset Bexp(AT)x(O)·</p><p>Note also that these sets, i.e. the elements of this partition, are d-dimensional</p><p>planes parallel to V ( see Fig.l.l).</p><p>Fig. 1.1.</p><p>4 1. Local Decompositions of Control Systems</p><p>An analysis similar to the one developed so far can be carried out by</p><p>examining the interaction between state and output. In this case one considers</p><p>instead a d-dimensional subspace W of !Rn characterized by the following</p><p>properties:</p><p>(i) W is invariant under A</p><p>(ii) W is contained in the kernel (the null-space) of the matrix C (i.e. is such</p><p>that Cx = 0 for all x E W)</p><p>(iii) W is the largest subspace which satisfies (i) and (ii) (i.e. contains any</p><p>other subspace of !Rn which satisfies both (i) and (ii)).</p><p>Properties (i) and (ii) imply the existence of a change of coordinates in</p><p>the state space which induces on the control system (1.1) a decomposition of</p><p>the form</p><p>x1 = Anx1 + A12x2 + B1u</p><p>±2 = A22X2 + B2u</p><p>y = C2X2</p><p>(in the new coordinates, the elements of W are the points having x2 = 0).</p><p>This decomposition shows that the set of coordinates denoted by x1 has no</p><p>influence on the output y. Thus, any two initial states whose x2 coordinates</p><p>are equal, produce identical outputs under any input, i.e. are indistinguish</p><p>able. Actually, since two states whose x2 coordinates that are equal are such</p><p>that their difference is an element of W, we deduce that any two states whose</p><p>difference is an element of W are indeed indistinguishable.</p><p>Condition (iii), in turns, guarantees that only the pairs of states charac-~</p><p>terized in this way (i.e. having a difference in W) are indistinguishable from</p><p>each other. As a matter of fact, it is known from the linear theory that the</p><p>condition (iii) is satisfied if and only if</p><p>W=ker ( gA )</p><p>CAn-1</p><p>(where ker(·) denotes the kernel of a matrix) and, if this is the case, the pair</p><p>(C2, A22) is an observable pair, i.e. satisfies the condition</p><p>k ( c~22 )</p><p>ran C2~ii-d-1</p><p>=n-d</p><p>or, what is the same, has the property that</p><p>1.2 Notations 5</p><p>As a consequence, any two initial states whose difference does not belong</p><p>to W are distinguishable from each other, in particular by means of the</p><p>output produced under zero input.</p><p>Again, we may synthesize the above discussion with the following con</p><p>siderations. Given a linear control system, Iet W be the largest subspace of</p><p>!Rn satisfying (i) and (ii). Associated with W there is a partition of !Rn into</p><p>subsets of the form</p><p>Sp={xEIRn :x=p+w,wEW}</p><p>characterized by the following property: the set of points indistinguishable</p><p>from a point p coincides exactly with the element - of the partition - which</p><p>contains p, i.e. with the set Sv itself. Note that again these sets - as in the</p><p>previous analysis - are planes parallel to W.</p><p>In the following sections of this Chapter and in the following Chapter we</p><p>shall deduce similar decompositions for nonlinear control systems.</p><p>1.2 Notations</p><p>Throughout these notes we shall study multivariable nonlinear control sys</p><p>tems with m inputs u1, ... , Um and p outputs Y1, ... , Yv described, in state</p><p>space form, by means of a set of equations of the following type</p><p>m</p><p>x = f(x) + L9i(x)ui</p><p>i=l</p><p>The state</p><p>X= (x1, ... ,xn)</p><p>is assumed to belong to an open set U of !Rn.</p><p>(1.2)</p><p>The mappings /, g1 , ..• , 9m which characterize the equation (1.2) are !Rn</p><p>valued mappings defined on the open set U; as usual, f(x),gl(x), ... ,gm(x)</p><p>denote the values they assume at a specific point x of U. Whenever con</p><p>venient, these mappings may be represented in the form of n-dimensional</p><p>vectors of real-valued functions of the real variables x1 , ... , Xn, namely</p><p>(1.3)</p><p>The functions ht, ... , hv which characterize the equation (1.2) are real-valued</p><p>functions also defined on U, and h1 (x), ... , hv(x) denote the values taken at</p><p>a specific point x. Consistently with the notation (1.3), these functions may</p><p>be represented in the form</p><p>6 1. Local Decompositions of Control Systems</p><p>(1.4)</p><p>In what follows, we assume that the mappings f, g1 , ... , Ym and the func</p><p>tions h1 , ... , hv are smooth in their arguments, i.e. that all entries of (1.3) and</p><p>(1.4) are real-valued functions of x 1 , ... , Xn with continuous partial deriva</p><p>tives of any order. Occasionally, this assumption may be replaced by the</p><p>stronger assumption that the functions in question are analytic on their do</p><p>main of definition.</p><p>The dass (1.2) describes a large number of physical systems of interest in</p><p>many engineering applications, including of course linear systems. The latter</p><p>have exactly the form (1.2), provided that f(x) is a linear function of x, i.e.</p><p>f(x) = Ax</p><p>for some n x n matrix A ofreal numbers, g1(x), ... ,gm(x) are constant func</p><p>tions of x, i.e.</p><p>Yi(x) = bi</p><p>where b1 , ... , bm are n x 1 vectors of real numbers, and h1 (x), ... , hv(x) are</p><p>again linear in x, i.e.</p><p>hi(X) = CiX</p><p>where c1 , ... , Cp are 1 x n (i.e. row) vectors of real numbers.</p><p>We shall encounter in the sequel many examples of physical control sys</p><p>tems that can be modeled by equations of the form (1.2). Note that, as a</p><p>state space for (1.2), we consider a subset U of !Rn rather than !Rn itself. This</p><p>Iimitation may correspond either to a constraint established by the equations</p><p>themselves ( whose solutions may not be free to evolve on the whole of !Rn) or</p><p>to a constraint specifically imposed on the input, for instance to avoid points</p><p>in the state space where some kind of "singularity" may occur. We shall be</p><p>more specific later on. Of course, in many cases one is allowed to set U = !Rn.</p><p>The mappings /, g1 , ... , Ym are smooth mappings assigning to each point</p><p>x of U a vector of !Rn, namely f(x), g1 (x), ... , Ym(x). For this reason, they</p><p>are frequently referred to as smooth vector fields defined on U. In many</p><p>instances, it will be convenient to manipulate - together with vector fields -</p><p>also dual objects called covector fields, which are smooth mappings assigning</p><p>to each point x (of a subset U) an element of the dual space (!Rn)*.</p><p>As we will see in a moment, it is quite natural to identify smooth covector</p><p>fields ( defined on a subset U of !Rn) with 1 x n (i.e. row) vectors of smooth</p><p>functions of x. For, recall that the dual space V* of a vector space V is</p><p>the set of all linear real-valued functions defined on V. The dual space of</p><p>an n-dimensional vector space is itself an n-dimensional vector space, whose</p><p>elements are called covectors. Of course, like any linear mapping, an element</p><p>w* of V* can be represented by means of a matrix. In particular, since w*</p><p>is a mapping from the n-dimensional space V to the I-dimensional space IR,</p><p>this representation is a matrix consisting of one row only, i.e. a row vector.</p><p>On these grounds, one can assimilate (!Rn)* with the set of all n-dimensional</p><p>1.2 Notations 7</p><p>row vectors, and describe any subspace of (!Rn)* as the collection of alllinear</p><p>combinations of some set of n-dimensional row ve~tors (for instance, the rows</p><p>of some matrix having n columns).</p><p>setting again</p><p>1</p><p>u = a(z) ( -b(z) + v) (4.25)</p><p>4.2 Exact Linearization Via Feedback 161</p><p>on the normal form of the equations, one obtains, if r (x) has a jacobian matrix with the following structure</p><p>4.3 The Zero Dynamics 165</p><p>c</p><p>0</p><p>D ( ... ) 1</p><p>{)ijj</p><p>* 8x</p><p>c</p><p>0</p><p>D G</p><p>0</p><p>D</p><p>1 0</p><p>0 0</p><p>and therefore nonsingular.</p><p>In the new coordinates we obtain equations in normal form, which, be</p><p>cause of the linearity of the system, have the following structure</p><p>i1 = Z2</p><p>i2 = zs</p><p>Zr-1 = Zr</p><p>Zr = Rf. + STJ +Ku</p><p>iJ = PE,+ QTJ</p><p>where R and S are row vectors and P and Q matrices, of suitable dimensions.</p><p>The zero dynamics of this system, according to our previous definition, are</p><p>those of</p><p>iJ = QTJ.</p><p>The particular choice of the last n - r new coordinates (i.e. of the elements</p><p>of TJ) entails a particularly simple structure for the matrix Q. As a matter of</p><p>fact, is easily checked that</p><p>dzr+l</p><p>dt</p><p>dZn-1</p><p>dt</p><p>dzn</p><p>dt</p><p>= dxn-r-1 ( ) ( ) dt = Xn-r t = Zn t</p><p>= dxn-r</p><p>~ = Xn-r+l(t) = -box1(t)- · · ·- bn-r-1Xn-r(t) + Z1(t)</p><p>= -boZrH(t)- ~ · ·- bn-r-1Zn(t) + Z1(t)</p><p>from which we deduce that</p><p>(_~</p><p>1 0</p><p>-bLJ</p><p>0 1</p><p>Q=</p><p>0 0</p><p>-b1 -b2</p><p>From the particular form of this matrix, it is clear that the eigenvalues of</p><p>Q coincide with</p><p>the zeros of the numerator polynomial of H(s), i.e. with the</p><p>166 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>zeros of the transfer function. Thus it is concluded that in a linear system</p><p>the zero dynamics are linear dynamics with eigenvalues coinciding with the</p><p>zeros of the transfer function of the system. r+l (x), 0 0 0, 4>n(x) with the property that L 9f/>;(x) = 0 (see Remark</p><p>4o1.3), one can still identify the zero dynamics of the system working on</p><p>equations of the form</p><p>i1 Z2</p><p>i2 = Z3</p><p>Zr-1 Zr</p><p>Zr = b(~, 17) + a(~, ry)u</p><p>iJ q(~, 17) + p(~, ry)u 0</p><p>As a matter of fact, having seen that the zero dynamics of the system</p><p>describe its behavior when the output is forced to be zero, we impose this</p><p>condition on the equations aboveo We obtain, as before, ~(t) = 0 and</p><p>0 = b(O, ry(t)) + a(O, ry(t))u(t) 0</p><p>Replacing u(t) from this equation into the last one, yields a differential equa</p><p>tion for 77( t)</p><p>0 ( ) ( )b(O,ry)</p><p>17 = q O,TJ - p O,ry a(O,ry)</p><p>which describes the zero dynamics in the new coordinates choseno</p><p>Example 403050 Suppose we want to calculate the zero dynamics ofthe system</p><p>already analyzed in the Example 401.5. In this case we don't have the normal</p><p>form, but the calculation of the zero dynamics is still very easy. Setting</p><p>z1 = z2 = 0 in the second equation yields</p><p>403 The Zero Dynamics 169</p><p>Z4- 4z.f</p><p>U=- o</p><p>2 + 2z3</p><p>Replacing this, and z1 = z2 = 0, in the third and fourth equation yields</p><p>Z4- 4z.f</p><p>Z3 = - Z3 - ----=-</p><p>2+ 2z3</p><p>Z4 = -2z~</p><p>which describes the zero dynamics of the systemo (t) = L'jh(x(t)) + L9 L'j-1h(x(t))u(t)</p><p>this turnsouttobe exactly the same constraint previously obtained for u(t),</p><p>but now expressed in terms of the functions which characterize the original</p><p>equationso</p><p>Fig. 4.6.</p><p>Note that, since the differentials dL}h(x), 0 $ i $ r -1, are linearly inde</p><p>pendent at X 0 (Lemma 4.1.1), the set Z* is a smooth manifold of dimension</p><p>n - r, near X 0 • The state feedback</p><p>170 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>-L/h(x)</p><p>u*(x)- ----=-...,....--</p><p>- L9 Lj- 1h(x)</p><p>by construction is such that</p><p>( d~~~~~) ) (f(x) + g(x)u*(x))</p><p>dLj- 1h(x)</p><p>= L}h(x) + L9 Lth(x)u*(x) _ '· .. (</p><p>Lth(x) + L9 h(x)u*(x) ) ( f~~~~~ )</p><p>L/h(x) + L9~~-1 h(x)u*(x) - Vt~h(x) .</p><p>Thus</p><p>( d~~~~~) ) (f(x) + g(x)u*(x)) = 0</p><p>dLj- 1h(x)</p><p>for all x E Z* (because h(x) = Lth(x) = · · · = Lj-1h(x) = 0 if x E Z*) and</p><p>therefore the vector field</p><p>f*(x) = f(x) + g(x)u*(x)</p><p>is tangent to Z*. As a consequence, any trajectory of the closed loop system</p><p>x = f*(x)</p><p>starting at a point of Z* remains in Z* (for small values oft). The restriction</p><p>f*(x)lz· of f*(x) to Z* is a well-defined vector field of Z*, which exactly</p><p>describes - in a coordinate-free setting - the zero dynamics of the system.</p><p>We will illustrate in the sequel a series of relevant issues in which the</p><p>notion of zero dynamics, and in particular its asymptotic properties, plays</p><p>an important role. For the time being we can show, for instance, how the</p><p>zero dynamics are naturally imposed as internal dynamics of a closed loop</p><p>system whose input-output behavior has been rendered linear by means of</p><p>state feedback. For, consider again a system in normal form and suppose the</p><p>feedback controllaw ( 4.25) is imposed, under which the input-output behav</p><p>ior becomes identical with that of a linear system consisting of a string of r</p><p>integrators between input and output (see Fig. 4.4). The closed loop system</p><p>thus obtained is described by the equations (4.26), that can be rewritten in</p><p>the form</p><p>~ = A~+Bv</p><p>iJ q(~, rJ)</p><p>Y = c~</p><p>4.3 The Zero Dynamics 171</p><p>with</p><p>A~ (~~~ ::~) B~m</p><p>c = (1 0 0).</p><p>If the linear subsystem is initially at rest and no input is applied, then y ( t) = 0</p><p>for all values of t, and the corresponding internal dynamics of the whole</p><p>(closed loop) system are exactly those of (4.28), namely the zero dynamics</p><p>of the open loop system.</p><p>We conclude the section by showing that the interpretation of</p><p>iJ(t) = q(O, 17(t)) ,</p><p>as of the dynamics describing the internal behavior of</p><p>the system when the</p><p>output is forced to track exactly the output y(t) = 0, can easily be extended</p><p>to the case in which the output to be tracked is any arbitrary function.</p><p>Consider the following problem, which is called the Problem of Reproducing</p><p>the Reference Output YR(t). Find, if any, pairs consisting of an initial state</p><p>X 0 and of an input function u0 (-), defined for all t in a neighborhood oft= 0,</p><p>suchthat the corresponding output y(t) of the system coincides exactly with</p><p>YR(t) for all t in a neighborhood oft= 0. Again, we are interested in finding</p><p>all such pairs (x 0 , U 0 ). Proceeding as before, we deduce that y(t) = YR(t)</p><p>necessarily implies</p><p>for all t and all 1 ~ i ~ r .</p><p>Setting</p><p>{R(t) = col(yR(t), y~) (t), ... , y~-l) (t))</p><p>we then see that the input u(t) must necessarily satisfy</p><p>y~)(t) = b({R(t),17(t)) + a({R(t),17(t))u(t)</p><p>where 17(t) is a solution of the differential equation</p><p>f7(t) = q({R(t),17(t)).</p><p>(4.29)</p><p>(4.30)</p><p>Thus, if the output y(t) has to track exactly YR(t), then necessarily the</p><p>initial state of the system must be set to a value such that {(0) = {R(O),</p><p>whereas 17(0) = 11° can be chosen arbitrarily. According to the value of 11°,</p><p>the input must be set as</p><p>u(t) = y~)(t)- b({R(t),17(t))</p><p>a({R(t), 17(t)}</p><p>(4.31)</p><p>172 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>where ry(t) denotes the solution of the differential equation ( 4.30) with initial</p><p>condition ry(O) = 1J0 • Note also that for each set of initial data ~(0) = ~R(O)</p><p>and ry(O) = 1J0 the input thus defined is the unique input capable of keeping</p><p>y(t) = YR(t) for all times.</p><p>The ( forced) dynamics ( 4.30) clearly correspond to the dynamics describ</p><p>ing the "internal" behavior of the system when input and initial conditions</p><p>have been chosen in such a way as to constrain the output to track exactly</p><p>YR(t). Note that the relations (4.30) and (4.31) describe a system with input</p><p>~R(t), output u(t) and state ry(t) that can be interpreted as a realization of</p><p>the inverse of the original system.</p><p>4.4 Local Asymptotic Stabilization</p><p>In this section we illustrate how the notion of zero dynamics can be helpful</p><p>in dealing with the problern of asymptotically stabilizing a nonlinear system</p><p>at a given equilibrium point. Suppose, as usual, a nonlinear system of the</p><p>form</p><p>x = f(x) + g(x)u</p><p>is given, with f(x) having an equilibrium point at X 0 that, without loss of</p><p>generality, we assume to be X 0 = 0. The problern we want to discuss is the</p><p>one of finding a smooth state feedback</p><p>u = a(x)</p><p>defined locally around the point X 0 = 0 and preserving the equilibrium, i.e.</p><p>such that a(O) = 0, with the property that the corresponding closed loop</p><p>system</p><p>x = f(x) + g(x)a(x)</p><p>has a locally asymptotically stable equilibrium at x = 0. We shall refer to it</p><p>as to the Local Asymptotic Stabilization Problem.</p><p>First of all, we review a rather well-known property, by discussing to</p><p>what extent the possibility of solving the problern in question depends on</p><p>the properties of the linear approximation of the system near x 0 = 0. To this</p><p>end, recall that the linear approximation of a system having an equilibrium</p><p>at X 0 = 0 is defined by expanding f(x) and g(x) as (see Remark 4.2.7)</p><p>f(x) Ax + h(x)</p><p>g(x) B+g1(x)</p><p>with</p><p>A- [ö!]</p><p>- ÖX x=O</p><p>and B = g(O).</p><p>From the point of view of the stability properties of the closed loop sys</p><p>tem, the importance of the linear approximation is essentially related to the</p><p>following result.</p><p>4.4 Local Asymptotic Stabilization 173</p><p>Proposition 4.4.1. Suppose the linear approximation is asymptotically sta</p><p>bilizable, i.e. either the pair (A, B) is controllable or- in case the pair (A, B)</p><p>is not controllable - the uncontrollable modes correspond to eigenvalues with</p><p>negative real part. Then, any linear feedback which asymptotically stabilizes</p><p>the linear approximation is also able to asymptotically stabilize the original</p><p>nonlinear system, at least locally. If the pair (A, B) is not controllable and</p><p>there exist uncontrollable modes associated with eigenvalues with positive real</p><p>part, the original nonlinear system cannot be stabilized at all.</p><p>Proof. Suppose the linear approximation is asymptotically stabilizable. Let</p><p>F be any matrix such that (A + BF) has all eigenvalues with negative real</p><p>part, and set</p><p>u=Fx</p><p>on the nonlinear system. The resulting closed loop system</p><p>x = f(x) + g(x)Fx = (A + BF)x + h(x) + g1 (x)Fx</p><p>has a linear approximation having all the eigenvalues in the left-half complex</p><p>plane. Thus, the Principle of Stability in the First Approximation proves that</p><p>the nonlinear closed loop system is locally asymptotically stable at x = 0.</p><p>Conversely, suppose the linear approximation has uncontrollable modes</p><p>associated with eigenvalues having positive real part. Let u = a(x) be any</p><p>smooth state feedback. The corresponding closed loop system has a linear</p><p>approximation of the form (recall that a(O) = 0)</p><p>x = [ß[f(x) + g(x)a(x)]] x = (A + B [ßa] )x</p><p>ßx x=O ßx x=O</p><p>which has eigenvalues with positive real part, irrespectively of what a is.</p><p>Thus, again by the Principle of Stability in the First Approximation, the</p><p>nonlinear closed-loop system is unstable at x = 0.</p><p>hold in the presence of</p><p>some eigenvalue of Q with zero real parto</p><p>In order to design the stabilizing control law there is no need to know</p><p>explicitly the expression of the system in normal form, but only to know</p><p>the fact that the system has a zero dynamics with a locally asymptotically</p><p>stable equilibrium at 17 = 00 Recalling how the coordinates z1, 0 0 0 , Zr and</p><p>the functions a(~, 17) and b(~, 17) are related to the original description of</p><p>the system, it is easily seen that, in the original coordinates, the stabilizing</p><p>control law assumes the form</p><p>which is particularly interesting, because expressed in terms of quantities</p><p>that can be immediately calculated from the original datao</p><p>By this method we can asymptotically stabilize also systems whose linear</p><p>approximation has uncontrollable modes corresponding to eigenvalues on the</p><p>imaginary axis, i.eo we can solve critical problems of local asymptotic stabi</p><p>lization, provided we know that for some choice of an "output" the system</p><p>has an asymptotically stable zero dynamicso</p><p>Example 404010 Consider the system already discussed in the Example 401.50</p><p>Its linear approximation at x = 0 is described by matrices A and B of the</p><p>form</p><p>0</p><p>0</p><p>-1</p><p>0</p><p>B~g(D)~ m</p><p>176 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>and has exactly one uncontrollable mode corresponding to the eigenvalue</p><p>A = 0. However, its zero dynamics (see Example 4.3.5)</p><p>Z4- 4zt</p><p>Z3 = -Z3 - ----=-</p><p>2+2z3</p><p>Z4 = -2z:</p><p>have an asymptotically stable equilibrium at Z3 = Z4 = 0. Thus, from our</p><p>previous discussion, we conclude that a control law of the form</p><p>u = LgL;h(x) ( -Ljh(x)- eoh(x)- c1L1h(x)}</p><p>locally stabilizes the equilibrium x = 0. 0. 0 there exist 8 > 0 and K > 0</p><p>suchthat</p><p>II x(O) II</p><p>output of the</p><p>180 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>system "tracks" the desired output YR(t), with an error which can be made</p><p>to converge to zero, as t --+ oo, with arbitrarily fast exponential decay.</p><p>Of course, an ever present concern in the design of control laws is that</p><p>the variables representing the internal behavior of the system remain bounded</p><p>when a specific controllaw is imposed. In the present situation, the asymp</p><p>totic analysis of the internal behavior of the system obtained by imposing</p><p>the controllaw (4.36) on (4.1) can be carried out in the following way.</p><p>First of all, note that if we consider, as we implicitly did, the reference</p><p>output YR(t) to be a fixed function of time, then the system (4.1) driven</p><p>by the input ( 4.36) can be interpreted as a time-varying nonlinear system.</p><p>In particular, looking at the behavior of the state variables in the Coordi</p><p>nates used for the normal form, it is easy to check that z1 , ... , Zr, satisfy the</p><p>identities</p><p>(i-1) + (i-1)</p><p>Zi = YR e</p><p>whereas 'TJ satisfies a differential equation of the form</p><p>iJ = q(~R(t) + x(t), TJ)</p><p>where, as in (4.29),</p><p>~R(t) = col(yR(t), y~) (t), ... , y~-l) (t))</p><p>and</p><p>x(t) = col(e(t),e(ll(t), ... ,e(r-ll(t)).</p><p>(4.38)</p><p>Equation ( 4.38), in view of the remarks at the end of section 4.3, can be seen</p><p>as an equation describing the "response" of the inverse system "driven" by</p><p>the function YR(t) + e(t).</p><p>Sufficient conditions for the boundedness of the zi(t)'s and TJ(t) are ex</p><p>pressed in the following statement.</p><p>Proposition 4.5.1. Suppose YR(t), y~) (t), ... , y~-l) (t) are defined for all</p><p>t :2:: 0 and bounded. Let TJR(t) denote the solution of</p><p>(4.39)</p><p>satisfying 'TJR(O) = 0. Suppose this solution is defined for all t ;:::: 0, bounded</p><p>and uniformly asymptotically stable. Finally, suppose the roots of the polyno</p><p>mial</p><p>Sr+ Cr-lSr-l + · · · + C1S +Co= 0</p><p>all have negative real part. Then, for sufficiently small a > 0, if</p><p>iziW)- y~-llW)I 0 there exists d > 0 such</p><p>that</p><p>iziW)- Y~-l)W)I izi(t)- Y~-l)(t)i IITJ(t)- 'TJR(t)ii</p><p>We see then that the solution of the problern of</p><p>asymptotically tracking the output of a reference model entails the use of a</p><p>more general type of state feedback than the one considered so far, in that</p><p>it includes also an internal dynamics. A feedback of this form is called a</p><p>dynamic state feedback.</p><p>Summarizing the whole discussion, we can conclude that, if the relative</p><p>degree ofthe model (4.42)-(4.43) is larger than or equal to the relative degree</p><p>of the system, there exists a dynamic feedback of the form ( 4.46) yielding an</p><p>output y(t) that converges asymptotically to the output YR(t) of the model,</p><p>for every possible input w(t), and for every possible initial state x(O), ((0).</p><p>184 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>The analysis of the internal asymptotic properties of the system thus</p><p>obtained is quite similar to the one developed earlier for the system (4.1)</p><p>(4.36). As a matter offact, it is immediate to check that, in the present case,</p><p>the closed loop system can be described in proper coordinates by equations</p><p>of the form</p><p>(</p><p>x</p><p>iJ</p><p>=</p><p>=</p><p>=</p><p>A(+Bw</p><p>Kx</p><p>q(F( + x,TJ)</p><p>with r = col(C, CA, ... , CAr-l ). The first one of these equations describes</p><p>the dynamics of the model (driven by its own input), the second one the</p><p>dynamics of the error (which is an autonomaus equation), the latter the</p><p>dynamics of the inversesystemdriven by the function C( + x1 .</p><p>4.6 Disturbance Decoupling</p><p>The normal form introduced in section 4.1 is also useful in understanding how</p><p>the output response of a given system can be protected from disturbances</p><p>affecting the state. Consider a system of the form</p><p>x = f(x) + g(x)u + p(x)w</p><p>y = h(x)</p><p>in which w represents an undesired input, or disturbance. We want to examine</p><p>under what conditions there exists a static state feedback control</p><p>u = a(x) + ß(x)v</p><p>yielding a closed loop system in which the output y is completely independent</p><p>of, or decoupled from, the disturbance w. This problern is commonly known</p><p>as Disturbance Decoupling Problem.</p><p>As usual, we discuss first the solution looking at the normal form of the</p><p>equations. Let the system have relative degree r at x 0 , and suppose the vector</p><p>field p( x) which multiplies the disturbance in the state equation being such</p><p>that</p><p>for all 0 :5 i :5 r - 1 and all x near X 0 •</p><p>If we write the state space equations choosing the same coordinates used</p><p>before to describe the normal form, we obtain</p><p>8z1 dx 8hdx</p><p>= = 8x dt 8x dt</p><p>= L1h(x(t)) + L9 h(x(t))u(t) + Lph(x(t))w(t)</p><p>= L1h(x(t)) = z2(t)</p><p>4.6 Disturbance Decoupling 185</p><p>because, by assumption, Lph(x) = 0 for all t suchthat x(t) is near x 0 • A</p><p>similar situation happens for all other subsequent equations, and thus we get</p><p>For Zr we still obtain</p><p>dz2</p><p>dt</p><p>dzr-1</p><p>dt</p><p>= zs(t)</p><p>= Zr(t) .</p><p>d:; = L'jh(x(t)) + L 9 L'j-1h(x(t))u(t)</p><p>because, again, LpL'j-1h(x) = 0. Thus, the first r equations are exactlythe</p><p>same as those of a normal form of a system without disturbance. This is not</p><p>anymore the case for the remaining ones, that will now appear depending</p><p>also on the disturbance w.</p><p>Using, as in the previous sections, a vector notation, we can' rewrite the</p><p>system in the form</p><p>Zr-1 = Zr</p><p>Zr = b(~, 71) + a(~, 7J)u</p><p>TJ = q(~, 7]) + k(~, 1J)W •</p><p>In addition we have, as usual</p><p>Y = Z1 •</p><p>Suppose now the following state feedback is chosen</p><p>b(~, 1J) V</p><p>u=---+--.</p><p>a(~, 71) a(~, 71)</p><p>This feedback yields a system which is described by the equations</p><p>Z1 = Z2</p><p>z2 = zs</p><p>Zr-1 = Zr</p><p>Zr = V</p><p>r, = q(~, 1J) + k(~, 1J)W</p><p>from which it is easily seen that the output, i.e. the state variable z1, has</p><p>been completely decoupled from the disturbance w.</p><p>186 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>The block-diagram interpretation of the closed loop system thus obtained,</p><p>described in Fig. 4.8, clearly explains what happened. The effect of the input</p><p>has been that of isolating from the output that part of the system on which</p><p>the disturbance has effect.</p><p>TJ = q({, 71) + k({, 1J)W</p><p>Fig. 4.8.</p><p>We have thus found a sufficient condition for the existence of solutions</p><p>to the problern of decoupling the output of a system from a disturbance, and</p><p>explicitly constructed a decoupling feedback. It is not difficult to prove that</p><p>the condition in question is also necessary, as we shall see in a moment. For</p><p>convenience, we summarize all the results of interest in a formal Statement</p><p>in which, for more generality, we specify the decoupling feedback in terms of</p><p>the functions f(x), g(x) and h(x) characterizing the original description of</p><p>the system.</p><p>Proposition 4.6.1. Suppose the system has relative degree r at X 0 • The</p><p>problern of finding a feedback u = o:(x) + ß(x)v, defined locally around X 0 ,</p><p>such that the output of the system is decoupled from the disturbance can be</p><p>solved if and only if</p><p>far all 0 ~ i ~ r - 1 and all x near X 0 • ( 4.4 7)</p><p>lf this is the case, then a solution is given by</p><p>L'jh(x) v</p><p>u =- L 9L/-1h(x) + L9L/-1h(x) .</p><p>Proof. We have only to prove the necessity. Let u = o:(x) + ß(x)v denote</p><p>any feedback decoupling the output from the disturbance, and consider the</p><p>corresponding closed loop system</p><p>:i: = f(x) + g(x)o:(x) + g(x)ß(x)v + p(x)w</p><p>y = h(x).</p><p>4.6 Disturbance Decoupling 187</p><p>By assumption, the output y has tobe independent of w, and this has to</p><p>be true also when v(t) = 0 for all t, i.e. for the system</p><p>x = f(x) + g(x)a(x) + p(x)w</p><p>y h(x) .</p><p>By repeating in the present case calculations similar to the ones clone in</p><p>section 4.1, we obtain</p><p>y{ll(t) = Lf+gah(x(t)) + Lph(x(t))w(t)</p><p>from which we see that y(t) can be independent from w(t) only if Lph(x) = 0</p><p>for all t such that x( t) is near X 0 • Assuming this condition satisfied, we</p><p>calculate y(2l(t) and get</p><p>y(2l(t) = LJ+90 h(x(t)) + LpLf+gah(x(t))w(t).</p><p>Again, we conclude that LpLf+gah(x) must be 0. The same arguments can</p><p>be repeated for all the higher derivatives of y(t), until we get</p><p>y(r)(t) = L'i+90h(x(t)) + LvL/f.~0h(x(t))w(t)</p><p>and see that also LvL/f.~0h(x) must be 0. We conclude that if the feedback</p><p>decouples y from w, necessarily</p><p>forallO~i~r-1 and all x near x 0 •</p><p>This is indeed the condition we wanted to prove because, as seen in the</p><p>proof of Lemma 4.2.1, L}+90h(x) = L}h(x) for all 0 ~ i ~ r- 1. 0 there exist</p><p>8 > 0 and K > 0 such that</p><p>II x(O) II</p><p>weaker than those established</p><p>beforeo Looking at the form of the closed loop system</p><p>x f(x) + g(x)a(x) + g(x)ß(x)v + (g(x)'Y(x) + p(x))w</p><p>y h(x)</p><p>it is immediately understood that what is needed is the possibility of finding</p><p>a function 'Y(x) suchthat</p><p>(g(x)'Y(x) + p(x)) E il.l(x) for all x near X 0</p><p>0 (4.49)</p><p>This condition is equivalent to</p><p>for all 0 :::::; i :::::; r - 1 and all x near X 0 , and this, recalling the definition of</p><p>relative degree, is in turn equivalent to</p><p>LpL}h(x)</p><p>LpL/- 1h(x)</p><p>0 for all 0 :::::; i :::::; r - 2</p><p>-L9 L/- 1h(x)'Y(x)</p><p>for all x near X 0</p><p>0 The second one of these can always be satisfied, by choosing</p><p>LpLr-lh(x)</p><p>"'(X) - - _ __.:,!--:-_</p><p>' - L 9 L/-1h(x) 0</p><p>Thus, the necessary and sufficient condition for solving the problern of decou</p><p>pling the disturbance from the output by means of a feedback that incorpo</p><p>rates measurements of the disturbance is simply the first oneo Note that this</p><p>4. 7 High Gain Feedback 189</p><p>condition weakens the condition (4.47) ofProposition 4.6.1, in thesensethat</p><p>now LpL}h(x) = 0 must be zero for all the values of i up to r- 2 included,</p><p>but not necessarily for i = r - 1.</p><p>If this is the case, a controllaw solving the problern of decoupling y from</p><p>w is clearly given by</p><p>L/h(x) v LpL/-1h(x)</p><p>u =- L9 L/-1h(x) + L9 L/-1h(x) - L9 L/-1h(x) w.</p><p>Note that, geometrically, the condition (4.49) has the following interpre</p><p>tation: at each x, the vector p(x) can be decomposed in the form</p><p>p(x) = c1(x)g(x) + Pl(x)</p><p>where c1(x) is a real-valued function and Pl(x) is a vector in n.L(x). This</p><p>can be expressed in the form</p><p>p(x) E D.L(x) + span{g(x)} for all x near x 0 (4.50)</p><p>thus showing to what extent the condition ( 4.48) can be weakened by allowing</p><p>a feedback which incorporates measurements of the disturbance.</p><p>4. 7 High Gain Feedback</p><p>In this section we consider again the problern of the design of a locally sta</p><p>bilizing feedback and we show that - under the stronger-- assuihpti.,on that</p><p>the zero dynamics are asymptotically stable in the first approximation - a</p><p>nonlinear system can be locally stabilized by means of output feedback. First</p><p>of all, we consider the case of a system having relative degree 1 at the point</p><p>X 0 = 0, and we show that asymptotic stabilization can be achieved by means</p><p>of a memoryless linear feedback.</p><p>Proposition 4.7.1. Consider a system oftheform (4.1}, with f(O) = 0 and</p><p>h(O) = 0. Suppose this system has relative degree 1 at x = 0, and suppose the</p><p>zero dynamics are asymptotically stable in the first approximation, i. e. that</p><p>all the eigenvalues of the matrix</p><p>Q _ [aq(e, 11)]</p><p>- 01J ((,fJ)=(O,O)</p><p>have negative real part. Consider the closed loop system</p><p>where</p><p>x = f(x) + g(x)u</p><p>u = -Kh(x)</p><p>(4.51)</p><p>190 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>{ K > 0 if L9 h(O) > 0</p><p>K</p><p>Ko the equilibrium x = 0 of (4.51} is asymptotically stable.</p><p>Proof. An elegant proof of this result is provided by the Singular Perturba</p><p>tions Theory (see Appendix B). Suppose L 9h(O) .(x) = L9 h(x).</p><p>At each x E E, the system x' = F(x, 0) has (n-1) trivial eigenvalues and one</p><p>nontrivial eigenvalue. Since by assumption >.(O) .(x) of Jx is negative.</p><p>We will show now that the reduced vector field associated with the system</p><p>( 4.52) coincides with the zero dynamics vector field. The easiest way to see</p><p>this is to express the first equation of (4.51) in normal form, that is</p><p>4.7 High Gain Feedback 191</p><p>i b(z,TJ)+a(z,TJ)u</p><p>iJ q(z,TJ)</p><p>in which z = h(x) E IR and TJ E JRn-l. Accordingly, system (4.52) becomes</p><p>E: (~) = (r::b(z,TJ)- a(z,TJ)Kz).</p><p>TJ r::q(z, TJ)</p><p>In the coordinates of the normal form, the set Eis the set of pairs (z, TJ) such</p><p>that z = 0. Thus</p><p>fR(x)=Px[ßF(x,r::)] =(0 I)(b(O,TJ))</p><p>ßc e=O, xEE q(O, TJ)</p><p>from which we deduce that the reduced system is given by</p><p>iJ = q(O,TJ)</p><p>i.e. by the zero dynamics of (4.1). Since, by assumption, the latter is asymp</p><p>totically stable in the first approximation at TJ = 0, it is possible to conclude</p><p>that there exists E:o > 0 such that, for each E: E (O,c0 ) the system (4.52) has</p><p>an isolated equilibrium point Xe near 0 which is asymptotically stable. Since</p><p>F(O,r::) = 0 for allE: E (O,co), we have necessarily Xe= 0, and this completes</p><p>the proof.</p><p>From the feedback (4.53), which actually is a state feedback, it is possible</p><p>to deduce an output feedback in the following way. Observe that the function</p><p>L}h(x(t)), for 0:::; i:::; r-1, coincides with the i-th derivative ofthe function</p><p>ytt) with respect to time. Thus the function w(t) is related to y(t) by</p><p>4. 7 High Gain Feedback 193</p><p>and can therefore be interpreted as the output of a system obtained by</p><p>cascade-connecting the original system with a linear filter having transfer</p><p>function exactly equal to the polynomial n(s). Clearly such a filter is not</p><p>physically realizable, but it is not difficult to replace it by a suitable physi</p><p>cally realizable approximation, without impairing the stability properties of</p><p>the corresponding closed loop system.</p><p>To this end, all we need is the following simple result.</p><p>Proposition 4.7.2. Suppose the system</p><p>x = f(x) - g(x)k(x)K</p><p>is asymptotically stable in the first approximation (at the equilibrium x = 0).</p><p>Then, if T is a sufficiently small positive number, also the system</p><p>x f(x)- g(x)(</p><p>( (1/T)( -( + k(x)K)</p><p>is asymptotically stable in the first approximation (at (x,() = (0,0)).</p><p>Proof. The proof of this result is another simple application of Singular Per</p><p>turbations Theory. For, change the variable ( into a new variable .z defined</p><p>by</p><p>z = -( + k(x)K</p><p>and note that the system in question becomes</p><p>Tz</p><p>f(x) - g(x)( -z + k(x)K)</p><p>ßk</p><p>-z + T K ßx [f(x) - g(x)( -z + k(x)K)] = -z + Tb(z, x) .</p><p>This system has exactly the structure of the system (B.16), with c = T.</p><p>There is only one nontrivial eigenvalue, which is equal to -1, and the reduced</p><p>system, which is given by</p><p>x = f(x)- g(x)k(x)K</p><p>is by assumption asymptotically stable in the first approximation. There</p><p>fore, for sufficiently small positive T the equilibrium (x, () = (0, 0) is indeed</p><p>asymptotically stable in the first approximation.</p><p>via state feedback and coordinates transformation,</p><p>we have to compute the vector fields adfg, ad}g, ad}g and test the conditions</p><p>of Theorem 402030 Appropriate calculations show that</p><p>ad1g(x) = ( ~I )</p><p>-2x3</p><p>ad}g(x) = ( ~ ) o</p><p>-2xi</p><p>Since</p><p>[g,ad,g](x) = UJ</p><p>one observes that</p><p>[g, adJg] ~ span{g, ad,g}</p><p>and therefore the distribution span{g, ad1g} is not involutiveo Thus, the con</p><p>ditions of Theorem 40203 are violated (see Remark 40208)0 However, in this</p><p>case</p><p>and</p><p>inv(span{g, adfg, ad}g}) = inv(span{g, adfg, [g, ad,g], ad}g})</p><p>='Pan{ m ' (D ' m ' m}</p><p>so that the conditions of Theorem 40802 are satisfied, with v = 3 and k = 30</p><p>Then, the maximal relative degree one can obtain for this system is r = v = 30</p><p>In order to find an output for which the relative degree is 3, one has to solve</p><p>the differential equation (4o57), which yields, in this case</p><p>-\(x) = x1 0</p><p>198 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>From the previous discussion, it is clear that choosing a feedback</p><p>-L}-\(x) +v 2</p><p>u = LgLJ.A(x) = -x1 + v</p><p>and new coordinates</p><p>Z1 = A(x) = Xt</p><p>Z2 = LJA(x) = x2 - x~</p><p>Z3 = LJ.A(x) = X3</p><p>one obtains a system which contains a linear subsystem of dimension 3. Con</p><p>pleting the choice of coordinates with</p><p>7J = 7J(X) = X4</p><p>yields</p><p>it = Z2</p><p>i2 = Z3</p><p>Z3 = V</p><p>r, = z1 + z~ .i(x)vi(x)</p><p>i=l</p><p>and note that</p><p>n n</p><p>0 = .~.)vj(x), ci(x)vi(x)] = L(Lv;Ci(x))vi(x) .</p><p>i=l i=l</p><p>Thus</p><p>(Lv 1 Ci(x) ... LvnCi(x)) = dci(x)(v1(x) ... Vn(x)) = 0.</p><p>i.e. dci(x) = 0, and ci(x) is independent of x for all 1 ::; i ::; n. Using this</p><p>property we deduce that</p><p>n</p><p>adjjj(x) = L cfadj-1g(x)</p><p>i=l</p><p>202 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>and, with a simple induction, also that</p><p>n</p><p>ad}g(x) = L::C~adj- 1 g(x)</p><p>i=l</p><p>for all k > n, where the c~ 's are real numbers. From this we have</p><p>(d5.(x),ad}g(x)) is independent of x for all k 2:0</p><p>and, using again the formula (4.2), it is easy to conclude that (4.65) holds</p><p>for any value of s, k.</p><p>(iv) Arrange (4.64) and (4.65) in the matrix relation</p><p>(</p><p>d1;~(~) ) (9(x) adig(x) ... adr;- 1g(x)) = constant.</p><p>dL"~- 1 A(x)</p><p>I</p><p>dh(x)</p><p>The last row of the matrix on the left-hand side is linearly dependent from</p><p>the first n ones through constant coefficients (because of the constancy of</p><p>the right-hand</p><p>side). Then, since the right factor of the left-hand side is</p><p>nonsingular for all x near x 0 , we deduce that</p><p>n-1</p><p>dh(x) = L bidLj5.(x)</p><p>i=O</p><p>where b0 , ... , bn_1 arereal numbers. This implies</p><p>n-1</p><p>h(x) = L biLj5.(x) + c (4.66)</p><p>i=O</p><p>where c is a constant. Moreover, this constant is zero if 5.(x0 ) = 0, because</p><p>of the assumptions h(x0 ) = 0 and f(x 0 ) = 0.</p><p>(v) We know for the theory developed in section 4.2 (see in particular</p><p>(4.23)-(4.24)) that the system (4.59), after the feedback</p><p>-Lj5.(x) 1 _</p><p>v= _ + - v</p><p>L9-L~- 1 A(x) LgL~-1 A(x)</p><p>I I</p><p>in the new coordinates</p><p>i-1-</p><p>Zi =Li A(x) o:::;i:::;n-1</p><p>is a controllable linear system. But in these new coordinates the output map</p><p>(4.60) also is linear, as (4.66) shows. Thus, the proof of the sufficiency is</p><p>completed.</p><p>The proof of the necessity requires only Straightforward calculations, and</p><p>is left to the reader. (x) under which</p><p>the vector field f and the output map h become respectively</p><p>[{}4> f(x)] = Az + k(Cz)</p><p>ßx z=-1 (z)) = Cz</p><p>where (A, C) is an observable pair and k is a n-vector valued function of a</p><p>real variable.</p><p>If this is the case, then an observer of the form</p><p>e = (A + GC)~ -Gy+ k(y)</p><p>yields an observation error (in the z coordinates)</p><p>e = ~- z = ~- 4>(x)</p><p>governed by the differential equation</p><p>e = (A+GC)e</p><p>which is linear and spectrally assignable (via the n-vector Gof real numbers).</p><p>Motivated by these considerations, we examine the following problem,</p><p>called the Observer Linearization Problem. Given a system without input</p><p>(4.67), and an initial state x 0 , find (if possible) a neighborhood U0 of X 0 , a</p><p>204 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>coordinates transformation z = 4l(x) defined on uo, a mapping k: h(U0 ) -</p><p>!Rn , such that</p><p>[</p><p>8 4> f(x)] = Az + k(Cz)</p><p>8x x=4i-l(z)</p><p>(4.68</p><p>h(4>-1 (z)) Cz (4.69</p><p>for all z E 4i(U0 ), for some suitable matrix A and row vector C, satisfyini</p><p>the condition</p><p>rank ( gA ) = n .</p><p>CAn-1</p><p>(4.70</p><p>The conditions for the solvability of this problern can be described a</p><p>follows.</p><p>Lemma 4.9.1. The Observer Linearization Problem is solvable only if</p><p>Proof. The observability condition ( 4. 70) implies the existence of a nonsin</p><p>gular n x n matrix T and a n x 1 vector G such that</p><p>T(A + GC)T-1 = (0 0 . . . 0 0)</p><p>1 0 . . . 0 0</p><p>. . . . . . .</p><p>0 0 . . . 1 0 (4.72</p><p>cr- 1 = ( o o . . . o 1) .</p><p>Suppose (4.68) and (4.69) hold, and set</p><p>z ~(x) = T4l(x)</p><p>k(y) T(k(y)- Gy).</p><p>Then, it is easily seen that</p><p>(o o ... o 1)z</p><p>[8~ f(x)]</p><p>8x z=.j-l(z)</p><p>(</p><p>0 0</p><p>1 0</p><p>0 0</p><p>-~- ~) z</p><p>1 0</p><p>+k((o o ... o 1)z).</p><p>4.9 Observers with Linear Error Dynamics 205</p><p>From this, we deduce that there is no loss of generality in assuming that</p><p>the pair (A, C) that makes (4.68) and (4.69) satisfied has directly the form</p><p>specified by the right-hand sides of (4.72). Now, set</p><p>z=P(x) =col(z1(x), ... ,zn(x)).</p><p>If (4.68) and (4.69) hold, we have, for all x E U0</p><p>h(x)</p><p>az1 f(x)</p><p>ÜI; f(x) =</p><p>ax</p><p>OZn f(x)</p><p>ax</p><p>Zn(x)</p><p>k1 (zn (x))</p><p>z1(x) + k2(zn(x))</p><p>where k1, ... , kn denote the n components of the vector k.</p><p>0 bserve that</p><p>~; f(x) = Zn-1(x) + kn(Zn(x))</p><p>L}h(x) OZn-1 J(x) + [Okn] OZn J(x)</p><p>OX Oy y=Zn OX</p><p>where</p><p>- okn okn</p><p>kn-1(Zn,Zn-d = ~Zn-1 + ~kn(Zn) + kn-1(Zn) ·</p><p>vZn vZn</p><p>Proceeding in this way one obtains for each L}h(x), for 1 ::; i ::; n -1, an</p><p>expression of the form</p><p>L}h(x) = Zn-i(x) + kn-i+1(zn(x), ... ,Zn-i+1(x)).</p><p>Differentiating with respect to x and arranging all these expressions to</p><p>gether, one obtains</p><p>ah ah</p><p>ax az</p><p>===U</p><p>0 0</p><p>.: ) ;; . aL1h aL1h 0 1</p><p>ax = az</p><p>8Ln-1h 8Ln-1h * * I I</p><p>ax az</p><p>206 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>This, because of the nonsingularity of the matrix on the right-hand side,</p><p>proves the claim. - 1 (z)) =Zn, we have</p><p>LadkiJh(x) = 0</p><p>I</p><p>for all 0 $ k $ n - 2 and</p><p>Using Lemma 4.1.1, we deduce that</p><p>for all 0 $ k $ n - 2 and</p><p>LsL/-1h(x) = 1.</p><p>Thus, the vector field 8 necessarily coincides with the unique solution of</p><p>(4.73).</p><p>Sufficiency. Suppose (i) holds and Iet r denote the solution of (4.73). Using</p><p>Lemma 4.1.1 one may immediately note (see (4.5)) that the matrix</p><p>208 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>( df~~~) ) ( r(x) ad1r(x) · · · ad1F 1r(x))</p><p>dL/- 1h(x)</p><p>has rank n for all x near X0 • Thus, the vector fields { r, adfT, ... , ad/-1r} are</p><p>linearly independent at all x near X 0 •</p><p>Let F denote a solution of the p.d.e. (4.74) and Iet Z 0 be a point such</p><p>that F(z 0 ) = X 0 • From the linear independence of the vector fields on the</p><p>right-hand side of (4.74) we deduce that the differential of F has rank n at Z 0 ,</p><p>i.e. that</p><p>Note also that if</p><p>is the column vector representing an element of V, and</p><p>w* = [w1 W2 • • • Wn]</p><p>is the row vector representing an element of V*, the "value" of w* at v is</p><p>given by the product</p><p>n</p><p>w*v = ~WiVi.</p><p>i=l</p><p>Most of the times, as often occurring in the literature, the value of w* at v</p><p>will be represented in the form of an inner product, writing (w*, v} instead</p><p>of simply w*v.</p><p>Suppose now that w1 , ... , Wn are smooth real-valued nmctions of the real</p><p>variables x1 , ... , Xn defined on an open subset U of !Rn, and consider the row</p><p>vector</p><p>w(x) ={wl(xl, ... ,xn) w2(x1, ... ,xn) ... Wn(Xl,···•xn)).</p><p>On the grounds of the previous discussion, it is natural to interpret the latter</p><p>as a mapping (a smooth one, because the wi's are smooth functions) assigning</p><p>to each point x of a subset U an element w( x) of the dual space (!Rn)*, i.e.</p><p>exactly the object that was identified as a covector field.</p><p>A covector field of special importance is the so-called differential, or gra</p><p>dient, of a real-valued function A defined on an open subset U of !Rn. This</p><p>covector field, denoted d..X, is defined as the 1 x n row vector whose i-th ele</p><p>ment is the partial derivative of A with respect to Xi· Its value at a point x</p><p>is thus</p><p>( a..x a..x a..x) d..X(x) = - - ... - .</p><p>axl 8x2 8xn</p><p>(1.5)</p><p>Note that the right-hand side of this expression is exactly the jacobian matrix</p><p>of .X, and that the more condensed notation</p><p>d..X(x) = a..x</p><p>ax</p><p>(1.6)</p><p>is sometimes preferable. Any covector field having the form (1.5)-(1.6), i.e.</p><p>the form of the differential of some real-valued function .X, is called an exact</p><p>differential.</p><p>8 1. Local Decompositions of Control Systems</p><p>We describe now three types of differential operation, involving vector</p><p>fields and covector fields, that are frequently used in the analysis of nonlinear</p><p>control systems. The first type of operation involves a real-valued function >.</p><p>and a vector field I, both defined on a subset U of JRn. From these, a new</p><p>smooth real-valued function is defined, whose value - at each x in U - is</p><p>equal to the inner product</p><p>8>. n 8>.</p><p>(d>.(x), l(x)} = 8xl(x) = ~ 8x/i(x) .</p><p>This function is sometimes called the derivative of >. along I and is often</p><p>written as L 1 >.. In other words, by definition</p><p>at each x of U.</p><p>Of course, repeated use of this operation is possible. Thus, for instance,</p><p>by taking the derivative of >. first along a vector field I and then along a</p><p>vector field g one defines the new function</p><p>8(L1>.)</p><p>L9 L1>.(x) = 8x g(x) .</p><p>If >. is being differentiated k times along I, the notation L'>. is used; in other</p><p>words, the function L'>.(x) satisfies the recursion</p><p>8 (Lk-1 >.)</p><p>Lk >.(x) = 1 l(x)</p><p>I 8x</p><p>with L~>.(x) = >.(x).</p><p>The second type of operation involves two vector fields I and g, both</p><p>defined on an open subset U of ]Rn. From these a new smooth vector field is</p><p>constructed, noted [I, g] and defined as</p><p>8g 81</p><p>[l,g](x) = 8xl(x)- 8xg(x)</p><p>at each x in U. In this expression</p><p>~ 1 ~ 2 ~ n ~ 1 ~ 2 U: n</p><p>8g ~ ~ ~ 81 ~ M; M:</p><p>8x</p><p>1 2 n</p><p>8x</p><p>1 2 n</p><p>~ 1 ~ 2 ~ n ~ 1 ~ 2 M: n</p><p>denote the Jacobian matrices of the mappings g and /, respectively.</p><p>1.2 Notations 9</p><p>The vector field thus defined is called the Lie product ( or bracket) ol f</p><p>and 9· Of course, repeated bracketing of a vector. field 9 with the same vector</p><p>field I is possible. Whenever this is needed, in order to avoid a notation of</p><p>the form [1, [1, ... , [1,9]]], that could generate confusion, it is preferable to</p><p>define such an Operation recursively, as</p><p>ad,9(x) = [l,ad~-1 9](x)</p><p>for any k ~ 1, setting acl}9(x) = 9(x).</p><p>The Lie product between vector fields is characterized by three basic</p><p>properties, that are summarized in the following statement. The proof of</p><p>them is extremely easy and is left as an exercise to the reader.</p><p>Proposition 1.2.1. The Lie product ol vector fields has the lollowin9 prop</p><p>erties:</p><p>(i) is bilinear over ~' i. e. il /I, h, 91, 92 are vector fields and r1, r2 real num</p><p>bers, then</p><p>[rd1 + r2!2, 91]</p><p>[!I,r191 +r292]</p><p>(ii) is skew commutative, i.e.</p><p>rdi1>9I] + r2[h,91]</p><p>ri[/I,9I] + r2[/I,92],</p><p>[!, 9] = -[9, I] '</p><p>(iii) satisfies the Jacobi identity, i.e., il j,9,p are vector fields, then</p><p>[1, [9,Pll + [9, [p, I]]+ [p, [1,9]] = 0.</p><p>The third type of Operation of frequent use involves a covector field w</p><p>and a vector field I, both defined on an open subset U of ~n. This operation</p><p>produces a new covector field, noted L JW and defined as</p><p>at each x of U, where the superscript "T" denotes transposition. This covector</p><p>field is called the derivative ol w alon9 I.</p><p>The operations thus defined are used very frequently in the sequel. For</p><p>convenience we Iist in the following statement a series of "rules" of major</p><p>interest, involving these Operations either separately or jointly. Again, proofs</p><p>are very elementary and left to the reader.</p><p>Proposition 1.2.2. The three types ol differential operations introduced so</p><p>lar are such that</p><p>(i) il a is a real-valued function, I a vector field and A a real-valued function,</p><p>then</p><p>(1.7)</p><p>10 1. Local Decompositions of Control Systems</p><p>(ii) i/ a, ß are real-valued functions and J, g vector fields, then</p><p>[af,ßg](x) = a(x)ß(x)[f,g](x) + (L,ß(x))a(x)g(x)- (Lga(x))ß(x)f(x),</p><p>(1.8)</p><p>(iii) i/ J, g are vector fields and A a real-valued function, then</p><p>(1.9)</p><p>(iv) if a, ß are real-valued functions, f a vector field and w a covector field,</p><p>then</p><p>LaJßw(x) = a(x)ß(x)(L,w(x)) + ß(x)(w(x), f(x))da(x)</p><p>+(L,ß(x))a(x)w(x) ,</p><p>( v) if f is a vector field and A a real-valued function, then</p><p>(vi) if J, g are vector fields and w a covector field, then</p><p>(1.10)</p><p>(1.11)</p><p>LJ(w,g)(x) = (L,w(x),g(x)) + (w(x), [f,g](x)). (1.12)</p><p>Example 1.2.1. As an exercise one can check, for instance, (1.7). By defini</p><p>tion, one has</p><p>n (BA) n (BA )</p><p>LaJA(x) = ~ Bxi (a(x)fi(x)) = ~ Bxi /;(x) a(x) = (LJA(x))a(x).</p><p>Or, as far as (1.10) is concerned,</p><p>To conclude the section, we illustrate another procedure of frequent use</p><p>in the analysis of nonlinear control systems, the change of coordinates in the</p><p>state space. As is well known, transforming the coordinates in the state space</p><p>is often very useful in order to highlight some properties of interest, like e.g.</p><p>reachability and observability, or to show how certain control problems, like</p><p>e.g. stabilization or decoupling, can be solved.</p><p>In the case of a linear system, only linear changes of coordinates are</p><p>usually considered. This corresponds to the substitution of the original state</p><p>vector x with a new vector z related to x by a transformation of the form</p><p>z=Tx</p><p>1.2 Notations 11</p><p>where T is a nonsingular n x n matrix. Accordingly, the original description</p><p>of the system</p><p>x = Ax+Bu</p><p>y = Cx</p><p>is replaced by a new description</p><p>in which</p><p>Ä = TAT- 1</p><p>z = Äz+Bu</p><p>y = Cz</p><p>c = cr-1.</p><p>If the system is nonlinear, it is more meaningful to consider nonlinear</p><p>changes of coordinates. A nonlinear change of coordinates can be described</p><p>in the form</p><p>z = ~(x)</p><p>where ~(x) represents a 1Rn-valued function of n variables, i.e.</p><p>(</p><p>tP1(x)) ( tP1(x1, ... ,xn))</p><p>~(x) = l/1~~~) = tP2(X1:::·,xn)</p><p>tPn(x) tPn(X1, ... ,Xn)</p><p>with the following properties</p><p>(i) ~(x) is invertible, i.e. there exists a function ~-1 (z) suchthat</p><p>~-1 (~(x)) = x</p><p>for all x in !Rn .</p><p>(ii) ~(x) and ~-1 (z) are both smooth mappings, i.e. have continuous partial</p><p>derivatives of any order.</p><p>A transformation of this type is called a global diffeomorphism on !Rn. The</p><p>first of the two properties is clearly needed in order to have the possibility of</p><p>reversing the transformation and recovering the original state vector as</p><p>while the second one guarantees that the description of the system in the</p><p>new coordinates is still a smooth one.</p><p>. Sometimes, a transformation possessing both these properties and defined</p><p>for all x is difficult to find and the properties in question are difficult to be</p><p>checked. Thus, in most cases one rather Iooks at transformations defined only</p><p>in a neighborhood</p><p>F is a diffeomorphism of a neighborhood of z 0 onto a neighborhood</p><p>of X 0 • Set iP = F-1 and</p><p>- {)i[J</p><p>f(z) = ( 8XJ(x))x=.P-'(z) · (4.77)</p><p>By definition, the mapping F is such that</p><p>so that</p><p>[ßiP k ] k -8 ad1r(x) = ( -1) ek+I</p><p>X x=.P- 1 (z)</p><p>(4. 78)</p><p>for all 0 :::; k :::; n - 1.</p><p>Using (4.77) and (4.78), one obtains, for all 0:::; k:::; n- 2</p><p>[ßiP ] -adk+Ir(x)</p><p>ßx f x=.P- 1 (z)</p><p>[ß{)iP [f, ad}r](x)]</p><p>X x=.P- 1 (z)</p><p>[/(z ), ( -1)kek+I]</p><p>that is</p><p>for i :f. k + 2 .</p><p>From these, one deduces that / 1 depends only on Zn, and that /;, for</p><p>2:::; i:::; n, is suchthat /;- z;_ 1 depends only on Zn· This proves that (4.68)</p><p>holds. Moreover, since</p><p>for 0 :::; k ( X) = col( tPl (X)' ... ' tPn (X)) is a</p><p>local diffeomorphism at X 0 • By construction</p><p>84></p><p>OX (Tl (x) · · · Tn(X)) = diag(cl (x), ... , Cn(X)} .</p><p>210 4. Nonlinear Feedback for Single-Input Single-Output Systems</p><p>Moreover, using again (4.80), it is easy to see that ci(~- 1 (~)) depends only</p><p>on ~i· Thus, there exist functions zi = /Li(~i) suchthat</p><p>(recall that ci(x0 ) "# 0). The composed function</p><p>z = T(x) = (col(p,1(6), ... ,P,n(~n)))(=~(z)</p><p>is suchthat</p><p>[8Ta ( r1(x) .. . Tn(x) )] =I</p><p>X z=T-l(z)</p><p>and therefore X= T- 1(z) solves the p.d.e. (4.79).</p><p>of a given point. A transformation of this type is called a</p><p>local diffeomorphism. In order to check whether or not a given transformation</p><p>is a local diffeomorphism, the following result is very useful.</p><p>12 1. Local Decompositions of Control Systems</p><p>Proposition 1.2.3. Suppose ~(x} is a smooth function defined on some sub</p><p>set U of JR.n . Suppose the jacobian matrix of ~ is nonsingular at a point</p><p>x = X 0 • Then, on a suitable open subset U0 of U, containing X 0 , cli( x) defines</p><p>a local diffeomorphism.</p><p>Example 1.2.2. Consider the function</p><p>which is defined for all (xi, x2) in JR2. Its jacobian matrix</p><p>a~ (1 1 )</p><p>ßx = 0 COSX2</p><p>has rank 2 at X 0 = (0, 0). On the subset</p><p>this function defines a diffeomorphism. Note that on a !arger set the function</p><p>does not anymore define a diffeomorphism because the invertibility property</p><p>is lost. For, observe that for each nurober x2 suchthat lx2l > (rr/2) , there</p><p>exists x~ suchthat lx~ I -1}.</p><p>This nmction is a diffeomorphism ( onto its image)' because ~(XI' X2) =</p><p>cli(x~, x~) implies necessarily XI = x~ and x2 = x~. However, this function is</p><p>not defined on all JR2 • f(x)]</p><p>8x x=-l(z)</p><p>g(z) [84> ] = -g(x)</p><p>8x x=-l(z)</p><p>h(z) [h(x)]x=-l(z) .</p><p>The latter are the expressions relating the new description of the system</p><p>to the original one. Note that if the system is linear, and if 4>(x) is linear as</p><p>well, i.e. if 4>(x) = Tx, then these formulas reduce to ones recalled before.</p><p>1.3 Distributions</p><p>We have observed in the previous section that a smooth vector field f, defined</p><p>on an open set U of JRn, can be intuitively interpreted as a smooth mapping</p><p>assigning the n-dimensional vector f(x) to each point x of U. Suppose now</p><p>that d smooth vector fields !I, ... , f d are given, all defined on the same open</p><p>set U and note that, at any fixed point x in U, the vectors !I ( x), ... , /d ( x)</p><p>span a vector space (a subspace of the vector space in which all the fi(x)'s</p><p>are defined, i.e. a subspace of ]Rn). Let this vector space, which depends an</p><p>x, be denoted by ..1(x), i.e. set</p><p>..1(x) = span{fi(x), ... , !d(x)}</p><p>and note that, in doing this, we have essentially assigned a vector space to each</p><p>point x of the set U. Motivated by the fact that the vector fields /I, ... , /d</p><p>are smooth vector fields, we can regard this assignment as a smooth one.</p><p>The object thus characterized, namely the assignment- to each point x</p><p>of an open set U of ]Rn - of the subspace spanned by the values at x of some</p><p>smooth vector fields defined on U, is called a smooth distribution. We shall</p><p>now illustrate a series of properties, concerning the notion of smooth distri</p><p>bution, that are of fundamental importance in all the subsequent analysis.</p><p>According to the characterization just given, a distribution is identified</p><p>by a set of vector fields, say {!I, ... , /d}; we will use the notation</p><p>..1 = span{JI, ... , /d}</p><p>to denote the assignment as a whole, and, as before, ..1(x) to denote the</p><p>"value" of ..1 at a point x.</p><p>Pointwise, a distribution is a vector space, a subspace of JRn. Based on</p><p>this fact, it is possible to extend to the notion thus introduced a number of</p><p>elementary concepts related to the notion of vector space. Thus, if ..11 and</p><p>14 1. Local Decompositions of Control Systems</p><p>L12 are distributions, their sum L1I + L12 is defined by taking pointwise the</p><p>sum of the subspaces L1I(x) and L12(x), namely</p><p>(L1I + L12)(x) = L1I(x) + .J2(x).</p><p>The intersection L1I n L12 is defined as</p><p>A distribution L1I contains a distribution L12, and is written L1I ::J L12, if</p><p>L1I(x) ::J L12(x) for all x. A vector field f belongs to a distribution L1, and is</p><p>written f E L1, if f(x) E L1(x) for all x. The dimension of a distribution at a</p><p>point x of U is the dimension of the subspace L1(x).</p><p>If F is a matrix having n rows and whose entries are smooth functions of</p><p>x, its columns can be considered as smooth vector fields. Thus any matrix of</p><p>this kind identifies a smooth distribution, the one spanned by its columns.</p><p>The value of such a distribution at each x is equal to the image of the matrix</p><p>F(x) at this point</p><p>.1(x) = Im(F(x)).</p><p>Clearly, if a distribution L1 is spanned by the columns of a matrix F, the</p><p>dimension of L1 at a point X 0 is equal to the rank of F(x 0 ).</p><p>Example 1. 3.1. Let U = JR3 , and consider the matrix</p><p>(</p><p>XI</p><p>F(x) = 1 ~ X3</p><p>XIX2</p><p>(1 + X3)X2</p><p>X2</p><p>Note that the second column is proportional to the first one, via the coefficient</p><p>x2 . Thus this matrix has at most rank 2. The first and third columns are</p><p>independent (and, accordingly, the matrix F has rank exactly equal to 2) if</p><p>XI is nonzero. Thus, we conclude that the columns of F span the distribution</p><p>characterized as follows</p><p>L1(x) apan{(+• } if XI = 0</p><p>( x, ,0)} L1(x) span{ 1 ~ x3 if XI -:j:. 0 .</p><p>The distribution has dimension 2 everywhere except on the plane XI = 0.</p><p>of U which is not a regular point is said to be a point of</p><p>singularity.</p><p>Example 1.3.4. Consider again the distribution defined in the Example 1.3.1.</p><p>The distribution in question has dimension 2 at each x such that x1 =f. 0 and</p><p>dimension 1 at each x such that x1 = 0. The plane { x E JR3 : x1 = 0} is the</p><p>set of points of singularity of Ll.</p><p>rows.</p><p>20 1. Local Decompositions of Control Systems</p><p>Sometimes, it is possible to construct codistributions starting from given</p><p>distributions, and conversely. The natural way to do this is the following one:</p><p>given a distribution ..1, for each x in U consider the annihilator of Ll(x), that</p><p>is the set of all covectors which annihilates all vectors in Ll(x)</p><p>Lll.(x) = {w* E (!Rn)*: (w*,v} = 0 for all v E Ll(x)}.</p><p>Since Ll.L(x) is a subspace of (!Rn)* , this construction identifies exactly a</p><p>codistribution, in that assigns - to each x of U - a subspace of (!Rn)*. This</p><p>codistribution, noted ..11., is called the annihilator of ..1.</p><p>Conversely, given a codistribution {}, one can construct a distribution,</p><p>noted [Jl. and called the annihilator of {}, setting at each x in U</p><p>[Jl.(x) = {v E !Rn: (w*,v} = 0 for all w* E {}(x)}</p><p>Some care is required, for distributionsfcodistributions constructed in this</p><p>way, about the quality of being smooth. As a matter of fact, the annihilator of</p><p>a smooth distribution may fail to be smooth, as the following simple example</p><p>shows.</p><p>Example 1. 3.1 0. Consider the following distribution defined on IR1</p><p>Then</p><p>..1 = span{x}.</p><p>Ll.L(x) = {0}</p><p>Ll.L(x) = (!R1 )*</p><p>if X :f: 0</p><p>if X= 0</p><p>and we see that ..11. is not smooth because it is not possible to find a smooth</p><p>covector field on IR1 which is zero everywhere but on the point x = 0. .::l2 is satisfied if and only if the inclusion</p><p>.::lf C .::l:f is satisfied. Finally, the annihilator [.:11 n .::l2].L of an intersection</p><p>of distributions is equal to the sum .::lf + .::l:f. If a distribution .::l is spanned</p><p>by the columns of a matrix F, whose entries are smooth functions of x, its</p><p>annihilator is identified, at each x in U, by the set of row vectors w* satisfying</p><p>the condition</p><p>w* F(x) = 0.</p><p>Conversely, if a codistribution [} is spanned by the rows of a matrix W, whose</p><p>entries are smooth functions of x, its annihilator is identified, at each x, by</p><p>the set of vectors v satisfying</p><p>W(x)v = 0.</p><p>Thus, in this case [J.L(x) is the kemel of the matrix W at the point x</p><p>n.L(x) = ker(W(x)) .</p><p>One can easily extend Lemmas 1.3.1 to 1.3.5. In particular, if X 0 is a</p><p>regular point of a smooth codistribution n, and dim(n(x0 )) = d, it is possible</p><p>to find an open neighborhood U 0 of X 0 and a set of smooth covector fields</p><p>{ w1, ... , Wd} defined on U 0 , such that the covectors w1, ... , Wd are linearly</p><p>independent at each x in U0 and</p><p>n(x) = span{wl(x), ... ,wd(x)}</p><p>at each x in U 0 • Moreover, every smooth covector field w betonging to [} can</p><p>be expressed, on U 0 , as</p><p>d</p><p>w(x) = L Ci(x)wi(x)</p><p>i=l</p><p>where c1, ... , cd are smooth real-valued functions of x, defined on U 0 •</p><p>In addition one can easily prove the following result.</p><p>Lemma 1.3.6. Let X 0 be a regular point of a smooth distribution .::l. Then</p><p>x 0 is a regular point of ,:j.L and there exists a neighborhood U 0 of X 0 such</p><p>that the restriction of ,:j.L to U is a smooth codistribution.</p><p>Example 1. 3.12. Let .::l be a distribution spanned by the columns of a matrix</p><p>F and [} a codistribution spanned by the rows of a matrix W, and suppose</p><p>the intersection [} n ,:j.L is to be calculated. By definition, a covector in</p><p>nn.::lj_{x) is an element of n(x) which annihilates all the elements of .::l{x).</p><p>A generic element in n(x) has the form 'YW(x), where 'Y is a row vector of</p><p>suitable dimension, and this (covector) annihilates all vectors of .::l(x) if and</p><p>only if</p><p>22 1. Local Decompositions of Control Systems</p><p>')'W(x)F(x) = 0. (1.14)</p><p>Thus, in order to evaluate {} n Ll..l.(x) at a point x, one can proceed in the</p><p>following way: first find a basis (say ')'1 , ..• , /'d) of the space of the solutions</p><p>of the linear homogeneaus equation (1.14), and then express {} n Ll..L(x) in</p><p>the form</p><p>{} n Ll..t.(x) = span{"YiW(x): 1 ~ i ~ d} .</p><p>Note that the "Yi's depend on the point x. If W(x)F(x) has constant rank for</p><p>all x in a neighborhood U, then the space of solutions of (1.14) has constant</p><p>dimension and the "Yi's depend smoothly on x. As a consequence, the row</p><p>vectors ')'1 W ( x), ... , ')' d W ( x) are smooth covector fields spanning {} n Ll..L.</p><p>of JR.n, is said to be completely integrable</p><p>if, for each point X 0 of U there exist a neighborhood U0 of x 0 , and n - d</p><p>real-valued smooth functions Al, ... , An-d, all defined on U 0 , such that</p><p>span{dAl, ... ,dAn-d} = Ll.L (1.17)</p><p>on U 0 (recall the notation (1.6) ). Thus, "complete integrability of the distri</p><p>bution spanned by the columns of the matrix F(x)" is essentially a synony</p><p>mous for "existence of n- d independent solutions of the differential equation</p><p>(1.16)" . The following result illustrates necessary and sufficient conditions</p><p>for complete integrability.</p><p>Theorem 1.4.1. (Frobenius) A nonsingular distribution is completely inte</p><p>grable if and only if it is involutive.</p><p>Proof. We shall show first that the property of being involutive is a necessary</p><p>condition, for a distributiontobe completely integrable. By assumption, there</p><p>exist functions Al, ... , An-d such that {1.17), or, what is the same, (1.16) is</p><p>satisfied. Now, observe that the equation (1.16) can also be rewritten as</p><p>24 1. Local Decompositions of Control Systems</p><p>ß)... a: fi(x) = (dAi(x), fi(x)} = 0 for all 1 :5: i :5: d, all x E uo (1.18)</p><p>and that the latter, using a notation established in section 1.2, can in turn</p><p>be rewritten as</p><p>for all 1 :5: i :5: d, all x E U 0 • (1.19)</p><p>Differentiating the function Ai along the vector field [/i, fk], and using (1.19)</p><p>and (1.9), one obtains</p><p>L[f,,fk]Ai(x) = LJ,LJkAi(x)- LJkL/iAi(x) = 0.</p><p>Suppose now the same operation is repeated for all the functions A1, ... , An-d.</p><p>We conclude that</p><p>(</p><p>L[!,,!k]Al(x) ) ( dA1(x) )</p><p>· · · = · · · [fi, !k](x) = 0</p><p>L[J;,fk]An-d(x) dAn-d(x)</p><p>for all x E U0 •</p><p>Since by assumption the differentials { dA1 , ... , dAn-d} span the distribution</p><p>Ll.L, we deduce from this that the vector field [Ii, fk] is itself a vector field</p><p>in ..:1. Thus, in view of the condition established in the Remark 1.3.5, we</p><p>conclude that the distribution L1 is involutive.</p><p>The proof of the sufficiency is constructive. Namely, it is shown how a</p><p>set of n- d functions satisfying (1.17) can be found. Recall, that, since L1 is</p><p>nonsingular and has dimension d, in a neighborhood uo of each point X 0 of</p><p>U there exist d smooth vector fields h, ... , f d, all defined on U 0 , which span</p><p>..:1, i.e. are such that</p><p>Ll(x) = span{ft(x), ... ,/d(x)}</p><p>at each x in U 0 • Let f d+l, ... , f n be a complementary set of vector fields, still</p><p>defined on U 0 , with the property that</p><p>span{ft(x), ... ,Jd(x),fd+l(x), ... ,fn(x)} = JR.n</p><p>at each x in U 0 •</p><p>Let { : x f-t 4'>{ (x)</p><p>is defined for all x in a neighborhood of x 0 , is a local diffeomorphism ( onto</p><p>its image), and [4'>{]- 1 = q;~t· Moreover, for any (sufficiently small) t, s</p><p>We show now that a solution of the partial differential equation (1.16) can</p><p>be constructed by taking an appropriate composition of the flows associated</p><p>with the vector fields h, ... , f n, i.e. of</p><p>4'>{; (X), ... , 4'>{: (X) .</p><p>To this end, consider the mapping</p><p>1/! : ]Rn</p><p>q;h 0 ••• 0 q;fn (xo)</p><p>Zt Zn</p><p>(1.20)</p><p>where uc = { z E ]Rn : lzi I a+I Ön</p><p>dd+l(x) = ~, ... ,dn(x) = äx</p><p>are annihilated by the vectors of Ll at each x in uo. These differentials ( which</p><p>are independent by construction) are therefore a solution of (1.16). At this</p><p>point, to complete the proof of the sufficiency we only have to show that (i)</p><p>and (ii) hold.</p><p>Proof of (i). It is known that, for all x E !Rn and sufficiently small !t!, the</p><p>flow P{ (x) of a vector field f is defined and this renders the mapping tli well</p><p>defined for all (z1 , ... , Zn) with !zil sufficiently small. Moreover, since a flow</p><p>is smooth, so is tli. We prove that tli is a local diffeomorphisms by showing</p><p>that the rank of tli at 0 is equal to n. Tothis purpose, let for simplicity (M)*</p><p>denote the jacobian matrix of a mapping M(x), i.e.</p><p>and note that, by the chain rule</p><p>(ph) ... (P/;- 1 ) ~ (cflli o ... o pfn (x0 ))</p><p>Zt * Zt-1 * 8zi Z 1 Zn</p><p>(ph) ... (Pfi-1) J· (pfi 0 ... 0 pfn (xo))</p><p>Zt * Z,-1 * Z Zt Zn</p><p>(ph) ... (Pfi-1) J· (q.f•-1 0 ... 0 q.h (tli(z))) .</p><p>Zt * Zt-1 * t -Zi-1 -Zt</p><p>In particular, at z = 0, since tli(O) = X 0</p><p>The tangent vectors fr (x0 ), ••• , fn(X 0 ) are by assumption linearly indepen</p><p>dent, and this proves that the n columns of (tli)* are linearly independent at</p><p>z = 0. Thus, the mapping tli has rank n at z = 0.</p><p>Proof of (ii). From the previous computations, we deduce also that, at</p><p>any z E Ur;,</p><p>(q.h) ... (cflli-1) /· (q.f•-1 0 ... 0 q.h (x)) = ätli</p><p>Zt * Zi-1 * t -Zi-1 -Zl azi</p><p>where x = tli(z). If we are able to prove that for all x in a neighborhood of</p><p>x 0 , for small !t! and for any two vector fields T and {) belonging to Ll,</p><p>(cflf}*r o P~t(x) E Ll(x)</p><p>i.e. that (4>t)*roP~t(x) is a (locally defined) vector field of Ll, then we easily</p><p>see that (ii) is true. To prove this, one proceeds as follows. Let {) be a vector</p><p>field of Ll and set</p><p>1.4 Frobenius Theorem 27</p><p>for i = 1, ... , d. Since</p><p>:!_(P{J) oP"(x) = -(P{J) {)f} oP"(x) dt -t * t -t * ox t</p><p>( differentiate the identity ( P~t) J Pf t = I with respect to t and interchange</p><p>dfdt with 8f8x) and</p><p>d ar</p><p>dt (fi o Pf(x)) = 0;{) o Pf(x)</p><p>the functions V;(t) just defined satisfy</p><p>Since both fJ and /i belang to L1 and L1 is involutive, there exist functions</p><p>Aij defined locally araund X 0 such that</p><p>d</p><p>[fJ, /i] = L Aij/i</p><p>j=l</p><p>and, therefore,</p><p>The functions V;(t) are seen as solutionsofalinear differential equation and,</p><p>therefore, it is possible to set</p><p>where X(t) is a d x d fundamental matrix of solutions. By multiplying on the</p><p>left both sides of this equality by (Pt)* we get</p><p>(!I (f!f(x)) ... /d(Pf(x))) = ( (Pfth(x) ... (f!t}Jd(x) )x(t)</p><p>and also, by replacing x by P~t(x)</p><p>Since X(t) is nonsingular for all t, we have that, for i = 1, ... , d</p><p>i.e.</p><p>28 1. Local Decompositions of Control Systems</p><p>This result, bearing in mind the possibility of expressing any vector field T</p><p>of .1 in the form</p><p>d</p><p>r= L:cdi</p><p>i=l</p><p>completes the proof of (ii).</p>Mais conteúdos dessa disciplina
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