Nonlinear Control Systems II (Communications and Control Engineering) (v. 2)

Published titles include: Parametrizations in Controt Estimation and Filtering Problems: Accuracy Aspects M. Gevers and G. Li Parallel Algorithms for Optimal Control ofLarge Scale Linear Systems Zoran Gajic and Xuemin Shen Loop Transfer Recovery: Analysis and Design A. Saberi, B.M. Chen and P. Sannuti Markov Chains and Stochastic Stability S.P. Meyn and R.L. Tweedie Robust Control: Systems with Uncertain Physical Parameters J. Ackermann in co-operation with A. Bartlett, D. Kaesbauer, W. Sienel and R. Steinhauser Optimization and Dynamical Systems U. Helmke and J.B. Moore Optimal Sampled-Data Control Systems Tongwen Chen and Bruce Francis Nonlinear Control Systems (3rd edition) Alberto Isidori Theory ofRobot Control C. Canudas de Wit, B. Siciliano and G. Bastin (Eds) Fundamental Limitations in Filtering and Control Marfa M. Seron, Julio Braslavsky and Graham C. Goodwin Constructive Nonlinear Control R. Sepulchre, M. Jankovic and P.V. Kokotovic

A Theory ofLearning and Generalization M. Vidyasagar Adaptive Control I.D. Landau, R. Lozano and M.M'Saad Stabilization ofNonlinear Uncertain Systems Miroslav Krstic and Hua Deng Passivity-based Control ofEuler-Lagrange Systems Romeo Ortega, Antonio Lorfa, Per Johan Nicklasson and Hebertt Sira-Ramirez Stability and Stabilization ofInfinite Dimensional Systems with Applications Zheng-Hua Luo, Bao-Zhu Guo and Orner Morgul Nonsmooth Mechanics (2nd edition) Bernard Brogliato

Alberto Isidori

Nonlinear Control Systems II With 17 Figures

~Springer

•. ,

-·

Alberto Isidori, Professor Universita di Roma "La Sapienza", Diparti.mento di Informatica e Sistemistica, Via Eudossiana 18,00184 Rome, Italy

Series Editors E.D. Sontag • M. Thoma

ISBN 1-85233-188-7 Springer-Verlag London Berlin Heidelberg British Library Cataloguing in Publication Data Isidori, A. Nonlinear control systems 2.- (Communications and control engineering) ...-(.Nonlinear control theory LTitle 629.8'36 1SBN185233 1887 Lillrary of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced. stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.

e Springer-Verlag London Limited 1999 Printed in Great Britain The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Camera ready by author Printed and bound by Athena:um Press Ltd, Gatesbead, Tyne & Wear 69/3830-543210 Printed on acid-free paper SPIN 10662236

For Maria Adelaide

Preface

The purpose of this book is to present a self-contained and coordinated description of several design methods for nonlinear control systems, with special emphasis on the problem of achieving stability, globally or on arbitrarily large domains, in the presence of model uncertainties. The book is intended to be a continuation of my earlier book Nonlinear Control Systems, dealing with the fundamentals of the theory of nonlinear control systems, whose third eqition was published in 1995. In this respect, it is written in the form of a "second volume" of a single work, and uses a numbering system that continues the one adopted in the earlier book, with which the overlap is essentially insignificant. The book is intended as a graduate text as well as a reference to scientists and engineers interested in the design of feedback laws for nonlinear control systems. In the last decade, methods for global stabilization of nonlinear systems have experienced a vigorous growth. The emergence of the concept of inputto-state stability, with its multi-faced consequences in terms of various versions of a small-gain criterion for the stability analysis of interconnected systems, the idea of making a system passive via feedback, with its far-reaching implications in terms of robustness, and the identification of special structures - such as those of the so-called systems in feedback or feedfoMJJard foro& - in which the design of feedback laws can be systematically addressed in a recursive way by dealing each time with a problem in "dimension one" (as in the so-called method of backstepping), have enormously increased our ability of designing feedback laws to the purpose of achieving stability, for various classes of systems modeled by nonlinear differential equations, in the presence of model uncertainties. The objective of this text is to render the reader familiar with major methods and ~esults, and enable him to follow the recent literature. The book is organized as follows. The first Chapter, namely Chapter 10, is devoted to the exposition of concepts and methods for the stability analysis of systems that can be viewed as a (cascade or feedback) interconnection of lower-dimensional subsystems. In particular, after a quick review of some basic tools for the analysis of the asymptotic behavior of nonlinear systems, the fundamental notion of input-to-state stability is described, a.long with various different characterizations and features, and it is shown how the so-

viii called input-to-state "gain function" of a system can be evaluated by means of a method which extends the classical method of Lyapunov for stability analysis. Then, this notion is used in the development of a small-gain theorem for input-to-state stable systems, extensively applied later in Chapters 11 and 12 to the purpose of achieving robust stability. The second part of Chapter 10 is dedicated to the exposition of the notion of dissipativity, with particular emphasis on the special cases of systems having a "finite £2 gain" and of "passive" systems. Again, it is shown how these properties can be tested and how they can be used in the stability analysis of interconnected systems. The last section of the Chapter discusses interesting (and classical) implications of these properties on the transfer function matrix of a linear system. Chapter 11 describes methods for robust global asymptotic stabilization. For systems modeled by equations in "lower-triangular" form containing unknown parameters, it is shown how to design robustly stabilizing feedback laws, for a series of cases of increasing complexity. The first case describes how the method of backstepping can be used in order to recursively design a feedback law to the purpose of imposing that a fixed (quadratic-like) positive definite function becomes a Lyapunov function for the closed-loop system. The feedback law so obtained is a function of a subset of the set of state variables, the equations modeling the systems are assumed to be linear in the unmeasured state variables, and the internal dynamics associated with the unmeasured state variables are assumed to be globally robustly asymptotically stable. Under appropriate additional hypotheses, this control scheme leads to the design of a dynamic feedback which uses only the output of the system as measured variable. Then, an extension of this method is described, that exploits the small-gain theorem for input-to-state stable systems, in which the hypotheses of linearity in the unmeasured variables and stability of the associated dynamics are no longer required. Finally, the Chapter is complemented by some results regarding the extensions of these methods to the case of systems having many inputs. Chapter 12 discusses the case in which, instead of looking at the problem of achieving global asymptotic stability, one rather seeks a feedback law able to impose the property that any trajectory corresponding to initial conditions in an arbitrarily large set (semiglobal stabiliz-ability) enters, in finite time, an arbitrarily small set (practical stabiliz-ability). This setup is particularly convenient in case only the output of the system is available for feedback. In fact, a system in "lower-triangular" form (containing unknown parameters) is globally diffeomorphic to a system in which the states which need to be measured for feedback coincide with the output and a number of its derivatives with respect to time. Such a diffeomorphism may depend on unknown parameters and unmeasured states, but this uncertainty is not a problem if, as explained in the course of the Chapter, output and its derivatives are replaced by estimates which, past a small initial time interval, become arbitrarily accurate. This idea is used to stabilize systems using dynamic output

ix

feedback, first under the hypothesis that the zero dynamics are globally robustly asymptotically stable and later also in the more challenging case in which this hypothesis does not hold. In Chapter 13, the issue of robust stability in the presence of unmodeled dynamics is dealt with via the small-gain theorem for systems having finite L 2 gain, by seeking feedback laws which render the gain of the system, between a fixed disturbance input and a fixed output, sufficiently small. This is the nonlinear version of the so-called problem of disturbance attenuation, and the methods by which the problem is solved are the nonlinear versions of the so-called methods of H 00 control for robust stabilization of linear systems. The Chapter presents a variety of situations, including the solution of the socalled problem of almost disturbance decoupling, and a comparison with the corresponding results which hold in the case oflinear systems is systematically carried out. Finally, Chapter 14 describes when and how it is possible to achieve global asymptotic stability by means of (state feedback) laws whose amplitude cannot exceed a fixed bound. In the first part of the Chapter, it is shown that a solution of this design problem, which of course exists only under special hypotheses, can be systematically achieved, for systems in "upper-triangular" form, via the recursive synthesis of appropriate (non-quadratic) Lyapunov functions. The second part of the Chapter describes an alternative, and appealing, recursive design procedure in which, at each stage, the desired asymptotic properties of the current subsystem are imposed by means of a feedback law which simply consists in the "saturation" of a suitable linear law. This yields a general design scheme with great potential, known as the scheme of the "nested-saturations", of which some applications are presented at the end of the Chapter. Although an attempt has been made to present, in a organized way, some of the leading ideas and results appeared in this area in the last decade or so, the exposition is far from being a complete overview of the intended subject, namely robust and global/semiglobal stabilization of nonlinear systems. In particular, the bibliography includes only the titles which have been actually used. Sincere apologies are owed for a number of omissions, unavoidable in any attempt like this one. I wish to express my deep gratitude to a number of people/institutions which have made this work possible. In particular, I am indebted to Washington University in St. Louis, its School of Engineering and Applied Science, its Department of Systems Science and Mathematics, and the funding agencies NSF and AFOSR, for the generous support, encouragement and advice received. I also wish to thank the NASA Research Center at Langley, VA, where in 1997 I had the opportunity to lecture for the first time, in a systematic manner, on the topics presented in this book. The financial support of the MURST is also gratefully acknowledged. I am indebted to Professors T. Basar, L. Praly, A. Teel and E. Sontag, for discussions, advice and ideas that

X

have helped me enormously in the understanding of the subject. I also wish to thank a number of collaborators and doctoral students with whom I had intensive exchanges of ideas: A. Astolfi, L. Marconi, A. Serrani, B. Schwartz, C. De Persis, R. DeSantis, also for their invaluable help in looking at various preliminary drafts of this work. Rome and St. Louis, June 1999

Alberto Isidori

Table of Contents

10. Stability of Interconnected Nonlinear Systems............ IO.I Preliminaries........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I0.2 Asymptotic Stability and Small Perturbations. . . . . . . . . . . . . . I0.3 Asymptotic Stability of Cascade-Connected Systems . . . . . . . . I0.4 lnput-t~State Stability ................................ : I0.5 Input-t~State Stability of Cascade-Connected Systems . . . . . . I0.6 The "Small-Gain" Theorem for Input-to-State Stable Systems IO. 7 Dissipative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I0.8 Stability of Interconnected Dissipative Systems . . . . . . . . . . . . . I0.9 Dissipative Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I I 11 I4 I7 3I 36 42 54 6I

11. Feedback Design for Robust Global Stability . . . . . . . . . . . . . II. I Preliminaries........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1.2 Stabilization via Partial State Feedback: a Special Case . . . . . Il.3 Stabilization via Output Feedback: a Special Case . . . . . . . . . . Il.4 Stabilization of Systems in Lower Triangular Form . . . . . . . . . . 11.5 Design for Multi-Input Systems ..........................

75 75 79 90 98 109

I2. Feedback Design for Robust Semiglobal Stability ......... I2.I Achieving Semiglobal and Practical Stability ............... I2.2 Semiglobal Stabilization via Partial State Feedback ......... I2.3 A Proof of Theorem 9.6.2 ................................ I2.4 Stabilization of Minimum-Phase Systems in Lower-Triangular Form ................................................. I2.5 Stabilization via Output Feedback Without a Separation Principle .............................................. I2.6 Stabilization via Output Feedback of Non-Minimum-Phase Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I2.7 Examples ..............................................

I25 I25 I35 I42

13. Disturbance Attenuation ................................. I3.I Robust Stability via Disturbance Attenuation .............. I3.2 The Case of Linear Systems ............................. I3.3 Disturbance Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I83 I83 I92 I99

I49 I57 I63 I72

xii

Table of Contents

13.4 13.5 13.6 13.7

Almost Disturbance Decoupling ..................... , .... 201 An Estimate of the Minimal Level of Disturbance Attenuation 207 £2-gain Design for Linear Systems ........................ 212 Global £ 2-gain Design for a Class of Nonlinear Systems ..... 216

14. Stabilization Using Small Inputs .......................... 14.1 Achieving Global Stability via Small Inputs ................ 14.2 Stabilization of Systems in Upper Triangular Form ......... 14.3 Stabilization Using Saturation Functions .................. 14.4 Applications and Extensions .............................

227 227 236 253 267

Bibliographical Notes ......................................... 281 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Index ............................... : ......................... 291

10. Stability of Interconnected Nonlinear Systems

10.1 Preliminaries For convenience of the reader, this section provides a quick review of the notion of comparison functions and their role in the well-known criterion of Lyapunov for determining stability and asymptotic stability. Definition 10.1.1. A continuous function a : [0, a) -+ [0, oo) is said to belong to class K if it is strictly increasing and a(O) = 0. If a = oo and limr-+oo a(r) = oo, the function is said to belong to class K 00 • Definition 10.1.2. A continuous function f3: [0, a) x [0, oo)-+ [0, oo) is said to belong to class K.C if, for each fixed s, the function a:

[0, a)

-+

r

f-t

[0, oo) f3(r, s)

belongs to class K and, for each fixed r, the function cp:

[0, oo)

-+

s

f-t

[0, oo) f3(r,s)

is decreasing and lims-+oo cp(s) = 0.

Class K and class K.C functions have some interesting features, that can be summarized as follows. The composition of two class K (respectively, class K 00 ) functions a 1(·) and a2(·), denoted a1(a2(·)) or a1 oa2(·), is a class K (respectively, class Koo) function. If a(·) is a class K function, defined on [0, a) and b = limr-+a a(r), there exists unique function, a- 1 : [0, b) -+ [0, a), such that a- 1 (a(r)) = r, for all r E [O,a) a(a- 1 (r)) = r, for all r E [O,b).

a

Moreover, a- 1 (·)is a class K function. If a(·) is a class Koo function, so is also a- 1 (· ). If /3(·, ·) is a class K..C function and a 1 (·), a 2 (·) are class K functions, the function thus defined 'Y:

[0, a) x [0, oo)

-+

(r,s)

f-t

[0, oo) a1(/3(a2(r),s))

2

10. Stability of Interconnected Nonlinear Systems

is a class KC function. It is also useful to know that any class KC function can always be estimated in terms of two other class Koo functions and of the exponential function, as indicated in the following result 1 .

Lemma 10.1.1. Assume /3(·, ·) is a class ICC function. Then, there exist two class Koo functions -y( ·) and 0( ·) such that

{3(r, s) :::; -y(e- 8 0(r)) for all (r,s) E [O,a) x [O,oo).

Finally, another important feature of the comparison functions is the following property, which is very useful in establishing the asymptotic convergence to zero of the trajectories of a nonlinear system 2 .

Lemma 10.1.2. Consider the differential equation

iJ

= -a(y)

where y E IR and a(·) is a locally Lipschitz class K function defined on [0, a). For all 0 :::; y 0
y(t) = cp(y 0 , t) where cp(·, ·) is a class KC function defined on [O,a) x [O,oo).

In what follows we denote, as usual, by llxll the Euclidean norm of a vector x E Ilr' and by B. (respectively, by h.) the open (respectively, closed) ball of radius c:, namely B.

= {X E !Rn : llxll < C:} ,

Now, consider a nonlinear system

x = f(x)

(10.1)

in which x E !Rn, f(O) = 0 and f(x) is locally Lipschitz. The stability, or asymptotic stability, properties of the equilibrium x = 0 of this system can be tested via the well known criterion of Lyapunov, which, in. terms of comparison function, can be expressed as follows.

1

2

For a proof of this result, see Sontag (1998). For a proof of this result, see Khalil (1996), page 656.

10.1 Preliminaries

3

Theorem 10.1.3. Let V : Bd -t lR be a C 1 function such that, for some class K functions g( ·), a(·), defined on [0, d),

for allllxll < d.

g(llxll) ~ V(x) ~ a(llxll) If

ov

(10.2)

for all llxll < d,

ox f(x) ~ 0

(10.3)

the equilibrium x = 0 of (10.1} is stable. If, for some class K function a{), defined on [0, d),

ov

ox f(x) ~ -a(llxll)

for allllxll < d,

(10.4)

the equilibrium x = 0 of (10.1} is locally asymptotically stable. If d = oo and, in the above inequalities, g( ·), a(·) are class K 00 functions, the equilibrium x = 0 of (10.1} is globally asymptotically stable. Proof. From (10.3) it is deduced that, so long as x(t) is defined, V(x(i)) is non increasing, i.e. V(x(t)) ~ V(x(O)) . Suppose c

< d and define 0 = a--l(g(c:)) .

Then, using (10.2}, observe that llx(O)II ~

g(llx(t)ll)

~

V(x(t))

~

V(x(O))

~

oimplies a(llx(O}II)

~ a(o)

= g(c)

,

i.e.

llx(t)ll ~ c . This, since f(x) is locally Lipschitz, shows that x(t) is defined for all t ~ 0 and proves the property of stability in the. sense of Lyapunov. Set V(t) = V(x(t)), 9(-) = a(a- 1 (·)) and observe that, if (10.4) holds, one has d~;t) ~ -O(V(t)) . Without loss of generality, suppose fl(·) is locally Lipschitz (if this is not the case, it can be replaced in wh.at follows by any locally Lipschitz class K function 0(-) such that O(r) ~ B(r)). Then, by Lemma 10.1.2, the differential equation iJ = -O(y) has a unique solution y(t) satisfying y(O)

= V(O), defined for all t ~ 0, and

y(t) = cp(V(O), t) for some class K.C function cp(·, ·). By the comparison Lemma,

V(t)

~

cp(V(O), t)

4

10. Stability of Interconnected Nonlinear Systems

and this yields

llx(t) II :::;

g-

1

( cp(a(llx(O) II),

t))

Since the right-hand side is a class K£ function in the arguments llx(O)II and

t, this proves local asymptotic stability. Global asymptotic stability is proven in the same way.


Remark 10.1.1. Note that the inequality on the right-hand side of (10.2) is redundant, since the existence of a function a(x) for which this inequality holds is an immediate consequence of the continuity of V(x) (and, of course, if g( ·) is of class Kco, so is necessarily a(·)). However, the inequality in question turns out to be useful in establishing bounds for the trajectories, as done for instance in the proof of Theorem 10.1.3. On the other hand, the inequality on the left-hand side of (10.2) is instrumental, together with (10.3), in establishing existence and boundedness of x(t). In particular, suppose the various inequalities considered in Theorem 10.1.3 hold ford= oo. The hypothesis (10.3) that V(x(t)) is non-increasing guarantees that x(t) is defined for all t ~ 0 and bounded, so long as x(O) is such that a(llx(O)II) belongs to the domain of the inverse of the function g(·). If the function g( ·) is a class Kco function, its inverse is defined on [0, oo) and therefore existence and boundedness of x(t) are guaranteed for any x(O).
= {x E !Rn

:

V(x):::; c}.

Note that, for any c in the image of the function g( ·) (i.e. for any c such that c = g(r) for some 0:::; r < oo), the inequality on the left-hand side of (10.2) yields x E flc => llxll :S g- 1 (c) i.e. the set flc is bounded. This property holds for all c > 0 if g( ·) is a class Kco function. If 8V(x) f(x) < 0 8x at each point x of the boundary of flc, it can be concluded that, for any initial condition in the interior of flc, the solution x(t) of (10.1) is defined for all t ~ 0 and such that x(t) E flc for all t ~ 0. Indeed, existence and uniqueness are guaranteed by the local Lipschitz property so long as x(t) E flc, because flc is a compact set. The fact that x(t) remains in flc for all t ~ 0 is proved by contradiction. For, suppose that, for some trajectory x(t), there is a time t 1 such that x(t) is in the interior of flc at all t < t 1 and x(t 1 ) is on the boundary of flc. Then,

10.1 Preliminaries

V(x(t)) V(x(tt))

5

< c for all t < t 1 =

c

and this contradicts the previous inequality, which shows that the derivative of V(x(t)) is strictly negative at t = t 1 . 0, the set fle is a compact set). The necessity of the latter property is an immediate consequence of (10.2), because x E De implies

which shows that De is bounded. Moreover, De is closed by definition, so it is compact. Conversely, suppose fle is compact and define, for all c ~ 0 p(c)

= zEnc max llxll ·

This function vanishes at c = 0, is positive for any nonzero c, is strictly increasing and lime-+ooP(c) = oo, because V(x) is continuous. Thus, p(·) is a class JC 00 function. Set g:(r) = p- 1 (r). Now, take any x, let c = V(x), and observe that indeed

Then, g:(llxll) :::; g:(m~ llxll) = g:(p(c)) = V(x) xEuc

i.e. this function satisfies the inequality on the left-hand side of (10.2).


Remark 10.1..4- In the inequality (10.4), the function a(·), was supposed to be of class JC. However, one can prove that, if for some V(x) defined for all x E Illn the inequalities (10.2) and (10.4) hold, with g:(·), a(·) of class JC 00 and a(·) of class JC, there is another function V(x) which satisfies similar inequalities but with g:(·), o(·) and a(·) all of class JC 00 • In other words, there is no loss of generality in assuming that a(·) is a class JC 00 function. To this end, let p( ·) be the function defined by an integral of the form p(r)

=for q(s)ds

in which q(·) is a smooth class }(00 function. Indeed, also the function p(·) is a class JC 00 function. Define V(x) as

V(x)

= p(V(x))

6

10. Stability of Interconnected Nonlinear Systems

and define

a(r)

= q(~(r))a(r)

.

By construction, the function V(x) is automatically C 1 and, as it easy to check, satisfies estimates of the form (10.2). Moreover, the function a(r) is a class Koo function, because so are the functions~(·) and q(·). Now, observe that

av av ax f(x) = q[V(x)]ax f(x) ::; -q[V(x)]a(ilxll) .

But

-q[V(x)]a(llxll) :S -q(~(llxll))a(llxll) = -a(llxll) and this completes the proof.

<1

It is well-known that the criterion for asymptotic stability provided by the previous Theorem has a converse, namely, the existence of a function V(x) having the properties indicated in Theorem 10.1.3 is implied by the property of asymptotic stability of the equilibrium'x = 0 of (10.1). In particular, the following result holds 3 . Theorem 10.1.4. Suppose the equilibrium x = 0 of {10.1) is locally asymptotically stable. Then, there exist d > 0, a C 1 function V : Bd -+ JR, and class K functions~(·), a(·), a(·), such that {10.2) and {10.4) hold. If the equilibrium x = 0 of {10.1) is globally asymptotically stable, there exist a C 1 function V : lRn -+ JR, and class Koo functions ~(·), a(·), a(·), such that {10.2) and {10.4) hold with d = oo. Remark 10.1.5. Note that, combining the result of Theorem 10.1.4 with an argument used in the proof of Theorem 10.1.3, it can be deduced that, if the equilibrium x = 0 is globally asymptotically stable, there exists a class KC function /3(·, ·)such that, for any X 0 , the solution x(t) of (10.1) with initial condition x(O) = X 0 satisfies an estimate of the form

for all t ;:::: 0. Note also that, using Lemma 10.1.1, this estimate can be replaced by an estimate of the form

in which 1U and 0(·) are class Koo functions. Since the inverse of 1U is defined on [0, oo) and is a class Koo function, this shows that, if the equilibrium x = 0 of (10.1) is globally asymptotically stable, any trajectory x(t) satisfies an estimate of the form

in which'){) and 0(·) are class K 00 functions. 3

For a proof of this result, see Kurzweil {1956).

<1

10.1 Preliminaries

7

It is well-known that, for a nonlinear system, the property of asymptotic stability of the equilibrium x = 0 does not necessarily imply e:cponential decay to zero of llx(t)ll· If the equilibrium x = 0 of system (10.1) is globally asymptotically stable and, moreover, there exist numbers d > 0, M > 0 and .>. > 0 such that

x(O) E Bd

=>

llx(t)ll

~ M e->.tllx(O)II

for all t ~ 0

it is said that this equilibrium is globally asymptotically and locally exponentially stable. In what follows, a characterization of those systems possessing a globally asymptotically and locally exponentially stable equilibrium is given. This characterization, and another interesting property that is presented immediately afterwards, are of great help in addressing certain problems of asymptotic stabilization, discussed in the next Chapters. Lemma 10.1.5. The equilibrium x = 0 of nonlinear system (10.1) is globally asymptotically and locally exponentially stable if and only if there exists a smooth function V(x) : !Rn -+ IR, and class JC 00 functions g(·), Ci(·), a(·), and real numbers > 0, a > 0, b > 0, such that

o

g(llxll)

~ V(x) ~

a(llxll)

av ax f(x) ~ -o(llxll) for all x E !Rn and

a(s) = bs 2 for all s E [O,o]. Proof. The equilibrium of the nonlinear system (10.1) is locally exponentially stable 4 if and only if there exists real numbers r > 0, g > 0, a > 0, "tl > 0 and a smooth function U(x) : Br -+ lR such that

.!!llxll 2 au

~ U(x) ~

ax f(x) ~

allxll 2 (10.5)

-.QIIxll 2

for all x E Br. Thus, to prove that the condition stated in the Theorem is sufficient, it is only needed to check that, for all x E B 0 and some a, V(x) ~ allxll 2 , which is indeed the case since V(x) is a smooth function vanishing at x = 0 together with its first derivatives. To prove the necessity, let U(x) be a function satisfying the conditions (10.5) and V(x) be a function satisfying (10.2) and (10.4) with d = oo. We claim that there exist real numbers k > 0, o > 0, p > 0, c 1 > 0, c2 > 0 sue~ that the set 4

See Khalil (1996), pages 140 and 149.

8

10. Stability of Interconnected Nonlinear Systems

satisfies and kV(x);::: U(x)

for all xES.

In fact, choose any p < r, and c 1 = ap

2

,

c2 =

2ap2 .

{x E IR.n : kV(x):::; c2} C

(10.6)

Then, choose k so that

Bp.

Such a k indeed exists because, if k ;::: c2 /g_(p)

In this way, S C

Bp

C Br and

XE S

=>

k V (X) ;:::

c: = ap2 ;::: U (X)

o

which proves (10.6). Finally, choose < min{kV{z)=cd llxiiNow, let a(·) be a smooth non-decreasing function, defined on [0, oo), and such that if s :::; cr a(s) = { if C2 :::; S.

~

Its derivative, denoted a' ( ·), satisfies

Set {3(x)

= a(kV(x))

and consider the function W(x) = f3(x)kV(x)

+ (1- f3(x))U(x),

which is well-defined because, by construction, Then, for all x E ]Rn , W(x) ;::: f3(x)kq(llxil)

Let 0

llxll ;::: r

implies 1-{3(x)

= 0.

+ (1- f3(x))gllxll 2 •

< a :::; g be such that as 2 < kq(s)

for all s E [c5, p] .

Then, from the previous inequality it can be concluded that there exists a class Koo function Q.(·), satisfying

10.1 Preliminaries

as 2 a(s) = { kg_(s) -

if s

9


ifs~p,

such that W(x) ~ Q.(!!x!l)

for all x E lRn . Moreover ew ax f(x)

ev

eu

a/3

= /3(x)k ox f(x) + (1- /3(x)) ax f(x) + (kV(x)- U(x)) axf(x).

0 bserve that o/3 (kV(x)- U(x)) ox f(x)

= (kV(x)- U(x))u'(kV(x))k ov ox f(x)

(10.7)

and that, by construction, a'(kV(x))(kV(x)- U(x)) ~ 0

for all x E lRn. Thus, the quantity (10.7) is always non-positive, and

oW f(x) < ax

ev f3(x)k ox f(x)

< -

+ (1- /3(x))

[,B(x)ka(llxll)

eu ox f(x)

+ (1- ,B(x))llllxll 2 ] •

Form this one can conclude, as before, that there exists a number 0 < b < Q and a class Koo function a(-), satisfying a(s) = bs 2 for all s E (0, o], such that

~~ f(x)

::; -a(l!x!l)

for all x E lRn. This completes the proof of the necessity.


The following lemma, which concludes the section, provides a useful estimate for the function a(l!x(t)ll) resulting from the composition of a. class K function a(·) with the norm of an integral curve x(t) of system (10.1), under the hypothesis that the latter has a globally asymptotically and locally exponentially stable at x = 0. ·'

Lemma 10.1.6. Consider the system (10.1}. Suppose the equilibrium x = 0 is globally asymptotically stable and locally exponentially stable. Let a(·) be a class K function which is differentiable at the origin. Then, there exists a class K function a(·) and a number A > 0 such that, for any X 0 E !Rn, the integral curve x(t) passing through X 0 at timet= 0 is such that

for all t

~

0.

10

10. Stability of Interconnected Nonlinear Systems

Proof. Set F(t) = a(llx(t)ll) .

By hypothesis, llx(t)ll is bounded by a class K..C function .B(IIxoll, t) and this, in view of Lemma 10.1.1, implies that, for some pair of class K.<XJ functions /{), 9(·), llx(t)ll ~ -y(e-tO(IIx 0 II)) . Note that -y(O(s)) as

~

s. Consider the class Koo functions o"(-) and i(·) defined

a(s) i(s)

=

2max{s,a(s)} ao-y(s).

Then, F(t) ~ a(Jix(t)ll) ~ i(e-t9(11xoll)).

Observe also that i(O(s)) > s for all s > 0. By hypothesis, the equilibrium x = 0 ,of the system is also locally exponentially stable. This means that there are numbers M > 0, .X > 0 and d > 0 such that, if llxoll ~ d, the integral curve passing through X 0 at time t = 0 satisfies llx(t)ll ~ Mllxolle->.t for all t ~ 0. Moreover, the function a(·) is by hypothesis differentiable at the origin. This implies the existence of numbers N > 0 and s > 0 such that, if s E [0, s], a(s) ~ Ns. Let

R = min{d,

.:r}.

Then, if llx 0 II ~ R, (10.8)

for all t ~ 0. Suppose, without loss of generality, N M > 1, define

A(s)

_ ( O(s) ) " = NM-y(O(s)) -r-l(R)

and consider the case llx 0 11 > R. Since i(O(s)) > s for all s > 0,

T ·=In O(llxoll) > 0 1.

-r-l(R)

and

A(llx 0 ll)e--'T1 = N Mi(O(IIxoll)) . As a consequence, we have, for all t E [0, T1 ], F(t)

~

~

i(O(IIx 0 11)) NM)"(O(IIx 0 11))

= A(llx

0

11)e--'T1 ~ A(llx0 11)e-,\t.

10.2 Asymptotic Stability and Small Perturbations

11

Moreover, by definition of T1.

and therefore, for all t

F(t)

~ ~

~

~

T1 ,

N Mllxo(Tt)lle->.(t-Tt) NMRe->...T,c>.t ~ A(llx 0 ll)e->.t

This proves that, for allllxoll

> R,

F(t) ~ A(llx 0 11)e->.t for all t ~ 0. On the other hand, for allllx 0 ll ~ R, (10.8) holds. Thus, let a{) be a class K. function satisfying a(s) a(s)

This proves the Lemma.

> ~

NMs A(s)

for s E [O,R] for s > R.

<J

10.2 Asymptotic Stability and Small Perturbations This section contains a result which analyzes the effect of a "small perturbation" affecting a system of the form (10.1), whose equilibrium x = 0 is assumed to be locally asymptotically stable. The theorem below, often referred to as the theorem of "total stability", shows that the trajectories of the perturbed system remain arbitrarily close to the equilibrium x = 0, although they may not necessarily asymptotically converge to it, provided that the perturbation as well as the initial state are sufficiently small. Theorem 10.2.1. Suppose the equilibrium x = 0 of {10.1} is locally asymptotically stable. Suppose g(x, t) is piecewise continuous in t and satisfies a Lipschitz condition llo(x',t)- g(x",t)ll ~ Lllx'- x"ll

=

for all t ~ 0 and all x', x" in some neighborhood U of x 0. Then, given any e > 0, there exist c51 > 0 and c52 > 0 {both possibly dependent one > 0} such . that, if llxoll ~ c51 llo(x, t)ll ~ cS2 for all llxll ~ e and all t ~ 0 , the solution x(t) of the perturbed system

x = f(x) + g(x, t)

(10.9)

12

10. Stability of Interconnected Nonlinear Systems

satisfying x(O)

=X

0

is such that

llx(t)ll ::; e for all t 2: 0.

Proof. Let V(x) be a C 1 function satisfying (10.2), (10.4). Since there exists a number M > 0 such that

~:

is C 0 ,

~~~:II::; M for all x E U. Suppose, without loss of generality, that e is such that BE C U. Fixe > 0 and let c > 0 be such that c::; q(e). Choose !52 such that -a(a- 1 (c)) By construction,

X

E

+ Ma2 < 0.

nc implies llxll::; e ·

In fact, q(llxll) ::; V(x) ::; c implies llxll::; Also, at each point

X

Q-

1

(c)::;

Q.-

1

(q(e)) =e.

of the boundary of

As a consequence, at each point

X

nc,

of the boundary of

nc,

av av 1 ax [f(x) + g(x, t)] ::; -a(llxll) +II ax 1162 ::; -a(a- (c)) + M 02 < 0 . From this, it can be concluded that, for any initial condition in the interior of ilc, the solution x(t) of (10.9) is defined for all t 2: 0 and is such that x(t) E ilc for all t 2: 0 (see Remark 10.1.2). To complete the proof it suffices to choose !51 < a- 1(c), for this guarantees that X 0 is in the interior of nc. In fact, implies V(x 0 ) < c.


Note that g(x, t) is not assumed to be vanishing at x = 0. In other words, the perturbed system may not have an equilibrium at x = 0. Or the perturbed system may have an unstable equilibrium at x = 0 (as shown in the following example). Nevertheless, in both cases the theorem of total stability guarantees that, if the perturbation is small enough, the trajectories of the perturbed system remain arbitrarily close to the equilibrium x = 0 of the unperturbed system.

10.2 Asymptotic Stability and Small Perturbatioras

13

Example 10.2.1. Consider the system 3

(10.10)

:i; = -x +'Yx

where 'Y > 0 is a small number. Indeed the "unperturbed" system :i; = -x3 has a globally asymptotically stable equilibrium at x = 0. On the other hand, the "perturbed system" has three equilibria, at x = 0, x = + fi, x .= -,ry. Elementary arguments, based on the principle of stability in the first approximation, show that, since 'Y is positive, the first one of these equilibria is unstable, while the other two are asymptotically stable. A glance at the graph of -x 3 + "fX suggests that

< < < <

< +fi => x(t) --+ +fi < +oo => x(t) --+ +fi => x(t) --+ -fi < 0 < -..rr => x(t) --+ -fi .

x(O) x(O) x(O) x(O)

Note that, even though the equilibrium x = 0 of the perturbed system is unstable, the boundedness properties indicated in the Theorem of total stability hold. In particular, given any e > 0, choose c\'1 = e and «52 = e 3 • Then, the perturbation term 'YX satisfies

I'Yxl ~ c\'2

for all

lxl

~ e

if fi ~ e. If this is the case, the two nonzero equilibria of the system lie in the interval [-e, +e], and we see form the previous analysis that any trajectory with initial condition lx(O)I ~ c\'1 = e remains confined to the set lxl ~e.
f(x,z)

i

g(z)

(10.11)

in which x E !Rn, z E !Rm, j(O, 0) = 0, g(O) = 0, and f(x, z), g(z) are locally Lipschitz on a neighborhood U of (x, z) = (0, 0). In view of the fact that the state z of the lower subsystem l,l.Cts as an input to the upper subsystem, we shall sometimes refer to the latter as to the "driven subsystem" and to the former as to the "driving subsystem". As an immediate corollary of the theorem of total stability, it can be deduced that if the equilibrium x = 0 of the upper subsystem, driven by z = 0, is locally asymptotically stable and the equilibrium z = 0 of the lower subsystem is stable, then the equilibrium (x,z) = (0,0) of the cascade is stable.

Corollary 10.2.2. Consider system (10.11}. Suppose the equilibrium x = 0 of :i; = f(x, O) is locally asymptotically stable and the equilibrium z = 0 of i = g(z) is stable. Then, the equilibrium (x,z) = (0,0) of {10.11} is stable.

14

10. Stability of Interconnected Nonlinear Systems

Proof. By hypothesis, for any E > 0, there exists J > 0 such that, if llzoll :::; J, the integral curve z (t) of i = g(z) satisfying Z (0) = Z is such that llz 0 (t)11 :::; E for all t ~ 0. Express f(x, Z 0 (t)) as 0

0

f(x, z 0 (t)) = f(x, 0)

0

+ g(x, t)

(10.12)

where

g(x, t) = f(x, Z 0 (t)) - f(x, 0) . Since f(x, z) is locally Lipschitz, there exist TJ > 0 and M > 0 such that llg(x, t)ll:::; Mllz 0 (t)ll:::; Me

for all llxll :::; TJ, all llzoll :::; J and t ~ 0. This bound for llg(x, t)ll can be rendered arbitrarily small by choosing a sufficiently small J and the result follows from Theorem 10.2.1.
10.3 Asymptotic Stability of Cascade-Connected Systems In this section we investigate the asymptotic stability of the equilibrium (x, z) = (0, 0) of a pair of cascade connected subsystems of the form

± i

= =

=

f(x,z) g(z) ,

in which x E IRn, z E IRm, /(0, 0) 0, g(O) Lipschitz on JR1l x JRm (see Fig. 10.1).

(10.13)

= 0, and f(x, z), g(z) are locally

Fig. 10.1. Cascade connection.

The main result of the analysis is that local asymptotic stability of the equilibrium x = 0 of the upper subsystem, driven by z = 0, and local asymptotic stability of the equilibrium z = 0 of the lower subsystem always imply local asymptotic stability of the equilibrium (x, z) = (0, 0) of the cascade. However, global asymptotic stability of the equilibrium x = 0 of the upper subsystem, driven by z = 0, and global asymptotic stability of the equilibrium z = 0 of the lower subsystem do not imply, in general, global asymptotic stability of the equilibrium (x, z) = (0, 0) of the cascade. To infer global asymptotic stability of the cascade, a (strong) extra condition is needed, as shown below. The key of the proof of these facts is the following result.

10.3 Asymptotic Stability of Cascade-Connected Systems

15

Theorem 10.3.1. Consider system (10.13). Suppose the equilibrium x = 0 of x = f(x, 0) is locally asymptotically stable. LetS be a set with the property that, for any X0 E S, the integral curve x(t) of i = f(x, 0) satisfying x(O) = X0 is defined for all t 2: 0 and is such that lim x(t)

t~oo

= o.

Pick any Z 0 and let Z 0 (t) denote the integral curve of i = g(z) satisfying 0 (0) = Z 0 • Suppose Z 0 ( t) is defined for all t 2: 0 and such that

Z

lim z 0 (t) = 0 .

t~oo

Pick any X 0 E S and let X 0 (t) denote the integral curve of x = f(x,z 0 (t)) satisfying X 0 (0) = X 0 • Suppose x 0 (t) is defined for all t 2: 0, is bounded and is such that X 0 ( t) E S for all t 2: 0. Then,

lim

t~oo

X

0

(t)

=0 .

Proof. First of all, observe that, using the theorem of total stability, it is easy to prove that, since the equilibrium x = 0 of x = f(x, 0) is locally asymptotically stable, given any e, there exist (hand 82 such that, if llxoll ~ 81 and llz(t)ll ~ 82 for all t 2: 0, the solution x(t) of

x = f(x, z(t)) that llx(t)ll ~ e

satisfying x(O) = X0 is such for all t 2: 0. To this end, it suffices to split f(x, z(t)) as in (10.12) and use arguments similar to those used in the proof of Corollary 10.2.2. As a consequence, the theorem will be proven if we can show that, for any e, it is possible to find a timeT such that llx 0 (T)II ~ 81, and llzo(t)ll ~ 82 for all t 2: T. Now, pick T1 > 0, set 0

ZT

1

and let

xJ. (t) 1

(

)

-

t -

ift ~ T1 if t > T 1 .

zo (t)

{

0

denote the integral curve of

x = }(x, zj. (t)) Indeed, xJ. (t) = X (t) for all 0 ~ t ~ T1 . Moreover, 1

satisfying xJ.1 {0) = X • for t > T 1 , x]., (t) is a solution of 0

0

1

x=

f(x,O)

and hence tends to 0 as t --+ oo, because exist T 2 such that

xJ. (Tl) 1

E

S. In particular, there. (10.14)

16

10. Stability of Interconnected Nonlinear Systems

The time T 2 thus characterized may depend on x!h (T1 ) and hence on T 1 but, since by hypothesis x 0 (t) is bounded, using a compactness argument one can conclude that there exists a T 2 depending only on X 0 for which the inequality (10.14) holds for any T1. Set T = T1 +T2. We will show now that, if T1 is large enough (a) llzo(t)ll :::::; tS2 for all t ~ T, and (b) llxo(T)- xr1 (T)II :::::;

i,

and this will conclude the proof. Property (a) is an immediate consequence ofthe fact that z 0 (t) converges to 0 as t-+ oo . To prove (b), we proceed as follows. Observe that x 0 (t) and (t) are integral curves of 1

Xr

and, respectively,

x = f(x,O)

satisfying x (T1) = xT, (Tl). Thus, 0

X0 (t)

fort~

= X (T1) + 0

t

lr1

T1,

f(x 0 (s), Z0 (s))ds

and

x!h (t)

X xr

=X

0

(T1)

+

Z

t

lrl

f(xr1 (s),O)ds.

1

Since 0 ( s), 1 ( s), 0 ( s) are defined for all s ~ T and bounded and f(x, z) is locally Lipschitz, there exist L > 0 and M > 0 such that

for all s

~

T 1 • Thus,

llxo(t)-

Xr (t)ll :$ L t

lr1

1

llxo(s)-

Xr (s)llds + M 1

{t llzo(s)llds.

lr1

Since z0 (s) converges to 0 as t-+ oo, given any c5 > 0, there is T 1 such that llz 0 (s)ll :::::; c5 for all t ~ T 1 . Thus, for all t ~ T1 we can write llxo(t)-

xr (t)ll:::::; L t 1

ir1

llxo(s)-

Xr (s)llds + Mc5(t- Tl). 1

Gronwall-Hellman's lemma yields llxo(t)-

Xr (t)ll:::::; ~t5 (eL(t-Tl) -1) 1

10.4 Input-to-State Stability

17

and, thus, llxo(T) -

Xr, (T)ll ~ ~<5 (eLT

2

-

1) .

In the right-hand side, M, L, T2 are fixed numbers, but <5 can be rendered arbitrarily small by choosing an appropriately large T1 . This proves (b) and completes the proof of the theorem. 0 one can choose S = Bli. By Corollary 10.2.2, the equilibrium (x, z) = (0, 0) is stable. Thus, there exist <51 and <52 such that, if X 0 and Z 0 satisfy

then x 0 (t) belongs to B 5 . From this, the result follows.

<1

Corollary 10.3.3. Consider the system {10.13). Suppose the equilibrium x = 0 of x = f(x, 0) is globally asymptotically stable and the equilibrium z = 0 of i = g(z) is globally asymptotically stable. Suppose the integral curves of the composite system are defined for all t ;:::_ 0 and boundul. Then, the equilibrium (x,z) = {0,0) of {10.13} is globally asymptotically stable. Proof. In this case, S = !Rn and the hypothesis needed to show, using Theorem 10.3.1, that for any initial condition (x 0 , z 0 ) the trajectory (x 0 (t), z 0 (t)) converges to zero as t -7 oo is simply the boundedness of x 0 (t). <1

10.4 Input-to-State Stability In the previous section, we have discussed the asymptotic behavior of the response x(t) of a system of the form

x = f(x,z) to an input z(t) which was, in turn, the response of the autonomous nonlinear system z = g(z). Since the latter was assumed to be locally (respectively, globally) asymptotically stable, the input z(t) driving the first system was a function of time

18

10. Stability of Interconnected Nonlinear Systems

which, for sufficiently small z(O) (respectively, for any z(O)), asymptotically decayed to 0 as t ~co. In this section we extend this type of analysis, to the case in which the input driving the first system is simply a bounded function of time. Of course, we cannot expect anymore that the state x(t) decays to 0 as t ~ co; rather, we are interested in the case in which x(t) remains bounded, and the bound on the state can be expressed as a (possibly nonlinear) function of the bound on the input. In the special case in which the input tends to 0 (in particular, when the input is identically zero), we still expect that x(t) converges to 0 as time tends to co, as it happened in the earlier situation. These requirements altogether lead to the notion of input-to-state stability, which was introduced by E. Sontag 5 and can be formally characterized as follows. Consider a nonlinear system

x = f(x,u)

(10.15)

with state x E !Rn, input u E !Rm, in which f(O, 0) = 0 and f(x, u) is locally Lipschitz on !Rn x !Rm. The input function u : [0, co) ~ !Rm of (10.15) can be any piecewise continuous bounded function. The set of all such functions, endowed with the supremum norm

llu(·)lloo =sup llu(t)ll t~O

is denoted by

L~.

Definition 10.4.1. System (10.15} is said to be input-to-state stable if there exist a class KC function /3(·, ·) and a class K function 1(·), called a gain function, such that, for any input u(·) E L~ and any X 0 E JR1l, the response x(t) of (10.15} in the initial state x(O) = X 0 satisfies (10.16) for all t

~

0.

Remark 10.4.1. Since, for any pair f3 > 0,1 > 0, max{/3,1} ~ f3 + 1 ~ max {2/3, 21}, it is seen that an alternative way to say that a system is inputto-state stable is that there exists a class KC function /3(·, ·) and a class K function 1(·) such that, for any input u(·) E L~ and any X 0 E !Rn, the response x(t) of (10.15) in the initial state x(O) = X 0 satisfies

(10.17) for all t ~ 0. Both estimates (10.16) and (10.17) will be indifferently used in the sequel. <1

5

See Sontag (1989).

10.4 Input-to-State Stability

19

In what follows, it will be shown that the property, for a given system, of being input-to-state stable, can be given a characterization which extends the well known criterion of Lyapunov for asymptotic stability. The key tool for this analysis is the notion of ISS-Lyapunov function, defined as follows.

Definition 10.4.2. A C 1 function V : !Rn --+ 1R is called an ISS-Lyapunov function for system {10.15) if there exist class K 00 functions Q(·), a(·), a{.), and a class K function x(-) such that

Q.(iixll) :S V(x) :S a(iixli)

(10.18)

for all x E IRn

and

llxll ~ x(llull)

:::}

aV ax f(x, u) ::; -a(ilxll)

for all

X

E IRn.

(10.19)

The reason why a function characterized in this way can play an important role in establishing the property of input-to-state stability can be e:xplaiQed as follows. First of all, observe that if V(x) is an ISS-Lyapunov function for (10.15), evaluation of (10.19) at u = 0 yields

av ax f(x, 0) :S -a(iixil)

for all

X

E !Rn ,

and therefore V(x) is a Lyapunov function in the usual sense of the term for the autonomous system x = f(x, 0). Thus, as seen in Section 10.1, the response of (10.15) with u(t) = 0 for all t ~ 0, and arbitrary initial state x(O) = X 0 , satisfies an estimate of the form

for all t ~ 0, where (3(·, ·) is a class K£ function, as expected from the definition of input-to-state stability. Consider now the case of a nonzero input u(·) E L~, let M

= liu(·)lloo,

and set

c = a(x(M)) .

.

Then, from the inequality on the right-hand side of (10.18), it is deduced that the set is such that Bx.(M)

C ilc.

As a consequence, llxll ~ x(M) at each x on the boundary of flc and therefore, at any t ~ 0 such that x(t) is on the boundary of ilc, llx(t)ll ~ x(i!u(t)li). Suppose now that (10.19) holds. Then,

20

10. Stability of Interconnected Nonlinear Systems

o~~x) f(x(t), u(t)) < 0 at any t 2: 0 such that x(t) is on the boundary of flc, and (see Section 10.1), it can be concluded that for any initial condition x' (0) in the interior of flc, the solution x'(t) of (10.15) is defined for all t 2: 0 and x'(t) E flc for all t 2: 0. In particular, for all t 2: 0, x' (t) satisfies

Setting

'Y(r)

=g-

1

(a(x(r))

=Q.-

1

oaox(r)'

we see that, for all t 2: 0, x' (t) satisfies llx'(t)ll ~ 'Y(IIu(·)lloo),

(10.20)

or, what is the same, llx'(-)lloo ~ 'Y(IIu(·)lloo) ·

Let now x(t) be the solution of (10.15), satisfying x(O) = X 0 • If X 0 E flc, the previous arguments show that llx(t)ll ~ 'Y(IIu(·)lloo) for all t 2: 0 and this proves that the estimate (10.16) holds. If not, i.e. if V(x 0 ) > c, observe that, so long as V(x(t)) > c, llx(t)ll > x(llu(t)ll) and this yields

dV~(t)) = ~~ f(x(t),u(t)) ~

-a(llx(t)ll)

< 0.

Thus, so long as V(x(t)) > c, the function V(x(t)) is decreasing, and this shows in particular that x(t) is bounded (in fact, llx(t)ll ~ Q.- 1 (V(x(t))) ~ g_- 1 (V(x(O))) ). Moreover, it is possible to show that, for some finite time t 0 , V(x(t))

> c, for all 0 ~ t < t 0

V(to) =c. By contradiction, suppose this is not the case. Then V(x(t))

> c for all

t 2: 0, and therefore,

dV~(t)) ~

-a(a- 1 (V(x(t))))

for all t 2: 0. In other words, the function V(t)

= V(x(t))

(10.21) is such that

V(t) ~ -a(a- 1 (V(t)) for all t 2: 0 and this shows (see proof of Theorem 10 .1.3) that, for some class KC function /3(·, ·), (10.22)

10.4 Input-to-State Stability

21

for all t ~ 0. In particular, x(t) tends to 0 as t-+ oo and thus V(x(t)) tends to 0 as t -+ oo, which contradicts the assumption that V(x(t)) was bounded from below by c > 0 for all t ~ 0. The inequality (10.21) holds for all t E [0, t 0 ). Then, an estimate of the form (10.22) holds for all t E [0, t 0 ). Since fort~ to we have

llx(t)JI ~ l(liu(·)lloo) we conclude that

for all t ~ 0, and this completes the proof of the fact that if V ( x) is an ISS-Lyapunov function for (10.15), then this system is input-to-state stable. Remark 10.4.2. The previous argument shows also how the function I(·) which appears in the estimate (10.17) can be computed from the functions g.(-), a"(-), and x(-) which characterize (10.18) and (10.19). As a matter of fact, 10 can be given the expression

l(r) =q- 1 oaox(r)

.
The arguments above show that the existence of an ISS-Lyapunov function is a sufficient condition for input-to-state stability. As in the case of asymptotic stability, this result has also a converse, namely the existence of a function V(x) having the properties indicated in (10.18) and (10.19) is implied by the property of input-to-state stability 6 Theorem 10.4.1. System (10.15} is input-to-state stable if and only if there exists an ISS-Lyapunov function.

There is an alternative way to check whether or not a function V(x) is an ISS-Lyapunov function for a given system, which is useful in many circumstances. Lemma 10.4.2. Consider system {10.15}. A C 1 function V: an -+a is an ISS-Lyapunov function for system {10.15} if and only if there exist class Koo functions q(·), a(·), a{), and a class K, function o"(-) such that {10.18} holds and ·

av ax f(x, u) ~ -a(llxll) +u(J]ull)

for all x E

an

and all u E am

Proof. Suppose (10.23) holds and define x(r)

= a- 1 (ku(r)),

with k > 1. Then, llxll ~ x(llull) implies 6

For a proof of this result, see Sontag and Wang (1995).

.

(10.23)

22

10. Stability of Interconnected Nonlinear Systems

which in turn implies

av k-1 ax f(x, u) ::; -a(llxll) +a(! lull) ::; --k-a(llxll)

0

This shows that a relation of the form (10.19) holds. Observe now that, if (10.19) holds and llxll ~ x(llull), then (10.23) holds for any a(·). To complete the proof, define

r/>(r) Then, for

av

= llull=r,llzll~x(r) max {-a f(x, u) + a(x(llull)} . X

llxll ::; x(llull),

av ax J(x, u) ::; -a(llxiD +if>( !lull) . Define

a(r)

= max{O, r/>(r)},

The function a(·) is continuous, nonnegative, and a(O) = 0. If a(·) is not a class K function, majorize it by a class K function, to have property (10.23) fulfilled. <1 Remark 10.4.3. From the proof of the previous Lemma, it is possible to deduce how the gain function "YO which appears in the estimate (10.17) can be computed from the functions _g_(·), a(·), a(·) and a(·) which characterize (10.18) and (10.23). As a matter of fact, recall that if V(x) is a (positive definite and proper) function satisfying (10.19), an estimate of the form (10.17) holds with a gain function')'(·) given by

"Y(r)

= .Q.- 1 0 Q 0 x(r)

0

On the other hand, the proof of the previous Lemma shows that if V(x) is a (positive definite and proper) function satisfying (10.23), then (10.19) holds for x(r) = a- 1 (ka(r)) and k > 1. Thus, an estimate of the form (10.17) holds with a gain function "YO given by ')'(r)

= Q.-l 0 a 0

where k is any number satisfying k > 1.

Q -

1

0

ka(r)

<1

The concepts introduced above are illustrated in the following simple examples.

10.4 Input-to-State Stability

23

Example 10.4.1. Consider a linear system

x =Ax+ Bu and suppose that all the eigenvalues of the matrix A have negative real part. Let P > 0 denote the unique solution of the Lyapunov equation P A+ AT P = -I, observe that the function V(x) = xT Px satisfies

for suitable g

> 0 and a > 0, and

av ax (Ax+ Bu) Pick any 0 <

~

that

-llxll 2 + 2llxiiiiPIIIIBIIIIull.

c:< 1 and set 2 c = 1- c: IIPIIIIBII

x(r) =cr.

J

Then

· llxll

~

x(llull)

=}

Thus, the system is input-to-state stable, with a gain function

'Y(r) = (ajg)cr which is a linear function.

Example 10..{2. Let n

<1

= 1, m =

1 and consider the system

x = -axk + bx"cp(u) , in which k Eli is odd, p E li satisfies p


J

(10.24)

a> 0, and cp(·) is a C1 function satisfying cp(O) = 0. Indeed, since a > 0 and k is odd, the system is globally asymptotically stable. Choose a candidate ISS-Lyapunov function as V(x) = ~x 2 , which yields = av f(x, u) ~ -axk+l + bx"+ 1cp(u) .

v

ax

Let 9( ·) be a class }( function satisfying

for all u E lR, and observe that, since k

Set

+ 1 is even

24

10. Stability of Interconnected Nonlinear Systems

v=k-p

and obtain

V~

+ lblfJ(Iul))

lxlp+l ( -alxlv

·

Thus, using the class Kco function a(r)

= crk+l

,

in which c > 0, it is deduced that

V~

-a(lxl)

if

-alxlv

+ lbiB(Iul)

~

-clxlv ,

i.e. if

(a- c)lxlv 2: lbiO,(Iul). Taking, without loss of generality, c (10.19) holds for the class K function x(r)

< a, it is concluded that condition

= ( lbl8(r)) t a-E

Thus, the system is input-to-state stable, with a gain function 1(·) which is bounded by this function x(
)

Example 10..{3. Let n

0

= 1, m = 1 and consider the system x = -axksgn(x) + bxPcp(u) ,

in which k Eli is even, p Eli satisfies p< k,

a > 0, and cp( ·) is a C 1 function satisfying cp(O) = 0. Note that the function xksgn(x) is a C 1 function, for any even integer k > 0. Choose the same candidate ISS-Lyapunov function as in the previous example, V(x) = !x 2 , to obtain

lf = av j(x, u) = -axk+lsgn(x) + bxP+lcp(u) .

ax

This, since k + 1 is odd, yields the same inequality found in the previous example, namely V ~ -alxlk+ 1 + lbllxl1'+ 1 8(lul) . from which identical conclusions follow.

<1

10.4 Input-to-State Stability

25

Example 10.4.4. An important feature of the two previous examples, which made it possible to prove that both systems were input-to-state stable, was the inequality (10.24). In fact, if this inequality does not hold, the system may fail to be input-to-state stable. This can be seen, for instance, in the simple example :i; = -x+xu. In fact, suppose u(t) = 2 for all t 2': 0. The state response of the system, to this input, from the initial state x(O) = X 0 coincides with that of the autonomous system

x=x' i.e. x(t) = etxo, which shows that the bound (10.16) cannot hold.

Example 10.4.5. Let n


= 2, m = 1 and consider the system .i = -z 3 + zy iJ = az 2 - y + u .

where a is a real parameter. To check whether or not this system might be input-to-state stable, for some value of a, choose the candidate ISS-Lya.punov function

to obtain

Recall that

1 z2y < -z4 - 2

and that, for any number

1 + -y2 2

o> 0, 0

-y2 Yu < - 2

1 + -u2 2o

Thus, it is seen that

.

J1 + aJ

Jl + aJ

o

1

2

2

2o

2 V<(-1+--)z 4 +(-1+--+-)y 2 +-u.

-

if

2

•

From this, it is easy to conclude that the system is input-to-state stable if this is the case, in the previous inequality the coefficient

JaJ < 1. In fact,

-I+ J1+aJ 2

o> 0 such that also the coefficient J1 +aJ o -1+--+-

of z 4 is negative, and it is possible to find 2

2

26

10. Stability of Interconnected Nonlinear Systems

of y 2 is negative. In other words, there exist numbers d1 that

> 0,

d2

> 0 such

Consider now the function

W(z, y) = d1 z 4

+ d2y 2 .

This function is positive definite and also proper because (see Remark 10.1.3), for any c > 0, the closed set {}c

= {(z,y) E JR?: W(z,y):::; c}

is bounded. In fact

\z!::;(;)!,!Y!::;(;)!.

=>

W(z,y)::;c

,1

2

Therefore, there exists a class /C 00 function a(·) such that

a(l!x!!) :::; W(z, y) for all x E JR.2 . As a consequence, .

V :::; -a(l!xl!)

1

+ 28 \u!

2

and it is concluded that an inequality of the form (10.23) holds, for u(r) =

1 2 28 r .

This shows that, if \a! < 1, the system is input-to-state stable.

<3

As shown, for instance, in the third of these examples, in a system of the form (10.15), the global asymptotic stability of the equilibrium x = 0 of :i; = f(x, 0) does not necessarily imply input-to state stability. Nevertheless, it is always possible to render such a system input-to-state stable by means of a ferrf.back transformation of the form

u

= f3(x)v

where /3( x) is an m x m matrix of smooth functions of x, defined and invertible for all x E !Rn. Note that feedback transformations of this kind, which actually correspond to x-dependent changes of coordinates in the space of input values, have been encountered several times before in the design of nonlinear control :systems.

10.4 Input-to-State Stability

27

Theorem 10.4.3. Consider system (10.15). Suppose the equilibrium x = 0 of x = f(x, 0) is globally asymptotically stable. Then, there exists an m x m matrix f3(x) of smooth functions of x, which is defined for all x E JR." and is nonsingular for all x, such that

x = f(x,f3(x)u)

(10.25)

is input-to-state stable. Proof. By the converse Lyapunov theorem, there exists a C 1 positive definite and proper function V(x) and a class /( 00 function o(·) such that

av ax f(x, 0) < -o(llxll)

for all nonzero x E IR.n.

We will prove the theorem by showing that, for some f3(x) having the properties indicated above and for some class JC function x( ·),

llxll

~

x(iiull)

~

av ax f(x, f3(x)u)

~

1

-2o(llxll)

for all

X

E ]Rn,

thus showing that V(x) is an ISS-Lyapunov function for (10.25). To this end, let g(·) be any smooth function g: [0, oo) -+ [0, oo), which is positive for all s ~ 0 and identically equal to 1 on the interval [0, 1], and set

f3(x) = g(llxii)I. The matrix f3(x) thus defined is a matrix of smooth functions of x, invertible for all x E IR.n . Define 1 8(s,r)= max {-;:;-/(x,v)+- o(s)}, 2 JlzJI=s,JivJI=r uX

av

and observe that, for aily choice of g(·),

av 1 ax f(x, f3(x)u) + 2o(llxll) ~ 8(llxll, g(llxll)llull) ,

(10.26)

for all x and u. The function 8(s, r), a continuous function defined for all (s, r) E (0, oo) x [0, oo), by hypothesis is such that d"(s, 0) < 0 for every s > 0. Thus, using continuity arguments, it is possible to prove the existence of a continuous function p(s), defined for all s ~ 0, with p(O) = 0 and p(s) > 0 for all s > 0, such that r~p(s)

~

8(s,r)
for all s > 0. In view of this and of (10.26), the result is proven if one can show the existence of a funCtion x(-) and complete the definition of g( ·) so · that (10.27) llxll ~ x(llull) ~ g(llxll)llull ~ p(llxll) .

28

10. Stability of Interconnected Nonlinear Systems

Let 8( ·) be any class

Koo

function which satisfies

B(s) < p(s) B(s)<s and set x(s) =

for s:::; 2 fors~2,

e- 1 (s).

Moreover, let g(·) be such that g ( s) :::; 1

for all s for s ~ 2.

g(s) < p(s) s

By construction, the functions B(·) and g(·) are such that for all s > 0.

g(s)B(s):::; p(s) Observe now that

llxll

~

x(iiuli)

=?

B(llxll)

~

iiuii .

Therefore,

llxll

~

x(iiuli)

g(llxll)liull:::; g(ilxii)B(IIxll):::; p(llxll)

=?

which shows that (10.27) holds and this completes the proof.


The notion of input-to-state stability lends itself to a number of alternative (equivalent) characterizations, among which the most relevant one can be derived as follows. Recall (see Remark 10.4.1) that a system is input-tostate stable if the response x(·) to an input u(·) E L~ satisfies an estimate of the form (10.17). Observe that, for any t ~ 0,

and ,8(·,0) is a class K function. Then, using (10.17), it is seen that, in an input-to-state stable system, the response x(·) to any input u(·) E L~ is bounded and, in particular, (10.28) for some class K function /o(-) (while r(·) is the same class K function for which the estimate ( 10 .17) holds). Moreover, since lim ,B(ilx 11, t) = 0 , t-+oo 0

in an input-to-state stable system the response x(t) to any input u(·) E L~ satisfies (10.29) lim sup llx(t)il:::; rCiiu(·)lloo)

t-+oo

10.4 lnput-tcr-State Stability

29

(where -y( ·) is again the same cla.ss K function for which the estimate (10.17) holds). The estimate (10.29) can be given an alternative characterization, which only involves the behavior of jju(t)JI for large t. In fact, the following result holds.

Lemma 10.4.4. Property (10.29) is equivalent to the property lim sup llx(t)JI ~ -y(limsup llu(t)JI). t-+oo t-+oo

(10.30)

Proof. Since -y(limsup jju(t)JI) ~ -y(Jju(·)JJoo), t-+oo then (10.30) implies (10.29), actually with the same function -y(-). Conversely, suppose (10.29) holds. Pick any X 0 E JRn, any u(·) E L~ and c > 0. Let

r = limsupllu(t)ll t-+oo and let h

>0

be such that

-y(r +h) - -y(r) < c . By definition of r, there exists T > 0 such that Jju(t)JI ~ r + h for all t ~ T. Let x(t) denote the response of system (10.15) from the initial state X0 = x(T) and input u(-) defined a.s u(t)

= u(t+T).

Clearly, x(t) = x(t + T), where x(·) is the response form the initial state and input u(·). By definition ofT, jju(t)jj ~ r

+h

for all t

~

X

0

0

i.e. Jju(·)lloo ~ r +h. Then (10.29) implies lim sup Jjx(t)JJ =lim sup Jjx(t)JJ ~ -y(Jju(·)lloo) ~ -y(r +h)< r(r) t-+oo t-+oo Letting c -t 0 yields (10.30), with the same

-rO a.s in (10.29).

+c

0

<1

Remark 10.4.4. From the proof of this Lemma it is deduced that, if (10.17) holds, then also (10.28) and (10.30) hold, with the same cla.ss K function -y(·). Thus, in particular, if it is known that V(x) is an ISS-Lyapunov function for system (10.15), from Remark 10.4.2 it follows that the response x(·) to any input u(·) E L~ is such that the estimates (10.28) and (10.30) hold, with

-r(·)

= ~-

1

o a ox(·).

<1

30

10. Stability of Interconnected Nonlinear Systems

The arguments above show that if the estimate (10.17) holds, then necessarily also (10.28) and (10.30) hold, for the same class K. function -y(·) and the class K function 'Yo(·)= 13(·,0). However, it turns out that the properties expressed by these two inequalities are not just implications, but equivalent characterizations, of the property of input-to-state stability 7 . Theorem 10.4.5. System (10.15} is input-to-state stable if and only if there exists class K. functions -y0 (·) and-y(·) such that, for any input u(·) E L~ and any x 0 E JRn, the response x(t) in the initial state x(O) = X 0 satisfies

llx(·)lloo lim sup llx(t)ll

< max{'Yo(llxoll),-y(llu(·)lloo)} < -y(limsup llu(t)ll).

t-400

t-400

Note that, if (10.17) holds, then - as shown before - also (10.28) and (10.30) hold, with the same class K function -y(·). Theorem 10.4.5 above says that the fulfillment of estimates of the form (10.28) and (10.30) implies input-to-state stability, i.e the fulfillment of an estimate of the form (10.17). However, is important to stress that this may occur for a possibly different function -y(·). This can be seen, for instance, in the following example. Example 10.4.6. Consider the system .

1

x = - - -2x 1 +u

for which

t

x(t) = x(O) exp(-

1 0

'

1 1 +u 2

( ) T

dT) .

Clearly, lx(t)l ~ lx(O)I

and

lim x(t) = 0 t-400

for any u(·) E L 00 and any x(O), so that (10.28) and (10.30) hold for -y0 (s) = s and -y(s) = 0. However, an estimate of the form (10.17) with-y(·)= 0 cannot hold, because the rate of converge of lx(t)l to zero decreases as llu(·)lloo increases. It is easy to see that V(x) = x 4 is an ISS-Lyapunov function for this system. In' faet., observe that, for any c > 0, ""'"'.. -·· .,,;..;::

lxl 2: clul

Thus, the two class K00 functions

X(r) 7

= Er

For a proof of this result, see Sontag and Wang (1996).

10.5 Input-to-State Stability of Cascade-Connected Systems

31

are such that

av 1 -(---x ) 1 + u2

ox

4

4x = -1< -a(lxl). +u2 -

Thus, an estimate of the form (10.17) holds, in which -y(-) can be given the expression -y(r) = g- 1 o a o x(r). Since one can pick g(r) = a(r) = r 4 , this yields -y(r)

= cr ,

where c is any small (but nonzero) positive number.

<1

10.5 Input-to-State Stability of Cascade-Connected Systems In this section we discuss a problem similar to that considered in Section 10.3, this time referred to the property of input-to-state stability. More precisely, we shall consider the cascade connection of two subsystems, each one being input-to-state stable, and we will prove that the composed system is still input-to-state stable. As a preliminary step in the analysis, we first discuss a property of input-to-state stable systems, which will be used later in the proof of the main result of this section, but that also has its own independent interest. Recall that, according to the results discussed in the previous section and, in particular, to Lemma 10.4.2, a system

x = f(x,u) is input-to-state stable if there exists a continuously differentiable function V(x) satisfying (10.18), a class /C 00 function a(·) and a classIC function a(·) such that av (10.31) f(x, u) ~ -a(llxll) +a( Ilull)

ox

for all x E !Rn and all u E !Rm . The main significance of the latter inequality is essentially the following one. Consider a bounded input u(·), let c > 0 be any small number, set

and c(d)

= llzll~d max V(x)

.

By definition, nc(d)

::>

iJd .

If (10.31) holds, the derivative of V(x(t)) is strictly negative at each x(t) such that llx(t)ll ~ d (thus, in particular, on the boundary of ilc(d))· In fact, at all such points,

32

10. Stability of Interconnected Nonlinear Systems

dV~(t))

:::; -o:(llx(t)ll)

+ u(iiu(t)ii):::;

-a:( d)+ u(iiu(·)lloo))

= -e ·

This shows that, under this input, for any trajectory x(t) there is a time

t 0 such that V(x(t)) :::; c(d) for all t ~ t 0 . In other words, the composed class K function o:- 1 (u(·)) characterizes how the bound llu(·)lloo on the input function determines the (finite) number c(d) by which V(x(t)) is guaranteed to be bounded for suitably large times. In view of this, it appears that only the composition of the two functions o:- 1 (·)and u(·) matters in establishing a correspondence between a bound on the input and a bound on the state, and that, of course, infinitely many other such pairs might yield the same result. Motivated by this observation, we will examine below the problem of constructing families of pairs of functions o:( ·) and u(·) rendering an inequality of the form (10.31) true for a given input-tostate system. For convenience, we will say that a pair {o:(·),u(·)}, in which the former is a class Koo function and the latter a class K function, is an ISSpair for system (10.15) if, for some C 1 function V(x) satisfying estimates of the form (10.18), the inequality (10.31) holds for all X E an and all U E am. Lemma 10.5.1. Assume {a:(·), cr(-)} is an ISS-pair for (10.15).

(i) Let a(-) be a class K function such that cr(r) = O[a(r)] as r -+ oo. Then, there exists a class Koo function a{) such that {a{), a(·)} is an ISS-pair. (ii} Let a{) be a class Koo function such that a(r) = O[o:(r)] as r -+ o+. Then, there exists a class K function a(-) such that {a(·), a(·)} is an ISS-pair. Proof. For both parts of the theorem, the proof will be conducted by considering a candidate ISS-Lyapunov function W(x) of the form

W(x) = p(V(x)) where p( ·) is a class Koo function defined by an integral of the form

1 8

p(s) =

q(t)dt

in which q(·) is a smooth function [0, oo) -+ [0, oo), which is non-decreasing and such that q(s) > 0 for s > 0 (the class of all such functions is often denoted as SN). For this function, we wish to obtain an inequality of the form (10.31). To. this end, observe that

aw

ax f(x, u)

= q[V(x)] av ax f(x, u) :::; q[V(x)][-o:(iixll) + u(iiuii)].

(10.32)

Set 8(s) =.a(o:- 1 (2u(s))). Then, it is easy to see that the right-hand side of (10.32) bounded by

is

1

- 2q[V(x)]o:(ilxli)

+ q[8(llull)]u(lluli)

.

(10.33)

10.5 Input-to-State Stability of Cascade-Connected Systems

33

In fact, this is indeed the case (no matter what 9(·) is) if a(llxll) ~ 2a(llull). If, a(llxll) :::; 2a(llull), observe that V(x) :::; o(llxll) :::; 9(11ull), in which case the right-hand side of (10.32) is bounded by the quantity -q[V(x))a(llxll) + q[B(IIuii)Ja(llull). The quantity (10.33) can in turn be bounded by 1

-2q[q(llxii)Ja(llxll)

+ q[9(lluii)Ja(llull),

and, therefore, part (i) is proven if one can show that it is possible to find q(·) and o{) such that

q(q(s))a(s) ~ 2a(s) q[8(r)]a(r) :::; O'(r) .

(10.34)

Assume, without loss of generality, that a(·) is class K.oo (if this is not the case, majorize it by a class K. 00 function), so that also 9(·) is class K. 00 and define

/3(r) = a(8- 1 (r)), Both these functions are class K. 00 and, since a(r) = O[O'(r)) as r --+ oo, also /3(r) = O[,B(r)) as r --+ oo. Using this property, it is easy to see that there exists a class SN function q(·) such that

q(r)/3(r) :::; ,B(r) for all r E [O,oo). Thus,

q[9(r))a(r) :::; O'(r) . Define

-

a(s)

= 21 q[q(s))a(s)

(10.35)

.

(10.36)

This is a class K. 00 function, because so is a(·) and q(·) is of class SN. Indeed, (10.35) and (10.36) prove (10.34) and this completes the proof of (i). To prove part (ii), we needto find q(·) and 0'(-) such that (10.34) holds. To this end, define

These functions are such that ,B(r) = 0[/3(r)) as r--+ o+. Using this property, it is easy to see that there exi~ts a class SN function q(·) such that

,B(s):::; q(s)/3(s) for all s E [O,oo). Thus

-

~q[9(s)]a(s):::;

-a(s).

(10.37)

Define (10.38) O'(r) = q[B(r)]a(s) , which is a class K. 00 function. Indeed, (10.37) and (10.38) prove (10.34) and this completes the proof of (ii). <3

34

10. Stability of Interconnected Nonlinear Systems

As an application of this result, we find the proof of the property that the cascade connection of two input-to-state stable systems is input-to-state stable. More precisely, consider a system of the form

x

f(x,z) g(z, u) ,

i

(10.39)

in which x E !Rn, z E !Rm, f(O, 0) = 0, g(O, 0) = 0, and f(x, z), g(z, u) are locally Lipschitz (see Fig. 10.2).

~ z~g(z,u)

I

z

;I•~J(z,z) I

Fig. 10.2. Cascade connection with input.

Theorem 10.5.2. Suppose that system

x = f(x,z),

(10.40)

viewed as a system with input z and state x is input-to-state stable and that system (10.41) i = g(z,u), viewed as a system with input u and state z is input-to-state stable as well. Then, system {1 0.39) is input-to-state stable. Proof. By hypothesis, there exist an ISS-pair {a{), cr(·)} for system (10.40) and an ISS-pair {,8(-},((·)} for system (10.41). Define a function iJ(·) in the following way for smalls jj(s) = { ,B(s) o-(s) for large s.

Then, by Lemma 10.5.1, part (ii), there exists((-) such that {iJ(·), ((·)}is an ISS-pair for system (10.41). Also, define a function c1(·) as

c1(s)

= 21 ,e(s).

Then, by Lemma 10.5.1, part (i), there exists a(·) such that {a(·), ~i3(·)} is an ISS-pair for system (10.40). As a consequence, there exist positive definite and proper functions V(x) and U(z), such that

10.5 Input-to-State Stability of Cascade-Connected Systems

av ax f(x, z)

< -a(llxll) + t.B(IIzll)

au

< -.B(IIzll) +((!lull) .

az g(z, u)

The composite function W(x, z) = V(x)

aw

ax f(x, z)

+

35

aw

az g(z, u) S

+ U(z)

satisfies

1-

-

-a(ll;z;ll) - 2.B(IIzll) +((!lull) .

and therefore W(x, z) is an ISS-Lyapunov function for the composite system (10.39).
x=

f(x,z),

viewed as a system with input z and state x is input-to-state stable and that the equilibrium z = 0 of i = g(z) , is globally asymptotically stable. Then, the equilibrium (x, z) tem (1 0.13} is globally asymptotically stable.

= (0, 0)

of sys-

Proof. The driving system is trivially input-to-state stable and so is the cascade (10.13), by Theorem 10.5.2. The latter, which has no input, is therefore globally asymptotically stable.
x=

f(x,u)

has an ISS-pair {a{), 0'( ·)} in which, for some integer q > 0, as r-+ oo. Then, for any K > 0, the system in question has an ISS-pair {a(-), u( ·)} in which iT(r) = Krq. Accordingly, for some positive definite and proper C 1 function V (x),

~~ f(x,u)

S -a(llxll) + Kllullq.

(10.42)

Suppose now the input function u : [0, oo) -+ !Rm is a piecewise continuous function satisfying lim T-+oo

f

Jo

T

llu(t)llqdt

< oo.

The set of all such functions, endowed with the norm

36

10. Stability of Interconnected Nonlinear Systems

roo

1

llu(·)llq = (Jo llu(t)llqdt) • , is denoted by L'J'. Integration of (10.42) on the interval [0, t] yields

1t llu(s)llqds:::; 1 llu(s)llqds 00

V(x(t))- V(x 0 )

:::;

K

K

and therefore V(x(t)):::; V(x

0 )

+ K[llu(·)llqr,

which in turn, using the estimates (10.18) yields

This shows that the response x( t) to any ,input u( ·) E L'J', in any initial state x 0 , exists for all t :::: 0 and is bounded, by a quantity which depends on the norm of the initial state and on the norm of the input function.

10.6 The "Small-Gain" Theorem for Input-to-State Stable Systems In this section we investigate the stability property of feedback interconnected nonlinear systems, and we will see that the property of input-to-state stability lends itself to a simple characterization of an important sufficient condition under which the feedback interconnection of two globally asymptotically stable systems remains globally asymptotically stable. Consider the following interconnected system

±1 :i:2

= fi(x1,x2) = /2(x1,x2,u),

(10.43)

in which x1 E 1Rn1 , x2 E 1Rn2 , u E IRm and h (0, 0) = 0, /2(0, 0, 0) = 0 (see Fig. 10.3). We suppose that the first subsystem, viewed as a system with internal state x 1 and input x 2 is input-to-state stable. Likewise, we suppose the second subsystem, viewed as a system with internal state x 2 and inputs x 1 and u is input-to-state stable. In view of the results discussed at the end of Section 10.4, the hypothesis of input-to-state stability of the first subsystem is equivalent to the existence of two class K functions -y01 (·), -y1(·) such that the response x 1(·) to any input x 2 ( ·) E £~ satisfies (10.44) for all

t:::: 0, and

10.6 The "Small-Gain" Theorem for Input-to-State Stable Systems

--

±1

= !I(x1,x2)

37

1-X1

-

:i:2 = h(xl, x2, u)

f.-

'11.

Fig. 10.3. Feedback connection with input.

lim sup llx1(t)ll::::; 'YI(limsup llx2(t)ll). t-+oo

(10.45)

t-+oo

Likewise the hypothesis of input-to-state stability of the second subsystem is equivalent to the existence of three class JC functions 1'02 ( ·), 1'2 ( ·), /u ( ·) ~uch that the response x 2 ( ·) to any input x 1( ·) E L~, u( ·) E L: satisfies (10.46) for all t

~

0, and

lim sup llx2(t)ll::::; max{1'2(limsup llxi(t)ll),-yu(limsup llu(t)ll)}. t-+oo

t-+oo

(10.47)

t-+oo

In what follows we shall prove that if t.he composite function -y1 o -y2(·) is a simple contraction, i.e. if (10.48) the composite system is input-to-state stable. This result is usually referred to as the small-gain theorem.

Theorem 10.6.1. If the condition {10.48) holds, system {10.43), viewed as a system with state x = (x 1 , x 2 ) and input u, is input-to-state stable. In particular, the class JC functions 'Yo(r)

=

max{2-yOI(r),~'Yo2(r),2-yl o-yo2(r),2-y2 o'Yoi(r)}

-y(r)

=

max{21'1 o "Yu(r), 2-yu(r)}

are such that response x(t) to any input u( ·) E L: is bounded and llx(·)lloo

<

max{'Yo(llxoll), 'Y(IIuOIIoo)}

lim sup llx(t)ll

::::;

-y(limsup llu(t)IJ).

t-+oo

t-+oo

38

10. Stability of Interconnected Nonlinear Systems

£-;;:,. We show first that ,the corresponding trajectories xl(t) and x 2 (t) exist for all t ~ 0 and are bounded. For, suppose this is not the case. Then, for every number R > 0, there exists a timeT > 0, such that the trajectories are defined on [0, T], and Proof. Pick x]' E 1Rn1 , x2 E 1Rn 2 , u(·) E

either

llxi(T)II > R

or

llx2(T)II > R.

(10.49)

Choose R such that

> max{"Yot(llx~ll),/'1 °'Yo2(llx211),/'1 °/'u(llu(·)lloo)} > max{"Yo2(llx211),/'2 ° 'Yol(llx]'ll),/'u(llu(·)lloo)},

R R

and let T be such that (10.49) hold. Define, for i x[(t)

= xi(t) =0

if

(10.50)

= 1, 2,

t E [0, T]

ift>T.

and let £ 1(t) denote the response of the top subsystem of (10.43), in the initial state x]', to the input xf (·). Since the latter is bounded on [0, oo), we have for all t ~ 0. Since, by causality, for all t E [0, T] we deduce that llxf<·)lloo

= tE[O,T] max llx1(t)ll ~ max{"YodllxriD,'Yl(llxf(-)lloo)}.

(10.51)

Similarly, let £ 2(t) denote the response of the bottom subsystem of (10.43), in the initial state x2, to the input xi(·), u(·). Since the latter are bounded on [O,oo), we have

for all t

~

0. This, in turn, since for all t E [0, T]

yields llxfOIIoo

= tE[O,T] max llx2(t)ll ~ max{"Yo2(llx~ll),/'2(11xf(·)lloo),'Yu(llu(·)lloo)} ·

- (10.52) Observe now that, if a~ max{b,c,O(a)} and 8(a)
10.6 The "Small-Gain" Theorem for Input-to-State Stable Systems

39

llxf(-)lloo ~ max{l'o1(llxriD,-r1 °1'o2(11x~ll),-y1 °1'u(llu(·)lloo)} · Observe also that if -y1 o -y2 ( ·) is a simple contraction, then also -y2 o -y1(·) is a simple contraction. In fact, let -r1 1(·) denote the inverse of the function -y1 (·), which is defined on an interval of the form [O,ri) where r~

= lim -y1(r) . r->oo

If -y1 o -y2(·) is a simple contraction, then -y2(r) < -r1 1(r) for all 0 < r < r~,

and this shows that 1'2b1(r)) < r for all r > 0, i.e. -y2 o -y1(-) is a simple contraction. Therefore, an argument identical to the one used before shows that llxiOIIoo ~ max{'Yo2(llx~ll),-y2 ° 'Yoi(IIxriD,-ru(llu(·)lloo)}. In particular, using (10.50), we have llxi(T)II llx2(T)II

~ ~

max{/'oi(IIxJ'II),-y1 o-yo2(llx~jl),-y1 o-yu(llu(·)lloo)} max{/'o2(llx211),-y2o1'01(llxJ'II),-yu(llu(·)lloo)}

which contradicts (10.49). Having shown that the trajectories are defined for all t (10.44) and (10.46) yield ~

llx1(-)lloo llx2(·) lloo

~

~

< R < R,

0 and bounded,

ma.x{/'OI(IIxJ'II),-yl(llx2(·)lloo)} ma.x{/'o2(11x211), 1'2 (jjx1 (·)lloo), 'Yu(llu( ·)lloo)} ,

combining which, and using the property that -y1o-y2(·) is a simple contraction, one obtains llx1(-)IJoo llx2(·)1ioo

~ ~

max{l'o1(11x)'jl),-y1 °1'o2(11x211),-y1 o-yu(llu(·)II<X>)} maxho2(llx211),-y2 o 'Yo1(11x)'jl),-yu(llu(·)lloo)} ·

In a similar way, combining now (10.45) and (10.47) (in which all the limits are finite since x 1(·) and x2 (·) are bounded) and using the property that -y1 o -y2( ·) is a simple contraction, one obtains

limsup llx1(t)ll t-too lim sup llx2(t)ll t-too From this, observing that

< 1'1 o-yu(limsupllu(t)ll) t-too < 'Yu(limsup llu(t)ll). t-too

and lim sup llx(t)ll S max{2limsup llx1(t)ll, 2limsup llx2(t)ll}, !-too t->oo t-too the result follows. <1

(10.53)

40

10. Stability of Interconnected Nonlinear Systems

The condition (10.48), i.e. the condition that the composed function ')'r o is a contraction, is usually referred to as the small gain condition. Of course, it can be written in different alternative ways depending on how the functions 'Yr (·) and 1'2 (·) are estimated. For instance, if it is known that Vr (xr) is an ISS-Lyapunov functions for the upper subsystem of (10.43), i.e. a function such ')'2 (-)

_qr (JJxrJJ) ~ Vi (xr) ~ or (JJxrJJ)

avr -a fr (xr, x2) xr

~ -a(JJxrJJ)

then (see Remark 10.4.4) estimates of the form (10.44) and (10.45) hold with

')'r (r)

= _q!r o or o Xr (r) .

Likewise, if V2 (x 2 ) is a function such that

_q2(JJx2JJ) ~ V2(x2) ~'o2(JJx2JI)

av2 -a h(xr,x2,u) X2

~

-a(JJx 2 JI),

then, by means of arguments similar to those used in Section 10.4, it is easy to deduce that estimates of the form (10.46) and (10.47) hold with

1'2(r) 'Yu(r)

=

_q2r o 02 o Xz(r) q2r o 02 o Xu(r) .

=

In this case the small-gain condition can be written in the form

_qlr

o

o 1 o Xr

o

_q2r o 02 o x2 (r) < r .

We conclude the section with a simple example. Example 10. 6.1. Consider the system -x~

±1

±2

=

+ xrx2 +u

axi - x2

.

(10.54)

in which a is a real parameter, viewed as a system with state (xr, x 2 ) E IR2 and input u. We regard this as interconnection of two one-dimensional systems, described by the upper and, respectively, lower equation, and we use the result of Theorem 10.6.1 to determine whether or not there are values of a such that the system is input-to-state stable. For the upper subsystem

viewed as a system with state xr and input x2, we consider the ISS-Lyapunov function V(x 1 ) =~xi and obtain

10.6 The "Small-Gain" Theorem for Input-to-State Stable Systems

.

V Choose any 0

8V

= -8X1 h(xl,xz)

4

S -lxll

41

2

+ lx1l lxzl.

< c < 1 and observe that (1- c)lxd 2 ~ lxzl

implies

VS

for the class /C 00 function a(r)

-a(jx11)

= er4 . Thus, the inequality (10.19} holds with x(r}

,;T. = r.--= . v1-c

Since we can take Q:(r}

1

= a(r} = 2r 2

it is deduced that the upper subsystem of (10.54) is input-to-state stable estimates of the form (10.44} and (10.45} hold, with ')'l(r}

,;r

= v1-e . r.--=

~d

.

For the lower subsystem of (10.54) X2

= axi -

x2

+u

viewed as a system with state x 2 and input (x 1 , u}, consider again a quadratic ISS-Lyapunov function V(xz) = ~x~ and obtain .

V

2 = -88V !z(xl,xz,u) S lxzl(iallxll -jx2l +lui). X2

Choose again any 0 < e

< 1 and observe that

(1- e)lxzl ~ lallx11 +lui 2

yields

VS

for the class /C 00

-,a(jx21) function a(r) = er . Thus, the class JC functions 2

Xz(r) = 21jajr2,

~

Xu(r) =

1- c

-€

are such that

lxzl ~ max{xz(lxli),xu(iui)}

V S -a(lxzi).

::?

As a consequence, estimates of the fonn (10.46) and (10.47) hold, with 2jajr 2 /2(r) = 1- e,

Xu(r)

2r

= 1- c

.

Checking thP small gain condition on these estimates yields 2jajr /2(/l(r}) = (1-c)2 which can be fulfilled, for all r > 0, if lal


< 1/2.


42

10. Stability of Interconnected Nonlinear Systems

10.7 Dissipative Systems In the previous section, we have characterized how, in a given input-to-state stable system, bounds on the input function determine bounds on the corresponding state response and we have described how such a characterization can be exploited in order to study the asymptotic stability of the interconnection of several subsystems. In the present and in the following section we conduct a somewhat similar analysis, but addressing the case in which also the outputs of the systems enter into the picture. To this end, consider a system described by equations of the form

x = y

=

f(x,u) h(x,u)

(10.55)

with state x E !Rn, input u E !Rm, output y E JRP, f(O, 0) = 0, h(O, 0) = 0, and where f(x, u), h(x, u) are locally Lipschitz. A notion which, in the case of systems of this form, plays a role having a number of similarities with the notion of inpuJ;-to-state stability is the notion of dissipativity, whose definition is essentially based on replacing, in the inequality (10.23), the class K function cr(ilull), with an arbitrary (thus not necessarily positive) continuous function q( u, y) of the input u and output y, with q(O, 0) = 0. This function is usually called a supply rate.

Definition 10.7.1. System (10.55) is said to be dissipative, with respect to the supply rate q( u, y), if there exists a C 1 function V (x) satisfying g(llxll) ~ V(x) ~ a(llxll)

(10.56)

for all x E !Rn, where g(·) and a(·) are class Koo functions, such that

ov ox f(x,u)

~ q(u,y)

(10.57)

for all x E !Rn, u E !Rm, and ally= h(x, u). The system is said to be strictly dissipative if, instead of {10.57}, /he inequality

ov ox f(x, u) ~ -a(llxll) + q(u, y)

(10.58)

holds for some class Koo function a(·). A function V(x), satisfying (10.56}, for which either {10.57} or (10.58} holds, is called a storage function for (10.55}. The inequalities {10.57} and ( 10.58) are called dissipation inequalities.

For convenience, it is useful to define the notion of dissipativity also in the case of a memoryless system, that is a system defined by an input-output map of the form y = cp(u), (10.59) where cp: !Rm --t JRP is continuous and cp(O) = 0.

10.7 Dissipative Systems

43

Definition 10.7.2. System (10.59) is dissipative, with respect to the supply rate q(u,y), if q(u,y) ~ 0 for all u E

~m

and ally=
Of course, an input-to-state stable system, with ISS-pair {a{), u( ·)}. is strictly dissipative with respect to the supply rate q(u,y)

= u(llull).

Conversely, if a system is strictly dissipative with respect to a supply rate q (u, y) satisfying

q(u, y) ~

u(llull)

for all u E

~m,

yE

~P,

for some class IC function u( ·), then this system is input-to-state stable. If a system is strictly dissipative with respect to a supply rate q(u, y) satisfying for all y E JRP , q(O,y) ~ 0 (10.60) then this system is globally asymptotically stable, in fact with Lyapunov function V(x). H a system is just dissipative (but possibly not strictly dissipative) with respect to a supply rate satisfying (10.60), V(x) is only guaranteed to satisfy

av ox f(x, 0) ::=; 0 and thus the system in question is only guaranteed to be stable in the sense of Lyapunov, maybe not globally asymptotically stable. The latter property holds if (10.60) is satisfied in the stronger sense q(O,y)

<0

for ally::/; 0,

(10.61)

and the system has the following property of zero-state detectability. Definition 10. 7.3. System (10.55) is zero-state detectable if, for any x 0 such that the solution X 0 (t) of x = f(x,O) satisfying x 0 (0) = X 0 is defined for all t ~ 0, the condition that h(x 0 (t),O) = 0 for all t ~ 0 implies limt-+oo X 0 (t) = 0. Remark 10. 7.1. In other words, a system is zero-state detectable if, whenever the input is identically zero on [0, oo ), any state trajectory which is defined for all t E [0, oo), and such that the associated output is identically zero on [0, oo), converges to zero as t-+ oo. <3

44

10. Stability of Interconnected Nonlinear Systems

To see why this property, in a system which is dissipative with respect to a supply rate satisfying (10.61), implies global asymptotic stability, one can proceed as follows. First of all, observe that

av ox f(x,O) ~ q(O,y) ~ 0.

(10.62)

Hence, V(x(t)) is non-increasing and this, since V(x) is positive definite and proper, implies that all trajectories are bounded. Set V00

= t4oo lim V(x(t)).

Using standard arguments, pick any trajectory x(t) and let r denote its wlimit set, which- since the trajectory x(t) is bounded- is nonempty, compact Since and invariant. By definition of w-limit set, V(x) = Voo at each X E r is invariant under x = f(x, 0), any trajectory x 0 (t) of x = f(x, 0) with initial condition X 0 in r is such that V(x 0 (t)) is constantly equal to V00 and (10.62) yields 0 ~ q(O,h(x 0 (t),O)) ~ 0.

r.

Because of (10.61), it follows that necessarily h(x 0 (t),O) = 0 for all t ~ 0. Using the property of zero-state detectability, this yields limt 4 oo X 0 (t) = 0 and, in turn, V00 = 0. Since V(x) is continuous and vanishing only at x = 0, this proves that also the trajectory x(t) tends to zero as t -4 oo. Besides input-to-state stable systems, there are other relevant classes of dissipative systems. One of these is that of the so-called finite Lq gain systems, which are defined as follows.

Definition 10.7.4. A system is said to have finite Lq gain if it is dissipative with respect a supply rate of the form q(u,y) for some K

= Kllullq- LIIYIIq,

(10.63)

> O,L > 0.

Note that, in a finite Lq gain system, the supply rate satisfies (10.61). Thus, finite Lq gain systems are globally asymptotically stable, if zero-state detectable. The terminology "finite Lq gain" is related to the following properties. First of all, observe that if a system is dissipative with respect to a supply rate of the form (10.63), it is also dissipative with respect to a supply rate of the form

(10.64) In fact, it is immediate to see that if V(x) renders the dissipation inequality (10.57) fulfilled for a supply rate q(x,y) of the form (10.63), the inequality

av axf(x,u):::; --rqllullq -iiYIIq

(10.65)

10.7 Dissipative Systems

45

holds, with V(x)

K

1

= LV(x),

"fq

= L,

and this shows that the system is dissipative with respect to a supply rate of the form (10.64). Integration of the inequality (10.65) on the interval [0, T] yields (see also the end of Section 10.5) that for any u( ·) E L';', and any initial state X 0 , . V(x(T))

::;

V(x 0 )

+ "fq

1T

JJu(t)JJqdt

-1T

<

V(x 0 )

+ "(q

1

Jlu(t)Jiqdt,

JJy(t)JJqdt

00

from which it is deduced that the response x(t) of the system is defined for all t E [0, oo) and bounded. Now, suppose X 0 = 0 and observe tha.t the previous inequality yields V(x(T))::; 'Yq

for any T

> 0. Since V(x(T))

1T ~

JJu(t)JJqdt

-1T

JJy(t)JJqdt

0, we deduce that

foT JJy(t)JJqdt::; 'Yq faT JJu(t)JJqdt::; 'Yq [llu(·)JJqr for any T

> 0 and therefore

i.e.

JJy(·)Jiq::; 'YJJu(·)Jiq. In other words, for any u( ·) E L';', the response of the system from the initial state x(O) = 0 is defined for all t ~ 0, produces an output y(·) which is a function in£:, and the ratio between the Lq norm of the output and the Lq norm of the input is bounded by "f. For this reason, the system is said to have a finite Lq gain, bounded from above by the number 'Y· The case q = 2 has a special interest, because functions in £'{' and L~ represent signals having finite energy over the infinite time interval [0, oo ), and therefore the number 'Yin the dissipation inequality (10.65) can be given the interpretation of ratio between the energies of output and input. Moreover, another interpretation, which does not necessarily require the consideration of inputs having finite energy, is possible. Suppose the input is a periodic function of time, with period T, i.e. that u(t

+ kT)

= u 0 (t),

for all t E [0, T), k

~

0

46

10. Stability of Interconnected Nonlinear Systems

for some piecewise continuous function U 0 (t), defined on [0, T). Also, suppose that, for some suitable initial state x(O) = X 0 , the state response X 0 (t) of the system is defined for all t E [0, TJ and satisfies X

0

(T) =

X

0 •

Then, it is obvious that x 0 (t) exists for all t ~ 0, and is a periodic function, having the same period T of the input, namely x 0 (t

+ kT) = x (t), 0

for all t E [O,T), k

~

0

and so is the corresponding output response y(t) = h(x 0 (t),u(t)). For the triplet { u(t), X 0 (t), y(t)} thus defined, integration of the inequality (10.65) (with q = 2) over an interval [t0 , t0 + T], with arbitrary t0 ~ 0, yields 0

V(x (to

+ T))- V(x

0

(to))::;

"·?

l.

to+T

.

//u(s)// 2 ds-

1t0 +T

to

0

i.e., since V(x (to

+ T))

l/y(s)l/ 2 ds,

to

0

= V(x (to)),

(10.66) Observe that the integrals on both sides of this inequality are independent of t 0 , because the integrands are periodic functions having period T, and recall that the root mean square value of any (possibly vector-valued) periodic function f(t) (which is usually abbreviated as r.m.s. and characterizes the average power of the signal represented by J(t) ) is defined as

1/J(-)1/r.m.s.

1 1to+T

= (T

2

l/f(s)l/ ds

)

!

.

to

With this in mind, (10.66) yields

1/y(·)l/r.m.s. ::5 -y/lu(·)/lr.m.s. ·

(10.67)

In other words, in a finite £ 2 gain system, the number "Y (which appears in the dissipation inequality (10.65)) happens to be also an upper bound for the ratio between the r.m.s. value of the output and the r.m.s value of the input, whenever a periodic input is producing (from an appropriate initial state) a periodic (state and output) response. We can therefore conclude that, in a system which satisfies a dissipation inequality of the form

the number "Y can be given these two interpretations. If the input represents a signal whose energy over the infinite interval [0, oo) is finite, then the corresponding output from the initial state x(O) = 0 is a function having finite

10.7 Dissipative Systems

47

energy over the interval [0, oo) and the ratio between the energies of output and input is bounded from above by the number I· On the other hand, if the input is a periodic function which produces, from some appropriate initial state x(O) = X 0 , a periodic state and output response, the number r provides a bound for the ratio between the average powers of the output and input. Example 10. 7.1. As an illustration of the last concept, suppose a system of the form (10.55) is strictly dissipative, with respect to a supply rate of the form (10.68)

By definition, this system is globally ~ymptotically stable and has finite L 2 gain. Suppose, in addition, that the functions q(·) and a{) which characterize the estimates (10.56) and the dissipation inequality (10.58), for some. {J > 0, a > 0, b > 0, are such that

(HJ.69) for all s E [0, fJ]. Then, (see Lemma 10.1.5) the equilibrium x = 0 of the system is also locally exponentially stable. Let the input be a sinusoidal function of time, of period T = 211" / w0 , e.g. u(t) = Uu 0 cos(w0 t)

(10.70)

where U > 0 and u 0 E lRm has unitary norm. Bearing in mind the results discussed in Section 8.1 it is easy to realize that, if U is sufficiently small, and the initial state is appropriately set, the system exhibits a response which is a periodic function of period T. In fact, observe that the input thus defined can be viewed as generated by an autonomous system of the form ( 8.2), with wE IR2 ,

s(w) p( w)

(

=

-~o ~0 ) w

u0 ( 1

in the initial state w(o), = (

0) w

~) u

.

Since this system is neutrally stable and the equilibrium x = 0 of x = f(x, 0) is locally exponentially stable, the hypotheses of Proposition 8.1.1 are fulfilled and there exists a mapping x = rr( w), defined in a neighborhood wo of the origin in JR2 , with rr(O) = 0, which satisfies

811" aw s(w)

= f(rr(w),p(w)) . If U is sufficiently small, w(t) E wo for all t :;:: system (10.55) in the initial state

(10.71)

0, and the response of

48

10. Stability of Interconnected Nonlinear Systems 0

x (0) = 1r( (

~) U)

(10.72)

is given by (10.73) which, indeed, is a periodic function (of period T). As a matter of fact, the latter is precisely the expression of the steady state response of the system to the input (10.70). Thus, in view of the interpretation presented above, one can conclude that, for any sufficiently small U, the (output) response jj(t) of system (10.55) to the input (10. 70) in the initial state (10. 72) is such that

llii(-)llr.m.s.

~

l'llu(·)llr.m.s.. <1

Another relevant class of dissipative systems is that of the so-called passive systems, which are defined as follows. Definition 10.7.5. A system in which m =pis said to be passive (respectively, strictly passive) if it is dissipative (respectively, strictly dissipative) with respect to the supply rate

(10.74) A system which is dissipative with respect to a supply rate of the form

(10.75) for some c

> 0,

it said to be output strictly passive.

Note that the supply rate of a passive system satisfies (10.60), and the supply rate of an output strictly passive system satisfies (10.61). Thus, strictly passive systems are globally asymptotically stable, and output strictly passive systems are globally asymptotically stable if zero-state detectable. Also, it is easy to see that output strictly passive systems are necessarily finite L 2 gain systems. In fact, observe that, for any 8 > 0

Suppose a system is output strictly passive, with supply rate (10.75) and choose 8 = c, to obtain

Thus, the system is dissipative with respect to the supply rate 1

q(u,y)

E:

= 2clluW- 2IIYl1 2

10.7 Dissipative Systems

49

and, therefore, also dissipative with respect to a supply rate of the form (10.68) with 'Y = c 1 . Observe that a strictly passive system is a system for which there exists a positive definite and proper C 1 function V(x) satisfying

av ax f(x,u)- yTu :$ -a(llxJI) for some class /C 00 function_g(·), i.e. such that the difference between the derivative of V(x) along trajectories and the supply rate yTu is bounded, from above, by a function which is negative for any nonzero x. A weaker version of this notion is that of output strict passivity, in which the bound in question is replaced by function of x and u, namely -.sllh(x,u)ll 2 , which is only non-positive. For the sake of completeness, we review below another variant of this notion, in which the bound in question is still a non-positive function x and u, but not necessarily proportional to the square of the norm of the output. Definition 10.7.6. A system of the form {10.55), in which m = p, is said to be weakly strictly passive if there exists a positive definite and proper function C 1 V(x), and a continuous function d(x,u), defined on lRn x lim and satisfying

d(O,O) such that

= 0,

d(x, u) ;::: 0 far all (x, u) E lin x lim ,

av ax f(x,u)

:$ -d(x,u)

+y

T

(10. 76)

u,

for all x E !Rn, u E !Rm, and, moreover, for any X 0 and any u 0 (-) such that the solution X 0 (t) of± = f(x,u 0 ) satisfying X 0 (0) = X 0 is defined for all t;::: 0,

lim

t-+oo

X

0

(t) = 0.

Remark 10. 7.2. Clearly, any strictly passive system is also weakly strictly passive, in the sense of the Definition above. An output strictly passive system is weakly strictly passive if, for any X 0 and any U 0 (-) such that the solution x 0 (t) of±= f(x,u 0 ) satisfying X 0 (0) = X 0 is defined for all t;::: 0, y(t) = 0 for all t ;::: 0

lim

t-+oo

X

0

(t) = 0.

It is also easy to see that a weakly strictly passive system has a globally asymptotically stable equilibrium at x = 0. In fact, by definition, its storage function V (x) satisfies

av axf(x,O) :$ -d(x,O) :$0,

50

10. Stability of Interconnected Nonlinear Systems

and this, as shown before, implies boundedness of all the trajectories. Let r denote thew-limit set of any trajectory x(t), and recall that, for each x E r, V(x) = V00 = limt--+oo V(x(t)). Since r is invariant under x = f(x,O), any trajectory X 0 (t) of x = f(x, 0) with initial condition X 0 in r is such that V(x 0 (t)) is constantly equal to V00 and the previous inequality yields

Therefore, X 0 (t) must converge to zero as t ~ oo. This shows that V00 and proves that x(t) tends to zero as t ~ oo. <J

=0

A third relevant class of dissipative systems is that of the so-called sector bounded systems.

Definition 10.7.7. A system in which m =pis said to be sector bounded if it is dissipative with respect to a supply rate of the form q(u,y)

= (y- au)T(bu- y),

(10. 77)

for some pair of real numbers a :$ b. Remark 10. 7.3. In the case of a single-input single-output memoryless system, modeled by a map y = cp(u) as in (10.59), to say that the system is dissipative with respect to a supply rate of the form (10. 77) is to say that (cp(u)- au)(bu- cp(u)) 2: 0

for all u E JR. The latter is equivalent to


a< cp(u)

-

u

-

and this is why systems dissipative with respect to a supply rate of the form (10. 77) are called "sector bounded". <J Note that all supply rates (10.68), (10.74), (10.75), (10.77) are quadratic forms in u, y, i.e. special cases of a function of the form q(u,y)

= (uT

yT)

(~ ~) (~)

=uTRu+2yTSu+yTQy

(10.78)

in which R, Q are symmetric matrices. In fact, finite L2 gain systems correspond to the case in which

= 0,

Q = -I,

= 21 I,

Q =0 ,

R =,,?I, S passive systems to the case in which R

= 0,

S

10.7 Dissipative Systems

51

output strictly passive systems to the case in which R

= 0,

S

= 21 1,

Q = -d,

sector-bounded systems to the case in which 1

R = -abl, S = 2,(a + b)I, Q =-I. If a system is dissipative with respect to a quadratic supply rate and the functions f(x, u) and h(x, u) which characterize the right-hand sides of (10.55) are affine forms in u, the dissipation inequality (10.57) can be given a more explicit characterization. More precisely, consider a nonlinear system described by equations of the form :i:

= f(x) + g(x)u

y

= h(x) + k(x)u

(10.79)

in which f(O) = 0 and h(O) = 0. This system is dissipative with respect to a given supply rate q(u, y) if there exists a positive definite and proper C 1 function V(x) such that

av . Bx [f(x) + g(x)u]- q(u, h(x) + k(x)u) S 0

(10.80)

for all x E !Rn, u E !Rm. If q(u,y) is a supply rate of the form (10.78), the left-hand side of this inequality can be viewed as a (complete) form of degree two in u, with coefficients depending on x, in fact equal to

av 8x f(x) -

h T (x)Qh(x)

av

+ [ax g(x)

- 2hT (x)(Qk(x)

+ S)]u

(10.81)

+ uT[-R- kT(x)S- STk(x)- e(x)Qk(x)]u. Thus, the input-affine system (10.79) is dissipative with respect to the quadratic supply rate (10. 78) if and only if there exists a positive definite and proper 0 1 function V(x) such that, for each x, (10.81) is a negative semi-definite form in u. This observation yields the following criterion for dissi pati vi ty. ·'

Proposition 10.7.1. System (10.79) is dissipative with respect to the supply rate (10. 78) if and only if: (i) the symmetric matrix

is positive semi-definite for all x E !Rn,

(ii) there exists a positive definite and proper C 1 function V(x) such that, for all x E !Rn, the set

52

10. Stability of Interconnected Nonlinear Systems

ov

1 U(x) = {u E !Rm : W(x)u = "2[ ox g(x)- 2hT(x)(Qk(x)

+ S)]T}

is not empty and, for all u E U(x),

(10.82) Proof. Set A(x)

ov ox f(x)- hT(x)Qh(x)

=

ov oxg(x)- 2hT(x)(Qk(x) + S).

B(x)

In view of the above observations, system (10. 79) is dissipative with respect to the supply rate (10.78) if and only if, for some V(x), at each fixed X E !Rn, (10.83) A(x) + B(x)u- u ~W(x)u is a negative semi-definite function of u. To prove the necessity observe that, if (10.83) is negative semi-definite, necessarily W(x) is positive semi-definite, i.e. condition (i) holds. Moreover, any u such that W(x)u = 0 is necessarily such that B(x)u = 0 and this shows that the vector BT(x) is in the image of the matrix W(x). As a consequence, the set U(x) is not empty. The value of (10.83) at any u E U(x) is equal to A(x)

+ uTW(x)u

and this, since the latter must be a non-positive number, completes the proof of the necessity of (ii). As far as the sufficiency is concerned, let x be fixed, say x = i. Pick any u E U(x), which is such that B(x) = 2uTW(x). Then, by standard completion of the squares, one obtains, for all u E !Rm, A(x)

+ B(x)u- uTW(x)u

= =

A(x)

+ 2uTW(x)u- uTW(x)u

A(x)

+ u.TW(x)u- (u- u)TW(x)(u- u)

which is non positive because, by assumption {i), W(x) is positive semidefinite and, by assumption (ii), A(x)

+ u.TW(x)u ::::; o.

Since x was any arbitrary point in !Rn, this completes the proof of the sufficiency. <1 Remark 10. 1.4. Note that the property of being strictly dissipative can be given a very similar characterization. The reader can easily check that system ( 10. 79) is strictly dissipative with respect to the supply rate (10. 78) if and only if conditions (i) and (ii) of the above Proposition hold, with (10.82) replaced by the inequality

10.7 Dissipative Systems

in which a{) is some class /C 00 function.

53


The criterion described in the previous Proposition lends itself to somewhat simpler versions in two extreme cases, the one in which W(x) is nonsingular for all x and the one in which W(x) is identically zero for all x. Corollary 10.7.2. Consider system {10.79} and let the supply rate {10.78} be given. Suppose W (x) is nonsingular for all x. Then, the system is dissipative if and only if W (x) is positive definite and there exists a positive definite and proper C 1 function V(x), such that, for all x E JRn,

(10.84)

Suppose W(x) = 0 for all x. Then, the system is dissipative if and only if there exists a positive definite and proper C 1 function V (x), such that, for all X E lRn,

av ax g(x) -

and

2hT(x)(Qk(x)

+ S) = 0

av ax f(x)- hT(x)Qh(x)

~ 0.

(10.85)

(10.86)

Proof. If W(x) is nonsingular, then the set U(x) is nonempty for all x E JRn, and coincides with the single element

u

8

= ~w- 1 (x)( 8 v g(x)- 2hT(x)(Qk(x) + S)]T. 2

X

Replacing this in (10.82) yields (10.84), and proves the first part of the corollary. On the other hand, if W(x) = 0, the set U(x) is nonempty if and only if (10.85) holds, in which case the set itself coincides with JRm. Since (10.82) reduces to (10.86), this proves the second part of the corollary.
and

av

1 U(x) = {u E JRm: Ru = "2[axg(x)- 2hT(x)S]T}.

For instance, consider the supply rate

54

10. Stability of Interconnected Nonlinear Systems

=R

= -y 2 I is nonsingular and positive definite. Thus, the system is dissipative (i.e. is a finite L 2 gain system) if and only if there exists

In this case W(x)

a positive definite and proper C 1 function V(x) such that, for all x E !Rn,

av ax f(x)

1 av

av

+ hT (x)h(x) + 4-y2 [OX g(x)][ ax g(x)]T

:::; 0.

Or, consider the supply rate

In this case W(x) = 0, and the system is dissipative (i.e. is a passive system) if and only if there exists a positive definite and proper C 1 function V(x) such that, for all x E !Rn,

av

8x f(x)

:S 0.

10.8 Stability of Interconnected Dissipative Systems We discuss in this section the stability properties of feedback interconnected dissipative systems. In particular, we consider the case in which the interconnection takes place at the level of inputs and outputs (as opposite to an interconnection occurring at the level of states, such as in (10.43)) and, since this is the most frequently considered case, we will study the case of negative feedback interconnection. More precisely, consider two systems E 1 and E 2 , described by equations of the form

Xi = /;(x;, u;) Yi = h;(xi, u;) with i = 1, 2, in which, in addition to the standard hypotheses considered so far for this class of systems (in particular /i(O, 0) = 0 and hi(O, 0) = 0) we assume also that dim( u 2 ) = dim(yl) dim(u!) = dim(y2) . Suppose that the functions h 1 (x 1 ,ut) and h2(x2,u2) are such that the constraint U2 = Yl Ut = -y2 makes sense. In other words, suppose that, for each x 1, x2 there is a unique pair u 1 , u 2 satisfying U2 = hl(Xt 1 Ut) u1 = -h2(x2, u2) .

10.8 Stability of Interconnected Dissipative Systems

55

:i:1 =fi(xi,ui)

Y2

Y1

= hi(xr,ui)

:i:2

= h(x2,u2)

Y2

= h2(x2,u2)

Fig. 10.4. Negative feedback connection.

If this is the case, the interconnected system obtained by imposing the constraint u 2 = y 1 and u 1 = -y 2 is the (negative) feedback interconnection of L'r and E 2 (see Fig. 10.4).

Remark 10.8.1. Of course, the feedback interconnection makes always sense if either one of the hi(x;, ui) is independent of u;. In the case of input affine systems, namely when

the negative feedback interconnection makes sense if the linear system h1 (xr)

u2 u1

=

+ kr (xr)u1

-h2(x2)- k2(x2)u2

as a unique solution u 1 , u 2 , and this occurs if and only if the matrix

is invertible, i.e. the matrix

is nonsingular for all x 1 , X2.


·'

Suppose now that both E 1 and E2 are strictly dissipative, with respect to quadratic supply rates q1 (u 1 ,yr) and q2 (u 2 ,y2 ). Then, if certain relations hold between the parameters which characterize these supply rates, the negative feedback interconnection of E 1 and E 2 is globally asymptotically stable, as shown below. In case the two systems are just dissipative, additional conditions of zero-state detectability are requested, for each one of the systems· or for their open loop interconnection E2 o Er, which is defined as

56

10. Stability of Interconnected Nonlinear Systems

=

±1

±2 =

=

Yz

h(x1,ul) h(xz,h1(x1,ul)) h2(xz,h1(x1,u1))

Theorem 10.8.1. Let two supply rates

= uT R.ui + 2yJ' Siui + yJ' QiYi ,

Qi(ui, yi)

be given and suppose that, for some a

i = 1,2

> 0, the symmetric matrix

is either:

(i) negative semi-definite, (ii) negative semi-definite, S 1

= aSi

and R1-+ aQz is nonsingular,

(iii) negative definite.

Then, if both E 1 and E 2 are strictly dissipative and (i) holds, the negative feedback interconnection is globally asymptotically stable. If both E 1 and Ez are dissipative, {ii) holds and E 2 o E 1 is zero-state detectable, the negative feedback interconnection is globally asymptotically stable. If both L\ and E2 are dissipative, {iii) holds and both L'1 and L'2 are zero-state detectable, the negative feedback interconnection is globally asymptotically stable. Proof. If L'1 and E 2 are strictly dissipative, with respect to the given supply rates, there exist positive definite and proper C 1 functions vl (xl) and V2(X2), such that

8V.·

- 'fi(xi, Ui) 8 Xi

:S -o:i(llxill) + Qi(ui, Yi)

where o:i (·) is a class /( 00 function. If a

> 0, the function

is still positive definite and proper, and such that aw. aw. ax x1+ ax X2 1 2

< -o:l(llxdl)-ao:2(llx211)+ql(ul,Yl)+aq2(u2,Y2) -0:1 (llx1ll) - ao:2(11x2ll) + Ql ( -y2, Yl) + aq2(Yb Y2) -al(llxlll)- aa2(llxziD + ( Yf

yJ) M

If M :S 0,

aaw Xl + aaw X2::::; -al(llxlll)- aa2(llx2ID X1

X2

and this proves global asymptotic stability.

(~~)

10.8 Stability of Interconnected Dissipative Systems

57

If E 1 and E 2 are just dissipative (i.e. not strictly dissipative), we have instead . + -8W Xz . ~ ( YrT YT ) M ( Yl ) . (10.87) -8W Xr 2 8~ 8 Xi ~

Thus, if M ~ 0 we conclude that all trajectories (of the negative feedback interconnection of E 1 and Ez) are bounded and that the equilibrium (x 1 , x 2 ) = (0, 0) is stable. Asymptotic stability is proved using the standard arguments of La Salle's invariance theorem. It is known that on each trajectory lime-. 00 W(x 1 (t),x 2 (t)) = W00 ~ 0 and that on thew-limit set r of this trajectory W(x 1 ,xz) = W 00 • Thus, using (10.87) and the fact that M ~ 0, it is concluded that

(yJCt)

yi(t))M

(~~~!D =o

along any trajectory with initial condition (x~, x~) E r. If (ii) holds, necessarily y 2 (t) = 0. Thus, each initial condition in produces a bounded trajectory (x~(t}, x~(t)) for which y 2{t) = u1(t) = 0. Since E 2 o E 1 is zero-state detectable, this trajectory necessarily converges to (0, 0) and therefore W 00 = 0. This proves global asymptotic stability. If (iii) holds, necessarily y 1 (t) = 0 and y2 (t) = .0 and, this time using zero-state detect ability of both E 1 and E 2 , global asymptotic stability follows again. <1

r

Remark 10.8.2. Note that the conclusions of this Theorem remain valid, with appropriate adaptations, if one of the two systems, e.g. E 2 , is a memoryless system, i.e. if y2 = cp(u 2 ), for some smooth function cp(·) satisfying cp(O) = 0. Recall that, in this case, E 2 is dissipative with respect to the supply rate qz (uz , Yz) if qz(uz, cpz(uz)) ~ 0

for all u 2 . Now, it is easy to check that the function V1 (x!) is such that 8Vr.

-xi 8x1

< -ar(llxrll) + Qr(ur,yr) + aqz(uz,yz)- aqz(uz,yz)

=

-ar (llxrll) - aqz(uz, cp(uz))

+ ( yJ yi) M

(

~~)

Thus, the negative feedback ii{terconnection is globally asymptotically stable if either: (i) E 1 is strictly dissipative and M is negative semi-definite, (ii) E 1 is dissipative, zero-state detectable, M is negative semi-definite, and

(iii) E 1 is dissipative, zero-state detectable, and M is negative definite.

<1

58

10. Stability of Interconnected Nonlinear Systems

There are numerous interesting consequences of this result. We discuss here just few of them. The first important special case occurs when both systems have finite £ 2 gain, and leads to an alternative version of the property, already seen in Section 10.6, that a "small gain" in a feedback interconnection does not destroy stability. Corollary 10.8.2. Suppose E 1 and E 2 are dissipative with respect supply rates of the form i

= 1, 2.

If both E 1 and E 2 are zero-state detectable, and /112

<

(10.88)

1

the negative feedback interconnection is globally asymptotically stable. Proof. To prove this result, observe that in M _ ( ( -1 + a1i)I 0

~his

case

(lr-0 a)I )

The matrix in question is negative definite if -1

+ a1i < 0,

i.e. if 2

1'1


1r- a< 0 1

2

f2

If the condition (10.88) holds, there exists an a > 0 which renders the latter conditions satisfied. Thus, the Corollary follows from case (iii) of Theorem 10.8.1.
Another important special case occurs when both systems are passive. In this case, for example, one can immediately obtain the following results. Corollary 10.8.3. Suppose E 1 and E 2 are dissipative with respect supply rates of the form i

= 1,2

where ei ;::: 0. If e1 > 0, e2 > 0 and both E 1 and E 2 are zero-state detectable, the negative feedback interconnection is globally asymptotically stable. If e 1 = 0, e2 > 0 and E2oE1 is zero-state detectable, the negative feedback interconnection is globally asymptotically stable. Proof. Set a= 1 in Theorem 10.8.1, to obtain M = (

If both e1 applies. <J

-=/;

0 and e 2

-=/;

-~1I -~2 I)

0, case (iii) applies. If e 1 = 0 and e 2 > 0, case (i)

10.8 Stability of Interconnected Dissipative Systems

59

Corollary 10.8.4. Consider the interconnection (see Fig. 10.5) of

f(x,u)

(10.89)

h(x)

y

and u = -cp(y). Suppose that system (10.89) is passive and zero-state detectable, and that the map cp(·) satisfies cp(O) = 0 and YT cp(y)

>0

for all y

f. 0.

Then, the interconnection is globally asymptotically stable. Proof. By hypothesis, the memoryless system y 2 = cp(u 2 ) is such that y'fu 2 = cp T (u2)u 2 > 0 for all u 2 f. 0. Thus, this system is dissipative with respect to the supply rate q2(u2,y2) = y'fu2 and the result follows from the property indicated in the Remark 10.8.2, case (ii). <J

u

x = f(x,u) y

y

= h(x)

Fig. 10.5. Memoryless negative feedback.

The third case considers slightly more elaborate supply rates. Corollary 10.8.5. Suppose E 1 is dissipative with respect to a supply rate of the form (10.90) q1(u1,yi) = ufu1 + ku[yl

and E 2 is dissipative with respect to a supply rate of the form

q2(u2, Y2)

= (Y2 ....:.' cu2)T ((k- t)u2 -

Y2) ,

(10.91)

with k > 2e > 0. If E 1 and E2 are zero-state detectable, the negative feedback interconnection is globally asymptotically stable. Proof. To prove this result, observe that in this case 2

M- ((-a.ek+ac: )1 t(a-l)kl This matrix is negative definite if

~(a-1)kl) :(1-a)J

60

10. Stability of Interconnected Nonlinear Systems

a> 1 E (k -

E)

1)k2 > -'(a------:-...:_____

4a

and these can be satisfied if a = 1 + o and o > 0 is sufficiently small. Thus, the Corollary follows from case (iii) of Theorem 10.8.1. <J Note that the hypothesis on system E 1 in the last corollary can be given a slightly different form. Set

and observe that ql (u1, YI)

= uJu1 + ku{yi = kuJ(~u1 + YI) = ku{yl

.

From this, it is easily deduced that system E 1 is dissipative with respect to the supply rate (10.90) if and only if the system E1 thus defined ±1 Y1

= fi(xi,ul) 1 = hi(XI,ul) + ku1

is dissipative with respect to the supply rate qi(ui,yi)

In fact, the function pation inequality

= u{y1 .

V1 (xi) = tV(xi)

(10.92)

satisfies by construction the dissi-

~v1 JI(x1,u1) ~ tii(ui,Yd.

UXJ

Thus, the hypothesis on E 1 in Corollary 10.8.5 can be rephrased by assuming that the auxiliary system E1 satisfies the property of being dissipative with respect to the supply rate (10.92). In other words, system E1 is assumed to be passive. In case both systems are single-input single-output and system E2 is memoryless, the last corollary specializes in the following well-known criterion for global asymptotic stability under memoryless, sector-bounded, negative feedback. Corollary 10.8.6. Consider the interconnection of f(x,u) y

h(x)

(10.93)

and u = -cp(y). Suppose {10.93) is zero-state detectable and that, for some k > 2£ > 0, the system

10.9 Dissipative Linear Systems

x =

f(x,u)

y

h(x)

1

c < cp(y) < k - c y

(10.94)

+ ku

is passive. Suppose the map cp(-) satisfies cp(O)

61

= 0 and

for ally

-:f. 0.

Then, the interconnection is globally asymptotically stable. Remark 10.8.3. Suppose that, in the above Corollary, f(x,u) is affine in u, i.e. f(x, u) = f(x) + g(x)u. Then, the criterion given by Corollary 10.7.2 can be used to test the property of passivity of (10.94). In fact, in this case

W(x) =

1

k

and the system is passive if and only if there exists a positive definite and proper C 1 function V(x) such that

av

8x j(x)

k

av

av

+ 4[ 8x g(x)- h T(x)j[ 8x g(x)- h T(x)]T::; 0.

for all X E !Rn . If, in particular, system (10.93) is a passive system, then (see end of Section 10. 7) the previous inequality holds for any k. In this case, the upper bound for cp(y)fy can be any arbitrarily large number. <1

10.9 Dissipative Linear Systems In the case of linear systems, the conditions for dissipativity presented in the previous section can be given alternative characterizations, in particular involving the frequency response matrix of the system. These characterizations are summarized in this section, beginning with the property, for a system, of having an L 2 gain which does not exceed a fixed number J. As a continuation of the Example 10.7.1, consider an asymptotically stable linear system, modeled by equatior..s of the form

y

Ax+Bu Cx+Du.

(10.95)

In this case, the function 1r(w) which solves the equation (10.71) is a linear function of w, i.e. 1r(w) = Ilw for some matrix

and the equation in question reduces to a Sylvester equation of the form

62

10. Stability of Interconnected Nonlinear Systems

II (

-~o ~0 )

= All

+ Bu0 ( 1 0 )

This equation, since all eigenvalues of A have negative real part (because the system is by hypothesis asymptotically stable), has a unique solution II. An elementary calculation (multiply first both sides on the right by the vector ( 1 j )T) yields lit + jll2 = (jwoi- A)- 1 Buo , i.e. II

= (Re[(jwoi- A)- 1 B]uo

Im[(jwoi- A)- 1 B]uo)

As shown in the Example 10.7.1, if the system is in the initial state X

0

(0) =II (

~) = Re[(jwoi- A)- 1 B]uo ,

the periodic input ii(t) = uo cos(wot) produces the periodic state response X

0

= llw(t) = JI1 cos(wot) -JI2 sin(wot),

(t)

(10.96)

and the periodic output response y(t)

=

Cxo(t) +Duo cos(wot) Re[T(jw0 )]u0 cos(w0 t) - Im[T(jwo)]uo sin(w0 t) ,

in which T(jw)

(10.97)

= C(jwi- A)- 1 B +D.

Observe now that

and that, in view of the specific form of II,

1

o

~

llii(s)ll 2 ds

71'

= -IIT(jwo)uoll 2 • wo

In other words

llii(·)ll~.m.s. = ~lluoW 2 lliiOII;.m.s. = ~IIT(jwo)uoll Thus, from the interpretation illustrated in the Example 10.7.1 one can conclude that, if the system is strictly dissipative with respect to a supply rate of the form (10.68), 2

2

2

IIT(jwo)uoll = 2llii(·)ll;.m.s. :::; 1 2llu(·)ll;.m.s. = 1 lluoll

2

10.9 Dissipative Linear Systems

63

i.e.

IIT(jwo)uall :::; rlluotl

0

Observing that both u 0 and w0 are arbitrary, it is seen from this that sup max

wERIIull=l

IIT(jw)ull :::; 1.

(10.98)

The quantity on the left-hand side is, by definition, the so-called H 00 norm of the matrix T(jw). Therefore, one can conclude that a linear system which is strictly dissipative with respect to a supply rate of the form (10.68), is asymptotically stable and the H 00 norm of its frequency response matrix is bounded from above by the number f. It will be shown now that also the converse of this property holds. To this end, using Proposition 10.7.1, condition (i), observe that the linear system (10.95) is dissipative with respect to a supply rate of the form (10.78) with R = 1 2 I and Q =-I, that is a supply rate of the form

only if the matrix

W

= 12I

-DTD

is positive semi-definite. Suppose that 1 is large enough so that this matrix is positive definite, and thus nonsingular. Then, using this time Corollary 10. 7.2, it is seen that the linear system (10.95) is strictly dissipative with respect to a supply rate of the form (10.99) if and only if there exists a positive definite and proper C 1 function V(x) such that, for all X E !Rn)

(10.100) where a(·) is a class Koo function. Now, suppose that V(x) is a quadratic form in x, i.e. a function of the form V(x) = XT Px

where Pis a symmetric matrix (wh~ch is necessarily positive definite if (10.56) holds). Suppose also that a(r) = 6r 2 for some 6 > 0. In this case, an easy calculation shows that the inequality (10.100) reduces to matrix inequality (10.101)

In this way, we have shown that if 1 is large enough so that the matrix W is positive definite and if there exists a symmetric positive definite solution P of the matrix inequality (10.101), the positive definite and proper function V(x) = xT Px satisfies the condition (10.100) and, consequently, the system is strictly dissipative, with respect to the supply rate (10.99). Inequalities

64

10. Stability of Interconnected Nonlinear Systems

of this kind are usually referred to as algebraic Riccati. inequalities. In other words, we can conclude that the existence of a positive definite solution of the algebraic lliccati inequality (10.101) implies strict dissipativity with respect to the supply rate (10.99) and this, bearing in mind the conclusions reached at the beginning of the section, in turn implies (10.98), i.e. the fact that 'Y is an upper bound for the H 00 norm of the frequency response matrix of the system. It will be proved below that if 'Y is an upper bound for the H00 norm of the frequency response matrix of the system, then there exists a positive definite solution of the algebraic lliccati inequality (10.101), thus showing that the properties in question are, actually, equivalent. This will be done by means of a circular proof involving another equivalent version of the property that 'Y is an upper bound for the H 00 norm of the frequency response matrix of the system, which is very useful for practical purposes, since it can be rather easily verified. More precisely, the fact that 'Y is an upper bound for the H 00 norm ofthe frequency response matrix of the system can be checked by looking at the spectrum of a matrix of the form (10.102)

in which Ro and Q0 are symmetric matrices which, together with A 0 , depend on the matrices A, B, C, D which characterize the system and on the number 'Y· A matrix with this structure is called an Hamiltonian matrix and has the property that its spectrum is symmetric with respect to the imaginary axis. In fact, since H (0 -I)= (-AJ Qo) = -HT (0I -I)-l OT I OT -Ro Ao

the matrices H and - HT are similar. Thus, if A is an eigenvalue of H, so is also -A. Since the entries of H are real numbers, this shows that if a+ jb is an eigenvalue of H, so is also -a+ jb. Theorem 10.9.1. Consider the linear system {10.95) and let 'Y > 0 be a fixed number. The following are equivalent:

(i) there exists a number i" < "(, such that the system is strictly dissipative with respect to the supply rate q(u,y) =

i"2 lluW -IIYII 2

,

(ii) all the eigenvalues of A have negative real part and the frequency response matrix of the system T(jw) = C(jwi- A)- 1 B + D satisfies sup max wERIIull=l

IIT(jw)ull < 'Y,

(10.103)

10.9 Dissipative Linear Systems

(iii) all the eigenvalues of A have negative real part, the matrix W DT D is positive definite, and the Hamiltonian matrix

A+ BW- 1 DTC H= ( -cTc-cTnw-lnTc

BW- 1 BT ) -AT-cTnw- 1 BT

65

= "1 2 I (10.104)

has no eigenvalues on the imaginary axis, (iv) the matrix W = 1'2 I- DT D is positive definite, and there exists a sym-

metric matrix P

>0

such that

PA+ ATP+ eTc+ [PB+CTD]W- 1 [PB +CTD]T < 0.

(10.105)

Proof. We have already proven, at the beginning of the section, that (i) implies that (10.95) is an asymptotically stable system, with a frequency response matrix· satisfying

sup max

wERIIull=l

Thus, (i) =? (ii). To show that (ii)

=?

IIT(jw)ull :::; i ·

(iii), first of all that note, since lim T(jw) = D

w-+oo

it necessarily follows that IIDull < "Y for all u with llull = 1 and this implies > DT D, i.e. the matrix W is positive definite. Now observe that the Hamiltonian matrix (10.104) can be expressed in the form H=L+MN

1' 2 I

for

L

= (

-~TC -~T) • M= (-~D)' N=

cw-lnTc w- 1 BT).

Suppose, by contradiction, that the matrix H has eigenvalues on the imaginary axis. By definition, there exist a 2n-dimensional vector x 0 and a number w 0 E lR such that (jwoi -'L)xo = MNxo. Observe now that the matrix L has no eigenvalues on the imaginary axis, because its eigenvalues coincide with those of A and -AT, and A is by hypothesis stable. Thus (jw 0 I- L) is nonsingular. Observe also that the vector u 0 = N x 0 is nonzero because otherwise x 0 would be an eigenvector of L associated with an eigenvalue at jw 0 , which is a contradiction. A simple manipulation yields (10.106) uo = N(jwoi- L)- 1 Mu 0 . It is easy to check that

66

10. Stability of Interconnected Nonlinear Systems

. (10.107) where T(s) function of

= C(sl- A)- 1 B +D. In fact, it suffices to compute the transfer i: = Lx+ Mu

y=Nx

and observe that N(sl- L)- M = w- 1 [TT( -s)T(s)- DT D]. Multiply (10.107) on the left by uJ"W and on the right by u0 , and use (10.106), to obtain 1

uJ"W uo

= uJ[TT (-jw0 )T(jwo) -

DT D]u0

,

which in turn, in view of the definition of W, yields

which contradicts (ii), thus completing the proof. To show that (iii) => (iv), set F

(A+BW- 1 DTC)T

Q

-BW- 1 BT

acT

cT(J

+ nw- 1 nT)c

(the latter is indeed possible because I + nw- 1 DT is a positive definite matrix, i.e. I+ nw- 1 DT = MT M for some nonsingular M). The pair (F, G) thus defined is stabilizable. In fact, suppose that this is not the case. Then, there is a vector x =f:. 0 such that

for some A with non-negative real part. Then,

and this implies Cx = 0, which in turn implies Ax = AX, and this is a contradiction because all the eigenvalues of A have negative real part. Moreover, it is easy to check that

and this matrix by hypothesis has no eigenvalues on the imaginary axis. Thus, by Lemma 13.6.2, there is a unique solution y- of y- F

+ pTy-- y-aaTy- + Q =

0'

the set of solutions Y of the inequality

u(F- GGTY-)

c c- '

(10.108)

10.9 Dissipative Linear Systems

YF+FTY- YGGTY +Q

>0

67

(10.109)

is nonempty and any Y in this set is such that Y < y-. Observe now that

o = y- F + FTy- =

=

y-ccTy- + Q y-(AT + cT nw- 1 BT) +(A+ sw- 1 nTc)Y- y-cT(I + nw- 1 nT)cv-- sw- 1 sT y- AT+ AY-- [Y-cTn- BJW- 1 [DTcy-- sTJ- y-cTcv-,

which yields y-AT +AY- 2:0.

Setting V(x)

= xTY-x, this inequality shows that, along the trajectories of (10.110)

the function V(x(t)) is non-decreasing, i.e. V(x(t)) 2: V(x(O)) for any _x(O) and any t 2: 0. On the other hand, system (10.110) is by hypothesis asymptotically stable, i.e. limt->co x(t) = 0. Therefore, necessarily, V(x(O)) ::; 0, i.e the matrix y- is negative semi-definite. From this, it is concluded that any solution Y of (10.109), that is of the inequality YAT

+ AY- [YCTD- B]W- 1 (DTCY- BT]- YCTCY > 0,

(10.111)

which necessarily satisfies Y < y- ::; 0, is a negative definite matrix. Take any of the solutions Y of (10.111) and consider P = -Y- 1 . By construction, this matrix is a positive definite solution of the inequality in (iv). To show that (iv) => (i) observe that the left-hand side of (10.105), which is negative definite, by continuity remains negative definite if 1 is replaced by i = 'Y - c: and c: is sufficiently small. Thus, for some 6 > 0, the. matrix P satisfies

As a consequence, the positive definite and proper function V(x) = satisfies the inequality (10.100).
xT Px

Also the property of strict passivity has important implications on the frequency response matrix of a linear system. In particular, we refer, in what follows, to the property of weak strict passivity. Observe (see Remark 10.7.2) that in a weakly strictly passive linear system (10.95) the matrix A has all the eigenvalues with negative real part and therefore, as shown at the beginning of the section, the periodic input

ii(t) = uo cos(wot) produces, from the initial state

68

10. Stability of Interconnected Nonlinear Systems

the periodic state response (10.96) and the periodic output response (10.97). Assume that rank(B) = m so that, for any nonzero u 0 , Bu0 =j:. 0 and x 0 (t) cannot be identically zero. Since X 0 (t) is defined for all t ?: 0, by definition of weak strict passivity the constraint implies limt--+oo x 0 (t) = 0, which is a contradiction because x 0 (t) is a nonzero periodic function. Thus, d(x 0 (t), u(t)) cannot be identically zero. As a consequence, since d(x, u) ?: 0 and d(x 0 (t), u(t)) is a periodic function of period ~:, it follows that 11!.

~o~o

d(x 0 (t),u(t))dt > 0.

Integration of the dissipation inequality

~~ f(x, u) ~ YT U- d(x, u) over the interval [0, ~:1 yields, bearing in mind the fact that X

0

(0)

=X

0

271" (-),

wo

the inequality h

271'

fo~o gT(t)~(t)dt?: ~o~o

d(x 0 (t),u(t))dt > 0.

In view of the specific form of u(t) and y(t), this inequality in turn yields uJRe[T(jwo)]uo

>0,

which indeed holds for every nonzero u 0 and every w0 . From this, bearing in mind the fact that T(s) is a matrix ofrational functions with real coefficients, standard manipulations show that the frequency response matrix T(jw) of a weakly strictly passive linear system is such that

for all w E JR. The result below shows that also the converse property is true.

10.9 Dissipative Linear Systems

69

Theorem 10.9.2. Consider the linear system (10.95). Suppose that B has rank m. Suppose also the pair (A, B) is controllable and the pair (C, A) is observable. The following are equivalent: (i) the system is weakly strictly passive, (ii) all the eigenvalues of A have negative real part and the frer;uency response

matrix of the system T(jw)

= C(jwl -

A) - l B

+D

is such that

T(jw)+TT(-jw) >0 for all w E JR., (iii) there exist an n x n symmetric positive definite matrix P, an m x m matrix K and an m x n matrix L such that ATP+PA=-LTL C=BTP+KTL

(10.112)

D+DT = KTK, and such that det for all s such that Re[s]

~

(A ~ sl

~) = n + m

0.

Proof. The implication (i) => (ii) has already been proven. To show that (ii) => (iii), we begin by recalling 8 that, if in the linear system (10.95) the matrix

A has all eigenvalues with negative real part and the transfer matrix T(s) is such that T(jw) + TT( -jw) ~ 0 for all w E JR., then there exists an r x m matrix of proper rational functions V(s), in which r is the rank of T(s) + TT(-s) over the field of rational functions of s, such that: rank[V(s)] = r for all sEc+, all poles of V(s) are inc-, and (10.113) for all sEC. In the present case, condition (ii) implies that the matrix T( s) +TT ( -s) is nonsingular for all s E C 0 , and therefore necessarily r m and rank[V(s)] m for all s with Re[s] ~ 0. Let F, G, H, K be a minimal realization of V (s). Since V (s) has all poles in c-, the matrix F has all eigenvalues inc- and, since the pair (F, H) is observable, the Lyapunov equation

=

=

(10.114) has a unique solution P > 0. 8

See e.g. Youla (1961)

70

10. Stability of Interconnected Nonlinear Systems

Observe that the transfer function matrix realization of the form

(:~) w

vT (-s) v (s) has a state-space

(-0 H -iT) ( ~~) + ( _;T K) ( KT H

G'f ) (

v

~~) + KT K v ,

and that the latter, via the change of coordinates

because of (10.114) can be transformed in

( ~~) = ( ~ w

_iT) (

( KT H

~~) + ( _HT% _ PG) v

+ GT p GT) ( ~~) + KT K v .

This shows that

vT (- s) v (s) = KT K

+ (KT H + GT P)(sl- F)- 1 G + [(KT H + GTJ5)( -sf- F)- 1 GjT

and (10.113) yields

+ DT + C(sl- A)- 1 B + [C(-sl- A)- 1 BjT = KT K + (KT H + GTJ5)(sl- F)- 1 G + [(KT H + GTJ5)(-sl- F)-lG]T. D

(10.115) This identity, equating the constant term and the term with poles in C- , yields D+DT =KTK, which is the last one of the {10.112) and

C(sl- A)- 1 B

= (KT H + GTJ5)(sl- F)- 1 G.

It is possible to show that the pair F, KT H end, observe that, since the matrix

V(s)

+ GTJ5 is observable. To this

= K + H(sl- F)- 1G

has rank m for all s such that Re[s] ( F

~

0 and

~ sl ~) = ( H(F! sl)-1 ~)

(F

~ sl V~s))

it follows that rank (

(10.116)

F- sl H

G)

K

= n +m

'

10.9 Dissipative Linear Systems

for all s such that Re[s] ~ 0. Now, suppose the pair F, KT H exists a nonzero vector v such that

for some >., i.e.

Fv

+ cTp

71

is not observable. Then, there

= ->.v

KT Hv

+ GTPv = 0.

Necessarily, Re[>.] > 0, because F has all eigenvalues with negative real part. Observe that

This shows (note that Pv

=f. 0) that the matrix

is singular at some >. with Re[>.] > 0, and this contradicts an earlier conclusion. Identity (10.116) shows that {A,B,C} and {F,G,KTH+GTP} are both minimal realizations of the same transfer function matrix, and therefore there must exist a nonsingular matrix T such that T AT- 1 = F,

Defining P

cr- 1 =

T B = G,

= TT PT and L = HT,

KT H

+ crp .

this shows that

PA+ATP=-LTL C= KTL+BTP,

which completes the proof of (10.112). Since the matrix

V(s) = K

+ H(sl-

Jf')- 1 G = K

has rank m for all s such that Re[s] before shows that rank

~

(A ~sf

+ L(sl- A)- 1 B

0, an argument identical to one used

~) = n + m

for all s such that Re[s] ~ 0 and this completes the proof of (iii). To show that (iii) ::::} (i), define V(x) = ~xT Px and, using (10.112); observe that

72

10. Stability of Interconnected Nonlinear Systems

2V

=

= = =

2xTPAx + 2xTPBu -xT LT Lx + 2uT(C- KT L)x -xT LT Lx + 2uT(C- KTL)x + 2uT Du- 2uTDu -xT LT Lx + 2uT y - 2uT KT Lx - u T KT K u .

Thus, where

d(x, u) =

1

2IILx + Kull 2 .

Now, d(x,u) = 0 implies Lx+Ku = 0. Thus, the system would be weakly strictly passive if so happens that any pair x( ·), u( ·) satisfying x(t)

=

0

=

Ax(t) + Bu(t) Lx(t) + Ku(t)

is such that limt-+oo x(t) = 0. The Laplace transform of such pair, denoted X(s),U(s), is a solution of the equation

and the matrix (

A~sl ~)

can lose rank only at values of s having negative real part. Thus, X(s) and U (s) are vectors of rational functions having all poles in C- and limt-+oo x(t) = 0. This completes the proof of the fact that (10.95) is weakly strictly passive. <1 We conclude the section with another interesting property of passive linear systems. Lemma 10.9.3. Consider a linear system

y

Ax+Bu Cx.

Suppose there exists a symmetric matrix P

(10.117)

> 0 such that

PA+ATP ~ 0 C=BTP. Then, the system is passive. Moreover, the pair (C, A) is detectable if and only if the pair (A, B) is stabilizable.

10.9 Dissipative Linear Systems

Proof. Set V(x) = ~xT Px, f(x) that V (x) satisfies

= Ax,

73

g(x) = B, h(x) = Cx and observe

av

ox j(x) :::; 0,

Therefore, as shown at the end of section 10.7, the system is passive. Observe now that if the pair (C, A) is detectable, the system is indeed zero-state detectable and therefore, using the result of Corollary 10.8.4, in which we take t.p(y) = y, it can be concluded that the system is globally asymptotically stabilized by the (linear) feedback law u = -Cx. This trivialiy shows that the pair (A, B) is stabilizable. To show that the converse is also true, observe that, as shown in the previous section, a feedback law of the form

is such that

av ox[Ax+Bu(x)] =

1

2xT(PA+ATP)x-xTPBBTPx < - xTPBBTPx = -JJu(x)JJ 2 .

(10.118)

Therefore, the system

· x =Ax+ Bu(x) = (A- BC)x

(10.119}

is stable in the sense of Lyapunov, and the w-limit set r of any trajectory satisfies T C {x E !Rn : BT Px = 0}. Any initial condition x E r produces a trajectory x(t) such that V(x(t)) is constant, and therefore xT(PA+ATP)x(t) =0 u(x(t)) = 0. The first one of these, since (P A +AT P) is symmetric and negative semidefinite, yields (PA +AT P)x(t) = 0, i.e. PAx(t)

= -AT Px(t)

.

·'

(10.120)

The second one, observing that x(t) is necessarily a trajectory of (because u(x(t)) = 0), yields, for any k ~ 0,

0 = dku1;(t))

x=

Ax

1

= :tkk~\ [~!] Ax(t) =- ::k~ 1 BT PAx(t)

and hence, using repeatedly (10.120), BT(AT)kPx(t) = 0. Putting all these constraints together yields

= -BT PAkx(t)

74

10. Stability of Interconnected Nonlinear Systems

(10.121) Now, suppose the pair (A, B) is stabilizable, decompose the system into controllable/uncontrollable parts

( X2~1) =

(Aux1

+ A12X2 + B1u) A22X2

and the matrix P accordingly. By hypothesis of stabilizability, all the eigenvalues of A22 have negative real part, and (Au, BI) is a controllable pair. Since AkB = ( At Bl)

0

and (A 11 ,B1) is a controllable pair, the constraint (10.121) implies

xT (t)Pu +xi (t)P12 = 0, i.e. (note that P11 is positive definite, and therefore nonsingular) xJ(t)

= -x:f(t)P12PiJ: 1 .

This, since x 2 (t) -+ 0 as t -+ oo, shows that also x 1 (t) -+ 0 as t -+ oo. As a consequence, V(x) = 0 on thew-limit set r of any trajectory of (10.119) and the equilibrium x = 0 of the latter is globally asymptotically stable. Suppose now that there exists a trajectory of system :i; = Ax for which y = Cx is identically zero. Since Cx coincides (modulo the sign) with the feedback law yielding the closed-loop system (10.119), the trajectory in question is necessarily also a trajectory of (10.119) and hence converges to zero as t -+ oo. This proves that system (10.117) is zero-state detectable.
11. Feedback Design for Robust Global Stability

11.1 Preliminaries The purpose of this Chapter is to describe some important tools for the design of feedback laws which globally asymptotically stabilize a nonlinear system in the presence of parameter uncertainties. We consider the case in which the mathematical model of the system to be controlled depends on a vector /1- E JRP of parameters, which are assumed to be constant, but whose actual values are unknown to the designer. The vector J1, of unknown parameters could be any vector in some a priori given set P, and the goal of the design is to find a feedback law (obviously independent of JJ) which globally asymptotically stabilizes the system for each value of JJ E P. A problem of this type is usually referred to as a problem of robust stabilization. In section 9.2, we have seen that a powerful method to address a problem of global stabilization, for certain classes of nonlinear systems, is a recursive procedure which- based on a repeated use of the result of Lemma 9.2.2consists in solving the problem in question for a sequence of subsystems of increasing dimension. This procedure is commonly referred to as the method of backstepping. In view of this, a natural point of departure for the analysis of a problem of robust stabilization seems to be the extension of the results of Lemma 9.2.1 and Lemma 9.2.2 to the case of systems affected by parameter uncertainties. In this section we outline the basic ideas which make such an extension possible, and in the subsequent sections we will address a number of specific situations, in order of increasing complexity. Consider a system modeled by equations of the form

z

= 'f(z,~,Jl,)

{ = q(z,{,p)+u

(11.1)

with state (z, {) E IR.n x IR, input u E IR, in which J1, E P C JRP is a vector of unknown parameters and P is a compact set. Assume, as usual, that f(z, {, p) and q(z, {, Jl) are smooth functions of their arguments, and also that f(O,O,J1,) = 0 and q(O,O,J1,) = 0 for every JJ E P, so that the point ' (z, {) = (0, 0) is an equilibrium (when u = 0) for every value of /1-· If the parameter J1, were known exactly, the problem of stabilizing such a system could be addressed using the result of Lemma 9.2.2, which says

76

11. Feedback Design for Robust Global Stability

that from the knowledge of a smooth function v*(z), with v*(O) globally asymptotically stabilizes the equilibrium z = 0 of

= 0, which

i = f(z,v*(z),p)

(11.2)

and the knowledge of a Lyapunov function V(z) for (11.2), it is possible to construct a feedback law u = u(z, ~), with u(O, 0) = 0, which globally asymptotically stabilizes the equilibrium (z,~) = (0,0) of (11.1). The result in question essentially reposes on the following two observations: (i) the (globally defined) change of variable y=~-v*(z)

(11.3)

transforms system (11.1) into a system having the same structure, namely a system of the form i /(z, y, p.) (11.4) q(z,y,p)+u, where

/(z, y, p)

=

f(z, v*(z)

q(z,y,p)

=

q(z,v*(z) +y,p)-

+ y, p)

~v: f(z,v*(z) +y,p),

but in which, as a consequence of the hypothesis on (11.2), the equilibrium z = 0 of (11.5) i = /(z,O,p) is globally asymptotically stable, with Lyapunov function V(z), (ii) once it is known that the equilibrium z = 0 of (11.5) is globally asymptotically stable, with Lyapunov function V(z), it is very easy to determine a feedback law u = u(z, y) which globally asymptotically stabilizes the equilibrium (z,y) = (0,0) of (11.4). In fact, knowing that V(z) is a Lyapunov function for (11.5), it is easy to determine the expression of a feedback law u = u(z,y) which renders

W(z,y)

= V(z) + 21 y 2

a Lyapunov function for

i

/(z, y, p) q(z, y, p)

+ u(z, y)

.

To this end, it suffices to expand /(z, y, p) as

/(z, y, p)

= /(z, 0, p) + p(z, y, p)y ,

(11.6)

11.1 Preliminaries

77

and to impose that (recall the proof of Lemma 9.2.1)

w

~: j(z, y, p,) + y ( q(z, y, p,) + u(z, y)) av-

oz f(z, 0, p,)

av

_

_

+ az yp(z, y, p,) + yq(z, y, p,) + yu(z, y)

av-

ozf(z,O,p,)-y 2 ,

which is achieved choosing u(z, y) as

_

u(z, y)

=-

av

_

az p(z, y, p,) - q(z, y, p,) - y .

This feedback law renders W negative definite and hence globally asymptotically stabilizes the equilibrium (z, y) = (0, 0) of (11.4). Reversing the change of variables (11.3) yields a feedback law

u(z, () = u(z, €- v*(z)) which globally asymptotically stabilizes the equilibrium (z, () = (0, 0) of (11.1). An immediate obstruction to the extension of this procedure to the case in which the parameter p, is unknown is the fact that the method in question relies upon the exact cancellation of certain terms in the derivative of the candidate Lyapunov function W (z, y). If the actual value of p, is not known, such a cancellation is no longer possible, and the procedure is not applicable. Therefore, in case p, is unknown, the stabilization problem of a system such as (11.4) must be addressed on different grounds. The issue is to determine a feedback law u(z,y) which globally asymptotically stabilizes the equilibrium (z,y) = (0,0) .of (11.6). Since in no way the tenn u(z,y) can be used to "cancel" the term q(z, y, p,) and the contribution of p(z, y, p,) to the derivative of the candidate Lyapunov function W(z, y), the best way to proceed seems to be the following one. Recall that system (11.5), which is obtained by setting y = 0 in the upper subsystem of (11.6), is globally asymptotically stable by hypothesis, and observe that the system

iJ

= q(O, ¥' p,)

+ u(O, y) ,

(11.7)

which is obtained by setting z = 0 in the lower subsystem of (11.6), can be rendered globally asymptotically stable, by appropriate choice of ii.(O, y), if the set P on which the uncertain parameter p, ranges is a compact set. To this end, note that the function q(O, y, p,), which by hypothesis vanishes at y = 0 and is a smooth function of y and p,, can be expressed in the form

q(O,y,J.L)

= k(y,p,)y

for some k(y, p,), and, ifP is a compact set, there exists a continuous function p (y), positive for all y, such that

78

11. Feedback Design for Robust Global Stability

lk(y,fl)l ~ p(y) for all

y

and /1· This being the case, it is immediate to check that the input u(o, y) = -2yp(y)

globally asymptotically stabilizes the equilibrium y = 0 of (11.7), actually with Lyapunov function y 2 • In fact, this choice yields, along the trajectories of (11.7), dy2 dt = 2y2 k(y, /1) - 4y2 p(y) ~ -2y 2 p(y) 0

This being the case, one may look at system (11.6) as to the interconnection of two stable systems, and try to appeal to the small gain theorem. Of course, global asymptotic stability of (11.5) and (11.7) is only a prerequisite for the application of the small gain theorem. What one should do is to check that the upper subsystem in (11.6), viewed as a system with input y and state z, is actually input-to-state stable and to determine an upper estimate of a "gain" function, namely a class IC function "Y(·) which renders the estimate llz(t)ll ~ max{,B(IIzoll, t), "YCIIYUIIoo)} fulfilled (for some class /C.C function ,8(·, ·)). Then, one should design a feedback law u(z, y) so as to obtain an input-to-state stable lower subsystem and to impose, on it, a "gain" function which would render the "small-gain" condition fulfilled. As difficult as it may seem at a first glance, this way of proceeding turns out to be pretty effective and helps in successfully addressing the robust stabilization problem. This procedure sketches how the robust stabilization problem can be addressed in case the upper system of (11.1) is already input-to-state stable. In some sense, if successful, it would produce a result which is the robust version of Lemma 9.2.1. If that system is not input-to-state stable, then a preliminary step is needed, which fortunately can always be accomplished if the subsystem in question is globally asymptotically stabilizable (i.e. if hypotheses similar to those used in Lemma 9.2.2 hold). In fact, suppose there exists a smooth function v*(z), with v*(O) = 0, which globally asymptotically stabilizes the equilibrium z = 0 of (11.2). Then, using Theorem 10.4.3, it follows that there exists also a smooth function ,B*(z), nowhere zero, such that the system z = f(z, v*(z) + ,B*(z)y, /1) , viewed as a system with state z and input y, is input-to-state stable. Since ,B*(z) is nowhere .zero, the procedure of step (i) can be repeated, considering now the change of variable y = [,B*(z))- 1 [~- v*(z)] , which transform system (11.1) into a system having a similar structure, namely

11.2 Stabilization via Partial State Feedback: a Special Case

i

=

iJ

=

79

/(z, y, p,) ij(z, y, p,) + [,B*(z)J- 1 u ,

but in which the upper subsystem is by construction input-to-state stable. In this way, the problem can be reduced to the problem considered above, namely the problem of designing a control u = u(z, y) able of making the lower subsystem input-to-state stable, with a "gain" function which would render the "small-gain" condition fulfilled. In this Chapter, we describe some important situations in which this design procedure can be successfully applied. We begin with the analysis of simple cases in which, due to the particular structure of the. system and to some hypotheses, the upper subsystem is already input-to-state stable, and, moreover, the actions needed to secure that the lower subsystem has the appropriate "gain" function are automatically build-in in the procedure used to check that a certain candidate Lyapunov function becomes an actual Lyapunov function, if a suitable feedback law is imposed. After this, we address substantially more general situations. \

11.2 Stabilization via Partial State Feedback: a Special Case We begin this section by addressing the case of a system modeled by equations of the form i = F(p,)z + G(y, p,)y (11.8) iJ = H(y, p,)z + K(y, p,)y + b(y, p,)u , in which z E !Rn, y E IR , u E IR , and p, E P C JRP is a vector of unknown parameters, with P a compact set. The functions F(p,), G(y, p,), H(y, p,), K(y, p,) and b(y, p,) are smooth functions of their arguments. A system of this form is perhaps one of the simplest examples of a. nonlinear system which - under appropriate hypotheses - can be globally robustly stabilized, actually by means of a feedback law of the form

u = a(y) . Note that, if the coefficient b(y,p,} is nowhere zero, this system can be regarded as a system having uniform relative degree one between input u and output y, and a zero dynamics characterized by

i = F(p,)z.

(11.9)

The hypotheses on which the robust stabilization method reposes are the asymptotic stability of the dynamics (11.9), and the existence of a positive lower bound for b(y, p,). To understand, before proceeding with a formal proof,' why these hypotheses are successful in the light of the general framework presented in the previous section, observe that if the dynamics (11.9) are

80

11. Feedback Design for Robust Global Stability

asymptotically stable, then automatically the upper subsystem is input-tostate stable. In fact, this system can be viewed as a linear system i

= F(J.L)Z + v

driven by the input v = G(y,J.L)Y and therefore, if (11.9) is asymptotically stable, its state response z(t) satisfies a bound of the form llz(t)ll

< Me-atllz(O)II + N max IIG(y(s),J.L)y(s)il o.:;s::;t

which indeed guarantees the property of input-to-state stability. On the other hand, the property that b(y, J.L) is bounded away from zero makes it possible to impose a suitably strong control action on the lower subsystem, so as to achieve input-to-state stability and a prescribed gain. Lemma 11.2.1. Consider system (11.8). Suppose that:

(i) for each J.L, the eigenvalues of F(J.L) have negative real part, or, what is the same, there exists a symmetric matrix P(J.L) > 0 (continuously depending on J.L) such that P(J.L)F(J.L) + F(J.L)P(J.L) =-I, (ii) there exists a real number b0 > 0 such that

b(y, J.L) 2: bo for all y E lR and all J.L E P. Then, there exists a smooth function !(Y) such that, if the system (11.8) is controlled by the input (11.10) u = -Y!(Y), the positive definite function

satisfies

av.

8z z

av. + 8y y < -c(llz112 + y2)

for some r:: > 0. As a consequence, the feedback law (11.10) globally asymptotically stabilizes the system for any value of J.L E P. Proof. Observe that

av.

av.

-z+-y= 8z 8y -zT z -

+ 2zT P(J.L)G(y, J.L)Y + 2yH(y, J.L)z + 2[K(y, J.L)

I ( yz)T( -GT(y,J.L)P(J.L)H(y,J.L)

- b(y, J.L)/(y)Jy 2 =

-P(J.L)G(y,J.L)-HT(y,J.L))(z) 2[b(y,J.L)/(y)- K(y,J.L)] y .

11.2 Stabilization via Partial State Feedback: a Special

C~

81

It is easy to see that, if !(Y) is appropriately chosen,

-P(J.L)G(y,J.L)-HT(y,J.L))-c:l>O(llll) 2[b(y, J.L}J(y)- K(y, J.L)] .

I ( -GT (y, J.L)P(J.L) - H(y, J.L)

for some c; > 0. To this end, recall that if Q 0 and and Q 0 is nonsingular

Ro are symmetric matrices

(~f ~) = (sl~o 1 ~) (~o Ro-SJQ0 So) (~ Qo-;so)' 1

so that the matrix on the left-hand side is positive definite if and only if Qo > 0 and Ro- S6Q 01 So > 0. Now, let c; be any number such that 0 < c; < 1. Then, the matrix in (11.11) is positive definite if and only if

2[b(y,J.L}J(y)- K(y,J.L)]- c: 1

> [P(J.L)G(y,J.L) + HT(y,J.L)F 1- c;[P(J.L)G(y,)L) + HT(y,p)]. Clearly, this inequality holds if !(Y) is such that

2b(y,J.L}J(y)

>

1

_c:(IIP(J.L)G(y,J.L) +HT(y,J.L)ii? +2K(y,J.L) +c:. (11.12)

1 Observe now that, since by hypothesis J.L ranges over a compact set, there exist continuous functions g(y),h(y),k(y), and a number PM> 0, such that IIG(y, J.L)II IIH(y,J.L)II IK(y, J.L)I IIP(J.L)II

< < < <

g(y) h(y) k(y)

(11.13)

PM

for all y and all J.L· Then, (11.12) holds if

!(Y)

1

> 2bo( 1 _c) [(PMg(y) + h(y)) 2 + 2k(y) + c:].

If !(Y) is chosen in this way

aavz z. + aavy iJ• < -c:(llzll2 + y2) and this concludes the proof.

,

<1

Remark 11.2.1. Of course, a similar result holds if, instead of condition (ii), a condition of the form b(y, J.L) ::; -bo < 0 , with b0 is a fixed number, is assumed. In this case, the stabilizing la.w would be u = Yl(y). If the set P is not compact, then the hypothesis that estimates of the form (11.13) hold must be explicitly invoked. <1

82

11. Feedback Design for Robust Global Stability

Remark 11.2.2. Note that the result remains valid in the more general case of a system of the form i

y

F(y, p,)z + G(y, p,)y H(y, p,)z + K(y, p,)y + b(y, p,)u,

so long as there exists a symmetric matrix P(p,) and not on y, such that

> 0, depending only on p,

P(p,)F(y,p,) +FT(y,p,)P(p,) =-I, for all y E IR and all p, E P.


Remark 11.2.3. The simple basic mechanism behind the proof of the Lemma is that the choice of a large !(Y) helps to (robustly) offset the effect of the off-diagonal terms in the matrix (11.11), so as to render it positive definite. This is indeed a "small-gain" argument, because such an action weakens the coupling between the upper and lower subsystem of (11.8). To see an alternative proof of this Lemma, which directly uses the small gain Theorem for input-to-state stable systems, one should proceed as follows. Set

Observe that :

1

i

= -zT z + 2zT P(p,)G(y, p,)y ~

-zT z + 6zT P(p,)P(p,)z + ~IIG(y, p,)IJ 2 y 2

for any 6 > 0. Recall also that, if .:1 > 0 and "X > 0 are the smallest and, respectively, the largest eigenvalue of a symmetric matrix M > 0,

for all x. If M depends continuously on a parameter p, and p, ranges over a compact set P, then an estimate of the form

in which 0 < !; < c are independent of p,, holds for all x and all p, E P. Bearing this fact and the estimates (11.13) in mind, choose 6 such that the matrix I- 6P(p,)P(p,) is positive definite for all p, and observe that the previous inequality yields 0 1

~ i ~ -allzll 2 + bl(Y)Y2

in which a > 0 and b > 0 are fixed numbers, and

11.2 Stabilization via Partial State Feedback: a Special Case

83

for some 0 < g < a. Without loss of generality, assume that g(y) = g( -y) and that g(lyi) is non decreasing, so that the latter inequality can be written in the form

av1.

8z z :$ -allzll

2

+ a(lyi)

where a ( ·) is the class IC 00 function a(r) = b[g(r)r] 2

•

As a consequence (see Remark 10.4.3), we see that the response z(·) to any bounded y(·) is such that

for some class IC£ function {31 ( ·, ·), where ')'1 ( ·) is a class IC 00 function of the form 11 (r)

= ~Jka(r) yg_ a

,

in which k > 1. In particular, for some d > 0, ')'1

(r) =_dg(r)r.

Let cp(y) be a continuous function satisfying cp(y) =
2bo')'(y) > h 2 (y)

+ 2k(y) + cp2 (y) .

Then, observe that 8V2.

-y 8y

+ 2K(y, p,)y 2 - 2b(y, p,)'y(y)y2 < ZT Z + h2 (y)y 2 + 2k(y)y2 - 2bo')'(y)y2 < -a(lyi) + llzll 2 , 2yH(y, f.L)Z

where a(·) is the class /C 00 function

a(r)

=, [cp(r)rJ 2

•

As a consequence, we see that the response y(·) to any bounded z(·) is such that 0 ly(t) I :$ max{f32{IY I, t), '/'2(11z0lloo)} for some class IC£ function f32 (·, ·), where 1'2 ( ·) is a class /C 00 function of the form in which k > 1.

84

11. Feedback Design for Robust Global Stability

Thus, system (11.8) with input (11.10) can be viewed as interconnection of two input-to-state stable systems. This system is globally asymptotically stable if the "small gain condition"

holds for all r > 0. This condition reduces to 1

dg(r)r < .jkrp(r)r and can indeed be satisfied by appropriate choice of rp(r).


Having shown how to stabilize a system of the form (11.8), we can use the methodology described in the previous section to prove a basic Lemma, which makes it possible to iteratively stabilize classes of systems having a "chained" structure.

Lemma 11.2.2. Consider a system o/the form

i Xr

iJ

= = =

F(J.t)z + G(xr,J.t)xr HI (xt, J.t)z + KI (xr, J.t)xr + bi (xi, J.t)Y H2(XI,Y,J.t)Z + K21(XIJY,J.t)XI + K22(XI,y,J,t)y + b2(xr, y, J.t)u

(11.14)

in which z E !Rn , x 1 E JRi , y E IR, u E IR, and J.t E P C JRP is a vector of unknown parameters, with P a compact set. The functions F(J.t), G(x 1 ,J.t), Hr(xi,J.t), KI(xi,J.t), bi(XI,J.t), H2(xi,y,J,t), K2I(Xt,Y,J.t), K22(xr,Y,J.t) and b2 (x 1 ,y,J.t) are smooth junctions of their arguments. Suppose that: (i) there exist a symmetric matrix P(J.t) > 0, continuously depending on J.t, an i x i matrix M 1 ( xr) of smooth functions, nonsingular for all xr, and a 1 x i vector f'I (x 1 ) of .smooth functions, such that

for some class Koo function q(·), and such that the positive definite function

satisfies 8Vr (

az

(11.15)

for some c > 0, (ii) there exists a real number b20 > 0 such that

11.2 Stabilization via Pa.rtial State Feedback: a Special Case

85

b2(xr,y,J-t) ~ ~o for all (xr,Y) E JRi+l and all J-t E P. Then, there exists a smooth function -y2 (x 1 , y) such that, if the system ( 11.14) is controlled by the input (11.16) the positive definite function

satisfies 2 8V2 . 8V2 . 8V2 . c 2 2 8zz+ x xr+ yy~-2(llzll +IIMr(xr)xrll +[y--yr(xr)xr] ). 8 1 8

Proof. For notational convenience, set

and

. _ 8V2 . 8V2 . V2 Z + Xt 8 Xr 8z

+

8V2 . Y· 8Y

Observe also that there exists a matrix W(xr) of smooth functions such that 8Ur T = 2x 1 W(xr) . 8 Xt Using the hypothesis (i), it is possible to obtain v2

~

8 1 xr

-c(llzll 2 + IIMr(xr)xrll 2 ) + (y --y1(X1)x1] 8u b1(Xr,J-t) + 2[y- 1'1 (xr )xr] (:ri --yr (xr)x1 - xT ~:~ x1)

=

-c(l!z!l 2 + I!M1 (x1 )xrll 2 )

+ 2[y --y1(xr)xr](xTW(x1)br(x1,J-t) + y- .:Y1(xr)x1), where we have set

= 1'i'(Xt ) + X1T -8-y{ · 8 Xt Bearing in mind the expressions of x1 and iJ, write xTW(xt)b1 (x1, J..t) + iJ- .:Y1 (xr):i:r = - ( 1'1 X1 )

A(xr, y, J-t)z + B(x1, y, J-t)Xt

where

+ C(x1 ,y, J-t)[y -1'1 (xr)xr] + b2(x1 ,y ,J-t)U

86

11. Feedback Design for Robust Global Stability

A(xt,Y,JL) B(x1, y, JL)

C(xt,Y,JL)

=

H2(Xt,Y,JL)- it(xt)Ht(Xt,JL) bf(xt,JL)WT(x!) +K21(Xt,Y,JL)

=

- it (xi) ( K1 (x1, JL) + b1 (x1, JL)'Yl (xt)) + K22(xt,Y,JLht(xt) K22(Xt,Y,JL)- it(xt)bt(Xt,JL).

Choosing the control u as in (11.16) yields

where 0

t:M!{xt)Mt (xt) -B(xt,y,JL) Since by hypothesis JL ranges over a compact set, there exist continuous functions a(x 1 ,y), b(x 1 ,y), and c(xt,Y) such that the matrices A(x 1 ,y,JL), B(xt,y,JL) and C(x 1 ,y,JL) satisfy

IIA(xt,y,JL)II IIB(xt,Y,JL)II IIC(xt, y, JL) II

~

a(x1,y) ~ b(xt,Y) ~ c(x1, y)

for all (x 1 , y) E JRi+ 1 and all JL E P. Therefore, by means of arguments identical to the ones used in the proof of the previous Lemma, it can be deduced that there exists a smooth function '"Y2(x 1 , y) such that

and this concludes the proof.

<1

Remark 11.2.{ Note that, setting

x2

= col(x1, y)

and

M2(x2)

= ( --y1(xt) Mt(xt)

0) 1

the function V2 (z, x 1 , y) can be written in the form

'

11.2 Stabilization via Partial State Feedback: a Special Case

87

which shows that this function has a structure identical to that of the function V1(z,x 1). Moreover, since by hypothesis M 1 (x!) is such that g.(llxt/1) ~ IIM1 (xi)xtll

2

for some class IC 00 function g.(-), it is possible to show that also M 2(x 2) is nonsingular for all x 2 and such that

for some class IC 00 function g( ·). To this end, it suffices to use the property (see Remark 10.1.3) that a positive definite function V(x) satisfies an estimate of the form g_(llxll) ~ V(x), with g(·) a class Koo function, if and only if, for any c > 0, the set of all x such that V(x) ~ cis a compact set. In particular, suppose (11.17)

Then, necessarily, IIMI(xi)xtll 2 IYI

~ ~

c 11'1 (xi)x1l

+ ..fi .

The first one of these, since by hypothesis IIM1(xi)x 1 ll 2 is bounded from below by a class Koo function of llx 1 ll, implies that Xt belongs to a compact set, say D1. Then, the second one yields

Since both x 1 and IYI are bounded, the set of all x2 which render (11.17) true is compact and therefore there exists a class Koo function of llx2ll which bounds IIM2(x2)x2ll 2 from below. <J The arguments used in the previous remark show that also the function

zT P(J..L)z + IIM1(xr)x1W

+ [y- 'Yt(xl)xt) 2 ,

for some class IC 00 function a(·), is bounded from below by a(llcol(z, x 1,y)ll). Thus, we can conclude, from the Lemma, that if the system (11.14) is controlled by the input (11.16), the fti~ction V2(z,xt,y) is such that

8V2 . 8V2 . Bz z + Bx x1 1

+

8V2 . E By y ~ -2a(llcol(z,xl,y)ll)

which enables us to assert that the feedback (11.16) is globally asymptotically stabilizing. Moreover, the function

V(t) = V2(z(t),x1(t),y(t)) is such that

88

11. Feedback Design for Robust Global Stability

for some a that

>

dV -<-aV dt 0 and this, because of standard comparison arguments, proves

V(t)

~

e-atV(O) ,

i.e. that V(t) decays to zero exponentially. Using the previous two Lemma, it is immediate to deduce the existence of a globally stabilizing feedback law for a system of the form

+ G(6, tL)6

i

F(IJ.)z

6

Hr(6,1J.)Z + Kr(6,1J.)~r + br(6,1J.)6

i2 =

H2(6,6,1J.)z+ LK2i(6,6,1J.)~i+b2(6,6,1J.)6

2

i=l

r

~r

=

Hr(6, ... ,~r,IJ.)z+ LKri({r, ... ,~r,IJ.)~i+br(6, ... ,~r,IJ.)U i=l

in which z E !Rn, ~i E IR, for i of unknown parameters.

= 1, ... , r, u E lR and /.L E P C JRP

(11.18) is a vector

Theorem 11.2.3. Consider system {11.18), with IJ. ranging over a compact set P. Suppose that: (i) for each IJ., the eigenvalues of F(tL) have negative real part,

(ii) there exist real numbers biD

> 0 such that

b;({r, ... ,~i,IJ.);::: bio

for all 1 ~ i ~ r, (~ 1 , ... , ~i) E JRi , and all IJ. E P. Then, there exists a smooth feedback law

u=u({!, ... ,~r), with u(O, . .. , 0), which globally asymptotically stabilizes the system for any value of IJ. E P. Proof. H r = 1, the hypotheses of the Theorem coincide with those of Lemma 11.2 .1, and therefore one can assert the existence of a smooth function '/'I (6) such that the positive definite function

satisfies

8Vr ( 8z

11.2 Stabilization via Partial State Feedback: a Special Case

89

for some E: > 0. Proceeding by induction, suppose r > I, set x1 =col({!, ... ,~;),

y = ~i+l,

rewrite the first n + i + 1 equations of (11.18), with ~i+ 2 replaced by u, i.n the form (11.14) and suppose hypothesis (i) of Lemma 11.2.2 holds. In other words, suppose there exist a 1 x i vector -y1(xr) of smooth functions and ani xi matrix M 1(x 1) of smooth functions, such that the positive definite function VI(z,xl) = zT P(JL)z + I/Ml(xl)x1W satisfies the inequality (11.15) for some E: > 0 (as shown above, this is the case if i = 1). Then, by Lemma 11.2.2, there exists a smooth function 1 2 (x 1,y) such that the positive definite function v2 (z, Xl' y)

= ZT P(JL)z + 1/Ml (xl)xlll 2 + [y- 'Yl (xl)x!] 2

satisfies, when u = --y2 (x 1,y)[y- -y1(x.)x 1], • E: 2 + IIMI(XI)xl/12 + [y- 'Y1(x1 )x1]). 2 V2 < -2(11zll

Writing

and

--y2(xl,~i+l)[~i+l- 'YI(x.)xd

=

('Y2(XI,~i+.)'Yl(xl) --y2(Xl,~i+I))(~~:J,

and bearing in mind Remark 11.2.4, it is concluded that the hypothesis (i) of Lemma 11.2.2 holds for the system consisting of the first n + i + 2 equations of (11.18). Thus, by induction, it is concluded that there exist a feedback law u=u({I, ... ,~r),

with u(O, ... , 0) = 0, and a positive definite function V(z, {!, ... , ~r), satisfying g_(/lcol(z,{I, ... ,~r)/1):::; V(z,{I, ... ,~r) for some class Koo function g_(·), with the property that

V:::;

-a(llcol(z,~l, ... ,~r)/1)

for some class Koo function a(·). This proves the Theorem.


90

11. Feedback Design for Robust Global Stability

We conclude this section with some remarks about the structure of system (11.18). As it can be seen, the state of this system is split into two subsets, namely z and 6, ... , (r, and the fact that the nonlinear terms of the model depend in a triangular manner on the individual (/s makes it possible to implement the recursive design procedure based on Lemma 11.2.2. Of course the class of systems modeled by equations of the form (11.18} includes, as a special case, the case of systems which - for a suitable choice of coordinates - exhibit a triangular dependence for all the state variables. In this respect, the set of top n equations in (11.18), namely

i=F(J.J.)z+g(6,J.J.)6, should be viewed as representing some subsystem in which a triangular dependence on the individual components of the state variables cannot be found. In order to cope with the fact that for this subsystem a triangular internal structure may not exist, the expressions or: the right-hand side of (11.18} have been assumed to be affine functions of z and this, as it is clear from the previous arguments, has simplified the synthesis of the stabilizing feedback. Another crucial hypothesis on (11.18} was the stability of the internal dynamics of the upper subsystem. In view of the general principles illustrated at the beginning of the Chapter, it may seem that this hypothesis could be removed, if the upper subsystem is stabilizable by some feedback law 6 = v•(z). However, it must be observed that the change of variables y=

6- v*(z),

needed to transform the system into one in which the top part is stable, in general destroys the particular structure of (11.18}, as the various expressions on the right-hand side may no longer be affine in z after the transformation. It is for this reason that the assumption in question was somewhat necessary in the present setup. We will see however in section 11.4 how, by appealing to the small-gain Theorem for input-to-state stable systems, such an assumption can eventually be removed.

11.3 Stabilization via Output Feedback: a Special Case In this section, we address the problem of robustly globally stabilizing a nonlinear system using output feedback only. If a system has uniform relative degree one and the special structure of system (11.8), we know from Lemma 11.2.1 that global stabilization is possible using a feedback which depends only on y, namely a memoryless output feedback. In this section, we consider a special class of systems having uniform relative degree larger than one, and we will show how, using the result of Theorem 11.2.3, global stabilization can be achieved using dynamic output feedback.

11.3 Stabilization via Output Feedback: a Special Cas€

91

The basic idea on which this stabilization method reposes is the following one. Suppose, without loss of generality, that the system (by hypothesis assumed to have relative degree r ~ 2) is modeled by equations of the form

x

=

iJ

=

f(x, y, J.L) + g(x, y, J.L)u h(x,y,J.L),

(11.19)

in which x E !Rn, y E IR, u E IR and J.L E P C JRI' is a vector of unknow·n parameters. Suppose to augment its dynamics by adding the following set of r - 1 equations

~2 ~3

->.16 + 6 ->.26 + ~4

~r

->-r-l~r

+ -y(y)u,

in which the >.i 's, for i = 1, ... , r- 1, are positive numbers, u and y are input and output of system (11.19) and -y(y) is a smooth function, bounded away from zero. In other words, consider the "augmented" system :i;

iJ

f(x,y,J.L) h(x, y, J.L)

~

A~+

+ g(x,y,J.L)u (11.20)

B-y(y)u

in which ~ =

and A=

col(6, ... • ~r) 0 0

1

C;'

->.2

. 0 0

->.r-2

0 0

Set also

0

c=

(1 0

jJ

B=m

... 0 0) ,

so that 6 = c~. Suppose, now, that there exists,a change of coordinates X=

X-

D(f.L)~

(11.21)

which transforms the augmented system (11.20) into a system of the form F(f.L)X + G(y, J.L)Y + d(f.L)C~ H(J.L)x + K(y, f.L)Y + b(f.L)C~

x

y

t with b(J..L) satisfying

=

A~

+ B-y(y )u ,

(11.22)

92

11. Feedback Design for Robust Global Stability

b(p.)

~

bo > 0

for all p.. If this is the case, the additional change of coordinates

z = x-

b(~) d(p.)y

transforms the latter system into a system of the form

z

F(p.)z

iJ

ii(p.)z + K(y, p.)y + b(p.)6

+ G(y,p.)y

-A16

+6

-A26

+ ~4

(11.23)

in which -

F(p.) G(y,p.)

ii(p.) -

K(y, p.)

1 = F(p.) - b(p.) d(p.)H(p.) 1 [ F(p.)d(p.)= G(y,p.) + b(p.) = H(p.) 1 = K(y, p.) + b(p.) H(p.)d(p.) .

1 b(p.)d(p.)H(p.)d(p.)d(p.)K(y,p.) ]

System (11.23) turns out to be a system having the same structure as system (11.18). If it satisfies the hypotheses of Theorem 11.2.3, global robust stabilization is possible, by means of a feedback law of the form u

= u(y,~).

Of course, this feedback stabilizes also system (11.20), which differs from (11.23) by a simple change of coordinates. But to say that this (memoryless) feedback law stabilizes (11.20) is the same as to say that the system

t u

A~+ B"{(y)u(y, ~)

=

u(y,~),

(11.24)

viewed as a system with input y, output u and internal state ~ E IR.r-l, is a dynamical feedback law which stabilizes system (11.19), and this is precisely the design goal that was desired to accomplish. In summary, bearing in mind the conditions of Theorem 11.2.3, it can be concluded that system (11.19) can be globally stabilized via dynamic output feedback if: (i) it is possible to transform system (11.20), via a change of coordinates of the form (11.21), into a system of the form (11.22),

11.3 Stabilization via Output Feedback: a Special Case

93

(ii) for each J.L, the eigenvalues of the matrix F(J.L) have negative real part, (iii) there exists a real number b0 > 0 such that

> bo

b(J.L) --y(y)

;:::

bo

for all y E lR and all J.L E P. We now address these points separately. In order to have condition (i) fulfilled, observe that this implies

f(x

+ D(J.L)~, y, J.L) + g(i: + D(J.L)~, y, J.L)u- D(J.L)A~- D(J.L)B--y(y)u = F(J.L)i: + G(y, J.L)Y + d(J.L)C~ .

Thus, necessarily,

f(i:

+ D(J.L)~, y, J.L) -

D(J.L)A~ =

F(J.L)i:

+ G(y, J.L)Y + d(J.L)C~

(11.25)

and

g(i: + D(J.L)~, y, J.L) = D(J.L)B--y(y) .

(11.26)

From the first one of these, we see that, necessarily,

f(x, 0, J.L) f(O,y,J.L) and

f(x,y,J.L)

=

F(J.L)x G(y,J.L)Y

= F(J.L)X + G(y,J.L)Y.

If this holds, then (11.25) holds if and only if

F(J.L)D(J.L) - D(J.L)A = d(J.L)C. On the other hand, (11.26) implies

g(x, y, J.L) = 9(J.L)'"Y(y) where Identical arguments prove, starting from

that necessarily

h(x,y,J.L)

= H(J.L)X + K(y,J.L)Y

and

H(J.L)D(J.L) = b(J.L)C. In summary, condition (i) can be fulfilled if and only if system (11.19) has the special form

94

11. Feedback Design for Robust Global Stability

±

=

iJ

=

F(J.L)X + G(y, J.L)Y + g(J.L)'y(y)u H(J.L)x + K(y,J.L)Y

(11.27)

and there exist D(J.L), d(J.L), b(J.L) such that

F(J.L)D(J.L) - D(J.L)A D(J.L)B H(J.L)D(J.L)

= =

d(J.L)C g(J.L) b(J.L)C.

(11.28)

It turns out that the latter equations can always be solved, and therefore it is possible to conclude that the necessary and sufficient condition which the system has to fulfill so that (i) holds is simply the condition that the system has the special form (11.27). For the sake of convenience, we state and prove separately the claim about the solvability of (11.28).

Lemma 11.3.1. Suppose system {11.27) has uniform relative degree r. Then the equations {11.28) have a unique soluti~n D(J.L), d(J.L), b(J.L). In particular

d(J.L)

= p(F(J.L))g(J.L),

(11.29)

where p(.\) is the polynomial (11.30)

and b(J.L) Proof. Partition D(J.L) into its

= H(J.L)r- 2 (J.L)9(J.L). r-

(11.31)

1 columns as

and observe that the second of (11.28) reduces to

dr-1 (J.L)

= g(J.L) .

The first condition, read column by column, yields, fori

and Thus, recursively, we get

9(J.L) [F(J.L)

+ Ar-II]g(J.L)

d(J.L) The last one of these proves (11.29).

= r- 1, r- 2, ... , 2

11.3 Stabilization via Output Feedback: a Special Case

95

So far, we have seen that the first two equations in (11.28) have a unique solution D(J.L), d(J.L). It remains to show that this solution is able to render the last equation fulfilled, for some b(J.L). To this purpose we use the hypothesis that system (11.27) has relative degree r. Computing the first r derivatives of y, we deduce that, necessarily,

and Using these, and bearing in mind the expressions previously found for the di(J.L)'s, we see that

and

H(J.L)dl (J.L) = H(J.L)Fr- 2 (J.L)g(J.L) which prove the last of (11.28).


We address now condition (ii) above. To this end, recall that by hypothesis system (11.27) has uniform relative degree r. Therefore, this system has a well-defined zero dynamics which can be identified in the following manner. Simple calculations show that the constraint y = 0 implies 0 0

= =

H(J.L)X H(J.L)F(J.L)x

0 0

= =

H(J.L)P- 2 (J.L)X H(p.)Fr-l(J.L)x

+ H(J.L)P- 2 (J.L)g(p.)"y(O)u.

From this it is seen that the zero dynamics manifold Z* of (11.27) is the subspace

z• = {x

E !Rn: H(J.L)Fi(J.L)x = O,i = 0, ... ,r- 2}

and, also, that the latter is

renden~d

(11.32)

invariant by the control

Assuming, without loss of generality, that 'Y(O) = 1 (if this is not the case, simply redefine g(J.L) and "Y(Y) so as to have this condition fulfilled), and bearing in mind the formula (11.31), this control can be rewritten as

96

11. Feedback Design for Robust Global Stability

We can conclude, therefore, that the zero dynamics of (11.27) are those of the restriction of the linear mapping (11.33) to its invariant subspace z·. It is possible to show that condition (ii) is actually equivalent to the property that the eigenvalues of the linear mapping (11.34) have negative real part. Lemma 11.3.2. Assume A1 > O, ... ,Ar-l > 0. The eigenvalues of the matrix F(J.L) have negative real part if and only if the eigenvalues of {11. 34} have ' negative real part.

Proof. The proof of this result is essentially based on the following property. Let A be a linear mapping !Rn -+ !Rn and let V be a k-dimensional invariant subspace. Then, it is well known that, choosing coordinates in such a way that vectors in V are vectors whose the last n-k entries are zero, the mapping A is represented by a matrix of the form

(11.35) and An is a matrix representation of the restriction of A to V. Note also that, since V is invariant for A, the mapping

A• : !Rn /V -+ !Rn /V {x}

t-+

{Ax}

(in which {x} denotes the equivalence class of x, namely the set of all x' such that ( x' - x) E V) is well defined, and A22 is a matrix representation of A •. Thus, the spectrum of A is the union of the spectra of Ajv and of A•. Consider now our case and set 2

V = span{9(J.L),F(J.L)9(J.L), ... ,F(J.Lr- 9(J.L)}.

This subspace is an invariant subspace of F(J.L). To see this, express the polynomial (11.30) in the form

p(A)

= Ar-l + ar-2Ar- 2 + ... + a1A + ao ,

and observe that a simple calculation yields

F(J.L) ( 9(J.L)

F(J.L)9(J.L)

11.3 Stabilization via Output Feedback: a Special Case

= ( 9(J.t)

F(J.t)9(J.t)

· · · F(J.tt- 2 9(J.t)) (

97

~~ 0 0

This proves that Vis invariant under F(J.t) and that

0 0 1 0

~ ~

Au=

(

(11.36)

0 0

is a matrix representation of F(J.t)lv· Consider now the subspace

This subspace is not necessarily invariant for F(J.t). However, it is complementary to V. In fact, no non-trivial linear combination of g(J.t), F(J.t)9(J.t), ... , F(J.tt- 2 9(J.t) can be annihilated by all H(J.t), H(J.t)F(J.t), ... , H(J.t)Fr- 2 (J.t)'s because, otherwise, this would contradict the fact that H(J.t)r- 2 (p)g(J.t) is nonzero. As a consequence of this, it is seen that, for each x E !Rn, there exists a unique z E z• such that

{x}=z+V, and, in particular, the mapping

P•: {x} t-7 {F(J.t)x} is such that

P•{x}

= {F(J.t)z}.

Now, it turns out that, by construction, for all z E z• ·'

mod V. Thus,

{F(J.t)z}

= {[F(J.t) -

and this shows that the mappings

1 1 b(J..L) g(J.t)H(J.t)pr- (J.t)]z}

P•

and (11.37)

98

11. Feedback Design for Robust Global Stability

are isomorphic. As a consequence, it is found that the matrix A22 in the bottom-right corner of the representation (11.35) of F(J.l.) is in fact also a matrix representation of (11.3 7). We can thus conclude the spectrum of F(J.l.) is the union of the spectra of (11.36) and of (11.37) and this proves the claim. <1 In summary, the arguments above show that the fulfillment of conditions (i), (ii) is possible if and only if the system (which was assumed to have uniform relative degree r) has the special form (11.27) and its zero dynamics are asymptotically stable. Therefore, it can be concluded the following result holds.

Proposition 11.3.3. Consider system

x = y =

F(J.l.)x + G(y,J.l.)Y + §(J.l.)r(y)u H(J.l.)X + K(y, J.l.),Y .

in which x E JR.n, y E JR., u E JR., J.1. E P C JR.P is a vector of unknown parameters ~nd P is a compact set. Assume the following (i) H(J.l.)Fi(J.l.)§(J.l.) = 0 fori = 1, ... , r- 3 and, for some bo > 0, H(J.l.)r- 2(J.l.)§(J.l.) ~ bo -y(y) ~ bo for all y and J.1. E P, (ii) for each J.1. E P, the eigenvalues of

have negative real part. Then, this system is globally asymptotically stabilizable by dynamic output feedback.

11.4 Stabilization of Systems in Lower Triangular Form We present in this section a method for robust global stabilization of systems described by equations having a lower-triangular structure, such as

i

=

f(z,{I,J.l.) ql(z,{I,J.l.) +bl(z,(l,J.l-)6 q2(z,{I,6,J.l.) + b2(z,{t,6,J.l-)6

(11.38)

11.4 Stabilization of Systems in Lower Triangular Form

99

in which z E ~n, ~i E lR, fori= 1, ... ,r, u E ~and J.1. E PC ~Pis a vector of unknown parameters. These systems are often referred to as systems in feedback form, in consideration of the fact that they correspond to a cascade interconnection of r + 1 subsystems, starting with the lower subsystem of (11.38) and ending with the upper subsystem of (11.38), in which the i-th subsystem is fed by the "outputs" ~;- 1 , . . . , 6, z of all subsequent subsystems in the cascade (see Fig. 11.1). This class of systems can be seen as an extension of the class of systems modeled by the equations (11.18), specifically because the expressions on the right-hand sides are no longer assumed to be affine functions. in z. Moreover, also the hypothesis that the upper subsystem, with ~ 1 = 0, has an asymptotically stable equilibrium at z = 0 can be dropped.

'------ · · ·

~r-1

Fig. 11.1. System in feedback form.

The fundamental result on which the possibility of robustly stabilizing a system of the form (11.38) is based is a Lemma which can be considered as the robust version of Lemma 9.2.1. Lemma 11.4.1. Consider the system f(z, ~, J.t) ¢(z, ~, J.t)

+ b(z, ~, J.t)u

(11.39)

in which (z, 0 E JRn x ~' f(O, 0, J.t) = 0 and ¢(0, 0, J.t) = 0. Suppose the following:

(i) for each J.1. the upper subsystem in (11.39) is input-to-state stable and, in particular, a class K 00 function 1{), independent of J.l., is known such that the response z(·) to any bounded~(·) satisfies /lz(·)lloo limsupllz(t)ll t--too

< <

for some class K£ function

max{,B(IIzoll,t),/'(ll~(·)lloo)} ')'(iimsupll~(t)/1) t-too

.8(·, ·),

(11.40)

100

11. Feedback Design for Robust Global Stability

(ii) there exists a number b0 > 0 such that b(z, (, p.) 2: bo for all (z, () and all p., (iii) there exist class K functions p 0 ( ·) and p1 ( ·), which are locally Lipschitz at the origin, such that max{I
for all (z, () and all p., (iv) the function p1 (I{)) is locally Lipschitz at the origin. Then, there exists a smooth function k((), with k(O) = 0, such that, under the control law (11.41) u = k(() + v J

the closed-loop system (11.39}- (11..41}, viewed as a system with input v and state (z,(), is input-to-state stable and, in particular, a class K00 function i{), independent of p., can be found such that the response x( ·) = (z( ·), (( ·)) to any bounded v ( ·) satisfies llxOIIoo

<

max{,B(IIxoll,t),i'(llv(·)lloo)}

lim sup llx(t) II

~

i'(lim sup llv(t)ll)

t-+oo

(11.42)

t-+oo

for some class KC function

/3(·, ·).

Proof. The first step in proving this Lemma consists in showing how, given any arbitrary class Koo function g( ·), it is possible to find a smooth function k(() such that the system

(11.43) viewed as a system with input (z, v) and state(, is input-to-state stable, and -in particular- such that the response to any bounded z(·) and v(·) satisfies l((t)l ~ max{/3(1( 1, t),g(llz(·)lloo),'"fv(llv(·)lloo)} 0

(11.44)

for some class K function 'Yv ( ·), independent of p., and some class KC function /3(·, ·). For convenience, this fact is stated and proved separately.

Lemma 11.4.2. Consider the system

x = 0 such that b(z,x,p.) 2: b0 for all (z,x) and all p.. Suppose there exist class K functions Po ( ·) and p 1 ( ·), which are locally Lipschitz at the origin, such that max{I
for all ( z, x) and all p.. Let g( ·) be some fixed class Koo function and suppose that p 1 (g- 1 ( ·)) is locally Lipschitz at the origin. Then there exists a smooth

11.4 Stabilization of Systems in Lower Triangular Form

101

feedback law u = k(x) such that the positive definite function V(x) satisfies JxJ

~ max{ dJvl, g(Jizll)}

=>

= x2

~~ [¢(z, x, p.) + b(z, x, p.)[k(x) + vJ] ~ -c-Jxl 2 , (11.45)

for some d > 0 and c > 0. Proof of Lemma lL/.2. Choose k(x) = -x- a(x) in which a(x) is a smooth strictly increasing function, with a(O) a( -x) = -a(x). Thus, xa(x) = lxla(Jxl). Observe that

= 0, and

av ax [t/>(z, x, p.) + b(z, x, p.)k(x) + b(z, x, p.)v] + 2x¢(z, x, p.) - 2x 2 b(z, x, p.) - 2xa(x)b(z, x, p.) + 2xvb(z, x, p.) ~ + 2JxiJ¢(z,x,p.)J- 2boJxl 2 - 2boJxJa(JxJ) + lxl 2 lb(z,x,p.) 12 + fvl 2 = + Jxi[2J¢(z,x,p.)J- 2boa(Jxl) + JxJJb(z,x,p.)J2]- 2boJxJ 2 + Jvl 2 . =

The inequality in (11.45) can be obtained if, for some c

-2bolxl 2

+ Jvl 2

> 0,

~ -c-Jxl 2

and if 2boa(JxJ) ~ 2J¢(z, x, p.)l

+ lxllb(z, x, p.)J 2

•

Choosing c < 2bo, first one of these conditions is indeed fulfilled if Jxl ~ dJvl, with d = 1/v2b0 - c-). The second one is fulfilled if

2boa(lxl) ~ 3 max{Po(Jxl), PI (liz II)} . This condition suggests to choose a(Jxl) ~

3 bo max{p0 (Jxi), pt(g 1 (Jxl))} , 2

{11.46)

so that, 2boa(Jxl) ~ 3po(Jxi) and Jxl ~ g(JizJJ)

2boa(Jxl) ~ 3pl (ilzll) .

This choice of a(Jxi) renders (11.45) satisfied. Finally, the hypotheses that Po(·) and p1 (g- 1 (·)) are locally Lipschitz at the origin guarantee that it is possible to choose a smooth a(x) such that (11.46) holds. This concludes the proof of Lemma 11.4.2. <1

Proof of Lemma 11..4.1 (continued). Property (11.45) shows that V(~) is an ISS-Lyapunov function for system

{ = ¢(z, {, p.) + b(z, {, p.)[k({) + v]

=~

2

·

(11.47)

102

11. Feedback Design for Robust Global Stability

viewed as a system with input (z, v) and state {. As a consequence (see Remark 10.4.2), its response to any bounded z(·) and v(·) satisfies an estimate of the form /~(t)/ ~

max{,B(/C/,t),!z(l/z(·)/1 00 ),/v(l/v(·)//oo)}

for some class KC function ,8(·, ·),where

/z(r) and

Q.,

= Q.- 1 o a o g(r)

a are such that g_(/W

~ V(~) ~

a(/W .

In the present case one can choose, trivially, g_(r) = o(r) = r 2 , and therefore lz(r) = g(r), and this proves that an estimate of the form (11.44) holds. The same argument also shows that lv(r) = dr, from which it is seen that this function is independent of J.L. From (11.44), as shown in section 10.4, it follows that the response of (11.47) to bounded z(·) and v(·) is such that

1/((·)1/oo

lim sup 1/~(t)/1

t--+oo

~

~

max{,B(I/C 1/, t), g(l/z( ·)1/oo), lv (1/v( ·)1/oo)} max{g(lim sup 1/z(t)/1), /v(lim sup 1/v(t)/1)} .

t-+oo

t--+oo

(11 .48)

Consider now the system i

=

f(z,~,J.L)

~

=

(z,~,J.L)

+ b(z,~,J.L)[k(~) + v].

(11.49)

Suppose the various hypotheses of Lemma 11.4.1 hold. In particular, the hypothesis (i) is that, in the upper subsystem, the response z(·) to bounded ~(·) satisfies the estimates (11.40). Define a class Koo function g(·) as

g(r)

1 1 = 21(r).

In view of the assumption (iv), the function

satisfies the hypotheses of Lemma 11.4.2 and therefore it is possible to choose k(~) so that, in the lower subsystem, the response~(·) to bounded z(·) and v(·) satisfies the estimates (11.48). Note also that g- 1 (r) > 1(r), i.e.

1og(r)
for all r

> 0.

Thus, system (11.49) satisfies the hypothesis of the small-gain Theorem and, consequently, the system - viewed as system with state x = (z, ~) and

11.4 Stabilization of Systems in Lower Triangular Form

"'-

103

input v- becomes input-to-state stable. In particular (see Theorem 10.6.1), the response x (·) = (z (·), ~ (·)) to any bounded v (·) satisfies estimates of the form (11.42), with

i(r)

= max{2'Y o 'Yv(r), 2rv(r)} .

This completes the proof of Lemma 11.4.1.

<1

Remark 1Lf..1. It is worth to stress that the main argument on which the proof of this Lemma is based is the fact that the lower subsystem of (11.49), which is a one-dimensional system, can be rendered - under the appropriate technical hypotheses (ii), (iii) and (iv) - input-to-state stable, with a prescribed gain function g(·), between input z and state~- As a consequence, no matter what the gain function r(·) of the upper subsystem is, it is possible to choose g(·) so as to have the small gain condition fulfilled, and the result of the Lemma immediately follows by the small gain Theorem. An equivalent argument wa.s also, a.s observed, the main ingredient of the proof of Lemma 11.2.1, which is a special case of the present result. <1 ·

Lemma 11.4.1 contains all the ingredients needed to set up a recursive design procedure for robust stabilization of a system having the form (11.38). In fact, it shows- under the technical hypotheses (ii), (iii) and (iv) -that, if the upper system of (11.39) is input-to-state stable and, in particular, estimates of the form (11.40) hold with a function 10 which is independent of J..L, then a feedback law k(~) can be found such that the entire system, with control u = k(O + v, i.e. system (11.49), is input-to-state stable and, in particular, estimates of the form (11.42) hold with a function i{) which is independent of J..L· To see how the recursion proceeds consider, for instance, a system modeled by equations of the form i

f(z,~,J..L)

+ b(z,~,J..L)( '1/J(z,~,(,J..L) + d(z,~,(,J..L)U. ¢(z,~,J..L)

(11.50)

·'

The change of variable v

= (- k(~)

transforms (11.50) into a system, modeled by equations of the form

i

f(z,~,J..L)

€ =

¢(z, ~, J..L)

v

+ b(z, ~, J..L)k(~) + b(z, ~, J..L)v i[J(z,~,v,J..L) + d(z,~,v,J..L)u,

(11.51)

104

11. Feedback Design for Robust Global Stability

in which the subsystem consisting of the upper two equations, viewed a system with state (z, ~) and input v, satisfies an hypothesis which exactly corresponds to the hypothesis (i) of Lemma 11.4.1. Thus, if the gain function i{) and the functions 1[1 (z, ~, v, J.L) and d( z, ~, v, J.L) which appear in the lower equation are such that the conditions corresponding to the technical hypotheses (ii), (iii) and (iv) of Lemma 11.4.1 hold, a control law

u=k(v)+w can be found, such that the corresponding closed loop system, viewed as a system with input w and state x = (z, ~' v), is input-to-state stable and, in particular, its response x(·) to any bounded w(·) will satisfy estimates of the form llx(·)lloo < max{,B(IIxoll, t),7'(11w(·)lloo)} limsupllx(t)ll

< 1-(limsupllw(t)ll) t~oo

t~oo

where 1-0 is a class K function which is independent of J.L· We conclude the section by showing how the hypothesis (i) of Lemma 11.4.1 can be weakened, assuming instead that there is a feedback law ~ = v*(z) such that system i = f(z, v*(z), J.L) is robustly globally asymptotically stable, with a Lyapunov function which is independent of J.L, and an extra technical condition is fulfilled. More precisely, suppose there exist a smooth function v*(z), with v*(O) = 0, and a smooth function V(z), satisfying estimates of the form g(llzll) :S V(z) :S a(llzll) , such that

(11.52)

aV(z) f(z, v*(z), J.L) :S-a( liz II) , a;-

where g( ·),a(·), a{) are class Koo functions (independent of J.L). Then, consider the function 6 : IR~o x IR --+ IR defined by

6(s, r)

= max [ 8 ~(z) f(z, v*(z) + r, J.L) + !a(s)] llzll=•

uz

2

.

By hypothesis, the value of this function at r = 0 is negative for any s > 0. What we need to assume in the following is that this function continues to be negative in a region of the form

R

= {(s,r)

E IR~o x IR: s

> O,lrl :S p(s)}

for some continuous function p( s), independent of J.L, defined for all s ;::: 0, with p(O) = 0 and p(s) > 0 for all s > 0.

11.4 Stabilization of Systems in Lower Triangular Form

105

Lemma 11.4.3. Consider the system i

= f(z,~,p.).

(11.53)

Suppose there exist a smooth function v*(z) = 0, with v*(O) = 0, a smooth function V (z), class Koo functions g_( ·),a(·), a{), and a continuous function p(s), defined for all s ~ 0, with p(O) = 0 and p(s) > 0 for all s > 0, all independent of p., such that (11.52) hold, and · 8 max [ V (z) f(z, v*(z) llzll=s 8 Z

+ r, p.) + -1 a(s)] < 0

(11.54)

2

for all s > 0 and alllrl :s; p(s). Then, there exist a smooth (positive-valued) function /3* (z), such that 0 < /3*(z) :s; 1 for all z, and a class Koo function x(·), both independent of p., such that

llzll

~ x(lvl)

8V~) ~f(z,v*(z)

=>

+/3*(z)v,p.) :s;

1

-2a(llzll)

for all z. Proof From the proof of Theorem 10.4.3, it is known that, starting from the function p(·), it is possible to construct a smooth function /3*(z), with 0 < /3*(z) :s; 1 and a class Koo function x(-) such that

llzll ~ x(lvl)

l!3*(z)vl :s; p(llzll)

=>

.

The functions f3*(z) and x(-) thus constructed are independent of p. because so is the function p( ·). Then, the result of the Lemma is a direct consequence of the hypothesis (11.54). <1 The use of this result in the problem of robustly stabilizing a system of the form (11.39) (or, more in general, a system of the form (11.38)), is straightforward. Suppose the top subsystem of (11.39) satisfies the hypothesis of the Lemma, and consider the change of variable

v

= [/3*(z)t

1

[~- v*(z)].

·'

This yields a system modeled by equations of the form

i i.J

=

f(z, v*(z) + /3*(z)v, p.) ~(z, v, p.) + b(z, v, p.)u

(11.55)

in which the upper subsystem satisfies an hypothesis which exactly corresponds to the hypothesis (i) of Lemma 11.4.1. In fact, in view of the result of Lemma 11.4.3, it can be asserted that this subsystem - viewed as a system' with state z and input v - is input-to-state stable, and in particular satisfies estimates of the form (11.40), with r{) = g:- 1 o a ox(·) independent of p..

106

11. Feedback Design for Robust Global Stability

> bo, then also the hypothesis corresponding to hypothesis (ii) of Lemma 11.4.1 holds, because

If b( z, ~, f.J.)

b(z, v, J.L) = [,B*(z)r 1 b(z, v*(z)

+ ,B*(z)v, p)

and J3*(z) ~ 1. Thus, if also the remaining two technical hypotheses hold, a result corresponding to that of Lemma 11.4.1 follows. The results presented in this section are illustrated in the following simple examples. Example 11.4 .1. Consider a system modeled by equations of the form

= { = i

-az 2 sgn(z)

+ bz({ +e)

z+u,

(11.56)

in which a and b are parameters satisfying

lbl :S b0

•

The upper subsystem, which is a special case of the system considered in the Example 10.4.3, is input-to-state stable. In particular, from the conclusions of this Example, it can be deduced that the subsystem in question satisfies condition (11.40) of Lemma 11.4.1 for a gain function

in which 0 < e < a. Since the uncertain parameters a and b are supposed to vary within fixed bounds (and the former, in particular, is always positive) it is possible to choose e and to find a number k > 0 such that

J!i_ < k a-e

for all admissible values of a and b, and therefore condition (11.40) of Lemma 11.4.1 holds for a gain function

7(r)

= k(r + r 2 ) ,

which is independent of (a, b). In other words, the hypothesis (i) of Lemma 11.4.1 is fulfilled. Condition (ii) of this Lemma trivially holds, and so does condition (iii), which reads as max{izl, 1{1} :S max{Po(IW,Pt0zl)}, for po(r) = PI(r) = r. Finally, also condition (iv) holds, since the function Pib(r)) = 7(r) is locally Lipschitz at r = 0. Following the proof of Lemma 11.4.1, choose a class K00 function g(·) such that the (small gain) condition

11.4 Stabilization of Systems in Lower Triangular Form 'Yo g(r)


for all r

107

>0

or, equivalently, the condition for all r > 0 hold. For example, choose

Suppose, without loss of generality, that 2k > 1, so that 2k(r + r 2 ) ~ r

and define

~f ~ 0 If € ~ 0 .

e

a(€) == { 3k(€ +e)

-a( -€)

-

By construction, this function satisfies the condition (11.46). Therefor:e,' in view of the result of the Lemma, it can be concluded that the control law

renders system {11.56) input-to-state stable.

<1

Example 11.4.2. Consider a system modeled by equations of the form

i

== az + b(l + z 2 )€

{11.57)

ez + u'

~

in which a and b are parameters satisfying

The upper subsystem is unstable, and therefore the first step in the design of a stabilizing feedback for (11.57) consists in finding a control law == v*(z) rendering this subsystem robustly globally asymptotically stable. Clearly, a law of the form v*(z) = -kz, if k > 0 is large enough, achieves this goal. In fact, this yields a system of the form

e

Z ==•-C!Zin which, if k

c3z

3

> 0 is large enough, c1

= bk- a,

C3

= bk

are both positive, for any value of a and b. The second step in the design consists in changing the state variable

v

= ~- v*(z)

which transforms the first subsystem in a system of the form

~

in

108

11. Feedback Design for Robust Global Stability Z=

-CIZ-

c3 z 3

+ b(1 + z 2 )v.

In order to be able to proceed with the design, this system must be inputto-state stable, which however is not obvious, because of the presence of the term (I+ z 2 ) multiplying the input v. Nevertheless, it is easy to render this system input-to-state stable, changing v into f3*(z)v, with

f3*(z)

= (1 ~ z2)

.

In other words, consider for (11.57) the change of variable

v

= [fJ*(z))-

1

[~- v*(z)]

= (1 + z

2

)(~

+ kz),

(11.58)

which yields a system of the form i

v =

-c 1 z- _c3z 3 + bv
(11.59)

in which c1 , c2 have the form indicated above, and


+ z2 )ki- kz 2 + ~2 2zz. 1+z

In this system, the upper subsystem is now input-to-state stable, with a quadratic ISS-Lyapunov function V(z) = z 2 and a gain function which is bounded by a linear function independent of a and b. In fact, routine ' calculations show that V ~ -2c1 z 2 + 2bzv from which it is deduced that

(with 0

< c < c1 ) if

izi ~ _lbl_lvl. C1-

c

Hence, the gain of the subsystem is bounded by a function of the form 1'(r) = dr where d is any number satisfying d>

_l_bi_ CJ-

0

c

From this point, the design proceeds as in the previous example. Observe that, in system (11.59), hypothesis (i) of Lemma 11.4.1 holds and so does hypothesis (ii). To check hypothesis (iii), observe that I
11.5 Design for Multi-Input Systems

109

Koo functions Po(·) and PI(·) for which hypothesis (iii) holds. Finally, also condition (iv) holds, since the function p1 (1'(r)) = p 1 (dr) is smooth at r = 0. Following the proof of the Lemma, choose a class Koo function g( ·) such that the (small gain) condition

>0

for all r holds. For example, choose

Then, choose a function a{) so as to satisfy the condition (11.46). In view of the result of the Lemma, it can be concluded that the control law u=-v-a(v) robustly globally asymptotically stabilizes system (11.59). Reversing the change of variable (11.58), we obtain a control law u

= -(1 + z2 )({ + kz)- o((1 + z

2

)(~

+ kz))

which robustly globally asymptotically stabilizes system (11.57).


11.5 Design for Multi-Input Systems The purpose of this section is to study how the stabilization procedures described in section 9.2 and in this Chapter can be implemented in the case of systems having m > 1 inputs. Of course, the point of departure is the study of conditions under which global normal forms similar to those discussed in section 9.1 exist for the system under consideration. For convenience, we skip the intermediate stage of multi variable systems having vector relative degree, i.e. the case of systems which, by means of suitable globally defined diffeomorphisms, can be changed into systems described by equations having the normal form studied in section 5.1, and we address directly the more general problem of the existence of a globally defined diffeomorphism yielding normal forms of the type introduced in section 6.1. The case of systems having vector relative degree will be briefly discu~sed later, as a special case. The existence of the normal forms in question relies upon the algorithm, described in section 6.1 in coordinate free terms as well as in the actual system coordinates, for the characterization of the zero dynamics of a multivariable nonlinear system. We begin by summarizing the basic steps of this algorithm, and we take this opportunity to strengthen the various "regularity" hypotheses already considered in section 6.1, so as to prepare ourselves, to the subsequent derivation of globally defined normal forms. Consider a nonlinear system described by equations of the form

110

11. Feedback Design for Robust Global Stability

x

=

=

y

f(x) h(x)

+ g(x)u

(11.60)

with the same number m of input and output components and state x defined in IRn. Assume that f(x) and them columns of g(x) are smooth vector fields, that the m entries of h(x) are smooth functions, and that /(0) = 0 and h(O) = 0. Step 0. Suppose that dh has constant rank, say s 0 , for all x and that possibly after reorder - the first so rows of dh are linearly independent at each x. Define H 0 (x) to be the vector consisting of the first so rows of h(x). Step 1. Suppose that L 9 Ho(x) has constant rank, say r 0 , for all x and thatpossibly after reorder- the first r 0 rows of L 9 H 0 (x) are linearly independent at each x. Then, there exists a unique (so- r 0 ) x r0 matrix Ro(x) of smooth functions of x, such that

(Ro(x) for all x. Set

Ro(x)

I)L 9 Ho(x)

= (Ro(x)

=0

I)

and define !1> 0 (x) to be the vector

!l>o(x)

= Ro(x)L,Ho(x) :

Suppose that co1(dH0 , d!li0 ) has constant rank, say s 0 + s 1 , for all x and that - possibly after reorder in the rows of d!l>0 - the first so + s1 rows of co1(dH0 , d!li0 ) are linearly independent at each x. Define H1(x) to be the vector consisting of the first so+ s 1 rows of col(Ho, !lio). Step 2. Suppose that L 9 H 1 (x) has constant rank, say r 1 , for all x and that -possibly after reorder in the last s 1 rows of L 9 H 1 (x) -the r 1 x m matrix consisting of the first r 0 rows of L 9 Ho(x) and of the first r1 - ro rows of L 9 !1>0 (x) has rank r 1 at each x. Then, there exists a unique (s 1 - r1 + r 0) x (r 1 - ro) matrix Qo(x) and a (s1 - r1 + ro) x so matrix Po(x) of smooth functions of x, such that

Ro(x) ( Po(x)

0

( Qo(x)

I)

)

LgH1 (x)

=0

for all x. Set

R 1(x)

Ro(x)

= ( Po(x)

0

(t;;?o(x)

I)

)

'

and define !li1(x) to be the vector

Suppose that col( dH1 , d!li 1) has constant rank, say so + s1 + 82, for all x and that - possibly after reorder in the rows of d!li1 -the first so+ 81 + s2 rows

11.5 Design for

Multi~Input

Systems

Ill

of col(dH1 , d~ 1 ) are linearly independent at each x. Define H 2 (x) to be the vector consisting of the first so + s1 + s2 rows of col(H1 , 41 1). Step k + 1 ;::: 3. Suppose that L 9 H~;(x) has constant rank, say rk! for all x and that- possibly after reorder in the last s~; rows of L 9 H~;(:i) -the r~; x m matrix consisting of the first r 0 rows of L 9 Ho(x), of the first r 1 - r 0 rows of L 9 41 0 (x), ... , and of the first r~;- r~o-1 rows of L 9 41k-1 (x) has rank r~; at each x. Then, there exists a unique (s~o- r~; + r~o-d x (r~o- rk-1) matrix Q~;_ 1 -(x) and a (sk- r~; + r~o-d x (so+···+ s~o-d matrix Pk-1 (x) of smooth functions of x, such that

for all x. Set

and define

R~;(x)

~k(x)

= (Rk-1(x)

Pk-1 (x)

to be the vector

41~;(x)

= (Pk-1(x)

( 0k-1(x)

I)) LtH~o(x).

Suppose that col(dH~;, d
< n.

it can be deduced (see again section 6.1) that so = m, s 1 =so- ro, and for all k ;::: 1, s~o+ 1 = s~; - r~; + r~o- 1 , and this, in turn, implies that

Ho(x)

= h(x)

and Moreover, the set

·'

is a smooth embedded submanifold of JRn, of codimension

The unique solution u*(x) of

112

11. Feedback Design for Robust Global Stability

renders Z* invariant with respect to the vector field f*(x)

= f(x)+g(x)u*(x).

As in section 6.1, define T 0 (x) = h(x) and Tk(x) to be them x 1 vector in which the first m- sk elements are zero, and the last sk elements are exactly those of PA:- 1 (x), for 1 :=; k :=; k*. Let T(x) be them x (k* + 1) matrix defined by T(x)

= { To(x)

T1 (x)

· · · Tk· (x) ) .

Let ni denote the number of nonzero entries in the i-th row of matrix T(x), and observe (see again section 6.1) that n 1 :=; n 2 :=; · · · :=; nm. Moreover

+ n2 + · · · + nm

n1

= so

+ s1 + · · · + Sk•

= r .

Let {~ (x) be the entry found at the i-th row and k- th column of the matrix T(x), for 1 :=; k :=; n;, and 1 :=; i :=; m. Set ai(x) bi(x)

for i

=

L9{~,(x)

=

L 1 {~, (x) ,

= 1, ... , m. Lengthy, but simple, manipulations show that L 9 {t{x)

L,{t(x)

for 1 :=; k :=; n 1

-

0 =

{l+ 1 (x)

(11.61)

1, and that there,are functions 8l,i(x) such that i-1

L 9 {~(x) = L 8L(x)ai (x)

(11.62)

j=1

and

i-1

Lt{i(x) = L8L(x)bi(x)

+{l+ 1 (x)

(11.63)

j=1

for 1 :=; k :=; ni - 1 , 2 :=; i :=; m. Finally, define

and observe that, by construction, the matrix A(x) is invertible for all x. Using the matrix A(x) and the vector b(x) thus defined, set ](x) g(x)

f(x)- g(x)A- 1 (x)b(x) g(x)A- 1 (x),

11.5 Design for Multi-Input Systems

113

and for 1

~

k

~

nm, and

lj1 . = (x)

"'"'I m

9;(x) -

n,

L...- L...-c5n,-i+I,;(x)Y,i (x), l=j+1i=2

= (-1)k- 1 adt 1 l?(x)

ljk(x)

for 1 ~ j ~ m- 1, 1 < k ~ n;. Then, the following result holds, which is the global version of Proposition 6.1.5. Proposition 11.5.1. Suppose that the vector fields Yf(x) ,

1 ~ j ~ m,

1 ~ k ~ n;

are complete. Then Z* is connected and there is a globally defined diffeomorphism l]i' : JR.R --..+ Z* X ar ' which changes system {11.60} into a system described by equations of the form

z fo(z, (e, ... , ~m)) + go(z, (~ d = {J ~1n,-1 ~In,

= =

~~I

=

{~ 2

=

{~

~l ~2n2-1 "2

~n2

~f

, ... , ~m))u

+ ai(x)u {~ + c5l1 (x)W(x) + a1(x)u)

bi(x)

+ c5;2 _ 1,1 (x)W(x) + a1(x)u) b2 (x) + a2 (x)u ·

(11.64)

i I · · · + 2::;: 1 c5~j(x)(b'(x) + a1(x)u)

i

~~;-1

t'n;

1

i-1 i . j + Lj=l c5n;-l,j(x)(b1(x) +a (x)u) bi(x) + ai(x)u

{n;

=

with Yi = ~f

fori= 1, ... ,m, where

e = (~L ... '~~J, and

X= l]i'-

1

(z, (e, ... ,~m)).

114

11. Feedback Design for Robust Global Stability

Remark 11.5.1. Note, comparing with the local normal form exhibited in Proposition 6.1.5, that the terms cri(x)u are missing. This is a consequence of the fact that, in the present situation, the ranks of the matrices L 9 Hk(x) were assumed to be constant for all x E !Rn and not just on the sets Mf, as in section 6.1. <J

J and g and from equations (11.61)

Proof. By definition of the vector fields and (11.62) we have that

1 :S i :S m, 1 :S k :S n; - 1 , 1:Si:Sm,

L J{i(x) Lf{~; (x)

and Lg{~(x)

= (15i, 1 (x) 15i,2 (x)

Lg{~; (x) Lg{Hx)

=

(Olx(i-1)

=

01xm

Set r,b~(x)

1 Olx(m-ij]

= Lj- 1 h;(x), for

(dL/r,b(x) , [!, gJ(x))

· · · 15L_ 1 (x)

Olx(m-i+l)]

'

2 :S i :S m 1 :S k :S n; - 1 , 1:Si:Sm, 1 :S k :S n; - 1 .

1 :S i :S m, 1 :S k :S n;. Then the identity

= L1 (dL/r,b(x)

, g(x))- ( dLj+lr,b(x) , g(x)) ,

the above equations and the definition of the vector fields ljk , 1 :S j :S m, 1 :S k :S nj, yield for i = j and s + k otherwise,

= ni + 1

(11.65)

for 1 :S i,j :S m, and 1 :S k :S ni, 1 :S s :S n;. Let ~ (X) deriote the flow of the vector field y, and define the map

r

~:Z*xlRr-tiRn

by

where (.i maps.

= ({f, ... , {~;).~is clearly smooth as it is the composition of smooth

Let x E !Rn and let q =

yk ~l;

(x). Thus, l

r,bf(q)- r,bi(x)

1 1

8

y_k

o_ 8t r,bf(~t' (x))dt

t Ly•r,bi(P;/ (x))dt

0

'

11.5 Design for Multi-Input Systems

115

Therefore, from equation (11.65), for i = j and s + k = n; + 1 otherwise, and ¢f(q)

={

for i = j and s + k otherwise.

t +
= n; + 1

Now, let ynm

t,O(X)

y,2

= ifJ -';~(x) o · · · o ifJ _ ¢; 2

y,2

2 _ 1 (z)

yl

yl

o ifJ -'¢~'-'(z) o ifJ -';;:."'(x) o · · · o ifJ -'.p~'(x)(x)

and observe that
= -
for 1:::; i:::; m, 1:::; s:::; n;. But this implies that t,O(x) E Z*, as
x

1--t ( t,O(x),

= ~f_(x)

(
maps r into Z* EZ*x!Rr,

X

e' e

llr" I and is clearly smooth. Further I for (z' (

= ~tn;+l-(n;+l-s)) -


I ••• I

~m))

= ~!

which implies, I[! o ifJ(z,(e,e, ... ,~m)) = (z,(e,e, ... ,~m)). Also, ifJ o r[l(x) = x by construction. Therefore, I[! = ifJ- 1 and since both P and I[! are smooth mappings and as !Rn is connected, it can be concluded that Z* is connected and I[! is a (globally defined) diffeomorphism. 'l'he special structure of the normal form (11.64) derives, as in section 6.1, from the special choice of the coordinates ~~ (x).
1 :::; j :::; m,

commute, i.e.

{Y/ l'~/1 I

=0

1 :::; k:::; ni

11. Feedback Design for Robust Global Stability

116

for 1:::; i,j:::; m, 1:::; s:::; ni, 1:::; k:::; nj. Then (11.60) is globally diffeomorphic to a system having the normal form (11.64) but with special structure

= fo(z, (~},a, ... , ~i)).

i

Moreover, the zero dynamics of the system, in the new coordinates, are described by i = f*(z) = fo(z, (0, 0, ... , 0)). Proof. First note that,

Hence,

by the properties of flows. Now write,

1

1 ) _...Y1 ) * 0 · · · 0 (_...Y/:.. ( ':1'~~1 ~{~~~~

(_...Y,'_, ~-(~~~~

v-1 *J.i

_...Y,'

0···0~-~!1

em)))) ( (C1 ("" ~ z, ~ , ... ,.,

for 1 :::; i :::; m. Recall that, if the vector fields T and X commute, function v(t) = (P~t)*X o
a


= Yi

1

o
for 1 :::; i :::; m, or in general,

P (~) * a~:.

= yn;-k+1

,

for 1 :::; i :::; m and 1 :::; k :::; ni. ~m)))D
(e, e, ... ,

D!li(
···

0

0

1

P

,

(e, e, ... ,f"))

(e, e, ... ,~m)))Yin;-k+ = (0

0

1

o
= I, it is clear

e, ... ,~m))

O)T,

where the element 1 appears in the (n- r + (n 1 + n2 + · · · + ni-d + k)-th position. But by construction,

9m(x) and

= Y~(x),

11.5 Design for Multi-Input Systems n1

m

L

gj(x) = Y/(x) +

117

l=j+1

L6~,-i+ 1 ,j(x)Yji(x), i=2

for 1 S j S m - 1. Therefore, (Dlli)go~ is a matrix in which the first n-r rows are identically zero and so (Dlli)g o ~

= (Dlli)gA o ~

is a matrix in which the first n - r rows are identically zero, i.e. in the new coordinates the first n - r equations are independent of the input u. Finally, by construction,

~ (~) = yn;-k+1 o ~ = (-1)n•-kad -Y} *

'

a~:,

1 ,

for 1 SiS m, and 1 S k S n;. Let

f = ~.; j o ~, 1

and let

Y;k

= ~;

Thus,

m

-

1

2

f=fo(z,(~ '~ , ... ,~

m))

and yn;-k+l

,

1

J!ik o ~.

n·

a "" ~ i a az + ~~~ka~L1,

= ~. a~k

Therefore,

ad-Y. 1

I But, as

~

m

= _ aJo(z,(~\e, ...• ~m)) a a~:m az

- 1

which implies

fo

-1 (

2

implying

)

a

-Ym o ~ =- acm

~n1'11.-l

aJo(z,(e.e, ...• ~m)) =O a,:m ,

is independent of ~!!!.. . So now,

But,

a a~!!!..- 1

is a morphism of Lie brackets, ad/Ym = ~.

i.e.

_

'

.

118

11. Feedback Design for Robust Global Stability

i.e. fo is independent of f~_ 1 . Proceeding inductively in this way, it can be shown that fa is independent of all f;.,, ... , ~L ~; 2 , ••• , ~~, ••• , ~~, ••• , ~f". i.e. fo is properly denoted fo(z, (~L ~?, ... ,~I")).
=

~t

~~

~1n,-1

~:..

t1n, €?

"2

{n2-1 {2n2

b1 (x)+a1 (x)u

~~

=

~~2 b2 (x)

d = €~;-1 {~.

fo(z,{(t,{?, ... ,~r))

(11.66)

+ a2 (x)u

{~

~~; bi(x)

+ ai(x)u

with Yi

= ~t

for i = 1, ... , m, in which the upper subsystem is driven only by Indeed, the feedback law u =A - 1 (x)( -b(x)

+ v)

{f, ... , ~I".

11.5 Design for Multi-Input Systems

119

changes system (11.66) into a new system decomposed into a set of m "chains of integrators" (where m is the number of input/output channels) that drive the (n- r)-dimensional subsystem

i=fo(z,(yl,···,Ym)), whose autonomous dynamics are precisely the zero dynamics of the original system. An immediate byproduct of such a decomposition is that the various stabilization techniques described in section 9.1 and in this Chapter can be implemented. Thus, for instance, if there exist smooth functions vi(z), ... ,v~(z), with vi(O) = ··· = v~(O) = 0, which globally asymptotically stabilize the equilibrium z = 0 of

i = fo(z, (vi(z), ... , v~(z))) , it is possible, "stepping-back" through them chains of integrators, to find a feedback law u*(x) which globally asymptotically stabilizes the equilibrium x = 0 of the full system. Example 11.5.1. As a simple illustration, consider (see Fig. 11.2) a system with two inputs, u 1 and u2, and two outputs, y 1 = {} and y2 = {~, described by equations of the form

i

=

j(z,y1,y2)

iJ1

V1

iJ2

{i

{i

V2.

(11.67)

Suppose there exist functions vi(z), v2(z), with vi(O) = v2(0) = 0, which globally asymptotically stabilize the equilibrium z = 0 of

i

= f(z, vt(z), v;(z)) .

Fig. 11.2. Two-input system having vector relative degree.

In order to find a globally stabilizing feedback for (11.67), construct first,· using Lemma 9.2.2, a feedback law ui(z,yl) which globally asymptotically stabilizes the equilibrium (z, yl) = (0, 0) of

120

11. Feedback Design for Robust Global Stability

z =

f(z,y~>vi(z))

iJ1

uj(z,y1).

.

Then, look at the system

z

f(z,yl,Y2) uj(z,y1)

iJ1 il2 ~i

{i

(11.68)

Vz'

a.s to a system in which a chain of two integrators drives, through y 2 , the subsystem

z

f(z,yl,yz) uj(z, Yl) ,

ih

which, by hypothesis, is globally asymptotically stabilized by y2 = vi(z). Using Theorem 9.2.3, it is possible to find, for (11.68), a globally asymptotically stabilizing feedback vz = u~(z, y1, Yz, {i), and this completes the design procedure. <1 The technique of backstepping, however, cannot a.s easily be implemented if the system does not have a vector relative degree, i.e. is not globally diffeomorphic to a system of the form (11.66), but- rather- is diffeomorphic to a system in the more general normal form considered by Proposition 11.5.2, i.e. form (11.64) with the additional constraint that

z=

fo(z, ({f, {r, ... , {f')) .

As a matter of fact, extra hypotheses are needed, a.s illustrated first in the following simple example.

Example 11.5.2. Consider a system (see Fig. 11.3) with two inputs, u 1 and u 2 , and two outputs, Y1 = {f and Y2 = {r, described by equations of the form

z

f(z,yl,Yz)

ih

v1

iJ2

{i

~~

v2.

+ a(z, Y1, Yz, {i)vl

(11.69)

Clearly, if a(z,y 1,yz,{i) =f. 0, the system in question does not have vector relative degree as the input u 1 appears "too soon" in the derivative of the output Y2 (see Example 5.4.1}. Hence, it is not immediately evident how to "step-back" a feedback law designed for z = j(z,y1,y2} all the way to the actual inputs u 1 , u 2 • As a matter offact, a normal form like {11.69} does not, in general, lend itself to the possibility of implementing a backstepping design procedure. However, there is a special situation in which this is possible.

ll.S Design for Multi-Input Syst~ms

121

Fig. 11.3. Two-input system not having vector relative degree.

Suppose

= o(z,y1,y2).

8(z,y1,y2,~~)

As in the previous example, suppose there exists functions vi(z), v2(z), with vi(O) = v:i(O) = 0, which globally asymptotically stabilize the equilibrium z = 0 of i = f(z,vt(z),v:i(z)). Backstepping once from a feedback y 1

= vt(z)

stabilizing the system

one can determine, as in the previous example, a feedback law u1(z,yt) stabilizing the system i = f(z,yi,v:i(z))

Y1

=

ui(z,yl).

Setting v1 = ut(z, yi) yields a system i

iJ1 i12

{~

= = = =

f(z, Y1, Y2) ui(z,yi) ~~ + J(z, YI, Y2)ui(z, yi) v2,

which can be regarded as a subsystem i

Y1

= ·' =

f(z,yi,Y2) ui(z, YI) ,

by hypothesis globally asymptotically stabilized by y2 = v:i(z), driven by

i/2 i~

= =

~~+8(z,yl,Y2)ui(z,yi) V2 ·

The latter is no longer a chain of two integrators (as in the previous example), but has anyway the lower-triangular structure that renders backstepping possible. Thus, in two iterations, from y2 = vi (z), it is possible to find a feedback law v2 = u;(z,y1,Y2,~n which stabilizes the system

122

11. Feedback Design for Robust Global Stability

f(z, YI, Y2) ur(z,y!) {~ + 6(z, YI. Y2)uj(z, yi)

i

ih iJ2 i~

u~(z, YI• Y2,

and this completes the design.

{n ,

<1

In general, if a system satisfies the hypotheses of Proposition 11.5.2, and hence admits a normal form of the type (11.64) with the additional property that the upper subsystem is driven only by €t, ... , €1", i.e. by YI, ... , Ym. a preliminary control law

u

= A-I(x)(-b(x) +v)

yields a normal form with the following structure i

=

n,-I

=

e

~In,

~l

fo(z,yi, · .,. ,ym) {~, VI

=

{~

+ 6li (x)vi

"2

e.2

€n2-I

~2n2

v2

~f

{i

{~.-I {~.

2

~i

+ ~2-I,I (x)vl

6i ( ) + z::i-I j=I Ij X Vj

=

'>n;

=

Vi

" i - I >i ( ) + L..Jj=I un;-I,j X Vj

with fori= 1, ... ,m.

In this system, it is possible to "step-back" feedback laws

YI

= vt(z),

... , Ym

= v:;.(z)

to feedback laws VI

= ut(z,~f, ... ,~f',e)

V2

= u;(z,~L ...•

aa .e .e)

11.5 Design for Multi-Input Systems if the various coefficients

aL

123

exhibit a "triangular" dependence on state

e::

oL

variables. That is, if the function depends only on variables belonging to the gtoup with superscript i - 1 and smaller (lp ::;: i - 1), or on variables having superscript i but with subscript that is less than or equal to k (lp = i & lb ::;: k), or on any of the "leading" variables in a group, namely any of the variables (1 ::;: lp ::;: m), i.e.

eip aL (z,eL ... ,ei ,e .ei> a~,l(z,{f, ... ,

er.e, a.e~)

el em tl 0ril,j ·( z,'>l> · · · >'>1 >'> '· · · tl tm tl 0ri2,j ( z,.,1, ... ,.,1 ,., , ...

ri

(

el

em el

ti-l ei) ,., >'>1 ti-l ti ti) ,., >'>1>'>2

ti-l ti ei

uni-l,j z,~1, ... ,~1 ,~ , ... ,~

'~l,\2·,

ti ) ... ,«.:,ni-l ·

It is easy to observe that if this type of "triangularity" property holds then the backstepping design methods can be directly applied to the system in normal form. The backstepping commences with the subsystem

z=

fo(z, (vt(z), v;(z), ... , v:;.(z))),

first backstepping n 1 times through the first group of variables (variables with the superscript I) to obtain the feedback law

ut(z,{i,v5(z), ... ,v:;.(z),e), then backstepping n 2 times through the second group of variables (variables with the superscript 2) to obtain the feedback law

u;(z,{i,{l,v3(z), ... ,v:;.(z),e,e>, continuing group by group in the numbered order 1 through m, backstepping times through group i (variables with the superscript i) to discover the feedback law • n;

u;(z,

eL a, ... ,eL v:+1 (z), ... , v:;.(z),e .e, ... ,€'> .

While considering group i for backstepping the leading variables {~+1, ... , {f' are of course fixed to the functions v;+l (z), ... , v:;. (z) respectively. It is simple to check whether the special dependence of the functions illustrated above holds. This dependence can be expressed by the geometric conditions

JL

124

11. Feedback Design for Robust Global Stability

= i and k < s. These conditipns mandat.e that ~m))) = JL(z, ~:I ~r e ~i-l, ~L {;, ... I~~)

for l > i, s -j:. 1 and for l JL(4>(z, (~\ ... I

••• I

I

I ••• I

for j = 1, ... , i -1, where the Yinz-s+l are the vector fields introduced earlier in this SeCtion, and 4>(z, 1 . , , I ~m)) iS the global diffeomorphism COnsidered in Proposition 11.5.2. Note that the condition thus found is coordinate free, for the functions JL were found from the zero dynamics algorithm.

ce

12. Feedback Design for Robust Semiglobal Stability

12.1 Achieving Semiglobal and Practical Stability In section 9.3 we have introduced the concept of semiglobal stabilizability, and we have shown (Theorem 9.3.1) how, using a linear feedback, it is possible to stabilize in a semiglobal sense (i.e. imposing that the domain of attraction of the equilibrium contains a prescribed compact set) a system of the form (9.23), under the hypothesis that the equilibrium z = 0 of its zero dynamics is globally a.symptotically stable. In this section, in preparation to the subsequent study of the problem of robust semiglobal stabilization using output feedback, we extend the result of Theorem 9.3.1 to the case of a system modeled by equations of the form

fo(z,6)

i

(12.1) (r

q(z,6, ... ,(r,J-L) + b(z,(l, ... ,(r,J-L)U,

in which z E !Rn, (i E lR, for i = 1,.,., r, u E IR and J-L E P C JRP is a vector of unknown parameters, ranging over a compact set P. As in Theorem 9.3.1, we assume that (i)

f 0 (0, 0)

= 0 and the equilibrium z = 0 of the subsystem

i

= fo(z, 0)

is globally asymptotically stable.

·'

Moreover, we also assume that (ii) for some b0

> 0,

Note that we do not make any specific assumption on q( z, 6, ... , (r, J-L), other than the standard hypothesis of smoothness. Thus, q(O, 0, ... , 0, J-L) is' not required to be 0 for all J-L, which means that (z, 6, ... , (r) = (0, 0, ... , 0) is not necessarily an equilibrium point of (12.1) for u = 0.

126

12. Feedback Design for Robust Semiglobal Stability

The notion of semiglobal stabilizability essentially requires that, for any given compact set, there exists a feedback law such that, in the corresponding closed-loop system, each trajectory with initial condition in this set asymptotically converges to the prescribed equilibrium. But, in the case of an uncertain system such as system (12.1), in which the equilibrium point may depend on the value of some unknown parameter, this notion has to be weakened, since it may no longer be possible that all trajectories converge to the same point regardless of the value of the unknown parameter. Thus, the next best alternative is to ask that all trajectories converge to a prescribed set, which hopefully will be some (small) neighborhood of a prescribed point, for instance the point (z, 6, ... , ~r) = (0, 0, ... , 0). To formalize this concept, the following terminology is often used: a trajectory is said to be captured by the set Q if it is defined for all t E [0, oo), enters the set Q at some finite time T and remains in this set for all t ~ T. ·A system is said to be semiglobally practically stabilizable if, for any (arbitrarily large) set K and any arbitrarily small set Q, there is a feedback law, which in general depends on both K and Q, such that any trajectory with initial condition in K is captured by the set Q. In what follows we will show that system (12.1) is semiglobally practically stabilizable, by means of a linear feedback law. For convenience, let Q~ denote the closed cube in ~n, centered at the origin, of side 2c, namely Q~::::;

{x E ~n: /x;/ S c,i = 1, ... ,n}.

Theorem 12.1.1. Consider system (12.1} and assume hypotheses (i) and (ii) hold. Given any arbitrarily large number R > 0 and any arbitrarily small number c > 0, there exists a feedback law of the form

(12.2) (in which the numbers k 1 , ... , kr depend on the choice of R and c) such that, in the closed loop system (12.1} - (12.2}, any initial condition in Q'lt+r produces a trajectory which is captured by the set Q;:+r. Proof. Let a 0 , a 1 , ... , ar_ 2 be such that the polynomial

Ar-t + ar-2Ar- 2 + · · · +at A +

ao

has all roots with negative real part, and set

= col(z,6, .. · ,~r-d, ~r + kr-lao6 + e- 2 at6 + ... + kar-2~r-1 x

( =

.

where k > 0 is a number to be determined later. Rewrite system (12.1) in the form

x (

=

F(x)+G( ij(x, (, /1)

+ b(x, (, J1)u

(12.3)

12.1 Achieving Semiglobal and Practical Stability

127

where

and q(x,(,f-t)

b(x, (, f-t)

=

q(z,6,. ·· ,~r-1,(- kr- 1ao6- ... - kar-2~r-1, f-t)

=

+ kr- 1 ao6 + · · · + k 2 ar-3~r-1 + kar-2((- kr-lllo~I - kr- 2 a16 - · · · - kar-2~r-d b(z, {I, ... , ~r-1, ( - k"- 1 ao6 - ... - kar-2~r-I, f-t) .

It is known from Theorem 9.3.1 that, given any R > 0, there is a number. k• > 0 such that, if k ;:::: k•, the equilibrium x = 0 of the system

x = F(x)

(12.4)

is locally asymptotically stable and, moreover, any initial condition X 0 E Q'R+r-I produces a trajectory which converges to this equilibrium as t-+ oo. In other words, Q'R+r-l is a subset of the domain of attraction A of the equilibrium x = 0 of this system. Pick, and fix, any k larger than or equal to k*. Recall that A is an open set and that there is a diffeomorphism,

cJ

: A X

-+ JRn+r-1 1--? X

with cJ(O) = 0. This diffeomorphism transform the restriction of system (12.4) to A into a system i = F(x) = cJ.F(cJ- 1 (x)), defined on JRn+r-I, whose equilibrium x = 0 is globally asymptotically stable. Hence, by the converse Lyapunov Theorem, there exists a smooth function V(x), which is positive definite and proper, such that

av. ox F(x) < o for all nonzero x. Now observe that the image Qof the set Q'R+r-I C A under the mapping cJ is a compact set. Therefore, there exists a number c > 1 such ili~ . V(x) ~c.

Consider now the smooth function, defined on A,

V(x) = V(cJ(x)) ,

128

12. Feedback Design for Robust Semiglobal Stability

and let L\c denote the set L\c

= {x E JRn+r-l

: V(x) ~ c}.

Clearly, by construction, -n+r-1

QR and

A

8V ax F(x) < 0,

Set

A

C "-'c C "-'c+l C

for all

X

A ,

E Llc+I,X

=f. 0.

e= (1 + e- 1 ao + ... + kar-2)

(recall that k and all a;'s are positive numbers) and

d = f 2 R2

•

Assume, without loss of generality that R > 1 so that d > 1, and note that

= 1, ... ,r

=> 1(1 2 ~ d ICI~b,l~;j~o for all i=1, ... ,r-1 => l~rl~fo~Jdo. 1~;1 ~ R for all i

(12.5)

Now, consider the function

cV(x) d( 2 W(x, () = c + 1- V(x) + d + 1- ( 2 which is defined in the interior of the set positive definite. Set, as usual

L\c+l

(12.6)

x { ( E lR : ( 2 ~ d + 1} and

ila = {(x,() E JRn+r: W(x,() ~a}. Observe that, if W(x, () ~ c2 2+~+1

+ ~ + 1,

V(x) ~ (c + 1) c2 + d2 + 1 + c ,

( 2

< (d 1) 2 + tf + 1 + c2+d2+1+d'

(12.7)

which shows that

Note also that

V(x) ~ c and ( 2 ~ d

=>

Choose now u = -k( in (12.3) and compute the derivative of the function W(x, () along the trajectories of the vector field thus obtained, namely of

x = (

F(x) +G( q(x, (, p,) - b(x, (, p,)k( .

(12.9)

12.1 Achieving Semiglobal and Practical Stability

129

This yields W(x,()

=

c(c + 1) av (c + 1 - V(x))2 ax [F(x) + G(]

+

d(d + 1) (d + _ ( 2 )2 2([q(x, (, J.l.) - b(x, (, J.l.)k(] . 1

From (12. 7) we obtain

c c+1 d d+1

< <

c(c + 1) (c + 1- V(x))2 d(d + 1) (d + 1- ( 2)2

< <

(2 + ~ + 1 + c) 2 c(c + 1) (2 + ~ + 1 + d) 2 d(d + 1)

Thus, if (x, () E f]c,+d,+l, W(x, () :S

Now, choose any arbitrarily small number p > 0 and consider the compact sets S = {(x,():p:SW(x,():S2+~+1} So = S n {(x, () : ( = 0}. By construction, (x,() = (0,0) is not in S0 , because (x,() = (0,0) and (see (12.8))

~

S,

SoC {(x,(): V(x) < c+ 1,( = 0} C Llc+l x {0}.

Observe that, by hypothesis, if x E Llc+l and x

. W(x, 0)

c(c + 1)

= (c + 1- V(x))2

f:. 0,

av

ax F(x) < 0.

Thus, the function W(x, () is negative at each point of the compact set S 0 . By continuity, W(x, () is negative at each point of some open set U :::) So. Consider now the compact setS= S \ U. Since ( f: 0 at each point of S there exists m > 0 such that (2 > m' for all (x, () E S. Also, there exists M > 0 such that

2(2+~+1+d)21_( ( )I]ICI M [ (2+~+1+c)21avl c(c+1) ax+ d(d+1) qx, ,J.l. :s for all (x, () E S and all J.l. E P. Thus, at each (x, () E S

130

12. Feedback Design for Robust Semiglobal Stability

.

d

-

W (x, () ~ - 2bo d + km 1

+M

.

This, as in the Proof of Theorem 9.3.1, shows that there is a number W(x, () is negative at each point of S. Note that k* depends on the choice of p. Since t: ... ,.,r t: ) E Q-n+r __. X E Q-nR+r-1 and l~il ~ R for all 1 ~ i ~ r ( z,.,r, R __,.._ k*

> 0 such that, if k > k*, the function

'*

V(x) ~ c,

and

1(1 2 ~ d

we have that

(z,6, ... ,~r)

E QR_+r

i.e.

Q'Jt+r C

f7c2+d2 .

Thus, if the initial condition of (12.9) iS in Q'Jt+r, the corresponding trajectory remains in f7c2+d2+I for all t ~ 0, because W(x(t), ((t)) is negative at each point of the boundary of f7c2+d2+1. Moreover, it is possible to see that at some finite time T the trajectory enters the set f7p and remains in f7P for all t ~ T. For, suppose this is not true. Then, the previous argument shows that W(x(t), ((t)) is always decreasing and converges to a nonnegative limit W0 ~ p. Let denote the w-limit set of the trajectory in question, and note that C S. It is known that W(x, () = W0 at each point of this set. Pick any initial condition in rand observe that the function W(x, () is constant along the corresponding trajectory. Thus, W(x, () 0 along this trajectory, which is a contradiction, because it was supposed that was in S and W(x, () < 0 on S. Thus, (x(t), ((t)) must enter the set f7p in finite time and it can never leave this set afterwards, because W(x, () is negative at each point of its boundary. In summary, we have shown that, given any p > 0, there is a number k* such that, if k ~ k*, any initial condition in Q'Jt+r produces a trajectory which is captured by the set

r

r

=

f7P

= {(x,(): W(x,()

~

r

p}.

To complete the proof, it remains to show that, if p is small enough,

np c

(J~+r.

Now, observe that W(x,()~p

V(x)

~

(cc:1Jp,

and

1(12 ~ ((dd:~p).

Since V(x) is positive definite, there exists a class K function g:(-) such that

g:(llxll)

~ V(x) .

12.1 Achieving Semiglobal and Practical Stability

131

Thus, if pis small enough, so that (c + 1)pf(c + p) is in the image of Q.(·),

W(x, () :::; p

llxll :S Q.-l((cc:1Jp) and

1(12 :S

(dd: 1Jp).

Both estimates for llxll and 1(1 on the right-hand side converge to 0 as p -t 0. Thus, there exists p such that W(x, () :::; p implies a-l((c+ 1)p) < ...:..__ ll xll < -c+p - .,fd'

and

1(1 < ( (d + 1)p) ~ < ~ -

This, using the second of (12.5) and the fact that d i.e. W(x,() :S p

-..jd

d+p

> 1, shows that

J~rl:::; E,

and completes the proof. Note that the feedback law which solves the problem has the form U

= -k[~r + kr-Iao6 + kr- 2at6 + · · · + kar-2~r-I]

which is precisely a feedback law of the form (12.2).

(12:10)

<1

Remark 12.1.1. The result of this Theorem can be extended to cover the case in which the upper subsystem of (12.1) contains an uncertain parameter JJ, namely has the form i = fo(z,6,JJ)

so long as it is known that there exists a function V(z), independent of JJ, satisfying g(llzll) :S V(z) :S a(IJzJI)

av oz f(z, 0, JJ-) :S -a(llzJI) for all z and JJ-, where Q.(·),a(·),a(-) are class Koo functions.

<J

As observed in the introduction, there is no guarantee that the closedloop system (12.1) - (12.2) possesses an equilibrium at (z,6, ... ,~r) = (0, 0, ... , 0), simply because it was not assumed that q(z, 6, ... , ~r, p) is vanishing at (z, 6, ... , ~r) = (0, 0, ... , 0), for all JJ-. On the contrary, if

q(O, 0,: .. , 0, JJ)

=0 ,

(12.11)

the point (z, ~ 1 , ... , ~r) = (0, 0, ... , 0) is indeed an equilibrium of the closedloop system. H this is the case, it might well happen that the trajectories of the closed-loop system, which are guaranteed to enter in finite time an arbitrary small neighborhood of the origin (which is invariant in positive time), actually converge, as t -t oo, to the equilibrium in question. In other words, it may happen that the feedback law (12.2) renders the equilibrium (z, 6, ... , ~r) = (0, 0, ... , 0) locally asymptotically stable, with a region of attraction which contains the set Q'Jtr. In what follows, it will be shown

132

12. Feedback Design for Robust Semiglobal Stability

that, in order to have this occurring, it suffices to assume that the equilibrium z = 0 of i = fo(z, 0) is not simply globally asymptotically, but also locally exponentially stable, i.e. that the eigenvalues of the matrix

Fo _ [8fo] - 8z (o,o) are all in the left-half plane. Under this extra hypothesis, in fact, the feedback law (12.10), for large k and k is a feedback law which asymptotically stabilizes the linear approximation of (12.1) at the equilibrium (z, 6, ... , (r, u) = (0, 0, ... , 0, 0), as the reader may easily verify on the basis of simple calculations. As a consequence, the arguments of the previous proof can be adapted to show that the law in question actually solves a problem of semiglobal stabilization (and not just a problem of semiglobal and practical stabilization). Corollary 12.1.2. Consider system (12.1) and assume hypotheses (i) and (ii) hold. Assume also that all the eigenvalues of the matrix F0 above have negative real part and that (12.11) holds. Given any arbitrarily large number R > 0, there exists a feedback law of the form (12.2), in which the numbers k 1 , ... , kr depend on the choice of R, such that, in the closed loop system (12.1) - (12.2), the equilibrium (z, () = (0, 0) is locally exponentially stable and, moreover, any initial condition in (J"Jtr produces a trajectory which asymptotically converges to (0, 0) as t -+ oo.

Proof. Consider the closed-loop system (12.1)- (12.2), and observe that the Jacobian matrix of the vector field F(x) at x = 0 has a block triangular structure

{)F ( [8fo] 8z0
:)

(12.12)

in which A is a matrix whose eigenvalues have by hypothesis negative real part. As a consequence, in the hypothesis of the Corollary, all the eigenvalues of (12.12) have negative real part and the equilibrium x = 0 of x = F(x) is locally exponentially stable. The same property holds for the equilibrium x = 0 of the system

i

= F(x)

obtained by means of the diffeomorphism of Theorem 12.1.1. In fact, since

[ {)~] OX

i=O

(12.13)

x = 4-i(x)

= 4-i. (O) [{)F] OX

considered in the proof

4-i;I (O) ' z=O

12.1 Achieving Semiglobal and Practical Stability

133

and . (0) is nonsingular, all the eigenvalues of this matrix have negative real part. As a consequence, in view of Lemma 10.1.5, it can be asserted that there exists a function V (x) such that fr(llxll)

:s: V(x),

for all X E !Rn' where frO and a(·) are class Koo functions which, for some

s0 > 0, ii > 0 and b > 0 satisfy for all s E (0, so]. It is not difficult to check that similar properties hold also for the function V(x) = V(
iiii
where Q > 0 and S(x) is a function satisfying lim S(x)

llzll-+0 llxll 2

= 0.

As a consequence, it is readily seen that, for some a> 0 and some sufficiently small s 0 , 2 allxll :::; V(x) for all x such that llxll :S: s 0 • An identical argument shows that, for some b > 0,

for all x such that llxll :S: so. Consider now again the Lyapunov function W(x, () introduced in the proof of Theorem 12.1.1 and observe that, for all llxll :S: so

> _c_V(x) > _c_allxll2 cV(x) · - c+ 1 c + 1- V(x) - c + 1 Likewise, for all

1(1 2 < d + 1

134

12. Feedback Design for Robust Semiglobal Stability

d(2 > - -d( 2 d+1-( 2 - d + 1 .

....,..-~--,::-

Thus, there is a number a such that, for all (x, () whose norm does not exceed s0 , the function W(x, () is bounded from below as

iljj(x,()JI 2

::;

W(x,().

(12.14)

The derivative of W(x, () along the trajectories of the closed loop system satisfies, for all (x, () whose norm does not exceed s0 ,

where

M(x,(,J..L)

c( c + 1)

= [ (c+ 1- V(x))2

Since

av

d( d + 1) ] 8x G; 2 d+ 1- (2q(x,(,J..L) .

[~~a] (o) = o,

q(O,O,J..L)

=0

it follows that the function M(x, (, J..L), a smooth function of (x, (, J..L), can be expressed in the form 2L 1 (x, (, J..L)x

+ L 2 (x, (, J..L)(

where L 1 (x, (, J..L) and L 2(x, (, J..L) are smooth functions. As a consequence, for all (x, () whose norm does not exceed s0 , W(x, () can be bounded from above as

.

W(x,():S-

(x)T ( (

cbl -L 1 (x,(,J..L)

-LJ(x,(,J..L)

~~~k-L2(x,(,J..L)

)

(x) (

·

Since J..L ranges over a compact set, given any positive number b < cb there is a number k* such that, for all k ;:::: k* and all (x, () whose norm does not exceed so

i.e.

W(x,()::; -bJI(x,()W.

(12.15)

Inequalities (12.14) and (12.15) prove, by Lemma 10.1.5, that, if k is large enough, the equilibrium (x, () = (0, 0) of the closed loop system (12.1) - (12.2) is locally exponentially stable. In particular, observe that these inequalities hold for all (x, () whose norm does not exceed a number s0 which is independent of k. Let eo > 0 be such that

12.2 Semiglobal Stabilization via Partial State Feedback

135

Since W(x,() is negative at each point of the boundary of the set nCO! the latter (which only depends on the function W(x, () and C{), and does not depend on the actual value of k) is invariant for positive time. Moreover, since W(x,() ~ cii(x,()ll 2 for some c and all (x,() E B 90 , any initial condition in nco produces a trajectory along which

.

W(x(t),((t))

~

b

-::W(x(t),((t)) c

:=

-a0 W(x(t),((t)).

As a consequence, any of such trajectories satisfies

W(x(t), z(t))

~

e-aotw(x(O), ((0)) ,

ll(x(t), ((t))ll

~

;.e-aotw(x(O), ((0))

i.e. satisfies

a

for all t 2:: 0. In particular, any of such trajectories converges to (0, 0) as t -too. Now, let c be such that

Theorem 12.1.1 shows that, if k is large enough, any trajectory with initial condition in Q~+r enters in finite time the set Q~+r, hence in particular the set nco· Therefore, it can be concluded that any trajectory with initial condition in Q~+r converges to (0, 0) as t -t oo and this completes the proof of the Corollary.
12.2 Semiglobal Stabilization via Partial State Feedback The key argument of the proof of Theorem 12.1.1 in the previous section was . essentially the fact that, in a system of the form

F(x) + G( q(x, () - b(x, ()k( (with x E

m.n

(12.16)

and ( E JR), under the hypotheses that:

(i) the equilibrium x = 0 of x = F(x) is locally asymptotically stable and its domain of attraction A contains the set Q'R,

> 0, arbitrarily small c > 0,

(ii) b(x, () 2:: bo for some bo

there is a number k* such that, if k 2:: k*, given any all trajectories with initial condition in Q~+l are captured by the set ()~+ 1 . As observed after the Theorem, this does not necessarily guarantee asymptotic convergence of all such trajectories to the point (x, () = (0, 0). Even assuming that the point in question is an equilibrium point for the system (hence assuming that q(O, 0) = 0), additional hypotheses are needed to be

136

12. Feedback Design for Robust Semiglobal Stability

able to conclude that all trajectories with initial condition in Q'R+ 1 , eventually converge to the equilibrium (x, () = (0, 0). For instance, as shown in Corollary 12.1.2, this is the case if the equilibrium x = 0 of :i; = F(x) satisfies the extra hypothesis of being locally exponentially stable. However, the proof of the Corollary only uses this extra hypothesis in order to guarantee that the equilibrium (x, () = (0, 0) of (12.16) is locally asymptotically stable, with a region of attraction which contains a neighborhood of the equilibrium which does not depend on k (as a matter of fact, the property that this equilibrium is locally exponentially stable, which actually holds under the said hypothesis if k is large enough, was not needed as such in the conclusion of the proof of the Corollary). Thus, one may argue that a result of asymptotic stability with prescribed domain of attraction such as that of Corollary 12.1.2 might hold under hypotheses weaker than those considered in this Corollary. In this section we wish to elaborate further on this· subject, and discuss when, in a system with an interconnected structure such as that of system (12.16), the equilibrium (x, () = (0, 0) is locally asymptotically stable with a region of attraction that contains a prescribed set. The need of extra hypotheses and the fact that the extra hypotheses of Corollary 12.1.2 might be weakened can be better understood with the aid of two simple examples. Example 12.2.1. Consider the system

(

-x3

+(

X-

k(.

(12.17)

In this system, the equilibrium (x, () = (0, 0) cannot be locally asymptotically stable, no matter how k is chosen. In fact, its linear approximation

has a characteristic polynomial A2 + kA -1 with one root having positive real part for any k. Note that this is not in contrast with the result of Theorem 12.1.1. In fact, system (12.17), for k > 0, has three equilibria (x,()

(0,0)

(x, ()

(1/Vk, 1/#)

(x,()

(-1/Vk, -1/Jk3).

The former of these is unstable, but the other two are locally asymptotically stable and, for any c > 0, there is a number k* such that, if k > k*, both the stable equilibria are in Q~ (see Fig. 12.1). This example is perhaps the simplest explanation of why, in system (12.16), the asymptotic stability of the equilibrium of x = F(x) is not sufficient to guarantee, even for large k > 0, local asymptotic stability of the

12.2 Semiglobal Stabilization via Partial State Feedback

137

Fig. 12.1. Pha.<>e portrait of the system.

equilibrium (x, () = (0, 0). To find an example that the extra condition of Corollary 12.1.2 is sufficient but not necessary, it suffices to look at the system :i;

(

=

-x3

=

3

x

+(

(12.18)

k(.

-

Consider the positive definite function

V(x, ()

1 4 1 2 = -x + -( • 4 2

Its derivative along the trajectories of (12.18) reads as

V=

-xa

+ x3( + (x3

- k(2 =- ( x3

() (

(

~1 ~1) ~)

and is negative definite. for k > 1. Thus, for k > 1, the equilibrium (x, () = (0, 0) of (12.18) is globally asymptotically stable. Another alternative way to check this conclusion is to use the small gain theorem. The upper subsystem, viewed as a system with state x and input (, is input to state stable, with gain function 11 (r)

= rs

.l_

.

If k > 0, the lower subsystem, viewed as a system with state ( and input x, is input to state stable, with gain function 1

3

!z(r) = // . Thus, if k > 1, the small gain condition is fulfilled.


138

12. Feedback Design for Robust Semiglobal Stability

The second part of this example suggests to use, in order to check the property of asymptotic stability of system (12.16) in situations other than that of local exponential stability of the equilibrium x = 0 of x = F(x), the small gain theorem. In the present context, however, only local asymptotic stability is of interest, because the convergence to an arbitrarily small neighborhood of the equilibrium is guaranteed by other arguments. Thus, it is expected that a sort of "local" version of the property of input-to-state stability, on which the small gain theorem is based, would suffice. This notion, together with an application to the present problem, is described hereafter. Consider a nonlinear system

x = f(x,u)

(12.19)

with state x E JRn, input u E JRm, in which /(0, 0) = 0 and f(x, u) is locally Lipschitz on lRn x .IR.m. Let X be an open subset of lRn containing the origin and let U be a positive number. Definition 12.2.1. System {12.19} is said to be input-to-state stable with restriction X on X 0 and restriction U on u( ·) if there exist class K functions 'YoU and '/'u(·) such that, for any X 0 EX and any input u(·) E L~ satisfying liu(·)lloo < U, the response x(t) in the initial state x(O) = X 0 satisfies

llx(·)lloo lim sup llx(t)il t-;oo

< max{'"Yo(llxoll),'f'u(liu(·)lloo)} < '"Yu(limsup liu(t)ii). t->oo

The function '"Yu(·) for which the estimates above hold is sometimes referred to as a local gain function, in recognition of the fact that the estimates in question hold only for those u(·) whose L 00 norm does not exceed the number U. In close analogy to the study presented in section 10.6, we show hereafter that, if the small gain condition is fulfilled, the feedback interconnection of two nonlinear systems that are both input-to-state stable, with restrictions on initial state and input, is -in turn -input-to-state stable, with restrictions on initial state and input. More precisely, consider the following interconnected system

Xt = It (xi, x2) ±2 = h(xt,X2,u),

(12.20)

in which XI E 1Rn 1 , X2 E 1Rn2 , u E lRm and ft (0, 0) = 0, /2(0, 0, 0) = 0. Suppose that the first subsystem, viewed as a system with internal state x 1 and input X2 is input-to-state Stable with restrictions X1 on and Ll2 on x 2(·). That is, suppose there exist class K functions '"Yot(·) and 1'10 such that for any x~ E X 1 and any 1ix2(·)lloo < Ll2, the response Xt(t) satisfies

xr

llxt(·)lloo lim sup llxt(t)il t-;oo

< maxbot(llx~ji),')'t(llx2(·)lloo)} :S: '"Yt(lifllsup llx2(t)jl). t->oo

12.2 Semiglobal Stabilization via Partial State Feedback

139

Likewise, suppose that the second subsystem, viewed as a system with internal State X2 and inputs XI and U is input-to-state stable With restrictions X2 on x2, Lli on xi(·) and U on u(·). That, is, suppose there exist class JC functions 'Yo2(·), 1'2(·) and-y,.(·) such that for any x2 E X2, any llxiOIIoo < Ll1 and any llu(-)lloo < U, the response x2(t) satisfies llx2(·)lloo lim sup llx2 (t) II

< max{'Yo2(llx211), 'Y2(IIxi( ·)lloo), 'Yu(llu( ·)lloo)} :S max{'Y2(lim sup llxi(t)ll), -y,.(lim sup llu(t) II)} .

t--too

t--too

t--too

The same arguments used in the proof of Theorem 10.6.1 show that, if the composite function -y1 o -y2 (·) is a simple contraction, i.e. if 1'1 (-y2(r))

< r for all r > 0,

(12.21)

the composite system (12.20) is input-to-stable, with appropriate restrictions on the initial states x~,x2 and the input u(·). The only extra precaution needed here is to carefully keep track of the restrictions that have to be. imposed in order to take advantage of the inequalities above, which characterize the property of input-to-state stability. To this end, recall that the key of the proof in question, used first to proof boundedness of all trajectories and then the property of input-to-state stability, was the substitution of the estimates provided by the second set of inequalities into the first set (and vice-versa). Now, suppose that x2 E x2, llxrOIIoo < Llr and llu(·)lloo < u, so that (12.22) Suppose also that x~ E Xr and llx2(·)iloo II xi( ·)lloo

< Ll2,

so that

:S maxhor (llxl'll), 1'1 (llx2 Olloo)} ·

(12.23)

Replacing (12.22) into (12.23) yields llx1 Olloo

:S max hoi (llx~jl), 1'1 o 'Yo2(11x211), 'YI o 'Y2(IIxi(·)IIoo),'YI o -y,.(llu(-)lloo)},

which, using the small gain condition (12.21), reduces to llxiOIIoo

:S max{'Yo1(11x~II);'YI 0 'Yo2(1lx~ll),-yi o-y,.(iluOIIoo)} ·

Since XI(·) was supposed to satisfy llxi(·)IIoo

< Ll 1, we need to assume that

'Yo1(llx~jl)

< Ll1 'YI o'Yo2(llx21i) < Ll1 'YI o 'Yu(llu(·)lloo) < Ll1 · Likewise, after having replaced (12.23) into (12.22), we see that the condition i1x2(·)1loo < Ll2 requires

140

12. Feedback Design for Robust Semiglobal Stability

'Yo2(11x21D 'Y2 o 'YOI (llx~ll) 'Yu(llu(·)lloo)

If these constraints have to hold, .XI= {xi E xl: 'YOI(IIxiiD

x~

< Ll2 < Ll2 < Ll2. must be restricted to the set

< Lli,'Y2 O'Yoi(IIxrll) < Ll2},

and x2 must be restricted to the set X2 = {x2 E X2: 'Yo2(llx2ll)

< Ll2,'Y1 o 'Yo2(llx2ll) < Llr}.

Moreover, u(·) must be such that llu(·)lloo for some

< [J

U satisfying [J ~

u,

The relations thus found identify appropriate candidates for the restrictions on the initial states x~,x2 and the input u(·) of system (12.20). Formally, following exactly the same procedure used to prove Theorem 10.6.1, it is possible to show that the following holds.

Theorem 12.2.1. If the condition (12.21} holds, system {12.20}, viewed as a system with state x = (x 1,x2) and input u is input-to-state stable, with restrictions XI on x~, X2 on x2 and U on u( ·). In particular, the class }C functions 'Yo(r)

=

max{2'YOI (r), 2'Yo2(r), 2'YI o 'Yo2(r), 2'Y2 o 'Yoi (r)}

'Y(r)

=

max{2'YI o 'Yu(r), 2'Yu(r)}

are such that response x(t) satisfies llx(·)lloo limsupllx(t)ll t-+oo

< max{'Yo(llxoii),'Y(IIu(·)lloo)} ~

'Y(limsupllu(t)ll). t-+oo

Remark 12.2.1. Also the property of input-to-state stability with restrictions can be checked by seeking the existence of appropriate Lyapunov functions. In fact, let V : !Rn --? lR be a CI function satisfying g:(llxll) ~ V(x) ~ o(llxll) , for some class Koo functions g:(·) and 0:(·). Suppose there exists numbers ~x and ~u and a class }C function x( ·) such that

llxll 2: x(llull)

=>

av

8X j(X, u) ~ -o:(llxll)

for all llxll ~ ~Z> llull ~ ~U 1 (12.24)

12.2 Semiglobal Stabilization via Partial State Feedback

where a{) is a class Koo function. Define c*

= _q(6.,)

< c*,

and

141

,

set and let U be such that a o x(U)

U

< 6u.

Then, it is readily seen that,

ilc· E B6 • . Moreover, if u is such that 1/u//

< U and xis on the boundary of {}c.,

a o x(//u/1) < c* ~ a(//x/1) , that V is negative at each point

i.e. 1/xl/ 2: x(l/ul/), so of the boundary of ilc. Thus, from arguments identical to those used at the beginning of section 10.4, it follows that for any X 0 EX and any u(·) satisfying 1/u(·)l/ 00 < U 1/x(t)l/ ~ ma.x{,B(I/x 0 1/,t),-y(l/u(·)l/oo)} for some class KC function ,8(·, ·), with -y(r)

= _q- 1 0 a 0 x(r) .

It follows that the system is input-to-state stable with restriction X on x 0 and restriction U on u( ·).
x = (

F(x) +G( q(x, (, jj) - b(x, (, jj)k(.

(12.25)

Suppose the upper subsystem, viewed as a system with state x and input (, is input-to-state stable with restrictions X for X 0 and Llc: for ((·), for some neighborhood X of x = 0 and some Ll.; > 0. Moreover, suppose the lower subsystem, viewed as a system with state ( and input x, is input-to-state stable with restrictions Z for CO and Ll., for x(·), for some neighborhood Z of ( = 0 and some Llx > 0. Suppose also that these restrictions as well as the estimates which characterize the property of input-to-state stability are independent of k and jj. If the (local) gain functions of the two subsystems satisfy the small gain condition (12.21), the equilibrium point (x, () = (0, 0) is' locally asymptotically stable, with a region of attraction that is guaranteed to contain a neighborhood Ao of the origin which does not depend on k. Thus,

142

12. Feedback Design for Robust Semiglobal Stability

8.'3 shown at the end of the proof of Corollary 12.1.2, if k is large enough, all trajectories with initial conditions in QR_+r Mymptotic converge to the equilibrium (x, () = (0, 0) 8.'3 t -t oo. The reader can eMily check that the hypotheses thus indicated are weaker than those of Corollary 12.1.2. In fact, if the equilibrium x = 0 ofX = F(x) is locally exponentially stable, the upper subsystem of (12.25), for some (small) neighborhood X of x = 0 and Ll, > 0, is indeed input-to-state stable with restrictions X and Ll,, and hM a (local) gain function 'YI (·) of the form

1 1 (r) =cr.

Also the lower subsystem, for some (small) neighborhood Z of ( = 0 and > 0, both independent of k and J.L, is input-to-state stable with restrictions Z and Ll,., and hM a (local) gain function that, for large k, can be bounded by a function 12 ( ·) of the form Ll,.

12(r)

< '_!_r - 2c

Therefore, for large k the small gain condition (12.21) is fulfilled.

12.3 A Proof of Theorem 9.6.2 Before proceeding with the description of how, for selected classes of nonlinear uncertain systems, the problem of semiglobal practical stabilization can be solved using output feedback, it is convenient to examine in detail how the problem in question can be solved in the absence of parameter uncertainties. We recall that this problem was addressed in section 9.6, where it was shown that, if a system x = f(x) + g(x)u (12.26) y

=

h(x),

is globally asymptotically stabilizable using a memoryless smooth state feedback and it is also uniformly obseroable, then semiglobal stabilization using dynamic output feedbaCk is possible (Theorem 9.6.2). In this section we provide a detailed proof of this result, which is interesting per se and also because it introduces some arguments that will be used in the later sections in dealing with similar stabilization problems for uncertain systems. Consider a system of the form (12.26), with state X E an, input u E 1R, output y E lR, in which it is assumed that f(x), g(x), h(x) are smooth functions of their arguments, and /(0) = 0, h(O) = 0. Suppose there exists a smooth feedback law u = a(x), with o(O) = 0 which renders the equilibrium x = 0 of the system x = f(x) + g(x)a(x) globally asymptotically stable. As a consequence (Theorem 9.2.3), there exists a smooth feedback law

12.3 A Proof of Theoretii"9.6.2

u = O(x,vo, ... ,Vn-d

143

(12.27)

which globally asymptotically stabilizes the extended 2n-d.imensional system :i;

vo

=

f(x) vl

+ g(x)vo (12,28)

Vn-2

Vn-1

Vn-1

u.

Moreover, suppose system (12.26) is uniformly observable and recall that this property, as shown in section 9.6, guarantees that the mapping w = tP(x, v) defined as in (9.58) has global inverse x = l[l(w, v). The controller which in section 9.6 was claimed to be able to semiglobally asymptotically stabilize system (12.26) was a dynamical system modeled by equations of the form

(

~~

~~-

) = (

Vn-2

Vn-1

Vn-1

fJ(l[I*(TJ, v), Vo, ... , Vn-d

)

(12.29)

(12.30)

and u

= v0 ,

(12.31)

in which: {i) v = col(vo, VI, •.• , V.i-2), 1J = col(TJI> 1]2, ••. , 1Jn), (ii) O(x, v0 , v1 , ... , Vn-d is the function (12.27), {iii) Cf'n(x,vo,VI, ... ,Vn-d is the,n-th function in the sequence (9.57), {iv) g > 0 is a design parameter which depends on the size of the set of initial conditions that must be asymptotically steered to the equilibrium, and eo, c1 , ... , Cn-I are coefficients of a Hurwitz polynomial

(v) IJi*(ry,v) is a function defined as follows

144

12. Feedback Design for Robust Semiglobal Stability

'ii(TJ, v) 'ii(TJ,V) { ii'ii(TJ,v)iiM

qi*(TJ,V) =

if ll'ii(TJ, v)ll < M if ll'ii(TJ,v)l/ ~ M,

(12.32)

where M > 0 is another design parameter which depends on the size of the set of initial conditions that must be asymptotically steered to the equilibrium. Note that 'ii*(17,v) is a function which coincides with 'ii(17,v) for all (TJ,V) such that the norm of qi (1J, v) is less than a fixed number M, and bounded (in norm) by M elsewhere. To prove Theorem 9.6.2, consider the closed-loop system (12.26) - (12.29) - (12.30) - (12.31), define

by e1 €i

= =

gn- 1 (cpo(x) -TJd gn-i(lt'i-1(X,Vo, · · · ,Vi-2) -TJi),

and observe that

e = D 9 (if>(x, v) -17)

where D9

Finally, set

= diag{gn-l, ... , g, 1}.

= col(x,vo,v1, ... ,vn-1).

z

This yields equations of the form i

=

e =

F(z)+Pt(z,e) gAe+P2(z,e)

in which -Cn-1 -Cn-2

A= (

. -c1

-eo PI(z,e)

=(

1 0 ... 0) 0

1 ... 0 . . ... . 0 0 ··· 1

'

o o ··· o

! ),

tPt(z,e)

(12.33)

P2(z,e)

=(

! ),

¢>2(z,e)

and

tPt(z,e) tP2 (z, e)

= 8(!1i*(4'(x,v)- D; 1 e,v),vo, ... ,Vn-d- 8(x,vo, ... ,vn-d = lt'n (x, vo, ... , Vn-d - 'Pn(!li* (4'(x, v) - D; 1e, v), vo, ... , Vn-1).

By construction, system

12.3 A Proof of Theorem 9.6.2

i

145

= F(z)

has a globally asymptotically stable equilibrium at z = 0. Thus, there exists a smooth real-valued function V(z) satisfying

q(llzll) ~ V(z) ~ n(llzll)

av Bz F(z) ~ -a:(V(z))

for all z, where q(·),n(·),a:(·) are class Koo functions. By hypothesis, {z(0),1J(O)} E Sz X S 11 where Sz and S 11 are fixed compact sets. Choose c such that ilc

= {z: V(z)

~

c} :J.Sz

and then choose the parameter M in the definition of the function lff* (7], v) M

= zEOc+l max llxll + 1.

With this choice, it is easy to check that the two vectors Pl (z, e) and P2(z, e) defined before vanish at e = 0, so long as z E ilc+l· In fact, recall that x = 1ff(!li(x,v),v). Thus, if z E ilc+l, we have IIIP(!I'(x,v),v)ll < M. As a consequence, if z E ilc+l

[w*(7J,v)]e=O = 1ff*(!li(x,v),v)

= 1ff(!li(x,v),v) = x,

and this shows that if z E ilc+l.

PI(z,O) = 0,

P2(z,O) = 0.

> 0,

Note that, for all (z, e) E ilc+l x lRn and for all g

illli*(!l'(x,v)-

D; 1 e,v)ll ~ M.

Thus, there exist positive numbers fh,{J 2 , independent of g, such that

IIPi(z, e)ll ~ /Ji, fOT all (z, e) E

lRn.

(12.34)

+ k1fJ1

(12.35)

ilc+l X

Let k1 be such that

---a;- II ~ k1 for

8V(z) 11

Then

8V(z) ----a;-(F(z)

for all (z, e) E ilc+l

X

lRn.

+ p 1 (z,e))

all z E ilc+l·

~ -a:(V(z))

146

12. Feedback Design for Robust Semiglobal Stability

Let P be the positive definite solution of P A + AT P = -I, and let k2 = IIPII. Then, using standard inequalities, observe that for all (z, e) E ne+l X !Rn, the function Q(e) = eT P e satisfies

where

II>

0 is any number. Let k 3 > 0 and k 4 > 0 be such that

and set

a(g)

= (g - ~)

:4 .

Assuming that g > 0 is large enough (so that a(g) > 0), this yields

8Q

oe (gAe + P2(z, e)) ~ -a(g)Q(e)

2

+ vk2/32

(12.36)

for all (z, e) E !]c+l x r. Inequalities (12.35) and (12.36) show that, if z(t) E !tc+1 and g is large enough,

dV(z(t)) < k /3 dt

-

dQ(e(t)) < vk /32 dt

1 1,

-

(12.37)

2 2 '

with k1 , /31 , v, k2, /32 independent of g. From these it can be seen that there is a fixed time T > 0 (independent of ~) such that, for every initial state z(O), 17(0) E Sz x S 11 , the solution z(t), 17(t) 18 defined for all t E [0, T] and, in particular, z(t) E !te+l for all t E [0, T]. In fact, integration of the first one of (12.37) on the interval [0, t] yields

V(z(t))- V(z(O))

~

k1f31t.

Choosing T (2k1{3I)- 1 , it is concluded that necessarily V(t) ~ V(O) + 1/2 ~or all t E [0, T], i.e. z(t) E De+l/ 2 for all t E [0, T], because otherwise this mequality would be contradicted. Then, the second one of (12.37) shows that also e(t), and thus 17(t), is defined for all t E [0, T]. We prove now the following Lemma.

=

Lemma 12.3.1. For any e there exists a number g* > 0 (independent of z(O), 1J(0)) such that, if g > g*,

lle(T)II

~

f,

and, moreover, for all T' > T z(t) E flc+1 far all t E [T, T'] => lle(t)ll·~ e far all t E [T, T'].

1203 A Proof of Theorem 90602

147

Proof. Consider again (12°36)0 Using the comparison Lemma, it is deduced

that

Q(e(t)) ~ e-a.(g)tQ(e(O))

+

1- e-a.(g)t a(g) vk2j3~

and this yields

Fix

€

and choose v to satisfy 2k2 {3~v ~ k3 €2 , so that, if a(g)

lle(t)ll 2

~ : : e-a.(g)tlle(O)II 2 + ;

> 1,

0

Observe that, for any fixed choice of initial conditions x(O),v(0),77(0), lim e-a(g)TIIe(O)II 2 = 0,

g--too

because lle(O)II = IID 9 (4>(x(O),v(O)) -77(0))II is bounded by a p!Jlynomial of degree n-1 in g. In particular, since x(O), v(O) and 77(0) range over a compact set, there is a number g• > 0 such that

:: e-a.(g)TIIe(O)II 2 for every g

~

;

> g• and every x(O), v(O), 77(0) and the result follows.


Using this Lemma, it can be proven that the trajectories of the system are bounded and, actually, converge to an arbitrarily small neighborhood of the origin. To this end, we observe first that there exists a positive non-decreasing function 'Y : [0, oo) --7 [0, oo ), with -y(O) = 0, which is independent of g (if g > 1), such that

IIP1(z,e)ll

~

'Y(IIell), for all (z,e) E

nc+l X

!Rn.

(12.38)

In fact, set 1/>1 (z, e)

= B(!li(tl>(x, v) -

D9-1o,e, v), vo, ... , Vn-1)- B(x, Vo, .. o, Vn-d

.

and observe that, since this is a smooth function of z, e which vanishes at e = 0, there is a positive non-decreasing function 7 : [0, oo) --7 [0, oo), with 7(0) = 0, independent of g (if g > 1), such that

I~I(z,e)l ~ i(llell), for all z E ilc+l and all e E lRn. For any z E nC+ll let Cz denote the set of all e E lRn such that ¢1(z,e)

= ~1(z,e),

148

12. Feedback Design for Robust Semiglobal Stability

and observe-that, in view of the specific choice of M, the the interior of £z. Thus, a bound of the form

poi~t

e

= 0 is in

IIP1(z, e)ll S 1'(11ell) holds for all z E Dc+l and all e E £z. On the other hand, we know that there exists a number fJ1 such that IIP1(z, e) II S /31 for all z E Dc+l and all e E !Rn. Thus, it is possible to find a function 'Y(·), with the properties indicated above, for which the bound (12.38) holds. Now, let o> 0 and p > 0 be such that the set

Bo = {z: llzll S 8} satisfies ilp

c Bo c

De.

Then, using the function 'Y(·) for which (12.38) holds, choose

f

so that

kn(t:) < o:(p) and consider the set

S = {z E Dc+1 : V(z) ~ p}. By construction, so long as

z(t) E S,

and

lle(t)ll S f

the function V(z(t)) satisfies

dV~(t)) S -o:(V(z(t))) + kl'Y(t:) < 0

(12.39)

i.e. is decreasing. We have already shown that, for all t E [0, T], the solution z(t), TJ(t) is defined and z(t) E int(Dc+l)· If g > g*, we know from the previous Lemma that lle(T)II S f. Then, using again this Lemma, we see that z(t) cannot leave the set Dc+l and lle(t)ll is bounded by f for all t ~ T. In fact, z(t) cannot reach the boundary of the set Dc+l> where V(z) > V(z(T)) without contradicting (12.39). It can also be shown that, in finite time, z(t) enters the set ilp and remains there for all subsequent times. For, suppose this is not true. Then, V(z(t)) is always decreasing and converges to a nonnegative limit Vo ~ p. Let r denote the w-limit set of the trajectory in question. It is well-known that V(z) = V0 at each point of r. Pick any initial condition in r and observe that the function V(z) is constant along the corresponding trajectory. Thus we have 0 S -o:(Vo) + kn(t:), i.e.

12.4 Stabilization of Minimum-Phase Systems in Lower-Triang\llar Forin

149

Vo < p which is a contradiction. Once z( t) ha.s entered f2p, it can never leave this set, because V(z(t)) is negative at each point of its boundary. So far, we have proven that, in finite time, (z(t), e(t)) enters an arbitrarily small neighborhood of the equilibrium (z,e) = (0,0). In order to complete the proof, observe that if g > 1, then IID; 1 ell S llell. Therefore, iff and are sufficiently small, ·

o

«Ji*(P(x,v)- D; 1 e,v)

= «Ji(P(x,v)- D; 1 e,v).

As a consequence, for allllzll So, llell Sf and g > 1, the function P:!(z,e), which vanishes at e = 0, can be bounded a.s IIP:!(z, e)ll = ksllell · Once z(t) ha.s entered the set B 0 , the function Q(e) satisfies

~~ e s -gllell 2 + 2k2k511ell 2 . and therefore, if g is large enough, lim e(t)

t-+oo

= 0.

At this point, a straightforward application of Theorem 10.3.1 shows that also z(t) -+ 0 a.s t-+ oo and this completes the proof.

12.4 Stabilization of Minimum-Phase Systems in Lower- Triangular Form We consider in this section the problem of robust stabilization of systems described by equations having a lower-triangular structure, such as equations (11.38), which are re-written here for convenience

i

=

6 = 6 = f.r

=

/o(z,f.I) QI(z,{I,JL) +b~(z,{I,JL)6 Q2(z,{I,f.2,JL) + ~(z,{I,f.2,JL)6 Qr(z,{I, ... ,f.r,JL)

(12.40)

+ br(z,{I, ... ,f.r,JL)U.

It is a.ssumed that z E !Rn, f.i E 1R, for i = 1, ... , r, u E lR and JL E P C JRP is a vector of unknown parameters. In particular, we address the problem of determining conditions under which such a system is semiglobally practically stabilizable by means of a (possibly dynamic) feedback law which uses a.s information only the output

150

12. Feedback Design for Robust Semiglobal Stability y ={I

of the system. The first step in the analysis consists in showing that there exists a globally defined partial change of coordinates, transforming system (12.40) into a system modeled by a simpler set of equations, which have the form

i

=

/o(z, (I)

(I

=

(2

(2

=

(3

(r

=

q(z,(I, ... ,(r,f.J.)+b(z,(I,···•(r,f.J.)U.

(12.41)

and in which (I= y.

The reason why this change of coordinates is useful is that, by construction, for each i = 1, ... , r, the (new) state variable(; coincides with the derivative of order i - 1 of the output y with respect to time. Moreover, these equations are precisely equations of the form (12.1), for which we have shown in section 12.1 that it is possible solve a problem of semiglobal and practical stabilization using a linear feedback in the variables (I, ... (r. In order to obtain the form (12.41), set Xo

1/Jo

= =

0 1,

define

XI(z,{I,f.J.) 1/JI(z,{I,f.J.)

Qr(z,{I,JL) bi(z,{I,f.J.),

and recursively, fori= 2, ... , r - 1, Xi ( z, ~I, ..• ,~;, J1. ) =

8[Xi-I + 1/Ji-I~i]

8

z

fo(z,{r

)

+ 1/Ji-I~i][QI(z,{I,f.J.) + bi(z,{I,f.J.)6 + ... 1 86 8[xi-1 + 1/Ji-1~i] [ + Qi-I(z,~I, · ..• ~i-I,JL)+b;-I(z,{r, ...• ~i-l,f.J.)~;] 8~.,_1 + 8[xi-1

+ 1/Ji-I (z, ~I. ... , ~i-1> JL)q;(z, 6, ... , ~;, JL) and 1/J;(z,~b

... ,~;,JL)

= 1/Ji-I(z,{I, ...• ~i-I,f.J.)b;(z,~l, ... ,~;,JL).

The way these formulas are constructed is such that, along the trajectories of system (12.40)

12.4 Stabilization of Minimum-Phase Systems in Lower-Triangular Form

d[xi-1 + 'I/Ji-1ed dt Therefore, for all i

151

c = Xi + ·'·'f'i'>i+1 ·

= 1, ... , r - 1, the variable

(; = Xi-1 (z, 6, · ·., ei-1 ,fl.) + '1/Ji-1 (z, e1, · .. , ei-1, f.L){; is such that

(; = (i+l which is precisely what is sought in the equations (12.41). Because of the special structure of the functions thus defined, the substitution of the state variables 6 J • • • J with the variables

er

6

(1 (3

= =

XI(z,6,f.L) +'¢1(z,6,f.L)6 X2(z,6,6,f.L) +'¢2(z,6,6,f.L)6

(r

=

Xr-I(z,{I, ... ,er-1,f.L)+'¢r-I(z,6, ... ,er-1,f.L){r

(2

(12.42)

is globally defined if the '¢;(z,6, ... ,{;,f.L)'s are nowhere zero. This is the case if - as assumed in Chapter 11 - there exist b;0 > 0 such that (12.43) for all i = 1, ... , r. Assuming this is the case, the variables {I' ... J pressed in the form ei = 9;(z, (I, ... 1 ( ; , f.L)

er can be uniquely ex-

where the 9;(z,(1, ... ,(;,f.L)'s is are smooth functions of their arguments. Therefore, it can be easily observed that (r = q(z, (I, ... , (r 1 f.L)

+ b(z, (1, ... , (r, f.L)U

which is precisely the last equation of (12.41). Moreover, for some

bo > 0, (12.44)

In summary, we have shown tliat, if there exists numbers b;o that (12.43) hold for all i = 1, ... ,r, there is a smooth mapping

JR.r

-t

ar

e

~

4>(z,e,f.L)

> 0 such

J

which has a globally defined smooth inverse, such that equations (12.40) are transformed into equations of the form (12.41), in which the property (12.44) holds. As shown in section 12.1, the form (12.41), if the subsystem

152

12. Feedback Design for Robust Semiglobal Stability .i

= fo(z, 0) ,

possesses a globally asymptotically stable equilibrium at z = 0, lends itself to the design of a very simple state feedback law, actually a linear feedback law of the form u = K(, able to solve a problem of semiglobal and practical robust stabilization. It is important, however, to stress that the change of coordinates (12.42) yielding the special form (12.41) explicitly depends on the vector z of state variables of (12.40) and on the vector f:.L of unknown parameters. In other words, the feedback law u = K (, in terms of the original state variables (z, ~), i~ a function of the form u = K.P(z, ~' f:.L)

which cannot be implemented in practice, because the vector f:.L is unknown (and perhaps the state z is not accessible for feedback). We shall see however, in the second half of this section, that this apparent inconvenience can be overcome, because the various compbnents of the vector ( needed for the implementation of this feedback law coincide with a certain number of derivatives with respect to time of the output y of the system, namely

and, as shown in the previous section, the latter can be reasonably estimated with the aid of a suitable auxiliary dynamical system which uses as information only the actual output y. In what follows we will show that, for systems of the form (12.40), semiglobal practical stabilization can be achieved by means of a dynamic, output feedback, controller of the form

~

~:. ::~:'

iJr

0

. ) =( ) +( ( TJr-1 1Jr g u = -O"t(k[TJr

)

(y -

TJd

gr Co

+ e- 1ao1JI + F- 2al1J2 + ... + kar-21Jr-Il)

in which the function CTt(·) is a saturation function, defined as

CTt(r) =

r { sgn(r)£

(12.45)

C!

if lrl < l if lrl ~ l,

the c; 's and a/s are (fixed) coefficients of two polynomials

(12.46)

12.4 Stabilization of Minimum-Phase Systems in Lower-Triangular Form

153

;.r + Cr-1;.r-1 + ... + c1>. +co ;.r-1 + ar-2;.r-2 + ... + a1>. + ao having all roots with negative real part, and k,k,g,f are design parameters, to be tuned in accordance with the data of the design problem. The standing hypotheses, on system (12.40), under which the proposed controller succeeds in securing semiglobal and practical stability are the following ones: (i)

f 0 (0,0) = 0 and the equilibrium z = 0 of the subsystem

z = fo(z,O) is globally asymptotically stable, (ii) for i = 1, ... , r, there exists biD

> 0 such that

b;(z,6, ... ,t;,;,J.L) >biD,

(iii) fori= 1, ... , r Qi(O,O, ... ,O,J.L) =0,

for all (z,6, ... ,t;,;) and all J.L E P. Theorem 12.4.1. Consider system {12.40) and assume hypotheses (i), (ii) and (iii) hold. Given any arbitrarily large number R > 0 and any arbitrarily small number£> 0, there are numbers k > O,k > O,g > 0,1! > 0 such that, in the closed loop system {12.40) - {12.45} - {12.46}, any initial condition in Q~+ 2 r produces a trajectory which is captured by the set Q~+ 2 r.

Proof. Define new state variables (; = Xi-1(z,6, .. · ,l;,i-l>J.L)

+ tPi-1(z,6, ·. · ,l;,;-1,J.L)(.;

and fori= 1, ... ,r. In a more convenient notation, these relations can be rewritten as
( =· e

=

D 9 ((-17)

where
D 9 =diag{g 7 Set

K =- ( kkr- 1ao

so that (12.46) becomes

1

, ...

kkr- 2 a1

,g,1}. ···

kkar-2

k)

154

12. Feedback Design for Robust Semiglobal Stability

u = ut(KTJ) = ut(K((- Diie)) .

Finally, set x

= col(z, (). This yields equations of the form :i; = F(x,J.L) + PI(x,e,J.L) e = gAe + P2(x, e, J.L)

in which /o(z,(I) (2

F(x,J.L)=

:

l

(12.47)

,

( q(z, (l, ... , (r, J.L) +'b(z, (I, ... , (r, J.L)K( -Cr-I -Cr-2

A=

. (

-ci

-eo PI(x,e,J.L)

1 0 ... 0) ~' ~ ::: ~

! ),

=(

,

(12.48)

0 0 ... 1 0 0 ... 0

P2(x,e,J.L)

=(

rh(x,e,J.L)

! ), rP2(x,e,J.L)

and rP1 (x, e, J.L) ¢J2 (x, e, J.L)

= =

b(z, (l, ... , (r, J.L)[ut(K((- D_gie)) - K(] q(z, (I, ... ,(,., J.L) + b(z, (I, ... ,(,., J.L)ut(K( ( - Diie)) .

Now, observe that, as a consequence of the hypothesis (iii), the function Xi(z, {I, ... , ~i, J.L) is such that Xi(O, 0, ... , 0, J.L) .

Therefore, the mapping

~(z,~,J.L)

satisfies

~(O,O,J.L)

= 0.

Since the mapping ~(z, ~. J.L) is a smooth mapping and f..L ranges over a compact set, given any R > 0 there exists R' > 0 such that

=> and, also, given any

£

> 0, there exists £ 1 > 0 =>

(z, ~) E

a:+r

and

11(11 ~

i.

12.4 Stabilization of Minimum-Phase Systems in Lower~Triangular Form

155

In view of this, the_proof of the Th_eorem amounts to show that, for every initial condition X 0 E Q';,.tr and 'T/ 0 E QR!. the trajectory of the system (12.4 7) i~ such that x(t) is captured by the set Q~,+r and e(t) is captured by the set Q;12 . Note, in fact, that this requirement on e(t) guarantees, if g > 1,.that 11(t) is captured by the set Q~, since

= (- D;

1/

Consider the system :i;

1

e.

= F(x,fJ),

and observe that this is precisely the system whose asymptotic properties were studied in the proof of Theorem 12.1.1. From that proof, it can be claimed that, given R' > 0, there is a choice of k > 0 and there exists· a positive definite function (the function (12.6), which in the present context will be rewritten, for consistency, as W(x)), defined on a bounded set which contains Q'lttr in its interior, such that, for some a> 0 satisfying Q-n+r R' C

the property

n J&a 1

aw

ax F(x, fJ)

<0

(12.49)

5:. a+ 1} ,

(12.50)

holds at each point of a set of the form S

= {x: p 5:. W(x)

where p is a number which can be rendered arbitrarily small by increasing the value of f.:. Bearing in mind the fact that W(x) is positive definite, choose p > 0 such that, for some 5 > 0 and p' > 0 {}PC

fhc

f1p' C Q~,+r,

and fix the value of k accordingly (i.e. such that (12.49) holds on (12.50)). The values of k and k thus chosen remain fixed throughout the rest of proof. Note also that there exists a class K. function a{) such that

aw

ax F(x, fJ) ~ -o(llxll),

for all x E f1a+l \ Bo .

(12.51)

Now, choose f as f.= max IK(I+l. :reaG+t

With this choice, it is easy to check there exist positive numbers independent of g, such that IIPi(x,e,fJ)II ~

/3i, for all

(x,e) E na+l X JR".

/3t, /h,

156

12. Feedback Design for Robust SemiglobaJ Stability

Moreover, the vector p 1 (x, e, JJ) defined before vanishes at e.=.O, so long as x E Da+l , and there exists a positive non-decreasing function -y(·), with -y(O) = 0, which is independent of g (if g > 1), such that

/IPt(x,e,JJ)II

~

--r(llell), for all (x,e) E Da+l

X

!Rr.

Let k1 be such that

I ~: II ~ kt Then

for all x E Da+l.

ow ox (F(x,JJ) + pt(x,e,JJ)) ~ -a(/lxll) + k ,8 1

1

(12.52)

for all (x,e) E (Da+l \B6) x JRr. Let P be the positive definite solution of PA + ATP = -I, and let k2 = IIPII· Then, as in the proof of Theore,m 9.6.2, we obtain, for any v > 0, that aQ (12.53) oe (gAe + p2(x, e, JJ)) ~ -a(g)Q(e) + vk2

.Bi

for all (x, e) E Da+l x !Rr, where

a(g)

~ (g- ~) k~.

Inequalities (12.52) and (12.53) show (see proof of Theorem 9.6.2) that there is a fixed time T > 0 (independent of g) such that the solution x(t), e(t) of (12.47) is defined for all t E (0, T] and, in particular, x(t) E Da+t for all t E (0, T]. Indeed, a Lemma identical to Lemma 12.3.1 holds, and thus one can claim that for any e there exists a number g• > 0 (independent of x(O), 77(0)) such that, if g > g*, lle(T)/1 ~ e, and, moreover, for all T' > T

x(t) E Da+t for all t E (T, T']

=?

/le(t)/1 ~ e for all t E (T, T'].

This, as in the proof of Theorem 9.6.2, shows that trajectories are bounded. In fact, so long as

x(t)

E

{x: p' ~ W(x) ~a+ 1},

and

lle(t)l] ~ e,

the function W (x ( t)) satisfies

dW~:(t)) ~ -a(llxll) + kn(e). Pick e such that a( J) > k 1--y( t-:). Then, so long as

12.5 Stabilization via Output Feedback Without a Separation Principle

x(t) E {x: p' :=:; W(x) :=:;a+ 1},

we have

and

1/e(t)ll :=:;

157

f,

dW(x(t)) dt <0,

i.e. W(x(t)) is decreaSing. This, as in the proof of Theorem 9.6.2, shows that in finite time the x(t) is captured by the set flp', and therefore captured qy the set Q~,+r. Iff< c:/2, then also e(t) is captured by the set Q~ 12 and this, as remarked before, completes the proof.
m~trix

[ofo] &

(0,0)

are all in the left-half plane, the proposed feedback law is able to locally exponentially stabilize the system. Corollary 12.4.2. Consider system (12.40) and assume hypotheses (i), (ii) and (iii) hold. Assume also that all the eigenvalues of the matrix F 0 above have negative real part. Given any arbitrarily large number R > 0 there are numbers k > 0, k > 0, g > 0, f > 0 such that, in the closed loop system (12.40) - (12.45)- (12.46), the equilibrium (z,~,TJ) = (0,0,0) is locally exponentially stable and, moreover, any initial condition in Q'Jt+ 2 r produces a trajectory which asymptotically converges to (0, 0, 0) as t-+ 00.

12.5 Stabilization via Output Feedback Without a Separation Principle In this section we illustrate a simple recursive design method for stabilizing a linear system by output feedback, which does not use any technique for pole assignment nor the separation principle , but rather uses concepts reminiscent of some classical design methods based on root-locus properties. Of course, this does not lead to any specific breakthrough in the case of linear systems, where pole assignment or the solution of a diophantine equation suffice to stabilize using output feedback, byt its nonlinear version, described in the next section, provides a design tool which may prove useful in the problem of robustly stabilizing, via output feedback, a possibly unstable and nonminimum phase nonlinear system. Consider a single-input single-output linear system

x y

=

Ax+ Bu Cx.

(12.54)

Let n denote the dimension of its state space and let r denote its relative degree. It is well-known, and it can be deduced- in particular- as a corollary

158

12. Feedback Design for Robust Semiglobal Stability

of Proposition 9.1.1 (because in the case of a linear system the completeness and commutativity hypotheses on the vector fields (9.2) are always fulfilled), that such a system can always be represented in the form

z

Foz

~1

+ Go6

6 (12.55)

{r

Hoz

+ a1{1 + · · · + ar{r + bu

6 in which z E IR.n-r and b :f:. 0. Set

( = col(z, {I, ... , {r-d, and

Go 0

0

0 0

0 0

a1

a2

ar-1) ,

1 0

0

0) ,

c

F~ ~

H= (Ho K= (0

V

!).

1

= {r'

G~m J= ar

and rewrite the system in the form (

v = y

=

F(+Gv H(+ Jv+bu K(,

(12.56)

in which ( E an- 1 and v coincides with y
( = Ya =

+ Gua H( + Jua.

F(

(12.57)

The main purpose of this section is to show that, if the auxiliary subsystem thus defined is stabilizable by (dynamic) output feedback, so is the full system (12.56). This result is the direct consequence of two Lemmas. The first Lemma shows how to stabilize (12.56) using a dynamic output feedback which uses vas input.

12.5 Stabilization via Output Feedback Without a Separation Principle

159

Lemma 12.5.1. Suppose system

tP

=

Ua

=

Lcp+ Mya Ncp

stabilizes system {12.57). Then there is a number k* such that, for all k the feedback law

rp u

> k*

Lcp + Mk(v- Ncp)

= ~ [N[Lcp + Mk(v- Ncp)]- k(v- Ncp)]

(12.58)

stabilizes system (12.56). Proof. Consider the interconnection of (12.56) with (12.58) and change the state variable v into a state variable f) defined as 8=v-Ncp,

which yields

+ G(Ncp + 8) Lcp + Mk8

(

F(

rp 0

=

H( + J(Ncp + 9)- k8.

Change now the state variables cp into new state variables u defined as u = cp + M8,

to obtain a system of the form

(~) 0

(:H =

L+G:JN)

(~)+(MJ~(~~~~N)M) 9

-(k-J+JNM)8+(H

JN)

(~)

(12.59) The latter can be viewed as the feedback interconnection of two subsystems. The upper subsystem, viewed as a system with state ((, u) and input 8, is stable by hypothesis and has' some finite L 2 -gain '"f, which is independent of k. The lower subsystem, viewed as a system with state 8 and input ((, u), is stable for large k, and its L 2 -gain decreases to 0 as k --+ oo. As a consequence, by the small gain theorem, if k is large enough, the system in question is stable. <1 The second Lemma shows how, in the previous stabilizer, v can be replaced by an estimate generated by a filter which receives y as input. Define

160

12. Feedback Design for Robust Semiglobal Stability

1 0 ... 0) 0

1 ...

0

. . .. .

. 1 0

0 0

0 0

. .. ...

'

(12.60)

R=(000···1), where co, c 1 , ... , Cr-I are the coefficients of a fixed polynomial having all roots with negative real part, and g > 0 is a parameter to be determined. Lemma 12.5 .2. Suppose system I{;

u

=

Acp + Bv Ccp +Dv

stabilizes system (12.56). Then, there is a r;umber g• such that, for all g > g* the feedback law r, PTJ + Qy (12.61) Acp+ BRTJ u Ccp+DRTJ stabilizes system (12.55). Proof. Define ei = gr-i(~i- 'T/i), fori= 1,2, ... ,rand observe that, since RTJ = v- er, the interconnection of (12.55) and (12.61) becomes

( = F(+Gv

v tP e:

= = =

H( + Jv + b(Ccp + Dv) - bDer Acp+ Bv- Ber gAe + B(H( + Jv + b(Ccp + Dv)- bDer),

(12.62)

in which A is the matrix (12.48), having all eigenvalues in c-, and B = col(O, 0, ... , 0, 1). This can be viewed as the feedback interconnection of two subsystems. One subsystem, modeled by the first three equations, with input er and internal state ((, v, cp), is stable by construction and has an L 2 -gain bounded by a fixed number which is independent of g. The other subsystem, modeled by the last equation, with input((, v, cp) and internal state e, is stable for large g and, as an elementary calculation shows, its L 2 -gain decreases to 0 as g --+ oo. Thus, again by the small gain theorem, the interconnection is stable for large g. <1 Indeed, the combination of these two Lemmas shows that if system

12.5 Stabilization via Output Feedback Without a Separation Principle

161

is a dynamic, output feedback, stabilizer for the auxiliary system (12.57), then the system

i!
u

=

PTJ + Qy Lt.p + Mk(RTJ- Ncp)

~ [N[Lcp + Mk(RTJ- Ncp)]- k(RTJ- Ncp)]

is a dynamic, output feedback, stabilizer for system (12.54). The hypothesis on which this stabilization scheme works is that the auxiliary system (12.57) is stabilizable by output feedback. Thus, it seems natural to pose the question of how much such an hypothesis is restrictive, in the context of the problem of stabilizing a linear system using output feedback. Well, it turns out that this hypothesis is not restrictive since it can be shown that, possibly up to a memoryless output feedback transformation, it must necessarily hold if the original system (12.56) is stabilizable by output feedback. Proposition 12.5.3. Suppose system {12.56) is stabilizable by output feedback and consider the memoryless feedback transformation u t--+ u

+ ky

which changes system {12.56) into system

( = v = y

=

F(+Gv

(12.63)

(H + bkK)( + Jv + bu K(.

Then, for every k E lR except possibly one single value k 0 , the auxiliary subsystem associated to {12.63), i.e. the system

( = Ya =

F(+Gua

(H

+ bkK)( +. Jua

.

is stabilizable by output feedback. Proof. Using the normal form recalled at the beginning of the section, observe that

(~

~) ~ (Z

G J 0

n~

0 0 0

0 0 0

0

1

a2

ar

0 b 0

0 1

Fo 0 0

Go 0 0

0 Ho 0

0 a1

1

0

0

162

12. Feedback Design for Robu.st Semiglobal Stability

A straightforward application of standard stabilizability / detectability tests, shows that if the pair (A, B) is stabilizabie, so is the pair (F0 , G0 ) and, likewise, if the pair (A, C) is detectable, so is the pair (F0 , H 0 ). Consider now the auxiliary system associated with (12.63) and define

(I; ~) = (H~kK

Fo 0 0

G)J -

0

Go 0 0

0 Ho

0

0 a1

0 0 0

1

+ kb

0 0 0

0

0

1

a2

ar-1

ar

Using the property that the pair (F0 ,G0 ) is stabilizable, it is not difficult to show that, for any A with nonnegative real part, rank ( F - AI G)

=n -

1

(12.64)

and therefore the pair (F, G) is stabilizable. Using the property that the pair (Fo, Ho) is detectable, it is not difficult to show that, if>. has nonnegative real part and >. =f. 0, (12.65) rank ( FH = n -1

AI)

for any k. The matrix in (12.65) has rank n- 1 at>.= 0 (and hence the pair (F, H) is detectable) if and only if

Fo det ( Ho

a!

Go

+ kb

)

=1-

0.

Observe that det (

~~

a

~\b)

1

= (a1 + kb)det(F0 ) + det (

~~ ~0 )

.

The right-hand side of this identity, a polynomial of first degree in k, cannot be identically vanishing because otherwise det(F0 ) = 0,

and

det (

~: ~0 = 0 )

would contradict the fact that the triplet (Fo, Go, Ho) is stabilizable and detectable. Thus, there is at most one single value of k for which (F, H) loses detectability. <J

Remark 12.5.1. System (12.63) is obtained from system (12.56) via a memoryless output feedback of the form u t-+ u+ky and this indeed leaves the property of stabilizability by output feedback unchanged. The previous Proposition shows that, in case the auxiliary system associated with (12.56) is not stabilizable by output feedback, the auxiliary system associated with (12.63) is stabilizable by output feedback for every nonzero k. <J

12.6 Stabilization via Output Feedback of Non-Minimum-Phase Systems

163

This Proposition shows that there is no loss of generality in approaching the problem of stabilization via output feedback in the manner suggested above, i.e. by "reducing" the problem to that of stabilizing an auxiliary lowerdimensional subsystem. Indeed, the process can be iterated all the way down to the case in which the auxiliary subsystem has dimension one (where a simple gain would suffice to stabilize). This yields a systematic method for stabilization via output feedback, which does not use any technique for pole assignment nor the idea of solving separately a problem of state-feedback stabilization and a problem of asymptotic state reconstruction. Rather, it is based on the recursive update of a sequence of "dynamic" output feedback stabilizers for a sequence of subsystems of increasing dimension.

12.6 Stabilization via Output Feedback of Non-Minimum-Phase Systems Many of the methods, described so far, for robust stabilization of a nonlinear system using partial state feedback rely upon the hypothesis that the system has a stable zero dynamics . The main reason why this hypothesis is assumed is that the methods in question use "high-gain" feedback in order to offset the presence of certain unwanted terms in the dynamics of the closed-loop system. The consequence of this high-gain control is to enforce a behavior whose asymptotic properties are directly influenced by the asymptotic properties of the zero dynamics. In particular, asymptotic stabilization may occur only if the latter is asymptotically stable, i.e. if the system is minimum-phase . For example, in the stabilization of a system of the form i

=

iJ =

fo(z, y, p.)

h (z, y, p.) + u

robust stability (with respect to the uncertain parameter p.) can be achieved by choosing the control u in such a way as to offset the (possibly unpleasant) presence on the term h (z, y, p.) in the derivative of a candidate Lyapunov function of the form

U(z,y)

=;

V(z) + (y- y•(z)) 2

,

where V(z) is a positive definite and proper function such that

~: fo(z,y•(z),p.) < 0

(12.66)

for all z ::j; 0. If the input u is only allowed to depend on y (and not on z), this goal, i.e. the fact that 8V

az fo(z,y,p.) + 2(y- y•(z))[f1 (z,y,p.) + u-

cy• & f 0 (z,y,p.)] < 0,

(12.67)

164

12. Feedback Design for Robust Semiglobal Stability

can only be achieved if y•(z) = 0 and V(z) is a Lyapunov function for the zero dynamics i = fo(z,O,p,).

If this is the case (i.e. if (12.66) holds for y•(z) = 0), the linear law u = -ky, where k is a sufficiently large number, solves the problem of semiglobal and practical stabilization, essentially because the term -ky 2 in (12.67) is able to dominate all other non-negative terms. If, on the contrary, the zero dynamics are unstable, some (output-based) estimate of a nontrivial y• (z) is needed, and this might not be compatible with a control action which, by dominating the term h (z, y, p,), would render z poorly observable from y. The method described in the previous section does not rely at all upon a similar procedure, but rather aims at taking explicit advantage of the term h (z, y, p,), to the purpose of determining a stabilizer for what was called the "auxiliary subsystem", in this case system

i Ya

= =

fo(z,ua,P,) fi(z,ua,P,).

We describe in this section a nonlinear version of that method. Consider a nonlinear system modeled by equations of the form i

fo(z,(I,p,)

6 6

q1(z,(I,p,) +b1(6)6

(r

y

q2(z,6,6,p,)

=

+~(6)6

(12.68)

qr(z,6, ... ,(r,Jl.) + br(6)u 6,

in which z E !Rn-r and p, is a (possibly vector-valued) unknown parameter, ranging over a compact set P. Assume that:

(i) fo(O, 0, p,)

= 0,

(ii) fori = 1, ... , r, there exists biO > 0 such that

(iii) fori= l, ... ,r

qi(O, 0, ... , 0, p,) = 0, for all (z, ( 17 ••• , (i) and all p, E P. Then, as shown in section 12.4, there exists a globally defined diffeomorphism ( r-t ( = P(z,(,p,),

12.6 Stabilization via Output Feedback of Non-Minimum-Phase Systems

165

such that <1>(0, 0, J.l-) = 0, transforming system (12.68) into a system described by equations of the form

i

fo(z, (t,JI-)

(1

(2

(2

(3

(r

q(z, (t, ... , (r,J.I-)

y

(12.69)

+ b((t)u

(1 .

=

Moreover,

q(O, 0, ... , 0, J.l-) = 0 b((t)

> bo

for some b0 > 0. As remarked before, the global diffeomorphism changing (12.68) _into (12.69) is J.~--dependent and z-dependent, but this is not an obstacle to semiglobal practical stabilization via output feedback. With system (12.69), we associate an auxiliary system Xa Ya

= =

fa(Xa, Ua, Jl-)

ha(Xa,Ua,JI-)

(12. 70)

in which

Xa=

(L)' (r-1

and ha(Xa 1 Jl-) =

q(z, (I. ... 1 (r-1, Ua, Jl-) ·

Motivated by the arguments presented in the previous section, we assume the following on the auxiliary system (12.70): (iv) there exists a dynamical

syst~m

<jJ

Ua

= =

of the form L(~tJ) N(~tJ)

+ Mya

(12. 71)

1

in which ltJ E IR" and a positive definite and proper smooth function V(xa, ~tJ) whose derivative along the trajectories of the interconnected system (12.70) - (12.71), namely system Xa

=

fa(Xa,N(~tJ),JI-)

<jJ

=

L(~tJ)+Mha(Xa,N(~tJ),JI-),

(12.72)

166

12. Feedback Design for Robust Semiglobal Stability

is negative definite, i.e. fJV

~ /a(Xa,N(cp),p.)

vXa

fJV

·

+ ~[L(cp) + Mha(Xa,N(cp),p.)] < 0 vcp

for all (xa, cp) =/: (0, 0). In other words, the system (12.71) is supposed to globally asymptotically stabilize the auxiliary system (12.70), and the interconnection (12.70) - (12.71) possesses a Lyapunov function V(xa,cp) which is independent of p.. Consider now, for the original plant (12.69), the dynamic feedback law

+ Mk[(r- N(cp)]

cp

L(cp)

u =

b(~I) [~~ ( L(cp) + Mk[(r- N(cp)J)- k[(r- N(cp)J]

(12.73)

where k is a positive number. We claim that a controller with this structure is able to robustly semiglobally practically stabilize the plant (12.69). To this end, observe that the feedback interconnection of (12.69) and (12.73), namely system Xa

=

/a(Xa,(r,!J.)

(r

ha(Xa, (r,!J.) +

cp

L(cp)

~~ ( L(cp) + Mk[(r- N(cp)J)

- k[(r- N(cp)]

+ Mk[(r- N(cp)], (12.74)

changing the state variable (r into the new variable

becomes

Xa

0

/a(Xa, () + N(cp),p.) ha(Xa,() + N(cp),p.)- k() L(cp) + Mk().

(12.75)

cp = This system can be viewed as the system resulting from the interconnection of a system with output () and input v, modeled by equations of the form fa(Xa,() + N(cp),p.) Xa (12.76) 0 = ha(Xa, () + N( cp), p.) + V L(cp)- Mv cp and a memoryless system (with input() and output v) v

= -k{}.

(12.77)

System (12.76) has uniform relative degree one between input v and output B, and its "high-frequency gain" coefficient is equal to one. Thus, the system has a (globally defined) zero dynamics manifold, the set

12.6 Stabilization via Output Feedback of Non-Minimum-Phase Systems

167

z• = {(xa,O,cp): 8 = 0}, which is rendered invariant by

Consequently, the zero dynamics of (12. 76) are those of Xa

rp

= =

fa(Xa,N(cp),J.L) L(cp) + Mha(xa, N(cp),J.L).

(12. 78)

These dynamics have by hypothesis a globally asymptotically stable equilibrium at (xa, cp) = (0, 0). Therefore, in view of the results discussed in section 12.1, a feedback law of the form (12.77) is able to solve a problem of semiglobal and practical stability. As a matter of fact, consider the positive definite and proper function

and let

na to denote the set

The following result holds. Lemma 12.6.1. For any R

such that

> 0 and

E:

> 0, and for any p > 0 and c > 0

np c Q~+v c Q'R+v c nc

there is a number k• such that, if k > k•, the derivative of the function W(xa, cp, 0) along the trajectories of (12. 75} is negative at each point of the set S = {(xa,cp,8): p :S W(xa,cp,8) :S c}. Proof. Let W(xa, cp, 8) denote the derivative of the function W(xa, rp, 8) along the trajectories of (12.75). Then,

.

W(xa,cp,8)

+

av =-a fa(xa,(l + N(cp),J.L) + 28[ha(xa,8 + N(cp),J.L)- kO] Xa

[~:L+M 8 [L(cp) +Mk8+Mha(xa,O+N(rp),J.L) -Mko].

The expression on the right-hand side can be put in the form

av av -a fa(xa, N(cp), J.L) + -a [L(cp) + Mha(Xa, N(cp), J.L)] Xa cp

2k0

2

+ R(x.._, cp, 0)8 ,

where R(xa, cp, 8) is an appropriate smooth function. In this expression, the sum of the first two terms is a negative definite function of (x a, cp), and therefore W(xa, cp, 0) is negative on the set

168

12. Feedback Design for Robust Semiglobal Stability

So

= S n {(xa, rp, 0) : 0 = 0} .

Thus, as observed in the proof of Theorem 12.1.1, it follows that the function W(xa, rp, 0) is negative at each point of some open subset U :J S0 . On the other hand, 0 2 is nowhere zero on the set S = S \ U. Thus, for some large k, W(xa, rp, 0) is negative on the entire set S. <1 The dynamic controller (12.73) uses the state variable (r, i.e. the derivative of order r- 1 of the output y of system (12.69), as input. Thus, in order to find an output feedback controller, this variable must be replaced by an appropriate estimate which, as seen in section 12.5, can be provided by a dynamical system of the form

iJ

= p"l + Qy

in which the matrices Q and P have the form (12.60). In this case, however, as shown in section 12.4, it is appropriate to saturate the resulting control law, so as to avoid the occurrence of finite escape times for large values of g. Thus, in the controller (12.73), the term k[(r- N(rp)]

is replaced by the term

O"t(k["lr- N(rp)]), in which CTt(·) is a saturation function

O"t(r)

={

r sgn(r)M

if if

lrl < f lrl ~f.

This yields a controller modeled by equations of the form

iJ cp

=

p"l + Qy L(rp) + Mo-t(k["lr- N(rp)])

b(~) [~: (L(rp) + Mo-t(k["lr- N(rp)]))- o-t(k["lr- N(rp)])].

u

(12.79) A controller of this type is able to robustly semiglobally practically stabilize the plant (12.69). To see that this is the case, consider the corresponding closed-loop system,

Xa

/a(Xa,(r,~)

(r

ha(Xa, (r,~)

cp

L(rp)

i}

P1J

+ ~: ( L(rp) + Mo-t{k["lr- N(rp)]))- O"t(k["lr- N(rp)])

+ Mo-t(k["lr- N(rp)])

+ Q(l

,

(12.80)

12.6 Stabilization via Output Feedback of Non-Minimum-Phase Systems

and again the change of variables 0

Xa

169

= (r- N(cp), to obtain

/a(Xa,O+N(cp),/1) ha(Xa,() + N(cp),/1)- at(k[17r- N(cp)J) L(cp) + Mo"t(k[17r- N(cp)]) Pry+ Q(1.

(12.81)

Define e;

= 9 r-i((; -ry;)

fori= 1, ... ,r (recall that ( 1, ... , (r-t are components of the vector xa), i.e e = D 9 (( -17) ,

in which D 9 is the matrix, already used before, . { g r~t , ... , g, 1} . D 9 = dtag

Then, system (12.81) can be rewritten in the form

Xa

iJ = cjJ

e

+ N(cp),/1) ha(Xa,O + N(cp),/1)- kO + cPt(O,e) fa(xa,O

L(cp) + MkO- M cPt (0, e) gAe+B¢2(xa,O,cp,e),

(12.82)

in which A is the matrix (12.48), having all eigenvalues in C- , B col(O, 0, ... , 0, 1), and

c/>1 (0, e) ¢2(Xa, 8, cp, e)

kO - at(kO- ker) ha(Xa, 0 + N(cp), !1)

+

-at(k()- ker).

8N

acp [L(cp)

+ M at(kO- ker)]

·

From this, using the result of Lemma 12.6.1, it is possible conclude the following. Theorem 12.6.2. Consider system {12.68} and assume hypotheses (i), (ii), (iii) and (iv) hold. Given any arbitrarily large number R > 0 and any arbitrarily small number E: > 0, there are numbers k > 0, g > 0, l > 0 such that, in the closed loop system {12.68} - {12. 79}, any initial condition in Q7z+v+r produces a trajectory which is captured by the set a:+v+r.

Proof. Since the changes of coordinates transforming system (12.68)- (12.79) into system (12.81) are globally defined diffeomorphisms which preserve the origin, it suffices to prove the Theorem for system (12.81). The proof uses the same arguments used in the proof of Theorem 12.4.1, and will be only summarily sketched. Let W(xa,cp,O) and fla be as in Lemma 12.6.1 and choose p,o,p',c such that

170

12. Feedback Design for Robust Semiglobal Stability

Jlp C Q~+v C Jlp' C Q:+v C QR_+v C Jlc C Jlc+l

and, using the result of Lemma 12.6.1, fix a number k such that the derivative of the function W(xa, cp, fJ) along the trajectories of (12. 75) is negative at each point of the set

S

= {(xa,cp,B): p ~ W(xa,ip,O)

~ c

+ 1}.

Then, choose the saturation level as l =

max

(z.,cp,6)E!Ic+l

Observe that, for all ((xa,cp,B),e) E

[}c+l

lkBI + 1 . x

!Rr

I¢I(B,e)l I¢2(Xa, fJ, cp, e)l I¢I(B,e)l '< in which {31 , /3 2 are fixed numbers, and -y(-) is a positive non-decreasing function such that -y(O) = 0, all independent of g. From this, the proof uses the same arguments as the proof of Theorem 12.4.1. <1 We conclude the section with some remarks about the structure of the controller used to stabilize the "auxiliary subsystem" (12.70). The previous Theorem considers the case in which the stabilizer for (12.70) is a strictly proper system, namely a system with no feed-through between Ya and Ua, and has the special structure (12.71), where the right-hand side is affine in the input Ya and the vector multiplying Ya is a constant vector. It is possible, however, to show that the structure thus assumed is not restrictive, from the point of view here considered. As a matter of fact, it is possible to prove that if a nonlinear system, such as system (12. 70), is globally asymptotically stabilized by an output feedback controller of a nonlinear general structure, then it is also semiglobally stabilizable by means of a system having the special structure (12. 71). Consider two subsystems :i;

y

=

cp u

f(x,u) h(x,u), 8(cp,v)

=

~~:( cp, v) '

(12.83)

(12.84)

in which x E !Rn, cp E !Rv, u E JR, y E lR, v E lR, f(O, 0) = 0, h(O, 0) fJ(O, 0) = 0 and ~~:(0, 0) = 0. Suppose that, for each x, cp, the equation

h(x, ~~:(cp, v))

=v

= 0,

12.6 Stabilization via Output Feedback of Non-Minimum-Phase Systems

has a unique solution

171

= v(x,cp),

v

which is a smooth function of (x, cp) (necessarily with v(O, 0) also that u = ~;,(cp, v(x, cp))

= 0).

Suppose

is the unique solution of the equation

= u.

~;,(cp, h(x,u))

In this case, the feedback interconnection of (12.83) and (12.84) via v is well-defined, and can be written as

±

=

f(x,

rp =

~;.(cp,

v(x, cp)))

=y

(12.85)

O(cp, v(x, cp)) .

Proposition 12.6.3. Suppose the equilibrium (x, cp) = (0, 0) of the interconnected system (12.85) is globally asymptotically and locally exponentially stable. Suppose also that, for all (x, cp) E llf' x lR" and all~ E JR,

{[{ + v(x, cp)

- h(x, ~;,(cp, ~

+ v(x, cp)))]

where a(·) is a class K function satisfying, for some

a(s)

= as2

~ a(~)

,

a> 0 and fJ > 0,

for all s E (0, fJ).

Consider the (strictly proper) system

rp v

=

-Mv+My

u

=

~;.(cp,v).

O(cp,v)

(12.86)

Then, given any arbitrarily large number R > 0, there exists a number M such that, in the closed loop system (12.83) - (12.86), the equilibrium (x,cp,v) = (0,0,0) is locally exponentially stable and, moreover, any initial condition in Ql/"'+ 1 produces a trajectory which asymptotically converges to (0,0,0) as t-+ oo. Proof To prove the result, change·v into

{ = v- v(x,cp) to obtain

= rp = { = :i;

f(x, ~;.(cp, {

+ v(x, cp)))

+ v(x, cp)) -M({ + v(x, cp)- h(x, ~;,(cp, { + v(x, cp))))

O(cp, ~

av 8v - axf(x,K-(cp,{+v(x,cp)))- 8cp(J(cp,{+v(x,cp)).

(12.87)

172

12. Feedback Design for Robust Semiglobal Stability

For ~ = 0 the upper two equations reduce to the feedback interconnection of (12.83) and (12.84), globally asymptotically and locally exponentially stable by hypothesis. On the other hand, it is easy to see that

de dt :::; -2M a(~) + (e + l~llxl + I~II'Pi)q(x, ~.p, ~) for some smooth function q(x, ~.p, ~). Then, the result follows from arguments identical to those used before in the proof of Theorem 12.1.1 and Corollary 12.1.2. <J In other words, this Proposition determines (sufficient) conditions under which a nonlinear system, such as system (12.70), which is globally asymptotically stabilizable by an output feedback controller of a nonlinear general structure, it is also semiglobally stabilizable by means of a system having the special structure (12.71). Note that the feedback interconnection of (12.83) and (12.86) differs from the feedback interconnection of (12.83) and (12.84) by the "insertion", between y and v, of a system characterized by the transfer function 1

T(s)= - 1 +rs in which the time constant T = 1/M, as shown in the proof of the Proposition, is appropriately small (see Fig. 12.2).

v

I

u

cp

=

11(~.p,

u

=

K(cp,v)

x

=

f(x,u)

y

=

h(x, u) ------::

v)

,--~

-

1 1 + TS

i y

Fig. 12.2. Block-diagram of the system.

12.7 Examples The purpose of the following example is to show that there are systems which are globally asymptotically stabilizable by static state feedback and uniformly observable , but nevertheless not globally asymptotically stabilizable by any

12.1 Exampies

173

continuous, static or dynamic, output feedback,. For these systems, the result of Theorem 9.6.2 is, in this respect, the best possible stabilization result available: global asymptotic stability via output feedback can never be achieved but local asymptotic stability with a domain of attraction which contains an; arbitrary large bounded set can. Example 12. 7.1. Consider the system

iJ

=

x =

X

2x 3

(12.88)

+u

with input u E IR, output y E JR. and state (y, x) E JR2 . This system is globally asymptotically stabilizable by static state feedback (the law u = -2x 3 - x- y does the job} and is also uniformly observable (in fact, the mapping (9.58) is the identity map). We will show that, no matter how a locally Lipschitz, static or dynamic, output feedback

~ = TJ({,y) u = O({,y)

(12.89)

is chosen, there are always initial conditions for which finite escape time occurs. . To this end, consider the differential equation

assume c > 0 and x(O) curve, given by

> 0, and observe that the corresponding integral

=

x(t)

x(O) 1. ,

(1 - 2c[x(O}j2t) 2

is monotonically increasing, and has finite escape time at T=

1

2c[x(O)j2

Observe also that, if x ~ 1 and lui S 1, x

3

S 2x3 + u S 3x3

(12.90)

.

Thus, if the input u(t) to system (12.88} is such that

ju(t)l S 1

for all t

~

0

and x(O) ~ 1, the response x(t) of (12.88), for all t for which it is defined, is always increasing and satisfies

Hence, in view of the comparison Lemma,

174

12. Feedback Design for Robust Semiglobal Stability

x t > x(O) ( ) - (1 - 2(x(0))2t)! '

from which it is seen that x(t) escapes to infinity at some time

r·

1

~ 2(x(0)] 2

(12.91)

·

On the other hand, using the right-hand side of (12.90), it is seen that, on the interval (0, r•), x(t) satisfies x(~) ~3x 3 (t),

and therefore, for any 0 < T < y•,

x(r) - (1- 6[x(r))2(t- r)]21

x(t) <

for all t

~ T

,

satisfying 1

t- T < 6[x(r)]2 · This inequality can be used to show that the integral of x(t) over the interval (0, r•) is finite. To this end, define a sequence of times to = 0 < t 1 < ... < t~c < ... in the following recursive way. Let Xk denote the value of x(t) at timet= t~c, i.e. and define tk+ 1 as t1c+1

where tk

= t~c

+ T~c

,

+ T~c is the finite escape time of the function

i.e. T~c

1

= 6~. k

Note that, since U~c(t) is an upper bound for the function x(t) on the time interval [t~c, tk+l), the value Xk+l is well defined and the recursion makes sense. Clearly, Since the function x(t), on the time interval

it follows that

[t~c, tk+I),

has a lower bound

12.7 Examples 2 x% xk+l ~ [1 - 2x%Tk]

with a

= .,j2f3 < 1.

3

1

2

= 2xk =:

175

2

a2 xk

As a consequence, Xk

~

1

kx(O) . a

(12.92)

Now, observe that Ak

=

i

t~+l

x(t)dt

tk

<

1Tk

k

l.

[1 - 6xi*

o

-

X

dt

=-

1

3xk

.

Therefore, using (12.92),

This, in turn, yields

1

oo

T"

x(t)dt

oo

1

k

~ {; 3:(0) = 3x(O) 1 ~a .

= {; Ak

which proves that the integral of x(t) over the interval [0, T•) is finite. As a consequence, if y(O) = 0 we obtain, for all t E [0, T*),

{t

1

a

Jo x(s)ds ~ 3x(O) 1- a ~ d V2f3) (recall that it was assumed

y(t) =

with d := -12f3/3(1 x(O) ~ 1). In summary, there is a number d > 0, independent of x(O), such that, if y(O) = 0, if x(O) ~ 1 and ju(t)j ~ 1, iy(t)l ~ d so long as x(t) is defined. Suppose system (12.88) is controlled by an output feedback of the form (12.89) and suppose, without loss of generality, that TJ(O, 0) = 0 and 0(0, 0) = 0. Let x(O) ~ 1, y(O) 0, ~(0) 0. Since u(O) = 0, using continuity arguments and the fact that jy(t)! is bounded by a quantity which is independent of x(O) so long as ju(t)l ~ 1, it is deduced that there is a time T 0 > 0, independent of x(O), such that ju(t)! ~ 1 for all t E [0, T0 ). If x(O) is such that

=

=

1-'

2x2(0)
the previous analysis (see (12.91)) shows that x(t) escapes to infinity in finite time. Thus, the feedback (12.89) cannot globally asymptotically stabilize the equilibrium (y, x, ~) = (0, 0, 0). The reason why system (12.88) cannot be globally asymptotically stabilized via output feedback is that there are internal states for which the trajectory escapes to infinity at some finite time t = T, while the corresponding output remains bounded on the interval [0, T). Thus, the phenomenon of finite escape time cannot be observed and the controller cannot react to it. <J

176

12. Feedback Design for Robust Semiglobal Stability

The second example is an application of the stabilization method described in section 12.6 to a system which is not minimum phase. Example 12. 7.2. Consider the three state approximation of the nonlinear Moore-Greitzer model of a compressor 1 oR(1 - lP 2

R)

-

R(O)

>0

-lJ! + lJ!c(lP) - 3lPR 1

,82 (lP + 1 -

(12.93)

,#) .

Here R == A 2 /4, where A is the rotating stall amplitude, lP is the scaled annuls-averaged flow, lJ! is the plenum pressure rise, o > 0 and ,B are fixed parameters, 1 is the throttle opening used as a control input, and 3

1

+ 1 + 2ip- 2ip

lJ!c(lP) = lJ!co

3

is the compressor characteristic. For any lP 0 , this system has an equilibrium at

(0, 'Po, lJ!c(lPo)) while, if

!~Pol

< 1, the system has one additional equilibrium at (1 - ~P6, 'Po, lJ!c( 'Po) - 3'Po (1 -fP6)) .

The equilibria in question are enforced by the constant input

'Po+ 1

#

[=

.

Consider, for instance, the case in which 'Po == 1, set 2+u

[==

#

and translate the coordinate so as to bring the equilibrium (0, 1, lJ!c(1)) to the origin, namely set ¢> == lP-1 1/J = lJ!- lJ!c(1) . This yields

R = ¢ = 1/J 1

=

-oR2

-

oR(2¢> + ¢ 2 )

-(3/2)¢2 1

-

(32(¢>-u),

See Krstic et al. (1998) for details.

(1/2)¢>3

-

3R¢- 3R -1/J

(12.94)

12.7 Examples

177

Note that the half-spaceR~ 0 is invariant, for every choice of u. Suppose !J!, i.e. '1/J in the model (12.94), is taken as output of the system. With such choice of output (and input), the system has relative degree 1 and has a zero dynamics

R

=

-uR2

-

uR(2¢J + ¢J2 )

¢ = -(3/2)¢J2 - (1/2)¢J3 -

3R¢J- 3R

which is not globally asymptotically stable, even in the invariant half-space R ~ 0 (for u = 1, there are two equilibria at (R, ¢J) = (0, 0) and (R, ¢J) = (0, -3), with the phase portrait depicted in Fig. 12.3).

Fig. 12.3. Phase portrait of the (unstable) zero dynamics of the compressor.

In order to apply the design methodology described above, consider the auxiliary system

R

=

-u R2

¢J

= =

-(3/2)
Ya

-

u R(2¢J + ¢J2 ) -

3R
Using the inequality R(2¢J + ¢2) $

observe that

~R2 + ~(2¢J + ¢J2)2

Ua

(12.95)

178

12. Feedback Design for Robust Semiglobal Stability

This shows that R(t) is decreasing whenever

R > 2¢+ ¢ 2

.

Thus, the upper subsystem of (12.95), viewed as a system with input ¢ and state R, is input-to-state stable, with gain function 1(r)

= 2r + r 2 •

As a consequence, one can think of using the result of Lemma 11.4.1 to globally asymptotically stabilize system (12.95). Condition (i) of this Lemma has been just checked, condition (ii) trivially holds, condition (iii) holds because 1- (3/2)¢ 2 - (1/2)¢ 3 - 3R¢- 3RI + 1¢1 is locally Lipschitz at the origin, and also condition (iv) holds, because 10 is locally Lipschitz at the origin. Therefore, the system in question can be globally asymptotically stabilized, by means of a feedback law of the form Ua = k(¢), which would be a nonlinear function of ¢. If just semiglobal stabilization is sought, a linear law suffices. To see why this is the case, observe that, given any number M > 0, it is possible to find a number a > 0 such that for all 1¢1 :::; M . Therefore, for all such ¢ we have

R -< -~R2 + ~a2,~.2 2 2 'P

,

and, consequently, R(t) is decreasing whenever R > al¢1. From this, it is possible to deduce (see section 12.2) that the upper subsystem of (12.95) is input-to-state stable, with restriction XR = {R E lR : R ~ 0} on R(O), restriction M on ¢(·) and linear gain function {q,(r) = ar. Note that the number a depends on M. Consider now the lower subsystem of (12.95), and set Ua = K¢ + v. The candidate ISS-Lyapunov function V(¢) = ~¢ 2 satisfies

V = - ~¢3 - ~¢4 2

2

-

3R¢2

-

3R¢ - K ¢ 2

-

¢v

3 3 9[ 4 2] 1 ) 2 02 2 1 2 < -2¢ + b2 ¢ +¢ +(2-K¢ +2R +2v' where b is any number. Again, it is possible to find a number L > 0 such that for all 1¢1 :S M , and therefore, choosing K such that L + ~- K:::; -1 we have, for all such¢,

12.7 Examples

179

2

1 V -< _A-.2 + -b2 R2 + -v2 . 'I' 2

Note that b is arbitrary, and K depends on b and M. From this property, is possible to deduce (see again section 12.2) that the lower subsystem of (12.95) with control Ua = K¢> + v, viewed as a system with input (R, v) and state ¢>, is input-to-state stable, with restriction X¢ = {¢ E IR: 1¢1 $ M} on ¢(0), restriction M/b on R(·), restriction M on v(·), and linear gain functions 'YR(r) =brand 'Yv(r)= r. Choose b so that ab < 1, b < 1, and fix K accordingly. Then, using Theorem 12.2.1, it is possible to conclude that system (12.95) with control Ua = K ¢> + v, viewed as a system with input v and state (R, ¢>) is input-to-state stable, with restriction

X= {(R,¢) E IR2

:

li(R,¢)11 $ M}

on (R(O), ¢(0)), restriction M on v(·), and linear gain function /v(r) = cr (for some c > 0). This proves, in particular, that for any M there exists K such that the equilibrium of ( 12.95), with control Ua = K ¢> is locally asymptotically stable, with a domain of attraction which contains the set {(R, ¢>) E IR2 : II(R, ¢>)11 $

M}. We have found in this way a "semiglobal" stabilizer for system (12.95). However, this is not yet of the required form. To find a controller of the form (12.71), an additional step is needed. Consider again system (12.95) and add an integrator on the input, i.e. set Ua


= =

r.p

u

Changing, as usual, the variable r.p into v

¢ = -(3/2)¢>2 -

(1/2)¢3

-

(12.96) 0

= r.p -

K ¢, we obtain

3R¢- 3R- K¢>- v

which is exactly the same equation found before, and iJ

= -K¢+u.

(12.97)

As it is clear from the previous analysis, choosing u = -Hv, it should be possible to enforce the desired properties of input-to-state stability also on the latter subsystem, viewed as a system with state v and input (R, ¢), with linear gains characterized by arbitrarily small gain factors. More precisely, set U (v) = v2 and observe that

U =3K¢>2 v + K¢3 v + 6KR¢>v + 6KRv + 2K 2 ¢v + (2K- 2H)v 2

.

180

12. Feedback Design for Robust Semiglobal Stability

As before, it is seen that given any number N > 0 and any (arbitrarily small) number e:, there is a choice of H (the number K has already been fixed) such that, if II(R(t),l/l(t))ll ~ N, for all t ~ 0, . U ~ -v2

e:2

E:2

+ 2R2 + 21/12.

Thus system (12.97), with control u = -Hv, is input-to-state stable, with no restriction on v(O), restriction Non (R(·), 1/J(·)), and linear gain function 1'(R,¢J(r) = e:r. Note that H depends on Nand c. Choose e: so that E:C < 1 . e: < 1, Then, using again Theorem 12.2.1, it can be proven that the system (12.95) - (12.96), with control u = -Hv, is locally asymptotically stable, with a domain of attraction which contains the set {(R,¢,v): ii(R,¢,v)ii < M}.

2

...

Fig. 12.4. Phase portrait of the stabilized auxiliary subsystem

Reversing the change of coordinates in (12.96), this shows that semiglobal stability, for system (12.95), can be obtained by means of a dynamic controller of the form Ua

=

(12.98) I{J,

which has precisely the required structure. The phase portrait of the corresponding interconnection (12.95)-(12.98) is shown in Fig. 12.4. Having determined a stabilizer for the auxiliary system (12.95), the theory presented in the previous section shows that one can construct a dynamic output feedback stabilizer for the original system (12.94). In this case, the

12.7 Examples

J m

0

0

0

:

5

0

0

:

10

' ~'O 15

i

m

20

! 25

181

0

OOOl 30

·_:E-.....-;...---:- :- :~·I

·_.:E--:- . ,.-:___,_· ---ll ::E--t-: . -'--: I 0

5

10

15

20

25

o

s

to

tS

20

25

0

5

10

15

20

25

30

..._._...:,.!

3D

' - - - - - - 1 !

30

Fig. 12.5. Time history of the stabilized compressor.

result of Lemma 12.6.1 suffices, since system (12.94) has relative degree r == 1 and there is no need to estimate derivatives of y. The resulting stabilizer has the form liJ == -Hcp + HK(3 2 k(y- cp) (12.99) u == (3 2 Hcp- (3 2 (HK/3 2 - +1)k(y- cp), with

y='I/J. Fig. 12.5 shows a trajectory of system (12.94) controlled by (12.99) (for f7 =1, K == 6, H =10, G = 0.5).

f3

= 1,

13. Disturbance Attenuation

13.1 Robust Stability via Disturbance Attenuation In this Chapter we will study problems of global stabilization of systems that can be modeled as feedback interconnection of two subsystems, one of which is accurately known while the other one is uncertain but has a finite £ 2 gain, for which an upper bound is available. More precisely, we consider systems modeled by equations of the form

±1 =

±2

fi(xt,h2(x2),u) h(x2,ht(Xt)),

(13.1)

which describe the feedback interconnection of a system

±1 y

=

ft(xt,w,u) ht(xt)

in which x 1 E JR.nt, w E lR, u E IR, y E lR, and fi (0, 0, 0) a system ±2 = h(x2,y) w = h2(x2)

(13.2)

= 0, h 1 (0) = 0, and (13.3)

in which X2 E IR.n 2 and h(O, 0) = 0, h2(0) = 0. The model of system (13.2) is supposed to be accurately known, while the model of system (13.3) is possibly unknown, but it is known that, for some 1'2 , this system is strictly dissipative, with respect to the supply rate

Global asymptotic stability of the uncertain system (13.1) can be achieved in the following way. Suppose u = u(xt) is a feedback law (assume, as usual, u(O) = 0) such that the system

±1 Y

= !I (x 1 , w, u(xr)) = ht(X~)

is strictly dissipative with respect to the supply rate

(13.4)

184

13. Disturbance Attenuation

and that 'i'l'Y2

< 1.

Then, by the Theorem 10.8.1, the system (see Fig. 13.1)

±1 = ±2

h(xl,h2(x2),u(x1)) h(x2,h1(x1))

=

(13.5)

is globally asymptotically stable. In other words, the feedback law u = u(xt) has globally robustly stabilized the uncertain system (13.1).

±2 =

w y ........___

w

±I

,..-----.

h(x2,y)

= h2(x2)

y

= h(xi,w,u) = h1 (xi)

u

-

...._.__ XI

II u(xi) IL Fig. 13.1. Robust stabilization via disturbance attenuation.

This observation motivates the following problem, known as Problem of Disturbance Attenuation (in the sense of the L 2 -gain), with Stability. Given a system modeled by equations of the form

x = y =

f(x,w,u) h(x)

(13.6)

in which x E Rn, wEIR, u E R, y E IR, and /(0,0,0) = 0, h(O) = 0, and a number 1' > 0, find, if possible, a feedback law u = u(x) such that the resulting closed-loop system is strictly dissipative with respect to the supply rate (13.7) In this setup, the input w of (13.6) is called the disturbance input, while the input u is called the control input. We describe hereafter a series of results, which are useful to address the problem of Disturbance Attenuation with Stability for an important class of nonlinear systems. Recall that, in view of the definition of dissipativity,

13.1 Robust Stability via Disturbance Attenuation

185

with respect to a supply rate of the form (13.7), the issue is to find a law u = u(x) such that, for some smooth function V(x) which is positive definite and proper, i.e. satisfies estimates of the form g_(llxll) :S V(x) :S a(llxli) ,

with g_( ·),a(·) class Koo functions, the dissipation inequality

av ox f(x, w, u(x)) :S -a(llxli) + -lw 2 -

h 2 (x)

(13.8)

holds for all x E llr' and all wE lR, with a(·) a class Koo function. In what follows, it will be shown that, for nonlinear systems that can be des~ ibed by equations having a special lower-triangular structure, the problem ot Disturbance Attenuation with Stability can be addressed by means of a technique of backstepping, which closely follows the procedure for global stabilization described in section 9.2, i.e. by recursively solving the problem in question for a sequence of systems of increasing dimension. The points. of departure are the appropriate versions of Lemma 9.2.1 and 9.2.2. Lemma 13.1.1. Consider a system described by equations of the form f(z,~,w)

i

q(z,~,w)

(13.9)

+u

h(z,O in which (z, ~) E !Rn x IR, f(O, 0, 0) that

= 0, q(O, 0, 0) = 0 and h(O, 0) = 0.

Assume

f(z,~,w)- f(z,~,O)

is independent

of~·

Assume also that, for some smooth real-valued function

R1(z,~), jq(z,~,w)- q(z,{,O)I

:S

R1(z,~)lwl

for all (z, {) and all w. Suppose there exist a number"'( > 0, a smooth real-valued function V(z), which is positive definite and proper, and a class Koo function a 0 (-) such that

av 2 2 oz f(z,O, w) :S -aQ(iizll) + "'( w -

2

h (z,O)

for all z and all w. Then, for every E > 0, there exist a smooth feedback law u = u(z,~), a smooth real-valued function W(z,~), which is positive definite and proper, and a class Koo function a(·) such that

aw

oz f(z,~,w)

+

aw

{)~ [q(z,~,w) +u(z,~)]

:::; -a(llxll) + ("'( + c) 2 w 2 for all (z, ~) and all w, where x

= col(z, ~).

-

h 2 (z,~)

(13.10)

186

13. Disturbance Attenuation

Proof. Set fo(z,(,) = f(z,(,,O), and Po(z,w) = q( z, (,, 0) and express q( z, (,, w) as

!I (z, (,)

q(z,(,,w)

= f(z,(,,w)-

f(z,(,,O). Set

. ·

= fr(z,(,) +PI(z,(,,w)

where, by hypothesis, jp 1 (z,(,,w)/ ~ R 1 (z,(,)/w/. System (13.9) becomes i

=

~ y

= =

fo(z, (,) + Po(z, w) fr(z,(,) +PI(z,(,,w) +u h(z,(,).

Set W(z, (.)

(13.11)

= V(z) + 21 e

and observe that

ow oz f(z,(,,w)

= = =

ow ow oz !o(z, ~) + oz Po(z, w) ow ow oz fo(z,O) + oz Po(z,w) + A(z,(,)(, ow oz f(z, 0, w) + A(z, (,)(,

where A(z,(,) is a suitable smooth function. Moreover,

ow o(, [q(z,(,,w) + u] = fr(z,(,)(, + p1 (z,(,,w)(, + u(,. Finally, note that

h 2 (z, (,)

= h 2 (z, 0) + B(z, (,)(,

where B(z, {) is a suitable smooth function. Thus, the left-hand side of (13.10), henceforth referred to as

W

W,

satisfies

=

ow oz f(z,O,w)+A(z,(,)(,+fi(z,(,)(,+p 1 (z,(,,w)(,+u(,

<

-ao(//z/1)

+ 12w2 -

+ B(z, (,)(, + A(z, (,)(,+/I (z, (,)(, + P1 (z, {, w){ + u(,. h2 (z, 0) - B(z, (,)(,

Choose

u = -df.- B(z, {)- A(z, {) - fr (z, (,) + v with d > 0, to obtain

W ~ -ao(//z/1) + 1 2 w2 -

h 2 (z, (,) -

df.2 + P1 (z, (,, w)(, + v(,.

To prove the Lemma, we choose now v so as to render v(, + PI(z,(,,w)(, ~ c 2w2

To this end, note that, for any c

>0

•

(13.12)

13.1 Robust Stability via Disturbance Attenuation

v{+Pt(z,{,w){ ~v{+Rt(z,{)Jwii{J ~v{+

R 21 (z {)e ; 4 2

187

+c 2w 2 ,

so that, to obtain (13.12), we choose

At this point, we have

W

~ ~

de +

-y2w2 + E:2w2 - h2(z, 0 2 -oo(JJzJJ)- d{ + ('Y + c) 2w 2 - h 2(z,{).

-oo(JJzJJ)-

(13.13)

Note that, since oo(·) is a class Koo function, the function o 0 (1izJJ) + d{2 is a positive definite and proper function of x = col(z, {). Thus (see Remark 10.1.3), there eXists a class Koo function a{) such that o(JJxJJ) ~ oo(llzll) + d{ This, in view of (13.13), completes the proof.

2

•

<1

Lemma 13.1.2. Consider a system described by equations of the form {13.9). Assume that f(z,{,w)- f(z,{,O)

is independent of{. Assume also that, for some smooth real-valued functions Ro(z), Rt (z, {), JJf(z,{,w)- f(z,{,O)II Jq(z, {, w) - q(z, {, O)J

< Ro(z)Jwl < R1 (z, {)JwJ

(13.14)

for all (z, {) and all w. Suppose there exist a number -y > 0, a smooth real-valued function v(z), with v(O) = 0, a smooth real-valued function V(z), which is positive definite and proper, and a class Koo function oo(·) such that (13.15)

for all z and all w. Then, for every•E: > 0, there exist a smooth feedback law u = u( z, 0, a smooth real-valued function W (z, {), which is positive definite and proper, and a class K 00 function o( ·) such that

8W

az f(z, {, w)

+

8W

8{ [q(z, {, w)

+ u(z, {)]

~ -o(JJxJJ) + ('Y + c) 2w 2 - h 2(z,{)

for all (z,{) and all w, where x

= col(z,{).

188

13. Disturbance Attenuation

Proof. Consider the globally defined change of coordinates 11 note that, in the new coordinates, system (13.9) becomes

=

i

e- v(z) and

f(z,T}+v(z),w):=-=f(z,T},w)

iJ = q(z, 1} + v(z), w)y

==-=

~~f(z, 1} + v(z), w) + u := ij(z, 1}, w) + u

h(z,T}+v(z)).

==-=

Indeed,

lii(z, 1}, w)- ij(z, 1}, 0)1 S R1 (z, 1} + v(z))iwl

+II~~ II.Ro(z)lwl ·

Thus, this system satisfies the hypotheses of Lemma 13.1.1, and the result follows. <1 This Lemma can be used repeatedly, to prove that the fulfillment of an inequality of the form (13.15) suffices to determine a solution of the problem of Disturbance Attenuation with Stability for systems having the following "lower-triangular" structure i

"""

e1 6

=

/{z,et,w) 6 +qt(z,6,w) ~3 + q2(z,~~.e2,w) (13.16)

er-l er y

=

~r

+ Qr-t(z,6, · · · ,er-I,w) u + qr(z,et, ... ,~r.w) h(z,6),

so long as certain assumptions hold. More precisely, assume, as before, that f(z, 6, w)- f(z, 6, 0) is independent of 6, and re":ritesystem (13.16) in the form

i

==-=

e1 = e2 """

/o(z,f.I) + Po(z,w) 6+ft(z,6)+Pt(z,6,w) 6 + h(z,6,6) + P2(z,6,6,w) (13.17)

er-1 er y where

and

= = =

er+fr-l(z,6, ... ,er-d+Pr-I(z,6, ... ,er-I,W) u+fr(z,6, ... ,er)+Pr(z,6, ... ,er,w) h(z,6),

/o(z,O Po(z,w)

.- /(z,6,0) .- f(z,6,w)- /(z,6,0)

13.1 Robust Stability via Disturbance Attenuation

189

q;(z,{l, ... ,{;,0) .- q;(z,{l,····{;,w) -q;(z,{l, ... ,{;,O),

f;(z,6, ... ,{;) p;(z,{l, ... ,{;,w)

·-

for i = 1, ... , r. Then, we have the following result.

Theorem 13.1.3. Consider system (13.17), in which fo(O, 0) = 0, h (0, 0) = 0, ... , fr(O, 0, ... , 0) = 0. Assume that, for some smooth functions Ro(z), R1(z,6), ... , Rr(z,{l, ... ,{r),

:S Ro(z)iwl

i!Po(z,w)ll

(13.18)

and (13.19)

for i = 1, ... , r. Suppose there exist a number 1 > 0, a smooth real-valued function 6 = v(z), with v(O) = 0, and a smooth real-valued function V(z), which is positive definite and proper, and a class /Coo function ao ( ·) such that

av oz (fo(z,v(z)) + Po(z,w)] :S -ao(llzll) + --·?w 2 -

h 2 (z,v(z))

(13.20)

for all z and all w. Then, for every e > 0, there exists a smooth feedback law u = u(z,{I, ... ,{r) such that the resulting closed-loop system is strictly dissipative with respect to the supply rate

q(w, y) Proof. Pick any 1 ::; i

= ('Y + e)2w2- y2 .

< r and define fo(z,{l) 6+!I(z,6)

i(z- >'>t+l c. )JO (

- (- ) Po z,w

=

and

{HI+ f,(z,6, · · · ,{;)

Po(z,w) Pl(z,{l,w) (

)

p;(z,6, ... ,{uw) ·'

i1 (z, {HI) PI(z,{Hl,w) h(z,{Hl)

= = =

fHl (z, {1, ... , {HI) Pi+l(z,{I, ... ,{HI,w) h(z,6).

Observe that system

z ~Hl y

io(z,{Hl) + fio(z,w)

= =

u+il(z,{Hl)+pl(z,{HI,W) h(z,{Hl)

)

190

l&:e Attenuation

is such tt

llfto(z,w)ll < Ro(z)lwl < RI(z,~i+I)iwl

Jpi(z,~r+I,w)i

for some(z,~i+ 1 ). Thus, a simple induction, based on Lemma 13.1.2, pr,ult. 0, Jw(t)J S K for all t 2: 0. In fact, since p;(,w) is smooth and vanishing at w = 0, there exists S;(z,6,.ch that

and R;(zsuch that

5';(z,6, ... ,~;,w)i S R;(z,6, ... ,~;) for all Jw In coe see from this Theorem that, in a system of the form (13.17), tie to be addressed in order to be able to solve the problem of Distumuation with Stability, is to find a smooth real-valued function 11(0) = 0, a smooth real-valued function V(z), which is positive cproper, rendering the inequality (13.20) satisfied, for all z and all It is to compare the dissipation inequality (13.20) with the original
f(z,v,w)

.i

y is

stricti~,

=

h(z,v)

(13.22)

with respect to the supply rate (13.7). The result of that, for a system of the form (13.17), the problem of finding a~w u = u(x) rendering the dissipation inequality (13.8) fulfilled cced to the problem of finding a feedback law v = v(z) Theorem~s

13.1 Robust Stability via Disturbance Attenuation

191

rendering the dissipation inequality (13.21) fulfilled, i.e. rendering the subsystem (13.22) strictly dissipative with respect to the supply rate (13. 7). This subsystem has a more general structure than subsystem (13.6), because its output is allowed to depend on the control input v, which was not the case for system (13.6), but has a lower dimension. In the next sections of the Chapter, we will often consider the special case of systems modeled by equations, such as (13.22), whose right-hand sides are affine functions of the disturbance input and of the control input. In these cases, the inequality (13.21) can be given a simpler expression, which does not call for the explicit existence of feedback laws, such as v(z), but just the existence of a function V(z). In fact, consider a system modeled by equations of the form x = f(x) + g(x)u + p(x)w with f(O) = 0, and set

y=(h(x))+( 0 0 r(x)

)u

with h(O) = 0 and r(x) =j:. 0 for all x. A feedback law u = u(x) renders this system strictly dissipative, with respect to the supply rate (13.7), if and only if there exists a positive definite and proper function V(x) such that av ax [f(x)+g(x)u(x)+p(x)w] ::; -a(llxll)+-y2w2-h2(x)-r 2(x)u 2(x), (13.23)

and the following holds.

Lemma 13.1.4. A function V(x) satisfies {13.23} for some u(x) if and only if av 1 [av ]2 1 [av ]2 2 axf(x)+h (x)- 4r2(x) axg(x) + 4-y2 axp(x) ::; -c.t(llxll). (13.24)

Proof. 0 bserve that, for any (v, w) E JR2 , av ax [j(x) + g(x)u + p(x)wj + h 2 (x) + r 2(x)u 2 - -y 2w 2 av 1 [av ]2 1 [av ] 2 = ax f(x) + h (x) - 4r 2(x) ax g(x) + 4-y 2 ax p(x)

1 av ]2 + [r(x)u + 2r(x) Bx g(x) -

[

')'W-

2

1 av ]2 2-y ax p(x) .

Suppose that, for some u(x), V(x) satisfies (13.23) for all x and w. Then, choosing 1 av w = 2-y2 axp(x) it is seen that (13.24) necessarily holds. Conversely, if the latter holds, then 1 av u(x) = - 2r 2 (x) ax g(x)

renders condition (13.23) satisfied for all x and w.

<1

192

13. Disturbance Attenuation

13.2 The Case of Linear Systems The purpose of this section is to show that the sufficient conditions of Theorem 13.1.3 for the solution of a problem of Disturbance Attenuation with Stability become, in the case of linear systems, also necessary, and that this fact can be exploited to the purpose of determining the optimal value of the achievable disturbance attenuation, namely a number 7* with the property that the problem in question is solvable for each 7 > 7* and is not solvable for 7 < 7*. Consider a linear system

x

=

y

Ax+Pw+Bu Cx

(13.25)

in which x E !Rn, w E IR, u E IR, y E JR. It follows from Theorem 10.9.1 that there exists a linear feedback law u = K x which renders the resulting closed-loop system strictly dissipative with respect to the supply rate

q(w,y) = ·iw2- y2 for some .:Y

< 7, if and only if the inequality

(A+ BK)TS + S(A + BK)

1

+ 2 SPPTS +eTC< 0

(13.26)

7

holds for some symmetric matrix S > 0. Thus, the ability of solving this matrix inequality for some K and some symmetric S > 0 determines the ability of solving the problem of Disturbance Attenuation with Stability for system (13.25), by means of a linear feedback law. In what follows, we will show that the inequality (13.26) can be solved if and only if 7 is larger than a fixed number 7*, which can be rather easily determined once system (13.25) has been expressed in special new coordinates. To this end, recall (see 12.55) that any linear system of the form (13.25) can always be expressed, by means of suitable changes of coordinates, in the following form, i

= Fz+G6 +Qw

~1

= {2 +p1w

~2

= 6 + P2W (13.27)

{r-1

{r y

= {r + Pr-1W =Hz+a16+ ... ar{r+bu+prW

=6,

(with z E !Rn-r) which is the linear counterpart of the form (13.17). The first step in the analysis consists in showing that the optimal level of achievable disturbance attenuation 7* only depends on the parameters which characterize the z-subsystem.

13.2 The Case of Linear Systems

193

Lemma 13.2.1. Consider the linear system (13.27) and let-y be a fixed number. The following are equivalent:

(i) there exists a linear feedback law which renders the resulting closed-loop system strictly dissipative with respect to the supply rate

q(w,y) = i2w2- y2 for some

i < -y,

(ii) there exist a matrix K and a symmetric matrix S holds,

>0

such that {13.26)

(iii) there exist a matrix L and a symmetric matrix Y

>0

such that

(13.28) (iv) there exists a symmetric matrix Z FZ

> 0 such

that

+ ZFT + -;.QQT- GGT < 0.

(13.29)

'Y

Proof. As observed before, (i)

=> (ii) by Theorem 10.9.1.

(ii) => (iii). Fori= 1, ... , r, set

Ai

= (

~ ~ ~ ~)

0 0 0 0 0 0

C;

= (1

0

0

1 0

B; ixi

· · · 0 ) 1 xi,

P;

Set also F; = (

~ ~~i) ,

G; = (

~{) ,

Q;

= ( ~) ,

H; = ( 0

C;) ,

and

K = (H

a1

···

ar ) .

With this choice, we see that system (13.27) can be rewritten in the form

:i:

Frx

y

Hrx

where x = col(z, ~1, ... , ~r).

+ Qrw + Gr(bu + kx)

194

13. Disturbance Attenuation

Suppose (ii) holds. Then, it is reaat. there exist .a matrix Kr and a matrix Xr =X'[> 0 which sat

where Mr = M'[ > 0. From this, by me can prove that, for 1 ::::; < r, there exist a matrix Ki and maX[ > 0 and Mi = M( > 0 such that

i

Xi(Fi + GiKi) + (Fi + GiKi)T Xi+-\+ H{ Hi 7 To this end, suppose (13.30) hold:n - r linear equation

+i

= -Mi .

(13.30)

and consider the

in the unknown Ki-l· Since Xi > 0, ithe bottom-right position is nonzero, and this equation has a uniq Define

Xi-1

=

(/(d;-1)x(d;-1)

(I(d; -1) x(d; -1)

J(d;-1)x(d;-1))

Ki-1 _I(d;-1)x(d;-1)) 1

Ki-1

multiply (13.30) on the left by

Ti

= ( I(d;-1)x.1) 01x(d;

and on the right by

Tl, and take the lock. This, since TiXiT( = (

Xi-)x 1 ) 01x(d;

,

and (I(d;-1)x(d;-1)

JHi-1

for i > 1, yields an identity of the fornith i replaced by i - 1. One last iteration of this process, 1) written for i = 1, shows that an inequality of the form (13.28) (iii) => (i). As observed, system

(13.~tructure

and

/o(z,{I) +Po(z,w) =+Qw. Define

of system (13.17),

13.2 The Case of Linear Systems

195

Suppose (13.28) holds and note that, since its left-hand side is sign definite, the same inequality continues to hold if 'Y is replaced by i' = 'Y - t: and t: > 0 is sufficiently small. Therefore, the functions V(z) and v(z) are such that av az [Fz

1 av

+ Gv(z)] + 41'2

avT 8z QQT 8z

+ [v(zW ~ -cllzW

for some c > 0, i.e. such that (recall Corollary 10.7.2 and subsequent observations)

~: [Fz + Gv(z) + Qw] ~ -cllzll 2 + i'2w2 -

[v(zW

for all z and all w. In other words, an inequality of the form (13.20) holds. Since all other hypotheses of Theorem 13.1.3 are satisfied, it is possible to conclude the existence of a feedback law u = u(z, {I, ... , ~r) such that the resulting closed-loop system is strictly dissipative with respect to the supply rate q(w,y) = i'2w2 _ y2. In particular, the construction underlying the various Lemmas used in the proof of Theorem 13.1.3 shows that u = u(z, {I, ... , ~r) is a linear feedback law. Thus, it is concluded that (i) must hold. (iii) <=> (iv). Add and subtract YGGTY to the left-hand side of (13.28) to obtain YF

+ FTY + ~YQQTY- YGGTY + (YG + LT)(YG + LT)T < 0. 'Y

Then, Z = y-I > 0 satisfies (13.29). Conversely, suppose Z satisfies (13.29). Set Y = z- 1 and L = -cTY, to obtain (13.28). <1 The implication (ii) => (iii) of this Lemma shows that, in the case of linear systems, the basic condition of Theorem 13.1.3, namely the existence of a function V ( z) (in this case a quadratic form z Ty z) and a function v( z) (in this case a linear function Lz) which satisfy the inequality (13.20) (equivalent, in this case, to an inequality of the form (13.28)) is also a necessary condition for the solution of the problem of Disturbance Attenuation with Stability. As a consequence, the optimal value of the achievable disturbance attenuation is precisely the infimum of all possible values of 'Y for which the inequality (13.28) can be fulfilled. This Lemma also shows that, in this specific setup, the quadratic matrix inequality (13.28) can actually be replaced by a simpler linear matrix inequality. As a matter of fact, the Lemma shows that the existence of a symmetric solution Z > 0 of the Sylvester inequality (13.29) is a necessary and sufficient condition for the existence of a linear feedback law yielding internal stability and a level i' < 'Y of attenuation between the disturbance input w and the output y in system (13.27). Thus, again, the optimal value of the achievable disturbance attenuation coincides with the

196

13. Disturbance Attenuation

infimum of all possible-values of 'Y for which the inequality (13.29) can be fulfilled. A further simplification is possible. Without loss of generality suppose that the z-subsystem of (13.27), after a suitable change of coordinates, is put in the form i1 = F1z1 +G16 +Q1w (13.31) i2 F2z2 + G26 + Q2w i3 = F3z3 + G36 + Q3w where a(F1 ) c c-, a(F2 ) that the following holds.

c

C 0 and a(F3 )

c c+.

Then, it is easy to realize

Lemma 13.2.2. There exists a symmetric matrix Z > 0 such that (13.29} holds if and only if there exists a symmetric matrix Z > 0 rendering

tz + ztT + ~QQT- acF < o, 'Y

(13.32)

'

where

Proof. Suppose (13.29) holds and set

where dim(ZI) = dim(FI). Then Z > 0 necessarily satisfies (13.32). Conversely, suppose the latter holds. Set

z=

(

~1 ~)

,

where the matrix Z 1 has to be determined. Rewrite the left-hand side of (13.29) as

where

{13.33)

13.2 The Case of Linear Systems

197

it is seen that (13.29) holds if and only if M 1 - N id- 1 NT is negative definite, i.e. T 1 T T -lT F1Z1 + Z1F1 < -2Q1Q1 +G1G 1 + NM- N . 'Y Let R > 0 be any symmetric matrix such that 1 'Y

T

T

-

1

T

R>2Q1Q 1 -G1G 1 -NM-N. Since a(FI)

c c-, the unique solution

Z1 of the Lyapunov equation

F1Z1 + Z1F(

= -R

is positive definite. Such a Z 1 renders the upper-left block of (13.33) negative definite and this completes the proof. <1 We see from this and from the previous Lemma that a level 'Y of disturbance attenuation can be achieved if and only if for this number 'Y the inequality (13.32) has a positive definite solution Z. This result can be ~sed in order to compute an explicit expression of the infimum of all possible values of 'Y for which the problem of Disturbance Attenuation with Stability can be solved. To this end, define

1oo e-Fas(G3G3T)e-(Fas)T ds 1oo e-Fa"(Q3Q3T)e-(Fas)T ds,

U3 Va

=

(13.34)

where the integrals are well-defined because by hypothesis all eigenvalues of - F 3 are in
2'Y

va < U3

Z>0

which solves

(13.35)

and

1 *Q 2 Q2 T X< X *G 2 G2T X 2X 'Y

(13.36)

for any (possibly complex) eigenvector x of- Fl. Proof. Rewrite the inequality in question as (13.37) with

198

13. Disturl:tation

and -

Ql

= --']sG3T

-

I

Q2

1 T = -2Q2Q2 + G2G2T

°

1 'Y Specializing tohe conditions of Theorem 13.6.3, it is ea.<>y to see that a solutionts if and only if the unique solution Y1 of

(13.38) is positive definite

x*C:!2x > 0 for any (possibly
1oo eF'f•(JieFisds

which, in view of tlefinitions given, is positive definite if and only if (13.35) holds. Tbndition is precisely condition (13.36). <1 The first one otitions can be further simplified. To this end, observe that if (13 be fulfilled for some 7, the matrix U3 needs to be invertible. 1112) is equivalent to (13.29), which in turn is equivalent tq (13.2&er holds, for some Y > 0, Land 7, the matrix F + G L must have nvalues in C-, because Y is a solution of the Lyapunov inequalitL) + (F + GL?Y < 0. Thus, the pair F, G must be stabilizabl·view of the decomposition (13.31), where by hypothesis F3 ha.<>
7 • := max { p (u3

max ' eigenvector of -Fi

{ x*Q2Q2 TT x*G2G2

x}}

0

(13.40)

X

The problem of Disttenuation with Stability is solvable for each 7 > 7• and is not so < 7•.

13.3 Disturbance Attenuation

199

13.3 Disturbance Attenuation We return now to the problem of Disturbance Attenuation with Stability for a nonlinear system of the special form (13.16), and we elaborate further on the possibility of fulfilling the main condition of Theorem 13.1.3, i.e. the existence of a smooth real-valued function v(z), with v(O) = 0, and a smooth, real-valued, positive definite and proper function V(z), such that 8V

oz f(z, v(z), w) ~

-a(llzl/) + -·,Pw2-

h 2(z, v(z))

(13.41)

for all z and all w, where a(·) is a class Koo function. Motivated by the analysis presented in the previous section for linear systems, we consider the case in which h(z,6)

=6

and we assume that the z-subsystem

i = f(z,6,w), can be split as

i1 = h(z1,z2,6,w) . (13.42) i2 = h(z2, 6, w) , where the z 1 -subsystem is viewed as the "stable component" in the sense that z1 = 0 is a globally asymptotically stable equilibrium for z2 = 0, 6 = 0, and w = 0, while the z2-subsystem represents a possibly "unstable" but "stabilizable component" in the sense that for some smooth function v(z2 ), the subsystem i2 = h(z2 , v(z2 ), 0) has a globally asymptotically stable equilibrium at z2 = 0. The decomposition in (13.42) can be viewed as a nonlinear version of the decomposition in (13.31). Our aim is to identify conditions under which the lower bound for the achievable level of disturbance attenuation is determined by properties of the unstable component of (13.42), namely the z2-subsystem. Lemma 13.3.1. Consider the system (13.42). Suppose that: (i) there exists a smooth real-valued function Vi (z!), that is positive definite ' and proper, such that 1 2 h(z1,z2,6,w) ~ -al(llzd/) +-ygw +-yJIIz2ll 2+-yg~f ~V UZ1

(13.43)

for some class K 00 function a 1 ( ·) and some positive real number 'Yo,

(ii) there exist a smooth real-valued function v2(z2), with v2(0) = 0, and a smooth real-valued function V2 (z2 ), that is positive definite and proper, such · that 2 2 2 (13.44) ~V h(z2,v2(z2),w) ~ -a2(llz2ID + -y w - v~(z2) UZ2

200

13. Disturbance Attenuation

for some class IC00 function a2 (·) and some positive .real number 'Y, (iii) for some r 1 > 0 and some constant a

Then for every E > 0 there exist a smooth function v(z), with v(O) and a real-valued, positive definite, proper function V(z) such that

= 0,

for all z and all w, where a(·) is a class IC00 function. Proof Suppose condition (i) is fulfilled for some 'Yo and multiply both sides of (13.43) by c 2 h5, where £ is any arbitrary positive number, to obtain a similar inequality, in which 'Yo is replaced by c. This shows that, without loss of generality, we can assume that (13.43) holds with 'Yo replaced by an arbitrary number c > 0. Define if a2(r) S c2 r 2 (13.45) if a2(r) > c2 r 2 . Since o2 ( ·) is a class IC00 function that satisfies (13.46) this o2(·) can replace a2(·) in (13.44). Also, (13.47)

which in turn implies, utilizing technical assumption (iii), that (13.48) So Lemma 10.5.1 can be applied to deduce the existence of a class IC00 function a 1 (-) and a smooth, positive definite, proper function 'C'I(zl), satisfying

8~

1 1 1 az1 ft(z1,z2,6,w) S -al(llz1ID + 2o2(lwl) + 2o2(llz2ID + 2o2CI61). {13.49)

13.4 Almost Disturbance Decoupling

Let V(z) =

av oz

V1 (zl) + "C2(z2)·

201

Then,

f (z, v2 (z), w)

:S -at(llztiD + ~a2()wl) + ~a2(llz2ID + ~a2(lv2(z2)1) - (1 +

~) a2(llz2ID +12 (1 + ~) lwl 2 -

(1 +

~)

lv2(z2)12

:S -at(llztiD + ~a2(lwl)- ~a2(llz2li) + ~a2(iv2(z2)1) _.:2 a2CIIz2ID + 1 2 ( 1 + 2

~) lwl 2 - ( 1 + ~) lv2Cz2W

:S -at(ilztiD + ~lwl 2 - ~a2(llz2ll) + ~lv2(z2)l 2

+1 2 ( 1 + .:;) lwl 2 - ( 1 + ~) )v2(z2)12 :S -a1 (ilztll) - ~a2 (llz2ll) + [12 + "2 (1 + 12 )] lwl 2 2

This concludes the proof.

-

lv2(z2W .

<1

Note that condition (i) expresses a special property of input-to-state stability for the first subsystem of (13.42), viewed as a system with state z 1 and inputs w, z2 and 6- The lemma just proven shows that this subsystem has no influence in determining the lower bound for the achievable level of disturbance attenuation. Note also that value of /o which renders (13.43) satisfied is, in this context, immaterial. The result described in this Lemma is, to some extent, a counterpart, for nonlinear systems modeled by equations of the form (13.16) and in which the z-subsystem admits a decomposition as in (13.42), of the result expressed by Lemma 13.2.2, which relates the achievable level of disturbance attenuation to a property of the "unstable" component of the z-subsystem. We will pursue later, in Section 13.5, the issue of seeking an explicit bound for 1 (which would provide a sort of counterpart of the bound determined, for linear systems, at the end of the previous section). In the meanwhile, we discuss the special case in which the problem of Disturbance Attenuation, with Stability, can be solved for any arbitrarily small value of f.

13.4 Almost Disturbance Decoupling Consider the following design problem. When is it possible, for a given system of the form (13.6), to find, for any (hence arbitrarily small) value of/, a

202

13. Disturbance Attenuation

feedback law (which indeed will depend on the chosen 1) which solves the problem of Disturbance Attenuation with Stability? This situation is often referred to as the Problem of Almost Disturbance Decoupling (with Stability). The analysis presented in section 13.2 immediately yields necessary and sufficient conditions for the solvability of such a problem in the case of a linear system. In fact, from Proposition 13.2.3, we see that it is possible to solve the problem of Disturbance Attenuation, with Stability, for any arbitrarily small value of 1, if and only if the number 1• given by (13.40) is 0, and this occurs if and only if V3 = 0, which in turn implies Q3 = 0, and Q2 T X = 0 for any eigenvector x of - F 2 T. In other words, Almost Disturbance Decou piing with Stability is possible if and only if the decomposition (13.31) has the special form F1z1 + G16 + Qlw (13.51) F2z2 + G26 + Q2w i3

F3Z3

+ G36

in which a(Fl) c c-, a(F¥ c C 0 , a(F3) c c+' and every eigenvector X of - F 2 T is annihilated by Q2 . In the special case· in which the matrix F has no eigenvalues in C 0 , the necessary and sufficient condition for Almost Disturbance Decoupling with Stability is simply that the unstable component of the z-subsystem is not affected at all by the disturbance w. A similar situation for the class of nonlinear systems considered in this Chapter is the one in which the z2 -subsystem of (13.42) is independent of the disturbance w, namely system (13.42) has the following special form i1

=

!I(z1,z2,6,w)

i2

=

h(z2,6).

(13.52)

In this case, we can derive from Lemma 13.3.1 the following result, which provides a sufficient condition for Almost Disturbance Decoupling with Stability.

Corollary 13.4.1. Consider the system {13.52). Suppose that condition (i) of Lemma 13. 3.1 holds and that (ii) there exist a smooth real-valued function v2(z2), with v2 (0) = 0, and a smooth real-valued function V2 (z2 ), that is positive definite and proper, such that av2 - h(z2,v2(z2)) +v22 (z2):::; -a2(llz21) (13.53) 1

8 Z2

for some class Koo function a2. Then for every 1 > 0 there exist a smooth function v(z), with v(O) and a real-valued, positive definite, proper function V(z) such that

av

2 2 az f(z, v(z), w) :::; -a(llzll) + 1 2w - v (z)

for all z and all w, where a(·) is a class Koo function.

= 0,

13.4 Almost Disturbance Decoupling

203

Proof. Let 0: 2 ( ·) be any class Koo function satisfying

for small r for larger.

{ o2(r)

_

r2 ja2

o2(r) =

Note that 0: 2(r) = O(o2(r)) as r -t o+. Thus, as shown in the proof of Lemma 10.5.1, there exists a smooth function q : JR.~ 0 -t lR.~o, positive and non decreasing, satisfying

Without loss of generality, suppose q(r) Define

p(r) and set V2(z2)

~

1.

=for q(s)ds

= p(V2(z2)). In view of condition

(ii), this yields

av2 av2 - f2(z2, v2(z2)) = q(V2(z2))- h(z2, v2(z2)) 8 8 Z2

Z2

< -0:2(ilz2ll)- q(V2(z2))iv2(z2W :::;

-0:2(ilz2ID -lv2(z2W ·

This shows that condition (ii) of Lemma 13.3.1 holds with 'Y = 0, for a function 0: 2 (r) which by construction satisfies condition (iii) of Lemma 13.3.1. Thus, the result follows.
i2

=

h{z2)

+ g2(z2){1 .

In this case, in view of Lemma 13.1.4, the inequality appearing in condition (ii) of Corollary 13.4.1 can be given the expression

av2 1 [av2 ]2 az h(z2)- 4 az 92(z2) :;; -o2(i1z2~1) . 2

2

To simplify the notation, let the latter be rewritten as

av 1 [av ]2 ax f(x)- 4 axg(x) :::; -o(llxll)'

(13.54)

(note that this is the particular form assumed by the inequality (13.24) when = 0, p(x) = 0 and r(x) = 1) and observe that, if an inequality of this form holds, necessarily V(x) must be such that

h(x)

av

axg(x)

=0

::::?

av

ax f(x) :::; -o(llxll) .

(13.55)

204

13. Disturbance Attenuation

In other words, a function V(x) which satisfies (13.54) is necessarily a control Lyapunov function (see section 9.4) for the system

x = f(x) + g(x)u.

(13.56)

In view of this, the problem naturally arises of when, i.e. under what hypotheses, from the knowledge of a control Lyapunov function V(x), it is possible to determine a function (perhaps different from V(x) itself) which satisfies the inequality (13.54). This problem is partially answered in the following result. Lemma 13.4.2. Suppose V(x) is a smooth control Lyapunov function for {13.56), i.e. a smooth real-valued positive definite and proper function satisfying {13. 55) for some positive definite function a(·). Suppose there exist a neighborhood S of x = 0 and a real number K > 0 such that

av [av, ax f(x)- K ax g(x)

)2 < 0

(13.57)

for all x E S \ {0}.' Then there exists a smooth real-valued positive definite and proper function U(x) satisfying, for all x,

au

1[au

-f(x)-- -g(x) ax 4 ax

)2 ~ -a(JJxJI) _

(13.58)

for some positive definite function a(·). If a(·) in {13.55} is a class IC 00 function, then there exists a smooth real-valued positive definite and proper function U(x) satisfying {13.58} for some class /Coo function a{). Proof. By hypothesis, the left-hand side of {13.57) is negative definite in a neighborhood of x = 0. Thus, after possibly having redefined the function a(·) for which {13.55) holds, it can be observed that, for some a> 0

llxll

.)2 ~ -a(llxli).

av [av ax f(x)- K ax g(x)



Set

M

= {x E IRn

av and observe that ax g(x)

u(a)

=

av : ax f(x)

f:. 0 on

{13.59)

+ a(iixli)?: 0},

M \ {0}. Define, for any a E {0, oo),

max {z:llzll=a}nM

av a f(x) + a(j)xj/)

x

[~~ g(x)]

2

By (13.59), u(a) ~ K for small a. Let g{) be any class IC 00 function satisfying Q(lxl) ~ V(x). Let q : IR~o --+ IR~o be any smooth, positive and

13.4 Almost Disturbance Decoupling

non decreasing, function such that q(r) at each x E M \ {0},

> 4uog_- 1 (r).

205

Then, by construction,

av -f(x) + n(llxll) q(V(x)) > 4u(llxll) 2:4 OX av 2 [axg(x)]

(13.60)

Consider the class Koo function p( ·) defined in the proof of Corollary 13.4.1 and set U(x) = p(V(x)). Then 8U f(x) - ~ [8U g(x)] 2 ox 4 ox

= q(V(x)) 8V f(x) ox

[q(V(x)) 8V g(x)] 2 2 OX

(13.61)

From this, it is easy to check that

au 1[au )2 ox f(x)- 4 ox g(x) ::; -q(V(x))n(llxll) .

(13.62)

In fact, if x rf_ M, then av q(V(x)) ox f(x)

< -q(V(x))n(llxll),

while, if x E M \ {0}, then (13.60) yields - [

q(V(x)) av ox g(x) 2

]2 < -q(V(x)) av ox f(x)- q(V(x))a(llxll) .

Since -q(V(x))n(llxll) ::; -[qog_.](llxll)n(llxll) and [qog_](·)n(·) is a positive definite function, (13.62) completes the proof. <1 Addressing the problem of Almost Disturbance Decoupling with Stability, this result can be used in order to check the fulfillment of condition (ii) of Corollary 13.4.1. In fact, if the second equation of (13.52) is affine in {I and a control Lyapunov function is known which satisfies a condition of the form (13.57), condition (ii) of the Corollary holds. From a different point of view, it may be worth observing that, if V(x) is a control Lyapunov function for (13.56) which satisfies the special condition (13.57), then this system can be globally asymptotically stabilized by smooth feedback (recall- from section 9.4 -,that, if no special assumptions are made, the knowledge of a control Lyapunov function implies stabilizability by means of a feedback law which is guaranteed to be smooth only on !Rn \ {0} ). Corollary 13.4.3. If there exists a control Lyapunov function V(x) satisfying the condition (13.57) of Lemma 13.4.2, then there exists a smooth realvalued positive definite and proper function U(x) such that, for any k 2: 1, the smooth feedback law k8U u(x) = ---g(x) (13.63) 4 OX globally asymptotically stabilizes the equilibrium x = 0 of (13.56).

206

13. Disturbance Attenuation

Proof. For any k

~

1

au ax [f(x)

+ g(x)u(x)]

k [au )2 = au ax f(x) - 4 ax g(x) < 0

for all nonzero x, and this completes the proof.

<1

Remark 13.4.1. Note that a smooth stabilizing feedback for (13.56) may exist under hypotheses weaker than (13.57). For, suppose there exist a neighborhood S of x = 0 and a smooth function k : S -+ lR, with k(O) = 0, such that av av (13.64) ax f(x) + k(x) ax g(x) < 0 for all xES\ {0}. Then, it is easy to show that there exists a smooth feedback law u = u*(x) which globally asymptotically stabilizes the equilibrium x = 0 of (13.56). In fact, in section 9.4 it was shown that, if V(x) is a control Lyapunov function for (13.56), then there,exists a smooth function k: IR" \ {0} -+ lR such that av ax [f(x) + g(x)k(x)] < 0 for all nonzero x. Let c : lRn -+ lR be a smooth function satisfying c(x) = 1

for xES' C S

0 :S: c(x) :S: 1 =0

forx¢S

c(x)

where S' is some neighborhood of x = 0. Then u*(x)

= c(x)k(x) + (1- c(x))k(x) .

defines a smooth function on lRn. The feedback law u av ax [f(x)

= u*(x) is such that

+ g(x)u*(x)]

= [I -

av

c(x)] ax [f(x)

-

av

+ g(x)k(x)] + c(x) ax [f(x) + g(x)k(x)] < 0

for all nonzero x. Thus, if V(x) is a control Lyapunov function for (13.56) which satisfies (13.64), there exists a smooth feedback which globally asymptotically stabilizes the equilibrium x = 0 of (13.56). Note also that {13.57) trivially implies (13.64). However, (13.57) implies- as shown in the previous corollary - the existence of a smooth stabilizing feedback of the special form (13.63).
13.5 An Estimate of the Minimal Level of Disturbance Attenuation

207

13.5 An Estimate of the Minimal Level of Disturbance Attenuation In this section, we address in part the issue of determining an estimate of the "minimal" value of 'Y for which an inequality of the form (13.44) can be solved. As explained in Section 13.3, such an estimate will provide also an estimate of the minimal level of disturbance attenuation achievable in a system of the form (13.17). For simplicity, we consider the case in which the underlying system .i2 = h(z2, 6, w)

is affine in both

6

and w, i.e. in which system (13.42) assumes the form i1

h(z1,z2,{i,w)

i2

h(z2)

+ g2(z2)6 + .P2(z2)w.

In view of Lemma 13.1.4, the inequality (13.44) is equivalent to the inequality av2 1 [av2 8z2h(z2)-4 8z2g2(z2)

]2 +4721 [av2 ]2 8z2P2(z2) ~-a2(llz2Ji).

which, in order to simplify the notation, will be rewritten as av f(x) - ~ [av g(x)] 2 + _1_ [av p(x) l2 ax 4 8x 4"(2 8x J

~ -a(llxll)

,

(13.65)

(note that this is the particular form assumed by the inequality (13.24) when h(x) = 0, and r(x) = 1). In what follows, we address the problem of constructing a solution V(x) of the inequality (13.65) from the knowledge of the solution of a similar inequality written for a system of lesser dimension. The price that we pay for this "reduction" of dimensionality , however, is that we cannot obtain a solution valid for all x E an, but valid only on some compact subset of lR". As we will see in a moment, this limitation is acceptable, from a certain point of view, if the set in question can be arbitrarily large. The approach is to some extent similar to the one that replaces the requirement of global stability by that of asymptotic stability with arbitrarily large basin of attraction. In the problem of asymptotic stabilization, the requirement was to obtain, via feedback, convergence to zero for all initial conditions in any prescribed (but arbitrarily large) compact set. In the problem of Disturbance Attenuation, the corresponding idea is to obtain, via feedback, a prescribed level "( of disturbance attenuation, for any bounded set of disturbances in L 2 [0, oo). In order to see that achieving disturbance attenuation for all disturbances in a bounded subset of £ 2 [0, oo) actually corresponds to solving a dissipation inequality for all x in a suitable compact subset of llln, consider again a system of the (general) form

208

13. Disturbance Attenuation

:i:

=

f(x,w,u)

y

=

h(x,u)

and suppose that, for some smooth feedback law u = u(x), the basic dissipation inequality for the estimation of the £ 2 gain, namely (13.66) holds for all w E JR, but only for all x E [}7 , K, where il7 2K

= {x E lRn: V(x):::; ·lK}

and K > 0 is a fixed number. Suppose also that the disturbance input satisfies

Then, it is easy to realize that any trajectory of

starting at x(O)

:i:

f(x,w,u(x))

y

h(x, u(x))

= 0 remains in il7 2K for all t 2: 0 and, consequently,

looo ly(r)l dr:::; '"( looo lw(rWdr. 2

2

(13.67)

In fact, integration of (13.66) along any of such trajectories yields

V(x(t)) :::; --y 2

lot lw(rWdr- lot iy(rWdr:::; --y lot lw(rWdr:::; --y K, 2

2

which shows that x(t) cannot leave il7 2K and therefore (13.66) again yields (13.67). Moreover, inequality (13.66) also shows that, if w = 0, any trajectory with x(O) E il7 2K remains in il7 2K for all t 2: 0 and converges to 0 as t ~ oo. Therefore, having an input u(x) which renders the inequality (13.66) fulfilled for all w and for all x E [}7 , K guarantees that in the resulting closedloop system: (i) the equilibrium x = 0 is asymptotically stable, with basin of attraction which contains []7 , K, (ii) for any disturbance input whose £ 2 [0, oo) norm does not exceed K, the L 2 gain between w(·) andy(·) does not exceed 'Y· With this and all prior results developed in this Chapter in mind, we consider now the following problem. Let 'Y > 0 be a fixed number. Given any K > 0, find a smooth function V(x), that is positive definite and proper, such that the inequality (13.65) holds for all x satisfying V(x):::; "f 2 K.

13.5 An Estimate of the Minimal Level of Disturbance Attenuation

209

Theorem 13.5.1. Consider the system

= =

i1 i2

fi(xi,X2) +p1(X1,x2)W h(xi,x2)+P2(XI,x 2)w+u,

(13.68)

with x 1 E IR"- 1 and X2 E IR, in which ft(x1JX2), h(x1,x2), PI(XI,X2), p2(x 1,x2) are smooth functions and h(O,O) = 0, /2(0,0) = 0. Set

and ft(X1JX2) h(x1,x2)

= =

fu(xi) + ft2(X1,X2)X2 h1(xi) + fn(x1,x2)x2,

and

= = =

Pt(xi) p;(xi) o* (xi)

P1(x1,0) P2(XI,0) o(xb 0)

and

!{1(xi) = 8*(xi)fu(xi) Suppose that, for some l

+ ..;.hl(xi)p;(xi)Pt(xi). 'Y

> 0, O(XI,X2)

~ f

(13.69)

for all x 1 , x 2 , and that there exist a smooth positive definite and proper function VI(xi), and a class K junction o 1 (·), satisfying the inequality 1 ( av1 • )2 av1 • 2 axl fu(xi) + 4"(2 ax/l(xi) + (hl(xi)) ~ -ol(llxliJ)

(13.70)

and such that, for some c > 0, (13.71)

for all x1. Then, given any K > 0, there exist a smooth positive definite and proper junction V(x) and a class K functi~n o(·) such that av f(x) ax where x

= (x 1 , x 2 )

f(x)

=(

~ [av g(x)] 2 + _1_ [av p(x)] 2 ~ -o(ilxiJ) 4

ax

4"(2

(13.72)

ax

and

h(x1Jx2) ) , p(x) f2(XI,X2)

for all x E {x E IRn : V(x) ~ "(2 K}.

=(

P1(x1,x2) ) , g(x) P2(XI,X2)

=(

0 ) , 1

210

13. Disturbance Attenuation

Proof. Fix K > 0. Let V(x) set such that

= V1(xl) + ,Bx~,

,B 2:: 1. Let B1 be any compact

and let

B2

= {x2 E IR: Jx2i :51../K} ·

Then, it is easily seen that V(x) :5 1 2 K implies x 1 E B1 and ,B ~ 1. Thus, {x E !Rn : V(x) :5 1 2 K} C B1 x 82 .

X2

E B2, for all

Simple, but lengthy, manipulations show 1 that there exists a number ,B* such that, for all ,B > ,B*, the function V(x) thus defined satisfies (13.72) for all X E 81 X 82 .
X1 y

= Jr1 (xl) + Pt(xl)u =

h1(xr),

has an L 2 -gain which is less than or equal to I· Remark 13.5.1. Note that the requirements that §*(xl) > 0 and that function vl (xl) satisfies the inequality 2 8Vi * 1 ( av1 * ) ) 2 8x1 fu (xl) + 4-y2 ax/1 (xi) + (!21 (xl ) < 0

for x 1

(13.73)

#- 0, imply that

8 6*(xl) Vi !11(x1) 8 Xt

for x 1

#- 0,

i.e. in particular

8Vi

- !11(xr) < 0, for x1 #- 0. 8 X1 This shows that the asymptotic stability of the subsystem

X1 = f11(xt) is a prerequisite for (13. 7b) to hold. 1


For more details, see Isidori et al. (1999).

(13.74)

13.5 An Estimate of the Minimal Level of Disturbance Attenuation

211

Theorem 13.5.1 also contains the technical condition (13.71). However, it is easy to see that such a condition is immaterial if the class K function o: 1 ( ·) which appears in (13.70) is quadratic. As a matter offact, since all functions on the left-hand-side of (13.71) vanish at x 1 = 0 together with their first derivatives, it is readily seen that, for any compact subset S C JRn-l , there exists c > 0 such that

max {

/h1/ , 2

1h1

av1 x jj8VJ. , ox fu 0 1

1

llavll , ox

2 }

1

:S cl/xiii

2

for all x 1 E S. This yields the following simplified version of Theorem 13.5.1. Corollary 13.5.2. Consider the system (13.68}. Suppose that, for some E> 0, a (x 1 , x2) 2: E for all x. Suppose that there exists a smooth positive definite and proper function (xi) satisfying the inequality (13. 70}, with 0:1 (/lXI 1/) = 2 f11/X1 1/ · Then, given any K > 0, there exist a smooth positive definite and prop.er function V(x) and a> 0 such that

vl

8V f(x) _ ![oV g(x)]2 8x 4 ox

+ _1_[8V p(x)]2 :S 2 4-y

ox

-a/lxl/ 2

for all x E {x E !Rn: V(x) :S -·?K}. As an example of how the result of Theorem 13.5.1 can be successfully utilized in order to determine a lower estimate of the minimal value of achievable disturbance attenuation, consider the simple case in which n = 2. Write fu (xi) = au (xdx1 h1 (xi) = a21 (xl)x1 . Let V1(x1) = 2 J0"'' A(s)sds, where A(s) is continuous, positive and bounded away from zero. This yields

8VJ.

2

- fu (xi) = 2A( x1 )au (xl)x 1 . 8 XJ In order to satisfy the necessary condition (13.74), assume that a11 (xi) < 0. Requirement (13. 70), with o: 1 (r) = f~r 2 , can be fulfilled when 2A(x1) {-a*(xl)au (xi) - a21 (x1 )pt(xi)p;(xih- 2) (13.75) --y- 2pt 2(xi)A 2 (xi)- a~ 1 (xi) > f~. A positive solution A(xr) to (13.75) can only be found if there are real, distinct, positive roots. This combined with the condition that §*(x 1 ) must be positive yields the following explicit estimate for 'Y 'Y

I

> /pt(xr)a 2I (xi)/ + ! lpt(xl)a21 (xi) + 2p;(xi) + /pt(xi)/ fi . ( 13 . 76) 2/au(xl)/

2

/au(xi)/

/au(xJ)/

212

13. Disturbance Attenuation

13.6 L 2 -gain Design for Linear Systems In the previous sections, we have considered a number of special cases of the dissipation inequality (13.23), or of the Hamilton-Jacobi-type inequality (13.24), which is equivalent to (13.23) if r(x) =/: 0 for all x: the inequality (13.8), which corresponds to the special case in which r(x) = 0, the inequality (13.54), which corresponds to the special case in which h(x) = 0, p(x) = 0 and r(x) = 1, and the inequality (13.65), which corresponds to the special case in which h(x) = 0 and r(x) = 1. In this and in the next section we discuss the possibility of determining solutions of the dissipation inequality (13.23) in the more general case, not examined so far, in which both h(x) and r(x) are nonzero. In other words, we consider a system modeled by equations of the form x f(x) + g(x)u + p(x)w (13.77) h(x) ) , y ( r(x)u with state x E !Rn, control input u E IR, disturbance input w E !Rr and output y E IR2 , with f(O) = 0, h(O) = 0 a.itd r(x) =/: 0 for all x, and we seek a feedback law u = u(x) rendering the system strictly dissipative with respect to the supply rate (13.78) where 1 is a fixed number, i.e. rendering the inequality (13.23) fulfilled, for some positive definite and proper V(x), for all w and x. For convenience, and also in order to present some material that will be used in the sequel, we examine first the case in which system (13.77) is a linear system and a linear feedback law is sought which renders the system dissipative with respect to the supply rate (13.78). Without any extra complication we can handle the case of a system in which u E !Rm, w E !Rr and y E JRP+m, i.e. the case of a system modeled by equations of the form

x =

Ax+Bu+Pw (13.79)

y where we assume that the m x m matrix R is nonsingular. First of all, it is important to observe that, from Theorem 10.9.1 and Lemma 13.1.4 it is easy to deduce the following result. Lemma 13.6.1. Consider the linear system {13. 79) and let 1 be a fixed number. The following are equivalent: (i) there exists a linear feedback law which renders the resulting closed-loop system strictly dissipative with respect to the supply rate

13.6 £2-gain Design for Linear Systems

213

for some i' < 'Y, (ii) there exist a matrix K and a symmetric matrix X > 0 such that

(iii) there exists a symmetric matrix X > 0 such that ATX

+ XA+ CTC- XB(RTR)- 1 BTX + \xppTX < 0.

(13.81)

'Y

Proof. The implication (i){::}(ii) is a direct consequence of Theorem 10.9.1. Moreover, the proof of Lemma 13.1.4 shows that if (13.80) holds for some K and some symmetric matrix X > 0, and R is a nonsingular matrix, then necessarily X satisfies (13.81). Conversely, if a symmetric matrix X > 0 satisfies (13.81), then this X satisfies (13.80) with

This completes the proof.


Thus, the problem we are considering can be solved if and only if there exists a symmetric matrix X > 0 which satisfies the matrix Riccati inequality (13.81). In what follows, we describe a set of necessary and sufficient conditions, all of which are easily verifiable, for the existence of the solution of a matrix inequality of this type. For convenience, in order to have inequality in question rewritten in a standard form, set G=CT, and Y

= x-1

(recall that the required solution X of (13.81) is nonsingular). Then, to find a solution X > 0 of the inequality (13.81) reduces to find a solution Y > 0 of an inequality of the form YF

+ FTY -

yGGTY

+ Q > 0.

(13.82)

Now, if the pair (F,G) is stabilizable, the existence of a solution of this inequality can ea.<>ily be determined by means of the following Lemma 2 , which wa.<> already used in the proof of Theorem 10.9.1, to prove the equivalence of different conditions ensuring that the £ 2 gain does not exceed a fixed number 'Y > 0.

2

For a proof, see Knobloch et al. (1993), Appendix A.

214

13. Disturbance Attenuation

Lemma 13.6.2. Suppose the pair (F, G) is stabilizable. The following three properties are equivalent: (i) there exists a symmetric matrix Y solving the inequality {13.82}, (ii) there exists a symmetric matrix y- such that y- F

+ pTy--

y-GGTy-

+ Q = 0,

a(F- GGTY-) C

c-

(13.83)

(iii) the Hamiltonian matrix H= ( F -Q

-GGT) _pT

(13.84)

has no eigenvalues on the imaginary axis. Suppose one of these conditions holds. Then, the solution y- in (ii) is unique and is such that the stable invariant subspace of H can be expressed as

Vs

= span ( /- )

.

(13.85)

Moreover, any solution Y of {13.82} is such that Y < y-. Finally, there exist a number co > 0 and a family of symmetric matrices Ye:, defined for c E (0, co) and continuously depending on c, such that lime:--.o ~ = y-, which satisfy

Thus, if the pair (F, G) is stabilizable, a positive definite symmetric solution Y of (13.82) exists if and only if the Hamiltonian matrix H has no eigenvalues and its stable invariant subspace can be expressed in the form (13.85) with y- a positive definite matrix. In the more general case in which the pair (F,G) is not stabilizable, a more elaborate test is needed, which can be described as follows. First of all, observe that the inequality remains unchanged under a transformation of the form G

f-t

SG,

with S a nonsingular matrix. Then, observe also that S can be chosen in such a way that (13.86)

in which the pair (Ft, Gl) is stabilizable, the eigenvalues of F2 are in C 0 , the eigenvalues of F3 are in c+ and K 2 , K 3 are suitable matrices. To this end, take any transformation highlighting the decomposition into controllable/uncontrollable parts

13.6 L2-gain Design for Linear Systems

TFT-1

=(Fe

215

Feu) Fu '

0

Then, using the fact that (Fe, Ge) is controllable, choose a matrix L such that the none of the eigenvalues of Fe +GeL is an eigenvalue of Fu and solve for M the Sylvester equation

Then, it easily seen that

In this way, we have found a transformation

T such that (13.87)

for some K. Now, split the Jordan form of Fu into three blocks, corresponding to eigenvalues in c-, C 0 and c+. Grouping together the first one of these blocks with the upper block of the decomposition (13.87) yields exactly a decomposition of the form {13.86) with the required properties. Let Q 1, Q 12 and Q2 be the (1,1), (1,2) and (2,2) blocks of the corresponding partition of (S- 1 )T QS- 1 • Then, the following Theorem 3 provides a general test for the solvability of the inequality (13.82). Theorem 13.6.3. There exists a symmetric matrix Y solving the inequality {13.82) if and only if there exist a symmetric _matrix Y1 and a matrix Y12 such that

+ Q1 = 0, G1GJY1) c c-

Y1F1_ + F(Y;-_ Y1G1GJY1

O"(F1 (F1- GIGfY1)Y12

+ y12F2 + Y!G!K2 + Q12 = 0

(13.88) (13.89)

x*[Q2 + KJ K2- (K2- GiY12)T(K2- GTY12)]x > 0 (13.90) hold for any (possibly complex) eigenvector x of F2. There exists a symmetric matrix Y > 0 solving the inequality {13.82) if and only if there exist a symmetric matrix Y1 > 0 and a matrix Y 12 such that the conditions above hold. As anticipated, the verification of the conditions in this Theorem is not difficult. First of all, observe that, since by hypothesis the pair (F1 , Gt) is stabilizable, one can use Lemma 13.6.2 to check the existence of Y1 satisfying (13.88). This occurs if and only if the Hamiltonian matrix 3

For a proof, see Scherer (1992).

216

13. Disturbance Attenuation

has no eigenvalues on the imaginary axis. If this is the case, then Y1 is unique and can be computed by expressing the stable invariant subspace of this Hamiltonian matrix in the form

Vs

= span (

JJ .

Then, since (F1 - G 1 GJY1) has all eigenvalues inc- and F 2 has all eigenvalues in C 0 , the Sylvester equation (13.89) has a unique solution Y12 . Using this matrix Y12 the third condition (13.90) must be checked, which involves only a finite number of tests.

13.7 Global L 2 -gain Design for a Class of Nonlinear Systems We return now to the case of a nonlinear system modeled by equations of the form (13.77), with state x E JRn, control input u E lR, disturbance input wE lR and output y E JR2 , with f(O) = 0, h(O) = 0 and r(x) -:f. 0 for all x, and we seek a feedback law u = u(x) rendering the system strictly dissipative with respect to the supply rate (13.78), where 1 is a fixed number, i.e. rendering the inequality (13.23) fulfilled, for some positive definite and proper V(x), for all wand x. As shown before, this is the same as finding a pair {u(x), V(x)} which renders the inequality 8

~~x) [f(x)+g(x)u(x)]+ 4 ~ 2 [ 8 ~~x) p(x)r +h2 (x)+r 2 (x)u 2 (x)::;

-a(llxiJ)

(13.91) fulfilled for all x. To find a pair {u(x), V(x)} which solves this inequality for all x in a (possibly small) neighborhood of the point x = 0 is not terribly difficult, and can be accomplished by determining the solution of a suitable Riccati inequality constructed from the parameters which characterize the linear approximation of the system (13.77) at this point. In fact, let

x

Ax+Bu+Pw (13.92)

y denote the linear approximation, at x

= 0, of the system in question, where

8 A= [ 8x x=o'

!]

B

= g(O),

C

= [88xh] x=o'

R

= r(O)

P .

= p(O)

13.7 Global L2-gain Design for a Class of Nonlinear Systems

217

Suppose there exist a 1 x n matrix K and an n x n symmetric and positive definite matrix X which render the strict lliccati inequality X(A

+ BK) +(A+ BK)T X+ ~xppT X+ CTC + R 2KT K < 0

(13.93)

I

satisfied. Then, an elementary calculation shows that the feedback law ll(x) = Kx and the positive definite and proper function V(x) = xT Xi satisfy (13.91) for all x in a neighborhood of x = 0, with o(r) = cr 2 and c > 0. Obviously, the linear feedback law u(x) = K x and the quadratic function V (x) = x T X x thus determined fail, in general, to provide a global solution of (13.91). One may conjecture, however, that in some special cases the information obtained from the solution of the lliccati inequality (13.93) suffices to determine a feedback law (which we indeed expect to be a nonlinear function of x) providing a global solution of the inequality (13.91) for some suitable V(x). The purpose of this section is to show that this conjecture is true for a special class of nonlinear systems, if the coefficient r(x) which weights the control u in the output y of (13.77) satisfies certain bounds. More specifically, we will prove that, for that class of nonlinear systems, starting from a solution pair {K,X} ofthe strict lliccati inequality (13.93), it is always possible to construct a pair {u(x), V(x)} which solves the inequality (13.91) if lr(x)l does not exceed a suitable bound r*(x), a continuous function which is equal toR in a neighborhood of x = 0 (as it should be) but possibly decays to zero as x grows. . More precisely, suppose the system in question can be described by equations of the form X1

=

x2

X2

=

X3-i-P2(XI,X2)W

+ pi(xi)w (13.94)

Xn-1

±,

Xn

+ Pn-1 (xi, X2, · ·., Xn_I)w + U + Pn(XI, X2, ... , Xn)W

fn(XI, X2, ... , Xn)

in which PI(xi), ... ,pn(XI,X2,····Xn), fn(XI,X2,···,:r:n) are smooth functions, fn(O, 0, ... , 0) = 0, and h(x)

=XI .

For convenience this system will be represented in the form

x = y = where

Ex+ Bfn(x)

Cx

+ Bu + p(x)w

(13.95)

218

13. Disturbance Attenuation

E

(

~~~

~) ,

0 0 0 0 0 0

1 0

m,

B

Pt(Xt)

c

(1

0

0

P2(x1, x2)

p(x)

0),

)

(

Pn(Xt,X2, ... ,xn)

and its linear approximation of (13.95) at x

x y

(E

=

= 0 in the form

+ BF)x + Bu + Pw

(13.96)

Cx,

where F _ [8Jn] p = p(O). 8x :z:=o' We will show that, if the Riccati inequality X(E+BF+BK)

+ (E+BF+BK)TX + 21 xppT X

+CTC+R2KT K

'Y

<0

(13.97) is solved by some pair {K, X}, it is possible to find a pair {u(x), V(x)} which solves the Hamilton-Jacobi inequality 8V(x) ----a;-[Ex + Bfn(x)

1 + Bu(x)] + 412

[8V(x) ]2 ~p(x) + h 2(x)

+ r 2(x)u 2(x)

~ 0

(13.98) provided lr(x)l is conveniently bounded. To this end, we need a few preliminary results. Set, for i = 1, ... , n,

it,

(1 0

0

0 )1 xi'

(~L

Bi

pi

=

(:),

and observe that An= E,Bn = B,Cn = C,Pn = P. It is easy to prove - by induction - that, if the inequality (13.97) holds, then, for each i = n- 1, n- 2 ... , 1, there exist K; and symmetric matrices X; > 0 and Qi > 0 such that

13.7 Global £2-gain Design for a Class of Nonlinear Systems

X; ( A;+ B;K;) +(A;+ B;K;) T X;+ 21 X;P;P;T X;+ C;T C; 'Y

= -Q; .

219

(13.99)

This depends on the following Lemma, whose proof uses arguments already used to prove the implication (ii) :::? (iii) in Lemma 13.2.1 and is not repeated here. Lemma 13.7.1. Suppose there exist a 1 xi matrix K; and i xi matrices X; = X{ > 0 and Q; = QT > 0 such that {13.99) holds. Let K;_ 1 be the unique solution of ( 01x(i-1)

1) X·t (

I(i-1)x(i-1)) _

K;_

-

1

O1x(t-1) . ·

Define

Then K;_ 1 , X;_ 1 X;-1

(Ai-1

( J(i-1) x(i-1)

KT 1 ) X

( l(i-1) x(i-1)

KT

•-

•- 1

(J(i-1)x(i-1))

K;-1

'

)Q·(J(i-1)x(i-1)) ' K;-1

> 0, Qi-1 > 0 are such that

+ B;-1Ki-d + (Ai-1 + B;-1K;_I)T X;-1

T T + 21 X;-1Pi-1pi-1Xi-1 + Ci-1 C;-1 'Y

-

-

(13.100) -Qi-1 .

For any C 1 function v(x) which vanishes at x = 0, let v1 1l(x) denote its first-order approximation at x = 0, i.e.

8 = [fJx v] z=D x.

v1 1l(x)

Likewise, for any C 2 real-valued function V(x) which vanishes at x = 0 together with all its first partial derivatives, let V1 2l(x) denote its secondorder approximation at x = 0, i.e.

Vl2l(x) =xT![~(fJV)T] 2 fJx

Set also

fJx

X. z=O

~=·(:}

and p;(xi)

~~~~1L) )

=(

p;(xl> ... , x;)

Then, the following result holds.

220

13. Disturbance Attenuation

Lemma 13.7.2. Suppose K;,X; = X{ > o,Q; = QT > 0 are such that {13.99) holds and let K;- 1 ,X;_ 1 ,Q;_ 1 be defined as in Lemma 13.7.1. Suppose there exist (i) a positive definite and proper smooth function Vi-I (x;_I) such that

~~ 11 (x;_I) = xT_ 1X;-Ixi-I , (ii) a set of smooth functions cp 1(x 1), cp 2(x 2), ... , cp;- 2(x;-2) vanishing at zero together with their first partial derivatives, (iii) a smooth function

Vi-I

= K;-IXi-lJ

(x;_ 1 ) vanishing at zero and such that v!~ 1 (Xi- I)

satisfying oVi-I (A;-IXi-1 {) Xi-1

+ 21 4"(

+ B;-IVi-1(x;_t)) + (Ci-1X;-d 2 .

(13.101)

(avi-1 )2 = -Si-1 (x;_t) ~Pi-1 (x;_I) UJI.l-1

for all X;-1, where

Then, there exist (iv) a positive definite and proper smooth function Vi(x;) such that

~[ 2 ! (x;) = xT X;x; ,

(v) a smooth function cp;- 1 (x;_I) vanishing at zero together with its first partial derivatives, 1

(vi) a smooth function v; (x;) vanishing at zero and such that v\ 1(x;) satisfying 2 8Vi (A;x; + B;v;(x;)) + 4"(2 1 ( 8Vi OX; ax/;(x;) ) + (C;x;) 2

for all

X;,

= -S;(x;)

= K;x;, (13.102)

where

S;(x;)

=

X2-

~1(xt)

)

T

Q,· (

X2-

~1(xt)

(

X; -

'Pi-1 (xi-1)

x; -

'Pi-1 (x;_I)

)

(13.103)

13.7 Global £2-gain Design for a Class of Nonlinear Systems

221

Proof. Set Z

= (Olx(i-1) l)Xi(Olx(i-1) T; = ( l(i-1) x(i-1) K'f__1) 01x(i-1)

1

1)T, '

and observe that

T;X;Tl

=(

X;-1 01 X (i-1)

o(i-1)x1)

z

.

(13.104)

The proof consists in choosing

Vi(x;)

= Vi-i{x;_i) + Z(x; -Vi-i (x;-1)) 2

and showing that one can find v;(x;) and cpi-l (xi_ I) such that (13.102) holds. To begin with, using (13.104), note that the function Vi(x;) thus defined satisfies the property indicated in (iv). To prove that (13.102) can be enforced, observe that, for this choice of V; (xi),

av:.

-a '(Aixi + B;v;(x;)) = X;

and

aV; aVi-1 ( -a Pi(x;) = -a--Pi-1 x;_I) X; Xi-1 avi-1 )) + 2Z(x;- Vi-:-I(X;-i)) ( p;(xi)- -a-. -Pi-1(Xi-l x,_1

·

Therefore the left-hand side of (13.102) reduces to an expression of the form

with a(x;) a smooth function which vanishes at Set

S;(x;)

=

Xi

= 0.

222

13. Disturbance Attenuation

and observe that, because of the special form of T;, this function has the required structure (13.103), in which IP•-1 (x,_I) = v,_1 (x;_I) - K,_1Xi- 1 + Ki- 1 (

cp}xi) ) ft't-2

(Xi-2)

is a smooth function vanishing at X;- 1 = 0 together with its first derivatives. Also, note that S;(x;) can be written in the form

with b(xd a smooth function which vanishes at x; = 0. Thus,

v;(xi)

= 21z(a(x;)- b(x;))

renders (13.102) satisfied. To complete the proof, it remains to show that the property indicated in (v) holds. Set vJ 1l(x;) = H;x; and consider the second order approximation of both sides of (13.102) near x; = 0. This yields X;(A;

+ B;H;) +(A;+ B;H;) T Xi+ 21 X;PiPiT X;+ C;T C; I

= -Q;.

Subtracting off from (13.99) one obtains X;B;(H;- K;)

+ (H;- Ki)TBJX; = 0,

which is a Sylvester equation with unique solution H; positive definite, and this completes the proof. <1

= K;,

because X; is

Using these two Lemmas it is possible to prove the following intermediate result. Lemma 13.7.3. Consider system {13.95). Suppose that, for some 1 > 0, there exist a 1 x n matrix K and n x n matrices X = XT > 0 and Q = QT > 0 such that

X(E+BF+BK)+(E+BF+BK)TX+~XPPTX+CTC = -Q. 'Y

(13.105)

Then there exist a smooth state feedback law u(x), a smooth function V(x) which is positive definite and proper, and a smooth function S(x), which is positive definite, such that u[ 1l(x)

=

V[ 2l(x)

=

S[ 2l(x)

Kx xT Xx xTQx,

(13.106)

13.7 Global £2-gain Design for a Class of Nonlinear Systems

223

which render the identity av 1 [av ]2 ax [Ex+ Bfn(x) + Bu(x)] + axp(x) + [Cx] 2 = -S(x) 412

(13.107)

satisfied for all x. Proof. Starting from (13.105), use Lemma 13.7.1 "backward" n times, arriving at an identity of the form 1

2

2

2X1(A1 +B1K1) + 2X1P 1 + 1 = -Ql. 1

xr

Set V1 (xi) = X 1 and observe that there exists a function v1 (xi), van(xi) = K 1x 1, which satisfies ishing at x 1 = 0 and such that

vPl

a Vi_ ax [A1x1 + B1v1(x!)] + 1

[a

1 Vi. ]2 ax p!(x!) 412 1

+ [C1x1] 2 = -Q 1x 21 .

From this, using Lemma 13.7.2 "forward" n times, arrive at an identity of the form av ax [Ex+ Bv(x)]

1 [av ]2 2 + 412 axp(x) + [Cx] = -S(x)

in which V(x) is positive definite and proper, S(x) is positive definite, and

vf 1l(x) y[2l(x) Sf 2l(x)

Setting u(x) = v(x)- fn(x)

proves the Lemma.

<J

We are now ready to draw the desired conclusion. Theorem 13.7.4. Consider system {13.95). Suppose that, for some pair of real numbers 1 > 0 and R > 0, there exist a 1 x n matrix K and n x n matrices X = XT > 0 and Y = yT' > 0 such that

(13.108) Set Q = Y + R 2 KT K and {based on Lemma 13. 7. 3) find a smooth state feedback u(x), a smooth positive definite and proper function V(x), and a. smooth positive definite function S (x) such that ( 13.106) holds and the identity {13.107) is fulfilled for all x. Set

224

13. Disturbance Attenuation

r*(x)

•

r (x)

=R =

y's(X5 lu(x)l

if

R 2 u 2 (x) ~ S(x)

if

R u (x) > S(x).

(13.109)

2 2

Then, the function r*(x) is well-defined and continuous at each x E !Rn, and satisfies r*(O)=R, O~r*(x)~R. Moreover, if lr(x)l ~ r*(x),

av 1 [av ]2 2 2 2 OX [Ex+ Bfn(x) + Bu(x)] + 41'2 oxp(x) + {Cx] + r (x)u (x) ~ 0 (13.110) for all x. Proof. Let

r

denote the set

r = {x E !Rn: R 2 u 2 (x) We first show that construction,

r

~ S(x)}.

contains an open neighborhood of x S[2 l(x) 1

u[ l(x)

= =

= 0.

In fact, by

xT(Y+R 2 KTK)x, Kx.

Thus, the function S(x)- R 2 u 2 (x) is positive definite in a neighborhood of x = 0 (recall that Y is by hypothesis positive definite) and x = 0 is in the interior of r. From this, since S(x) > 0 for any nonzero x, we can deduce that the entire set To= {x E !Rn : u(x) = 0} is in the interior of r. Thus, the function r* (x) is well defined at each x E !Rn. The same argument also shows that the function

y's(X5 lu(x)l is continuous on some open set rl :::J !Rn - r and since its value on ar is equal to R, the function r* (x) is continuous at each x E !Rn. Indeed, r* (0) = Rand O~r*(x)~R.

Finally, by construction, u(x), V(x), Q(x) are such that, for any function r*(x),

av 1 [ov ] OX [Ex+ Bfn(x) + Bu(x)] + 41'2 oxp(x)

2

=- S(x)

+ [Cx] 2 + r* 2 (x)u 2 (x)

+ r* 2 (x)u 2 (x).

The choice of r* (x) indicated in the theorem yields

-S(x)

+ r* 2 (x)u2 (x)

for all x and this completes the proof.

<J

~ 0

13.7 Global L2-gain Design for a Class of Nonlinear Systems

225

Remark 13. 7.1. Note that, in the previous Theorem, the inequality (13.91) has been solved for a(·) = 0. Thus, the resulting closed loop system, namely system x Ex+ Bfn(x) + Bu(x) + p(x)w y

(

r(x~:(x))

'

is dissipative but possible not strictly dissipative with respect to the supply rate q(w,y) = -y 2 w 2 - jjyjj 2 . Nevertheless, the system is guaranteed to be globally asymptotically stable, since it is zero-state detectable, as a simple check shows (see section 10.7).
14. Stabilization Using Small Inputs

14.1 Achieving Global Stability via Small Inputs In this Chapter, we describe methods for global (robust) stabilization of nonlinear systems, by means of memoryless feedback, in cases in which the amplitude of the control input cannot exceed a fixed bound. Of course, if such a hard constraint is imposed on the amplitude of the control input, one cannot expect - in general - that global asymptotic stability is possible, unless the uncontrolled system already possesses this property to a certain extent. The simplest case in which so happens is when there exists a positive definite and proper function, whose derivative along the trajectories of the uncontrolled system is negative semi-definite but, possibly, not negative definite. In this case, in fact, under mild hypotheses, it is possible to find a smooth feedback law, whose amplitude does not exceed any (arbitrarily small) a priori fixed number, yielding global asymptotic stability. We discuss this special case first, as a point of departure for the analysis of more general structures that will be presented in the subsequent sections of the Chapter. As an introduction to the main result of the section, we begin by describing a simple, but important, stability result which is a special case of Corollary 10.8.4. Proposition 14.1.1. Consider the system

x = f(x) + g(x)u

(14.1)

in which x E ~n, u E JR, f(x) and g(x) are smooth vector fields and f(O) = 0. Suppose there exists a smooth positive definite and proper function V(x) satisfying (14.2) LtV(x) ~ 0 for all x. Suppose also that the system f(x) y

is zero-state detectable. Then, for any c: u

(14.3)

L 9 V(x)

> 0,

the feedback law

= -c:L 9 V(x)

globally asymptotically stabilizes the equilibrium x

(14.4)

= 0.

228

14. Stabilization Using Small Inputs

Proof. Observe that, by construction, system (14.1), with output y

= L 9 V(x)

is passive (see end of section 10. 7) and zero-state detectable. Thus, the result follows from Corollary 10.8.4, taking cp(y) = &y.
Then, for any£

JR

-+

JR

s

f-t

v'f+S2.

s

(14.5)

> 0, the feedback law u

= -£u(L9 V(x))

globally asymptotically stabilizes the equilibrium x

(14.6)

= 0.

Proof. The result follows again from Corollary 10.8.4, taking this time cp(y) w(y).
=

The feedback law shown in this Proposition is such that

iu(x)l < £ for all x , and therefore particularly suited in all practical cases in which the control input u of (14.1) is required to satisfy a constraint of the form

iu(t)i < U for all t. Without any special extra effort, it is possible to extend the result of this Proposition to the case of systems which are modeled by equations that are not necessarily affine in the control input, i.e. systems described by equations of the general form

x = f(x,u),

with x E JRn, u E JR, in which f(x, u) is a smooth function and /(0, 0) = 0. To this end, note that, without loss of generality, the function f(x, u) can be expressed in the form

f(x,u)

= f(x) + 9I(x)u + g2(x,u)u 2

where f(x) and g 1 (x) are smooth vector fields, g2(x, u) is a smooth function, and f(O) = 0. Then the following holds.

14.1 Achieving Global Stability via Small Inputs

229

Theorem 14.1.3. Consider the system

x = f(x) + 9I(x)u + g2(x,u)u 2 ,

(14.7)

in which x E JRn, u E JR, f(x) and g1 (x) are smooth vector fields, g2 (x,u) is a smooth function, and f(O) = 0. Suppose there exists a smooth positive definite and proper function V(x) satisfying ~

L1V(x)

jar all x.

0

(14.8)

Suppose also that the system f(x) L 9 , V(x)

y

(14.9)

is zero-state detectable. Let a(-) be the function defined in Proposition 14.1.2, let c be a fixed number satisfying 0 < c < 1 and let p(x) be any smooth function such that p(x) 2: max IIY2(x,u)ll · lui:S<

The feedback law u(x)

= ->..(x)a(L9 , V(x)),

in which >..(x) =

c

av

2 ( 1 +p (x)l/ ax

2

II)

globally asymptotically stabilizes the equilibrium x

,

= 0.

Proof. Suppose >..(x) is a smooth function satisfying

< >..(x)

0

~

E

for all x, and set

u(x) = ->..(x)a(£ 9 , V(x)) ·= Since la(s)l

->..(x)~=L:;=9 '::=V=:(x~)~ .j1 + [L 9 , V(x)F

< 1 for all s, we have iu(x)l <

E

·'

for all x, and IIY2(x,u(x))ll ~ p(x) for all x. Along the trajectories of the closed loop system, we have

. V = L 1V(x)

av

+ L 9 , V(x)u(x) + axg2(x,u(x))u 2 (x)

< L 1V(x)- >..(x) [Lg, V(xW .j1 + [L 9 , V(x)J2 - L V( ) -

I

X

+

>..( ) X

[Lg, l'(xW

+II avax llp(x)>..2(x) 1+[Lg,[LV(xW V(x)J2 9,

( 1

,\(x)

I av II ( ))

.j1 + [Lg, V(x)J2 - + .jl + [Lg, V(x)J2 ax p X

•

230

14. Stabilization Using Small Inputs

If

(14.10) we have

for all x, and therefore

V
V(x)

+ x(x)

I

2

[Lg, V(x)] <0 y'1 + [L 9 , V(x)J2 -

(14.11)

for all x. Condition (14.10) is indeed satisfied with

and this shows that the feedback law indicated in the theorem renders (14.11) fulfilled. Since V(x) is a proper function, this proves boundedness of all trajectories and stability in the sense of Lyapunov. As a consequence, any trajectory x(t) has a nonempty w-limit set r, on which the function V(x) is constant. We see from (14.11) that, necessarily, Lg, V(x)

=0

for all

X

E

r

r.

and this in turn implies that u(x) = 0 at each point of This shows also that any trajectory x(t) entirely contained in r is necessarily a trajectory of

± = f(x) for which L 9 , V(x(t))

= 0.

The assumption that system (14.9) is zero-state detectable implies that r. Thus, also x(t) --+ 0

x(t) --+ 0 as t --+ oo and this proves that V(x) is 0 on as t--+ oo, because V(x) vanishes only at x = 0.
Note that also the feedback law derived in this Theorem is such that

iu(x)l < f: for all x.

14.1 Achieving Global Stability via Small Inputs

231

Remark 14.1.1. The results of the previous two Propositions can be immediately extended to a system

± = f(x) + g(x)u

(14.12}

with m > 1 inputs. In this case, in fact, since system (14.12} with output y = [L 9 V(x)JT is by construction passive, if the hypothesis that system (14.3) is zero-state detectable is replaced by the hypothesis that

±

f(x)

(14.13)

[L 9 V(x)F

=

y

is zero-state detectable, the result of Corollary 10.8.4 still applies. Thus, system (14.12) is stabilized by the feedback law

u

= -c-[L9 V(x)JT

or by the bounded feedback law

u

= -c-a([L 9 V(xW)

where now the function a(·) is defined as

a:

!Rm

--*

!Rm

(sl>···,sm)

f-t

(~, ... , 1+st

ffl).

(14.14}

1+s~

Also a multi-input version of Theorem 14.1.3 holds, whose derivation is left to the reader.
Lemma 14.1.4. Let f(x) and g(x) be smooth vector fields of'fill..n, with f(O) = 0, and suppose there exists a smooth positive definite and proper function V(x) such that Lt V(x) ::; 0 for all x. Then, system

±

f(x)

y

L 9 V(x)

is zero-state detectable if and only if so is system :i;

f(x)

y

L 9 V(x)) ( LtV(x) .

(14.15)

232

14. Stabilization Using Small Inputs

Proof. The "only if' part is trivially true by definition. To prove the "if' part, observe that, using the same arguments of the proof of the previous Theorem, system (14.16) ± = f(x)- g(x)L9 V(x)

is globally asymptotically stable. In fact

av ox [f(x) -

g(x)L 9 V(x)]

= L, V(x) -

[L 9 V(xW :::; 0

and on thew-limit set of each trajectory L 9 V(x) = L,V(x)

= 0.

Now, consider any trajectory x(t) of (14.15) for which L 9 V(x(t)) = 0. This is indeed a trajectory of (14.16) and hence x(t) ~ 0 as t ~ oo. This shows that (14.15) is zero-state detectable.
± =Ax+Bu,

(14.17)

and suppose that (A, B) is stabilizable and there exists a symmetric matrix > 0 such that (14.18)

P

= BT Px, is detectable. Hence, by Proposition 14.1.2 and its multi-input version (see Remark 14.1.1), it can be globally asymptotically stabilized by the feedback law u(x) = -Ea(BT Px) = w( -BT Px) (14.19)

It is known from Lemma 10.9.3 that this system, with output y

in which a : am

~

am

is the function

and E > 0 is any fixed number. It is possible to show that the stabilizing property of the feedback law (14.19) with respect to system (14.17) holds not just for the specific function a(·) indicated above, but also for any other function in a more general class of functions, called saturation functions, and, moreover, that the resulting closed loop system turns out to have a property of input-to-state stability with restrictions (in the sense of Definition 12.2.1), with respect to additive disturbances. As a matter of fact, observe first that the function a(·) is just a special member of the class of functions characterized by the following definition.

14.1 Achieving Global Stability via Small Inputs

Definition 14.1.1. A locally Lipschitz function u(·) : IR saturation function if:

~

233

IR is said to be a

- u(O) = 0 and su(s) > 0 for all s -=f. 0, - there exist lf and k such that ju(s)l ::::; k for all s E IR lim inf ju(s)l 2: lf, 1•1-+oo - u( s) is differentiable in a neighborhood of s u' (0) A function u : Il~m

and, for all i

~

= 0 and

=1.

Rm is an Rm -valued saturation function if

= 1, ... , m,

u i ( ·) is a saturation function.

Remark 14.1.2. It is easily verified that, if u(·) is a saturation function, then there exists a number K > 0 such that

ju(s)- si::::; Ksu(s)

(14.20)

for all s E IR. In fact, observe that, since u'(O) = 1 and su(s) > 0 for all s -=f. 0, there exist J > 0 and A > 0 such that ju(s) - sl ::::; Asu(s) ,

for all lsi

< 6.

Moreover, for all s,

ju(s)- sl::::; k +lsi, and there exists c: > 0 such that ju(s)i 2: c: ' This yields, for any K

for all lsi 2: 15.

> 0,

Ku(s)s 2:

Kc:l~l,

for all lsi 2: 6.

Choosing K so that K 2: A and K c:l61 2: k + 161 completes the proof of (14.20). Using (14.20) it is also easy to see that there exists a number H 2: 1 such that (14.21) iu(s)i ::::; min{Hisl, k} for all s E R

<1

Then, the following result can be proven.

234

14. Stabilization Using Small Inputs

Proposition 14.1.5. Consider system (Lf.17). Suppose (A, B) is stabilizable and there exists a symmetric matrix P > 0 such that (14.18) holds. Let o-(·) be any !Rm -valued saturation function, and consider the closed loop system (14.22) x =Ax+ Bo-( -BT Px + v) + w . Then, there exists a number J > 0 such that (14.22) is input-to-state stable, with no restriction on X 0 , and restriction on v(·) and w(·). In particular, there exist a class K function ")'o(·) and numbers g" > 0, gw > 0 such that, for any X 0 E !Rn, for any input v(·) E L~ satisfying llvOIIoo < and any input w(·) E L~ satisfying llw(·)lloo < J, the response x(t) in the initial state x(O) = X 0 satisfies

a

a,

llxOIIoo lim sup llx(t)ll

< <

max{'"Yo(llxoll), ""Yv(llv(·)lloo), ""Yw(llw(·)lloo)} max{'"Yv(limsup llv(t)ll),")'w(limsup llw(t)ll)}, t-too t-too

t--+oo

where

= gvr,

T'v(r)

T'w(r)

= gwr

for all r. Proof. From the previous discussion, it is known that system

is asymptotically stable. Thus, there exists a symmetric matrix Q that Set

z

> 0 such

= -BTPx+v.

Along the trajectories of (14.22), the quadratic form

V0 (x)

1 T = 2x Qx

satisfies

V0 =

= =

+ Bo-(z) + w] -llxll + xTQ[Bo-(z)- Bz + Bv + w] -llxll 2 + xTQB[o-(z)- z] + xTQBv + xTQw.

xTQ[(A- BET P)x +BET Px 2

Using the fact that llo-(z)- zll:::; KzTo-(z) for some K > 0 (see (14.20)), and standard estimates for quadratic forms, it is deduced that there exist positive numbers a 1 ,a2 ,a3 , only depending on Q and B, such that

Consider now the positive definite and proper function

14.1 Achieving Global Stability via Small Inputs

235

Along the trajectories of (14.22),

= jxT Pxj 112 xT P[Ax + Bu(z) + w]

~·i

= ~lxT Pxj 112 xT(PA +AT P)x + jxT Pxj 112[xT PBu(z) + xT Pw]· 2

< jxT Pxj 1 12[-z T u(z) + vT u(z) + xT Pw]

.

Using the fact that zTa(z) ~ 0, that llu(z)il ~ k for some k > 0 and standard estimates for quadratic forms, it is deduced that there exist positive numbers b1 , ~, b3 , only depending on P, such that

Set V(x)

= Vo(x) + .AV1 (x)

with ,\ > 0. Then, along the trajectories of (14.22),

~

V

+ .Ab3llwll)llxll 2+ (a1K- .Abi)iixiizTu(z) + (a2 + .A~k)llxllllvll + asllxllllwll ·

(-1

Choose ,\ so that and

~

such that

Then, if

llw(t)il we obtain

~ ~

for all t ~ 0,

V ~ llxll (- ~llxll + (a2 + .Ab2k)jlvll + aallwll) .

We have in this way shown that, setting, for instance

the function V(x) is such that

llxll

~

max{c,iivll,cwllwll} BV [Ax+ Bu( -BT Px

ax

+ v) + w] < _llx11

-

2

(14.23)

4

for all x E !Rn, all v E IlF and all w E IRn satisfying llwll ~ ~Observe that there exist positive numbers g_ 0 , ao, a1 , such that the function V(x) satisfies

236

14. Stabilization Using Small Inputs

g(l/xll) ::; V(x) ::; a(llxll) , with

g(r) =Qor 2,

a(r) =aor 2 +alr 3 .

Thus, the inequality (14.23) above shows that the system is input-to-state stable, with no restriction on X 0 , no restriction on v(·), restriction§ on w(·), and "gain" functions

Since, for any d > 0, there exists

a0 > 0 such that

a(r) ::; iior 2 for all r::; d, it follows that for r E [O,d] the functions !v(·) and !wO can be estimated by linear functions. This proves the Proposition, if it is assumed that also v(·) is restricted by§, i.e. llv(·)lloo::; §. <J

14.2 Stabilization of Systems in Upper Triangular Form In Chapters 11 and 12, we have discussed the problem of global asymptotic stabilization of certain classes of nonlinear systems described by equations having a lower-triangular structure, namely systems which are modeled by equations of the form (11.38). We examine in this section the case of systems described by equations having an upper-triangular structure, such as XI

x2

JI(x1,x2, ... ,Xn,u) f2(x2,X3, ... ,xn,u) (14.24)

Xn-1 Xn

fn-I(Xn-I,Xn,u) fn(Xn, u) '

in which the functions j;(x;,Xi+I, ... ,xn,u) are supposed to satisfy appropriate hypotheses, which will be introduced in the course of the exposition. These systems are often referred to as systems in feedforward form , in consideration of the fact that they correspond to a cascade interconnection of n subsystems, starting with the lower subsystem of (14.24) and ending with the upper subsystem of (14.24), in which the i-th subsystem is fed by the "outputs" x;+I, ... , Xn of all previous subsystems in the cascade (see Fig. 14.1).

Because of this triangular structure, the design of stabilizing feedback can be achieved in a recursive way, based on the following procedure. Suppose that a feedback law u = an(x) is known which stabilizes the lower subsystem of the chain

14.2 Stabilization of Systems in Upper Triangular Form

237

Fig. 14.1. System in feedforward form.

and set (14.25) This yields a system described by equations whose structure is the same as the structure of (14.24), with u replaced by un, but- in addition- the subsystem corresponding to the last equation of the chain is stable, by construction, if Un = 0. Consider now the subsystem formed by the last two equations, which have the form Xn-1 Xn

fn-l(Xn-l,Xn,nn(Xn) +un) := jn-i(Xn-1,Xn,Un) fn(Xn,nn(x)

+ Un)

:= jn(Xn,Un),

and suppose of being able to design a feedback law Un which renders it stable. Set

= nn-1 (xn, Xn-1)

which, actually, is the same as setting U = O!n(Xn)

+ O!n-1 (Xn

1

Xn-1)

+ Un-1

on the original system (14.24). By doing this, a system is obtained in which the upper n- 2 equations have the same structure as that of (14.24), with u replaced by Un-1, and in which the subsystem corresponding to the lower two equations of the chain is stable, by construction, if Un- 1 = 0. At this point, the last two equations of the chain can be considered as a single subsystem, with state xn-.:1 = (xn, Xn-1) and input Un- 1 , which is stable when Un- 1 = 0. The entire system can still be regarded as a system modeled by equations of the form (14.24), with n replaced by n- 1, Xn- 1 replaced by Xn-l, and u replaced by Un-l, and - in addition - the last subsystem of the chain is stable by construction when the input is zero. In other words, the entire system is precisely in the same conditions as the original system (14.24) was after the imposition of the control (14.25), and one can proceed in the same manner. It is seen from these considerations that a system modeled by equations of the form (14.24) can be stabilized if:

238

14. Stabilization Using Small Inputs

- it is known how to stabilize the lower subsystem of the chain, - it is known how to stabilize a system of the form '1/l(z,~,u)

i

(14.26)

¢(~, u) '

knowing that the lower subsystem is stable when the input is zero. In this section we describe conditions under which the second one of these two design problems can be successfully addressed. We assume, as usual, that '1/J(z, €, u) and ¢(z, u) are smooth functions of their arguments and that '1/J(O, 0, 0) = 0, ¢(0, 0) = 0, and - to begin with - also that their are affine in the control input u. The more general case of an arbitrary nonlinear dependence on u will be dealt with, using the result of Theorem 14.1.3, at the end of the section. This being the case, system (14.26) is rewritten, for convenience, in the form i = a(z) + p(z, 0 + b(z, ~)u (14.27) ~ = cp(~) + O({)u , with z E !Rn, ~ E IR", u E IR, and where a(O) = 0, p(z, 0) = 0, cp(O) = 0. In line with the previous discussion, it is assumed that the equilibrium ~ = 0 of

~

= cp(~)

(14.28)

is globally asymptotically stable. Note that, setting

x= (

~) '

f(x) = ( a(z) ;(~)(z,

€)) '

g(x)

= ( b~(ef))

'

system (14.27) can be regarded as a system of the form

x = f(x) + g(x)u.

(14.29)

Thus, the problem of stabilizing system (14.27) could seem as difficult as any other nonlinear stabilization problem. However, in the present setup, one may take advantage of the upper triangular structure of f(x) and of the fact that (14.28) is by hypothesis globally asymptotically stable. Of course, it would be trivial to assume that the entire system

x= f(x) is globally asymptotically stable (in which case u = 0 would solve the stabilization problem), but it is not trivial- and relevant in many important cases - to consider the "next easier" assumption, namely the one in which, as a consequence of the hypothesis of the existence of a suitable Lyapunov function, this system is guaranteed to have bounded trajectories and to be stable in the sense of Lyapunov (but possibly not globally asymptotically stable). H this is the case, in fact, the result of Proposition 14.1.1 can be invoked,

14.2 Stabilization of Systems in Upper Triangular Form

239

which shows that, if a positive definite and proper function V(x) is known such that Lf V(x) ~ 0 for all x, it is not terribly difficult to globally asymptotically stabilize a system of the form (14.29), provided that appropriate additional conditions hold. In view of this, we discuss now the problem of identifying situations in which the main hypothesis of Proposition 14.1.1, namely the existence of a smooth function V(x) such that (14.2) holds and (14.3) is zero-state detectable, is fulfilled. As shown below, the existence of a function V (x) satisfying (14.2) is guaranteed when, in system (14.27): - the function a(z) is linear in z, namely a(z) symmetric matrix P > 0 such that

=

Az, and there exists a

(i.e. the equilibrium i = 0 of the associated linear system i = Ai is stable in the sense of Lyapunov), - the equilibrium ~ = 0 of the system ~ = cp(~) is globally asymptotically stable and also locally exponentially stable, · - the function p(z, ~) has a linear growth in z, nainely, is such that

llp(z, ~)II

~ 1(11~11) (1

+ liz II)

for all z, {

where -y(·) is a class K function, differentiable at the origin. Before addressing the issue of the existence (and possibly the construction) of such a function under the said hypotheses, it is useful to start with the analysis of a special case, the one in which, in system (14.27),

a(z) p(z, ~) cp({)

+ AT

=0

Az

and A

p(~)

and p(~) is a polynomial in the components of~ and all eigenvalues of F have negative real part .

F{

(i.e. A is a skew - symmetric matrix)

In other words, we examine the special case in which the system :i: has the form i = Az + p({)

~

=

= f(x) (14.30)

F{

with A, p({), F satisfying the hypotheses indicated above. In this case, a proper function V (z) such that L 1V (z) ~ 0 can be constructed by appealing to the following interesting result 1 . Lemma 14.2.1. Let A E !Rnxn and F E lR"x" be fixed matrices. Let Q denote the vector space consisting of all polynomials of degree p in 6, (2, ... , {,, with coefficients in IR, which vanish at { = 0. Let :F denote the linear operator 1

See Byrnes et al. (1997), page 13, for a proof.

240

14. Stabilization Using Small Inputs

q(~)

Suppose that none of the eigenvalues ofF is an eigenvalue of A. Then, .'liven any element p( ~) of Qn, there is a unique element 1r( ~) of Qn such that

a1r

a~F~

= A1r(~) + p(~).

(14.31)

By means of thf function 1r(~), it is possible to transform system (14.30) into a decoupled system. In fact, set

( = z- 7r(~) and, using (14.31), observe that system {14.30) is transformed into

This system.is stable in the sense of Lyapunov. In particular, if U(~) is any Lyapunov function for the lower subsystem, i.e. a positive definite and proper function such that

au

a~ F~

< 0 . for all

~

;t 0 ,

the (positive definite and proper) function thus defined W((, ~)

= (T ( + U(~)

is sucl1 that (recall that A by hypothesis satisfies A= -AT) {14.32)

Consider now the (positive definite and proper) function obtained from W(z, ~) by reversing the change of coordinates, namely the function W(z, ~)

= W(z- 1r{~), ~)

which can be written in the form W(z, ~)

= zT z + U(~) + !li(z, ~)

{14.33)

with a "cross-term" !li(z, ~) defined as

Indeed, we expect that the derivative of this {positive definite and proper) function along the trajectories of system (14.30) is equal to the expression

14.2 Stabilization of Systems in Upper Triangular Form

241

obtained by reversing the change of coordinates in the rir,ht-hand side of (14.32), i.e. just equal to 8U(() F( 8( . Thus, since 8(zTz) A -0 8z z- '

we expect that the cross-term 1/i(z, () satisfies 8(zT z) ----az-P(()

81/i

81/i

+ Bz [Az + p(()] + B( F( =

0.

(14.34)

Of course this is the case, as a straightforward calculation, which uses (14.31) and the property that A= -AT, shows.

Remark 14.2.1. Note that the same conclusions remain valid if the lower subsystem of (14.30), instead of being a linear system, is a nonlinear system, namely { = c,o(() , so long as there is a (globally defined) solution 1r((} of

a~~() c,o(() =

A1r(()

+ p(()

.

(14.35)

The set

M = {(z,() E an

X

IR"' : z = 7r(()}

is the stable manifold of =

{

Az + p(() c,o(() .

(14.36)

Thus, to say that equation (14.35) has a globally defined solution is equivalent to say that the stable manifold of (14.36) can be represented in the form above, namely as the graph of a globally defined function 1r((). <1 The fact that, in this particular case, it was possible to find a Lyapunov function for (14.30) with a cross~term 1/i(z, () whose derivative along the trajectories cancels out with the sign indefinite term 8(zT z) p(() 8z

suggests seeking a similar result for the more general case in which the system x = f(x) has the form Az + p(z,{) (14.37) c,o(() .

242

14. Stabilization Using Small Inputs

> 0 such that

For this system, suppose there is a symmetric matrix P

PA+ATP::;O, and suppose there exists a positive definite and proper function U(~) such that au a~
W(z,

~)

such that the function

0 = zT Pz + U(O + I.P(z,~)

(14.38)

is positive definite and proper and such that

a(zTPz) az p(z, ~)

+

a.P az [Az

+ p(z, ~)] +

ai.P a~
Then, the derivative of the function' W(z, (14.37) would satisfy

aw az [Az

~)

(14.39)

along the trajectories of

aw

au

+ p(z,O] + a~
i.e. would be negative semi-definite on !Rn x IR", thus complying with the hypothesis (14.2) of Proposition 14.1.1. If this function would also be such that system (14.3) is zero-state detectable, then the stabilizing feedback provided by Proposition 14.1.1 could be used. Of course, existence of such a cross-term implies, as a byproduct, that the trajectories of system (14.37) are bounded for any initial condition. Thus, we check first that this property holds. Instrumental, in this part of the analysis, are the hypotheses of linear growth in z for the coupling term in (14.37), and of local exponential stability of the equilibrium ~ = 0 of the lower subsystem (in addition to the already assumed existence of P > 0 and U(~), which reflects the stability in the sense of Lyapunov of i = Az, and the global asymptotic stability of the lower subsystem). Lemma 14.2.2. Consider system (14.37). Suppose: (i) there exists a symmetric matrix P

> 0 such that P A+ AT P:::; 0,

(ii) the equilibrium ~ = 0 of the system ~ stable and also locally exponentially stable,

=
is globally asymptotically

(iii) the function p(z, ~) is such that

llp(z,0/1:::; 1(11~11)(1 + llzll) far all

z,~,

where rO is a class K function, differentiable at the origin. Then, for every (z 0 , ~ 0 ) E !Rn x IR", the integral curve of (1.{37) passing through (z 0 ,e) at timet= 0 is bounded on [O,oo).

14.2 Stabilization of Systems in Upper 'Triangular Form

243

Proof. Of course, we only need to establish boundedness of z(t). Let V(t)

= zT(t)Pz(t)

and observe that, by hypothesis

V(t)

= :=:;

2zT(t)P[Az(t) + p(z(t),~(t))] 2llz(t)I/I/PIIIIP(z(t), ~(t))ll :=:; 2llz(t)111/Pih'(ll~(t)ll)(1 + llz(t)ll).

From hypothesis (ii), using the property indicated in Lemma 10.1.6, we know that there exist a class K function 0(-) and a number b > 0, such that 1'(1/~(t) II) :=:; O(IIC ll)e-bt

for all t ;::: 0. Assuming - without loss of generality - that llz(t)ll ;::: 1, and using the fact that llzll 2 :=:; czTpz for some c > 0, we obtain

for some d > 0. This, by the Gronwall-Bellman inequality, yields V(t)

:S V(O)edO(IICI/) fo' ,-••ds :=:; V(O)e(d/b)O(IICI/)

which shows that V(t) is bounded on [0, oo). Thus, since from below by (1/c)llzll 2 , also z(t) is bounded.
zT Pz

is bounded

Having shown that, under the hypotheses indicated in this Lemma, the trajectories of system (14.37) are bounded, we proceed with the determination of a function lli(z,~) satisfying (14.39). To explain how this can be achieved, set again, as before, X

=

and also

G(x)

(z) , ~

!() _ (Az+p(z,~))

= lli(z,~),

X

-

F(x)

tp(~)

,

= 2zTPp(z,~).

In these notations, (14.39) reduces to the partial differential equation

8G , ox f(x)

= -F(x)

(14.40)

in the unknown G(x). Let x(x, t) denote the value at timet;::: 0 of the integral curve of :i; = f(x) passing through X at time t = 0. As shown before, the function x(x, t) is defined for all (x, t) E (lRn x lR") x [0, oo). It is possible to show that the function F(x(x, t)) has the following interesting properties.

244

14. Stabilization Using Small Inputs

Lemma 14.2.3. Suppose the hypotheses (i), (ii), (iii) of Lemma 14.1!.1! hold. Then, then function F(x(x, t)) is such that

= 0,

lim F(x(x, t))

t-+oo

(14.41)

fooo F(x(x,s))ds < oo,

(14.42)

for all x E (JRn x IR.v). Moreover, the function G(x)

=

l)O F(x(x, s))ds,

(14.43)

is a smooth function. Proof. Let z(z, {, t) and(({, t) denote the upper and, respectively, lower components of x(x, t) (note that the argument z is not present in the lower component ofx(x, t) because the lower subsystem of (14.37) is independent of z), and observe that JF(x(x, t))i ::; 211PIIIIz(z, {, t)li(1

+ iiz(z, {, t)llhCII(({, t)li)

.

We have shown in Lemma 14.2.2 that, for every (z, {), there is a number M > 0, depending on (z, {),such that llz(z, {, t)ll :::; M for all t ~ 0. Also, we know that, for some b > 0 and class K function 0(·), !(II(({, t)li) ::; O(il{li)e-bt for all t ~ 0 and all {. Thus JF(x(x, t))l ::; 2IIPIIM(1 + M)O(II€il)e- 6t

and both (14.41) and (14.42) immediately follow

2 .
Now, set X

0

(t) = x(x 0 , t)

and, using the well-know semi-group property of x(x0 , t), note that

Taking derivatives with respect to time yields

°

8 [ 8 X f(x)] z=z•(t)

Since

X

0

= -F(x(x

0 ,

t))

= -F(x

0

(t)).

was arbitrary, this shows that 8G ox f(x)

= -F(x)

i.e. that the function (14.43) solves the p.d.e. (14.40). 2

The proof that G(x) is a smooth function can be found in Jankovic et al. (1996).

14.2 Stabilization of Systems in Upper Triangular Form

245

z,'

We have in this way found a function which renders the condition (14.39) fulfilled. In terms of arguments, this function is expressed as (14.44) In addition, it is also possible to show that the resulting function (14.38) is positive definite and proper 3 .

Lemma 14.2.4. Suppose the hypotheses (i), (ii), (iii) of Lemma 14.2.2 hold. Then, then function (14.45)

is positive definite and proper. Remark 14.2.2. It is left to the reader, as an exercise, to see that, going through the same arguments of the previous discussion, similar conclusions can be reached in case the linear term Az in system (14.37) is substituted by a nonlinear term a(z), so long as there exists a positive definite and proper. function V(z) such that LaV(z)::; 0 and such that

II~~ lillziJ :S cV(z). for some c > 0. In this case, the cross-term !li(z, ') would have the expression 00

!li(z,,)=

1 0

av

[~] uz

_

z=z(z,€,s)

-

p(z(z,,,s),,(,,s))ds.

However, to establish continuous differentiability of !li(z, ') appropriate additional hypotheses on a(z) are needed 4 •
f(x)

y

L 9 V(x)) ( L,V(x)

Observe that the system we are dealing with, namely system (14.27), in the present case (i.e. when a(z) = Az), can be written as 3

4

For the proof, see Jankovic et al. (1996). See Sepulchre et al. (1996), for further details.

246

14. Stabilization ·using Small Inputs

Az + p(z,()
+ b(z,()u

(14.46)

+ 8(()u

and therefore the issue is to establish whether or not the system

z

Az + p(z,()

( =


YI Y2

(14.47)

is zero-state detectable. It will be shown that this is the case if: - the linear approximation of system (14.46) at the equilibrium (z, (, u) = (0, 0, 0) is stabilizable, , - the functions p( z, () and b( z, () are such that

~p] (z,O) [ v(

= M =constant,

b(z, 0) = B = constant.

(14.48)

Note, to this end, that the linear approximation of system (14.46) at the equilibrium (z,(,u) = (0,0,0) is a linear system of the form Az+M(+Bu

(14.49)

F(+Gu in which

M

8 = [8( P] (o,o)'

B

= b(O, 0),

F-- [8t.p] 8(

(o)'

G = 9(0).

Lemma 14.2.5. Consider system {14.46). Suppose the hypotheses (i), (ii), (iii) of Lemma 14.2.2 hold. Suppose also that the linear approximation {14.49) of system (14.46) at the equilibrium (z, ~' u) = (0, 0, 0) is stabilizable and that {14.48) hold. Then, system {14.47} is zero-state detectable. Proof Consider the linear system (14.49) and note that it satisfies all the hypotheses of Lemma 14.2.4. Thus, formula (14.45) provides the expression of a positive definite function whose derivative along the trajectories of (14.49) with u = 0 is negative semi-definite. In particular, choose U(~) to be a quadratic function (which is indeed possible because the system in question is linear) and observe that the cross-term reduces to the function

1 2[h(z,~,s)]TPMeF ~ds 00

tJiL(z,() =

where 'h (z, ~, t) is the solution of

8

14.2 Stabilization of Systems in Upper Triangular Form

.i(t)

= Az(t) + MeFt~,

with z(O)

247

=z .

Since the latter is a linear function in (z, ~), the cross-term IP(z, ~) is quadratic. Therefore, the entire function (14.45) is a quadratic form, which can be written as

with PL > 0. By construction, the derivative of this function along the trajectories of (14.49) with u = 0 is a negative semi-definite quadratic form. Thus, system (14.49), which is assumed to be stabilizable, satisfies the hypotheses of Lemma 10.9.3. As a consequence, the system Az+M~

.i

F~

y

(BT

GT )PL

(14.50)

(n

is detectable. Note that particular trajectories of this system are those of the form (z(t), 0) in which z(t) satisfies .i = Az. Thus, since system (14.50) is detectable, so is also the system .i

Az

(14.51)

w

Now, return to the nonlinear system (14.46), and recall that the issue is to establish that system (14.47) is zero-state detectable. Since by construction Y2

= zT(PA +AT P)z +

au

au

a~ cp(() ~ a~ cp(~)

and the term on the right-hand side is a negative definite function of E, Y2 = 0 implies ~ = 0. Thus, to study the zero-state detectability of (14.47) reduces (recall that p(z,O} = 0} to study the zero-state detectability of .i

=

Az

y

=

[aw] az

b(z, o)

+

(z,O)

[aw] a~

(14.52)

O(O\ . (z,O)

·

Now, it is easy to check that the output of (14.52} coincides (modulo a constant factor) with the output of (14.51}, which was proven to be detectable. Thus also (14.52) is detectable and this proves the Lemma. To see that this is true, observe that the output w of (14.51} is w

= (z T

O)PL

(B) -2 _1[awL] 8z G

(z,o)B

1[awL] 8f

+2

(z,o)G,

248

14. Stabilization Using Small Inputs

and therefore, in view of the second one of hypotheses (14.48) and the definition of B and G

2w

= [-awL] b(z,O) + [awL] - 8 .,c (z,O) 9(0). (z,O) 8 Z

Comparing with the output of (14.52), we see that it suffices to check that

[aw] 8z

[aw] 8~

[awL]

(z,O)

= 8z

{z,O)'

{z,O)

=

[awL] 8f

{z,O) .

The first one is trivially true, because

WL(z, 0)

= W(z, 0) = zT Pz.

As far as the second one is concerned, we need to check that

[8{)~h] 1

(z,O)

=

[81ft] {)~

(14.53)

(z,O) .

Using the first one of the hypotheses (14.48), it is seen that

[-81ft] 8~ (z,O) =

1

00

0

_ T P[8p] 2[z(z,O,s)] c

8·"'

= roo 2[z(z,O,s)]TPM[ 88 f.] ~

fo

ds (OJ

=

(z,O)

[8{] 8c

ds

., (0)

roo 2[z(z,O,s)]TPMeF ds 8

fo

where, in the last passage, we have used the fact that Fs 8{] [ 8~ (O) = e ·

On the other hand,

81ft

[ 8 ~L] {z,O) Since z(z,O,s) proof. <J

=

lao 2[zL(z,o,sWPMeF ds. 8

0

= zL(z,O,s), this shows that (14.53) holds and completes the

We have thus shown that, under the hypotheses of this Lemma, system (14.46), thanks to the result of Proposition 14.1.1, can be globally asymptotically stabilized, and this accomplishes one of the main design goals stated at the beginning on the section (namely, stabilization of a system of the form (14.26)). One of the hypotheses of the last Lemma is that the equilibrium ~ = 0 of the lower subsystem of (14.46), when u = 0, is also locally exponentially stable. Thus, in view of seeking a recursive use of the previous results in order to be able to stabilize a system of the form (14.24), we should check that the globally asymptotically stabilizing law proposed for (14.46) also achieves local exponential stability. This is indicated in the following result, which for convenience - repeats all the hypotheses introduced in the course of the previous analysis.

14.2 Stabilization of Systems in Upper Triangular Form

249

Theorem 14.2.6. Consider system

i

=

+ p(z, ~) + b(z, Ou cp(~) + B(~)u .

Az

~

(14.54)

Assume the following: (i) there exists a symmetric matrix P

> 0 such that P A+ AT P:::; 0,

(ii) the equilibrium ~ = 0 of the system ~ = cp(~) is globally asymptotically stable and also locally exponentially stable, (iii) the function p(z, ~) is such that

Jlp(z, ~)II where

10

:::; 1(11~11)(1 + JlziJ)

for all z, ~,

is a class }( function, differentiable at the origin.

(iv) the functions p(z, ~) and b(z, ~) are such that

=constant, [{)~] ., {z,O) 8

b(z,O) =constant,

(v) the linear approximation of (14.54) at the equilibrium (z, ~, u) = (0, 0, 0) is stabilizable.

Let

U(~)

all~,

be a smooth function such that, for g.(II~IJ)

:::; U(~)

L'PU(~)

:::; a(ll~ll)

< -a(il~ll)

where g(·), a(·), a(·) are class JC 00 functions with the property that, for some 6 > O,a > O,b > 0, g(s) = as 2 and a(s) = bs 2 for all s E [0,8]. Let IP(z,~) be the function defined in {14.44) and set •

u (z, ~)

= 2z

T

Pb(z, ~)

o.P

+

{)z

b(z, ~)

a.P au + 0~ 8(~) + 0 ~ 0(~) .

(14.55)

The feedback law u = -cu*(z,~)

for any E: > 0 globally asymptotical;y and locally exponentially stabilizes the equilibrium (z, 0 = (0, 0) of (14.54). Proof. All has already been proven, except the local exponential stability. This follows from arguments similar to those used in the proof of the previous Lemma. The proof of this Lemma has shown that u*(z,O) = [

w]

8 8Z

(z,O)

b(z,O)

8

+ [ ~] 8

.,

{z,O)

O(O)

and that the latter is a linear function of z, which can be given the form

250

14. Stabilization Using Small Inputs

In a similar way, one can show that the linear approximation of the function u • (0, ~) at ~ = 0 can be given the form 2 ( BT

(iT ) PL (

~)

.

From this, it is easily seen that the linear approximation of -cu*(z, 0 at (z, ~) = 0 coincides with the law UL(z,()

= -2c(BT

GT)PL ({) ,

which is known (using Proposition 14.1.1 and the property that (14.50) is detectable) to asymptotically stabilize the linear approximation of (14.54) at the equilibrium (z, ~' u) = (0, 0, 0). This proves the result. <1 As shown at the beginning of the section, a result of this kind lends itself to the implementation of a recursive design procedure for the synthesis of a stabilizing feedback law for a system of the form XI ±2

= =

AIXI +PI (xi ,x2, ... ,xn) + bi (xi ,x2, ... , Xn)U A2x2 + P2(X2, xa, ... , Xn) + b2(x2, xa, ... , Xn)u

(14.56)

Xn-I Xn

= =

An-IXn-I + Pn-I (xn-I, Xn) + bn-I (Xn-I, Xn)U c,o(xn) + O(xn)u ,

in which it is assumed that the matrices A;, for i = 1, ... , n - 1, are such that the associated system i; = A;x; is stable in the sense of Lyapunov (but possibly not asymptotically stable), and p;(x;, 0, ... , 0) = 0. If the lower system of the chain, when u = 0, is globally asymptotically and locally exponentially stable, if the terms Pn-l (Xn-I, Xn) and bn-l (xn-l, Xn) satisfy hypotheses (iii) and (iv), and if the linear approximation of

Xn-I Xn

= =

An-IXn-I + Pn-l(Xn-l,Xn) c,o(xn) + O(xn)u ,

+ bn-l(Xn-l,Xn)U

is stabilizable, one can find a feedback law Un-1 (xn-l, Xn) which renders the system

= An-lXn-l +Pn-l(Xn-l,Xn)+bn-l(Xn-!,Xn)(Un-l(Xn-l,Xn)+u) = c,o(xn) + O(xn)(un-l(Xn-l,Xn)+u), globally asymptotically and locally exponentially stable, when u = 0, and Xn-l Xn

then iterate the process. Note that, by proceeding in this way, i.e. replacing u

14.2 Stabilization of Systems in Upper Triangular Form

251

by u(xn-I, Xn) +u, the interconnection terms p;(x;, ... , xi) and b;(x;, ... , xi) are changed and therefore the hypotheses (iii) and (iv) must be re-checked (the fulfillment of hypothesis (v) is guaranteed, at each stage, by the hypothesis that the linear approximation of the entire system (14.56) is stabilizable, thanks to the triangular structure and to the fact that a state feedback does not destroy stabilizability). One situation which the fulfillment of hypotheses (iii) and (iv) is guaranteed in advance, is when the interconnection terms satisfy

< {;(II (xi+ I,···, Xn)/1)(1 + 1/x;/1) //b;(x;, Xi+ I,···, Xn)l/ < X;(//(x;+I, ·. ·, Xn)li)(l + 1/x;/1)

\lp;(x;, Xi+ I,···, Xn)ll

[

8pi

]

8(Xi+I, ... , Xn)

(x;,O, ... ,o)

b;(x;, 0, ... , 0)

=

constant

=

constant,

for appropriate class K functions {;(·) and X;(·). In fact, consider for instance the case in which n = 3. In this case, after having designed the feedback law u2(x 2 ,x 3) and replaced u by u 2 (x 2 , x3) +u one obtains a system of the form

XI

=

±2 = ±a

=

AI xi+ PI(XI,x2,xa) + bi(xi>x2,xa)u A2x2 + P2(x2,x3) + ~(x2,xa)u ]3(x2,x3,u),

in which

Pl(Xl,x2,xa) bi(XI,X2,X3)

= =

PI(x1,x2,x3) + bi(xi,x2,xa)u2(x2,x3) bl(XI,X2,X3).

Trivially, if bi (x 1 , 0, 0) is constant so is bi (xi, 0, 0). Moreover, since = 0, also

u2(0, 0)

8p1 ] _ [ 8p1 ] [ O(X2, X3) (xt,O,O)- 8(x2,X3) (xt,O,O)

+

b ( 0 O) [ 8u2 ] l XI, ' 0(x2,X3) (0,0)

is constant. Finally,

//PI (xi, X2, X3)1/ ~ 'YI(I/(x2,X3)/I)(l +

1/xd/) + XI(I/(x2,xa)ll)(l + llxdi)llui(X2,X3)11

which establishes the desired growth condition for ji1 (x 1 ,x2 ,x3 ). To conclude the section, we observe that the results described above can also be used for global asymptotic stabilization of systems described by equations of the form

252

14. Stabilization Using Small Inputs

XI

A1x1 +gi(XI,X2,···,xn,u)

=

i:2 Xn-l

A2x2

+ 92(x2, X3, ... , Xn, u)

An-lXn-l

Xn

9n(Xn,

+ 9n-l (Xn-l, Xn, u)

u) ,

in which the right-hand side is not an affine function of the control input u. To this end, in fact, it suffices to split the g; (x;, ... , Xn, u) 's as

in which p;(x;, ... , Xn)

and use the result of Theorem 14.1.3. At each iteration of the recursive procedure, one has to deal with a system of the form

i

e

=

Az+p(z,{)+b(z,e)u+q(z,{,u)u2 cp({) + 9({)u + -y({, u)u 2 .

(14.57)

If A,p(z, {), b(z, {), cp({), 9({) are such that the hypotheses of Theorem 14.2.6 hold, it is possible to find a positive definite proper function W(z, {) such that 8W 8W z [Az + p(z, {)) + { cp({) ~ 0 8 8

and such that system i

e y

= = =

Az + p(z,{) cp({) 8W 8W 8z b(z, {) + 8{ 9({) ,

is zero-state detectable. Thus, using Theorem 14.1.3, it is possible to find a feedback law which globally asymptotically stabilizes (14.57). As shown before, at each stage of the iteration, the hypotheses of Theorem 14.2.6 have to be re-checked, unless particular hypotheses are set on the various functions which characterize the system, so as to guarantee these properties in advance. The details on how this can be achieved are left as an exercise to the reader.

14.3 Stabilization Using Saturation Functions

253

14.3 Stabilization Using Saturation Functions The basic structure analyzed in the previous section (see (14.26)) was that of a nonlinear system ( =
z = 1/J(z,(,u) in which 1/J(z, 0, 0)

= Az.

Under the hypothesis that P A+ AT P :S 0 for some symmetric matrix P > 0, and appropriate additional conditions on the function 'lj;(z, (, u), it was shown that a system of this form can be globally asymptotically stabilized, using a control input with bounded amplitude. In this section, we describe a different approach to the stabilization of systems having this structure, which is in a certain sense more appealing, because the control law that will result is not based on the explicit availability of a Lyapunov function, as it happens for the feedback law (14.55) determined in the previous section. It must be stressed, in fact, that the explicit calculation of the law (14.55) requires the prior determination of the term IP(z, () via the integral (14.44), and this may result, in general, in a difficult computational task. The first stage in the analysis consists in the study of the case in which the dynamics of a linear system

z = Az+Bu, in which the pair (A, B) stabilizable and P A+ AT P :S 0 for some symmetric matrix P > 0, are perturbed by the addition of a nonlinear term of the form g((, u), with (the state of a nonlinear system

( = f((,u), i.e. the case of a system having the following structure '

i

Az + Bu

+ g((,u)

f((, u) .

(14.58)

Indeed, this can be viewed as a system the form (14.57), if it is assumed that b(z,() and q(z,(,u) are independent of z. In what follows, it will be described how system (14.58) can be stabilized by means of a feedback law of the form

u

= u(z,v),

(14.59)

254

14. Stabilization' Using Small Inputs

in which v represents an additional control input, introduced here for the purpose of setting the stage for a recursive design in the case' of systems having a feedforward structure. Note that this feedback depends only on the state z and not on the state~, as opposite to the feedback derived in the previous section, which was depending on both z and~- We shall see, however, that this structure of the control is not particularly restrictive in case the lower subsystem of (14.58) possesses a globally asymptotically equilibrium at (~, u) = (0, 0), as a natural intuition seems to suggest. The control law (14.59) induces a feedback coupling between the two subsystems, which would be otherwise only cascade coupled. As a matter of fact, system (14.58) with feedback law (14.59) can be regarded as the interconnection of a system with inputs u 1 and v, state x1 and output y 1, modeled by equations of the form

Jr(xl,u1,v) hr (x1, p) ,

Y1

(14.60)

and a system with input u 2, state x 2 and output y 2, modeled by equations of the form h(x2,u2) ±2 (14.61) h2(x2,u2), Y2 via (see Fig. 14.2)

u2

= Y1

u1

= Y2

(14.62)

·

v ±1 = fr(xi,ur,v) Y1 = hl(xr,v)

Y1

-

U!

Y2

±2 = h(x2, u2) Y2 = h2(x2, u2)

U2

Fig. 14.2. Feedback connection with input v.

To see that this is the case, it suffices to set x1 = z, x2 =

fr(xr, ur, v) h1(X1,v) h(x2,u2) h2(x2, u2)

Ax 1 +Bu(x1,v) +u1 u(x1,v) j(x2, u2) g(x2, u2) .

~,

and

(14.63)

14.3 Stabilization Using Saturation Function.s

255

Stability of such an interconnected system can be studied via the smallgain theorem. In the present case, taking advantage of the fact that the control laws that are to be used have bounded amplitude, it is convenient to appeal to a modified version of this theorem, which is explained below and uses a concept of "gain", introduced by A. Tee! 5 , which considers only bounds on the asymptotic behavior of the response, as t -+ oo. For a piecewise-continuous function u : [0, oo) -+ !Rm, define llu(-)lla =lim sup{ max lui(t)l}. t-+oo

l::Ot::Om

The quantity thus introduced is referred to as the asymptotic "norm" ofu(·). For a system having inputs and outputs, modeled by equations of the form ± = f(x,u) (14.64) y = h(x,u), with x E !Rn, u E !Rm, y E JR.P, and f(O, 0) = 0, h(O, 0) 0, the definition which follows characterizes the notion of "gain" in terms of asymptotic bounds on the input and on the corresponding output response.

Definition 14.3.1. System (14.64) is said to satisfy an asymptotic (input-. output) bound, with restriction X on X 0 and ·restriction U on u(·), if there exists a class K function 'Yu ( ·), called a gain function, such that, for any 0 X E X and for any piecewise~continuous input u( ·) satisfying llu( ·)II a < U, the response x(t) in-4he initial state x(O) = X 0 exists for all t ~ 0 and is such that y(t) = h(x(t), u(t)) satisfies jjy(·)lla ::; 'Yu(llu(·)lla) ·

Remark 14.3.1. It is obvious that the above definition, in the way it is expressed, only characterizes how the asymptotic bound on the output is affected by the asymptotic bound on the input and no special care is taken of bounds on the state, if any. In order to compare this with earlier definitions, it is convenient to specialize it to the case in which h(x, u) = x, i.e. the output of (14.64) coincides with its state. In this case, the definition says that, for any x 0 EX, llx(·)lla::; 'Yu(llu(·)lla)

but does not prescribe any special relation between llx( t) II and llxo II (as the definition of input-to-state stability, with restrictions, does). If, in particular, X is a neighborhood of the origin and u(t) = 0, the definition implies that lim llx(t)ll = 0,

t--+oo

but does not necessarily imply local asymptotic stability of the origin, since this point is not required to be stable in the sense of Lyapunov. It is because of this fact that, in the subsequent applications, in order to check that 5

See Tee) (1996a}.

256

14. Stabilization ·using Small Inputs

certain systems have a (global) asymptotically stable equilibrium, the issue of local asymptotic stability will be addressed separately via other methods, specifically the principle of stability in the first approximation. <1 Suppose now that system (14.60) satisfies an asymptotic input-output bound, with restriction XI on xJ', restriction UI on UI (·) and restriction v on v(·), that is, suppose there exist gain functions "YI(·) and "YvO such that, for any XI (0) E XI, the response XI (t) to piecewise-continUOUS UI (·) and v(·) (each one satisfying the indicated restriction) exists for all t ~ 0 and

Likewise, suppose that system (14.61) satisfies an asymptotic input-output bound, with restriction X2 On x2 and restriction U2 on U2(·), that is, SUppose there exists a gain function 1'2(·) such that, for any x 2(0) E X 2, the response x 2 (t) to a piecewise-continuous u 2 (·) (satisfying the indicated restriction) exists for all t 2: 0 and

The following Theorem shows that, if the gain functions "YI ( ·) and 1'2 (·) satisfy the small-gain condition, the interconnected system (14.60) - (14.61) - (14.62) satisfies an asymptotic input-output bound, with appropriate restrictions. Theorem 14.3.1. Consider the interconnected system of Fig. 14.2 and suppose both subsystems satisfy asymptotic input-output bounds, with restrictions and gain functions as indicated above. Suppose UI = oo. Suppose also that lim 1'1 (r) < oo,

r-+oo

Let

ii

be any number satisfying

ii :-:; v, Suppose that, for all (xi(O),x2(0)) E xl X x2 and any piecewise-continuous v(·) satisfying llv(·)l/a :<:; V, the response (xi (t), x2(t)) exists for all t 2: 0. Then, if "YI o 1'2(r) < r for all r > 0, the interconnected system satisfies an asymptotic input-output bound, with restriction XI x X 2 on (xJ',x2), restriction ii on v(·) and

1/YIOIIa < I/Y2(·)1/a

<

"Yv(llv01/a) 1'2 o "Yv(liv(·)l/a) ·

(14.65)

14.3 Stabilization Using Saturation Functions

Proof. If 1/v(·)/la:::;

V:::; V,

since there is no restriction on

llu 1(·)lla.,

257

we have

Assuming that 1/v(-)1/a is finite (otherwise, there is nothing to prove), this shows - in view of the hypothesis on /l (·) and the definition of V - that IIY1 Oil a is finite and Therefore, IIY2(·)ila ::S /2(/IYl Olla) , and also /ly 2(·)/la is finite. · Combining the inequalities and using the small-gain condition yields (14.65). 0 such that (14.18) holds. Then, the matrix A - BET P has all eigenvalues in c-. Let o"(-) be any !Rm -valued saturation function and consider the system :i;

y

= =

Ax+Bu(-BTPx+v)+w

(14.66)

X.

Then, there exists a number tS' > 0, such that {14.66) satisfies an asymptotic (input-output) bound, with no restriction on X 0 and restriction tS' on v(·) and w(·), with linear gain functions !v(·) and !w(·). Proof. If v(·) and w(·) are piecewise-continuous, the response x(t) exists for all t ~ 0, for any x(O). Set c = max{.Jm,yn} and 6' = oj2c. Then, for any v(·) and w(-) such that

max{/lvOIIa, llw(·)lla} :::;

o'

there is T > 0 such that

l!v(t)/1 :::; tS,

!lw(t)l! :::;

o,

for all t E [T, oo).

The estimates in Proposition 14.1.5 hold for any X 0 • Thus, no matter what the value of x(T) is, the result follows from the fact that the restrictions of v(·) and w(·) to the interval [T,oo) satisfy the required bounds.
258

14. Stabilization Using Small Inputs

In other words, this Corollary says that, if (A, B) is stabilizable and there exists a symmetric matrix P > 0 such that (14.18) holds, the set of matrices K such that A+ BK has all eigenvalues inc- and Ax+Ba(Kx+v)+w y

X

(14.67)

satisfies an asymptotic (input-output) bound, with no restriction on X 0 and restriction o' > 0 on v(·) and w(·), with linear gain functions, is not empty. This fact is exploited in the derivation of the next result, which describes how a control law for a system of the form (14.58) can be effectively designed, simply using linear functions and saturation functions. Theorem 14.3.3. Consider the system

Az+Bu+g({,u) f({, u) '

(14.68)

in which x E IR.n, { E JR.", u E IR.m, g({,u) and f({,u) are locally Lipschitz, and g(O, 0) = 0, f(O, 0) = 0. Assume that: (i) (A, B) is stabilizable and there exists a symmetric matrix P > 0 such that PA+ATP~O,

(ii) the system

{ = f({,u) y = { satisfies an asymptotic {input-output) bound, with restriction E on { 0 and restriction U > 0 on u(-), with linear gain function. (iii) the function g({, u) is such that lim

llg({, u) II

il((,u)ll->0

II({, u)ll

= 0.

Let a ( ·) be any Rm -valued saturation function. Pick an n x m matrix K such that A+ BK has all eigenvalues inc- and, for some o' > 0, system (14.67) satisfies an asymptotic (input-output) bound, with no restriction on 0 X and restriction o' on v(·) and w(·), with linear gain functions. Pick two m X m matrices F and fl. Then, there exist numbers >. > 0 and V > 0 such that system (14.68), with control u • (z, v) = >.a (Kz + >. + flv , (14.69)

rv)

and output y = col(z, {) satisfies an asymptotic {input-output) bound, with restriction ]Rn x E on (z 0 , {0 ) and restriction V on v(·), with linear gain function.

14.3 Stabilization Using Saturation Functions

v

259

Yl

= Az + Bu*(z,v) + u1 YI = u*(z,v) i

r--

u1

~

= !(~, u2)

Y2 =g(~,u2)

Y2

u2

Fig. 14.3. Feedback connection with input v.

Proof. The feedback law (14.69) yields the closed loop system of Fig. 14.3, which is a structure to which it is possible to apply the small-gain Them:em 14.3.1. For convenience, the proof is split into a few steps. (a) We first show that, if v(·) is piecewise continuous, the trajectories are defined for all t. To this end, suppose the trajectories are defined on [0, T) and observe that, from the properties of a saturation function,

IIYI(t)ll

~

m>.k + llf1v(t)11.

Thus, y 1 (t) is bounded on [0, T). Consider now, for the lower subsystem of Fig. 14.3, the input thus defined

U2(t)

Yl(t) 0

if tE [O,T) if t?_T,

and let ~(t) denote the response from the initial state ~ 0 • By hypothesis (ii), since llu2 (·)lla = 0, ~2 (t) is defined for all t?. 0 and hence bounded on [0, T). By causality, ~(t) = ~(t) for all t E [0, T) and therefore ~(t) is bounded on [O,T). Now, consider the upper subsystem of Fig. 14.3. Since both v(·) and y 2 (t) are bounded on [0, T), an identical argument shows that z(t) is bounded on [0, T). This shows that no finite escape time is possible, i.e. the trajectories are defined for all t ?. 0. (b) Observe that, if K is chosen as indicated in the Theorem, then, for each >. > 0, the system

z. = y

=

Az+B>.u (Kz+v) -->.- +w

(14.70)

z

satisfies an asymptotic (input-output) bound, with no restriction on Z 0 and restriction >.0' on v(·) and w(·), with the same linear gain functions, say

260

14. Stabilization Using Small Inputs

'Yv(r)

= Lvr,

= Lwr.

'Yw(r)

To check that this is the case, it suffices to define the new coordinate

x= z/>., which changes the previous system into x = Ax+Bcr(Kx+*)+I y

=

>.x.

As a consequence, if

IIFv(·)lla :S >.8', for any

Z

0

,

+ u10IIa :S

IIB!2v(·)

>.8',

the response z(t) of the upper subsystem in Fig. 14.3 satisfies llz(·)lla

:S max{LviiFv(·)lla, LwiiB!2v(·) + U1 (-)lla} .

Indeed, it is possible to find a number J" that, if !lv(·)lla :S >.8",

> 0 and

a number N

> 0 such

for any z 0 , the response z(t) of the upper subsystem in Fig. 14.3 satisfies

!lz(·)lla :S Nmax{!lv(-)lla,llul(·)lla} · Now observe, using the property (14.21), that, for each i l[u*(z,v)]il

(14.71)

= 1, ... , m

+ l[!2v]il

<

>.icri([Kz:rv]i)l

<

>.min{HI(Kz: rv]il, k}

<

max{min{2>.k,L max lzii},L max lvil}}

+ l(!2v]il

1:5J:5n

(14.72)

l~J:5m

r,

where L > 0 is a constant depending on H, !2. This, using (14.71), shows that the output response y 1(·) of the upper subsystem of Fig. 14.3, if llv(·)lla for any

Z

0

:S >.8",

llul(·)lla

:S >.J",

(14.73)

satisfies

IIYIOIIa

:S max{min{2>.k, LN max{llu1 (·)lla, llvOIIa}}, LllvOIIa} .

Since, for any triplet of nonnegative real numbers a, b, c min{a,max{b,c}}:::; max{c,min{a,b}}, the latter estimate yields IIYI(·)IIa

:S max{min{2>.k,LN!Iui(·)IIa},L(N + 1)llv(·)lla}.

(14.74)

14.3 Stabilization Using Saturation Functions

261

The estimate in question holds if (14. 73) holds, in which llu 1(·)II a. ::; >..o". To show that this holds for any llu 1(·)lla, it suffices to assume, without loss of generality, that N ~ 2k/ Lo", which yields min{2>..k,LNIIu1(·)lla}

= 2..\k.

If this is the case, in fact, (14.74) reduces to

IIY10IIa::; ma.x{2>.k,L(N + 1)llv(-)lla}, which indeed holds because (14.72) implies l[u*(z,v)]il::; ma.x{2>..k,L m~ lvil}. 1~3~m

It can therefore be concluded that (14.74) holds for any llu1(·)lla and for llv(·)lla::; >..o". Now, consider a class K function 'Yl (-) satisfying

LNr,

11(r) 11(r)

< LNr,

lim 1 1 (r)

3>..k,

r--+oo

X

0

,

for any

if r ::; 2..\kj LN for all r

and set

'Yv(r) = L(N + 1)r . Then, it is concluded that the upper system of Fig. 14.3 satisfies an asymptotic input-output bound, with no restriction on Z 0 , restriction U1 = oo on u 1 (·), restriction M" on v(·), and gain functions 11(·), 'Yv(·), with lim 11 (r) = 3>..k

r--+oo

and i'vO a linear function. (c) By assumption (ii), the response ~0 of the lower subsystem of Fig. 14.3, for any ~o E E and for any u2(·) such that llu2(·)lla::; U, satisfies (14.75) for some G > 0. Moreover, using assumption (iii), given any number o> 0, such that

1~j~m

f

>

0, there exists

l~j$m

Thus, using (14.75), it is concluded that, given any number M > 0, there exists U2 > 0 such that the response Y2 (·) of the lower subsystem of Fig. 14.3, for any C E E and for any u2(·) such that llu2(·)lla::; U2, satisfies

262

14. Stabilization Using Small Inputs

with 'Y2(r)

= Mr .

Choose M such that MLN< 1

(14. 76)

and fix u2 accordingly. (d) In the previous two steps we have shown that the two subsystems of Fig. 14.3 satisfy the hypotheses assumed for (14.60) and (14.61). In order to be able to use the result of Theorem 14.3.1, we have to choose V such that V:::; M" and max{ lim rdr),1v(V)}:::; max{3.-\k,L(N + 1)M"}:::; U2, r--+oo

which is indeed always possible. Since, by (14.76), the small gain condition

r! 0 'Y2(r) < T holds for all r > 0, Theorem 14.3.1 shows that, for any (z 0 , (0 ) E !Rn x 2, and all v(·) satisfying llv(·)lla:::; V, we have

!IYIOIIa < 1v(llv(·)lla) IIY20IIa < r2 ° iv(llvOIIa) ' where the gain functions on the right-hand sides are linear functions. Combining these inequalities with those provided by hypothesis (ii) and by (14.71) it is seen that there exist a number V :::; V and linear gain function for which the result of the theorem holds. <1 If the lower subsystem of Fig. 14.3 has an equilibrium, at ({, u) = (0, 0), which is asymptotically stable in the first approximation, then the feedback law (14.69), with v = 0, is such that the equilibrium (z, ~) = (0, 0) of the corresponding closed loop system is asymptotically stable, and all trajectories with initial conditions in !Rn x 2 converge to zero as t ~ oo. In particular, if 2 = IR", the equilibrium in question is globally asymptotically stable.

Corollary 14.3.4. Consider the system (14.68). Suppose the hypotheses of Theorem 14.3.3 hold with 2 = IRv and choose K as indicated in the Theorem. Suppose, also, that f({, u) is differentiable at ({, u) = (0, 0) and that the equilibrium ( = 0 of ( = f({,O) is asymptotically stable in the first approximation. Then, the interconnected system {14.68) - (1.{69) is globally asymptotically stable, when v = 0.

14.3 Stabilization Using Saturation Functions

263

Proof Note that the linear approximation of the system at the equilibrium (z, ~, v) = (0, 0, 0) is characterized by a matrix of the form

( A+*BK

~)

in which A + B K and F have all eigenvalues in C -. Thus, this equilibrium is locally asymptotically stable. By Theorem 14.3.3, if v(t) = 0, · lim (x(t), €(t))

t_.oo

= (0, 0)

for any (x(O), ~(0)). Thus, the result follows.


The result of Theorem 14.3.3 can also be used to the purpose of recursively stabilizing systems in feedforward form. To this end, some additional hypotheses are needed, as is indicated in the result which follows. Lemma 14.3.5. Consider the system

= =

.i ~i

Az + g;(€;, u)

(14. 77)

!;(€;, u)

in which z E JRn, (; E JR", u E JRm, g;((i,u) and J;((;,u) are locally Lipschitz, differentiable at((;, u) = (0, 0), and g;(O,O) = 0, J;(O,O) = 0. Assume that: (i) there exists a symmetric matrix P > 0 such that P A+ AT P ::::; 0, (ii) the linear approximation of (14.77) at the equilibrium (z,(;,u) is stabilizable, Moreover, assume that there exists a function

lR"

a; :

X ]Rm

((;, v)

--+

]Rm

r-+

a;((;, v) ,

with a;(O,O) = 0, which is locally Lipschitz, differentiable at with the following properties: (iiia) the matrix 8a;((;,v)] [ a:v (o,o) is nonsingular, (iiib) the matrix [

a/; ( €;, a; (€;' v))] 8(; (0,0)

has all eigenvalues in C- , (iiic) the system

(;

==

/;((;,a;((;, v))

y

==

€i

= (0,0,0)

(~;,v)

= (0,0),

264

14. Stabilization Using Small Inputs

satisfies an asymptotic {input-output) bound, with restriction Xi on striction V > 0 on v( ·), with linear gain function. Set {i+l = col(z,~i), ;; = n + v,

and [

~i,

re-

8h+I (~i+I, ai (~i, v))] 8~i+r

co,o)

. [ 8fi+t(~i+l,ai(~i,v))] 8v (o,o)

Then, the pair (Fi+ 1 , Gi+I) satisfies the hypotheses of Corollary 1.{3.2. Let a(·) be any JRm -valued saturation function. Pick a ;; x m matrix Ki+ 1 such that (Fi+ 1 +Gi+ 1 Ki+I) has all eigenvalues inc- and, for some <5' > 0, system :i: = Fi+tX + Gi+ta(Ki+tX + v) + w (14.78) y = X satisfies an asymptotic {input-output) bound, with no restriction on X 0 and restriction <5' on v(·) and w(·), with linear gain functions. Pick two m x m matrices r and fl such that r + fl is nonsingular. Consider the function Qi+ 1

:

JRii

!Rm

-+

!Rm

(~i+t,v)

f--?

ai(~i,>.a(Ki+I(i~r+Fv) +ilv).

X

Then, there exist numbers).> 0 and (a) the matrix [

8ai+l (~i+l,

8v

V > 0 such

(14. 79)

that

v)) (o,o)

is nonsingular, (b) the matrix

8 /i+r (~i+l, ai+l (~i+l, v))) [ a~i+l co.o) has all eigenvalues in C-, (c) the system (i+l = y

h+I (~i+l, ai+l ((i+l, v)) €i+l

satisfies an asymptotic (input-output} bound, with restriction xi+l on ~i+r, restriction V > 0 on v(·), with linear gain function.

{14.80)

= Rn X xi

14.3 Stabilization Using Saturation Functions

265

Proof. It is easy to check that the matrix Fi+ 1 has the following structure

of;~

ra~J

+ ( ofiQ;

)

) (

[au Lo.o)

(0,0)

) o [00:;] 0~; (0,0)

'

that the matrix Gi+ 1 has the following structure Gi+r

=(

Q; [at;] au

[00:;] Tv

) (0,0)

oo , ( ' )

and that

[of;] [8o:;] [8/;(t:,;,o:;(t:,;,v))] [aJ;] 8f.i (0,0) + au (0,0) 8~; (0,0) = of,; (0,0)

•

Thus, using the hypotheses (i), (ii), (iiia) and (iiib), it is immediate to check that the pair (Fi+I, Gi+ 1) is stabilizable and that there exists a symmetric matrix Pi+ I > 0 such that Pi+ 1Fi+ 1 + Fl~. 1 Pi+ 1 ::; 0, i.e. that the p1!-ir (Fi+ 1 , G;+I) satisfies the hypotheses of Corollary 14.3.2. Since = [oo:;) (r + n) , [ ao:Hr]

8v (O,o) property (a) indeed holds. Since [

ov

(o,o)

8fi+r(f.i+t,O:i+r(f.Hr,v))] c ao.,i+l

(0,0)

_ "'· +G· K· - L"t+I •+I •+1 ·

then also property (b) holds. To prove (c), set

9i+r(E,;,w)

= h+r(E.i+r,o:;(f,;,w))- Fi+tE.i+r- Gi+IW,

where the notation on the left-hand side reflects the fact that the function on the right-hand side depends only on f,; and w and not on the z-component of f.Hr, and consider the auxiliary system (

~

Fi+r(+Gi+tW+9i+t(f.,w)

=

driven by W

=

(14.81)

/i(E,, o:;(E,, w)) ' , (KH;.(A+

A(j

rv) + n

uV ,

in the initial condition (( ,C) = (E;'+ 1 ,E,i). By construction, the response ((t) coincides with the response f.H 1 (t) of system (14.80). Now, system (14.81) has precisely the structure of the system considered in Theorem 14.3.3. Since 0

lim

ll9i+t(f.,w)ll

li({,w)ll-+0

ll(~,w)ll

=0 ,

all hypotheses of this Theorem hold and thus property (c) follows.

<J

266

14. Stabilization Using Small Inputs

Clearly, this result can be repeatedly used to globally asymptotically stabilize a system in feedforward form, such as a system modeled 'by the equations XI = A1x1 +gi(x2, ... ,Xn,u) ±2 = A2x2 +g2(x3, ... ,Xn,u) (14.82) Xn-1 Xn

=

An-lXn-1 + gn-1 (xn, u) fn(Xn,U),

under the hypotheses that each of the upper n - 1 subsystems, when the corresponding input (i.e. (xi+ 1, ... , Xn, u) for the i-th subsystem) is zero, is stable in the sense of Lyapunov, and the n-th subsystem, by means of some feedback law u = on(Xn, v), can be changed into a system which satisfies an asymptotic (input-output) bound, with some nonzero restriction on v(·) and linear gain function. Theorem 14.3.6. Consider a system o/ the form {14.82), in which x; E !Rn•, u E !Rm, the g;(Xi+I, ... , Xn, u) 's are locally Lipschitz, differentiable at (xi+ I, ... , Xn, u) = (0, ... , 0, 0), and g;(O, ... , 0, 0) 0. Assume that:

=

(i) for each i = 1, ... , n- 1, there exists a symmetric matrix P; > 0 such that P;A; + A[P; ~ 0, (ii) the linear approximation of {14.82) at the equilibrium (xJ, ... ,Xn,u) (0, ... , 0, 0) is stabilizable. If there exists a function

=

with on(O, 0) = 0, which is locally Lipschitz and differentiable at (xn, v) (0, 0), with the following properties: (iiia) the matrix

=

is nonsingular, (iiib) the matrix [

8fn(Xn 1 On(Xn 1 8xn

v))] (0,0)

has all eigenvalues in C- , (iiic) the system Xn

Y

=

fn(Xn, On(Xn, v)) Xn

satisfies an asymptotic {input-output) bound, with no restriction on striction V > 0 on v(·), with lznear gain function.

x~,

re-

14.4 Applications and Extensions

267

Then, there exists a feedback law

which globally asymptotically stabilizes the equilibrium point (x 1 , ... , xn) (0, ... , 0).

14.4 Applications and Extensions The design methods described in the previous sections lend themselves to a variety of applications and extensions. For instance, the result of Theorem 14.3.6 can be used to show that a linear system

:i:

= Ax+Bu

in which (A, B) is stabilizable, if the matrix A does not have eigenvalues with positive real part, can be globally stabilized using a feedback law whose ainplitude does not exceed any (arbitrarily small) fixed bound. The special case in which the eigenvalues of A on the imaginary axis have unitary geometric multiplicity, since there exists a matrix P > 0 such that P A+ AT P::; 0, can be dealt with using the feedback law described in Proposition 14.1.5. H the matrix A does not have eigenvalues with positive real part and the eigenvalues on the imaginary axis have higher geometric multiplicity, a preliminary change of coordinates in the state space can reduce the system to a form in which Theorem 14.3.6 is immediately applicable. For instance, consider the case of a stabilizable single-input system in which the matrix A has one eigenvalue at zero, of multiplicity r - 1, and all remaining eigenvalues with negative real part. Then, in suitable coordinates, the system can be described by equations of the form Xt

AtXl

X2

X3

+ A12X2 + · · · + AlrXr + B1u (14.83)

Xr-1

Xr u

in which A 1 has all the eigenvalues in c-. This system can be regarded as a system of the form (14.82), in which A 2 = · · · = Ar-1 = 0. Indeed, the hypotheses (i) and (ii) of Theorem 14.3.6 hold. As far as the hypotheses (iii) are concerned, observe that the last equation characterizes a subsystem to which Corollary 14.3.2 is applicable. In particular, the feedback law O:r(Xr,

v) =a( -Xr + v) ,

in which a(·) is any saturation function, is such that

268

14. Stabilization Using Small Inputs

satisfies an asymptotic (input-output) bound, with no restriction on x~, restriction V > 0 on v(·), with linear gain function. Thus, this feedback law is such that three hypotheses (iiia), (iiib) and (iiic) of Theorem 14.3.6 hold. As a consequence, the system in question can be globally asymptotically stabilized, by means of a feedback law which has the form u

= u( -Xr + v(xn ... , x2)) .

As shown in the proof of Lemma 14.3.5, the function v(xr, ... , x2) is a composition of saturation functions, which has the form

v(xr, ... ,x2) ,

A!CT

=

(k6xr+ktxr-1 't

+ >.2 (

A

-(T

At

2

...

Ar-2 (k~- xr+···+k;=~x2))) + --, CT \ Ar-3

•

"r-2

Note that the recursive process does not need to involve the first subsystem, since the latter is a linear stable system driven by an input which, under the indicated feedback law, asymptotically decays to zero as t tends to oo. Remark 14.4.1. It is useful to observe that the class of systems considered above is the largest class of linear systems which can be globally stabilized using bounded feedback. In fact, if the matrix A has an eigenvalue with positive real part and the control is bounded, there are initial states that can never be driven to the origin. This can be easily seen, for instance, if the system has a positive eigenvalue, by changing coordinates in such a way as to isolate a one-dimensional subsystem of the form

x=ax+bu with a> 0. If lui : : ; U, any initial state X 0 satisfying x 0 2: (jbUI/a) produces a trajectory that diverges as t tends to oo. <1 As an example of how the results of the previous two sections can be extended to more general classes of systems, we consider here the case of systems modeled by a "perturbed" version of the equations (14.56), such as equations of the form

+ bt (xr, X2, ... , Xn)u + qr (xt, x2, ... , Xn, u)u 2 a2(x2,X3, ... ,xn) + b2(x2,X3, ... ,xn)u + q2(Xt,X2, ... ,Xn,u)u2

Xt x2

= =

a1 (x1 ,x2, ... , Xn)

Xn-1

= =

an-1 (Xn-1, Xn)

Xn

+ bn-1 (Xn-1, Xn)U + qn-1 (Xt, X2, ... , Xn, u)u2

an(Xn) + b(Xn)U + qn(Xt,X2 7 ••• 7 Xn, u)u 2 7 (14.84)

14.4 Applications and Extensions

269

in which u E JR, and the functions a;(x;, ... ,xn), b;(x;, ... ,xn) sa.tisfy hypotheses similar to those considered for the corresponding functions in the form (14.56). These equations no longer have an upper-triangular structure, but the extra terms which destroy this structure contain u 2 as a factor. Thus, it is reasonable to expect that their presence could be rendered, to some extent, "ineffective" by choosing an input having sufficiently small amplitude (and this can be done by taking explicit advantage of the fact that arbitrarily small inputs are, in this case, allowed by the hypotheses assumed on the various functions which characterize the upper-triangular part of (14.84)). For simplicity, as an illustration on how to proceed in more general situations, we restrict the analysis to the case in which the decomposition (14.84) consists only of two subsystems, of which the lower one is one-dimensional. More precisely, we consider the case of systems modeled by equations of the form z = Az + Mf. + Bu + p(f.) + g(f.)u + q(z, f., u)u2 ~ = u + ¢>(z, f., u)u 2 in which z E JRn, f. E lR, A is such that P.4 +AT P ~ 0 for some P > 0, p(f.) vanishes as f.= 0 together with its first derivative p'(f.), and g(f.) vanishes as f.= 0. Moreover, we assume that, for some K > 0 and for some continuous positive-valued function -y(f.),

//q(z,f.,u)ll

~

K-y(f.)(1 + llzll)

(14.86)

and

/¢>(z, f., u)/ ~ K(1 +/f./) ,

(14.87)

for all (z, f.) E JRn x 1R and all /u/ ~ 1. Let c be a number satisfying 0 < c < 1, let f(·) : IR -t IR be any continuous function satisfying 0 < f(s) let

~

1,

for all s E lR,

J.LO : Ill -t IR denote the function 1

J.L(s) =

v'l + s2

'

set x = (z, f.) and define

-\(x)

= c!(llzii)J.L(f.) .

We begin discussing the effect, on system (14.85) of a control law of the form (14.88) u = -\(x)u(,\(!) +

v)

in which -\(x) is a function of the type indicated above and the function u(·) is the saturation function

270

14. Stabilization Using Small Inputs

a(s)

s

= ..Jf+S22 1+s

already considered in section 14.1. As shown below, this input is able to induce certain properties of input-to-state stability, with restriction on v, on the lower subsystem of (14.85). Lemma 14.4.1. There is a numberc 1 for all E :::; c 1 , there exists a number M

2K/~/(1 +

(

/W :::;

< 1, depending only on K, such that, > 0 satisfying

-~

1 + A(x) + v

)2 :::; M(1+e) f(//z/1)

(14.89)

for all (z,~) E JRn x lR and all/v/:::; 1. Proof. The inequality on the left-hand side is equivalent to an inequality of the form

with

~

a= A(x) '

b = 2K/~/(1 + /~/) .

If b :::; 1 there is nothing to prove. Thus, there exists ~0 > 0, depending only on K, such that the left-hand side of (14.89) holds for all/~/ :S ~o- Forb> 1, an elementary calculation shows that the inequality in question holds, for all /v/ :S 1, if /a/ 2: ..jb2 - 1 and /a± ../b2 - 1/ 2: 1. The latter are implied by Ia/ 2: lb/ + 1, i.e.

~1~)

2:

2K/~/(1 + /~/) + 1

for

all/~/ 2: ~o-

Since /f(//z/1)/ :::; 1, the latter holds if

/~/ > E 2K/~/(1 + /~/) + 1 ~

for all /~/ 2: ~o,

which indeed holds if E is small enough. The inequality on the right-hand side is straightforward, if /v/ :S 1. <1 Lemma 14.4.2. Consider the system

{ = u + ¢(z(t), ~, u)u2

(14.90)

in which ¢(z, ~, u) satisfies the hypothesis {L/.87}, for all (z, {) E an x lR and all lui :S 1. Moreover, suppose z(t) is a continuous function defined for all t 2: 0. Choose the input {14.88}, withE :S c 1 . Then, if /vi :S 1, the function U(~) is such that

=e

(14.91)

14.4 Applications and Extensions·

where

271

e

a(~)= 1 + (2 is a class/( function. In particular, since ..\(x) :S

E,

l~(t)l :S max{I~(O)I, J2E}.

(14.92)

Moreover, if f(llz(t)li) is bounded away from zero on [O,oo), for any v(·) satisfying llv(·)lloo :S 1, l~(t)l :S max{O(I~(O)i)e-bt, v'2cllv(·)lloo} ,

(14.93)

in which 8(·) is a class/( function and b > 0. Proof Observe that, in view of the specific form of the function a(-),

-e + ..\(x)v~

)2 1+ (..\(x) +v -~

+ 2 --r=K===I~I=(1=+==1=~1==)=

'. )2 1 + (..\(x) + v

( -~ )2 1+ ..\(x) +v

-~

which, in view of the left-hand side of (14.89), yields

u. :s

c-e/2-e/2+..\2 (x)v 2 )

)2 1+ (..\(x) +v -~

.

This, using the right-hand side of (14.89), yields (14.91). From (14.91) it is immediate to deduce (14.92). Finally, if f(llz(t)ll) is bounded away from zero on [O,oo), we have that

1~1 2:

hEivl

=?

U :S -oo(~)

for some fixed c > 0. This, since o:(O in a neighborhood of~ = 0 can be bounded from below by ae for some a> 0, shows (in view of Lemma 10.1.5 and 10.1.6) that if v = 0 l~(t)l

:S ~(I~(O)i)e-bt

and this, bearing in mind the properties of an input-to-state stable system described in section 10.4, completes the proof. <1 As a first consequence of this Lemma, it is possible to prove the following stabilization result.

272

14. Stabilization Using Small Inputs

Proposition 14.4.3. Consider system (1.4.85). Suppose that PA+AT P ~ 0 for some P > 0, that p( ~), p' (~), and g( 0 vanish at ~ = 0 and that the bounds (14.86) and (14.87) hold for all (z, ~) E IRn x lR and all lui ~ 1. Suppose also that the pair

is stabilizable. Then, there exists a feedback law u = u • ( z, ~) which globally asymptotically stabilizes the equilibrium (z, ~) = (0, 0). Proof. Set

u(~) = EJL(~)O" C:;(~)) ,

and consider system (14.85) with feedback law u

= u(~) + v.

This yields a system of the form Az + p(z, ~)

+ b(z, ~)v + q(z, ~' v)v 2 rp(z, ~) + B(z, ~)v + J>(z, ~' v )v 2

i

(14.94)

in which p(z, ~)

M~ + Bu(~)

b(z, ~)

B

rp(z, 0

u(~)

+ c/!(z, {, u({))u

B(z,O

1+

[~¢] u 2 (~) + (z,~,u(~))2u(~). uU u=u(O

+ g(~) +

+ p(~) + g(~)u(O + q(z, ~' u(~))u

[af)q J U

u=u({) 2

u

2

2

(()

(~) + q(z, ~' u(~))2u(~)

(~)

The result of the Proposition can be proven by showing that the system obtained from (14.94) by neglecting the terms of order higher than 1 in v, namely the system

i

= . Az + p(z, ~) + b(z, Ov

~

=

rp(z, 0

+ B(z, ~)v ,

(14.95)

satisfies hypotheses identical (or equivalent to) those of Theorem 14.2.6. As a consequence, there exists a positive definite proper function W(z, ~) such that

aw

az [Az

+ p(z, OJ +

aw

a~ rp(z, 0 ~ 0

and such that system i

Az + p(z,~)

~

rp(z,~)

y

=

aw

az b(z,~)

+

aw

a~ B(z,~),

14.4 Applications and Extensions

273

is zero-state detectable. Thus, using Theorem 14.1.3, it is possible to find a feedback law which globally asymptotically stabilizes (14.94). An immediate check shows that hypotheses (i), (iii), (iv) and (v) of Theorem 14.2.6 hold. As far as hypothesis (ii) is concerned, observe that u(~) can be viewed as an input of the form (14.88), with v = 0 and f(z) = 1. Thus, in the system i = Az + p(z,~) (14.96) ~ = cp(z,~), so long as z(t) is defined, the response ~(t) satisfies the estimates of Lemma 14.4.2. In particular, ~(t) is bounded by max{I~(O)I, v'2c-}. This, using the growth condition llp(z,~)ll ~ /'(IW(l + llzll), shows that, so long as z(t) is defined, IIAz(t) + p(z(t),~(t))ll ~Co+ C\llz(t)ll for some C0

> 0 and C1 > 0. Thus llz(t)ll

~ llz(O)II + 1t[Co + CIIIz(s)ll]ds

from which, using Gronwall-Beilman's inequality, it can be concluded that z(t) cannot escape to oo in finite time. In other words, z(t) is defined for all t and therefore, by Lemma 14.4.2, the function U(~) = is such that

au

8~ cp(z, ~) ~ -c1

e

e

+ (2

and l~(t)l ~ 9(I~(O)I)e-bt ,

for some c > 0, b > 0 and class K function 9 (·). This shows that system (14.95) satisfies an hypothesis equivalent to the hypothesis (ii) of Theorem 14.2.6 and also that the function U(~) has the properties indicated in that Theorem. From this, it is not difficult to prove - adjusting to the present case the arguments presented in section 14.2- that the function W(z, eJ

= zT Pz + U(~) + IP(z, e) ,

in which the cross-term IP(z, ~) has the expression .P(z,()

= fooo 2[z(z,~,s)]TPp(z(z,~,s),{(z,~,s))ds,

with z(z, ~, s) and ((z, ~, s) denoting the value at time t = s of the integral curve of system (14.96) passing through (z,~) at timet = 0, is a positive definite and proper smooth function which possesses the properties indicated above.
274

14. Stabilization Using Small Inputs

The actual construction of the stabilizing feedback law whose existence has been proven in the previous Proposition requires the explicit calculation of the Lyapunov function W(z, ~), which in turn requires the explicit integration of the ordinary differential equation (14.96) for any initial condition (z, ~). This may prove to be a difficult computational task. A way to avoid this problem is to look at the possibility of recursive use of saturated controls, following the pattern of ideas explained in section 14.3. In what follows, we show how this can be achieved in the (easy) case in which the upper subsystem of (14.85) is one-dimensional. In more general cases, appropriate extensions can be worked-out. Proposition 14.4.4. Consider the system ~

+ u + p(~) + g(~)u + q(z, ~, u)u 2

(14.97)

u+ ¢(z,~,u)u 2

in which z E lR ,~ E lR, p(~), p'W and g'(~) vanish at~= 0, and q(z,~,u), ¢(z, ~, u) satisfy the hypotheses (14.86) and (14.87}, for all (z, ~) E lR x lR and all lui s; 1. Choose the input u in which

= A(x)£T(- A~) + ~£7(- JL~z)))

,

1 1 A(x) = cJL(z)JL(O = c ~2 ~.

(14.98)

v1+z y1+~

Then, there is a number c• > 0 such that, for all 0 < c s; c•, the equilibrium of the closed loop system (14.97} - (14.98) is globally asymptotically stable.

Proof. Observe, first of all, that the control (14.98) has the form

u

= A(x)£T(- A~)

+v),

with

For convenience, the proof is split in three steps. (a) In the first step it is shown that, if c is small enough, ~(t) and z(t) are defined for all t ~ 0, and there exists a finite timeT* > 0 such that l~(t)i s;

To this end, assume c

A(x(t))

for all t

~

T*.

(14.99)

< c 1 and note that, so long as z(t) is defined, since

lvl s; 1, from the proof of Lemma 14.4.2, with f(s)

= JL(s), one obtains

i~(t)i s;L

for some number L > 0. Thus, using the fact that lui that j£T(s)l s; 1 and (14.86), we have

<

1, that

JL(~)

s; 1,

14.4 Applications and Extensions

i

< 1~1 + 1 + IP(~)I + lg(~)l + < C1 + C2(1 + lzl)tL2(z) ,

275

K/'(~)(1 + lzl)c: 2 t.t?(~)tL 2 (z)

where cl and c2 are such that 1~1 + 1 + IP(OI + lg(~)l :::; c1 K !'(~)c- 2 tL 2 (~) :::; C2

for all 1~1 :::; L. Thus, lz(t)l $lz(O)I + C1t +

l

0

t

1 + lz(s)l ( ) ds:::; Co+ C1t, 1 + z2 s

for suitable C0 ,C1. This shows that z(t) cannot escape to oo in finite time. Observe that, if lsi ~ 1 and lvl < ~'

sgn(s)u(-s+v) <

-21 -a

for some a > 0. Observe also that, since lu(s)l $ 1, tL(z) :::; 1, and (I + IWtL(~) $ v'2, then 2 2 2 lr/>(z, ~' u)u 1 S K(l + l~l)c- 2 A (x) S K(1 + IWc- tL(~)J.t(Z)A(x) :::; J2Kc: 2 A(x). Now, suppose that l~(t)l ~

A(x(t))

for all t

~

0.

(14.100)

Then, we have

Let c:2 be such that v"2Kc:~

=a. Then, for all c: S

1 -dl~l < - -A(x(t))

dt -

which implies l~(t)l-1~(0)1 S

2

!1t

-2

min{ciJc2}, we have

'

A(x(s))ds.

(14.101)

(14.102)

0

Now, recall that l~(t)l $ L for all t ~ 0 and that

A(x(t)) = c: which yields

1

1

1 1 > f1"":"T?2 2 y'I + ~ (t) v'I + z (t) - v 1 + L .j1 +(Co+ C1 t)2 2

,

276

14. Stabilization Using Small Inputs

lim t-+oo

rt A.(x(s))ds =

Jo

00.

Thus, (14.102) cannot hold for all t ?: 0, i.e. (14.100) cannot hold for all t?: 0. This being the case, letT be any time such that j~(T)j = A.(x(T)), observe that

l~(t)l )] 1 (di~IJ I~(T)I ~(x(T)) [dtd(A.(x(t)) t=T = A.(x(T)) dt t=T- A.(x(T)) A.(x(T)) and therefore, using (14.101), that d ( l~(t)l )] [ dt A.(x(t)) t=T::;

1

-2 +

I

i.X(x(T)) A.(x(T)) ·

(14.103)

On the other hand

I

.X(x(t)) < i A.(x(t)) -

Jill_+ _E)_ 1+e 1+z2 ·

Using the hypotheses that the functions p(~),g(~) vanish at~= 0, and the bounds (14.86) and (14.87), it is easy to check that there is a number c: 3 such that, if 1~1 ::; c: 3 and lui ::; C:3 l~llu

+ r/>(z, ~. u)ul + 1+e

.:....:..o..:._-....:.-::"----'-~

lzll~

+ u + p(~) + g(~)u + q(z, ~. u)u 2 1 < -1 1+ z 2 2.

This, using the facts that lui ::; E and, at timet= T, shows that, if c:::; min{Et,E2,c:3},

.X(x(t)) IJ

[i A(x(t))

I~(T)I

= A.(x(T)) ::;

€,

1

t=T

<2

and hence, using (14.103), that

i (_@2]_)] <0 [dt A.(x(t)) t=T · It is concluded from this that, for some T* ?: 0, (14.99) necessarily holds. (b) Consider now the upper equation of (14.97) on the time interval [T*, oo ). Add and subtract -A(x)v, to obtain i

= A(x)v + [~- A(x)v + u] + p(~) + g(~)u + q(z,~,u)u

2

.

(14.104)

Observe- using the property (14.20) of the saturation function- that

14.4 Applications and Extensions

I~- A(x)v + ul

I~- .A(x)v + A(x)uC(!) + v) I =

+v)- C(!) +v)l

A(x)lu(.x(!)

-~ -~ < .A(x)u ( .A(x) + v) ( .A(x) + v) Since p(~),

277

p'(~)

and g(O vanish

at~=

(

-~

:c::; .A(x) .A(x)

+ v)

2

.

0, there exists functions PW and

9(~) such that

p(~)

+ g(~)u = fi(~)e + 9(~)~u

and hence, using the fact that

1~1 :c::;

.A(x) and lui :c::; .A(x)

IPW + g(~)ul :c::; .A(x)(lfi(~)l + 19(~)1)1~1

·

Using similar arguments, and the bound (14.86), observe also that

lq(z,~, u)lu2

<

K-y(~){1 + lzl)c 2 JL(z)JL(~).A(x)u 2 C(!)

<

v'2K1'(~)c 2 A(x) (.x(!) + v

r.

1~1 :c::;

Finally, using the facts that

( -~

.A(x) .A(x) + v

+ v)

.A{x) and .A(x) :c::; c, note that

)2 = ( -~ + .A(x)v)

2

:c::;

.A(x)[l~l + 2clvl + c-v

2

] .

Putting all these estimates into (14.104) yields

z=

.A(x)v +
(14.105)

in which l
with

a(~, c)

(I+ .;'2K-y(0c-2 ]2c-

b(~,c)

(1 + .;'2K-y{0c-2 ]c-

c(~,c)

=

1 + .;'2K-y(~)c 2 + lfi(~)l + 19(~)1.

Clearly, given any t5 > 0, and any k > 1 + lfi(O)I + 19(0)1, there exists c4 such that, for all c :c::; c- 4 , l
v

= ~uC(;)).

Suppose c < min {c-i}, so that, if lz(t)l -

l:Si$4

> 0,

278

14. Stabilization Using Small Inputs

dlzl < ..X(x) dt - 8

[-a(~)+ !(8 + 8kc)] f.L(Z)

2

.

Observing that the saturation function a(·) has an inverse a- 1 (-) defined on the interval ( -1, 1), choose 8 and c such that 8 + 8kc < 1.

Then, we have that dlzl

dt< 0 , i.e, for all

for all

J:.L > a-

1

f.L(Z)

(1/2)

= Jl73,

lzl~ > .Jl73 ·

This shows that lz(t)l is decreasing whenever lz(t)l is large and hence lz(t)l is bounded. Moreover, it is easy to see that there exists a finite time T 0 ~ T* such that lz(t)l ~ 1 for all t

~

T 0 • In fact, if lz(t)l

> 1 for all t ~ T*, then

lz(t)l ) a ( f-L(z(t)) >a(!) ' and

dl:l ~ A~x) [-a(l) + 1/2] .

Since the function ..X(x(t)) is bounded away from zero (because both {(t) and z(t) are bounded), the right-hand side of this inequality is negative and bounded away from zero on [T*, oo). Integrating on [T•, oo) yields a contradiction, because the integral of the left-hand side on this interval is finite. (c) Having shown that z(t) is bounded, the function f-L(iz(t)l) is bounded away from zero. Hence, the conclusion (14.93) of Lemma 14.4.2 holds for {(t). As far as z(t) is concerned, observe that the function V(z) = ~z 2 , if 8 and c are chosen as indicated in the previous step, since v is an odd function of z, satisfies

14.4 Applications and Extensions Since, for some

k > 0, kJJ.t2(z)

and J~J ~

E,

279

+ z2 ~ k~ ~

k(1

+ Jz/)

we have

JzJkJ~JJJ.t 2(z) + z 2 ~ kJzJJ~J + Ekz 2

.

This yields

V < -r==E=J.t(=~)'===[-az 2 + kJzJJ~J], 1

with

If

oand E

+

C~z)r 1

0

-

a=----kE. 8 16 are small enough, a> 0. Moreover, on the time interval [T ,oq), 0

1~

z

1 + ( - () J.t z

)2 ~ v'3 ,

and

1 1 + £2

·~ ~ J.t(~) ~ 1 .

Thus, the previous inequality shows (see e.g. the proof of Lemma 14.4.2) that Jz(t)J ~ max{B(Jz(O)/)e-ct,LIJ~(·)Jioo},

(14.106)

in which B(·) is a class K function and c > 0, L > 0 are fixed numbers. This, together with (14.93), the fact that v(·) respects the restriction Jv(t) J ~ 1 and the fact that Jv(t)J ~ Jz(t)J on the time interval [T 0 , oo ), shows, via the small-gain theorem, that if E is small enough (so that ../2cL < 1), all trajectories of the closed loop system converge to the equilibrium (z, ~) = (0, 0) as t -+ oo and that the latter is stable in the sense of Lyapunov. This completes the proof. <1

Bibliographical Notes

Chapter 10. Additional material on the comparison functions and their role in the study of the asymptotic properties of the trajectories of a nonlinear system can be found in the books of Hahn (1967) and Khalil (1996). In particular, the first of these books refers to Theorem 10.2.1 as to the "theorem of total stability". Theorem 10.3.1 is due to Sontag (1990). The notion of "input-to-state" stability and that of ISS-Lyapunov function have been introduced and thoroughly studied by E. Sontag, in a series of fundamental papers beginning with Sontag (1989): An overview of the main features of the notion of input-to-state stability can be found in Sontag (1995). The full proof of Theorem 10.4.1 can be found in Sontag-Wang (1995). The proof of Theorem 10.4.3 is taken from Sontag (1990). The use of the alternative characterization of the notion of input-to-state stability provided by the pair of inequalities (10.28) and (10.29) was introduced by A. Teel (see also, in this respect, Jiang et al. (1994) and Coron et al. (1995)). The proof of Lemma 10.4.4 is taken from Sontag-Wang (1996). The full proof of the equivalence of the pair of inequalities (10.28) and (10.29) with the property of input-to-state stability, namely Theorem 10.4.5, can be found in Sontag-Wang (1996). The proof of Lemma 10.5.1 and its application in Theorem 10.5.2 are due to Sontag-Tee! (1995). The proof of the small-gain theorem for input-to-state stable systems, namely Theorem 10.6.1, is the proof suggested by Tee! (1996a) (see also Coron et al. (1995)). The notions given here of a "dissipative" nonlinear system, and the related notions of "finite L 2 gain" and of "passive" system, follow - with minor modifications - the seminal work of Willems (1972). The stability of nonlinear interconnected dissipative systems has been the subject of many papers, beginning with the work of Hill-Moylan (1976). In particular, Theorem -10.8.1 is from Hill-Moylan (1977). Section 10.9 summarizes a number of classical results, from the theory of linear systems, concerning the relation (Theorem 10.9.1) between the property of being stable and having a finite L2 gain not exceeding i' < 'Y and the property of having a transfer function matrix whose Hoo norm does not exceed "f, a.D.d the relation (Theorem 10.9.2) between the property of being weakly strictly passive and the property of having a transfer functiori matrix which is positive real. These results are usually referred to as "bounded-real lemma" and "positive-real lemma", or Kalman-Yakubovic-Popov lemma. The notion of "weak strict passivity" used here was suggested by Lozano-Joshi (1990), and helps to obtain the characterization given in Theorem 10.9.2. Also Lemma 10.9.3 is a classical result in linear system theory. Chapter 11. The recursive design method, outlined in the first section of this Chapter, and known as the method of backstepping, has become a very popular tool for the design of feedback laws in the presence of parameter uncertainties. The' potential of the idea on which this method reposes has been successfully demonstrated in the work of Kanellakopoulos et al. (1991), where a systematic method

282

Bibliographical Notes

for the design of adaptive control laws was introduced. The results presented here in section 11.2 are essentially based on the works of Freeman-Kokotovic (1993) (1996a). The results presented in section 11.3 are essentially based on the works of Marino-Tomei (1993a) - (1993b), though the derivation described here is slightly different. Abundant sources of additional material on the method of backstepping are the books of Krstic et al. (1995) and Marino-Tomei (1995). The results presented here in section 11.4, in which at each stage of the recursion the goal of the design is to impose - via feedback - the "gain function" of an input-to-state stable system, are based on the work of Jiang et al. (1994) (see also Coron et al. (1995)). The derivation of normal forms for multi-input systems, for the general case in which the system in question does not possess a vector relative degree, is based on our earlier zero-dynamics algorithm presented in Isidori (1995), and is taken from the recent work of Schwartz et al. (1999). Chapter 12. The material described in this Chapter is taken in large part from the fundamental work of Teel-Praly (1995). In particular, from this work is taken Theorem 12.1.1, which is an extension of Theorem 9.3.1, Corollary 12.1.2, the extended version of this Corollary presented at the end of section 13.2, and their respective proofs. One of the key features oLTheorem 12.1.1 is the fact that, using high-gain feedback, it is possible to steer the trajectories to an arbitrarily small neighborhood of the origin, despite of the presence of "matched" uncertainties. The proof of this fact uses an argument which extends, to the setup of semiglobal and practical stabilizability, the argument suggested by Bacciotti (1992) for the proof of Theorem 9.3.1. The notion of input-to-state stability with restrictions and the corresponding small-gain Theorem 12.2.1 have been introduced by Tee! (1996a). Section 13.3 provides a proof of Theorem 9.6.2, which is different from the proof originally given by Teel-Praly (1994). The arguments introduced in this alternative proof are used also in the proofs of some other results presented later in the Chapter, such as Theorem 12.4.1 and Theorem 12.6.2. Lemma 12.3.1 captures the fundamental idea of Khalil-Esfandiari (1993) that, if the first n- 1 derivatives of the output of the system are estimated by an "observer" such as the one defined by equations of the form (12.30), past an initial fixed time interval (on which the trajectories of the system are guaranteed to exist because the effect of possibly large estimation errors is neutralized by the "saturation" included in the feedback law), the estimation error can be rendered arbitrarily small. The results presented in sections 13.5 and 13.6 are taken from Isidori (1999). In particular, this work seems to be - to the best of our knowledge - the first one to have called the attention on the importance, in problems of nonlinear stabilization via output feedback, of being able of stabilizing the so-called auxiliary system (12.70), a property that turns out to be necessary in the case of linear systems. The first example in section 13.7 is taken from Mazenc et al. (1994). The second example, which is taken from Isidori (1999), seems to suggest that the (semiglobal) stabilization approach, based on the results presented here in section 13.6, compares favorably with other methods presented in the literature, and summarized in Krstic et al. (1998). Chapter 13. The method, described in section 13.1, for the design of a feedback law to the purpose of achieving asymptotic stability and a prescribed upper bound on the £2 gain, between disturbance and output in a system of the form (13.16), is based on the works of Marino et al. (1989), (1994). The role played, in the case of linear systems, by the properties of the "unstable component" of the zero dynamics on establishing the minimal level of disturbance attenuation, was indeed widely acknowledged in the literature, but the explicit characterization given here has been - to the best of our knowledge - only pointed out recently by Isidori et al. (1999). The same work introduces the analysis presented here in sections 13.3

Bibliographical Notes

283

and 13.5, which in some way extends the results given in section 13.2 to the case of nonlinear systems. The results on almost disturbance decoupling presented in section 13.4, an extension of the results of Marino et al. (1994), are based on the work of Isidori (1996b). Section 13.6 describes classical, necessary and sufficient, conditions for the solution - in the case of a linear system - of the problem of achieving internal stability and a prescribed upper bound on the £2 gain, or -what is the same- a prescribed bound on the Hoc norm of the transfer function matrix. Theorem 13.6.3, which is taken from the fundamental work of Scherer (1992), shows how the existence of the solution of the characterizing Riccati inequality (13.81) can be tested in a simple way. The result presented in section 13.7 are due to Isidori-Lin (1998). Chapter 14. Propositions 14.1.1 and 14.1.2, which show that a system whose uncontrolled dynamics possesses a Lyapunov function whose derivative is only semidefinite can be stabilized - if zero-state detectable - by means of a feedback law given by the derivative of this function along the vector field which weights the input, are corollaries of Corollary 10.8.4 and due to Hill-Moylan (1977). The extension given by Theorem 14.1.3 is due toW. Lin (1996). Proposition 14.1.5, a special case of a more general property which holds for the indicated class of systems, is due to Liu et al. (1996). The proof given here of this Proposition is the one suggested in that paper. The possibility of recursively stabilizing systems in upper-triangular, or "feedforward" form, whose antecendents can be found in the work of Tee! (1992), was independently investigated by Tee! (1996a), by Mazenc-Praly (1996) and by Jankovic et al. (1996). The exposition here in section 14.2 is based on the work of Jankovic et al. (1996). Additional material on this specific approach, based on the construction of a cross-term for a Lyapunov function via a line-integral along the trajectories of the uncontrolled system, can be found in the book of Sepulchre et al. (1997). The results presented here in section 14.3, which lead to the synthesis of globally stabilizing laws consisting in the composition of linear laws and saturation functions, are due to Tee! (1996a). The exposition of these results follows very closely that in the original source. The first application presented in section 14.4 is actually a corollary from Tee! (1996a), while the extension to systems (14.84), not in upper-triangular form, is based - with slight modifications - on the work of Lin-Li (1999).

References

Astolfi A. (1997): On the relationship between state feedback and full information regulators in nonlinear singular Hoo control, IEEE TI-ans.Aut. Contr. AC-42, pp. 984-988. Astolfi A. (1998): New results on the global stabilization of minimum phase nonlinear systems, Automatica 34, pp. 783-788. Bacciotti A. (1992): Linear feedback: the local and potentialy global stabilization of cascade systems, Proc. !FAG Nonlinear Control Systems Design, pp. 21-25. Bacciotti A. (1992): Local Stabilizability of Nonlinear Control Systems, World Scientific (Singapore). Ball J., J.W. Helton, M.L. Walker (1993): Hoo control for nonlinear systems via output feedback, IEEE TI-ans.Aut. Contr. AC-38, pp. 546-559. Barbu C., R. Sepulchre, W. Lin, P.V. Kokotovic (1997): Global asymptotic stabilization of the ball-and-beam system, Proc. 36th IEEE Conf. on Decis. and Contr, pp. 2351-2354. Basar T., P. Bernhard (1990): Hoo-optimal control and related Minimax design problems, Birkhauser (Boston). Battilotti S. (1996): Global output regulation and disturbance attenuation with global stability via measurement feedback for a class of nonlinear systems, IEEE TI-ans.Aut. Contr. AC-41, pp. 315-327. Byrnes C.l., F. Delli Priscoli, A. Isidori (1997): Output Regulation of Uncertain Nonlinear Systems, Birkhauser (Boston). Byrnes C.l., A. Isidori (1991): Asymptotic stabilization of minimum-phase nonlinear systems, IEEE TI-ans. Aut. Contr. AC-36, pp. 1122-1137. Byrnes C.I., A. Isidori, J.C. Willems (1991): Passivity, feedback equivalence and the global stabilization of minimum-phase nonlinear systems, IEEE 7rans. Aut. Contr. AC-36, pp. 1228-1240. Coron J.M., L. Praly, A.R. Teet (1995): Feedback stabilization of nonlinear systems: sufficient conditions and Lyapunov and input~utput techniques, TI-ends in Control, A. Isidori ed., Springer Verlag (London), pp. 293-348. Esfandiari F., H.K. Khalil (1992): Output feedback stabilization of fully linearizable systems, Int. J. Contr. 56, pp. 1007-1037. Freeman R.A., P.V. Kokotovic (1993): Design of 'softer' robust nonlinear control laws, Automatica 29, pp. 1425-1437. Freeman R.A., P.V. Kokotovic (1996a): Tracking controllers for systems linear i1;1 the unmeasured states, Automatica 32, pp. 733-746. Freeman R.A., P.V. Kokotovic (1996b): Inverse optimality in robust stabilization, SIAM J. Contr. Optimiz. 34, pp. 1365-1391.

286

References

Freeman R.A., P.V. Kokotovic (1996c): Robust Nonlinear Control,Design, Birkhauser (Boston). Freeman R.A., L. Praly (1998): Integrator backstepping for bounded controls and control rates, IEEE 11-ans. Aut. Contr. AC-43, pp. 258-262. Hahn W. (1967): Stability of motion, Springer Verlag (Berlin). Helton J.W., M.R. James (1998): Extending Hoo Control to Nonlinear Systems, SIAM Frontiers in Applied Mathematics. Hill D., P.J. Moylan (1976): The stability of nonlinear dissipative systems, IEEE 1Tans. Aut. Contr. AC-21, pp. 708-711. Hill D., P.J. Moylan (1977): Stability results for nonlinear feedback systems, Automatica 13, pp. 377-382. Hill D., P.J. Moylan (1980): Connections between finite gain and asymptotic stability, IEEE 1Tans. Aut. Contr. AC-25, pp. 931-935. Imura J., T. Sugie, T. Yoshikawa (1994): Global robust stabilization of nonlinear cascaded systems, IEEE 1Tans. Aut. Contr. AC-39, pp. 1084-1089. Isidori A. (1995): Nonlinear Control Systems_, 3rd ed., Springer Verlag (London). Isidori A. (1996a): A note on almost disturbance decoupling for nonlinear minimum phase systems, Syst. Contr. Lett. 27, pp. 191-194. Isidori A. (1996b): Global almost disturbance decoupling with stability for nonminimum phase single-input single-output nonlinear systems, Syst. Contr. Lett. 28, pp. 115-122. Isidori A. (1999): A tool for semiglobal stabilization of uncertain non-minimumphase nonlinear systems via output feedback, IEEE 1Tans. Aut. Contr., to appear. Isidori A., A. Astolfi (1992): Disturbance attenuation and H 00 -control via measurement feedback in nonlinear systems, IEEE 11-ans. Aut. Contr. 37, pp. 1283-1293. Isidori A., W. Lin (1998): Global £2-gain state feedback design for a class of nonlinear systems, Syst. Contr. Lett. 29, pp. 295-302. Isidori A., W. Kang (1995): Hoo Control via Measurement Feedback for General Nonlinear Systems, IEEE 1Tans. on Automatic Contr. AC-40, pp. 466-472. Isidori A., B. Schwartz, T.J. Tarn (1999): Semi-global £2 performance bounds for disturbance attenuation in nonlinear systems, IEEE 1Tans. on Automatic Contr., to appear. Jankovic M., R. Sepulchre, P. Kokotovic (1996): Constructive Lyapunov stabilization of nonlinear cascade systems, IEEE 1Tans. Aut. Contr. AC-41, pp. 1723-1735. Jakubczyk B., W. Respondek (1998): Geometry of Feedback and Optimal Control, Marcel Dekker (New York). Jiang Z.P., I. Mareels (1997): A small gain control method for nonlinear cascaded systems with dynamic uncertainties, IEEE 1hms. Aut. Contr. AC-42, pp. 292-308. Jiang Z.P., I.M.Y. Mareels, Y. Wang (1996): A Lyapunov formulation of the nonlinear-small gain theorem for interconnected ISS systems, Automatica 32, pp. 1211-1215. Jiang Z.P., A.R. Teel, L. Praly (1994): Small-gain theorem for ISS systems and applications, Math. Contr. Sign. Syst. 7, pp. 95-120.

References

287

Kanellakopoulos I., P.V. Kokotovic, A.S. Morse (1991): Systematic design of adaptive controllers for feedback linearizable systems, IEEE 1hms. Aut. Contr. AC-36, pp. 1241-1253. Kanellakopoulos I., P.V. Kokotovic, A.S. Morse (1992): A toolkit for nonlinear feedback design, Syst. Contr. Lett. 18, pp. 83-92. Khalil H. (1994): Robust servomechanism output feedback controllers for feedback linearizable systems, Automatica 30, pp. 1587-1599. Khalil H.K. (1996): Nonlinear Systems, 2nd ed., McMillan (Nnew York). Khalil H.K., F. Esfandiari (1993): Semiglobal stabilization of a class of nonlinear systems using output feedback, IEEE Trans. Aut. Contr., AC-38, pp. 14121415. Knobloch H.W., A. Isidori, D. Flockerzi (1993): Topics in Control Theory, Birkhauser (Basel). Kokotovic P.V., H.K. Khalil, J. O'Reilly (1986): Singular perturbation methods in control: analysis and design, Academic Press. Krener A.J. (1997): Necessary and sufficient conditions for nonlinear worst case (Hoo) control and estimation, J. Math. Syst. Estim. Contr. 7, pp. 81-106. Krstic M., D. Fontaine, P.V. Kokotovic, J.D. Paduano (1998): Useful nonlinearities and global stabilization of bifurcations in a model of jet engine surge and stall, IEEE Trans. Aut. Contr. AC-43, pp. 1739-1745. Krstic M., I. Kanellakopoulos, P. Kokotovic (1995): Nonlinear Adaptive Control Design, J.Wiley (New York). Krstic M., H. Deng (1998): Stabilization of Nonlinear Uncertain Systems, SpringerVerlag (London). Kurzweil J. (1956): On the inverse of Lyapunov's second theorem on stability of motion, Amer. Math. Soc. Trans/., Ser 2, 24, pp. 19-77. Lin W. (1995a): Input saturation and global stabilization of nonlinear systems via state and output feedback, IEEE Trans. Aut. Contr. AC-40, pp. 776-782. Lin W. (1995b): Feedback stabilization of general nonlinear control systems: a passive systems approach, Syst. Contr. Lett. 25, pp. 41-52. Lin W. (1996): Global asymptotic stabilization of general nonlinear systems with stable free dynamics via passivity and bounded feedback, Automatica 32, pp. 915-924. . Lin W., X. Li (1999): Synthesis of nonlinear systems with marginally unstable free dynamics using state-dependent saturation, Syst. Contr. Lett. , to appear. Lin Z. (1999): Low Gain Feedback, Springer-Verlag (London). Lin Z., A. Saberi (1995): Robust semi-global stabilization of minimum-phase input-output linearizable systelllB via partial state and output feedback, IEEE Trans. Aut. Contr. AC-40, pp. 1029-1041. Liu W., Y. Chitour, E.D. Sontag (1996): On finite-gain stabilizability of linear systems subject to saturations, SIAM J. Contr. Optimiz., 34, pp. 1190-1219. Lozano R., B. Brogliato, I.D. Landau (1992): Passivity and global stabilization of cascaded nonlinear systems, IEEE Trans. Aut. Contr. AC-37, pp. 1386-1388. Lozano R., S.M. Joshi (1990): Strictly positive real functions revisited, IEEE Trans. Aut. Contr. AC-35, pp. 1243-1245. Marino R., W. Respondek, A.J. Van der Schaft (1989): Almost disturbance de~ coupling for single-input single-output nonlinear systems, IEEE 1rans. Aut. Contr. 34, pp. 1013-1017.

288

References

Marino R., W. Respondek, A.J. Vander Schaft, P. Tomei (1994): Nonlinear Hco almost disturbance decoupling, Syst. Contr. Lett. 23, pp. 159-168. Marino R., P. Tomei (1993a): Global adaptive output feedback control of nonlinear systems, part I: linear parametrization, IEEE 7rans. Aut. Contr. AC-38, pp. 17-32. Marino R., P. Tomei (1993b): Global adaptive output feedback control of nonlinear systems, part II: nonlinear parametrization, IEEE 7rans. Aut. Contr. AC-38, pp. 33-48. Marino R., P. Tomei (1995): Nonlinear Control Design: geometric, adaptive, robust, Prentice-Hall (New York). Mazenc F., L. Praly (1996): Adding integrations, saturated controls and stabilization for feedforward systems, IEEE TI-ans. Aut. Contr. AC-41, pp. 1559-1577. Mazenc F., L. Praly, W.P. Dayawansa (1994): Global stabilization by output feedback: examples and counterexamples, Syst. Contr. Lett. 23, pp. 119-125. Ortega R. (1991): Passivity properties for stabilization of cascaded nonlinear systems, Automatica 21, pp. 423-424. Praly L., Z.P. Jiang (1993): Stabilization by output feedback for systems with ISS inverse dynamics, Syst. Contr. Lett. 21, pp. 19-33. Rantzer A. (1996): On the Kalman-Yakubovic-Popov lemma, Syst. Contr. Lett. 28, pp. 7-10. Saberi A., P.V. Kokotovic, H. Sussmann (1990): Global stabilization of partially linear composite systems, SIAM J. Contr. Optimiz., 28, pp. 1491-1503. Scherer C. (1992): Hco-control by state feedback for plants with zeros on the imaginary axis, SIAM J. Contr. and Optimiz. 30, pp. 123-142. Schwartz B., A. Isidori, T.J. Tarn (1999): Global normal forms for MIMO nonlinear systems with application to stabilization and disturbance attenuation, Math. Contr. Sig. Syst. 12. Sepulchre R. (1999): Slow peaking and low-gain designs for global stabilization of nonlinear systems, IEEE 7rans. Aut. Contr., to appear. Sepulchre R., M. Jankovic, P.V. Kokotovic (1997): Constructive Nonlinear Control, Springer Verlag (London). Seibert· P., Suarez R. (1990): Global stabilization of nonlinear cascade systems, Syst. Contr. Lett. 14, pp. 347-352. Sontag E.D. (1989): Smooth stabilization implies coprime factorization, IEEE 1!-ans. Aut. Contr. AC-34, pp. 435-443. Sontag E.D. (1990): Further facts about input to state stabilization, IEEE 11-ans. Aut. Contr. AC-35, pp. 473-476. Sontag E.D. (1995): On the input-to-state stability property, European J. Contr. 1, pp. 24-36. Sontag E.D. (1998): Comments of integral variants of ISS, Syst. Contr. Lett. 34, pp. 93-100. Sontag E., A.R. Teel (1995): Changing supply functions in input/state stable systems, IEEE 7rans.Aut. Contr. AC-40, pp. 1476-1478. Sontag E.D., Y. Wang (1995): On characterizations of the input-to-state stability property, Syst. Contr. Lett. 24, pp. 351-359. Sontag E.D., Y. Wang (1996): New characterizations of input-to-state stability, IEEE 7rans. Aut. Contr. AC-41, pp. 1283-1294.

References

289

Teel A. (1992): Semiglobal stabilization of minimum phase nonlinear systems in special normal forms, Syst. Contr. Lett. 19, pp. 187-192. Teel A. (1996a): A nonlinear small gain theorem for the analysis of control systems with saturations, IEEE Trans. Aut. Contr. AC-41, pp. 1256-1270. Tee) A. (1996b): On graphs, conic relations and input-output stability of nonlinear feedback systems, IEEE 1rans. Aut. Contr. AC-41, pp. 702-709. Tee! A.R., L. Praly (1994): Global stabilizability and observability imply semiglobal stabilizability by output feedback, Syst. Cont. Lett. 22, pp. 313-325. Teel A.R., L. Praly (1995): Tools for semi-global stabilization by partial state and output feedback, SIAM J. Contr. Optimiz. 33, pp. 1443-1488. Tsinias J. (1989): Sufficient Lyapunov-like conditions for stabilizability, Math. Contr. Sig. Syst. 2, pp. 343-357. Tsinias J. (1993): Sontag's "input to state stability condition" and the global stabilization using state detection, Syst. Contr. Lett. 20, pp. 219-226. Tsinias J. (1993): Versions of Sontag's "input to state stability condition" and the global stabilization problem, SIAM J. Contr. Optimiz. 31, pp. 928-941. Vander Schaft A.J. (1991): A state-space approach to nonlinear Hoo control, Syst. Contr. Lett. 16, pp. 1-8. Van der Schaft A.J. (1992): L2-gain analysis of nonlinear systems and nonlinear Hoo control, IEEE 1rans. Aut. Contr., 37, pp. 770-784. Vander Schaft A.J. (1996): L2 Gain and Passivity Techniques in Nonlinear Control, Springer-Verlag (London). Vidyasagar M. (1980): Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability, IEEE 1rans. Aut. Contr. AC-25, pp. 773-779. Willems, J.C. (1972): Dissipative dynamical systems, Archive for Rational Mechanics and Analysis 45, pp. 321-393. Youla, D.C. (1961): On the factorization of rational matrices, IRE Trans. Information Th. IT-7, pp. 172-189.

Index

Affine function - of the control input 191, 201, 238 - of the disturbance input 191 - of the z variables 90 Almost Disturbance Decoupling, problem of 202 - for linear systems 202 - for nonlinear systems 202 Amplitude constraint 228, 230 Asymptotic - bound 255 - gain 255 -norm 255 - stability 3 Augmented system 91 Auxiliary - inequality 210 - subsystem linear 158 nonlinear 165, 170 stabilization of 170

B,,iJ.

2

Backstepping 75 - for disturbance attenuation 185 - in multi-input systems 119, 120, 123 Ball in Rn 2 Bounded control 227 Cancellation 77 Captured by 126 Chains of integrators 119 Compact - set of initial conditions 125 - set of parameters 75 Comparison - function 1 -lemma 3 Compressor 176 Constraint, on control input 230 Contraction, simple 37

Control input 184 Control Lyapunov function Cross-term 240, 245, 253 Cube in Rn 126

204

Detectability - hypothesis, in a linear system -162 - zero-state 43, 228, 231 Detectable pair 162 Diffeomorphism 127 - globally defined 109, 113, 116 - parameter-dependent 151, 164 Diophantine equation 157 Dissipative system 42 - strictly 42, 183 Disturbance - attenuation -- arbitrarily small value of 202 -- optimal value of 192, 198,207,211 -- problem of 184 -- with cost on control input, for linear systems 212 -- with cost on control input, for nonlinear systems 216 - input 184 -- bounded set of 207 Domain of attraction 127 Eigenvalue - multiplicity of 267 - with positive real part Escape time, finite 173

268

Feed-through 170 Feedback form 99 Feedback law, linear 152, 192, 212 Feedforward form 236 Finite Lq gain 44 Frequency response 61 Function - class /C

292

Index

- class Koo 1 - class KC 1 - comparison 1 - control Lyapunov 204 - gain 18 -- asymptotic 255 - ISS-Lyapunov 19 - positive definite 5 - positive semi-definite 227 - proper 5 - saturation 232, 258 - storage 42 Gain - asymptotic 255 - factor 228 - function 18 -- characterization of 21 -- characterization of, alternative -- design of 100, 103 -- local 138 -- prescribed 103 Hamiltonian matrix High-frequency gain High-gain 163 Hoo norm 63

64,214 166

Inequality - dissipation 42 - Hamilton-Jacobi 212,218 -- local solution of 217 - Riccati 64, 213, 216, 218 -- test for solvability 215 - Sylvester 195 Input - average power of 46 - bounded 18 - finite energy of 45 - periodic 45 Integrators 119 Interconnection - cascade 13, 34 - feedback 36 -- negative 54 lnvariance 4 Iterative design 84

L':,

18 Lq-gain 44 L2-gain 45 Linear growth 239, 242, 269 Lower-triangular system 149, 188 Lyapunov

- criterion 2 -- converse of 6 - function 2 -- for input-to-state stability

19

Memoryless system 42, 142 Minimum-phase system 163 Non-affine 228, 252 Non-stabilizable pair 214 Normal form - of a linear system 158, 192 - of multi-input multi-output systems 109, 113, 122 -- with special structure 116

22

0 bservability - uniform 142, 172 Output - derivatives of 152 - feedback 142,173 -- dynamic 90, 142, 149, 163 -- memory less 90 Parameter - set P 75 -unknown 75 Passive 228 Passive system 48 - output strictly 49 - strictly 48 - weakly strictly 49 Perturbations 11 Perturbed system 11,268 Pole assignment 157 Positive real 68

Q'R

126

Recursive design - for disturbance attenuation 185 - for robust stability 75, 103 - of dynamic output feedback 163 Reduction - of dimensionality 207 Regularity hypotheses 109 Relative degree - larger than one 90 -one 79,166 - uniform 118 - vector 109 r.m.s. 46 Robust stabilization 75 Root mean square 46

Index Saturation 144, 152, 168 - function 232, 258 -- composition of 268 - - properties of 233 -- .1Rm-valued 233 - level 145,155,170 Sector bounded 50 Separation principle - for linear systems 157 - for nonlinear systems 142 Small - control amplitude 228 - gain factor 228 Small-gain - argument 82 - condition 40, 256 - theorem for input-to-state stable systems 37 for input-to-state stable systems, with restrictions 140 -- for systems satisfying asymptotic bounds 256 -- for finite L2 gain systems 58 -- robust stability via 78, 99, 184 Spectral radius 198 Stability - asymptotic -- global 3 -- local 3 - exponential -- local 7 - input-to-state 18 · -- equivalent characterization of 30 -- with restrictions 138, 234 - total 11 - under small perturbations 11 Stabilizability - hypothesis, in a linear system 162 - practical 126 - semiglobal 126 Stabilizable pair 162,213,232 Stabilizable via state feedback 142 Stabilization - robust 75 - via linear gains and saturation functions 258 Stable - globally asymptotically 7 - locally exponentially 7, 132, 171, 239 Stable manifold 241 Supply rate 42

Time constant, effect of a small Triangular - dependence 123 - lower 98, 149 - upper 236

llu(·)lla 255 Uncertain - parameter 75 - subsystem 183 Unstable component - of the zero dynamics 199 Upper-triangular system 236 Vector fields, commuting

118

Zero dynamics 95, 119, 163 - algorithm 109 -- regularity hypotheses of - manifold 166 ilc 4

109

293

172