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 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).

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

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. 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. 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 ~ ~ 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)

. 1.

Q -

1

0

ka(r)

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
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.

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) 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.

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

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.

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). 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 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. 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.. 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

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. 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). 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. 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. 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

< < <
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.

,

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) = 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.

.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).

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. 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

<
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.

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).

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 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

<
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)

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.

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( ·). 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.

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 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 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 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. . =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 (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. 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)
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(,


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

•

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. 0 for any (possibly 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 .

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. 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 . 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

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.

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 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. 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). 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. ..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 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

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

<
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. 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. 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 ( = 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. 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}


.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. 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 .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. (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. 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. 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. 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


(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

0, and any k > 1 + lfi(O)I + 19(0)1, there exists c4 such that, for all c :c::; c- 4 , l 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.