Communications and Control Engineering

Springer-Verlag London Ltd.

Published titles include:

Parametrizations in Control, Estimation and Filtering Problems: Accuracy Aspects M. Gevers and G. Li

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 of Robot Control C. Canudas de Wit, B. Siciliano and G. Bastin (Eds)

Fundamental Limitations in Filtering and Control Maria M. Seron, Julio Braslavsky and Graham C. Goodwin

Constructive Nonlinear Control R. Sepulchre, M. Jankovic and P.V. Kokotovic

A Theory of Learning and Generalization M. Vidyasagar

Adaptive Control I.D. Landau, R. Lozano and M.M'Saad

Stabilization of Nonlinear Uncertain Systems Miroslav Krstic and Hua Deng

Passivity-based Control of Euler-Lagrange Systems Romeo Ortega, Antonio Loria, Per Johan Nicklasson and Hebertt Sira-Ramirez

Stability and Stabilization of Infinite Dimensional Systems with Applications Zheng-Hua Luo, Bao-Zhu Guo and Orner Morgul

Nonsmooth Mechanics Bernard Brogliato

Nonlinear Control Systems II Alberto Isidori

Arjan van der Schaft

L2 -Gain and Passivity Techniques in Nonlinear Control With 27 Figures

, Springer

Arjan van der Schaft. Dr University ofTwente. Faculty of Mathematical Sciences. PO Box217. 7500 AE Enschede. The Netherlands

Series Editors E.D. Son tag • M. Thoma

ISBN 978-1-4471-1154-2

British Library Cataloguing in Publication Data Schaft, A.J. van der

L2-gain and passivity techniques in nonlinear control. -2nd ed - (Communications and control engineering) l.Nonlinear control theory 2.Automatic control I.Title 629.8'36 ISBN 978-1-4471-1154-2

Library of Congress Cataloging-in-Publication Data Schaft, A.J. van der.

L2-gain and passivity techniques in nonlinear control I A.J. van der Schaft. p. cm -- (Communications and control engineering series)

Includes bibliographical references. ISBN 978-1-4471-1154-2 ISBN 978-1-4471-0507-7 (eBook) DOI 10.1007/978-1-4471-0507-7 1. Feedback control systems. 2. Nonlinear control theory. I. Title. 11. Series.

TJ216.S34 1999 629.8'321--dc21 99-045693

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Cl Springer-Yedag London 2000 Originally published by Springer-Verlag London Berlin Heidelberg in 2000 Sofu:over reprint of the hardcover 2nd edition 2000

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Preface to the Second Edition

With respect to the first edition as Volume 218 in the Lecture Notes in Con­trol and Information Sciences series the basic idea of the second edition has remained the same: to provide a compact presentation of some basic ideas in the classical theory of input-output and closed-loop stability, together with a choice of contributions to the recent theory of nonlinear robust and 1foo control and passivity-based control. Nevertheless, some parts of the book have been thoroughly revised and/or expanded, in order to have a more balanced presen­tation of the theory and to include some of the new developments which have been taken place since the appearance of the first edition. I soon realized, how­ever, that it is not possible to give a broad exposition of the existing literature in this area without affecting the spirit of the book, which is precisely aimed at a compact presentation. So as a result the second edition still reflects very much my personal taste and research interests. I trust that others will write books emphasizing different aspects.

Major changes with respect to the first edition are the following:

• A new section has been added in Chapter 2 relating L2-gain and passivity via scattering, emphasizing a coordinate-free, geometric, treatment.

• The section on stability in Chapter 3 has been thoroughly expanded, also incorporating some recent results presented in [182J.

• Chapter 4 has been largely rewritten and expanded, incorporating new developments. The first part of Section 4.1 is based on Section 2.3 in the first edition.

• A new Chapter 5 has been added on the topic of feedback equivalence to a passive system, based on the paper [29].

Furthermore, minor changes and expansions have been made throughout the book. The references have been updated and expanded, but nevertheless they are still far from being complete.

VI PREFACE TO THE SECOND EDITION

I believe the theory of passivity and Lz-gain techniques in nonlinear con­trol is a rich mixture of classical results and recent state space developments, which is of interest for applications, actually and potentially. Definitely, much research remains to be done and some aspects have been hardly considered at all. The present book aims at further stimulating this research by bringing together in a single monograph well-established contributions to the area, plus a choice of the recent developments. As such this book is not only intended for researchers in the area, but can be also used in an advanced course for students specializing in systems and control. The required baCkground is some basic knowledge of control theory and stability theory, as well as some understanding of linear robust control theory.

The contents of the book are organized as follows: Chapter 1 summarizes the classical notions of input-output and closed-loop stability. The presentation is very much based on (and is sometimes almost literally taken from) Vidyasagar's excellent "Nonlinear Systems Analysis" [203]. Chapter 2 also largely follows the treatment of small-gain and passivity the­orems from Vidyasagar, [203], with some additions from e.g. Desoer & Vidyasagar, [43]. Section 2.3 gives a geometric treatment of scattering in this context. Chapter 3 gives a rather detailed treatment of the theory of dissipative systems based on the fundamental paper (Willems, [208]), emphasizing the applica­tions towards finite Lz-gain, passivity, and stability. Chapter 3 can be regarded as the state space synthesis of Chapters 1 and 2. Chapter 4 puts the theory of passive state space systems into the perspective of Euler-Lagrange equations and a generalized form of Hamiltonian dynam­ics, called port-controlled Hamiltonian systems with dissipation. Properties of these systems apart from passivity are investigated, and explored for sta­bilization purposes by means of energy-shaping. Passivity-based control is treated from this point of view. The notion of control by interconnection is emphasized. Chapter 5 deals with the problem of rendering a nonlinear system passive by the use of state feedback. These tools are further used for the stabilization of cascaded systems. Chapter 6 deals with the nonlinear analogs of the linear notions of left- and right factorization of transfer matrices, and with nonlinear all-pass (inner­outer) factorization. These are used for constructing nonlinear uncertainty

PREFACE TO THE SECOND EDITION VII

models, for obtaining a nonlinear Youla-Kucera parametrization of stabilizing controllers, and for deriving the minimum-phase factor of nonlinear systems. Chapter 7 treats the theory of nonlinear state feedback 1foo control, and derives necessary conditions for the output feedback Jfoo control problem. Finally, Chapter 8 is devoted to checking (local) solvability of Hamilton-Jacobi inequalities and to the structure of their solution set. Emphasis is on the re­lations between nonlinear dissipation and Hamilton-Jacobi inequalities on the one hand and linearized dissipation and Riccati inequalities on the other hand, with applications towards nonlinear optimal and Jfoo control.

At the end of each chapter some notes have been added referring to the main sources which have been used in writing the chapter, and containing some ad­ditional remarks concerning references and related developments which have not been treated in the main text.

The relation between the chapters can be explained by the following diagram:

Acknowledgements Many people have contributed in some way or another to the genesis of this book. Chapter 3 is based on the work of my former thesis advisor Jan C. Willems, who also otherwise has shaped my scientific attitude and taste in a deep way. Chapter 4 owes a lot to an inspiring and fruitful co­operation with Bernhard Maschke, as well as with Romeo Ortega and Morten Dalsmo, while Chapter 6 is based on joint research with Andrew Paice, Joe Ball and Jacquelien Scherpen. Also I acknowledge useful and stimulating dis­cussions with many other people, including Peter Crouch, Bill Helton, David

viii PREFACE TO THE SECOND EDITION

Hill, Alberto Isidori, Gjerrit Meinsma, Carsten Scherer, Hans Schumacher, Rodolphe Sepulchre, Stefano Stramigioli, and my colleague Henk Nijmeijer. I thank the graduate students of the spring trimester of 1994 for being an attentive audience. Gjerrit Meinsma is furthennore gratefully acknowledged for his patient way of handling all sorts of Jb.TEX problems. Finally, I thank the Faculty of Mathematical Sciences for providing me with secretarial support for the preparation of this book; in particular I sincerely thank Marja Langkamp, and for the first edition Marjo Mulder, for their great efforts in bringing the manuscript to its present fonn.

Enschede, August 1999,

Arjan van der Schaft

From the Preface to the First Edition

The first version of these lecture notes were prepared for part of a graduate course taught for the Dutch Graduate School of Systems and Control in the spring trimester of 1994.

My main goal when writing the first version of these lecture notes was to pro­vide some kind of synthesis between the classical theory of input-output and closed-loop stability on the one hand, and recent work on nonlinear 1ioo control and passivity-based control on the other hand. Apart from my own research interests in nonlinear 1ioo control and in passive and Hamiltonian systems, this motivation was further triggered by some discussions with David Hill (Syd­ney, Australia), Romeo Ortega, Rogelio Lozano (both Compiegne, France) and Olav Egeland (Trondheim, Norway), at a meeting of the GR Automatique du CNRS in Compiegne, November 1993, devoted to passivity-based and 1ioo control. During these discussions also the idea came up to organize a pre-CDC tutorial workshop on passivity-based and nonlinear 1ioo control, which indeed took place - with remarkable success - at the 1994 CDC under the title "Non­linear Controller Design using Passivity and Small-Gain techniques". Some improvements of the contents and presentation of Chapter 2 of the final ver­sion of these lecture notes are directly due to the lecture presented by David Hill at this workshop [65]. I was also fortunate to receive from Rogelio Lozano a set of handwritten lecture notes [106] concerning positive real transfer func­tions and passivity, which helped me to put the material of Chapters 2 and 3 into a proper perspective.

As said before, the main aim of the lecture notes is to provide a synthesis between classical input-output and closed-loop stability theory, in particular the small-gain and passivity theorems, and the recent developments in passivity­based and nonlinear 1ioo control. From my point of view the trait d'union between these two areas is the theory of dissipative systems, as laid down by Willems in the fundamental paper [208], and further developed by Hill and

x FROM THE PREFACE TO THE FIRST EDITION

Moylan in a series of papers [66, 67, 68, 69]. Strangely enough, this theory has never found its place in any textbook or research monograph; in fact I have the impression that the paper [208J is still relatively unknown. Therefore I have devoted Chapter 3 to a detailed treatment of the theory of dissipative systems, although primarily geared towards L2-gain and passivity supply rates.

One of the nice aspects of classical input-output and closed-loop stability the­ory, as well as of dissipative systems theory, is their firm rooting in electrical network analysis, with the physical notions of passivity, internal energy and supplied power. Furthermore, using the scattering transformation a direct link is established with the finite gain property. Passivity-based control, on the other hand, used these same physical notions but draws its motivation pri­marily from the control of mechanical systems, especially robotics. Indeed, a usual approach is via the Euler-Lagrange equations of mechanical systems. In Chapter 4 of the lecture notes my aim is to show that the passivity properties of electrical networks, of mechanical systems described by Euler-Lagrange equations, and of constrained mechanical systems, all can be unified within a (generalized) Hamiltonian framework. This leaves open, and provokes, the question how other properties - apart from the passivity property - inherent in the generalized Hamiltonian structure, may be exploited in stability anal­ysis and design. .. . the rest of the lecture notes is mainly devoted to the use of L2-gain techniques in nonlinear control, with an emphasis on nonlinear 1100 control. The approach mimics to a large extent similar developments in robust linear control theory, while the specific choice of topics is biased by my own recent research interests and recent collaborations, in particular with Joe Ball and Andrew Paice. The application of these L2-gain techniques relies on solving (stationary) Hamilton-Jacobi inequalities, and sometimes on their nonlinear factorization. This constitutes a main bottleneck in the application of the theory, which is similar to the problems classically encoun­tered in nonlinear optimal control theory (solving Hamilton-1acobi-Bellman equtions), and, more generally, in nonlinear state space stability analysis (the construction of Lyapunov functions). In some cases, e.g. for passive systems (!), the structure of the system naturally leads to (candidate) solutions of the Hamilton-Jacobi inequalities, but in general explicit globally-defined solutions of these Hamilton-Jacobi inequalities are hard to obtain. On the other hand, a first-order approach (linearization) may already yield useful information about the local solvability of Hamilton-Jacobi inequalities.

Enschede, January 1996,

Arjan van der Schaft

Contents

1 Input-Output Stability 1 1.1 Lq-spaces and their extensions; input-output maps . 1.2 Lq-stability and Lq-gain; closed-loop stability 4 1.3 Notes for Chapter 1 . ... .... . . . . . . 9

2 Small-gain and Passivity of Input-Output Maps 11 2.1 The small-gain theorem . . . ... .... 11 2.2 Passivity and the passivity theorem . . . 14 2.3 Relation between passivity and L2-gain 23 2.4 Notes for Chapter 2 . . .. . ... . . . 27

3 Dissipative Systems Theory 31 3.1 Dissipative systems . . ... . . 31 3.2 Stability of dissipative systems . 37 3.3 Stabilization of passive systems 44 3.4 The small-gain and passivity theorems revisited 45

3.4.1 Interconnection of passive systems . . 46 3.4.2 The small-gain theorem 50

3.5 Dissipativity and optimal control 56 3.6 Notes for Chapter 3 . . . . . . . 58

4 Hamiltonian Systems as Passive Systems 63 4.1 Euler-Lagrange equations and passivity 63

4.1.1 Tracking control of a robot manipulator 68 4.1.2 Passivity and Riemannian geometry 69

4.2 Hamiltonian control systems . .. . .. .. . . 72 4.2.1 Port-controlled Hamiltonian systems .. 72 4.2.2 Properties of port-controlled Hamiltonian systems 80 4.2.3 Port-controlled Hamiltonian systems with dissipation. 85 4.2.4 Stabilization by damping injection . . . . . . . . . . . 90

XII CONTENTS

4.3 Control of port-controlled Hamiltonian systems with dissipation 92 4.3.1 Control by interconnection . . . . . . . . . . . . . .. 92 4.3.2 Passivity-based control of port-controlled Hamiltonian

systems with dissipation . . . . . . . 102 4.4 Implicit port-controlled Hamiltonian systems .... 109

4.4.1 Power-conserving interconnections ..... 109 4.4.2 Implicit port-controlled Hamiltonian systems 112 4.4.3 Scattering representations of power-conserving inter-

connections 117 4.5 Notes for Chapter 4 121

5 Passivity by Feedback 125 5.1 Feedback equivalence to a passive system 125 5.2 Stabilization of cascaded systems. 130 5.3 Notes for Chapter 5 . . . . . . . 136

6 Factorizations of Nonlinear Systems 137 6.1 Stable kernel and image representations; L2-gain perturbation

models. . . . . . . . . . . . . . . . . . . . . . 137 6.1.1 Stable kernel and image representations . . . . . . . . 137 6.1.2 L2-gain perturbation models . . . . . . . . . . . . . . 144

6.2 Stable kernel representations and parametrization of stabiliz-ing controllers. . . . . 146

6.3 All-pass factorizations 154 6.4 Notes for Chapter 6 159

7 Nonlinear Jfoo Control 163 7.1 State feedback Jfoo control 164 7.2 Output feedback Jfoo control 175 7.3 Notes for Chapter 7 . . . 189

8 Hamilton-Jacobi Inequalities 193 8.1 Solvability of Hamilton-Jacobi inequalities ...... 193 8.2 An aside on optimal control. . . . . . . . . . . . . . . 205 8.3 Dissipativity of a nonlinear system and its linearization 211 8.4 Jfoo control of a nonlinear system and its linearization. 216 8.5 Notes for Chapter 8 . . . . . . . . . . . . . . . . . . . 226

Bibliography 229

Index 247