Switching in Systems and Control.pdf

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Systems & Control: Foundations & Applications Series Editor Tamer Ba~ar, University of Illinois at Urbana-Champaign Editorial Board Karl Johan Astrom, Lund Institute of Technology, Lund, Sweden Han-Fu Chen, Academia Sinica, Beijing William Helton, University of California, San Diego Alberto Isidori, University of Rome (Italy) and Washington University, St. Louis Petar V. Kokotovic, University of California, Santa Barbara Alexander Kurzhanski, Russian Academy of Sciences, Moscow and University of California, Berkeley H. Vincent Poor, Princeton University Mete Soner, Ko~ University, Istanbul

Daniel Liberzon Switching in Systems and Control Springer Science+Business Media, LLC

Daniel Liberzon Coordinated Science Laboratory University of lllinois at Urbana-Champaign Urbana, IL 61801 U.S.A. library of Congress Cataloging-in-Publication Data Liberzon, Daniel, 1973 Switching in systems and control I Daniel Liberzon. p. cm. - (Systems and control) Includes bibliographical references and index. ISBN 978-1-4612-6574-0 DOI 10.10071978-1-4612-0017-8 ISBN 978-1-4612-0017-8 (eBook) 1. Switching theory. 2. Automatic control. I. Title. II. Series. TK7868.S9L53 2003 621.3815'372-dc21 2003050211 CIP AMS Subject Classifications: Primary: 93B12, 93D05, 93D09, 93D15, 93D20; Secondary: 34D20, 34030, 34H05 Printed on acid-free paper © 2003 Springer Science+Business Media New York Originally published by Birkhauser Boston in 2003 Softcover reprint of the hardcover 1st edition 2003 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this pUblication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. ISBN978-1-4612-6574-0 SPIN 10888361 Typeset by the author. 987654321

Contents Preface I Introduction 1 Basic Concepts l.1 Classes of hybrid and switched systems. 1.1.1 State-dependent switching . . . . l.l.2 Time-dependent switching . . . . l.l.3 Autonomous and controlled switching 1.2 Solutions of switched systems . . . . . 1.2.1 Ordinary differential equation!:> 1.2.2 Zeno behavior ... . 1.2.3 Sliding modes ... . 1.2.4 Hy!:>teresis switching II Stability of Switched Systems 2 Stability under Arbitrary Switching 2.1 Uniform stability and common Lyapunov functions 2.1.1 Uniform !:>tahility concepts ... 2.1.2 Common Lyapunov functions . 2.1.3 A converse Lyapunov theorem 2.1.4 Switched linear systems . . . . ix 1 3 3 5 6 8 9 9 10 12 14 17 21 21 21 22 24 26

VI Contents 2.1.5 A counterexample . . . . . . 2.2 Commutation relations and stability 2.2.1 Commuting systems . . . . . 2.2.2 Nilpotent and solvable Lie algebras. 2.2.3 More general Lie algebras . . . . . . 2.2.4 Discussion of Lie-algebraic stability criteria 2.3 Systems with special structure 2.3.1 Triangular systems ... . 2.3.2 Feedback systems . . . . . 2.3.3 Two-dimensional systems 3 Stability under Constrained Switching 3.1 Multiple Lyapunov functions . 3.2 Stability under slow switching. 3.2.1 Dwell time . . . . . . . 3.2.2 Average dwell time .. . 3.3 Stability under state-dependent switching 3.4 Stabilization by state-dependent switching . 3.4.1 Stable convex combinations .. 3.4.2 Unstable COnvex combinations III Switching Control 4 Systems Not Stabilizable by Continuous Feedback 4.1 Obstructions to continuous stabilization 4.1.1 State-space obstacles . 4.1.2 Brockett's condition . . . . . . . 4.2 Nonholonomic systems . . . . . . . . . . 4.2.1 The unicycle and the nonholonomic integrator 4.3 Stabilizing an inverted pendulum . . . . . . . . 5 Systems with Sensor or Actuator Constraints 5.1 The bang-bang principle of time-optimal control 5.2 Hybrid output feedback . . . . . . . . . . . 5.3 Hybrid control of systems with quantization 5.3.1 Quantizers . . . . . . . . . 5.3.2 Static state quantization .. 5.3.3 Dynamic state quantization 5.3.4 Input quantization . . . . . 5.3.5 Output quantization ... . 5.3.6 Active probing for information 6 Systems with Large Modeling Uncertainty Introductory remarks . . . . . . . . . . . . . 6.1 28 30 30 34 37 41 42 43 45 51 53 53 56 56 58 61 65 65 68 73 77 77 77 79 81 83 89 93 93 96 100 100 103 108 116 121 124 129 129

Contents vii 6.2 First pass: basic architecture . . . . . 6.3 An example: linear supervisory control 6.4 Second pass: design objectives. . . . . 6.5 Third pass: achieving the design objectives . . . 6.5.1 Multi-estimators 6.5.2 Candidate controllers . . . . 6.5.3 Switching logics. . . . . . . . 6.6 Linear supervisory control revisited. 6.6.1 Finite parametric uncertainty 6.6.2 Infinite parametric uncertainty 6.7 Nonlinear supervisory control . . . . . 6.8 An example: a nonholonomic system with uncertainty IV Supplementary Material A Stability A.1 Stability definitions . . . . . . . . . A.2 Function classes K, K oc , and K£ . A.3 Lyapunov's direct (second) method A.4 LaSalle's invariance principle ... A.5 Lyapunov's indirect (first) method A.6 Input-to-state stability . . . . . . . B Lie Algebras B.l Lie algebras and their representations B.2 Example: sl(2, IR) and gl(2, IR) . . . . . B.3 Nilpotent and solvable Lie algebras .. B.4 Semisimple and compact Lie algebras. B.5 Subalgebras isomorphic to 81(2, IR) B.6 Generators for gl(n, IR) . . . . . . . Notes and References Bibliography Notation Index 131 134 137 139 139 142 145 154 156 159 160 163 167 169 169 170 171 173 174 175 179 179 180 181 181 182 183 185 203 229 231

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