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Communications and Control Engineering

Published titles include: Stability and Stabilization of Infinite Dimensional Systems with Applications Zheng-Hua Luo, Bao-Zhu Guo and Omer Morgul Nonsmooth Mechanics (Second edition) Bernard Brogliato Nonlinear Control Systems II Alberto Isidori L2-Gain and Passivity Techniques in Nonlinear Control Arjan van der Schaft Control of Linear Systems with Regulation and Input Constraints Ali Saberi, Anton A. Stoorvogel and Peddapullaiah Sannuti Robust and H∞ Control Ben M. Chen Computer Controlled Systems Efim N. Rosenwasser and Bernhard P. Lampe Dissipative Systems Analysis and Control Rogelio Lozano, Bernard Brogliato, Olav Egeland and Bernhard Maschke Control of Complex and Uncertain Systems Stanislav V. Emelyanov and Sergey K. Korovin Robust Control Design Using H∞ Methods Ian R. Petersen, Valery A. Ugrinovski and Andrey V. Savkin Model Reduction for Control System Design Goro Obinata and Brian D.O. Anderson Control Theory for Linear Systems Harry L. Trentelman, Anton Stoorvogel and Malo Hautus Functional Adaptive Control Simon G. Fabri and Visakan Kadirkamanathan Positive 1D and 2D Systems Tadeusz Kaczorek Identification and Control Using Volterra Models F.J. Doyle III, R.K. Pearson and B.A. Ogunnaike Non-linear Control for Underactuated Mechanical Systems Isabelle Fantoni and Rogelio Lozano Robust Control (Second edition) Ju¨rgen Ackermann Flow Control by Feedback Ole Morten Aamo and Miroslav Krstic´ Learning and Generalization (Second edition) Mathukumalli Vidyasagar Constrained Control and Estimation Graham C. Goodwin, Marı´a M. Seron and Jose´ A. De Dona´ Randomized Algorithms for Analysis and Control of Uncertain Systems Roberto Tempo, Giuseppe Calafiore and Fabrizio Dabbene

Zhendong Sun and Shuzhi S. Ge Switched Linear Systems Control and Design With 52 Figures

Zhendong Sun, PhD Hamilton Institute, National University of Ireland, Maynooth, County Kildare, Ireland Shuzhi Sam Ge, PhD, DIC Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576 Series Editors E.D. Sontag • M. Thoma • A. Isidori • J.H. van Schuppen British Library Cataloguing in Publication Data Sun, Zhendong, 1968– Switched linear systems : control and design. - (Communication and control engineering) 1. Linear control systems I. Title 629.8′32 II. Ge, S. S. (Shuzhi S.) 2. Switching theory ISBN 1852338938 Library of Congress Cataloging-in-Publication Data p. cm. — (Communications and control engineering, ISSN 0178-5354) ISBN 1-85233-893-8 (alk. paper) 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 repro- duced, 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. Communications and Control Engineering Series ISSN 0178-5354 ISBN 1-85233-893-8 Springer London Berlin Heidelberg Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag London Limited 2005 Printed in the United States of America 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 infor- mation 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 authors 69/3830-543210 Printed on acid-free paper SPIN 10990476

For our parents with love and gratitude. For Hui and Jinlan, and our children with love and pride.

I keep the subject constantly before me and wait till the ﬁrst dawnings open little by little into the full light. — Sir Isaac Newton

Preface A switched linear system is a hybrid system which consists of several linear subsystems and a rule that orchestrates the switching among them. Switched linear systems provide a framework which bridges the linear systems and the complex and/or uncertain systems. On one hand, switching among linear systems may produce complex system behaviors such as chaos and multiple limit cycles. On the other hand, switched linear systems are relatively easy to handle as many powerful tools from linear and multilinear analysis are available to cope with these systems. Moreover, the study of switched linear systems provides additional insights into some long-standing and sophisticated problems, such as intelligent control, adaptive control, and robust analysis and control. Switched linear systems have been investigated for a long time in the control literature and have attracted increasingly more attention since the 1990s. The literature grew exponentially and quite a number of fundamental concepts and powerful tools have been developed from various disciplines. Despite the rapid progress made so far, many fundamental problems are still either unexplored or less well understood. In particular, there still lacks a uniﬁed framework that can cope with the core issues in a systematic way. This motivated us to write the current monograph. The book presents theoretical explorations on several fundamental prob- lems for switched linear systems. By integrating fresh concepts and state-of- the-art results to form a systematic approach for the switching design and feedback control, a basic theoretical framework is formed towards a switched system theory which not only extends the theory of linear systems, but also applies to more realistic problems. The book is primarily intended for researchers and engineers in the sys- tem and control community. It can also serve as complementary reading for linear/nonlinear system theory at the post-graduate level. The book contains seven chapters which exploit several independent yet related topics in detail.

viii Preface Chapter 1 introduces the system description, background and motivation of the study, and presents several general concepts and fundamental observa- tions which provide a sound base for the book. Chapter 2 concisely reviews some necessary facts from Wonham’s geomet- ric method, linear algebra and stability theory, as well as other useful tools such as diﬀerential inclusion theory, Lie algebra, and automata theory. Chapter 3 deals with stabilizing switching design for switched autonomous systems. Besides stability, the switching frequency and robustness are also considered. First, general results are presented for pointwise/consistent sta- bilization. Second, based on the average method, periodic switching laws are designed to steer the switched systems stable. Third, a state-feedback switch- ing law is presented based on an appropriate partition of the state space. To avoid possible chattering induced by the perturbations, a modiﬁed strategy is proposed by introducing a positive level set. Fourth, to further reduce the switching frequency, a combined switching law is developed by integrating the time-driven switching mechanism with the state-feedback switching scheme. An observer-based switching law is also formulated for the case when the state information is not available. Finally, the discrete-time counterparts are brieﬂy discussed. In Chapter 4, we address the controllability, observability, feedback equiv- alence and canonical forms for switched control systems. For continuous-time systems, we prove that both the controllable set and the unobservable set are subspaces of the total space. Veriﬁable geometric characterizations are pre- sented for the controllable/unobservable subspaces. Based on these, a switched control system can be brought into the canonical decomposition via suitable coordinate and feedback transformations. Parallel results are presented for discrete-time switched systems. The chapter also investigates the problem of sampling without loss of controllability. Sampling criteria are obtained and various regular switching and digital control schemes are discussed in detail. In addition, we discuss in depth several other issues including the controllabil- ity under constrained switching/input, local controllability, and decidability of controllability/observability. The problem of feedback stabilization is investigated in Chapter 5. Stabi- lizing design for switched control systems is challenging since both the control input and the switching law are design variables, and their interaction must be fully understood. Based on the canonical decomposition, we are able to design stabilizing switching/input laws for several classes of switched control systems. In particular, for a linear system controlled by multiple controllers and measured by multiple sensors, an elegant and complete treatment is pre- sented for the problem of dynamic output feedback stabilization. We also solve the stabilization problem for the switched linear system where the summation of the controllability subspaces of the individual subsystems is the total state space. For the case that the system is completely controllable, several suﬃ- cient conditions are presented for quadratic stabilizability and non-quadratic stabilizability as well. For the most general setting where both controllable

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