Introduction to Control of Oscillations and Chaos PDF
Preview Introduction to Control of Oscillations and Chaos
* i WORLD SCIENTIFIC SERIES ON ^ Alexander L. Fradkov NONLINEAR SCIENCE Series Editor: Leon O. Chua INTRODUCTION TO CONTROL OF OSCILLATIONS AND CHAOS Alexander L. Fradkov & Alexander Yu. Pogromsky World Scientific INTRODUCTION TO CONTROL OF OSCILLATIONS RNA CHAOS WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A. MONOGRAPHS AND TREATISES Published Titles Volume 15: One-Dimensional Cellular Automata B. Voorhees Volume 16: Turbulence, Strange Attractors and Chaos D. Ruelle Volume 17: The Analysis of Complex Nonlinear Mechanical Systems: A Computer Algebra Assisted Approach M. Lesser Volume 19: Continuum Mechanics via Problems and Exercises Edited by M. E. Eglit and D. H. Hodges Volume 20: Chaotic Dynamics C. Mira, L. Gardini, A. Barugola and J.-C. Cathala Volume 21: Hopf Bifurcation Analysis: A Frequency Domain Approach G. Chen and J. L. Moiola Volume 22: Chaos and Complexity in Nonlinear Electronic Circuits M. J. Ogorzalek Volume 23: Nonlinear Dynamics in Particle Accelerators R. Dila"o and R. Alves-Pires Volume 25: Chaotic Dynamics in Hamiltonian Systems H. Dankowicz Volume 30: Quasi-Conservative Systems: Cycles, Resonances and Chaos A. D. Morozov Volume 31: CNN: A Paradigm for Complexity L. O. Chua Forthcoming Titles Volume 4: Methods of Qualitative Theory in Nonlinear Dynamics (Part I) L. Shilnikov, A. Shilnikov, D. Turaev and L. O. Chua Volume 18: Wave Propagation in Hydrodynamic Flows A. L. Fabrikant and Y. A. Stepanyants Volume 27: Thermomechanics of Nonlinear Irreversible Behaviours G. A. Maugin Volume 32: From Order to Chaos II L. P. Kadanoff a | WORLD SCIENTIFIC SERIES ON rm Series A Vol. 35 NONLINEAR SCIENCE Series Editor: Leon 0. Chua INTRODUCTION TO CONTROL OF OSCILLOTIONS ONO CHAOS A. L. Fradkov A. Yu. Pogromsky Russian Academy of Sciences X YP World Scientific Singapore •NewJersey•London •HongKong Published by World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, Farrer Road, Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Libraryof Congress Cataloging-in-Publication Data Fradkov, A. L. (Aleksandr L'vovich) Introduction to control ofoscillationsand chaos / Alexander L. Fradkov, Alexander Yu. Pogromsky. p. cm. -- (World Scientific series on nonlinearscience, Series A ; vol. 35) Includes bibliographical references and index. ISBN 9810230699 1. Nonlinear control theory. 2. Chaotic behavior in systems. I. Pogromsky, Alexander Yu. II. Series: World Scientific series on nonlinearscience. Series A, Monographs and treatises ; v. 35. QA402.35.F73 1998 629.8'36--dc2l 98-40055 CIP British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright m 1998 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording oranyinformation storage and retrieval systemnow known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. This book is printed on acid-free paper. Printed in Singapore by Uto-Print Preface During the last quarter of the century, numerous studies in different coun- tries confirmed that a complex and particularly chaotic behavior is observed in systems of various types in physics, mechanics, engineering, chemistry and chemical technology, biology, economics, etc., see [223, 329, 214, 217, 87, 66]. Chaos may serve as an indicator of the complexity of the system as well as an instrument of its further evolution. A dramatic growth of interest in the problem of controlling chaotic systems has been recently ob- served. An indication of this is the increase in the number of publications. For instance, the bibliography on control and synchronization of chaos [59] contains almost 800 titles, with about 700 published in 1993-1996. One of the reasons for such an interest is that control promises both a better understanding of chaotic behavior and the means of influencing and modifying it. Various applications were reported, such as eliminating chaotic regimes in lasers, increasing the reaction rate in chemical technolo- gies by means of chaotic stirring, providing secure communications by using chaotic carrier signals and treating cardiac arrhythmia. Another reason is the interdisciplinary nature of the problem which at- tracts the attention of different scientific communities and makes it attrac- tive to even wider audience. Evidence of this is given by the surprisingly large number of publications in the scientific mass media . The headings speak for themselves: • "Putting gentle reins on unruly systems" [11], • "Mastering chaos" [79], • "Do chaos-control techniques offer hope for epilepsy?" [127], • "Keeping chaos at bay" [150], V vi Preface • "Chaos in harness" [181], • "Chaos under control" [216], • "How to get order out of stirring things up" [242]. It seems it is expected that "Coping with chaos" would cause a kind of revolution in science and technology! However, analysis and control of oscillatory and chaotic systems is ex- tremely difficult due to their intricate nonlinear dynamics. There are some interesting questions that still require unbiased scientific investigation: • Is it possible to control chaotic systems? • What are the most efficient methods of controlling oscillatory sys- tems? • What are the possibilities and limitations of controlling oscillatory and chaotic systems? It is important to answer these questions in view of the numerous poten- tial applications in laser and plasma technologies, communications, biology and medicine, economics, ecology, etc. The first question has a positive answer. A few approaches to control- ling chaos were proposed; linearization of the Poincare map [232]; periodic forcing [174],[149], linear and optimal control [306],[142] and others. How- ever, the most promising methods for controlling nonlinear dynamics are those of nonlinear control theory [152], [227]. Also, the uncertainty which is always present in real problems demands special control methods which can in fact be provided by the existing theory of nonlinear and adaptive control (e.g. [89, 94, 168] ). However, many of the published papers on the control of chaos do not make full use of the existing nonlinear control theory; many results are ob- tained by means of computer simulations. On the other hand, most control theorists and engineers are not familiar with the potential applications of control of chaotic systems. That is why the authors' primary goal was to write a book which would give a reasonably rigorous exposition of mod- ern nonlinear control theory as applied to various oscillatory and chaotic systems. The authors' approach is based on the consideration of chaotic systems in the broader context of oscillatory systems, keeping in mind that oscilla- tions may be either periodic or nonperiodic, e.g. chaotic. We demonstrate that the classical concepts of Lyapunov function, Poincare map and G. Preface vu Birkhoff 's recurrence are suitable both for the analysis and for the design of systems with oscillatory behavior. Such a view may simplify the analysis and facilitate the usage of modern control theory. On the other hand, we do not use the concept of probability and the probabilistic theory of random processes staying within the deterministic framework. The book begins with an introduction where the idea of control and the motivating examples from the different fields of science and technology are discussed. Chapter 2 presents the main concepts and results of nonlinear and adap tive control theory which serve as a solid mathematical background for the control of oscillatory systems. The necessary mathematical concepts and tools for the analysis of os cillatory and chaotic systems are introduced in Chapter 3. Chapter 4 contains nonlinear and adaptive control algorithms and their applicability conditions with respect to different assumptions about the structure, parameters and measured outputs of the controlled oscillatory systems. The control design methods are based on the concepts of Lya- punov functions, passivity, Poincare maps, speed gradient, and gradient algorithms. The control objectives include the excitation or suppression of oscillations to the desired energy level, transformation of the oscillation mode from chaotic to periodic, synchronization, etc. The chapter contains a number of theorems which establish the system stability, performance and robustness under disturbances. The described methods and algorithms are illustrated in Chapter 5 by a number of model examples, including classical models of oscillatory and chaotic systems: pendulums, the Lorenz, Duffing, Henon, Chua systems. The performance of the proposed algorithms is derived theoretically from the results of the previous chapters and also evaluated by computer simu lation. Chapter 6 contains application problems from different fields of science and technology. These problems include: • Synchronization of chaotic generators based on tunnel diodes; • Stabilization of swings in power systems; • Control of thin films growth process; • Control of oscillatory behavior of populations in ecology; • Synchronization of business cycles. Most of the results from Chapter 6 are based on the joint research of the viii Preface authors and experts in the relative fields. To make the book more suitable for the teaching process a number of exercises are placed at the end of the book. In fact, the fields of nonlinear control and nonlinear oscillations were developing suprisingly independently. The present book is perhaps the first one to bring together these two important branches of nonlinear science. The original methods presented in the book were first published in [96, 110], and summarized in [99, 111]. The additional feature of the book is that it incorporates some results in nonlinear control, adaptive control and nonlinear oscillations obtained by the researchers of the former Soviet Union who had good scientific schools in those fields. These results were published in Russian and were not well known in the West. The authors hope that the book may be interesting and useful to re- searchers, to theoretically oriented engineers, teachers and students from the fields of electrical and mechanical engineering , physics, chemistry, bi- ology, economics. Control engineers in various fields of technology dealing with complex oscillatory systems may also apply the methods given in the book. Finally, mathematicians might find there some new results , unsolved problems and motivating examples. The prospective reader should have some degree of familiarity with stan- dard university courses of calculus, linear algebra and ordinary differential equations. Knowledge of deterministic chaos and linear control theory will also help in the understanding some parts of the book. The authors would like to acknowledge the valuable help of their col- leagues associated with the Laboratory "Control of Complex Systems" of the Institute for Problems of Mechanical Engineering of Russian Academy of Sciences: B.R. Andrievsky, M.V. Druzhinina, P.Yu. Guzenko, I.A. Makarov, A.Yu. Markov, V.O. Nikiforov, E.V. Panteley, I.G. Polushin, A.A. Stotsky, A.S. Shiriaev. We are pleased to thank our colleagues and friends from other universi- ties who contributed a lot in shaping our view of the subject: I.I. Blekhman, F.L. Chernousko, K. Furuta, D.J. Hill, H. Nijmeijer, R. Ortega, A.A. Per- vozvansky, V.Ya. Yakubovich. We are also thankful to our colleagues from the different fields: E.G. Dymkin, S.A. Kukushkin, A.V. Lyamin, A.V. Os- ipov, G.S. Simin and V.P. Shiegin, for their contributions. Our research was significantly supported by the Institute for Problems of Mechanical Engineering. The results of researches supported by the Preface IX Russian Foundation of Basic Research (grants 93-013-16322, 96-01-01151), INTAS (project 94-0965), ISF (J58100-1995) and by the St. Petersburg Scientific and Educational Center for the Problems of Machine Building, Mechanics and Control Processes (project 2.1-589 of the Russian Federal Program "Integration") are also reflected in the book. Important contribu tions stem from contacts with students and from the lecture course "Con trol of Oscillations and Chaos" delivered by one of the authors at the Baltic State Technical University and the St. Petersburg University in 1996-1997. Finally it is our pleasure to acknowledge valuable comments made during our invited visits and seminars given in more than 50 universities in 15 countries. Alexander Fradkov, Alexander Pogromsky St. Petersburg, 1997.
The list of books you might like
