Table Of ContentCONTROL OF CHAOS IN
NONLINEAR CIRCUITS AND SYSTEMS
WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE
Editor: Leon O. Chua
University of California, Berkeley
Series A. MONOGRAPHS AND TREATISES
Volume 47: Chaos, Bifurcations and Fractals Around Us
W. Szempli´nska-Stupnicka
Volume 48: Bio-Inspired Emergent Control of Locomotion Systems
M. Frasca, P. Arena & L. Fortuna
Volume 49: Nonlinear and Parametric Phenomena
V. Damgov
Volume 50: Cellular Neural Networks, Multi-Scroll Chaos and Synchronization
M. E. Yalcin, J. A. K. Suykens & J. P. L. Vandewalle
Volume 51: Symmetry and Complexity
K. Mainzer
Volume 52: Applied Nonlinear Time Series Analysis
M. Small
Volume 53: Bifurcation Theory and Applications
T. Ma & S. Wang
Volume 54: Dynamics of Crowd-Minds
A. Adamatzky
Volume 55: Control of Homoclinic Chaos by Weak Periodic Perturbations
R. Chacón
Volume 56: Strange Nonchaotic Attractors
U. Feudel, S. Kuznetsov & A. Pikovsky
Volume 57: A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science
L. O. Chua
Volume 58: New Methods for Chaotic Dynamics
N. A. Magnitskii & S. V. Sidorov
Volume 59: Equations of Phase-Locked Loops
J. Kudrewicz & S. Wasowicz
Volume 59: Equations of Phase-Locked Loops
J. Kudrewicz & S. Wasowicz
Volume 60: Smooth and Nonsmooth High Dimensional Chaos and
the Melnikov-Type Methods
J. Awrejcewicz & M. M. Holicke
Volume 61: A Gallery of Chua Attractors (with CD-ROM)
E. Bilotta & P. Pantano
Volume 62: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids
A. Berezovski, J. Engelbrecht & G. A. Maugin
Volume 63: Advanced Topics on Cellular Self-Organizing Nets and Chaotic
Nonlinear Dynamics to Model and Control Complex Systems
R. Caponetto, L. Fortuna & M. Frasca
Lakshmi - Control of Chaos.pmd 2 10/21/2008, 1:23 PM
NOWONRLLDI NSCEIEANTRIFI CS SCERIIEESN OCNE Series A Vol. 64
Series Editor: Leon O. Chua
CONTROL OF CHAOS IN
NONLINEAR CIRCUITS AND SYSTEMS
Edited by
Bingo Wing-Kuen Ling
King’s College London, UK
Herbert Ho-Ching Iu
The University of Western Australia
Hak-Keung Lam
King’s College London, UK
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
Published by
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UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
CONTROL OF CHAOS IN NONLINEAR CIRCUITS AND SYSTEMS
World Scientific Series on Nonlinear Science, Series A — Vol. 64
Copyright © 2009 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 or any information storage and retrieval
system now 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.
ISBN-13 978-981-279-056-9
ISBN-10 981-279-056-X
Printed in Singapore.
Lakshmi - Control of Chaos.pmd 1 10/21/2008, 1:23 PM
PREFACE
Nonlinear circuits and systems, such as electronic circuits (Chapter 5),
power converters (Chapter 6), human brains (Chapter 7), phase lock
loops (Chapter 8), sigma delta modulators (Chapter 9), etc, are found
almost everywhere. Understanding nonlinear behaviours as well as
control of these circuits and systems are important for real practical
engineering applications.
Control theories for linear circuits and systems are well developed
and almost complete. However, different nonlinear circuits and systems
could exhibit very different behaviours. Hence, it is difficult to unify a
general control theory for general nonlinear circuits and systems. Up to
now, control theories for nonlinear circuits and systems are still very
limited. The objective of this book is to review the state-of-the-art chaos
control methods for some common nonlinear circuits and systems,
such as those listed in the above, and stimulate further research and
development in chaos control for nonlinear circuits and systems.
This book consists of three parts. The first part of the book consists of
reviews on general chaos control methods. In particular, a time-delayed
approach written by H. Huang and G. Feng is reviewed in Chapter 1.
A master slave synchronization problem for chaotic Lur’e systems is
considered. A delay independent and delay dependent synchronization
criteria are derived based on the H performance. The design of the time
∞
delayed feedback controller can be accomplished by means of the
feasibility of linear matrix inequalities. In Chapter 2, a fuzzy model
based approach written by H.K. Lam and F.H.F. Leung is reviewed. The
synchronization of chaotic systems subject to parameter uncertainties is
considered. A chaotic system is first represented by the fuzzy model. A
switching controller is then employed to synchronize the systems. The
stability conditions in terms of linear matrix inequalities are derived
v
vi Preface
based on the Lyapunov stability theory. The tracking performance and
parameter design of the controller are formulated as a generalized
eigenvalue minimization problem which is solved numerically via some
convex programming techniques. In Chapter 3, a sliding mode control
approach written by Y. Feng and X. Yu is reviewed. Three kinds of
sliding mode control methods, traditional sliding mode control, terminal
sliding mode control and non-singular terminal sliding mode control, are
employed for the control of a chaotic system to realize two different
control objectives, namely to force the system states to converge to zero
or to track desired trajectories. Observer based chaos synchronizations
for chaotic systems with single nonlinearity and multi-nonlinearities are
also presented. In Chapter 4, an optimal control approach written by
C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao is reviewed. Systems with
nonparametric regression with jump points are considered. The rough
locations of all the possible jump points are identified using existing
kernel methods. A smooth spline function is used to approximate each
segment of the regression function. A time scaling transformation is
derived so as to map the undecided jump points to fixed points. The
approximation problem is formulated as an optimization problem and
solved via existing optimization tools.
The second part of the book consists of reviews on general chaos
controls for continuous-time systems. In particular, chaos controls for
Chua’s circuits written by L.A.B. Tôrres, L.A. Aguirre, R.M. Palhares
and E.M.A.M. Mendes are discussed in Chapter 5. An inductorless
Chua’s circuit realization is presented, as well as some practical
issues, such as data analysis, mathematical modelling and dynamical
characterization, are discussed. The tradeoff among the control objective,
the control energy and the model complexity is derived. In Chapter 6,
chaos controls for pulse width modulation current mode single phase
H-bridge inverters written by B. Robert, M. Feki and H.H.C. Iu are
discussed. A time delayed feedback controller is used in conjunction
with the proportional controller in its simple form as well as in its
extended form to stabilize the desired periodic orbit for larger values of
the proportional controller gain. This method is very robust and easy to
implement. In Chapter 7, chaos controls for epileptiform bursting in
the brain written by M.W. Slutzky, P. Cvitanovic and D.J. Mogul are
Preface vii
discussed. Chaos analysis and chaos control algorithms for manipulating
the seizure-like behaviour in a brain slice model are discussed. The
techniques provide a nonlinear control pathway for terminating or
potentially preventing epileptic seizures in the whole brain.
The third part of the book consists of reviews on general chaos
controls for discrete-time systems. In particular, chaos controls for phase
lock loops written by A.M. Harb and B.A. Harb are discussed in Chapter
8. A nonlinear controller based on the theory of backstepping is designed
so that the phase lock loops will not be out of lock. Also, the phase lock
loops will not exhibit Hopf bifurcation and chaotic behaviours. In
Chapter 9, chaos controls for sigma delta modulators written by B.W.K.
Ling, C.Y.F. Ho and J.D. Reiss are discussed. A fuzzy impulsive control
approach is employed for the control of the sigma delta modulators. The
local stability criterion and the condition for the occurrence of limit cycle
behaviours are derived. Based on the derived conditions, a fuzzy
impulsive control law is formulated so that the occurrence of the limit
cycle behaviours, the effect of the audio clicks and the distance between
the state vectors and an invariant set are minimized supposing that the
invariant set is nonempty. The state vectors can be bounded within any
arbitrary nonempty region no matter what the input step size, the initial
condition and the filter parameters are.
The editors are much indebted to the Editor of the World Scientific
Series on Nonlinear Science, Prof. Leon Chua, and to Senior Editor
Ms. Lakshmi Narayan for their help and congenial processing of the
edition.
Bingo Wing Kuen Ling
Herbert Ho Ching Iu
Hak Keung Lam
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CONTENTS
Preface v
Section A: General Chaos Control Methods
1. Robust Synchronization of Chaotic Systems based on 3
Time-delayed Feedback Control
H. Huang and G. Feng
2. Synchronization of Uncertain Chaotic Systems based on 35
Fuzzy-model-based Approach
H.K. Lam and F.H.F. Leung
3. Sliding Mode Control of Chaotic Systems 55
Y. Feng and X. Yu
4. A New Two-stage Method for Nonparametric Regression with 79
Jump Points
C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao
Section B: Chaos Control for Continuous-time Systems
5. Chaos Control for Chua’s Circuits 97
L.A.B. Tôrres, L.A. Aguirre, R.M. Palhares and
E.M.A.M. Mendes
6. Chaos Control for a PWM H-bridge Inverter 165
B. Robert, M. Feki and H.H.C. Iu
ix
