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Zhong Li Fuzzy Chaotic Systems

[Studies in Fuzziness and Soft Computing] Fuzzy Chaotic Systems Volume 199 ||

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  • Zhong Li

    Fuzzy Chaotic Systems

  • Studies in Fuzziness and Soft Computing, Volume 199

    Editor-in-chiefProf. Janusz KacprzykSystems Research InstitutePolish Academy of Sciencesul. Newelska 601-447 WarsawPolandE-mail: [email protected]

    Further volumes of this seriescan be found on our homepage:springer.com

    Vol. 183. Larry Bull, Tim Kovacs (Eds.)Foundations of Learning Classifier Systems,2005ISBN 3-540-25073-5

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    Vol. 186. Radim Belohlvek, VilmVychodilFuzzy Equational Logic, 2005ISBN 3-540-26254-7

    Vol. 187. Zhong Li, Wolfgang A. Halang,Guanrong Chen (Eds.)Integration of Fuzzy Logic and ChaosTheory, 2006ISBN 3-540-26899-5

    Vol. 188. James J. Buckley, Leonard J.JowersSimulating Continuous Fuzzy Systems, 2006ISBN 3-540-28455-9

    Vol. 189. Hans BandemerMathematics of Uncertainty, 2006ISBN 3-540-28457-5

    Vol. 190. Ying-ping ChenExtending the Scalability of LinkageLearning Genetic Algorithms, 2006ISBN 3-540-28459-1

    Vol. 191. Martin V. ButzRule-Based Evolutionary Online LearningSystems, 2006ISBN 3-540-25379-3

    Vol. 192. Jose A. Lozano, Pedro Larraaga,Iaki Inza, Endika Bengoetxea (Eds.)Towards a New Evolutionary Computation,2006ISBN 3-540-29006-0

    Vol. 193. Ingo GlcknerFuzzy Quantifiers: A Computational Theory,2006ISBN 3-540-29634-4

    Vol. 194. Dawn E. Holmes, Lakhmi C. Jain(Eds.)Innovations in Machine Learning, 2006ISBN 3-540-30609-9

    Vol. 195. Zongmin MaFuzzy Database Modeling of Imprecise andUncertain Engineering Information, 2006ISBN 3-540-30675-7

    Vol. 196. James J. BuckleyFuzzy Probability and Statistics, 2006ISBN 3-540-30841-5

    Vol. 197. Enrique Herrera-Viedma, GabriellaPasi, Fabio Crestani (Eds.)Soft Computing in Web InformationRetrieval, 2006ISBN 3-540-31588-8

    Vol. 198. Hung T. Nguyen, Berlin WuFundamentals of Statistics with Fuzzy Data,2006ISBN 3-540-31695-7

    Vol. 199. Zhong LiFuzzy Chaotic Systems, 2006ISBN 3-540-33220-0

  • Zhong Li

    Fuzzy Chaotic SystemsModeling, Control, and Applications

    ABC

  • Dr. Zhong LiFaculty of Electricaland Computer EngineeringFernUniversitaet in Hagen58084 HagenGermanyE-mail: [email protected]

    Library of Congress Control Number: 2006XXXXXX

    ISSN print edition: 1434-9922ISSN electronic edition: 1860-0808ISBN-10 3-540-33220-0 Springer Berlin Heidelberg New YorkISBN-13 978-3-540-33220-6 Springer Berlin Heidelberg New York

    This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violations areliable for prosecution under the German Copyright Law.

    Springer is a part of Springer Science+Business Mediaspringer.comc Springer-Verlag Berlin Heidelberg 2006

    Printed in The Netherlands

    The use of general descriptive names, 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 protective lawsand regulations and therefore free for general use.Typesetting: by the author and techbooks using a Springer LATEX macro packagePrinted on acid-free paper SPIN: 11353348 89/techbooks 5 4 3 2 1 0

  • To Juan and Yifan

  • Preface

    Bringing together the two seemingly unrelated concepts, fuzzy logic and chaostheory, is primarily motivated by the concept of soft computing (SC), initiatedby Lot A. Zadeh, the founder of fuzzy set theory. The principal constituentsof SC are fuzzy logic (FL), neural network theory (NN) and probabilisticreasoning (PR), with the latter subsuming parts of belief networks, geneticalgorithms, chaos theory and learning theory. What is important to note isthat SC is not a melange of FL, NN and PR. Rather, it is an integration inwhich each of the partners contributes a distinct methodology for addressingproblems in their common domain. In this perspective, the principal contri-butions of FL, NN and PR are complementary rather than competitive.

    SC diers from conventional (hard) computing in that it is tolerant ofimprecision, uncertainty and partial truth. In eect, the role model for softcomputing is the human mind. From the general SC concept, we extract FLand chaos theory as the object of this book to study their relationships orinteractions.

    Over the past few decades, fuzzy systems technology and chaos theoryhave received ever increasing research interests from, respectively, systemsand control engineers, theoretical and experimental physicists, applied math-ematicians, physiologists, and other communities of researchers. Especially,as one of the emerging information processing technologies, fuzzy systemstechnology has achieved widespread applications around the globe in manyindustries and technical elds, ranging from control, automation, and articialintelligence (AI) to image/signal processing and pattern recognition. On theother hand, in engineering systems chaos theory has evolved from being simplya curious phenomenon to one with real, practical signicance and utilization.We are now standing at the threshold of major advances in the control andsynchronization of chaos with new applications across a broad range of engi-neering disciplines, where chaos control promises to have a major impact onnovel time- and energy-critical engineering applications.

    Notably, studies on fuzzy systems and chaos theory have been carryingout separately. Although there have been some eorts on exploring the inter-

  • VIII Preface

    actions between fuzzy logic and chaos theory, it is still far away from fullyunderstanding their mutual relationships. Intuitively, fuzzy logic resembleshuman reasoning in its use of approximate information and uncertainty togenerate decisions, and chaotic dynamics could be a fundamental reason forhuman brain to produce and process massive information instantly. It is be-lieved that they have a close relationship in human reasoning and informationprocessing, and it has a great potential to combine fuzzy systems with chaostheory for future scientic research and engineering applications.

    This book does not intend to in fact, is not able to give an in-depthexplanation of the interactions or intrinsic relationships between fuzzy logicand chaos theory, but tries to provide some heuristic research achievementsand insightful ideas to attract more attention on the topic. Although thisbook may raise more questions than it can provide answers, We hope that itnevertheless contains seeds for future brooming research.

    More precisely, this book is written in the following way: it starts with thefundamental concepts of fuzzy logic and fuzzy control, chaos theory and chaoscontrol, as well as the denition of chaos on the metric space of fuzzy sets,followed by fuzzy modeling and (adaptive) fuzzy control of chaotic systems, allbased on both Mamdani fuzzy models and Takagi-Sugeno (TS) fuzzy models;then, it discusses some other topics, such as synchronization, anti-control ofchaos, intelligent digital redesign, all for TS fuzzy systems, and spatiotemporalchaos and synchronization in complex fuzzy systems; nally, it ends with apractical application example of fuzzy-chaos-based cryptography.

    I am very grateful to Prof. Wolfgang A. Halang and Prof. Guanrong Chenfor their long-term support and friendship in various aspects of my work andlife, without which this book would not have been completed. Special thanksare given to Dr. Hojae Lee for providing me with the materials presentedin Chapters 6, 9, and 12. Thanks also go to the following individuals whoprovided some original gures or helpful comments: Oscar Calvo, FedericoCuesta, Patrick Grim, Z.H. Guan, K.-Y. Lian, Domenico M. Porto, M. LaRosa, Hua O. Wang, and H.B. Zhang. I am very appreciative of the discussionsand collaborations with Peter Kloeden, Zongyuan Mao, Ping Ren, Bo Zhang,Yaobin Mao, Wenbo Liu, Shujun Li, Martin Skambraks, Jutta During, PingLi, and Hong Li. I owe my deepest thanks to my parents for their fostering andeducation. Finally, I wish to express my appreciation to Prof. Janusz Kacprzykfor recommending this book to the Springer series, Studies in Fuzziness andSoft Computing, and the editorial and production sta of Springer-Verlag inHeidelberg for their ne work in producing this new monograph.

    Hagen, Germany Zhong LiFebruary 2006

  • Contents

    1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Fuzzy Logic and Fuzzy Control Systems . . . . . . . . . . . . . . . . . . . . 1

    1.1.1 Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Fuzzy Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.2 Chaos and Chaos Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Chaos Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    1.3 Interactions between Fuzzy Logic and Chaos Theory . . . . . . . . . 101.4 About This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    2 Fuzzy Logic and Fuzzy Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Fuzzy Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    2.2.1 Crisp Sets and Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.2 Fundamental Operations of Fuzzy Sets . . . . . . . . . . . . . . . 172.2.3 Properties of Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.4 Some Other Fundamental Concepts of Fuzzy Sets . . . . . 202.2.5 Extension Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    2.3 Fuzzy Relations and Their Compositions . . . . . . . . . . . . . . . . . . . 212.3.1 Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Operations of Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . 222.3.3 Composition of Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . 22

    2.4 Fuzzy Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 Generalized Modus Ponens and Modus Tollens . . . . . . . . 232.4.2 Fuzzy Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4.3 Fuzzy Rule Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4.4 Fuzzy Inference Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.5 Fuzzier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.6 Defuzzier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    2.5 Fuzzy Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.6 Fuzzy Systems as Universal Approximators . . . . . . . . . . . . . . . . . 28

  • X Contents

    3 Chaos and Chaos Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1 Logistic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Hopf Bifurcation of Higher-dimensional Systems . . . . . . . . . . . . . 373.4 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5 Routes to Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.6 Center Manifold Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.7 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.8 Control of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    3.8.1 Ott-Grebogi-Yorke Method . . . . . . . . . . . . . . . . . . . . . . . . . 443.8.2 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.8.3 Pyragas Time-delayed Feedback Control . . . . . . . . . . . . . . 513.8.4 Entrainment and Migration Control . . . . . . . . . . . . . . . . . 52

    4 Denition of Chaos in Metric Spaces of Fuzzy Sets . . . . . . . . 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Chaos of Dierence Equations in n with a Saddle Point . . . . . 57

    4.2.1 Sucient Conditions for Chaos in n . . . . . . . . . . . . . . . . 574.2.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    4.3 Chaotic Maps in Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4 Chaos of Discrete Systems in Complete Metric Spaces . . . . . . . . 644.5 Chaos of Dierence Equations in Metric Spaces

    of Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.5.1 Chaotic Maps on Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . 684.5.2 Example of a Chaotic Map on Fuzzy Sets . . . . . . . . . . . . 70

    5 Fuzzy Modeling of Chaotic Systems I (Mamdani Model) . 735.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 Double-scroll Chaotic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3 Single-scroll Chaotic System: Logistic Map . . . . . . . . . . . . . . . . . 785.4 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.5 Two-dimensional Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    6 Fuzzy Modeling of Chaotic Systems II (TS Model) . . . . . . 916.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.2 TS Fuzzy Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.3 Preliminary Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.4 TS Fuzzy Modeling of Chaotic Systems: Examples . . . . . . . . . . . 97

    6.4.1 Continuous-time Chaotic Lorenz System . . . . . . . . . . . . . 976.4.2 Continuous-time Flexible-joint Robot Arm . . . . . . . . . . . 996.4.3 Dung-like Chaotic Oscillator . . . . . . . . . . . . . . . . . . . . . . 1026.4.4 Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

    6.5 Discretization of Continuous-time TS Fuzzy Models . . . . . . . . . . 1056.6 Bifurcation Phenomena in TS Fuzzy Systems . . . . . . . . . . . . . . . 106

    6.6.1 Bifurcation in TS Fuzzy Systems . . . . . . . . . . . . . . . . . . . . 107

  • Contents XI

    6.6.2 TS Fuzzy Systems with Linear Consequents . . . . . . . . . . 1116.7 Appendix: Bifurcation Analysis for = 1 . . . . . . . . . . . . . . . . . . . 115

    7 Fuzzy Control of Chaotic Systems I (Mamdani Model) . . 1217.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.2 Design of Fuzzy Logic Controllers . . . . . . . . . . . . . . . . . . . . . . . . . 122

    7.2.1 Selection of Variables and the Universe of Discourse . . . 1227.2.2 Fuzzication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.2.3 Discretization and Normalization of a Universe

    of Discourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.2.4 Construction of the Rule-base . . . . . . . . . . . . . . . . . . . . . . . 1247.2.5 Defuzzication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.2.6 Fuzzy Inference Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 129

    7.3 Fuzzy Control of Chuas Chaotic Circuit . . . . . . . . . . . . . . . . . . . . 1297.4 Fuzzy Control of Chaotic Lorenz System . . . . . . . . . . . . . . . . . . . 135

    8 Adaptive Fuzzy Control of Chaotic Systems(Mamdani Model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.2 Stable Directly Adaptive Fuzzy Control of Chaotic Systems . . . 144

    8.2.1 Design of the Supervisory Controller us . . . . . . . . . . . . . . 1468.2.2 Design of the Controller uc . . . . . . . . . . . . . . . . . . . . . . . . . 146

    8.3 Design of Directly Adaptive Fuzzy Controllers . . . . . . . . . . . . . . 1478.4 Adaptive Fuzzy Control of the Dung Oscillator . . . . . . . . . . . . 148

    9 Fuzzy Control of Chaotic Systems II(TS Model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1539.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1539.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1549.3 Parallel-Distributed Compensation . . . . . . . . . . . . . . . . . . . . . . . . . 1559.4 Lyapunov Stability of TS Fuzzy Systems . . . . . . . . . . . . . . . . . . . 1569.5 Stability Analysis and Controller Design Based

    on LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.5.1 Continuous-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.5.2 Discrete-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

    9.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1689.6.1 Continuous-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1689.6.2 Discrete-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

    9.7 TS Fuzzy-model-based Adaptive Control . . . . . . . . . . . . . . . . . . . 181

    10 Synchronization of TS Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . 18910.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18910.2 Exact Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19010.3 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

    10.3.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19410.3.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

  • XII Contents

    10.4 Synchronization of Chens Chaotic Systems . . . . . . . . . . . . . . . . . 19610.5 Synchronization of Hyperchaotic Systems . . . . . . . . . . . . . . . . . . . 199

    11 Chaotifying TS Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20511.2 Chaotifying Discrete-time TS Fuzzy Systems . . . . . . . . . . . . . . . . 206

    11.2.1 Discrete-time TS Fuzzy System via Mod-Operation . . . . 20611.2.2 Anti-controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20811.2.3 Verication of the Anti-control Design . . . . . . . . . . . . . . . 20911.2.4 A Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

    11.3 Chaotifying Discrete-time TS Fuzzy Systemsvia a Sinusoidal Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

    11.4 Chaotifying Continuous-time TS Fuzzy Systemsvia Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

    11.5 Chaotifying Continuous-Time TS Fuzzy Systemsvia Time-delay Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22311.5.1 PDC Controller for Locally Controllable TS Fuzzy

    Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22411.5.2 Controller Design for General TS Fuzzy Systems . . . . . . 22611.5.3 Verication of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22811.5.4 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

    12 Intelligent Digital Redesign for TS Fuzzy Systems . . . . . . . . . 23912.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23912.2 Digital Fuzzy Systems and Their Discretization . . . . . . . . . . . . . 24112.3 Global State-matching Intelligent Digital Redesign . . . . . . . . . . . 24312.4 Digital Redesign for Dung-like Chaotic Oscillator . . . . . . . . . . 24612.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

    13 Spatiotemporal Chaos and Synchronizationin Complex Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25513.2 Complex Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

    13.2.1 Single-unit: Chaotic Fuzzy Oscillator . . . . . . . . . . . . . . . . . 25613.2.2 Macro-system: Chain of Fuzzy Oscillators . . . . . . . . . . . . 257

    13.3 Synchronization Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26113.4 Complex Networks: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 26713.5 Collective Behavior versus Network Topology . . . . . . . . . . . . . . . 270

    14 Fuzzy-chaos-based Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . 27514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27514.2 Working Principle of the Cryptosystem . . . . . . . . . . . . . . . . . . . . . 27714.3 Decryption by Fuzzy-model-based Synchronization . . . . . . . . . . . 279

    References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

  • 1Introduction

    At rst glance, fuzzy logic and chaos theory may seem two totally dierentareas with merely marginal connections to each other. In this introduction,after reviewing the evolution of fuzzy set theory and chaos theory, respectively,we explain briey the ideas why we bring them together, and we shall showthat the understanding of the interactions between fuzzy systems and chaostheory lays a solid foundation for better applications of the two promisingnew technologies, and their integration oers a great number of interestingpossibilities in their interplay and future developments.

    1.1 Fuzzy Logic and Fuzzy Control Systems

    1.1.1 Fuzzy Logic

    The precision of mathematics owes, to a high extent, its success to the ef-forts of Aristotle and the philosophers who preceded him. In their eorts todevise a concise theory of logic, and later mathematics, the so-called Lawsof Thought were posited [1]. One of those, the Law of the Excluded Mid-dle, states that every proposition must either be True or False. Even whenParminedes proposed the rst version of this law (around 400 B.C.) therewere strong and immediate objections: for example, Heraclitus proposed thatthings could be simultaneously True and not True.

    It was Plato who laid the foundation for what would become fuzzy logic, byindicating that there was a third region (between True and False) where theseopposites appeared. Other more modern philosophers echoed his sentiments,notably Hegel, Marx, and Engels. But it was Lukasiewicz who rst proposeda systematic alternative to the bi-valued logic of Aristotle [2].

    In the early 1900s, Lukasiewicz described a three-valued logic, along withthe mathematics to accompany it. The third value he proposed can best betranslated as the term possible, and he assigned it a numeric value between

    Zhong Li: Fuzzy Chaotic Systems, StudFuzz 199, 111 (2006)www.springerlink.com c Springer-Verlag Berlin Heidelberg 2006

  • 2 1 Introduction

    True and False. Eventually, he proposed an entire notion and axiomatic systemfrom which he hoped to derive modern mathematics.

    Later, he explored four-valued logic, ve-valued logic, and then declaredthat, in principle, there was no diculty in deriving an innite-valued logic.Lukasiewicz felt that three- and innite-valued logics were the most intriguing,but he ultimately settled on a four-valued logic, because it seemed to be themost easily adaptable to the Aristotelian logic.

    In about the same time, Knuth proposed a three-valued logic similar tothat of Lukasiewicz, from which he speculated that mathematics would be-come even more elegant than that in traditional bi-valued logic. His insight,apparently missed by Lukasiewicz, was to use the integral range [1, 0,+1]rather than [0, 1, 2]. Nonetheless, this alternative failed to gain acceptance,and has fallen into relative obscurity.

    It was not until relatively recently that the notion of an innite-valuedlogic was brought forward. In 1965, Lot A. Zadeh published his seminalwork Fuzzy sets [3, 4], which described the mathematics of what is calledfuzzy set theory today. This theory proposed a membership function (or thevalues False and True) to operate over the range of real numbers [0.0, 1.0].New operations for the calculus of logic were formulated, and showed to be,in principle, a generalization of classic logic.

    Since then, fuzzy set theory has received tremendous interest in researchand applications. This is mainly due to the immense success of Japaneseconsumer products that made use of fuzzy technology in the 1980s, and sometheoretical breakthrough on the stability analysis of fuzzy systems in the1990s.

    Fuzzy systems, including fuzzy logic and fuzzy set theory, are an alterna-tive to traditional notions of set theory and logic that have their origins inancient Greek philosophy, and applications at the leading edge of ArticialIntelligence. The main idea of fuzzy systems is to extend the classical two-valued modeling of concepts and attributes like tall, fast or old in a sense ofpartial truth. This means that a person is not just viewed as tall or not tall,but as tall to a certain degree between 0 and 1.

    Classical models usually try to avoid vague, imprecise or uncertain infor-mation, because it is considered to have a negative inuence in an inferenceprocess. Fuzzy systems, on the other hand, deliberately make use of this kindof information. This usually leads to simpler, more suitable models, which areeasier to handle and are more familiar to human beings.

    Zadehs proposal of modeling the mechanism of human thinking, withlinguistic fuzzy values rather than numbers [5], led to the introduction offuzziness into systems theory and to the development of a new class of math-ematical systems called fuzzy systems. In general, we shall refer to fuzzy sys-tems as those resulted from fuzzication of a conventional system. A centralcharacteristic of fuzzy systems is that they are based on the concept of fuzzycoding (partitioning) of information. Fuzzy systems operate with fuzzy setsinstead of numbers. Each fuzzy set has more expressive power than a single

  • 1.1 Fuzzy Logic and Fuzzy Control Systems 3

    number. The use of fuzzy sets permits a generalization of information. Thisgeneralization is associated with the introduction of imprecision. In many realproblems the imprecision is admissible, even useful, because the categories ofhuman thinking are vague ideas which are very hard to quantify. In essence,the representation of the information in fuzzy systems imitates the mecha-nism of approximate reasoning performed in the human mind. The precisionof conventional systems theory is obtained as a limiting case in the continuityof varying levels of abstraction.

    Fuzzy system models basically fall into two categories, which dier fun-damentally in their abilities to represent dierent types of information. Therst category includes linguistic models, which have been referred to so faras Mamdani fuzzy models. They are based on collections of IF-THEN ruleswith vague predicates and use fuzzy reasoning [6, 7]. In these models, fuzzyquantities are associated with linguistic labels, and a fuzzy model is essen-tially a qualitative expression of the underlying system. Models of this typeform a basis for qualitative modeling that describes the system behavior byusing natural language [8]. A corresponding fuzzy logic controller is a proto-typical example of such a linguistic model, in which its rules give a linguisticexpression of the control strategy in a common sense.

    The second category of fuzzy models is based on the Takagi-Sugeno (TS)method of reasoning [9, 10, 11]. These models are formed by logical rules thathave a fuzzy antecedent part and a functional consequent. They are combina-tions of fuzzy and nonfuzzy models. Fuzzy models based on the TS method ofreasoning integrate the ability of linguistic models for qualitative knowledgerepresentation with great potential for expressing quantitative information.In addition, this type of fuzzy models permits a relatively easy application ofvarious powerful learning techniques for system identication from data andcontroller design.

    Because of the linear dependence of each rule on the input variables of theunderlying system, the TS method is capable of acting as an interpolatingsupervisor of multiple linear controllers that are to be applied, respectively,under dierent operating conditions of a dynamic nonlinear system. For exam-ple, the performance of an aircraft may change dramatically with altitude andMach number. Linear controllers, though easy to compute and well-suited toany given ight condition, must be updated regularly and smoothly to keepup with the changing state of the airplane. A TS fuzzy inference model isextremely well suited to the task of smoothly interpolating the linear controlgains that would be applied across the input space; it is a natural and e-cient gain scheduler. Similarly, a TS fuzzy system is suited to model nonlinearsystems by interpolating between multiple linear models.

    Since a TS fuzzy system is a more compact and computationally ecientrepresentation than a Mamdani fuzzy system, TS fuzzy system typically usesadaptive techniques for model construction. These adaptive techniques canbe used to customize the membership functions so that the fuzzy system canbest t the data available.

  • 4 1 Introduction

    Here are some general comparisons of the two dierent fuzzy models:

    Advantages of the Mamdani fuzzy model:

    i) Intuitive;ii) Widespread acceptance;iii) Well-suited to human linguistic input.

    Advantages of the TS fuzzy model

    i) Computationally ecient;ii) Works well with linear techniques (e.g., PID control);iii) Works well with optimization and adaptive techniques;iv) Guaranteed continuity of the output surface;v) Well-suited to mathematical analysis.

    1.1.2 Fuzzy Control Systems

    Fuzzy control is the most successful and active branch of fuzzy system tech-nology, in terms of both theoretical analysis and practical applications. Theprimary thrust of this novel control paradigm, created in the early 1970s,was to utilize the knowledge and experience extracted from a human controloperator to intuitively construct controllers so that the resulting controllerswere able to emulate human control behavior to a certain extent. Comparedto the traditional control paradigm, the advantages of the fuzzy control par-adigm are twofold. First, a precise mathematical model of the system to becontrolled is not required; second, a satisfactory nonlinear controller can oftenbe developed empirically without using complicated mathematics [12].

    Industrial automation and commercial production have successfully beendeveloped worldwide using fuzzy control. In this regard, Japan has led the way.Its success includes Hitachi automated train operation of the Sendai subwaysystem that has been in daily operation since 1987. The trains, controlledby fuzzy predictive controllers, consume less electric energy, and ride morecomfortably than the ones controlled by conventional controllers. AnotherHitachi product is the group fuzzy control operation for elevators. The waitingtime and idle time of the elevators are both reduced during the rush hours;and riding and stopping are smoother.

    In the late 1980s, a real-time fuzzy control drug delivery system was suc-cessfully developed and clinically implemented to regulate blood pressure inpostsurgical open-heart patients at cardiac surgical intensive care units [13].This is the worlds rst real-time fuzzy control application in medicine. Fuzzysystems have also been applied to the control of muscle immobility and hy-pertension during general anesthesia, assessment of cardio-vascular dynamicsduring ventricular assistance, diagnosis of artery lesions and coronary steno-sis, support for seriodiagnosis, intelligent medical alarms, and multineuronstudies. Other successful medical applications are the detection of coronary

  • 1.2 Chaos and Chaos Control 5

    artery disease, classication of tissues and structures in electrocardiograms,and classication of normal and cancerous tissues in brain magnetic resonanceimages.

    The theory of fuzzy systems has advanced signicantly along with therapid increase of their practical applications. In early days, most fuzzy con-trollers were used as black-box controllers in that their internal mathematicalstructures were unknown. Since the late 1980s, signicant progress has beenmade to mathematically explore the analytic structures of various types offuzzy controllers [14]. Fuzzy control has been related to PID control, slidingmode control, adaptive control, relay control, etc. in conventional control sys-tems, resulting in insightful understanding of fuzzy control within the contextof classical control. The advances have also been used to analyze some impor-tant aspects of fuzzy control systems, including system stability and controlperformance, and to better design fuzzy controllers.

    System modeling and system control are two closely related problems.Fuzzy modeling is a new modeling paradigm, and fuzzy models are nonlin-ear dynamic models. Comparing with the conventional black-box modelingtechniques that can only utilize numerical data, the uniqueness of the fuzzymodeling approach lies in its ability to utilize both qualitative and quanti-tative information. This advantage is practically important and even crucialin many circumstances. Fuzzy models are often intuitive, as fuzzy sets, fuzzylogic and fuzzy rules are intuitive and linguistic. However, fuzzy models arenot as simple as those models that can be expressed in mathematical for-mulae. In general, fuzzy models are black-box models. Nevertheless, undercertain conditions, analytical structures of some fuzzy models can be derivedand, hence, can be used conveniently.

    A system capable of uniformly approximating any continuous function ona compact set is called a functional approximator or a universal approxima-tor. The issue of universal approximation is crucial to fuzzy systems. In thecontext of control, the question is whether a fuzzy controller can always beconstructed to uniformly approximate a desired continuous nonlinear controlsolution with sucient accuracy. For modeling, the question is whether alwaysa fuzzy model can be established, which is capable of uniformly approximat-ing any continuous nonlinear physical system. Recent theoretical work has ledto armative answers to these qualitative questions for both Mamdani fuzzymodels and TS fuzzy models.

    1.2 Chaos and Chaos Control

    1.2.1 Chaos

    Chaos, dened in the Encyclopaedia Britannica, originates from the Greeko, which means the primeval emptiness of the universe before thingscame into being of the abyss of Tartarus, the under world.. . . . In the later

  • 6 1 Introduction

    cosmology, chaos generally designated the original state of things, howeverconceived. The modern meaning of the word is derived form Ovid, who sawchaos as the original disordered and formless mass, from which the maker ofthe Cosmos produced the ordered universe [15]. Yet, there is another inter-pretation of chaos in the ancient Chinese vocabulary, where chaos generallyrefers to an unusual phenomenon that is disorderly and irregular. A moderndictionary denition of chaos provides two meanings: (i) utter disorder andconfusion, and (ii) the unformed original state of the universe. In the modernscientic terminology, however, chaos has a rather precise but fairly compli-cated denition by means of the dynamics of a generally nonlinear system.

    Historically, at the turning of last century (around 1900), French math-ematician Henry Poincare had already observed that fully deterministic dy-namics do not necessarily imply explicit predictions on the evolution of adynamical system. This can be considered as a milestone in the approachto the study of dynamical chaos. Although in the 1930s and 1940s strangebehavior was observed in many physical systems, the notion that this phe-nomenon was inherent to deterministic systems was never suggested. Evenwith the powerful results of Steve Smale in the 1960s, complicated behaviorof deterministic systems remained no more than a mathematical curiosity.Scientically, chaos implies the existence of undesirable randomness, but theself-organization concept at the edge of chaos denotes the order out of chaos.The American essayist and historian Henry Adams (1858-1918) expressed thescientic meaning of chaos succinctly: Chaos often breeds life, when orderbreeds habit [16].

    Not until the late 1970s, with the advent of fast and powerful computers,was it recognized that chaotic behavior was prevalent in almost all domains ofscience and technology. In the development of chaos theory, the rst evidenceof physical chaos seems to be the discovery of Edward Lorenz, the Lorenz at-tractor or buttery eect, in 1963. The rst underlying mechanism withinchaos was observed by American physicist Mitchell Feigenbaum, who in 1976discovered that when an ordered system begins to break down into chaos,a consistent pattern of rate doubling occurs. Later on, the works of MichelHenon and Carl Heiles [17] and Boris Chirikov [18] provided new insights intothe origin of chaotic behaviors in dissipative as well as in conservative systems.To that end, the term chaos was rst formally introduced into mathematicsby Li and Yorke in their paper period-3 implies chaos [19]. Ever since then,there have been several alternative but closely related denitions of chaos,among which that of Devaney is perhaps the most popular one [20]. Never-theless, a unied, universally accepted, and rigorous denition of chaos is notyet available in the current scientic literature [21].

    Devaney states that a map f : S S, where S is a set, is chaotic if(i) f is transitive on S: for any pair of nonempty open sets U and V in S,

    there is an integer k > 0 such that fk(U) V is nonempty;

  • 1.2 Chaos and Chaos Control 7

    (ii) f has sensitive dependence on initial conditions : there is a real number > 0, depending only on f and S, such that in any nonempty opensubset of S there is a pair of points, whose eventual iterates under f areseparated by a distance of at least ;

    (iii) the periodic points of f are dense in S.

    Another denition requires the set S to be compact, but drops condition(iii) from the above [22]. It is even believed that only the transitive propertyis essential in this denition. Although a precise and rigorous mathematicaldenition of chaos does not seem to be available any time soon, some funda-mental features of chaos are well accepted, which can be used to signify oridentify chaos in most cases. A fundamental property of chaos is its extremesensitivity to initial conditions. Other features of chaos include the embed-ding of a dense set of unstable periodic orbits in its strange attractor, positiveleading Lyapunov exponent, nite Kolmogorov-Sinai entropy or positive topo-logical entropy, continuous power spectrum, positive algorithmic complexity,ergodicity, Arnolds cat map, Smale horseshoe map, a statistically orienteddenition of Shilnikov, and some other unusual limiting properties [21, 25].In other words, chaotic motion is an unstable bounded stationary motion (i.e.locally unstable but globally bounded). This denition unfolds the two aspectsof chaotic motion: instability and boundedness. Or, in other words, chaoticmotion is a bounded stationary motion without equilibrium, periodicity andalmost-periodicity [27].

    In fact, our real world is essentially nonlinear, thus chaos is ubiquitous.Besides the above mathematical descriptions of chaos, some scientists viewsmay help understand the meaning of chaos from various perspectives. For in-stance, Physicist Paul C. W. Davies said that reality without chaos is veryrare, yet scientists have developed their understanding of the world by study-ing only a small piece of orderly and predictable reality, but ignoring chaosor dismissing it as noise in an otherwise well-dened system, even thoughchaos is the rule much more than the exception. Biochemist Linda Jean Shep-herd believes that chaos science is shifting how we see the world, becauseit is changing the central metaphor by which science understands things. In-stead of the metaphor of the Newtonian clock with its cogwheels and levers,all being rational and predictable, chaos theory suggests metaphors that aremuch more indeterminate, unpredictable and random, e.g., turbulent rivers,weather, or smoke, and closer to how the world really works. Shepherd believesthat chaos theory describes a real universe and demonstrates that precise pre-diction is impossible in complex systems [23]. Physicist Stephen H. Kellertagrees and believes that chaos theory oers a way to understand this unpre-dictability and randomness, allowing us to see how randomness arises, what itlooks like, and how, at large scale and long term, randomness shows a complexnon-repetitive pattern. Ultimately, physicist M. Mitchell Waldrop sees chaostheory as widening our view of order in the world. No longer will scientistshave to focus only on the small portion of reality that is predictable; now they

  • 8 1 Introduction

    can also study the great wildness beyond simplistic traditional order. So tospeak, chaos theory reverts the primary face of the real world.

    Mathematician James Gleick describes three essential features of chaos.First, chaos is characterized by sensitive dependence on initial conditions, orwhat has become known as the buttery eect the ap of a butteryswings in a Brazilian rainforest may produce a tornado in Texas next week.In chaotic systems, like weather, turbulence, or smoke, small disturbancescan have dramatic eects due to the underlying webs of interconnectednessthat can amplify small changes. Second, chaotic systems are aperiodic or neverundergo a regular repetition of values. They never settle into a precise pattern,because nothing repeats in any systematic way. They are no-repeat systems.And third, chaotic systems are focused on strange attractors attractorsthat not only attract but also repel. An attractor is the central organizer of asystem, but the attractor in a chaotic system is aected by the buttery eectand the no-repeat characteristic; so, the attraction is paradoxically a strangeattraction toward buttery unpredictability and no-repeat randomness thatcreate a complex, non-repetitive pattern which can only be described in longterm and in large scale.

    Chaos is considered together with relativity and quantum mechanics as oneof the three monumental discoveries of the twentieth century. For physicistsand philosophers, relativity and quantum mechanics may rank above chaosfor their impacts on the way we view the world. As for science in general, it isnot clear that these theories have had any distinct eect on medicine, biology,and geology. Yet, chaotic dynamics are having an important impact on all ofthe elds, particularly, chemistry, physics, economics and sociology [26].

    1.2.2 Chaos Control

    Understanding chaos has long been the main focus of research in the eldof nonlinear dynamics. Although chaos is a very attractive subject for study,due to its intrinsic topological complexity it was once believed to be neitherpredictable nor controllable. However, recent research eorts have shown thatnot only (short term) prediction but also control of chaos are possible. It isnow well known that most conventional control methods and many specialtechniques can be used for chaos control [25, 28].

    For many years, the feature of the extreme sensitivity to initial conditionsmade chaos undesirable, and most experimentalists considered such charac-teristic as something to be denitely avoided. Besides this feature, chaotic sys-tems have two other important properties. Firstly, there is an innite numberof unstable periodic orbits embedded in the underlying chaotic attractor. Inother words, the skeleton of a chaotic attractor is a collection of an innitenumber of unstable periodic orbits. Secondly, the dynamics in the chaoticattractor are ergodic, which implies that during its temporal evolution thesystem ergodically visits a small neighborhood of every point in each one ofthe unstable periodic orbits embedded within the chaotic attractor [29].

  • 1.2 Chaos and Chaos Control 9

    In their attempt of controlling chaos, physicists and mathematiciansbrought about some fresh ideas and novel techniques that utilize the very na-ture of chaos for controls. For instance, the so-called OGY control method [30]employs the classical feedback control idea, which might be understood as akind of pole-placement method and is technically quite simple, but virtuallyit takes advantage of chaos itself in the sense of using its structural stabilityand its basic property of having a dense set of periodic orbits near a saddle.Such special properties are not available for non-chaotic systems; therefore,such a control methodology was not rst recommended by control theoristsand engineers who usually if not always try to use brute force to regulateand stabilize unstable dynamics. When brute force type of controls is not al-lowed, e.g., in fragile and microscopic biological control systems such as humanbrain and heart regulations, new control methods utilizing the extreme sensi-tivity of chaos to tiny variations are very desirable. This usually leads to somenon-conventional approaches. Today, there are some non-traditional controlideas and methods developed in the eld of chaos control, including systemparameter tuning, bifurcation monitoring, entropy reduction, state pinning,phase delay, weak oscillation input, disorder input, and some special-purposefeedback and adaptive controls, to list just a few [31].

    In contrast, recent research has shown that chaos can actually be usefulunder certain circumstances, and there is growing interest in utilizing the verynature of chaos. Today, the traditional trend of understanding and analyzingchaos has evolved to ordering and utilizing chaos. A new research direction inthe eld of applied chaos technology not only includes controlling chaos, whichmeans to weaken or completely suppress chaos when it is harmful, but alsoincludes chaotication, called anti-control of chaos, which refers to enhancingexisting chaos or purposely creating chaos when it is useful and benecial.

    For example, uid mixing is not only useful but actually important in thesituation that two uids are to be thoroughly mixed by consuming minimalenergy. The uid mixing problem is a curiosity in your coee cup, a minorproblem when baking in the kitchen, but a $20-billion problem for the U.S.chemical industry: Why does it take so long for stirred liquids to mix? Partof the answer is that regular stirring causes some streams to return to theirstarting points. The path that the stream travels may be highly complex,but eventually it meets up with itself, forming regular islands essentiallyunmixed pockets of liquid woven intricately through the mixture. Mixing isone of the nicest applications of exploiting chaos, Julio M. Ottino thereforesaid, in many cases, you would like to get rid of chaos, but in the case ofmixing, we would often like to enhance it, and we would certainly like tounderstand it [24]

    Scientists have always been trying to unravel the mechanism of how ourbrains endow us with inference, thoughts, perception, reasoning and, mostfascinating of all, emotion such as happiness and sadness. The fundamentalreason for the ability of human brains to process massive information instantlymay lie in the controlled chaos of the brains. Other potential applications

  • 10 1 Introduction

    of chaos control in biological systems have reached out from the brain toelsewhere, such as the human heart. In contrast to the common belief thathealthy heartbeats are completely regular, a normal heart rate may uctuatesin an erratic fashion, even at rest, and may actually be somewhat chaotic [47].

    Chaos control and anti-control methodologies promise to have a majorimpact on many novel, time- and energy-critical applications, such as high-performance circuits and devices (e.g., delta-sigma modulators and powerconverters), liquid mixing, chemical reactions, biological systems (e.g., in thehuman brain, heart, and perceptual processes), crisis management (e.g., in jet-engines and power networks), secure information processing (e.g., chaos-basedencryption), and decision-making in critical events. This new and challengingresearch area has become a scientic inter-discipline, involving engineers incontrols, systems, electronics, and mechanics, as well as applied mathemati-cians, theoretical and experimental physicists, physiologists, and above all,nonlinear dynamics specialists [25].

    1.3 Interactions between Fuzzy Logic and Chaos Theory

    Although the relationship between fuzzy logic and chaos theory is not yetcompletely understood at the moment, the study on their interactions hasbeen carried out for more than two decades, at least with respect to thefollowing aspects: fuzzy control of chaos [32, 33], adaptive fuzzy systems fromchaotic time series [34], theoretical relations between fuzzy logic and chaostheory [35, 36], fuzzy modeling of chaotic systems with assigned properties [37,38], chaotifying Takagi-Sugeno (TS) fuzzy models [39, 40, 41, 42], and fuzzy-chaos-based cryptography [43].

    At about the same time, fuzzy logic and chaos theory entered into the kenof science. Fuzzy logic was originally introduced by Lot Zadeh in 1965 inhis seminal paper fuzzy sets [3], while the rst evidence of physical chaoswas Edward Lorenzs discovery in 1963 [44], although the study of chaos canbe traced back to hundreds of years ago to some philosophical pondering [15]and to the work of the French mathematician Jules Henri Poincare at thebeginning of the last century [45]. Is it only a coincidence?

    Roughly speaking, fuzzy set theory resembles human reasoning using ap-proximate information and inaccurate data to generate decisions under un-certain environments. It is designed to mathematically represent uncertaintyand vagueness, and to provide formalized tools for dealing with imprecision inreal-world problems. On the other hand, chaos theory is a qualitative study ofunstable aperiodic behavior in deterministic nonlinear dynamical systems. Re-search reveals that it is due to the drastically evolving and changing chaoticdynamics that the human brain can process massive information instantly.The controlled chaos of the brain is more than an accidental by-product ofthe brain complexity, including its myriad connections, but rather, it may bethe chief property that makes the brain dierent from an articial-intelligence

  • 1.4 About This Book 11

    machine [46]. Therefore, it is believed that both fuzzy logic and chaos theoryare related to human reasoning and information processing. It is also believedthat to understand the complex information processing within the humanbrain, fuzzy data and fuzzy logical inference are essential, since precise math-ematical descriptions of such models and processes are clearly out of questionwith todays scientic knowledge.

    Based on the above-mentioned observations, the study on the interactionsbetween fuzzy logic and chaos theory allows a better understanding of their re-lation, and may provide a new and promising, although challenging, approachfor theoretical research and simulational investigation of human intelligence.

    1.4 About This Book

    This book is organized as follows: Chapters 2 4 introduce the fundamentalconcepts of fuzzy logic and fuzzy control, chaos theory and chaos control, aswell as the denition of chaos on the metric space of fuzzy sets, respectively.Then, fuzzy modeling and (adaptive) fuzzy control of chaotic systems, allbased on both Mamdani fuzzy models and Takagi-Sugeno (TS) fuzzy mod-els, will be given in Chapters 5 9. A very important topic, synchronizationof fuzzy systems, will be discussed in Chapter 10. In Chapter 11, system-atic anti-control approaches will be studied. Chapter 12 discusses intelligentdigital redesign for fuzzy systems. More complicated spatiotemporally chaoticphenomena and synchronization in complex fuzzy systems will be investigatedin Chapter 13. Finally, an application of fuzzy-chaos-based control method incryptography will be illustrated in Chapter 14.

    With this book, we try to bridge the apparent gap between fuzzy logic andchaos theory. It is expected that the studies within the book can be useful fora wide range of challenging applications and that the ideas derived from thestudies would lay a foundation on which further research in this promisingarea can be based.

    This book can serve as a reference book for researchers working in theinterdisciplinary areas related to, among others, both fuzzy logic and chaostheory.

  • 2Fuzzy Logic and Fuzzy Control

    Over the past few decades, there has developed a tremendous amount of lit-erature on the theory of fuzzy set and fuzzy control. This chapter attemptsto sketch the contours of fuzzy logic and fuzzy control for the readers, whomay have no knowledge in this eld, with easy-to-understand words, avoidingabstruse and tedious mathematical formulae.

    2.1 Introduction

    Fuzzy logic is in nature an extension of conventional (Boolean) logic to handlethe concept of partial truth truth values between completely true andcompletely false. It was introduced by Lot Zadeh in 1965 as a means tomodel the uncertainty of natural language.

    Fuzzy logic in the broad sense, which has been better known and exten-sively applied, serves mainly as a means for fuzzy control, analysis of vague-ness in natural language and several other application domains. It is one of thetechniques of soft-computing, i.e. computational methods tolerant to subopti-mality and impreciseness (vagueness) and giving quick, simple and sucientlygood solutions [48, 49, 50, 51].

    Fuzzy logic in the narrow sense is a symbolic logic with a comparative no-tion of truth developed fully in the spirit of classical logic (syntax, semantics,axiomatization, truth-preserving deduction, completeness, and propositionaland predicate logic). It is a branch of many-valued logic based on the paradigmof inference under vagueness. This fuzzy logic is a relatively young discipline,not only serving as a foundation for the fuzzy logic in a broad sense but alsoof independent logical interest, since it turns out that strictly logical investi-gation of this kind of logical calculi can go rather far [52, 53, 54, 55].

    Fuzzy logic has several unique features that make it a particularly goodchoice for many control problems:

    i) It is inherently robust, since it does not require precise, noise-free inputsand can, thus, be fault-tolerant if a feedback sensor quits or is destroyed.

    Zhong Li: Fuzzy Chaotic Systems, StudFuzz 199, 1329 (2006)www.springerlink.com c Springer-Verlag Berlin Heidelberg 2006

  • 14 2 Fuzzy Logic and Fuzzy Control

    The output control is a smooth control function despite a wide range ofinput variations.

    ii) Since a fuzzy logic controller processes user-dened rules governing thetarget control system, it can be modied and tweaked easily to improveor drastically alter system performance. New sensors can easily be in-corporated into the system simply by generating appropriate governingrules.

    iii) Fuzzy logic is not limited to a few feedback inputs and one or two controloutputs, nor is it necessary to measure or compute rate-of-change of para-meters. It is sucient with any sensor data to provide some indication ofa systems actions and reactions. This allows the sensors to be inexpensiveand imprecise thus keeping overall system cost and complexity low.

    iv) Because of the rule-based operation, any reasonable number of inputscan be processed (18 or more) and numerous outputs (14 or more)generated, although dening a rule-base quickly becomes complex if toomany inputs and outputs are chosen for a single implementation, sincerules dening their interrelations must be dened, too. Then, it would bebetter to break the control system into smaller chunks and use severalsmaller fuzzy logic controllers distributed on the system, each one withmore limited responsibilities.

    v) Fuzzy logic can control nonlinear systems that would be dicult or im-possible to model mathematically. This opens doors for control systemsthat would normally be deemed unfeasible for automation.

    In summery, fuzzy logic was conceived as a better method for sorting andhandling data, and has proven to be an excellent choice for many controlsystem applications, since it mimics human control logic. It can be built intoanything from small, hand-held products to large computerized process controlsystems. It uses an imprecise but very descriptive language to deal with inputdata more like a human operator. It is very robust and forgiving of operatorand data input, and often works when rst implemented with little or notuning.

    2.2 Fuzzy Set Theory

    2.2.1 Crisp Sets and Fuzzy Sets

    Normal sets in classic set theory are called crisp sets as compared with thefuzzy sets in fuzzy set theory. Let C be a crisp set dened on the universe ofdiscourse U , then for any element x of U , either x C or x C. For a crispset C its characteristic function C : U {0, 1} is dened as [56, 57]:Denition 2.1. C : U {0, 1} is a characteristic function of the set C ifor all x (i stands for if and only if):

  • 2.2 Fuzzy Set Theory 15

    C(x) ={

    1, when x C,0, when x C. (2.1)

    In fuzzy set theory this property is extended. Thus, in a fuzzy set F , itis not necessary that either x F or x F . The characteristic function isextended to a so-called membership function, which assigns to each x U avalue from the unit interval [0, 1] instead of from the two-element set {0, 1} inclassic set theory. The set dened with such an extended membership functionis called a fuzzy set.

    Denition 2.2. The membership function F of a fuzzy set F is a function

    F : U [0, 1]. (2.2)

    Therefore, each element x in U has a membership degree F (x) [0, 1]. Fis completely determined by the set of tuples

    F = {(x, F (x))|x U}. (2.3)

    It should be noticed that a fuzzy set is actually a generalized subset of aclassical set, and a universe of discourse is never fuzzy [58].

    Example 2.3. Membership functions of three fuzzy sets, namely,slow,medium and fast, for the speed of a car are shown in Fig. 2.1. Here, theuniverse of discourse is all speeds of the car, i.e., U = [0, Vmax], where Vmax isthe maximum possible speed of the car. At the speed of 45 km/h, for instance,the fuzzy set slow has membership degree 0.5, namely, slow(45) = 0.5, thefuzzy set medium has membership degree 0.5, namely, medium(45) = 0.5,and the fuzzy set fast has membership degree 0, namely, fast(45) = 0 [59].

    35 55 750

    speed(mph)

    F (speed)

    slow medium fast1

    0.5

    Fig. 2.1. Membership functions of three fuzzy sets, namely, slow, medium andfast, for the speed of a car

  • 16 2 Fuzzy Logic and Fuzzy Control

    A fuzzy set can be expressed as:

    1. Discrete case (when the universe of discourse is nite): Let the universeof discourse U be U = {u1, u2, . . . , un}. Then, a fuzzy set F on U can berepresented as follows:

    F = F (u1)/u1 + F (u2)/u2 + + F (un)/un =n

    i=1

    F (ui)/ui. (2.4)

    2. Continuous case (when the universe of discourse is innite): When theuniverse of discourse U is an innite set, a fuzzy set F on U can berepresented as follows:

    F =u

    F (ui)/ui. (2.5)

    Membership functions of fuzzy sets may have various forms. The most im-portant one is the triangular shape, like the fuzzy set medium in Fig. 2.1.Other common forms are the trapezoidal as shown in Fig. 2.2, the Gaussianmembership functions in Fig. 2.3, as well as the irregularly shaped and arbi-trary membership functions.

    1

    x

    A

    0

    Fig. 2.2. Trapezoidal type membership function

    Each kind of membership function has its own merits and disadvantages.For instance, the triangular membership function is extensively used in prac-tice for their simplicity, because it is dened with a minimal amount of infor-mation (data). It can easily be modied with their parameters, and its linearsegments ease the derivation of a models input/output map. On the otherhand, it lacks continuous dierentiability. The Gaussian function is expressedas

    F (x) = exp

    [

    (x ba

    )2]. (2.6)

  • 2.2 Fuzzy Set Theory 17

    A

    1

    x0

    Fig. 2.3. Gaussian membership function

    It is determined by the two parameters, a and b, where b is the modal orcentral value, and a determines its width. The Gaussian function is of anygrade of smooth and continuous dierentiability, which eases the theoreticalanalysis of fuzzy systems.

    The actual shape of a fuzzy set depends completely on the semantics ofthe concept intended to be represented. In other words, there are no universalor pre-dened fuzzy sets [57]. A fuzzy set makes no sense without the contextof a system or model, which means that certain shapes are representative ofparticular classes of knowledge.

    The shape of a fuzzy set is almost always a continuous line from left tothe right edge of the set. The contours of a fuzzy set represent the semanticproperties of the underlying concept, so the closer the set surface maps to thebehavior of a physical or conceptual phenomenon, the better our model willreect the real world. When we turn to building fuzzy models, fuzzy systemsare tolerant of approximations not only in their problem spaces but also inthe representation of fuzzy sets. This means that they perform well even whena fuzzy set does not map exactly with its model concept.

    2.2.2 Fundamental Operations of Fuzzy Sets

    Union of fuzzy sets A and B, AB, is a fuzzy set dened by the membershipfunction:

    AB(x) = A(x) B(x), (2.7)where

    A(x) B(x) ={

    A(x), A(x) B(x)B(x), A(x) < B(x)

    =max{A(x), B(x)}.

  • 18 2 Fuzzy Logic and Fuzzy Control

    Intersection of fuzzy sets A and B, A B, is a fuzzy set dened by themembership function:

    AB(x) = A(x) B(x), (2.8)where

    A(x) B(x) ={

    A(x), A(x) B(x)B(x), A(x) > B(x)

    =min{A(x), B(x)}.Complement of fuzzy set A, A, is a fuzzy set dened by the membership

    function:A(x) = 1 A(x). (2.9)

    It should be noted that union, intersection, and complement of crisp setsare special cases of union, intersection, and complement of fuzzy sets, respec-tively.

    The above denition of the classical operators was introduced by LotZadeh, which is only one possible choice of operations for union, intersection,and complement, as it is by no means unique. More general terms, t-normand t-conorm, are dened as follows:

    A t-norm is a function t : [0, 1] [0, 1] [0, 1] that satises the followingfour conditions:

    i) Boundary condition: t(0, 0) = 0, t(x, 1) = t(1, x) = x;ii) Commutativity: t(x, y) = t(y, x);iii) Monotonicity: t(x, y) t(, ), if x and y ;iv) Associativity: t(t(x, y), z) = t(x, t(y, z)).

    Besides the classical min operator proposed by Lot Zadeh, other t-normstend to compensate for the strict upper bound of the minimum, such as those:

    i) Algebraic product: ta : (x, y) xy;ii) Bounded product: tb : (x, y) max(0, x + y 1);

    iii) Drastic product: td : (x, y)

    x, if y = 1y, if x = 10, otherwise

    .

    A t-conorm (or s-norm) is a function s : [0, 1][0, 1] [0, 1] that satisesthe following four conditions:

    i) Boundary condition: s(1, 1) = 1, s(x, 0) = s(0, x) = x;ii) Commutativity: s(x, y) = s(y, x);iii) Monotonicity: s(x, y) s(, ), if x and y ;iv) Associativity: s(s(x, y), z) = s(x, s(y, z)).

    Besides the above given classical max operator, other t-conorms include:

    i) Algebraic sum: sa : (x, y) x + y xy;

  • 2.2 Fuzzy Set Theory 19

    ii) Bounded sum: sb : (x, y) min(1, x + y);

    iii) Drastic sum: sd : (x, y)

    x, if y = 1y, if x = 10, otherwise

    .

    2.2.3 Properties of Fuzzy Sets

    Let A, B, and C be fuzzy sets on the universe of discourse U [60].

    i) Properties valid for both fuzzy and crisp sets.Idempotency law:

    A A = A,A A = A; (2.10)Commutativity law:

    A B = B A,A B = B A; (2.11)

    Associativity law:

    A (B C) = (A B) C,A (B C) = (A B) C;

    Distributivity law:

    A (B C) = (A B) (A C), A (B C) = (A B) (A C);

    The law of double negation:A = A; (2.12)

    De Morgans law:

    A B = A B, A B = A B.

    ii) Properties valid for crisp sets, but in general not for fuzzy sets.The law of excluded middle:

    A A = U ; (2.13)

    The law of contradiction:A A = , (2.14)

    where means an empty set.

  • 20 2 Fuzzy Logic and Fuzzy Control

    2.2.4 Some Other Fundamental Concepts of Fuzzy Sets

    Here, we introduce some concepts of fuzzy sets, which may not be applicablein classical set theory [61].

    i) A fuzzy set F of U is called normal if there exists at least one elementx U such that F (x) = 1. A fuzzy set that is not normal is thus calledsubnormal.Note that all crisp sets except for the null set are normal.

    ii) The height of a fuzzy set F is the largest membership degree of any ele-ment in F . It is denoted by height(F ), hence, height(F ) = maxx{F (x)}.

    iii) The support of a fuzzy set F , denoted by Supp(F ), is the crisp set of allpoints x U such that F (x) > 0.

    iv) The center of a fuzzy set F is the point(s) x U at which F (x) achievesits maximum value.

    v) If the support of a fuzzy set F is a single point in U at which F = 1,the F is called a fuzzy singleton.

    vi) Assume A and B are two fuzzy sets of U . A is said to be a subset of (orcontained in) B, denoted by A B, if B(x) A(x) for each x U .

    vii) Assume A and B are two fuzzy sets of U . A and B are said to be equal,denoted by A = B, if A B and B A, or A(x) = B(x) for eachx U .

    viii) -cuts of the fuzzy set F on the universe of discourse U is dened as:

    strong -cut: F = {x|F (x) > }, [0, 1),weak -cut: F = {x|F (x) }, (0, 1].

    Weak -cuts are also called -level sets.

    2.2.5 Extension Principle

    Under many circumstances we can only characterize and deal with numericinformation imprecisely. For instance, we use such terms as, about 3, near zero,and more or less than 15. These are examples of the so-called fuzzy numbers.In fuzzy set theory, we represent fuzzy numbers as fuzzy sets of real numbers.Fuzzy numbers are used to perform arithmetic operation in intelligent systems,where the extension principle plays a fundamental role in extending any pointoperations to operations involving fuzzy sets.

    Denition 2.4 (Extension Principle). Let U and V be two universes ofdiscourse, and f be a map from U to V . For a fuzzy set A on U , the extensionprinciple denes a fuzzy set B on V by

    B(v) ={

    supv=f(u) A(u), if f1(v) = ,0 if f1(v) = .

    (2.15)

    When f is a one-to-one map, it is recast as:

    B(v) = A(u). (2.16)

  • 2.3 Fuzzy Relations and Their Compositions 21

    2.3 Fuzzy Relations and Their Compositions

    Fuzzy relations can be explained as extensions of relations in classical settheory, and the compositions of fuzzy relations allow us to perform fuzzyreasoning, which plays a key role in clustering, pattern recognition, inference,and control. They are also applied in the so-called soft sciences, such aspsychology, medicine, economics and sociology.

    2.3.1 Fuzzy Relations

    Ambiguous relationships in daily conversations like x and y are almost equaland x is much more beautiful than y are dicult to be expressed in termsof ordinary relations. Fuzzy relations are what makes it possible to expressthose frequently used ambiguous relationships.

    Denition 2.5 (Fuzzy relation). Let U and V be two universes of dis-course. A fuzzy relation R is a fuzzy set in the product space U V , which ischaracterized by a membership function R:

    R : U V [0, 1]. (2.17)Especially when U = V , R is known as a fuzzy relation on U .

    As a generalization of fuzzy relations, the n-ary fuzzy relation R in U1 U2 . . . Un is dened by

    R =U1U2...Un

    R(x1, x2, . . . , xn)/(x1, x2, . . . , xn), xi Ui, (2.18)

    where R : U1 U2 . . . Un [0, 1].Note that when n = 1, R is an unary fuzzy relation, and a fuzzy set in U1;

    when n = 2, it is the fuzzy relation dened in (2.17).The converse fuzzy relation of fuzzy relation R, R1, is dened as

    R1(y, x) = R(x, y). (2.19)

    The following are the most basic fuzzy relations: For any x, y UIdentity relation

    I I(x, y) ={

    1; x = y0; x = y . (2.20)

    Zero relationO O(x, y) = 0. (2.21)

    Universe relationE E(x, y) = 1. (2.22)

    Example 2.6. The following are examples of these three relations.

    I =

    1 0 00 1 00 0 1

    , O =

    0 0 00 0 00 0 0

    , E =

    1 1 11 1 11 1 1

    .

  • 22 2 Fuzzy Logic and Fuzzy Control

    2.3.2 Operations of Fuzzy Relations

    Since fuzzy relations are expressed by fuzzy sets on a Cartesian product space,we can apply the same operations on fuzzy relations as on fuzzy sets.

    Let R and S be fuzzy relations on U V . The operations are dened asfollows:

    Union of fuzzy relations: R S

    RS(x, y) = R(x, y) S(x, y). (2.23)

    Intersection of fuzzy relations: R S

    RS(x, y) = R(x, y) S(x, y). (2.24)

    Complement of fuzzy relations:

    R = 1 R(x, y). (2.25)

    Inclusion of fuzzy relations: R S

    R S R(x, y) S(x, y),x U,y V. (2.26)

    2.3.3 Composition of Fuzzy Relations

    Let R and S be fuzzy relations on U V and V W , respectively. The so-called Sup-Star composition of R and S, R S, is a fuzzy relation on U Wdened as:

    R S RS(x, z) = supyV

    {R(x, y) S(y, z)}, (2.27)

    where x U and z W , and can be any t-norm operator. It is possible thatS is just a fuzzy set in V ; in this case, the S(y, z) in (2.27) becomes S(y),then RS(y, z) becomes RS(y), and the others remain the same.

    2.4 Fuzzy Reasoning

    Fuzzy reasoning is sometimes called fuzzy inference or approximate reasoning.It is used in a fuzzy rule to determine the rule outcome from the given ruleinput information. Fuzzy rules represent control strategy or modeling knowl-edge/experience. When specic information is assigned to input variables inthe rule antecedent, fuzzy inference is needed to calculate the outcome foroutput variables in the rule consequence.

  • 2.4 Fuzzy Reasoning 23

    2.4.1 Generalized Modus Ponens and Modus Tollens

    In classical logic, reasoning is based on modus ponens and modus tollens,which are complementary. In modus ponens, when the statement If A, thenB is true, we infer If A is true, then B is true. This reasoning can bewritten as

    premise 1: A Bpremise 2: Aconsequence: B

    where A and B are crisp sets, and means implication.It can also be recast in an IF-THEN form as

    premise 1: If x is A Then y is Bpremise 2: x is Aconsequence: y is B.

    On the other hand, in modus tollens, when the statement If A, then B istrue, we infer If B is not true, then A is not true. The reasoning process isdescribed as

    premise 1: A Bpremise 2: not Bconsequence: not A.

    Fuzzy reasoning is, however, based on the generalized modus ponens (GMP),which can be written in IF-THEN form as

    premise 1: If x is A Then y is Bpremise 2: x is A

    consequence: y is B

    where A, A, B, and B are fuzzy sets.It is noted that other than modus ponens based on crisp sets, in the GMP

    the fuzzy sets A in premise 1 and A in premise 2 can be dierent. On theother hand, even though in the GMP the fuzzy sets A in premise 1 and A

    in premise 2 do not have to be the precisely the same, we can still infer theconclusion y is B from the premise y is B by their similarity. Thus, fuzzyreason is sometime called approximately reasoning.

    Accordingly, the generalized modus tollens is described as the followinginference procedure:

    premise 1: If x is A Then y is Bpremise 2: y is B

    consequence: x is A

    which, when B = not B and A = not A, reduces to modus tollens.

  • 24 2 Fuzzy Logic and Fuzzy Control

    2.4.2 Fuzzy Implications

    Denition 2.7 (Fuzzy implications). Let A and B be fuzzy sets on U andV , respectively. A fuzzy implication, denoted by A B, is a special kind offuzzy relation on U V satisfying the same conditions known as t-norm, suchas:

    Mamdanis methodAB(x, y) = A(x) B(y)

    Algebraic productAB(x, y) = A(x) B(y)

    Bounded product

    AB(x, y) = 0 (A(x) + B(y) 1)

    Drastic product

    AB(x, y) =

    A(x), if B(y) = 1B(y), if A(x) = 10, otherwise

    Zadehs method (Lukasiewiczs implication)

    AB(x, y) = 1 (1 A(x) + B(y))

    Boolean logic implication

    AB(x, y) = (1 A(x)) B(y)

    Godel logic implication

    AB(x, y) ={

    1, A(x) B(y)B(y), A(x) > B(y)

    Goguens implication

    AB(x, y) ={

    1, A(x) = B(y)B(y)/A(x), A(x) > B(y)

    2.4.3 Fuzzy Rule Base

    Fuzzy reasoning is performed with inference rules, which are expressed in IF-THEN format, called fuzzy IF-THEN rules. A fuzzy rule base is composed ofa collection of fuzzy IF-THEN rules. The most popular fuzzy rules are Mam-dani fuzzy rules and Takagi-Sugeno (TS) fuzzy rules, which are our concernthroughout this book.

  • 2.4 Fuzzy Reasoning 25

    Mamdani Fuzzy Rules

    A general Mamdani fuzzy rule, for either fuzzy control or fuzzy modeling, canbe expressed as,

    RlM : IF x1 is Fl1 and and xn is F ln, THEN y is Gl, (2.28)

    where F li and Gl are fuzzy sets, x = (x1, . . . , xn)T U and y V are input

    and output linguistic variables, respectively, and l = 1, 2, . . . , q. This kind offuzzy IF-THEN rules provides a convenient framework to incorporate humanexperts knowledge.

    Example 2.8. A Mamdani fuzzy rule used to describe the air conditioner is:

    IF room temperature is a little high AND humidity is quite highTHEN air conditioner setting is high,

    where room temperature and humidity are input variables and air condi-tioner setting is an output variable; a little high, quite high, and highare fuzzy sets.

    Takagi-Sugeno (TS) Fuzzy Rules

    Instead of using fuzzy sets in the consequence part of (2.28), TS fuzzy rulesadopt linear functions, which represent input-output relations.

    A TS fuzzy rule is described as,

    RlT : IF x1 is Fl1 and and xn is F ln,

    THEN yl = cl0 + cl1x1 + + clnxn, (2.29)

    where F li are fuzzy sets, ci are real-valued parameters, yl is the system output,

    and l = 1, 2, . . . , q. That is, in the TS fuzzy rules, the IF part (premise) is fuzzybut the THEN part (consequence) is crisp the output is a linear combinationof input variables.

    Example 2.9. The preceding air conditioner rule can be written as:

    IF room temperature x is about 20C AND humidity y is about 80%THEN air conditioner setting z = 0.2x + 0.05y.

    The equation means that the room temperature has four times weight as thatof humidity. In general, it is dicult to determine the linear equations forthe consequence part empirically. Therefore, it is assumed that the rules areobtained with modeling techniques by using input-output data.

  • 26 2 Fuzzy Logic and Fuzzy Control

    2.4.4 Fuzzy Inference Engine

    In a fuzzy inference engine, the fuzzy IF-THEN rules in the fuzzy rule baseare converted to a map from a fuzzy set in U = U1 Un to fuzzy setsin V . Mamdani fuzzy rules and TS fuzzy rules use dierent fuzzy inferenceapproaches, which will be introduced in the sequel, respectively.

    Mamdani Fuzzy Inference Method

    The reasoning method uses fuzzy relations and their composition. A fuzzyIF-THEN rule (2.28) is interpreted as a fuzzy implication F l1 F ln Glon U V . Let a fuzzy set A on U be the input to a fuzzy inference engine;then each fuzzy IF-THEN rule (2.28) determines a fuzzy set B on V usingthe sup-star composition (2.27), which writes

    B(y) = supxU

    {F l1F lnGl(x, y) A(x)}. (2.30)

    Denote F l1 F ln = A and Gl = B, the rule (2.28) is therefore denotedby A B. Thus, the fuzzy implications given in Denition 2.7 can be applied.

    Takagi-Sugeno (TS) Inference Method

    The Mamdani fuzzy model is a general framework in which linguistic informa-tion from human experts is quantied and dealt with. Its main disadvantageis that its inputs and outputs are fuzzy sets, whereas in many engineering sys-tems the inputs and outputs of a system are real-valued variables. To overcomethis disadvantage, the TS fuzzy model proposes a solution with real-valuedinputs and outputs.

    For TS fuzzy model (2.29), fuzzy reasoning is carried out by the weightedmean,

    y(x) =

    Ml=1

    lyl

    Ml=1

    l

    , (2.31)

    where the weight l is the overall truth value of the premise of rule RlT forthe input, and is given by

    l =n

    i=1

    F li (xi), (2.32)

    where F li (xi) is the membership degree of the fuzzy set Fli .

    The advantage of the TS inference method lies in its compact systemequation (2.32). Therefore, parameters can be estimated and order can be

  • 2.4 Fuzzy Reasoning 27

    determined with systematic approaches. On the other hand, its disadvantageis also the no-fuzzy THEN part, which does not provide a natural frameworkto incorporate fuzzy rules from human experts.

    In order to deal with real-valued inputs and outputs for fuzzy systems, it isstraightforward to add a fuzzier to the input and a defuzzier to the output.The fuzzier maps crisp points in U to fuzzy sets in U , and the defuzziermaps fuzzy sets in V to crisp points in V .

    2.4.5 Fuzzier

    A fuzzier performs a map from a crisp point x = (x1, . . . , xn)T into a fuzzyset A in U [59]. There are (at least) two possible choices of this map:

    Singleton fuzzier : A is a fuzzy singleton with support x, i.e., A(x) = 1 forx = x and A(x) = 0 for all other x U with x = x.

    Nonsingleton fuzzier : A(x) = 1 and A(x) decreases from 1 as x movesaway from x, for example, A(x) = exp

    [ (xx)T (xx)2

    ], where 2 is a

    parameter characterizing the shape of A(x).

    It seems that only the singleton fuzzier has been applied in practice. However,the nonsingleton fuzzier can be very useful in cases where the input data arecorrupted by noise.

    2.4.6 Defuzzier

    A defuzzier performs a map from fuzzy sets in V to a crisp point in V . Here,we adopt the most popular centroid defuzzier (also called center-averagedefuzzier), which maps the fuzzy set A R in V to a crisp point,

    y =

    Ml=1

    ylAR(yl)

    Ml=1

    AR(yl)

    , (2.33)

    where yl is a point in V at which Gl(y) achieves its maximum value, and Blis given by (2.30).

    When A = Ax is a fuzzy singleton, i.e., Ax(x) = 1 and Ax(x) = 0 for

    x = x, (2.33) can be expressed as,

    y = y(x) =

    Ml=1

    ylAl(x)

    Ml=1

    Al(x)

    . (2.34)

  • 28 2 Fuzzy Logic and Fuzzy Control

    2.5 Fuzzy Control

    Based on the descriptions above, a fuzzy control system can thus be con-structed as shown in Fig. 2.4. It is composed of the following four elements:

    A fuzzy rule base: It contains a fuzzy logic quantication of the expertslinguistic description of how to achieve good control.

    A fuzzy inference engine: It emulates the experts decision-making in inter-preting and applying knowledge about how to control the plant.

    A fuzzier: It converts controller inputs into information that the inferencemechanism can easily use to activate and apply rules.

    A defuzzier: It converts the conclusions of the inference mechanism intoactual outputs for the process.

    Fuzzy Inference Engine

    Fuzzier Defuzzier

    Fuzzy Rule Base

    x in U y in V

    Fuzzy setsin U

    Fuzzy setsin V

    Fig. 2.4. Block diagram of fuzzy logic control systems

    In summary, a fuzzy controller incorporates knowledge of human expertsin a form of logical inference rules, enabling it to act in a human-like fashion.

    2.6 Fuzzy Systems as Universal Approximators

    Fuzzy logic systems have very strong functional capabilities, which means thatif properly constructed, they can perform very complex operations (e.g., muchmore complex than those performed by a linear map) [59, 62]. Actually, it isknown that fuzzy logic systems possess the universal approximation property.

    Suppose that a fuzzy logic system f(x) adopts center-average defuzzi-cation, product for the premise and implication, and Gaussian membershipfunctions. Then, the universal approximation theorem as given below holds.

  • 2.6 Fuzzy Systems as Universal Approximators 29

    Theorem 2.10 (The universal approximation theorem [59]). For anygiven real continuous function (x) dened on a compact set U n and anarbitrary > 0, there exists a fuzzy system f(x) in the form of (2.33) suchthat

    supxU

    |f(x) (x)| < . (2.35)

    It is remarked that this theorem just guarantees that there exists a wayto dene a fuzzy logic system, which can uniformly approximate any givenfunction to arbitrary accuracy. It does not, however, say how to nd such afuzzy logic system, which is indeed very dicult. Moreover, the more accurateit is, the larger is the number of rules the fuzzy logic system needs. It is alsoremarked that the theorem is only a justication for using the fuzzy logicsystems. But what we need and are concerned about is the capability of fuzzylogic systems to incorporate linguistic information in a natural and systematicway, constituting their unique advantage.

  • 3Chaos and Chaos Control

    For the abstruse and vast nonlinear dynamical and control systems it is dif-cult, if not impossible, to cover all the concepts within one chapter. In thischapter, through exploring the simplest logistic map, we sketch some basicbut important concepts and some related essential ones in the theory of non-linear dynamical and control systems, as well as review some now popularmethodologies of chaos control.

    3.1 Logistic Map

    We begin with the logistic map to introduce some basic concepts as it possessesbifurcations, stable and unstable periodic orbits, periodic windows, ergodicand mixing behaviors, homoclinic connections, chaotic orbits and some kindsof universality.

    The logistic map is denoted as a dierence equation in the form of

    xn+1 = f(, xn) = xn(1 xn), (3.1)which was discovered by Robert May and Mitchell Feigenbaum in 1976 asa population model. It represents the population as a fraction, xn, of themaximum population that is supported by a habitat. represents the pop-ulation growth ratio of the species. Thus, xn [0, 1] and 0 < 4. It iseasy to see that, if > 4, the map (3.1) does not map the interval [0, 1] intoitself [26, 66, 67].

    Solving the xed-point equation

    x = f(, x), (3.2)

    we can derive the maps xed point(s). Now, let us investigate the stabilityof the xed points. When 0 < < 1, x1 = 0 is the unique xed point(x2 = ( 1)/ is out of [0, 1]). Because of |f (, x1)| = |f (, 0)| = < 1,the xed point x1 = 0 is stable and is an attracting point, thus, all points

    Zhong Li: Fuzzy Chaotic Systems, StudFuzz 199, 3152 (2006)www.springerlink.com c Springer-Verlag Berlin Heidelberg 2006

  • 32 3 Chaos and Chaos Control

    map into 0 under the iterations of f no matter where the point starts, seeFig. 3.1 (a). Further, when 1, there are two xed points x1 = 0 andx2 =

    ( 1)

    . Due to |f (, 0)| = 1, the xed point x1 becomes unstablejust as the second xed point x2 is born. The second xed point x2 is stablefor 1 < < 3, because |f (, x2)| = |f (, ( 1)/)| = |1 | < 1, then allpoints converge to x2, see Fig. 3.1 (b). Thereafter, what happens when bothxed points become unstable for > 3?

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    xn

    x n+

    1

    =1

    (a)

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    xn

    x n+

    1

    =2.78

    (b)

    Fig. 3.1. The logistic map with various parameter values of

    It is clear that the stability of the xed points changes as the parameter varies. To see this, we plot the attracting set of f as a function of witha value = 2.8 where the xed point x2 is still an attracting (or stable)xed point. This numerical simulation is shown in Fig. 3.2, which can displaysome characteristic properties of the asymptotic solution of the logistic map,allowing one to see at a glance, where qualitative changes in the asymptoticsolution occur. Such changes are termed as bifurcation.

    We can see that initially the attracting set consists of a single point thatbifurcates into two at = 3.0. Subsequently, these points bifurcate again intofour points, which bifurcate into eight, and so on. The interval in betweenbifurcations decreases until eventually what looks like a chaotic set appears.The chaotic region appears interspersed with bands, which consist of only asmall number of points termed periodic windows.

    Let us now look closely at the mechanism behind these phenomena. Asnoted above, the rst bifurcation occurs at = 1, with which the xed pointx1 = 0 becomes unstable and the xed point x2 = ( 1)/ is born andbecomes stable. The second bifurcation takes place at = 3, where the xedpoint x2 becomes unstable and an attracting 2-cycle is born, as shown inFig. 3.3. Similar to deriving the xed points of the logistic map, the 2-cyclepoints x1 and x

    2 can be obtained by solving the equations f(, f(, x)) = x.

    This is a fourth order equation:

  • 3.1 Logistic Map 33

    2.8 3 3.2 3.4 3.6 3.8 40

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    x n

    Fig. 3.2. Bifurcation diagram of the logistic map for 2.8 < < 4

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    xn

    x n+

    1

    =3.1

    Fig. 3.3. Periodic-2 orbits of the logistic map at = 3.1

  • 34 3 Chaos and Chaos Control

    3x4 + 23x3 (2 + 3)x2 + (2 1)x = 0. (3.3)

    Already knowing the two xed points of the iteration, (3.3) can be rstreduced to a third-order equation with x1 = 0,

    3x3 + 23x2 (2 + 3)x + (2 1) = 0, (3.4)

    and further dividing (3.4) by (x ( 1)/) yields:

    3x2 + (2 + 3)x (2 + ) = 0. (3.5)

    The solutions of (3.5) are

    x1,2 =1 +

    2 2 32

    , (3.6)

    which form a 2-cycle for the logistic map f with each a xed point of f2 = ff .Let the points x1, x2, . . . , xp denote the points of a p-cycle of the logistic

    map, where f(xi) = xi+1 and f(xp) = x1. Each of the points xi is a xedpoint of fp. By the chain rule of dierentiation, one has

    dfp

    dx

    xi

    = f (x1)f (x2) f (xp) =p

    j=1