Li Fuzzy Chaotic Systems

Modeling, Control, and Applications
1. Auflage 2006
ISBN: 978-3-540-33221-3
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 199, 323 Seiten

Reihe: Studies in Fuzziness and Soft Computing

"Transition between Fuzzy and Chaotic Systems" provides original heuristic research achievements and insightful ideas on the interactions or intrinsic relationships between fuzzy logic and chaos theory. It presents the fundamental concepts of fuzzy logic and fuzzy control, chaos theory and chaos control, as well as thedefinition of chaos on the metric space of fuzzy sets. This monograph discusses and illustrates fuzzy modeling and fuzzy control of chaotic systems, synchronization, anti-control of chaos, intelligent digital redesign, spatiotemporal chaos and synchronization in complex fuzzy systems; as well as a practical application example of fuzzy-chaos-based cryptography. Like other very good books, this book may raise more questions than it can provide answers. It therefore generates a great potential to attract more attention to combine fuzzy systems with chaos theory and contains important seeds for future scientific research and engineering applications.

Written for: Researchers, engineers, graduate students in Soft computing, Fuzziness and Complexity/Nonlinear Systems

Keywords:
Chaos
Fuzzy Systems

Autoren/Hrsg.

Li, Zhong

Inhaltsverzeichnis

1;1 Introduction;13
1.1;1.1 Fuzzy Logic and Fuzzy Control Systems;13
1.1.1;1.1.1 Fuzzy Logic;13
1.1.2;1.1.2 Fuzzy Control Systems;16
1.2;1.2 Chaos and Chaos Control;17
1.2.1;1.2.1 Chaos;17
1.2.2;1.2.2 Chaos Control;20
1.3;1.3 Interactions between Fuzzy Logic and Chaos Theory;22
1.4;1.4 About This Book;23
2;2 Fuzzy Logic and Fuzzy Control;24
2.1;2.1 Introduction;24
2.2;2.2 Fuzzy Set Theory;25
2.2.1;2.2.1 Crisp Sets and Fuzzy Sets;25
2.2.2;2.2.2 Fundamental Operations of Fuzzy Sets;28
2.2.3;2.2.3 Properties of Fuzzy;30
2.2.4;2.2.4 Some Other Fundamental Concepts of Fuzzy Sets;31
2.2.5;2.2.5 Extension Principle;31
2.3;2.3 Fuzzy Relations and Their Compositions;32
2.3.1;2.3.1 Fuzzy Relations;32
2.3.2;2.3.2 Operations of Fuzzy Relations;33
2.3.3;2.3.3 Composition of Fuzzy Relations;33
2.4;2.4 Fuzzy Reasoning;33
2.4.1;2.4.1 Generalized Modus Ponens and Modus Tollens;34
2.4.2;2.4.2 Fuzzy Implications;35
2.4.3;2.4.3 Fuzzy Rule Base;35
2.4.4;2.4.4 Fuzzy Inference Engine;37
2.4.5;2.4.5 Fuzzifier;38
2.4.6;2.4.6 Defuzzifier;38
2.5;2.5 Fuzzy Control;39
2.6;2.6 Fuzzy Systems as Universal Approximators;39
2.6.1;Theorem 2.10 (The universal approximation theorem [59]).;40
3;3 Chaos and Chaos Control;41
3.1;3.1 Logistic Map;41
3.2;3.2 Bifurcations;45
3.3;3.3 Hopf Bifurcation of Higher-dimensional Systems;47
3.4;3.4 Lyapunov Exponents;48
3.5;3.5 Routes to Chaos;49
3.6;3.6 Center Manifold Theory;50
3.7;3.7 Normal Forms;53
3.8;3.8 Control of Chaos;53
3.8.1;3.8.1 Ott-Grebogi-Yorke Method;54
3.8.2;3.8.2 Feedback Control;58
3.8.3;3.8.3 Pyragas Time-delayed Feedback Control;61
3.8.4;3.8.4 Entrainment and Migration Control;62
4;4 Definition of Chaos in Metric Spaces of Fuzzy Sets;63
4.1;4.1 Introduction;63
4.2;4.2 Chaos of Di.erence Equations in with a Saddle Point;67
4.2.1;4.2.1 Su.cient Conditions for Chaos in;67
4.2.2;4.2.2 Some Examples;71
4.3;4.3 Chaotic Maps in Banach Spaces;73
4.4;4.4 Chaos of Discrete Systems in Complete Metric Spaces;74
4.5;4.5 Chaos of Difference Equations in Metric Spaces of Fuzzy Sets;78
4.5.1;4.5.1 Chaotic Maps on Fuzzy Sets;78
4.5.2;Theorem 4.18. (Kaleva);79
4.5.3;Theorem 4.19. (Kloeden);79
4.5.4;4.5.2 Example of a Chaotic Map on Fuzzy Sets;80
5;5 Fuzzy Modeling of Chaotic Systems – I (Mamdani Model);83
5.1;5.1 Introduction;83
5.2;5.2 Double-scroll Chaotic System;85
5.3;5.3 Single-scroll Chaotic System: Logistic Map;88
5.4;5.4 Lyapunov Exponents;91
5.5;5.5 Two-dimensional Map;93
6;6 Fuzzy Modeling of Chaotic Systems – II (TS Model);100
6.1;6.1 Introduction;100
6.2;6.2 TS Fuzzy Models;101
6.3;6.3 Preliminary Theorem;105
6.4;6.4 TS Fuzzy Modeling of Chaotic Systems: Examples;106
6.4.1;6.4.1 Continuous-time Chaotic Lorenz System;106
6.4.2;6.4.2 Continuous-time Flexible-joint Robot Arm;108
6.4.3;6.4.3 Dufing-like Chaotic Oscillator;111
6.4.4;6.4.4 Other Examples;112
6.5;6.5 Discretization of Continuous-time TS Fuzzy Models;114
6.6;6.6 Bifurcation Phenomena in TS Fuzzy Systems;115
6.6.1;6.6.1 Bifurcation in TS Fuzzy Systems;116
6.6.2;6.6.2 TS Fuzzy Systems with Linear Consequents;120
6.6.3;x;120
6.6.4;x,;120
6.6.5;x;120
6.6.6;x,;120
6.6.7;x;121
6.6.8;x,;121
6.6.9;x;121
6.6.10;x,;121
6.7;6.7 Appendix: Bifurcation Analysis for;124
6.8;= 1;124
6.8.1;Lemma 6.5.;127
6.8.2;Lemma 6.6.;127
7;7 Fuzzy Control of Chaotic Systems – I (Mamdani Model);129
7.1;7.1 Introduction;129
7.2;7.2 Design of Fuzzy Logic Controllers;130
7.2.1;7.2.1 Selection of Variables and the Universe of Discourse;130
7.2.2;7.2.2 Fuzzi.fication;131
7.2.3;7.2.3 Discretization and Normalization of a Universe of Discourse;131
7.2.4;7.2.4 Construction of the Rule-base;132
7.2.5;7.2.5 Defuzzification;136
7.2.6;7.2.6 Fuzzy Inference Mechanisms;137
7.3;7.3 Fuzzy Control of Chua’s Chaotic Circuit;137
7.4;7.4 Fuzzy Control of Chaotic Lorenz System;143
8;8 Adaptive Fuzzy Control of Chaotic Systems (Mamdani Model);150
8.1;8.1 Introduction;150
8.2;8.2 Stable Directly Adaptive Fuzzy Control of Chaotic Systems;151
8.2.1;8.2.1 Design of the Supervisory Controller;153
8.2.2;8.2.2 Design of the Controller;153
8.3;8.3 Design of Directly Adaptive Fuzzy Controllers;154
8.4;8.4 Adaptive Fuzzy Control of the Du.ng Oscillator;155
9;9 Fuzzy Control of Chaotic Systems – II (TS Model);159
9.1;9.1 Introduction;159
9.2;9.2 Preliminaries;160
9.3;9.3 Parallel-Distributed Compensation;161
9.4;9.4 Lyapunov Stability of TS Fuzzy Systems;162
9.5;9.5 Stability Analysis and Controller Design Based on LMIs;163
9.5.1;9.5.1 Continuous-time Case;163
9.5.2;9.5.2 Discrete-time Case;168
9.6;9.6 Simulations;174
9.6.1;9.6.1 Continuous-time Case;174
9.6.2;9.6.2 Discrete-time Case;181
9.7;9.7 TS Fuzzy-model-based Adaptive Control;187
10;10 Synchronization of TS Fuzzy Systems;194
10.1;10.1 Introduction;194
10.2;10.2 Exact Linearization;195
10.3;10.3 Synchronization;199
10.3.1;10.3.1 Case 1;199
10.3.2;10.3.2 Case 2;200
10.4;10.4 Synchronization of Chen’s Chaotic Systems;201
10.5;10.5 Synchronization of Hyperchaotic Systems;204
11;11 Chaotifying TS Fuzzy Systems;209
11.1;11.1 Introduction;209
11.2;11.2 Chaotifying Discrete-time TS Fuzzy Systems;210
11.2.1;11.2.1 Discrete-time TS Fuzzy System via Mod-Operation;210
11.2.2;11.2.2 Anti-controller Design;212
11.2.3;11.2.3 Veri.cation of the Anti-control Design;213
11.2.4;11.2.4 A Simulation Example;213
11.3;11.3 Chaotifying Discrete-time TS Fuzzy Systems via a Sinusoidal Function;214
11.4;11.4 Chaotifying Continuous-time TS Fuzzy Systems via Discretization;222
11.5;11.5 Chaotifying Continuous-Time TS Fuzzy Systems via Time-delay Feedback;227
11.5.1;11.5.1 PDC Controller for Locally Controllable TS Fuzzy Systems;228
11.5.2;11.5.2 Controller Design for General TS Fuzzy Systems;230
11.5.3;11.5.3 Veri.cation of Chaos;232
11.5.4;11.5.4 Simulation Examples;234
11.5.4.1;A. Approximate Linearization Approach;238
11.5.4.2;B. Globally Exact Linearization Approach;238
12;12 Intelligent Digital Redesign for TS Fuzzy Systems;243
12.1;12.1 Introduction;243
12.2;12.2 Digital Fuzzy Systems and Their Discretization;245
12.3;12.3 Global State-matching Intelligent Digital Redesign;247
12.4;12.4 Digital Redesign for Du.ng-like Chaotic Oscillator;250
12.5;12.5 Appendix;252
13;13 Spatiotemporal Chaos and Synchronization in Complex Fuzzy Systems;258
13.1;13.1 Introduction;258
13.2;13.2 Complex Fuzzy Systems;259
13.2.1;13.2.1 Single-unit: Chaotic Fuzzy Oscillator;259
13.2.2;13.2.2 Macro-system: Chain of Fuzzy Oscillators;260
13.3;13.3 Synchronization Index;264
13.4;13.4 Complex Networks: Preliminaries;270
13.5;13.5 Collective Behavior versus Network Topology;273
14;14 Fuzzy-chaos-based Cryptography;277
14.1;14.1 Introduction;277
14.2;14.2 Working Principle of the Cryptosystem;279
14.3;14.3 Decryption by Fuzzy-model-based Synchronization;281
15;References;286

Leseproben

2 Fuzzy Logic and Fuzzy Control (Sp. 13-14)

Over the past few decades, there has developed a tremendous amount of literature on the theory of fuzzy set and fuzzy control. This chapter attempts to sketch the contours of fuzzy logic and fuzzy control for the readers, who may have no knowledge in this ?eld, with easy-to-understand words, avoiding abstruse and tedious mathematical formulae.

2.1 Introduction

Fuzzy logic is in nature an extension of conventional (Boolean) logic to handle the concept of partial truth – truth values between “completely true” and “completely false”. It was introduced by Lot? Zadeh in 1965 as a means to model the uncertainty of natural language.

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

Fuzzy logic in the narrow sense is a symbolic logic with a comparative notion of truth developed fully in the spirit of classical logic (syntax, semantics, axiomatization, truth-preserving deduction, completeness, and propositional and predicate logic). It is a branch of many-valued logic based on the paradigm of 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 also of independent logical interest, since it turns out that strictly logical investigation 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 good choice for many control problems:

i) It is inherently robust, since it does not require precise, noise-free inputs and can, thus, be fault-tolerant if a feedback sensor quits or is destroyed. The output control is a smooth control function despite a wide range of input variations.

ii) Since a fuzzy logic controller processes user-de?ned rules governing the target control system, it can be modi?ed and tweaked easily to improve or drastically alter system performance. New sensors can easily be incorporated into the system simply by generating appropriate governing rules.

iii) Fuzzy logic is not limited to a few feedback inputs and one or two control outputs, nor is it necessary to measure or compute rate-of-change of parameters. It is sufficient with any sensor data to provide some indication of a system’s actions and reactions. This allows the sensors to be inexpensive and imprecise thus keeping overall system cost and complexity low.

iv) Because of the rule-based operation, any reasonable number of inputs can be processed (1–8 or more) and numerous outputs (1–4 or more) generated, although de?ning a rule-base quickly becomes complex if too many inputs and outputs are chosen for a single implementation, since rules de?ning their interrelations must be de?ned, too. Then, it would be better to break the control system into smaller chunks and use several smaller fuzzy logic controllers distributed on the system, each one with more limited responsibilities.

v) Fuzzy logic can control nonlinear systems that would be difficult or impossible to model mathematically. This opens doors for control systems that would normally be deemed unfeasible for automation. In summery, fuzzy logic was conceived as a better method for sorting and handling data, and has proven to be an excellent choice for many control system applications, since it mimics human control logic. It can be built into anything from small, hand-held products to large computerized process control systems. It uses an imprecise but very descriptive language to deal with input data more like a human operator. It is very robust and forgiving of operator and data input, and often works when ?rst implemented with little or no tuning.