Integration of Fuzzy Logic and Chaos Theory (eBook)

Zhong Li, Wolfgang A. Halang, Guanrong Chen, City University Of Hong Kong, China (Eds.) (Herausgeber)

eBook Download: PDF

2006 | 1. Auflage
641 Seiten
Springer-Verlag
9783540325024 (ISBN)

Infos

This book attempts to present some current research progress and results on the interplay of fuzzy logic and chaos theory. More specifically, this book includes a collections of some state-of-the-art surveys, tutorials, and application examples written by some experts working in the interdisciplinary fields overlapping fuzzy logic and chaos theory.



The content of the book covers fuzzy definition of chaos, fuzzy modeling and control of chaotic systems using both Mamdani and Takagi-Sugeno models, fuzzy model identification using genetic algorithms and neural network schemes, bifurcation phenomena and self-referencing in fuzzy systems, complex fuzzy systems and their collective behaviours, as well as some applications of combining fuzzy logic and chaotic dynamics, such as fuzzy-chaos hybrid controllers for nonlinear dynamic systems, and fuzzy-model-based chaotic cryptosystems.



This book can serve as a handy reference for researchers working in the interdisciplines related, among others, to both fuzzy logic and chaos theory.



Written for:

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



Keywords:

Chaos Theory

Fuzzy Logic

Inhalt

Preface 6
Contents 8
Beyond the Li–Yorke Definition of Chaos 10
1 Introduction 10
2 Background 14
3 Chaos of Difference Equations in with a Saddle Point 16
4 Chaotic Mappings in Banach Spaces 23
5 Chaos of Discrete Systems in Complete Metric Spaces 24
6 Chaos of Difference Equations in Metric Spaces of Fuzzy Sets 28
7 Conclusions 31
References 32
Chaotic Dynamics with Fuzzy Systems 34
1 Introduction 34
2 A Brief Review of Chaos 34
3 Fuzzy Modeling of Chaotic Behaviors 35
4 Two-Dimensional Maps 46
5 Conclusions 52
References 52
Fuzzy Modeling and Control of Chaotic Systems 54
1 Introduction 54
2 Fuzzy Modeling of Chaotic Systems 55
3 Stabilization 59
4 Synchronization 68
5 Chaotic Model Following Control 77
6 Concluding Remarks 88
References 89
Fuzzy Model Identification Using a Hybrid mGA Scheme with Application to Chaotic System Modeling 90
1 Introduction 90
2 Takagi–Sugeno Fuzzy Systems 91
3 Fuzzy Model Identification by Using mGA Hybrid Scheme 92
4 An Example: The Chaotic Mackey–Glass Time Series 99
5 Conclusions 105
References 106
Fuzzy Control of Chaos 108
1 Control of Chaos 108
2 Fuzzy Control of Chaos 111
3 Fuzzy Logic Controller 113
4 Fuzzy Chaos Control in Electronic Circuits: An Introductory Example 127
5 Conclusions 132
Acknowledgments 132
References 132
Chaos Control Using Fuzzy Controllers ( Mamdani Model) 136
1 Introduction 136
2 Fuzzy Logic Control Preliminaries and Background 137
3 Mathematical Models 152
4 Numerical Simulations 152
5 Conclusion 161
References 161
Digital Fuzzy Set-Point Regulating Chaotic Systems: Intelligent Digital Redesign Approach 166
1 Introduction 166
2 Preliminaries 167
3 Intelligent Digital Redesign of Fuzzy Set- Point Regulator 175
4 Examples 179
5 Closing Remarks 187
Appendix 187
References 191
Anticontrol of Chaos for Takagi–Sugeno Fuzzy Systems 194
1 Introduction 194
2 Chaotifying Discrete-Time TS Fuzzy Systems 196
3 Anticontrol of Chaos via Sinusoidal Function 203
4 Anticontrol of Chaos for Continuous-Time TS Fuzzy Systems via Discretization 206
5 Anticontrol of Chaos for Continuous-Time TS Fuzzy Systems via Time- Delay Feedback 218
6 Concluding Remarks 234
References 234
Chaotification of the Fuzzy Hyperbolic Model 238
1 Introduction 238
2 Chaotification of the Fuzzy Hyperbolic Model by Impulsive Control Method 239
3 Chaotification of the Fuzzy Hyperbolic Model by Inverse Optimal Control Method 246
4 Chaotification of the Original System 255
5 Summary 265
References 265
Fuzzy Chaos Synchronization via Sampled Driving Signals 268
1 Introduction 268
2 Fuzzy Modeling of Dynamical Systems 270
3 Fuzzy Logic Controller Design 272
4 Fuzzy Logic Observer Design 276
5 Chaos Synchronzation Via Fuzzy Observer Design 278
6 Digitally Redesigned Takagi–Sugeno Fuzzy Observers 284
7 Concluding Remarks 290
References 291
Bifurcation Phenomena in Elementary Takagi – Sugeno Fuzzy Systems 294
1 Introduction 294
2 Fuzzy Systems and Bifurcation Theory 295
3 Examples 300
4 Summary 320
Appendix: Bifurcation Analysis of Example 1 for ß= 1 321
References 323
Self-Reference, Chaos, and Fuzzy Logic 326
1 Introduction 326
2 A Simple Fuzzy Logic 327
3 Self-Reference as Iteration: The Example of the Liar 332
4 Attractor and Repeller Fixed Points in the Phenomena of Self- Reference 337
5 Fuzzy Chaos 344
6 Fuzzy Self-Reference in Two Dimensions 349
7 Fuzzy Triplists Modeled in Three Dimensions 359
8 Conclusion 363
Acknowledgments 366
References 366
Chaotic Behavior in Recurrent Takagi–Sugeno Models 370
1 Introduction 370
2 On the Nature of Chaos 371
3 Modeling Chaos by Takagi–Sugeno Rule Bases with One- Time Delay Case 374
4 Modeling of Chaos by Takagi–Sugeno Rule Bases with High- Order Time Delay Case 392
5 Summary 398
References 398
Theory of Fuzzy Chaos for the Simulation and Control of Nonlinear Dynamical Systems 400
1 Basic Concepts of Dynamical Systems 400
2 Controlling Chaos 404
3 Towards a Theory of Fuzzy Chaos 419
4 Controlling Chaotic Behavior Using Fuzzy Chaos 420
5 Conclusions 422
References 422
Complex Fuzzy Systems and Their Collective Behavior 424
1 Introduction 424
2 Complex Fuzzy System 426
3 The Collective Dynamics Through the Syncronization Index 432
4 The Collective Behavior Versus the Network Topology 438
5 Conclusions 441
Appendix 443
Acknowledgments 445
References 445
Real-Time Identification and Forecasting of Chaotic Time Series Using Hybrid Systems of Computational Intelligence 448
1 Introduction 448
2 Identification of Chaotic Signals in Real Time Using the Hurst Exponent 452
3 Dynamic Reconstruction of Chaotic Signals with Known Structure 454
4 Dynamic Reconstruction and Forecasting of Chaotic Signals with Radial Basis Function Networks 458
5 Forecasting of Chaotic Sequences Using Neuro- Fuzzy Networks 468
6 Modeling and Forecasting of Chaotic Sequences Using Neo- Fuzzy Kolmogorov’s Networks 479
7 Conclusions 487
References 487
Fuzzy–Chaos Hybrid Controllers for Nonlinear Dynamic Systems 490
1 Introduction 490
2 Review of Chaos and Fuzzy Systems 492
3 Concept of the Controller 494
4 Fuzzy–Chaos Hybrid Controller 498
5 Stability of the Closed-Loop Controller 499
6 Design Example 1: Henon Map 501
7 Design Example 2: Lorenz Attractor 504
8 Design Example 3: Two-link Robot Arm 508
9 Conclusions 513
References 514
Fuzzy Model Based Chaotic Cryptosystems 516
1 Introduction 516
2 Chaotic Cryptosystem Structure 518
3 Takagi–Sugeno Fuzzy Modeling for Chaotic Systems 520
4 Chaotic Cryptosystem Using Discrete-time Systems 523
5 DSP-Based Experiments 530
6 Conclusions 533
Acknowledgment 533
References 534
Evolution of Complexity 536
1 Introduction 536
2 Self-Organization and Adaptation of Complex Systems 542
3 Basic Principle of Evolutionary Computation 544
4 Biologically Inspired Computing 552
5 Order vs. Complexity in the Question of Information 554
6 Overview of Evolutionary Algorithms 558
7 Parallel Grammatical Evolution with Sexual Selection 559
8 Origin of Complexity 564
9 Parallel Evolutionary Optimization of Controllers with a System’s Identi . cation 566
10 Parallel Grammatical Evolution with Backward Processing 572
11 Conclusions 585
References 586
Problem Solving via Fuzziness-Based Coding of Continuous Constraints Yielding Synergetic and Chaos- Dependent Origination Structures 588
1 Introduction 588
2 Arti.cial Systems and Natural Systems 589
3 Layered Problem Solving System Architecture Based on Fuzzy Coding of Continuous Constraints 590
4 Two Approaches in the Proposed System Architecture 594
5 Discussion and Conclusion 607
References 610
Some Applications of Fuzzy Dynamic Models with Chaotic Properties 612
1 Introduction 612
2 Reconstruction of Chaotic Orbits with Takagi – Sugeno Recurrent Rule Bases 613
3 Business–Cycles Modeling in Multiproject Systems Based on Recurrent Mamdani Models 620
4 Summary 633
References 633

Textauszug

Chaos Control Using Fuzzy Controllers (Mamdani Model) (p. 127)

Ahmad M. Harb and Issam Al-Smadi

Abstract.

Controlling a strange attractor, or say, a chaotic attractor, is introduced in this chapter. Because of the importance to control the undesirable behavior in systems, researchers are investigating the use of linear and nonlinear controllers either to get rid of such oscillations (in power systems) or to match two chaotic systems (in secure communications).

The idea of using the fuzzy logic concept for controlling chaotic behavior is presented. There are two good reasons for using the fuzzy control: .rst, mathematical model is not required for the process, and second, the nonlinear controller can be developed empirically, without complicated mathematics. The two systems are well-known models, so the first reason is not a big deal, but we can take advantage from the second reason.
1 Introduction

Modern nonlinear theories, such as bifurcation and chaos, have been widely used in many fields. Many researchers have used such theories to investigate and analyze the stability problem. Abed and Varaiya [1], Dobson et al. [2], and Harb et al. [3] used the bifurcation theory to analyze the stability of voltage collapse and SSR phenomena in electrical power systems. Endo and Chua [4] and Harb and Harb [5] analyzed the stability of phase-looked loop (PLL) in communication systems.

Nayfeh and Balachandran [6] and Harb et al. [7] analyzed the stability of Dufing oscillator in mechanical systems. Recently, research has been devoted toword the bifurcation and chaos control of such mentioned systems. The main goal of bifurcation and chaos control is stabilizing bifurcation branches, changing the type of bifurcation from subcritical to supercritical Hopf bifurcation, and delaying the bifurcations. Abed et al. [8–10] used state feedback nonlinear controllers to change the type of the Hopf bifurcation and to suppress the amplitude of the limit cycles at the vicinity of the Hopf bifurcation points.

Ikhouane and Krstic [11], Harb et al. [12, 13], and Zaher et al. [14–17] used recursive backstepping algorithms to design nonlinear controllers to stabilize systems of chaotic behavior. Fuzzy set theory has been used successfully in virtually all technical fields, including modeling, control, and signal/image processing. Fuzzy control is a rule-base system that is based on fuzzy logic. Since fuzzy is described as computing with words rather than numbers, then fuzzy control can be described as control with sentences rather than equations.

In 1974, Professor Mamdani was the first to develop the concept of the fuzzy controller. Driankov et al. [18] and Calvo and Cartwright [19] introduced the idea of fuzzy in chaos control. Tang et al. [20], Mann et al. [21], Hu et al. [22], and Gradjevac [23] used the PID fuzzy controller, while Hsu and Cheng [24] and Toliyat et al. [25] designed a fuzzy controller to enhance power system stability.

Produkt-Details

• Erscheint lt. Verlag: 1.1.2006
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Theorie / Studium
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik
• ISBN-13: 9783540325024 / 9783540325024