Li / Chen / (Eds.) Integration of Fuzzy Logic and Chaos Theory

1. Auflage 2006
ISBN: 978-3-540-32502-4
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 641 Seiten

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

Autoren/Hrsg.

Li, Zhong

Chen, Guanrong

(Eds.), China

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;Beyond the Li–Yorke Definition of Chaos;10
3.1;1 Introduction;10
3.2;2 Background;14
3.3;3 Chaos of Difference Equations in with a Saddle Point;16
3.4;4 Chaotic Mappings in Banach Spaces;23
3.5;5 Chaos of Discrete Systems in Complete Metric Spaces;24
3.6;6 Chaos of Difference Equations in Metric Spaces of Fuzzy Sets;28
3.7;7 Conclusions;31
3.8;References;32
4;Chaotic Dynamics with Fuzzy Systems;34
4.1;1 Introduction;34
4.2;2 A Brief Review of Chaos;34
4.3;3 Fuzzy Modeling of Chaotic Behaviors;35
4.4;4 Two-Dimensional Maps;46
4.5;5 Conclusions;52
4.6;References;52
5;Fuzzy Modeling and Control of Chaotic Systems;54
5.1;1 Introduction;54
5.2;2 Fuzzy Modeling of Chaotic Systems;55
5.3;3 Stabilization;59
5.4;4 Synchronization;68
5.5;5 Chaotic Model Following Control;77
5.6;6 Concluding Remarks;88
5.7;References;89
6;Fuzzy Model Identification Using a Hybrid mGA Scheme with Application to Chaotic System Modeling;90
6.1;1 Introduction;90
6.2;2 Takagi–Sugeno Fuzzy Systems;91
6.3;3 Fuzzy Model Identification by Using mGA Hybrid Scheme;92
6.4;4 An Example: The Chaotic Mackey–Glass Time Series;99
6.5;5 Conclusions;105
6.6;References;106
7;Fuzzy Control of Chaos;108
7.1;1 Control of Chaos;108
7.2;2 Fuzzy Control of Chaos;111
7.3;3 Fuzzy Logic Controller;113
7.4;4 Fuzzy Chaos Control in Electronic Circuits: An Introductory Example;127
7.5;5 Conclusions;132
7.6;Acknowledgments;132
7.7;References;132
8;Chaos Control Using Fuzzy Controllers ( Mamdani Model);136
8.1;1 Introduction;136
8.2;2 Fuzzy Logic Control Preliminaries and Background;137
8.3;3 Mathematical Models;152
8.4;4 Numerical Simulations;152
8.5;5 Conclusion;161
8.6;References;161
9;Digital Fuzzy Set-Point Regulating Chaotic Systems: Intelligent Digital Redesign Approach;166
9.1;1 Introduction;166
9.2;2 Preliminaries;167
9.3;3 Intelligent Digital Redesign of Fuzzy Set- Point Regulator;175
9.4;4 Examples;179
9.5;5 Closing Remarks;187
9.6;Appendix;187
9.7;References;191
10;Anticontrol of Chaos for Takagi–Sugeno Fuzzy Systems;194
10.1;1 Introduction;194
10.2;2 Chaotifying Discrete-Time TS Fuzzy Systems;196
10.3;3 Anticontrol of Chaos via Sinusoidal Function;203
10.4;4 Anticontrol of Chaos for Continuous-Time TS Fuzzy Systems via Discretization;206
10.5;5 Anticontrol of Chaos for Continuous-Time TS Fuzzy Systems via Time- Delay Feedback;218
10.6;6 Concluding Remarks;234
10.7;References;234
11;Chaotification of the Fuzzy Hyperbolic Model;238
11.1;1 Introduction;238
11.2;2 Chaotification of the Fuzzy Hyperbolic Model by Impulsive Control Method;239
11.3;3 Chaotification of the Fuzzy Hyperbolic Model by Inverse Optimal Control Method;246
11.4;4 Chaotification of the Original System;255
11.5;5 Summary;265
11.6;References;265
12;Fuzzy Chaos Synchronization via Sampled Driving Signals;268
12.1;1 Introduction;268
12.2;2 Fuzzy Modeling of Dynamical Systems;270
12.3;3 Fuzzy Logic Controller Design;272
12.4;4 Fuzzy Logic Observer Design;276
12.5;5 Chaos Synchronzation Via Fuzzy Observer Design;278
12.6;6 Digitally Redesigned Takagi–Sugeno Fuzzy Observers;284
12.7;7 Concluding Remarks;290
12.8;References;291
13;Bifurcation Phenomena in Elementary Takagi – Sugeno Fuzzy Systems;294
13.1;1 Introduction;294
13.2;2 Fuzzy Systems and Bifurcation Theory;295
13.3;3 Examples;300
13.4;4 Summary;320
13.5;Appendix: Bifurcation Analysis of Example 1 for ß= 1;321
13.6;References;323
14;Self-Reference, Chaos, and Fuzzy Logic;326
14.1;1 Introduction;326
14.2;2 A Simple Fuzzy Logic;327
14.3;3 Self-Reference as Iteration: The Example of the Liar;332
14.4;4 Attractor and Repeller Fixed Points in the Phenomena of Self- Reference;337
14.5;5 Fuzzy Chaos;344
14.6;6 Fuzzy Self-Reference in Two Dimensions;349
14.7;7 Fuzzy Triplists Modeled in Three Dimensions;359
14.8;8 Conclusion;363
14.9;Acknowledgments;366
14.10;References;366
15;Chaotic Behavior in Recurrent Takagi–Sugeno Models;370
15.1;1 Introduction;370
15.2;2 On the Nature of Chaos;371
15.3;3 Modeling Chaos by Takagi–Sugeno Rule Bases with One- Time Delay Case;374
15.4;4 Modeling of Chaos by Takagi–Sugeno Rule Bases with High- Order Time Delay Case;392
15.5;5 Summary;398
15.6;References;398
16;Theory of Fuzzy Chaos for the Simulation and Control of Nonlinear Dynamical Systems;400
16.1;1 Basic Concepts of Dynamical Systems;400
16.2;2 Controlling Chaos;404
16.3;3 Towards a Theory of Fuzzy Chaos;419
16.4;4 Controlling Chaotic Behavior Using Fuzzy Chaos;420
16.5;5 Conclusions;422
16.6;References;422
17;Complex Fuzzy Systems and Their Collective Behavior;424
17.1;1 Introduction;424
17.2;2 Complex Fuzzy System;426
17.3;3 The Collective Dynamics Through the Syncronization Index;432
17.4;4 The Collective Behavior Versus the Network Topology;438
17.5;5 Conclusions;441
17.6;Appendix;443
17.7;Acknowledgments;445
17.8;References;445
18;Real-Time Identification and Forecasting of Chaotic Time Series Using Hybrid Systems of Computational Intelligence;448
18.1;1 Introduction;448
18.2;2 Identification of Chaotic Signals in Real Time Using the Hurst Exponent;452
18.3;3 Dynamic Reconstruction of Chaotic Signals with Known Structure;454
18.4;4 Dynamic Reconstruction and Forecasting of Chaotic Signals with Radial Basis Function Networks;458
18.5;5 Forecasting of Chaotic Sequences Using Neuro- Fuzzy Networks;468
18.6;6 Modeling and Forecasting of Chaotic Sequences Using Neo- Fuzzy Kolmogorov’s Networks;479
18.7;7 Conclusions;487
18.8;References;487
19;Fuzzy–Chaos Hybrid Controllers for Nonlinear Dynamic Systems;490
19.1;1 Introduction;490
19.2;2 Review of Chaos and Fuzzy Systems;492
19.3;3 Concept of the Controller;494
19.4;4 Fuzzy–Chaos Hybrid Controller;498
19.5;5 Stability of the Closed-Loop Controller;499
19.6;6 Design Example 1: Henon Map;501
19.7;7 Design Example 2: Lorenz Attractor;504
19.8;8 Design Example 3: Two-link Robot Arm;508
19.9;9 Conclusions;513
19.10;References;514
20;Fuzzy Model Based Chaotic Cryptosystems;516
20.1;1 Introduction;516
20.2;2 Chaotic Cryptosystem Structure;518
20.3;3 Takagi–Sugeno Fuzzy Modeling for Chaotic Systems;520
20.4;4 Chaotic Cryptosystem Using Discrete-time Systems;523
20.5;5 DSP-Based Experiments;530
20.6;6 Conclusions;533
20.7;Acknowledgment;533
20.8;References;534
21;Evolution of Complexity;536
21.1;1 Introduction;536
21.2;2 Self-Organization and Adaptation of Complex Systems;542
21.3;3 Basic Principle of Evolutionary Computation;544
21.4;4 Biologically Inspired Computing;552
21.5;5 Order vs. Complexity in the Question of Information;554
21.6;6 Overview of Evolutionary Algorithms;558
21.7;7 Parallel Grammatical Evolution with Sexual Selection;559
21.8;8 Origin of Complexity;564
21.9;9 Parallel Evolutionary Optimization of Controllers with a System’s Identi . cation;566
21.10;10 Parallel Grammatical Evolution with Backward Processing;572
21.11;11 Conclusions;585
21.12;References;586
22;Problem Solving via Fuzziness-Based Coding of Continuous Constraints Yielding Synergetic and Chaos- Dependent Origination Structures;588
22.1;1 Introduction;588
22.2;2 Arti.cial Systems and Natural Systems;589
22.3;3 Layered Problem Solving System Architecture Based on Fuzzy Coding of Continuous Constraints;590
22.4;4 Two Approaches in the Proposed System Architecture;594
22.5;5 Discussion and Conclusion;607
22.6;References;610
23;Some Applications of Fuzzy Dynamic Models with Chaotic Properties;612
23.1;1 Introduction;612
23.2;2 Reconstruction of Chaotic Orbits with Takagi – Sugeno Recurrent Rule Bases;613
23.3;3 Business–Cycles Modeling in Multiproject Systems Based on Recurrent Mamdani Models;620
23.4;4 Summary;633
23.5;References;633

Leseproben

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.