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About this book

Bringing together the two seemingly unrelated concepts,fuzzy logic andchaos theory,isprimarilymotivatedbytheconceptofsoft computing (SC),initiated by Lot? A. Zadeh, the founder of fuzzy set theory. The principal constituents of SC are fuzzy logic (FL), neural network theory (NN) and probabilistic reasoning (PR), with the latter subsuming parts of belief networks, genetic algorithms, chaos theory and learning theory. What is important to note is that SC is not a melange of FL, NN and PR. Rather, it is an integration in which each of the partners contributes a distinct methodology for addressing problems in their common domain. In this perspective, the principal cont- butions of FL, NN and PR are complementary rather than competitive. SC di?ers from conventional (hard) computing in that it is tolerant of imprecision, uncertainty and partial truth. In e?ect, the role model for soft computing is the human mind. From the general SC concept, we extract FL and chaos theory as the object of this book to study their relationships or interactions. Over the past few decades, fuzzy systems technology and chaos theory have received ever increasing research interests from, respectively, systems and control engineers, theoretical and experimental physicists, applied ma- ematicians, physiologists, and other communities of researchers. Especially, as one of the emerging information processing technologies, fuzzy systems technology has achieved widespread applications around the globe in many industriesandtechnical?elds,rangingfromcontrol,automation,andarti?cial intelligence (AI) to image/signal processing and pattern recognition. On the otherhand,inengineeringsystemschaostheoryhasevolvedfrombeingsimply a curious phenomenon to one with real, practical signi?cance and utilization.

Table of Contents

Frontmatter

1. Introduction

Abstract
At first glance, fuzzy logic and chaos theory may seem two totally different areas with merely marginal connections to each other. In this introduction, after reviewing the evolution of fuzzy set theory and chaos theory, respectively, we explain briefly the ideas why we bring them together, and we shall show that the understanding of the interactions between fuzzy systems and chaos theory lays a solid foundation for better applications of the two promising new technologies, and their integration offers a great number of interesting possibilities in their interplay and future developments.
Zhong Li

2. Fuzzy Logic and Fuzzy Control

Abstract
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 field, with easy-to-understand words, avoiding abstruse and tedious mathematical formulae.
Zhong Li

3. Chaos and Chaos Control

Abstract
For the abstruse and vast nonlinear dynamical and control systems it is difficult, if not impossible, to cover all the concepts within one chapter. In this chapter, through exploring the simplest logistic map, we sketch some basic but important concepts and some related essential ones in the theory of nonlinear dynamical and control systems, as well as review some now popular methodologies of chaos control.
Zhong Li

4. Definition of Chaos in Metric Spaces of Fuzzy Sets

Abstract
This chapter reviews the development of the definitions of chaos from the wellknown Li-Yorke definition of chaos for difference equations in 葶1 to those for difference equations in 葶n with either a snap-back repeller or saddle point as well as for maps in Banach spaces and complete metric spaces, among which Devaney’s definition will be used in this book in the proof that chaos exists in anti-controlled fuzzy systems. Finally, a definition of chaos for maps in a space of fuzzy sets, namely, the metric space (ξn,D) of fuzzy sets on the base space 葶n, is given, aiming to lay a theoretical foundation for further studies on the interactions between fuzzy logic and chaos theory. Some illustrative examples are presented.
Zhong Li

5. Fuzzy Modeling of Chaotic Systems – I (Mamdani Model)

Abstract
In this chapter we introduce an approach to model chaotic dynamics in a linguistic manner based on the Mamdani fuzzy model. This approach allows to design robust chaotic generators by means of few fuzzy sets and using a small number of fuzzy rules. The generated chaotic signals can be of assigned characteristics (e.g., Lyapunov exponents). As examples, fuzzy descriptions of well-known discrete chaotic maps, such as the logistic map, a double-scroll attractor and the 2-dimensional Hénon map, are given to illustrate the effectiveness of the proposed approach.
Zhong Li

6. Fuzzy Modeling of Chaotic Systems – II (TS Model)

Abstract
In this chapter, another typical fuzzy modeling approach, based on the TS fuzzy model, for chaotic dynamics is introduced. The TS fuzzy model has a rigorous mathematical expression and, thus, eases the stability analysis and controller design. A couple of examples will be given to show the procedure of how to construct TS fuzzy models of chaotic systems.
Zhong Li

7. Fuzzy Control of Chaotic Systems – I (Mamdani Model)

Abstract
In this chapter, to stabilize chaotic dynamics a design method for fuzzy controllers based on the Mamdani model is introduced, which is also called a model-free approach. As illustrative examples, the chaotic Lorenz system and Chua’s circuit will be controlled using this approach.
Zhong Li

8. Adaptive Fuzzy Control of Chaotic Systems (Mamdani Model)

Abstract
In this chapter, methodologies of adaptive fuzzy control for chaotic systems will be introduced, and some illustrative examples will be presented.
Zhong Li

9. Fuzzy Control of Chaotic Systems – II (TS Model)

Abstract
In this chapter, we introduce a TS-model-based fuzzy control method to stabilize chaotic dynamics with parametric uncertainties, also called model-based approach, by using the Linear Matrix Inequalities (LMI) techniques. In the end, this approach will be applied to control the chaotic Lorenz system and Chua’s chaotic circuit.
Zhong Li

10. Synchronization of TS Fuzzy Systems

Abstract
In this chapter, synchronization of TS fuzzy systems is discussed, where a fuzzy feedback law is adopted and realized via exact linearization (EL) techniques and by solving LMI problems. Two examples, synchronization of Chen’s systems and hyperchaotic systems, are given for illustration.
Zhong Li

11. Chaotifying TS Fuzzy Systems

Abstract
In this chapter, chaotifying both discrete-time and continuous-time Takagi-Sugeno (TS) fuzzy systems is introduced. To chaotify discrete-time TS fuzzy systems, the parallel distributed compensation (PDC) method is employed to determine the structure of a fuzzy controller so as to make all Lyapunov exponents of the controlled TS fuzzy system strictly positive. But for continuous-time ones, the chaotification approach is based on fuzzy feedback linearization and a suitable approximate relationship between a time-delay differential equation and a discrete map. The time-delay feedback controller, chosen among several candidates, is a simple sinusoidal function of the delay states of the system, which can have an arbitrarily small amplitude. These anti-control approaches are all proved to be mathematically rigorous in the sense of Li and Yorke. Some examples are given to illustrate the effectiveness of the proposed anti-control methods.
Zhong Li

12. Intelligent Digital Redesign for TS Fuzzy Systems

Abstract
This chapter introduces digital control of chaotic systems represented by TS fuzzy systems using intelligent digital redesign (IDR) techniques. The term, intelligent digital redesign, involves converting an existing analog TS fuzzymodel- based controller into an equivalent digital counterpart in the sense of state-matching. The IDR problem is viewed as a minimization problem of norm distances between nonlinearly interpolated linear operators to be matched. The main features of this method are that its constructive condition, with global rather than local state-matching for given chaotic systems, is formulated in terms of linear matrix inequalities (LMI). The stability property is preserved by the proposed IDR algorithm. A set-point regulation example of a chaotic system is demonstrated to visualize the feasibility of the developed methodology, which implies safe digital implementation of chaos control systems.
Zhong Li

13. Spatiotemporal Chaos and Synchronization in Complex Fuzzy Systems

Abstract
In this chapter, spatiotemporal chaos in arrays of coupled fuzzy-logic-based chaotic oscillators is observed, and the effects of network topology on synchronization through de.ning a synchronization index are investigated in the framework of complex networks.
Zhong Li

14. Fuzzy-chaos-based Cryptography

Abstract
As an application example of integrating fuzzy logic and chaos theory, a fuzzymodel-based chaotic cryptosystem is introduced in this chapter.
Zhong Li

Backmatter

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