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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.

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.

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.

This chapter reviews the development of the definitions of chaos from the wellknown Li-Yorke definition of chaos for difference equations in 葶 to those for difference equations in 葶 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 (ξ,) of fuzzy sets on the base space 葶, 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.

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.

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.

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.

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

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.

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.