Integration of Fuzzy Logic and Chaos Theory

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.