Fuzzy Adaptive Controller for Synchronization of Uncertain Fractional-Order Chaotic Systems

Introduction

Fractional calculus is a mathematical topic with more than three centuries old history and can be considered as a natural generalization of ordinary differentiation and integration to arbitrary (non-integer) order. During the last 3 decades, it has attracted increasing attentions of physicists as well as engineers from an application point of view. In fact, it has been found that many systems in interdisciplinary fields can be accurately modeled by fractional-order differential equations, such as heat diffusion systems, batteries, neurons, viscoelastic systems (Bagley and Calico, 1991), dielectric polarization (Sun et al., 1984), electrode-electrolyte polarization (Ichise et al., 1971), some finance systems, electromagnetic waves (Heaviside, 1971), and so on.

Chaos (or hyperchaos) is a nonlinear phenomenon which has been observed in physical, mechanical, medical, biological, economical and electrical systems to name a few. Chaotic systems are nonlinear and deterministic rather than probabilistic (Yin-He et al., 2012; Ginarsa et al., 2013). They have some important characteristics such that: 1) they have an unusual sensitivity to initial states (therefore they are not predictable in the long run). 2)They are not periodic. 3) They have fractal structures, 4) Such systems are governed by one or more control parameters. The principal feature used to identify a chaotic behaviour is the well-known Lyapunov exponent criteria. In fact, a system that has one positive Lyapunov exponent is known as a chaotic system. However, a hyperchaotic circuit, which is usually a 4-dimensional system, is characterised by more than one positive Lyapunov exponent. It is worth mentioning that higher dimensional chaotic systems with more than one positive Lyapunov exponent can indeed show more complex dynamics. It has been recently demonstrated that many fractional-order systems can display chaotic (or heperchaotic) behaviours such that: fractional-order Duffing system (Gao and Yu, 2005), fractional-order Chua’s system (Hartley et al., 1995), fractional-order Lorenz system (Yu et al., 2009), fractional-order Chen system (Li and Peng, 2004), fractional-order Rössler system (Li and Chen, 2004), fractional-order Liu system (Daftardar-Gejji and Bhalekar, 2010), fractional-order Arneodo system (Lu, 2005) to name a few.

The synchronization is defined as a problem that consists in designing a system (slave or response system) whose behaviour mimics another one (drive or master system). In the literature, several synchronization techniques have been already developed, namely: complete synchronization (Carroll et al., 1996; Sun and Zhang, 2004; Bowonga et al., 2006), phase synchronization (Rosenblum et al., 1996; Pikovsky et al., 1997), lag synchronization (Cailian et al., 2005), generalized synchronization (Morgul and Solak, 1996; Morgul and Solak, 1997), generalized projective synchronization (Li and Xu, 2004; Yan and Li, 2005; Li, 2006), and so on. However, all these synchronization methods focus on integer-order chaotic systems that represent a very special case of the non-integer (fractional-order) chaotic systems. In addition, it has been assumed in (Carroll et al., 1996; Sun and Zhang., 2004; Bowonga et al., 2006; Rosenblum et al., 1996; Pikovsky et al., 1997; Cailian et al., 2005; Morgul and Solak, 1996; Morgul and Solak, 1997; Li and Xu, 2004; Yan and Li, 2005; Li, 2006) that models of the chaotic systems are almost known. Therefore, it is very interesting to extend these fundamental results to uncertain fractional-order chaotic systems and to incorporate an online function approximator (such as a fuzzy system) to deal with model uncertainties.