Adaptive Hybrid Intelligent Tracking Control for Uncertain Fractional Order Chaotic Systems

This paper presents an adaptive hybrid fuzzy controller to achieve prescribed tracking performance of fractional order chaotic systems. Depending on plant knowledge and control knowledge, a weighting factor can be adjusted by combining the indirect adaptive fuzzy control effort and the direct fuzzy adaptive control effort. Nonlinear fractional order chaotic response system is fully demonstrated to track the trajectory generated from fractional order chaotic drive system. The numerical results show that tracking error and control effort can be made smaller and the proposed hybrid intelligent control scheme is more flexible during the design process. DOI: 10.4018/ijsda.2012010101 2 International Journal of System Dynamics Applications, 1(1), 1-16, January-March 2012 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. systems and related phenomena is receiving growing attention (Kinai et al., 2009; Grigorenko, 2003). The tracking problem of fractional order chaotic systems is first investigated by Deng and Li (2005) who carried out tracking in case of the two fractional Lü systems. Afterwards, they studied chaos tracking of the Chen system with a fractional order in a different manner (Li & Peng, 2004; Li & Chen, 2003; Deng & Lin, 2005). Furthermore, the advent of fuzzy set techniques provides us with a powerful tool to solve demanding real word problem with uncertain and unpredictable environments. Based on the universal approximation theorem (Wang, 1993, 1994; Wang & Mendel, 1992; Hwang & Lin, 1992) (fuzzy logic controllers are general enough to perform any nonlinear control actions), there is rapidly growing interest in systematic design methodologies for a class of nonlinear systems using fuzzy adaptive control schemes. By equipping with a training algorithm an adaptive fuzzy controller is synthesized from a collection of fuzzy IF-THEN rules and the parameters of the membership functions characterizing the linguistic terms in the IF-THEN rules changed according to some adaptive law for the purpose of controlling a plant to track a reference trajectory. Like the conventional adaptive control, the adaptive fuzzy control is classified into direct and indirect fuzzy adaptive control categories (Wang et al., 2002a, 2002b; Lin et al., 2004). A direct adaptive fuzzy controller uses fuzzy logic systems as controller in which linguistic fuzzy control rules can be directly incorporated into the controller. On the other hand, an indirect adaptive fuzzy controller uses fuzzy descriptions to model the plant in which fuzzy IF-THEN rules describing the plant can be directly incorporated into the indirect fuzzy controller. Moreover, a hybrid adaptive fuzzy controller can be constructed using a weighting factor to sum together the control efforts from indirect adaptive fuzzy controller and direct adaptive fuzzy controller (Lin & Lee, 2011; Lin & Kuo, 2011). In this paper, by combining the approximate mathematical model, linguistic model description and linguistic control rules into a single adaptive fuzzy controller, an adaptive hybrid fuzzy) controller is proposed to achieve prescribed tracking performance of fractional order chaotic systems. A new adaptive H fuzzy control algorithm incorporated Lyapunov stability criterion is proposed so that not only the stability of adaptive fuzzy control system is guaranteed but also the influence of the approximation error and external disturbance on the tracking error can be attenuated to an arbitrarily prescribed level. This paper is organized as follows: In Section 2, an introduction to fractional derivative and its relation to the approximation solution will be addressed. Section 3 generally proposes adaptive hybrid fuzzy control of uncertain fractional order systems in presence of uncertainty and its stability analysis. In Section 4, application of the proposed method on fractional order expression chaotic system is investigated. Finally, the simulation results and conclusion will be presented in Section 5. 2. BASIC DEFINITION AND PRELIMINARIES FOR FRACTIONAL ORDER SYSTEMS The fractional calculus has been known more than 300 years, since the development of regular calculus. It is a generalization of integration and differentiation to non-integer order fundamental operator, denoted by a t q D , where a and t are the limits of the operator. This operator is a notation for taking both the fractional integral and functional derivative in a single expression defined as