Adaptive H∞ fuzzy control for a class of uncertain discrete-time nonlinear Systems

In this paper, an adaptive H interval type-2 fuzzy controller is proposed for a class of unknown nonlinear discrete-time systems with training data corrupted by noise or rule uncertainties involving external disturbances. Adaptive interval type-2 fuzzy control scheme and H control approach are incorporated to implement the main objective of controlling the plant to track a reference trajectory. The Laypunov stability theorem has been used to testify the asymptotic stability of the whole system and the free parameters of the adaptive fuzzy controller can be tuned on-line by an output feedback control law and adaptive laws. The overall adaptive scheme guarantees the global stability of the resulting closed-loop system in the sense that all signals involved are uniformly bounded. The simulation example is given to confirm validity and tracking performance of the advocated design methodology. DOI: 10.4018/jalr.2010100104 International Journal of Artificial Life Research, 1(4), 48-67, October-December 2010 49 Copyright © 2010, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. robust stabilization and disturbance rejection problems (Lin, Wang, et al., 2004; Wang, Liu, et al., 2002; Wang, Lin, et al., 2002). In the conventional H optimal control, the plant model must be known beforehand. In this study, the H optimal control design will be extended toward the nonlinear systems with unknown or uncertain models via fuzzy technique. However, several adaptive fuzzy sliding mode control systems have been developed for continuous-time systems (Chang, Park, et al., 2002; Lee, Kim, et al., 2001; Yu, Man, et al., 1998; Utkin, 1977), which cannot be expanded into discrete-time systems directly. Only a few of them are devoted to discrete-time systems (Shi, 2007; Sarpturk, Istefanopulos, et al., 1987; Han, Su, et al., 2000; Shaocheng & Tianyou, 1998; Fuchun, Zengqi, et al., 1998; Guo, Shi, et al., 2009; Spooner, Ordonez, et al., 1997a, 1997b). Nevertheless, because of the environment changes and the associated uncertainties, linguistic uncertainty and noisy training data, the chosen type-1 sets might not be appropriate anymore. A type-2 fuzzy set is characterized by a fuzzy membership function (Mendel & John, 2002; Mendel & Wu, 2007; Liang & Medel, 2000a, 2000b; Hagras, 2007; Medel, 2007). For each element of this set is a fuzzy set in [0, 1], unlike a type-1 fuzzy set where the membership grade is a crisp value in [0, 1]. A type-2 FLS is characterized by IF-THEN rules, but its antecedent or consequent sets are type-2 Hence, type-2 FLSs can be used when the circumstances are too uncertain to determine exact membership grades such as when training data are corrupted by noise. The interval type-2 FLS has been successfully applied to fuzzy neural network (Lin, Kuo, et al., 2010; Lin, Liu, et al., 2009; Lin, 2010a, 2010b, 2010c; Lin, Mehdi, et al., 2010), VLSI testing and fuzzy controller designs (Lin, 2010d). The Lyapunov stability theorem has been used to testify the asymptotic stability of the whole system and the free parameters of the adaptive fuzzy controller can be tuned on-line by an output feedback control law and adaptive laws. In this paper, the proposed design method attempts to combine the attenuation technique, interval type-2 fuzzy logic approximation method, and adaptive control algorithm for the robust tracking control design of the nonlinear discrete-time systems with a large uncertainty or unknown variation in plant parameters and structures. The remaining of this paper is organized by giving problem formulation and a brief of the interval type-2 fuzzy logic system is presented. Adaptive H interval type-2 fuzzy control scheme for a class of nonlinear discretetime system is presented next. A simulation example to demonstrate the performances of the proposed method is provided and then a conclusion of the advocated design methodology is given.