Enhanced Fuzzy Controller for Nonlinear Systems: Membership-Function-Dependent Stability Analysis Approach

Abstract

Fuzzy-model-based (FMB) control approach provides a systematic and effective way to control nonlinear systems. It has been shown by various applications (Lam et al., 1998; Lian et al., 2006; (b)Tanaka et al., 1998) that FMB control approach performs superior to some traditional control approaches. Based on the T-S fuzzy model (Sugeno & Kang, 1988; Takagi & Sugeno, 1985) the system dynamics of the nonlinear can be represented by some local linear models in the form of linear state-space equations. With the fuzzy logic technique, the overall system dynamics of the nonlinear plant is a fuzzy combination of the local linear models. Consequently, the fuzzy model offers a systematic way and general framework to represent the nonlinear plants in the form of averaged weighted sum of local linear systems. This particular structure exhibits favourable property to facilitate the system analysis and control synthesis. In general, the stability analysis for FMB control systems can be classified into two categories, i.e., membership function (MF)-independent (Chen et al.,1993; Tanaka & Sugeno, 1992) and MF-dependent (Fang et al., 2006; Feng, 2006; Kim & Lee 2000; Liu & Zhang, 2003a; Liu & Zhang, 2003b; Tanaka et al., 1998a; Teixeira et al.,2003) stability analysis approaches. Under the MF-independent stability analysis, the membership functions of both fuzzy model and fuzzy controller are not considered during stability analysis. The system stability is guaranteed to be asymptotically stable if there exists a common positive definite matrix to a set of stability conditions in the form of Lyapunov inequalities (Chen et al.,1993; Tanaka & Sugeno, 1992). The main advantages under the MF-independent analysis approach are 1). The membership functions of the fuzzy controller can be designed freely. Some simple and easy-to-implement membership functions can be employed to realize the fuzzy controller to lower the implementation cost. 2). The grades of membership functions of the fuzzy model are not necessarily known which implies parameter uncertainties of the nonlinear plant are allowed. Consequently, the fuzzy controller exhibits an inherent robustness property for nonlinear plant subject to parameter uncertainties. However, the membership function mismatch (both fuzzy model and fuzzy controller do not share the same membership functions) leads to very conservative stability analysis results. Furthermore, it can be shown that the fuzzy controller designed based on the stability conditions in (Chen et al.,1993; Tanaka & Sugeno, 1992) can be replaced by a liner controller.