Robust Fuzzy Control of Parametric Uncertain Nonlinear Systems Using Robust Reliability Method

Abstract

Stability is of primary importance for any control systems. Stability of both linear and nonlinear uncertain systems has received a considerable attention in the past decades (see for example, Tanaka & Sugeno, 1992; Tanaka, Ikeda, & Wang, 1996; Feng, Cao, Kees, et al. 1997; Teixeira & Zak, 1999; Lee, Park, & Chen, 2001; Park, Kim, & Park, 2001; Chen, Liu, & Tong, 2006; Lam & Leung, 2007, and references therein). Fuzzy logical control (FLC) has proved to be a successful control approach for a great many complex nonlinear systems. Especially, the well-known Takagi-Sugeno (T-S) fuzzy model has become a convenient tool for dealing with complex nonlinear systems. T-S fuzzy model provides an effective representation of nonlinear systems with the aid of fuzzy sets, fuzzy rules and a set of local linear models. Once the fuzzy model is obtained, control design can be carried out via the so called parallel distributed compensation (PDC) approach, which employs multiple linear controllers corresponding to the locally linear plant models (Hong & Langari, 2000). It has been shown that the problems of controller synthesis of nonlinear systems described by the T-S fuzzy model can be reduced to convex problems involving linear matrix inequalities (LMIs) (Park, Kim, & Park, 2001). Many significant results on the stability and robust control of uncertain nonlinear systems using T-S fuzzy model have been reported (see for example, Hong, & Langari, 2000; Park, Kim, & Park, 2001; Xiu & Ren, 2005; Wu & Cai, 2006; Yoneyama, 2006; 2007), and considerable advances have been made. However, as stated in Guo (2010), many approaches for stability and robust control of uncertain systems are often characterized by conservatism when dealing with uncertainties. In practice, uncertainty exists in almost all engineering systems and is frequently a source of instability and deterioration of performance. So, uncertainty is one of the most important factors that have to be taken into account rationally in system analysis and synthesis. Moreover, it has been shown (Guo, 2010) that the increasing in conservatism in dealing with uncertainties by some traditional methods does not mean the increasing in reliability. So, it is significant to deal with uncertainties by means of reliability approach and to achieve a balance between reliability and performance/control-cost in design of uncertain systems. In fact, traditional probabilistic reliability methods have ever been utilized as measures of stability, robustness, and active control effectiveness of uncertain structural systems by Spencer et al. (1992,1994); Breitung et al. (1998) and Venini & Mariani (1999) to develop