Control Design of Fuzzy Systems with Immeasurable Premise Variables

Abstract

The Takagi-Sugeno fuzzy model is described by fuzzy if-then rules which represent local linear systems of the underlining nonlinear systems (Takagi & Sugeno, 1985; Tanaka et al., 1996; Tanaka & Sugeno, 1992) and thus it can describe a wide class of nonlinear systems. In the last decade, nonlinear control design methods based on Takagi-Sugeno fuzzy system have been explored. Since the stability analysis and state feedback stabilization first made in (Tanaka & Sugeno, 1992), system theory and various control schemes for fuzzy systems have been developed. Parallel to state feedback control design, observer problems were also considered in (Tanaka & Sano, 1994; Tanaka & Wang, 1997; Yoneyama et al., 2000; Yoneyama et al., 2001a). When the observer is available, it is known in (Yoneyama et al., 2000) that for fuzzy systems, the separation property of designing state feedback controller and observer is established. Thus, an output feedback controller for fuzzy systems was proposed in (Yoneyama et al., 2001a). Theory was extended to H∞ control (Cao et al., 1996; Chen et al., 2005; Feng et al., 1996; Hong & Langari, 1998; Katayama & Ichikawa, 2002; Yoneyama et al., 2001b). In spite of these developments in fuzzy system control theory, the separation property holds only for a limited class of fuzzy systems where the premise variable is measurable. When we consider a fuzzy system, the selection of the premise variables plays an important role in system representation. The premise variable is usually given, and hence the output is natually selected. In this case, however, a class of fuzzy systems is limited. If the premise variable is the state of the system, a fuzzy system can represent the widest class of nonlinear systems. In this case, output feedback controller design is difficult because the state variable is immeasurable and it is not available for the premise variable of an output feedback controller. For this class of fuzzy systems, output feedback control design schemes based on parallel distributed compensator (PDC) have been considerd in (Ma et al., 1998; Tanaka & Wang, 2001; Yoneyama et al., 2001a) where the premise variable of the controller was replaced by its estimate. Furthermore, Linear Matrix Inequality (Boyd et al, 1994) approach was introduced in (Guerra et al., 2006). Uncertain system approach was taken for stabilizatin and H∞ control of both continuous-time and discrete-time fuzzy systems in (Assawinchaichote, 2006; Tanaka et al., 1998; Yoneyama, 2008a; Yoneyama, 2008b). However, controller design conditions given in these approaches are still conservative. This chapter is concerned with robust H∞ output feecback controller design for a class of uncertain continuous-time Takagi-Sugeno fuzzy systems where the premise variable is the immeasurable state variable. This class of fuzzy systems covers a general nonlinear system Source: Fuzzy Systems, Book edited by: Ahmad Taher Azar, ISBN 978-953-7619-92-3, pp. 216, February 2010, INTECH, Croatia, downloaded from SCIYO.COM