Switching Takagi-Sugeno Systems

Abstract

IntroductionThe objective of this chapter is to study discrete-time switching nonlinear systems . Each subsystem is written as an equivalent T-S fuzzy models. Since the introduction of T-S fuzzy models by Takagi and Sugeno [147] in 1985, fuzzy model control has been extensively studied because T-S fuzzy models provide an effective representation of complex nonlinear systems [28, 31, 38, 39, 41, 81, 83, 134, 135]. In the literature, few works were interested on this kind of systems as pointed out in the general introduction. One can consult [148, 166] where a detailed study of the problem of stability and controller switching for switching fuzzy systems is presented. In all these previous works, only a common Lyapuniv function is used for all the T-S fuzzy subsystems. Even for switching linear systems, it was proven that the use of such a common Lyapunov function leads to conservative results [54, 146]. In this chapter, sufficient conditions of stabilizability, by state feedback control, are obtained by using two types of multiple Lyapunov function. The first one called switching Lyapunov function is multiple for the switching system while common for the T-S fuzzy subsystems. The second, called switched Lyapunov function is multiple for both the T-S fuzzy subsystems and the switching modes. This choice allows a general approach which may be less conservative than the one obtained with switching Lyapunov function. These conditions are then worked out to be presented under LMIs form. The case of saturated switching T-S fuzzy systems is also investigated by using only unsaturating controllers.