Contents I. Introduction
The interest in the T-S fuzzy system started when Takagi and Sugeno introduced their T-S system. Since then, the T-S system has become one of the most active and fruitful areas of fuzzy control. In fact, this system has been successfully applies to a wide range of processes namely, batch chemical reactors, cement kilns, and so on. Thanks to the use of T-S fuzzy model, a complex dynamic model can be made up of a set of local linear subsystems by means of fuzzy inference ([1]–[4]). These models have become a useful tool to deal with a class of nonlinear systems. A set of if-then rules is used to describe the models. This gives local linear approximations of an underlying system. For decades, researchers have been increasingly attracted by the stability analysis and control design for T-S fuzzy systems ([5]–[9]). The main approach to deal with these kinds of problems is Lyapunov stability theory. We obtain the stability condition of fuzzy systems from the Lyapunov direct method. To satisfy the Lyapunov equation, we must find a common positive matrix P for all local linear models. If there is a large number of rules of a fuzzy system, it is often difficult to find a common definite matrix. Bends, the existence of this matrix makes it very difficult to design a fuzzy controller for many fuzzy systems. Many scholars have made problems, in particular the concept of the parallel distributed compensation (PDC) technique to design a stabilizing controller. Yet, one of the main difficulties in stability analysis consists in the way to study the behavior of the solutions, especially when the convergence to a small neighborhood of the origin.
