Robust stability and stabilization for uncertain Takagi–Sugeno fuzzy time-delay systems

Abstract

In this paper, we consider robust stability and stabilization of uncertain Takagi–Sugeno fuzzy time-delay systems where uncertainties come into the state and input matrices. Some stability conditions and robust stability conditions for fuzzy time-delay systems have already been obtained in the literature. However, those conditions are rather conservative and do not guarantee the stability and robust stability for a wide class of fuzzy systems. This is true in case of designing stabilizing controllers for fuzzy time-delay systems. We first consider robust stability conditions of uncertain fuzzy systems. Conditions we obtain here are delay-dependent conditions that depend on the upper bound of time delay, and are given in linear matrix inequalities(LMIs). An appropriate selection of Lyapunov–Krasovskii function and introduction of free weighting matrices generalize robust stability conditions. Next, we consider the stabilization problem with memoryless and delayed feedback controllers. Based on our generalized robust stability conditions, we obtain delay-dependent sufficient conditions for the closed-loop system to be robustly stable, and give a design method of robustly stabilizing controllers. Finally, we give two examples that illustrate our results. We compare our conditions with other stability conditions and show our conditions are rather generalized.