Stability and Stabilization of a Class of Discrete-Time Fuzzy Systems With Semi-Markov Stochastic Uncertainties

Abstract

This paper addresses the problems of stability and stabilization for a class of discrete-time Takagi–Sugeno fuzzy systems with semi-Markov stochastic uncertainties. By means of the semi-Markov kernel, the probability density function of the sojourn time for different modes in describing the underlying semi-Markov stochastic uncertainties can be mode-dependent in contrast with the previous studies. Both the sojourn-time-independent and sojourn-time-dependent Lyapunov functions are proposed, by which the stability criteria are obtained in terms of a new $\boldsymbol {\sigma }$ -error mean square stability concept, and the latter is demonstrated to be less conservative and more practical than some existing results. Then, control synthesis problem is investigated and the sufficient conditions on the existence of admissible mode-dependent state-feedback stabilizing controller are developed. A numerical example and a cart-pendulum system are given to show the effectiveness and potential of the new design techniques.