Fuzzy $$H_\infty $$ Sliding Mode Control Under Phase-Type Distribution

Abstract

The issue of sliding mode control for phase-type stochastic nonlinear semi-Markovian switching systems (S-MSSs) via the T-S fuzzy strategy is investigated. One unrealistic assumption, that is, the sojourn time in stochastic switching systems follows an exponential distribution, is removed by S-MSSs model, which is one of the main features compared to the existing literature. First, by using the plant transformation method and the supplementary variable technique, phase-type S-MSSs are equivalent to associated Markovian switching systems (MSSs). Then, an integral sliding surface function is established to cope with the influence of the switching phenomenon in the plant. By using Lyapunov functions and inequality optimization problems, sufficient conditions are provided for stochastic stability of the system with a prescribed $$H_\infty $$ performance index. In addition, a fuzzy sliding mode control law is synthesized to guarantee that the associated T-S fuzzy MSSs fulfill the reaching condition in bounded time. Finally, the validity of the conclusions is verified by the single-link robot arm model.