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This note studies the design of sliding mode control (SMC) for discrete-time hybrid stochastic switched systems with repeated scalar nonlinearities. The weighed ℋ ∞ gain performance is considered for the system dynamics to optimize its transient state performance. First, sufficient conditions are given to guarantee the corresponding system is exponentially stable while achieving a desired weighed ℋ ∞ performance. A new switching surface function is constructed by the average dwell time technique and the positive diagonally dominant Lyapunov functional method to further reduce the conservativeness induced by the repeated scalar nonlinearity. Then, the corresponding sliding mode dynamics are obtained and the solvability condition for the desired switching surface function is derived. Furthermore, the synthesis of the proposed SMC law is proposed to force the resulting closed-loop system trajectories onto the pre-specified sliding mode region with a desired level of accuracy. Finally, the feasibility and the effectiveness of the presented new design techniques are illustrated by examples and simulations.
Su et al. (2016) studied this question.