Robust H_∞ control of stochastic linear switched systems with dwell time

The theory of H_∞ control of switched systems is extended to stochastic systems with state-multiplicative noise. Sufficient conditions are obtained for the mean square stability of these systems where dwell time constraint is imposed on the switching. Both nominal and uncertain polytopic systems are considered. A Lyapunov function, in a quadratic form, is assigned to each subsystem that is nonincreasing at the switching instants. During the dwell time, this function varies piecewise linearly in time following the last switch, and it becomes time invariant afterwards. Asymptotic stochastic stability of the set of subsystems is thus ensured by requiring the expected value of the infinitesimal generator of this function to be negative between switchings, resulting in conditions for stability in the form of LMIs. These conditions are extended to the case where the subsystems encounter polytopic-type parameter uncertainties. The method proposed is applied to the problem of finding an upper bound on the stochastic L₂-gain of the system. A solution to the robust state-feedback control problem is then derived, which is based on a modification of the L₂-gain bound result. Two examples are given that demonstrate the applicability of the proposed theory.