H_∞ Control of Stochastic Discrete-Time Switched Systems with Dwell

Linear discrete-time switched stochastic systems are considered, where the issues of mean square stability, stochastic l₂ -gain, and state-feedback control design problems are treated and solved. Solutions are obtained for both: nominal and polytopic uncertain systems. In all these problems, the switching obeys a dwell time constraint. In our solution, to each subsystem of the switched system, a Lyapunov function is assigned that is nonincreasing at the switching instants. The latter function is allowed to vary piecewise linearly, starting at the end of the previous switch instant, and it becomes time invariant as the dwell proceeds. In order to guarantee asymptotic stability, we require the Lyapunov function to be negative between consecutive switchings. We thus obtain a linear matrix inequality condition. Based on the solution of the stochastic l₂-gain problem, we derive a solution to the state-feedback control design, where we treat a variety of special cases. Being affine in the system matrices, all the above solutions are extended to the uncertain polytopic case. The proposed theory is demonstrated by a flight control example.