Robust state-feedback control of stochastic state-multiplicative discrete-time linear switched systems with dwell time

Linear discrete-time switched stochastic systems are considered, where the problems of mean square stability, stochastic l₂-gain and state-feedback control design are treated and solved. Solutions are obtained for both nominal and polytopic-type 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 after the dwell. In order to guarantee asymptotic stability, we require the Lyapunov function to be negative between consecutive switchings. We thus obtain Linear Matrix Inequalities conditions. 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 aforementioned solutions are extended to the uncertain polytopic case. The proposed theory is demonstrated by a practical example taken from the field of flight control.