Robust control of time-delayed stochastic switched systems with dwell

Linear, state-delayed, discrete-time, stochastic, switched systems are considered, where the problems of stochastic $l_2$-gain and state-feedback control designs are treated and solved. We first develop a special version of a bounded real lemma for the said systems for the nominal case.

Based on the this lemma we derive state-feedback gains for nominal systems where in our solution method, to each subsystem of the switched system, a Lyapunov function is assigned that is non-increasing at the switching instants and where a dwell time constrain is imposed on the system. The assigned Lyapunov function is allowed to vary piecewise linearly in time, starting at the end of the previous switch instant, and it becomes time-invariant after the dwell. Based on the solution of the state-feedback control for nominal systems and exploiting the fact that this solution is affine in the system matrices, a state-feedback control is derived for the polytopic case. We bring a numerical example that demonstrates the solvability and tractability of our solution method.