Robust H2/H∞ control for discrete-time systems with Markovian jumps and multiplicative noise: Infinite horizon case

This paper is focused on an infinite horizon H2/H∞ control problem for a broad class of discrete-time Markov jump systems with (x, u, v)-dependent noise. Above all, we develop a stochastic Popov-Belevich-Hautus (PBH) criterion for checking exact detectability. By which, an extended Lyapunov stability theorem is established in terms of a generalized Lyapunov equation. Further, a necessary and sufficient condition is presented for the existence of a state feedback H2/H∞ controller on the basis of four coupled matrix Riccati equations, which can be solved numerically by a backward iterative algorithm. Finally, a numerical example is supplied to illustrate the proposed theoretical results.