Stochastic H2/H∞ control of discrete-time periodic Markov jump systems with detectability

This paper studies an infinite horizon H₂/H_∞ optimal control problem for discrete-time periodic Markov jump systems involving (x, u, v)-dependent noise. The system coefficients and transition probability of Markov jump parameter are all set to be periodically time-varying. By means of the detectability and its spectral criterion, a Lyapunov equation based stability theorem is firstly developed for the asymptotic mean square stability of considered systems. Then, a game theoretic approach is employed to derive a necessary and sufficient condition for the existence of state-feedback optimal H₂/H_∞ controller, whose feedback gains can be constructed in terms of the solution to a group of coupled periodic difference equations. Finally, a numerical example is presented to illustrate our proposed theoretical results.