Infinite horizon <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$H_2/H_\infty $$</EquationSource> </InlineEquation> optimal control for discrete-time Markov jump systems with (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$x,u,v$$</EquationSource> </InlineEquation>)-dependent noise

In this paper, an infinite horizon <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$H_2/H_\infty $$</EquationSource> </InlineEquation> control problem is addressed for a broad class of discrete-time Markov jump systems with (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$x,u,v$$</EquationSource> </InlineEquation>)-dependent noises. First of all, under the condition of exact detectability, the stochastic Popov–Belevich–Hautus (PBH) criterion is utilized to establish an extended Lyapunov theorem for a generalized Lyapunov equation. Further, a necessary and sufficient condition is presented for the existence of state-feedback <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$H_2/H_\infty $$</EquationSource> </InlineEquation> optimal controller on the basis of two coupled matrix Riccati equations, which may be solved by a backward iterative algorithm. A numerical example with simulations is supplied to illustrate the proposed theoretical results. Copyright Springer Science+Business Media New York 2013