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Robust $$\mathcal H_{\infty}$$ control of descriptor discrete-time Markovian jump systems. (English) Zbl 1120.93057
Summary: This paper considers the stochastic stability and the robust $$\mathcal H_{\infty}$$ control of descriptor discrete-time systems with Markovian jumping parameters. In terms of linear matrix inequalities, a necessary and sufficient condition is proposed, which ensures a discrete-time descriptor Markovian jump system to be regular, causal and stochastically stable. A robust admissibility condition and a robust bounded real lemma are also developed. Based on these, a sufficient condition on the existence of a state-feedback controller which guarantees the robust admissibility and the $$\mathcal H_{\infty}$$ performance is also given by employing the linear matrix inequality technique. A robustly stabilizing $$\mathcal H_{\infty}$$ state feedback controller can be constructed through the numerical solutions of linear matrix inequalities. Finally, an example is provided to demonstrate the effectiveness of the proposed approach.

##### MSC:
 93E20 Optimal stochastic control 93E03 Stochastic systems in control theory (general) 93C55 Discrete-time control/observation systems 93B35 Sensitivity (robustness) 93E15 Stochastic stability in control theory 93D21 Adaptive or robust stabilization 93B50 Synthesis problems
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