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Robust variance-constrained $H_{\infty }$ control for stochastic systems with multiplicative noises. (English) Zbl 1117.93068
Summary: In this paper, the robust variance-constrained $H_{\infty }$ control problem is considered for uncertain stochastic systems with multiplicative noises. The norm-bounded parametric uncertainties enter into both the system and output matrices. The purpose of the problem is to design a state feedback controller such that, for all admissible parameter uncertainties, (1) the closed-loop system is exponentially mean-square quadratically stable; (2) the individual steady-state variance satisfies given upper bound constraints; and (3) the prescribed noise attenuation level is guaranteed in an $H_{\infty }$ sense with respect to the additive noise disturbances. A general framework is established to solve the addressed multiobjective problem by using a linear matrix inequality (LMI) approach, where the required stability, the $H_{\infty }$ characterization and variance constraints are all easily enforced. Within such a framework, two additional optimization problems are formulated: one is to optimize the $H_{\infty }$ performance, and the other is to minimize the weighted sum of the system state variances. A numerical example is provided to illustrate the effectiveness of the proposed design algorithm.

##### MSC:
 93E03 General theory of stochastic systems 93B36 $H^\infty$-control 93B35 Sensitivity (robustness) of control systems 93B52 Feedback control 93C55 Discrete-time control systems
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