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Robust ${H}_{\infty }$ control for a class of nonlinear stochastic systems with mixed time delay. (English) Zbl 1128.93015
Summary: This paper is concerned with the problem of robust ${H}_{\infty }$ control for a class of uncertain nonlinear Itô-type stochastic systems with mixed time delays. The parameter uncertainties are assumed to be norm bounded, the mixed time delays comprise both the discrete and distributed delays, and the sector nonlinearities appear in both the system states and delayed states. The problem addressed is the design of a linear state feedback controller such that, in the simultaneous presence of parameter uncertainties, system nonlinearities and mixed time delays, the resulting closed-loop system is asymptotically stable in the mean square and also achieves a prescribed ${H}_{\infty }$ disturbance rejection attenuation level. By using the Lyapunov stability theory and the Itô differential rule, some new techniques are developed to derive the sufficient conditions guaranteeing the existence of the desired feedback controllers. A unified linear matrix inequality is proposed to deal with the problem under consideration and a numerical example is exploited to show the usefulness of the results obtained.
##### MSC:
 93B35 Sensitivity (robustness) of control systems 93B36 ${H}^{\infty }$-control 93E03 General theory of stochastic systems 93C10 Nonlinear control systems 93C41 Control problems with incomplete information 93B52 Feedback control 93E15 Stochastic stability 93D05 Lyapunov and other classical stabilities of control systems