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Exponential $$H_\infty$$ filter design for stochastic Markovian jump systems with both discrete and distributed time-varying delays. (English) Zbl 1309.93146
Summary: This paper is concerned with the exponential $$H_{\infty }$$ filter design problem for stochastic Markovian jump systems with time-varying delays, where the time-varying delays include not only discrete delays but also distributed delays. First of all, by choosing a modified Lyapunov-Krasovskii functional and employing the property of conditional mathematical expectation, a novel delay-dependent approach is developed to deal with the mean-square exponential stability problem and $$H_{\infty }$$ control problem. Then, a mean-square exponentially stable and Markovian jump filter is designed such that the filtering error system is mean-square exponentially stable and the $$H_{\infty }$$ performance of estimation error can be ensured. Besides, the derivative of discrete time-varying delay $$h(t)$$ satisfies $$\dot{h}(t)\leq \eta$$ and simultaneously the decay rate $$\beta$$ can be finite positive value without equation constraint. Finally, a numerical example is provided to illustrate the effectiveness of the proposed design approach.
##### MSC:
 93E03 Stochastic systems in control theory (general) 93B36 $$H^\infty$$-control
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