# Article

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Keywords:
stochastic systems; distributed time-varying delay; $H_{\infty }$ filter; linear matrix inequality
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.
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