Research On The Robust Stability Of Grey Stochastic Time Delays Systems

This paper investigates the robust stability of grey stochastic time-delay systems by applying the methods of Lyapunov function, Lyapunov-Krasovskii functional and transformation model and employing the tools of Ito ? formula, matrix inequalities, Ho lder inequality and Schur complement and using the decomposition technique of the continuous matrix-covered sets of grey matrix, Especially, the problems of robust stability for grey stochastic time-delay systems of neutral type, distributed-delay type and neutral distributed-delay type are deeply studied. Some efficiency criteria are established. Significant results are obtained.1. The investigated background, significance and value for grey uncertain stochastic time-delay systems are illustrated. The development history and research status of robust stability of uncertain time-delay systems are reviewed. The differences and similarities are pointed out on the study methods of robust stability between grey uncertain time-delay systems and uncertain time-delay systems, and the problems what need to be investigated and focused are proposed.2. The problem of robust stability of grey stochastic linear time-delay systems is investigated. Firstly, based on the presented lemma of asymptotic stability for stochastic linear time-delay systems and the decomposition technique of the continuous matrix-covered sets of grey matrix, the robust asymptotic stability of grey stochastic linear time-delay systems is analyzed, and the delay- independent algebraic criteria of robust asymptotic stability are given. The method is presented to select the free parameters in the delay-independent algebraic criteria. Secondly, the p-moment exponential robust stability of grey stochastic linear time-delay systems is investigated by applying the Lyapunov function method and the transformation model method respectively. The algebraic criteria for p-moment exponential robust stability are obtained. Finally, numerical examples show the convenience and effectiveness of the criteria got in this paper.3. A class of grey stochastic time-delay systems with a distributed-delay term is considered. Firstly, based on the Lyapunov- Krasovskii functional method, the stochastic robust stability and the exponential robust stability in mean square for grey stochastic time-delay systems with a distributed-delay term is explored. The delay-dependent criteria of stochastic robust stability are shown. Secondly, the p-moment exponential robust stability of the described system is investigated by the Lyapunov function and the transformation model respectively. The delay- dependent algebraic criteria for p-moment exponential robust stability are obtained. Finally, the examples indicate the effectiveness of the obtained criteria.4. A class of grey stochastic control systems in which the deterministic portion and the stochastic portion have a distributed-delay term respectively is considered. The stochastic robust stability and the exponential robust stability of the close-loop systems under a state feedback control law are probed by the Lyapunov-Krasovskii functional method. The conditions for the stochastic robust stability and the exponential robust stability are derived in the forms of a nonlinear matrix inequality and a linear matrix inequality. When the parametric matrix of control input vector is a zero matrix and the time tags of states are mutual difference, the p-moment exponential robust stability of the described systems are studied by using the Lyapunov-Krasovskii functional method and the transformation model method respectively. Two delay- dependent algebraic criteria are presented. The effectiveness of the proposed criteria is verified.5. The problem of robust stability of grey neutral stochastic time-delay systems is investigated. Firstly, based on the Lyapunov-Krasovskii functional method, the exponential robust stability in mean square for grey neutral stochastic time-delay systems is probed. The delay-independent criteria of stochastic robust stability are shown, and then a sufficient condition for almost sure exponential robust stability is proposed. Secondly, by applied the Lyapunov function and the transformation model respectively, and used several algebraic inequalities, the p-moment exponential robust stability of the described system is investigated. The delay-dependent algebraic criteria for p-moment exponential robust stability are established. Finally, Numerical examples indicate the effectiveness of the presented criteria. The conservation and practicality of these criteria are compared.6. The problem of robust stability of grey neutral stochastic time-delay systems with distributed- delays is studied. Based on the Lyapunov-Krasovskii functional, Ito ? formula, Ho lder inequality and Schur complement theorem, the criteria of exponential robust stability in mean square for two classes of grey neutral stochastic time-delay systems with distributed-delays are derived in the forms of a nonlinear matrix inequality and a linear matrix inequality. The effectiveness and practicality of the obtained criteria are verified. The conservation of these criteria is compared.7. The problem of robust stability of grey stochastic time-delay systems with impulsive effect is studied. By using the proposed lemma of stochastic stability of stochastic time-delay systems with impulsive effect, the criterion of stochastic robust stability for grey stochastic time-delay systems with impulsive effect is obtained. Based on the Lyapunov-Krasovskii functional method, the delay-independent algebraic condition of p-moment robust stability is established. Numerical example shows the effectiveness of the criteria derived in this paper.