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Exponential stability and exponential stabilization of singularly perturbed stochastic systems with time-varying delay

机译：具有时变时滞的奇摄动随机系统的指数稳定性和指数镇定

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摘要

In this paper, the problems of exponential stability and exponential stabilization for linear singularly perturbed stochastic systems with time-varying delay are investigated. First, an appropriate Lyapunov functional is introduced to establish an improved delay-dependent stability criterion. By applying free-weighting matrix technique and by equivalently eliminating time-varying delay through the idea of convex combination, a less conservative sufficient condition for exponential stability in mean square is obtained in terms of ε-dependent linear matrix inequalities (LMIs). It is shown that if this set of LMIs for ε=0 are feasible then the system is exponentially stable in mean square for sufficiently small ε≥0. Furthermore, it is shown that if a certain matrix variable in this set of LMIs is chosen to be a special form and the resulting LMIs are feasible for ε=0, then the system is ε-uniformly exponentially stable for all sufficiently small ε≥0. Based on the stability criteria, an ε-independent state-feedback controller that stabilizes the system for sufficiently small ε≥0 is derived. Finally, numerical examples are presented, which show our results are effective and useful.
机译：研究了具有时变时滞的线性奇摄动随机系统的指数稳定性和指数稳定性问题。首先，引入适当的Lyapunov泛函以建立改进的时延相关稳定性准则。通过应用自由加权矩阵技术并通过凸组合的思想等效地消除时变延迟，就可以得到关于ε依赖的线性矩阵不等式（LMI）的保守度较低的均方指数稳定性的充分条件。结果表明，如果这套ε= 0的LMI是可行的，那么对于足够小的ε≥0，系统的均方指数稳定。此外，表明如果将这组LMI中的某个矩阵变量选择为一种特殊形式，并且所得的LMI对于ε= 0是可行的，则对于所有足够小的ε≥0，系统都是ε均匀指数稳定的。根据稳定性标准，得出一个独立于ε的状态反馈控制器，该控制器可将系统稳定在足够小的ε≥0的水平。最后，给出了数值例子，表明我们的结果是有效和有用的。

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