Exponential Mean-Square Stability of Linear Systems subject to Slowly Varying Delays with Known Stochastic Distribution

Time delay is a critical aspect concerning stability and robustness of controlled systems. This paper considers a class of linear time-delayed systems where the distribution and the maximum change rate of the delay are known. For these systems, it proposes a method to investigate their stability. Therefore, the delay is partitioned into intervals with occurrence probabilities to approximate the delay distribution. The number of intervals can be freely chosen to tradeoff between complexity and the quality of the approximation of the distribution. Considering this delay distribution approximation, the system is analysed for exponential stability in mean-square sense (ESMSS), and the benefits of this method are shown in numeric examples. It is revealed that by better approximating the delay distribution the maximum allowable delay can be increased. Further, if the delay change rate bound gets small, the conservatism is reduced even more. The result of this stability analysis is a statement for the expected value of the states at infinite time. Thus, no statement about stability for short time frames is made, which needs to be considered when choosing this approach.