Assessing the Stability of Linear Systems with Random Delays

This paper explores the stability problems of discrete-time linear time-invariant (LTI) systems subject to random time delays. We first develop a general mean-square small-gain stability condition for feedback systems containing structured stochastic multiplicative uncertainties. Subsequently, we apply this mean-square small-gain condition to certain LTI random delay systems, where delays exhibit random delay lengths. Necessary and sufficient mean-square stability criteria are derived. These criteria, applicable to systems with either single or multiple sources of random delays, typically involve computing the spectral radius of a constant matrix. This computation allows us to determine whether a system's state variance matrix converges asymptotically despite the presence of random delays.