Contraction Analysis of Almost Surely Monotone Discrete-Time Systems with Unknown Time Delay

In this paper, we utilize monotonicity to simplify contraction analysis of discrete-time nonlinear time-delay systems with parameters following stochastic processes. First, we extend the concept of almost sure monotonicity to the time-delay systems, which can be verified via analysis of the corresponding prolonged systems. Next, we introduce a novel notion of uniform incremental asymptotic stability (UIAS) in the first moment to the time-delay systems and develop its sufficient condition. By virtue of almost sure monotonicity, if this condition holds, the time-delay systems are UIAS in the first moment for any finite length of delay. Moreover, the proposed result suggests that UIAS in the first moment has a strong connection with stability of the averaged deterministic systems. We formalize this observation for almost surely cooperative systems by establishing the equivalence between UIAS and uniform incremental asymptotic mean stability.