Contraction Analysis of Differentially Positive Discrete-Time Stochastic Systems

In this article, we examine the role of differential positivity in the contraction analysis of discrete-time nonlinear systems with parameters that follow stochastic processes. We investigate the concept of almost sure differential positivity, which parallels the positivity of linear systems, and utilize it to derive vector-type sufficient conditions for uniform incremental exponential stability in the first moment. Employing these conditions, we demonstrate that network stability analysis can be made scalable by decomposing it into analysis of subcomponents and interconnection structures. Our results are applicable to a range of systems, including nonlinear Markov jump systems, nonlinear deterministic systems, and linear stochastic systems, by specifying the system structures. In addition, we perform converse analysis for almost surely cooperative systems, i.e., almost surely differentially positive systems with respect to the positive orthant cones.