Compositional Verification of Large-Scale Stochastic Systems via Relaxed Small-Gain Conditions

In this paper, we provide a compositional framework for the construction of finite abstractions (a.k.a. finite Markov chains (MCs)) for networks of not necessarily stable discrete-time stochastic systems. The proposed scheme is based on a notion of finite-step stochastic simulation functions, using which one can employ an abstract system as a substitution of the original one in the verification process with guaranteed error bounds. To this end, we first develop a new type of small-gain conditions which are less conservative than the existing ones in compositionally quantifying the probabilistic distance between the interconnection of stochastic subsystems and that of their finite abstractions. We then propose an approach to construct finite MCs together with their corresponding finite-step simulation functions for discrete-time nonlinear stochastic systems satisfying a finite-step version of an incremental input-to-state stability (δ-ISS) property. We also construct finite MCs for a particular class of nonlinear stochastic systems whose 1-step δ-ISS can be easily verified using the semidefinite programming. Finally, we demonstrate the effectiveness of the proposed results to a network of four subsystems such that one of them is not stable.