Compositional Synthesis of not Necessarily Stabilizable Stochastic Systems via Finite Abstractions

Abstract

In this paper, we propose a compositional framework for the construction of finite abstractions (a.k.a. finite Markov decision processes (MDPs)) for networks of not necessarily stabilizable discrete-time stochastic control systems. The proposed scheme is based on a notion of finite-step stochastic simulation function, using which one can employ an abstract system as a substitution of the original one in the controller design process with guaranteed error bounds. In comparison with the existing notions of simulation functions, a finite-step stochastic simulation function needs to decay only after some finite numbers of steps instead of at each time step. In the first part of the paper, we develop a new type of small-gain conditions which are less conservative than the existing ones. The proposed condition compositionally quantifies the distance in probability between the interconnection of stochastic control subsystems and that of their (finite or infinite) abstractions. In particular, using this relaxation via finite-step stochastic simulation functions, it is possible to construct finite abstractions such that stabilizability of each subsystem is not necessarily required. In the second part of the paper, for the class of linear stochastic control systems, we construct finite MDPs together with their corresponding finite-step stochastic simulation functions. Finally, we demonstrate the effectiveness of the proposed results by compositionally constructing finite MDP of a network of four subsystems such that one of them is not stabilizable.