Compositional Synthesis of Finite Abstractions for Continuous-Space Stochastic Control Systems: A Small-Gain Approach

Abstract

This paper is concerned with a compositional approach for constructing finite abstractions (a.k.a. finite Markov decision processes) of interconnected discrete-time stochastic control systems. The proposed framework is based on a notion of so-called stochastic simulation function enabling us to use an abstract system as a substitution of the original one in the controller design process with guaranteed error bounds. In the first part of the paper, we derive sufficient small-gain type conditions for the compositional quantification of the distance in probability between the interconnection of stochastic control subsystems and that of their (finite or infinite) abstractions. In the second part of the paper, we construct finite abstractions together with their corresponding stochastic simulation functions for the class of linear stochastic control systems. We apply our results to the temperature regulation in a circular building by constructing compositionally a finite abstraction of a network containing 1000 rooms. We use the constructed finite abstractions as substitutes to synthesize policies compositionally regulating the temperature in each room for a bounded time horizon.