Compositional abstraction-based synthesis for continuous-time stochastic hybrid systems

Abstract

In this paper, we propose a compositional framework for the construction of discrete-time finite abstractions, also known as finite Markov decision processes, from continuous-time stochastic hybrid systems by quantifying the distance between their outputs in a probabilistic setting. The proposed scheme is based on the notion of stochastic simulation functions, which is used to relate continuous-time stochastic systems with their discrete-time counterparts. Accordingly, one can employ discrete-time abstract systems as substitutions of the continuous-time ones in the controller design process with guaranteed error bounds on their output trajectories. To this end, we first derive sufficient small-gain type conditions for the compositional quantification of the probabilistic distance between the interconnection of original continuous-time stochastic hybrid systems and their discrete-time (finite or infinite) abstractions. We then construct finite abstractions together with their corresponding stochastic simulation functions for a particular class of nonlinear stochastic hybrid systems having some stability property. We illustrate the effectiveness of the proposed results by applying our approaches to the temperature regulation in a circular building and constructing compositionally a discrete-time abstraction from its original continuous-time dynamics in a network containing 1000 rooms. We employ the constructed discrete-time abstractions as substitutes to compositionally synthesize policies regulating the temperature of each room for a bounded time horizon.