Approximations of Stochastic Hybrid Systems: A Compositional Approach

TL;DR

This paper introduces a compositional framework for approximating interconnected stochastic hybrid systems, enabling easier controller design by quantifying and managing approximation errors through stochastic simulation functions.

Contribution

It develops a systematic method to construct approximations of stochastic hybrid systems and their interconnections, including jump linear systems, with error quantification.

Findings

Framework effectively constructs approximations for interconnected systems.

Sufficient conditions enable compositional error analysis.

Constructive scheme for jump linear stochastic systems approximations.

Abstract

In this paper we propose a compositional framework for the construction of approximations of the interconnection of a class of stochastic hybrid systems. As special cases, this class of systems includes both jump linear stochastic systems and linear stochastic hybrid automata. In the proposed framework, an approximation is itself a stochastic hybrid system, which can be used as a replacement of the original stochastic hybrid system in a controller design process. We employ a notion of so-called stochastic simulation function to quantify the error between the approximation and the original system. In the first part of the paper, we derive sufficient conditions which facilitate the compositional quantification of the error between the interconnection of stochastic hybrid subsystems and that of their approximations using the quantified error between the stochastic hybrid subsystems and their corresponding approximations. In particular, we show how to construct stochastic simulation functions for approximations of interconnected stochastic hybrid systems using the stochastic simulation function for the approximation of each component. In the second part of the paper, we focus on a specific class of stochastic hybrid systems, namely, jump linear stochastic systems, and propose a constructive scheme to determine approximations together with their stochastic simulation functions for this class of systems. Finally, we illustrate the effectiveness of the proposed results by constructing an approximation of the interconnection of four jump linear stochastic subsystems in a compositional way.