Compositional Abstractions of Interconnected Discrete-Time Stochastic Control Systems

TL;DR

This paper introduces a compositional framework for creating abstractions of interconnected discrete-time stochastic control systems using stochastic simulation functions, enabling probabilistic analysis and synthesis.

Contribution

It develops new small-gain conditions for compositional probabilistic distance quantification and proposes a computational scheme for linear stochastic systems.

Findings

Successfully constructed a 4-dimensional abstraction of a 100-dimensional interconnected system.

Validated the effectiveness of the approach through a case study.

Provided a scalable method for analysis of complex stochastic control systems.

Abstract

This paper is concerned with a compositional approach for constructing abstractions of interconnected discrete-time stochastic control systems. The abstraction framework is based on new notions of so-called stochastic simulation functions, using which one can quantify the distance between original interconnected stochastic control systems and their abstractions in the probabilistic setting. Accordingly, one can leverage the proposed results to perform analysis and synthesis over abstract interconnected systems, and then carry the results over concrete ones. 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 abstractions. In the second part of the paper, we focus on the class of discrete-time linear stochastic control systems with independent noises in the abstract and concrete subsystems. For this class of systems, we propose a computational scheme to construct abstractions together with their corresponding stochastic simulation functions. We demonstrate the effectiveness of the proposed results by constructing an abstraction (totally 4 dimensions) of the interconnection of four discrete-time linear stochastic control subsystems (together 100 dimensions) in a compositional fashion.