Compositional Abstractions of Interconnected Discrete-Time Switched Systems

TL;DR

This paper presents a compositional method for creating finite abstractions of interconnected discrete-time switched systems, enabling scalable control design and performance analysis.

Contribution

It introduces a novel compositional approach using alternating simulation functions and small-gain conditions for interconnected switched systems.

Findings

Successfully applied to a road traffic network with 50 cells.

Constructed finite abstractions for switched subsystems under stability assumptions.

Enabled control design to maintain traffic density below a threshold.

Abstract

In this paper, we introduce a compositional method for the construction of finite abstractions of interconnected discrete-time switched systems. Particularly, we use a notion of so-called alternating simulation function as a relation between each switched subsystem and its finite abstraction. Based on some small-gain type conditions, we use those alternating simulation functions to construct compositionally an overall alternating simulation function as a relation between an interconnection of finite abstractions and that of switched subsystems. This overall alternating simulation function allows one to quantify the mismatch between the output behavior of the interconnection of switched subsystems and that of their finite abstractions. Additionally, we provide an approach to construct finite abstractions together with their corresponding alternating simulation functions for discrete-time switched subsystems under standard assumptions ensuring incremental input-to-state stability of a switched subsystem. Finally, we apply our results to a model of road traffic by constructing compositionally a finite abstraction of the network containing $50$ cells of $1000$ meters each. We use the constructed finite abstractions as substitutes to design controllers compositionally keeping the density of traffic lower than $30$ vehicles per cell.