Compositional Synthesis of Finite Abstractions for Networks of Systems: A Small-Gain Approach

TL;DR

This paper presents a novel compositional method for constructing finite symbolic models of interconnected control systems using small-gain reasoning, enabling scalable controller synthesis for large networks.

Contribution

It introduces a small-gain based compositional scheme using alternating simulation functions for finite abstractions of interconnected systems, applicable to large networks.

Findings

Successfully applied to temperature regulation in a 1000-room building.

Demonstrated effectiveness on a nonlinear fully connected network.

Outperformed existing compositional techniques in case studies.

Abstract

In this paper, we introduce a compositional scheme for the construction of finite abstractions (a.k.a. symbolic models) of interconnected discrete-time control systems. The compositional scheme is based on small-gain type reasoning. In particular, we use a notion of so-called alternating simulation functions as a relation between each subsystem and its symbolic model. Assuming some small-gain type conditions, we construct compositionally an overall alternating simulation function as a relation between an interconnection of symbolic models and that of original control subsystems. In such compositionality reasoning, the gains associated with the alternating simulation functions of the subsystems satisfy a certain "small-gain" condition. In addition, we introduce a technique to construct symbolic models together with their corresponding alternating simulation functions for discrete-time control subsystems under some stability property. Finally, we apply our results to the temperature regulation in a circular building by constructing compositionally a finite abstraction of a network containing $N$ rooms for any $N\geq3$. We use the constructed symbolic models as substitutes to synthesize controllers compositionally maintaining room temperatures in a comfort zone. We choose $N=1000$ for the sake of illustrating the results. We also apply our proposed techniques to a nonlinear example of fully connected network in which the compositionality condition still holds for any number of components. In these case studies, we show the effectiveness of the proposed results in comparison with the existing compositionality technique in the literature using a dissipativity-type reasoning.