From dissipativity theory to compositional synthesis of symbolic models

TL;DR

This paper develops a compositional framework for creating finite symbolic models of interconnected discrete-time control systems using dissipativity properties, enabling scalable abstraction with quantifiable error bounds.

Contribution

It introduces a novel compositional approach based on storage functions and dissipativity to construct finite abstractions of interconnected systems, including a method for systems with incremental passivity.

Findings

Successfully applied to a network of linear systems

No restrictions on subsystem gains or number of subsystems

Provides a scalable way to quantify abstraction errors

Abstract

In this work, we introduce a compositional framework for the construction of finite abstractions (a.k.a. symbolic models) of interconnected discrete-time control systems. The compositional scheme is based on the joint dissipativity-type properties of discrete-time control subsystems and their finite abstractions. In the first part of the paper, we use a notion of so-called storage function as a relation between each subsystem and its finite abstraction to construct compositionally a notion of so-called simulation function as a relation between interconnected finite abstractions and that of control systems. The derived simulation function is used to quantify the error between the output behavior of the overall interconnected concrete system and that of its finite abstraction. In the second part of the paper, we propose a technique to construct finite abstractions together with their corresponding storage functions for a class of discrete-time control systems under some incremental passivity property. We show that if a discrete-time control system is so-called incrementally passivable, then one can construct its finite abstraction by a suitable quantization of the input and state sets together with the corresponding storage function. Finally, the proposed results are illustrated by constructing a finite abstraction of a network of linear discrete-time control systems and its corresponding simulation function in a compositional way. The compositional conditions in this example do not impose any restriction on the gains or the number of the subsystems which, in particular, elucidates the effectiveness of dissipativity-type compositional reasoning for networks of systems.