Compositional Construction of Approximate Abstractions of Interconnected Control Systems

TL;DR

This paper presents a compositional method for constructing approximate abstractions of interconnected control systems, enabling scalable controller synthesis with formal guarantees, especially for linear systems, using simulation functions and small gain conditions.

Contribution

It introduces a novel compositional framework using simulation functions and small gain conditions for interconnected systems, with a specific focus on linear control systems and their abstractions.

Findings

Successfully constructed abstractions for interconnected linear systems.

Demonstrated controller synthesis for linear temporal logic specifications.

Validated the approach on a system with four interconnected linear subsystems.

Abstract

We consider a compositional construction of approximate abstractions of interconnected control systems. In our framework, an abstraction acts as a substitute in the controller design process and is itself a continuous control system. The abstraction is related to the concrete control system via a so-called simulation function: a Lyapunov-like function, which is used to establish a quantitative bound between the behavior of the approximate abstraction and the concrete system. In the first part of the paper, we provide a small gain type condition that facilitates the compositional construction of an abstraction of an interconnected control system together with a simulation function from the abstractions and simulation functions of the individual subsystems. In the second part of the paper, we restrict our attention to linear control system and characterize simulation functions in terms of controlled invariant, externally stabilizable subspaces. Based on those characterizations, we propose a particular scheme to construct abstractions for linear control systems. We illustrate the compositional construction of an abstraction on an interconnected system consisting of four linear subsystems. We use the abstraction as a substitute to synthesize a controller to enforce a certain linear temporal logic specification.