Dissipativity of system abstractions obtained using approximate input-output simulation

TL;DR

This paper investigates how QSR dissipativity properties are preserved or transformed when creating system abstractions via approximate input-output simulation, aiding in control design and stability analysis.

Contribution

It provides conditions for dissipativity preservation between continuous systems and their abstractions, and analyzes how dissipativity matrices change during abstraction and composition.

Findings

Conditions for dissipativity of continuous systems from abstractions

Characterization of dissipativity matrix changes during abstraction

Analysis of dissipativity in system-controller compositions

Abstract

This work focuses on the invariance of important properties between continuous and discrete models of systems which can be useful in the control design of large-scale systems and their software implementations. In particular, this paper discusses the relationships between the QSR dissipativity of a continuous state dynamical system and of its abstractions obtained through approximate input-output simulation relations. First, conditions to guarantee the dissipativity of the continuous system from its abstractions are provided. The reverse problem of determining the Q, S and R dissipativity matrices of the abstract system from that of the continuous system is also considered. Results characterizing the change in the dissipativity matrices are provided when the system abstraction is obtained. Since, under certain conditions, QSR dissipative systems are known to be stable, the results of this paper can be used to construct stable system abstractions as well. In the second part of this paper, we analyze the dissipativity of the approximate feedback composition of a continuous dynamical system and a discrete controller. We present illustrative examples to demonstrate the results of this paper.