Relationship between dissipativity concepts for linear time-varying port-Hamiltonian systems

TL;DR

This paper explores the relationships between dissipativity concepts in linear time-varying systems, showing how port-Hamiltonian systems relate to passivity and supply conditions, with conditions for conversion and practical applications.

Contribution

It provides a comprehensive analysis of the connections between dissipativity, passivity, and port-Hamiltonian structures in time-varying systems, including conditions for representation conversion.

Findings

Linear time-varying port-Hamiltonian systems are passive.

They have nonnegative supply rates.

Conditions for converting other dissipativity concepts into port-Hamiltonian form are established.

Abstract

The relationship between different dissipativity concepts for linear time-varying systems is studied, in particular between port-Hamiltonian systems, passive systems, and systems with nonnegative supply. It is shown that linear time-varying port-Hamiltonian systems are passive, have nonnegative supply rates, and solve (under different smoothness assumptions) Kalman-Yakubovich-Popov differential and integral inequalities. The converse relations are also studied in detail. In particular, sufficient conditions are presented to obtain a port-Hamiltonian representation starting from any of the other dissipativity concepts. Two applications are presented.