Port-Hamiltonian systems and monotonicity

TL;DR

This paper explores the extension of port-Hamiltonian systems to include maximal cyclically monotone relations, linking physical modeling and convex optimization, and providing new methods for analyzing and computing system equilibria.

Contribution

It introduces a novel class of incrementally port-Hamiltonian systems based on maximal cyclically monotone relations and develops methods for their composition and equilibrium computation.

Findings

Maximal cyclically monotone relations can be generated via convex functions.

Incrementally port-Hamiltonian systems with convex Hamiltonians are equilibrium independent passive.

A convex optimization approach is proposed for computing system equilibria.

Abstract

The relationships between port-Hamiltonian systems modeling and the notion of monotonicity are explored. The earlier introduced notion of incrementally port-Hamiltonian systems is extended to maximal cyclically monotone relations, together with their generating functions. This gives rise to new classes of incrementally port-Hamiltonian systems, with examples stemming from physical systems modeling as well as from convex optimization. An in-depth treatment is given of the composition of maximal monotone and maximal cyclically monotone relations, where in the latter case the resulting maximal cyclically monotone relation is shown to be computable through the use of generating functions. Furthermore, connections are discussed with incremental versions of passivity, and it is shown how incrementally port-Hamiltonian systems with strictly convex Hamiltonians are (maximal) equilibrium independent passive. Finally, the results on compositionality of monotone relations are employed for a convex optimization approach to the computation of the equilibrium of interconnected incrementally port-Hamiltonian systems.