Energy matching in reduced passive and port-Hamiltonian systems

TL;DR

This paper explores energy matching in port-Hamiltonian systems by proposing a structure-preserving decomposition and a method for Hamiltonian approximation in reduced models, ensuring minimal energy deviation while preserving input-output behavior.

Contribution

It introduces a Kalman-like decomposition for pH systems and formulates a convex optimization approach for Hamiltonian matching in model reduction.

Findings

The decomposition separates controllable and observable parts of pH systems.

The Hamiltonian approximation problem is convex and solvable via semidefinite programming.

Numerical examples demonstrate the effectiveness of the proposed methods.

Abstract

It is well known that any port-Hamiltonian (pH) system is passive, and conversely, any minimal and stable passive system has a pH representation. Nevertheless, this equivalence is only concerned with the input-output mapping but not with the Hamiltonian itself. Thus, we propose to view a pH system either as an enlarged dynamical system with the Hamiltonian as additional output or as two dynamical systems with the input-output and the Hamiltonian dynamic. Our first main result is a structure-preserving Kalman-like decomposition of the enlarged pH system that separates the controllable and zero-state observable parts. Moreover, for further approximations in the context of structure-preserving model-order reduction (MOR), we propose to search for a Hamiltonian in the reduced pH system that minimizes the $\mathcal{H}_2$-distance to the full-order Hamiltonian without altering the input-output dynamic, thus discussing a particular aspect of the corresponding multi-objective minimization problem corresponding to $\mathcal{H}_2$-optimal MOR for pH systems. We show that this optimization problem is uniquely solvable, can be recast as a standard semidefinite program, and present two numerical approaches for solving it. The results are illustrated with three academic examples.