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 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 $H_2$-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.
%0 Journal Article
%1 holicki2023energy
%A Holicki, Tobias
%A Nicodemus, Jonas
%A Schwerdtner, Paul
%A Unger, Benjamin
%D 2023
%K exc2075 myown pn4 preprint
%T Energy matching in reduced passive and port-Hamiltonian systems
%X 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 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 $H_2$-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.
@article{holicki2023energy,
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 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.},
added-at = {2023-09-26T20:56:45.000+0200},
archiveprefix = {arXiv},
author = {Holicki, Tobias and Nicodemus, Jonas and Schwerdtner, Paul and Unger, Benjamin},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2a4ab02f43b234754df6ecd3dfc02de4a/benjaminunger},
eprint = {2309.05778},
interhash = {fef9d3e8aaf8abe4be1f249a0e8e10bc},
intrahash = {a4ab02f43b234754df6ecd3dfc02de4a},
keywords = {exc2075 myown pn4 preprint},
primaryclass = {math.OC},
timestamp = {2023-09-26T20:56:45.000+0200},
title = {Energy matching in reduced passive and port-Hamiltonian systems},
year = 2023
}
