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 H2-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 H2-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 Generic
%1 holicki2023energy
%A Holicki, T.
%A Nicodemus, J.
%A Schwerdtner, P.
%A Unger, B.
%D 2023
%K PN4-3(II) PN4 EXC2075 selected
%R 10.48550/arXiv.2309.05778
%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 H2-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 H2-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.
@misc{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 H2-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 H2-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 = {2024-03-26T11:56:19.000+0100},
author = {Holicki, T. and Nicodemus, J. and Schwerdtner, P. and Unger, B.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2d848ffa73bdd7b6271778c13e46d321a/exc2075},
doi = {10.48550/arXiv.2309.05778},
howpublished = {ArXiV},
interhash = {fef9d3e8aaf8abe4be1f249a0e8e10bc},
intrahash = {d848ffa73bdd7b6271778c13e46d321a},
keywords = {PN4-3(II) PN4 EXC2075 selected},
timestamp = {2024-03-26T11:56:19.000+0100},
title = {Energy matching in reduced passive and port-Hamiltonian systems},
year = 2023
}