Energy Matching in Reduced Passive and Port-Hamiltonian Systems

Authors

Tobias Holicki, Jonas Nicodemus, Paul Schwerdtner, Benjamin Unger

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 multiobjective minimization problem corresponding to \( \mathcal{H}_2 \)-optimal MOR for pH systems. We show that this optimization problem is uniquely solvable and can be recast as a standard semidefinite program, and we present two numerical approaches for solving it. The results are illustrated with three academic examples.

Citation

• Journal: SIAM Journal on Control and Optimization
• Year: 2025
• Volume: 63
• Issue: 3
• Pages: 2154–2176
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/23m1600931

BibTeX

@article{Holicki_2025,
  title={{Energy Matching in Reduced Passive and Port-Hamiltonian Systems}},
  volume={63},
  ISSN={1095-7138},
  DOI={10.1137/23m1600931},
  number={3},
  journal={SIAM Journal on Control and Optimization},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Holicki, Tobias and Nicodemus, Jonas and Schwerdtner, Paul and Unger, Benjamin},
  year={2025},
  pages={2154--2176}
}

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