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Energy matching in reduced passive and port-Hamiltonian systems

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 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.

Energy matching in reduced passive and port-Hamiltonian systems

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 -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 -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.
Paper Structure (21 sections, 9 theorems, 69 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 21 sections, 9 theorems, 69 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review and state-of-the-art
  3. Organization of the manuscript
  4. Notation and abbreviations
  5. Preliminaries
  6. Controllability and Observability
  7. Port-Hamiltonian systems and the Kalman-Yakubovich-Popov inequality
  8. Extended port-Hamiltonian systems
  9. Minimality of extended pH systems
  10. Energy matching algorithm for surrogate models
  11. Analysis of the energy matching optimization problem
  12. A special case: Positive-real balanced truncation
  13. Numerical approach
  14. Model reduction workflow
  15. Numerical experiments
  16. ...and 6 more sections

Key Result

Theorem 2.2

Consider the dynamical system $\Sigma$ in eq:LTI and the associated KYP inequality eq:KYP.

Figures (3)

  • Figure 1: $\mathcal{H}_2$-error of the input-output dynamic and the Hamiltonian dynamic over the reduced orders in the mass-spring-damper example.
  • Figure 2: Error trajectory of the output and the Hamiltonian.
  • Figure 3: $\mathcal{H}_2$-error of the input-output dynamic and the Hamiltonian dynamic over the reduced orders in the poroelasticity example.

Theorems & Definitions (25)

  • Definition 2.1: Port-Hamiltonian system SchJ14
  • Theorem 2.2
  • proof
  • Example 4.1
  • Theorem 4.2
  • proof
  • Corollary 4.3
  • proof
  • Example 4.4
  • Theorem 4.5
  • ...and 15 more