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Symplectic model order reduction of port-Hamiltonian systems

Silke Glas, Mir Mamunuzzaman, Hongliang Mu, Hans Zwart

TL;DR

This work develops a structure-preserving model order reduction method for linear, time-invariant port-Hamiltonian systems by leveraging symplectic MOR concepts. The proposed symplectic-pH MOR preserves the port-Hamiltonian form and the block structure of the full-order model, ensuring the reduced model remains physically interpretable; in the special case of mass–spring–damper systems it recovers the second-order Arnoldi method. The method achieves transfer-function interpolation at prescribed frequencies and demonstrates competitive or superior accuracy (H2 and H∞) on an LRCR circuit, while highlighting potential instability issues for alternative structure-preserving approaches like SO-Arnoldi in some regimes. Overall, symplectic-pH MOR offers a novel and practically relevant reduction strategy for electrical networks and other block-structured pH systems, with future work including derivative matching to further improve fidelity.

Abstract

This work proposes a novel structure-preserving model order reduction (MOR) method for linear, time-invariant port-Hamiltonian (pH) systems. Our goal is to construct a reduced order pH system, which can still be interpreted in the physical domain of the full order model. By this we mean, that if an electrical circuit is the initial high-dimensional pH system, we want the reduced order model to be still interpretable as an electronic circuit. In the case of the well-known mass spring damper (MSD) system, there are MOR methods available, which already guarantee the preservation of this particular structure. Moreover, we show that our new structure-preserving MOR method, which is based on symplectic MOR methods, will recover the known second-order Arnoldi method in the case of MSD systems. However, for the example of an electrical circuit pH model (and more models of similar block structure), our method yields a novel model reduction method. We present numerical results on the aforementioned electronic circuit model, highlighting the advantages of the proposed method.

Symplectic model order reduction of port-Hamiltonian systems

TL;DR

This work develops a structure-preserving model order reduction method for linear, time-invariant port-Hamiltonian systems by leveraging symplectic MOR concepts. The proposed symplectic-pH MOR preserves the port-Hamiltonian form and the block structure of the full-order model, ensuring the reduced model remains physically interpretable; in the special case of mass–spring–damper systems it recovers the second-order Arnoldi method. The method achieves transfer-function interpolation at prescribed frequencies and demonstrates competitive or superior accuracy (H2 and H∞) on an LRCR circuit, while highlighting potential instability issues for alternative structure-preserving approaches like SO-Arnoldi in some regimes. Overall, symplectic-pH MOR offers a novel and practically relevant reduction strategy for electrical networks and other block-structured pH systems, with future work including derivative matching to further improve fidelity.

Abstract

This work proposes a novel structure-preserving model order reduction (MOR) method for linear, time-invariant port-Hamiltonian (pH) systems. Our goal is to construct a reduced order pH system, which can still be interpreted in the physical domain of the full order model. By this we mean, that if an electrical circuit is the initial high-dimensional pH system, we want the reduced order model to be still interpretable as an electronic circuit. In the case of the well-known mass spring damper (MSD) system, there are MOR methods available, which already guarantee the preservation of this particular structure. Moreover, we show that our new structure-preserving MOR method, which is based on symplectic MOR methods, will recover the known second-order Arnoldi method in the case of MSD systems. However, for the example of an electrical circuit pH model (and more models of similar block structure), our method yields a novel model reduction method. We present numerical results on the aforementioned electronic circuit model, highlighting the advantages of the proposed method.
Paper Structure (17 sections, 8 theorems, 68 equations, 4 figures, 1 algorithm)

This paper contains 17 sections, 8 theorems, 68 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and problem setting
  3. Model order reduction by projection
  4. Review of the port-Hamiltonian representation
  5. Model reduction of port-Hamiltonian systems
  6. Symplectic model reduction
  7. Symplectic geometry on complex vector spaces
  8. Symplectic model reduction of port-Hamiltonian systems
  9. Port-Hamiltonian system without dissipation
  10. Port-Hamiltonian system with dissipation
  11. Construction of matrices for symplectic-pH with dissipation
  12. Mass spring damper systems
  13. Numerical Results
  14. Problem statement
  15. Second-order form of the LRCR circuit
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose $G(s) = C(s I_{2n} - A)^{-1}B + D$. Given a set of distinct interpolation points $\{s_m\}_{m= 1}^{M} \subset {\mathbb C}$ and right tangent directions $\{\mu_m\}_{m = 1}^{M} \subset {\mathbb C}^{p}$, define $V \in {\mathbb C}^{2n \times M}$ as Then for any $W \in {\mathbb C}^{2n \times M}$ satisfying $W^{*}V = I_{M}$, the reduced systems $G_{red}(s) = C_{red}(s I_{M}- A_{red})^{-1}B_{red}

Figures (4)

  • Figure 1: LRCR circuit
  • Figure 2: Magnitude (left) and phase (right) of the transfer function (FOM) corresponds to the LRCR circuit where $C_i= 1$ pF, $L_{i} = 1$ nH, $R_{i,1} = 1$ m$\Omega$, and $R_{i,2} = 1\,$k$\Omega$.
  • Figure 3: $H_2$ (left) and $H_{\infty}$ (right) errors of ROMs obtained by different structure-preserving MOR methods.
  • Figure 4: Poles of ROMs obtained by the SO-Arnoldi method with reduced dimension $r=28$ (left) and $r=36$ (right).

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2: Gugercin2012IRKA
  • proof
  • Definition 1
  • Proposition 3
  • proof
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • ...and 7 more