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Model reduction techniques for linear constant coefficient port-Hamiltonian differential-algebraic systems

Sarah-Alexa Hauschild, Nicole Marheineke, Volker Mehrmann

TL;DR

The paper addresses reducing large port-Hamiltonian differential-algebraic systems (pHDAEs) while preserving energy-based structure and all algebraic constraints. It adapts two reduction paradigms—power-conservation based methods (ECRM, FCRM) and moment matching (MM)—to the pHDAE setting using a structure-preserving regularization that decouples dynamic and algebraic variables, enabling index reduction up to two. Numerical experiments on semi-discretized flow problems (Stokes and Oseen) and a holonomically constrained damped mass-spring system show that ECRM generally provides robust $H_\infty$ and $H_2$ performance, MM offers excellent local moment matching, and FCRM can be competitive but is limited by invertibility requirements and potential high-frequency feed-through effects. Overall, the work delivers practical, structure-preserving reduced-order models for complex multi-physics networks and guides method choice based on problem characteristics and frequency emphasis.

Abstract

Port-based network modeling of multi-physics problems leads naturally to a formulation as port-Hamiltonian differential-algebraic system. In this way, the physical properties are directly encoded in the structure of the model. Since the state space dimension of such systems may be very large, in particular when the model is a space-discretized partial differential-algebraic system, in optimization and control there is a need for model reduction methods that preserve the port-Hamiltonian structure while keeping the (explicit and implicit) algebraic constraints unchanged. To combine model reduction for differential-algebraic equations with port-Hamiltonian structure preservation, we adapt two classes of techniques (reduction of the Dirac structure and moment matching) to handle port-Hamiltonian differential-algebraic equations. The performance of the methods is investigated for benchmark examples originating from semi-discretized flow problems and mechanical multibody systems.

Model reduction techniques for linear constant coefficient port-Hamiltonian differential-algebraic systems

TL;DR

The paper addresses reducing large port-Hamiltonian differential-algebraic systems (pHDAEs) while preserving energy-based structure and all algebraic constraints. It adapts two reduction paradigms—power-conservation based methods (ECRM, FCRM) and moment matching (MM)—to the pHDAE setting using a structure-preserving regularization that decouples dynamic and algebraic variables, enabling index reduction up to two. Numerical experiments on semi-discretized flow problems (Stokes and Oseen) and a holonomically constrained damped mass-spring system show that ECRM generally provides robust and performance, MM offers excellent local moment matching, and FCRM can be competitive but is limited by invertibility requirements and potential high-frequency feed-through effects. Overall, the work delivers practical, structure-preserving reduced-order models for complex multi-physics networks and guides method choice based on problem characteristics and frequency emphasis.

Abstract

Port-based network modeling of multi-physics problems leads naturally to a formulation as port-Hamiltonian differential-algebraic system. In this way, the physical properties are directly encoded in the structure of the model. Since the state space dimension of such systems may be very large, in particular when the model is a space-discretized partial differential-algebraic system, in optimization and control there is a need for model reduction methods that preserve the port-Hamiltonian structure while keeping the (explicit and implicit) algebraic constraints unchanged. To combine model reduction for differential-algebraic equations with port-Hamiltonian structure preservation, we adapt two classes of techniques (reduction of the Dirac structure and moment matching) to handle port-Hamiltonian differential-algebraic equations. The performance of the methods is investigated for benchmark examples originating from semi-discretized flow problems and mechanical multibody systems.

Paper Structure

This paper contains 11 sections, 6 theorems, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model framework of port-Hamiltonian differential-algebraic systems
  3. Port-Hamiltonian systems
  4. Structure-preserving regularization
  5. Model reduction techniques
  6. Power conservation based model order reduction
  7. Moment Matching
  8. Numerical results
  9. Flow problems
  10. Damped mass-spring system
  11. Conclusion

Key Result

Theorem 1

Consider a pHDAE of the form BEAMXZ18. If for given input function $u$ the system has a (classical) solution $x\in \mathcal{C}^1(\mathbb{I},\mathbb{R}^n)$ in $\mathbb{I}$, then Furthermore, if $W=0$, then $\frac{d}{dt}{\mathcal{H}}(x)=u^Ty$.

Figures (4)

  • Figure 4.1: Flow problems: Stokes (top) and Oseen (bottom). Relative errors for ECRM, FCRM as well as MM at $s_0=\infty$ and at $s_0=0$, plotted over frequency (left) and in $H_\infty$-norm over reduced state size $r$(right).
  • Figure 4.2: Error behavior of FCRM for very high frequencies due to the additional feed-through term in the reduced Oseen model.
  • Figure 4.3: Damped mass-spring system with holonomic constraint MehS05
  • Figure 4.4: Mass-spring system: index-reduced formulation \ref{['MultiBodyPH']}, $n=2g+1$(top) and index-reduced formulation via minimal extension \ref{['MultiBodyME']}, $n=2g+2$(bottom). Relative errors of reduced transfer functions plotted over frequency (left) and in $H_\infty$-norm over reduced state size $r$(right). (MM is performed at $s_0=\infty$ as well as at $s_0=10^{-10}$ for \ref{['MultiBodyPH']} and at $s_0=0$ for \ref{['MultiBodyME']}.)

Theorems & Definitions (14)

  • Definition 1: pHDAE
  • Theorem 1
  • Lemma 1
  • Theorem 2: Decoupled pHDAE
  • Remark 1
  • Theorem 3: Reduced model by ECRM
  • proof
  • Theorem 4: Reduced model by FCRM
  • proof
  • Remark 2
  • ...and 4 more