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Structure-preserving model reduction on manifolds of port-Hamiltonian systems

Silke Glas, Hongliang Mu

Abstract

This paper considers structure-preserving model order reduction (MOR) techniques for port-Hamiltonian (pH) systems, which are typically derived from energy-based modelling. To keep favorable properties of pH systems such as stability and passivity in a reduced order model (ROM), we use structure-preserving methods in the reduction process. There exists an extensive literature on structure-preserving MOR methods of pH systems, however, to the best of our knowledge, there does not exist an intrusive structure-preserving MOR method for nonlinear pH systems on the base of general nonlinear approximation maps. To close this gap, we propose a MOR method for pH systems based on the idea of the generalized manifold Galerkin (GMG) reduction. The resulting MOR method can be applied to both linear and nonlinear pH systems resulting in ROMs, which are again of pH form. For the numerical examples, we employ a linear and a nonlinear mass-spring-damper system and the results show that the proposed MOR methods have lower relative reduction error compared to existing methods.

Structure-preserving model reduction on manifolds of port-Hamiltonian systems

Abstract

This paper considers structure-preserving model order reduction (MOR) techniques for port-Hamiltonian (pH) systems, which are typically derived from energy-based modelling. To keep favorable properties of pH systems such as stability and passivity in a reduced order model (ROM), we use structure-preserving methods in the reduction process. There exists an extensive literature on structure-preserving MOR methods of pH systems, however, to the best of our knowledge, there does not exist an intrusive structure-preserving MOR method for nonlinear pH systems on the base of general nonlinear approximation maps. To close this gap, we propose a MOR method for pH systems based on the idea of the generalized manifold Galerkin (GMG) reduction. The resulting MOR method can be applied to both linear and nonlinear pH systems resulting in ROMs, which are again of pH form. For the numerical examples, we employ a linear and a nonlinear mass-spring-damper system and the results show that the proposed MOR methods have lower relative reduction error compared to existing methods.
Paper Structure (16 sections, 1 theorem, 51 equations, 4 figures, 1 table, 3 algorithms)

This paper contains 16 sections, 1 theorem, 51 equations, 4 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Introduction to port-Hamiltonian systems
  4. Generalized manifold Galerkin
  5. Discrete empirical interpolation method for pH systems
  6. Generalized manifold Galerkin for pH systems
  7. State approximation and corresponding ROMs
  8. Constructing pH systems with state approximation
  9. State-approximations and resulting ROMs
  10. Linear state approximation and GMG-POD-ROM
  11. Quadratic state approximation and GMG-QM-ROM
  12. Numerical results
  13. Implementation details and error measures
  14. Linear mass-spring-damper system
  15. Nonlinear mass-spring-damper system
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $\Sigma(\bm{J},\bm{R},\mathcal{H},\bm{B}, \bm{u},{\bm y},{\bm x}_0)$ be an N-dimensional pH system equ:Nonlinear_FOM_pH, and let $\varphi\colon\mathbb{R}^{r}\mapsto\mathbb{R}^{N}$ be an approximation map. If for all $\check{\bm x}\in\mathbb{R}^{r}$, ${\rm span}(\bm{B})\subseteq {\rm span}(\bm{D}

Figures (4)

  • Figure 1: Linear mass-spring-damper system
  • Figure 2: Reduction and output error obtained by SP1-POD-ROM, GMG-POD-ROM, and GMG-QM-ROM for the linear mass-spring-damper system with the inputs $\bm{u}(t) = 0.1$ and $\bm{u}(t)=0.1\sin(t)$. Linear-app represents the projection error of corresponding linear embedding maps. The lower bound for the ROMs resulting from the quadratic embeddings maps is computed by \ref{['equ:non_app_lower_bound']}.
  • Figure 3: Reduction and output error obtained by SP2-POD-ROM, GMG-POD-ROM, and GMG-QM-ROM for the nonlinear mass-spring-damper system with the inputs $\bm{u}(t) = 0.1$ and $\bm{u}(t)=0.1\sin(t)$. Linear-app represents the projection error of corresponding linear embedding maps. The lower bound for the ROMs resulting from the quadratic embeddings maps is computed by \ref{['equ:non_app_lower_bound']}.
  • Figure 4: Error in the energy balance equation \ref{['equ:def_energy_conserve_error']} of the FOM and ROMs with a reduced order of 16 obtained by the SP2-POD-ROM, GMG-POD-ROM, and GMG-QM-ROM for the nonlinear mass-spring-damper system with two different inputs.

Theorems & Definitions (2)

  • Theorem 3.1
  • proof