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Structure-Preserving Discretization and Model Order Reduction of Boundary-Controlled 1D Port-Hamiltonian Systems

Jesus-Pablo Toledo-Zucco, Denis Matignon, Charles Poussot-Vassal, Yann Le Gorrec

TL;DR

This work targets the discretization and reduction of one-dimensional Boundary-Controlled Port-Hamiltonian Systems (BC-PHSs) with boundary actuation. It combines a structure-preserving Partitioned Finite Element Method (PFEM) for spatial discretization with the Loewner framework for passive model order reduction, along with a projector to recover physical states from reduced coordinates. The authors explicitly formulate discretized port-Hamiltonian matrices in terms of the PDE coefficients and boundary-parameterization matrices, and show that passivity (or impedance energy preservation) is preserved in the discretized model and can be retained in the reduced model using spectral-zero based interpolation and a suitable shift. The approach is demonstrated on a 1D wave equation and a Timoshenko beam, illustrating accurate input/output behavior within a target frequency range and energy-consistent dynamics, with a practical projector to interpret ROM states physically. The methodology yields efficient, stable reduced models suitable for real-time control and simulation of boundary-driven hyperbolic PDEs, with potential extensions to higher-order PDEs, parametric/MIMO cases, and broader boundary control formulations.

Abstract

This paper presents a systematic methodology for the discretization and reduction of a class of one-dimensional Partial Differential Equations (PDEs) with inputs and outputs collocated at the spatial boundaries. The class of system that we consider is known as Boundary-Controlled Port-Hamiltonian Systems (BC-PHSs) and covers a wide class of Hyperbolic PDEs with a large type of boundary inputs and outputs. This is, for instance, the case of waves and beams with Neumann, Dirichlet, or mixed boundary conditions. Based on a Partitioned Finite Element Method (PFEM), we develop a numerical scheme for the structure-preserving spatial discretization for the class of one-dimensional BC-PHSs. We show that if the initial PDE is passive (or impedance energy preserving), the discretized model also is. In addition and since the discretized model or Full Order Model (FOM) can be of large dimension, we recall the standard Loewner framework for the Model Order Reduction (MOR) using frequency domain interpolation. We recall the main steps to produce a Reduced Order Model (ROM) that approaches the FOM in a given range of frequencies. We summarize the steps to follow in order to obtain a ROM that preserves the passive structure as well. Finally, we provide a constructive way to build a projector that allows to recover the physical meaning of the state variables from the ROM to the FOM. We use the one-dimensional wave equation and the Timoshenko beam as examples to show the versatility of the proposed approach.

Structure-Preserving Discretization and Model Order Reduction of Boundary-Controlled 1D Port-Hamiltonian Systems

TL;DR

This work targets the discretization and reduction of one-dimensional Boundary-Controlled Port-Hamiltonian Systems (BC-PHSs) with boundary actuation. It combines a structure-preserving Partitioned Finite Element Method (PFEM) for spatial discretization with the Loewner framework for passive model order reduction, along with a projector to recover physical states from reduced coordinates. The authors explicitly formulate discretized port-Hamiltonian matrices in terms of the PDE coefficients and boundary-parameterization matrices, and show that passivity (or impedance energy preservation) is preserved in the discretized model and can be retained in the reduced model using spectral-zero based interpolation and a suitable shift. The approach is demonstrated on a 1D wave equation and a Timoshenko beam, illustrating accurate input/output behavior within a target frequency range and energy-consistent dynamics, with a practical projector to interpret ROM states physically. The methodology yields efficient, stable reduced models suitable for real-time control and simulation of boundary-driven hyperbolic PDEs, with potential extensions to higher-order PDEs, parametric/MIMO cases, and broader boundary control formulations.

Abstract

This paper presents a systematic methodology for the discretization and reduction of a class of one-dimensional Partial Differential Equations (PDEs) with inputs and outputs collocated at the spatial boundaries. The class of system that we consider is known as Boundary-Controlled Port-Hamiltonian Systems (BC-PHSs) and covers a wide class of Hyperbolic PDEs with a large type of boundary inputs and outputs. This is, for instance, the case of waves and beams with Neumann, Dirichlet, or mixed boundary conditions. Based on a Partitioned Finite Element Method (PFEM), we develop a numerical scheme for the structure-preserving spatial discretization for the class of one-dimensional BC-PHSs. We show that if the initial PDE is passive (or impedance energy preserving), the discretized model also is. In addition and since the discretized model or Full Order Model (FOM) can be of large dimension, we recall the standard Loewner framework for the Model Order Reduction (MOR) using frequency domain interpolation. We recall the main steps to produce a Reduced Order Model (ROM) that approaches the FOM in a given range of frequencies. We summarize the steps to follow in order to obtain a ROM that preserves the passive structure as well. Finally, we provide a constructive way to build a projector that allows to recover the physical meaning of the state variables from the ROM to the FOM. We use the one-dimensional wave equation and the Timoshenko beam as examples to show the versatility of the proposed approach.
Paper Structure (20 sections, 5 theorems, 75 equations, 7 figures)

This paper contains 20 sections, 5 theorems, 75 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem Formulation
  3. Boundary-controlled port-Hamiltonian systems in a one-dimensional domain
  4. Partial Differential Equation
  5. Hamiltonian
  6. Boundary inputs and outputs
  7. Proposed methodology
  8. Partitioned Finite Element Method
  9. Spatial discretization through the projections.
  10. Structure Preservation of the Hamiltonian
  11. Model Order Reduction in the Loewner Framework
  12. Standard solution in the Loewner approach
  13. Realization with real values:
  14. Order reduction via Singular Value Decomposition
  15. Discussion about the structure-preserving ROM
  16. ...and 5 more sections

Key Result

Theorem 1

Let $V_\mathcal{B},V_\mathcal{C} \in \mathbb{R}^{n \times 2n}$ be two full-rank matrices such that Define the boundary input and boundary output as The system Eq:PDE-InputOutput is a boundary control system. Furthermore, for all inputs $u(t) \in L_{loc}^2((0,\infty),\mathbb{R}^n)$, initial data$x_0(\zeta)\in H^{1}((a,b);\mathbb{R}^n)$ with $u(0) = V_\mathcal{B}\left(\right)$, the following bal

Figures (7)

  • Figure 1: Diagram of the article contributions. Read as follows: from the physical laws, a class of boundary control systems is modeled using PDEs. The PDEs have state $x(\zeta,t)$ with $\zeta \in[a,b]$ and $t\geq 0$. The dynamics of the PDE is completely described by the matrices $(P,G,\mathcal{H},V_\mathcal{B},V_\mathcal{C})$. The spatial variable $\zeta$ is discretized using the FEM obtaining an $ODE$ with state $x_d(t)$ of size $N$. The matrices of the obtained ODE are defined as a function of the PDE matrices $(P,G,\mathcal{H},V_\mathcal{B},V_\mathcal{C})$ and the basis functions $\Phi(\zeta)$ (required for discretization). Finally, the ODE is reduced using the Loewner framework, obtaining another $ODE$ with state $x_r(t)$ with size $r \ll N$. We provide the projector $'T'$ that enables to approximate the state $x_d(t)$ of the FOM using the state $x_r(t)$ of the ROM.
  • Figure 2: Example of basis functions using $N=4$.
  • Figure 3: Magnitude Bode diagrams of the discretized model and the reduced one.
  • Figure 4: Energy of the discretized model, preliminary ROM and final ROM.
  • Figure 5: Bode magnitude diagrams of the discretized Timoshenko beam model and the ROM.
  • ...and 2 more figures

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Theorem 1
  • Lemma 2
  • Example 1
  • Remark 3
  • Proposition 1
  • Proposition 2
  • ...and 6 more