Structure-preserving discretization and model order reduction of boundary-controlled 1D port-Hamiltonian systems

Authors

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

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.

Keywords

distributed port-hamiltonian systems, finite element method, loewner framework, structure-preserving discretization methods

Citation

Journal: Systems & Control Letters
Year: 2024
Volume: 194
Issue:
Pages: 105947
Publisher: Elsevier BV
DOI: 10.1016/j.sysconle.2024.105947

BibTeX

@article{Toledo_Zucco_2024,
  title={{Structure-preserving discretization and model order reduction of boundary-controlled 1D port-Hamiltonian systems}},
  volume={194},
  ISSN={0167-6911},
  DOI={10.1016/j.sysconle.2024.105947},
  journal={Systems \&amp; Control Letters},
  publisher={Elsevier BV},
  author={Toledo-Zucco, Jesus-Pablo and Matignon, Denis and Poussot-Vassal, Charles and Le Gorrec, Yann},
  year={2024},
  pages={105947}
}

Download the bib file

References

van der Schaft AJ, Maschke BM (2002) Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42(1–2):166–194. https://doi.org/10.1016/s0393-0440(01)00083- – 10.1016/s0393-0440(01)00083-3
Le Gorrec Y, Zwart H, Maschke B (2005) Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM J Control Optim 44(5):1864–1892. https://doi.org/10.1137/04061167 – 10.1137/040611677
Rashad R, Califano F, van der Schaft AJ, Stramigioli S (2020) Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information 37(4):1400–1422. https://doi.org/10.1093/imamci/dnaa01 – 10.1093/imamci/dnaa018
Seslija M, van der Schaft A, Scherpen JMA (2012) Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems. Journal of Geometry and Physics 62(6):1509–1531. https://doi.org/10.1016/j.geomphys.2012.02.00 – 10.1016/j.geomphys.2012.02.006
Golo G, Talasila V, van der Schaft A, Maschke B (2004) Hamiltonian discretization of boundary control systems. Automatica 40(5):757–771. https://doi.org/10.1016/j.automatica.2003.12.01 – 10.1016/j.automatica.2003.12.017
Kotyczka P (2016) Finite Volume Structure-Preserving Discretization of 1D Distributed-Parameter Port-Hamiltonian Systems. IFAC-PapersOnLine 49(8):298–303. https://doi.org/10.1016/j.ifacol.2016.07.45 – 10.1016/j.ifacol.2016.07.457
Trenchant V, Ramirez H, Le Gorrec Y, Kotyczka P (2018) Finite differences on staggered grids preserving the port-Hamiltonian structure with application to an acoustic duct. Journal of Computational Physics 373:673–697. https://doi.org/10.1016/j.jcp.2018.06.05 – 10.1016/j.jcp.2018.06.051
Cardoso-Ribeiro FL, Matignon D, Lefèvre L (2018) A structure-preserving Partitioned Finite Element Method for the 2D wave equation ⁎ ⁎This work is supported by the project ANR-16-CE92-0028, entitled Interconnected Infinite-Dimensional systems for Heterogeneous Media, INFIDHEM, financed by the French National Research Agency (ANR). Further information is available at https://websites.isae-supaero.fr/infidhem/the-project/. IFAC-PapersOnLine 51(3):119–124. https://doi.org/10.1016/j.ifacol.2018.06.03 – 10.1016/j.ifacol.2018.06.033
Serhani A, Matignon D, Haine G (2019) Partitioned Finite Element Method for port-Hamiltonian systems with Boundary Damping: Anisotropic Heterogeneous 2D wave equations. IFAC-PapersOnLine 52(2):96–101. https://doi.org/10.1016/j.ifacol.2019.08.01 – 10.1016/j.ifacol.2019.08.017
Cardoso-Ribeiro FL, Matignon D, Lefèvre L (2020) A partitioned finite element method for power-preserving discretization of open systems of conservation laws. IMA Journal of Mathematical Control and Information 38(2):493–533. https://doi.org/10.1093/imamci/dnaa03 – 10.1093/imamci/dnaa038
Brugnoli A, Alazard D, Pommier-Budinger V, Matignon D (2019) Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates. Applied Mathematical Modelling 75:940–960. https://doi.org/10.1016/j.apm.2019.04.03 – 10.1016/j.apm.2019.04.035
Haine G, Matignon D, Monteghetti F (2022) Structure-preserving discretization of Maxwell’s equations as a port-Hamiltonian system. IFAC-PapersOnLine 55(30):424–429. https://doi.org/10.1016/j.ifacol.2022.11.09 – 10.1016/j.ifacol.2022.11.090
Brugnoli A, Cardoso-Ribeiro FL, Haine G, Kotyczka P (2020) Partitioned finite element method for structured discretization with mixed boundary conditions. IFAC-PapersOnLine 53(2):7557–7562. https://doi.org/10.1016/j.ifacol.2020.12.135 – 10.1016/j.ifacol.2020.12.1351
Brugnoli A, Haine G, Matignon D (2022) Explicit structure-preserving discretization of port-Hamiltonian systems with mixed boundary control. IFAC-PapersOnLine 55(30):418–423. https://doi.org/10.1016/j.ifacol.2022.11.08 – 10.1016/j.ifacol.2022.11.089
Mayo AJ, Antoulas AC (2007) A framework for the solution of the generalized realization problem. Linear Algebra and its Applications 425(2–3):634–662. https://doi.org/10.1016/j.laa.2007.03.00 – 10.1016/j.laa.2007.03.008
Antoulas AC, Lefteriu S, Ionita AC (2017) Chapter 8: A Tutorial Introduction to the Loewner Framework for Model Reduction. Model Reduction and Approximation 335–37 – 10.1137/1.9781611974829.ch8
Benner P, Goyal P, Van Dooren P (2020) Identification of port-Hamiltonian systems from frequency response data. Systems & Control Letters 143:104741. https://doi.org/10.1016/j.sysconle.2020.10474 – 10.1016/j.sysconle.2020.104741
Poussot-Vassal, Data-driven port-Hamiltonian structured identification for non-strictly passive systems. (2023)
Cherifi K, Brugnoli A (2021) Application of data-driven realizations to port-Hamiltonian flexible structures. IFAC-PapersOnLine 54(19):180–185. https://doi.org/10.1016/j.ifacol.2021.11.07 – 10.1016/j.ifacol.2021.11.075
Moreschini A, Simard JD, Astolfi A (2024) Data-driven model reduction for port-Hamiltonian and network systems in the Loewner framework. Automatica 169:111836. https://doi.org/10.1016/j.automatica.2024.11183 – 10.1016/j.automatica.2024.111836
Villegas, (2007)
Macchelli A, Le Gorrec Y, Ramirez H (2020) Exponential Stabilization of Port-Hamiltonian Boundary Control Systems via Energy Shaping. IEEE Trans Automat Contr 65(10):4440–4447. https://doi.org/10.1109/tac.2020.300479 – 10.1109/tac.2020.3004798
Willems JC (1972) Dissipative dynamical systems Part II: Linear systems with quadratic supply rates. Arch Rational Mech Anal 45(5):352–393. https://doi.org/10.1007/bf0027649 – 10.1007/bf00276494
Beattie, (2022)
Sorensen DC (2005) Passivity preserving model reduction via interpolation of spectral zeros. Systems & Control Letters 54(4):347–360. https://doi.org/10.1016/j.sysconle.2004.07.00 – 10.1016/j.sysconle.2004.07.006
Gugercin S, Polyuga RV, Beattie C, van der Schaft A (2012) Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica 48(9):1963–1974. https://doi.org/10.1016/j.automatica.2012.05.05 – 10.1016/j.automatica.2012.05.052
Mehrmann V, Unger B (2023) Control of port-Hamiltonian differential-algebraic systems and applications. Acta Numerica 32:395–515. https://doi.org/10.1017/s096249292200008 – 10.1017/s0962492922000083
Gosea, Model reduction of linear and nonlinear systems in the Loewner framework: A summary. (2015)
Antoulas AC (2008) On the Construction of Passive Models from Frequency Response Data (Konstruktion passiver Modelle auf der Basis von Frequenzbereichsdaten). auto 56(8):447–452. https://doi.org/10.1524/auto.2008.072 – 10.1524/auto.2008.0722