Beyond Singular Perturbation for linear port-Hamiltonian systems

Authors

Mario Spirito, Bernhard Maschke, Yann Le Gorrec

Abstract

This paper is concerned with the structure/passivity-preserving model order reduction of linear port-Hamiltonian systems. We first present, with the help of the implicit port-Hamiltonian formulation, some issues related to the standard Singular Perturbation technique. We use a simple example to analyze these aspects, paving the way for a recent approach developed to reduce the model order, called Beyond Singular Perturbation. This technique leverages the description of the dominant evolution of the original system whenever a time-scale separation is present among its dynamics. With this tool, we study the system’s Hamiltonian time evolution, showing that we can successfully reconstruct the dominant behavior of the Hamiltonian by slightly modifying the storage function of the resulting reduced-order model. We conclude the work with a discussion about the class of input signals that allow a good performance of the reduced-order model.

Keywords

Model Order Reduction; Beyond Singular Perturbation; Port-Hamiltonian systems; Lagrange subspace; Structure-preserving

Citation

• Journal: European Journal of Control
• Year: 2025
• Volume:
• Issue:
• Pages: 101237
• Publisher: Elsevier BV
• DOI: 10.1016/j.ejcon.2025.101237

BibTeX

@article{Spirito_2025,
  title={{Beyond Singular Perturbation for linear port-Hamiltonian systems}},
  ISSN={0947-3580},
  DOI={10.1016/j.ejcon.2025.101237},
  journal={European Journal of Control},
  publisher={Elsevier BV},
  author={Spirito, Mario and Maschke, Bernhard and Le Gorrec, Yann},
  year={2025},
  pages={101237}
}

Download the bib file

References

• Achleitner, F., Arnold, A. & Mehrmann, V. Hypocoercivity and controllability in linear semi‐dissipative Hamiltonian ordinary differential equations and differential‐algebraic equations. Z Angew Math Mech 103, (2021) – 10.1002/zamm.202100171
• Antoulas, A. C., Sorensen, D. C. & Gugercin, S. A survey of model reduction methods for large-scale systems. Contemporary Mathematics 193–219 (2001) doi:10.1090/conm/280/04630 – 10.1090/conm/280/04630
• Beattie, Structure-preserving model reduction for nonlinear port-Hamiltonian systems. (2011)
• Beattie, Structure-preserving interpolatory model reduction for port-Hamiltonian differential-algebraic systems. (2022)
• Beattie, C., Mehrmann, V., Xu, H. & Zwart, H. Linear port-Hamiltonian descriptor systems. Math. Control Signals Syst. 30, (2018) – 10.1007/s00498-018-0223-3
• Benner, P., Gugercin, S. & Willcox, K. A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems. SIAM Rev. 57, 483–531 (2015) – 10.1137/130932715
• Benner, (2017)
• Cervera, J., van der Schaft, A. J. & Baños, A. Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica 43, 212–225 (2007) – 10.1016/j.automatica.2006.08.014
• Duindam, (2009)
• Fortuna, (2012)
• FUJIMOTO, K. Balanced Realization and Model Order Reduction for Port-Hamiltonian Systems. JSDD 2, 694–702 (2008) – 10.1299/jsdd.2.694
• Güdücü, C., Liesen, J., Mehrmann, V. & Szyld, D. B. On Non-Hermitian Positive (Semi)Definite Linear Algebraic Systems Arising from Dissipative Hamiltonian DAEs. SIAM J. Sci. Comput. 44, A2871–A2894 (2022) – 10.1137/21m1458594
• Gugercin, S. An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems. Linear Algebra and its Applications 428, 1964–1986 (2008) – 10.1016/j.laa.2007.10.041
• Gugercin, S. & Antoulas, A. C. A Survey of Model Reduction by Balanced Truncation and Some New Results. International Journal of Control 77, 748–766 (2004) – 10.1080/00207170410001713448
• Gugercin, Interpolation-based H2 model reduction for port-Hamiltonian systems. (2009)
• Gugercin, S., Polyuga, R. V., Beattie, C. & van der Schaft, A. Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica 48, 1963–1974 (2012) – 10.1016/j.automatica.2012.05.052
• Hao, J., Chen, C., Shi, L. & Wang, J. Nonlinear Decentralized Disturbance Attenuation Excitation Control for Power Systems With Nonlinear Loads Based on the Hamiltonian Theory. IEEE Trans. On Energy Conversion 22, 316–324 (2007) – 10.1109/tec.2005.859977
• Hartmann, C., Vulcanov, V.-M. & Schütte, C. Balanced Truncation of Linear Second-Order Systems: A Hamiltonian Approach. Multiscale Model. Simul. 8, 1348–1367 (2010) – 10.1137/080732717
• Hauschild, S.-A., Marheineke, N. & Mehrmann, V. Model reduction techniques for port‐Hamiltonian differential‐algebraic systems. Proc Appl Math and Mech 19, (2019) – 10.1002/pamm.201900040
• Horn, (2012)
• Ionescu, Moment matching for linear port Hamiltonian systems. (2011)
• Ionescu, T. C. & Astolfi, A. Families of moment matching based, structure preserving approximations for linear port Hamiltonian systems. Automatica 49, 2424–2434 (2013) – 10.1016/j.automatica.2013.05.006
• Jardón-Kojakhmetov, H., Muñoz-Arias, M. & Scherpen, J. M. A. Model reduction of a flexible-joint robot: a port-Hamiltonian approach. IFAC-PapersOnLine 49, 832–837 (2016) – 10.1016/j.ifacol.2016.10.269
• Kawano, Y. & Scherpen, J. M. A. Structure Preserving Truncation of Nonlinear Port Hamiltonian Systems. IEEE Trans. Automat. Contr. 63, 4286–4293 (2018) – 10.1109/tac.2018.2811787
• Kokotović, (1999)
• Kumar, R. & Ezhilarasi, D. A state-of-the-art survey of model order reduction techniques for large-scale coupled dynamical systems. Int. J. Dynam. Control 11, 900–916 (2022) – 10.1007/s40435-022-00985-7
• Lopezlena, R., Scherpen, J. M. A. & Fujimoto, K. Energy-Storage Balanced Reduction of Port-Hamiltonian Systems. IFAC Proceedings Volumes 36, 69–74 (2003) – 10.1016/s1474-6670(17)38869-9
• Mamunuzzaman, (2022)
• Maschke, Port-controlled Hamiltonian systems: modelling origins and systemtheoretic properties. (1993)
• Mehl, C., Mehrmann, V. & Sharma, P. Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations. SIAM J. Matrix Anal. & Appl. 37, 1625–1654 (2016) – 10.1137/16m1067330
• Mehl, C., Mehrmann, V. & Wojtylak, M. Linear Algebra Properties of Dissipative Hamiltonian Descriptor Systems. SIAM J. Matrix Anal. & Appl. 39, 1489–1519 (2018) – 10.1137/18m1164275
• Mehrmann, Differential-algebraic systems with dissipative Hamiltonian structure. Mathematics of Control, Signals, and Systems (2023)
• Moser, (2022)
• Polyuga, R. V. & van der Schaft, A. Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity. Automatica 46, 665–672 (2010) – 10.1016/j.automatica.2010.01.018
• Polyuga, R. V. & van der Schaft, A. J. Effort- and flow-constraint reduction methods for structure preserving model reduction of port-Hamiltonian systems. Systems & Control Letters 61, 412–421 (2012) – 10.1016/j.sysconle.2011.12.008
• Reato, F. M., Ricci, C., Misfatto, J., Calzaferri, M. & Cinquemani, S. Multi-physics model of DC micro motors for dynamic operations. Sensors and Actuators A: Physical 361, 114570 (2023) – 10.1016/j.sna.2023.114570
• van der Schaft, The Hamiltonian formulation of energy conserving physical systems with external ports. AEÜ International Journal of Electronics and Communications (1995)
• Spirito, (2023)
• Spirito, M., Maschke, B. & Le Gorrec, Y. Singular Perturbations for Implicit port-Hamiltonian systems. 2024 European Control Conference (ECC) 2071–2076 (2024) doi:10.23919/ecc64448.2024.10590942 – 10.23919/ecc64448.2024.10590942
• Suman, Investigation and implementation of model order reduction technique for large scale dynamical systems. Archives of Computational Methods in Engineering (2022)
• Van der Schaft, (2000)
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. FnT in Systems and Control 1, 173–378 (2014) – 10.1561/2600000002
• van der Schaft, A. & Maschke, B. Generalized port-Hamiltonian DAE systems. Systems & Control Letters 121, 31–37 (2018) – 10.1016/j.sysconle.2018.09.008
• van der Schaft, A. & Mehrmann, V. Linear port-Hamiltonian DAE systems revisited. Systems & Control Letters 177, 105564 (2023) – 10.1016/j.sysconle.2023.105564
• van der Schaft, Structure-preserving model reduction of complex physical systems. (2009)
• Vasil’eva, A. B. ASYMPTOTIC BEHAVIOUR OF SOLUTIONS TO CERTAIN PROBLEMS INVOLVING NON-LINEAR DIFFERENTIAL EQUATIONS CONTAINING A SMALL PARAMETER MULTIPLYING THE HIGHEST DERIVATIVES. Russ. Math. Surv. 18, 13–84 (1963) – 10.1070/rm1963v018n03abeh001137
• Willems, J. C. Dissipative dynamical systems part I: General theory. Arch. Rational Mech. Anal. 45, 321–351 (1972) – 10.1007/bf00276493
• Wolf, T., Lohmann, B., Eid, R. & Kotyczka, P. Passivity and Structure Preserving Order Reduction of Linear Port-Hamiltonian Systems Using Krylov Subspaces. European Journal of Control 16, 401–406 (2010) – 10.3166/ejc.16.401-406