Finite Volume Structure-Preserving Discretization of 1D Distributed-Parameter Port-Hamiltonian Systems

Authors

Paul Kotyczka

Abstract

A family of finite-dimensional approximate models is proposed which preserves the port-Hamiltonian structure of a class of open systems of conservation laws. The approach is based on conservative generalized leapfrog schemes with given consistency orders in terms of their stencil. The finite volume perspective fits naturally to the formulation of the conservation laws on staggered grids. Some observations on current structure-preserving discretization methods are discussed and related to the proposed approach. A frequently used benchmark example highlights some of the method’s properties and differences to existing structure-preserving schemes.

Keywords

Port-Hamiltonian systems; systems of conservation laws; distributed-parameter systems; semi-discretiziation; finite volume methods

Citation

• Journal: IFAC-PapersOnLine
• Year: 2016
• Volume: 49
• Issue: 8
• Pages: 298–303
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2016.07.457
• Note: 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE 2016- Bertinoro, Italy, 13—15 June 2016

BibTeX

@article{Kotyczka_2016,
  title={{Finite Volume Structure-Preserving Discretization of 1D Distributed-Parameter Port-Hamiltonian Systems}},
  volume={49},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2016.07.457},
  number={8},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Kotyczka, Paul},
  year={2016},
  pages={298--303}
}

Download the bib file

References

• Eymard, Finite volume methods. Handbook of numerical analysis (2000)
• Farle, A port-Hamiltonian finite-element formulation for the Maxwell equations. Proc. IEEE Int. Conf. on Electromagnetics in Advanced Applications (ICEAA), Torino, Italy (2013)
• Fornberg, B. & Ghrist, M. Spatial Finite Difference Approximations for Wave-Type Equations. SIAM Journal on Numerical Analysis vol. 37 105–130 (1999) – 10.1137/s0036142998335881
• Golo, Hamiltonian discretization of boundary control systems. Auto-matica (2004)
• Hamroun, Passivity based control of a reduced port-controlled Hamiltonian model for the shallow water equations. Proc. 47th IEEE Conf. Decision and Control, Cancun, Mexico (2008)
• ISERLES, A. Generalized Leapfrog Methods. IMA Journal of Numerical Analysis vol. 6 381–392 (1986) – 10.1093/imanum/6.4.381
• Iserles, (2009)
• Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
• Kotyczka, P. & Blancato, A. Feedforward control of a channel flow based on a discretized port-Hamiltonian model. IFAC-PapersOnLine vol. 48 194–199 (2015) – 10.1016/j.ifacol.2015.10.238
• LeVeque, (1992)
• LeVeque, (2002)
• Moulla, R., Lefévre, L. & Maschke, B. Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws. Journal of Computational Physics vol. 231 1272–1292 (2012) – 10.1016/j.jcp.2011.10.008
• Ramirez, H., Le Gorrec, Y., Macchelli, A. & Zwart, H. Exponential Stabilization of Boundary Controlled Port-Hamiltonian Systems With Dynamic Feedback. IEEE Transactions on Automatic Control vol. 59 2849–2855 (2014) – 10.1109/tac.2014.2315754
• Seslija, M., Scherpen, J. M. A. & van der Schaft, A. Explicit simplicial discretization of distributed-parameter port-Hamiltonian systems. Automatica vol. 50 369–377 (2014) – 10.1016/j.automatica.2013.11.020
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Kane Yee. Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media. IEEE Transactions on Antennas and Propagation vol. 14 302–307 (1966) – 10.1109/tap.1966.1138693