Hamiltonian discretization of boundary control systems

Authors

Goran Golo, Viswanath Talasila, Arjan van der Schaft, Bernhard Maschke

Abstract

A fundamental problem in the simulation and control of complex physical systems containing distributed-parameter components concerns finite-dimensional approximation. Numerical methods for partial differential equations (PDEs) usually assume the boundary conditions to be given, while more often than not the interaction of the distributed-parameter components with the other components takes place precisely via the boundary. On the other hand, finite-dimensional approximation methods for infinite-dimensional input–output systems (e.g., in semi-group format) are not easily relatable to numerical techniques for solving PDEs, and are mainly confined to linear PDEs. In this paper we take a new view on this problem by proposing a method for spatial discretization of boundary control systems based on a particular type of mixed finite elements, resulting in a finite-dimensional input–output system. The approach is based on formulating the distributed-parameter component as an infinite-dimensional port-Hamiltonian system, and exploiting the geometric structure of this representation for the choice of appropriate mixed finite elements. The spatially discretized system is again a port-Hamiltonian system, which can be treated as an approximating lumped-parameter physical system of the same type. In the current paper this program is carried out for the case of an ideal transmission line described by the telegrapher’s equations, and for the two-dimensional wave equation.

Keywords

Boundary control; Spatial discretization; Finite elements; Port-Hamiltonian systems

Citation

• Journal: Automatica
• Year: 2004
• Volume: 40
• Issue: 5
• Pages: 757–771
• Publisher: Elsevier BV
• DOI: 10.1016/j.automatica.2003.12.017

BibTeX

@article{Golo_2004,
  title={{Hamiltonian discretization of boundary control systems}},
  volume={40},
  ISSN={0005-1098},
  DOI={10.1016/j.automatica.2003.12.017},
  number={5},
  journal={Automatica},
  publisher={Elsevier BV},
  author={Golo, Goran and Talasila, Viswanath and van der Schaft, Arjan and Maschke, Bernhard},
  year={2004},
  pages={757--771}
}

Download the bib file

References

• Bossavit, Differential forms and the computation of fields and forces in electromagnetism. European Journal of Mechanics, B/Fluids (1991)
• Bossavit, (1998)
• Courant TJ (1990) Dirac manifolds. Trans Amer Math Soc 319(2):631–661. https://doi.org/10.1090/s0002-9947-1990-0998124-1 – 10.1090/s0002-9947-1990-0998124-1
• Dalsmo M, van der Schaft A (1998) On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM J Control Optim 37(1):54–91. https://doi.org/10.1137/s0363012996312039 – 10.1137/s0363012996312039
• Dorfman, (1993)
• Golo G, van der Schaft A, Stramigioli S (2003) Hamiltonian Formulation of Planar Beams. IFAC Proceedings Volumes 36(2):147–152. https://doi.org/10.1016/s1474-6670(17)38882-1 – 10.1016/s1474-6670(17)38882-1
• Golo G, Talasila V, van der Schaft AJ (2004) Approximation of the Telegrapher’s equations. In: Proceedings of the 41st IEEE Conference on Decision and Control, 2002. IEEE, pp 4587–4592 – 10.1109/cdc.2002.1185099
• Maschke BMJ, van der Schaft AJ (2000) Port Controlled Hamiltonian Representation of Distributed Parameter Systems. IFAC Proceedings Volumes 33(2):27–37. https://doi.org/10.1016/s1474-6670(17)35543-x – 10.1016/s1474-6670(17)35543-x
• Maschke BM, Van Der Schaft AJ, Breedveld PC (1992) An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute 329(5):923–966. https://doi.org/10.1016/s0016-0032(92)90049-m – 10.1016/s0016-0032(92)90049-m
• Olver, (1993)
• (2001) Putting energy back in control. IEEE Control Syst 21(2):18–33. https://doi.org/10.1109/37.915398 – 10.1109/37.915398
• Ortega R, van der Schaft A, Maschke B, Escobar G (2002) Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38(4):585–596. https://doi.org/10.1016/s0005-1098(01)00278-3 – 10.1016/s0005-1098(01)00278-3
• van der Schaft, A. J. (1999). The mathematics of systems and control: From intelligent control to behavioral systems, Interconnection and Geometry. Groningen: Foundation Systems and Control.
• van der Schaft, (2000)
• van der Schaft, Implicit port-controlled Hamiltonian systems. Journal of Society of Instrument and Control Engineers of Japan (2000)
• van der Schaft, The Hamiltonian formulation of energy conserving physical systems with external ports. Archiv fur Electronik und Ubertragungstechnik (1995)
• 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)00083-3 – 10.1016/s0393-0440(01)00083-3
• Talasila, V., Golo, G., & van der Schaft, A. J., 2002. The wave equation as a port-Hamiltonian system and a finite dimensional approximation. In D.S. Gilliam, J. Rosenthal (Eds.), Proceedings of 15th international symposium mathematical theory of networks and systems (MTNS), South Bend.