Port-Hamiltonian Representation and Discretization of Undamped Wave Equation System

Authors

Abstract

The port-Hamiltonian representation of undamped wave equation system captures the essential energy and dynamical features in an unified framework. In this paper, an undamped wave equation system is given in the port-Hamiltonian formulation. The important issue of obtaining discrete version of the port-Hamiltonian system representation is explored. The exact discretization method is applied to the port-Hamiltonian system which transforms the system from a continuous to a discrete state space setting. The development of the discretized port Hamiltonian system formulation is necessary foundation for the discrete and computer based regulator and model realizations. The results are illustrated by numerical simulations.

Keywords

Port-Hamiltonian; Exact Discretization; Undamped Wave Equation System

Citation

Journal: IFAC-PapersOnLine
Year: 2016
Volume: 49
Issue: 8
Pages: 309–314
Publisher: Elsevier BV
DOI: 10.1016/j.ifacol.2016.07.459
Note: 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE 2016- Bertinoro, Italy, 13—15 June 2016

BibTeX

@article{Xu_2016,
  title={{Port-Hamiltonian Representation and Discretization of Undamped Wave Equation System}},
  volume={49},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2016.07.459},
  number={8},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Xu, Qingqing and Dubljevic, Stevan},
  year={2016},
  pages={309--314}
}

Download the bib file

References

Bassi, L., Macchelli, A. & Melchiorri, C. An Algorithm to Discretize One-Dimensional Distributed Port Hamiltonian Systems. Lecture Notes in Control and Information Sciences 61–73 doi:10.1007/978-3-540-73890-9_4 – 10.1007/978-3-540-73890-9_4
Duindam, (2009)
Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica vol. 40 757–771 (2004) – 10.1016/j.automatica.2003.12.017
Havu, V. & Malinen, J. The Cayley Transform as a Time Discretization Scheme. Numerical Functional Analysis and Optimization vol. 28 825–851 (2007) – 10.1080/01630560701493321
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
Kazantzis, N. & Kravaris, C. Time-discretization of nonlinear control systems via Taylor methods. Computers & Chemical Engineering vol. 23 763–784 (1999) – 10.1016/s0098-1354(99)00007-1
Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
Maschke, B. & van der Schaft, A. 4 Compositional Modelling of Distributed-Parameter Systems. Lecture Notes in Control and Information Sciences 115–154 (2005) doi:10.1007/11334774_4 – 10.1007/11334774_4
Maschke, B. M. & van der Schaft, A. J. Port-Controlled Hamiltonian Systems: Modelling Origins and Systemtheoretic Properties. IFAC Proceedings Volumes vol. 25 359–365 (1992) – 10.1016/s1474-6670(17)52308-3
Alizadeh Moghadam, A., Aksikas, I., Dubljevic, S. & Forbes, J. F. Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs. Automatica vol. 49 526–533 (2013) – 10.1016/j.automatica.2012.11.016
Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
Seslija, M., van der Schaft, A. & Scherpen, J. M. A. Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems. Journal of Geometry and Physics vol. 62 1509–1531 (2012) – 10.1016/j.geomphys.2012.02.006
Staffans, (2005)
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
Zwart, (1995)