Port Controlled Hamiltonian Representation of Distributed Parameter Systems

Authors

B.M.J. Maschke, A.J. van der Schaft

Abstract

A port controlled Hamiltonian formulation of the dynamics of distributed parameter systems is presented, which incorporates the energy flow through the boundary of the domain of the system, and which allows to represent the system as a boundary control Hamiltonian system. This port controlled Hamiltonian system is defined with respect to a Dirac structure associated with the exterior derivative and based on Stokes’ theorem. The definition is illustrated on the examples of the telegrapher’s equations, Maxwell’s equations and the vibrating string.

Keywords

distributed parameter systems; Hamiltonian systems; Dirac structures; boundary control

Citation

• Journal: IFAC Proceedings Volumes
• Year: 2000
• Volume: 33
• Issue: 2
• Pages: 27–37
• Publisher: Elsevier BV
• DOI: 10.1016/s1474-6670(17)35543-x
• Note: IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Princeton, NJ, USA, 16-18 March 2000

BibTeX

@article{Maschke_2000,
  title={{Port Controlled Hamiltonian Representation of Distributed Parameter Systems}},
  volume={33},
  ISSN={1474-6670},
  DOI={10.1016/s1474-6670(17)35543-x},
  number={2},
  journal={IFAC Proceedings Volumes},
  publisher={Elsevier BV},
  author={Maschke, B.M.J. and van der Schaft, A.J.},
  year={2000},
  pages={27--37}
}

Download the bib file

References

• Abraham, (1978)
• Breedveld, Physical Systems Theory in Terms of Bond Graphs. (1984)
• Courant, T. J. Dirac manifolds. Trans. Amer. Math. Soc. 319, 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM J. Control Optim. 37, 54–91 (1998) – 10.1137/s0363012996312039
• Dorfman, (1993)
• Dworsky, (1979)
• Fattorini, H. O. Boundary Control Systems. SIAM Journal on Control 6, 349–385 (1968) – 10.1137/0306025
• Holm, D. D., Marsden, J. E., Ratiu, T. & Weinstein, A. Nonlinear stability of fluid and plasma equilibria. Physics Reports 123, 1–116 (1985) – 10.1016/0370-1573(85)90028-6
• Ingarden, (1985)
• Lewis, D., Marsden, J., Montgomery, R. & Ratiu, T. The Hamiltonian structure for dynamic free boundary problems. Physica D: Nonlinear Phenomena 18, 391–404 (1986) – 10.1016/0167-2789(86)90207-1
• Maschke, B. M., Van Der Schaft, A. J. & Breedveld, P. C. An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute 329, 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
• Maschke, Interconnection of systems: the network paradigm. (1996)
• Maschke, Interconnected mechanical systems, part II: the dynamics of spatial mechanical networks. (1997)
• Maschke, Note on the dynamics of LC circuits with elements in excess (1998)
• Olver, (1993)
• Ortega, Stabilization of port controlled Hamiltonian systems. (1999)
• Ortega, Stabilization ofportcontrolled hamiltonian systems: Passivation and energy-balancing (1999)
• Paynter, (1961)
• Saintellier, (1993)
• van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
• van der Schaft, The Hamiltonian formulation of energy conserving physical systems with external ports. Archiv für Elektronik und übertragungstechnik (1995)