Reduction of Stokes-Dirac structures and gauge symmetry in port-Hamiltonian systems

Authors

Marko Seslija, Arjan van der Schaft, Jacquelien M.A. Scherpen

Abstract

Stokes-Dirac structures are infinite-dimensional Dirac structures defined in terms of differential forms on a smooth manifold with boundary. These Dirac structures lay down a geometric framework for the formulation of Hamiltonian systems with a nonzero boundary energy flow. Simplicial triangulation of the underlaying manifold leads to the so-called simplicial Dirac structures, discrete analogues of Stokes-Dirac structures, and thus provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The port-Hamiltonian systems defined with respect to Stokes-Dirac and simplicial Dirac structures exhibit gauge and a discrete gauge symmetry, respectively. In this paper, employing Poisson reduction we offer a unified technique for the symmetry reduction of a generalized canonical infinite-dimensional Dirac structure to the Poisson structure associated with Stokes-Dirac structures and of a fine-dimensional Dirac structure to simplicial Dirac structures. We demonstrate this Poisson scheme on a physical example of the vibrating string.

Keywords

Port-Hamiltonian systems; Poisson structures; Dirac structures; distributed-parameter systems; symmetry reduction

Citation

• Journal: IFAC Proceedings Volumes
• Year: 2012
• Volume: 45
• Issue: 19
• Pages: 114–119
• Publisher: Elsevier BV
• DOI: 10.3182/20120829-3-it-4022.00030
• Note: 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control

BibTeX

@article{Seslija_2012,
  title={{Reduction of Stokes-Dirac structures and gauge symmetry in port-Hamiltonian systems}},
  volume={45},
  ISSN={1474-6670},
  DOI={10.3182/20120829-3-it-4022.00030},
  number={19},
  journal={IFAC Proceedings Volumes},
  publisher={Elsevier BV},
  author={Seslija, Marko and van der Schaft, Arjan and Scherpen, Jacquelien M.A.},
  year={2012},
  pages={114--119}
}

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References

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