On Casimir functionals for field theories in Port-Hamiltonian description for control purposes

Authors

Abstract

We consider infinite dimensional Port-Hamiltonian systems in an evolutionary formulation. Based on this system representation conditions for Casimir densities (functionals) will be derived where in this context the variational derivative plays an extraordinary role. Furthermore the coupling of finite and infinite dimensional systems will be analyzed in the spirit of the control by interconnection problem. Our Hamiltonian representation differs significantly from the well-established one using Stokes-Dirac structures that are based on skew-adjoint differential operators and the use of energy variables. We mainly base our considerations on a bundle structure with regard to dependent and independent coordinates as well as on differential-geometric objects induced by that structure.

Citation

Journal: IEEE Conference on Decision and Control and European Control Conference
Year: 2011
Volume:
Issue:
Pages: 7759–7764
Publisher: IEEE
DOI: 10.1109/cdc.2011.6160430

BibTeX

@inproceedings{Schoberl_2011,
  title={{On Casimir functionals for field theories in Port-Hamiltonian description for control purposes}},
  DOI={10.1109/cdc.2011.6160430},
  booktitle={{IEEE Conference on Decision and Control and European Control Conference}},
  publisher={IEEE},
  author={Schoberl, Markus and Siuka, Andreas},
  year={2011},
  pages={7759--7764}
}

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References

Luo, Z.-H., Guo, B.-Z. & Morgul, O. Stability and Stabilization of Infinite Dimensional Systems with Applications. Communications and Control Engineering (Springer London, 1999). doi:10.1007/978-1-4471-0419-3 – 10.1007/978-1-4471-0419-3
Giachetta, G., Mangiarotti, L. & Sardanashvily, G. New Lagrangian and Hamiltonian Methods in Field Theory. (1997) doi:10.1142/2199 – 10.1142/2199
Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38, 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42, 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
maschke, Compositional Modelling of Distributed-parameter Systems Ser Advanced Topics in Control Systems Theory Springer Lect Notes in Control and Information Sciences (2005)
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
Pasumarthy, R. & van der Schaft, A. J. Achievable Casimirs and its implications on control of port-Hamiltonian systems. International Journal of Control 80, 1421–1438 (2007) – 10.1080/00207170701361273
Schlacher, K. Mathematical modeling for nonlinear control: a Hamiltonian approach. Mathematics and Computers in Simulation 79, 829–849 (2008) – 10.1016/j.matcom.2008.02.011
Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (Springer New York, 1986). doi:10.1007/978-1-4684-0274-2 – 10.1007/978-1-4684-0274-2
macchelli, Port hamiltonian formulation of infinite dimensional systems: Part ii boundary control by interconnection. Proc 43rd IEEE Conf Decision and Control (CDC) (2004)
macchelli, Port hamiltonian formulation of infinite dimensional systems: Part i modeling. Proc 43rd IEEE Conf Decision and Control (CDC) (2004)
Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM J. Control Optim. 43, 743–767 (2004) – 10.1137/s0363012903429530
Schöberl, M., Ennsbrunner, H. & Schlacher, K. Modelling of piezoelectric structures–a Hamiltonian approach. Mathematical and Computer Modelling of Dynamical Systems 14, 179–193 (2008) – 10.1080/13873950701844824