Topological geometry and control for distributed port-Hamiltonian systems with non-integrable structures

Authors

Gou Nishida, Bernhard Maschke, Masaki Yamakita

Abstract

This paper discusses topological geometrical aspects and a control strategy for a distributed port-Hamiltonian system with a non-integrable structure called a distributed energy structure. First, we show a geometrical structure of port variables determined by differential forms. Next, we state the necessary condition for regarding the distributed energy structure as a boundary energy structure which is boundary integrable. From these results, we define the fundamental form that generates the distributed port-Hamiltonian system with distributed energy structures in a variational problem. Finally, we present a new concept of boundary controls for the distributed port-Hamiltonian system with distributed energy structures in space-time coordinates.

Citation

• Journal: 2008 47th IEEE Conference on Decision and Control
• Year: 2008
• Volume:
• Issue:
• Pages: 1291–1297
• Publisher: IEEE
• DOI: 10.1109/cdc.2008.4738896

BibTeX

@inproceedings{Nishida_2008,
  title={{Topological geometry and control for distributed port-Hamiltonian systems with non-integrable structures}},
  DOI={10.1109/cdc.2008.4738896},
  booktitle={{2008 47th IEEE Conference on Decision and Control}},
  publisher={IEEE},
  author={Nishida, Gou and Maschke, Bernhard and Yamakita, Masaki},
  year={2008},
  pages={1291--1297}
}

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References

• Saunders, D. J. The Geometry of Jet Bundles. (1989) doi:10.1017/cbo9780511526411 – 10.1017/cbo9780511526411
• lifshitz, the classical theory of fields. Course of Theoretical Physics Series (1980)
• Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (Springer New York, 1993). doi:10.1007/978-1-4612-4350-2 – 10.1007/978-1-4612-4350-2
• Burke, W. L. Applied Differential Geometry. (1985) doi:10.1017/cbo9781139171786 – 10.1017/cbo9781139171786
• choquet-bruhat, Analysis Manifolds and Physics Part I Basics (1982)
• Warner, F. W. Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics (Springer New York, 1983). doi:10.1007/978-1-4757-1799-0 – 10.1007/978-1-4757-1799-0
• flanders, Differential Forms with Applications to the Physical Sciences (1963)
• jost, Riemannian Geometry and Geometric Analysis (2005)
• madsen, Form Calculus to Cohomology (1997)
• Cheeger, J. On the Hodge theory of Riemannian pseudomanifolds. Proceedings of Symposia in Pure Mathematics 91–146 (1980) doi:10.1090/pspum/036/573430 – 10.1090/pspum/036/573430
• 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
• 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
• Morita, S. Geometry of Differential Forms. Translations of Mathematica Monographs (2001) doi:10.1090/mmono/201 – 10.1090/mmono/201
• Stramigioli, S. Geometric modeling of mechanical systems for interactive control. Lecture Notes in Control and Information Sciences 309–332 doi:10.1007/bfb0110389 – 10.1007/bfb0110389
• 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
• Nishida, G. & Yamakita, M. Disturbance structure decomposition for distributed-parameter port-Hamiltonian systems. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) 2082-2087 Vol.2 (2004) doi:10.1109/cdc.2004.1430355 – 10.1109/cdc.2004.1430355
• Gaffney, M. P. The Harmonic Operator for Exterior Differential Forms. Proceedings of the National Academy of Sciences vol. 37 48–50 (1951) – 10.1073/pnas.37.1.48
• (0)
• Ortega, R., van der Schaft, A. J. & Maschke, B. M. Stabilization of port-controlled Hamiltonian systems via energy balancing. Lecture Notes in Control and Information Sciences 239–260 (1999) doi:10.1007/1-84628-577-1_13 – 10.1007/1-84628-577-1_13