On the Interconnection Structures of Irreversible Physical Systems

Authors

Damien Eberard, Bernhard Maschke, Arjan J. van der Schaft

Abstract

An energy balance equation with respect to a control contact system provides port outputs which are conjugated to inputs. These conjugate variables are used to define the composition of port contact systems in the framework of contact geometry. We then propose a power-conserving interconnection structure, which generalizes the interconnection by Dirac structures in the Hamiltonian formalism. Furthermore, the composed system is again a port contact system, as illustrated on the example of a gas-piston system undergoing some irreversible transformation.

Keywords

contact manifold, contact structure, contact system, hamiltonian system, port output

Citation

• ISBN: 9783540738893
• Publisher: Springer Berlin Heidelberg
• DOI: 10.1007/978-3-540-73890-9_16

BibTeX

@inbook{Eberard_2007,
  title={{On the Interconnection Structures of Irreversible Physical Systems}},
  ISBN={9783540738909},
  ISSN={0170-8643},
  DOI={10.1007/978-3-540-73890-9_16},
  booktitle={{Lagrangian and Hamiltonian Methods for Nonlinear Control 2006}},
  publisher={Springer Berlin Heidelberg},
  author={Eberard, Damien and Maschke, Bernhard and van der Schaft, Arjan J.},
  year={2007},
  pages={209--220}
}

Download the bib file

References

• V.I. Arnold, Equations Diffrentielles Ordinaires (1978)
• Arnold, V. I. Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics (Springer New York, 1989). doi:10.1007/978-1-4757-2063-1 – 10.1007/978-1-4757-2063-1
• 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
• Eberard, D., Maschke, B. & van der Schaft, A. J. CONSERVATIVE SYSTEMS WITH PORTS ON CONTACT MANIFOLDS. IFAC Proceedings Volumes 38, 342–347 (2005) – 10.3182/20050703-6-cz-1902.00711
• Eberard, D., Maschke, B. M. & van der Schaft, A. J. Port contact systems for irreversible thermodynamical systems. Proceedings of the 44th IEEE Conference on Decision and Control 5977–5982 doi:10.1109/cdc.2005.1583118 – 10.1109/cdc.2005.1583118
• J.W. Gibbs, Collected Works (1928)
• R. Herman, Geometry, Physics and Systems (1973)
• Jongschaap, R. & Öttinger, H. C. The mathematical representation of driven thermodynamic systems. Journal of Non-Newtonian Fluid Mechanics 120, 3–9 (2004) – 10.1016/j.jnnfm.2003.11.008
• Libermann, P. & Marle, C.-M. Symplectic Geometry and Analytical Mechanics. (Springer Netherlands, 1987). doi:10.1007/978-94-009-3807-6 – 10.1007/978-94-009-3807-6
• 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
• MrugaŁa, R. Geometrical formulation of equilibrium phenomenological thermodynamics. Reports on Mathematical Physics 14, 419–427 (1978) – 10.1016/0034-4877(78)90010-1
• R. Mrugala, Bull. of the Polish Academy of Sciences (1980)
• Ortega, J.-P. & Planas-Bielsa, V. Dynamics on Leibniz manifolds. Journal of Geometry and Physics 52, 1–27 (2004) – 10.1016/j.geomphys.2004.01.002
• Mrugala, R., Nulton, J. D., Christian Schön, J. & Salamon, P. Contact structure in thermodynamic theory. Reports on Mathematical Physics 29, 109–121 (1991) – 10.1016/0034-4877(91)90017-h
• R.W. Brockett, Geometric Control Theory, volume 7 of Lie groups: History, Frontiers and Applications (1977)
• van der Schaft, A. J. System theory and mechanics. Lecture Notes in Control and Information Sciences 426–452 (1989) doi:10.1007/bfb0008472 – 10.1007/bfb0008472
• 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
• A.J. Schaft van der, Archiv für Elektronik und Übertragungstechnik (1995)
• A.J. Schaft van der, Modelling and Control of Mechanical Systems (1997)