Authors

A. Favache, V.S. Dos Santos Martins, D. Dochain, B. Maschke

Abstract

The dynamics of open irreversible thermodynamic systems, that is systems including both the balance equation of the energy and the entropy, has been formulated as contact vector fields with generating functions depending on some external (control) variable and called conservative port contact systems. In this paper we relate the dynamical properties of these systems (equilibrium points, asymptotic stability) to properties of the generating functions (the contact Hamiltonian functions). We show that the equilibrium points of the system satisfy certain conditions involving the contact Hamiltonian function. We also consider Lyapunov’s first theorem to emphasize a stability criterion for the equilibrium points in terms of this contact Hamiltonian function and relate it to some thermodynamical properties. These results are then related to the physical phenomena that are taking place in the system.

Citation

  • Journal: IEEE Transactions on Automatic Control
  • Year: 2009
  • Volume: 54
  • Issue: 10
  • Pages: 2341–2351
  • Publisher: Institute of Electrical and Electronics Engineers (IEEE)
  • DOI: 10.1109/tac.2009.2028973

BibTeX

@article{Favache_2009,
  title={{Some Properties of Conservative Port Contact Systems}},
  volume={54},
  ISSN={1558-2523},
  DOI={10.1109/tac.2009.2028973},
  number={10},
  journal={IEEE Transactions on Automatic Control},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Favache, A. and Dos Santos Martins, V.S. and Dochain, D. and Maschke, B.},
  year={2009},
  pages={2341--2351}
}

Download the bib file

References

  • sandler, Chemical and Engineering Thermodynamics (1999)
  • callen, Thermodynamics and an Introduction to Thermostatics (1985)
  • godbillon, Gomtrie diffrentielle et mcanique analytique (1969)
  • Mrugała, R. Continuous contact transformations in thermodynamics. Reports on Mathematical Physics 33, 149–154 (1993) – 10.1016/0034-4877(93)90050-o
  • eberard, on the interconnection structures of open physical systems. Proc 3rd IFAC Workshop Lagrangian Hamiltonian Methods Nonlin Control (2006)
  • de groot, Non-equilibrium thermodynamics (1984)
  • kreyszig, Introductory Functional Analysis With Applications (1989)
  • Mrugaa̵, R. On a special family of thermodynamic processes and their invariants. Reports on Mathematical Physics 46, 461–468 (2000) – 10.1016/s0034-4877(00)90012-0
  • jeltsema, on mechanical mixed potential, content and co-content. Proc Eur Control Conf (2003)
  • Jeltsema, D. & Scherpen, J. M. A. A power-based description of standard mechanical systems. Systems & Control Letters 56, 349–356 (2007) – 10.1016/j.sysconle.2006.10.015
  • Ortega, R., Jeltsema, D. & Scherpen, J. M. A. Power shaping: A new paradigm for stabilization of nonlinear RLC circuits. IEEE Trans. Automat. Contr. 48, 1762–1767 (2003) – 10.1109/tac.2003.817918
  • eberard, energy-conserving formulation of rlc-circuits with linear resistors. Proc 17th Int Symp Math Theory Netw Syst (2006)
  • Eberard, D., Maschke, B. M. & van der Schaft, A. J. An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes. Reports on Mathematical Physics 60, 175–198 (2007) – 10.1016/s0034-4877(07)00024-9
  • 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
  • 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
  • Willems, J. C. Dissipative dynamical systems part I: General theory. Arch. Rational Mech. Anal. 45, 321–351 (1972) – 10.1007/bf00276493
  • gibbs, The Collected Works of J Willard Gibbs (1931)
  • Carathéodory, C. Untersuchungen über die Grundlagen der Thermodynamik. Math. Ann. 67, 355–386 (1909) – 10.1007/bf01450409
  • Grmela, M. Reciprocity relations in thermodynamics. Physica A: Statistical Mechanics and its Applications 309, 304–328 (2002) – 10.1016/s0378-4371(02)00564-2
  • Maschke, B., Ortega, R. & Van Der Schaft, A. J. Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans. Automat. Contr. 45, 1498–1502 (2000) – 10.1109/9.871758
  • favache, contact structures: application to interconnected thermodynamical systems. Proc Eur Control Conf (2007)
  • Jeltsema, D., Ortega, R. & M.A. Scherpen, J. An energy-balancing perspective of interconnection and damping assignment control of nonlinear systems. Automatica 40, 1643–1646 (2004) – 10.1016/j.automatica.2004.04.007
  • 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
  • Mrugała, R. Submanifolds in the thermodynamic phase space. Reports on Mathematical Physics 21, 197–203 (1985) – 10.1016/0034-4877(85)90059-x
  • Putting energy back in control. IEEE Control Syst. 21, 18–33 (2001) – 10.1109/37.915398
  • Lozano, R., Brogliato, B., Egeland, O. & Maschke, B. Dissipative Systems Analysis and Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-3668-2 – 10.1007/978-1-4471-3668-2
  • kugi, Nonlinear Control Based on Physical Models (2001)
  • Schaft, A. L2-Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Sciences (Springer Berlin Heidelberg, 1996). doi:10.1007/3-540-76074-1 – 10.1007/3-540-76074-1
  • garcia-canseco, power-based control of physical systems: two case studies. Proc 17th IFAC World Congress (2008)
  • khalil, Nonlinear Systems (2002)
  • 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
  • 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
  • MrugaŁa, R. Geometrical formulation of equilibrium phenomenological thermodynamics. Reports on Mathematical Physics 14, 419–427 (1978) – 10.1016/0034-4877(78)90010-1
  • 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
  • abraham, Foundations of Mechanics (1994)
  • eberard, Extension des systmes hamiltoniens ports aux systmes irrversibles une approche par la gomtrie de contact (2006)
  • eberard, conservative systems with ports on contact manifolds. Proc 16th IFAC World Congress (2005)