Feedback equivalence of input–output contact systems

Authors

Hector Ramirez, Bernhard Maschke, Daniel Sbarbaro

Abstract

Control contact systems represent controlled (or open) irreversible processes which allow us to represent simultaneously the energy conservation and the irreversible creation of entropy. Such systems systematically arise in models established in Chemical Engineering. The differential-geometric of these systems is a contact form in the same manner as the symplectic 2-form is associated to Hamiltonian models of mechanics. In this paper we study the feedback preserving the geometric structure of controlled contact systems and render the closed-loop system again as a contact system. It is shown that only a constant control preserves the canonical contact form, hence a state feedback necessarily changes the closed-loop contact form. For strict contact systems, arising from the modelling of thermodynamic systems, a class of state feedback that shapes the closed-loop contact form and contact Hamiltonian function is proposed. The state feedback is given by the composition of an arbitrary function and the control contact Hamiltonian function. The similarity with structure preserving feedback of input–output Hamiltonian systems leads to the definition of input–output contact systems and to the characterization of the feedback equivalence of input–output contact systems. An irreversible thermodynamic process, namely the heat exchanger, is used to illustrate the results.

Keywords

Nonlinear control; Input–output contact systems; Contact geometry; Irreversible Thermodynamics

Citation

Journal: Systems & Control Letters
Year: 2013
Volume: 62
Issue: 6
Pages: 475–481
Publisher: Elsevier BV
DOI: 10.1016/j.sysconle.2013.02.008

BibTeX

@article{Ramirez_2013,
  title={{Feedback equivalence of input–output contact systems}},
  volume={62},
  ISSN={0167-6911},
  DOI={10.1016/j.sysconle.2013.02.008},
  number={6},
  journal={Systems &amp; Control Letters},
  publisher={Elsevier BV},
  author={Ramirez, Hector and Maschke, Bernhard and Sbarbaro, Daniel},
  year={2013},
  pages={475--481}
}

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References

Eberard, D., Maschke, B. & van der Schaft, A. J. CONSERVATIVE SYSTEMS WITH PORTS ON CONTACT MANIFOLDS. IFAC Proceedings Volumes vol. 38 342–347 (2005) – 10.3182/20050703-6-cz-1902.00711
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 vol. 60 175–198 (2007) – 10.1016/s0034-4877(07)00024-9
MrugaŁa, R. Geometrical formulation of equilibrium phenomenological thermodynamics. Reports on Mathematical Physics vol. 14 419–427 (1978) – 10.1016/0034-4877(78)90010-1
Mrugala, R., Nulton, J. D., Christian Schön, J. & Salamon, P. Contact structure in thermodynamic theory. Reports on Mathematical Physics vol. 29 109–121 (1991) – 10.1016/0034-4877(91)90017-h
Hermann, (1973)
Hermann, (1974)
Mrugaa̵, R. On a special family of thermodynamic processes and their invariants. Reports on Mathematical Physics vol. 46 461–468 (2000) – 10.1016/s0034-4877(00)90012-0
Favache, A., Dochain, D. & Maschke, B. An entropy-based formulation of irreversible processes based on contact structures. Chemical Engineering Science vol. 65 5204–5216 (2010) – 10.1016/j.ces.2010.06.019
Favache, A., Dos Santos Martins, V. S., Dochain, D. & Maschke, B. Some Properties of Conservative Port Contact Systems. IEEE Transactions on Automatic Control vol. 54 2341–2351 (2009) – 10.1109/tac.2009.2028973
van der Schaft, On feedback control of Hamiltonian systems. (1986)
van der Schaft, System theory and mechanics. (1989)
Brockett, Control theory and analytical mechanics. (1977)
Marsden, (1992)
Couenne, F., Jallut, C., Maschke, B., Breedveld, P. C. & Tayakout, M. Bond graph modelling for chemical reactors. Mathematical and Computer Modelling of Dynamical Systems vol. 12 159–174 (2006) – 10.1080/13873950500068823
Arnold, (1989)
Libermann, (1987)
Ramirez, H., Maschke, B. & Sbarbaro, D. On the Hamiltonian formulation of the CSTR. 49th IEEE Conference on Decision and Control (CDC) 3301–3306 (2010) doi:10.1109/cdc.2010.5717317 – 10.1109/cdc.2010.5717317
Ramirez, H., Maschke, B. & Sbarbaro, D. Irreversible port-Hamiltonian systems: A general formulation of irreversible processes with application to the CSTR. Chemical Engineering Science vol. 89 223–234 (2013) – 10.1016/j.ces.2012.12.002
van der Schaft, (1984)
Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
Abbott, (1966)
Evans, (1998)
Myint-U, (2007)
Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
Fujimoto, K. & Sugie, T. Stabilization of Hamiltonian systems with nonholonomic constraints based on time-varying generalized canonical transformations. Systems & Control Letters vol. 44 309–319 (2001) – 10.1016/s0167-6911(01)00150-5