Derivation of Input-State-Output Port-Hamiltonian Systems from bond graphs

Authors

Abstract

This paper presents methods to obtain models in the form of Input-State-Output Port-Hamiltonian Systems from causal nonlinear bond graph models. This is done first establishing equivalences among key variables in both domains through the comparison of the expressions of the stored system energy in both formalisms. Later, with the help of the general field-representation of bond graphs and its associated standard implicit form, the functions characterizing this class of Port-Hamiltonian Systems, i.e., interconnection, dissipation and input/output matrices, as well as their properties, are immediately expressed in terms of bond graphs parameters. Under suitable assumptions, the method supports the direct derivation of Input-State-Output Port-Hamiltonian Systems – which is an explicit type of PHS – even from bond graphs having causally coupled dissipators and storages in derivative causality, which are known to imply algebraic and implicit differential equations. The methods are illustrated with some application examples covering different causal situations. Besides its intrinsic interest as a technique for model conversion, the contribution is seen as a useful step towards implementing Port-Hamiltonian based control system design methods with the support of BG techniques.

Keywords

Bond graphs; State-Input-Output Port-Hamiltonian Systems; Model equivalences

Citation

Journal: Simulation Modelling Practice and Theory
Year: 2009
Volume: 17
Issue: 1
Pages: 137–151
Publisher: Elsevier BV
DOI: 10.1016/j.simpat.2008.02.007
Note: Bond Graph Modelling

BibTeX

@article{Donaire_2009,
  title={{Derivation of Input-State-Output Port-Hamiltonian Systems from bond graphs}},
  volume={17},
  ISSN={1569-190X},
  DOI={10.1016/j.simpat.2008.02.007},
  number={1},
  journal={Simulation Modelling Practice and Theory},
  publisher={Elsevier BV},
  author={Donaire, Alejandro and Junco, Sergio},
  year={2009},
  pages={137--151}
}

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References

Achir, Bond graph and flatness based backstepping control of a salient permanent magnet synchronous motor. Journal of System and Control Engineering Part I (2005)
Borutzky, (2004)
Dauphin-Tanguy, G., Rahmani, A. & Sueur, C. Bond graph aided design of controlled systems. Simulation Practice and Theory vol. 7 493–513 (1999) – 10.1016/s0928-4869(99)00009-9
Delgado, M. & Sira-Ramı́rez, H. A bond graph approach to the modeling and simulation of switch regulated DC-to-DC power supplies. Simulation Practice and Theory vol. 6 631–646 (1998) – 10.1016/s0928-4869(98)00011-1
Donaire, Energy shaping and interconnection and damping assignment control in the bond graph domain. (2006)
Donaire, A. & Junco, S. Energy shaping, interconnection and damping assignment, and integral control in the bond graph domain. Simulation Modelling Practice and Theory vol. 17 152–174 (2009) – 10.1016/j.simpat.2008.02.012
Fujimoto, K., Sakurama, K. & Sugie, T. Trajectory tracking control of port-controlled Hamiltonian systems via generalized canonical transformations. Automatica vol. 39 2059–2069 (2003) – 10.1016/j.automatica.2003.07.005
Gawthrop, P. J. Physical model-based control: A bond graph approach. Journal of the Franklin Institute vol. 332 285–305 (1995) – 10.1016/0016-0032(95)00044-5
Golo, Hamiltonian Formulation of Bond Graphs, Nonlinear and Hybrid Systems in Automotive Control. (2003)
Huang, S. Y. & Youcef-Toumi, K. Zero Dynamics of Physical Systems From Bond Graph Models—Part I: SISO Systems. Journal of Dynamic Systems, Measurement, and Control vol. 121 10–17 (1999) – 10.1115/1.2802426
Junco, S. Stability analysis and stabilizing control synthesis via Lyapunov’s second method directly on bond graphs of nonlinear systems. Proceedings of IECON ’93 - 19th Annual Conference of IEEE Industrial Electronics 2065–2069 doi:10.1109/iecon.1993.339394 – 10.1109/iecon.1993.339394
Junco, Speed control of series dc-motor: a bond graph based backstepping design. (2002)
Junco, Virtual prototyping of bond graphs models for controller synthesis through energy and power shaping. (2004)
Karnopp, (2000)
Lanczos, (1960)
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
Ould Bouamama, B., Medjaher, K., Samantaray, A. K. & Staroswiecki, M. Supervision of an industrial steam generator. Part I: Bond graph modelling. Control Engineering Practice vol. 14 71–83 (2006) – 10.1016/j.conengprac.2005.01.004
Sueur, C. & Dauphin-Tanguy, G. Bond-graph approach for structural analysis of MIMO linear systems. Journal of the Franklin Institute vol. 328 55–70 (1991) – 10.1016/0016-0032(91)90006-o
van der Schaft, (2000)
van der Schaft, A. Port-Hamiltonian systems: an introductory survey. Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006 1339–1365 (2007) doi:10.4171/022-3/65 – 10.4171/022-3/65
Wells, (1967)
Willems, J. C. Dissipative dynamical systems part I: General theory. Archive for Rational Mechanics and Analysis vol. 45 321–351 (1972) – 10.1007/bf00276493
Yeh, T.-J. Backstepping control in the physical domain. Journal of the Franklin Institute vol. 338 455–479 (2001) – 10.1016/s0016-0032(01)00015-1