A port-Hamiltonian formulation of physical switching systems with varying constraints

Authors

Claire Valentin, Miguel Magos, Bernhard Maschke

Abstract

This paper extends a generic method to design a port-Hamiltonian formulation modeling all geometric interconnection structures of a physical switching system with varying constraints. A non-minimal kernel representation of this family of structures (named Dirac structures) is presented. It is derived from the parameterized incidence matrices which are a mathematical representation of the primal and dual dynamic network graphs associated with the system. This representation has the advantage of making it possible to model complex physical switching systems with varying constraints and to fall within the framework of passivity-based control.

Keywords

Modeling; Port-Hamiltonian systems; Network graph; Family of geometric interconnection structures; Incidence matrix; Energy exchanges

Citation

• Journal: Automatica
• Year: 2007
• Volume: 43
• Issue: 7
• Pages: 1125–1133
• Publisher: Elsevier BV
• DOI: 10.1016/j.automatica.2006.12.022

BibTeX

@article{Valentin_2007,
  title={{A port-Hamiltonian formulation of physical switching systems with varying constraints}},
  volume={43},
  ISSN={0005-1098},
  DOI={10.1016/j.automatica.2006.12.022},
  number={7},
  journal={Automatica},
  publisher={Elsevier BV},
  author={Valentin, Claire and Magos, Miguel and Maschke, Bernhard},
  year={2007},
  pages={1125--1133}
}

Download the bib file

References

• Bloch, Representation of dirac structures on vector spaces and nonlinear LC circuits. (1999)
• Buisson, J. Analysis of switching devices with bond graphs. Journal of the Franklin Institute vol. 330 1165–1175 (1993) – 10.1016/0016-0032(93)90070-b
• Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM Journal on Control and Optimization vol. 37 54–91 (1998) – 10.1137/s0363012996312039
• Escobar, G., van der Schaft, A. J. & Ortega, R. A Hamiltonian viewpoint in the modeling of switching power converters. Automatica vol. 35 445–452 (1999) – 10.1016/s0005-1098(98)00196-4
• Haddad, W. M., Nersesov, S. G. & Chellaboina, V. Energy-based control for hybrid port-controlled Hamiltonian systems. Automatica vol. 39 1425–1435 (2003) – 10.1016/s0005-1098(03)00113-4
• Jeltsema, D., Scherpen, J. M. A. & Klaassens, J. B. Energy control of multi-switch power supplies; an application to the three-phase buck type rectifier with input filter. 2001 IEEE 32nd Annual Power Electronics Specialists Conference (IEEE Cat. No.01CH37230) vol. 4 1831–1836 – 10.1109/pesc.2001.954388
• Karnopp, (1990)
• Lozano, (2000)
• Manon, P., Valentin-Roubinet, C. & Gilles, G. Optimal control of hybrid dynamical systems: application in process engineering. Control Engineering Practice vol. 10 133–149 (2002) – 10.1016/s0967-0661(01)00123-x
• 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 vol. 329 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
• Maschke, B. M., van der Schaft, A. J. & Breedveld, P. C. An intrinsic Hamiltonian formulation of the dynamics of LC-circuits. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications vol. 42 73–82 (1995) – 10.1109/81.372847
• Narayanan, (1997)
• 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
• Paynter, (1961)
• Recski, (1989)
• Scherpen, J. M. A., Jeltsema, D. & Klaassens, J. B. Lagrangian modeling of switching electrical networks. Systems & Control Letters vol. 48 365–374 (2003) – 10.1016/s0167-6911(02)00290-6
• Sussmann, H. J. A maximum principle for hybrid optimal control problems. Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304) vol. 1 425–430 – 10.1109/cdc.1999.832814
• Valentin, C., Magos, M. & Maschke, B. Hybrid port–Hamiltonian systems: From parameterized incidence matrices to hybrid automata. Nonlinear Analysis: Theory, Methods & Applications vol. 65 1106–1122 (2006) – 10.1016/j.na.2005.12.013
• Van der Schaft, (2000)
• Van der Schaft, (2000)