Port-Hamiltonian Systems on Graphs

Authors

A. J. van der Schaft, B. M. Maschke

Abstract

In this paper we present a unifying geometric and compositional framework for modeling complex physical network dynamics as port-Hamiltonian systems on open graphs. Basic idea is to associate with the incidence matrix of the graph a Dirac structure relating the flow and effort variables associated to the edges, internal vertices, as well as boundary vertices of the graph, and to formulate energy-storing or energy-dissipating relations between the flow and effort variables of the edges and internal vertices. This allows for state variables associated to the edges, and formalizes the interconnection of networks. Examples from different origins such as consensus algorithms are shown to share the same structure. It is shown how the identified Hamiltonian structure offers systematic tools for the analysis of the resulting dynamics.

Citation

• Journal: SIAM Journal on Control and Optimization
• Year: 2013
• Volume: 51
• Issue: 2
• Pages: 906–937
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/110840091

BibTeX

@article{van_der_Schaft_2013,
  title={{Port-Hamiltonian Systems on Graphs}},
  volume={51},
  ISSN={1095-7138},
  DOI={10.1137/110840091},
  number={2},
  journal={SIAM Journal on Control and Optimization},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={van der Schaft, A. J. and Maschke, B. M.},
  year={2013},
  pages={906--937}
}

Download the bib file

References

• Arcak, M. Passivity as a Design Tool for Group Coordination. IEEE Transactions on Automatic Control vol. 52 1380–1390 (2007) – 10.1109/tac.2007.902733
• Brayton, R. K. & Moser, J. K. A theory of nonlinear networks. I. Quarterly of Applied Mathematics vol. 22 1–33 (1964) – 10.1090/qam/169746
• Bürger M., Orlando, FL (2011)
• Camlibel M. K., The Netherlands (2012)
• Cervera, J., van der Schaft, A. J. & Baños, A. Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica vol. 43 212–225 (2007) – 10.1016/j.automatica.2006.08.014
• 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
• Davies, T. H. Mechanical networks—I Passivity and redundancy. Mechanism and Machine Theory vol. 18 95–101 (1983) – 10.1016/0094-114x(83)90100-3
• De Persis, C. & Kallesoe, C. S. Pressure Regulation in Nonlinear Hydraulic Networks by Positive and Quantized Controls. IEEE Transactions on Control Systems Technology vol. 19 1371–1383 (2011) – 10.1109/tcst.2010.2094619
• Goldin D., GA (2010)
• Maschke B. M., London (1997)
• 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
• Olfati-Saber, R., Fax, J. A. & Murray, R. M. Consensus and Cooperation in Networked Multi-Agent Systems. Proceedings of the IEEE vol. 95 215–233 (2007) – 10.1109/jproc.2006.887293
• Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
• Rahmani, A., Ji, M., Mesbahi, M. & Egerstedt, M. Controllability of Multi-Agent Systems from a Graph-Theoretic Perspective. SIAM Journal on Control and Optimization vol. 48 162–186 (2009) – 10.1137/060674909
• A., The Netherlands (1999)
• van der Schaft, A. Characterization and partial synthesis of the behavior of resistive circuits at their terminals. Systems & Control Letters vol. 59 423–428 (2010) – 10.1016/j.sysconle.2010.05.005
• A., Italy (2011)
• van der Schaft A. J., China (2009)
• van der Schaft A. J., Arch. Elek. Übertr. (1995)
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• van der Schaft A. J., Mexico (2008)
• van der Schaft A. J., Heidelberg (2009)
• van der Schaft A. J., France (2010)
• van der Schaft A. J., Italy (2012)
• Smale, S. On the mathematical foundations of electrical circuit theory. Journal of Differential Geometry vol. 7 (1972) – 10.4310/jdg/1214430827
• Smith, M. C. Synthesis of mechanical networks: the inerter. IEEE Transactions on Automatic Control vol. 47 1648–1662 (2002) – 10.1109/tac.2002.803532
• Vankerschaver J., GA (2010)
• The Behavioral Approach to Open and Interconnected Systems. IEEE Control Systems vol. 27 46–99 (2007) – 10.1109/mcs.2007.906923
• Willems, J. Terminals and Ports. IEEE Circuits and Systems Magazine vol. 10 8–26 (2010) – 10.1109/mcas.2010.938635
• Zelazo, D. & Mesbahi, M. Edge Agreement: Graph-Theoretic Performance Bounds and Passivity Analysis. IEEE Transactions on Automatic Control vol. 56 544–555 (2011) – 10.1109/tac.2010.2056730