Modeling of physical network systems

Authors

Abstract

Conservation laws and balance equations for physical network systems typically can be described with the aid of the incidence matrix of a directed graph, and an associated symmetric Laplacian matrix. Some basic examples are discussed, and the extension to k -complexes is indicated. Physical distribution networks often involve a non-symmetric Laplacian matrix. It is shown how, in case the connected components of the graph are strongly connected, such systems can be converted into a form with balanced Laplacian matrix by constructive use of Kirchhoff’s Matrix Tree theorem, giving rise to a port-Hamiltonian description. Application to the dual case of asymmetric consensus algorithms is given. Finally it is shown how the minimal storage function for physical network systems with controlled flows can be explicitly computed.

Keywords

available storage, laplacian matrix, matrix tree theorem, physical network, port-hamiltonian system

Citation

Journal: Systems & Control Letters
Year: 2017
Volume: 101
Issue:
Pages: 21–27
Publisher: Elsevier BV
DOI: 10.1016/j.sysconle.2015.08.013
Note: Jan C. Willems Memorial Issue, Volume 2

BibTeX

@article{van_der_Schaft_2017,
  title={{Modeling of physical network systems}},
  volume={101},
  ISSN={0167-6911},
  DOI={10.1016/j.sysconle.2015.08.013},
  journal={Systems \&amp; Control Letters},
  publisher={Elsevier BV},
  author={van der Schaft, Arjan},
  year={2017},
  pages={21--27}
}

Download the bib file

References

Willems JC (1972) Dissipative dynamical systems part I: General theory. Arch Rational Mech Anal 45(5):321–351. https://doi.org/10.1007/bf0027649 – 10.1007/bf00276493
Willems JC (1972) Dissipative dynamical systems Part II: Linear systems with quadratic supply rates. Arch Rational Mech Anal 45(5):352–393. https://doi.org/10.1007/bf0027649 – 10.1007/bf00276494
(2007) The Behavioral Approach to Open and Interconnected Systems. IEEE Control Syst 27(6):46–99. https://doi.org/10.1109/mcs.2007.90692 – 10.1109/mcs.2007.906923
Willems J (2010) Terminals and Ports. IEEE Circuits Syst Mag 10(4):8–26. https://doi.org/10.1109/mcas.2010.93863 – 10.1109/mcas.2010.938635
Bollobas, (1998)
Godsil, (2004)
van der Schaft AJ, Maschke BM (2013) Port-Hamiltonian Systems on Graphs. SIAM J Control Optim 51(2):906–937. https://doi.org/10.1137/11084009 – 10.1137/110840091
Kirchhoff G (1847) Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Annalen der Physik 148(12):497–508. https://doi.org/10.1002/andp.1847148120 – 10.1002/andp.18471481202
van der Schaft, The Hamiltonian formulation of energy conserving physical systems with external ports. Arch. Elektron. Uebertrag.tech. (1995)
van der Schaft A, Jeltsema D (2014) Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control 1(2–3):173–378. https://doi.org/10.1561/260000000 – 10.1561/2600000002
Mesbahi, (2010)
van der Schaft A, Maschke B (2009) Conservation Laws and Lumped System Dynamics. Model-Based Control: 31–4 – 10.1007/978-1-4419-0895-7_3
Seslija M, van der Schaft A, Scherpen JMA (2014) Hamiltonian perspective on compartmental reaction–diffusion networks. Automatica 50(3):737–746. https://doi.org/10.1016/j.automatica.2013.12.01 – 10.1016/j.automatica.2013.12.017
Rao S, van der Schaft A, Jayawardhana B (2013) A graph-theoretical approach for the analysis and model reduction of complex-balanced chemical reaction networks. J Math Chem 51(9):2401–2422. https://doi.org/10.1007/s10910-013-0218- – 10.1007/s10910-013-0218-8
van der Schaft A, Rao S, Jayawardhana B (2015) Complex and detailed balancing of chemical reaction networks revisited. J Math Chem 53(6):1445–1458. https://doi.org/10.1007/s10910-015-0498- – 10.1007/s10910-015-0498-2
Chapman A, Mesbahi M (2011) Advection on graphs. IEEE Conference on Decision and Control and European Control Conference 1461–146 – 10.1109/cdc.2011.6161471
Rantzer A (2015) Scalable control of positive systems. European Journal of Control 24:72–80. https://doi.org/10.1016/j.ejcon.2015.04.00 – 10.1016/j.ejcon.2015.04.004
Cortés J (2008) Distributed algorithms for reaching consensus on general functions. Automatica 44(3):726–737. https://doi.org/10.1016/j.automatica.2007.07.02 – 10.1016/j.automatica.2007.07.022
Zhang H, Lewis FL, Qu Z (2012) Lyapunov, Adaptive, and Optimal Design Techniques for Cooperative Systems on Directed Communication Graphs. IEEE Trans Ind Electron 59(7):3026–3041. https://doi.org/10.1109/tie.2011.216014 – 10.1109/tie.2011.2160140
Mirzaev I, Gunawardena J (2013) Laplacian Dynamics on General Graphs. Bull Math Biol 75(11):2118–2149. https://doi.org/10.1007/s11538-013-9884- – 10.1007/s11538-013-9884-8
Sontag ED (2001) Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction. IEEE Trans Automat Contr 46(7):1028–1047. https://doi.org/10.1109/9.93505 – 10.1109/9.935056
Chopra, Passivity-based control of multi-agent systems. (2006)
Willems JC (2013) Power and energy as systemic properties — Part II: Mechanical systems. 52nd IEEE Conference on Decision and Control 175–18 – 10.1109/cdc.2013.6759878
van der Schaft AJ, Maschke BM (2002) Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42(1–2):166–194. https://doi.org/10.1016/s0393-0440(01)00083- – 10.1016/s0393-0440(01)00083-3
Seslija M, van der Schaft A, Scherpen JMA (2012) Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems. Journal of Geometry and Physics 62(6):1509–1531. https://doi.org/10.1016/j.geomphys.2012.02.00 – 10.1016/j.geomphys.2012.02.006