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Physical Network Systems and Model Reduction

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Mathematical Control Theory II

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 462))

Abstract

The common structure of a number of physical network systems is identified, based on the incidence structure of the graph, the weights associated to the edges, and the total stored energy. State variables may not only be associated to the vertices, but also to the edges of the graph; in clear contrast with multiagent systems. Structure-preserving model reduction concerns the problem of approximating a complex physical network system by a system of lesser complexity, but within the same class of physical network systems. Two approaches, respectively, based on clustering and on Kron reduction, are explored.

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Notes

  1. 1.

    See for a further discussion [24].

  2. 2.

    The setup can be easily extended (i.e., by using Kronecker products) to the situation that the scalar variable \(x_i\) is replaced by a vector in some higher dimensional physical space, e.g., \(\mathbb {R}^3\); see the remarks later on.

  3. 3.

    Note that \(\mathscr {K}\), and therefore the Laplacian matrix \(\mathscr {L}= B \mathscr {K}B^T\), is dependent on the choice of the thermodynamic equilibrium \(x^*\). However, this dependence is minor: for a connected complex graph the matrix \(\mathscr {K}\) is, up to a positive multiplicative factor, independent of the choice of \(x^*\) [25].

  4. 4.

    For an extension of these results to complex-balanced mass action kinetics reaction networks we refer to [15].

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Authors and Affiliations

  1. Jan C. Willems Center for Systems and Control, Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands

    Arjan van der Schaft

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  1. Arjan van der Schaft
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Correspondence to Arjan van der Schaft .

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Editors and Affiliations

  1. Department of Electrical Engineering, Indian Institute of Technology, Mumbai, India

    Madhu N. Belur

  2. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands

    M. Kanat Camlibel

  3. Department of Electronics and Computer Science, University of Southampton, Southampton, United Kingdom

    Paolo Rapisarda

  4. Engineering and Technology Institute Groningen, University of Groningen, Groningen, The Netherlands

    Jacquelien M.A. Scherpen

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van der Schaft, A. (2015). Physical Network Systems and Model Reduction. In: Belur, M., Camlibel, M., Rapisarda, P., Scherpen, J. (eds) Mathematical Control Theory II. Lecture Notes in Control and Information Sciences, vol 462. Springer, Cham. https://doi.org/10.1007/978-3-319-21003-2_11

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  • DOI: https://doi.org/10.1007/978-3-319-21003-2_11

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-21003-2

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