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Linear port-Hamiltonian descriptor systems

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Abstract

The modeling framework of port-Hamiltonian systems is systematically extended to linear constrained dynamical systems (descriptor systems, differential-algebraic equations) of arbitrary index and with time-varying constraints. A new algebraically and geometrically defined system structure is derived. It is shown that this structure is invariant under equivalence transformations, and that it is adequate also for the modeling of high-index descriptor systems. The regularization procedure for descriptor systems to make them suitable for simulation and control is modified to preserve the port-Hamiltonian form. The relevance of the new structure is demonstrated with several examples.

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Acknowledgements

We acknowledge many interesting discussions with Robert Altmann and Philipp Schulze from TU Berlin and Arjan Van der Schaft from RU Groningen. We also thank two anonymous reviewers for helpful comments to improve the presentation. The first author has been supported by Einstein Foundation Berlin, through an Einstein Visiting Fellowship. The second author has been supported by Deutsche Forschungsgemeinschaft for Research support via Project A02 in CRC 1029 TurbIn, and project B03 in CRC-TR154 as well as by Einstein Foundation Berlin within the Einstein Center ECMath.

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Authors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    Christopher Beattie

  2. Institut für Mathematik MA 4-5, TU Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany

    Volker Mehrmann

  3. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Hongguo Xu

  4. Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands

    Hans Zwart

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  1. Christopher Beattie
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  2. Volker Mehrmann
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  4. Hans Zwart
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Correspondence to Volker Mehrmann.

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Christopher Beattie: Supported by Einstein Foundation Berlin, through an Einstein Visiting Fellowship. Volker Mehrmann: Supported by Einstein Foundation Berlin via the Einstein Center ECMath and by Deutsche Forschungsgemeinschaft via Project A02 within CRC 1029 ’TurbIn’. Hongguo Xu : Partially supported by Alexander von Humboldt Foundation and by Deutsche Forschungsgemeinschaft, through the DFG Research Center Matheon Mathematics for Key Technologies in Berlin.

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Beattie, C., Mehrmann, V., Xu, H. et al. Linear port-Hamiltonian descriptor systems. Math. Control Signals Syst. 30, 17 (2018). https://doi.org/10.1007/s00498-018-0223-3

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