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Port-Hamiltonian Differential-Algebraic Systems

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Surveys in Differential-Algebraic Equations I

Part of the book series: Differential-Algebraic Equations Forum ((DAEF))

Abstract

The basic starting point of port-Hamiltonian systems theory is network modeling; considering the overall physical system as the interconnection of simple subsystems, mutually influencing each other via energy flow. As a result of the interconnections algebraic constraints between the state variables commonly arise. This leads to the description of the system by differential-algebraic equations (DAEs), i.e., a combination of ordinary differential equations with algebraic constraints. The basic point of view put forward in this survey paper is that the differential-algebraic equations that arise are not just arbitrary, but are endowed with a special mathematical structure; in particular with an underlying geometric structure known as a Dirac structure. It will be discussed how this knowledge can be exploited for analysis and control.

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Notes

  1. 1.

    In the port-Hamiltonian formulation there is a clear preference for taking the charges to be the state variables instead of the voltages V i . Although this comes at the expense of the introduction of extra variables, it will turn out to be very advantageous from a geometric point of view.

  2. 2.

    Note that this separation is already present in the geometric description of Hamiltonian dynamics in classical mechanics; see e.g. [1]. There the dynamics is defined with the use of the Hamiltonian and the symplectic structure on the phase space of the system. Dirac structures form a generalization of symplectic structures, and allow the inclusion of algebraic constraints. Note furthermore that the symplectic structure in classical mechanics is commonly determined by the geometry of the configuration space, while the Dirac structure of a port-Hamiltonian system captures its network topology.

  3. 3.

    Throughout we adopt the convention that \(\frac{\partial H}{\partial x}(x)\) denotes the column vector of partial derivatives of H.

  4. 4.

    In many cases, F will be derivable from a so-called Rayleigh dissipation function \(\mathfrak{R}:\mathbb {R}^{m_{r}}\rightarrow \mathbb {R}\), in the sense that \(F(e_{R})=\frac{\partial\mathfrak{R}}{\partial e_{R}}(e_{R})\).

  5. 5.

    The Whitney sum of two vector bundles with the same base space is defined as the vector bundle whose fiber above each element of this common base space is the product of the fibers of each individual vector bundle.

  6. 6.

    On the Dutch version of the euro.

  7. 7.

    This is very much like the multi-port description of a passive linear circuit, where it is known that although it is not always possible to describe the port as an admittance or as an impedance, it is possible to describe it as a hybrid admittance/impedance transfer matrix, for a suitable selection of input voltages and currents and complementary output currents and voltages [3].

  8. 8.

    The equality \(0=b^{T}(x)\frac{\partial H}{\partial x}(x)\) also has the interpretation (well-known in a mechanical system context) that the constraint input b(x)λ is ‘workless’; i.e., the evolution of the value of the Hamiltonian H is not affected by this term.

  9. 9.

    For more details regarding the precise form of the integrability conditions see Sect. 7.

  10. 10.

    However it can be shown [26] that if (4.33) holds for some non-degenerate resistive relation then it has to hold for all.

  11. 11.

    I thank Stephan Trenn for an enlightening discussion on this issue.

  12. 12.

    See [2] for a direct approach to the composition of multiple Dirac structures.

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Acknowledgements

This survey article is based on joint work with many colleagues, whom I thank for a very stimulating collaboration. In particular I thank Bernhard Maschke for continuing joint efforts over the years.

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  1. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, PO Box 407, 9700 AK, Groningen, The Netherlands

    A. J. van der Schaft

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Correspondence to A. J. van der Schaft .

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  1. Institut für Mathematik, Technische Universität Ilmenau, Weimarer Straße 25, Ilmenau, 98693, Germany

    Achim Ilchmann

  2. Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, Hamburg, 20146, Germany

    Timo Reis

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van der Schaft, A.J. (2013). Port-Hamiltonian Differential-Algebraic Systems. In: Ilchmann, A., Reis, T. (eds) Surveys in Differential-Algebraic Equations I. Differential-Algebraic Equations Forum. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34928-7_5

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