Composition of Dirac structures and control of Port-Hamiltonian systems

Arjan van der Schaft, J. Cervera

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    Key feature of Dirac structures (as opposed to Poisson or symplectic structures) is the fact that the standard composition of two Dirac structures is again a Dirac structure. In particular this implies that any power-conserving interconnection of port-Hamiltonian systems is a port-Hamiltonian system itself. This constitutes a fundamental property in the port-Hamiltonian approach to modeling, simulation and control of complex physical systems. Furthermore, the composed Dirac structure directly determines the algebraic constraints of the interconnected system, as well as its Casimir functions. Especially the Casimirs are of prime importance in the set-point regulation of port-Hamiltonian systems. It is therefore of importance to characterize the set of achievable Dirac structures when a given plant port-Hamiltonian system is intercon-nected with an arbitrary controller port-Hamiltonian system. The set of achievable Dirac structures in a restricted sense has been recently characterized in [1, 2]. Here we extend this theorem to the present situation occurring in the interconnection of a plant and controller Hamiltonian system. Furthermore, wegive an insightful procedure for the construction of the controller Dirac structure. This procedure works for the general case of (non-closed) Dirac structures on manifolds. In this way we also fully characterize the set of achievable Casimir functions of the interconnected ("closed-loop") system. This yields a fundamental limitation to the design of stabilizing controllers for underactuated mechanical systems by interconnection with a port-Hamiltonian controller.
    Original languageUndefined
    Title of host publicationProceedings of the 15th International Symposium on the Mathematical Theory of Networks and Systems
    EditorsD.S. Gilliam, J. Rosenthal
    Place of PublicationSouth Bend, Indiana, USA
    PublisherUniversity of Notre Dame
    Number of pages13
    ISBN (Print)not assigned
    Publication statusPublished - 2002
    Event15th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2002 - University of Notre Dame, Notre Dame, United States
    Duration: 12 Aug 200216 Aug 2002
    Conference number: 15

    Publication series

    PublisherUniversity of Notre Dame


    Conference15th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2002
    Abbreviated titleMTNS 2002
    Country/TerritoryUnited States
    CityNotre Dame


    • EWI-16744
    • METIS-210921
    • IR-69143

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