Distributed port-Hamiltonian formulation of infinite dimensional systems

Alessandro Macchelli, A. Macchelli, Arjan van der Schaft, Claudio Melchiorri

    Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

    Abstract

    In this paper, some new results concerning the modeling and control of distributed parameter systems in port Hamiltonian form are presented. The classical finite dimensional port Hamiltonian formulation of a dynamical system is generalized in order to cope with the distributed parameter and multi-variable case. The resulting class of infinite dimensional systems is quite general, thus allowing the description of several physical phenomena, such as heat conduction, piezoelectricity and elasticity. Furthermore, classical PDEs can be rewritten within this framework. The key point is the generalization of the notion of finite dimensional Dirac structure in order to deal with an infinite dimensional space of power variables. In this way, also in the distributed parameter case, the variation of total energy within the spatial domain of the system can be related to the power ow through the boundary. Since this relation deeply relies on the Stokes theorem, these structures are called Stokes-Dirac structures. As far as concerns the control problem, it seems natural that also finite dimensional control methodologies developed for finite dimensional port Hamiltonian systems can be extended in order to cope with infinite dimensional systems. In this paper, the control by interconnection and energy shaping methodology is applied to the stabilization problem of a distributed parameter system by means of a finite dimensional controller interconnected to its boundary. The key point is the generalization of the definition of Casimir function to the hybrid case, that is the dynamical system to be considered results from the power conserving interconnection of an infinite and a finite dimensional part. A simple application concerning the stabilization of the one-dimensional heat equation is presented.
    Original languageEnglish
    Title of host publicationProceedings of the 16th International Symposium on Mathematical Theory of Networks and Systems
    Place of PublicationLeuven
    PublisherKatholieke Universiteit Leuven
    Pages-
    Number of pages25
    ISBN (Print)9056825178
    Publication statusPublished - 2004
    Event16th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2004 - Leuven, Belgium
    Duration: 5 Jul 20049 Jul 2004
    Conference number: 16

    Conference

    Conference16th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2004
    Abbreviated titleMTNS
    CountryBelgium
    CityLeuven
    Period5/07/049/07/04

    Keywords

    • METIS-220139
    • EWI-16812
    • IR-69099

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