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Preprint 5-2019

Structure-preserving discretization for port-Hamiltonian descriptor systems

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Author(s) : Volker Mehrmann , Riccardo Morandin

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 5-2019

MSC 2000

65L80 Methods for differential-algebraic equations
65P10 Hamiltonian systems including symplectic integrators
93D05 Lyapunov and other classical stabilities

Abstract :
We extend the modeling framework of port-Hamiltonian descriptor systems to include under-~and over-determined systems and arbitrary differentiable Hamiltonian functions. This structure is associated with a Dirac structure that encloses its energy balance properties. In particular, port-Hamiltonian systems are naturally passive and Lyapunov stable, because the Hamiltonian defines a Lyapunov function. The explicit representation of input and dissipation in the structure make these systems particularly suitable for output feedback control. It is shown that this structure is invariant under a wide class of nonlinear transformations, and that it can be naturally modularized, making it adequate for automated modeling. We investigate then the application of time-discretization schemes to these systems and we show that, under certain assumptions on the Hamiltonian, structure preservation is achieved for some methods. Numerical examples are provided.

Keywords : port-Hamiltonian system, descriptor system, differential-algebraic equation, passivity, stability, system transformation, Dirac structure, geometric numerical integration, symplectic methods

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