Discrete Port Hamiltonian Systems

Abstract

Either from a control theoretic viewpoint or from an analysis viewpoint it is necessary to convert smooth systems to discrete systems, which can then be implemented on computers for numerical simulations. Discrete models can be obtained either by discretizing a smooth model, or by directly modeling at the discrete level itself. One of the goals of this paper is to model port-Hamiltonian systems at the discrete level. We also show that the dynamics of the discrete models we obtain exactly correspond to the dynamics obtained via a usual discretization procedure. In this sense we offer an alternative to the usual procedure of modeling (at the smooth level) and discretization.

Stabilization of Discrete Port-Hamiltonian Dynamics via Interconnection and Damping Assignment

“…As a consequence, the feedback solution to (17) assigns the target dynamics (11) to (9). As far as the explicit form is concerned, plugging u =ū + v into (10), withū as in (18), one gets (12).…”

Structure Preserving Discretization of 1D Nonlinear Port-Hamiltonian Distributed Parameter Systems

“…In the following, we define a power balance for the expansions ( 24) and ( 27) that matches (22) and the inclusion (23). For this, first observe that, by (19) and (22), (e p ab , e q ab , e b ab ) | ( f p ab , f q ab , f b ab…”

“…The discretization in space and time of distributed parameter port-Hamiltonian systems involves fundamental questions on energy conservation, power balances, modularity in system composition and the preser-vation of Hamiltonian properties in the discretization process. Specific for this model class, discrete differential geometries were introduced in [21,22] to define discrete Dirac structures and discrete port-Hamiltonian dynamics on discrete manifolds. Collocation based methods and simplicial Dirac structures are employed in [23,24,3,25,26] to derive a discrete exterior calculus in which numerical integration preserves Dirac structures, dynamics and constitutive equations.…”

Discrete IDA-PBC design for 2D port-Hamiltonian systems

“…Among the discrete Hamiltonian systems formulation that can be found in the literature (e.g. structure discretization [Talasila et al, 2006], time-discretisation [Laila and Astolfi, 2006b], sampled-data [Stramigioli et al, 2005, Monaco et al, 2009), the discrete Hamiltonian dynamics (11) encodes two fundamental properties: energy conservation and passivity. Proposition 3.…”

“…Otherwise, discrete controller synthesis can be tackled from a direct design viewpoint. In the Hamiltonian framework, this approach is based on the definition of a discrete port-Hamiltonian dynamics (e.g., structure discretization [Talasila et al, 2006], time-discretisation [Laila and Astolfi, 2006b], sampled-data [Stramigioli et al, 2005, Monaco et al, 2009). All these discrete Hamiltonian dynamics have been derived with respect to some initial intrinsic properties, such as passivity, (almost) energy balance.…”