Explicit Simplicial Discretization of Distributed-Parameter Port-Hamiltonian Systems

TL;DR

This paper introduces a finite-dimensional framework using simplicial Dirac structures to model distributed-parameter port-Hamiltonian systems, facilitating their analysis and control through matrix representations and energy shaping techniques.

Contribution

It develops a novel simplicial discretization approach for port-Hamiltonian systems, linking topological structures with energy-based modeling on simplicial manifolds.

Findings

Matrix representations of simplicial Dirac structures are derived.

Structural invariants related to energy shaping are identified.

Framework enables finite-dimensional analysis of distributed systems.

Abstract

Simplicial Dirac structures as finite analogues of the canonical Stokes-Dirac structure, capturing the topological laws of the system, are defined on simplicial manifolds in terms of primal and dual cochains related by the coboundary operators. These finite-dimensional Dirac structures offer a framework for the formulation of standard input-output finite-dimensional port-Hamiltonian systems that emulate the behavior of distributed-parameter port-Hamiltonian systems. This paper elaborates on the matrix representations of simplicial Dirac structures and the resulting port-Hamiltonian systems on simplicial manifolds. Employing these representations, we consider the existence of structural invariants and demonstrate how they pertain to the energy shaping of port-Hamiltonian systems on simplicial manifolds.