Stokes-Lagrange and Stokes-Dirac representations of $N$-dimensional port-Hamiltonian systems for modelling and control

TL;DR

This paper introduces the Stokes-Lagrange structure, extending port-Hamiltonian systems to N-dimensional domains, enhancing modeling and control through new energy port integration and multiple system representations.

Contribution

The paper develops the Stokes-Lagrange structure as a novel extension of port-Hamiltonian systems, enabling implicit Hamiltonian definitions and energy port incorporation in N-dimensional settings.

Findings

Multiple equivalent system representations demonstrated

Advantages for numerical simulation shown

Enhanced control design tools provided

Abstract

In this paper, we extend the port-Hamiltonian framework by introducing the concept of Stokes-Lagrange structure, which enables the implicit definition of a Hamiltonian over an $N$-dimensional domain and incorporates energy ports into the system. This new framework parallels the existing Dirac and Stokes-Dirac structures. We propose the Stokes-Lagrange structure as a specific case where the subspace is explicitly defined via differential operators that satisfy an integration by parts formula. By examining various examples through the lens of the Stokes-Lagrange structure, we demonstrate the existence of multiple equivalent system representations. These representations provide significant advantages for both numerical simulation and control design, offering additional tools for the modelling and control of port-Hamiltonian systems.