[Paper Review] Port-Hamiltonian systems on discrete manifolds

Marko Šešlija, Jacquelien M.A. Scherpen+1 more|arXiv (Cornell University)|Jan 27, 2012

Control and Stability of Dynamical Systems7 references4 citations

TL;DR

This paper introduces a geometric framework for port-Hamiltonian systems on discrete manifolds using simplicial Dirac structures, enabling structure-preserving spatial discretization of distributed-parameter systems. By modeling cochains on primal and dual simplicial complexes with coboundary operators and Hodge star matrices, the method preserves key physical properties like energy conservation and symplecticity, and successfully discretizes the wave and telegraph equations with 1/n accuracy.

ABSTRACT

This paper offers a geometric framework for modeling port-Hamiltonian systems on discrete manifolds. The simplicial Dirac structure, capturing the topological laws of the system, is defined in terms of primal and dual cochains related by the coboundary operators. This finite-dimensional Dirac structure, as discrete analogue of the canonical Stokes-Dirac structure, allows for the formulation of finite-dimensional port-Hamiltonian systems that emulate the behaviour of the open distributed-parameter systems with Hamiltonian dynamics.

Motivation & Objective

• To develop a geometric framework for modeling port-Hamiltonian systems on discrete manifolds that preserves intrinsic system structures.
• To define a discrete analogue of the infinite-dimensional Stokes-Dirac structure using cochains on primal and dual simplicial complexes.
• To enable finite-dimensional, structure-preserving spatial discretizations of distributed-parameter systems with boundary energy flow.
• To ensure preservation of physical properties such as energy conservation, symplecticity, and differential gauge symmetry in numerical schemes.
• To demonstrate the method on the wave equation and telegraph equations with quantified accuracy.

Proposed method

• Representing the spatial domain as a simplicial complex and its circumcentric dual to model discrete differential forms via cochains.
• Defining the discrete exterior derivative as the coboundary operator on primal and dual complexes.
• Constructing a simplicial Dirac structure using primal and dual cochains linked by the coboundary and Hodge star operators.
• Introducing a modified dual boundary operator to preserve the integration-by-parts formula in the discrete setting.
• Formulating finite-dimensional port-Hamiltonian systems using matrix representations of the Hodge star, coboundary, and trace operators.
• Deriving discrete system dynamics via matrix equations that mirror the continuous port-Hamiltonian structure.

Experimental results

Research questions

• RQ1 How can a finite-dimensional Dirac structure be constructed on discrete manifolds to model distributed-parameter systems?
• RQ2 What is the discrete analogue of the Stokes-Dirac structure in the context of simplicial complexes?
• RQ3 How can boundary conditions be consistently incorporated in the discrete setting to preserve energy flow?
• RQ4 Can the resulting discretization preserve key physical properties such as energy conservation and symplecticity?
• RQ5 What is the convergence rate of the proposed structure-preserving discretization for wave and telegraph equations?

Key findings

• The proposed simplicial Dirac structure provides a finite-dimensional, structure-preserving discretization of distributed-parameter port-Hamiltonian systems.
• The method successfully models the wave equation and telegraph equations on a bounded domain using discrete cochains and matrix operators.
• The discretization preserves key physical properties such as energy conservation, symplecticity, and differential gauge symmetry.
• The accuracy of the method is quantified as 1/n, with n being the number of elements in the discretization.
• The discrete system can be physically realized with inductors, capacitors, and transformers, reflecting the non-Poisson nature of the finite-dimensional Dirac structure.
• Stabilization is achieved by terminating boundary ports with resistive elements, confirming passivity and practical applicability.