Modeling of Physical Network Systems

TL;DR

This paper presents a mathematical framework for modeling physical network systems using Laplacian matrices, extending to complex networks, and provides methods for system conversion, analysis, and explicit computation of storage functions.

Contribution

It introduces a method to convert non-symmetric Laplacian systems into balanced forms using Kirchhoff's theorem, enabling port-Hamiltonian modeling and analysis.

Findings

Conversion of asymmetric Laplacian systems to balanced form

Explicit computation of minimal storage functions

Application to dual asymmetric consensus algorithms

Abstract

Conservation laws and balance equations for physical network systems typically can be described with the aid of the incidence matrix of a directed graph, and an associated symmetric Laplacian matrix. Some basic examples are discussed, and the extension to $k$-complexes is indicated. Physical distribution networks often involve a non-symmetric Laplacian matrix. It is shown how, in case the connected components of the graph are strongly connected, such systems can be converted into a form with balanced Laplacian matrix by constructive use of Kirchhoff's Matrix Tree theorem, giving rise to a port-Hamiltonian description. Application to the dual case of asymmetric consensus algorithms is given. Finally it is shown how the minimal storage function for physical network systems with controlled flows can be explicitly computed.