Port-Hamiltonian Systems on Graphs

In this paper we present a unifying geometric and compositional framework for modeling complex physical network dynamics as port-Hamiltonian systems on open graphs. The basic idea is to associate with the incidence matrix of any directed graph a Dirac structure relating the flow and effort variables associated to the edges and vertices of the graph, and to formulate energy-storing or energy-dissipating relations between the flow and effort variables of the edges and the internal vertices. This allows for state variables associated to the edges and formalizes the interconnection of networks. Examples from different origins, such as consensus algorithms, that share the same structure are shown. It is shown how the identified Hamiltonian structure offers systematic tools for the analysis and control of the resulting dynamics.