Port-Hamiltonian systems on graphs (1107.2006v2)

Abstract: In this paper we present a unifying geometric and compositional framework for modeling complex physical network dynamics as port-Hamiltonian systems on open graphs. Basic idea is to associate with the incidence matrix of the graph a Dirac structure relating the flow and effort variables associated to the edges, internal vertices, as well as boundary vertices of the graph, and to formulate energy-storing or energy-dissipating relations between the flow and effort variables of the edges and 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 are shown to share the same structure. It is shown how the identified Hamiltonian structure offers systematic tools for the analysis of the resulting dynamics.

• The paper introduces a geometric framework for modeling complex physical systems using port-Hamiltonian systems defined on open graphs and Dirac structures.
• This framework provides a unified approach applicable across diverse fields, including electrical circuits, mechanical systems, and network algorithms like consensus.
• The research establishes a scalable method for analyzing and interconnecting heterogeneous systems, bridging classical mechanics and modern network theory with potential for multi-physics and control applications.

Port-Hamiltonian Systems on Graphs

The paper "Port-Hamiltonian Systems on Graphs" by A.J. van der Schaft and B.M. Maschke introduces a geometric framework for modeling complex physical systems using port-Hamiltonian systems on open graphs. By leveraging the incidence matrix of a graph, the authors construct Dirac structures to relate flow and effort variables among edges, vertices, and boundary vertices. This formulation enables the modeling of network dynamics with energy-storing or dissipative characteristics, offering systematic methods for analysis and interconnection.

The core idea revolves around associating Dirac structures with discrete topological constructs, such as graphs, which encapsulate the conservation laws intrinsic to various physical systems. The paper categorizes these structures into flow-continuous and effort-continuous Dirac structures, differing in their treatment of boundary interactions. Additionally, a separable Dirac structure termed the Kirchhoff-Dirac structure is derived for scenarios without storage or dissipation at vertices, encapsulating classical Kirchhoff laws.

Significantly, this formulation finds applications across diverse fields, including electrical circuits, mechanical systems like mass-spring-damper systems, hydraulic networks, and consensus algorithms. For instance, in electrical circuits, the Kirchhoff-Dirac structure captures current and voltage laws, while the port-Hamiltonian approach provides a unified portrayal of capacitors, inductors, and resistors from an energy perspective. Similarly, mass-spring-damper systems are modeled by associating energy storage with springs and dissipation with dampers, revealing analogies to electrical components.

The paper further explores the dynamical analysis of port-Hamiltonian systems, highlighting stability conditions and the role of invariant subspaces under specific structural choices. A key contribution is the identification of common mathematical structures across examples, allowing for unified analysis and insights into the dynamics of non-physical systems, such as consensus and clustering algorithms.

The theoretical implications of this research are profound, establishing a bridge between classical Hamiltonian mechanics and modern network theory. Practically, this framework provides a scalable approach to modeling heterogeneous systems, accommodating both energy-conserving and dissipative interactions. As port-Hamiltonian systems inherently possess compositional properties, they hold promise for applications in multi-physics systems and cyber-physical network design.

Future research may explore extensions to dynamic graphs and higher-dimensional complexes, offering pathways for spatial discretization of distributed parameter systems. Additionally, the paper hints at the potential integration with control paradigms, suggesting fruitful avenues for marrying passivity-based control strategies with the network-centric models developed herein. This aligns with ongoing developments in systems theory, affording robust tools for simulating, analyzing, and designing intricate interconnected systems.