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Port-Hamiltonian systems on graphs

A. J. van der Schaft, B. M. Maschke

arXiv:1107.2006v2math.OCeess.SYmath.DSmath.SG

TL;DR

The paper addresses how complex physical network dynamics can be modeled within one geometric framework on open graphs. It associates graph incidence structure with Dirac structures and combines these with energy-storing or energy-dissipating relations. The resulting port-Hamiltonian structure is shared by examples including consensus algorithms and supplies tools for analysis, simulation, and control.

  • Problem

    Network dynamics from physical and non-physical origins require a common framework that represents graph interconnection, conservation, storage, and dissipation.

  • Method

    The paper constructs Dirac structures from graph incidence operators and relates edge, internal-vertex, and boundary flows and efforts through energy-storing or energy-dissipating relations.

  • Results

    The framework identifies a shared port-Hamiltonian structure across physical networks, consensus and clustering algorithms, with compositional interconnection yielding port-Hamiltonian systems.

  • Takeaways & Limitations

    The identified structure provides tools for stability analysis, simulation, and control while supporting open-system, heterogeneous, and multi-scale network modeling.

  • Takeaways & Limitations

    The paper considers only basic building blocks, while extensions to dynamical graphs, switching systems, and broader spatial-discretization settings remain further work.

Abstract

from arXiv · show

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.

1 Introduction

The paper develops a geometric framework for physical network dynamics on directed open graphs, linking graph structure with generalized Hamiltonian dynamics. It identifies a common port-Hamiltonian structure across physical and non-physical network models and emphasizes compositionality, scalability, and analysis tools.

  • Framework: The framework models physical network dynamics on directed open graphs using energy-derived constitutive relations and a generalized Hamiltonian structure.The framework treats storage at vertices and/or edges through constitutive relations derived from an energy function.
  • Framework: The graph incidence matrix directly defines a Dirac structure that captures network conservation laws.The paper identifies three canonically defined Dirac structures on vertex and edge spaces, differing in the role of boundary vertices.
  • Common structure: Consensus and clustering algorithms share the same port-Hamiltonian structure as the paper’s physical network examples.The authors use this connection to unify network dynamics from different origins.
  • Scope: The paper’s examples are simple, while extensions to arbitrary k-complexes and structure-preserving spatial discretization are described as future or related work.The authors also exclude random evolution of the graphs themselves from the framework considered here.
  • Compositionality: The identified common structure formalizes networks as open systems and supports compositional modeling of heterogeneous and multi-scale systems.The paper attributes this scalability to the compositionality properties of port-Hamiltonian systems.

2 From directed graphs to Dirac structures

This section builds the graph-based geometric ingredients of the framework, beginning with directed graphs, incidence operators, and open-graph boundary spaces. It then defines Dirac structures and their composition, which preserves the port-Hamiltonian modeling framework under interconnection.

  • Directed graphs and spaces: A directed graph uses vertices for nodes and directed edges for links, with its incidence matrix encoding edge orientation.The matrix entry is −1 at a head vertex, 1 at a tail vertex, and 0 otherwise.
  • Directed graphs and spaces: Vertex and edge spaces assign variables to vertices and edges, while flows and efforts occupy these spaces and their duals.The incidence matrix induces an incidence operator from the edge space to the vertex space, with an adjoint co-incidence operator.
  • Open graphs: An open graph separates internal vertices from boundary vertices, allowing boundary vertices to interconnect with other open graphs.Boundary spaces are isomorphic to boundary-vertex spaces but retain different physical interpretations, such as external forces versus momenta.
  • Dirac structures: A Dirac structure is a power-conserving geometric relation between flow and effort variables with maximal dimension.The paper defines flows in a vector space and efforts in its dual, using an indefinite inner product based on duality pairings.
  • Composition: Dirac structures compose to form another Dirac structure, and this property makes interconnections of port-Hamiltonian systems compositional.Under standard interconnection, the resulting system has the composed Dirac structure and a Hamiltonian equal to the sum of component Hamiltonians.

3 Port-Hamiltonian systems on graphs

The paper defines port-Hamiltonian systems with canonical graph Dirac structures and illustrates the framework across physical and algorithmic network examples.

  • Examples: Port-Hamiltonian systems on graphs are introduced through canonical graph Dirac structures and illustrated on mass-spring-damper, mechanism, consensus, and clustering examples.The section ranges from physical systems to coordination and agreement-style network dynamics.

3.1 Definition of port-Hamiltonian systems with regard to the graph Dirac structures

Port-Hamiltonian systems on graphs combine graph-induced Dirac structures with energy-storing and dissipative constitutive relations. This supports flexible storage assignments, algebraic constraints, energy accounting, and extensions to changing graphs.

  • Energy-storing relations: Energy-storing relations derive conjugate efforts from a Hamiltonian function of energy variables.The energy variables have the same dimension as the paired flow and effort variables.
  • Dissipative relations: Dissipative relations satisfy nonnegative dissipation, expressed by < e | −f >≥0 for every admissible flow-effort pair.For dampers, the paper gives f1d = −D(e1d) with (e1d)T D(e1d) ≥0.
  • Definition: A graph Dirac structure supplies the linear flow-effort relations, while constitutive relations specify energy storage or dissipation for internal variables.Storage and dissipation may be assigned to vertices, edges, or both.
  • Constraints: When efforts derived from the Hamiltonian are constrained by the Dirac structure, the resulting port-Hamiltonian model generally has algebraic constraints on its state variables.The constraints depend on the Hamiltonian H.
  • Energy balance: The total energy increases by externally supplied power minus dissipated power, reflecting the power-conserving property of the Dirac structure.The balance is expressed using the boundary power < eb | fb > and dissipative power −< eR | fR >.
  • Extensions: The framework can be extended to dynamically changing graphs, producing switching port-Hamiltonian systems on graphs.The paper presents this as a direct extension of the definition.

3.2 Mass-spring-damper systems

Mass-spring-damper networks are modeled by assigning masses to vertices and springs or dampers to edges, with graph incidence coupling their port variables. Boundary treatment determines whether forces or velocities serve as inputs.

  • Network model: Mass-spring-damper systems assign masses to vertices and springs and dampers to edges of a directed graph.The incidence matrix separates spring-edge and damper-edge contributions.
  • Mass-spring systems: For mass-spring systems, vertex momenta and edge elongations form the state variables, with a Hamiltonian combining kinetic and potential energies.In one dimension, p ∈Λ0 = R^N and q ∈Λ1 = R^M.
  • Boundary interaction: Boundary masses introduce external forces and boundary velocities through the effort-continuous graph Dirac structure.The matrix E associates boundary vertices with the corresponding boundary masses.
  • Boundary interaction: With massless boundary vertices, boundary velocities are inputs and boundary forces are outputs; with boundary masses, the roles reverse.A hybrid case can combine massive and massless boundary vertices.
  • Mass-spring systems: Ordinary spring elongations satisfy q = B^T qc, placing them in the invariant subspace im B^T ⊂Λ1.Here qc denotes the vector of vertex positions.
  • Mass-damper systems: Replacing springs with linear dampers gives f1 = −Re1, where R is a positive diagonal matrix of damping constants.Damper flows represent damping forces and efforts represent velocities.

3.3 Spatial mechanisms

Spatial mechanisms extend the graph-based port-Hamiltonian framework from point masses to rigid bodies and kinematic pairs. The overall mechanism is assembled by composing the rigid-body, pair, and graph Dirac structures.

  • Primary graph: Spatial mechanisms model rigid bodies as graph vertices and kinematic pairs as edges of a primary graph.The vertex dynamics are rigid-body dynamics rather than point-mass dynamics.
  • Rigid-body dynamics: Each rigid body has state space SE(3) × se∗(3), with body-frame momentum and kinetic and potential energies contributing to its Hamiltonian description.The inertia operator defines kinetic energy, while U(Q) defines potential energy.
  • Rigid-body dynamics: External wrenches and conjugate twists provide the force-effort port variables for each rigid body.The wrench is expressed in the fixed frame, as is its conjugate twist.
  • Kinematic pairs: Kinematic pairs impose constraints on relative twists and transmitted wrenches between rigid bodies.The pair constitutive relations are expressed in a common configuration-dependent frame.
  • Kinematic pairs: The kinematic-pair Dirac structure is non-constant on SE(3).This configuration dependence is explicitly noted in the paper.
  • Composition: The kinestatic model is obtained by composing the graph Dirac structure with the Dirac structures of all kinematic pairs.This composition yields the Dirac structure DKS.
  • Composition: The complete mechanism is formed by composing the kinestatic Dirac structure with the rigid-body Dirac structures over the product state space of all bodies.The complete state space is X = (SE(3) × se∗(3))^nRB.

3.4 Hydraulic networks

Hydraulic networks use graph flows, stored fluids, and pressures to represent conservation, storage, and dissipation. With reservoirs, the resulting dynamics are port-Hamiltonian and extend naturally to boundary interactions.

  • Network model: Hydraulic networks represent pipes as directed edges and fluid reservoirs or connection points as vertices.The section focuses on vertices associated with fluid reservoirs.
  • Conservation: The incidence matrix relates stored fluid at vertices to flows through edges, reducing to Kirchhoff’s current law Bν = 0 without reservoirs.Here ν denotes the vector of edge flows.
  • Pipe constitutive relations: Pipe-flow models combine energy storage and dissipation, with stored energy based on flow and λe(νe) representing a damping force.For edge e, the stored energy is given as 1/2 Je νe^2.
  • Port-Hamiltonian formulation: With fluid reservoirs, vertex pressures derive from reservoir Hamiltonians, making the dynamics port-Hamiltonian with state variables xv and ϕe.The formulation extends to controlled reservoirs and direct inflows or outflows at boundary vertices.

3.5 Port-Hamiltonian formulation of consensus algorithms

Consensus and clustering dynamics, despite their non-physical origins, can be represented within the same port-Hamiltonian graph structure as mass-damper systems. This representation also exposes outputs, dissipation, and conditions for subnetworks to reach consensus.

  • 3.5 Port-Hamiltonian formulation of consensus algorithms: Its equations coincide with those of the mass-damper system when leader variables are treated as boundary efforts.
  • 3.5 Port-Hamiltonian formulation of consensus algorithms: Consensus agents are vertices of an undirected interaction graph, with leaders represented as boundary vertices and followers as internal vertices.
  • 3.5 Port-Hamiltonian formulation of consensus algorithms: Follower consensus dynamics use positive edge weights and can be collected into a matrix form using the graph incidence matrix and a diagonal edge-weight matrix.
  • 3.5 Port-Hamiltonian formulation of consensus algorithms: The resulting consensus model is a port-Hamiltonian system for the flow-continuous graph Dirac structure with Hamiltonian H(x).
  • 3.5 Port-Hamiltonian formulation of consensus algorithms: The artificial boundary-flow output measures the discrepancy between leader and follower variables.
  • 3.5 Port-Hamiltonian formulation of consensus algorithms: The Laplacian matrix BGBT is independent of the arbitrary orientation chosen for the graph incidence matrix.
  • 3.5.1 Network clustering dynamical models: Network clustering models add objective-function dissipation and edge energy functions, yielding a port-Hamiltonian system with Hamiltonian H(x,z) = 1/2∥x∥2 + V(z).
  • 3.5.1 Network clustering dynamical models: When the edge energy functions define bounded constitutive relations, consensus among the x_i variables may be restricted to subnetworks.

4 Dynamical analysis

The paper analyzes mass-spring-damper dynamics through equilibria, Casimirs, energy dissipation, and invariant-subspace conditions. Under stated connectivity and damping conditions, trajectories converge to uniquely determined equilibria, including equilibria shifted by constant disturbances.

  • 4 Dynamical analysis: The analysis uses a mass-spring-damper graph whose vertices represent masses, spring edges represent springs, and damper edges represent dampers.
  • 4 Dynamical analysis: The graph is assumed connected, with the analysis extendable componentwise when it is not connected.
  • 4.1 Equilibria and Casimirs: Equilibria require the Hamiltonian momentum gradient to satisfy consensus conditions on the spring-damper graph and the position gradient to lie in the spring-graph cycle space.
  • 4.1 Equilibria and Casimirs: Casimirs include the total momentum 1T p and linear functions kT q with k ∈ ker Bs, constraining trajectories to affine spaces.
  • 4.1 Equilibria and Casimirs: For all time, q(t) − q0 remains in the spring-graph co-cycle space while 1T p(t) remains equal to 1T p0.
  • 4.2 Stability analysis: For quadratic Hamiltonians, every initial condition has a unique equilibrium in its invariant affine space, and asymptotic convergence is characterized by the largest GLs-invariant subspace contained in ker BTd.
  • 4.2 Stability analysis: Dissipation causes the energy derivative to be nonpositive, while LaSalle analysis identifies the invariant subspace governing the limiting behavior.
  • 4.2 Stability analysis: Pervasive damping means damper influence spreads through the whole system, corresponding to the relevant invariant subspace being span 1.

5 Port-Hamiltonian systems on graphs obtained by symmetry reduction

The paper relates port-Hamiltonian mass-spring and mass-spring-damper models on graphs to canonical Hamiltonian systems through symmetry reduction. The reduction identifies graph-based state spaces and connects consensus results with classical mechanics.

  • Symmetry reduction: Mass-spring port-Hamiltonian systems arise by symmetry reduction from a symplectic formulation that exploits invariance of the Hamiltonian.The symmetry acts by translating all configuration coordinates by a common constant.
  • Mass-spring formulation: The edge state q represents spring elongations through q = B^Tq_c, while p represents the momentum associated with the mass coordinates.The canonical formulation uses configuration coordinates q_c and momenta p before reduction.
  • Reduced state space: For connected graphs, the reduced mass-spring state space is im B^T × Λ0, with im B^T ⊂ Λ1.For disconnected graphs, reduction can be performed independently on each connected component, yielding the same state-space form.
  • Dissipative systems: The same reduction procedure applies to mass-spring-damper systems, even though they are not Hamiltonian systems in the standard symmetry-reduction framework.The resulting equations reduce to the previously obtained port-Hamiltonian equations on the reduced state space.
  • Consensus and convergence: Double-integrator networks are identified as linear mass-spring-damper systems with unit masses, spring constants, and damping coefficients.The paper presents its theorem as a direct extension of a velocity-consensus result.
  • Consensus and convergence: For connected spring graphs, convergence of positions and momenta to span 1 requires the stated invariant-subspace condition and ker B_s = 0.The additional condition ker B_s = 0 corresponds to the absence of cycles in the spring graph.

6 The Kirchhoff-Dirac structure on graphs and its port-Hamiltonian dynamics

The Kirchhoff-Dirac structure models graph dynamics using edge and boundary flow-effort variables while constraining internal vertex flows to zero. It supports circuit and mechanical examples and yields structural constraints on boundary variables.

  • Definition and dynamics: The Kirchhoff-Dirac structure constrains internal vertex flows to zero, so vertices carry no energy storage or dissipation.Energy-storing and dissipative relations are consequently defined only for edge variables.
  • Definition and dynamics: The Kirchhoff-Dirac structure is a separable Dirac structure obtained by composing a graph Dirac structure with a trivial internal-vertex constraint.It involves only edge and boundary flow-effort variables, unlike flow- or effort-continuous structures.
  • Electrical circuits: In RLC circuits, edge flows and efforts are currents and voltages, and the structure expresses Kirchhoff’s current and voltage laws.Capacitors and inductors contribute energy-storage relations, while resistors contribute dissipative relations satisfying V_eI_e ≤ 0.
  • Boundary variables: Boundary flows in a connected open graph sum to zero, while boundary efforts are defined only up to a common translation.Fixing one boundary vertex as reference reduces the boundary flow-effort space by two dimensions and expresses efforts as voltage differences.
  • Closing open graphs: An open graph can be closed by adding a virtual ground vertex and virtual edges connecting it to every boundary vertex.The resulting closed-graph Kirchhoff-Dirac structure extends the open-graph structure through transformed boundary efforts and flows.
  • Mechanical analogy: A spring is strictly analogous to an inductor, whereas a moving mass corresponds to a grounded capacitor rather than an ordinary capacitor.The strict mechanical analog of a capacitor is identified as an inerter.

7 Conclusions

The paper develops a geometric and compositional framework for port-Hamiltonian dynamics on graphs from incidence-matrix conservation laws. It connects network models to Hamiltonian analysis and identifies extensions beyond the basic graph setting.

  • Conclusions: The incidence matrix defines three canonical Dirac structures on vertex, edge, and boundary spaces, including a Kirchhoff-Dirac structure with no vertex storage or dissipation.Energy-storing and energy-dissipating relations then produce port-Hamiltonian network dynamics.
  • Scope and extensions: The treatment is limited to the basic building blocks of port-Hamiltonian systems on graphs.The paper frames this as a limitation of exposition rather than a restriction on the compositional framework’s possible extensions.
  • Conclusions: The framework’s compositionality supports interconnection of heterogeneous and multi-scale port-Hamiltonian systems.The paper also points to dynamical graphs and switching port-Hamiltonian systems as extensions.
  • Conclusions: The identified port-Hamiltonian structure provides tools for analyzing, simulating, and controlling the resulting network dynamics.The paper specifically relates these tools to passivity-based control, control by interconnection, and network synthesis.
  • Scope and extensions: A companion framework extends directed graphs to k-complexes for spatially discretized models of 2-D Maxwell and general diffusive systems.The paper notes the relation of these methods to structure-preserving discretizations of partial differential equation models.
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