Port-Hamiltonian Unification

• Port-Hamiltonian unification is a rigorous framework that models, interconnects, and simulates multiphysics systems by preserving energy balance and stability.
• It employs Dirac structures, skew-symmetric interconnections, and modular decomposition to integrate diverse domains like mechanical, electrical, and thermal networks.
• The approach supports both monolithic and distributed simulations, ensuring passivity and stability through advanced operator splitting and iterative methods.

Port-Hamiltonian unification is a rigorous, structure-preserving framework for modeling, interconnecting, and simulating complex multiphysics networks across scales and domains. It systematically encodes both the conservation and dissipation of energy through networked "ports," yielding a class of models that encompasses mechanical, electrical, fluid, and thermal systems under one algebraic-geometric umbrella. Central to this unification are the concepts of Dirac structures, skew-symmetric interconnection, modular subsystem decomposition, and compatibility with both monolithic and distributed computational approaches. The framework ensures that key invariants—energy (or exergy) balance, passivity, and Lyapunov stability—are preserved at each step, regardless of whether models are aggregated or split for simulation and analysis (Ehrhardt et al., 25 Nov 2025, Schaft et al., 2011, Hauschild et al., 2019).

1. Core Port-Hamiltonian System Structure

A (standard) port-Hamiltonian ODE is defined by the state $x\in\mathbb{R}^n$, input $u\in\mathbb{R}^m$, output $y\in\mathbb{R}^m$, Hamiltonian $H\in C^1(\mathbb{R}^n,\mathbb{R})$, skew-symmetric structure matrix $J(x)=-J(x)^{T}$, symmetric positive semidefinite dissipation matrix $R(x)=R(x)^T\succeq 0$, and input matrix $G(x)$. The dynamics is

$$\dot{x} = [J(x) - R(x)] \nabla H(x) + G(x) u, \qquad y = G(x)^T \nabla H(x).$$

The "dissipation inequality" (power balance) is

$$\frac{d}{dt}H(x) = -\nabla H(x)^T R(x) \nabla H(x) + u^T y,$$

which implies strict output passivity. This structure generalizes classical Hamiltonian systems and forms the algebraic backbone for describing coupled subsystems, DAEs, and infinite-dimensional PDEs in a unified fashion (Ehrhardt et al., 25 Nov 2025, Bartel et al., 2023, Hauschild et al., 2019, Schaft, 2024).

2. Interconnection and Monolithic Formulation

The unification is achieved by encoding interconnection at the port level using a skew-symmetric coupling matrix $W=-W^T$. Consider two pH-subsystems with state $x_i$, energy $H_i(x_i)$, and corresponding $J_i$, $R_i$, $G_i$. Linear interconnection at ports leads to

$$\begin{pmatrix} u_1 \ u_2 \end{pmatrix} = W \begin{pmatrix} y_1 \ y_2 \end{pmatrix}.$$

The global (monolithic) port-Hamiltonian system is then obtained by stacking states, energies, and port maps: $$X = \begin{pmatrix} x_1 \ x_2 \end{pmatrix}, \quad \mathcal{H}(X) = H_1(x_1) + H_2(x_2), \quad \mathcal{G}(X) = \text{block-diag}(G_1, G_2),$$

$$\mathcal{J}(X) = \begin{pmatrix} J_1 & 0 \ 0 & J_2 \end{pmatrix} - \mathcal{G}(X) W \mathcal{G}(X)^T, \quad \mathcal{R}(X) = \text{diag}(R_1, R_2),$$

$$\dot{X} = [\mathcal{J}(X) - \mathcal{R}(X)] \nabla \mathcal{H}(X), \qquad Y = \mathcal{G}(X)^T \nabla \mathcal{H}(X).$$

This form preserves passivity and stability globally, and provides a unified Lyapunov framework: if $\mathcal{R} \succ 0$ (except at $X=0$), then the unique equilibrium is globally asymptotically stable (Ehrhardt et al., 25 Nov 2025). The unification is independent of the subsystem domains—mechanical, electrical (e.g., MNA circuits), hydraulic, thermal, or others—so long as the interfaces are port-Hamiltonian (Schaft et al., 2011, Bartel et al., 2023).

3. Structure-Preserving Decoupling: Splitting and Distributed Simulation

The monolithic system can be systematically inverted back to weakly coupled modules via operator splitting or dynamic iteration:

• Operator splitting: Decompose the structure matrix into block-diagonal ("local") and off-diagonal ("coupling") components:

$$\mathcal{J} = \mathcal{J}_{\rm diag} + \mathcal{J}_{\rm off}, \quad \mathcal{R} = \mathcal{R}_{\rm diag} + \mathcal{R}_{\rm off}.$$

Apply Lie-Trotter or Strang splitting to alternate between integrating uncoupled local flows and coupling-induced skew-Hamiltonian flows, with rigorous local/global error bounds and structure preservation.

• Dynamic iteration: At each iteration $k$, update subsystem states independently, using the most recent coupled port values from other blocks until convergence. Provided the interconnection satisfies a small-gain criterion, convergence is linear and the method preserves the pH structure (Ehrhardt et al., 25 Nov 2025, Günther et al., 2020).

The decoupled view enables scalable, parallel, or multi-rate integration, and re-use of subsystem-specific legacy solvers—without loss of passivity or energy consistency. Monolithic and split forms are systematically interchangeable, enabling both stability analysis and efficient computation across networks (Ehrhardt et al., 25 Nov 2025).

4. Geometric and Graph-Theoretic Foundations

Port-Hamiltonian unification is deeply rooted in Dirac structures and their composition, including:

• Dirac structures from graphs: Incidence matrices give separable Dirac structures mediating the flow/effort coupling between edges and vertices, enforcing conservation laws (e.g., Kirchhoff, mass balance) at every scale (Schaft et al., 2011).
• Boundary and interconnection ports: Open systems (with boundaries) carry explicit port variables, through which energy is exchanged or through which network interconnections are realized.
• Compositionality: Composing Dirac structures by matching ports guarantees that the combined network is again a Dirac (thus port-Hamiltonian) system, ensuring modular extensibility.
• Reduction and multi-scale coupling: From discrete graphs (mass-spring, RLC networks, consensus) to continuum PDEs (hydrodynamics, elasticity), algebraic and topological compatibility is achieved via such structures (Schaft et al., 2011, Rashad et al., 2024, Bendimerad-Hohl et al., 2024).

Unification at the structure level underpins robust analytical techniques, e.g., via identification of Casimir invariants, energy-based Lyapunov functions, and graph-induced constraints.

5. Applications and Multiphysics Modeling

Port-Hamiltonian unification enables a broad spectrum of complex system modeling, including:

• Coupled circuit-electromagnetic-thermal networks: Modified nodal analysis, Maxwell equations, EM-circuit interactions, and energy networks can all be written and interconnected using pH-DAE (port-Hamiltonian differential-algebraic equations) (Bartel et al., 2023).
• Distributed-parameter and PDE systems: Structure-preserving discretization of PDEs on meshes or networks (using, e.g., discrete exterior calculus) naturally integrates with the unification framework, and boundary conditions, including moving boundaries, carry over through the time-dependent Dirac structure (Seslija et al., 2011, Meijer et al., 24 Jan 2025).
• Thermodynamics and exergy: Extending the Hamiltonian from energy to exergy, and embedding GENERIC formalism, ties irreversible thermodynamic systems (multidomain, non-isothermal) into port-Hamiltonian unification (Lohmayer et al., 2020, Hauschild et al., 2019).
• Control and systems theory: Controller design via interconnection (e.g., passivity-based, Casimir shaping), set-point stabilization, and Lyapunov-based analysis rely on the preserved structure and global passivity (Schaft, 2024, Ehrhardt et al., 25 Nov 2025).

The framework extends seamlessly from lumped-parameter systems to infinite-dimensional cases, provides modularity for large-scale networked control, and is compatible with graph-theoretic representation and machine learning-based structure inference (Salnikov et al., 2022).

6. Numerical, Algebraic, and Algorithmic Aspects

Unification has critical implications for numerical analysis and automated modeling:

• Operator splitting and structure-preserving integration: Splitting schemes (Lie, Strang) are compatible with the port-Hamiltonian structure, support parallelization, and preserve global stability and passivity, with quantifiable error bounds (Ehrhardt et al., 25 Nov 2025).
• Automated model generation: Explicit input-state-output port-Hamiltonian forms can be automatically realized from multi-bond graphs, subject to algebraic rank conditions ensuring the existence of a pH representation (Pfeifer et al., 2019).
• DAE and iterative (Jacobi, Gauss-Seidel) methods: Distributed solvers for coupled pH-DAEs are again pH-DAEs with modified ports, ensuring preservation of the Hamiltonian structure at every iteration (Günther et al., 2020).
• Structure-preserving discretization: Techniques such as finite-volume, DG finite element, and discrete exterior calculus methods yield numerical schemes that inherit exact (discrete) Dirac structures, optimal convergence, and power-preserving properties (Kumar et al., 2022, Seslija et al., 2011).

Systematic modularity and structure preservation translate into robust computational pipelines for large multiphysics systems.

7. Broad Implications and Generalization

Port-Hamiltonian unification provides a single structural and geometric language for modeling, simulation, reduction, and control of energy-based dynamic networks. Its reach encompasses:

• All classical and modern multiphysics domains (mechanical, electrical, thermal, fluid, continuum mechanics, thermodynamically consistent flows).
• Balanced and hierarchical coupling of finite-dimensional, DAE, and infinite-dimensional systems.
• A template for both theoretical analysis and practical engineering synthesis, control, and optimization.
• Harmony with compositionality principles (via Dirac and Stokes–Dirac/Lagrange structures), ensuring preservation of physical laws at each scale and interface.

The unification thus enables a systematic, globally structure-preserving framework that is modular, extensible, and mathematically robust, reflecting the actual interconnectivity and energy/dissipation mechanisms of contemporary large-scale engineered and natural systems (Ehrhardt et al., 25 Nov 2025, Schaft et al., 2011, Rashad et al., 2024, Bendimerad-Hohl et al., 2024, Philipp et al., 2023, Hauschild et al., 2019).