Port-Hamiltonian formulation of energy-based modeling frameworks

Abstract: The energy-based modelling framework proposed by Altmann and Schulze is demonstrated to have two distinct representations as port-Hamiltonian systems. The ambiguity is expressed in the role of the algebraic variable, and is distinct from the usual ambiguity in the geometric representation of coordinate representations of classical port-Hamiltonian systems. A straightforward extension to systems with feedthrough is given.

• The paper introduces two distinct port-Hamiltonian representations for energy-based models, addressing latent variable ambiguity in both geometric and algebraic formulations.
• It presents implicit and explicit energy formulations that clarify the roles of free covectors and resistive port variables, enhancing system interconnection strategies.
• The study extends port-Hamiltonian theory to include both non-feedthrough and feedthrough systems, ensuring structure preservation in complex modular designs.

Port-Hamiltonian Formulation of Energy-Based Modeling Frameworks

Overview

The paper "Port-Hamiltonian formulation of energy-based modeling frameworks" (2606.21544) addresses the geometric and algebraic foundations of energy-based modeling systems, specifically those introduced by Altmann and Schulze. It formally establishes that these frameworks admit two distinct port-Hamiltonian system representations, uncovering new sources of ambiguity in the algebraic variable that are fundamentally different from classical geometric ambiguities. The analysis encompasses both feedthrough and non-feedthrough system classes, providing a unified geometric lens and broadening the scope of port-Hamiltonian theory to encompass these emergent models.

Classical Port-Hamiltonian Systems: Geometric Structure

Port-Hamiltonian systems are inherently geometric, employing Dirac structures as models for lossless energy transport and interconnection. The system architecture consists of:

• Conservative elements: Modeled by Lagrangian submanifolds in cotangent bundles, directly linked to the Hamiltonian function.
• Dissipative elements: Captured via non-negative, linear relations within Lagrangian subspaces.
• Connective parts: External ports encoded in free variables of Dirac structures.

This geometric split facilitates automated modeling and structure-preserving interconnection, with the total system retaining the port-Hamiltonian structure through composition, as evidenced in the preservation of energy balances and the underlying graph-theoretic interconnection rules.

Energy-Based Modeling Frameworks: Algebraic Formulation

Altmann and Schulze's energy-based models extend classical port-Hamiltonian systems by introducing distinguished algebraic variables and facilitating energy-balanced models with generalized state spaces. The models are cast as differential-algebraic systems in coordinates:

• Hamiltonian $H$
• Structured antisymmetric and symmetric matrices $J, R$
• Input/output matrices $B, B^\top$

The algebraic ambiguity arises specifically in the interpretation of latent variables (e.g., $z_3$), leading to multiple geometric representations with differing fiber structures for the state and port variables.

Dual Port-Hamiltonian Representations

The paper rigorously demonstrates two port-Hamiltonian formulations for the energy-based system class:

1. Implicit Energy Formulation: $z_3$ is treated as a free covector in an enlarged Lagrangian submanifold, essentially introducing auxiliary states with constant values, and viewing $z_3$ as a coordinate in the cotangent fiber. This formulation leverages implicit Hamiltonian constraints and enables latent variable elimination, leading to equivalence with the original energy-based model.
2. Explicit Energy Formulation: $z_3$ is re-interpreted as a component of the resistive port variables, namely as a constrained vector subject to port constraints but non-contributing to dissipation. This provides a formulation with explicitly defined energy, yielding transparent input/output relations and explicit latent port structure.

The paper further explores the ambiguity in this duality—specifically, the non-canonical choice in identifying $z_3$ as either vector or covector—emphasizing its geometric and algebraic roots.

Structure-Preserving Interconnection

A key feature is the preservation of structure under interconnection, reflecting the composability central to port-Hamiltonian theory. The paper details sufficient conditions for structure-preserving interconnection, leveraging abstract characterizations of Dirac structures and Lagrangian subspaces. It formalizes composition rules ensuring that the interconnected system retains the energy-based modeling structure, even under dimension-changing port transformations and resistive port augmentations.

Extension to Feedthrough Systems

The analysis is extended to include feedthrough terms. The augmented systems are shown to admit both explicit and implicit port-Hamiltonian representations analogous to the non-feedthrough case. The generalization recovers recent energy-stable port-Hamiltonian frameworks [Buchfink et al., 2026] and ensures that the expanded system class forms a true generalization of prior models. Feedthrough inclusion does not disrupt the geometric structure, underlining the robustness of the port-Hamiltonian paradigm.

Numerical Results and Claims

The paper does not focus on simulation or direct numerical performance evaluation, but instead provides rigorous algebraic and geometric equivalences. The bold claim is the existence of two fundamentally distinct port-Hamiltonian representations for energy-based modeling frameworks, with detailed structural criteria for equivalence and ambiguity resolution. The constructive proofs provide explicit algebraic transformations and characterizations, substantiating the theoretical developments.

Implications and Future Directions

Theoretical implications are significant for the geometric modeling of multi-domain physical systems, particularly in automatic interconnection, structure-preserving discretization, and model reduction [Altmann et al., 2026]. The port-Hamiltonian formulation enhances modularity, model composability, and clarifies latent variable roles. Future developments are anticipated in:

• Higher-dimensional Dirac and Lagrangian structure analysis for complex interconnected networks
• Algorithmic synthesis of port-Hamiltonian representations for generalized energy-based systems
• Numerical integration, stability analysis, and control design exploiting dual representations and ambiguities

The framework invites further investigation into canonical coordinate choices, port-variable interpretation, and categorical approaches to energy-based modeling in the context of both deterministic and stochastic dynamics.

Conclusion

This paper establishes that energy-based modeling frameworks, including those with feedthrough, admit dual port-Hamiltonian system representations distinguished by the role of algebraic variables. The analysis provides explicit geometric and algebraic criteria for equivalence, acknowledges new sources of ambiguity, and generalizes the structure-preserving properties of port-Hamiltonian systems to a broader class. The results pave the way for advanced compositional modeling, feedthrough inclusion, and port-variable abstraction, reinforcing the theoretical foundation for modular and robust physical system design (2606.21544).