Port-Hamiltonian Systems

• Port-Hamiltonian systems are defined as geometric, energy-based frameworks that model interconnections and energy exchanges across diverse physical domains.
• They employ Dirac structures for lossless interconnection and resistive mappings to capture dissipation, ensuring passivity and Lyapunov stability.
• Applications span electrical circuits, mechanical systems, and distributed-parameter systems, enabling robust control, model reduction, and structure-preserving discretization.

A port-Hamiltonian system (PHS) is a geometric, energy-based framework for the modeling, analysis, and control of open nonlinear, linear, finite- or infinite-dimensional, and multi-physical dynamical systems. PHS generalize classical Hamiltonian systems to allow for energy dissipation, constraints, boundary interactions, port-based interconnection, and coupling across physical domains. The structure is grounded in the use of Dirac structures for expressing energy-conserving interconnection laws and Lagrangian or resistive structures for constitutive and dissipative relations. At its core, a PHS encodes the dynamics and energetics of complex systems through the careful assignment of energy (Hamiltonian function), skew-adjoint interconnection, and dissipative mappings, often in state space or exterior-algebraic/generalized geometric representations. The theory unifies diverse domains—mechanics, electromagnetics, circuits, fluids, thermodynamics—under a compositional paradigm that is robust to modularity, discretization, and feedback interconnection.

1. Mathematical Framework and Dirac Structures

The port-Hamiltonian paradigm is built on three key components:

1. Hamiltonian Function: $H: X \to \mathbb{R}$, the total stored energy (generalized potential plus kinetic, electromagnetic, thermal, etc.), typically smooth and bounded below.
2. Interconnection via Dirac Structures: An energy-conserving relation $D \subseteq TX \oplus T^*X$ (in finite dimensions, $J^T=-J$), enforcing power-conservation $$\langle e, \dot{x} \rangle + \langle e^p, f^p \rangle = 0$$ and allowing for ports that interface subsystems or the environment (Krhac et al., 2024, Schaft, 2024, Schaft et al., 2011, Bartel et al., 2023). In networked or discretized settings, this extends naturally to open graphs or discrete manifolds using incidence matrices or discrete exterior calculus (Schaft et al., 2011, Seslija et al., 2012).
3. Dissipative or Resistive Structure: $R(x)=R(x)^T \succeq 0$ captures irreversible (dissipative) effects, with positive semi-definite quadratic form or generally monotone relations (Schaft, 2024, Schaft et al., 2022, Camlibel et al., 2022). The combination of Dirac (lossless interconnection) and resistive coupling yields a maximally monotone structure, crucial for Lyapunov stability and passivity (Schaft et al., 2022).

The generic (finite-dimensional) state-space representation,

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

captures energy flow, with $J(x)$ skew-symmetric, $R(x)$ positive semi-definite, and $G(x)$ the port matrix for input $u$ and output $D \subseteq TX \oplus T^*X$0 (Bartel et al., 2023).

Energy Balance

The fundamental dissipation (power-balance) inequality holds: $D \subseteq TX \oplus T^*X$1 demonstrating the system is (input–output) passive, with $D \subseteq TX \oplus T^*X$2 as a storage or Lyapunov function (Schaft, 2024, Bartel et al., 2023, Schaft et al., 2022).

2. Extensions: DAEs, Boundary Ports, and Infinite-Dimensional Systems

PHS generalize to systems with algebraic constraints, infinite-dimensional PDEs, and systems on time-varying domains:

• PH-DAEs: Descriptor forms account for algebraic constraints (e.g., circuits, constrained mechanics) by considering singular or rectangular mass matrices and algebraic constraint equations. The port-Hamiltonian-DAE structure, $D \subseteq TX \oplus T^*X$3, unifies modeling for general networked, possibly index-2, systems (Schaft et al., 2022, Mehrmann et al., 2019, Günther et al., 2020).
• Boundary/Distributed Parameter Systems: Distributed-parameter PHS are governed by skew-adjoint differential (Stokes–Dirac) structures encoding spatial/temporal energy exchanges and include boundary ports for energy flow at domain boundaries. In PDEs, power-balance is ensured by integrating over the spatial domain and accounting for boundary fluxes (Rashad et al., 2024, Meijer et al., 24 Jan 2025). The Dirac formalism also applies to moving boundaries, allowing dynamic meshing and energy-stable discretizations (Meijer et al., 24 Jan 2025).
• Discrete Manifolds and Graphs: The structure extends naturally to graph-based and discretized systems via discrete Dirac structures. For example, the coupling of flow and effort variables across simplicial complexes captures topological constraints and energy balances exactly in finite-dimensional settings (Schaft et al., 2011, Seslija et al., 2012).

3. Modular Interconnection, Control, and Coupling

Modular Coupling

PHS are closed under power-conserving interconnection: the composition of Dirac structures remains a Dirac structure, whether interconnecting subsystems, implementing boundary feedback, or assembling large-scale multiphysics models (Ehrhardt et al., 25 Nov 2025, Bartel et al., 2023, Günther et al., 2020). This guarantees that properties such as passivity, energy conservation/dissipation, and structural integrity are globally preserved.

Control by Interconnection

Compositional control strategies exploit PHS structure:

• Set-point stabilization and shaping: Passivity-based feedback, Casimir function shaping, and interconnection with controller-structure PHS yield simple, physically interpretable, and robust control laws (Schaft, 2024).
• Energy/Power Ports: Newer extensions distinguish between power ports (effort × flow = power) and energy ports (conjugate variables entering directly in the Hamiltonian), facilitating direct energy shaping (Krhac et al., 2024, Schaft, 2024).
• Distributed and Decoupled Simulation: Structure-preserving couplings (linear/skew-symmetric interconnection of subsystems) and operator splitting for distributed simulation retain PHS properties at all scales (Ehrhardt et al., 25 Nov 2025).

4. Discretization, Numerical Methods, and Stochastic PHS

Structure-Preserving Discretization

Time discretization using symplectic Runge–Kutta methods (collocation, Gauss–Legendre, discrete gradients) preserves discrete Dirac structures and yields discrete-time PHS with exact or high-order discrete-assured energy balances (Kotyczka et al., 2018, Mehrmann et al., 2019, Cherifi et al., 2023). The passage from continuous to discrete PHS is systematically achieved by discretizing the underlying Dirac relation and enforcing energy contracts at each time step.

Energy-Stable and Exergetic Formulations

Recent generalizations include energy-stable PHS (es-pH), which blend port-Hamiltonian and energy-stable system formalisms, and exergetic PHS, in which the Hamiltonian expresses exergy (maximum available work) in thermodynamic settings (Buchfink et al., 6 Jun 2025, Lohmayer et al., 2020). These formulations support model reduction, optimal control, and extend to nonequilibrium thermodynamics (GENERIC structure correspondence).

Stochastic Port-Hamiltonian Systems

PHS with stochastic ports enable power-conserving modeling under random perturbations. Each port can embed a noise process (semimartingale), and the stochastic Dirac structure ensures energy-balance and interconnection properties are maintained in expectation. The resulting S(PHS) encompass noise-driven systems, stochastic optimal control, and uncertainty quantification in physical networks (Cordoni et al., 2019).

5. Theoretical Properties: Passivity, Monotonicity, and Generic Controllability

Passivity and Lyapunov Stability

Passivity—the property that the system cannot deliver more energy than it receives—follows from the structural conditions $D \subseteq TX \oplus T^*X$4, $D \subseteq TX \oplus T^*X$5, and under mild convexity of $D \subseteq TX \oplus T^*X$6 ensures Lyapunov or asymptotic stability for $D \subseteq TX \oplus T^*X$7 configurations (Schaft, 2024, Schaft et al., 2022, Camlibel et al., 2022).

Monotonicity and Incremental Passivity

PHS can be cast as monotone or maximally monotone operators. Incrementally port-Hamiltonian systems admit relations that are cyclically or maximally monotone, enabling convex optimization approaches to equilibrium computation and facilitating the analysis of incremental and differential passivity (Camlibel et al., 2022).

Generic Controllability

In the linear case, controllability is a generic property of the port-Hamiltonian class: the set of controllable linear PHS is a relative generic subset of all such models, meaning that for almost all choices of parameters, the system is controllable (Kirchhoff, 2021). Exceptional uncontrollable cases form a set of measure zero.

6. Physical and Engineering Applications

PHS are foundational in modeling:

• Electrical and electronic circuits: Including nonlinear networks, interconnecting components via Kirchhoff-Dirac structures, and subsuming MNA equations into an energy-structured DAE model (Gernandt et al., 2020, Bartel et al., 2023).
• Mechanical and multi-body systems: Mass–spring–damper networks, constrained multibody dynamics.
• Distributed-parameter systems: Continuum mechanics, Navier–Stokes fluids, Maxwell's equations, and coupled fluid–structure or electromagnetics–circuit models (Rashad et al., 2024, Meijer et al., 24 Jan 2025, Bartel et al., 2023).
• Thermodynamics and exergy-based systems: Incorporating exergy as Hamiltonian, bond-graph methods, and facilitating thermodynamic optimization (Lohmayer et al., 2020).

Beyond direct modeling, PHS provide systematic methodologies for structure-preserving model reduction, energy-based optimal control, structure-preserving numerical schemes, modular system construction, and robust fault-tolerant control for large multi-physics systems (Ehrhardt et al., 25 Nov 2025, Buchfink et al., 6 Jun 2025, Kotyczka et al., 2018, Bartel et al., 2023).

7. Generalizations and Recent Developments

Research continues to expand the PHS framework:

• Singular and Beyond-Passivity Systems: Including formulations where the vector field admits singularities, yielding cyclo-dissipative systems capable of active power injection, relevant for power electronics and microgrid control (Sandberg et al., 11 Feb 2026).
• Descriptor and Under/Overdetermined Systems: Generalizations to under- and over-determined systems, higher-index DAEs, and arbitrary differentiable Hamiltonians, with Dirac structures accommodating algebraic constraints and invariance under nonlinear coordinate transformations (Mehrmann et al., 2019, Schaft et al., 2022).
• Discrete- and Time-Varying Domains: Extension to moving boundary problems, discrete manifolds, and adaptive mesh settings, leveraging geometric structure for robust spatial-temporal discretization (Meijer et al., 24 Jan 2025, Seslija et al., 2012).

A unifying thread across this body of work is that the port-Hamiltonian structure provides a geometric, modular, and physically interpretable framework that transcends disciplinary boundaries, ensuring that core energetic properties—conservation, dissipation, interconnection passivity—are maintained at every level of modeling, discretization, reduction, and control (Krhac et al., 2024, Schaft, 2024, Ehrhardt et al., 25 Nov 2025).