Identification of Port-Hamiltonian Differential-Algebraic Equations from Input-Output Data

TL;DR

This work tackles data-driven identification of port-Hamiltonian differential-algebraic equations (pH-DAEs) from input-output data, enabling models that respect interconnection and passivity while accommodating algebraic constraints. It introduces a framework that couples port-Hamiltonian neural networks with a DAE solver, using a backward-Euler discretization and Newton iterations within an output-error SUBNET scheme to identify parameters without index reduction. The parameter set $\theta$ preserves the pH structure through constrained forms of $E_\theta$, $J_\theta$, $R_\theta$, $Q_\theta$, and $G_\theta$, with the training driven by differentiable solvers and gradient-based optimization. A DC power-network case study demonstrates accurate state and parameter recovery under noisy IO data, with performance degrading gracefully as measurement noise increases, highlighting the method’s robustness and practical relevance for large-scale energy networks.

Abstract

Many models of physical systems, such as mechanical and electrical networks, exhibit algebraic constraints that arise from subsystem interconnections and underlying physical laws. Such systems are commonly formulated as differential-algebraic equations (DAEs), which describe both the dynamic evolution of system states and the algebraic relations that must hold among them. Within this class, port-Hamiltonian differential-algebraic equations (pH-DAEs) offer a structured, energy-based representation that preserves interconnection and passivity properties. This work introduces a data-driven identification method that combines port-Hamiltonian neural networks (pHNNs) with a differential-algebraic solver to model such constrained systems directly from noisy input-output data. The approach preserves the passivity and interconnection structure of port-Hamiltonian systems while employing a backward Euler discretization with Newton's method to solve the coupled differential and algebraic equations consistently. The performance of the proposed approach is demonstrated on a DC power network, where the identified model accurately captures system behaviour and maintains errors proportional to the noise amplitude, while providing reliable parameter estimates.

Figures (5)

• Figure 1: Computational pipeline of the simulated response using SUBNET and the DAE solver. The encoder $\psi_\eta$ estimates the initial state from past inputs and outputs, which is propagated by the DAE solver over the truncation length $T$.
• Figure 2: DC power network example as proposed by Mehrmann2019Structure-preservingSystems.
• Figure 3: Example of the power network system behaviour for the first 25 seconds of the training dataset. The first subplot shows the external generator voltage taken as an input, $u=E_G$. The second subplot shows the corresponding noisy output measurement of the generator current, $y=I_G$.
• Figure 4: Simulation of the identified model. In the first subplot, the dots represent the sampled true output values and the solid line indicates the simulated model response, for five seconds for visibility. The second subplot shows the error between the predicted and measured outputs for the entire time period of 50 seconds, where the std of the noise is given by the dotted line.
• Figure 5: Boxplot of the normalised parameter deviations for 10 independent runs with noisy data sets containing measurement of $(I_G, V_1, V_2)$ with noise level of 40 dB.