Identifiability of Differential-Algebraic Systems

TL;DR

This work addresses the identifiability problem for nonlinear differential-algebraic equations (DAEs) by introducing a practical, rank-based identifiability test that relies solely on the model structure and output derivatives, avoiding index reduction or integration. By augmenting the system with parameter dynamics, the authors derive an identifiability matrix whose rank condition determines local identifiability, and they show how this framework subsumes nonlinear ODEs and linear DAEs as special cases. The method is demonstrated on a chemical reactor, a pendulum with index-3 DAEs, and a linear DAE system, highlighting the crucial roles of sensor placement, excitation, and system structure in identifiability. The approach provides actionable guidance for experimental design and data-driven DAE methods, with code and examples available to facilitate broader adoption in structure-preserving modeling domains.

Abstract

Data-driven modeling of dynamical systems often faces numerous data-related challenges. A fundamental requirement is the existence of a unique set of parameters for a chosen model structure, an issue commonly referred to as identifiability. Although this problem is well studied for ordinary differential equations (ODEs), few studies have focused on the more general class of systems described by differential-algebraic equations (DAEs). Examples of DAEs include dynamical systems with algebraic equations representing conservation laws or approximating fast dynamics. This work introduces a novel identifiability test for models characterized by nonlinear DAEs. Unlike previous approaches, our test only requires prior knowledge of the system equations and does not need nonlinear transformation, index reduction, or numerical integration of the DAEs. We employed our identifiability analysis across a diverse range of DAE models, illustrating how system identifiability depends on the choices of sensors, experimental conditions, and model structures. Given the added challenges involved in identifying DAEs when compared to ODEs, we anticipate that our findings will have broad applicability and contribute significantly to the development and validation of data-driven methods for DAEs and other structure-preserving models.

Figures (5)

• Figure 1: Observability of descriptor systems and its special cases.
• Figure 2: Identifiable regions of the chemical reactor model for measurement signals given by: (a)$y = x_1$, (b)$y = x_2$, and (c)$y = x_3$. The blue (red) colors correspond to states $\bm x$ in which the parameter $T_c$ is identifiable (unidentifiable). The black solid line represents the state trajectory $\bm x(t)$ starting at the initial condition $\bm x(t_0) = [0.5 \,\, 350 \,\, 0.4995]^\mathsf{T}$ and converging to a limit cycle. The system parameters were set to $(c_0,T_0,T_c) = (1, 350,305)$, and $(k_1,k_2,k_3,k_4,k_5) = (1, 209.205, 2.0921, 8.7503\cdot 10^3,7.2\cdot 10^{10})$. The tolerance for the numerical computation of the rank condition \ref{['eq.identifiabilitytest']} is set to $\max_{\bm z} \sigma n \epsilon(\norm{\mathcal{I}(\theta,\bm z)}_2)$, where $\epsilon(b)$ is the floating-point relative accuracy of $b$. The numerical simulations are shown using the ode15s solver in MATLAB.
• Figure 3: (a) State evolution of the pendulum equation as a function of time. (b, c) Identifiable (blue) and unidentifiable (red) regions in the phase plane $(x_1,x_2)$ depending on the parameter set $\theta$ sought to be identified: (b) parameter sets $m$, $g$, $L$, $[m \,\, g]$, $[g \,\, L]$, and $[m \,\, L]$ are all (locally) identifiable everywhere in the phase space; (c) parameter set $[m \,\, g \,\, L]$ is unidentifiable almost everywhere in the phase space. The parameters are set in the simulation as $L=6.25$, $g=9.81$, and $m=0.3$.
• Figure 4: Parameter estimation of the pendulum model using a prediction-error method for nonlinear DAE identification. Boxplots of the estimated parameter values for the sets of parameters predicted to be: (a) identifiable versus (b) unidentifiable. The x-axis shows the number of data points used for the parameter estimation. Each box comprise the estimated values for a 100 independent experiments. The dashed lines indicate the true values of the parameters. In each experiment, the mean square error between a measured noisy trajectory and a trajectory predicted by the model is minimized over the model parameters. The minimization is performed using the Levenberg-Marquardt algorithm, using the same settings as in the paper proposing the method Bereza2022. The initial estimates for the parameters are set to $m'=0.1$, $g'=8.21$, and $L'=5.41$.
• Figure 5: Identifiability of linear descriptor systems for different model structures and parameter sets. The network structure of the underlying model is shown on the left panels: each node represents a (differential or algebraic) variable and an edge points from node $j$ to $i$ if $A_{ij}\neq 0$. The parameters (represented by edges) sought to be identified in each case are highlighted in green. The identifiable (unidentifiable) states in the simulated trajectory are indicated in blue (red) on the right panels. The initial conditions of differential variables $\bm x_1(t_0)$ were randomly drawn from a normal distribution. (a) Identifiability of submatrices $A_{12}$ and $A_{21}$ in a dense DAE model. (b) Identifiability of matrix $A$ in a dense DAE model. (c) Identifiability of matrix $A$ in a sparse DAE model. (d) Identifiability of matrix $A$ in a dense ODE model.

Theorems & Definitions (27)

• Lemma 1
• Definition 1
• Remark 1
• Definition 2
• Lemma 2
• Theorem 1
• Remark 2
• Remark 3
• Proposition 1
• Definition 3