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Subspace Based Identification of Errors-in-Variables Linear Descriptor Systems

Deepanjhan Das, Shankar Narasimhan

arXiv:2608.30259v1eess.SY

TL;DR

The paper addresses two persistent gaps in noisy linear descriptor-system identification: requiring an input-output partition and pre-specifying the DAE index. It extends subspace-based iterative PCA behaviorally, treating measured variables symmetrically and estimating system structure from data. Simulations on index-0 and index-1 systems confirm accurate parameter estimation and the framework's effectiveness.

  • Problem

    Existing EIV identification methods retain a priori input-output classification and may require pre-specification of whether a DAE is index-0 or index-1.

  • Method

    The framework extends SMI-IPCA to a behavioral formulation that jointly estimates noise variances, constraint counts, minimal order, algebraic constraints, and the dynamic kernel.

  • Results

    Simulation studies on index-0 and index-1 systems confirm that the algorithm correctly estimates all system parameters.

  • Takeaways & Limitations

    The identification strategy can be used for causal discovery in linear descriptor systems without prior input-output or index specification.

Abstract

from arXiv · show

The identification of linear descriptor systems (DAEs) from noise-corrupted data makes two critical assumptions: requirement of an \textit{a priori} classification of variables into inputs and outputs, and a pre-specified structural assumption with respect to the index of the system. This paper proposes a data-driven methodology for identifying index-0 and index-1 DAEs within an errors-in-variables framework. We extend a subspace-based iterative PCA (SMI-IPCA) approach to the behavioral setting, treating all measured variables as a unified augmented vector to avoid classification bias. This method enables systematic estimation of the noise variances, the number of algebraic and differential output variables, while simultaneously identifying the algebraic constraints and kernel representation of the dynamic system corresponding to its minimal realization order without prior structural knowledge. Simulation studies on index-0 and index-1 systems demonstrate the effectiveness of the proposed approach and its practical applicability.

I. INTRODUCTION

Linear descriptor-system identification from noisy data is complicated by unknown variable roles and DAE index. The proposed behavioral subspace framework addresses both without prior structural classification.

  • Descriptor systems couple differential and algebraic equations, representing physical laws and conservation principles directly in model structure.
  • Classical subspace methods assume noise-free inputs, while errors-in-variables methods address bias but commonly retain input-output classification.
  • Existing approaches also require practitioners to pre-specify whether the system is index-0 or index-1 before identification.
  • The proposed method unifies all measured signals symmetrically and infers the input-output partition from data.
  • It detects algebraic constraints to determine the index and estimates noise variances, system order, constraint counts, and implicit constraint matrices in one procedure.
  • The framework represents dynamic behavior through a minimal-order kernel or dynamic constraint matrix, with algebraic constraints encoded separately.

B. Measurement error model under EIV setting

Under the errors-in-variables model, every measured variable equals its true value plus additive measurement noise. The noise is modeled as Gaussian white noise with diagonal covariance.

  • All measured variables are modeled as true values corrupted by additive random noise.
  • The measurement noise is assumed to be a Gaussian white-noise sequence with a non-singular diagonal covariance matrix Σ_ez.

C. Problem statement

The identification problem is to recover noise statistics, constraint structure, and minimal dynamic representation directly from noisy measurements, without user-specified variable or index classifications.

  • The task estimates the diagonal measurement-noise covariance matrix Σez from noisy measurements of nz variables.
  • It determines the number of algebraic equations ma and the number of inputs ℓ.
  • The framework identifies algebraic constraints Aa when algebraic equations are present.
  • It estimates the reduced dynamical-system order nd and identifies a minimal-realization basis of the dynamic constraint matrix Ad,nd.
  • Identification uses only the measured sequence, without a priori y/u partition or index-0/index-1 structural specification.

III. FOUNDATIONS

SMI-IPCA converts lagged noisy input-output data into a constraint-identification problem and uses scaled PCA with iterative noise estimation to recover model structure.

  • SMI-IPCA maps a state-space identification problem into static constraint identification by stacking lagged input and output measurements.
  • The foundational construction uses a priori input-output classification when forming the lagged data matrix ˜Zf.
  • The lagged true variables satisfy mf − n independent linear dynamic equations, represented using observability and block Toeplitz matrices.
  • Measurement noise in the lagged vector depends only on input and output noise variances and has a block-diagonal covariance structure.
  • SMI-IPCA scales lagged data using the lagged noise covariance and applies PCA to estimate system order, noise variances, and system matrices.
  • The number of constraints is inferred by testing whether the smallest singular values equal unity, with corresponding right singular vectors estimating the constraint matrix.
  • The iterative algorithm alternates model estimation and noise-variance optimization while searching over admissible constraint counts.
  • The state-space matrices are recovered from the estimated constraint matrix up to rotational ambiguity through systematic algebraic manipulations.

IV. PROPOSED METHODOLOGY

The proposed EIV-DAE framework is organized as a hierarchical methodology with three sequential steps for solving the identification problem.

  • The complete identification pipeline is divided into three sequential steps.

A. Estimation of the measurement noise variances

The first methodology step constructs a lagged augmented-data representation and iteratively estimates measurement-noise variances and lagged constraints, while deferring state-dimension estimation.

  • The method maps the dynamic DAE formulation into an equivalent static constraint problem using IPCA on a lagged data matrix built from z(t).
  • The dynamic constraint model uses minimal order Ad,nd when the lagging length is f1 = nd + 1.
  • Noise covariance estimation starts from Σez,f1 = If1 ⊗ Σez, parameterized by nz unknown measurement-noise variances.
  • The noise variances are estimated by iteratively solving the IPCA optimization problem, with the identifiability bound determined by the total number of variables.
  • The procedure estimates the total number of constraints relating lagged variables, then determines whether algebraic relations exist among the variables.
  • Because the number of output variables is unknown, the estimated lagged-constraint count cannot directly provide the state dimension as in SMI-IPCA.

B. Discovery of algebraic constraints

Algebraic constraints are identified as a left-null-space problem using PCA on measurement data scaled by estimated noise variances. Unity eigenvalues determine the constraint dimension, after which algebraic output variables and a regression representation are obtained.

  • Algebraic-constraint identification is formulated as finding the left null space of the true unlagged data matrix.
  • PCA is applied to the noise-scaled unlagged data matrix ZS to estimate the algebraic constraint matrix Âa.
  • The smallest ma covariance eigenvalues should be equal to unity, as used in maximum likelihood PCA to estimate the null-space dimension.
  • Candidate algebraic output variables are selected by testing whether the corresponding ma × ma submatrix Âa,D is invertible.
  • The remaining columns form Âa,I, whose implicit algebraic kernel representation yields a regression form using the algebraic regression matrix.

C. Identification of differential constraints

Differential constraints and minimal dynamic order are inferred by comparing constraint counts from two different lag lengths. A final PCA step then estimates the dynamic constraint matrix using the least lag length encoding all dynamic relations.

  • The number of differential constraints is obtained by removing the algebraic relations from the constraints of the first lagged data matrix.
  • A second lagged data matrix with a different stacking length exposes structural redundancy and provides a second constraint count.
  • The reduced dynamic order is estimated by equating the order inferred from two lag lengths, and the input count is ℓ̂ = nz − m̂.
  • PCA on a scaled lagged matrix constructed with f = n̂d + 1 estimates the minimal-order dynamic constraint matrix.
  • The index-0 scree plots display eigenvalues of covariance matrices for the scaled lagged and unlagged data matrices.

V. SIMULATIONS STUDIES

Two simulations evaluate the framework on index-0 and index-1 systems using noisy data. The index-0 study recovers noise levels, system structure, and dynamic behavior, while the simulations validate the stated estimation objectives.

  • The simulations target index-0 and index-1 systems and explicitly validate the four estimation objectives of the proposed framework.
  • The index-0 experiment uses a fourth-order 2×2 dynamic process driven by two full-band RBS inputs over N = 6095 samples.
  • At SNR = 10, estimated noise variances closely match the true values, and zero unity eigenvalues correctly identify the system as index-0.
  • The index-0 algorithm correctly estimates 2 differential output variables, dynamic order 4, and 2 input variables.
  • Across 200 bootstrap runs, estimated state-space models have unbiased pole-zero estimates within the reported 95% confidence intervals.

B. Index-1 DAE system: Two-tank liquid-level process

The two-tank process provides an index-1 DAE simulation with noisy measurements and algebraic constraints. The procedure estimates noise variances, identifies two algebraic outputs, and recovers a reduced dynamic representation.

  • Process model: The non-interacting two-tank system has flow through each valve determined only by its respective upstream liquid level.
  • Simulation setup: The simulation uses full-band RBS excitation, N = 6095 samples, sampling interval 1s, and Gaussian measurement noise at SNR = 10.
  • Constraint discovery: With f1 = 9, the algorithm finds 34 lagged constraints and estimates noise variances [0.099, 0.056, 0.010, 0.031, 0.024].
  • Constraint discovery: The two unity eigenvalues indicate 2 algebraic constraints, with h1 and h2 selected as the most robust algebraic output variables at condition number 1.149.
  • Dynamic identification: Using f2 = 6 yields 22 constraints, 2 dynamical output variables, and reduced system order 2.
  • Dynamic identification: The final global constraint model is estimated with f = n̂d + 1 = 3, producing Âd,3 ∈ R10×15.
  • Dynamic identification: The estimated system poles are [0.764±0.008, 0.286±0.012], close to true poles [0.758, 0.298], and the zero set is empty.

VI. CONCLUSION

The proposed EIV subspace identification framework removes a priori input-output partitioning and DAE index specification. Simulations confirm correct estimation for both index-0 and index-1 systems.

  • The framework eliminates a priori input-output variable partitioning and structural index pre-specification.
  • Simulation results confirm correct estimation of all system parameters for index-0 and index-1 DAE systems.
  • The identification strategy can support causal discovery in linear descriptor systems.
  • Future work includes systems with process noise and higher-index DAEs.
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