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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.

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.Source: §I · p. 1
  • Classical subspace methods assume noise-free inputs, while errors-in-variables methods address bias but commonly retain input-output classification.Source: §I · p. 1
  • Existing approaches also require practitioners to pre-specify whether the system is index-0 or index-1 before identification.Source: §I · p. 1
  • The proposed method unifies all measured signals symmetrically and infers the input-output partition from data.Source: §I · p. 1, A. State-space form of index-0 and… · p. 2
  • It detects algebraic constraints to determine the index and estimates noise variances, system order, constraint counts, and implicit constraint matrices in one procedure.Source: §I · p. 1
  • The framework represents dynamic behavior through a minimal-order kernel or dynamic constraint matrix, with algebraic constraints encoded separately.Source: A. State-space form of index-0 and… · p. 2

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.

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.Source: C. Problem statement · p. 2
  • It determines the number of algebraic equations ma and the number of inputs ℓ.Source: C. Problem statement · p. 2
  • The framework identifies algebraic constraints Aa when algebraic equations are present.Source: C. Problem statement · p. 2
  • It estimates the reduced dynamical-system order nd and identifies a minimal-realization basis of the dynamic constraint matrix Ad,nd.Source: C. Problem statement · p. 2
  • Identification uses only the measured sequence, without a priori y/u partition or index-0/index-1 structural specification.Source: C. Problem statement · p. 3

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.

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.Source: §IV · p. 3

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.

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.

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.

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.

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

Simulation setup

Constraint discovery

Dynamic identification

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.Source: §VI · p. 6
  • Simulation results confirm correct estimation of all system parameters for index-0 and index-1 DAE systems.Source: §VI · p. 6
  • The identification strategy can support causal discovery in linear descriptor systems.Source: §VI · p. 6
  • Future work includes systems with process noise and higher-index DAEs.Source: §VI · p. 6
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