System representations in subspaces of finite-sample signals and their application to data-driven fault detection

Jiahao Wang; Linlin Li; Maiying Zhong; Steven X. Ding; Wei Cheng

arxiv: 2604.17444 · v1 · submitted 2026-04-19 · 📡 eess.SY · cs.SY

System representations in subspaces of finite-sample signals and their application to data-driven fault detection

Linlin Li , Steven X. Ding , Jiahao Wang , Maiying Zhong , Wei Cheng This is my paper

Pith reviewed 2026-05-10 05:39 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords finite-sample signalsimage subspaceresidual subspacefundamental lemmadata-driven fault detectionsingular value decompositionorthogonal projectionsystem representation

0 comments

The pith

The fundamental lemma is equivalent to the image subspace of finite-sample signals, enabling SVD-based projections for data-driven fault detection.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that finite-sample input-output signals admit an image subspace that captures nominal system dynamics and a residual subspace that captures uncertainties. It proves this image subspace is equivalent to the fundamental lemma of subspace identification. The equivalence supports a fault detection method that applies singular value decomposition to realize orthogonal projections onto both subspaces, generating residuals that flag deviations from nominal behavior. A reader would care because the approach works directly from collected data without first building an explicit model of the system.

Core claim

The paper claims that the image representation of finite-sample signals models the nominal system dynamics and is equivalent to the fundamental lemma, while the residual representation describes uncertainties essential for fault detection. Orthogonal projections realized via singular value decomposition in a low-rank matrix approximation produce projection-based residuals whose evaluation detects faults, with performance bounds derived from matrix perturbation theory.

What carries the argument

Finite-sample image and residual subspaces obtained from low-rank approximation via singular value decomposition, which enable orthogonal projections to separate nominal dynamics from faults.

If this is right

The equivalence extends the fundamental lemma to finite-sample settings for direct system representation.
Fault detection proceeds by generating residuals through orthogonal projection onto the residual subspace without intermediate model identification.
Matrix perturbation analysis supplies quantitative bounds on how uncertainties affect detection thresholds.
The method relates to and can be compared against other data-driven fault detection techniques that rely on subspace or parity-space ideas.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same subspaces could be updated recursively when new data arrives, supporting online monitoring without recomputing the full decomposition each time.
The approach might combine with existing subspace identification routines so that the same data batch serves both modeling and fault detection.
In high-dimensional sensor arrays the low-rank structure could reduce computational cost by working only with the dominant singular vectors.
Testing on systems whose dynamics slowly drift would check whether the subspace separation remains useful beyond the paper's static assumptions.

Load-bearing premise

The collected finite-sample signals must possess well-defined low-rank image and residual subspaces that singular value decomposition can reliably separate to isolate faults from normal operation.

What would settle it

Apply the SVD projection procedure to input-output data from a system with an injected fault of known size and timing; if the resulting residual signal shows no statistically significant increase above the fault-free case, the separation claim is falsified.

read the original abstract

This paper deals with system representations in finite-sample signal subspaces and their application to data-driven fault detection. The first part addresses concepts of finite-sample image and kernel system representations and, associated with them, image and residual subspaces of finite-sample signals. On this basis, the equivalence between the fundamental lemma and finite-sample image subspace is demonstrated. While the image representation models the nominal system dynamics, the residual representation describes uncertainties in the input-output data and is essential for fault detection. This result extends the fundamental lemma and builds the basis for exploring data-driven fault detection. In the second part, a data-driven projection-based fault detection approach is developed. By means of a singular value decomposition, orthogonal projections onto the image and residual subspaces are realized in the context of a low-rank matrix approximation, leading to projection-based residual generation and evaluation. Finally, analysis of detection performance in the framework of matrix perturbation theory and comparison with existing data-driven fault detection methods are explored.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

It gives a finite-sample version of the fundamental lemma via image and residual subspaces, then builds an SVD projection method for fault detection, but the noise robustness rests on unverified low-rank gaps.

read the letter

The punchline for this paper is that it establishes an equivalence between the fundamental lemma and the image subspace of finite-sample signals, then uses that to develop a projection-based method for data-driven fault detection. What stands out is how they define finite-sample image and residual subspaces and show that the image one captures the nominal dynamics while the residual handles uncertainties. The SVD-based orthogonal projections for residual generation are a practical step, and the perturbation analysis provides some theoretical backing for detection performance. This extends the data-driven control literature in a logical way without overclaiming a complete overhaul. The soft spots are around the practical applicability under noise. The equivalence holds for exact data, but when noise is present, the low-rank property of the data matrix and the singular value gap needed for reliable subspace separation aren't guaranteed without additional conditions on sampling length or noise intensity. The paper uses general matrix perturbation results instead of tailored bounds for the structured case, which leaves the detection guarantees somewhat loose. There's also no mention of empirical tests in the abstract, so we don't see how it performs on real or simulated data compared to existing methods. This paper is for people already working in subspace-based fault detection and data-driven methods in systems and control. A reader who knows the fundamental lemma will appreciate the finite-sample extension and the direct application to fault detection. It deserves a serious referee because the core technical steps are coherent and the problem it addresses is relevant, even if some assumptions need more scrutiny and validation would strengthen it.

Referee Report

2 major / 3 minor

Summary. The paper introduces finite-sample image and kernel system representations along with their associated image and residual subspaces. It demonstrates an equivalence between the fundamental lemma and the finite-sample image subspace representation. This is then used to develop a data-driven fault detection scheme that employs SVD-based low-rank approximation to construct orthogonal projections onto the image and residual subspaces, enabling residual generation and evaluation, with detection performance analyzed via matrix perturbation theory.

Significance. If the central equivalence and projection construction hold under the stated conditions, the work extends the fundamental lemma to finite-sample settings and supplies a concrete SVD-based method for model-free fault detection. The explicit use of image/residual subspace projections and the perturbation-theoretic performance bounds represent a clear technical contribution that could strengthen data-driven approaches in systems and control.

major comments (2)

[§3] §3 (equivalence demonstration): The equivalence between the fundamental lemma and the finite-sample image subspace is established for exact (noise-free) data where the Hankel-like matrix has exact low rank; the fault-detection extension in §4 applies the same subspaces to noisy data without deriving explicit conditions (e.g., on sampling length T or noise variance) that guarantee a sufficient singular-value gap persists after finite sampling.
[§4.3] §4.3 (perturbation analysis): The detection-performance bounds rely on generic matrix perturbation results applied to the SVD of the data matrix; because the underlying matrix is a structured (Hankel) matrix formed from finite-sample trajectories, it is not shown that the generic bounds remain tight or that fault-induced perturbations dominate finite-sample effects in the residual subspace.

minor comments (3)

[§2] Notation for the image subspace projector P_I and residual projector P_R is introduced in §2 but used without an explicit matrix expression until §4; adding the explicit formulas early would improve readability.
[§5] The comparison with existing data-driven fault-detection methods (e.g., parity-space or subspace-based approaches) is mentioned in the abstract and conclusion but lacks a quantitative table or explicit metric-by-metric discussion; a short comparative table would strengthen the claims.
[§2.1] A few typographical inconsistencies appear in the definition of the finite-sample Hankel matrix (e.g., row/column indexing) between §2.1 and the appendix; these should be unified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the constructive review and for recognizing the technical contributions of the work. We address each major comment point by point below and indicate the revisions we intend to make.

read point-by-point responses

Referee: [§3] §3 (equivalence demonstration): The equivalence between the fundamental lemma and the finite-sample image subspace is established for exact (noise-free) data where the Hankel-like matrix has exact low rank; the fault-detection extension in §4 applies the same subspaces to noisy data without deriving explicit conditions (e.g., on sampling length T or noise variance) that guarantee a sufficient singular-value gap persists after finite sampling.

Authors: The equivalence in Section 3 is derived under the exact, noise-free setting with the data matrix having exact low rank, as explicitly stated. In Section 4 the fault-detection scheme applies SVD-based low-rank approximation to noisy trajectories, which is the standard practical extension used throughout the data-driven control literature. We agree that explicit, quantitative conditions on T and noise variance guaranteeing a sufficient singular-value gap are not derived. In the revised manuscript we will insert a clarifying paragraph (likely in Section 4.1) that states the working assumptions required for the approximation to remain valid and that points to existing finite-sample results in the literature. revision: partial
Referee: [§4.3] §4.3 (perturbation analysis): The detection-performance bounds rely on generic matrix perturbation results applied to the SVD of the data matrix; because the underlying matrix is a structured (Hankel) matrix formed from finite-sample trajectories, it is not shown that the generic bounds remain tight or that fault-induced perturbations dominate finite-sample effects in the residual subspace.

Authors: The performance bounds in Section 4.3 are obtained from standard, matrix-agnostic perturbation results; these results remain valid for any matrix, including the structured Hankel matrix generated by finite trajectories. The bounds are therefore conservative and are not claimed to be tight. We acknowledge that a structure-exploiting analysis that would prove dominance of fault-induced perturbations over finite-sample effects is absent. In the revision we will add a short discussion in Section 4.3 that (i) clarifies the applicability of generic bounds to structured matrices and (ii) states the implicit assumption that the fault is sufficiently strong relative to the residual subspace dimension and data length; we will also include a brief numerical illustration of this dominance. revision: partial

Circularity Check

0 steps flagged

No circularity: equivalence shown to external fundamental lemma; SVD projections and perturbation analysis are independent constructions.

full rationale

The derivation begins from the external fundamental lemma (Willems et al.), demonstrates its equivalence to finite-sample image subspaces via direct matrix arguments on Hankel-like data matrices, then applies standard SVD for orthogonal projections onto image/residual subspaces and matrix perturbation theory for detection bounds. None of these steps reduce to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations; the low-rank assumptions are stated explicitly as conditions rather than derived from the target result. The central claims remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the work references the fundamental lemma as background and introduces subspace concepts without detailing new postulates or fitted quantities.

pith-pipeline@v0.9.0 · 5474 in / 966 out tokens · 31220 ms · 2026-05-10T05:39:31.499233+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

, & author Nikiforov, I

author Basseville, M. , & author Nikiforov, I. ( year 1993 ). title Detection of Abrupt Changes - Theory and Application \/ . address New Jersey : publisher Prentice-Hall

work page 1993
[2]

, author Liu, Z

author Chen, H. , author Liu, Z. , & author Huang, B. ( year 2023 ). title Data-driven fault detection for lipschitz nonlinear systems: From open to closed-loop systems . journal Automatica \/ , volume 155 \/ , pages 111161

work page 2023
[3]

author Ding, S. X. ( year 2008 ). title Model-Based Fault Diagnosis Techniques - Design Schemes, Algorithms, and Tools \/ . publisher Springer-Verlag

work page 2008
[4]

author Ding, S. X. ( year 2014 ). title Data-driven design of monitoring and diagnosis systems for dynamic processes: A review of subspace technique based schemes and some recent results . journal Journal of Process Control \/ , volume 24 \/ , pages 431--449

work page 2014
[5]

author Ding, S. X. ( year 2020 ). title Advanced Methods for Fault Diagnosis and Fault-tolerant Control \/ . address Berlin : publisher Springer-Verlag

work page 2020
[6]

author Ding, S. X. , author Li, L. , & author Liu, T. ( year 2026 ). title An alternative paradigm of fault diagnosis in dynamic systems: orthogonal projection-based methods . journal Automatica \/ , volume 183 \/ , pages 112637

work page 2026
[7]

author Ding, S. X. , author Yang, Y. , author Zhang, Y. , & author Li, L. ( year 2014 ). title Data-driven realization of kernel and image representations and their application to fault detection and control system design . journal Automatica \/ , volume 50 \/ , pages 2615--2623

work page 2014
[8]

, author Coulson, J

author Doerfler, F. , author Coulson, J. , & author Markovsky, I. ( year 2023 ). title Bridging direct and indirect data-driven control formulations via regularizations and relaxations . journal IEEE Transactions on Automatic Control \/ , volume 68 \/ , pages 883--897

work page 2023
[9]

, & author Young, G

author Eckart, C. , & author Young, G. ( year 1936 ). title The approximation of one matrix by another of lower rank . journal Psychometrika \/ , volume 1 \/ , pages 211–218

work page 1936
[10]

, & author Kadali, R

author Huang, B. , & author Kadali, R. ( year 2008 ). title Dynamic Modelling, Predictive Control and Performance Monitoring, a Data-Driven Subspace Approach \/ . address London : publisher Springer-Verlag

work page 2008
[11]

, author Luo, H

author Huo, M. , author Luo, H. , author Cheng, C. , author Li, K. , author Yin, S. , author Kaynak, O. , author Zhang, J. , & author Tang, D. ( year 2023 ). title Subspace-aided sensor fault diagnosis and compensation for industrial systems . journal IEEE Transactions on Industrial Electronics \/ , volume 70 \/ , pages 9474--9482

work page 2023
[12]

( year 1995 )

author Kato, T. ( year 1995 ). title Perturbation Theory for Linear Operators \/ . address Berlin : publisher Springer-Verlag

work page 1995
[13]

, & author D \"o rfler, F

author Markovsky, I. , & author D \"o rfler, F. ( year 2021 ). title Behavioral systems theory in data-driven analysis, signal processing, and control . journal Annual Reviews in Control \/ , volume 52 \/ , pages 42--64

work page 2021
[14]

, author Muixí, A

author Markovsky, I. , author Muixí, A. , author Zlotnik, S. , & author Diez, P. ( year 2026 ). title A behavioral approach to direct data-driven fault detection . journal Mechanical Systems and Signal Processing \/ , volume 245 \/ , pages 113802

work page 2026
[15]

author Patton, R. J. , author Frank, P. M. , & author (Eds.), R. N. C. ( year 2000 ). title Issues of Fault Diagnosis for Dynamic Systems \/ . address London : publisher Springer

work page 2000
[16]

author Polderman, J. W. , & author Willens, J. C. ( year 1998 ). title Introduction to mathematical systems theory: a behavioral approach \/ . publisher Springer

work page 1998
[17]

, author Platt, J

author Schoelkopf, B. , author Platt, J. , author Shawe-Taylor, J. , author Smola, A. , & author Williamson, R. ( year 2001 ). title Estimating the support of a high-dimensional distribution . journal Neural Computation \/ , volume 13 \/ , pages 1443--1471

work page 2001
[18]

author Sheikhi, M. A. , author Gleizer, G. d. A. , author Keviczky, T. , & author Esfahani, P. M. ( year 2025 ). title Fault diagnosis in dynamical systems: Geometric interpretation and tractable algorithms . journal Annual Review of Control, Robotics, and Autonomous Systems \/ ,

work page 2025
[19]

author Stewart, G. W. , & author Sun, J.-G. ( year 1990 ). title Matrix Perturbation Theory \/ . address San Diego, CA : publisher Academic Press

work page 1990
[20]

( year 2000 )

author Vinnicombe, G. ( year 2000 ). title Uncertainty and Feedback: H_ Loop-Shaping and the Gap Metric \/ . publisher World Scientific

work page 2000
[21]

author van Waarde, H. J. , author De Persis, C. , author Camlibel, M. K. , & author Tesi, P. ( year 2020 a ). title Willems’ fundamental lemma for state-space systems and its extension to multiple datasets . journal IEEE Control Systems Letters \/ , volume 4 \/ , pages 602--607

work page 2020
[22]

author van Waarde, H. J. , author Eising, J. , author Trentelman, H. L. , & author Camlibel, M. K. ( year 2020 b ). title Data informativity: A new perspective on data-driven analysis and control . journal IEEE Transactions on Automatic Control \/ , volume 65 \/ , pages 4753--4768

work page 2020
[23]

, author Zhou, J

author Wang, J. , author Zhou, J. , & author Chen, X. ( year 2021 ). title Data-Driven Fault Detection and Reasoning for Industrial Monitoring \/ . publisher Springer

work page 2021
[24]

, author You, K

author Wang, Y. , author You, K. , author Huang, D. , & author Shang, C. ( year 2025 ). title Data-driven output prediction and control of stochastic systems: An innovation-based approach . journal Automatica \/ , volume 171 \/ , pages 111897

work page 2025
[25]

author Willems, J. C. , author Rapisarda, P. , author Markovsky, I. , & author De Moor , B. L. ( year 2005 ). title A note on persistency of excitation . journal Systems & Control Letters \/ , volume 54 \/ , pages 325--329

work page 2005
[26]

author Willsky, A. S. ( year 1976 ). title A survey of design methods for failure detection in dynamic systems . journal Automatica \/ , volume 12 \/ , pages 601--611

work page 1976
[27]

, author Yang, H

author Zhao, S. , author Yang, H. , author Kareck, T. L. , author Khan, F. , & author Wang, Q. ( year 2026 ). title Data-driven fault detection and diagnosis in industrial process systems: A systematic review and perspective . journal Reliability Engineering & System Safety \/ , volume 270 \/ , pages 112159

work page 2026

[1] [1]

, & author Nikiforov, I

author Basseville, M. , & author Nikiforov, I. ( year 1993 ). title Detection of Abrupt Changes - Theory and Application \/ . address New Jersey : publisher Prentice-Hall

work page 1993

[2] [2]

, author Liu, Z

author Chen, H. , author Liu, Z. , & author Huang, B. ( year 2023 ). title Data-driven fault detection for lipschitz nonlinear systems: From open to closed-loop systems . journal Automatica \/ , volume 155 \/ , pages 111161

work page 2023

[3] [3]

author Ding, S. X. ( year 2008 ). title Model-Based Fault Diagnosis Techniques - Design Schemes, Algorithms, and Tools \/ . publisher Springer-Verlag

work page 2008

[4] [4]

author Ding, S. X. ( year 2014 ). title Data-driven design of monitoring and diagnosis systems for dynamic processes: A review of subspace technique based schemes and some recent results . journal Journal of Process Control \/ , volume 24 \/ , pages 431--449

work page 2014

[5] [5]

author Ding, S. X. ( year 2020 ). title Advanced Methods for Fault Diagnosis and Fault-tolerant Control \/ . address Berlin : publisher Springer-Verlag

work page 2020

[6] [6]

author Ding, S. X. , author Li, L. , & author Liu, T. ( year 2026 ). title An alternative paradigm of fault diagnosis in dynamic systems: orthogonal projection-based methods . journal Automatica \/ , volume 183 \/ , pages 112637

work page 2026

[7] [7]

author Ding, S. X. , author Yang, Y. , author Zhang, Y. , & author Li, L. ( year 2014 ). title Data-driven realization of kernel and image representations and their application to fault detection and control system design . journal Automatica \/ , volume 50 \/ , pages 2615--2623

work page 2014

[8] [8]

, author Coulson, J

author Doerfler, F. , author Coulson, J. , & author Markovsky, I. ( year 2023 ). title Bridging direct and indirect data-driven control formulations via regularizations and relaxations . journal IEEE Transactions on Automatic Control \/ , volume 68 \/ , pages 883--897

work page 2023

[9] [9]

, & author Young, G

author Eckart, C. , & author Young, G. ( year 1936 ). title The approximation of one matrix by another of lower rank . journal Psychometrika \/ , volume 1 \/ , pages 211–218

work page 1936

[10] [10]

, & author Kadali, R

author Huang, B. , & author Kadali, R. ( year 2008 ). title Dynamic Modelling, Predictive Control and Performance Monitoring, a Data-Driven Subspace Approach \/ . address London : publisher Springer-Verlag

work page 2008

[11] [11]

, author Luo, H

author Huo, M. , author Luo, H. , author Cheng, C. , author Li, K. , author Yin, S. , author Kaynak, O. , author Zhang, J. , & author Tang, D. ( year 2023 ). title Subspace-aided sensor fault diagnosis and compensation for industrial systems . journal IEEE Transactions on Industrial Electronics \/ , volume 70 \/ , pages 9474--9482

work page 2023

[12] [12]

( year 1995 )

author Kato, T. ( year 1995 ). title Perturbation Theory for Linear Operators \/ . address Berlin : publisher Springer-Verlag

work page 1995

[13] [13]

, & author D \"o rfler, F

author Markovsky, I. , & author D \"o rfler, F. ( year 2021 ). title Behavioral systems theory in data-driven analysis, signal processing, and control . journal Annual Reviews in Control \/ , volume 52 \/ , pages 42--64

work page 2021

[14] [14]

, author Muixí, A

author Markovsky, I. , author Muixí, A. , author Zlotnik, S. , & author Diez, P. ( year 2026 ). title A behavioral approach to direct data-driven fault detection . journal Mechanical Systems and Signal Processing \/ , volume 245 \/ , pages 113802

work page 2026

[15] [15]

author Patton, R. J. , author Frank, P. M. , & author (Eds.), R. N. C. ( year 2000 ). title Issues of Fault Diagnosis for Dynamic Systems \/ . address London : publisher Springer

work page 2000

[16] [16]

author Polderman, J. W. , & author Willens, J. C. ( year 1998 ). title Introduction to mathematical systems theory: a behavioral approach \/ . publisher Springer

work page 1998

[17] [17]

, author Platt, J

author Schoelkopf, B. , author Platt, J. , author Shawe-Taylor, J. , author Smola, A. , & author Williamson, R. ( year 2001 ). title Estimating the support of a high-dimensional distribution . journal Neural Computation \/ , volume 13 \/ , pages 1443--1471

work page 2001

[18] [18]

author Sheikhi, M. A. , author Gleizer, G. d. A. , author Keviczky, T. , & author Esfahani, P. M. ( year 2025 ). title Fault diagnosis in dynamical systems: Geometric interpretation and tractable algorithms . journal Annual Review of Control, Robotics, and Autonomous Systems \/ ,

work page 2025

[19] [19]

author Stewart, G. W. , & author Sun, J.-G. ( year 1990 ). title Matrix Perturbation Theory \/ . address San Diego, CA : publisher Academic Press

work page 1990

[20] [20]

( year 2000 )

author Vinnicombe, G. ( year 2000 ). title Uncertainty and Feedback: H_ Loop-Shaping and the Gap Metric \/ . publisher World Scientific

work page 2000

[21] [21]

author van Waarde, H. J. , author De Persis, C. , author Camlibel, M. K. , & author Tesi, P. ( year 2020 a ). title Willems’ fundamental lemma for state-space systems and its extension to multiple datasets . journal IEEE Control Systems Letters \/ , volume 4 \/ , pages 602--607

work page 2020

[22] [22]

author van Waarde, H. J. , author Eising, J. , author Trentelman, H. L. , & author Camlibel, M. K. ( year 2020 b ). title Data informativity: A new perspective on data-driven analysis and control . journal IEEE Transactions on Automatic Control \/ , volume 65 \/ , pages 4753--4768

work page 2020

[23] [23]

, author Zhou, J

author Wang, J. , author Zhou, J. , & author Chen, X. ( year 2021 ). title Data-Driven Fault Detection and Reasoning for Industrial Monitoring \/ . publisher Springer

work page 2021

[24] [24]

, author You, K

author Wang, Y. , author You, K. , author Huang, D. , & author Shang, C. ( year 2025 ). title Data-driven output prediction and control of stochastic systems: An innovation-based approach . journal Automatica \/ , volume 171 \/ , pages 111897

work page 2025

[25] [25]

author Willems, J. C. , author Rapisarda, P. , author Markovsky, I. , & author De Moor , B. L. ( year 2005 ). title A note on persistency of excitation . journal Systems & Control Letters \/ , volume 54 \/ , pages 325--329

work page 2005

[26] [26]

author Willsky, A. S. ( year 1976 ). title A survey of design methods for failure detection in dynamic systems . journal Automatica \/ , volume 12 \/ , pages 601--611

work page 1976

[27] [27]

, author Yang, H

author Zhao, S. , author Yang, H. , author Kareck, T. L. , author Khan, F. , & author Wang, Q. ( year 2026 ). title Data-driven fault detection and diagnosis in industrial process systems: A systematic review and perspective . journal Reliability Engineering & System Safety \/ , volume 270 \/ , pages 112159

work page 2026