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

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.

• The paper develops a novel framework that unifies finite-sample image and kernel representations with the fundamental lemma for robust fault detection.
• It introduces a projection-based method using SVD to separate nominal trajectories from residuals induced by faults, ensuring optimal detectability.
• The approach rigorously quantifies the impact of finite data and noise via matrix perturbation theory, providing practical performance guarantees.

System Representations in Subspaces of Finite-Sample Signals and Data-Driven Fault Detection

Problem Context and Motivation

This work provides a comprehensive system-theoretic framework for analyzing and detecting faults in LTI systems using finite-sample data, leveraging model-free and data-driven perspectives. While model-based fault detection (FD) relies on accurate first-principles models, practical systems often present model uncertainties and operate in data-rich environments. Existing data-driven approaches (e.g., subspace methods, ML-based latent space encoding) typically reconstruct system output or form nominal datasets for projections, but lack a unified theoretical foundation connecting system representations, the fundamental lemma, and fault detectability—especially under uncertainty.

The paper systematically develops representations of system behavior in finite-sample subspaces, generalizing the theory of image and kernel (nullspace) representations, and links these with the behavioral systems framework, in particular, Willems' fundamental lemma. These developments lay the groundwork for robust, theoretically justified, data-driven FD algorithms.

Finite-Sample Image and Kernel Representations

The classical image and kernel representations, defined via left/right coprime factorizations and the associated Bézout identities, are extended from infinite sequences ($\ell_2$ space) to finite-sample signals. The key result is that for an arbitrary $s$-length input-output trajectory of a controllable LTI system ($x(k+1) = Ax(k)+Bu(k)$, $y(k) = Cx(k)+Du(k)$), there exist matrices $M_s,N_s$ such that any stacked sample vector $[u_s(k)^\top~ y_s(k)^\top]^\top$ can be written as

$$\begin{bmatrix} u_s(k) \ y_s(k) \end{bmatrix} = \begin{bmatrix} M_s \ N_s \end{bmatrix} v_{s+n}(k-n),$$

where $v_{s+n}$ denotes a latent excitation vector and $n$ is the system order. This “image representation” parametrizes all nominal finite trajectories and establishes a corresponding image subspace $\mathcal{I}_{G,s}$ in the finite-sample domain.

The authors show that the finite-sample image representation is constructively equivalent to the trajectory space described in the fundamental lemma (i.e., the column space of the block Hankel data matrix, $s$0 constructed from persistently exciting data). Therefore,

$s$1

providing an intuitive and algebraic basis for the behavioral characterization of systems and offering a new proof of the fundamental lemma. This result demonstrates that system structure and excitation requirements for data-driven methods are encoded in the image subspace properties.

The kernel representation and the associated residual subspace are formed analogously. The kernel subspace captures system dynamics driven by uncertainty, sensor/process noise, or faults, via a paired representation that is parameterized with explicit relation to observer gain choices and feedback—an extension not covered by classical behavioral theory.

Residual Generation and Subspace Fault Detection

Faults and other uncertainties result in a departure of observed trajectories from the image subspace. The framework enables defining, for any finite data, residual subspaces that are theoretically orthogonal complements (under suitable conditions, e.g., $s$2, where $s$3 is the window length and $s$4 is system lag/observability index). The kernel representation becomes:

$s$5

with $s$6 designed such that it annihilates the image subspace and fully captures the complementary residual subspace induced by faults and uncertainties. The constructive equivalence between the kernel representation and the classical parity matrix is established, showing that the approach subsumes parity-space FD methods.

Importantly, the necessary and sufficient conditions for the existence and dimension of these subspaces are carefully derived, exposing that for $s$7 sufficiently large (greater than the system's observability index), the image and residual subspaces fully span the data space—critical for practical FD design.

Data-Driven Projection-Based Fault Detection: Theory and Implementation

The second major contribution is the formulation and rigorous analysis of a data-driven fault detection method, underpinned by the previous subspace constructions. The detection algorithm consists of

1. Subspace Identification: Construct a block Hankel matrix $s$8 from persistently exciting, healthy system trajectories.
2. Low-rank Approximation via SVD: Compute $s$9 and extract the leading left singular subspace $x(k+1) = Ax(k)+Bu(k)$0 of appropriate dimension—this approximates the normalized image subspace $x(k+1) = Ax(k)+Bu(k)$1.
3. Orthogonal Subspace Projection: Any test trajectory is split into image and residual components via orthogonal projections:

$x(k+1) = Ax(k)+Bu(k)$2

where $x(k+1) = Ax(k)+Bu(k)$3 spans the residual subspace.

1. Residual Evaluation and Thresholding: The norm (or a statistical function) of the projection onto the residual subspace is monitored. For Gaussian noise models, a $x(k+1) = Ax(k)+Bu(k)$4-type statistic is used. For nonparametric settings, support vector data description (SVDD) is employed for robust decision thresholds.

The theoretical novelty is in quantifying the effect of finite data and noise: matrix perturbation theory (notably the Davis-Kahan theorem) is used to bound the deviations between the true and estimated subspaces, linking detection performance (FAR, MDR) to the spectral properties of the noise and data. Explicit empirical formulas for estimating these bounds using only data-derived singular values are provided.

The method is compared to classical least-squares output prediction approaches, emphasizing that the projection-based residual is optimal in the input-output subspace, rather than being limited to the output (or latent) space. The former is shown to have higher detectability, especially under multiplicative uncertainties.

Theoretical Implications and Main Claims

Key Theoretical Results:

• The equivalence of the finite-sample image representation to the fundamental lemma and a constructive proof of the lemma using explicit system-theoretic objects.
• The explicit construction and dimensional characterization of kernel and residual subspaces for finite data—enabling robust FD even in the presence of system uncertainties.
• The rigorous connection between the projection-based residual and parity/LS-based residuals, situating the methodology within a broader theory, and demonstrating conditions under which each is preferable.

Contrasts with Prior Work:

• Unlike contemporary data-driven FD methods, the projection-approach models the structure of both nominal and uncertain dynamics, directly incorporates the effect of finite data length and system order, and provides a posteriori performance guarantees using gap metric and SVD spectra.
• The extension of the fundamental lemma to data corrupted with uncertainties (not just nominal trajectories) is established, a result rarely addressed in existing behavioral or FD literature.

Implications for Theory and Practice

This analysis establishes a new unifying paradigm for designing data-driven FD schemes that are theoretically grounded, parameter-free, and robust to system uncertainty and finite data limitations—addressing a major gap in the practical deployment of model-free FD in industrial systems. The explicit algebraic conditions for subspace existence, residual generator construction, and performance quantification facilitate real-world implementation, with applications ranging from industrial process monitoring to safety-critical embedded systems.

For the theoretical community, the results clarify the relationship between classical model-based representations, the behavioral perspective, and data-driven approaches, and provide new insights into the algebraic-geometric underpinnings of data-driven system analysis—including the role of persistent excitation and explicit latent variable construction.

Future research may extend these concepts to nonlinear systems, input-saturated scenarios, or online/adaptive variants. The subspace framework also offers a template for extensions to data-driven control (not just detection), optimization of excitation signals, and quantification of fundamental limitations of data-driven system analysis.

Conclusion

This work rigorously develops the system-theoretic foundations, algebraic constructions, and statistical analysis for data-driven fault detection in finite-sample settings. The equivalence of image representations and the fundamental lemma is established, kernel and residual subspaces are systematically characterized, and a practical, theoretically-justified orthogonal projection-based FD method is proposed and rigorously analyzed. The results provide both a robust practical tool and a framework for extending data-driven systems theory and FD to broader classes of uncertain and stochastic systems.

Reference: “System representations in subspaces of finite-sample signals and their application to data-driven fault detection” (2604.17444)