Fault detection on manifolds of nonlinear dynamical systems with dual autoencoders

Abstract: Autoencoders are commonly used for unsupervised data-driven fault detection in nonlinear dynamical systems. Despite their widespread success and often favorable performance compared with traditional approaches, most applications rely on heuristic reconstruction of measured data using features learned from nominal training data, without explicit insight into the underlying nonlinear dynamics. This lack of interpretability limits the extension of autoencoder-based fault detection methods to higher levels of fault diagnosis, e.g., fault localization and quantification, and confines their use largely to application-oriented studies. To address this limitation, we propose a strategy for detecting parametric faults in nonlinear stochastic mechanical systems. A mathematical representation of the output data is developed using Koopman operator theory, which motivates their embedding on a manifold and its subsequent approximation with a two-stage autoencoder. Fault detection is formulated within a hypothesis-testing framework, in which new data are tested for consistency with a neighborhood of the manifold identified from nominal observations. The proposed method is validated through Monte Carlo simulations of a toy mechanical system with two types of nonlinearity and applied to two well-known real benchmarks, where it provides favorable fault-detection performance compared with standard autoencoders.

• The paper develops a Koopman-theoretic dual autoencoder that learns a nominal manifold from delay output covariance Hankel matrices and detects faults through complementary residual directions.
• The method achieves 5% false positives and 94.7% detection for cubic nonlinearities, 7% false positives and 100% detection for periodic nonlinearities, and strong results on bridge and wind-turbine benchmarks.
• The framework offers interpretable, dynamics-informed monitoring but relies on local Koopman spectral assumptions, heuristic thresholds, and an unproven separation between stochastic variability and fault-induced changes.

This paper develops a fault detection framework for parametric faults in nonlinear stochastic mechanical systems, built on a Koopman-theoretic characterization of delay output covariance Hankel matrices and a two-stage (dual) autoencoder that learns both a nominal manifold and its complementary directions. The work addresses a recognized weakness of autoencoder-based structural health monitoring: while reconstruction-based detectors perform well empirically, they lack an explicit connection to system dynamics, which limits interpretability and extension to fault localization and quantification.

Motivation and positioning

The authors situate their contribution between physics-based and data-driven fault detection. Physics-based approaches—nonlinear state observers with residual statistics, joint state–parameter Bayesian filtering, and physics-informed neural networks—require accurate nominal models or correct fault parameterizations at significant engineering or computational cost. Data-driven one-class methods, including support-vector novelty detection and deep Koopman-based residual generation, avoid this cost but either depend on kernel choices or, in existing Koopman formulations, treat deterministic approximations only. Standard autoencoders learn nominal latent representations without interpretable dynamics. The proposed method combines two prior threads: subspace-based fault detection for linear time-invariant systems, where residuals monitor directions complementary to a nominal Hankel covariance subspace, and delay-embedding learning of nonlinear systems via Koopman theory. The central idea is to replace the fixed linear subspace of classical subspace detection with a locally nonlinear manifold learned from data.

Theoretical development

The monitored system is modeled as a parametric discrete-time nonlinear state-space model driven by i.i.d. process and measurement noise, with a fault-sensitive parameter vector $\theta$. A stochastic Koopman operator is defined as a conditional-expectation operator on observables, and the measurement map is assumed to admit a truncated discrete spectral representation in $n$ dominant eigenfunctions and modes—an assumption appropriate for systems operating near an attracting hyperbolic equilibrium within a bounded region. The paper is explicit that this finite spectral approximation applies to regimes characterized by closed orbits or local linearizability, and it concedes that arbitrary observables need not admit such expansions.

The core theoretical result establishes that the empirical past–future output covariance Hankel matrix factorizes as

$$\hat{\mathcal{H}}(x,\theta) \approx \mathcal{O}(\theta)\,\Gamma(x,\theta),$$

up to truncation-error terms of order $O(\bar\varepsilon) + O(\bar\varepsilon^2)$ and finite-sample terms $o(1)$, under bounded fourth moments of the eigenfunctions and ergodicity of the state Markov chain. Here $\mathcal{O}(\theta)$ stacks observability-like blocks $V(\theta)\Lambda(\theta)^i$ built from Koopman modes and eigenvalues, and $\Gamma(x,\theta)$ contains empirical second-order Koopman feature correlations. Vectorizing this factorization shows that each finite realization yields a point $\zeta_h(x,\theta)$ in $\mathbb{R}^{(pr)^2}$, and a second proposition establishes—under smoothness of the retained eigenfunctions and constant rank of the realization-to-Hankel map—that collections of such points are samples from a smooth embedded manifold $n$0. This provides the formal justification for manifold-based learning of nominal behavior: variability across stochastic realizations corresponds to motion along the manifold, whereas parameter changes are assumed to move the Hankel matrix transversally away from it.

Dual autoencoder architecture

Manifold learning proceeds in two stages. The first stage trains a standard encoder–decoder pair on vectorized nominal Hankel matrices, learning a characteristic latent space of dimension $n$1 that parametrizes $n$2. The second stage freezes the first-stage encoder and augments the network with complementary encoder and decoder maps trained on samples from an ambient hyper-rectangle containing the manifold; these maps capture coordinates transverse to $n$3, which vanish ideally for on-manifold data. Fault detection then uses the complementary map $n$4 as the residual generator: under an idealized training assumption that its zero set intersected with the domain equals exactly the nominal manifold, zero residual implies membership in $n$5. In practice the residual norm is compared against a threshold set from nominal calibration data, e.g., a 95% quantile. The authors note plainly that a rigorous statistical characterization of the residual—including uncertainty from finite data, noise, approximation error, and learned mappings—is left open, so the hypothesis test lacks formal false-alarm guarantees.

Numerical validation

The method is validated by Monte Carlo simulation of a 6-DOF mass–spring chain with cubic (Duffing) and periodic stiffness nonlinearities, excited by white noise, with acceleration outputs at three DOFs corrupted by Gaussian sensor noise. Faults are modeled as 5% and 10% stiffness reductions in one spring. Using $n$6 lags, hidden width 4096, bottleneck $n$7, and ADAM optimization, training takes roughly 43 s and 92 s for the two stages. With thresholds at the 95% nominal quantile, the method achieves false positive rates of 7% (periodic) and 5% (cubic), and false negative rates of 0% and 5.3%, respectively.

Benchmarking against fully connected AE, VAE, and beta-VAE baselines (each applied either to Hankel matrices or raw time series) shows a favorable trade-off:

Nonlinearity	Metric	Best baseline	DAE
Cubic	False alarm / Detection	VAE/BVAE: ~2% / ~75%; AE 1: 12.5% / 100%	5.0% / 94.7%
Periodic	False alarm / Detection	BVAE 1: 0.5% / 81.5%; AE 1: 19.9% / 100%	7.0% / 100%

Variational variants achieve lower false alarms but substantially lower detection rates, while the plain AE attains perfect detection at high false-alarm cost. Notably, all Hankel-matrix-trained variants outperform their raw-time-series counterparts, supporting the covariance-domain representation as the more informative feature space.

Applications to experimental benchmarks

On the S101 bridge benchmark (15 accelerometers, ambient excitation, progressive damage including pier cuts, settlements, and tendon cuts), the method detects all damage scenarios with a 10% false alarm rate and 99% detection rate—comparable to established subspace-based results on the same campaign. Against the autoencoder baselines, the proposed detector is slightly more sensitive (99% vs. 97.3% for AE 1) but has a higher false alarm rate (10% vs. 1%), a trade-off the authors acknowledge directly rather than claim superiority across both metrics.

On the Vestas V27 wind turbine blade (11 accelerometers over 3.5 months, trailing-edge cracks of 15/30/45 cm, varying wind speed, temperature, and rotor speed), the method detects all damage levels using low-frequency data without explicit environmental or operational condition correction, achieving a 14% false alarm rate and 8% no-detection rate. The authors concede these rates are somewhat high and attribute part of the gap to unmodeled operating variability. Against the baselines, the proposed method dominates on false alarms (14% vs. 23.9–30.7% for Hankel-trained variants) but trails AE 1's 100% detection rate at 92%, again reflecting a balance rather than uniform dominance.

Limitations and open questions

Several assumptions bound the validity of the framework. The Koopman spectral truncation presumes bounded operating regions near stable regimes with finitely dominant components; systems outside this class are not covered. The manifold separation assumption—that parameter changes move Hankel matrices transversally off $n$8 while stochastic variability moves them along it—is asserted rather than proven for general fault types, and non-membership is attributed to parameter change only by assumption. The change-detection lemma relies on idealized autoencoder training ($n$9), which cannot hold exactly in practice. Thresholds are heuristic quantiles without distributional theory, and the residual's statistical characterization under finite data, noise, approximation error, and mapping uncertainty remains undeveloped. Environmental and operational variability in the wind turbine application is not explicitly compensated, and the authors identify accounting for operating parameters and estimation uncertainty in the test design as open work.

Conclusion

The paper contributes a mathematically grounded alternative to purely heuristic autoencoder fault detection: a Koopman-based proof that delay output covariance Hankel matrices populate a low-dimensional manifold, a dual autoencoder that learns this manifold together with its normal bundle, and a residual test that constitutes a nonlinear analogue of subspace-based fault detection. Empirical results on simulated nonlinear chains and two established experimental benchmarks show competitive detection performance with reasonable false alarm rates, though not uniformly superior to simpler autoencoders on every metric. The principal unresolved issue is statistical: converting the heuristic threshold test into a provably calibrated hypothesis test under realistic uncertainty.