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Data-Driven Probabilistic Fault Detection and Identification via Density Flow Matching

Joshua D. Ibrahim, Mahdi Taheri, Soon-Jo Chung, Fred Y. Hadaegh

Abstract

Fault detection and identification (FDI) is critical for maintaining the safety and reliability of systems subject to actuator and sensor faults. In this paper, the problem of FDI for nonlinear control-affine systems under simultaneous actuator and sensor faults is studied. We model fault signatures through the evolution of the probability density flow along the trajectory and characterize detectability using the 2-Wasserstein metric. In order to introduce quantifiable guarantees for fault detectability based on system parameters and fault magnitudes, we derive upper bounds on the distributional separation between nominal and faulty dynamics. The latter is achieved through a stochastic contraction analysis of probability distributions in the 2-Wasserstein metric. A data-driven FDI method is developed by means of a conditional flow-matching scheme that learns neural vector fields governing density propagation under different fault profiles. To generalize the data-driven FDI method across continuous fault magnitudes, Gaussian bridge interpolation and Feature-wise Linear Modulation (FiLM) conditioning are incorporated. The effectiveness of our proposed method is illustrated on a spacecraft attitude control system, and its performance is compared with an augmented Extended Kalman Filter (EKF) baseline. The results confirm that trajectory-based distributional analysis provides improved discrimination between fault scenarios and enables reliable data-driven FDI with a lower false alarm rate compared with the augmented EKF.

Data-Driven Probabilistic Fault Detection and Identification via Density Flow Matching

Abstract

Fault detection and identification (FDI) is critical for maintaining the safety and reliability of systems subject to actuator and sensor faults. In this paper, the problem of FDI for nonlinear control-affine systems under simultaneous actuator and sensor faults is studied. We model fault signatures through the evolution of the probability density flow along the trajectory and characterize detectability using the 2-Wasserstein metric. In order to introduce quantifiable guarantees for fault detectability based on system parameters and fault magnitudes, we derive upper bounds on the distributional separation between nominal and faulty dynamics. The latter is achieved through a stochastic contraction analysis of probability distributions in the 2-Wasserstein metric. A data-driven FDI method is developed by means of a conditional flow-matching scheme that learns neural vector fields governing density propagation under different fault profiles. To generalize the data-driven FDI method across continuous fault magnitudes, Gaussian bridge interpolation and Feature-wise Linear Modulation (FiLM) conditioning are incorporated. The effectiveness of our proposed method is illustrated on a spacecraft attitude control system, and its performance is compared with an augmented Extended Kalman Filter (EKF) baseline. The results confirm that trajectory-based distributional analysis provides improved discrimination between fault scenarios and enables reliable data-driven FDI with a lower false alarm rate compared with the augmented EKF.

Paper Structure

This paper contains 28 sections, 6 theorems, 78 equations, 4 figures, 2 tables.

Table of Contents

  1. Problem Formulation
  2. System Model
  3. Density Propagation
  4. Problem Statement
  5. Fault Detectability and Identifiability in the Density Space
  6. Contraction Theory in Density Space
  7. Contraction of Trajectories
  8. Contraction of Distributions
  9. Flow Matching on Trajectories for FDI
  10. Neural Flows
  11. Neural Network Training and Fault Estimation
  12. Numerical Case Study
  13. Performance Evaluation
  14. Conclusion
  15. Problem Formulation
  16. ...and 13 more sections

Key Result

Lemma 3

System eq:diff_ode is contracting (i.e., all the solution trajectories exponentially converge to a single trajectory globally from any initial condition), if there exists a uniformly positive definite metric $\mathbf{M}(\mathbf{x}, t)=\mathbf{H}(\mathbf{x}, t)^{\top} \mathbf{H}(\mathbf{x}, t), \ \ma where $\alpha>0$. If eq:diff_ode is contracting, one has

Figures (4)

  • Figure 3: Plot of Type 1 faults comparing true actuator effectiveness $\boldsymbol{\eta} = [0.20, 0.65, 0.40, 0.15]$ and onset times $\mathbf{t}_{\mathrm{start}} = [7.0, 18.0, 29.0, 36.0]$ with estimated $\hat{\boldsymbol{\eta}} = [0.188, 0.689, 0.454, 0.209]$ and $\hat{\mathbf{t}} = [8.1, 39.8, 29, 39.6]$ using our method and augmented EKF estimates.
  • Figure 4: Plot of confusion matrix of Type 2 faults comparing 10 different random fault hypotheses with 10 trajectories for each with noise.
  • Figure 5: Plot of Type 1 faults comparing true actuator effectiveness $\boldsymbol{\eta} = [0.20, 0.65, 0.40, 0.15]$ and onset times $\mathbf{t}_{\mathrm{start}} = [7.0, 18.0, 29.0, 36.0]$ with estimated $\hat{\boldsymbol{\eta}} = [0.188, 0.689, 0.454, 0.209]$ and $\hat{\mathbf{t}} = [8.1, 39.8, 29, 39.6]$ using our method and augmented EKF estimates.
  • Figure 6: Plot of confusion matrix of Type 2 faults comparing 10 different random fault hypotheses with 10 trajectories for each with noise.

Theorems & Definitions (12)

  • Definition 1: $\varepsilon$-Detectability
  • Definition 2: $\varepsilon$-Identifiability
  • Lemma 3: Deterministic Contraction
  • Definition 4: 2-Wasserstein Distance
  • Theorem 5: Wasserstein Contraction
  • Theorem 6: Wasserstein FDI
  • Definition 7: $\varepsilon$-Detectability
  • Definition 8: $\varepsilon$-Identifiability
  • Lemma 9: Deterministic Contraction
  • Definition 10: 2-Wasserstein Distance
  • ...and 2 more