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A Robust Fault Detection Filter for Linear Time-Varying System with Non-Gaussian Noise

Zhemeng Zhang, Yifei Nie, Le Yin

TL;DR

The paper tackles robust fault detection for linear time-varying (LTV) systems under non-Gaussian noise by reformulating the $H_{\infty}$ filter in a regularized least-squares (RLS) framework and deriving a least-squares estimate of the fault vector $\theta$. It then extends the generalized likelihood ratio (GLR) approach with a generalized innovation ratio (GIR) to improve detection when Gaussian-based Kalman filtering is unreliable. The key contributions are the RLS-based $H_{\infty}$ filtering, a LS method for fault-vector estimation, and the GIR detector that remains effective under non-Gaussian disturbances and unknown fault onset times. Numerical examples demonstrate that the GIR with the $H_{\infty}$ filter outperforms traditional GLR-KF methods, offering enhanced fault detection reliability in practical noisy environments.

Abstract

This paper addresses the problem of robust fault detection filtering for linear time-varying (LTV) systems with non-Gaussian noise and additive faults. The conventional generalized likelihood ratio (GLR) method utilizes the Kalman filter, which may exhibit inadequate performance under non-Gaussian noise conditions. To mitigate this issue, a fault detection method employing the $H_{\infty}$ filter is proposed. The $H_{\infty}$ filter is first derived as the solution to a regularized least-squares (RLS) optimization problem, and the effect of faults on the output prediction error is then analyzed. The proposed approach using the $H_{\infty}$ filter demonstrates robustness in non-Gaussian noise environments and significantly improves fault detection performance compared to the original GLR method that employs the Kalman filter. The effectiveness of the proposed approach is illustrated using numerical examples.

A Robust Fault Detection Filter for Linear Time-Varying System with Non-Gaussian Noise

TL;DR

The paper tackles robust fault detection for linear time-varying (LTV) systems under non-Gaussian noise by reformulating the filter in a regularized least-squares (RLS) framework and deriving a least-squares estimate of the fault vector . It then extends the generalized likelihood ratio (GLR) approach with a generalized innovation ratio (GIR) to improve detection when Gaussian-based Kalman filtering is unreliable. The key contributions are the RLS-based filtering, a LS method for fault-vector estimation, and the GIR detector that remains effective under non-Gaussian disturbances and unknown fault onset times. Numerical examples demonstrate that the GIR with the filter outperforms traditional GLR-KF methods, offering enhanced fault detection reliability in practical noisy environments.

Abstract

This paper addresses the problem of robust fault detection filtering for linear time-varying (LTV) systems with non-Gaussian noise and additive faults. The conventional generalized likelihood ratio (GLR) method utilizes the Kalman filter, which may exhibit inadequate performance under non-Gaussian noise conditions. To mitigate this issue, a fault detection method employing the filter is proposed. The filter is first derived as the solution to a regularized least-squares (RLS) optimization problem, and the effect of faults on the output prediction error is then analyzed. The proposed approach using the filter demonstrates robustness in non-Gaussian noise environments and significantly improves fault detection performance compared to the original GLR method that employs the Kalman filter. The effectiveness of the proposed approach is illustrated using numerical examples.

Paper Structure

This paper contains 12 sections, 1 theorem, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Typical fault
  4. Kalman filter
  5. $H_{\infty}$ filter
  6. Fault effect on the innovation
  7. Solving the fault vector
  8. Fault detection
  9. Fault effect on generalized innovation ratio
  10. Fault detection with unknown onset time
  11. Numerical Examples
  12. Conclusion

Key Result

Proposition 5.1

The GIR defined in hkdefine can be expressed in terms of $E_k$ and $d_k$ as follows

Figures (4)

  • Figure 1: The innovation of Kalman filter (top) and $H_{\infty}$ filter (bottom) for the impulsive fault with $\theta = {\left[\, 1.5 ,\, 0 \,\right]}^T$
  • Figure 2: The innovation of Kalman filter (top) and $H_{\infty}$ filter (bottom) for the impulsive fault with $\theta = {\left[\, 0.6 ,\, 0 \,\right]}^T$
  • Figure 3: The GIR $h_k$ of Kalman filter (top) and the GIR $h_k$ of $H_{\infty}$ filter (bottom) for the impulsive fault with $\theta = {\left[\, 1.5 ,\, 0 \,\right]}^T$
  • Figure 4: The GIR $h_k$ of Kalman filter (top) and the GIR $h_k$ of $H_{\infty}$ filter (bottom) for the impulsive fault with $\theta = {\left[\, 0.6 ,\, 0 \,\right]}^T$

Theorems & Definitions (5)

  • Remark 2.1
  • Remark 4.1
  • Proposition 5.1
  • proof
  • Remark 5.2