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Distributionally Robust Fault Detection Trade-off Design with Prior Fault Information

Yulin Feng, Hailang Jin, Steven X. Ding, Hao Ye, Chao Shang

TL;DR

The paper tackles fault detection under distributional uncertainty and the need to protect known critical faults, extending distributionally robust fault detection (DRFD) with a prior fault information mechanism. It introduces a new DRFD trade-off based on a relaxed distributionally robust chance constraint and a Wasserstein ambiguity set $𝔇_W(θ,N)$, augmented by a robustness metric parameter $η$ to quantify sensitivity to a known fault profile $\bar{f}$; the objective becomes $ρ(P)+γ η$ subject to a constraint linking FAR under $𝔇_W$ and a deviation term $d_W(·,\hat{𝔓}_N)$. The authors derive an exact reformulation into bilinear matrix inequalities (BMIs) and propose a tailored sequential minimization algorithm with an initialization strategy to solve the problem efficiently. Validated on a simulated three-tank system and a realistic lithium-ion battery cell, the approach achieves a more balanced detection performance by strengthening detection of the known fault while preserving detectability for unknown faults, under distributional uncertainty and controlled FAR.

Abstract

The robustness of fault detection algorithms against uncertainty is crucial in the real-world industrial environment. Recently, a new probabilistic design scheme called distributionally robust fault detection (DRFD) has emerged and received immense interest. Despite its robustness against unknown distributions in practice, current DRFD focuses on the overall detectability of all possible faults rather than the detectability of critical faults that are a priori known. Henceforth, a new DRFD trade-off design scheme is put forward in this work by utilizing prior fault information. The key contribution includes a novel distributional robustness metric of detecting a known fault and a new relaxed distributionally robust chance constraint that ensures robust detectability. Then, a new DRFD design problem of fault detection under unknown probability distributions is proposed, and this offers a flexible balance between the robustness of detecting known critical faults and the overall detectability against all possible faults. To address the resulting semi-infinite chance-constrained problem, we first reformulate it to a finite-dimensional problem characterized by bilinear matrix inequalities. Subsequently, a tailored heuristic solution algorithm is developed, which includes a sequential minimization procedure and an initialization strategy. Finally, case studies on a simulated three-tank system and a real-world battery cell are carried out to showcase the effectiveness of the proposed heuristic algorithm and the advantages of our DRFD method.

Distributionally Robust Fault Detection Trade-off Design with Prior Fault Information

TL;DR

The paper tackles fault detection under distributional uncertainty and the need to protect known critical faults, extending distributionally robust fault detection (DRFD) with a prior fault information mechanism. It introduces a new DRFD trade-off based on a relaxed distributionally robust chance constraint and a Wasserstein ambiguity set , augmented by a robustness metric parameter to quantify sensitivity to a known fault profile ; the objective becomes subject to a constraint linking FAR under and a deviation term . The authors derive an exact reformulation into bilinear matrix inequalities (BMIs) and propose a tailored sequential minimization algorithm with an initialization strategy to solve the problem efficiently. Validated on a simulated three-tank system and a realistic lithium-ion battery cell, the approach achieves a more balanced detection performance by strengthening detection of the known fault while preserving detectability for unknown faults, under distributional uncertainty and controlled FAR.

Abstract

The robustness of fault detection algorithms against uncertainty is crucial in the real-world industrial environment. Recently, a new probabilistic design scheme called distributionally robust fault detection (DRFD) has emerged and received immense interest. Despite its robustness against unknown distributions in practice, current DRFD focuses on the overall detectability of all possible faults rather than the detectability of critical faults that are a priori known. Henceforth, a new DRFD trade-off design scheme is put forward in this work by utilizing prior fault information. The key contribution includes a novel distributional robustness metric of detecting a known fault and a new relaxed distributionally robust chance constraint that ensures robust detectability. Then, a new DRFD design problem of fault detection under unknown probability distributions is proposed, and this offers a flexible balance between the robustness of detecting known critical faults and the overall detectability against all possible faults. To address the resulting semi-infinite chance-constrained problem, we first reformulate it to a finite-dimensional problem characterized by bilinear matrix inequalities. Subsequently, a tailored heuristic solution algorithm is developed, which includes a sequential minimization procedure and an initialization strategy. Finally, case studies on a simulated three-tank system and a real-world battery cell are carried out to showcase the effectiveness of the proposed heuristic algorithm and the advantages of our DRFD method.
Paper Structure (10 sections, 3 theorems, 41 equations, 10 figures, 2 tables, 2 algorithms)

This paper contains 10 sections, 3 theorems, 41 equations, 10 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Residual Generation and Evaluation
  4. Problem Formulation of DRFD
  5. Main result
  6. A New Trade-off Formulation of DRFD
  7. Reformulation and Solution Algorithm
  8. Case Study on A Simulated Three-tank System
  9. Case Study on A Realistic Lithium-ion Battery Cell
  10. Conclusion

Key Result

Lemma 1

The following optimization problem on an arbitrary non-empty set $\mathcal{S} \in \Xi$, where the decision variable $\mathbb{P}_{\xi}$ is an arbitrary distribution on probability space $\mathcal{M}\left(\Xi\right)$ is equivalent to the following problem:

Figures (10)

  • Figure 1: Schematic of the three-tank system.
  • Figure 2: Cross-validation results under different Wasserstein radii and overall detectability
  • Figure 3: Trade-off between $\rho(\cdot)$ and $\eta$ in NDR$1$
  • Figure 4: Trade-off between $\rho(\cdot)$ and $\eta$ in NDR$2$
  • Figure 5: FD results on the simulated three-tank system
  • ...and 5 more figures

Theorems & Definitions (14)

  • Remark 1
  • Definition 1: FAR and FDR, ding2008model
  • Remark 2
  • Definition 2: Wasserstein distance, kantorovich1958space
  • Definition 3: Wasserstein ambiguity set, mohajerin2018data
  • Remark 3
  • Remark 4
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 4 more