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Robust Multivariate Detection and Estimation with Fault Frequency Content Information

Jingwei Dong, Kaikai Pan, Sergio Pequito, Peyman Mohajerin Esfahani

TL;DR

A thresholding rule is proposed to guarantee both the false alarm rate (FAR) and the fault detection rate (FDR) and an exact reformulation of the optimal estimation filter design using the restricted Hinf performance index is derived, which is inherently non-convex.

Abstract

This paper studies the problem of fault detection and estimation (FDE) for linear time-invariant (LTI) systems with a particular focus on frequency content information of faults, possibly as multiple disjoint continuum ranges, and under both disturbances and stochastic noise. To ensure the worst-case fault sensitivity in the considered frequency ranges and mitigate the effects of disturbances and noise, an optimization framework incorporating a mixed H_/H2 performance index is developed to compute the optimal detection filter. Moreover, a thresholding rule is proposed to guarantee both the false alarm rate (FAR) and the fault detection rate (FDR). Next, shifting attention to fault estimation in specific frequency ranges, an exact reformulation of the optimal estimation filter design using the restricted Hinf performance index is derived, which is inherently non-convex. However, focusing on finite frequency samples and fixed poles, a lower bound is established via a highly tractable quadratic programming (QP) problem. This lower bound together with an alternating optimization (AO) approach to the original estimation problem leads to a suboptimality gap for the overall estimation filter design. The effectiveness of the proposed approaches is validated through applications of a non-minimum phase hydraulic turbine system and a multi-area power system.

Robust Multivariate Detection and Estimation with Fault Frequency Content Information

TL;DR

A thresholding rule is proposed to guarantee both the false alarm rate (FAR) and the fault detection rate (FDR) and an exact reformulation of the optimal estimation filter design using the restricted Hinf performance index is derived, which is inherently non-convex.

Abstract

This paper studies the problem of fault detection and estimation (FDE) for linear time-invariant (LTI) systems with a particular focus on frequency content information of faults, possibly as multiple disjoint continuum ranges, and under both disturbances and stochastic noise. To ensure the worst-case fault sensitivity in the considered frequency ranges and mitigate the effects of disturbances and noise, an optimization framework incorporating a mixed H_/H2 performance index is developed to compute the optimal detection filter. Moreover, a thresholding rule is proposed to guarantee both the false alarm rate (FAR) and the fault detection rate (FDR). Next, shifting attention to fault estimation in specific frequency ranges, an exact reformulation of the optimal estimation filter design using the restricted Hinf performance index is derived, which is inherently non-convex. However, focusing on finite frequency samples and fixed poles, a lower bound is established via a highly tractable quadratic programming (QP) problem. This lower bound together with an alternating optimization (AO) approach to the original estimation problem leads to a suboptimality gap for the overall estimation filter design. The effectiveness of the proposed approaches is validated through applications of a non-minimum phase hydraulic turbine system and a multi-area power system.
Paper Structure (25 sections, 11 theorems, 73 equations, 15 figures, 1 table, 2 algorithms)

This paper contains 25 sections, 11 theorems, 73 equations, 15 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Model Description and Problem Statement
  4. Problem 1: Fault detection
  5. Problem 1a (Fault detection filter design)
  6. Problem 1b (Thresholding with guarantees on FAR and FDR)
  7. Problem 2: Fault estimation
  8. Problem 2 (Fault estimation filter design)
  9. Fault Detection: Optimal Design and Thresholding
  10. Fault detection filter design
  11. Thresholding rule
  12. Fault Estimation: Optimal Design and Suboptimality Gap
  13. Fault estimation filter design
  14. Convex approximation with suboptimality gap
  15. Problem 2r (Fault estimation with finite frequency content)
  16. ...and 10 more sections

Key Result

Theorem 3.1

Consider the system eq:SS model, the structure of the filter eq: FD filter, and the state-space realizations $(\mathcal{A}_r,\mathcal{B}_{\omega r},\mathcal{C}_r)$ and $(\mathcal{A}_r,\mathcal{B}_{fr},\mathcal{C}_r)$. Given the degree $d_N$, $d_a=d_N$, the dimension of the residual $n_r$, a scalar $ where for each frequency range $\Theta_m = \{\theta_f : \theta_{1_m} \leq \theta_f \leq \theta_{2_m

Figures (15)

  • Figure 1: Geometric illustration of the multi-dimensional residual.
  • Figure 2: Fault and its estimates generated using different methods.
  • Figure 3: Errors of fault estimates.
  • Figure 4: Errors of fault estimates with different degrees.
  • Figure 5: Detection results for $f_{agc_2}$ and $f_{tie_{12}}$.
  • ...and 10 more figures

Theorems & Definitions (35)

  • Definition 2.3: $\mathcal{H}_2$ norm scherer1997multiobjective
  • Definition 2.4: $\mathcal{H}_{\_}$ index liu2005lmi
  • Remark 2.6: Difficulty in FDR computation
  • Definition 2.7: Restricted $\mathcal{H_{\infty}}$ norm gao2011h
  • Remark 2.8: Differences between Problem 1a and 2
  • Theorem 3.1: Optimal detection: exact finite reformulation
  • proof
  • Remark 3.2: The auxiliary matrix $V_m$
  • Remark 3.3: Residuals with arbitrary dimensions
  • Lemma 3.4: Sub-Gaussian concentration vershynin2018high
  • ...and 25 more