Probabilistic Performance Analysis of Fault Diagnosis Schemes

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Contents List of Figures List of Tables List of Algorithms vi vii viii 1 Introduction 1 1.1 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Background 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.3 Expectation, Mean, and Variance . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.4 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.5 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.6 Common Probability Distributions . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Reliability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.1 Basic Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.2 Brief Survey of Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Designing for Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5.1 Physical Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5.2 Analytical Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 Existing Performance Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6.1 Standard Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6.2 Probabilistic Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6.3 Quickest Detection Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 23 ii
3 Probabilistic Performance Analysis 24 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Quantifying Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.1 Fault Detection and Hypothesis Testing . . . . . . . . . . . . . . . . . . . 26 3.3.2 Probabilistic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.3 Aggregate Measures of Performance . . . . . . . . . . . . . . . . . . . . . . 29 3.4 Characterizing the Range of Achievable Performance . . . . . . . . . . . . . . . 30 3.4.1 Randomized Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4.2 Receiver Operating Characteristic . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 Certifying and Visualizing Performance . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5.1 Bounds on Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5.2 Bound on Bayesian Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.6 Extension to Fault Isolation and Identification . . . . . . . . . . . . . . . . . . . 40 3.6.1 Quantifying Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6.2 Bayesian Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.6.3 ROC Curves for Multiple Hypotheses . . . . . . . . . . . . . . . . . . . . . 43 4 Computational Framework 44 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Fault Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2.1 Limiting Complexity with Structured Markov Chains . . . . . . . . . . . 47 4.2.2 Special Case: Fault Model Based on Component Failures . . . . . . . . . 55 4.3 System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.1 Assumptions Regarding the System Dynamics . . . . . . . . . . . . . . . 57 4.3.2 Computing the Conditional Mean and Variance . . . . . . . . . . . . . . 59 4.3.3 Special Case: Models with Only Additive Faults . . . . . . . . . . . . . . . 60 4.4 Decision Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4.1 Threshold Decision Functions . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4.2 Dynamic Decision Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.5 Algorithms for Computing Performance . . . . . . . . . . . . . . . . . . . . . . . 68 4.5.1 Sufficiently Structured Systems . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.5.2 LTV Special Case Based on Component Failures . . . . . . . . . . . . . . 70 4.5.3 LTI Special Case Based on Component Failures . . . . . . . . . . . . . . . 74 4.6 Comments on Continuous-Time Models . . . . . . . . . . . . . . . . . . . . . . . 76 5 Worst-Case Performance Analysis 77 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.1.2 Types of Uncertainty Considered . . . . . . . . . . . . . . . . . . . . . . . 78 iii
Page 1 and 2: Probabilistic Performance Analysis
Page 3 and 4: Abstract Probabilistic Performance
Page 5: Soli Deo gloria. i
Page 9 and 10: List of Figures 2.1 “Bathtub” s
Page 11 and 12: List of Tables 4.1 Time-complexity
Page 13 and 14: Acknowledgements When I started wri
Page 15 and 16: tic metrics that rigorously quantif
Page 17 and 18: tion. • Complexity of Markov Chai
Page 19 and 20: Given an event B ∈ F with P(B) >
Page 21 and 22: Note that E ( f (x) | y ) is a rand
Page 23 and 24: for all c ∈ R, where erf(c) := 2
Page 25 and 26: Hazard Rate λ 0 0 break-in 0 t 1 t
Page 27 and 28: malfunction — an intermittent irr
Page 29 and 30: where the matrix Q is chosen to app
Page 31 and 32: v w u G θ y F r δ V d Figure 2.3.
Page 33 and 34: These relationships can be used to
Page 35 and 36: ε Residual 0 0 T f T d Time Figure
Page 37 and 38: Chapter 3 Probabilistic Performance
Page 39 and 40: the worst-case performance under a
Page 41 and 42: For example, P tp,k = P(D 1,k ∩ H
Page 43 and 44: where the subscript k has been omit
Page 45 and 46: 1 (α 3 ,β 3 ) Pd,k Probability of
Page 47 and 48: Definition 3.9. The upper boundary
Page 49 and 50: 1 ε increasing ε = 0 Pd,k Probabi
Page 51 and 52: 1 P d,k P f,k β Q 0,k Probability
Page 53 and 54: In general, as Q 0,k decreases, the
Page 55 and 56: from J k by taking column-sums. If
Page 57 and 58:
Chapter 4 Computational Framework 4
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Fact 4.1. Given a Markov chain θ
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v 1 v 2 v 3 v 4 Figure 4.1. Simple
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for some p ∈ (0,1). Then, the cor
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Proof. Let ϑ 0:l be a possible pat
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the matrix A as in Theorem 4.12. Th
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every 1-bit of b i is a 1-bit of b
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Assumed Structure of the Residual G
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Therefore, conditional on the event
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In the non-scalar case (i.e., r k
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probability matrix is given by ( Λ
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s 2 . .. s 0 s 1 s q Figure 4.4. St
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metrics at time k are defined as Ĵ
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Proposition 4.32. Let N be the fina
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Algorithm 4.2. Procedure for comput
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Second, we show that the running ti
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Table 4.1. Time-complexity of compu
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For p ∈ [1,∞], the l p -norm ba
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linear time-varying uncertainties
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for the appropriate choice of k f a
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Hence, it is straightforward to sho
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v w f (θ) u G θ y . F r Figure 5.
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Maximizing the Probability of False
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β ∆ α v f (θ) u G θ y . F r F
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β ∆ α β ∆ 1 . .. ∆q α (a)
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if and only if [ T (βi ) T T (M i
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Fix a parameter sequence θ = ϑ an
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As in Section 5.2.2, if the matrix
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Chapter 6 Applications 6.1 Introduc
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p t p s v t + f t (θ) φ γ ˆV ĥ
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300 18 V (m/s) 200 h (km) 12 Airspe
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1 P tn,k 0.8 P fp,k P fn,k (a) Prob
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0.4 Probability of False Alarm, P
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The following matrices correspond t
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6.4.2 Applying the Framework The ma
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P ⋆ f Probability of False Alarm,
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Chapter 7 Conclusions & Future Work
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measurement noise. Hence, the appli
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[12] R. H. Chen, D. L. Mingori, and
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[40] R. L. Graham, D. E. Knuth, and
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[68] L. A. Mironovski, Functional d
Page 139 and 140:
[95] H. B. Wang, J. L. Wang, and J.
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Probabilistic Performance Analysis of Fault Diagnosis Schemes

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