Probabilistic Performance Analysis of Fault Diagnosis Schemes

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5.1.3 Worst-case Optimization Problems . . . . . . . . . . . . . . . . . . . . . . 80 5.2 Formulating Tractable Optimization Problems . . . . . . . . . . . . . . . . . . . 81 5.2.1 Simplifying Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2.2 Simplified Worst-case Optimization Problems . . . . . . . . . . . . . . . . 84 5.3 Problems with No Model Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.4 Problems with Model Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.4.1 Interpolation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.4.2 Using the Interpolation Results to Find Worst-case Performance . . . . 95 6 Applications 100 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2 Types of Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.3 Air-Data Sensor Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.3.2 Applying the Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4 VTOL Aircraft Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.4.2 Applying the Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7 Conclusions & Future Work 116 References 119 iv
List of Figures 2.1 “Bathtub” shape of a typical hazard rate curve . . . . . . . . . . . . . . . . . . . 12 2.2 General fault diagnosis problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 General parametric fault diagnosis problem . . . . . . . . . . . . . . . . . . . . . 18 2.4 System of four physically redundant sensors . . . . . . . . . . . . . . . . . . . . . 19 2.5 System of four analytically redundant sensors . . . . . . . . . . . . . . . . . . . . 21 2.6 Typical plot of the residual due to a particular fault . . . . . . . . . . . . . . . . 22 3.1 General parametric fault diagnosis problem . . . . . . . . . . . . . . . . . . . . . 25 3.2 Performance achievable by randomizing a collection of deterministic tests . . 32 3.3 Visual summary of facts about the range of achievable performance . . . . . . 33 3.4 Set of performance points achieved by a parameterized family of tests . . . . . 36 3.5 Bound on availability over time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.6 Bound on the performance metrics P f and P d over time . . . . . . . . . . . . . 38 3.7 Bound on the performance metrics P f and P d visualized in roc space . . . . . 39 3.8 Bound on Bayesian risk visualized in roc space . . . . . . . . . . . . . . . . . . . 39 4.1 Simple example of a directed graph . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 State-transition diagram of an up-down counter . . . . . . . . . . . . . . . . . . 63 4.3 Comparison of an up-down counter and a threshold decision function . . . . 65 4.4 State-transition diagram for a system that reconfigures . . . . . . . . . . . . . . 66 5.1 Uncertain fault diagnosis problem with no model uncertainty . . . . . . . . . . 86 5.2 Uncertain fault diagnosis problem with model uncertainty . . . . . . . . . . . . 90 5.3 Block diagrams for the interpolation results . . . . . . . . . . . . . . . . . . . . . 92 6.1 Air-data sensor system with a fault diagnosis scheme . . . . . . . . . . . . . . . 102 6.2 Air-data sensor equations for subsonic flight in the troposphere . . . . . . . . . 104 6.3 Performance metrics for the air-data sensor system . . . . . . . . . . . . . . . . 106 6.4 Performance metrics for the air-data sensor system in roc space . . . . . . . . 107 6.5 Worst-case probability of false alarm for the air-data sensor system with an uncertain input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 v
Page 1 and 2: Probabilistic Performance Analysis
Page 3 and 4: Abstract Probabilistic Performance
Page 5 and 6: Soli Deo gloria. i
Page 7: 3 Probabilistic Performance Analysi
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
Page 59 and 60:
Fact 4.1. Given a Markov chain θ
Page 61 and 62:
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
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[95] H. B. Wang, J. L. Wang, and J.
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