Probabilistic Performance Analysis of Fault Diagnosis Schemes

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Page 1 and 2:
Probabilistic Performance Analysis
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Abstract Probabilistic Performance
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Soli Deo gloria. i
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3 Probabilistic Performance Analysi
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List of Figures 2.1 “Bathtub” s
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List of Tables 4.1 Time-complexity
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Acknowledgements When I started wri
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tic metrics that rigorously quantif
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tion. • Complexity of Markov Chai
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Given an event B ∈ F with P(B) >
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Note that E ( f (x) | y ) is a rand
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for all c ∈ R, where erf(c) := 2
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Hazard Rate λ 0 0 break-in 0 t 1 t
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malfunction — an intermittent irr
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where the matrix Q is chosen to app
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v w u G θ y F r δ V d Figure 2.3.
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These relationships can be used to
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ε Residual 0 0 T f T d Time Figure
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Chapter 3 Probabilistic Performance
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the worst-case performance under a
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For example, P tp,k = P(D 1,k ∩ H
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where the subscript k has been omit
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1 (α 3 ,β 3 ) Pd,k Probability of
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Definition 3.9. The upper boundary
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1 ε increasing ε = 0 Pd,k Probabi
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1 P d,k P f,k β Q 0,k Probability
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In general, as Q 0,k decreases, the
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from J k by taking column-sums. If
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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
Page 87 and 88: Second, we show that the running ti
Page 89 and 90: Table 4.1. Time-complexity of compu
Page 91 and 92: For p ∈ [1,∞], the l p -norm ba
Page 93 and 94: linear time-varying uncertainties
Page 95 and 96: for the appropriate choice of k f a
Page 97 and 98: Hence, it is straightforward to sho
Page 99 and 100: v w f (θ) u G θ y . F r Figure 5.
Page 101 and 102: Maximizing the Probability of False
Page 103 and 104: β ∆ α v f (θ) u G θ y . F r F
Page 105 and 106: β ∆ α β ∆ 1 . .. ∆q α (a)
Page 107 and 108: if and only if [ T (βi ) T T (M i
Page 109 and 110: Fix a parameter sequence θ = ϑ an
Page 111 and 112: As in Section 5.2.2, if the matrix
Page 113 and 114: Chapter 6 Applications 6.1 Introduc
Page 115 and 116: p t p s v t + f t (θ) φ γ ˆV ĥ
Page 117 and 118: 300 18 V (m/s) 200 h (km) 12 Airspe
Page 119 and 120: 1 P tn,k 0.8 P fp,k P fn,k (a) Prob
Page 121 and 122: 0.4 Probability of False Alarm, P
Page 123 and 124: The following matrices correspond t
Page 125 and 126: 6.4.2 Applying the Framework The ma
Page 127 and 128: P ⋆ f Probability of False Alarm,
Page 129 and 130: Chapter 7 Conclusions & Future Work
Page 131 and 132: measurement noise. Hence, the appli
Page 133 and 134: [12] R. H. Chen, D. L. Mingori, and
Page 135 and 136: [40] R. L. Graham, D. E. Knuth, and
Page 137: [68] L. A. Mironovski, Functional d
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