Probabilistic Performance Analysis of Fault Diagnosis Schemes

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6.6 Worst-case probability of detection for the air-data sensor system with an uncertain fault signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.7 Linearized vtol aircraft model with additive model uncertainty . . . . . . . . . 109 6.8 Performance metrics for the vtol aircraft example . . . . . . . . . . . . . . . . . 113 6.9 Worst-case probability of false alarm for the vtol aircraft example with additive model uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.10 Worst-case probability of detection for the vtol aircraft example with additive model uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 vi
List of Tables 4.1 Time-complexity of the performance analysis algorithms . . . . . . . . . . . . . 76 5.1 Interpolation results for linear operators with and without feedback . . . . . . 95 6.1 Steady-state performance of the air-data sensor system . . . . . . . . . . . . . . 106 6.2 Steady-state performance of the vtol aircraft example . . . . . . . . . . . . . . . 113 vii
Page 1 and 2: Probabilistic Performance Analysis
Page 3 and 4: Abstract Probabilistic Performance
Page 5 and 6: Soli Deo gloria. i
Page 7 and 8: 3 Probabilistic Performance Analysi
Page 9: List of Figures 2.1 “Bathtub” s
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
Page 63 and 64:
for some p ∈ (0,1). Then, the cor
Page 65 and 66:
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
Page 71 and 72:
Assumed Structure of the Residual G
Page 73 and 74:
Therefore, conditional on the event
Page 75 and 76:
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
Page 83 and 84:
Proposition 4.32. Let N be the fina
Page 85 and 86:
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
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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
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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)
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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
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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
Page 125 and 126:
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
Page 135 and 136:
[40] R. L. Graham, D. E. Knuth, and
Page 137 and 138:
[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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