Probabilistic Performance Analysis of Fault Diagnosis Schemes

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[ u1 ] v 1 w 1 f 1 S 1 y 1 [ u2 ] v 2 w 2 f 2 [ u3 ] v 3 w 3 f 3 S 2 S 3 y 2 y 3 y 4 mux y F g ŷ − r δ d [ u4 ] v 4 w 4 f 4 S 4 Figure 2.5. System of four analytically redundant sensors. Each sensor S i is affected by a distinct input u i , noise v i , disturbance w i , and fault signal f i . The block labeled g represents a set of analytical relationships, which use the vector of measurements y to produce a residual vector r . Then, the decision function δ uses the residual vector r to produce a decision d. can be used as a substitute. Also, the ability of the system to detect other sensor failures is reduced, because y 1 enters into all four of the residuals. 2.6 Existing Performance Analyses 2.6.1 Standard Approaches In the fault detection literature, there are two primary ways to assess the performance of a fault detection scheme: simulation and design criteria. The simulation-based approach, used in [12, 15, 25, 32, 42, 62, 89, 107–109], involves simulating a number of realizations of the residual r given that a particular fault occurs at a particular time (see Figure 2.6 for a typical plot of a single simulation). From these simulation data, one can generally get a sense of how well the fault detection scheme detects the fault in question. However, the number of simulations—usually just one—is often too small to say anything statistically meaningful about the performance. Moreover, it is impractical to produce such a plot for every possible fault that may affect the system. By simulating the effect that a particular fault has on the residual, these simulation-based performance assessments assume that either the residual has reached steady-state when the fault occurs or, for some other reason, the time at which the fault occurs is irrelevant. Such assumptions are only meaningful when the residual is completely decoupled from the known inputs, unknown disturbances, and noise signals. The second approach to assessing the performance of fault detection schemes is to quote the numerical value of design criteria. Examples of design criteria are given in Section 2.4.2. This approach, used in [10, 12, 14, 26, 39], is most useful for comparing fault detection schemes designed using similar criteria. Although it may be possible to produce a scheme using one set of design criteria and then assess their performance with respect to another set, 21
ε Residual 0 0 T f T d Time Figure 2.6. Typical plot of the response of the residual to the occurrence of a particular fault at time T f . The residual crosses the threshold ε at time T d , giving a detection delay of T d − T f . the actual values of the criteria may be hard to interpret in terms of the desired system-level performance (e.g., overall reliability, false alarm rate). 2.6.2 Probabilistic Approaches Recognizing the need for more rigorous and informative performance metrics, some authors in the fault diagnosis community (e.g., [8, 24, 100]) have proposed the probability of false alarm as a performance metric. For a fixed time k, a false alarm is defined as the event that the fault detection scheme indicates a fault at time k, given that no fault has occurred at or before time k. Conditional on the event that no fault has occurred, the only source of randomness in the residual {r k } is the noise signal {v k }. In many cases, the distribution of the stochastic process {r k } is easily computed, and the probability of a false alarm can be evaluated (or at least bounded above). However, the probability of false alarm alone cannot characterize the performance of a fault detection scheme. Consider, for example, the trivial decision function defined as δ 0 : (k,r k ) → 0, for all k and r k . Paired with any residual generator F , the fault detection scheme V = (F,δ 0 ) will have zero probability of false alarm, but V is incapable of detecting faults. Hence, it is also necessary quantify the probability of detection, which is the probability that the fault detection scheme correctly detects a fault when one is present. In general, the probability of detection must be computed for each fault or each class of faults. Performing these computations can be intractable unless special care is taken. For example, the class of fault signals considered in [100] is restricted to the set of randomly occurring biases, which are easily parameterized by the time of occurrence and the magnitude of the bias. More commonly, authors use simulation or design criteria, as in the previous section, to complement the probability of false alarm (e.g., [8]). One of the main objectives of this 22
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 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: These relationships can be used to
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
Page 67 and 68: the matrix A as in Theorem 4.12. Th
Page 69 and 70: 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
Page 77 and 78: probability matrix is given by ( Λ
Page 79 and 80: s 2 . .. s 0 s 1 s q Figure 4.4. St
Page 81 and 82: metrics at time k are defined as Ĵ
Page 83 and 84: 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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Probabilistic Performance Analysis of Fault Diagnosis Schemes

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