Probabilistic Performance Analysis of Fault Diagnosis Schemes

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v w u G θ y F r δ V d Figure 3.1. General parametric fault diagnosis problem. Faults affect the physical parameters of the system, which in turn affect the system model parameter θ. The plant G is subject to a known deterministic input u, a random input v, and a deterministic disturbance w. The residual generator uses the plant input u and output y to produce a residual r , and the decision function δ uses the residual r to produce a decision d about the current value of θ. Together, F and δ form a fault diagnosis scheme, denoted V = (F,δ). corresponding set of possible decisions is defined as D := {0,1,..., q}. The purpose of the fault diagnosis scheme V = (F,δ) is to produce a decision d k ∈ D, at each time k, indicating which subset Θ dk ⊂ Θ most likely contains the parameter θ k . Of course, the scheme V does not have direct access to the parameter. Instead, V must make a decision based on the known input {u k } and the measured output {y k }, which is corrupted by the noise signal {v k } and the disturbance {w k }. Therefore, the performance of the scheme V is quantified by the probability that the correct decision is made. The number of partitions q determines what type fault diagnosis problem the scheme V is designed to address. If q = 1, the set Θ 1 contains all faulty parameter values, and V is interpreted as a fault detection scheme. If q > 1, each subset Θ i ⊂ Θ represents a different class of faulty behavior, and V is interpreted as a fault isolation scheme. If the parameter space Θ is finite and each partition Θ i a singleton set, then V achieves fault identification, as well. In Section 3.3, we define probabilistic performance metrics for the fault detection problem (q = 1). Then, in Section 3.6, these results are extended to the more general fault isolation problem (q > 1). In this chapter and in Chapter 4, we assume that the deterministic input {u k } is known and fixed, that there is no deterministic disturbance {w k }, and that G θ is a known function of the parameter {θ k }. Chapter 5 extends these results by considering how uncertainty impacts the performance metrics. In particular, Chapter 5 presents some techniques for computing 25
the worst-case performance under a given uncertainty model. 3.3 Quantifying Accuracy Our performance analysis of fault detection is rooted in the theory of statistical hypothesis testing. This approach not only allows us to utilize the tools and terminology of hypothesis testing, it also allows us to draw connections between fault detection and other fields, such as signal detection [54, 61, 75, 93], medical diagnostic testing [31, 73, 111], and pattern recognition [34, 57]. For a standard mathematical treatment of statistical hypothesis testing, see Lehmann and Romano [60]. 3.3.1 Fault Detection and Hypothesis Testing For the sake of simplicity, this section focuses on the problem of fault detection, while the more general fault isolation problem is treated in Section 3.6. Hence, the parameter space is partitioned into two sets: the set containing the nominal parameter, Θ 0 = {0}, and the set containing all faulty parameter values, Θ 1 = Θ c 0 . At each time k, define the hypotheses H 0,k : θ k ∈ Θ 0 , H 1,k : θ k ∈ Θ 1 , and let H i ,k be the event that hypothesis H i ,k is true, for each i . Since exactly one hypothesis is true at each time, the sets H 0,k and H 1,k form a partition of the sample space Ω. The fault detection scheme V is interpreted as a test that decides between the hypotheses H 0,k and H 1,k . Although the input data u 0:k = {u 0 ,...,u k } are known and deterministic, the distribution of the output data y 0:k = {y 0 ,..., y k } clearly depends on which hypothesis is true. Together, u 0:k and y 0:k are interpreted as a test statistic, which is used by the test V to produce a decision d k in D = {0,1}, at time k. Let D 0,k be the event that d k = 0 and let D 1,k be the event that d k = 1. Of course, exactly one of these events is true at each time, so the sets D 0,k and D 1,k form another partition of the sample space Ω. 3.3.2 Probabilistic Analysis Let the prior probabilities of the hypotheses be denoted Q 0,k := P(H 0,k ), Q 1,k := P(H 1,k ). Since exactly one hypothesis is true and exactly one decision is made at each time k, the performance of the test V is characterized by the probability that the events D i ,k and H j,k 26
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 and 34: These relationships can be used to
Page 35 and 36: ε Residual 0 0 T f T d Time Figure
Page 37: Chapter 3 Probabilistic Performance
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
Page 85 and 86: Algorithm 4.2. Procedure for comput
Page 87 and 88: 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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