Probabilistic Performance Analysis of Fault Diagnosis Schemes

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• Sample time: T s = 0.05s • Time horizon: N = 72,000 (i.e., N T s = 1hour) • Flight path angle: 0.5 ◦ (constant) • Airspeed: V = 45 m/s (constant) • Initial Altitude: h(0) = 200m • Noise standard deviations: σ t = 2.5Pa, σ s = 2.5Pa • Bias fault magnitudes: b t = −0.04kPa, b s = 0.05kPa • Continuous failure time models: λ t = λ s = 0.001hr −1 = 2.78 × 10 −7 s −1 • Discrete failure time models: q t = q s = 1.389 × 10 −7 • Filter pole: a = 0.003 (before discretization) • Threshold: ε = 2m The resulting performance metrics are plotted in Figures 6.3(a) and 6.3(b). Note that, in this case, the component failure rates are so small that the plots of {P fn,k } and {P tp,k } are barely distinguishable from zero. Next, we plot the roc curves as the threshold ε varies from 0.1 m to 50 m. The curves shown in Figure 6.4 correspond to times ranging from 1 minute to 1 hour. Note that, in Figure 6.3(b), the probability of detection {P d,k } dips at about 7 minutes. Hence, some of the roc curves in Figure 6.4 cross over one another. However, the general trend is that the roc curves pass closer to the ideal point (0,1) as time increases. For our third numerical experiment, we observe that the probability of detection, plotted in Figure 6.3(b), converges to a steady-state value. To better understand the effects of changing the washout filter pole a and the noise standard deviation σ, we compute the steady-state values of {P d,k } as a and σ vary. In Table 6.1, these steady-state values are tabulated for a ranging from 0.0005 to 0.004 and σ ranging from 2Pa to 10Pa. Note that the value of a listed in Table 6.1 corresponds to the continuous-time washout filter W before discretization. Also, the same standard deviation σ is used for both noise signals, {v t,k } and {v s,k }. All other parameters remain the same as in the previous experiments. In our fourth experiment, we seek to find the worst-case flight path, with respect to the probability of false alarm. For these optimizations, we use the values a = 0.003 and σ t = σ s = 2.5Pa, as in the first two experiments. We assume that there is no disturbance w or model uncertainty ∆ affecting the system. The class of uncertain inputs considered is B 2 (u ◦ ,γ) = { ũ + u ◦ : ‖ũ‖ 2 < γ } , where u ◦ = (V,h) is the flight path described in the first experiment. Since we only consider additive input faults, the conditional variance of the residual, {Σ k } does not depend on the fault parameter sequence θ or the uncertain input. Hence, we can fulfill Assumptions 1–3 of 105
1 P tn,k 0.8 P fp,k P fn,k (a) Probability 0.6 0.4 P tp,k 0.2 0 1 0 10 20 30 40 50 60 P d,k 0.8 P f,k Q 0,k (b) Probability 0.6 0.4 0.2 0 0 10 20 30 40 50 60 Time (min) Figure 6.3. Performance metrics for the air-data sensor system. Plot (a) shows the joint probability performance metrics, and plot (b) shows the conditional probability performance metrics. Note that the sequences {P fn,k } and {P tp,k } have small values and are barely distinguishable from zero. Table 6.1. Steady-state performance of the air-data sensor system for various values of the washout filter pole a and the noise standard deviation σ. Note that the values of the pole a refer to the continuous-time dynamics before discretization, but the standard deviation σ refers to the discretized iid Gaussian noise sequences (i.e., σ s = σ t = σ). Noise Standard Deviation, σ (Pa) Pole, a 2 4 6 8 10 0.0005 0.9742 0.9482 0.9216 0.8943 0.8662 0.001 0.9739 0.9469 0.9183 0.8875 0.8534 0.0015 0.9736 0.9454 0.9137 0.8756 0.8211 0.002 0.9732 0.9435 0.9064 0.8423 0.7303 0.0025 0.9729 0.9410 0.8879 0.7631 0.5968 0.003 0.9725 0.9373 0.8427 0.6517 0.4692 0.0035 0.9720 0.9291 0.7687 0.5387 0.3680 0.004 0.9715 0.9104 0.6790 0.4411 0.2933 106
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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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Page 99 and 100: v w f (θ) u G θ y . F r Figure 5.
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Page 129 and 130: Chapter 7 Conclusions & Future Work
Page 131 and 132: measurement noise. Hence, the appli
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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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