Probabilistic Performance Analysis of Fault Diagnosis Schemes

• 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
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