Probabilistic Performance Analysis of Fault Diagnosis Schemes

1
ε increasing
Pd,k
Probability of Detection,
0.5
0
0 0.5 1
Probability of False Alarm,
P f,k
Figure 6.4. Performance metrics for the air-data sensor system plotted in roc space. Each roc curve
represents the performance of the fault detection scheme shown in Figure 6.1 at a particular time step
as the threshold ε on the decision function δ is varied.
Section 5.2.1 by using the proportional threshold
ε k = ν √ Σ k ,
where ν = 2.25. We use the yalmip interface [63] to SeDuMi [90] to solve the optimization
problem. The resulting worst-case values Pf ⋆ (γ) are plotted in Figure 6.5 for γ ranging from 0
to 10.
Finally, we compute the worst-case fault signal, with respect to the probability of detection.
For this computation, we assume that there are no other sources of uncertainty. Let ϑ
be the fault parameter sequence in which both sensors fail at k = 18,000 (15 minutes). The
class of uncertain fault signals considered is
B 2 (f ◦ ,γ) = { ˜ f + f ◦ (ϑ) : ‖ ˜ f ‖ 2 < γ } ,
where f ◦ (ϑ) is the nominal bias fault with magnitudes b s and b f defined above. The time
horizon of the simulation is shortened to 17 minutes (i.e., N = 20,400 time steps). Hence, the
signal f ˜ must decrease the probability of detection (i.e., suppress the effect of the nominal
fault f ◦ (ϑ)) over a 2 minute interval. Again, we use the yalmip interface [63] to formulate the
optimization problem and SeDuMi [90] to solve it. The resulting worst-case values Pd ⋆ (γ) are
plotted in Figure 6.6 for γ ranging from 1.5 to 2.0. Note that, for each γ, the value of Pd ⋆(γ)
would increase as the number of time steps N is increased, because the perturbation f ˜
would have to suppress the effect of f ◦ (ϑ) over a longer time span. That is, increasing N
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