Probabilistic Performance Analysis of Fault Diagnosis Schemes

Fix a parameter sequence θ = ϑ and let the residual generator F be partitioned as F = [ F 1 F 2
]
.
Divide the residual into its non-random and random parts, as follows:
where
r = r unc + r rnd ,
r unc = F 2 G 21,θ β + F 2 G 23,θ f (θ) + (F 2 G 24,θ + F 1 )u
r rnd = F 2 G 22 v.
Since v is zero-mean by assumption, the conditional mean of the residual at time k is
and the conditional variance at time k is
ˆr k = E(r k | θ 0:k = ϑ 0:k ) = r unc
k
,
Σ k = E ( (r k − ˆr k ) 2) ( (r ) )
rnd 2
= E .
Note that, as desired, the sequence {Σ k } does not depend on β or θ.
k
Maximizing the Probability of False Alarm
Assume that no faults have occurred (ϑ = 0). Recall that the worst-case probability of false
alarm is
Pf ⋆ = 1 − min min P( )
|r k | < ε k | θ 0:k = 0 0:k .
∆∈P ∆ 0≤k≤N
As explained in Section 5.2.2, the crux of computing P ⋆ f
ˆr ⋆ k = max
∣ ∣r unc∣ ∆∈P
k ,
∆
is solving
for k = 0,1,..., N . There are two cases to consider: P ∆ = ∆ 2,lti (γ) and P ∆ = ∆ 2,ltv (γ).
Case 1. Suppose that ∆ belongs to the set ∆ 2,lti (γ) and assume that G 11,0 is an lti operator
with ‖G 11,0 ‖ i 2 < 1 γ . Then, for k = 0,1,..., N, applying Theorem 5.6 yields the following
optimization:
ˆr ⋆ k = maximize
β
∣ r
unc∣
k
subject to r unc = F 2 G 21,0 β + (F 2 G 24,0 + F 1 )u
z = G 13,0 f (0) +G 14,0 u
J (β) ≽ 0,
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