Probabilistic Performance Analysis of Fault Diagnosis Schemes

If the residual generator is partitioned as
then the residual can be written as
r = F 1 y + F 2 u
]
F =
[F 1 F 2 ,
= (F 1 G 1,ϑ + F 2 )u + F 1 G 2,ϑ v + F 1 G 3,ϑ w + F 1 G 4,ϑ f (ϑ)
= (F 1 G 1,ϑ + F 2 )(u ◦ + ũ) + F 1 G 2,ϑ v + F 1 G 3,ϑ w + F 1 G 4,ϑ
(
f ◦ (ϑ) + ˜ f ) .
Divide the residual into the sum of its nominal, uncertain, and random parts as follows:
where
r = r nom + r unc + r rnd ,
r nom = (F 1 G 1,ϑ + F 2 )u ◦ + F 1 G 4,ϑ f ◦ (ϑ),
r unc = (F 1 G 1,ϑ + F 2 )ũ + F 1 G 3,ϑ w + F 1 G 4,ϑ ˜ f ,
r rnd = F 1 G 2,ϑ 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 nom
k
+ r unc
k
,
Σ k = E ( (r k − ˆr k ) 2) ( (r ) )
rnd 2
= E .
Note that Assumption 1 holds because the variance Σ is not affected by any of the uncertain
signals ũ, w, or ˜ f . However, Assumption 2 only holds if the operator G 2,ϑ does not depend
on the fault parameter ϑ. That is,
]
G ϑ =
[G 1,ϑ G 2 G 3,ϑ G 4,ϑ .
A convenient choice is to take G 2 = I , which corresponds to additive measurement noise
injected between the plant G ϑ and the residual generator F .
k
87