Probabilistic Performance Analysis of Fault Diagnosis Schemes

Assumed Structure of the Residual Generator
Given the Gaussian assumptions on x 0 and {v k } and clg structure of the model G θ , the
conditional density p y|θ (y k | θ 0:k ) is Gaussian, for all k. To ensure that p r |θ (r k | θ 0:k ) is also
Gaussian, assume that the residual generator F is a linear time-varying (ltv) system of the
form
⎧
⎨ ξ k+1 = Ã k ξ k + ˜B u,k u k + ˜B y,k y k ,
F
⎩ r k = ˜C k ξ k + ˜D u,k u k + ˜D y,k y k .
(4.13)
Note that this system is unaffected by changes in the parameter {θ k }, except through the
measured output {y k }.
Combined Dynamics
Assuming that G θ is clg and F is linear, the interconnection of the two systems is a single
clg system that takes {u k }, {v k }, and {f k } as its inputs and outputs the residual {r k }. For
each k, let η k := (x k ,ξ k ) be the combined state of the system. The combined dynamics can
be written as
η k+1 = A k (θ k )η k + B u,k (θ k )u k + B v,k (θ k )v k + B f f k (θ 0:k ), (4.14)
r k = C k (θ k )η k + D u,k (θ k )u k + D v,k (θ k )v k + D f f k (θ 0:k ), (4.15)
where
A k (θ k ) :=
B u,k (θ k ) :=
[ ]
Âk (θ k ) 0
,
˜B y,k Ĉ k (θ k ) Ã k
[
]
[
ˆB u,k (θ k )
, B v,k (θ k ) :=
˜B u,k + ˜B y,k ˆD u,k (θ k )
[ ]
C k (θ k ) := ˜D y,k Ĉ k (θ k ) ˜C k ,
] [
ˆB v,k (θ k )
, B f :=
˜B y,k ˆD v,k (θ k )
]
ˆB f
,
˜B y,k ˆD f
D u,k (θ k ) := ˜D u,k + ˜D y,k ˆD u,k (θ k ), D v,k (θ k ) := ˜D y,k ˆD v,k (θ k ), D f := ˜D y,k ˆD f .
At this point, some remarks about the initial condition of F are in order. Intuitively, the
expected value of the residual at time k = 0 should be zero. Hence, assuming that θ 0 = 0
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