Probabilistic Performance Analysis of Fault Diagnosis Schemes

β
∆
α
v
f (θ)
u
G θ
y
.
F
r
Figure 5.2. Uncertain fault diagnosis problem with model uncertainty. The uncertain operator ∆
is constrained to lie in some bounded, convex uncertainty set. For simplicity, we assume that the
signals u and f (θ) are known.
well-posed (i.e., the inverse of I −G 11,ϑ ∆ exists for all ϑ ∈ Θ and all admissible ∆), then
α = (I −G 11,θ ∆) −1( G 12,θ v +G 13,θ f (θ) +G 14,θ u ) ,
which implies that
T v→y = G 21,θ ∆(I −G 11,θ ∆) −1 G 12,θ +G 22,θ .
Therefore, Assumptions 1 and 2 hold if the noise v does not pass through the uncertain
operator ∆ (i.e., G 12,θ = 0), and the map G 22,θ does not depend on the parameter θ. That is,
G θ =
[
]
G 11,θ 0 G 13,θ G 14,θ
.
G 21,θ G 22 G 23,θ G 24,θ
The important special case G 22 = I corresponds to additive measurement noise.
Fix a parameter sequence ϑ and an input u. Assuming that G 12,θ = 0 and θ = ϑ, the
signals α and β are given by the equations
α = (I −G 11,θ ∆) −1( G 13,θ f (ϑ) +G 14,θ u ) ,
β = ∆α = ∆ ( G 13,θ f (ϑ) +G 14,θ u ) . (5.7)
Since the signals f (ϑ) and u are known and ∆ is constrained to be a member of the set P ∆ ,
these equations can be interpreted as a constraint on the signal β. Hence, our approach
to computing the worst-case performance is to compute the worst-case β, such that equation
(5.7) is satisfied by some ∆ ∈ P ∆ . The theoretical results that yield such constraints on
β can be found in the literature on interpolation theory and model invalidation.
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