Probabilistic Performance Analysis of Fault Diagnosis Schemes

scalar-valued, we can write ˆr ⋆ k
as the solution of the optimization
ˆr ⋆ k = −min {
min
ρ ∈P (•)
− ˆr k (ρ),
min
ρ ∈P (•)
}
ˆr k (ρ) .
This problem is convex if P (•) is a convex set and both ˆr k (ρ) and − ˆr k (ρ) are convex functions
of ρ (i.e., ˆr k (ρ) is affine in ρ). Once optimum values k ⋆ and ρ ⋆ have been obtained, the
worst-case probability of false alarm is given by
P ⋆ f = 1 − P( |r k ⋆(ρ ⋆ )| < ε k ⋆ | θ 0:k ⋆ = 0 0:k ⋆)
.
To summarize, the problem of computing Pf
⋆
and ˆr k is affine in ρ, for all k.
is a convex optimization if P (•) is a convex set
Minimizing the Probability of Detection
Suppose that Assumptions 1–3 hold. Let ϑ be a fault parameter sequence such that ϑ N ≠ 0,
and let k f be the fault time, as defined in equation (5.2). The worst-case probability of
detection is
Pd ⋆ = 1 − max min P( )
|r k (ρ)| < ε k | θ 0:k = ϑ 0:k .
ρ ∈P (•) k f ≤k≤N
By Proposition 5.2, optimum values of ρ and k are obtained by solving
ˆµ ⋆ = min
ρ ∈P (•)
max
k f ≤k≤N
| ˆr k (ρ)|
Σk
.
If we define the vector
and the diagonal matrix
⎡ ⎤
ˆr k f
(ρ)
ˆr
ˆR(ρ) :=
k f +1(ρ)
⎢ ⎥
⎣ . ⎦
ˆr N (ρ)
{
1
W := diag ,
Σ k f
1
Σ k f +1
}
1
,..., , (5.6)
Σ N
then we may write
ˆµ ⋆ = min
ρ ∈P (•)
∥ ∥W 1/2 ˆR(ρ) ∥ ∥
∞
.
Since the matrix W is fixed, taking the ∞-norm is equivalent to taking the weighted pointwise
maximum of ˆr k f
(ρ),..., ˆr N (ρ). Because the pointwise maximum of convex functions is
convex [5], computing P ⋆ d is a convex optimization if P (•) is convex and each ˆr k is a convex
function of ρ, for k = k f ,..., N. Once an optimum value ρ ⋆ has been computed, let k ⋆ be
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