Probabilistic Performance Analysis of Fault Diagnosis Schemes

Hence, it is straightforward to show that
P ( (∣ ∣
) ∣∣∣ r k (ρ) ∣∣∣
|r k (ρ)| < ε k | θ 0:k = ϑ 0:k = P <
ε ∣ )
k ∣∣
θ0:k = ϑ 0:k
Σk Σk
= P ( |µ k (ρ)| < ν | θ 0:k = ϑ 0:k
)
= L ( ν, ˆµ k (ρ,ϑ 0:k ) ) .
Let k 1 ,k 2 ∈ N be any two time points in the interval [k f , N ]. By Lemma 5.3,
P ( |r k1 (ρ)| < ε k1 | θ 0:k1 = ϑ 0:k1
)
< P
(
|rk2 (ρ)| < ε k2 | θ 0:k2 = ϑ 0:k2
)
if and only if
∣ ˆµk1 (ρ,ϑ 0:k1 ) ∣ ∣ >
∣ ∣ ˆµk2 (ρ,ϑ 0:k2 ) ∣ ∣ .
5.2.2 Simplified Worst-case Optimization Problems
The section demonstrates how Assumptions 1–3 and Proposition 5.2 are applied to the
problems of computing P ⋆ f and P ⋆ d .
Maximizing the Probability of False Alarm
Suppose that Assumptions 1–3 hold and assume that no faults have occurred (i.e., ϑ = 0).
The worst-case probability of false alarm is
Pf ⋆ = 1 − min min P( )
|r k (ρ)| < ε k | θ 0:k = 0 0:k
ρ ∈P (•) 0≤k≤N
By Proposition 5.2, optimum values of ρ and k are obtained by solving
ˆµ ⋆ = max
ρ ∈P (•)
| ˆr k (ρ)|
max = max
0≤k≤N Σk 0≤k≤N
max | ˆr k (ρ)|
.
ρ ∈P (•) Σk
Because Σ k does not depend on ρ, this optimization may be solved in two separate stages.
First, for k = 0,1,..., N , solve the optimization
and then compute
ˆr ⋆ k = max
ρ P (•)
| ˆr k (ρ)|, (5.5)
ˆµ ⋆ = max
0≤k≤N
ˆr ⋆ k
Σk
.
At this point, we must consider what additional assumptions are needed to ensure
that the optimization (5.5) can be written as a convex program. Because the residual is
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