Probabilistic Performance Analysis of Fault Diagnosis Schemes

where 0 0:k denotes the sequence of k +1 zeros. Clearly, uncertainty has a negative impact on
performance when the probability of false alarm increases. Hence, a worst-case parameter
ρ ⋆ ∈ P (•) , with respect to the probability of a false alarm, is defined as an optimum point of
the following optimization problem:
P ⋆ f = max
ρ ∈P (•)
where N ≥ 0 is a fixed final time.
max
0≤k≤N
P f,k(ρ)
= 1 − min min P( ) (5.1)
|r k (ρ)| < ε k | θ 0:k = 0 0:k ,
ρ ∈P (•) 0≤k≤N
Minimizing the Probability of Detection
We analyze the effect of uncertainty conditional on the occurrence of particular fault. Fix a
final time N , and let ϑ 0:N ∈ Θ N+1 be a possible fault parameter sequence, such that ϑ N ≠ 0.
Define
k f := min{k ≥ 0 : ϑ k ≠ 0}. (5.2)
That is, the fault represented by the sequence ϑ 0:N occurs at time k f . For any ρ ∈ P (•) , the
probability of detecting the fault at time k is
P d,k (ρ,ϑ 0:N ) = P ( |r k (ρ)| ≥ ε k | θ 0:k = ϑ 0:k
)
= 1 − P ( |r k (ρ)| ≤ ε k | θ 0:k = ϑ 0:k
)
With respect to the probability of detecting the fault parameterized by ϑ 0:N , a worst-case
parameter ρ ⋆ ∈ P (•) is defined as an optimum point of the following optimization problem:
Pd ⋆ (ϑ 0:N ) = min max P d,k(ρ,ϑ 0:N )
ρ ∈P (•) k f ≤k≤N
= 1 − max min P( ) (5.3)
|r k (ρ)| < ε k | θ 0:k = ϑ 0:k .
ρ ∈P (•) k f ≤k≤N
In other words, a worst-case parameter ρ ⋆ ∈ P (•) diminishes the effect of the fault parameterized
by ϑ 0:N as much as or more than any other parameter ρ ∈ P (•) .
5.2 Formulating Tractable Optimization Problems
Both optimization problems (5.1) and (5.3) involve the expression
min P( )
|r k (ρ)| < ε k | θ 0:k = ϑ 0:k , (5.4)
k f ≤k≤N
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