Probabilistic Performance Analysis of Fault Diagnosis Schemes

for the appropriate choice of k f and ϑ 0:N . The chief difficulty in solving (5.1) and (5.3) is
expressing the minimum (5.4) as a function of ρ, which can then be minimized or maximized
to compute Pf
⋆ or P d ⋆ , respectively. To properly address this difficulty, we must make some
additional assumptions about the sequence {r k (ρ)}. Then, under these assumptions, we
develop a heuristic that allows us to write the minimization (5.4) in a more tractable form.
5.2.1 Simplifying Assumptions
Fix ρ ∈ P (•) , and let ˆr k (ρ,ϑ 0:k ) and Σ k (ρ,ϑ 0:k ) be the mean and variance, respectively, of
the residual r k (ρ) conditional on the event {θ 0:N = ϑ 0:N }. To make the minimization (5.4)
tractable, we make the following assumptions:
Assumption 1. The variance {Σ k } does not depend on the uncertain parameter ρ.
Assumption 2. The variance {Σ k } does not depend on the sequence ϑ 0:N .
Assumption 3. The threshold ε k is chosen in proportion to the variance Σ k . That is, for
some fixed ν > 0, ε k = νΣ k , for all k.
Remark 5.1. The purpose of Assumption 1 is to simplify the relationship between the uncertain
parameter ρ and the function being minimized in (5.4). Similarly, Assumption 3
simplifies the minimization (5.4) by removing the effect of the time-varying threshold {ε k }.
Because the sequence of thresholds {ε k } must be chosen a priori, Assumption 3 is only
possible when Assumptions 1 and 2 hold. An important special case where Assumptions 1
and 2 hold is the case where the noise signal v is added directly to the system output y
before it enters the residual generator F .
Proposition 5.2. Let ρ ∈ P (•) , 0 ≤ k f < N , and ϑ 0:N ∈ Θ N+1 . If Assumptions 1–3 hold, then
argmin
k f ≤k≤N
P ( |r k (ρ)| < ε k | θ 0:k = ϑ 0:k
)
= argmax
k f ≤k≤N
∣ ˆrk (ρ,ϑ 0:k ) ∣ ∣
Σk
.
To facilitate the proof of this proposition, we first establish the following lemma:
Lemma 5.3. Let the function L: [0,∞) × R → [0,1) be defined as
For any ν > 0 and all µ 1 ,µ 2 ∈ R,
∫ ν
)
1 (s − µ)2
L(ν,µ) := exp
(− ds.
−ν 2π 2
|µ 1 | < |µ 2 | ⇐⇒ L(ν,µ 1 ) > L(ν,µ 2 ).
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