Probabilistic Performance Analysis of Fault Diagnosis Schemes

Algorithm 4.1. General procedure for computing the performance metrics, where the decision function
δ is a time-varying threshold.
Require: A final time N ∈ N, a Gaussian initial state η 0 ∼ N (ˆη 0 ,Λ 0 ), a sequence of thresholds
{ε k } such that ε i > 0, and a fault model θ ∼ ( Θ,{Π k },π 0
)
.
1 for all ϑ 0:N ∈ Θ N+1 with nonzero probability do
2 for k = 0,1,..., N do
3 if k = 0 then
4 P(θ 0 = ϑ 0 ) = π 0 (ϑ 0 )
5 else
6 P(θ 0:k = ϑ 0:k ) = Π k−1 (ϑ k−1 ,ϑ k ) P(θ 0:k−1 = ϑ 0:k−1 )
7 end if
8 ˆη k+1 = A k (ϑ k )ˆη k + B u,k (ϑ k )u k + B f f k (ϑ 0:k )
9 ˆr k = C k (ϑ k )ˆη k + D u,k (ϑ k )u k + D f f k (ϑ 0:k )
10 Λ k+1 = A k (ϑ k )Λ k A T k (ϑ k) + B v,k (ϑ k )B T v,k (ϑ k)
11 Σ k = C k (ϑ k )Λ k C T k (ϑ k) + D v,k (ϑ k )D T v,k (ϑ k)
12 Compute P ( )
D 0,k | θ 0:k = ϑ 0:k
13 if ϑ k ∈ Θ 0 then
14 P tn,k = P tn,k + P ( ) ( )
D 0,k | θ 0:k = ϑ 0:k P θ0:k = ϑ 0:k
15 P fp,k = P fp,k +
(1 − P ( ) )
D 0,k | θ 0:k = ϑ 0:k P ( )
θ 0:k = ϑ 0:k
16 else
17 P fn,k = P fn,k + P ( ) ( )
D 0,k | θ 0:k = ϑ 0:k P θ0:k = ϑ 0:k
18 P tp,k = P tp,k +
(1 − P ( ) )
D 0,k | θ 0:k = ϑ 0:k P ( )
θ 0:k = ϑ 0:k
19 end if
20 end for
21 end for
• Line 12 computes the conditional probability P(D 0,k | θ 0:k = ϑ 0:k ), and then Lines 13–19
use this probability to update the performance metrics. Note that Line 18 is technically
superfluous, because the performance metrics must sum to one.
Remark 4.31. While most of the computation is straightforward, Line 1 is the most difficult
portion of this algorithm, as it requires all possible parameter sequences to be generated.
One option is to generate and store all the sequences in an array. However, this size of
such an array would be prohibitively large. Another option is to dynamically generate the
sequences while bookkeeping which sequences have already been considered. This is the
approach taken with the special cases in Sections 4.5.2 and 4.5.3. However, we have not yet
discovered a practical implementation for this portion of the algorithm.
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