Probabilistic Performance Analysis of Fault Diagnosis Schemes

of the algorithm, and it is a straightforward matter to write a version of Algorithm 4.3 for
any finite number of components. Algorithm 4.2 consists of two parts:
• Lines 1–7 simulate the portion of the conditional mean of the residual due to the
initial condition η 0 and the known input u 0:N . Lines 1–7 also simulate the conditional
variance of the residual, which does not depend on the fault input ∑ j ϕ j (k − κ j ).
• Lines 8–16 simulate the portion of the conditional mean of the residual due to each
component failing at each possible time.
Algorithm 4.3, on the other hand, consists of four parts:
• Lines 2–4 compute the performance metrics P tn,k and P fp,k .
• Lines 5–10 update the performance metrics P fn,k and P tp,k by considering all possible
cases where component 1 fails but component 2 does not.
• Lines 11–16 update the performance metrics P fn,k and P tp,k by considering all possible
cases where component 2 fails but component 1 does not.
• Lines 17–24 update the performance metrics P fn,k and P tp,k by considering all possible
cases where both components fail.
Proposition 4.33. Assume that the probability P(κ j = k) can be computed in O(1) time,
for all j and k. Also, assume that the decision function δ is such that P(D 0,k | θ 0:k = ϑ 0:k )
can be computed in O(1) time for any ϑ 0:N ∈ Θ N+1 and all k ≥ 0. Then, the running time
of Algorithm 4.2 is O(LN 2 ) and the running time of Algorithm 4.3 is O(LN L ). Therefore,
computing the performance metrics requires a total of O ( LN max{2,L}) time.
Proof. First, we show that the running time of Algorithm 4.2 is O(LN 2 ). Since updating the
recurrences in Lines 3–6 takes O(1) time, Lines 2–7 take O(N + 1) time to compute. Similarly,
Lines 12–13 take O(1) time to compute. The number of times that Lines 12–13 must be
executed is
L∑
N∑
j =1 ˆκ j =1
N∑
k=ˆκ j
1 =
=
L∑
N∑
j =1 ˆκ j =1
N − ˆκ j + 1
L∑ N (N + 1)
j =1
2
= O(LN 2 ).
Therefore, Lines 8–16 take O(LN 2 ) to compute, and the total running time of Algorithm 4.2
is O(LN 2 ).
71