Probabilistic Performance Analysis of Fault Diagnosis Schemes

Second, we show that the running time of the L-component version of Algorithm 4.3
is O(LN L ). For i = 0,1,...,L, we must consider all cases in which i components fail at or
before time N . There are ( L) i ways to choose which i components fail, and each component
can fail at any time κ ∈ {1,2,..., N }. By the binomial theorem [40], the total number of cases
to consider is
( )
L∑ L
N i = (1 + N ) L = O(N L ).
i
i=0
In Algorithm 4.3, Lines 2-4, 6–9, 12–15, and 19–22 are essentially identical. In general, these
four lines must be executed for each possible case. By assumption, the probabilities of the
form
P(D 0,k | κ j = s j , j = 1,...L),
as well as the component failure probabilities P(κ j = s j ) and P(κ j > k), can be evaluated
in O(1) time. Since we must compute L such component failure probabilities in each
possible case, the running time of Algorithm 4.3 is O(LN L ). Therefore, the total time required
to compute the performance metrics is O(LN 2 ) +O(LN L ) = O ( LN max{2,L}) .
Remark 4.34. At first glance, the combined running time of Algorithms 4.2 and 4.3, seems little
better than the polynomial running time of the general procedure given in Algorithm 4.1.
However, as shown in Section 4.2.2, a system with L components leads to a Markov chain
with state space Θ = {0,1,...,2 L − 1}. Therefore, the running time of Algorithm 4.1 would be
O ( N 2L −1 ) , which is significantly worse than O(LN L ) for practical values of L and N .
4.5.3 LTI Special Case Based on Component Failures
The special case considered in the previous section can be simplified further by assuming
that the dynamics are time-invariant. That is, we assume the combined dynamics are of the
form
η k+1 = Aη k + B u u k + B v v k + B f
r k = Cη k + D u u k + D v v k + D f
L∑ (
ϕ j k − κj (θ 0:k ) ) ,
j =1
L∑ (
ϕ j k − κj (θ 0:k ) ) ,
As in the ltv case, superposition is used to reduce the amount of computation required.
However, because the system is now lti, the portion of the conditional mean of the residual
due to component j failing at time κ j can be obtained by time-shifting the portion due to
component j failing at time 1. For all n ∈ N, let the n-shift operator z n be defined by
j =1
z n : x 0:N → { }
0,...,0, x
} {{ } 0 , x 1 ,..., x N−n ,
n zeros
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