Probabilistic Performance Analysis of Fault Diagnosis Schemes

Proposition 4.32. Let N be the final time used in Algorithm 4.1, and let Θ = {0,1,...,m}.
In additions to the assumptions on {θ k }, G θ , F , and δ made above, assume that the fault
input f k (ϑ 0:k ) can be computed in O(1) time, for any k and ϑ 0:k . Then, the total running time
of Algorithm 4.1 is O(N m+1 ).
Proof. Because θ is assumed to be a tractable Markov chain, the for all-loop over possible
sequences ϑ 0:N executes O(N m ) times. Line 4 is a simple look-up and Line 6 is a single
multiplication, so Lines 3–7 take O(1) time to compute. Since f k (ϑ 0:k ) can be computed
in O(1) time, Lines 8–11 can be computed in O(1) time, as well. By assumption, the decision
function δ is such that Line 12 can be computed in O(1) time. Clearly, the remaining
computations (Line 13–19) can also be computed in O(1) time. Since each individual line
takes O(1) time, we conclude that each iteration of the for-loop over k takes O(1) time.
Therefore, the total running time of Algorithm 4.1 is O(N m+1 ).
4.5.2 LTV Special Case Based on Component Failures
In this section we present a special system structure, based on Sections 4.2.2 and 4.3.3, that
permits a more straightforward implementation of Algorithm 4.1. Suppose that the system
consists of L components that fail independently at random, and assume that system is
only affected by additive faults. Hence, the combined dynamics of the system G θ and the
residual generator F are given by
η k+1 = A k η k + B u,k u k + B v,k v k + B f
r k = C k η k + D u,k u k + D v,k v k + D f
L∑ (
ϕ j k − κj (θ 0:k ) ) ,
j =1
L∑ (
ϕ j k − κj (θ 0:k ) ) ,
where κ j (θ 0:k ) is the random time at which the j th component fails. Because θ 0:k only affects
the system via the random failure times, specifying a particular parameter sequence ϑ 0:N is
equivalent to specifying the corresponding failure times ˆκ j := κ j (ϑ 0:N ), for j = 1,2,...,L.
Another important feature of this special case is the additive structure of the fault input.
Since each ϕ j enters additively, the portion of the residual due to each ϕ j can be computed
separately and then combined using the principle of superposition. Similarly, the portion
of the residual due to the initial condition η 0 and the known input u 0:N can be computed
separately. Because ϕ j has no effect until the j th component fails (i.e., ϕ j (k − ˆκ j ) = 0,
for k < ˆκ j ), we only need to compute the portion of the residual due to ϕ j for k ≥ ˆκ j .
The procedure for computing the performance metrics for this special case is split
into two parts: Algorithm 4.2 computes each portion of the residual, while Algorithm 4.3
computes the performance metrics. Although Algorithm 4.2 applies to any system of L
components, Algorithm 4.3 focuses on the case L = 2. This greatly simplifies the presentation
j =1
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