Probabilistic Performance Analysis of Fault Diagnosis Schemes

probability matrix is given by
(
Λk (ϑ 0:k ) ) i j := P(z k = j | z k−1 = i ,θ 0:k = ϑ 0:k )
⎧
⎪⎨
P(d k = 0 | θ 0:k = ϑ 0:k ), if j = max{0,i −C d },
= P(d k = 1 | θ 0:k = ϑ 0:k ), if j = min{z max ,i +C u },
⎪⎩
0, otherwise,
for all i , j ∈ Z. The conditional distribution of z k , defined as
(
λk (ϑ 0:k ) ) i = P(z k = i | θ 0:k = ϑ 0:k ), i ∈ Z,
is computed via the equation
λ T k (ϑ 0:k) = λ T −1 Λ 0(ϑ 0 ) Λ 1 (ϑ 0:1 )···Λ k (ϑ 0:k ).
The probability that the up-down counter exceeds the threshold τ is
P( d ˆ
z∑
max
k = 1 | θ 0:k = ϑ 0:k ) =
i=τ
P(z k = i | θ 0:k = ϑ 0:k ) =
z∑
max
i=τ
(
λk (ϑ 0:k ) ) i .
Suppose that, for some k 1 , the underlying decision function δ decides that a fault has
occurred in such a way that d l = 1, for l ≥ k 1 . If z k1 = 0, then the decision sequence { d ˆ k }
will remain at 0 until ⌈τ/C u ⌉ time steps have passed. That is, the up-down counter has an
inherent detection delay, specified by the ratio τ/C u . Of course, this delay provides a degree
of robustness when the underlying decision function is prone to false alarms. When a
false alarm does occur, ⌈C u/C d ⌉ time steps with no further false alarms must pass before the
counter state {z k } falls below its original value. Hence, the ratio C u/C d specifies how long it
takes for a spurious up-count to be “forgotten.”
Similarly, suppose that for some k 2 , the effect of a fault subsides and d l = 0, for all
l ≥ k 2 . If z k2 happens to be at z max , then the decision sequence { d ˆ k } will not return to 0 until
⌈(z max −τ)/C d ⌉ time steps have elapsed. As in the previous scenario, the up-down counter has
an inherent delay, specified by the ratio (z max −τ)/C d . This particular delay provides a degree of
robustness against missed detections.
Although the up-down counter seems to have inherent delays in these idealized scenarios,
the robustness provided by the up-down counter can actually lead to a more responsive
fault detection scheme. Figures 4.3(a) and 4.3(b) show the realizations of the counter state
{z k } and the residual {r k }, respectively, for a typical up-down counter based on a ε-threshold
decision function. In this particular simulation, a fault occurs at time k 1 and subsides at
time k 2 . The delay in the up-down counter can clearly be seen in Figure 4.3(a). However, the
original decision function has a large number of false alarms. If the threshold ε is increased
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