Probabilistic Performance Analysis of Fault Diagnosis Schemes

z max
(a)
Up−Down Counter State, z
τ
0
(b)
Residual, r
ε
0
−ε
0 k 1
k 2
Time Step, k
Figure 4.3. Comparison of the behavior of an up-down counter (a) and the behavior of the underlying
threshold decision function (b). The horizontal blue lines indicated the threshold regions, and the
vertical shaded bands indicate the ranges of time where the respective decision function signals that a
fault has occurred. The actual fault starts at time k 1 and stops at time k 2 .
to the point where the number of false alarms is reasonable, the delay of the original threshold
decision function would be even greater. Therefore, in this case, the up-down counter
actually responds more quickly.
Note that for α > 0, the parameters (C d α,C u α,τα, z max α) define an equivalent up-down
counter with state space
Z α := {0,α,2α,..., z max α}.
In the special case where
C d = C u = τ = z max ,
the decisions produced by the up-down counter are identical to those produced by the
original decision function (i.e., dk ˆ = d k , for all k).
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