Environmental Engineering Reference
In-Depth Information
A recursive algorithm that performs this choice is easily obtained by noticing
that the likelihood ratio of r ð k Þ associated to the above problem is the inverse of
the likelihood ratio for the initial problem. In terms of log-likelihood ratio, this
corresponds to a sign change. Hence, the following strategy is proposed to detect
possible fault disappearance. From time instant t a þ 1, run the following recursive
CUSUM algorithm
g dis ð k Þ¼ max ð 0 ; g dis ð k 1 Þ s ð k ÞÞ
g dis ð t a þ 1 Þ¼ 0
ð 10 : 14 Þ
d ð k Þ¼ d ð k 1 Þ 1 f g ð k 1 Þ [ 0 g þ 1
d ð t a þ 1 Þ¼ 0
ð 10 : 15 Þ
t a ; dis ¼ min f k : g dis ð k Þ [ h a g
ð 10 : 16 Þ
k 1 ¼ d ð t a ; dis Þ
ð 10 : 17 Þ
On the other hand, when the detection/isolation algorithm given by Eqs. 10.11 -
10.13 is used, the detection of fault disappearance corresponds to a new hypothesis
test, which cannot be translated into a simple transformation of this algorithm, like
a sign change. Therefore, the easiest way to proceed is to re-initialize all the n f
decision functions computed from Eq. 10.11 to zero each time a threshold is
crossed in Eq. 10.13 , and to keep issuing an alarm as long as periodic crossing of
the threshold is observed.
Let us now consider the application of the above tools to handle sensor fault
detection first.
10.3 Individual Signal Monitoring
Usual alarm systems check whether the signal issued by a sensor lies within the
measurement range. Yet somewhat more involved verifications can be made on a
single signal before resorting to hardware or analytical redundancy to look for
small magnitude faults. We will illustrate this claim by using a recursive CUSUM
algorithm for variance change detection in order to detect the appearance of an
excessive measurement noise on a signal. The same type of algorithm can also be
used to detect a flat signal.
10.3.1 Excessive Noise
Let us assume that the measured signal can be modeled as:
y ð t Þ¼ cx ð t Þ
ð 10 : 18 Þ
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