Environmental Engineering Reference
In-Depth Information
y m ð kT s Þ¼ y ð kT s Þþ v ð kT s Þ
ð 10 : 19 Þ
where the first equation corresponds to the measurement process with t 2
the
continuous time, x(t) is the physical signal which is measured, c is the sensor gain
and y(t) the sensor output. The second equation models the data acquisition system
operating at the sampling period T s . v ð kT s Þ denotes the measurement noise, sup-
posed to be a normally distributed zero mean white noise sequence with variance
r 2 . kT s will be replaced by k below, for the sake of concision.
An excessive noise is characterized by an increase in the noise variance to cr 2
where c [ 1. Such an increase may occur notably due to a poor contact in an
electric connection. It induces an increase in the variance of y(k). Hence monitoring
this variance is a natural way to detect excessive noise. To determine an empirical
estimate of the variance, an empirical estimate of the mean of y(k) is needed. Yet
this estimation step will be avoided here by exploiting typical properties of x(t).
Indeed, the spectrum of x(t) is normally concentrated in the low frequency range
with regard to measurement noise. Hence, it is reasonable to assume that, within a
time window around time kT s , the physical signal can be described as
R
x ð t Þ¼ a 1 t þ a 0
with
t 2½ð k W = 2 Þ T s ; ð k þ W = 2 Þ T s
where a 0 ; a 1 2
and W is an even integer corresponding to the window size.
Under this hypothesis, the trends within the signal y(t) can be filtered out by
considering its second derivative d 2 y ð t Þ
R
d t 2 . Translating this procedure on the mea-
surement sequence amounts to computing
r ð k Þ¼ y m ð k þ 1 Þ 2y m ð k Þþ y m ð k 1 Þ
T s
ð 10 : 20 Þ
which is an approximation of the second derivative of y(t) at time kT s . Given the
white noise hypothesis, the variance of r(k) is equal to r r ¼ 6r 2 = T s . Notice that
division by T s can be omitted in Eq. 10.20 without affecting the removal of the
trend. This corresponds the approach used in Sect. 10.4.2 .
It is then straightforward to detect the change in the variance of r r by using the
recursive form of the CUSUM algorithm given by Eqs. 10.5 - 10.8 applied to the
following hypothesis test:
Choose between the following two hypotheses:
L ð r ð i ÞÞ ¼ N ð 0 ; 6r 2 = T s Þ
H 0
for i ¼ 1 ; ... ; k
L ð r ð i ÞÞ ¼ N ð 0 ; 6r 2 = T s Þ
for i ¼ 1 ; ... ; k 0 1
H 1
¼ N ð 0 ; 6cr 2 = T s Þ
for i ¼ k 0 ; ... ; k and c [ 1
where k 0 is the unknown fault occurrence time.
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