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where the sample vector for normal operating conditions is denoted by X
, f rep-
resents the magnitude of the fault and
N
is a fault direction vector. Necessary and
suf
cient conditions for detectability are:
N ¼
ð
0, with N
PP
T
I
Þ
N
6
¼
the projection of
N
on the residual subspace;
¼
ð
[
f
, with f the projection of f on the residual subspace.
PP
T
I
Þ
f
2
d
The drawbacks of SPE index for fault detection are mainly two: the first is
related to the assumption of normal distribution to estimate the threshold of this
index, the second is that the SPE is a weighted sum, with unitary coef
cients, of
quadratic residues X
i
. To improve the fault detection, these two drawbacks are
faced assuming that the process is faultless if, for each i:
X
i
d
i
i
¼
1
; ...;
m
;
ð
16
Þ
dence limit for X
i
. To estimate the con
where
d
i
, even if the
normality assumption of X
i
is not valid, the solution is to estimate the PDF directly
from X
i
through a non parametric approach. In Yu (
2011a
,
b
) and Odiowei and Cao
(
2010
), KDE is considered because it is a well established non parametric approach
to estimate the PDF of statistical signals and evaluate the control limits. Assume y is
a random variable and its density function is denoted by p
d
i
is a con
dence limit
ð
y
Þ
. This means that:
Z
k
P
ð
y
k
Þ
¼
p
ð
y
Þ
dy
:
ð
17
Þ
\
1
ð
Þ
Hence, by knowing p
y
, an appropriate control limit can be given for a speci
c
con
dence bound
a
, using Eq. (
17
). Replacing p
ð
y
Þ
, in Eq. (
17
), with the estimation
of the probability density function of X
i
, called
ð
X
i
Þ
^
p
, the control limits will be
estimated by:
Z
d
i
ð
X
i
Þ
d X
i
¼
a:
ð
18
Þ
^
p
1
Fault isolation and diagnosis are performed by the PCA contributions: de
ning
m
, the total contribution of the ith process vari-
the new observation vector x
j
2 R
able Xi
i
is
X
N
j¼1
x
ij
CONT
i
¼
i ¼ 1
; ...;
m
:
ð
19
Þ
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