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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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