TELKOM
NIKA
, Vol. 11, No. 12, Decem
ber 20
13, pp.
7605
~76
1
0
e-ISSN: 2087
-278X
7605
Re
cei
v
ed
Jun
e
29, 2013; Revi
sed Aug
u
st
17, 2013; Accepted Sept
em
ber 4, 201
3
A Novel Sensor Fault Detection Method
C
h
e
n
Yo
n
gqi*
1
,
Zhao Yiming
2
Coll
eg
e of Scie
nce an
d T
e
chnolo
g
y
, N
i
ng
bo
Univers
i
t
y
, CHI
NA
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: lingfen
77
81
@16
3
.com
1
, zhao
yi
min
g
@n
b
u
.edu.cn
2
A
b
st
r
a
ct
T
o
detect the sensor fau
l
ts o
f
the plant w
i
th mi
n-
distur
ban
ce, a w
e
ighted
least squar
es
interva
l
regressi
on
mo
del is pr
opos
e
d
. T
he output
of the propos
e
d
mo
de
l is an i
n
te
rval b
a
n
d
which ca
n resist th
e
disturb
ance
i
n
fl
uenc
e a
n
d
giv
e
correct s
ens
or fau
l
t al
ar
m.
Additi
ona
lly, th
e ti
me c
o
mp
le
xity of this
mo
del
is low
becaus
e only a set of
linear e
q
u
a
tio
n
s can det
er
mine the p
a
ra
meters. T
he exp
e
ri
ments of fa
ult
instanc
e de
mo
nstrate the fea
s
ibil
ity and effe
ctiveness of th
e interva
l
regre
ssion
mo
del.
Ke
y
w
ords
:
fau
l
t Detection, le
ast squares, int
e
rval re
gressi
o
n
;
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
To dete
c
t the
sen
s
o
r
faults, one ca
n obt
ain
the regre
ssi
on mo
del
of
the plant
without
faults. The se
nso
r
faults ca
n be dete
c
ted
by co
mpa
r
in
g the estimat
ed output
s of the reg
r
e
ssi
o
n
model a
nd th
e observe
d o
u
tputs of the
plant. S
upp
ort vector
re
gre
ssi
on (SV
R
), which owns
high ge
ne
rali
zation p
e
rfo
r
mance, is an
effective
method to con
s
truct this
reg
r
e
ssi
on mo
del [1].
But, when the sy
stem i
s
a plant with
min-di
stur
bance, outputs will locate in
an interval
band.
The re
gressi
on mod
e
l ba
sed o
n
tradit
i
onal SVR o
n
ly pre
s
ent
s cri
s
p outp
u
ts and
ca
n n
o
t
descri
be the
effect of min-di
sturban
ce
in pl
ant wh
ose
output i
s
an inte
rval
band. Fo
r this
rea
s
on,
traditi
onal SV
R i
s
not fit to d
e
te
ct the
se
ns
or
faults of th
e p
l
ant with
min
-
disturban
ce. I
t
is li
kely to
re
gard
the
di
sturba
nc
e a
s
sensor faults a
nd give
a
wro
ng al
arm. In
orde
r to
resol
v
e
this problem,
interval re
gression m
odel
who
s
e
reg
r
e
ssi
on outp
u
ts is an inte
rva
l
band mo
del
must be p
r
op
ose
d
to detect the sensor f
aults of the pl
ant with min-disturban
ce.
For the
s
e years, many interval re
gre
ssi
on mod
e
l
s
are p
r
o
p
o
s
ed. Supp
ort
vector
interval
regression networks (
SVIRNs) i
s
present
ed. Thi
s
mo
del utilizes t
w
o
radial
basi
s
function net
works to identify the upper a
nd lowe
r si
d
e
s
of the data interval [2]. Suppo
rt vector
interval regre
ssi
on ma
chi
n
e (SVIRM) i
s
desi
gne
d
for
cri
s
p in
put an
d output d
a
ta
[3]. SVIRM is
robu
st in the
sen
s
e that outliers do n
o
t a
ffect the resultin
g interval reg
r
e
ssion.
-su
ppo
rt
vector i
n
terv
al re
gre
s
sion
netwo
rks
are pr
opo
se
d to evaluate
i
n
terval lin
ear and
nonlin
e
a
r
reg
r
e
ssi
on m
odel
s for crisp input
and
output dat
a
[4]. Experime
n
tal re
sult
s
manifest th
at
-
sup
port
vect
or i
n
terval
re
gre
ssi
on
net
works i
s
use
f
ul in p
r
a
c
tice, espe
cially
wh
en
noi
se
is
hetero
s
ced
a
stic. However, like SVM with
inequality
co
nstrai
nts, the weig
ht
vectors and the bia
s
term are worked o
u
t by a compli
cate
d
quad
ratic
pro
g
rammi
ng p
r
oblem. Tho
u
gh by modifyi
ng
the solution,
the time com
p
lexity of SVMR
ba
sed o
n
seq
uential
minimal opti
m
ization
(SM
O
)
algorith
m
is
high[5]. Du
e
to this, in
this st
u
d
y, a novel weig
hted lea
s
t squares i
n
terv
al
reg
r
e
ssi
on (WLS_I
R)
i
s
p
r
opo
se
d
by a
pplying
th
e
fu
zzy set prin
ci
ple
to wei
ght vectors
fo
r
th
e
purp
o
se of
e
s
timating th
e
interval
of impre
c
i
s
e o
b
servatio
ns.
Compa
r
ed
wit
h
the exi
s
tin
g
sup
port vecto
r
fuzzy regression mo
del
s, only a se
t of linear e
quatio
ns are nee
de
d to determin
e
the weight vector a
nd bia
s
term of WL
S_
IR. Con
s
e
quently, WLS
_
IR own
s
the
advantage o
f
low time
com
p
lexity. Additionally, the bo
und
s of
the in
terval reg
r
e
ssion mod
e
l are
influenced b
y
outliers in
tra
i
ning
data. In
this
pap
er, t
he
rewei
ghting sc
heme [6] is
intr
od
uc
ed
to
r
e
s
i
s
t
the
influenc
e
of outliers
.
The rest of t
h
is p
ape
r is
orga
nized a
s
the
followin
g
.
least squa
res su
ppo
rt vector
machi
n
e
s
is i
n
trodu
ce
d in se
ction 2. WLS_IR is
stud
ied in Section
3. Section 4 introdu
ce
s th
e
reweightin
g schem
e to
re
sist o
u
tliers i
n
fluen
ce
. Fa
ult instan
ce
Experiment i
s
p
r
e
s
ente
d
in
Section 6. Se
ction 7 put
s forw
ard the co
nclu
ding rem
a
rks.
Evaluation Warning : The document was created with Spire.PDF for Python.
e-ISSN: 2
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NIKA
Vol. 11, No
. 12, Dece
mb
er 201
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7606
2.
Leas
t Squar
es Suppor
t Vector Ma
ch
ines
In this se
ction, least squares
sup
p
o
rt vector
machi
n
e
s
（
LS_SVM
）
is b
r
i
e
fl
y
introdu
ce
d [7]
.
Given traini
ng data
set
i
i
Y
X
,
,
n
i
,
,
1
LS-SVM sup
poses th
e
Hyperpl
ane
s
as the follo
wi
ng:
b
y
i
T
i
)
(
x
(1)
W
h
er
e
i
x
,
i
y
are
input
varia
b
l
e
an
d
output
varia
b
le,
)
(
i
x
is
a no
nline
a
r functio
n
whi
c
h m
a
p
s
the feature
space of i
nput
into
a
high
e
r
dim
e
n
s
ion f
eature
spa
c
e
and
can b
e
rea
c
he
d by t
he
kernel
strategy.
is a
coefficient
det
ermini
ng the
margi
n
of
su
pport ve
cto
r
s
and
b i
s
a
b
i
as te
rm. T
h
e coefficie
n
ts
)
,
(
b
a
r
e
dete
r
m
i
ned
by mini
mizing
the
fo
llowing
regul
ari
z
ed
ri
sk fun
c
tion a
nd usi
ng the
equality co
nst
r
aints.
)
0
(
2
1
2
1
min
1
2
,
c
e
c
J
n
i
i
T
b
(2)
s.t.
i
i
T
i
e
b
x
w
y
)
(
,
n
i
1
Whe
r
e
i
e
is the
error va
riabl
e and
used t
o
co
nst
r
u
c
t a
soft ma
rgin
hyper
plane.
In
Equation
(2),
the first term,
measures th
e inverse
of t
he ma
rgin di
stance. In o
r
d
e
r to obtai
n the
minimum st
ru
ctural
risk, the first term should b
e
mini
mized.
c
is the regula
r
i
z
atio
n para
m
eter
determi
ning t
he fitting erro
r minimizatio
n
and sm
ooth
ness.
Finally, the
deci
s
io
n fun
c
tion of the
cla
ssifie
r
of
LS_SVM can be
expre
s
sed a
s
following:
n
i
i
i
b
a
f
1
)
(
)
(
)
(
x
x
x
(3)
The fun
c
tion
al form of
)
(
i
x
n
eed n
o
t to b
e
kn
own si
nce it is defin
e
d
by the
kern
el
function
),
(
)
(
)
,
(
j
t
i
j
i
K
x
x
x
x
,
1
n
i
n
j
1
. Different kern
el function
s pre
s
ent differen
t
mappin
g
s fro
m
the i
nput
spa
c
e
to the
high
dime
nsion featu
r
e
space. Th
e
commonly
use
d
kernel
s for re
gre
ssi
on p
r
ob
lem are give
n
as follows:
Linea
r ke
rn
el:
y
x
y
x
t
K
)
,
(
Polynomial kernel:
d
t
K
)
1
(
)
,
(
y
x
y
x
RBF ke
rnel:
)
,
(
y
x
K
)
2
exp(
2
2
y
x
D
u
e to
th
e eq
u
a
lity co
ns
tra
i
n
t
s
i
n
the
fo
rmulatio
n, LS
_SVM is solved by
a
set
o
f
linea
r
equalitie
s in
st
ead of
a com
p
licate
d
qu
ad
ratic
pro
g
ram
m
ing p
r
obl
em
. For thi
s
rea
s
on, LS
_SVM
is a
lo
w
comp
utational
com
p
lexity metho
d
. But the
reg
r
essio
n
o
u
tpu
t
of LS_SVM i
n
only
a
cri
s
p
data. Whe
n
Available informatio
n is
uncertain
a
n
d
impre
c
i
s
e,
LS_SVM ca
n not solve this
probl
em. For
this rea
s
o
n
, Weig
hted Le
ast Squa
res i
n
terval Re
gre
ssi
on is p
r
op
ose
d
based the
theory of LS
_SVM and th
e interval
reg
r
essio
n
of
Ta
naka. Thi
s
in
terval re
gre
s
sion
model
i
s
pre
s
ente
d
as
follows:
3.
The De
sign of WLS_I
R
In this sectio
n, a
novel
WLS_IR m
odel
is p
r
o
p
o
s
ed
b
a
se
d o
n
LS_
SVM and
the
interval
reg
r
e
ssi
on th
eory. This m
odel can be
resolved by
a set of line
a
r equ
ation
s
as agai
nst t
h
e
compli
cate
d quad
ratic pro
g
rammi
ng.
th
e
interval
reg
r
essio
n
probl
em is to find
the paramete
r
s
)
,
,
(
b
c
ω
whi
c
h is the
solution of the obje
c
tion fun
c
tion a
s
follo
ws:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
e-ISSN:
2087
-278X
A Novel Se
nsor Fault Dete
ction Metho
d
(Ch
en Yon
gqi
)
7607
)
(
2
)]
(
[
2
1
)
,
(
min
2
2
1
2
2
1
1
2
i
n
i
i
i
i
T
T
v
v
b
c
c
w
w
w
J
(4)
Subjec
t to:
,
,
2
,
1
,
)
1
(
))
(
)(
1
(
)
)
(
(
)
1
(
))
(
)(
1
(
)
(
2
1
n
i
e
H
y
x
c
H
b
x
w
e
H
y
x
c
H
b
x
w
i
i
i
i
i
i
i
i
i
i
(
5
)
Whe
r
e
i
1
and
i
2
are
sla
ck v
a
riabl
es. Thi
s
optimi
z
atio
n pro
b
lem, i
n
clu
d
ing the
con
s
trai
nts, can be solved
by the Lagra
n
ge functio
n
a
s
follows:
]
)
1
(
)
)
(
)(
1
(
)
)
(
(
[
]
)
1
(
)
)
)((
1
(
)
)
(
[(
)
(
2
)]
(
[
2
1
)
,
,
,
,
,
,
(
2
1
2
1
1
1
2
2
1
2
1
2
2
1
2
1
i
i
i
i
n
i
i
i
i
i
i
n
i
i
i
i
i
n
i
i
T
T
i
i
i
i
e
H
y
b
x
c
H
b
x
w
e
H
y
b
x
c
H
b
x
w
b
c
c
w
w
c
b
w
L
(5)
Whe
r
e
i
i
2
1
,
are L
a
g
ran
ge multi
p
liers. Com
p
uting t
he pa
rtial derivative
s
of (5
), one
can d
e
rive:
)
(
)
(
0
2
1
1
i
i
n
i
i
x
w
w
L
)
(
)
1
)(
(
0
2
1
1
i
i
n
i
i
x
H
c
c
L
n
i
i
n
i
i
H
H
b
b
L
1
2
1
1
)
2
(
0
0
)
1
(
)
)(
1
(
]
)
(
[
0
1
1
i
i
i
i
i
i
e
H
y
b
x
c
H
b
x
w
L
0
)
1
(
)
)(
1
(
]
)
(
[
0
2
2
i
i
i
i
i
i
e
H
y
b
x
c
H
b
x
w
L
(6)
As me
ntione
d by Va
pnic [
1
], the ma
p f
unctio
n
does
not ne
ed to
b
e
kno
w
n
sin
c
e it is
defined by
the choice
of kernel
function.
For this re
aso
n
, two kernel fun
c
tions,
)
(
)
(
)
,
(
j
T
i
j
i
x
x
x
x
k
and
)
(
)
(
)
,
(
j
T
i
j
i
x
x
x
x
k
, are used to re
pl
ace
. Lagra
n
ge
multipliers
i
i
2
1
,
and bias te
rm
b
ca
n be obtai
ned
. Then, the u
pper
bou
nd a
nd lower b
o
u
nd of
LS_SVFR are derived a
s
follows:
Evaluation Warning : The document was created with Spire.PDF for Python.
e-ISSN: 2
087-278X
TELKOM
NIKA
Vol. 11, No
. 12, Dece
mb
er 201
3: 760
5 – 7610
7608
]
)
2
[(
2
)
,
(
)
1
)(
(
2
)
2
4
(
)
(
)
,
(
)
(
)
(
)
(
)
(
_
1
2
1
1
2
1
1
1
2
1
1
2
1
2
1
1
n
i
i
n
i
i
i
i
n
i
i
n
i
i
n
i
i
i
i
i
i
n
i
i
H
H
x
x
k
H
H
H
x
x
k
x
c
x
w
x
up
F
)
,
(
)
1
)(
(
)
(
)
,
(
)
(
)
(
)
(
)
(
_
2
1
1
2
1
2
1
1
x
x
k
H
x
x
k
x
c
x
w
x
down
F
i
i
n
i
i
i
i
i
i
n
i
i
(7)
From the
con
d
itions fo
r opt
imality, this r
egre
s
sion
pro
b
lem can be
solved by the
matrix
Equation
(5
): The
choi
ce
of the
weig
hts
i
v
1
and
i
v
2
is
determi
ned
b
a
se
d u
pon
the e
r
ror
variable
s
/
1
1
i
i
,
/
2
2
i
i
.
4.
Re
w
e
i
ghting
Scheme to
Resist Outliers Influence
Rob
u
st estim
a
tes are
o
b
ta
ined by
u
s
ing
the
sam
e
ite
r
atively re
wei
ghting
app
ro
ach
in
referen
c
e [6]. The iterative approa
ch is
summari
ze
d a
s
follows:
1) Set
1
1
i
v
,
1
2
i
v
,
1
i
The prop
ose
d
int
e
rval re
gre
s
si
on is u
s
ed to
obtain the
estimated o
u
tputs. The
n
, the reg
r
e
ssi
on
errors
/
,
/
2
2
1
1
i
i
i
i
are calcul
ated.
2) Re
peat:
3)
])
,
ith
ot the
errors
([
*
483
.
1
2
1
i
i
MAD
s
4) Determi
n
e
the weig
hts
)
,
(
)
(
2
)
(
1
i
k
i
k
v
v
base
d
up
on
,
/
2
,
/
)
(
)
2
)
(
1
)
(
1
s
r
s
r
i
k
Ii
k
i
k
i
k
and
the logisti
c
weights fun
c
tio
n
)
(
1
)
(
1
1
)
tanh((
)
(
i
k
i
k
k
r
r
r
v
,
)
(
2
)
(
2
2
)
tanh((
)
(
i
k
i
k
k
r
r
r
v
.
5) Solve the weig
hted inte
rval reg
r
e
ssi
o
n
model with
weig
ht
k
v
1
and
k
v
2
.
6) Set
1
i
i
7) Until La
grange multipli
ers
)
1
(
2
)
1
(
1
,
i
k
i
k
and
)
(
2
)
(
1
,
i
k
i
k
,
m
k
,
1
,
0
are
sufficientl
y
clo
s
e to ea
ch
other.
5. Experiments
In the first example, We a
pply the unified WL
S_I
R
to the data se
t of crisp inp
u
ts and
interval outpu
ts sho
w
n in T
able 1.
To illustrate the pro
p
o
s
ed
method, the
se
co
n
d
exam
ple [22] are
pre
s
ente
d
. Beca
use
this
func
tion is
not affec
t
ed by
o
u
tliers, t
he
weig
ht pa
rameters
i
v
1
i
v
2
is a
s
sume
d F
r
om
Figure 1,
LS_SVFR performs
fairly well for this
func
tion.
No
w, LS_SVFR is appli
e
d
in sensor fau
l
ts
detection f
o
r the plant with min-distu
r
ban
ce.
Senso
r
is im
portant for th
e plant to achieve it
s opti
m
al perfo
rma
n
ce. All sen
s
or faults mu
st
be
detecte
d accurately and
ra
pidly to preve
n
t
serio
u
s a
c
cide
nts. Co
nsider the pla
n
t:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
e-ISSN:
2087
-278X
A Novel Se
nsor Fault Dete
ction Metho
d
(Ch
en Yon
gqi
)
7609
Table 1. Crisp Inputs an
d Interval Outpu
t
s
No(i)
Crisp input
i
x
Fuzz
y
output
)
,
(
i
i
e
y
1 0.1
(2.25,0.78
5)
2 0.2
(2.875,0.8
75)
3 0.3
(2.5,1)
4 0.4
(4.25,1.75
)
5 0.5
(4,1.5)
6 0.6
(5.25,1.25
)
7 0.7
(7.5,2)
8 0.8
(8.5,1.5
)
)
(
0
1
)
(
1
1
)
(
1
5
.
0
0
1
)
(
)
(
0
1
)
(
2
2
1
0
)
(
t
f
t
d
t
x
t
y
t
u
t
x
t
x
(
8
)
Figure 1. Simulation Result for the Seco
nd Example
Whe
r
e
n
R
x
is th
e
state
of the
plant,
m
R
u
is the
co
ntrol i
nput
,
p
R
y
is the
measurable
output of th
e
plant,
l
R
d
is th
e un
kn
own
min-di
sturban
ce,
l
R
f
is
sensor
fault.
f
and
d
are t
he un
cou
p
led
forms.
The un
kno
w
n
min-di
sturb
a
n
ce i
s
assum
ed as follo
win
g
:
)
rand(
*
1
*
0.2
-
0.1
)
(
t
d
(23)
It denotes no
ise
s
gen
erate
d
in the interval [-
0.1, 0.1] at random.
The se
nsor f
ault is
given as:
s
t
s
t
s
t
s
t
t
f
8
,
4
0
8
4
1
.
0
)
sin(
3
.
0
)
(
(24)
In this experi
m
ent,
s
t
2
.
0
, m=4, n=4. Simulati
on time is chosen a
s
10
second.
TWh
en se
nsor faults o
c
cur, output of the
plant will beyond the interval. Figure 2
-
3 sh
o
w
estimate
s of the interval re
gre
ssi
on an
d output of
pla
n
t with sen
s
o
r
faults. As shown in Figu
re
2-3, be
cau
s
e
there a
r
e sen
s
or fault
s
in the pl
ant bet
ween 4
s
and 8
s
, the outputs of plant also
is
beyond
the i
n
terval bet
wee
n
4
s
a
nd
8s.
LS_SVFR i
s
su
ccessful
i
n
detectin
g
the
sen
s
o
r
fault
s
as ea
rly as p
o
ssible.
Evaluation Warning : The document was created with Spire.PDF for Python.
e-ISSN: 2
087-278X
TELKOM
NIKA
Vol. 11, No
. 12, Dece
mb
er 201
3: 760
5 – 7610
7610
Figure 2. Estimates of Interval Bounds a
nd the
Firs
t Output Parameter of the Plant with
Senso
r
Fault
s
and
Distu
r
b
ance
Figure 3. Estimates of Interval Bounds a
nd
the Secon
d
O
u
tput Parame
ter of the Plant
with Sensor F
aults an
d Di
sturba
nce
6. Conclu
sion
In order to
p
r
eserve
the
advantag
es
of
LS_SVM
and fu
zzy re
gre
ssi
on,
WL
S_IR i
s
pre
s
ente
d
by inco
rpo
r
atin
g the con
c
e
p
t of fuzzy
set theo
ry. By choo
sing
different ke
rnel
function
s, WL
S_IR can d
e
n
o
te differe
nt type nonli
nea
r reg
r
e
ssi
on m
odel to a
dapt
different d
a
ta
sets.
The
ex
perim
ents of
fault in
stan
ce de
mon
s
trat
e the
fea
s
ibil
ity and
effect
iveness
of th
e
interval re
gre
ssi
on mod
e
l.
Ackn
o
w
l
e
dg
ement
This
wo
rk
was
sup
porte
d
by the Twel
fth
Five-Year Plan of Zhe
jiang p
r
ovin
ce key
discipline, S
c
ientific
Re
sea
r
ch Fu
n
d
of
Zheji
ang Provincial Educatio
n De
partm
e
n
t
(Y201
119
567
), The
Zh
ejia
ng Excellent
Yong
Te
ac
h
e
rs in
University and
Coll
ege
s F
undin
g
prog
ram (No.
20101
215
).
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ces
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port
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w
or
k.
Ma
ch
i
n
e
Le
arn
i
ng
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5; 20
(3): 273-2
97.
[2]
J Jen
g
, C C
h
uan
g, Sh
un-F
eng
Su. Su
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ort Vector Int
e
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Re
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e
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o
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essio
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w
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n
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Se
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Evaluation Warning : The document was created with Spire.PDF for Python.