TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5261 ~ 52
6
7
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.483
0
5261
Re
cei
v
ed O
c
t
ober 1
8
, 201
3; Revi
se
d Febru
a
ry 6, 20
14; Accepted
March 2, 201
4
Identification of Nonlinear System Based on Fuzzy
Model with Enhanced Gradient Search
Arbab Nigha
t
Khizer*
1
, Dai Yaping
2
, Amir Mahmood Soomro
3
,
Xu Xiang Yang
4
1,2,
3,4
School of Automatio
n
, Beiji
ng Institute
of
T
e
chnol
og
y,
Beiji
ng 10
00
8
1
, P. R. China
1,3
Mehran Univ
ersit
y
of En
gii
n
eeri
ng an
d T
e
chno
log
y
, Jams
horo, Sin
dh, P
a
kistan
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: arbab
nig
hat
@gmai
l
.com1
A
b
st
r
a
ct
The ide
n
tificati
on an
d
mod
e
li
ng the
o
ry of nonli
n
e
a
r syste
m
s h
a
s alw
a
y
s
been c
hal
le
ngi
ng t
o
researc
hers. Fu
z
z
y
syste
m
due to its l
ang
uag
e descr
ipti
ve w
a
y simi
lar
to hu
man
bra
i
n an
d de
al w
i
t
h
qua
litative
i
n
fo
rmati
o
n
inte
lli
g
ently
prov
es b
e
tter cho
i
ce
for n
onl
in
ear s
ystem mod
e
li
n
g
ov
er
last fe
w
deca
des.
Th
e
f
u
z
z
y
syste
m
t
h
eory itself also
has no
nli
near
character
i
stics therefor
e
w
hen
estab
lish
i
n
g
th
e
fu
z
z
y
model of
nonlinear syst
em
; it s
hou
ld
be able to well
describe the
nonlinear characte
ristics. Takagi-
Sugeno (TS) fu
z
z
y
system
s
are not only s
u
itable for mo
deling t
he nonlinear system
due
to combination of
the g
ood
perfo
rma
n
ce w
i
th th
e si
mp
le l
i
n
ear
express
i
o
n
s, but als
o
us
eful
to des
ign
the
fu
zz
y
c
ontrol
l
e
r
.
T
h
is p
aper
pr
opos
ed
a
new
opti
m
i
z
a
t
io
n
alg
o
rith
m
na
med
as E
nha
nc
ed Gra
d
ie
nt S
earch
(EGS) fo
r
ide
n
tificatio
n
of
nonl
ine
a
r system
base
d
on T
S
fu
z
z
y
syste
m
. In propose
d
EGS, par
ameter
s of me
mb
ersh
ip
functions
are
train
ed
ada
ptiv
ely so
as to
c
a
lcul
ate th
e gr
adi
ent of c
o
st
function
w
h
ich
is n
e
cessary
for
mi
ni
mi
z
i
ng
the
error. Usi
ng
g
r
adi
ent i
n
for
m
a
t
ion of c
o
st fun
c
tion, EGS ap
plies
in
an innovative way s
u
ch
that it ke
eps
a
nd
upd
ates th
e b
e
st
searc
h
results
at ever
y traini
ng
st
ep
duri
n
g
the
opt
imi
z
a
t
io
n
proc
ess
.
T
he a
ppl
icab
ili
ty of EGS for T
S
fu
zz
y
mo
del s
how
s sp
l
end
id
perfor
m
ance
esp
e
cia
l
l
y
in
mode
lin
g
of
nonlinear system
.
Ke
y
w
ords
:
e
n
hanc
ed gr
ad
ie
nt search, p
a
r
a
meter esti
mation,
Gauss
i
an
me
mbers
h
ip fu
nction, n
onl
in
e
a
r
system
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Dynami
cs
m
odelin
g for u
n
kn
own no
nli
near
plant b
e
com
e
s ch
all
enge
duri
n
g
re
cent
years. M
any
method
s hav
e bee
n propo
sed
and im
pl
emented
to cope with
this deman
ding
i
s
sue.
Adaption of accurate
p
a
rameters
fo
r nonlin
ear
mo
del throug
h t
r
ainin
g
i
s
on
e of the
mo
st
comm
on techniques. Am
ong
all, fuzzy modeling
proves as a dominant
modeling tool for
nonlin
ear
system ide
n
tification [1
-4]. Ho
wever,
th
e nonlin
ear
system
s are
mostly com
p
lex
enou
gh to
b
e
ide
n
tified;
therefo
r
e th
e
accu
rate
m
odelin
g h
a
s
to don
e by
estimating
the
para
m
eters o
f
membership
functions a
n
d
fuzzy ru
le
s.
The estimati
on of such param
eters is an
importa
nt task be
cau
s
e th
ese p
a
ra
met
e
rs
sho
w
no
n
linearity in the output of fuzzy mo
del. Two
major step
s are con
s
id
ered
when de
sign
a
fu
zzy
model from I/O (in
put an
d
output)
data
e.g.
stru
cture ide
n
tification
(e
stimation of n
u
mbe
r
of req
u
ired
fuzzy rules
and
me
mbershi
p
fun
c
tion
with
cente
r
s and width
s
) a
nd
pa
ram
e
ter
identificat
io
n
(lea
rnin
g process of the
co
nse
que
nt) [5-6].
The num
ber
of rules in th
e fuzzy sy
ste
m
can
be d
e
t
ermine
d by dividing the i
nput and o
u
tput
data sp
ace into many partitions. Thi
s
ste
p
is kn
ow
n as stru
cture id
e
n
tification. After the sel
e
cti
on
of
st
ru
ct
ure o
f
f
u
zzy
sy
st
e
m
,
the estimation of para
m
eters for b
o
th membe
r
ship functio
n
an
d
rule
s are ne
ed to be d
e
termin
ed un
d
e
r pa
ram
e
ter estimation
step. This i
s
consi
dered a
s
an
essential p
a
rt
for fuzzy mo
deling
sin
c
e the stru
ctu
r
e i
s
no
rmally selecte
d
ba
se
d upon
a pri
o
ri
system kno
w
l
edge an
d accuracy of fuzzy m
odel ma
inly depend
s on both the fuzzy rul
e
s a
n
d
membe
r
ship
function. Parameter e
s
tim
a
tion can b
e
con
s
ide
r
ed
as an optimi
z
ation p
r
obl
e
m
,
whi
c
h is
use
d
for finding t
he prope
r pa
ramete
rs
so
as to redu
ce
the co
st
function that dire
ctly
descri
b
e
s
th
e a
c
cura
cy
of model. S
e
veral
meth
ods are b
e
i
ng u
s
e
d
for estimatin
g
the
para
m
eters
su
ch a
s
g
e
n
e
tic alg
o
rith
ms [7-9], least me
an
sq
uare
(LMS
)
and evol
utio
nary
algorith
m
[10
-
11]. Sepa
rat
ed e
s
timatio
n
stag
es
are
req
u
ire
d
to
find the pa
ra
meters optim
al
values a
nd th
at is the mai
n
dra
w
ba
ck o
f
t
hese meth
ods. Also gu
aranty ab
out the conve
r
g
e
n
ce
of optimizatio
n pro
c
e
ss
co
uld not be giv
en.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5261 – 52
67
5262
This pa
pe
r propo
sed an e
n
han
ced optim
ization al
g
o
rit
h
m usin
g gra
d
ient se
arch i
n
su
ch
a way that it
kee
p
s the be
st trac
ks of m
any sea
r
che
s
and also u
p
d
a
tes the be
st sea
r
ch with fast
conve
r
ge
nce. The incorp
oration
of enhan
cin
g
co
nce
p
t in the
gradi
ent se
arch bo
und
s the
optimizatio
n
pro
c
e
s
s to
keep th
e b
e
st
re
sults a
n
d
at the
sam
e
time ig
no
re the
un
sati
sfied
solutio
n
. At first, TS fuzzy
model i
s
b
u
ilt with the i
n
itia
lly guesse
d st
ructu
r
e
and
p
a
ram
e
ters for a
certai
n n
onlin
ear
syste
m
. After that, EGS is a
pplied
to find o
u
t the
best
paramet
ers for
nonli
n
ear
system b
a
se
d on its inp
u
t-output d
a
ta. The propo
se
d EGS prove
s
better
choi
ce for finding t
he
optimal parameters
of the TS fuz
z
y
models
with better acc
u
rac
y
.
The pap
er is organi
ze
d a
s
follows: Section-2 de
scribed fuzzy modelin
g of nonlinea
r
system
and
the ap
pro
p
ria
t
e co
st fun
c
ti
on u
s
ed
for
identificatio
n. Section
-
3 i
n
trodu
ced
EGS
algorith
m
fo
r tuning
the
unkno
wn
parameter
s
of
nonlin
ear sy
stem. Se
ctio
n-4
sho
w
s the
simulatio
n
re
sults of pa
ra
meter e
s
timat
i
on usi
ng
EG
S. Finally, section-5 di
scu
ssed
con
c
lu
sio
n
of
t
h
is re
sea
r
ch.
2. Fuzz
y
Modeling
2.1. Nonlinear
Sy
stem
Single in
put
singl
e outp
u
t (SISO) i
s
u
s
ed
to
sho
w
the n
online
a
r
dyna
mic sy
stem in
disc
rete time
as
:
,
(1)
Whe
r
e
is th
e cu
rrent o
u
tput,
sh
ows t
he no
nline
a
r
mappin
g
b
e
twee
n outp
u
t
and
input
:
1
2
…..
1
2
…
.
.
Nonli
nea
r system identi
f
ication can
be defined
as to find
out the nonline
a
r
relation
shi
p
betwee
n
the o
u
tput and inp
u
t, repre
s
e
n
ting as:
,
(
2
)
So
that
is close to
and could b
e
co
nsid
ere
d
as
estimation of
. This
phen
omen
on
is sh
own by Figure 1.
Figure 1. Non
linear System
and Model
2.2. TS Fuzzy
Model
The mod
e
l introdu
ce
d by Takagi
-Sug
eno ha
s gain
ed intere
st in
fuzzy model
ing an
d
control a
ppli
c
ations [12-13
]. IF.....THEN rule
s fo
r
re
a
s
oni
ng i
s
th
e
ba
se
structu
r
e
of TS mo
d
e
l
whi
c
h
con
s
ist
s
of ante
c
ed
ents a
s
a fu
zzy set
s
and
con
s
e
que
nts
as a lin
ea
r function
s. Due
to
this arran
gem
ent, a comple
x affine nonlinear
system
coul
d be ap
proximated by TS fuzzy mod
e
l.
Non
lin
ear
System
M
odel
,
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Identification of
Nonlin
ear System
Base
d
on Fu
zzy M
odel with
… (Arbab
Nigh
at Khize
r
)
5263
For TS fuzzy modelin
g, ch
oose the line
a
r
dynami
c
a
u
toreg
r
e
s
sive
moving average with
Exogeneo
us i
nput (ARMAX
) model to
de
scribe the n
o
n
linea
r syste
m
. For
th
rule:
:
If
is
,
and
1
is
,
and ……. a
nd
is
,
and
is
,
and
1
is
,
and …
…
. and
is
,
Then
,
,
1
……..
,
,
,
1
……..
,
or
∑
,
∑
,
Whe
r
e
……,
and
,……,
are p
r
e
s
e
n
t and p
a
st p
l
ant output
s
and
inputs. After center av
e
r
ag
e defuzzification, the estim
a
ted output is:
∑
,…..,
,
,…
,..,
∑
,…
..,
,
,…..,
(3)
Whe
r
e
is unkn
o
wn parameter vect
or, rep
r
e
s
en
ting as
,
,
. The
nonlin
ear
system identifica
t
ion usin
g TS
fuzzy
mod
e
l
coul
d be d
o
n
e
by estimati
ng the un
kn
o
w
n
para
m
eter ve
ctor
through
the optimizati
on of co
st function.
3.
Enhance
d
G
r
adient Se
ar
ch (EGS)
Gene
rally, gradient info
rm
ation of co
st
f
unction i
s
use
d
to cal
c
ulate the nex
t update
dire
ction in g
r
adie
n
t sea
r
ch without u
s
i
ng the
ori
g
in
al function v
a
lue [14
-
15].
The ba
sic i
dea
behin
d
the
g
r
adie
n
t search is to
mov
e
the p
a
ra
m
e
ters in
su
ch directio
n t
hat it sh
ould
be
oppo
site to g
r
adie
n
t (o
r sl
ope)
of the e
rro
r
surfa
c
e. This ensures
that
error sh
ould
al
way
s
be
decrea
s
e
d
when
a n
e
w p
a
ram
e
ter up
d
a
tes
are in
itia
ted. Whil
e in
prop
osed E
n
han
ced
G
r
adi
ent
Search (E
GS
), co
st fun
c
tio
n
and
slo
pe (gradi
ent
)
bot
h are
consi
d
e
r
ed in
order t
o
ke
ep the
b
e
st
sea
r
che
s
du
ri
ng optimization.
3.1. Problem
Formulation
The optimization probl
e
m
starts wit
h
minimizati
on of the cost function
0
,
whe
r
e
∈
⊆
, is a vector of a
d
justable
paramet
ers. Th
erefore to find
∗
∈
s
u
ch that:
∗
1
2
∑
(
4
)
To find a parameter
(
∗
that
sha
p
e
s
a fun
c
tion
,
in such
a way wh
en
,
and
matc
hes
,
then c
o
s
t
func
tion c
an be minimiz
ed as
:
(
5
)
3.2.
EGS
w
i
th TS Fuzzy
Model
The main i
d
e
a
of EGS in TS fuzzy mod
e
ling is that th
e gra
d
ient en
han
ced
sea
r
ch start
s
from initialization, selection,
ca
lculation a
nd upd
ating. In first st
ep, in
itialize the pa
ramete
rs
su
ch
as lea
r
nin
g
rate for gra
d
ie
nt search,
ce
nters a
nd spread
s of Gau
s
sian me
mbe
r
ship fun
c
tion
for
the optimization process.
In the next ste
p
, sele
ct the
vectors
for pa
ramete
rs. Du
ring cal
c
ulati
on,
cri
s
p
outp
u
t of
fuzzy syst
em has to b
e
determi
n
e
d
.
Finally updating the parameter value
s
by
applying
the
EGS and
obt
ained
be
st p
a
r
amete
r
va
lu
es. T
he
overall EGS al
gorithm for
TS fu
zzy
model can be
summa
rized
by Figure 2.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5261 – 52
67
5264
Figure 2. Pro
posed EGS Algorithm for T
S
Fuzzy Mo
d
e
l
4.
Parameter Estimation us
ing EGS
The p
r
op
ose
d
EGS alg
o
rithm is
used
for p
a
ra
met
e
r e
s
timation
of TS fuzzy
model
explained in
section
-
2. To e
s
timate the p
a
ram
e
ter, the
following n
o
n
linear fun
c
tio
n
is used.
cos
1.5
s
i
n
0
.
5
The o
u
tput
is a n
onlin
ear function
of input
.
To f
i
nd f
u
zzy
sy
st
e
m
|
that
approximate
s
the nonline
a
r
functio
n
over a certai
n time interval.
Gau
ssi
an fuzzy set
s
are
u
s
ed
becau
se it
en
sures g
r
eate
s
t possibl
e g
e
nerali
z
atio
n o
f
the sy
stem.
For
Gau
s
sian
fuzzy sets, t
he
level of contri
bution of nonl
inear fun
c
tion
to overall output can b
e
de
termine
d
as:
(
6
)
Whe
r
e
and
are ce
nter an
d width for
membership function (
input). The fuzzy input-
output ch
ara
c
teristic i
s
then
describ
ed by
:
Star
t
In
itialize train
i
n
g
dat
a
In
itialize th
e
′
,
’
and
’
val
u
e
Yes
N
o
Select vectors
for
param
e
ter
estim
a
tion
Ap
pl
y
E
nha
nc
ed
G
r
ad
ie
n
t
s
e
ar
ch
alg
o
rith
m
Ob
tain
ed
best
param
e
ter
l
En
d
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Identification of
Nonlin
ear System
Base
d
on Fu
zzy M
odel with
… (Arbab
Nigh
at Khize
r
)
5265
|
∑
∏
∑∏
(
7
)
Suppo
se th
ere are five rul
e
s for five Gau
ssi
an in
put T
S
fuzzy m
ode
l, therefo
r
e fo
r the
s
e
arrang
ement,
fifteen para
m
eters ne
ed
to be a
d
ju
ste
d
(five inp
u
t
cente
r
s an
d
spread
s
and
five
output
singlet
on). Initially,
para
m
eter va
lues are
cho
s
en
ra
ndo
mly within th
e
ra
nge
of traini
n
g
data. Choose five initial values
ran
d
o
m
ly for cent
ers
(
,
1
….
.
5
in the range
s [0 6] and
spread
s
,
1
…
.
5
from [0 2].
Similarly, initial va
lues are cho
s
en rand
omly for singl
eton
,
1
…
.
.
5
in the range
[0 15]. For EGS algorithm
, the require
d param
eters are sele
cted
as
0
.
0001
,
1
…
3
.
Ran
domly chosen initial
para
m
eter
s v
a
lue for
Gau
ssi
an me
mb
ership
function
s are distrib
u
ted eq
ually in the ra
nge
[0 6] as shown in Figu
re 3 and Tabl
e
1.
Figure 3. Initial Input Mem
bership F
u
n
c
tion
Table 1. Parameters Initial Value
1
2 3 4
5
2.2310
0.7078
13.1423
9.4320
9.7337
0 1.5
3
4.5
6
0.64
0.64 0.64 0.64
0.64
The m
odelin
g re
sult of
a
ce
rtain n
onl
inear fun
c
tio
n
after
apply
i
ng EGS p
a
rameters
estimation
al
gorithm
a
r
e showi
ng throu
gh Figures
4
-
8. From the simulated
results, it is cleared
that the fuzzy
model which
is estimate
d throug
h EGS
can cl
osely match the n
o
nlinea
r functi
on
very well.
Th
e value
of th
e cost fu
nctio
n
in te
rm
s of
measurement
of mo
deling
error is showi
n
g
in Figure 9. The be
st estim
a
ted paramet
ers valu
es
of
membe
r
ship functio
n
are
shown in Tabl
e
2. It is obse
r
ved from the
simulate
d re
sults t
hat
so
me paramete
r
s a
r
e
conve
r
ging to different
values
even
though th
eir i
n
itial sta
r
ting
values
are same. This mean
s
that EGS
optimization
algorith
m
ma
ke
partition
s
of the inp
u
t a
nd o
u
tput do
main a
u
tomat
i
cally a
c
cordi
ng to th
e de
si
red
input a
nd
out
put data.
Esti
mated
param
eters a
r
e
ove
r
lapp
ed i
n
th
e form
of i
n
p
u
t memb
ersh
ip
function
s as sho
w
n
in Fig
u
re
5 mean
s that
pr
opo
se
d optimi
z
atio
n p
r
o
c
e
s
s te
nds to
red
u
ce the
stru
cture
com
p
lexity in a
si
gnifica
nt way
and
doe
s n
o
t
depe
nd u
pon
the inp
u
t an
d
output
spa
c
e
s
( in thi
s
case
whi
c
h a
r
e
divided into five
partition
s). T
he cost fu
ncti
on value i
n
te
rms
of ab
solu
te
error lie
s bet
wee
n
0-0.15
as sho
w
n in
Figure 9
cl
ea
rly depi
cted the excell
ent
perfo
rman
ce
of
prop
osed EG
S.
Table 2. Para
meters Final
Value
n
1 2 3
4
5
9.3198
8.8581
9.9612
-1.5783
13.7114
1.3237
1.6316
4.9389
0.5769
0.2455
0.3852
1.3457
1.2294
1.0126
1.4696
0
1
2
3
4
5
6
0
0.
2
0.
4
0.
6
0.
8
1
I
n
i
t
i
a
l
G
aus
s
i
an
M
e
m
e
b
e
r
s
h
i
p F
unc
t
i
on
x
M
e
m
b
e
r
s
h
i
p
fu
n
c
ti
o
n
1
2
3
4
5
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Vol. 12, No. 7, July 201
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67
5266
Figure 4. Non
linear F
u
n
c
tio
n
and its
Fuzzy Approx
imation
/
∗
Figure 5. Input Membershi
p
after Fun
c
tion
Approximatio
n
Figure 6. Estimates of Inpu
t Membership
F
u
n
c
tion
C
ente
r
Figure 8. Estimates of Inpu
t Membership
Functio
n
Spread
s
Figure 7. Estimate of Outp
ut Membershi
p
F
u
n
c
tion
C
ente
r
s
Figure 9. Erro
r Value
s
between Nonline
a
r
and
Fuzzy Approx
imated Fun
c
ti
on
5. Conclu
sion
This pa
per p
r
esents a
si
mple
and
a
c
curate m
e
tho
d
to i
dentify a
nonlin
ea
r syste
m
throug
h para
m
eter estim
a
tion.
Enha
nced
G
r
a
d
ient
Search (EGS
) whi
c
h
i
s
b
a
si
cally
infe
rred
from gra
d
ient
descent met
hod is p
r
op
osed for pa
ram
e
ter estim
a
tio
n
of TS fuzzy modeling. T
he
prop
osed EG
S uses the
origin
al functi
on value
and
gradie
n
t of co
st function
for updating
it
durin
g o
p
timization
process in
enh
an
cin
g
way, t
heref
ore th
e p
r
op
o
s
ed
identifi
c
a
t
ion metho
d
i
s
a
hybridi
z
ation
of optimization and com
p
lexity r
eduction. The si
mulated resu
lt for param
eter
estimation
of nonline
a
r fu
nction
sho
w
e
d
that the
propo
sed al
go
ri
thm is capa
bl
e to provid
e the
excelle
nt solu
tions.
0
1
2
3
4
5
6
-2
-1
0
1
2
3
4
5
6
7
8
x
f
(
x)
& g
(
x)
g(
x
)
f(
x
)
0
1
2
3
4
5
6
0
0.
2
0.
4
0.
6
0.
8
1
1
2
3
4
5
x
M
e
m
b
e
r
s
h
i
p
fu
nc
t
i
on
0
20
0
40
0
60
0
80
0
1
000
1
200
-4
-2
0
2
4
6
8
c1
c2
c3
c4
c5
T
r
ai
ni
ng t
i
m
e
P
a
r
a
m
e
t
e
r
e
s
ti
m
a
te
(
c
)
0
20
0
400
600
800
100
0
1200
0
0.
5
1
1.
5
2
2.
5
3
3.
5
4
4.
5
1
2
3
4
5
T
r
ai
ni
ng T
i
m
e
P
a
r
a
m
e
ter
es
ti
m
a
tes
(
s
)
0
20
0
40
0
60
0
80
0
10
00
12
00
-4
-2
0
2
4
6
8
10
12
14
16
b1
b2
b3
b4
b5
T
r
ai
n
i
ng
T
i
m
e
P
a
ra
m
e
t
e
r e
s
t
i
m
a
t
e
s
(b
)
0
1
2
3
4
5
6
0
0.
0
5
0.
1
0.
1
5
0.
2
0.
2
5
x
E
rro
r a
b
s
(f
-g
(x
))
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
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ISSN:
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046
Identification of
Nonlin
ear System
Base
d
on Fu
zzy M
odel with
… (Arbab
Nigh
at Khize
r
)
5267
Referen
ces
[1]
Cerrad
a
M, J
Atuilar, E Coli
na, A
T
i
tli. D
y
nam
ic
al memb
ershi
p
function
s: An approac
h for adapti
v
e
fuzz
y
mo
del
lin
g.
F
u
zz
y
Sets and Syste
m
s.
200
5; 513-
533.
[2]
Kosko B. F
u
zzy s
y
stems as u
n
ivers
a
l Appr
o
x
im
ators.
IEEE Trans. Comput.,
1994; 43: 13
29-1
333.
[3]
Pedr
y
cz W, M
Reformat. Ev
olutionar
y
f
u
zz
y
m
o
de
lling. I
EEE T
r
ans. Fuzz
y
S
y
st., 2003; 11(5): 652-
665.
[4]
T
T
a
kagi, M Suge
no. F
u
zz
y id
entific
atio
n
of sy
st
ems a
nd its app
lic
ati
on to mod
e
li
n
g
and co
ntrol.
IEEETrans. Sys. Man and Cy
ber.,
1985; 1
5
(
1
): 116–
13
2.
[5]
M Land
aj
o, MJ Río, R Pére
z.
A note on s
m
ooth ap
pro
x
i
m
ation ca
pa
bil
i
t
ies of fuzz
y
s
y
stems.
IEEE
Trans. Fu
z
z
y
Syst.,
2001; 9(2): 229-23
7.
[6]
Castell
a
n
o
G, MA F
anel
li.
An
appr
oac
h to structure i
dent
ifi
c
ation
of fu
zz
y
mo
de
ls
. Proc. 6th IEEE Int
.
Conf. Fuzz
y
S
yst., Amendola, Italy
. 1997; 1: 531-
536.
[7]
Cord
on O, MJ Del J
e
sus, F
Herre
r
a
. Gen
e
tic le
arni
ng
o
f
fuzz
y
ru
le
b
a
sed c
l
assific
a
tion s
y
stems
coop
eratin
g
w
i
t
h
fuzz
y
re
ason
i
ng metho
d
s.
Int. Jr. Intellige
n
t Syst.,
1998; 13
: 1025-1
0
5
3
.
[8]
Lee
CW
, YC Shin. C
onstructi
on of fuzz
y s
y
s
t
ems
usin
g le
a
s
t-squares m
e
thod
an
d ge
neti
c
alg
o
rithm
.
Elsevier.
2003;
137(3): 29
7-3
23.
[9]
Sanchez L, J Otero. A fast
genetic meth
od
fo
r i
n
du
cti
ng d
e
scr
iptiv
e
fuzz
y
models.
F
u
zz
y
S
e
ts and
System
s.
20
04
; 141(1): 33-4
6
.
[10]
T
Back, U Hammel, HP Sch
w
ef
el. Evol
utio
nar
y comp
utati
on: comme
nts
on the h
i
stor
y and curr
en
t
state.
IEEE Tra
n
s. Evolut. Com
p
ut.,
1997; 1(
1): 3-17.
[11]
T
Hatanaka, Y Ka
w
a
guch
i
, K Uosaki. N
onl
i
near
s
y
stem i
dentific
atio
n b
a
sed
on ev
olu
t
ionar
y fuz
z
y
mode
lin
g.
IEEE Congr. Evol
u
t
ionary C
o
mput
ation.
20
04; 1: 646-
651.
[12] M Sugen
o, K
T
anaka. Successive ide
n
ti1c
a
t
ion of
a fuzz
y
mode
l and its app
licati
on to pred
iction of
a
complex
s
y
stem.
F
u
zz
y
Sets
and Syste
m
s.
199
1; 42: 315-
334.
[13]
T
T
a
kagi, M Suge
no. F
u
zz
y i
dentific
atio
n of
s
y
stem
s
and
its app
licati
on to
mode
lin
g a
nd
control.
IE
EE
transactio
n
on
Man, Cyber
neti
cs.,
1985; 15: 116-
132.
[14]
LD Berkov
itz. Conv
e
x
it
y
and optimiz
ation
i
n
.
John W
ile
y &
Sons, Inc., New
York. 2
002.
[15]
CT
Kelley
.
Iterative
Methods for Op
timization.
SIAM.
1999.
Evaluation Warning : The document was created with Spire.PDF for Python.