TELKOM
NIKA
, Vol.14, No
.1, March 2
0
1
6
, pp. 171~1
8
0
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v14i1.2663
171
Re
cei
v
ed Se
ptem
ber 9, 2015; Re
vi
sed
Jan
uar
y 9, 20
16; Accepted
Jan
uary 21, 2
016
Volterra Series identification Based on
State Transition
Algorithm with Orthogonal Transformation
Cong Wang*
1
, Hong-Li Z
h
ang
1
, Wen
-
hui Fan
2
1
Colle
ge of Ele
c
trical Eng
i
ne
e
r
ing,
Xin
jia
ng U
n
iversit
y
Urum
qi Xin
jia
ng
，
83
004
7, Chi
n
a
2
Departme
n
t of Automation, T
s
ing
hua U
n
iv
er
sit
y
, Beij
in
g 10
008
4, Chi
n
a
*Corres
p
o
ndi
n
g
author, em
ail
:
64108
73
85@
qq.com
A
b
st
r
a
ct
A Volterr
a
ke
rnel
id
entificati
on
met
hod
b
a
sed
on
state
transitio
n
alg
o
rith
m w
i
th or
thogo
na
l
transformatio
n
(calle
d OT
ST
A) w
a
s propose
d
to solve th
e
hard pr
ob
le
m i
n
ide
n
tifyin
g V
o
lterra ker
n
e
l
s of
non
lin
ear syste
m
s. Firstly, the popu
lati
on w
i
th chaotic s
equ
ences w
a
s in
iti
a
li
z
e
d
by us
ing
chaotic strate
gy.
T
hen the ortho
gon
al transfor
m
ati
on w
a
s used to finish
th
e mutati
on o
p
e
rator of the selecte
d
ind
i
vid
ual
.
OT
ST
A w
a
s used
on th
e i
d
e
n
tificatio
n
of V
o
lterra s
e
ri
es, and
co
mp
are
d
w
i
th particl
e s
w
arm o
p
ti
mi
z
a
t
i
o
n
(calle
d PSO) and state transit
ion a
l
gor
ith
m
(ST
A
). T
he simulati
on resu
lts show
ed that OT
ST
A has bette
r
ide
n
tificatio
n
pr
ecisio
n an
d co
nverg
ence th
a
n
PSO
and ST
A under n
on-n
o
ise int
e
rferen
ce. And w
hen there
is no
ise, the
id
entificati
on
pre
c
ision, c
onv
erg
ence
an
d a
n
ti-i
nterfer
enc
e of
OT
ST
A are als
o
sup
e
ri
or to P
S
O
and ST
A.
Ke
y
w
ords
:
State T
r
ansl
a
ti
on Al
gorit
h
m
; Orthogon
al T
r
ansfor
m
ati
on;
Nonl
in
ear Syst
em; V
o
lterra
S
e
ries;
System
Identification
Copy
right
©
2016 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
More a
nd mo
re hig
h
co
upl
ing nonli
nea
r system a
p
p
aere
d
for the
developm
ent
of high
techn
o
logy,
how to d
e
scribe the
s
e
m
odel
s h
a
s be
en b
e
come
a rese
arch
h
o
tspot.
With
the
developm
ent of
the nonli
n
ear
theo
ry, Volterra se
rie
s
has bee
n wi
dely applied i
n
the modeli
n
g
and faults di
agno
si
s of nonline
a
r sy
stem [1]. Volte
rra fun
c
tion
al
seri
es
con
s
i
ders the dyn
a
mic
cha
r
a
c
ters of
the sy
stem,
and it
s kern
el ha
s a
disti
n
ct phy
sical
meanin
g
. So
the serie
s
can
approximate arbitrary precision
c
ontinu
ous fun
c
tion
on the set, and de
scribe
s
the categ
o
rie
s
of
nonlin
ear p
h
e
nomen
on [2].
The key of
bu
ilding nonlin
e
a
r system
by Volterra
seri
e
s
m
odel
is to
identify the
structu
r
e
and p
a
ra
met
e
r of the
mo
del [3]. So, there
is
an u
r
gent
dema
n
d
for effe
ctive identificatio
n
method
s.
The tra
d
ition
a
l identificati
on metho
d
s on Vo
lterra
seri
es
gen
erally ad
opt
the least
squ
a
re
s alg
o
rithm [4], but the lea
s
t squ
a
r
es’ i
dentific
a
t
ion efficien
cy is relatively low an
d ea
sy to
fall into lo
cal
minimum. In
re
cent ye
ars, in
telligent o
p
timization
m
e
thod
s ha
s
b
een int
r
od
uced
into the
ke
rn
el identificatio
n on
the Volt
erra
seri
es problem
s, like
geneti
c
alg
o
ri
thm [5], adap
tive
ant colony
algorithm
[6], quantum
pa
rt
icle
swarm
o
p
timization
[7
-8], cro
s
s-correlation
meth
od
[9], etc. Those algo
rithm
s
can
overcom
e
the draw
ba
cks of the tra
d
itional ide
n
tification meth
ods,
su
ch a
s
the
requireme
nt o
n
the continu
ous
diffe
renti
able o
b
jectiv
e functio
n
, a
nd the
sen
s
iti
v
ity
the measurement noi
se.
However, they still have th
eir own limitations on solving the problem of
optimizatio
n [10]. Con
s
eq
u
ently, none o
f
these algo
ri
thms can a
c
curately solve
the probl
em
of
Volterra se
rie
s
identificatio
n.
In orde
r to overcome th
e sho
r
tco
m
in
gs of
Volterra seri
es id
e
n
tification, this pap
er
prop
osed the
Orthog
onal
Tran
sfo
r
mati
on State Tr
a
n
sition Alg
o
rit
h
m (OTSTA
). OTSTA is a
new
intelligen
ce al
gorithm. And
it is easy to underst
an
d, due to the less num
bers of the paramete
r
s
and the simpl
e
algorithm
structu
r
e. Firstl
y, in
the initialization ph
ase
chaotic
seq
u
ence wa
s used
to initialize th
e pop
ulation.
Then the
ort
hogo
nal tra
n
s
form
ation m
e
ch
ani
sm wa
s intro
d
u
c
ed
to
mutate
some
individu
al
with poo
r fitne
s
s in
the
process of th
e
se
arch to
in
cre
a
se
the
diversity
and
give mo
re op
portu
nity to jump
out
of local optim
um. Finally, T
he n
e
w meth
od i
s
comp
ared
with traditio
n
a
l state tran
sition algorith
m
and PSO
method throu
gh sim
u
lation
verification.
The
results sho
w
that OTSTA a global opti
m
ization al
go
rithm with st
rong ro
bu
stne
ss. It can resolve
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ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 14, No. 1, March 2
016 : 171 – 1
8
0
172
the co
nflict b
e
twee
n conv
erge
nce spee
d and
glob
al
sea
r
ch capa
b
ility efficiently and
so fa
cilitate
diversity withi
n
the popul
ation, improvin
g
the global se
arch ability of the algorithm
.
2. Volterra Series
The
singl
e in
put an
d o
u
tp
ut nonli
nea
r
system
can b
e
exp
r
esse
d by
Volterra serie
s
[11]
as
the follows:
1
1
10
0
1
y,
,
k
nk
k
m
ki
i
m
hi
i
u
n
i
LL
(1)
Her
e
,
un
and
yn
are
the inp
u
t an
d output
of th
e sy
stem respectively,
1
,,
kk
hi
i
L
is
the
k
th-o
rd
er tim
e
domain
ke
rnel of the system [7].
The first th
re
e o
r
de
rs with
the Volterra
serie
s
i
s
gene
rally used to
d
e
scrib
e
the
d
y
namics
cha
r
a
c
teri
stics of
nonli
nea
r syste
m
. The
k
th-o
rd
er tim
e
dom
ain i
s
u
n
ique
and
sy
mmetric.
With
its
symmetry, the Volterra
se
ries is
sho
w
n
as Equatio
n (2):
12
3
12
1
11
12
00
1
11
3
0
,,
,,
,,
]
i
N
NN
ii
j
i
N
NN
ij
i
k
j
yn
h
i
un
i
A
i
j
h
i
j
u
n
i
u
n
j
B
i
jk
h
i
jk
u
n
i
u
n
j
u
n
k
e
n
(2)
Her
e
,
1
,
2,
if
i
j
Ai
j
if
i
j
，
,
1
,,
6
,
(
3
,
if
i
j
k
Bi
j
k
i
f
i
j
j
k
i
k
if
e
l
s
e
II
，
,
1,
2
,
3
p
Np
is the Volterra kernel mem
o
ry length, an
d
en
is the trun
ca
tion error.
2
23
2
3
,
,
1,
,
2
1,
,
(1
)
,
,
3
(
)
1
,
,
(
1
)
T
X
n
xn
xn
N
x
n
x
n
x
n
xn
N
x
n
x
n
x
n
x
n
N
LL
L
(3)
11
2
2
2
3
33
3
1
h
(
0
)
,
,
(
1
),
0,
0
,
0,
1
,
,
1
.
1
),
h
0
,
0
,
0
,
0,
0,
1
,
,
0
,
0
,
1
,
,
1
,
1
,
1
T
M
Hh
N
h
h
h
N
N
hh
N
h
N
N
N
LL
LL
（
(4)
Whe
r
e,
N
re
pre
s
ent
s the kernel mem
o
ry lengt
h. The system inp
u
t vector is
X
(
n
), and
kernel
vec
t
or
is
H
.
Equation (2)
descri
b
e
s
the
relation
sh
i
p
betwe
en inp
u
t
and output
of the nonlin
ear
system, which can b
e
expressed a
s
the
vec
t
or form, as
follows
:
T
yn
H
X
n
e
n
(5)
It can
be
se
e
n
from
Equati
on (5) that th
e out
p
u
t of a
nonlin
ear sy
stem ca
n b
e
e
x
presse
d
as a
linea
r
combi
nation
of each
ele
m
ent of
the
input vecto
r
X
(n).
Th
e
Volterr
a
s
e
ries
model
ba
se
d
nonlin
ear sy
stem ide
n
tifica
tion is
u
s
e
d
t
o
solve the
kernel
ve
ctor
H
wh
en th
e i
nput
and output seque
nce
of
t
he system
are given.
The
essen
c
e
of th
e
ide
n
tificatio
n
is a
param
eter
optimizatio
n pro
c
e
ss.
In this paper,
state transiti
on algo
rithm with
ortho
gon
al transfo
rm
wa
s use
d
to solve the
ker
nel v
e
ct
o
r
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Volterra Se
rie
s
identificatio
n Based o
n
State Tran
sitio
n
Algorithm
with… (Co
ng
Wan
g
)
173
3. Orthog
on
al Transform
a
tion Sta
t
e T
r
ansition
Algorithm
3.1. State Tr
ansition Alg
o
rithm
The
state t
r
a
n
sition
alg
o
rit
h
m
wa
s p
r
op
ose
d
by
YANG in
20
11 [1
2
-
14]. A
sol
u
tion to
the
spe
c
ific optim
ization
p
r
oble
m
can
be
de
scrib
e
d
a
s
a
st
ate, and
the
o
p
timization
al
gorithm
can
b
e
treated a
s
st
ate tran
sition.
Then the p
r
o
c
e
ss to
solve
the optimizat
ion
problem
can be rega
rd
ed
as a state tra
n
sition p
r
o
c
e
ss.
The state tra
n
sition alg
o
rit
h
m is ea
sy to unde
rsta
nd
, due to the
less numb
e
rs of the
para
m
eters a
nd the sim
p
l
e
algo
rithm
st
ructu
r
e. Th
e
state tran
sitio
n
is defin
ed
as the follo
wi
ng
form:
1
1
()
kk
k
k
k
kk
x
Ax
B
u
yf
x
(6)
Her
e
,
n
k
x
R
stand
s for a state and co
rrespo
nds to
a sol
u
tion of the optimizatio
n
probl
em.
n
n
kk
A
BR
，
are
state transiti
on matrixe
s
whi
c
h ca
n be
rega
rded a
s
the operato
r
s
of optimizatio
n algo
rithm.
n
k
uR
is the fun
c
tio
n
of the st
ate
k
x
and it
s hi
story state.
f
is
the
obje
c
tive function.
3.2. The Tra
n
sition Ope
r
ators
There a
r
e th
ree op
erators
calle
d rotatio
n
trans
f
o
rmati
on (RT
)
, tran
slation t
r
an
sf
ormatio
n
(TT),
expa
nsi
on tran
sfo
r
m
a
tion (ET
)
in
STA.
Rotation
tran
sform
a
tion is u
s
e
d
to improve t
he
global
se
arch
ability, tran
sl
ation tra
n
sfo
r
mation
can
i
m
prove
local
sea
r
ch a
b
ility, and exp
a
n
s
i
o
n
transfo
rmatio
n can b
a
lan
c
e the relatio
n
s
between
th
e two. Besid
e
s, refe
ren
c
e
[15] propo
sed
axesio
n tran
sformation to simplify t
he search ability of one dimen
s
i
onal.
The detail
s
of the four ope
rators a
r
e
sho
w
n a
s
follows [15]:
(1)
Rotation transfo
rmatio
n:
1
2
1
kk
r
k
k
x
xR
x
nx
(7)
(2) T
r
an
slatio
n transf
o
rmati
on:
1
1
1
2
kk
kk
t
kk
xx
xx
R
xx
(8)
(3) Expan
sio
n
Tran
sfo
r
ma
tion:
1
kk
e
k
x
xR
x
(9)
(4) Axe
s
ion T
r
an
sform
a
tion
:
1a
kk
k
x
xR
x
(10)
Her
e
,
n
k
x
R
,
，，
，
are all
positive con
s
tants, calle
d
ro
tation fac
t
or, trans
l
ation fac
t
or,
expan
sion fa
ctor, an
d axesio
n facto
r
respe
c
tively.
nn
r
R
R
is a random matrix with its
element
s b
e
l
ongin
g
to the
ran
ge
of [-1,
1] an
d
2
k
x
is 2
-
n
o
rm
of a ve
ct
or.
t
RR
is a
ra
ndo
m
variable
with i
t
s eleme
n
ts b
e
longi
ng to the ran
ge of [0,1].
nn
e
R
R
is a
rand
o
m
diagon
al m
a
trix
with its
eleme
n
ts ob
eying t
he Ga
ussia
n
distrib
u
tion.
nn
a
R
R
is a
ran
dom
d
i
agon
al matri
x
with
its element
s obeying the
Gau
ssi
an di
stribution a
nd o
n
ly one ran
d
o
m
index has
value.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 14, No. 1, March 2
016 : 171 – 1
8
0
174
The procedu
re of the origin
al state tran
si
tion algorith
m
can be o
u
tlin
ed as follo
ws.
1: Initialize feasibl
e
sol
u
tion x(0) rando
mly, set
，，
，
and k
←
0
2: repeat
3:
k=
k+
1
4: while
≤
error do
5: State
←
Ro
tation transfo
rmation (
(1
)
xk
, times
of s
e
arc
h
enforcement,
α
)
6: if
mi
n
(
)
(
(
1
))
f
S
ta
te
f
x
k
then
7: Updatin
g
(1
)
xk
8: State
←
Transl
a
tion tran
sform
a
tion (
(1
)
xk
, times of search enforcem
e
n
t,
β
)
9: if
mi
n
(
)
(
(
1
))
f
S
ta
te
f
x
k
then
10: Upd
a
ting
(1
)
xk
11: end if
12: end if
13:
←
c
f
14: end while
15: State
←
Expansi
on Tra
n
sformation (
(1
)
xk
, times of sea
r
ch enfo
r
ceme
nt,
)
16: if
mi
n
(
)
(
(
1
))
f
S
ta
te
f
x
k
then
17: Upd
a
ting
(1
)
xk
18: State
←
Tran
slation tra
n
sformation (
(1
)
xk
, times
of s
e
arc
h
enforcement,
β
)
19: if
mi
n
(
)
(
(
1
))
f
S
ta
te
f
x
k
then
20: Upd
a
ting
(1
)
xk
21: end if
22: end if
23: State
←
Axesion T
r
an
sf
ormatio
n
(
(1
)
xk
, times of search enforcem
e
n
t
,
)
24: if
mi
n
(
)
(
(
1
))
f
S
ta
te
f
x
k
then
25: Upd
a
ting
(1
)
xk
26: State
←
Tran
slation tra
n
sformation (
(1
)
xk
, times
of s
e
arc
h
enforcement,
β
)
27: if
mi
n
(
)
(
(
1
))
f
S
ta
te
f
x
k
then
28: Upd
a
ting
(1
)
xk
29: end if
30: end if
31:
()
x
k
←
(1
)
xk
32: until the specifie
d termi
nation criteri
o
n is met
3.3. Orthog
o
n
al Trans
f
or
mation Stra
teg
y
In orde
r to
further
enha
nce the
alg
o
rithm'
s searchin
g ability, the state tran
sition
algorith
m
ba
sed on the o
r
thogo
nal tran
sform (O
TS
TA) is propo
se
d
.
In OTSTA chaotic
strate
g
y
is used to i
n
itialize the
popul
ation fo
r its non
re
p
eatability an
d erg
odi
city. The orth
og
onal
transfo
rmatio
n op
eratio
n i
s
a
pplie
d on
the p
oor in
dividual
s du
ri
ng the
proce
ss,
whi
c
h
ca
n
effectively avoid pre
m
ature conve
r
ge
nc
e and imp
r
ov
e the global
search ability.
3.3.1. Initiali
z
i
ng
In non
-line
a
r system,
ch
a
o
s i
s
a
com
m
on motio
n
phen
omen
on
with
su
ch
e
x
cellent
cha
r
a
c
teri
stics as e
r
go
dicit
y
, randomne
ss and “reg
ula
r
ity”.Cha
otic
motion ca
n e
x
perien
c
e all
the
states in the
state sp
ac
e without rep
e
tition acco
rdin
g
to cert
ai
n “rul
e” within
ce
rtain motion ra
nge
[16]. Durin
g
initialization, f
i
rstly ch
aotic
stra
tegy i
s
u
s
ed to
rand
o
m
ly generate
M dimen
s
io
nal
vec
t
or
11
1
1
2
1
,,
,
)
M
Xx
x
x
（
. Then,
the model ite
r
ative ch
aotic
seq
uen
ce co
ntaining N
ve
ctors
is obtain
ed b
y
the Logistic
map [17], sho
w
n in Equatio
n (11
)
.
1
=
(
1
)
,
0
,
1
,
...
,
1
kk
k
XX
X
k
N
(11)
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TELKOM
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Volterra Se
rie
s
identificatio
n Based o
n
State Tran
sitio
n
Algorithm
with… (Co
ng
Wan
g
)
175
Her
e
,
0,
4
]
,
[
0
,
1
]
x
（
.In this pape
r, we u
s
e the same p
a
ram
e
ter setting
=4
as [17, 18]
.
The
fitne
s
s values
of all
the state
s
are
calculate
d
by fitne
ss functio
n
. Th
en
better
perfo
rman
ce
part is
cho
s
e
n
as the initial
solution.
Usi
ng
cha
o
tic seq
uen
ce to
initialize
stat
es im
p
r
ove
s
the diversity of the state
s
without
lose of ra
ndo
mness.
3.3.2. Orthog
onal Trans
f
o
r
mation Stra
teg
y
In order to
maintain
the
diversity a
nd b
r
e
adth
o
f
sea
r
ch, thi
s
p
ape
r
ado
pts the
orthog
onal t
r
ansfo
rmatio
n
along
with t
he o
r
igi
nal fo
ur o
perators.
About 10%
individual
s
()
D
t
with the p
o
o
r
fitness valu
e
are
ch
ose
n
from th
e o
v
erall
size
P
(
t
) after e
a
ch
compl
e
ted
st
ate
transfo
rmatio
n.
Then o
r
th
ogon
al matrix
X
is gotten t
h
rou
gh
o
r
tho
gonal tran
sfo
r
mation
u
n
d
e
r
the orth
ogo
n
a
l ba
si
s. Fo
r
any
()
x
Dt
, if the
ort
hogo
nal
x
’ has
better fitness
value,
x
would be
repla
c
e
d
by
x
’. Otherwi
se,
x would b
e
p
r
eserve
d. Th
e
ortho
gon
al operation i
s
repeate
d
until
all
cho
s
e
n
poo
r individual a
r
e
repla
c
e
d
.
For any
,
k
x
, there is
',
'
,
xx
.
x
’ i
s
call
ed the
o
r
thog
onal tran
sformation
of
k
x
. And
''
x
x
.Here
is the orth
og
onal ba
si
s of
k
x
.
4. Volterra Series Identi
fication b
y
OTSTA
The e
s
sen
c
e
of
Volterra
seri
es ide
n
tificati
on
is tha
t
it coul
d
co
nvert the
pa
ramete
r
identificatio
n
probl
em to
o
p
timization
p
r
oblem. STA i
s
u
s
e
d
to find
functio
n
opti
m
al solution
and
to get the
minimum
eva
l
uation
fun
c
tion
value. Th
e kern
el ve
ctor
H
of
Volterra
s
e
ri
es,
whi
c
h
ne
ed
s
to be i
dentified, is seen
as th
e
state
X
k
of OTST
A, and the
state tran
sitio
n
is
rega
rd
ed a
s
the pro
c
e
s
s of identification
algorith
m
.
For Volte
rra
seri
es i
dentifi
c
ation
pro
b
le
m,
the squ
a
re of the differen
c
e
between
the
actual
output
and the p
a
rameter m
ode
l output is
se
t to be the
e
v
aluation fun
c
tion of
Volterra
seri
es id
entification, sho
w
n
in Equation (12):
2
1
L
i
J
hy
k
i
y
k
i
%
(12)
Her
e
,
L
i
s
th
e length
of t
he
windo
w, a
nd
y
k
%
is the
e
s
timated outp
u
t value. Th
e
algo
rithm
demon
strates the whole p
r
oce
s
s of OTSTA:
OTS
T
A
Step 1 Initialization:
Initialize
initial
st
ate b
y
chaotic se
quence
Set paramete
r
s
Calculate the fitness value based on equation(8
)
Step 2 Iteration
Execute strateg
y
: RT,ET,A
T
If get better
fitne
ss value, execute TT, else maint
a
in
Step 3 Updating
the status
1
()
k
f
x
<
()
k
f
x
,
1
k
x
instead of
k
x
,else
k
x
maintain
Step 4 Use Orth
ogonal transform
ation
OT used
on 10
%
individual w
i
th poor values
maintain better st
ate
best
x
Step 5 Replace
be
s
t
x
replaces the curr
ent state
step 6 End
Meet the re
quire
ment, end
Else back to step
2
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016 : 171 – 1
8
0
176
5. Performan
ce Ev
aluation
We con
s
ide
r
e
d
the followin
g
se
con
d
ord
e
r nonli
nea
r
model a
s
the experim
ental
subj
ect
[19]:
22
y
(
)
0
.
5
(
)
0.
4
(
1
)
0.
9
(
2)
+1.
2
(
)
0
.
2
(
1
)
0
.
8
(
1
)
(
2
)
nu
n
u
n
u
n
u
n
u
n
u
n
u
n
(13)
Acco
rdi
ng to the Volterra theory, the ke
rn
el vector of th
e nonlin
ear
system wa
s
H
= [0.5,-0.4, 0.9,
1.2, 0, 0, 0.2,-0.8, 0].
The white n
o
i
s
e si
gnal was cho
s
en a
s
th
e system inp
u
t. The varia
n
ce
wa
s set
as 1 an
d
the length wa
s set a
s
20. In orde
r to verify t
he search ability and sea
r
ch sp
eed
of the OTSTA,
the input a
n
d
output with
or
without noi
se
were
both
con
s
id
ere
d
whe
n
u
s
ing t
he second
orde
r
Volterra
to
bu
ild the
nonli
n
ear mod
e
l. In
orde
r to
te
st the p
e
rfo
r
ma
n
c
e
of the
p
r
op
ose
d
al
gorith
m
,
STA, PSO and Refe
ren
c
e
[19], which
were re
co
gni
zed a
s
di
stin
guished al
gorithms for Volt
erra
seri
es id
entification, were
use
d
for co
m
pari
s
on
with OTSTA.
The OTSTA
was
used to identify
the kernel ve
ct
or of Volterra se
rie
s
with
out noise
who
s
e p
a
ra
m
e
ters
we
re se
t as follows
o
n
the basi
s
of
experime
n
tal
method:
a)
Times of sea
r
ch enfo
r
ceme
nt : 500,
b)
The num
ber
of epoch: 100
,
c)
Comm
uni
cati
on frequ
en
cy : 50Hz,
d)
: 1 to e-5,
e)
、、
: 1,
f)
c
f
: 5.
In orde
r to e
n
su
re fai
r
ne
ss, the STA wa
s set the same
pa
ram
e
ters
as
OT
STA. The
para
m
eters o
f
PSO were set through
m
any times test as follows:
a)
The numb
e
r
of particle : 1
00,
b)
The numb
e
r
of iterations :
500,
c)
Contra
ction f
a
ctor
s
: 0.72,
d)
Ac
c
e
lerating fac
t
or:
c
1=
c
2
=
1.49.
We u
s
ed the
averag
e devi
a
tion to evalu
a
te
the stabili
ty of
three alg
o
rithm
s
.
10
i1
av
.dev
=
*
TT
(14)
*
T
is the value that the actual
value minus
simulatio
n
value,
T
is the a
c
tual value.
5.1. Under No Noise Inte
r
f
eren
ce
Program
s we
re run in
dep
e
ndently for 2
0
tra
ils for e
a
c
h al
gorith
m
in MATLAB
R20
10a
The compa
r
i
s
on
re
sults f
o
r OTSTA, S
T
A, PSO and QPSO [19]
were listed i
n
Table
1. The
conve
r
ge
nce
curve
of OTS
T
A unde
r no
noise interf
e
r
ence was
sh
own
as Fi
gure 1. Figu
re 2
and
Figure 3
sho
w
ed
the
co
n
v
ergen
ce
curves
of th
e V
o
lterra
kernel
vecto
r
h
1
(0) and
h
2
(0,0)
of
PSO
，
OTST
A and STA re
spe
c
tively. The truth-valu
es were
h
1
(0
)=
0
.
5
，
h
2
(0,0)=1.
2.
Table 1. The
results un
de
r the free-noi
se
interfere
n
ce
Kernel
H
truth-
value
STA O
T
S
T
A
PSO
Q
PSO
[1
9
optimal
value
av.dev time
/
optimal
value
av.dev time
/
optimal
value
av.dev time
/
optimal
value
h
1
(0)
0.5
0.5
1.3e-5
61.3
0.5 1.9e-6
65.8
0.5 2.1e-5
168.
2
0.5
h
1
(1)
-0.4
-0.4
0.9e-7
-0.4
2.1e-8
-0.4
1.1e-7
-0.4
h
1
(2)
0.9
0.9 3.1e-10
0.9 0.9e-10
0.9
0.9e-9
0.9
h
2
(0,0
)
1.2
1.2 0.3e-6
1.2 2.2e-7
1.2 1.9e-6
1.2
h
2
(0,1
)
0
0 1.2e-8
0 1.6e-9
0 1.7e-8
0
h
2
(0,2
)
0
0 0.4e-11
0 0.5e-11
0 3.5e-11
0
h
2
(1,1
)
0.2
0.2 2.7e-9
0.2 1.2e-9
0.2 0.1e-9
0.2
h
2
(1,2
)
-0.8
-0.8
2.6e-8
-0.8
1.7e-8
-0.8
1.3e-7
-0.8
h
2
(2,2
)
0
0 3.1e-12
0 0.8e-12
0 2.4e-12
0
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
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ISSN:
1693-6
930
Volterra Se
rie
s
identificatio
n Based o
n
State Tran
sitio
n
Algorithm
with… (Co
ng
Wan
g
)
177
It can be se
e
n
from the Table 1 that O
T
STA is sup
e
rio
r
to STA
and PSO in Volterra
seri
es i
dentification u
nde
r
no noi
se. It not only had
fa
st conve
r
g
e
n
c
e spee
d, but
also ha
d stro
ng
global
search
ability. Un
de
r no
noi
se
int
e
rfer
en
ce, O
T
STA had
no
obviou
s
adv
antage
comp
ared
with QPSO in Reference [19].
Figure 1. The con
v
ergen
ce
cu
rve under n
o
n
o
ise inte
rfere
n
ce
The p
r
efe
r
en
ce of th
e alg
o
r
ithms wa
s j
u
dged
by
the
para
m
eters i
dentificatio
n
error [2
0].
It can be
se
e
n
from Fig
u
re 1, Figu
re 2
and Fig
u
re
3 that OTST
A can g
e
t the optimal
sol
u
tion
whe
n
the
ite
r
ation
num
be
r u
p
to
100,
whi
c
h
mea
n
s
O
T
STA h
a
d
a
better converg
e
n
c
e
and
highe
r preci
s
i
on in identificat
ion on the
Volterra
se
rie
s
.
Figure
2. The
convergen
ce
curve
s
of the
Volterra
ke
rn
el v
e
ctor h
1
(0
)und
er the n
o
noise
interferen
ce
Figure 3. The
convergen
ce
curve
s
of the
Volterra
ke
rn
el vector h2
(0
,0)und
er the
no
noise interfe
r
ence
5.2. Under
Noise Inter
f
er
ence
For the noi
sy case, the noise of sup
e
rp
os
ition ad
ded
on the input and the output wa
s
indep
ende
nt stationa
ry whi
t
e noise a
nd i
t
s sign
al SNR was 2
0
dB.
The
noi
se
wa
s a
dde
d o
n
th
e inp
u
t an
d
o
u
tput respe
c
tively. We
use
d
the
same
m
e
thod
s.
The
re
sults of
the th
ree
alg
o
rithm
s
a
nd
Referen
c
e
[1
9] we
re
sho
w
n in
Tabl
e 2.
The
co
nverg
e
n
ce
curve
of
OTS
T
A und
er noi
se i
n
terfe
r
en
ce
wa
s
sh
own in
Figu
re
4. It can
be
se
en from o
p
timal
value, av.dev
and
sim
u
lati
on time i
n
ta
ble2 t
hat OT
STA
still
ha
d
fast conve
r
g
ence spee
d and
stron
g
glo
bal
sea
r
ch a
b
ility than STA and PSO u
n
der
noise int
e
rferen
ce. A
nd OTSTA
wa
s
notable
sup
e
rior than Refe
rence [19] in av.dev.
0
50
10
0
15
0
20
0
25
0
30
0
35
0
40
0
45
0
50
0
0
0.
0
2
0.
0
4
0.
0
6
0.
0
8
0.
1
0.
1
2
0.
1
4
0.
1
6
0.
1
8
T
h
e n
u
m
b
e
r
of
i
t
er
a
t
i
o
n
s
T
h
e
ou
t
p
ut
e
r
ro
r
0
50
100
15
0
200
25
0
30
0
35
0
400
450
500
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
T
h
e
num
be
r
o
f
it
e
r
a
tio
ns
h1(
0
)
ST
A
OT
ST
A
PS
O
0
50
10
0
150
20
0
25
0
300
35
0
40
0
450
50
0
0.6
0.8
1
1.2
1.4
1.6
1.8
2
T
h
e
num
b
e
r
o
f
it
e
r
a
t
io
ns
h2(
0,
0
)
OT
S
T
A
ST
A
PS
O
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93-6
930
TELKOM
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Vol. 14, No. 1, March 2
016 : 171 – 1
8
0
178
Table 2 Th
e result
s of thre
e algorith
m
s
unde
r the noi
se interfe
r
e
n
ce
Kernel
H
truth-
value
STA O
T
S
T
A
PSO
Q
PSO
[1
9]
optimal
value
av.dev time
/
optimal
value
av.dev time
/
optimal
value
av.dev time
/
optimal
value
av.dev time
/
h
1
(0)
0.5
0.49
2.1
e-3
76.3
0.49 1.8e-4
79.1
0.51 2.2e-3
188.
5
0.49 9.7e-3
NaN
h
1
(1)
-0.4
-0.41
0.3e-3
-0.40
0.1e-4
-0.4
1.2e-3
-0.40
3.7e-3
h
1
(2)
0.9
0.90
1.4e-2
0.90 2.2e-3
0.90 1.9e-3
0.90 7.0e-3
h
2
(0,0
)
1.2
1.18
4.3e-2
1.21 1.1e-3
1.18 2.7e-2
1.18 2.0e-2
h
2
(0,1
)
0
0
1.5e-3
0 3.6e-4
0 1.1e-3
0 1.6e-3
h
2
(0,2
)
0
0
0.7e-3
0 0.4e-3
0 0.5e-3
0 0.4e-3
h
2
(1,1
)
0.2
0.21
2.9e-3
0.2
4.3e-4
0.19
0.1e-2
0.19 0.6e-3
h
2
(1,2
)
-0.8
-0.8
1.8e-3
-0.8
1.7e-3
-0.83
2.3e-2
-0.78
1.7e-2
h
2
(2,2
)
0
0
3.1e-4
0 1.5e-4
0 1.9e-3
0 1.4e-3
Figure 4. The
convergen
ce
cu
rve un
de
r noise interfe
r
ence
Figure 5
and
6
sho
w
e
d
t
he
cha
nge
s i
n
Volterra
ke
rnel ve
cto
r
h
1
(0) and
h
2
(0,0) with
numbe
r of
iteration
s
u
s
ing the three al
go
rith
ms, whi
c
h
descri
bed th
e co
nverg
e
n
ce
cha
r
a
c
teri
stics of the three
algorithm
s in
the
optimizat
ion pro
c
e
s
s. The OTSTA
algorith
m
co
u
l
d
conve
r
ge a
n
d
prod
uce good o
p
timization
results after a small numb
e
r of iteration
s
,
demon
stratin
g
convergen
ce
ch
ara
c
te
ri
stics
signifi
ca
ntly better. P
S
O and
STA
had
fluctu
ated
obviou
s
ly and
slowly conve
r
gen
ce
spe
e
d
by noise influ
enced.
Figure 5. The
convergen
ce
curve
s
of the
Volterra
ke
rn
el v
e
ctor h
1
(0
)und
er the n
o
i
se
interferen
ce
Figure 6. The
convergen
ce
curve
s
of the
Volterra
ke
rn
el v
e
ctor h
2
(0
,0)und
er the
noise
interferen
ce
0
50
100
150
200
25
0
30
0
35
0
400
450
500
0
0.
02
0.
04
0.
06
0.
08
0.
1
0.
12
0.
14
0.
16
T
h
e
n
u
m
be
r o
f
it
e
r
a
tio
ns
T
h
e
out
pu
t
e
r
r
o
r
0
50
100
15
0
20
0
250
30
0
350
40
0
45
0
500
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
T
h
e
numbe
r
o
f
ite
r
a
tio
ns
h1
(
0
)
PS
O
OT
ST
A
STA
0
50
10
0
15
0
20
0
25
0
30
0
35
0
40
0
45
0
50
0
0
0.
2
0.
4
0.
6
0.
8
1
1.
2
1.
4
1.
6
1.
8
2
T
h
e
n
u
m
b
e
r
o
f
i
t
era
t
i
o
n
s
h2
(
0
,
0
)
PS
O
OT
ST
A
ST
A
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Volterra Se
rie
s
identificatio
n Based o
n
State Tran
sitio
n
Algorithm
with… (Co
ng
Wan
g
)
179
From
the
an
alysis of the
comp
ari
s
o
n
of
the
above
figures an
d
t
ables, we
know
that
OTSTA is a suitabl
e tool to solve the volterra
seri
e
s
identificatio
n unde
r no n
o
ise a
s
well
as
noise interfe
r
ence. It is not only has hig
h
pre
c
isi
on, b
u
t also ha
s Strong
rob
u
stn
e
ss.
6. Conclusio
n
Based o
n
the
the analysi
s
on the prin
cip
l
e and t
he ki
n
e
matical
cha
r
acteri
stics of
Volterra
seri
es, a ne
w improved int
e
lligent algo
ri
thm is pro
p
o
s
ed
with orth
ogon
al strate
gy. The chao
tic
strategy i
s
use
d
to initialize the
p
opulat
io
n wit
h
ch
aotic
seque
nces, a
nd orth
ogo
n
a
l
transfo
rmatio
n is
use
d
to t
r
an
sform
mut
a
tion on
so
m
e
poo
r in
divid
uals,
whi
c
h
can imp
r
ove t
h
e
global
se
arch ability. Thi
s
imp
r
oved
state tra
n
si
ti
on alg
o
rithm
is a
pplie
d t
o
identify Vo
lterra
seri
es, a
nd t
he re
sult
s a
r
e analy
z
ed
compa
r
ing
wit
h
STA and
PSO. Throu
g
h
the si
mulat
i
on
experim
ent, OTSTA algo
ri
thm achi
eves highe
r identi
f
ication p
r
e
c
i
s
ion tha
n
ST
A and PSO, and
has
highe
r id
entification
speed th
an P
S
O unde
r no
noise as
we
ll as noi
se i
n
terfere
n
ce. State
transitio
n al
g
o
rithm
with o
r
thogo
nal tra
n
sformati
on i
s
u
s
ed
on th
e identificatio
n on th
e Volt
erra
seri
es. T
h
is
method
can
not only improve th
e
global
sea
r
ch ca
pability effectively, avoid
prem
ature co
nverge
nce,
b
u
t
also ca
n maintain
sim
p
le structu
r
e
and h
a
s
high
sea
r
ch effici
ency
of state tran
sition. This pa
per verifie
s
t
hat t
he OTS
T
A is feasi
b
l
e
on no
nline
a
r sy
stem Vo
lterra
kernel i
dentif
ication. T
h
e
method
pro
v
ides
a
ne
w effective m
e
thod fo
r no
nlinea
r
syste
m
identificatio
n.
Ackn
o
w
l
e
dg
ements
The work wa
s su
ppo
rted
by the Nation
al Na
tu
ral Science Fou
n
d
a
tion of Chi
n
a (G
rant
No. 51
575
46
9) a
nd the
O
u
tstandi
ng
Doctor Grad
u
a
t
e Student In
novation Proj
ect of Xinjia
ng
University (No. XJUBSCX
-
2015
014
).
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hou J, Xi
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