Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 1
,
Febr
u
a
r
y
201
6,
pp
. 16
7
~
17
6
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
1.8
694
1
67
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Identification of Nonlinear Sy
stem
s Struct
ured by Wiener-
Hamm
erstein M
o
del
A Br
ouri,
S Sl
assi
ENS
A
M,
AEEE Departm
,
L2MC,
Moula
y
Ism
a
il
Universit
y
,
Mek
n
es, Moroc
c
o
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
J
u
l 12, 2015
Rev
i
sed
O
c
t 25
, 20
15
Accepted Nov 16, 2015
Wiener-Hammerstein s
y
stems consist
of a series connection including
a
nonline
a
r s
t
ati
c
elem
ent s
a
nd
wiched with t
w
o linear s
ubs
ys
tem
s
. The
problem of identif
y
i
ng Wien
er
-Hammers
tein m
odels is addressed in the
presence of hard
nonlinearity
an
d two linear sub
s
y
s
tems of structure entir
e
ly
unknown (as
y
mptotically
s
t
able)
.
Furtherm
ore, the static nonlinearity
is not
required
to be invertib
le. Given th
e s
y
s
t
em nonparametric nature, the
identif
ic
ation
pr
oblem
is presen
tl
y de
alt with
b
y
d
e
veloping
a two-stage
frequency
id
entification
met
hod
,
involving simple inputs.
Keyword:
Hard
no
n
lin
earity
Freq
u
e
n
c
y syst
e
m
id
en
tificatio
n
W
i
en
er
mo
d
e
ls
Hammerstein
m
o
d
e
ls
Copyright ©
201
6 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Ad
il Brou
ri,
Depa
rtem
ent of AEEE,
ENSAM, L2
M
C
, Mou
l
ay Ismail Un
iv
ersity,
EN
SA
M, Mar
j
an
e
2
,
B
P
4
024, Mekn
es, Moro
cco.
Em
a
il: a.b
r
ou
ri
@en
s
am
-u
m
i
.a
c.m
a
& b
r
o
u
ri_ad
il@yaho
o.fr
1.
INTRODUCTION
W
i
en
er-Hammerstein
syste
m
s co
n
s
ist o
f
a
series
connecti
on incl
udi
ng a
nonlinear static ele
m
ent
sand
wich
ed
wi
th
two lin
ear su
b
s
ystem
s
(Fig
ure
1
)
. Acco
rd
ingly, this structureof m
ode
ls can be
vie
w
ed as a
gene
ral
i
zat
i
o
n
of
Ham
m
erst
ei
n an
d
W
i
e
n
er
m
odel
s
an
d s
o
i
t
i
s
e
x
pect
ed t
o
feat
ure
a
su
peri
or
m
odel
i
n
g
cap
ab
ility. Th
i
s
h
a
s b
e
en
confirm
e
d
b
y
sev
e
ral practical ap
p
licatio
n
s
e.g.
p
a
ralyzed sk
el
etal
m
u
scle d
y
n
a
m
i
cs
[1
]. No
te th
at, th
e in
tern
al sig
n
a
ls:
v
(
t
),
w
(
t
),
x
(
t
) an
d
ξ
(
t
) are not accessible to m
e
a
s
urem
ents. The only
m
easurabl
e
si
g
n
al
s are
t
h
e
sy
st
em
i
nput
u
(
t
)
an
d
ou
tpu
t
y
(
t
).
As a
m
a
tter o
f
fact,
W
i
en
er-Hammerstein
syste
m
s are
mo
re d
i
fficu
lt to
id
en
tify th
an
th
e si
m
p
ler
Hammerstein
an
d
Wien
er mo
d
e
ls. Th
e com
p
lex
i
t
y
o
f
the fo
rm
er lies i
n
th
e fact th
at th
ese syste
m
s
in
vo
lv
e
tree internal si
gnals
not acce
ssible to m
eas
urem
ents,
whe
r
eas the latter only invol
ve two. The
n
, it is not
surprisi
ng that
only a
fe
w m
e
thods
ar
e
a
v
aila
ble
that deal with W
i
e
n
er-H
ammerstein
syst
e
m
id
en
tificatio
n.
The a
v
ai
l
a
bl
e m
e
t
hods
ha
ve
been
de
vel
o
pe
d f
o
l
l
o
wi
n
g
t
h
r
ee
m
a
i
n
app
r
o
aches i
.
e. i
t
e
rat
i
ve n
onl
i
n
ea
r
opt
i
m
i
zati
on
pr
oced
u
r
es e.
g.
[
2
]
-
[
3
]
;
st
oc
hast
i
c
m
e
t
hods
e.
g.
[
4
]
-
[
5
]
;
fre
q
u
e
n
cy
m
e
t
hods
[
6
]
-
[
9
]
.
R
o
u
g
h
l
y
, t
h
e i
t
e
rat
i
v
e m
e
t
h
o
d
s (e
.g
. [
2
]
-
[
3
]
)
necessi
t
a
t
e
a l
a
rge am
ount
of
dat
a
;
si
nce com
put
at
i
o
n
tim
e
and m
e
mory usa
g
e drast
i
cally increase, and ha
ve
local
convergence properties
wh
ich
n
ecessitates th
at
a
fairly accurate param
e
ter esti
mates are avail
a
ble to initia
liz
e the search process. Th
is pri
o
r knowle
dge i
s
not
requ
ired
in
stoch
a
stic
m
e
th
o
d
s b
u
t
th
ese are
g
e
n
e
rally relied
on
sp
ecific assu
m
p
tio
n
on
th
e in
pu
t sign
als (e.g.
g
a
ussian
ity, p
e
rsisten
t
ex
citatio
n....) and
on
syste
m
m
o
d
e
l (e.g
. M
A
lin
ear su
bsyste
m
s
, sm
o
o
t
h
n
o
n
lin
earity).
The f
r
eq
ue
ncy
m
e
t
hods are
ge
neral
l
y
appl
i
e
d
t
o
no
npa
ram
e
tri
c
sy
st
em
s under m
i
nim
a
l
assum
p
t
i
ons an
d onl
y
requ
ire
sim
p
le
p
e
ri
o
d
i
c ex
citatio
n
s
. Bu
t, t
h
ey so
m
e
ti
m
e
necessi
t
a
t
e
several
dat
a
gene
rat
i
o
n e
xpe
ri
m
e
nt
s.
The p
r
ese
n
t
i
d
ent
i
f
i
cat
i
o
n
m
e
t
hod i
s
q
u
i
t
e di
ffe
rent
of
pre
v
i
o
us f
r
e
q
uency
m
e
t
hod
s. I
n
[
3
]
t
h
e
i
d
ent
i
f
i
cat
i
o
n
m
e
t
hods
re
qui
r
e
a s
p
eci
al
desi
gn
o
f
t
h
e i
n
p
u
t
si
gnal
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 1, Feb
r
uar
y
20
1
6
:
16
7 – 17
6
16
8
In [
9
]
t
h
e i
d
en
t
i
f
i
cat
i
on
m
e
t
hod i
s
base
d o
n
t
h
e best
l
i
n
ear app
r
o
x
i
m
ati
on t
echni
que
us
i
ng cl
ass o
f
Gaus
sian (-like) signals. In [10], th
e a
u
thors show that there are m
a
ny
local minim
a
, the estim
a
tion m
u
st
to
be re
peat
ed s
e
veral
t
i
m
es
wi
t
h
di
ffe
rent
st
art
i
ng
va
lues to increa
se
the cha
n
ces
of
finding a
m
odel
cor
r
es
po
n
d
i
n
g
t
o
a g
o
o
d
l
o
cal
m
i
nim
u
m
.
In [
11]
, a
n
a
p
pr
oa
ch
based
o
n
t
h
e st
anda
r
d
S
V
M
fo
r re
g
r
essi
on
was
p
r
esen
ted. Th
e qu
ite poo
r resu
lts ob
tain
ed
i
n
t
h
at work
hi
ghl
i
g
ht
ed
s
o
m
e
o
f
t
h
e l
i
m
it
ati
ons
o
f
t
h
e m
e
tho
d
.
I
n
part
i
c
ul
a
r
, o
n
l
y
a NFIR
m
o
del
st
ruct
ure
was t
a
ke
n i
n
t
o
acco
u
n
t
,
w
h
i
c
h di
d
not
pe
rf
orm
wel
l
si
nce t
h
e
con
s
i
d
ere
d
sy
st
em
has a l
o
ng
im
pul
se res
p
o
n
se.
A
not
her
p
r
o
b
l
e
m
was gi
ven
by
t
h
e
hi
g
h
c
o
m
put
at
i
o
n
a
l
t
i
m
e
and m
e
m
o
ry
usa
g
e, w
h
i
c
h
m
a
de i
t
di
ffi
c
u
l
t
t
o
w
o
r
k
wi
t
h
a l
a
r
g
e
am
ount
of
dat
a
. Seve
ral
S
V
M
-
l
i
k
e
approaches (e.g.
[12]-[13]), base
d
on the l
east squa
res SVM (LSSVM
),
are c
h
aracterized by a ve
ry
high
num
ber of
pa
ra
m
e
t
e
rs.
Fi
gu
re 1.
W
i
e
n
er-
H
am
m
e
rst
e
i
n
M
o
del
st
ruct
ure
Fig
u
re
2
.
Hard
n
o
n
lin
earity with
prelo
a
d
In
t
h
i
s
pa
per,
a f
r
eq
ue
ncy
-
dom
ai
n i
d
e
n
t
i
f
i
cat
i
on
sche
m
e
i
s
desi
g
n
e
d
fo
r
Wi
ene
r
-
H
am
m
e
rst
e
i
n
sy
st
em
s i
nvol
v
i
ng t
w
o
l
i
n
ea
r
sub
s
y
s
t
e
m
s
(asym
p
t
o
t
i
cal
l
y
stabl
e
)
o
f
e
n
t
i
r
el
y
un
k
n
o
w
n st
r
u
ct
u
r
e,
u
n
l
i
k
e
m
a
ny
pre
v
i
o
us
w
o
r
k
s. Q
u
i
t
e
a fe
w
pre
v
i
o
us
st
u
d
i
es have
deal
t
wi
t
h
bl
oc
k-
or
i
e
nt
ed sy
st
em
s (o
f a
n
y
t
y
pe)
t
h
at
in
vo
lv
e p
i
ecewise affin
e
non
lin
earities (Fig
ure
2
)
th
at
are, po
ssi
b
l
y d
i
sco
n
tinuo
us and
o
f
a
p
r
i
o
ri
un
kno
wn
structure. T
h
e
syste
m
nonline
a
rity can have several e
ffect
s
[6]. In
particul
ar, it
m
a
y contain a saturation effect
or
dead z
one
. One
key
co
nt
ri
but
i
o
n o
f
t
h
e p
r
esent
w
o
rk i
s
to
sho
w
th
at the syste
m
id
en
tificatio
n
is po
ssib
le
wi
t
h
o
u
t
passi
n
g
by
an o
r
t
h
og
onal
seri
es e
x
p
a
nsi
o
n o
f
t
h
e (
pos
si
bl
y
)
di
sc
o
n
t
i
n
u
o
u
s i
n
put
no
nl
i
n
ea
ri
t
y
. Gi
ven
th
e syste
m
n
o
n
p
a
ram
e
tric n
a
tu
re, th
e id
en
tificatio
n
prob
le
m is p
r
esen
tly d
ealt with
b
y
dev
e
lop
i
ng
a two
-
stag
e
fre
que
ncy
i
d
e
n
t
i
f
i
cat
i
on m
e
t
hod
. Fi
rst
,
t
h
e i
d
ent
i
f
i
cat
i
o
n o
f
sy
st
em
nonl
i
n
eari
t
y
can
be
achi
e
ve
d
by
us
i
ng a
set o
f
co
nstan
t
p
o
i
n
t
s. Th
en, th
e lin
ear subsyste
m
s
can
be d
ealt b
y
d
e
velo
p
i
ng
a
frequ
en
cy id
en
tificatio
n
m
e
t
hod.
Th
e
o
u
tlin
e
o
f
th
e rem
a
in
in
g
p
a
rt
o
f
t
h
is p
a
p
e
r con
s
ists of 4
section
s
. The id
en
tification p
r
ob
lem
is
fo
rm
all
y
descr
i
bed i
n
Sect
i
o
n
2. Sect
i
on
3
i
s
dev
o
t
e
d
to
t
h
e id
en
tification
o
f
t
h
e system
n
o
n
lin
ear el
e
m
en
t.
Th
e lin
ear sub
s
yste
m
s
id
en
tificatio
n
is
d
i
scu
s
sed
i
n
Section
4
.
Sim
u
latio
n
s
are p
r
esen
ted
in
Section
5
.
2.
IDENTIFICATION
PROBLEM ST
ATE
M
ENT
We a
r
e i
n
terest
ed i
n
system
s that ca
n
be
des
c
ribe
d
by the
W
i
e
n
e
r
-Hammerstein st
ruct
ure (Fi
g
ure
1)
with
h
a
rd
no
n
l
i
n
earity
(Figu
r
e
2)
w
ith
k
now
n
seg
m
en
ts
nu
mb
er
q
. Th
is m
o
d
e
l is an
alytically d
e
scrib
e
d
by th
e
fo
llowing
equ
a
tio
n
s
:
v
(
t
)=
g
i
(
t
)*
u
(
t
) (
1
a)
()
()
*
(
)
()
i
vt
g
t
u
t
wt
f
f
(1
b)
()
()
()
*
(
)
(
)
(
)
*
(
)
oo
vt
yt
g
t
w
t
t
g
t
f
t
(1c
)
whe
r
e
g
i
(
t
)
L
-1
(
G
i
(
s
))
an
d
g
o
(
t
)
L
-1
(
G
o
(
s
)) a
r
e the i
nve
rse L
a
place tra
n
sform
of
G
i
(
s
) a
n
d
G
o
(
s
) (res
p
ectively
)
;
*
r
efers t
o
th
e co
nvo
lu
tion
op
eratio
n.
The linear subsystem
s
are of e
n
tirel
y unknown st
ructure. T
h
ere are only
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Id
en
tifica
tio
n
of
No
n
lin
ea
r S
y
stems S
t
ru
ct
u
r
ed
b
y
Wien
er-Hammerstein
Mod
e
l
(Ad
il Bro
u
r
i)
16
9
suppose
d to be asym
ptotically stable
(because syste
m
ide
n
tification is ca
rried
out in open
loop) and with
no
nze
r
o st
at
i
c
gai
n
(i
.e
.
0
)
0
(
i
G
and
0
)
0
(
o
G
). Also, not
e
that the nonzer
o st
at
i
c
-
g
ai
n re
qui
rem
e
nt i
s
satisfied by m
o
st real life syste
m
s.
In fact, only derivative
syste
m
s
m
a
ke
an exce
ption that can be coped with
usi
n
g a
d
-
h
oc a
d
apt
a
t
i
o
ns
of
t
h
e m
e
t
hod
de
v
e
l
ope
d i
n
t
h
i
s
p
a
per
.
Fo
r a
p
r
o
b
l
em
o
f
id
en
tifiab
ility, at least o
n
e o
f
non
lin
eari
t
y seg
m
en
t h
a
s no
n
z
ero slop
e. Th
e ex
tern
al
n
o
i
se
()
t
i
s
su
p
pos
ed
t
o
be a
zer
o-m
ean st
at
i
o
nary
se
que
nce
o
f
i
n
de
pen
d
e
n
t
ra
n
d
o
m
vari
abl
e
s an
d e
r
g
o
d
i
c
.
Let
mM
uu
b
e
th
e work
i
n
g
in
terv
al.
Th
e prob
lem
c
o
m
p
lex
ity a
l
so
lies
in
th
e fact th
at th
e in
ternal
si
gnal
s
a
r
e
n
o
t
uni
quel
y
de
fi
ne
d f
r
o
m
an i
n
p
u
t
-
o
u
t
p
ut
vi
ew
poi
nt
.
In
effect
,
i
f
()
,
(
)
,
()
io
Gs
f
v
G
s
is
represen
tativ
e o
f
th
e system th
en
, an
y
m
o
d
e
l o
f
the fo
rm
12
1
2
(
)
/
,
(
)
,
(
)
/
io
Gs
k
k
f
k
v
G
s
k
is
also
represe
n
tative whate
v
er the real num
bers
1
0
k
and
2
0
k
(Fig
ure 3). To
g
e
t b
e
n
e
fit o
f
m
o
d
e
l p
l
u
r
ality,
t
h
ese c
onst
a
nt
s
can
be c
h
ose
n
as f
o
l
l
o
w
s
:
2
(0)
o
kG
and
1
(0
)
i
kG
.
Accord
ing
l
y, th
e system
to
be id
en
tified
is
d
e
scri
b
e
d b
y
t
h
e tran
sfer fun
c
tio
n
s
:
()
()
/
(
0
)
ii
i
Gs
Gs
G
;
()
()
/
(
0
)
oo
o
Gs
Gs
G
(2a
)
and the
nonli
n
earity:
()
(
0
)
(
0
)
oi
f
xG
f
G
x
(2
b)
The
n
, t
h
e foc
u
s m
odel is cha
r
acteri
zed
b
y
the fo
llo
wi
n
g
pro
p
e
rties:
(0
)
(
0
)
1
io
GG
(3
)
Equ
a
tio
n
(3) i
m
p
l
ies th
at, if
u
(
t
) i
s
c
o
nst
a
nt
t
h
en t
h
e st
e
a
dy
-st
a
t
e
u
ndi
st
ur
bed
o
u
t
p
ut
x
(
t
) de
pe
nd
s
o
n
l
y
o
n
th
e i
n
pu
t v
a
l
u
e an
d the no
n
lin
earity
f
(.). Sp
ecifically, let:
()
ut
U
fo
r
0
r
kT
t
, with
k>
1
(4
)
whe
r
e
k
is an
y in
teg
e
r and
T
r
is co
m
p
arab
le to
th
e syste
m
rise t
i
m
e.
Th
en
, th
e in
tern
al sig
n
a
l
x
(
t
), in
th
e
steady-state, is
of the
form
of:
()
(
)
x
tf
U
(5
)
Fig
u
re 3
.
W
i
ener-Hammerstein
Mod
e
l
m
u
lti
p
licity
3.
IDENTIFICATION OF SYSTEM NONL
INEARIT
Y
Th
e
Wien
er-Ha
m
m
e
rstein
syste
m
is ex
cited b
y
a set of con
s
tan
t
i
n
pu
ts
1
()
;
;
N
ut
U
U
, whe
r
e
the num
b
er
N
is arbitrarily c
hos
en by t
h
e
user and
12
N
UU
U
. Afterward, using the
fact that
()
()
()
yt
x
t
t
, it is read
ily o
b
t
ain
e
d fro
m
(5
) t
h
at,
t
h
e
steady state of the system
output
y
(
t
) ca
n
be
expresse
d as
follows:
()
(
)
()
w
h
e
r
e
1
;
;
j
yt
f
U
t
j
N
(6
)
x
(
t
)
v
(
t
)
u
(
t
)
(
t
)
f
(.)
w
(
t
)
y
(
t
)
k
1
k
2
The nonlinearity of
the
transformed syste
m
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 1, Feb
r
uar
y
20
1
6
:
16
7 – 17
6
17
0
Hence
,
t
h
e es
tim
a
te of
()
j
f
U
, f
o
r a
n
y
i
n
put
j
U
, can
be
rec
ove
red
by
a
v
e
r
ag
i
ng
y
(
t
) on
a
su
fficien
tly larg
e in
terv
al (the n
o
i
se
()
t
is zero-m
ean). T
h
e above res
u
lts
su
gg
est th
e fo
llo
wi
n
g
estim
at
o
r
fo
r
f
(.
)
:
0
1
ˆ
ˆ
()
(
)
(
)
r
jj
r
kT
f
UX
k
y
t
d
t
kT
(7
)
with
1
j
N
. Acc
o
r
d
i
ngl
y
,
a n
u
m
b
er of
poi
nt
s o
f
t
h
e n
onl
i
n
ea
r fu
nct
i
o
n
f
(.) can thus
be accurately
esti
m
a
ted
(i.e. th
e resu
ltin
g
syste
m
, in
th
e stead
y state, b
o
ils d
o
w
n
to
t
h
e lin
earity p
a
rt
f
(.))
. T
h
is y
i
elds th
e
fo
llowing
statemen
t:
Prop
osi
t
i
o
n 1
The c
o
upl
e
of
p
o
i
n
t
s
ˆ
()
,
jj
UX
k
,
fo
r
1
j
N
, d
e
term
in
ed
i
n
th
e Non
lin
earity Esti
m
a
to
r,
co
nv
erg
e
(in pro
b
a
b
ility) to
the traj
ect
o
r
y
o
f
f
(.)
.
Accord
ing
l
y, fro
m
(7
), on
e
gets esti
m
a
tes o
f
N
po
in
ts
,(
)
jj
Uf
U
bel
o
n
g
i
n
g t
o
f
(.
). Furt
herm
ore,
t
h
e
larg
er th
e
p
a
rameter
N
is, the
better estim
a
t
ion accuracy.
The
n
,
by
succ
essively conne
cting all avail
a
bl
e
poi
nt
s
(,
(
)
)
;
1
jj
Uf
U
k
N
, a pi
e
cewi
s
e affi
ne
app
r
oxi
m
a
t
i
on of
f
(.
) is o
b
tained.
If the
num
ber o
f
o
b
t
ain
e
d seg
m
en
t is less th
an
q
,
th
e n
o
n
lin
ear
system
is
excited
b
y
o
t
her co
nstan
t
in
pu
ts. Fin
a
lly, let
c
h
oo
se
any se
gm
ent
l
o
f
th
e i
d
en
tified
n
o
n
lin
earity with
n
o
n
z
ero slo
p
e
.
4.
LINEAR SUB
S
YSTE
MS IDENTIFICATION
In t
h
i
s
sect
i
o
n
,
an i
d
e
n
t
i
f
i
cat
i
on m
e
t
hod i
s
pr
op
ose
d
t
o
obt
ai
n est
i
m
at
es of t
h
e c
o
m
p
l
e
x gai
n
co
rresp
ond
ing
to
th
e two
lin
ear sub
s
ystem
s
()
i
Gj
and
()
o
Gj
for a
set of
fre
quencies
1
;
...
;
m
. Le
t
()
a
r
g
(
)
ii
Gj
and
()
a
r
g
(
)
oo
Gj
. Fo
r si
m
p
l
i
city, we
p
r
esen
tly su
ppo
se th
at th
e no
n
lin
earity
id
en
tificatio
n hav
e
b
een exactly d
e
term
in
ed
.
Th
en
, let
d
e
fi
ne th
e
v
a
riab
les
(fo
r
an
y
ω
):
()
()
()
io
(8a
)
()
()
()
io
Gj
G
j
G
j
(8
b)
Th
e sub
s
ystem id
en
tification
can
b
e
im
p
l
e
m
en
ted
i
n
two
stag
es:
Firstly, an ac
curate estim
ates of
()
k
Gj
and
()
k
,
fo
r a
n
y
f
r
eq
uency
1
;.
.
.
;
km
, can
be
d
e
term
in
ed
.
The i
d
e
n
t
i
f
i
cat
i
on
pr
obl
em
unde
r st
u
d
y
i
s
deal
t
usi
n
g a
m
e
t
hod
base
d
on t
h
e f
r
eq
ue
n
c
y
appr
oac
h
.
Th
e Wien
er-Ha
m
m
e
rstein
syste
m
is
ex
cited
with
a g
i
v
e
n
sin
e
inpu
t:
()
s
i
n
(
)
ok
ut
u
U
t
(9
)
whe
r
e t
h
e am
plitude
0
U
is a priori sm
all value.
The c
h
oice of
u
o
ca
n
be
per
f
o
r
m
e
d usi
n
g t
h
e
ex
peri
m
e
nt
al
dat
a
o
f
n
onl
i
n
e
a
ri
t
y
est
i
m
a
ti
on.
It
ca
n t
a
ke
a
n
y
val
u
e i
n
t
h
e
vi
ci
ni
t
y
fr
om
t
h
e ce
nt
er
of
se
gm
ent
l
. The
n
,
as the
lin
ear su
bsystem
G
i
(
s
) is asy
m
p
t
o
tical
ly sta
b
le, it fo
llows
fro
m
(3)-(9),
one h
a
s in
t
h
e stead
y
state:
()
(
)
s
i
n
(
(
)
)
oi
k
k
i
k
vt
u
U
G
j
t
(1
0)
If
v
(
t
)
spa
n
s
o
n
l
y
t
h
e ch
ose
n
s
e
gm
ent
,
one
ge
t
s
:
w
(
t
)
S
*
v
(
t
)
P
*
(1
1)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Id
en
tifica
tio
n
of
No
n
lin
ea
r S
y
stems S
t
ru
ct
u
r
ed
b
y
Wien
er-Hammerstein
Mod
e
l
(Ad
il Bro
u
r
i)
17
1
whe
r
e
S
*
i
s
t
h
e
sl
ope of se
gm
ent
l
and
P
*
is th
e v
a
lu
e o
f
w
(
t
) whe
n
()
0
vt
. In practice, this case can be
easily d
e
tected
b
y
a si
m
p
le in
sp
ection
of th
e o
u
t
pu
t sign
al.
Fo
r a sm
all v
a
lu
e o
f
the am
p
litu
d
e
U
, the st
eady
st
at
e of sy
st
em
out
put
y
(
t
) i
s
a si
ne si
g
n
al
(
up t
o
n
o
i
s
e)
. A
s
so
on as
,
v
(
t
)
spa
n
s at least two se
gm
ents,
y
(
t
) is
not
a si
ne si
g
n
a
l
.
Acc
o
r
d
i
n
gl
y
,
a j
udi
ci
o
u
s
choi
ce
fo
r
U
c
a
n be
gi
ven
p
r
act
i
cal
l
y
. Then
, fr
om
(10
)
-
(
1
1)
, t
h
e
in
tern
al si
gn
al
w
(
t
) is
written
i
n
th
e fo
llowing fo
rm
:
**
*
()
(
)
s
i
n
(
(
)
)
ik
k
i
k
o
wt
U
S
G
j
t
S
u
P
(1
2)
As the lin
ear su
b
s
ystem
()
o
Gs
is asym
p
t
o
tical
ly sta
b
le, it fo
llows
fro
m
(1
2)
an
d
(
8
a-
b)
t
h
at, t
h
e
steady state
un
di
st
u
r
be
d o
u
t
put
x
(
t
) ca
n
be
expres
sed as
follows:
**
*
()
(
)
s
i
n
(
)
ok
k
k
xt
S
u
P
U
S
G
j
t
(1
3)
Fin
a
lly, as
()
()
()
yt
x
t
t
, one i
m
m
e
d
i
atel
y g
e
ts:
*
()
(
)
s
i
n
(
)
(
)
kk
k
o
yt
U
S
G
j
t
y
t
(1
4)
whe
r
e:
**
oo
yS
u
P
(1
5)
On th
e
o
t
h
e
r
han
d
,
recall th
at sin
e
si
g
n
a
ls
that o
s
cillate at th
e sam
e
freq
u
e
n
c
y as
si
n
(
)
kk
t
an
d
ha
vi
n
g
th
e am
p
litu
d
e
U
a
r
e
of t
h
e
form
:
()
s
i
n
k
zt
U
t
(1
6)
whe
r
e
IR
is arbitrary and
IR
de
notes the set
of real
num
b
er
s.
It is
readily seen that:
()
sin
(
)
(
)
k
kk
Ut
z
t
(1
7)
Let
,,
ko
Uu
C
is t
h
e
p
a
rameterized
lo
cus con
s
titu
ted of all
p
o
i
n
t
s
o
f
co
ord
i
n
a
tes
()
,
(
)
zt
x
t
. T
h
ese c
u
rves
are
viewe
d
as
a
ge
neralization of
the Lissa
jous c
u
rves
use
d
i
n
l
i
n
ear
sy
st
em
freque
ncy
a
n
al
y
s
i
s
[
5
]
.
Th
ese i
d
eas are fo
rm
alized
in
th
e
fo
llowing
p
r
op
o
s
ition
:
Prop
osi
t
i
o
n 2
C
onsi
d
er t
h
e
W
i
e
n
e
r
-
H
am
m
e
rst
e
i
n
sy
st
em
descri
bed
by
equat
i
o
ns
(1a
-
c) an
d exci
t
e
d
by
t
h
e i
n
p
u
t
(9
), with
u
0
and
U
are
judiciously chose
n
s
o
t
h
at
t
h
e sy
st
e
m
out
put
y
(
t
) i
s
si
ne si
gnal
(
u
p t
o
n
o
i
s
e)
. T
h
en,
o
n
e
has:
The l
o
cus
,,
ko
Uu
C
is a
lin
ear cu
rv
e if
an
d on
ly if
)
(
k
(m
od
ul
o
π
).
Ano
t
h
e
r k
e
y idea o
f
th
e propo
sed
ap
pro
ach
(g
etting
b
e
n
e
fi
t fro
m
Pro
p
o
s
i
tio
n
2
)
is to
d
e
termin
e th
e
gai
n
m
odul
us
()
(
)
()
ki
k
o
k
Gj
G
j
G
j
and t
h
e phase
()
()
(
)
ki
k
o
k
by
t
uni
ng t
h
e p
a
ram
e
t
e
r
u
n
til th
e lo
cus
,,
ko
Uu
C
sh
ows lin
ear curv
e. Th
e
po
in
t is th
at th
e lo
cu
s
,,
ko
Uu
C
depe
n
d
s
on t
h
e si
g
n
al
()
x
t
whic
h is not accessible to m
e
asurem
ent. Thi
s
is pr
ese
n
tly cope
d
with m
a
king full use
of t
h
e inform
ation
at h
a
nd
,
n
a
m
e
l
y
th
e p
e
riod
icity (with
p
e
riod
k
/
2
) o
f
bot
h
)
(
t
z
and
()
x
t
and the e
r
godicity of the noi
s
e
()
t
. Bearing
t
h
ese in
m
i
n
d
,
the relatio
n
(
)
()
()
yt
x
t
t
sug
g
e
st
s th
e
fo
llowing esti
m
a
to
r:
1
1
ˆ
(,
)
(
)
M
j
k
x
tM
y
t
j
T
M
;
[0
,
)
k
tT
(1
8a)
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S
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:
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08
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o
l
.
6, No
. 1, Feb
r
uar
y
20
1
6
:
16
7 – 17
6
17
2
ˆˆ
(,
)
(
,
)
k
x
tj
T
M
x
t
M
fo
r an
y in
teg
e
r
0
j
(
18b
)
whe
r
e
2/
kk
T
and
M
is a su
fficien
tly larg
e i
n
teg
e
r. Sp
ecifically,
for a fixed ti
me in
stan
t
t
, th
e
q
u
a
n
tity
ˆ
(,
)
x
tM
tu
rn
s ou
t to
b
e
th
e m
ean
v
a
lu
e of t
h
e (m
easure
d
) s
e
que
n
ce
()
;
0
1
.
.
.
k
yt
j
T
j
. The
n
, a
n
estim
ate
,,
,
ˆ
ko
Uu
M
C
of
,,
ko
Uu
C
is si
m
p
ly
o
b
t
ain
e
d
substitu
tin
g
ˆ
(,
)
x
tM
to
()
x
t
whe
n
c
o
n
s
t
r
uc
t
i
ng
,,
.
ko
Uu
C
Thes
e
rem
a
rk
s lead
t
o
th
e fo
llowing propo
sitio
n
:
Prop
osi
t
i
o
n 3
Co
n
s
i
d
er th
e
prob
lem
state
m
e
n
t of
Pro
p
o
s
itio
n2
. Th
en,
o
n
e
h
a
s:
1)
ˆ
(,
)
x
tM
conve
rge
s
i
n
probability to
()
x
t
(as
M
).
2)
,,
,
ˆ
ko
Uu
M
C
co
nv
erg
e
s in
p
r
ob
ab
ility to
,,
ko
Uu
C
(as
M
)
.
On t
h
e ot
her
h
a
nd
, l
e
t
*
th
e co
rr
esp
ond
ing
valu
e of
and
()
k
s
the
slop of
t
h
e obtained curve
*
,,
.
ko
Uu
C
Kn
o
w
i
n
g t
h
e s
i
gn o
f
*
S
, the phase
()
k
can be re
cove
re
d m
odul
o 2
π
. Let
us c
onsi
d
er t
h
e pa
ram
e
t
e
r
defi
ned
as
fol
l
ows:
0
if
*
sig
n
(
)
sig
n
(
)
k
Ss
else
1
(1
9)
Let
ˆ
()
M
k
s
d
e
no
tes th
e esti
m
a
te
o
f
()
k
s
. T
h
en, an estim
a
t
e
ˆ
ˆ
()
,
(
)
Mk
M
k
Gj
of
()
,
(
)
kk
Gj
can be det
e
rm
ined
, one
ha
s
t
h
us, f
o
r any
fre
q
u
ency
k
:
*
,,
ˆˆ
ˆ
(
)
()
()
Mk
i
k
o
k
MM
(m
odul
o
π
)
(2
0a)
*
,,
ˆ
()
ˆˆ
ˆ
()
(
)
(
)
Mk
Mk
i
k
o
k
MM
s
Gj
G
j
G
j
S
(
20b
)
5.
SIMULATION
The ide
n
tification m
e
thod descri
bed i
n
this pa
per
will now beillustrated by sim
u
lation usi
ng
Matlab
/
Si
m
u
li
n
k
. Presen
tly, th
e
system
is characterized by:
0.
2
()
(
1
)
(
0.
2)
i
Gs
ss
and
0.
05
()
(0
.
1
)
(
0
.
5
)
o
Gs
ss
(2
1)
Th
e co
n
s
i
d
ered
non
lin
ear elemen
tis illu
strat
e
d
b
y
Fi
g
u
re
4. Th
e
no
ise
ξ
(
t
) is a seque
nce
of
norm
ally
di
st
ri
b
u
t
e
d (
p
s
e
ud
o
)
ran
d
o
m
num
bers,
wi
t
h
zero
-
m
ean and
st
andar
d
de
vi
at
i
on
0.
02
. Firstly, th
e ai
m
is
to
esti
m
a
te th
e
syste
m
n
o
n
linearity. Th
e id
en
tificatio
n
m
e
t
h
od
d
e
scri
b
e
d
in
Sectio
n
3
will n
o
w
b
e
applied
and, accordi
ngly, the syste
m
is successively
excited by
11
N
const
a
nt
i
n
put
s
;1
j
Uj
N
,
w
h
er
e
th
e
val
u
es
j
U
an
d the ob
tain
ed
esti
mates o
f
()
j
f
U
are sh
own
in Fi
g
u
re 5
.
Th
e tru
e
no
n
lin
earity and th
e set
of
poi
nt
s
ˆ
()
,
jj
L
Uf
U
(1
1
1
)
j
, are
rep
r
esen
ted in
Fig
u
re
6
.
By co
nn
ecting
th
e
set o
f
co
llin
ear po
in
ts
(
F
igu
r
e 6)
. T
h
e
3
q
se
gm
ents
are then obtaine
d
.
The
no
nl
i
n
ea
r
sy
st
em
i
s
exci
ted by
(9
).
Fi
g
u
r
e 7
sh
o
w
s exa
m
pl
e of obt
ai
n
e
d res
u
l
t
s
w
h
e
n
v
(
t
) sp
an
s
at least two segm
ents. Then,
y
(
t
) is no
t a sine sig
n
a
l. Th
is
co
nfirm
s
th
e resu
lt alread
y obtain
e
d
using
the p
l
o
t
,,
,
ˆ
ko
Uu
M
C
. Th
is latter, t
u
rn
s
ou
t to
b
e
a
n
o
n
static curve, wh
atev
er
[0
2
)
(Fig
ure 9
a
-b
). For a sm
all v
a
lu
e
of
U
, e.
g.
0.
25
,
U
1
0.
01
(
/
)
rd
s
and
0.7
5
o
u
, Fi
gu
re 9a s
h
o
w
s t
h
e m
easured o
u
t
p
ut
y
(
t
). This
sig
n
a
l
tu
rn
s
o
u
t
t
o
be a sin
e
si
g
n
al (u
p to
no
ise). Th
en,
u
s
ing
th
e
esti
m
a
to
r (18
a
-b), t
h
e filtered
o
u
t
p
u
t
ˆ
(,
)
x
tM
is
gene
rat
e
d
.
ˆ
(,
)
x
tM
is rep
r
esen
ted in
(Fig
ure
9
b
). The lo
cu
s
1
,,
,
ˆ
o
Uu
M
C
i
s
pl
ot
t
e
d f
o
r
di
f
f
ere
n
t
02
and
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208
8-8
7
0
8
Id
en
tifica
tio
n
of
No
n
lin
ea
r S
y
stems S
t
ru
ct
u
r
ed
b
y
Wien
er-Hammerstein
Mod
e
l
(Ad
il Bro
u
r
i)
17
3
a sam
p
l
e
of t
h
e obt
ai
ne
d c
u
r
v
es i
s
sh
o
w
n
b
y
Fi
gure
s
1
0
a-
b. It
i
s
seen t
h
at
1
,,
,
ˆ
o
Uu
M
C
associated to
2
i
s
not
lin
ear
((
)
k
). F
u
rt
h
e
r, the cu
r
v
e
1
,,
,
ˆ
o
Uu
M
C
associated to
*
3.
0
1
is affin
e
portio
n. Ad
d
ition
a
l
l
y, it
i
s
seen t
h
at the sign
of
*
S
is di
ffe
rent
fr
om
that o
f
1
,,
,
ˆ
o
Uu
M
C
.
We h
a
v
e
t
h
us show
n th
at
*
1
ˆ
(
)
6.
15
(
)
M
rd
(m
odul
o
2
π
).
From
Fi
g
u
re 1
0
b
,
one
ha
s
ˆ
(
)
0.
97
Mk
s
.
The est
i
m
at
or (2
0
b
) i
s
u
s
ed t
o
get
est
i
m
a
t
e
s of t
h
e m
odul
u
s
()
k
Gj
. Accordi
n
gly,
ˆ
()
0
.
9
7
.
Mk
Gj
Othe
r
res
u
lts s
h
o
w
i
n
Ta
ble
1
.
Fi
gu
re 4.
N
o
nl
i
n
ear ha
rd element
f
(.) co
nsid
ered in
sim
u
lati
o
n
Fi
gu
re 5.
u
(
t
),
y
(
t
) an
d th
e
und
istu
rb
ed
o
u
t
pu
t estim
a
t
e
Fi
gu
re
6.
The
t
r
ue
N
.
L a
n
d
se
t
of
est
i
m
a
t
e
d poi
nt
s
-0
.
8
-0
.
6
-0
.
4
-0.
2
0
0.
2
0.
4
0.
6
0.
8
-0.
8
-0.
6
-0.
4
-0.
2
0
0.
2
0.
4
0.
6
0.
8
1
v
2
v
3
*
*
S =
-
1
2
0
200
400
600
800
100
0
1200
1400
1600
-1
-0
.
5
0
0.
5
1
Th
e i
n
p
u
t
s
i
g
n
a
l
u
(
t
)
0
200
400
600
800
100
0
1200
1400
1600
-1
-0
.
5
0
0.
5
1
Ti
m
e
(
s
)
y(
t
)
an
d
X
(
L
)
U
=
0
.
4
10
U =
0
.
4
8
6
U =
-
0
.
8
2
1
2
U =
0
X
=
-
0.
79
4
X =
0
.
1
9
5
3
X =
0
.
0
1
6
X =
0
.
4
0
1
X
=
-
0.
21
7
X =
-
0
.
6
0
2
8
X =
-
0
.
3
9
X =
0
.
6
1
9
11
X
=
0
.
8
1
10
X =
-
1
.
0
1
j
^
^
^
X
=
0
.
9
8
^
^
^
^
^
^
^
^
^
-1
-0
.
8
-0
.
6
-0
.
4
-0
.
2
0
0.
2
0.
4
0.
6
0.
8
1
-1
-0.
8
-0.
6
-0.
4
-0.
2
0
0.
2
0.
4
0.
6
0.
8
1
Th
e
t
r
u
e
N
.
L
s
e
t
of
po
i
n
t
s
(
U
,
X
(
L
)
)
j
j
^
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I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 1, Feb
r
uar
y
20
1
6
:
16
7 – 17
6
17
4
Figure
7. The
s
t
eady-state out
put
y
(
t
) o
b
t
a
i
n
e
d
ove
r o
n
e peri
od
Figure
8. a. T
h
e locus
1
,,
,
ˆ
o
Uu
M
C
fo
r
5.
1
(
)
k
; b
.
1
,,
,
ˆ
o
Uu
M
C
fo
r
6.
1
(
)
k
Figure
9. a. T
h
e steady-state
of
y
(
t
);
b.
O
n
e
pe
ri
o
d
of
ˆ
(,
)
x
tM
Figure 10.
a
.
1
,,
,
ˆ
o
Uu
M
C
fo
r
2(
)
rd
; b
.
1
,,
,
ˆ
o
Uu
M
C
for
3.
01
(
)
rd
13
00
140
0
150
0
16
00
17
00
18
00
-2
.
5
-2
-1
.
5
-1
-0
.
5
0
0.
5
1
Ti
m
e
(
s
)
T
h
e t
r
u
e
sy
st
em
ou
t
p
u
t
y
(
t
)
-2
-1.
5
-1
-0
.
5
0
0.
5
1
1.
5
2
-2.
5
-2
-1.
5
-1
-0.
5
0
0.
5
1
-2
-1.
5
-1
-0.
5
0
0.
5
1
1.
5
2
-2.
5
-2
-1.
5
-1
-0.
5
0
0.
5
1
2000
2100
2
200
2300
2400
2500
2600
2700
2800
2900
-1
-0.
9
5
-0.
9
-0.
8
5
-0.
8
-0.
7
5
-0.
7
-0.
6
5
-0.
6
-0.
5
5
-0.
5
Ti
m
e
(
s
)
T
h
e sy
st
em
ou
t
p
u
t
y
(
t
)
100
20
0
300
400
50
0
600
-1
-0
.
9
5
-0
.
9
-0
.
8
5
-0
.
8
-0
.
7
5
-0
.
7
-0
.
6
5
-0
.
6
-0
.
5
5
-0
.
5
Ti
me
(
s
)
T
h
e
fil
te
r
e
d
o
u
tp
ut
-0
.
2
5
-0
.
2
-0
.
1
5
-0
.
1
-0
.
0
5
0
0.
05
0.
1
0.
1
5
0.
2
0.
25
-1
-0
.
9
5
-0
.
9
-0
.
8
5
-0
.
8
-0
.
7
5
-0
.
7
-0
.
6
5
-0
.
6
-0
.
5
5
-0
.
5
-0
.
2
5
-0
.
2
-0
.
1
5
-0
.
1
-0
.
0
5
0
0.
05
0.
1
0.
1
5
0.
2
0.
2
5
-1
-0
.
9
5
-0
.
9
-0
.
8
5
-0
.
8
-0
.
7
5
-0
.
7
-0
.
6
5
-0
.
6
-0
.
5
5
-0
.
5
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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ECE
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:
208
8-8
7
0
8
Id
en
tifica
tio
n
of
No
n
lin
ea
r S
y
stems S
t
ru
ct
u
r
ed
b
y
Wien
er-Hammerstein
Mod
e
l
(Ad
il Bro
u
r
i)
17
5
Tabl
e 1. Fre
q
u
e
ncy
gai
n
est
i
m
a
t
e
s
F
requency gain est
i
m
a
tes
ˆ
()
M
k
Gj
k
(
rd/s
)
0.
01
0.
05
0.
1
()
k
(
rd
)
6.
1 5.
42
4.
74
ˆ
()
M
k
(
rd
)
6.
15
5.
38
4.
78
()
k
Gj
0.
99
0.
86
0.
62
ˆ
()
M
k
Gj
0.
97
0.
88
0.
65
6.
CO
NCL
USI
O
N
We ha
ve de
v
e
l
ope
d a new
freq
u
e
n
cy
i
d
ent
i
f
i
cat
i
o
n
meth
od
to
d
eal
with
W
i
en
er-Hammerstein
syste
m
s; th
e id
en
tificatio
n
pro
b
l
em
is ad
dressed
i
n
p
r
ese
n
ce
of hard nonlinearity
an
d two lin
ear sub
s
yste
m
s
o
f
st
ru
ct
u
r
e en
tirely u
nkn
own. Th
e
p
r
esen
t stud
y c
onstitu
tes a sig
n
ifican
t prog
ressin
frequ
e
n
c
y-d
o
m
ain
id
en
tificatio
n of b
l
o
c
k-o
r
ien
t
ed
n
o
n
lin
ear syste
m
id
en
tifi
cati
o
n. Th
e
o
r
i
g
inality o
f
th
e
p
r
esen
t stud
y lies
in
the
fact th
at th
e syste
m
is n
o
t
necessarily p
a
rametric a
nd o
f
st
ruct
u
r
e t
o
t
a
l
l
y
un
kn
o
w
n
.
A
not
her
feat
u
r
e of t
h
e
m
e
thod is the fact that the exciting signals a
r
e easily
generated and the es
tim
a
tion algori
thm
s
can be sim
p
ly
im
pl
em
ent
e
d.
The
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p
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gai
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s
an
d
pha
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f
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I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
6, No
. 1, Feb
r
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20
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6
17
6
BIOGRAP
HI
ES OF
AUTH
ORS
Adil Brouri In
2000, he obtain
e
d the Agrég
a
tion of
Electrical Engineer
ing fr
om the ENSET,
Rabat, Morocco and, in 2012, he
obtained a Ph.D. in Automatic Control from th
e University
of
Mohammed 5,
Morocco. He has been Professeu
r-A
grégé for several
y
e
ars. Since 2013 he joined
the ENS
A
M
,
Univers
i
t
y
of M
y
Is
m
a
il in M
e
knes
,
M
o
rocco and
a
m
e
m
b
er of the L
2
M
C
Lab. His
res
earch in
ter
e
s
t
s
include nonlin
ear s
y
s
t
em
iden
tification and no
nlinear
control. He published
several pap
e
rs o
n
these topics.
E
-
ma
i
l
:
a.
brouri@e
n
sa
m-umi
.
a
c
.
ma
& brouri_
adil@
y
ahoo.fr
Sm
Slassi Teach
er a
t
high s
c
hoo
l. His r
e
sear
ch
i
n
terests in
clud
e
nonline
a
r s
y
st
e
m
identifi
c
a
tion
and nonlinear
co
ntrol.
Evaluation Warning : The document was created with Spire.PDF for Python.