Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
5, N
o
. 1
,
Febr
u
a
r
y
201
5,
pp
. 10
2
~
11
0
I
S
SN
: 208
8-8
7
0
8
1
02
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
Robust Synchronization of
the Unified Chaotic System
Hatem Tr
abel
si*,
Mohamed Benrejeb*
* Labor
atoir
e
d
e
Rech
erch
e
en A
u
tom
a
tique
(LA.
R.A),
ENIT
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Oct 6, 2014
Rev
i
sed
No
v
31
, 20
14
Accepted Dec 20, 2014
This
paper investigates the
s
y
n
c
hroni
zation pro
b
lem of the un
ified ch
aotic
s
y
stem. The cas
e
of identical,
bu
t
unknow
n, master and
slave unif
i
ed
chaotic
s
y
stems
is
considered. Based
o
n
co
mpound matrices formalis
m, a unif
i
ed
s
y
nchronization
control sch
e
me is
proposed independen
t
ly
of th
e unknown
s
y
stem
par
a
m
e
te
r. Sim
u
lation
res
u
lts ar
e provided to show the
ef
fectiveness
of the presen
ted
scheme.
Keyword:
C
o
m
pou
nd
M
a
t
r
i
ces
Robustne
ss
Sy
nch
r
oni
zat
i
o
n
Uni
f
ied C
h
aoti
c System
Copyright ©
201
5 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
:
Hatem
Trabelsi,
Laboratoi
r
e de Recherc
h
e
e
n
Aut
o
m
a
tique (LA.R.A),
Ecol
e Nat
i
onal
e
d
’
I
ngé
ni
eu
rs de
T
u
ni
s,
BP 37
,
100
2 Tu
n
i
s Belv
éd
èr
e, Tu
n
i
sia
Em
a
il: 7
a
te
m
.
t
r
ab
elsi@g
m
a
il.
co
m
1.
INTRODUCTION
Ch
ao
s sy
n
c
hron
izatio
n is an
attractiv
e ph
eno
m
en
o
n
i
n
vol
v
e
d
i
n
a vari
et
y
o
f
real
-l
i
f
e
pr
ocesses
.
In
19
9
0
, Pec
o
ra a
nd C
a
rr
ol
p
r
o
v
e
d [
1
]
t
h
at
t
w
o
chaot
i
c
sy
st
e
m
s can sy
nchr
oni
ze.
Thi
s
m
eans t
h
at
one
s
y
ste
m
(slav
e
syste
m
),
can
fo
llo
w
th
e
traj
ect
o
r
ies of an
o
t
h
e
r
on
e
(master system
), whe
n
a
n
a
p
pr
o
p
ri
at
e c
ont
rol
l
a
w i
s
appl
i
e
d
.
Si
nce
t
h
en,
m
a
ny
sy
nch
r
oni
zat
i
o
n schem
e
s
hav
e
been
pr
o
pose
d
[
2
]
,
[
3
]
,
[4]
,
[
5
]
s
u
ch as
nonlinea
r
co
n
t
r
o
l
[
6
],
n
o
n
lin
ear
ob
server
[4
],
[7
],
[8
]
ad
ap
tiv
e con
t
r
o
l [9
],
[
1
0
]
,
[1
1
]
activ
e contr
o
l [12
]
,
[1
3
]
, [
1
4
]
,
fuzzy
c
o
ntrol [15], [16]
, a
n
d
backstepping c
ont
rol
[17]
,
[18]. More
recently
, in 2002
,
L
ü
a
n
d Che
n
et
al. [19]
in
v
e
stig
ated
some sp
ecific chao
tic system
s an
d
d
e
scri
b
e
d the
m
in
a un
ified fo
rm
kn
own
as th
e
u
n
i
fied
ch
ao
tic
sy
st
em
.
Thi
s
s
y
st
em
pl
ay
s
a
ve
ry
i
m
port
a
nt
r
o
l
e
i
n
t
h
e
s
t
udy
of
t
h
e
ge
n
e
ral
i
zed L
o
re
n
z
sy
st
em
fam
i
ly
[2
0]
.
Dif
f
eren
t resu
lts related
t
o
t
h
e
u
n
i
fied
ch
ao
tic syste
m
are repo
rted
in literatu
re [
2
1
]
,
[2
2
]
, [23
]
, [24
]
, [25
]
.
In t
h
i
s
pap
e
r
,
we p
r
o
p
o
se a
sy
nch
r
o
n
i
zat
i
o
n co
nt
r
o
l
sche
m
e
based o
n
t
h
e co
nce
p
t
of
com
pou
nd
matrices, in
ord
e
r t
o
sy
n
c
hron
ize two
id
en
t
i
cal b
u
t
u
nkn
own un
ified
chao
tic syste
m
s. Co
m
p
o
u
n
d
m
a
trices
[26
]
, [27
]
, h
a
ve
in
teresting
sp
ectral p
r
op
ert
i
es
m
a
k
i
n
g
o
f
th
em
a
p
o
werfu
l
too
l
for stabilit
y
stu
d
y
[26], [
28
].
In
[2
7
]
, ex
istence o
f
Hop
f
Bifu
rcation
i
n
d
y
na
m
i
cal syst
e
m
s
an
alysis and
st
ab
ility o
f
m
a
tri
ces are i
n
v
e
sti
g
ated
using t
h
e c
o
m
pound m
a
trices form
alis
m
.
The
pa
pe
r i
s
or
gani
ze
d
as
f
o
l
l
o
ws
.
I
n
Sect
i
o
n
2,
we
i
n
t
r
od
uce
b
r
i
e
fl
y
t
h
e
uni
fi
ed
cha
o
t
i
c
sy
st
em
an
d
t
h
e t
h
e
o
ret
i
cal
t
ool
use
d
i
n
t
h
i
s
w
o
r
k
,
n
a
m
e
l
y
t
h
e com
poun
d m
a
t
r
ix m
e
t
hod.
I
n
Sect
i
on
3,
r
o
b
u
s
t
sy
nch
r
o
n
i
zat
i
o
n c
ont
rol
sche
m
e
i
s
pr
o
pose
d
fo
r i
d
ent
i
cal
b
u
t
un
k
n
o
w
n
m
a
st
er and
sl
ave
uni
fi
ed
ch
aot
i
c
syste
m
s. Ob
tai
n
ed resu
lts are
tested
th
rou
g
h
n
u
m
erical si
mu
latio
n
s
, in Sectio
n
4
.
2.
PROBLEM STATEMENT
The
u
n
i
f
i
e
d
ch
aot
i
c
sy
st
em
[19]
can
be
ex
pre
ssed
by
t
h
e
f
o
l
l
o
wi
ng
di
f
f
e
r
en
t
i
a
l
equat
i
o
ns:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Rob
u
s
t
S
y
n
c
hro
n
i
za
tion
o
f
the Un
ified
Chaotic S
y
stem
(Hatem Tr
abelsi)
10
3
12
1
21
3
2
31
2
3
(2
5
1
0
)
(
)
(2
8
3
5
)
(2
9
1
)
(1
/
3
)
(
8
)
xx
x
x
xx
x
x
xx
x
x
(1
)
whe
r
e
1
x
,
2
x
and
3
x
are state varia
b
le
s and
a c
onst
a
nt
pa
ram
e
t
e
r
.
For
v
a
r
y
ing co
n
tinuo
usly in
[
0
,
1
]
, t
h
e whole f
a
m
i
l
y
o
f
sy
ste
m
s is ch
ao
tic [29
]
.
It includ
es, in
p
a
rticu
l
ar
,
the
canonical Lore
nz [30],
C
h
en
[
3
1]
an
d L
ü
[2
9]
cha
o
tic
syste
m
s respec
tively for
0
,
1
an
d 0.8.
Let syste
m
(1)
be the
m
a
ster syste
m
and
defi
ne t
h
e
sl
ave
sy
st
em
as
12
1
1
21
3
2
2
31
2
3
3
(2
5
1
0
)
(
)
(2
8
3
5
)
(2
9
1
)
(1
/
3
)
(
8
)
yy
y
u
yy
y
y
y
u
yy
y
y
u
(2
)
whe
r
e
1
y
,
2
y
and
3
y
are state vari
ables of the sl
ave system
,
th
e p
a
ram
e
ter i
n
tro
d
u
c
ed
fo
r
th
e m
a
ster
syste
m
. Given
the error
vector
12
3
(,
,
)
T
ee
e
e
defi
ned
by
ii
i
ey
x
,
1..3
i
,
(3
)
1
u
,
2
u
and
3
u
are th
e co
n
t
ro
l laws to
b
e
d
e
sign
ed
such
th
at t
h
e
foll
owi
n
g
er
ro
r
dy
nam
i
cal sy
stem
(4) is stable
12
1
2
1
1
21
1
3
2
1
1
3
2
2
31
2
3
1
2
3
3
(
2
5
10)(
)
(
2
5
1
0)(
)
(2
8
3
5
)
(2
9
1
)
(
2
8
3
5
)
(
2
9
1
)
(1
/
3
)
(
8
)
(1
/
3
)
(
8
)
ey
y
x
x
u
ey
y
y
y
x
x
x
x
u
ey
y
y
x
x
x
u
(4
)
Defi
ne t
h
e
ext
e
nde
d
st
at
e vect
or
()
i
as
12
3
1
2
3
(,
,
,
,
,
)
T
x
xx
y
y
y
(5
)
and the m
a
trices
T
and
N
by
10
0
1
0
0
01
0
0
1
0
00
1
0
0
1
T
(6
)
11
11
25
10
2
5
10
0
2
5
1
0
2
5
1
0
0
(.)
2
8
3
5
2
9
1
28
3
5
29
1
81
8
1
00
33
3
3
Nx
y
xy
(7
)
suc
h
that t
h
e e
r
ror
vector (3) a
n
d the
dynam
i
cal
err
o
r
cha
o
tic sy
stem
(4)
ca
n
be e
x
p
r
esse
d
by
eT
(8
)
(.
)
eN
u
(9
)
with
(.
)
uK
(1
0)
and
36
(.)
(
(.))
ij
Kk
R
is a non constant control gain m
a
trix to
be calculated such that (9) is stable.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
ECE Vo
l.
5
,
No.
1
,
Febru
a
ry
2
015
:
10
2
–
11
0
10
4
Ass
u
m
e
, furt
he
rm
ore, that
t
h
ere exists a m
a
trix
36
(.)
(
(.)
)
ij
Aa
R
such that
(.
)
(
.
)
(
.
)
NK
A
T
(1
1)
There
f
ore, the
dynam
i
cal error
sy
stem
(4) ca
n
be
red
u
ce
d to
(.
)
eA
e
(1
2)
Our aim
consists on expressi
ng m
a
trix
A
(.
)
en
tr
ies, wh
ich dep
e
nd on
th
o
s
e of
m
a
tr
ix
K
(.), then calculating the
gain m
a
trix
K
(.)
s
u
ch
that sy
s
t
em
(12
)
is st
able. In the
sequel, we use, for sim
p
lici
t
y
,
the notation
ij
k
instead
of
(.)
ij
k
.
3.
PROP
OSE
D
ROBU
ST S
Y
NC
HR
ONIZ
ATIO
N S
C
H
E
ME OF
TH
E M
A
STER
-S
LAVE
U
N
IFI
E
D
CH
AOTI
C S
Y
STEM
3.
1.
B
a
si
c Ide
a
Synchronization between
the master
sy
ste
m
(1) and t
h
e slave system
(2
) i
s
equivalent t
o
the stability
of the
dynam
i
cal error system
(12).
The stabilit
y study
of
the the character
istic m
a
trix
(.
)
A
of system
(12) is
per
f
o
r
m
e
d bas
e
d
on
the
co
m
poun
d m
a
trix m
e
thod.
R
e
lated p
r
elim
inary
n
o
tio
ns a
r
e intr
od
uces
in th
e
following.
Let
()
n
M
R
be the line
a
r space
of m
a
trices of size
n
x
n
with entri
e
s in
R
and let
A
be a m
a
trix in
()
n
M
R
and
k
an integer in [1,
n
]
.
W
e
note by
the exterior
product in
n
Rn
.
Defi
nition 1
[26], [27]:
The additiv
e com
p
ound m
a
trix
[]
k
A
of
A
, with
respect to
the
canonical basis in the
th
k
exterior product
space
kn
R
is a linear
operat
or
on
kn
R
and ca
n
be d
e
fine
d o
n
a
de
com
posable el
em
ent
12
..
.
k
vv
v
by
[]
11
1
(.
.
.
)
.
.
.
.
.
.
k
k
ki
k
i
A
vv
v
A
v
v
,
1
..
.
n
k
vv
R
(1
3)
Relations between entries
(
ij
a
) o
f
A
a
n
d th
ose
o
f
[]
k
A
(
ij
a
) are line
a
r
.
Let
i
be
an
i
n
teger
in
[1
,
k
n
C
]
.
I
f
w
e
n
o
t
e
b
y
(
i
) =
(
1
,
...
,
k
ii
) t
h
e
th
i
me
mb
er in th
e lex
i
co
gr
ap
h
i
c ord
e
r
i
ng
of
integer
k-t
u
ples such t
h
at
1
1.
.
.
k
ii
n
, we can obtain the additive com
pound m
a
trix
entries
from
the followi
ng result.
Proposition 1 [26], [27]:
11
..
.
,
(
)
(
)
,
(1
)
,
(
)
(
)
()
,
0(
)
(
)
.
kk
rs
ii
i
i
rs
ji
s
ij
r
aa
i
f
i
j
a
i
f
exactly
one
e
n
try
i
of
i
d
oe
s
n
o
t
occ
u
r
i
n
j
a
and
j
d
oe
s
n
o
t
occ
u
r
i
n
i
i
f
i
d
if
fe
rs
fr
om
j
i
n
t
wo
or
m
o
re
e
n
t
r
ie
s
(1
4)
In
pa
rticular
,
w
e
ha
ve
[1]
A
A
,
[]
()
n
At
r
a
c
e
A
and
for
A
3
()
M
R
11
22
23
1
3
[2
]
32
11
33
12
31
21
22
3
3
aa
a
a
Aa
a
a
a
aa
a
a
(1
5)
Defi
nition 2
[27]: Let
.
a vecto
r
no
rm
on
()
n
M
R
and
A
a m
a
trix in
()
n
M
R
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
7
0
8
Rob
u
s
t
S
y
n
c
hro
n
i
za
tion
o
f
the Un
ified
Chaotic S
y
stem
(Hatem Trabelsi)
10
5
The L
o
zinski
ǐ
m
easure (logarith
m
i
c
m
easure)
of
A
with respect to
.
is defi
ne
d
by
0
1
()
l
i
m
h
Ih
A
A
h
(
16)
As e
x
am
ples, Lozinski
ǐ
m
eas
ure
of a m
a
trix
A
with respect t
o
the
three c
o
m
m
on vect
or
norm
s
1
i
i
x
x
,
2
2
i
i
x
x
and
su
p
i
i
x
x
are
1
,
()
s
u
p
(
)
jj
ij
j
ii
j
A
aa
,
2
()
(
)
2
T
A
A
As
and
,
()
s
u
p
(
)
ii
i
j
i
jj
i
A
aa
(1
7)
whe
r
e
()
s
A
de
not
es
t
h
e m
a
xim
u
m
real
part
of
t
h
e
ei
gen
v
al
ues
o
f
A
.
Co
m
p
o
u
n
d
m
a
trices presen
t
a po
werfu
l
too
l
for th
e st
ab
ility stud
y of m
a
tri
ces.
Th
e
fo
llowing
resu
lt
will b
e
u
s
ed
in th
e
sequ
el.
The
o
rem
1
[27
]
: if
(1
)d
e
t
(
)
0
n
A
th
en
A
is stab
le if an
d on
ly
if
t
h
ere
ex
ists
a
Lo
zin
s
ki
ǐ
m
easure
on
()
m
M
R
suc
h
that
[2
]
()
0
A
,
2
n
mC
.
Accord
ing
to
th
eorem
1
,
th
e stab
ility o
f
th
e ch
aracteristic matrix
(.)
A
of sy
st
em
(12
)
can
be
st
udi
ed t
h
r
o
ug
h
its d
e
term
in
an
t and
its seco
nd co
m
p
o
und
m
a
trix
.
3.2.
Dynamical E
rror System Stability Study Ba
sed on Com
p
ound
Matrices
By so
lv
i
n
g equ
a
tio
n (1
1
)
,
we ob
tain
th
e ch
aracteristic m
a
trix
(.)
A
of
t
h
e
dy
na
m
i
cal
error
sy
s
t
em
11
12
13
21
22
1
2
3
31
1
3
2
3
3
25
10
25
10
(.
)
2
8
3
5
2
9
1
81
33
kk
k
Ak
k
x
k
kx
k
k
(1
8)
fr
om
which is
ded
u
ce
d the
se
con
d
c
o
m
pou
n
d
m
a
trix as e
x
p
r
esse
d in
(
1
4
)
11
2
2
1
2
3
1
3
[2
]
13
2
1
13
3
1
2
31
21
2
2
33
41
1
76
38
(.)
2
5
1
0
33
86
1
1
28
35
33
kk
x
k
k
Ax
k
k
k
k
kk
k
k
(1
9)
In addition,
we obtain
relatio
ns bet
w
een entri
e
s of m
a
trix
(.
)
K
w
h
ich is
a
bloc
k
interde
p
e
nde
nt
m
a
trix
14
11
15
1
2
1
6
1
3
24
2
1
25
2
2
26
1
1
2
3
3
4
31
35
1
1
3
2
36
33
kk
k
k
kk
kk
k
k
k
y
x
k
kk
k
x
y
k
kk
(2
0)
Referri
ng to t
h
e com
pound matrix en
tries and the determ
in
ant of m
a
trix
A
(.
), we pr
o
p
o
se,
by
t
h
e use of
theorem
1, the
following
results.
The
o
rem
2:
G
l
obal
sy
nc
hr
o
n
i
zation is
ac
hieved
betwee
n
uni
fied
cha
o
tic sy
stem
s descr
i
bed
by
(
1
) a
n
d
(
2
)
inde
pen
d
e
n
tly
of
the
param
e
ter
, if t
h
e foll
owi
n
g control l
a
w is applied
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
ECE Vo
l. 5
,
No. 1
,
Febru
a
ry
2
015
:
10
2
–
11
0
10
6
11
1
1
21
2
2
1
1
3
31
1
2
()
(
)
(
1
.16
2
.
2
8
1
.16
)
(
)
(
)
()
ux
x
y
ux
x
y
y
x
y
ux
y
y
(2
1)
whe
r
e
15.
33
and
44.
50
.
Pr
oof
:
All diag
o
n
al elem
ents of t
h
e c
o
m
pou
nd m
a
trix
A
[2
]
de
pe
nd
o
n
k
11
and
k
22
. L
e
t loo
k
f
o
r a
ga
in m
a
trix invol
vin
g
only
k
11
and
k
22
and c
o
nse
que
ntly
k
14
and
k
25
. By substituting all other
ij
k
elem
ents in
A
by
0,
m
a
trices
A
and
A
[2
]
bec
o
m
e
11
22
1
1
2
5
10
25
1
0
0
(.
)
2
8
3
5
2
9
1
81
0
33
k
Ak
x
x
(2
2)
11
22
1
[2
]
11
1
22
41
1
0
76
38
(.
)
2
5
1
0
33
86
1
1
02
8
3
5
33
kk
x
Ax
k
k
(2
3)
The
determ
inant of m
a
trix
A
(.
)
is give
n by
32
11
22
2
11
1
2
2
1
1
2
2
22
1
1
1
1
22
11
22
11
29
25
det(
(.)
)
50
(
1
9
5
)
33
1
(
2
5
1730
77
70
)
3
80
8
8
10
720
33
3
Ak
k
xk
k
k
k
x
k
x
kk
kk
(2
4)
By ap
p
l
ying theo
r
e
m
1
,
using th
e Lozin
s
k
i
ǐ
m
easure
with respect t
o
|.|
1
, syste
m
(1
1) is stable if the followi
ng
inqualities are
satisfied
11
22
1
41
1
0
kk
x
(2
5a)
11
1
76
38
28
35
0
33
xk
(
25b
)
22
11
86
25
10
0
33
k
(2
5c)
32
11
22
2
11
1
2
2
1
1
2
2
22
1
1
1
1
22
11
22
1
1
29
2
5
50
(
1
95
)
33
1
(
(
25
17
30
)
7
7
7
0
)
3
80
8
8
10
7
2
0
0
33
3
kk
xk
k
k
k
xk
x
k
k
k
k
(
25d
)
Inequalities (25a),
(25b) and
(25c) are suf
f
icient
conditions
guarantying t
h
at
and (25d)
is related
to t
h
e
determ
inant of
matrix
A
(.
)
.
Left-hand si
des of inequal
ities (25a),
(25b)
and
(25c), can
be maj
o
rated
gi
ven that 0
1.
Furt
herm
ore,
polynom
i
al in
equality (25d) can
be sa
tisfied
whe
n
a
ll
m
onom
ials
are non
positive
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
7
0
8
Rob
u
s
t
S
y
n
c
hro
n
i
za
tion
o
f
the Un
ified
Chaotic S
y
stem
(Hatem Tr
abelsi)
10
7
C
o
n
d
itions
(
2
5
a
),
(2
5
b
)
,
(
2
5c)
an
d
(2
5
d
) ca
n
be the
r
ef
o
r
e r
e
duce
d
t
o
11
1
46
/
3
kx
(2
6a)
22
60
k
(
26b
)
11
22
29
2
5
57
kk
(2
6c)
The
gain m
a
tri
x
entry
k
11
ca
n
be c
h
osen in t
h
e form
11
1
kx
with
46
15.33
3
(2
7)
Substituting (26c) in
(27), it com
e
s
22
1
1
29
2
9
57
1
.
16
1.16
2
.
2
8
25
25
2
5
kx
x
(2
8)
and
a possible choice of
the gain m
a
trix entry
k
22
is
22
1
1.
16
1.16
2.
28
kx
with
0
(2
9)
Give
n the c
o
n
s
traint o
n
the
param
e
ter
an
d th
e n
e
w ex
pressio
n
of
k
22
, (2
6c) h
o
lds fo
r
eve
r
y
>
44
.5
0.
Finally
, by calculating the
ot
her entries of the gai
n
m
a
trix
K
(.), acc
o
r
di
n
g
to (
2
0), a
n
d
usin
g the
relation t
h
e
cont
rol la
w e
x
pressi
o
n
of t
h
e
o
rem
2 is
retrie
ved
.
6
1
ii
j
j
j
uk
(3
0)
Note that
an
d
rep
r
ese
n
t tunin
g
para
m
e
ters for t
h
e desi
g
n
e
d
c
ont
roller
use
d
to en
hance
sy
stem
perform
a
nces.
An optim
a
l
choice of t
h
ese
param
e
te
rs is done through tria
l and error
process.
4.
SIMULATION RESULTS
In this
section, 3 cases
are
c
onsi
d
ere
d
to
sh
ow th
e ef
f
ectiv
en
ess
o
f
th
e pr
opo
sed
m
e
t
h
od
:
=
0
(Lore
n
z chaoti
c syste
m
),
= 0.
8
(Lü
cha
o
tic sy
stem
) and
=
1
(
C
hen
ch
ao
tic syste
m
)
.
Co
rr
espo
nd
ing
sim
u
lation res
u
lts are re
pre
s
en
ted res
p
ectivel
y
in fi
gu
re
1,
2
an
d
3.
Figure 1.
State trajectories of
m
a
ster and sla
v
e system
s for
=
0. C
ont
rol
is activated at t
i
m
e
t=1.
Differential equations are
solved
u
s
in
g
th
e fou
r
th
-o
rder
R
u
ng
e–
Ku
tt
a m
e
thod wi
th a tim
e step size
equal t
o
0.
00
1.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
ECE Vo
l. 5
,
No. 1
,
Febru
a
ry
2
015
:
10
2
–
11
0
10
8
Figure 2.
State trajectories of
m
a
ster and sla
v
e system
s for
=
0.8. C
ont
rol is activated at
ti
m
e
t=1.
In all sim
u
lat
i
ons, the constant param
e
ters
and
are
res
p
ectively chosen e
qual t
o
15.5 a
n
d
55.
Differe
n
t
initial conditions are c
o
nside
r
ed for t
h
e m
a
ster and the
slave system
s and
ar
e res
p
ectivel
y fixed t
o
(-2,
-1, 2)
and
(1,
0,
0.6). In the
thre
e cases,
we
c
a
n
notice th
at
the traj
ectories
of th
e co
ntr
o
lle
d s
l
a
v
e sys
t
e
m
synchronize
with those
of t
h
e m
a
ster system
. Num
e
ri
cal sim
u
lations
ha
ve s
h
ow
n
the
ef
fective
n
ess
of the
pr
o
pose
d
m
e
thod
.
Figure 3.
State trajectories of
m
a
ster and sla
v
e system
s for
=
1. C
ont
rol
is activated at t
i
m
e
t=1.
Unlike
othe
r r
e
po
rted
results
, as in [2
4]
, [
3
2]
and
[3
3]
,
the control law
designed in
thi
s
work is independent
of t
h
e chaotic
syste
m
param
e
ter
. For t
h
is reason it’s qualified as
robu
st. More
ove
r, for the
speci
fic
cases
of
Lore
nz, C
h
en
and L
ü
chaoti
c system
s, the
pe
rform
e
d
sim
u
la
tions i
ndi
cate that
sy
nchronization is
achieved
faster t
h
an
in
o
t
her
pre
v
io
us
wo
rk
s [
2
3]
, [
2
4]
, [
3
2]
.
5.
CO
NCL
USI
O
N
In this
paper, i
s
investigated
th
e syn
c
h
r
on
izatio
n
of
id
en
tical, b
u
t
unk
nown, m
a
s
t
er
an
d
slav
e un
if
ied
chaotic sy
stem
s. The
pr
op
os
ed sy
nc
hr
oniz
a
tion sche
m
e
is based
on c
o
m
pou
nd m
a
trices form
alism
.
The
obtaine
d c
o
ntr
o
l law is
in
dep
e
nde
nt
of
the
u
n
k
n
o
w
n
sy
ste
m
param
e
ter and
is c
onse
q
ue
ntly efficient
for all the
fam
i
ly of consi
d
ered chaotic syste
m
s. Num
e
rical sim
u
lat
i
ons are
provided
to illustrate
the capability of the
pr
o
pose
d
m
e
thod
w
h
ich ca
n
be ap
plied to
a lar
g
e class
o
f
cha
o
tic sy
stem
s, with or w
ithout
unce
r
tainties.
A
p
o
ssible e
x
te
nsio
n
of
this
w
o
r
k
is t
h
e sy
nc
hr
o
n
iza
tion o
f
two dif
f
ere
n
t u
n
k
n
o
w
n
unified chaotic system
s.
REFERE
NC
ES
[1]
L.M
.
P
ecora
and
T
.
L.
Carrol
l
,
“
S
ynchron
i
z
a
tion
i
n
chao
tic
s
y
s
t
em
s
”
.
Ph
ys. Re
v
.
L
e
tt
. 64, pp. 821–8
25, 1990
.
[2]
L.M
.
P
ecor
a
an
d
T
.
L.
Carro
ll,
“
Driving
s
y
s
t
em
s
with
chaot
i
c
s
i
gnals
”.
Ph
ys
. Re
view
,
A
44
(4),
pp. 2374-2383,
1991.
[3]
E. Ot
t,
C.
Grebo
g
y
,
and
J.A.
Y
o
r
k
e, “
C
ontro
lling
chaos”,
Ph
ys. Re
v
.
L
e
tt
., 64 (1
1),
pp. 1
196-1
199, 1
990.
[4]
H. Nijmeijer
an
d I
.
M.Y
.
Mar
eels, “A
n observ
e
r
looks at s
y
n
c
hro
n
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IEEE
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r
ans. Cicu
its
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y
st
,
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882
–
890, 1997
.
[5]
P
.
P
.
Singh and
H. Handa, “V
arious S
y
nchroniza
tion
Schemes
for Chaoti
c D
y
namical S
y
stems”,
Int
e
rnation
a
l
Journal of
Sci
e
n
tifi
c
Engineering
and T
echno
logy
,
V
o
lume No.1, I
ssue No.3, pp
. 2
9
-33, 2012
.
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IJECE
ISS
N
:
2088-8708
Rob
u
s
t
S
y
n
c
hro
n
i
za
tion
o
f
the Un
ified
Chaotic S
y
stem
(Hatem Tr
abelsi)
10
9
[6]
E.
El
abbas
y
,
H.
Agiza
and
M
.
El-Des
s
o
k
y
,
“
G
l
obal
chao
s
s
y
nchronization
for
four scroll attractor
b
y
nonlin
ear
control”,
Sc
i.
Res,
Essay
1
,
pp
. 6
5–71, 2006
.
[7]
J. Mata-Machu
ca, R. Martín
ez-Gue
rra and
R.
Aguilar
-
Lóp
ez, “An e
xpon
ential po
ly
nomial observ
e
r fo
r
s
y
nchronization
of chao
tic s
y
s
t
ems”,
Commun.
Nonlinear S
c
i. N
u
mer
.
Simul
., 15
, pp
. 41
14–4130, 2010.
[8]
Z.
Zhang
,
H.
S
h
ao,
Z.
W
a
ng
and H.
S
h
en,
“
R
educed
-ord
er
observer design
for th
e s
y
nchr
onization of
th
e
genera
liz
ed
Lore
nz ch
aoti
c s
y
s
t
e
m
s
”
,
Appl. Math
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,
pp. 7614–7621
,
2012.
[9]
Y
.
Y
u
, “Adaptiv
e s
y
n
c
hronizatio
n
of a un
ified
c
h
aoti
c s
y
s
t
em
”,
Chaos, Solitons
and Fracta
ls
,
36, pp. 329-333
,
2008.
[10]
M. Y
a
n
,
X. Zh
eng and
J.
Zhen
,
“Sy
n
chron
i
zatio
n of
H
y
per
c
haotic S
y
stems under Acti
ve Adap
tive Sliding
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e
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esian Journal
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l
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n
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[11]
E.A. Umoh, “Adaptiv
e H
y
b
r
id
S
y
nchronization
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Loren
z
-84 S
y
stem with
Unce
rt
ai
n Pa
ra
me
te
rs”
,
TELKOMNIKA
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,
pp
. 52
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.
[12]
E.W
.
Bai
and K.E.
Lonngren
, “Sy
n
chron
i
zation
o
f
two Lor
e
nz s
y
stems using active con
t
rol”,
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BIOGRAP
HI
ES OF
AUTH
ORS
Hatem Trabelsi
was born in Tun
i
sia in 1980. He
obtain
e
d the Master of Automatic Control and
Signal Processing from the “Eco
le Nationale d’I
ngni
eurs de Tun
i
s” in
2007. He
is curren
t
ly
a
PhD student in
Automatic Control and Sign
al P
r
oces
s
i
ng in
LARA ENIT. His
c
u
rrent r
e
s
ear
ch
inter
e
sts ar
e con
t
rol and
s
y
n
c
hron
ization of
con
tin
uous and discrete ch
aotic s
y
stem
s.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
ECE Vo
l. 5
,
No. 1
,
Febru
a
ry
2
015
:
10
2
–
11
0
11
0
Mohamed BENREJEB has obtained
his Diploma of “Ingénieur I
DN” (French “Grande Ecole”)
in 1973,
a Master degr
ee of Au
tomatic Con
t
rol
in 1974,
a Ph.D
. in Automatic
Control of th
e
University
of Lille in
1976 and the
DSc
of the
sa
me University
in 1980. H
e
is
currently
a fu
ll
profes
s
o
r at
the
“
E
cole N
a
tion
a
l
e
d’Ingén
i
eurs
d
e
Tun
i
s
”
and
an
invited
P
r
ofes
s
o
r at
the
“
E
col
e
Centra
le d
e
Lil
l
e”.
His
res
e
arch
int
e
res
t
s
are
in
the
ar
ea
of an
a
l
y
s
is
and s
y
nth
e
s
i
s
of com
p
lex
s
y
stem
s based on classica
l and
non conventio
nal
approa
ches and recen
tl
y
in
discrete ev
ent
s
y
stems domain.
Evaluation Warning : The document was created with Spire.PDF for Python.