TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5362 ~ 53
6
7
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.526
3
5362
Re
cei
v
ed
De
cem
ber 1, 20
13; Re
vised
Ma
rch 19, 20
14; Accepted
April 10, 201
4
Controlled Synchroniza
tion by Limited Capacity
Communication Channel
Ziqi Dong*, Yongchun Li
u, Da Lin
Schoo
l of Auto
matic and El
ec
tronic Informati
on, Sichu
an U
n
iversit
y
of Sci
ence a
nd T
e
chnol
og
y, Sich
ua
n
643
00
0, Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: dongz
q2
009
@12
6
.com
A
b
st
r
a
ct
The w
o
rk focuses on sync
h
ron
i
z
a
tio
n
of
time-d
el
ay chaos syste
m
s
under i
n
for
m
ati
o
n
constrai
nts. A theoretic
al a
nalysis for ti
me-
del
ay
e
d
s
ystem by
li
mit
ed co
mmu
n
ic
ation c
han
ne
l
is
provi
ded. By
usin
g the
Ly
a
pun
ov-Krasovs
kii a
ppr
oac
h,
a sync
h
ron
i
z
a
t
i
on c
o
n
d
itio
n i
s
first obtai
ne
d.
Encod
e
r of b
i
n
a
ry cod
i
ng
is d
e
sig
ned
by this
cond
it
ion,
and
synchro
ni
z
a
ti
o
n
error
of this
systems te
nd
s
to
z
e
r
o
. Finally
, analytica
l
co
nditi
ons are
us
ed for
synchro
ni
z
a
ti
on of time-de
l
ay cha
o
s
systems u
nder
infor
m
ati
on co
nstraints, and s
i
mulati
ons
ver
i
fy the valid
ity of obtain
ed res
u
l
t
.
Ke
y
w
ords
: ch
aos synchr
oni
zation, li
mited c
apac
ity chan
ne
l, coder
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
Cha
o
s i
s
a
n
unu
sual
so
phisti
c
ated
n
online
a
r b
e
h
a
vior. Tho
u
g
h
ch
ao
s sy
stems a
r
e
determi
nisti
c
system
s,
t
hei
r beh
avior i
s
extremely se
nsitive to
initi
a
l facto
r
s
and
is un
predicta
b
le.
Cha
o
s
sy
n
c
h
r
oni
zat
i
on i
s
t
o
cont
r
o
l a
cha
o
s
sy
st
e
m
(nam
ed
r
e
sp
on
siv
e
sy
st
em)
so t
h
a
t
it
follows a
noth
e
r
cha
o
s sy
stem (name
d
d
r
ive sy
st
em)
[1]. Recently, the a
ppli
c
ations are
widel
y
applie
d in a variety of area
s, see e.g. [2-15].
Und
e
r ide
a
l condition
s, the
standa
rd a
s
sumpti
o
n
is that the com
m
unication chann
el is
faultless, that is, the obtained out
put is t
he sam
e
as t
he input of
the observe
r [16]. Neverthel
ess
it is just
not the situ
ation i
n
so
me a
c
tua
l
con
d
ition
s
. For exa
m
ple,
in networke
d
cont
rol
syste
m
s,
physi
cal pla
n
t
and co
ntroll
er are not sit
uated in
sam
e
place, and
measured co
ntrol si
gnal
s
are
transmitted via informatio
n netwo
rks [17]. Recent
ly, limitations of controlli
ng
synch
r
oni
zat
i
on
unde
r con
s
traints imp
o
se
d by a finite
cap
a
city
information chan
nel have b
e
e
n
well
analyzed.
Con
s
id
erin
g t
he limited
inf
o
rmatio
n
cha
nnel,
con
d
itio
ns
are u
s
e
d
to a
nalyze
cha
o
s sy
ste
m
s
synchro
n
ization. In this job,
synchroni
za
tion in time-d
elayed ch
ao
s systems by a
limited capa
city
informatio
n chann
el is inv
e
stigate
d
. We pre
s
e
n
t
a theoretical
an
alysis fo
r the
cou
p
led
syst
ems
with time
del
ay. Assu
ming
that input
si
gnal
s a
r
e co
d
ed
a
nd sent unde
r
info
rm
ation con
s
trai
nts,
the de
co
de
r
gets th
e finite
co
ded
signal
s by
no
ni
deal
inform
ation
cap
a
city
cha
nnel. Sin
c
e
o
n
ly
the finite-valued si
gnal
s
are tra
n
smitted,
ide
a
l an
alysis
can
n
o
t
be applied
to impleme
n
t
stabili
zation. The
Lyap
uno
v-Krasovskii approa
ch
a
n
d
encodin
g
pro
c
e
s
ses
are
di
scusse
d to d
eal
with
synchro
n
izatio
n for the time
-delay
ed
cha
o
ti
c system
by
th
e nonid
eal cap
a
city
informat
ion
cha
nnel.
2. Descrip
tio
n
of Problem
Con
s
id
erin
g the co
uple
d
system with time delay is o
f
the form:
)
(
1
i
x
f
m
ax
x
N
i
i
(1)
R
N
i
i
u
y
x
K
y
f
n
ay
y
i
)
(
)
(
1
(2)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Controlled S
y
nch
r
oni
zatio
n
by Lim
i
ted Capa
city Com
m
unication Chann
el (Ziqi
Don
g
)
5363
W
h
er
e
i
m
and
i
n
a
r
e
para
m
eters,
K
re
pre
s
ent
s
co
upling
stren
g
th.
R
u
me
an
s erro
r
comp
en
sat
i
o
n
.
i
is
a time f
unc
tion of the form:
N
t
t
i
i
i
1,2,
i
,
sin
0
(3)
And
0
mea
n
s zero-frequ
en
cy
comp
one
n
t,
i
represents amp
litudes.
A flow
chart t
o
illustrate
the system i
s
given in Figu
re 1.
Figure 1. Flow Ch
art for M
a
ster-slave Contro
lle
d Synchroni
zation
unde
r Inform
ation Co
nstra
i
nts
Definin
g
syn
c
hroni
zatio
n
error a
s
follo
wi
ng:
)
(
)
(
)
(
t
y
t
x
t
e
(4)
No
w que
stion
arises
whe
n
transmissio
n error
i
s
co
nsi
dere
d
. At the transmitter side the
sign
al
)
t
(
x
sho
u
l
d
be
co
ded
a
nd
cod
e
wo
rd
s a
r
e
se
nt via the limite
d
cap
a
city
com
m
unication
cha
nnel
at di
screte
sampli
ng time
in
sta
n
ts
0,1,2,
k
,
kT
t
s
k
, as
s
T
r
e
pre
s
en
ts
s
a
mp
lin
g
time. To sim
p
lify this anal
ysis, we a
ssume that ch
a
nnel noi
se a
nd tran
smi
ssi
on delay ma
y be
ignored. Sin
c
e co
ded
sig
n
a
l be
sent via
the noni
deal
cap
a
city cha
nnel, nam
ely,
)
t
(
x
]
k
[
x
k
,
transmissio
n error o
c
curs.
Assum
e
the
code
d si
gna
ls ca
n be o
n
the receivin
g end by sim
ilar
sampli
ng time
k
t
, zero-ord
er extrapolation
can be appli
ed to convert
the digital sequen
ce
]
[
k
x
to the cont
rolle
r si
de
)
t
(
x
. So, we
can
obtain
]
k
[
x
)
t
(
x
,
s
s
T
k
t
kT
)
1
(
. And
transmissio
n error can be
descri
be in th
e form of:
)
(
)
(
)
(
t
x
t
x
t
x
(
5
)
From (4) a
nd
(5), syn
c
h
r
oni
zation
can b
e
desig
ned a
s
:
)
(
)
(
)
(
)
(
t
t
y
t
x
t
e
x
(6)
A positive definite Lyapun
o
v
-Krasovskii functio
nal is d
e
fined a
s
:
d
t
e
t
p
t
e
t
V
N
i
t
i
1
0
)
(
2
2
)
(
)
(
)
(
2
1
)
(
(7)
Then,
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: 5362 – 53
67
5364
N
i
i
i
N
i
t
t
t
e
t
e
t
e
t
p
d
t
e
t
p
t
e
t
e
t
V
i
1
2
2
2
1
0
)
(
2
)]
(
)
(
)
(
)
(
[
)
(
)
(
)
(
)
(
)
(
)
(
(8)
A
ssu
ming t
h
a
t
0
)
(
t
p
for all
t
.
)
(
)
(
)
(
)
(
)
1
)
(
)
(
4
1
)
(
)
(
)
(
)
(
)
(
)
1
)(
(
2
)
(
)
(
)
1
(
)
(
)
(
)
1
)(
(
4
)
(
)
(
)]
(
[
)
(
)
(
)
(
)
(
)
1
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
2
1
2
2
2
1
2
2
2
2
2
1
2
1
2
t
e
u
t
e
t
K
t
e
y
f
n
t
p
t
p
K
a
t
e
u
t
e
t
K
t
e
t
p
y
f
n
t
e
t
p
t
e
t
p
y
f
n
t
e
t
p
K
a
t
e
u
t
t
Ke
t
e
t
p
t
e
t
p
t
e
t
e
y
f
n
t
e
K
a
t
V
R
x
N
i
i
i
R
x
i
i
i
N
i
i
i
R
x
i
N
i
i
i
N
i
i
i
i
i
i
(9)
We c
hoo
se e
rro
r co
mpe
n
s
a
tion
R
u
as:
)
(
t
K
u
x
R
(10)
Then,
)
(
),
(
)
(
)
1
)
(
)
(
4
1
)
(
(
)
(
2
2
1
2
2
t
e
Q
t
p
F
t
e
y
f
n
t
p
t
p
K
a
t
V
N
i
i
i
i
(11)
W
h
er
e
N
1
i
i
2
2
i
1
)
y
(
f
n
Q
i
and
)
(
4
)
(
),
(
t
p
Q
t
p
K
a
Q
t
p
F
. To
s
h
ow
0
)
t
(
V
, that is
enou
gh to
prove
0
min
F
. It only ha
ppe
n
s
whe
n
2
)
(
Q
t
p
i
s
sat
i
sf
ie
d wit
h
Q
K
a
F
min
. At last, a ne
ce
ssary co
ndi
tion for syn
c
h
r
oni
zation i
s
descri
bed of the form:
2
1
1
2
1
)
(
sup
N
i
i
i
i
y
f
n
K
a
(12)
In this way, the system
ca
n transmit finite in
formatio
n unde
r the nonid
eal ca
p
a
city informat
ion
cha
nnel.
Based
on the
stable
con
d
ition, encodin
g
pro
c
e
s
ses fo
r the ne
ce
ssa
r
y tran
smissi
on rate
can b
e
de
sig
ned. Introdu
ce memoryle
ss bina
ry
cod
e
r
to be a discretized ma
p a
s
follows:
)
sgn(
)
(
x
M
x
q
M
(13)
The ra
nge int
e
rval is
]
,
[
M
M
and
)
sgn(
of the form:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Controlled S
y
nch
r
oni
zatio
n
by Lim
i
ted Capa
city Com
m
unication Chann
el (Ziqi
Don
g
)
5365
0
1
0
1
)
sgn(
x
x
x
(14)
Evidently,
M
x
q
x
M
)
(
for all
x
su
ch t
h
at
M
x
x
2
:
.
What
i
s
more, every cod
e
word
sign
al in
clude
s
1
R
bit. Then o
u
tput of this
coder is
de
scri
bed a
s
)
(
x
q
x
M
. C
o
ns
id
er
the
seq
uen
ce of the cent
ral nu
mbers
0,1,2,
k
,
]
k
[
c
, where prima
r
y co
ndition is
0
0
c
. A
t
step
k
the cod
e
r comp
ares the
current
o
b
tained outp
u
t
]
[
k
x
with the number
]
[
k
c
, forming
the deviation
symbol
]
[
]
[
]
[
k
c
k
x
k
x
. It
can b
e
discretized
with a given
]
[
k
M
M
.
Then outp
u
t signal:
])
[
(
]
[
]
[
k
x
q
k
x
k
M
(15)
R
e
pr
es
e
n
t
s
an
R
-bit com
m
u
n
icatio
n symb
ol. Then we
can de
scribe
]
1
[
k
c
,
]
[
k
M
as
follow:
,
0,1
k
,
0
]
0
[
c
],
k
[
x
]
k
[
c
]
1
k
[
c
(16)
,
0,1,
k
,
)
(
]
[
0
M
M
M
k
M
k
(17)
1
0
re
pre
s
e
n
ts d
e
cay parame
t
er,
0
M
is initial
value and
M
st
and
s fo
r extremum
of
]
[
k
M
. To m
eet all
the
ran
g
e
of
initial valu
es of
0
x
,
0
M
mu
st b
e
la
rge
en
ou
gh. Equ
a
tion
s
(15
)
, (1
6),
an
d (1
7)
discu
s
s the
en
cod
e
r pro
c
e
d
u
r
e. T
he p
r
o
c
ed
ure
s
for this de
coder a
r
e
simil
a
r
to the code
r.
In orde
r to
study the re
lationship be
tween tran
smissi
on rate
and the a
c
hievable
accuracy of the co
der a
n
d
deco
der, we sup
p
o
s
e that gro
w
th rate o
f
)
(
t
x
is uniformly
bounde
d.
The a
c
curate
boun
d
x
L
for the rate of
)
(
t
x
is
x
C
L
x
x
sup
, where
x
is
from (
1
).
We
assume
the
uppe
r b
oun
d
of the tran
smissi
on
erro
r is
)
(
sup
t
x
t
. The total trans
m
is
s
i
on
err
o
r for ea
c
h
interv
al
]
[
1
,
k
k
t
t
sh
ould sat
i
sf
y
s
x
x
T
L
M
t
)
(
.
M
t
x
2
)
(
is the
s
u
ffic
i
ent c
ondition for all
t
. Sampling i
n
terval
s
T
must
meet
x
s
L
T
. If above con
d
itions
hold, codi
ng
i
n
terval
shoul
d satisfy
s
x
T
L
2
2
. We ho
pe t
r
an
smissi
on
rate
i
n
shorte
st
sampli
ng tim
e
is large en
o
ugh to achiev
e high efficie
n
cy. Tran
smi
ssi
on rate
R
ne
ed be g
r
eater
than or eq
ual
to
)
1
(
log
2
s
x
T
L
M
, so it satisfies the inequality.
)
1
(
log
2
s
x
T
L
M
R
(18)
If
s
T
is small and
R
is large e
n
ough, then a
n
arbitra
r
ily sm
all value of
can be a
s
sure
d.
3. Simulation Resul
t
s
For si
mulatio
n
we thin
k ab
out the time delayed sy
ste
m
in the following form:
)
(
1
1
sin
t
x
m
ax
x
(19)
R
t
u
y
x
K
y
t
n
ay
y
)
(
sin
)
(
)
(
1
1
(20)
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: 5362 – 53
67
5366
Followin
g
pa
ramete
r valu
es of this
co
upled
system
are cho
s
en:
0
.
1
a
,
0
.
4
1
m
,
0
.
2
0
,
05
.
0
1
a
,
001
.
0
1
,
01
.
2
K
,
s
t
fin
1000
,
45
x
L
,
5
0
M
,
2
M
,
)
1
.
0
exp(
s
T
, and
0
.
3
,
,
4
.
0
,
2
.
0
. The system exhib
i
ts a synchro
n
izatio
n beha
vior,
see in Fig
u
re 2.
Figure 2. Time Histo
r
ie
s of State Variabl
es
of Drive and Maste
r
System
s (19
)
an
d (20
)
Figure 3 sh
o
w
s the
state of t
he synch
r
onization erro
r system.
Figure 3. The
Synchro
n
ization Erro
r
4. Conclusio
n
Que
s
tion
s of
synchro
n
ization of the time
delaye
d
cha
o
tic syste
m
s
un
de
r
informatio
n
con
s
trai
nts are con
s
ide
r
e
d
. Based on th
e Lyapunov
-krasovskii app
roa
c
h, a sim
p
le con
d
ition is
obtaine
d
to ensure syn
c
hroni
zatio
n
.
Furthe
rmo
r
e,
encode
r an
alysis
deal
with the
s
e
chaotic
sy
st
em
s cle
a
r
ly
t
o
make sur
e
sy
n
c
h
r
o
n
izat
io
n
error tends to 0. Above prop
o
s
ed meth
od
is
successfully
used to the
system
with ti
me dela
y. M
o
reover, num
e
rical
simul
a
tions illustrate
the
feasibility.
Ackn
o
w
l
e
dg
ements
This
re
sea
r
ch is
sup
p
o
r
te
d by the You
t
h F
oun
datio
n
of Sich
uan
Provinci
al E
d
ucation
Dep
a
rtme
nt (No. 1
1
ZB09
7), the Tale
nts Proj
e
c
t
of Sichua
n University of Scien
c
e an
d
Enginee
ring
(No. 20
11
RC0
7
), the Key p
r
oject of A
r
tificial Intelligence Key Laboratory of Sic
h
uan
Province (No. 2011
RZJ02
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Controlled S
y
nch
r
oni
zatio
n
by Lim
i
ted Capa
city Com
m
unication Chann
el (Ziqi
Don
g
)
5367
Referen
ces
[1]
Yanb
o Ga
o,
Xi
aome
i
Z
h
a
ng,
Guopi
ng
Lu.
D
i
ssipat
iv
e s
y
n
c
hron
izatio
n
of
non
lin
ear
cha
o
t
ic s
y
st
ems
und
er informati
on constra
i
nts.
Information Sci
ences
. 20
13; 2
25: 81-9
7
.
[2]
Kazu
yuk
i
Aihar
a. Chaos a
nd it
s appl
icat
io
ns, Proced
ia IUT
A
M, 2012; 5: 19
9-20
3.
[3]
BR An
drievsk
y, AL
F
r
adk
o
v
. Contro
l a
n
d
o
b
servati
o
n
via c
o
mmu
ni
cation
cha
n
n
e
l
s
w
i
th
limite
d
band
w
idth.
Gyroscopy a
nd N
a
vigati
o
n
. 20
10; 1: 126-1
33.
[4]
Ale
x
a
nder
LF
r
adkov, Bor
i
s A
ndri
e
vsk
y
,
Rob
i
n J Eva
n
s. Co
ntroll
ed s
y
nc
hr
oniz
a
tion
un
de
r informati
o
n
constrai
ns.
Phys. Rev.E,
2008; 78: 036
210.
[5]
Ale
x
a
nder
L F
r
adkov, Bor
i
s
Andri
e
vsk
y
,
R
J
Evans. Ch
a
o
tic obs
erver-b
ased s
y
n
c
hro
n
i
zatio
n
un
der
informati
on co
nstraints.
Phys
. Rev.E,
2006; 73: 066
20
9.
[6]
M Eisencraft, RD Fanganiello, JMV Grzy
bow
ski, DC
Soriano, R Attu, AM
Batista, EEN Macau, LHA
Monteir
o
, JMT
Roma
no, R Su
yam
a
, T
Yone
yama. Ch
a
o
s-b
a
sed c
o
mmun
i
cation s
y
stems
in no
n-id
ea
l
chan
nels.
Comm
un Nonlinear Sci Num
e
r Simula
. 20
12; 17:
470
7-47
18.
[7]
Cunfa
ng F
e
ng,
Yan Z
h
a
ng, Ji
ntu Sun, W
e
i
Qi, Yi
ngh
ai W
a
ng. Gener
al
ize
d
pr
o
j
ective s
y
nchro
n
izati
o
n
in time-d
ela
y
e
d
chaotic s
y
ste
m
s,
Chaos, Sol
i
tons an
d F
r
actals.
200
8; 38: 743-
747.
[8]
Dibak
ar Ghos
h, Santo Ba
nerj
ee. Ada
p
ti
ve
schem
e for s
y
nchr
oniz
a
tion-
base
d
multip
aramete
r
estimatio
n
from a singl
e cha
o
tic time series
and its ap
plic
a
t
ion.
Phys.Rev.
E.
2008; 78: 0
562
11.
[9]
Gexi
a
W
a
ng. Cha
o
s
s
y
nc
hr
oniz
a
tion
of d
i
screte-time d
y
n
a
mic
s
y
st
e
m
s
w
i
th a li
mited
c
apac
it
y
communic
a
tio
n
chann
el. No
nli
near D
y
n. 2
0
1
1
; 63: 277-
283.
[10] CK.Ahn,
An
H
a
ppro
a
ch to
anti
-
s
y
nc
hro
n
izati
o
n for cha
o
tic s
y
stems.
Phys
ic
s Letter A
. 20
0
9
; 373:
172
9-17
33.
[11]
Vidh
ya
char
an
Bhaskar,
La
urie
L Jo
iner.
A
daptiv
e rat
e
cod
i
n
g
usi
ng co
nvo
l
utio
nal c
o
d
e
s fo
r
as
ynchr
ono
us
code
divis
i
o
n
multipl
e
acc
e
s
s
commun
i
cati
on ov
er sl
o
w
l
y
fadin
g
ch
an
ne
ls.
Co
mp
uters
and El
ectrical
Engi
neer
in
g
. 2005; 31: 2
17-2
40.
[12]
Ale
x
a
nder
L F
radkov, Boris
Andri
e
vsk
y
,
R
o
bi
n J Evans. S
y
nc
hro
n
izati
on
of nonl
in
ear s
ystems under
informati
on co
nstraints, Cha
o
s
. 2008; 18: 03
710
9.
[13] Yanb
oGao,
Xi
aome
i
Z
hang,
Guopi
ng Lu, Y
u
fan Z
hen
g. Impulsiv
e
s
y
nc
hron
iz
atio
n of discrete-tim
e
chaotic s
y
stem
s under comm
unic
a
tion co
nst
r
aints.
Co
mmu
n
Non
lin
ear Sc
i Nu
mer Si
mu
l
a
t.
2011; 16:
158
0-15
88.
[14]
Bing
xian
g Li
u,
Yan W
u
, Xin
g
Xu, Na H
u
, Xi
an
g Ch
en
g. Cha
o
s ad
aptiv
e improv
ed
pa
rticle s
w
ar
m
alg
o
rithm for
solvin
g m
u
lti-o
b
jectiv
e o
p
timi
zation.
T
E
LK
OMNIKA Indo
nesi
an J
our
al
of El
ectrica
l
Engi
neer
in
g
. 2014; 12: 7
03-7
10.
[15]
Hon
g
she
ng S
u
, Ying Qi. A chaos clo
ud
parti
cle s
w
arm a
l
g
o
rithm bas
ed a
v
aila
bl
e transfe
r capab
ilit
y.
T
E
LKOMNIKA Indon
esi
an Jou
r
al of Electrica
l
Engin
eeri
n
g
. 2
014; 12: 3
8
-47.
[16]
Lei Z
h
ou, Gu
o
p
in
g L
u
. Detec
t
ion
and
stabi
li
zation
for discr
ete-time
descriptor s
y
stems v
i
a a lim
ited
capac
it
y
comm
unic
a
tion c
han
nel.
Autom
a
tica
. 2009; 4
5
: 22
72-2
277.
[17]
Lei Zh
ou, Gu
opi
ng
Lu. Sta
b
iliz
atio
n for n
onli
n
e
a
r s
y
ste
m
s via a
limit
ed ca
pac
it
y
c
o
mmunic
a
tio
n
chan
nel
w
i
t
h
d
a
ta packet dr
o
pout.
J Control
T
heory App
l
. 2010; 8: 11
1-11
6.
Evaluation Warning : The document was created with Spire.PDF for Python.