TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5251 ~ 52
6
0
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.585
3
5251
Re
cei
v
ed Fe
brua
ry 4, 201
4; Revi
se
d Ma
rch 15, 201
4
;
Accepte
d
March 29, 201
4
Adaptive Hybrid Synchroniza
tion of Lorenz-84 System
with Uncertain Parameters
Ed
w
i
n Albert
Umoh
Dep
a
rtment of Electrical E
ngi
neer
ing T
e
chn
o
lo
g
y
F
edera
l
Pol
y
te
chnic, Kaur
a N
a
mod
a
, Niger
ia
email: e
d
d
y
um
oh@
gmai
l.com
A
b
st
r
a
ct
This paper
presents the adaptive
control
and hy
brid syn
chroni
z
a
t
i
on of Loren
z
-
8
4 c
h
aotic syst
e
m
using a
master-slave to
pology. The Loren
z
-
84 is
an
11-t
e
rm
dissipativ
e system
that
possess
ed f
our
qua
dratic n
o
n
l
i
near
ities i
n
its coup
led
alg
ebr
aic struct
ure w
h
ich res
u
lts to
the evo
l
utio
n o
f
a dense c
h
a
o
ti
c
attractors in
bo
th 2-D
an
d 3-
D
spac
es. F
i
rstly, an a
d
a
p
tive
n
onli
n
e
a
r fee
d
b
a
ck co
ntroll
er
w
a
s desi
gne
d t
o
suppr
ess the c
haotic dy
na
mic
s
of the
system. By using Lya
pun
ov stabi
lity cr
iterion, the
a
sym
ptotic sta
b
i
lity
of the err
o
r sta
t
es w
a
s guar
a
n
teed
an
d the
state dyn
a
m
ics
w
e
re stabi
li
z
e
d. Seco
ndly,
a
daptiv
e n
onl
in
e
a
r
feedb
ack c
ont
rollers
w
e
re
desi
gne
d to
guar
ante
e
th
e
co-ex
i
stence
of sync
h
ro
ni
z
a
ti
on
a
nd
a
n
ti-
synchroni
z
a
tion of the system
. By s
u
itable select
ion
of feedback c
o
efficient
s and
Lyapunov f
unction
candidat
e, the uncertain parameters of the slave
system
were estimated. Nu
meric
a
l sim
u
lations
via
MAT
L
AB show
the conver
gen
ce of the
uncer
tain par
a
m
eter
s to their true valu
es after a transi
ent time w
h
i
l
e
the two system
s synchroni
z
ed com
p
letely.
Ke
y
w
ords
:
ad
aptive co
ntrol, hybri
d
synchro
ni
z
a
ti
on, Lor
en
z
-
8
4
, Lyap
un
o
v
stability theor
y
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
Re
sea
r
ch int
e
re
st in cha
o
tic ph
enom
ena h
a
s
rise
n astrono
mically durin
g the la
st
decade
s. Thi
s
is owi
ng to
the fact that
inqui
sition int
o
ch
aotic
phe
nomen
a
conti
nue
s to reve
al
new
way
s
t
hat chao
s i
s
embe
dde
d
in man
-
ma
d
e
and
natu
r
al sy
stems,
leadin
g
to b
e
tter
unde
rsta
ndin
g
s of their u
s
efuln
e
ss in solving
no
n-t
r
ivial challe
n
ges in en
gin
eerin
g and n
on-
engineeri
ng sciences.
T
he breakthr
ough by Ott, Grebogi and Y
o
rk
e [1] gave im
petus to
chaos
control
re
sult
ing in
dive
rse
metho
d
s of
supp
re
ssing
chao
s i
n
expe
rime
n
t
al and
real-life
scena
rio
s
. Wi
th the su
cce
s
sful coupli
ng
of two
chaoti
c
sy
stem
s by
Peco
ra a
nd
Caroll [2], the
r
e
has bee
n a
conve
r
ge
nce
of multidi
sci
plinar
y
app
ro
ach
e
s on
st
udying m
e
th
ods of coupl
ing
almost all evo
l
ved cha
o
tic systems.
Synchroni
zati
on is
a p
r
o
c
e
ss
wh
ere
b
y the traje
c
to
rie
s
of two
iden
tical o
r
no
n-i
dentical
system
s are
co
uple
d
u
n
i
d
ire
c
tionally or bidi
rectio
n
a
lly usi
ng
suitably de
sig
ned li
nea
r
a
nd
nonlin
ear
co
n
t
rollers. In th
e literatu
r
e,
most
synchr
o
n
izat
io
n s
c
h
e
m
es f
a
ll
s int
o
t
w
o
cla
sse
s,
v
i
z.
maste
r
-slave
type and m
u
tual syn
c
h
r
o
n
i
z
ation. In
ma
ster-sl
a
ve type, an o
r
igin
al
cha
o
tic
syst
em
serve
s
a
s
the drive syste
m
to provide
couplin
g
dynamics to re
gulate the st
ate trajecto
ri
es of
anothe
r syste
m
termed the
resp
on
se sy
stem into
syn
c
hrony in tran
sient time. Chao
s and
cha
o
s
synchro
n
ization of cha
o
s
have f
ound a
pplication in different types of comm
uni
cation
s sy
ste
m
s
[3], power
system
s [4], biological sy
stems [5] and
oscillators [6] amongst
others. Different types
of
synchroni
zation sche
mes hav
e b
een p
r
op
ose
d
in the lite
r
ature such
as g
ene
rali
zed
synchro
n
ization [7], hybrid
synchronization [8],
gene
ralize
d
proje
c
tive synchro
n
i
z
ation [9], an
d
hybrid fun
c
tio
n
synchro
n
ization [10].
Method
s
of
synchro
n
ization
inclu
de a
daptive cont
rol [11], sliding mode cont
rol [12],
fuzzy
co
ntrol
[13], ada
ptive feed
ba
ck [
14], ob
serve
r
-ba
s
ed
control [15], ba
cksteppi
ng d
e
si
gn
[16], and imp
u
lsive syn
c
h
r
onization [17]
among ot
h
e
r
s. Adaptive
method
s of synchro
n
ization
have gain
ed
accepta
n
ce d
ue to their p
r
actical rel
e
v
ance in
real
-life system
whe
r
e mo
st or all
the
system p
a
ra
meters may be un
kno
w
n
or un
ce
rtain.
Unli
ke mo
st other
synchro
n
izatio
n sche
me
s
whe
r
e the
co
ntrol obje
c
tives a
r
e contin
gent upo
n
the availability of all state p
a
ram
e
ters, the
adaptive m
e
thod
s
can
be
use
d
to
estim
a
te un
kn
ow
n para
m
eters o
f
the
sy
stem. The obje
c
tive
of
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: 5251 – 52
60
5252
this wo
rk i
s
to
desig
n ada
ptive controll
ers via
feedba
ck co
ntrol te
ch
nique
s to co
n
t
rol and hyb
r
i
d
-
synchro
n
ize the com
p
lex d
y
namics of the Lore
n
z-8
4
system.
2.
The Loren
z-84 Cha
o
tic S
y
stem
The L
o
ren
z
-84
system
[18] is an
11
-ter
m
dissip
a
t
ive system
that po
ssessed fou
r
quad
ratic n
o
n
linea
rities in
its couple
d
algeb
rai
c
st
ructure whi
c
h
result
s to the evolution o
f
a
den
se chaoti
c
attra
c
tors i
n
both 2
-
D
and 3
-
D
sp
a
c
e
s
. The L
o
renz-84 i
s
to
pologi
cally n
on-
equivalent to
the Lore
n
z-63 [19] whi
c
h evolves
th
e well-kn
own
butterfly attractor.
Ho
wev
e
r
,
unlike the
Lo
ren
z
-63
syst
em which is
argu
ably
o
n
e
of
t
he mo
st
st
udie
d
cha
o
t
i
c
sy
st
em,
t
h
e
Lore
n
z-8
4
sy
stem ha
s received very scanty inte
re
st in the literatu
r
e
even tho
u
gh it dynami
cs
and pro
p
e
r
ties have
tre
m
endo
us
ap
plicatio
ns
in engin
eeri
ng and non
-en
g
i
neeri
ng syst
ems
desi
gn. Thu
s
,
the motivatio
n
for thi
s
stu
d
y is
to st
udy the co
ntroll
a
b
ility and syn
c
hroni
zability
of
the system. T
he governing
equatio
ns of
the Loren
z-84
system is giv
en by:
(
1
)
Whe
r
e
12
3
,,
x
xx
a
r
e
states of the
syste
m
.
,,
,
are po
sitive
co
nstant
s. Fo
r
values of
0.25
,
8
,
1
,
4
, the system evolves the st
ate dynamics in Figure 1.
Figure 1. 2-D
Portrait
s of the O
pen
-loo
p Lore
n
z-8
4
Ch
aotic System
By linearizi
n
g
(1) at the poi
nt
(
0
,0
,0
)
E
, we obtain
ed the Ja
cob
i
an given by:
(0
,
0
,
0
)
0
0
0.25
0
0
01
0
0
1
0
00
1
0
0
1
E
J
(
2
)
-1
0
1
2
3
-4
-2
0
2
4
x1
x2
-1
0
1
2
3
-4
-2
0
2
4
x1
x3
-3
-2
-1
0
1
2
3
-4
-2
0
2
4
x2
x3
'2
2
12
3
1
'
21
2
1
3
2
'
31
2
1
3
3
xx
x
x
xx
x
x
x
x
xx
x
x
x
x
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Adaptive
Hyb
r
id Syn
c
h
r
oni
zation of Lo
re
nz-84 Sy
stem
with Un
certai
n… (Ed
w
in Albert Um
oh)
5253
The ch
aracte
ristic e
quatio
n is
32
1.75
0.5
0
.25
. This
gives
the following
eigenvalu
e
s
1,
2
,
3
(1,
1
,
0
.
2
5
)
.
3.
Adap
tiv
e
Control of the L
o
ren
z
-8
4 Sy
stem
w
i
th
Uncertain Par
a
meter
s
In ord
e
r to
asymptoticall
y
y stabili
ze
t
he dyn
a
mi
cs of the
Lo
ren
z
-84
syst
ems
with
uncertain parameters
at equillibrium point
0
e
x
, we add a
d
aptive feedba
ck
cont
rolle
rs and the
controlled
system (1) b
e
co
mes:
(3)
Whe
r
e
,1
,
2
,
3
i
ui
are a
daptive feedb
ack co
ntrolle
rs
to be desi
g
ned u
s
ing the
states
of the system
and
ˆ
ˆ
ˆˆ
,,
,
are e
s
timated pa
ram
e
ters of
,,
,
. Th
e adaptive fe
edba
ck
controlle
rs
ca
n be rep
r
e
s
e
n
ted as:
(4)
Whe
r
e
,1
,
2
,
3
i
i
is given as:
1
1
2
2
3
3
x
x
x
(5)
And
is a diag
onal matrix whose diag
ona
ls eleme
n
ts
11
2
2
33
[,
,
]
di
ag
c
ons
titutes
the feedba
ck coeffici
ents of
the controllers, su
ch that:
1
11
2
22
3
33
00
00
00
x
x
x
(6)
By inserting (6) into (4
), Equation (3
) be
comes:
'
11
1
1
1
'
21
3
1
3
2
2
'
31
2
1
2
3
3
ˆˆ
ˆ
ˆ
ˆ
ˆ
x
xx
x
x
xx
xx
x
xx
x
x
x
x
(7)
After expandi
ng, (7) b
e
co
mes:
10
pt
'2
2
12
3
1
1
'
21
2
1
3
2
2
'
31
2
1
3
3
3
x
xx
x
u
x
xx
x
x
x
u
xx
x
x
x
x
u
22
1
12
3
1
2
21
2
1
3
2
3
31
2
1
3
3
ˆˆ
ˆ
ˆ
ˆ
ˆ
ux
x
x
ux
x
x
x
x
ux
x
x
x
x
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: 5251 – 52
60
5254
'
11
11
'
21
3
2
2
'
31
2
3
3
ˆˆ
ˆ
ˆ
ˆˆ
ˆ
()
()
(
)
()
(
)
ˆ
ˆ
()
ˆ
()
x
xx
xx
x
x
xx
x
x
(8)
Let,
ˆ
ˆ
ˆ
ˆ
(9)
By using (9
) in (8), the eq
u
a
tion red
u
ces to:
'
11
1
1
'
21
3
1
2
'
31
2
3
3
3
x
xx
x
xx
x
xx
x
x
(10)
Inorde
r to d
e
rive the rel
a
tio
n
shi
p
for th
e
para
m
eter up
date la
w, we
cho
o
se a Lya
punov fun
c
tio
n
can
d
idate [20
]
:
22
2
2
2
2
2
12
3
1
2
3
(,
,
,
,
,
,
)
(
)
2
Vx
x
x
x
x
x
(11)
For a
s
ymptoti
c
stabili
zatio
n
of the system,
(0
)
0
;
(
.
)
0
;
VV
11
2
2
3
3
(.)
(
)
Vx
x
x
x
x
x
(
12)
From (9), it is noted that:
ˆ
ˆ
ˆˆ
;;
;
(13)
Putting (10) a
nd (13
)
into (12) an
d solvi
ng gives the f
o
llowin
g
:
11
1
1
2
1
3
1
2
3
1
2
3
3
(.)
(
(
3
)
(
)
(
)
ˆ
ˆ
ˆˆ
()
(
)
(
)
(
)
)
Vx
x
x
x
x
x
x
x
x
x
x
(14
)
Rea
r
rangin
g
(14) give
s:
22
2
2
11
1
2
3
3
1
2
1
12
3
ˆ
ˆ
ˆ
(.
)
(
(
)
(
)
(
3
)
ˆ
(2
)
)
Vx
x
x
x
x
x
xx
x
(15)
From (15
)
, the para
m
eter
update la
ws become
s
:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Adaptive
Hyb
r
id Syn
c
h
r
oni
zation of Lo
re
nz-84 Sy
stem
with Un
certai
n… (Ed
w
in Albert Um
oh)
5255
2
14
12
3
5
16
27
ˆ
ˆ
2
ˆ
3
ˆ
x
xx
x
x
x
(16)
Whe
r
e
,
4
,
5
,6
,7
i
i
are po
sitive con
s
tan
t
s.
Theorem
1
: The
co
ntro
lled Lo
re
nz-84
system
(3) with
u
n
certain param
eters
is
asymptoticall
y
stabilize
d
in the sen
s
e
of Ly
apuno
v for all initi
a
l con
d
itions by the adaptive
feedba
ck co
n
t
rol law (4) where the p
a
ra
meter up
date
law is given
by (16).
Proof:
By inserting (16) int
o
(15
)
, it is observe
d that:
22
2
2
2
2
2
11
1
2
3
3
4
5
6
7
(.
)
(
)
Vx
x
x
(17)
Whi
c
h is n
e
g
a
tive definite function o
n
7
. Therefore, th
e para
m
eter
estimation e
r
rors
woul
d
conve
r
ge exp
onentially to zero a
s
0
t
.
4. Numerical
Simulations
The Lo
ren
z
-84 syste
m
(3), ada
ptive f
eedba
ck co
ntrol laws (4
) and the p
a
ram
e
ter
update l
a
ws (16
)
were
simulate
d in
MATLAB e
n
vironm
ent for the foll
o
w
ing
parame
t
ers
0.25
,
8
,
1
,
4
and i
n
itial co
ndition
s for
system
12
3
[
(
0)
,
(
0),
(
0)]
[
2
,
6
,
10]
xx
x
,
para
m
eter e
s
timates
ˆ
ˆ
ˆˆ
[
(
0)
,
(
0)
,
(
0)
,
(
0)
]
[
4
,
7
,
12
,
2
]
. The
resulta
n
t plots are give
n in
the following figures
.
(a) Stabili
zed
state dynami
c
s
(b)
Conve
r
ge
d control laws
0
0.
05
0.
1
0.
15
0.
2
0
2
4
6
8
10
t(
s
)
x1
,
x
2,
x3
x1
x2
x3
0
0.
05
0.
1
0.
15
0.
2
-
600
-
400
-
200
0
200
t(s
)
u1
,
u
2
,
u3
u1
u2
u3
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: 5251 – 52
60
5256
(c) Co
nverg
e
d
estimation
error dyna
mics
(d) Conve
r
ge
d
para
m
eter estimate
s
Figure 2. Simulated Results
of the Loren
z-8
4
System
5.
Adap
tiv
e
H
y
brid Sy
nchroniza
tion of
the Lor
enz-8
4
Chao
tic Sy
stem
In this
se
ctio
n, the o
b
je
ctive of compl
e
te
syn
c
h
r
on
ization
of id
entical
Lo
ren
z
-8
4 i
s
reali
z
ed via
the de
sig
n
of
linear and
no
nlinea
r cont
rollers. In hyb
r
id
synchro
n
i
z
ation, the
r
e
is a
co-existen
ce
of com
p
lete
synchroni
zatio
n
and
ant
i
-
sy
nch
r
oni
zatio
n
[21]. In this
pape
r, the
sl
ave
system
is ad
opted
as the
contro
lled
sy
stem
with
un
certai
n p
a
ra
meters. Th
us, the two
sy
stems
are represent
ed as follo
ws:
'2
2
12
3
1
'
21
2
1
3
2
'
31
2
1
3
3
xx
x
x
xx
x
x
x
x
xx
x
x
x
x
(18)
'2
2
1
12
3
1
'2
21
2
1
3
2
'3
31
2
1
3
3
ˆˆ
ˆ
ˆ
ˆ
ˆ
L
L
L
yy
yy
u
yy
y
y
y
y
u
yy
y
y
y
y
u
(19)
Let the hybrid
synch
r
oni
zati
on error b
e
d
e
fined a
s
:
11
1
22
2
33
3
ey
x
ey
x
ey
x
(20)
By using (2
0), the erro
r dynamic
s of the
two system
s
become
s
:
'2
2
2
2
1
12
3
1
2
3
1
'
2
21
2
1
3
2
1
2
1
3
2
'3
31
2
1
3
3
1
2
1
3
3
ˆˆ
ˆ
ˆ
ˆ
ˆ
L
L
L
ey
y
y
x
x
x
u
e
y
y
y
y
y
xx
xx
x
u
e
y
y
y
y
y
xx
xx
x
u
(21)
Equation (21) can be
simpli
fied to:
0
0.
05
0.1
0.
15
0.
2
-6
-4
-2
0
2
4
t(
s
)
P
a
ra
m
e
te
r e
s
ti
m
a
te
e
rro
r
a
.
t
ild
e
b
.
t
ild
e
f.
ti
l
d
e
g
.
t
ild
e
0
0.
0
5
0.
1
0.
1
5
0.2
-5
0
5
10
15
t(
s
)
P
a
ra
m
e
t
e
r e
s
ti
m
a
t
e
s
bhat=4.0
ghat=1.0
ahat=0.25
fhat=8
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Adaptive
Hyb
r
id Syn
c
h
r
oni
zation of Lo
re
nz-84 Sy
stem
with Un
certai
n… (Ed
w
in Albert Um
oh)
5257
'
12
2
2
3
3
3
1
1
1
1
'2
21
2
1
2
1
3
1
3
2
'3
31
2
1
2
1
3
1
3
3
ˆˆ
()
(
)
(
)
(
)
ˆ
ˆ
3(
)
(
)
ˆ
ˆ
ˆ
L
L
L
ee
y
x
e
y
x
e
e
y
u
ey
y
x
x
y
y
x
x
e
u
ey
y
x
x
y
y
x
x
e
u
(22)
Let,
ˆ
ˆ
ˆ
ˆ
(23)
By using (2
3) in (22), the e
quation redu
ces to:
'
1
12
2
2
3
3
3
1
1
1
'2
21
2
1
2
1
3
1
3
2
'3
31
2
1
2
1
3
1
3
3
()
(
)
3
ˆ
ˆ
ˆ
L
L
L
ee
y
x
e
y
x
e
e
y
u
ey
y
x
x
y
y
x
x
e
u
ey
y
x
x
y
y
x
x
e
u
(24)
And the adap
tive control la
w be
come
s:
1
22
2
3
3
3
1
1
1
1
1
2
12
1
2
1
3
1
3
2
2
2
3
12
1
2
1
3
1
3
3
3
3
()
(
)
3
ˆ
ˆ
ˆ
L
L
L
ue
y
x
e
y
x
e
e
y
e
uy
y
x
x
y
y
x
x
e
e
uy
y
x
x
y
y
x
x
e
e
(25)
Then by in
serting
(25
)
i
n
(24
)
, we
have a n
e
w relation
shi
p
for the
syn
c
hroni
zation
error
dynamics:
'
11
1
'
12
2
'
13
3
ee
ee
ee
(26)
We can al
so
note from (23
)
that:
ˆ
ˆ
ˆˆ
;;
;
(27
)
Inorde
r to derive the relatio
n
shi
p
for the para
m
et
er u
p
date law, we cho
o
se a Lya
punov fun
c
tio
n
can
d
idate:
222
2
2
2
2
12
3
1
2
3
ˆ
(,
,
,
,
,
,
)
(
)
2
V
e
e
e
eee
(28
)
The pa
rtial de
rivative of (28
)
along the
trajecto
rie
s
of the syste
m
be
come
s:
11
2
2
3
3
(.
)
(
)
2
Ve
e
e
e
e
e
(29
)
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: 5251 – 52
60
5258
By using (2
6) and (27
)
in (29), the rel
a
tionship (2
9) b
e
com
e
s:
22
2
11
1
2
3
3
ˆ
ˆ
ˆˆ
(.
)
(
)
Ve
e
e
(30
)
It can also b
e
obse
r
ved fro
m
(30), that
the paramete
r
update la
w is given by:
4
5
6
7
ˆ
ˆ
ˆ
ˆ
(31)
For all
0,
4,
5
,
6,
7
i
i
Theorem
2
: The ma
ste
r
Lore
n
z-8
4
sy
stem (18
)
an
d the controll
ed sl
ave sy
stem with
uncertain p
a
rameters are hybrid
-syn
ch
ronized fo
r all
initial conditi
ons by the a
daptive cont
ro
l
law (2
5) where the param
e
t
er update la
w is given
by (31) while the synch
r
o
n
ization errors a
n
d
para
m
eter e
s
timation errors co
nverg
ed
asymptoticall
y
in transient
time.
Proof:
By inserting (31) int
o
(30
)
, it is observe
d that:
22
2
2
2
2
2
11
1
2
3
3
4
5
6
7
(.)
(
)
Ve
e
e
(32)
Whi
c
h is n
e
gative definite function o
n
7
. Therefo
r
e, the synch
r
oni
zation a
n
d
paramete
r
estimation e
r
rors
woul
d co
nverge exp
o
n
entially to zero as
0
t
.
6. Numerical
Simulations
(a) Syn
c
hroni
zed x1-y1
sta
t
es
(b) Antisyn
c
h
r
oni
zed x2
-y2
states
(c) Synch
r
oni
zed x3-y3
sta
t
es
(d)
Conve
r
ge
d error dyna
mics
0
0.
1
0.2
0.
3
0.
4
0.
5
-1
5
-1
0
-5
0
5
t(
s
)
x1,
y1
x1
y1
0
0.
1
0.
2
0.3
0.4
0.5
-1
0
-5
0
5
10
t(
s
)
x2,
y2
x2
y2
0
0.
1
0.
2
0.
3
0.
4
0.
5
-1
5
-1
0
-5
0
5
10
t(s
)
x3
,
y
3
x3
y3
0
0.00
5
0.01
0.0
1
5
0.02
-1
0
0
10
20
30
t(
s
)
e1
,
e
2
,
e3
e1
e2
e3
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Adaptive
Hyb
r
id Syn
c
h
r
oni
zation of Lo
re
nz-84 Sy
stem
with Un
certai
n… (Ed
w
in Albert Um
oh)
5259
(e) Conve
r
ge
d
para
m
eter estimate
s
(f) Co
nverg
e
d
estimation e
r
rors
(g)
Conve
r
ge
d adaptive la
ws
Figure 3. Dyn
a
mics of the
Synchroni
zed
Loren
z-84 S
y
stems
The ma
ster
L
o
ren
z
-84
syst
em (1
8), ad
a
p
tive
controll
ed re
sp
on
se
system
(19
)
, adaptive
control laws (25) an
d the p
a
ram
e
ter up
d
a
te law
(3
1)
were sim
u
lat
ed in MATLA
B
environme
n
t fo
r
the following
para
m
eters
0.25
,
8
,
1
,
4
and initial co
ndition
s for master
system
12
3
[
(
0)
,
(
0)
,
(
0)]
[
2
,
1
,
14]
xx
x
,
slav
e sy
st
e
m
12
3
[
(
0),
(
0)
,
(
0)]
[
3
,
4
,
9]
yy
y
paramete
r
es
timates
ˆ
ˆ
ˆˆ
[
(
0)
,
(
0)
,
(
0)
,
(
0)
]
[
10
,
8
,
1
4
,
5
]
. The initial co
ndit
i
ons
of the sy
nch
r
oni
zatio
n
error dynami
c
s
be
com
e
s
12
3
[
(
0)
,
(
0),
(
0)]
[
5
,
3
,
23
]
ee
e
.
The
re
sultant
plots
are giv
en in th
e
Figure 3.
7. Conclu
sion
Adaptive con
t
rol an
d hyb
r
id syn
c
h
r
oni
zation
of the
Lore
n
z-8
4
sy
stem
with u
n
c
ertai
n
para
m
eters i
s
rep
o
rted
in t
h
is p
ape
r. By app
rop
r
iately
sele
cting th
e
feedba
ck
co
efficients
of the
control law, the state
dynamics
of
the system were
asymptoti
c
al
ly stabilized i
n
transient ti
me
and the
esti
mated p
a
ra
m
e
ters converged to thei
r
t
r
ue
valu
es. Appro
p
riate control
la
ws were
equally de
sig
ned for
co
-e
xistent co
upli
ng of i
denti
c
al Loren
z-8
4
system in
a maste
r
-sla
ve
topology via hybrid syn
c
hroni
zatio
n
schem
e. Pr
op
er sel
e
ctio
n of the feedb
ack co
efficie
n
ts
enge
nde
red
a compl
e
te synchroni
zatio
n
and anti-sy
nch
r
oni
zatio
n
of the st
ates of the system
while the e
s
ti
mated pa
ram
e
ters
of the controlle
d
slav
e system
con
v
erged to the
i
r true value
s
in
trans
ient time.
Referen
ces
[1]
Ott E
,
Grebogi C, Yorke JA. Controlling chaos.
Physical Review Letter
. 1990; 64:1
196-
11
99.
[2]
Pecora LM, Ca
roll T
L
. Synchr
oniz
a
tion i
n
ch
aotic s
y
st
ems.
Physical Review Letter
. 1990; 64:821-
82
4.
[3]
Li F
Z
.
Secure
communic
a
tio
n
and
imp
l
eme
n
t
ation for
a ch
aotic a
u
ton
o
m
ous s
y
stem.
T
E
LKOMNIKA
Indon
esi
an Jou
r
nal of Electric
al Eng
i
ne
eri
n
g
.
2014; 1
2
(1): 3
61-3
70.
[4]
Harb AM, A
b
e
d
-Jab
ar N. C
o
ntrolli
ng
ho
pf bi
furcati
on
and
chaos
in
a s
m
all
po
w
e
r s
ystem.
Chaos
,
Solito
n
s an
d F
r
actals.
200
3; 1
8
:105
5-1
063.
[5]
Al-Khe
dha
iri A
.
T
he nonl
in
e
a
r co
ntrol
of
food c
hai
n m
ode
l us
ing
n
o
n
lin
ear
fee
dba
ck.
Appl
ie
d
Mathe
m
atic
al Scienc
es.
200
9; 13(12): 5
91-
604.
0
0.
0
0
5
0.
01
0.
015
0.
02
-1
0
-5
0
5
10
15
t(s
)
P
a
r
a
m
e
te
r
e
s
ti
m
a
t
e
s
g
hat
aha
t
b
hat
f
hat
0
0.
005
0.01
0.01
5
0.
02
-1
5
-1
0
-5
0
5
10
t(s
)
P
a
ra
m
e
t
e
r e
s
t
i
m
a
t
i
o
n
e
rro
rs
at
il
de
bt
il
de
ft
il
de
gt
il
de
0
0
.
005
0.
0
1
0
.
015
0.
02
-
1
5000
-
1
0000
-
5000
0
5000
t(
s
)
uL1
,
uL2
,
uL3
uL
1
uL
2
uL
3
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: 5251 – 52
60
5260
[6]
Gang
F
Y
, Xi
a
n
HH. Cha
o
s on phas
e
n
o
is
e of V
an
der
Pol Oscil
l
ator.
T
E
LKOMNIKA Indo
nesi
a
n
Journ
a
l of Elec
trical Eng
i
ne
eri
n
g
. 201
0; 8(3): 301-
308.
[7]
Abarb
a
n
e
l H
D
I, Rulkov
NF, Sushch
ik MM. Gener
aliz
ed s
y
n
c
hron
izatio
n of
chaos: th
e a
u
xi
lliar
y
s
y
ste
m
appr
oach.
Phy
s
ical Rev
i
ew
E
. 1996; 5
3
(5): 4
528-
453
5.
[8]
Lan Y, Li Q. Hybr
id s
y
nc
hron
i
z
ation i
n
h
y
per
chaotic s
y
stem
s.
Journal of C
o
mputati
o
n
a
l Info. Systems
.
201
2; 8(18): 76
99-7
707.
[9]
Yun-Pi
ng S, J
un-Min
L, Ji
an
g-An W
,
Hu
i-L
i
n W
.
Gener
ali
z
ed
proj
ective
s
y
nchro
n
iz
ati
on
of cha
o
ti
c
s
y
stems via
ad
aptive l
earn
i
n
g
control.
Ch
ines
e Phys. B
. 201
0; 19(2)0
2
0
505
-1-8.
[10]
Z
hang
C-L, L
i
J-M. H
y
brid f
unc
tio
n
pr
oject
i
ve s
y
nc
hron
iz
ation
of cha
o
ti
c s
y
stems
w
i
t
h
time-var
yi
ng
param
eters vi
a
fouri
e
r ser
i
es
expa
nsio
n
. Int. Jour
nal
of A
u
t
o
matio
n
a
n
d
c
o
mputi
ng.
201
2;
9(4): 38
8-
394.
[11]
Vaid
ya
nat
han
S, Raja
go
pal
K. Globa
l ch
a
o
s s
y
nc
hron
iz
ation
of PAN
and
LU
cha
o
ti
c s
y
stem
via
ada
ptive co
ntrol.
Int. J.
Info.
Tech. Conv. Serv.
2011; 1(
3): 22-33.
[12]
Roo
pae
i M, Jahromi MZ
. Syn
c
hr
on
izatio
n of a class of chao
tic s
y
stems w
i
t
h
full
y
unk
no
w
n
param
eter
s
usin
g ad
aptive
slidi
ng mo
de a
ppro
a
ch.
Ch
ao
s.
2008; 18: 43
112-
7.
[13]
Sargo
l
zae
i
M,
Yagh
oo
bi M, Yazdi RAG. Modell
i
n
g
and s
y
n
c
hron
izatio
n of chaotic g
y
rosc
ope us
ing T
S
fuzz
y
approach.
Advances in
Electron
ic and
Electric Eng
i
ne
erin
g.
201
3;
3
(3): 339-3
46.
[14]
El-Dessok
y
, M
M
. Yassen MT
. Adaptive fee
dback
co
ntrol
for chaos co
ntrol an
d s
y
nchr
oniz
a
tion f
o
r
ne
w
ch
aotic d
y
namic
s
y
stem.
Mathe
m
atic
al Pro
b
le
ms in En
gi
neer
ing
. 20
12; 34
7
210-
1-12.
[15]
Morgul O, Sol
a
k E. Observer-base
d
s
y
nchr
oniz
a
tion
of chaotic s
y
stems.
Physical Review E
. 1996
;
54(5): 48
03-
48
11.
[16]
Njah AN, Ojo KS, Adeba
yo
GA.
Generaliz
ed contro
l and
s
y
nchr
oniz
a
tio
n
of chaos in
RCL-sh
unte
d
jose
phso
n
ju
nc
tion usi
ng b
a
ck
steppi
ng d
e
sig
n
.
Ph
y
s
ica C. Superc
o
n
d
. 20
10; 470: 5
58-5
64.
[17]
Yang T
,
Chu
a
LO. Impulsi
ve stabi
lizati
o
n for
contro
l
and s
y
n
c
hro
n
i
z
ation
of cha
o
tic s
y
st
ems:
Appl
icatio
ns to
secure c
o
mm
unic
a
tions.
IE
EE T
r
ansactio
n
on
Circ
u
its
and Syst
ems-I
:
F
unda
me
ntal
T
heory an
d Ap
plicati
ons.
1
9
9
7
; 44(10):9
76-
988.
[18]
Lore
n
z EN. Irregul
arit
y
:
a fun
d
a
menta
l
pro
per
t
y
of the atmos
pher
e.
T
e
llus A. 1984; 36: 98-
110.
[19]
Lore
n
z EN. De
terministic n
o
n
peri
odic flo
w
.
J. Atm
o
s. Sci
. 196
3;
20:13
0-1
41.
[20]
Wong LK, Le
u
ng FHF,
T
a
m
PKS. An improved L
y
a
p
unov
function-b
a
se
d stabil
i
t
y
a
n
a
l
y
s
is meth
od for
fuzz
y
logic control s
y
stems.
Electronics Letters.
2000; 36(
12
): 1085-1
0
9
6
.
[21]
Lan Y, Li Q. Hybr
id s
y
nc
hron
i
z
ation i
n
h
y
per
chaotic s
y
stem
s
. Journal of C
o
mputati
o
n
a
l Info. Systems
.
201
2; 8(18): 76
99-7
707.
Evaluation Warning : The document was created with Spire.PDF for Python.