In e na ional Jou nal on Bioin o ma ics & Biosciences (IJBB) Vol.3, No.1, Ma ch 2013
DOI : 10.5121/ijbb.2013.3103 21
ANTI-SYNCHRONIZING BACKSTEPPING CONTROL
DESIGN FOR ARNEODO CHAOTIC SYSTEM
Sunda apandian Vaidyana han1
1Resea ch and De elopmen Cen e, Vel Tech D . RR & D . SR Technical Uni e si y
A adi, Chennai-600 062, Tamil Nadu, INDIA
[email p o ec ed]
ABSTRACT
In his pape , we de i e new esul s o backs epping con olle design o he an i-synch oniza ion o
A neodo chao ic sys em (1980). Backs epping con ol is a ecu si e p ocedu e ha combines he choice o
a Lyapuno unc ion wi h he design o a eedback con olle . In an i-synch oniza ion o chao ic sys ems,
he s a es o he synch onized sys ems ha e he same absolu e alues, bu opposi e signs. Fi s , we de i e
an ac i e backs epping con olle o he an i-synch oniza ion o iden ical A neodo chao ic sys ems. Nex ,
we de i e an adap i e backs epping con olle o he an i-synch oniza ion o iden ical A neodo chao ic
sys em, when he sys em pa ame e s a e unknown. The an i-synch oniza ion esul s o A neodo chao ic
sys ems ha e been p o ed using Lyapuno s abili y heo y. Nume ical simula ions ha e been shown o
illus a e he backs epping con olle s de i ed in his pape o A neodo chao ic sys em.
KEYWORDS
Backs epping Con ol; Chaos; An i-Synch oniza ion; A neodo Sys em.
1. INTRODUCTION
Chaos heo y deals wi h he beha iou o nonlinea dynamical sys ems ha a e highly sensi i e
o ini ial condi ions, an e ec which is popula ly known as he bu e ly e ec [1]. Small
di e ences in ini ial condi ions esul in widely di e ging ou comes o chao ic sys ems, ending
long- e m p edic ion impossible in gene al. The chaos phenomenon was i s obse ed in wea he
models by he Ame ican scien is , Lo enz ([2], 1963). Since hen, chaos heo y has ound
applica ions in a a ie y o ields in science and enginee ing [3-9].
The p oblem o con olling a chao ic sys em was i s in oduced by O e al. ([10], 1990). The
p oblem o chaos synch oniza ion occu s when wo o mo e chao ic oscilla o s a e coupled o
when a chao ic oscilla o d i es ano he chao ic oscilla o ([11], 1990). The idea o chaos an i-
synch oniza ion is o use he ou pu o he mas e sys em o con ol he ou pu o he sla e sys em
so ha he s a es o he mas e and sla e sys ems ha e he same absolu e alues, bu opposi e
signs, i.e. he sum o he ou pu signals o he mas e and sla e sys ems can con e ge o ze o
asymp o ically.
Since he pionee ing wo k by Peco a and Ca oll [11], a ious me hods ha e been de eloped in
he chaos li e a u e o he synch oniza ion o chao ic sys ems such as ac i e con ol me hod [12-
15], adap i e con ol me hod [16-20], ime-delay eedback con ol me hod [21], sampled-da a
con ol me hod [22-23], sliding mode con ol me hod [24-30], backs epping con ol me hod [31-
33], e c.
In his pape , we deploy backs epping con ol me hod o he an i-synch oniza ion o iden ical
A neodo chao ic sys ems ([34], 1980). Backs epping con ol me hod is a ecu si e p ocedu e ha
combines he choice o a Lyapuno unc ion wi h he design o a eedback con olle .
In e na ional Jou nal on Bioin o ma ics & Biosciences (IJBB) Vol.3, No.1, Ma ch 2013
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The o ganiza ion o his esea ch pape is as ollows. In Sec ion 2, we design an ac i e
backs epping con olle o he an i-synch oniza ion o iden ical A neodo sys ems when he
sys em pa ame e s a e known. In Sec ion 3, we design an adap i e backs epping con olle o
he an i-synch oniza ion o iden ical A neodo sys ems when he sys em pa ame e s a e unknown.
Sec ion 4 con ains he conclusions o his wo k.
2. ACTIVE BACKSTEPPING CONTROLLER DESIGN FOR THE ANTI-
SYNCHRONIZATION OF ARNEODO SYSTEMS
2.1 Theo e ical Resul s
A neodo sys em ([34], 1980) is one o he classical 3-D chao ic sys ems as i cap u es many
ea u es o chao ic sys ems. In his sec ion, we in es iga e he p oblem o ac i e backs epping
con olle design o he an i-synch oniza ion o iden ical A neodo chao ic sys ems, when he
sys em pa ame e s a e known.
As he mas e sys em, we conside he 3-D A neodo dynamics
1 2
2 3
2
3 1 2 3 1
,
,
,
x x
x x
x ax bx x x
=
=
= − − −
(1)
whe e
12 3
, ,x x x
a e he s a es and
,a b
a e posi i e, known pa ame e s o he sys em.
Figu e 1. S ange Chao ic A ac o o he A neodo Sys em
In e na ional Jou nal on Bioin o ma ics & Biosciences (IJBB) Vol.3, No.1, Ma ch 2013
23
The A neodo sys em (1) unde goes chao ic beha iou when he sys em pa ame e alues a e
chosen as
7.5a=
and
3.8.b=
The s ange chao ic a ac o o he A neodo sys em (1) is shown in Figu e 1.
As he sla e sys em, we conside he con olled 3-D A neodo dynamics
1 2
2 3
2
3 1 2 3 1
,
,
,
y y
y y
y ay by y y u
=
=
= − − − +
(2)
whe e
1 2 3
, ,y y y
a e he s a es and
u
is he ac i e con ol o be designed.
The an i-synch oniza ion e o be ween he mas e sys em (1) and he sla e sys em (2) is de ined
as
1 1 1
2 2 2
3 3 3
( ) ( ) ( ),
( ) ( ) ( ),
( ) ( ) ( ).
e y x
e y x
e y x
= +
= +
= +
(3)
The design p oblem is o ind a con ol
( )u
so ha he e o con e ges o ze o
asymp o ically, i.e.
( ) 0
i
e →
as
→ ∞
o
1,2,3.i=
The e o dynamics is easily de i ed as
1 2
2 3
2 2
3 1 2 3 1 1
,
,
.
e e
e e
e ae be e y x u
=
=
= − − − − +
(4)
In his sec ion, we apply he ac i e backs epping con ol me hod o design a con olle
( ).u
Theo em 1. The iden ical A neodo chao ic sys ems (1) and (2) a e globally and exponen ially
an i-synch onized o all ini ial condi ions by he ac i e backs epping con olle
2 2
1 2 3 1 1
( ) (3 ) (5 ) 2 .u a e b e e y x= − + − − − + +
(5)
P oo . Fi s , we de ine a Lyapuno unc ion
2
1 1
1,
2
V z=
(6)
whe e
1 1.z e=
(7)
I s ime de i a i e along he solu ions o sys ems (1) and (2) is ob ained as
2
1 1 1 1 1 1 2 1 1 1 2
( ).V z z e e e e z z e e= = = = − + +
(8)
In e na ional Jou nal on Bioin o ma ics & Biosciences (IJBB) Vol.3, No.1, Ma ch 2013
24
Nex , we de ine
2 1 2.z e e= +
(9)
F om (9), i ollows ha
2
1 1 1 2.V z z z= − +
(10)
Secondly, we de ine he Lyapuno unc ion
( )
2 2 2
2 1 2 1 2
1 1 .
2 2
V V z z z= + = +
(11)
The ime de i a i e o
2
V
is gi en by
2 2
2 1 2 2 1 2 3
(2 2 ).V z z z e e e= − − + + +
(12)
Nex , we de ine
3 1 2 3
2 2 .z e e e= + +
(13)
F om (13), i ollows ha
2 2
2 1 2 2 3.V z z z z= − − +
(14)
Finally, we de ine he Lyapuno unc ion
( )
2 2 2 2
2 3 1 2 3
1 1 .
2 2
V V z z z z= + = + +
(15)
Clea ly,
V
is a posi i e de ini e unc ion on
3.R
The ime de i a i e o
V
is ob ained as
( )
2 2 2 2
1 2 2 3 3 2 3 1 2 3 1 1
2 2V z z z z z e e ae be e y x u= − − + + + + − − − − +
(16)
A simple calcula ion gi es
2 2 2 2 2
1 2 3 3 1 2 3 1 1
(3 ) (5 ) 2 .V z z z z a e b e e y x u
= − − − + + + − + − − +
(17)
Subs i u ing he backs epping con olle
u
de ined by (5) in (17), we ge
222
1 2 3 .V z z z= − − −
(18)
Clea ly,
V
is a nega i e de ini e unc ion on
3.R
Hence, by Lyapuno s abili y heo y [35], he e o dynamics (4) is globally exponen ially s able.
This comple es he p oo .
In e na ional Jou nal on Bioin o ma ics & Biosciences (IJBB) Vol.3, No.1, Ma ch 2013
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2.2 Nume ical Resul s
Fo nume ical simula ions using MATLAB, he ou h o de Runge-Ku a me hod wi h ini ial
s ep
8
10h−
=
is used o sol e he A neodo sys ems (1) and (2) wi h he backs epping con olle
u
de ined by (5). The pa ame e s o he A neodo chao ic sys ems a e selec ed as
7.5a=
and
3.8.b=
The ini ial alues o he mas e sys em (1) a e chosen as
1 2 3
(0) 14, (0) 5, (0) 6x x x= = − =
The ini ial alues o he sla e sys em (2) a e chosen as
1 2 3
(0) 18, (0) 12, (0) 16y y y= = = −
Figu e 2 depic s he an i-synch oniza ion o A neodo chao ic sys ems (1) and (2).
Figu e 3 depic s he ime-his o y o he an i-synch oniza ion e o s
1 2 3
, , .e e e
Figu e 2. An i-Synch oniza ion o A neodo Chao ic Sys ems
In e na ional Jou nal on Bioin o ma ics & Biosciences (IJBB) Vol.3, No.1, Ma ch 2013
26
Figu e 3. Time-His o y o he An i-Synch onizing E o s
1 2 3
, ,e e e
3. REGULATING ACTIVE BACKSTEPPING CONTROLLER DESIGN FOR THE
ANTI-SYNCHRONIZATION OF ARNEODO SYSTEMS
3.1 Theo e ical Resul s
In his sec ion, we de i e new esul s o he adap i e backs epping con olle design o an i-
synch oniza ion o A neodo sys ems when he pa ame e s
a
and
b
a e unknown.
As he mas e sys em, we conside he 3-D A neodo dynamics
1 2
2 3
2
3 1 2 3 1
,
,
,
x x
x x
x ax bx x x
=
=
= − − −
(19)
whe e
12 3
, ,x x x
a e he s a es and
,a b
a e unknown pa ame e s o he sys em.
As he sla e sys em, we conside he con olled 3-D A neodo dynamics
1 2
2 3
2
3 1 2 3 1
,
,
,
y y
y y
y ay by y y u
=
=
= − − − +
(20)
whe e
1 2 3
, ,y y y
a e he s a es and
u
is he adap i e con ol o be designed.
In e na ional Jou nal on Bioin o ma ics & Biosciences (IJBB) Vol.3, No.1, Ma ch 2013
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The an i-synch oniza ion e o be ween he mas e sys em (19) and he sla e sys em (20) is
de ined as
1 1 1
2 2 2
3 3 3
( ) ( ) ( ),
( ) ( ) ( ),
( ) ( ) ( ).
e y x
e y x
e y x
= +
= +
= +
(21)
The design p oblem is o ind a con ol
( )u
so ha he e o con e ges o ze o
asymp o ically, i.e.
( ) 0
i
e →
as
→ ∞
o
1,2,3.i=
The e o dynamics is easily de i ed as
1 2
2 3
2 2
3 1 2 3 1 1
,
,
.
e e
e e
e ae be e y x u
=
=
= − − − − +
(22)
In his sec ion, we apply he adap i e backs epping con ol me hod o design a con olle
( ).u
Inspi ed by he con ol law de ined by Eq. (5) in he ac i e backs epping con olle design, we
may conside he adap i e backs epping con olle design law gi en by
2 2
1 2 3 1 1
ˆ
ˆ
( ) (3 ) (5 ) 2 ,u a e b e e y x= − + − − − + +
(23)
whe e
ˆ( )a
and
ˆ( )b
a e es ima es o he unknown pa ame e s
a
and
,b
espec i ely.
We de ine he pa ame e es ima ion e o s as
ˆ
( ) ( )
a
e a a = −
and
ˆ
( ) ( )
b
e b b = −
(24)
No e ha
ˆ
( ) ( )
a
e a = −
and
ˆ
( ) ( )
b
e b = −
(25)
Nex , we shall s a e and p o e he second main esul o his pape .
Theo em 2. The iden ical A neodo chao ic sys ems (19) and (20) wi h unknown pa ame e s
a
and
b
a e globally and exponen ially an i-synch onized o all ini ial condi ions by he
adap i e backs epping con olle
2 2
1 2 3 1 1
ˆ
ˆ
( ) (3 ) (5 ) 2 ,u a e b e e y x= − + − − − + +
(26)
whe e
ˆ( )a
and
ˆ( )b
a e es ima es o
a
and
,b
espec i ely, and he pa ame e upda e law
is gi en by
1 2 3 1
1 2 3 2
ˆ( ) (2 2 ) ,
ˆ( ) (2 2 ) ,
a a
b b
a e e e e k e
b e e e e k e
= + + +
= − + + +
(27)
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wi h posi i e con ol gains
a
k
and
.
b
k
P oo . Fi s , we de ine he Lyapuno unc ion
2
1 1
1,
2
V z=
(28)
whe e
1 1.z e=
(29)
The ime de i a i e o
1
V
is gi en by
2
1 1 1 1 1 1 2 1 1 1 2
( ).V z z e e e e z z e e= = = = − + +
(30)
Nex , we de ine
2 1 2.z e e= +
(31)
F om (30), i ollows ha
2
1 1 1 2.V z z z= − +
(32)
Secondly, we de ine he Lyapuno unc ion
( )
2 2 2
2 1 2 1 2
1 1 .
2 2
V V z z z= + = +
(33)
The ime de i a i e o
2
V
is gi en by
2 2
2 1 2 2 1 2 3
(2 2 ).V z z z e e e= − − + + +
(34)
Nex , we de ine
3 1 2 3
2 2 .z e e e= + +
(35)
F om (34), i ollows ha
2 2
2 1 2 2 3.V z z z z= − − +
(35)
Finally, we de ine he Lyapuno unc ion
( ) ( )
2 2 2 2 2 2 2 2
2 3 1 2 3
1 1 1 .
2 2 2
a b a b
V V z e e z z z e e= + + + = + + + +
(36)
The ime de i a i e o
V
is ob ained as
2 2 2 2 2
1 2 3 3 1 2 3 1 1 ˆ
ˆ
(3 ) (5 ) 2 .
a b
V z z z z a e b e e y x u e a e b
= − − − + + + − + − − + − −
(37)
Subs i u ing he backs epping con olle
u
de ined by (26) in (37), we ge
( )
( )
222
1 2 3 1 3 2 3 ˆ
ˆ.
a b
V z z z e e z a e e z b= − − − + − + − −
(38)
In e na ional Jou nal on Bioin o ma ics & Biosciences (IJBB) Vol.3, No.1, Ma ch 2013
29
Subs i u ing he pa ame e law (27) in (38) and no ing ha
3 1 2 3
2 2 ,z e e e= + +
we ge
2 2 2 2 2
1 2 3 ,
a a b b
V z z z k e k e= − − − − −
(39)
which is a nega i e de ini e unc ion on
5.R
Thus, by Lyapuno s abili y heo y [35], he p oo is comple e.
3.2 Nume ical Resul s
Fo nume ical simula ions wi h MATLAB, he ou h-o de Runge-Ku a me hod wi h ini ial s ep
8
10h−
=
is used o sol e he A neodo sys ems (19) and (20) wi h he backs epping con olle
u
de ined by (26) and he pa ame e upda e law de ined by (27).
The pa ame e s o he A neodo chao ic sys ems a e chosen as
7.5a=
and
3.8.b=
The ini ial alues o he pa ame e es ima es a e chosen as
ˆ(0) 16a=
and
ˆ(0) 9.b=
The con ol gains a e chosen as
6
a
k=
and
6.
b
k=
The ini ial alues o he mas e sys em (19) a e chosen as
1 2 3
(0) 4, (0) 5, (0) 8x x x= = = −
The ini ial alues o he sla e sys em (20) a e chosen as
1 2 3
(0) 2, (0) 6, (0) 5y y y= = = −
Figu e 4 depic s he an i-synch oniza ion o A neodo chao ic sys ems. Figu e 5 depic s he ime-
his o y o he an i-synch oniza ion e o s
1 2 3
, , .e e e
Figu e 6 depic s he ime-his o y o he
pa ame e es ima ion e o s
, .
a b
e e
