ADAPTIVE CONTROLLER DESIGN FOR THE ANTI-SYNCHRONIZATION OF HYPERCHAOTIC YANG AND HYPERCHAOTIC PANG SYSTEMS

7/30/2019 ADAPTIVE CONTROLLER DESIGN FOR THE ANTI-SYNCHRONIZATION OF HYPERCHAOTIC YANG AND HYPERCHAOTIC

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International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 2013

DOI : 10.5121/ijics.2013.3201 1

ADAPTIVE CONTROLLERDESIGN FOR THE

ANTI-SYNCHRONIZATION OF HYPERCHAOTIC

YANG AND HYPERCHAOTIC PANG SYSTEMS

Sundarapandian Vaidyanathan1

1Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University

Avadi, Chennai-600 062, Tamil Nadu, [email protected]

ABSTRACT

In the anti-synchronization of chaotic systems, a pair of chaotic systems called drive and responsesystems

are considered, and the design goal is to drive the sum of their respective states to zero asymptotically. This

paper derives new results for the anti-synchronization of hyperchaotic Yang system (2009) andhyperchaotic Pang system (2011) with uncertain parameters via adaptive control. Hyperchaotic systems

are nonlinear chaotic systems withtwo or more positive Lyapunov exponents and they have applications in

areas like neural networks, encryption, secure data transmission and communication. The main results

derived in this paper are illustrated with MATLAB simulations.

KEYWORDS

Hyperchaos, Adaptive Control, Anti-Synchronization, Hyperchaotic Systems.

1. INTRODUCTION

Since the discovery of a hyperchaotic system by O.E.Rssler ([1], 1979), hyperchaotic systems

are known to have characteristics like high security, high capacity and high efficiency.Hyperchaotic systems are chaotic systems having two or morepositive Lyapunov exponents. They

are applicable in several areas like oscillators [2], neural networks [3], secure communication [4-5], data encryption [6], chaos synchronization [7], etc.

The synchronization problem deals with a pair of chaotic systems called the drive and responsechaotic systems, where the design goal is to drive the difference of their respective states to zero

asymptotically [8-9].

The anti-synchronization problem deals with a pair of chaotic systems called the drive and

response systems, where the design goal is to drive the sum of their respective states to zeroasymptotically.

The problems of synchronization and anti-synchronization of chaotic and hyperchaotic systems

have been studied via several methods like active control method [10-12], adaptive control

method [13-15],backstepping method [16-19], sliding control method [20-22] etc.

This paper derives new results for the adaptive controller design for the anti-synchronization ofhyperchaotic Yang systems ([23], 2009) and hyperchaotic Pang systems ([24], 2008) with
mailto:[email protected]:[email protected]


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unknown parameters. The main results derived in this paper were proved using adaptive control

theory [25] and Lyapunov stability theory [26].

2. PROBLEM STATEMENT

The drive system is described by the chaotic dynamics

( )x Ax f x= + (1)

whereA is the n n matrix of the system parameters and : n nf R R is the nonlinear part.

The response system is described by the chaotic dynamics

( )y By g y u= + + (2)

whereB is the n n matrix of the system parameters, :n n

g R R is the nonlinear part andn

u R is the active controller to be designed.

For the pair of chaotic systems (1) and (2), the design goal of the anti-synchronization problemis

to construct a feedback controller ,u which anti-synchronizes their states for all (0), (0) .n

x y R

Theanti-synchronization erroris defined as

,e y x= + (3)

Theerror dynamics is obtained as

( ) ( )e By Ax g y f x u= + + + + (4)

The design goal is to find a feedback controller uso that

lim ( ) 0t

e t

= for all (0)e Rn

(5)

Using the matrix method, we consider a candidate Lyapunov function

( ) ,T

V e e Pe= (6)

where P is a positive definite matrix.

It is noted that : nV R R is a positive definite function.

If we find a feedback controlleru

so that

( ) ,T

V e e Qe= (7)

whereQ is a positive definite matrix, then : nV R R is a negative definite function.Thus, by Lyapunov stability theory [26], the error dynamics (4) is globally exponentially stable.


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When the system parameters in (1) and (2) are unknown, we need to construct a parameter update

law for determining the estimates of the unknown parameters.

3. HYPERCHAOTIC SYSTEMS

The hyperchaotic Yang system ([23], 2009) is given by

1 2 1

2 1 1 3 4

3 3 1 2

4 1 2

( )x a x x

x cx x x x

x bx x x

x dx x

=

= +

= +

=

(8)

where , , ,a b c d are constant, positive parameters of the system.

The Yang system (8) exhibits a hyperchaotic attractor for the parametric values

35, 3, 35, 2, 7.5a b c d = = = = = (9)

The Lyapunov exponents of the system (8) for the parametric values in (9) are

1 2 3 40.2747, 0.1374, 0, 38.4117 = = = = (10)

Since there are two positive Lyapunov exponents in (10), the Yang system (8) is hyperchaotic forthe parametric values (9).

The phase portrait of the hyperchaotic Yang system is described in Figure 1.

The hyperchaotic Pang system ([24], 2011) is given by

1 2 1

2 2 1 3 4

3 3 1 2

4 1 2

( )

( )

x x xx x x x x

x x x x

x x x

=

= +

= +

= +

(11)

where , , , are constant, positive parameters of the system.

The Pang system (11) exhibits a hyperchaotic attractor for the parametric values

36, 3, 20, 2 = = = = (12)

The Lyapunov exponents of the system (9) for the parametric values in (12) are

1 2 3 41.4106, 0.1232, 0, 20.5339 = = = = (13)

Since there are two positive Lyapunov exponents in (13), the Pang system (11) is hyperchaotic

for the parametric values (12).

The phase portrait of the hyperchaotic Pang system is described in Figure 2.


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Figure 1. The Phase Portrait of the Hyperchaotic Yang System

Figure 2. The Phase Portrait of the Hyperchaotic Pang System


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4. ADAPTIVE CONTROL DESIGN FOR THE ANTI-SYNCHRONIZATION OF

HYPERCHAOTIC YANG SYSTEMS

In this section, we design an adaptive controller for the anti-synchronization of two identical

hyperchaotic Yang systems (2009) with unknown parameters.

Thedrive system is the hyperchaotic Yangdynamicsgiven by

1 2 1

2 1 1 3 4

3 3 1 2

4 1 2

( )x a x x

x cx x x x

x bx x x

x dx x

=

= +

= +

=

(14)

where , , , ,a b c d are unknown parameters of the system and 4x R is the state.The response system is the controlled hyperchaotic Yangdynamics given by

1 2 1 1

2 1 1 3 4 2

3 3 1 2 3

4 1 2 4

( )y a y y u

y cy y y y u

y by y y u

y dy y u

= +

= + +

= + +

= +

(15)

where4

y R is the state and 1 2 3 4, , ,u u u u are the adaptivecontrollers to be designed.

For the anti-synchronization, the error e is defined as

1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (16)

Then we derive the error dynamics as

1 2 1 1

2 1 4 1 3 1 3 2

3 3 1 2 1 2 3

4 1 2 4

( )e a e e u

e ce e y y x x u

e be y y x x u

e de e u

= +

= + +

= + + +

= +

(17)

The adaptive controller to achieve anti-synchronization is chosen as

1 2 1 1 1

2 1 4 1 3 1 3 2 2

3 3 1 2 1 2 3 3

4 1 2 4 4

( ) ( )( )

( ) ( )

( ) ( )

( ) ( ) ( )

u t a t e e k e

u t c t e e y y x x k e

u t b t e y y x x k e

u t d t e t e k e

=

= + +

=

= +

(18)


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In Eq. (18), , ( 1, 2, 3, 4)i

k i = are positive gains and ( ), ( ), ( ), ( ), ( )a t b t c t d t t are estimates for the

unknown parameters , , , ,a b c d respectively.

By the substitution of (18) into (17), the error dynamics is simplified as

1 2 1 1 1

2 1 2 2

3 3 3 3

4 1 2 4 4

( ( ))( )

( ( ))

( ( ))

( ( )) ( ( ))

e a a t e e k e

e c c t e k e

e b b t e k e

e d d t e t e k e

= =

=

=

(19)

As a next step, we define the parameter estimation errors as

( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )

a b c d e t a a t e t b b t e t c c t e t d d t e t t = = = = = (20)

Upon differentiation, we get

( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c d

e t a t e t b t e t c t e t d t e t t = = = = = (21)

Substituting (20) into the error dynamics (19), we obtain

1 2 1 1 1

2 1 2 2

3 3 3 3

4 1 2 4 4

( )a

c

b

d

e e e e k e

e e e k e

e e e k e

e e e e e k e

=

=

=

=

(22)

We consider the candidate Lyapunov function

( )2 2 2 2 2 2 2 2 21 2 3 41

2a b c d

V e e e e e e e e e= + + + + + + + + (23)

Differentiating (23) along the dynamics (21) and (22), we obtain

( )( ) ( ) ( )

2 2 2 2 2

1 1 2 2 3 3 4 4 1 2 1 3

1 2 1 4 2 4

( )

a b

c d

V k e k e k e k e e e e e a e e b

e e e c e e e d e e e

= + +

+ + +

(24)

In view of (24), we choose the following parameter update law:

1 2 1 5 1 4 8

2

3 6 2 4 9

1 2 7

( ) ,

,

a d

b

c

a e e e k e d e e k e

b e k e e e k e

c e e k e

= + = +

= + = +

= +

(25)

Next, we prove the following main result of this section.


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Theorem 4.1 The adaptive control law defined by Eq. (18) along with the parameter update law

defined by Eq. (25) achieveglobal and exponential anti-synchronization of the identicalhyperchaotic Yang systems (14) and (15)with unknown parameters for all initial conditions

4(0), (0) .x y R Moreover, the parameter estimation errors ( ), ( ), ( ), ( ), ( )a b c d e t e t e t e t e t

globally and exponentially converge to zero for all initial conditions.

Proof.The proof is via Lyapunov stability theory [26] by taking V defined by Eq. (23) as the

candidate Lyapunov function. Substituting the parameter update law (25) into (24), we get

2 2 2 2 2 2 2 2 2

1 1 2 2 3 3 4 4 5 6 7 8 9( )

a b c d V e k e k ek e k e k e k e k e k e k e=

(26)

which is a negative definite function on9.R This completes the proof.

Next, we illustrate our adaptive anti-synchronization results with MATLAB simulations.The

classical fourth order Runge-Kutta method with time-step8

10h

= has been used to solve the

hyperchaotic Yang systems (14) and (15) with the nonlinear controller defined by (18).

The feedback gains in the adaptive controller (18) are taken as 4, ( 1, ,9).ik i= =

The parameters of the hyperchaotic Yang systems are taken as in the hyperchaotic case, i.e.

35, 3, 35, 2, 7.5a b c d = = = = =

For simulations, the initial conditions of the drive system (14) are taken as

1 2 3 4(0) 7, (0) 16, (0) 23, (0) 5x x x x= = = =

Also, the initial conditions of the response system (15) are taken as

1 2 3 4(0) 34, (0) 8, (0) 28, (0) 20y y y y= = = =

Also, the initial conditions of the parameter estimates are taken as

(0) 12, (0) 8, (0) 7, (0) 5, (0) 4a b c d = = = = =

Figure 3 depicts the anti-synchronization of the identical hyperchaotic Yang systems.

Figure 4 depicts the time-history of the anti-synchronization errors 1 2 3 4, , , .e e e e

Figure 5 depicts the time-history of the parameter estimation errors , , , , .a b c d e e e e e


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Figure 3. Anti-Synchronization of Identical Hyperchaotic Yang Systems

Figure 4. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e e


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Figure 5. Time-History of the Parameter Estimation Errors , , , ,a b c d e e e e e


HYPERCHAOTIC PANG SYSTEMS

In this section, we design an adaptive controller for the anti-synchronization of two identical

hyperchaotic Pang systems (2011) with unknown parameters.Thedrive system is the hyperchaotic Pangdynamics given by

1 2 1

2 2 1 3 4

3 3 1 2

4 1 2

( )

( )

x x x

x x x x x

x x x x

x x x

=

= +

= +

= +

(27)

where , , , are unknown parameters of the system and4

x R is the state.

The response system is the controlled hyperchaotic Pangdynamics given by

1 2 1 1

2 2 1 3 4 2

3 3 1 2 3

4 1 2 4

( )

( )

y y y u

y y y y y u

y y y y u

y y y u

= +

= + +

= + +

= + +

(28)


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where4

y R is the state and 1 2 3 4, , ,u u u u are the adaptivecontrollers to be designed.


1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (29)


1 2 1 1

2 2 4 1 3 1 3 2

3 3 1 2 1 2 3

4 1 2 4

( )

( )

e e e u

e e e y y x x u

e e y y x x u

e e e u

= +

= + +

= + + +

= + +

(30)


1 2 1 1 1

2 2 4 1 3 1 3 2 2

3 3 1 2 1 2 3 3

4 1 2 4 4

( )( )

( )

( )

( )( )

u t e e k e

u t e e y y x x k e

u t e y y x x k e

u t e e k e

=

= + +

=

= +

(31)

In Eq. (31), , ( 1, 2,3, 4)i

k i = are positive gains and ( ), ( ), ( ), ( )t t t t are estimates for the

unknown parameters , , , respectively.


1 2 1 1 1

2 2 2 2

3 3 3 3

4 1 2 4 4

( ( ))( )

( ( ))

( ( ))

( ( ))( )

e t e e k e

e t e k e

e t e k e

e t e e k e

=

=

=

= +

(32)


( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )e t t e t t e t t e t t

= = = = (33)


( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )e t t e t t e t t e t t = = = = (34)



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1 2 1 1 1

2 2 2 2

3 3 3 3

4 1 2 4 4

( )

( )

e e e e k e

e e e k e

e e e k e

e e e e k e

=

=

=

= +

(35)


( )2 2 2 2 2 2 2 21 2 3 41

2V e e e e e e e e = + + + + + + + (36)


( )( ) ( )

2 2 2 2 2

1 1 2 2 3 3 4 4 1 2 1 3

2

2 1 4 1 2

( )

( )

V k e k e k e k e e e e e e e

e e e e e e e

= + +

+ + +

(37)


1 2 1 5

2

3 6

2

2 7

1 4 1 2 8

( )

( )

e e e k e

e k e

e k e

e e e e k e

= +

= +

= +

= + +

(38)

Next, we prove the following main result of this section.

Theorem 5.1 The adaptive control law defined by Eq. (31) along with the parameter update lawdefined by Eq. (38) achieve global and exponential anti-synchronization of the identicalhyperchaotic Pang systems (27) and (28) with unknown parameters for all initial conditions

4(0), (0) .x y R Moreover, the parameter estimation errors ( ), ( ), ( ), ( )e t e t e t e t globally and

exponentially converge to zero for all initial conditions.



2 2 2 2 2 2 2 2 2

1 1 2 2 3 3 4 4 5 6 7 8 9( )

a b c d V e k e k ek e k e k e k e k e k e k e=

(39)

which is a negative definite function on 9.R This completes the proof.

Next, we illustrate our adaptive anti-synchronization results with MATLAB simulations. The


10h


hyperchaotic Pang systems (27) and (28) with the nonlinear controller defined by (31).

The feedback gains in the adaptive controller (31) are taken as 4, ( 1, ,8).ik i= =


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The parameters of the hyperchaotic Pang systems are taken as in the hyperchaotic case, i.e.

36, 3, 20, 2 = = = =


1 2 3 4(0) 17, (0) 22, (0) 11, (0) 25x x x x= = = =


1 2 3 4(0) 24, (0) 18, (0) 24, (0) 17y y y y= = = =


(0) 3, (0) 4, (0) 27, (0) 15 = = = =

Figure 6depicts the anti-synchronization of the identical hyperchaotic Pang systems.

Figure 7depicts the time-history of the anti-synchronization errors 1 2 3 4, , , .e e e e

Figure 8 depicts the time-history of the parameter estimation errors , , , .e e e e

Figure 6. Anti-Synchronization of Identical Hyperchaotic Pang Systems


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Figure 8. Time-History of the Parameter Estimation Errors , , ,e e e e


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HYPERCHAOTIC YANG AND HYPERCHAOTIC PANG SYSTEMS

In this section, we design an adaptive controller for the anti-synchronization of non-identical

hyperchaotic Yang (2009) and hyperchaotic Pang systems (2011) with unknown parameters.Thedrive system is the hyperchaotic Yangdynamics given by

1 2 1

2 1 1 3 4

3 3 1 2

4 1 2

( )x a x x

x cx x x x

x bx x x

x dx x

=

= +

= +

=

(40)

where , , , ,a b c d are unknown parameters of the system and4

x R is the state.The response system is the controlled hyperchaotic Pangdynamics given by

1 2 1 1

2 2 1 3 4 2

3 3 1 2 3

4 1 2 4

( )

( )

y y y u

y y y y y u

y y y y u

y y y u

= +

= + +

= + +

= + +

(41)

where , , , are unknown parameters,4

y R is the state and 1 2 3 4, , ,u u u u are the

adaptivecontrollers to be designed.


1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (42)


1 2 1 2 1 1

2 2 4 1 1 3 1 3 2

3 3 3 1 2 1 2 3

4 1 2 1 2 4

( ) ( )

( )

e y y a x x u

e y e cx y y x x u

e y bx y y x x u

e y y dx x u

= + +

= + + +

= + + +

= + +

(43)


1 2 1 2 1 1 1

2 2 4 1 1 3 1 3 2 2

3 3 3 1 2 1 2 3 3

4 1 2 1 2 4 4

( )( ) ( )( )

( ) ( )

( ) ( )

( )( ) ( ) ( )

u t y y a t x x k e

u t y e c t x y y x x k e

u t y b t x y y x x k e

u t y y d t x t x k e

=

= + +

= +

= + + +

(44)


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In Eq. (44), , ( 1, 2,3, 4)i

k i = are positive gains, ( ), ( ), ( ), ( ), ( )a t b t c t d t t are estimates for the

unknown parameters , , , ,a b c d respectively, and ( ), ( ), ( ), ( )t t t t are estimates for the

unknown parameters , , , respectively.


1 2 1 2 1 1 1

2 2 1 2 2

3 3 3 3 3

4 1 2 1 2 4 4

( ( ))( ) ( ( ))( )

( ( )) ( ( ))

( ( )) ( ( ))

( ( ))( ) ( ( )) ( ( ))

e t y y a a t x x k e

e t y c c t x k e

e t y b b t x k e

e t y y d d t x t x k e

= +

= +

=

= +

(45)


( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )

( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )

a b c d e t a a t e t b b t e t c c t e t d d t e t t

e t t e t t e t t e t t

= = = = =

= = = = (46)


( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )

( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )

a b c d e t a t e t b t e t c t e t d t e t t

e t t e t t e t t e t t

= = = = =

= = = =

(47)


1 2 1 2 1 1 1

2 2 1 2 2

3 3 3 3 3

4 1 2 1 2 4 4

( ) ( )

( )

a

c

b

d

e e y y e x x k e

e e y e x k e

e e y e x k e

e e y y e x e x k e

= +

= +

=

= +

(48)


( )2 2 2 2 2 2 2 2 2 2 2 2 21 2 3 41

2a b c d

V e e e e e e e e e e e e e = + + + + + + + + + + + + (49)


2 2 2 21 1 2 2 3 3 4 4 1 2 1 3 3 2 1

4 1 4 2 1 2 1 3 3

2 2 4 1 2

( )

( )

+ ( )

a b c

d

V k e k e k e k e e e x x a e e x b e e x c

e e x d e e x e e y y e e y

e e y e e y y

= + + +

+ + + +

+ +

(50)


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1 2 1 5 1 2 1 10

3 3 6 3 3 11

2 1 7 2 2 12

4 1 8 4 1 2 13

( ) , ( )

,

,

, ( )

a

b

c

d

a e x x k e e y y k e

b e x k e e y k e

c e x k e e y k e

d e x k e e y y k

= + = +

= + = +

= + = +

= + = + +

4 2 9

e

e x k e

= +

(51)

Theorem 6.1 The adaptive control law defined by Eq. (44) along with the parameter update law

defined by Eq. (51) achieve global and exponential anti-synchronization of the non-identical

hyperchaotic Yang system (40) and hyperchaotic Pang system (41) with unknown parameters for

all initial conditions4

(0), (0) .x y R Moreover, all the parameter estimation errors globally and

exponentially converge to zero for all initial conditions.



2 2 2 2 2 2 2 2 2

1 1 2 2 3 3 4 4 5 6 7 8 9

2 2 2 2

10 11 12 13

( )a b c d V e k e k ek e k e k e k e k e k e k e

k e k e k e k e

=

(52)

which is a negative definite function on13

.R This completes the proof.

Next, we illustrate our adaptive anti-synchronization results with MATLAB simulations. The


10h


hyperchaotic systems (40) and (41) with the nonlinear controller defined by (44). The feedback

gains in the adaptive controller (31) are taken as 4, ( 1, ,8).ik i= =

The parameters of the two hyperchaoticsystems are taken as in the hyperchaotic case, i.e.

35, 3, 35, 2, 7.5, 36, 3, 20, 2a b c d = = = = = = = = =


1 2 3 4(0) 29, (0) 14, (0) 23, (0) 9x x x x= = = =


1 2 3 4(0) 14, (0) 18, (0) 29, (0) 14y y y y= = = =



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(0) 9, (0) 4, (0) 3, (0) 8, (0) 4,

(0) 6, (0) 2, (0) 11, (0) 9

a b c d

= = = = =

= = = =

Figure 9depicts the anti-synchronization of the non-identical hyperchaotic Yang and hyperchaotic

Pang systems. Figure 10depicts the time-history of the anti-synchronization errors 1 2 3 4, , , .e e e e

Figure 11 depicts the time-history of the parameter estimation errors , , , ,e .a b c d e e e e Figure 12

depicts the time-history of the parameter estimation errors , , , .e e e e

Figure 9. Anti-Synchronization of Hyperchaotic Yang and Hyperchaotic Pang Systems



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Figure 11. Time-History of the Parameter Estimation Errors , , , ,a b c d e e e e e

Figure 12. Time-History of the Parameter Estimation Errors , , ,e e e e


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7. CONCLUSIONS

In this paper, we have used adaptive control to derive new results for the anti-synchronization ofhyperchaotic Yang system (2009) and hyperchaotic Pang system (2011) with unknown

parameters. Main results of anti-synchronization design results for hyperchaotic systemsaddressed in this paper were proved using adaptive control theory and Lyapunov stability theory.Hyperchaotic systems have important applications in areas like secure communication, data

encryption, neural networks, etc.MATLAB simulations have been shown to validate anddemonstrate the adaptive anti-synchronization results for hyperchaotic Yang and hyperchaotic

Pang systems.

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Author

Dr. V. Sundarapandian earned his D.Sc. in Electrical and Systems Engineeringfrom Washington University, St. Louis, USA in May 1996. He is Professor and

Dean of the R & D Centre at Vel Tech Dr. RR & Dr. SR Technical University,

Chennai, Tamil Nadu, India. So far, he has published over 300 research works in

refereed international journals. He has also published over 200 research papers in

National and International Conferences. He has delivered Key Note Addresses at

many International Conferences with IEEE and Springer Proceedings. He is an India

Chair of AIRCC. He is the Editor-in-Chief of the AIRCC Control Journals

International Journal of Instrumentation and Control Systems,International Journal

of Control Theory and Computer Modeling,International Journal of Information Technology, Control and

Automation,International Journal of Chaos, Control, Modelling and Simulation, andInternational Journal

of Information Technology, Modeling and Computing. His research interests are Control Systems, Chaos

Theory, Soft Computing, Operations Research, Mathematical Modelling and Scientific Computing. He has

published four text-books and conducted many workshops on Scientific Computing, MATLAB and

SCILAB.

Documents

ADAPTIVE CONTROLLER DESIGN FOR THE ANTI-SYNCHRONIZATION OF HYPERCHAOTIC YANG AND HYPERCHAOTIC PANG SYSTEMS