International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 2013DOI : 10.5121/ijics.2013.3201...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20132unknown parameters. The main...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20133When the system parameters i...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20134Figure 1. The Phase Portrait...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201354. ADAPTIVE CONTROL DESIGN F...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20136In Eq. (18), , ( 1,2,3,4)ik ...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20137Theorem 4.1 The adaptive con...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20138Figure 3. Anti-Synchronizati...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20139Figure 5. Time-History of th...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201310where 4y∈R is the state and...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 2013111 2 1 1 12 2 2 23 3 3 34 1 ...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201312The parameters of the hyper...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201313Figure 7. Time-History of t...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 2013146. ADAPTIVE CONTROL DESIGN ...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201315In Eq. (44), , ( 1,2,3,4)ik...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201316In view of (50), we choose ...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201317ˆ ˆ ˆˆ ˆ(0) 9, (0) 4, (0) 3...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201318Figure 11. Time-History of ...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 2013197. CONCLUSIONSIn this paper...
International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201320[19] Sundarapandian, V. (20...
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ADAPTIVE CONTROLLER DESIGN FOR THE ANTI-SYNCHRONIZATION OF HYPERCHAOTIC YANG AND HYPERCHAOTIC PANG SYSTEMS

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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) and
hyperchaotic 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.

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

  1. 1. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 2013DOI : 10.5121/ijics.2013.3201 1ADAPTIVE CONTROLLER DESIGN FOR THEANTI-SYNCHRONIZATION OF HYPERCHAOTICYANG AND HYPERCHAOTIC PANG SYSTEMSSundarapandian Vaidyanathan11Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical UniversityAvadi, Chennai-600 062, Tamil Nadu, INDIAsundarvtu@gmail.comABSTRACTIn the anti-synchronization of chaotic systems, a pair of chaotic systems called drive and responsesystemsare considered, and the design goal is to drive the sum of their respective states to zero asymptotically. Thispaper derives new results for the anti-synchronization of hyperchaotic Yang system (2009) andhyperchaotic Pang system (2011) with uncertain parameters via adaptive control. Hyperchaotic systemsare nonlinear chaotic systems withtwo or more positive Lyapunov exponents and they have applications inareas like neural networks, encryption, secure data transmission and communication. The main resultsderived in this paper are illustrated with MATLAB simulations.KEYWORDSHyperchaos, Adaptive Control, Anti-Synchronization, Hyperchaotic Systems.1. INTRODUCTIONSince the discovery of a hyperchaotic system by O.E.Rössler ([1], 1979), hyperchaotic systemsare known to have characteristics like high security, high capacity and high efficiency.Hyperchaotic systems are chaotic systems having two or morepositive Lyapunov exponents. Theyare 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 zeroasymptotically [8-9].The anti-synchronization problem deals with a pair of chaotic systems called the drive andresponse 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 systemshave been studied via several methods like active control method [10-12], adaptive controlmethod [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
  2. 2. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20132unknown parameters. The main results derived in this paper were proved using adaptive controltheory [25] and Lyapunov stability theory [26].2. PROBLEM STATEMENTThe drive system is described by the chaotic dynamics( )x Ax f x= + (1)where A 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)where B is the n n× matrix of the system parameters, : n ng →R R is the nonlinear part andnu ∈R is the active controller to be designed.For the pair of chaotic systems (1) and (2), the design goal of the anti-synchronization problemisto construct a feedback controller ,u which anti-synchronizes their states for all (0), (0) .nx y ∈RTheanti-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 thatlim ( ) 0te t→∞= for all (0)e ∈Rn(5)Using the matrix method, we consider a candidate Lyapunov function( ) ,TV 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 controller uso that( ) ,TV 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.
  3. 3. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20133When the system parameters in (1) and (2) are unknown, we need to construct a parameter updatelaw for determining the estimates of the unknown parameters.3. HYPERCHAOTIC SYSTEMSThe hyperchaotic Yang system ([23], 2009) is given by1 2 12 1 1 3 43 3 1 24 1 2( )x a x xx cx x x xx bx x xx 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 values35, 3, 35, 2, 7.5a b c d = = = = = (9)The Lyapunov exponents of the system (8) for the parametric values in (9) are1 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 by1 2 12 2 1 3 43 3 1 24 1 2( )( )x x xx x x x xx x x xx x x= −= − += − += − +(11)where , , ,    are constant, positive parameters of the system.The Pang system (11) exhibits a hyperchaotic attractor for the parametric values36, 3, 20, 2   = = = = (12)The Lyapunov exponents of the system (9) for the parametric values in (12) are1 2 3 41.4106, 0.1232, 0, 20.5339   = = = = − (13)Since there are two positive Lyapunov exponents in (13), the Pang system (11) is hyperchaoticfor the parametric values (12).The phase portrait of the hyperchaotic Pang system is described in Figure 2.
  4. 4. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20134Figure 1. The Phase Portrait of the Hyperchaotic Yang SystemFigure 2. The Phase Portrait of the Hyperchaotic Pang System
  5. 5. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201354. ADAPTIVE CONTROL DESIGN FOR THE ANTI-SYNCHRONIZATION OFHYPERCHAOTIC YANG SYSTEMSIn this section, we design an adaptive controller for the anti-synchronization of two identicalhyperchaotic Yang systems (2009) with unknown parameters.Thedrive system is the hyperchaotic Yangdynamicsgiven by1 2 12 1 1 3 43 3 1 24 1 2( )x a x xx cx x x xx bx x xx 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 by1 2 1 12 1 1 3 4 23 3 1 2 34 1 2 4( )y a y y uy cy y y y uy by y y uy dy y u= − += − + += − + += − − +(15)where4y∈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 as1 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 as1 2 1 12 1 4 1 3 1 3 23 3 1 2 1 2 34 1 2 4( )e a e e ue ce e y y x x ue be y y x x ue de e u= − += + − − += − + + += − − +(17)The adaptive controller to achieve anti-synchronization is chosen as1 2 1 1 12 1 4 1 3 1 3 2 23 3 1 2 1 2 3 34 1 2 4 4ˆ( ) ( )( )ˆ( ) ( )ˆ( ) ( )ˆ ˆ( ) ( ) ( )u t a t e e k eu t c t e e y y x x k eu t b t e y y x x k eu t d t e t e k e= − − −= − − + + −= − − −= + −(18)
  6. 6. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20136In Eq. (18), , ( 1,2,3,4)ik i = are positive gains and ˆ ˆ ˆˆ ˆ( ), ( ), ( ), ( ), ( )a t b t c t d t t are estimates for theunknown parameters , , , ,a b c d  respectively.By the substitution of (18) into (17), the error dynamics is simplified as1 2 1 1 12 1 2 23 3 3 34 1 2 4 4ˆ( ( ))( )ˆ( ( ))ˆ( ( ))ˆ ˆ( ( )) ( ( ))e a a t e e k ee c c t e k ee b b t e k ee d d t e t e k e = − − −= − −= − − −= − − − − −(19)As a next step, we define the parameter estimation errors asˆ ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c de 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 getˆ ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c de 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 obtain1 2 1 1 12 1 2 23 3 3 34 1 2 4 4( )acbde e e e k ee e e k ee e e k ee e e e e k e= − −= −= − −= − − −(22)We consider the candidate Lyapunov function( )2 2 2 2 2 2 2 2 21 2 3 412a b c dV e e e e e e e e e= + + + + + + + + (23)Differentiating (23) along the dynamics (21) and (22), we obtain( )( ) ( ) ( )2 2 2 2 21 1 2 2 3 3 4 4 1 2 1 31 2 1 4 2 4ˆˆ( )ˆ ˆˆa bc dV k e k e k e k e e e e e a e e be 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 4 823 6 2 4 91 2 7ˆˆ ( ) ,ˆ ˆ,ˆa dbca e e e k e d e e k eb e k e e e k ec e e k e= − + = − += − + = − += + (25)Next, we prove the following main result of this section.
  7. 7. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20137Theorem 4.1 The adaptive control law defined by Eq. (18) along with the parameter update lawdefined by Eq. (25) achieveglobal and exponential anti-synchronization of the identicalhyperchaotic Yang systems (14) and (15)with unknown parameters for all initial conditions4(0), (0) .x y ∈R Moreover, the parameter estimation errors ( ), ( ), ( ), ( ), ( )a b c de 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 thecandidate Lyapunov function. Substituting the parameter update law (25) into (24), we get2 2 2 2 2 2 2 2 21 1 2 2 3 3 4 4 5 6 7 8 9( ) a b c dV 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 on 9.R This completes the proof. Next, we illustrate our adaptive anti-synchronization results with MATLAB simulations.Theclassical fourth order Runge-Kutta method with time-step 810h −= has been used to solve thehyperchaotic 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 as1 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 as1 2 3 4(0) 34, (0) 8, (0) 28, (0) 20y y y y= = − = = −Also, the initial conditions of the parameter estimates are taken asˆ ˆ ˆˆ ˆ(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 eFigure 5 depicts the time-history of the parameter estimation errors , , , , .a b c de e e e e
  8. 8. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20138Figure 3. Anti-Synchronization of Identical Hyperchaotic Yang SystemsFigure 4. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e e
  9. 9. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 20139Figure 5. Time-History of the Parameter Estimation Errors , , , ,a b c de e e e e5. ADAPTIVE CONTROL DESIGN FOR THE ANTI-SYNCHRONIZATION OFHYPERCHAOTIC PANG SYSTEMSIn this section, we design an adaptive controller for the anti-synchronization of two identicalhyperchaotic Pang systems (2011) with unknown parameters.Thedrive system is the hyperchaotic Pangdynamics given by1 2 12 2 1 3 43 3 1 24 1 2( )( )x x xx x x x xx x x xx x x= −= − += − += − +(27)where , , ,    are unknown parameters of the system and 4x∈ R is the state.The response system is the controlled hyperchaotic Pangdynamics given by1 2 1 12 2 1 3 4 23 3 1 2 34 1 2 4( )( )y y y uy y y y y uy y y y uy y y u= − += − + += − + += − + +(28)
  10. 10. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201310where 4y∈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 as1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (29)Then we derive the error dynamics as1 2 1 12 2 4 1 3 1 3 23 3 1 2 1 2 34 1 2 4( )( )e e e ue e e y y x x ue e y y x x ue e e u= − += + − − += − + + += − + +(30)The adaptive controller to achieve anti-synchronization is chosen as1 2 1 1 12 2 4 1 3 1 3 2 23 3 1 2 1 2 3 34 1 2 4 4ˆ( )( )ˆ( )ˆ( )ˆ( )( )u t e e k eu t e e y y x x k eu t e y y x x k eu t e e k e= − − −= − − + + −= − − −= + −(31)In Eq. (31), , ( 1,2,3,4)ik i = are positive gains and ˆ ˆˆ ˆ( ), ( ), ( ), ( )t t t t    are estimates for theunknown parameters , , ,    respectively.By the substitution of (31) into (30), the error dynamics is simplified as1 2 1 1 12 2 2 23 3 3 34 1 2 4 4ˆ( ( ))( )ˆ( ( ))ˆ( ( ))ˆ( ( ))( )e t e e k ee t e k ee t e k ee t e e k e    = − − −= − −= − − −= − − + −(32)As a next step, we define the parameter estimation errors asˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )e t t e t t e t t e t t          = − = − = − = − (33)Upon differentiation, we getˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )e t t e t t e t t e t t      = − = − = − = −      (34)Substituting (33) into the error dynamics (32), we obtain
  11. 11. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 2013111 2 1 1 12 2 2 23 3 3 34 1 2 4 4( )( )e e e e k ee e e k ee e e k ee e e e k e= − −= −= − −= − + −(35)We consider the candidate Lyapunov function( )2 2 2 2 2 2 2 21 2 3 412V e e e e e e e e   = + + + + + + + (36)Differentiating (36) along the dynamics (34) and (35), we obtain( )( ) ( )2 2 2 2 21 1 2 2 3 3 4 4 1 2 1 322 1 4 1 2ˆˆ( )ˆˆ ( )V k e k e k e k e e e e e e ee e e e e e e     = − − − − + − − + − − + − + − + −(37)In view of (37), we choose the following parameter update law:1 2 1 523 622 71 4 1 2 8ˆ ( )ˆˆˆ ( )e e e k ee k ee k ee 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 conditions4(0), (0) .x y ∈R Moreover, the parameter estimation errors ( ), ( ), ( ), ( )e t e t e t e t    globally andexponentially converge to zero for all initial conditions.Proof.The proof is via Lyapunov stability theory [26] by taking V defined by Eq. (36) as thecandidate Lyapunov function. Substituting the parameter update law (38) into (37), we get2 2 2 2 2 2 2 2 21 1 2 2 3 3 4 4 5 6 7 8 9( ) a b c dV 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. Theclassical fourth order Runge-Kutta method with time-step 810h −= has been used to solve thehyperchaotic 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= = 
  12. 12. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201312The parameters of the hyperchaotic Pang systems are taken as in the hyperchaotic case, i.e.36, 3, 20, 2   = = = =For simulations, the initial conditions of the drive system (27) are taken as1 2 3 4(0) 17, (0) 22, (0) 11, (0) 25x x x x= = − = − =Also, the initial conditions of the response system (28) are taken as1 2 3 4(0) 24, (0) 18, (0) 24, (0) 17y y y y= = − = = −Also, the initial conditions of the parameter estimates are taken asˆ ˆˆ ˆ(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 eFigure 8 depicts the time-history of the parameter estimation errors , , , .e e e e   Figure 6. Anti-Synchronization of Identical Hyperchaotic Pang Systems
  13. 13. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201313Figure 7. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e eFigure 8. Time-History of the Parameter Estimation Errors , , ,e e e e   
  14. 14. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 2013146. ADAPTIVE CONTROL DESIGN FOR THE ANTI-SYNCHRONIZATION OFHYPERCHAOTIC YANG AND HYPERCHAOTIC PANG SYSTEMSIn this section, we design an adaptive controller for the anti-synchronization of non-identicalhyperchaotic Yang (2009) and hyperchaotic Pang systems (2011) with unknown parameters.Thedrive system is the hyperchaotic Yangdynamics given by1 2 12 1 1 3 43 3 1 24 1 2( )x a x xx cx x x xx bx x xx dx x= −= − += − += − −(40)where , , , ,a b c d  are unknown parameters of the system and 4x∈ R is the state.The response system is the controlled hyperchaotic Pangdynamics given by1 2 1 12 2 1 3 4 23 3 1 2 34 1 2 4( )( )y y y uy y y y y uy y y y uy y y u= − += − + += − + += − + +(41)where , , ,    are unknown parameters,4y∈R is the state and 1 2 3 4, , ,u u u u are theadaptivecontrollers to be designed.For the anti-synchronization, the error e is defined as1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (42)Then we derive the error dynamics as1 2 1 2 1 12 2 4 1 1 3 1 3 23 3 3 1 2 1 2 34 1 2 1 2 4( ) ( )( )e y y a x x ue y e cx y y x x ue y bx y y x x ue y y dx x u = − + − += + + − − += − − + + += − + − − +(43)The adaptive controller to achieve anti-synchronization is chosen as1 2 1 2 1 1 12 2 4 1 1 3 1 3 2 23 3 3 1 2 1 2 3 34 1 2 1 2 4 4ˆ ˆ( )( ) ( )( )ˆ ˆ( ) ( )ˆˆ( ) ( )ˆˆ ˆ( )( ) ( ) ( )u t y y a t x x k eu t y e c t x y y x x k eu t y b t x y y x x k eu t y y d t x t x k e = − − − − −= − − − + + −= + − − −= + + + −(44)
  15. 15. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201315In Eq. (44), , ( 1,2,3,4)ik i = are positive gains, ˆ ˆ ˆˆ ˆ( ), ( ), ( ), ( ), ( )a t b t c t d t t are estimates for theunknown parameters , , , ,a b c d  respectively, and ˆ ˆˆ ˆ( ), ( ), ( ), ( )t t t t    are estimates for theunknown parameters , , ,    respectively.By the substitution of (44) into (43), the error dynamics is simplified as1 2 1 2 1 1 12 2 1 2 23 3 3 3 34 1 2 1 2 4 4ˆ ˆ( ( ))( ) ( ( ))( )ˆ ˆ( ( )) ( ( ))ˆˆ( ( )) ( ( ))ˆˆ ˆ( ( ))( ) ( ( )) ( ( ))e t y y a a t x x k ee t y c c t x k ee t y b b t x k ee t y y d d t x t x k e      = − − + − − −= − + − −= − − − − −= − − + − − − − −(45)As a next step, we define the parameter estimation errors asˆ ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c de t a a t e t b b t e t c c t e t d d t e t te t t e t t e t t e t t           = − = − = − = − = −= − = − = − = −(46)Upon differentiation, we getˆ ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c de t a t e t b t e t c t e t d t e t te t t e t t e t t e t t      = − = − = − = − = −= − = − = − = −            (47)Substituting (46) into the error dynamics (45), we obtain1 2 1 2 1 1 12 2 1 2 23 3 3 3 34 1 2 1 2 4 4( ) ( )( )acbde e y y e x x k ee e y e x k ee e y e x k ee e y y e x e x k e = − + − −= + −= − − −= − + − − −(48)We consider the candidate Lyapunov function( )2 2 2 2 2 2 2 2 2 2 2 2 21 2 3 412a b c dV e e e e e e e e e e e e e    = + + + + + + + + + + + + (49)Differentiating (49) along the dynamics (47) and (48), we obtain2 2 2 21 1 2 2 3 3 4 4 1 2 1 3 3 2 14 1 4 2 1 2 1 3 32 2 4 1 2ˆˆ ˆ( )ˆ ˆˆ ˆ( )ˆˆ+ ( )a b cdV k e k e k e k e e e x x a e e x b e e x ce e x d e e x e e y y e e ye e y e e y y          = − − − − + − − + − − + −           + − − + − − + − − + − −          − + − + −       (50)
  16. 16. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201316In view of (50), we choose the following parameter update law:1 2 1 5 1 2 1 103 3 6 3 3 112 1 7 2 2 124 1 8 4 1 2 13ˆˆ ( ) , ( )ˆ ˆ,ˆˆ ,ˆ ˆ, ( )abcda e x x k e e y y k eb e x k e e y k ec e x k e e y k ed e x k e e y y k= − + = − += − + = − += + = += − + = − + +  4 2 9ˆee x k e = − +(51)Theorem 6.1 The adaptive control law defined by Eq. (44) along with the parameter update lawdefined by Eq. (51) achieve global and exponential anti-synchronization of the non-identicalhyperchaotic Yang system (40) and hyperchaotic Pang system (41) with unknown parameters forall initial conditions 4(0), (0) .x y ∈R Moreover, all the parameter estimation errors globally andexponentially converge to zero for all initial conditions.Proof.The proof is via Lyapunov stability theory [26] by taking V defined by Eq. (49) as thecandidate Lyapunov function. Substituting the parameter update law (51) into (50), we get2 2 2 2 2 2 2 2 21 1 2 2 3 3 4 4 5 6 7 8 92 2 2 210 11 12 13( ) a b c dV e k e k ek e k e k e k e k e k e k ek e k e k e k e   = − −− − − − − − −− − − −(52)which is a negative definite function on 13.R This completes the proof. Next, we illustrate our adaptive anti-synchronization results with MATLAB simulations. Theclassical fourth order Runge-Kutta method with time-step 810h −= has been used to solve thehyperchaotic systems (40) and (41) with the nonlinear controller defined by (44). The feedbackgains 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     = = = = = = = = =For simulations, the initial conditions of the drive system (40) are taken as1 2 3 4(0) 29, (0) 14, (0) 23, (0) 9x x x x= = = − = −Also, the initial conditions of the response system (41) are taken as1 2 3 4(0) 14, (0) 18, (0) 29, (0) 14y y y y= = − = = −Also, the initial conditions of the parameter estimates are taken as
  17. 17. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201317ˆ ˆ ˆˆ ˆ(0) 9, (0) 4, (0) 3, (0) 8, (0) 4,ˆ ˆˆ ˆ(0) 6, (0) 2, (0) 11, (0) 9a b c d    = = = − = = −= = = = −Figure 9depicts the anti-synchronization of the non-identical hyperchaotic Yang and hyperchaoticPang systems. Figure 10depicts the time-history of the anti-synchronization errors 1 2 3 4, , , .e e e eFigure 11 depicts the time-history of the parameter estimation errors , , , ,e .a b c de e e e  Figure 12depicts the time-history of the parameter estimation errors , , , .e e e e   Figure 9. Anti-Synchronization of Hyperchaotic Yang and Hyperchaotic Pang SystemsFigure 10. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e e
  18. 18. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201318Figure 11. Time-History of the Parameter Estimation Errors , , , ,a b c de e e e eFigure 12. Time-History of the Parameter Estimation Errors , , ,e e e e   
  19. 19. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 2013197. CONCLUSIONSIn 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 unknownparameters. 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, dataencryption, neural networks, etc.MATLAB simulations have been shown to validate anddemonstrate the adaptive anti-synchronization results for hyperchaotic Yang and hyperchaoticPang systems.REFERENCES[1] Rössler, O.E. (1979) “An equation for hyperchaos,” Physics Letters A, Vol. 71, pp 155-157.[2] Machado, L.G., Savi, M.A. & Pacheco, P.M.C.L. (2003) “Nonlinear dynamics and chaos incoupled shape memory oscillators,” International Journal of Solids and Structures, Vol. 40, No.19, pp. 5139-5156.[3] Huang, Y. & Yang, X.S. (2006) “Hyperchaos and bifurcation in a new class of four-dimensionalHopfield neural networks,” Neurocomputing, Vol. 69, pp 13-15.[4] Tao, Y. (1999) “Chaotic secure communication systems – history and new results”, Telecommun.Review, Vol. 9, pp 597-634.[5] Li, C., Liao, X. & Wong, K.W. (2005) “Lag synchronization of hyperchaos with applications tosecure communications,” Chaos, Solitons & Fractals, Vol. 23, No. 1, pp 183-193.[6] Prokhorov, M.D. & Ponomarenko, V.I. (2008) “Encryption and decryption of information inchaotic communication systems governed by delay-differential equations,” Chaos, Solitons &Fractals, Vol. 35, No. 5, pp 871-877.[7] Yassen, M.T. (2008) “Synchronization hyperchaos of hyperchaotic systems”, Chaos, Solitons andFractals, Vol. 37, pp 465-475.[8] Pecora, L.M. & Carroll, T.L. (1990) “Synchronization in chaotic systems”, Phys. Rev. Lett., Vol.64, pp 821-824.[9] Ott, E., Grebogi, C. & Yorke, J.A. (1990) “Controlling chaos”, Phys. Rev. Lett., Vol. 64, pp 1196-1199.[10] Huang, L. Feng, R. & Wang, M. (2004) “Synchronization of chaotic systems via nonlinearcontrol,” Physics Letters A, Vol. 320, No. 4, pp 271-275.[11] Lei, Y., Xu, W. & Zheng, H. (2005) “Synchronization of two chaotic nonlinear gyros using activecontrol,” Physics Letters A, Vol. 343, pp 153-158.[12] Sarasu, P. & Sundarapandian, V. (2011) “Active controller design for generalized projectivesynchronization of four-scroll chaotic systems”, International Journal of System Signal Controland Engineering Application, Vol. 4, No. 2, pp 26-33.[13] Sarasu, P. & Sundarapandian, V. (2012) “Generalized projective synchronization of two-scrollsystems via adaptive control,” International Journal of Soft Computing, Vol. 7, No. 4, pp 146-156.[14] Sundarapandian, V. (2012) “Adaptive control and synchronization of a generalized Lotka-Volterrasystem,” Vol. 1, No. 1, pp 1-12.[15] Sundarapandian, V. (2013) “Adaptive controller and synchronizer design for hyperchaotic Zhousystem with unknown parameters,” Vol. 1, No. 1, pp 18-32.[16] Bowong, S. &Kakmeni, F.M.M. (2004) “Synchronization of uncertain chaotic systems viabackstepping approach,” Chaos, Solitons & Fractals, Vol. 21, No. 4, pp 999-1011.[17] Suresh, R, & Sundarapandian, V. (2012) “Global chaos synchronization of WINDMI and Coulletchaotic systems by backstepping control”, Far East J. Math. Sciences, Vol. 67, No. 2, pp 265-287.[18] Suresh, R. & Sundarapandian, V. (2012) “Hybrid synchronization of n-scroll Chua and Lur’echaotic systems via backstepping control with novel feedback”, Arch. Control Sciences, Vol. 22,No. 3, pp 255-278.
  20. 20. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 201320[19] Sundarapandian, V. (2013) “Anti-synchronizing backstepping design for Arneodo chaoticsystem”, International Journal on Bioinformatics and Biosciences, Vol. 3, No. 1, pp 21-33.[20] Senejohnny, D.M. & Delavari, H. (2012) “Active sliding observer scheme based fractional chaossynchronization,” Comm. Nonlinear Sci. Numerical Simulation, Vol. 17, No. 11, pp 4373-4383.[21] Sundarapandian, V. (2012) “Anti-synchronization of hyperchaotic Xu systems via sliding modecontrol”, International Journal of Embedded Systems, Vol. 2, No. 2, pp 51-61.[22] Sundarapandian, V. (2013) “Anti-synchronizing sliding controller design for identical Pansystems,” International Journal of Computational Science and Information Technology, Vol. 1,No. 1, pp 1-9.[23] Yang, Q. & Liu, Y. (2009) “A hyperchaotic system from a chaotic system with one saddle and twostable node-foci,” J. Mathematical Analysis and Applications, Vol. 360, pp 293-306.[24] Pang, S. & Liu, Y. (2011) “A new hyperchaotic system from the Lü system and its control,”Journal of Computational and Applied Mathematics, Vol. 235, pp 2775-2789.[25] Aström, K.J. & Wittenmark, B. (1994) Adaptive Control, 2ndedition, Prentice Hall, N.Y., USA.[26] Hahn, W. (1967) The Stability of Motion, Springer, Berlin.AuthorDr. V. Sundarapandian earned his D.Sc. in Electrical and Systems Engineeringfrom Washington University, St. Louis, USA in May 1996. He is Professor andDean 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 inrefereed international journals. He has also published over 200 research papers inNational and International Conferences. He has delivered Key Note Addresses atmany International Conferences with IEEE and Springer Proceedings. He is an IndiaChair of AIRCC. He is the Editor-in-Chief of the AIRCC Control Journals –International Journal of Instrumentation and Control Systems, International Journalof Control Theory and Computer Modeling,International Journal of Information Technology, Control andAutomation, International Journal of Chaos, Control, Modelling and Simulation, and International Journalof Information Technology, Modeling and Computing. His research interests are Control Systems, ChaosTheory, Soft Computing, Operations Research, Mathematical Modelling and Scientific Computing. He haspublished four text-books and conducted many workshops on Scientific Computing, MATLAB andSCILAB.

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