Anti-Synchronization of Tigan and Li Systems with Unknown Parameters via Adaptive Control

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1 An Intrnational Journal of Optimization and Control: Thoris & Applications Vol., No.1, pp.17-8 (01) IJOCTA ISSN: ISSN: Anti-Synchronization of Tigan and Li Systms with Unknown Paramtrs via Adaptiv Control Sundarapandian VAIDYANATHAN a and Karthikyan RAJAGOPAL b a Rsarch and Dvlopmnt Cntr, Vl Tch Dr. RR & Dr. SR Tchnical Univrsity - India sundarvtu@gmail.com b School of Elctronics and Elctrical Enginring, Singhania Univrsity - India rkarthikyan@gmail.com (Rcivd Octobr 9, 011; in final form Dcmbr 1, 011) Abstract. In this papr, th adaptiv nonlinar control mthod has bn dployd to driv nw rsults for th anti-synchronization of idntical Tigan systms (008), idntical Li systms (009) and nonidntical Tigan and Li systms. In adaptiv anti-synchronization of idntical chaotic systms, th paramtrs of th mastr and slav systms ar unknown and th fdback control law has bn drivd using th stimats of th systm paramtrs. In adaptiv anti-synchronization of non-idntical chaotic systms, th paramtrs of th mastr systm ar known, but th paramtrs of th slav systm ar unknown and accordingly, th fdback control law has bn drivd using th stimats of th paramtrs of th slav systm. Our adaptiv synchronization rsults drivd in this papr for th uncrtain Tigan and Li systms ar stablishd using Lyapunov stability thory. Numrical simulations ar shown to dmonstrat th ffctivnss of th adaptiv anti-synchronization schms for th uncrtain chaotic systms addrssd in this papr. Kywords: Adaptiv control, Anti-synchronization, Chaos, Tigan systm, Li systm. AMS Classification: 4H10, 9C10, 9C15 1. Introduction Chaotic systms ar nonlinar systms that ar highly snsitiv to initial conditions. This snsitivity is popularly known as th buttrfly ffct [1]. Th first chaotic systm was discovrd by Lornz [] whn h was studying wathr pattrns. Sinc th pionring work by Pcora and Carroll [] chaos synchronization and antisynchronization problms hav bn studid xtnsivly and intnsivly in th chaos litratur [-1]. Chaos thory has bn applid succssfully to a varity of filds such as physical systms [4], chmical systms [5], cological systm [6], scur communications [7-8], tc. In th last two dcads, various schms hav bn applid for chaos synchronization such as th OGY mthod [9], th activ control mthod [10-17], th adaptiv control mthod [18-], th tim-dlay fdback mthod [4], th backstpping dsign mthod [5-6], th sampld-data fdback synchronization mthod [7], th sliding mod control mthod [8-1] and othrs. In most of th chaos synchronization approachs, th mastr-slav or driv-rspons formalism is usd. If a particular chaotic systm is calld a mastr or driv systm and anothr chaotic systm is calld a slav or rspons systm, thn th goal of anti-synchronization is to us th output of th mastr systm to control th slav systm so that th stats of th slav systm hav th sam amplitud but opposit signs as th stats of th mastr systm asymptotically. Corrsponding Author. rkarthikyan@gmail.com 17

2 18 V. Sundarapadian, R. Karthikyan / Vol., No.1, pp.17-8 (01) IJOCTA In this papr, w discuss th antisynchronization of idntical hyprchaotic Tigan systms [], idntical Li systms [], and nonidntical Tigan and Li systms. Our synchronization rsults ar stablishd using th Lyapunov stability thory [4]. In adaptiv synchronization of idntical chaotic systms, th paramtrs of th mastr and slav systms ar unknown and w dvis fdback control laws using th stimats of th systm paramtrs. In adaptiv synchronization of non-idntical chaotic systms, th paramtrs of th mastr systm ar known, but th paramtrs of th slav systm ar unknown and w dvis fdback control laws using th stimats of th paramtrs of th slav systm. This papr has bn organizd as follows. In Sction, w discuss th adaptiv antisynchronization of idntical Tigan systms []. In Sction, w discuss th adaptiv antisynchronization of idntical Li systms []. In Sction 4, w discuss th adaptiv antisynchronization of non-idntical Tigan and Li systms. In Sction 5, w summariz th main rsults obtaind in this papr. y a( y y ) u y ( c a) y ay y u 1 1 y by y y u 1 () whr y1, y, y ar th stat variabls and u1, u, u ar th nonlinar control inputs to b dsignd. Th Tigan systms (1) and () ar chaotic whn th paramtr valus ar chosn as a.1, b 0.6, c 0 Th strang chaotic attractor of th systm (1) is dpictd in Figur 1.. Adaptiv Anti-Synchronization of Idntical Tigan Systms This sction dtails th adaptiv antisynchronization of idntical Tigan systms [], whn th paramtrs of th mastr and slav systms ar unknown..1. Thortical Rsults As th mastr systm, w considr th Tigan dynamics dscribd by x a( x x ) 1 1 x ( c a) x ax x 1 1 x bx x x 1 (1) Figur 1. Strang Attractor of th Tigan Systm Th anti-synchronization rror is dfind as y x, ( i 1,,). () i i i Th rror dynamics is obtaind as whr x1, x, x ar th stat variabls and abcar,, unknown paramtrs of th systm. As th slav systm, w considr th controlld Tigan dynamics dscribd by a( ) u ( c a) ay y ax x u b y y x x u. 1 1 (4)

3 Anti-Synchronization of Tigan and Li Chaotic Systms with Unknown Paramtrs via Adaptiv Control 19 W dfin th adaptiv control functions as u ( t) aˆ ( ) k u ( t) ( cˆ aˆ ) ay ˆ y ay ˆ y k u () t b ˆ y y x x k 1 1 (5) whr aˆ, bˆ, cˆar stimats of abcrspctivly,,, and k1, k, k ar positiv constants. Substituting (5) into (4), th closd-loop rror dynamics is obtaind as ( a aˆ )( ) k ( c cˆ) ( a aˆ)( y y x x ) k ( b bˆ ) k. (6) W dfin th paramtr stimation rrors as a aˆ, b bˆ, c cˆ. (7) a b c Using (7), th rror dynamics is simplifid as ( ) k 1 a ( y y x x ) k c 1 a k. b (8) For th drivation of th updat law for adjusting th stimats of paramtrs, th Lyapunov mthod is usd. W considr th quadratic Lyapunov function dfind by 1 V 1 a b c, (9) which is a positiv dfinit function on R W not that aˆ, bˆ, cˆ (10) a b c Diffrntiating (9) along th trajctoris of (8) and noting (10), w find that V k k k 1 1 ˆ a 1 ( y1 y x1x ) a ˆ ˆ b b c 1 c (11) In viw of (11), th stimatd paramtrs ar updatd by th following law: aˆ ( y y x x ) k a bˆ k 5 b cˆ k 1 6 c (1) whr k4, k5, k6ar positiv constants. Now, w stat and prov th following rsult. Thorm 1. Th idntical uncrtain Tigan systms (1) and () ar globally and xponntially antisynchronizd by th adaptiv control law (5), whr th updat law for th paramtr stimats aˆ, bˆ, cˆis givn by (1) and ki,( i 1,,,6) ar positiv constants. Th rrors for paramtr stimats a, b, c dcay to zro xponntially as t. Proof. This rsult is a simpl consqunc of th Lyapunov stability thory. W know that V as dfind in (9) is a positiv dfinit function on R Substituting (1) into (11), w obtain V k k k k k k (1) a 5 b 6 c, which is a ngativ dfinit function on R Hnc, by th Lyapunov stability thory [4], it follows that i ( t) 0 xponntially as t for i 1,, and a 0, b 0, c 0 as t. This complts th proof.

4 0 V. Sundarapadian, R. Karthikyan / Vol., No.1, pp.17-8 (01) IJOCTA.. Numrical Rsults For th numrical simulations, th fourth-ordr 6 Rung-Kutta mthod with tim-stp h 10 is usd to solv th two systms of diffrntial quations (1) and () with th adaptiv nonlinar controllr (5) and updat law of stimats (1). W tak k 4 for i 1,,,6. i Th paramtrs of th Tigan systms ar chosn so that th systm (1) and () ar chaotic, i.. a.1, b0.6 and c 0. Th initial valus of th paramtr stimats ar chosn as aˆ(0) 1, bˆ (0) and cˆ(0) 5. Th initial valus of th mastr systm (1) ar chosn as x1(0) 7, x(0) 1 and x (0) 8. Th initial valus of th slav systm () ar chosn as y (0) 4, y (0) 8 and y (0) 7. 1 Figur shows th anti-synchronization of th Tigan systms (1) and (). Figur shows th tim-history of th antisynchronization rrors 1,,. Figur 4 shows th tim-history of th paramtr stimats aˆ, bˆ, c ˆ. Figur 5 shows th tim-history of th paramtr stimation rrors a, b, c. Figur 6 shows th tim-history of th applid control inputs u1, u, u. Figur. Anti-Synchronization of Idntical Tigan Chaotic Systms Figur. Tim History of th Error Stats 1,, Figur 4. Tim History of th Estimats aˆ, bˆ, c ˆ

5 Anti-Synchronization of Tigan and Li Chaotic Systms with Unknown Paramtrs via Adaptiv Control 1 x ( x x ) 1 1 x x x x 1 x x x x 1 (14) whr x1, x, x ar th stat variabls and,, ar unknown paramtrs of th systm. As th slav systm, w considr th controlld Li dynamics dscribd by Figur 5. Tim History of th Estimation Errors y ( y y ) u y y y y u 1 y y y y u 1 (15) whr y1, y, y ar th stat variabls and u1, u, u ar th nonlinar control inputs to b dsignd. Th Li systms (14) and (15) ar chaotic whn th paramtr valus ar chosn as; 5, 16, 1. Th strang chaotic attractor of th systm (14) is dpictd in Figur 7. Figur 6. Tim History of th Applid Control Inputs u, u, u 1. Adaptiv Anti-Synchronization of Idntical Li Systms This sction dtails th adaptiv antisynchronization of idntical Li systms [], whn th paramtrs of th mastr and slav systms ar unknown..1. Thortical Rsults As th mastr systm, w considr th Li dynamics dscribd by Figur 7. Strang Attractor of th Li Systm Th anti-synchronization rror is dfind as y x, ( i 1,,). (16) i i i

6 V. Sundarapadian, R. Karthikyan / Vol., No.1, pp.17-8 (01) IJOCTA Th rror dynamics is obtaind as ( ) u y y x x u 1 1 y y x x u. 1 1 W dfin th adaptiv control functions as u ( t) ˆ ( ) k u () t y y x x k 1 1 u ( t) ˆ y y x x k 1 1 (17) (18) whr ˆ, ˆ, ˆ ar stimats of,,, rspctivly and k1, k, k ar positiv constants. Substituting (18) into (17), th closd-loop rror dynamics is obtaind as ( ˆ )( ) k k ( ˆ ) ( ˆ ) k. W dfin th paramtr stimation rrors as; (19) ˆ, ˆ, ˆ. (0) Using (0), th rror dynamics is simplifid as ( ) k k k. (1) For th drivation of th updat law for adjusting th stimats of paramtrs, th Lyapunov mthod is usd. W considr th quadratic Lyapunov function dfind by 1 V 1, () which is a positiv dfinit function on R W not that ˆ, ˆ, ˆ () Diffrntiating () along th trajctoris of (1) and noting (), w find that V k11 k ˆ k 1 ( 1 ) (4) ˆ ˆ. In viw of (4), th stimatd paramtrs ar updatd by th following law: ˆ ( ) k ˆ ˆ k 5 k 6 whr k4, k5, k6ar positiv constants. (5) Now, w stat and prov th following rsult. Thorm. Th idntical uncrtain Li systms (14) and (15) ar globally and xponntially antisynchronizd by th adaptiv control law (18), whr th updat law for th paramtr stimats ˆ, ˆ, ˆ is givn by (5) and ki,( i1,,,6) ar positiv constants. Th rrors for paramtr stimats,, dcay to zro xponntially as t. Proof. This rsut is a simpl consqunc of th Lyapunov stability thory. W know that V as dfind in () is a positiv dfinit function on R Substituting (5) into (4), w obtain V k k k k k k (6) , which is a ngativ dfinit function on R Hnc, by th Lyapunov stability thory [4], it follows that i ( t) 0 xponntially as t for i 1,, and 0, 0, 0, as t.

7 Anti-Synchronization of Tigan and Li Chaotic Systms with Unknown Paramtrs via Adaptiv Control.. Numrical Rsults For th numrical simulations, th fourth-ordr 6 Rung-Kutta mthod with tim-stp h 10 is usd to solv th two systms of diffrntial quations (14) and (15) with th adaptiv nonlinar controllr (18) and updat law of stimats (5). W tak ki 4 for i 1,,,6. Th paramtrs of th Li systms ar chosn so that th systm (14) and (15) ar chaotic, i.. 5, 16 and 1. Th initial valus of th paramtr stimats ar chosn as ˆ(0) 4, ˆ (0) 5 and ˆ(0) 1. Th initial valus of th mastr systm (14) ar chosn as x1(0) 4, x(0) 8 and x (0) 10. Th initial valus of th slav systm (15) ar chosn as y1(0) 1, y(0) 15 and y (0) 7. Figur 8 shows th anti-synchronization of th Li systms (14) and (15). Figur 9 shows th tim-history of th anti-synchronization rrors 1,,. Figur 10 shows th tim-history of th paramtr stimats ˆ, ˆ, ˆ. Figur 11 shows th tim-history of th paramtr stimation rrors,,. Figur 1 shows th tim-history of th applid control inputs u1, u, u. Figur 9. Tim History of th Error Stats 1,, Figur 10. Tim History of th Estimats ˆ, ˆ, ˆ Figur 8. Anti-Synchronization of Idntical Li Chaotic Systms Figur 11. Tim History of th Estimation Errors

8 4 V. Sundarapadian, R. Karthikyan / Vol., No.1, pp.17-8 (01) IJOCTA y ( y y ) u y y y y u 1 y y y y u 1 (8) whr y1, y, yar th stat variabls,,, ar unknown paramtrs of th systm and u1, u, u ar th nonlinar control inputs to b dsignd. Th anti-synchronization rror is dfind as y x, ( i 1,,). (9) i i i Th rror dynamics is obtaind as Figur 1. Tim History of th Applid Control Inputs u, u, u 1 4. Adaptiv Anti-Synchronization of Tigan and Li Chaotic Systms This sction dtails th adaptiv antisynchronization of Tigan and Li systms. Hr, w considr th Tigan systm [] as th mastr systm, whos paramtrs ar known. Also, w considr th Li systm [] as th slav systm, whos paramtrs ar unknown Thortical Rsults As th mastr systm, w considr th Tigan dynamics dscribd by x a( x x ) 1 1 x ( c a) x ax x 1 1 x bx x x 1 (7) whr x1, x, xar th stat variabls and abc,, ar known paramtrs of th systm. As th slav systm, w considr th controlld Li dynamics dscribd by ( y y ) a( x x ) u y ( c a) x y y ax x u y bx y y x x u. 1 1 W dfin th adaptiv control functions as u ( t) ˆ ( y y ) a( x x ) k u ( t) y ( c a) x y y ax x k u () t ˆ ˆ y bx y y x x k 1 1 (0) (1) whr ˆ, ˆ, ˆ ar stimats of,,, rspctivly and k1, k, k ar positiv constants. Substituting (1) into (0), th closd-loop rror dynamics is obtaind as ( ˆ )( y y ) k k ( ˆ ) ( ˆ ) y k. () W dfin th paramtr stimation rrors as ˆ ˆ ˆ. ()

9 Anti-Synchronization of Tigan and Li Chaotic Systms with Unknown Paramtrs via Adaptiv Control 5 Using (), th rror dynamics is simplifid as ( y y ) k k y k. (4) For th drivation of th updat law for adjusting th stimats of paramtrs, th Lyapunov mthod is usd. W considr th quadratic Lyapunov function dfind by 1 V 1, (5) Thorm. Th Tigan systm (7) with known paramtrs and th Li systm (8) with unknown paramtrs ar globally and xponntially antisynchronizd by th adaptiv control law (1), whr th updat law for th paramtr stimats ˆ, ˆ, ˆ is givn by (8) and ki,( i1,,,6) ar positiv constants. Th rrors for paramtr stimats,, dcay to zro xponntially as t. Proof. This rsut is a simpl consqunc of th Lyapunov stability thory. W know that V as dfind in (5) is a positiv dfinit function on R Substituting (8) into (7), w obtain which is a positiv dfinit function on W not that ˆ ˆ ˆ. R (6) V k k k k k k (9) , which is a ngativ dfinit function on R Hnc, by th Lyapunov stability thory [4], it follows that i ( t) 0 xponntially as t for i 1,, and a 0, b 0, c 0, as t. This complts th proof. Diffrntiating (5) along th trajctoris of (4) and noting (6), w find that V k k k 1 1 ˆ 1 ( y y1 ) ˆ ˆ y. (7) In viw of (7), th stimatd paramtrs ar updatd by th following law: ˆ ( y y ) k ˆ ˆ y k k 6 whr k4, k5, k6ar positiv constants. Now, w stat and prov th following rsult. (8) 4.. Numrical Rsults For th numrical simulations, th fourth-ordr 6 Rung-Kutta mthod with tim-stp h 10 is usd to solv th two systms of diffrntial quations (7) and (8) with th adaptiv nonlinar controllr (1) and updat law of stimats (8). W tak ki 4 for i 1,,,6. Th paramtrs of th Tigan systm (7) ar chosn so that th systm is chaotic, namly, a.1, b 0.6 and c 0. Th paramtrs of th Li systm (8) ar chosn so that th systm is chaotic, namly, 5, 16 and 1. Th initial valus of th paramtr stimats ar chosn as ˆ(0), ˆ(0) 1and ˆ(0) 7. Th initial valus of th mastr systm (7) ar chosn as x1(0) 4, x(0) 8 and x (0) 5.

10 6 V. Sundarapadian, R. Karthikyan / Vol., No.1, pp.17-8 (01) IJOCTA Th initial valus of th slav systm (8) ar chosn as y (0) 6, y (0) 4 and y (0) 9. 1 Figur 1 shows th anti-synchronization of th Tigan systm (7) and th Li systm (8). Figur 14 shows th tim-history of th antisynchronization rrors 1,,. Figur 15 shows th tim-history of th paramtr stimats ˆ, ˆ, ˆ. Figur 16 shows th tim-history of th paramtr stimation rrors,,. Figur 17 shows th tim-history of th applid control inputs u1, u, u. Figur 15. Tim History of th Estimats ˆ, ˆ, ˆ Figur 1. Anti-Synchronization of Non-Idntical Tigan and Li Chaotic Systms Figur 16. Tim History of th Estimation Errors Figur 14. Tim History of th Error Stats 1,, Figur 17. Tim History of th Applid Control Inputs u, u, u 1

11 Anti-Synchronization of Tigan and Li Chaotic Systms with Unknown Paramtrs via Adaptiv Control 7 5. Conclusion In this papr, th adaptiv control mthod has bn applid in th study of global chaos antisynchornization of idntical Tigan systms [] idntical Li systms [] and non-idntical Tigan systm with known paramtrs and th Li systm with unknown paramtrs. For th adaptiv antisynchronization of idntical chaotic systms, it was assumd that th systm paramtrs ar unknown. For th adaptiv anti-synchronization of diffrnt chaotic systms, it was assumd that th paramtrs of th mastr systm ar known, but th paramtrs of th slav systm ar unknown. Our throtical rsults hav bn fully stablishd using th Lyapunov stability thory. Numrical simulations ar also shown for th antisynchronization of idntical and non-idntical Tigan and Li chaotic systms to dmonstrat th ffctivnss of th adaptiv anti-synchronization schms drivd in this papr. Acknowldgmnts Th authors thank th many anonymous rviwrs for thir valuabl commnts and suggstions concrning this papr. Rfrncs [1] Alligood, K.T., Saur, T. & York, J.A., Chaos: An Introduction to Dynamical Systms, Springr, Nw York (1987). [] Lornz, E., Dtrministic nonpriodic flow, J. Atmos. Scincs, 0, (196). [] Pcora, L.M. & Carroll, T.L., Synchronization in chaotic systms, Phys. Rv. Lttrs, 64, (1990). [4] Lakshmanan, M. & Murali, K., Chaos in Nonlinar Oscillators: Controlling and Synchronization, World Scintific, Singapor (1996). [5] Han, S.K., Krrr, C. & Kuramoto, Y., Dphasing and bursting in coupld nural oscillators, Phys. Rv. Lttrs, 75, (1995) [6] Blasius, B., Hupprt, A. & Ston, L., Complx dynamics and phas synchronization in spatially xtndd cological systm, Natur, 99, (1999). [7] Cuomo, K.M. & Oppnhim, A.V. Circuit implmntation of synchronizd chaos with applications to communications, Phys. Rv. Lttrs, 71, (199). [8] Li, Z., Li, K., Wn, C. & Soh, Y.C., A nw chaotic scur communication systm, IEEE Trans. Comm, 51 (8), (00). [9] Ott, E., Grbogi, C. & York, J.A., Controlling chaos, Phys. Rv. Ltt., 64, (1990). [10] Bai, E.W. & Longrn, K.E., Synchronization of two Lornz systms using activ control, Chaos, Solit. Fractals, 8, (1997). [11] Ho, M.C. & Hung, Y.C., Synchronization of two diffrnt chaotic systms using gnralizd activ control, Phys. Ltt. A, 01, (00). [1] Huang, L., Fng, R. & Wang, M., Synchronization of chaotic systms via nonlinar control, Phys. Ltt. A, 0, (005). [1] Li, Y., Xu, W., Shn, J. & Fang, T., Global synchronization of two paramtrically xcitd systms using activ control, Chaos Solit. Fract., 8, (006). [14] Chn, H.K., Global chaos synchronization of nw chaotic systms via nonlinar control, Chaos Solit. Fract.,, (005). [15] Vincnt, U.E., Synchronization of idntical and non-idntical 4-D systms via activ control, Chaos Solit. Fract., 1, (007). [16] Sundarapandian, V. & Karthikyan, R., Global chaos synchronization of hyprchaotic Liu and hyprchaotic Chn systms by activ nonlinar control, CIIT Int. J. Digital Signal Procssing, (), (011). [17] Sundarapandian, V. & Karthikyan, R., Global chaos synchronization of Chn and Cai systms by activ nonlinar control, CIIT Int. J. Digital Signal Procssing, (), (011). [18] Lu, J., Wu, X., Han, X. & Lü, J., Adaptiv fdback stabilization of a unifid chaotic systm, Phys. Ltt. A, 9, 7- (004). [19] Chn, S.H. & Lü, J., Synchronization of an uncrtian unifid systm via adaptiv control, Chaos Solit. Fract., 14, (00). [0] Aghababa, M.P. & Aghababa, H.P., Adaptiv finit-tim stabilization of uncrtain nonautonomous chaotic lctromchanical gyrostat systms with unknown paramtrs, Mch. Rsarch Commun., 8, (011). [1] Aghababa, M.P., A novl adaptiv finit-tim controllr for synchronizing chaotic gyros with

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