ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU S FOUR-WING CHAOTIC SYSTEM WITH CUBIC NONLINEARITY

Transcription

1 ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU S FOUR-WING CHAOTIC SYSTEM WITH CUBIC NONLINEARITY Sunrpnin Viynthn 1 1 Reserch n Development Centre, Vel Tech Dr. RR Dr. SR Technicl University Avi, Chenni , Tmil Nu, INDIA sunrvtu@gmil.com ABSTRACT This pper investigtes the ptive chos control n synchroniztion of Liu s four-wing chotic system with cuic nonlinerity (Liu, 009) n unknown prmeters. First, we esign ptive control lws to stilize the Liu s four-wing chotic system with cuic nonlinerity to its unstle equilirium t the origin se on the ptive control theory n Lypunov stility theory. Next, we erive ptive control lws to chieve glol chos synchroniztion of ienticl Liu s four-wing chotic systems with cuic nonlinerity n unknown prmeters. Numericl simultions re shown to emonstrte the effectiveness of the propose ptive chos control n synchroniztion schemes. KEYWORDS Aptive Control, Chos Synchroniztion, Chos, Four-Wing System. 1. INTRODUCTION Chotic systems re ynmicl systems tht re highly sensitive to initil conitions. The sensitive nture of chotic systems is commonly clle s the utterfly effect [1]. Since chos phenomenon in wether moels ws first oserve y Lorenz in 1961, lrge numer of chos phenomen n chos ehviour hve een iscovere in physicl, socil, economicl, iologicl n electricl systems. The control of chotic systems is to esign stte feeck control lws tht stilizes the chotic systems roun the unstle equilirium points. Active control technique is use when the system prmeters re known n ptive control technique is use when the system prmeters re unknown [-4]. Synchroniztion of chotic systems is phenomenon tht my occur when two or more chotic oscilltors re couple or when chotic oscilltor rives nother chotic oscilltor. Becuse of the utterfly effect, which cuses the exponentil ivergence of the trjectories of two ienticl chotic systems strte with nerly the sme initil conitions, synchronizing two chotic systems is seemingly very chllenging prolem in the chos literture [5-16]. In 1990, Pecor n Crroll [5] introuce metho to synchronize two ienticl chotic systems n showe tht it ws possile for some chotic systems to e completely synchronize. From then on, chos synchroniztion hs een wiely explore in vriety of fiels incluing physicl systems [6], chemicl systems [7], ecologicl systems [8], secure communictions [9-10], etc. DOI : /ijcse

2 In most of the chos synchroniztion pproches, the mster-slve or rive-response formlism hs een use. If prticulr chotic system is clle the mster or rive system n nother chotic system is clle the slve or response system, then the ie of synchroniztion is to use the output of the mster system to control the slve system so tht the output of the slve system trcks the output of the mster system symptoticlly. Since the seminl work y Pecor n Crroll [5], vriety of impressive pproches hve een propose for the synchroniztion of chotic systems such s the smple-t feeck synchroniztion metho [11], OGY metho [1], time-ely feeck metho [13], ckstepping metho [14], ptive esign metho [15], sliing moe control metho [16], etc. This pper is orgnize s follows. In Section, we erive results for the ptive chos control of the Liu s four-wing chotic system with cuic nonlinerity (Liu, [17], 009) n unknown prmeters. In Section 3, we erive results for the ptive synchroniztion of ienticl Liu s four-wing chotic systems with cuic nonlinerity n unknown prmeters. In Section 4, we summrize the min results otine in this pper.. ADAPTIVE CHAOS CONTROL OF LIU S FOUR-WING CHAOTIC SYSTEM WITH CUBIC NONLINEARITY.1 Theoreticl Results The four-wing chotic system with cuic nonlinerity (Liu, [17], 009) is one of the importnt moels of 3D four-wing chotic systems. Using spectrl nlysis, Liu showe tht this chotic system hs extremely wie frequency nwith compre with tht of the Lorenz system n other four-wing chotic systems. The ynmics of Liu s four-wing system is escrie y = ( x x ) + x x = ( x + x ) x x = cx + x + x x x (1) where x1, x, x3 re the stte vriles n,, c, re positive constnts. The system (1) is chotic when the prmeter vlues re tken s = 50, = 13, c = 13 n = 6. () When the prmeter vlues re tken s in (), the system (1) is chotic n the system lineriztion mtrix t the equilirium point E 0 = (0,0,0) is given y 0 A = 0 0 c which hs the eigenvlues 109

3 λ = 13, λ = n λ 3 = 1 Since λ 3 > 0, it is immeite from Lypunov stility theory [18] tht the system (1) is unstle t the equilirium point E 0 = (0, 0, 0). The stte portrit of the chotic system (1) is escrie in Figure 1. Figure 1. Stte Orits of Liu s Four-Wing Chotic System In this section, we esign ptive control lw for glolly stilizing the chotic system (1) when the prmeter vlues re unknown. Thus, we consier the controlle chotic Liu system s follows. = ( x x ) + x x + u = ( x + x ) x x + u = c x + x + x x x + u (3) where u1, u n u3 re feeck controllers to e esigne using the sttes n estimtes of the unknown prmeters of the system. In orer to ensure tht the controlle system (3) glolly converges to the origin symptoticlly, we consier the following ptive control functions u = ( x x ) x x k x u = ( x + x ) + x x k x u = cx x x x x k x (4) where,, c n re estimtes of the prmeters,, c n, respectively, n k1, k, k 3 re positive constnts. 110

4 Sustituting the control lw (4) into the Liu ynmics (1), we otin = ( ) ( x x ) k x = ( )( x + x ) k x 1 = ( c c ) x + ( ) x k x (5) Let us now efine the prmeter errors s e = e = e = c c c e = (6) Using (6), the close-loop ynmics (5) cn e written compctly s = e ( x x ) k x = e ( x + x ) k x 1 = e x + e x k x 3 c (7) For the erivtion of the upte lw for justing the prmeter estimtes,, c,, the Lypunov pproch is use. Consier the qurtic Lypunov function 1 V = ( x1 + x + x3 + e + e + ec + e ) (8) which is positive efinite function on Note lso tht R 7. = e = c = c e = (9) Differentiting V long the trjectories of (7) n using (9), we otin V = k1 x1 k x k 3 x3 + e x1 ( x x1 ) e x( x1 x) e c x3 c e + xx3 (10) 111

5 In view of Eq. (10), the estimte prmeters re upte y the following lw: = x1 ( x x1 ) + k4e = x( x1 + x) + k5e c = x3 + k6ec = x x + k e 3 7 (11) where k4, k5, k6, k7 re positive constnts. Sustituting (11) into (10), we get V = k x k x k x k e k e k e k e (1) c 7 which is negtive efinite function on R 7. Thus, y Lypunov stility theory [18], we otin the following result. Theorem 1. The Liu s four-wing chotic system (1) with cuic nonlinerity n unknown 3 prmeters is glolly n exponentilly stilize for ll initil conitions x(0) R y the ptive control lw (4), where the upte lw for the prmeters is given y (11) n ki, ( i = 1, K,7) re positive constnts.. Numericl Results For the numericl simultions, the fourth orer Runge-Kutt metho is use to solve the chotic system (1) with the ptive control lw (4) n the prmeter upte lw (11). The prmeters of the Liu s four-wing system (1) re selecte s = 50, = 13, c = 13 n = 6. For the ptive n upte lws, we tke k =, ( i = 1,, K,7). Suppose tht the initil vlues of the estimte prmeters re (0) = 1, (0) = 8, c (0) = 6 n (0) = 16 The initil vlues of the Liu s four-wing system (1) re tken s x (0) = (4,36,45). i When the ptive control lw (4) n the prmeter upte lw (11) re use, the controlle Liu system converges to the equilirium E 0 = (0, 0, 0) exponentilly s shown in Figure. The prmeter estimtes re shown in Figure 3. 11

6 Figure. Time Responses of the Controlle Liu s Four-Wing System Figure 3. Prmeter Estimtes ( t), ( t), c ( t), ( t ) 113

7 3. ADAPTIVE SYNCHRONIZATION OF IDENTICAL LIU S FOUR-WING CHAOTIC SYSTEMS WITH CUBIC NONLINEARITY 3.1 Theoreticl Results In this section, we iscuss the ptive synchroniztion of ienticl Liu s four-wing chotic systems (Liu, [17], 009) with cuic nonlinerity n unknown prmeters. As the mster system, we consier the Liu s four-wing chotic system escrie y = ( x x ) + x x = ( x + x ) x x = cx + x + x x x (13) where x1, x, x3 re the stte vriles n,, c, re unknown system prmeters. The system (13) is chotic when the prmeter vlues re tken s = 50, = 13, c = 13 n = 6. As the slve system, we consier the controlle Liu s four-wing chotic system escrie y y = ( y y ) + y y + u y = ( y + y ) y y + u y = cy + y + y y y + u (14) where y1, y, y3 re the stte vriles n u1, u, u3 re the nonliner controllers to e esigne. The synchroniztion error is efine y e = y x, ( i = 1,,3) (15) i i i Then the error ynmics is otine s = ( e e ) + y y x x + u = ( e + e ) y y + x x + u = ce + e + y y y x x x + u (16) Let us now efine the ptive control functions u1 ( t), u( t), u3( t) s u = ( e e ) y y + x x k e u = ( e + e ) + y y x x k e u = ce e y y y + x x x k e (17) 114

8 where,, c n re estimtes of the prmeters,, c n respectively, n k1, k, k3 re positive constnts. Sustituting the control lw (17) into (16), we otin the error ynmics s = ( )( e e ) k e = ( )( e + e ) k e 1 = ( c c ) e + ( ) e k e (18) Let us now efine the prmeter errors s e = e = e = c c c (19) Sustituting (19) into (18), the error ynmics simplifies to = e ( e e ) k e = e ( e + e ) k e 1 = e e + e e k e 3 c (0) For the erivtion of the upte lw for justing the estimtes of the prmeters, the Lypunov pproch is use. Consier the qurtic Lypunov function 1 V = ( e1 + e + e3 + e + e + ec + e ) (1) which is positive efinite function on Note lso tht R 7. = e = c = c e = () Differentiting V long the trjectories of (0) n using (), we otin 115

9 V = k1 e1 k e k3 e 3 k4 e4 + e e1 ( e e1 ) e e ( e1 e ) + + (3) + e c e3 c e + ee3 + In view of Eq. (3), the estimte prmeters re upte y the following lw: = e1 ( e e1 ) + k4e = e ( e1 + e ) + k5e c = e3 + k6ec = e e + k e 3 7 (4) where k4, k5, k6, k7 re positive constnts. Sustituting (4) into (3), we get V = k e k e k e k e k e k e k e (5) c 7 which is negtive efinite function on R 7. Thus, y Lypunov stility theory [18], it is immeite tht the synchroniztion error n the prmeter error ecy to zero exponentilly with time for ll initil conitions. Hence, we hve prove the following result. Theorem. The ienticl Liu s four-wing systems (13) n (14) with cuic nonlinerity n unknown prmeters re glolly n exponentilly synchronize for ll initil conitions y the ptive control lw (17), where the upte lw for prmeters is given y (4) n ki,( i = 1, K,7) re positive constnts. 3. Numericl Results For the numericl simultions, the fourth orer Runge-Kutt metho is use to solve the two systems of ifferentil equtions (13) n (14) with the ptive control lw (17) n the prmeter upte lw (4). In the chotic cse, the prmeters of the Liu s four-wing chotic systems (13) n (14) re = 50, = 13, c = 13 n = 6, We tke the positive constnts ki, ( i = 1, K,7) s k i = for i = 1,, K,7. Suppose tht the initil vlues of the estimte prmeters re n (0) = 1. (0) =, (0) = 6, c (0) = 4 We tke the initil vlues of the mster system (13) s x (0) = (5,15,0). 116

10 We tke the initil vlues of the slve system (14) s y (0) = (1,36,4). Figure 4 shows the ptive chos synchroniztion of the ienticl Liu s four-wing chotic systems. Figure 5 shows tht the estimte vlues of the prmeters,, c n converge to the system prmeters = 50, = 13, c = 13 n = 6. Figure 4. Aptive Synchroniztion of the Ienticl Liu s Four-Wing Systems Figure 5. Prmeter Estimtes ( t), ( t), c ( t), ( t ) 117

11 4. CONCLUSIONS In this pper, we eploye ptive control theory for the chos control n synchroniztion of the Liu s four-wing chotic system (Liu, 009) with cuic nonlinerity n unknown system prmeters. First, we esigne ptive control lws to stilize the Liu s four-wing system to its unstle equilirium point t the origin se on the ptive control theory n Lypunov stility theory. Then we erive ptive synchroniztion scheme n upte lw for the estimtion of system prmeters for ienticl Liu s four-wing chotic systems with cuic nonlinerity n unknown prmeters. Our synchroniztion schemes were estlishe using Lypunov stility theory. Since the Lypunov exponents re not require for these clcultions, the propose ptive control theory metho is very effective n convenient to chieve chos control n synchroniztion of Liu s four-wing chotic systems. Numericl simultions re shown to emonstrte the effectiveness of the propose ptive chos control n synchroniztion schemes. REFERENCES [1] Alligoo, K.T., Suer, T. Yorke, J.A. (1997) Chos: An Introuction to Dynmicl Systems, Springer, New York. [] Ge, S.S., Wng, C. Lee, T.H. (000) Aptive ckstepping control of clss of chotic systems, Internt. J. Bifur. Chos, Vol. 10, pp [3] Wng, X., Tin, L. Yu, L. (006) Aptive control n slow mnifol nlysis of new chotic system, Internt. J. Nonliner Science, Vol. 1, pp [4] Sun, M., Tin, L., Jing, S. Xun, J. (007) Feeck control n ptive control of the energy resource chotic system, Chos, Solitons Frctls, Vol. 3, pp [5] Pecor, L.M. Crroll, T.L. (1990) Synchroniztion in chotic systems, Phys. Rev. Lett., Vol. 64, pp [6] Lkshmnn, M. Murli, K. (1996) Nonliner Oscilltors: Controlling n Synchroniztion, Worl Scientific, Singpore. [7] Hn, S.K., Kerrer, C. Kurmoto, Y. (1995) Dephsing n ursting in couple neurl oscilltors, Phys. Rev. Lett., Vol. 75, pp [8] Blsius, B., Huppert, A. Stone, L. (1999) Complex ynmics n phse synchroniztion in sptilly extene ecologicl system, Nture, Vol. 399, pp [9] Feki, M. (003) An ptive chos synchroniztion scheme pplie to secure communiction, Chos, Solitons n Frctls, Vol. 18, pp [10] Murli, K. Lkshmnn, M. (1998) Secure communiction using compoun signl from generlize synchronizle chotic systems, Phys. Rev. Lett. A, Vol. 41, pp [11] Yng, T. Chu, L.O. (1999) Control of chos using smple-t feeck control, Internt. J. Bifurct. Chos, Vol. 9, pp [1] Ott, E., Greogi, C. Yorke, J.A. (1990) Controlling chos, Phys. Rev. Lett., Vol. 64, pp [13] Prk, J.H. Kwon, O.M. (003) A novel criterion for elye feeck control of time-ely chotic systems, Chos, Solitons n Frctls, Vol. 17, pp [14] Yu, Y.G. Zhng, S.C. (006) Aptive ckstepping synchroniztion of uncertin chotic systems, Chos, Solitons n Frctls, Vol. 7, pp [15] Lio, T.L. Tsi, S.H. (000) Aptive synchroniztion of chotic systems n its pplictions to secure communictions, Chos, Solitons n Frctls, Vol. 11, pp

12 [16] Konishi, K.., Hiri, M. Kokme, H. (1998) Sliing moe control for clss of chotic systems, Phys. Lett. A, Vol. 45, pp [17] Liu, X. (009) A new four-wing chotic system with cuic nonlinerity n its circuit implementtion, Chin. Phys. Lett., Vol. 6, pp [18] Hhn, W. (1967) The Stility of Motion, Springer, New York. Author Dr. V. Sunrpnin is Professor (Systems n Control Engineering), Reserch n Development Centre t Vel Tech Dr. RR Dr. SR Technicl University, Chenni, Ini. His current reserch res re: Liner n Nonliner Control Systems, Chos Theory, Dynmicl Systems n Stility Theory, Soft Computing, Opertions Reserch, Numericl Anlysis n Scientific Computing, Popultion Biology, etc. He hs pulishe over 170 reserch rticles in interntionl journls n two text-ooks with Prentice-Hll of Ini, New Delhi, Ini. He hs pulishe over 50 ppers in Interntionl Conferences n 90 ppers in Ntionl Conferences. He is the Eitor-in-Chief of the AIRCC journls Interntionl Journl of Instrumenttion n Control Systems, Interntionl Journl of Control Systems n Computer Moelling. He hs elivere severl Key Note Lectures on Control Systems, Chos Theory, Scientific Computing, SCILAB, etc. 119