ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTOR

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1 ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTOR Sundrpndin Vidynthn 1 1 Reserch nd Development Centre, Vel Tech Dr. RR & Dr. SR Technicl University Avdi, Chenni , Tmil Ndu, INDIA sundrvtu@gmil.com ABSTRACT Recently, novel three-dimensionl highly chotic ttrctor hs been discovered by Srisuchinwong nd Munmungsen (2010). This pper investigtes the dptive control nd synchroniztion of this highly chotic ttrctor with unknown prmeters. First, dptive control lws re designed to stbilize the highly chotic system to its unstble equilibrium point t the origin bsed on the dptive control theory nd Lypunov stbility theory. Then dptive control lws re derived to chieve globl chos synchroniztion of identicl highly chotic systems with unknown prmeters. Numericl simultions re shown to demonstrte the effectiveness of the proposed dptive control nd synchroniztion schemes. KEYWORDS Adptive Control, Stbiliztion, Chos Synchroniztion, highly chotic system, Srisuchinwong system. 1. INTRODUCTION Chotic systems re nonliner dynmicl systems, which re highly sensitive to initil conditions. The sensitive nture of chotic systems is usully clled s the butterfly effect [1]. In 1963, Lorenz first observed the chos phenomenon in wether models. Since then, lrge number of chos phenomen nd chos behviour hve been discovered in physicl, socil, economicl, biologicl nd electricl systems. The control of chotic systems is to design stte feedbck control lws tht stbilizes the chotic systems round the unstble equilibrium points. Active control technique is used when the system prmeters re known nd dptive control technique is used when the system prmeters re unknown [2-4]. Chos synchroniztion is phenomenon tht my occur when two or more chotic oscilltors re coupled or when chotic oscilltor drives nother chotic oscilltor. Becuse of the butterfly effect, which cuses the exponentil divergence of the trjectories of two identicl chotic systems strted with nerly the sme initil conditions, synchronizing two chotic systems is seemingly very chllenging problem in the chos literture [5-16]. In 1990, Pecor nd Crroll [5] introduced method to synchronize two identicl chotic systems nd showed tht it ws possible for some chotic systems to be completely synchronized. From then on, chos synchroniztion hs been widely explored in vriety of fields including physicl systems [6], chemicl systems [7], ecologicl systems [8], secure communictions [9-10], etc. In most of the chos synchroniztion pproches, the mster-slve or drive-response formlism hs been used. If prticulr chotic system is clled the mster or drive system nd nother DOI : /ijist

2 chotic system is clled the slve or response system, then the ide 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 by Pecor nd Crroll [5], vriety of impressive pproches hve been proposed for the synchroniztion of chotic systems such s the OGY method [11], ctive control method [12-16], dptive control method [17-21], smpled-dt feedbck synchroniztion method [22], time-dely feedbck method [23], bckstepping method [24], sliding mode control method [25-28], etc. In this pper, we investigte the dptive control nd synchroniztion of n uncertin novel threedimensionl highly chotic ttrctor discovered by B. Srisuchinwong nd B. Munmungsen ([29], 2010). First, we devise dptive stbiliztion scheme using stte feedbck control for the highly chotic system bout its unstble equilibrium t the origin. Then, we devise dptive synchroniztion scheme for identicl highly chotic systems with unknown prmeters. The stbility results derived in this pper re estblished using Lypunov stbility theory. This pper is orgnized s follows. In Section 2, we give system description of the highly chotic system (Srisuchinwong nd Munmungsen, 2010). In Section 3, we derive results for the dptive stbiliztion of the highly chotic system with unknown prmeters. In Section 4, we derive results for the dptive synchroniztion of identicl highly chotic systems with unknown prmeters. In Section 5, we summrize the min results obtined in this pper. 2. SYSTEM DESCRIPTION The highly chotic system ([29], 2010) is one-prmeter fmily of three-dimensionl chotic systems, which is described by the dynmics = 10( x x ) = x 40x x = 10x x x where xi, ( i = 1,2,3) re the stte vribles nd is constnt positive prmeter of the system. The system (1) is highly chotic when the prmeter vlue is tken s = (2) The stte orbits of the highly chotic system (2) re described in Figure 1. In [29], it hs been shown tht when is ner 296.5, the mximum Lypunov exponent L (mx) = nd the mximum Kpln-Yorke dimension D KY (mx) = (1) 2

3 Figure 1. Stte Orbits of the Highly Chotic System When the prmeter vlues re tken s in (2), the system (1) is highly chotic nd the system lineriztion mtrix t the equilibrium point E 0 = (0,0,0) is given by A = which hs the eigenvlues λ = 1, λ = nd λ 3 = (4) 1 2 Since λ 3 is positive eigenvlue, it is immedite from Lypunov stbility theory [30] tht the system (1) is unstble t the equilibrium point E 0 = (0,0,0). 3. ADAPTIVE CONTROL OF THE HIGHLY CHAOTIC SYSTEM 3.1 Theoreticl Results In this section, we design dptive control lw for globlly stbilizing the highly chotic system (1) when the prmeter vlue is unknown. Thus, we consider the controlled highly chotic system s follows. = 10( x x ) + u = x 40x x + u = 10x x x + u where u1, u2 nd u3 re feedbck controllers to be designed using the sttes nd estimtes of the unknown prmeter of the system. In order to ensure tht the controlled system (5) globlly converges to the origin symptoticlly, we consider the following dptive control functions 3 (5)

4 u = ˆ( x x ) k x u = x ˆ + 40x x k x u = 10x x + x k x where â is the estimte of the prmeter nd ki,( i = 1, 2,3) re positive constnts. Substituting the control lw (6) into the highly chotic dynmics (5), we obtin = k x = ( ˆ ) x k x = k x Let us now define the prmeter estimtion error s e = ˆ (8) Using (8), the closed-loop dynmics (7) cn be written compctly s = k x = e x k x = k x For the derivtion of the updte lw for djusting the prmeter estimte ˆ, the Lypunov pproch is used. Consider the qudrtic Lypunov function V = ( x1 + x2 + x3 + x4 + e ), (10) 2 which is positive definite function on Note lso tht R 5. = ˆ & (11) Differentiting V long the trjectories of (9) nd using (11), we obtin V& = k1x ˆ 1 k2x2 k3x3 + e x1x2 & (12) In view of Eq. (12), the estimted prmeters re updted by the following lw: ˆ & = x x + k e (13) where k 4 is positive constnt. Substituting (13) into (12), we get V& = k x k x k x k e (14) which is negtive definite function on R 4. Thus, by Lypunov stbility theory [27], we obtin the following result. (6) (7) (9) 4

5 Theorem 1. The highly chotic system (5) with unknown prmeters is globlly nd 3 exponentilly stbilized for ll initil conditions x(0) R by the dptive control lw (6), where the updte lw for the prmeter is given by (13) nd ki, ( i = 1, K,4) re positive constnts. 2.2 Numericl Results For the numericl simultions, the fourth order Runge-Kutt method is used to solve the highly chotic system (5) with the dptive control lw (6) nd the prmeter updte lw (13). The prmeter of the highly chotic system (5) is selected s = For the dptive nd updte lws, we tke k = 4, ( i = 1,2,3, 4). Suppose tht the initil vlue of the estimted prmeter is tken s ˆ(0) = 7. The initil vlues of the highly chotic system (5) re tken s x (0) = (14,15,22). i When the dptive control lw (6) nd the prmeter updte lw (13) re used, the controlled highly chotic system (5) converges to the equilibrium E 0 = (0, 0, 0) exponentilly s shown in Figure 2. The prmeter estimte â is shown in Figure 3, which converges to = Figure 2. Time Responses of the Controlled Highly Chotic System 5

6 Figure 3. Prmeter Estimte ˆ( t ) 4. ADAPTIVE SYNCHRONIZATION OF IDENTICAL HIGHLY CHAOTIC SYSTEMS 4.1 Theoreticl Results In this section, we discuss the dptive synchroniztion of identicl highly chotic systems with unknown prmeter. As the mster system, we consider the highly chotic dynmics described by = 10( x x ) = x 40x x = 10x x x (15) where xi, ( i = 1,2,3) re the stte vribles nd is the unknown system prmeter. As the slve system, we consider the controlled highly chotic dynmics described by y& = 10( y y ) + u y& = y 40y y + u y& = 10y y y + u where yi, ( i = 1,2,3) re the stte vribles nd ui, ( i = 1,2,3) re the nonliner controllers to be designed. (16) 6

7 The synchroniztion error is defined by e = y x, ( i = 1,2,3) (17) i i i Then the error dynmics is obtined s = 10( e e ) + u = e 40( y y x x ) + u = 10( y y x x ) e + u Let us now define the dptive control functions u1( t), u2( t), u3( t) s (18) u = 10( e e ) k e u = e ˆ + 40( y y x x ) k e u = 10( y y x x ) + e k e where â is the estimte of the prmeter, nd k, k, 1 2 k 3 re positive constnts. Substituting the control lw (19) into (18), we obtin the error dynmics s = k e = ( ˆ ) e k e = k e Let us now define the prmeter estimtion error s (19) (20) e = ˆ (21) Substituting (21) into (20), the error dynmics simplifies to = k e = e e k e = k e For the derivtion of the updte lw for djusting the estimte of the prmeter, the Lypunov pproch is used. Consider the qudrtic Lypunov function V = ( e1 + e2 + e3 + e ) (23) 2 which is positive definite function on Note lso tht R 4. = ˆ & (24) Differentiting V long the trjectories of (22) nd using (24), we obtin V& = k1e ˆ 1 k2e2 k3e3 + e e1e 2 & (25) (22) In view of Eq. (25), the estimted prmeter is updted by the following lw: 7

8 ˆ & = e e + k e (26) where k 4 is positive constnts. Substituting (24) into (23), we get V& = k e k e k e k e (27) , which is negtive definite function on R 4. Thus, by Lypunov stbility theory [30], it is immedite tht the synchroniztion error nd the prmeter error decy to zero exponentilly with time for ll initil conditions. Hence, we hve proved the following result. Theorem 2. The identicl highly chotic systems (15) nd (16) with unknown prmeters re globlly nd exponentilly synchronized for ll initil conditions by the dptive control lw (19), where the updte lw for prmeter is given by (26) nd ki,( i = 1,2,3,4) re positive constnts. 3.2 Numericl Results For the numericl simultions, the fourth order Runge-Kutt method is used to solve the two systems of differentil equtions (15) nd (16) with the dptive control lw (19) nd the prmeter updte lw (26). Here, we tke the prmeter vlue s = nd the gins s k = 4 for i = 1,2,3, 4. We tke the initil vlue of the estimted prmeter s ˆ(0) = 10. We tke the initil stte of the mster system (15) s x (0) = (2,15,10) nd the slve system (16) s y (0) = (18,6,4). Figure 4 shows the dptive chos synchroniztion of the identicl highly chotic systems. Figure 5 shows tht the estimted vlue â converges to the system prmeter = i Figure 4. Adptive Synchroniztion of the Highly Chotic Systems 8

9 5. CONCLUSIONS Figure 5. Prmeter Estimte ˆ( t ) In this pper, we pplied dptive control theory for the stbiliztion nd synchroniztion of the highly chotic system (Srisuchinwong nd Munmungsen, 2010) with unknown system prmeters. First, we designed dptive control lws to stbilize the highly chotic system to its equilibrium point t the origin bsed on the dptive control theory nd Lypunov stbility theory. Then we derived dptive synchroniztion scheme nd updte lw for the estimtion of system prmeters for identicl highly chotic systems with unknown prmeters. Our synchroniztion schemes were estblished using Lypunov stbility theory. Since the Lypunov exponents re not required for these clcultions, the proposed dptive control method is very effective nd convenient to chieve chos control nd synchroniztion of the highly chotic system. Numericl simultions re shown to vlidte nd illustrte the effectiveness of the dptive stbiliztion nd synchroniztion schemes derived in this pper. REFERENCES [1] Alligood, K.T., Suer, T. & Yorke, J.A. (1997) Chos: An Introduction to Dynmicl Systems, Springer, New York. [2] Ge, S.S., Wng, C. & Lee, T.H. (2000) Adptive bckstepping control of clss of chotic systems, Internt. J. Bifur. Chos, Vol. 10, pp [3] Wng, X., Tin, L. & Yu, L. (2006) Adptive control nd slow mnifold nlysis of new chotic system, Internt. J. Nonliner Science, Vol. 21, pp [4] Sun, M., Tin, L., Jing, S. & Xun, J. (2007) Feedbck control nd dptive control of the energy resource chotic system, Chos, Solitons & Frctls, Vol. 32, 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 nd Synchroniztion, World Scientific, Singpore. 9

10 [7] Hn, S.K., Kerrer, C. & Kurmoto, Y. (1995) Dephsing nd bursting in coupled neurl oscilltors, Phys. Rev. Lett., Vol. 75, pp [8] Blsius, B., Huppert, A. & Stone, L. (1999) Complex dynmics nd phse synchroniztion in sptilly extended ecologicl system, Nture, Vol. 399, pp [9] Feki, M. (2003) An dptive chos synchroniztion scheme pplied to secure communiction, Chos, Solitons nd Frctls, Vol. 18, pp [10] Murli, K. & Lkshmnn, M. (1998) Secure communiction using compound signl from generlized synchronizble chotic systems, Phys. Rev. Lett. A, Vol. 241, pp [11] Ott, E., Grebogi, C. & Yorke, J.A. (1990) Controlling chos, Phys. Rev. Lett., Vol. 64, pp [12] Wu, Y., Zhou, X. & Chen, J. (2009) Chos synchroniztion of new chotic system, Chos, Solitons & Frctls, Vol. 42, pp [13] Sundrpndin, V. & Krthikeyn, R. (2011), Globl chos synchroniztion of four-scroll chotic systems by ctive nonliner control, Interntionl Journl of Control Theory nd Applictions, Vol. 4, No. 1, pp [14] Sundrpndin, V. (2011) Hybrid chos synchroniztion of hyperchotic Liu nd hyperchotic Chen systems by ctive nonliner control, Interntionl Journl of Computer Science, Engineering nd Informtion Technology, Vol. 1, No. 2, pp [15] Sundrpndin, V. & Suresh, R. (2011) Globl chos synchroniztion of hyperchotic Qi nd Ji systems by nonliner control, Interntionl Journl of Distributed nd Prllel Systems, Vol. 2, No. 2, pp [16] Sundrpndin, V. (2011) Globl chos synchroniztion of Shimizu-Moriok nd Liu-Chen chotic systems by ctive nonliner control, Interntionl Journl of Advnces in Science nd Technology, Vol. 2, No. 4, pp [17] Smuel, B. (2007) Adptive synchroniztion between two different chotic systems, Adptive Commun. Nonliner Sci. Num. Simultion, Vol. 12, pp [18] Lio, T.L. & Tsi. S.H. (2000) Adptive synchroniztion of chotic systems nd its pplictions to secure communictions, Chos, Solitons nd Frctls, Vol. 11, pp [19] Sundrpndin, V. (2011) Adptive control nd synchroniztion of hyperchotic Newton-Leipnik system, Interntionl Journl of Advnced Informtion Technology, Vol. 1, No. 3, pp [20] Sundrpndin, V. (2011) Adptive control nd synchroniztion of hyperchotic Ci system, Interntionl Journl of Control Theory nd Computer Modeling, Vol. 1, No. 1, pp [21] Sundrpndin, V. (2011) Adptive synchroniztion of uncertin Sprott H nd I chotic systems, Interntionl Journl of Computer Informtion Systems, Vol. 1, No. 5, pp 1-7. [22] Yng, T. & Chu, L.O. (1999) Control of chos using smpled-dt feedbck control, Internt. J. Bifurct. Chos, Vol. 9, pp [23] Prk, J.H. & Kwon, O.M. (2003) A novel criterion for delyed feedbck control of time-dely chotic systems, Chos, Solitons nd Frctls, Vol. 17, pp [24] Yu, Y.G. & Zhng, S.C. (2006) Adptive bckstepping synchroniztion of uncertin chotic systems, Chos, Solitons nd Frctls, Vol. 27, pp [25] Konishi, K., Hiri, M. & Kokme, H. (1998) Sliding mode control for clss of chotic systems, Phys. Lett. A, Vol. 245, pp [26] Sundrpndin, V. (2011) Globl chos synchroniztion of Pehlivn systems by sliding mode control, Interntionl Journl on Computer Science nd Engineering, Vol. 3, No. 5, pp [27] Sundrpndin, V. & Sivperuml, S. (2011) Globl chos synchroniztion of hyperchotic Chen systems by sliding mode control, Interntionl Journl of Engineering Science nd Technology, Vol. 3, No. 5, pp

11 [28] Sundrpndin, V. (2011) Globl chos synchroniztion of four-wing chotic systems by sliding mode control, Interntionl Journl of Control Theory nd Computer Modeling, Vol. 1, No. 1, pp [29] Srisuchinwong, B. & Munmungsen, B. (2010) A highly chotic ttrctor for dul-chnnel singlettrctor, privte communiction system, Proceedings of the Third Chotic Modeling nd Simultion Interntionl Conference, Chni, Crete, Greece, June 2010, pp [30] Hhn, W. (1967) The Stbility of Motion, Springer, New York. Author Dr. V. Sundrpndin is Professor (Systems nd Control Engineering), Reserch nd Development Centre t Vel Tech Dr. RR & Dr. SR Technicl University, Chenni, Indi. His current reserch res re: Liner nd Nonliner Control Systems, Chos Theory, Dynmicl Systems nd Stbility Theory, etc. He hs published over 170 reserch rticles in interntionl journls nd two text-books with Prentice-Hll of Indi, New Delhi, Indi. He hs published over 50 ppers in Interntionl Conferences nd 90 ppers in Ntionl Conferences. He is Senior Member of AIRCC nd is the Editor-in-Chief of the AIRCC journls Interntionl Journl of Instrumenttion nd Control Systems, Interntionl Journl of Control Theory nd Computer Modeling, etc. He is n Associte Editor of the journls Interntionl Journl of Control Theory nd Applictions, Interntionl Journl of Advnces in Science nd Technology, Interntionl Journl of Computer Informtion Systems, etc. He hs delivered severl Key Note Lectures on Control Systems, Chos Theory, Scientific Computing, MATLAB, SCILAB, etc. 11