ADAPTIVE CONTROL AND SYNCHRONIZATION OF RÖSSLER PROTOTYPE-4 SYSTEM

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1 ADAPTIVE CONTROL AND SYNCHRONIZATION OF RÖSSLER PROTOTYPE-4 SYSTEM 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 This pper investigtes the dptive control nd synchroniztion of Rössler prototype-4 system with unknown prmeters. The Rössler prototype-4 system is clssicl three-dimensionl chotic system studied y O.E. Rössler (1979). First, dptive control lws re designed to stilize the Rössler prototype-4 system to its unstle equilirium t the origin sed on the dptive control theory nd Lypunov stility theory. Then dptive control lws re derived to chieve glol chos synchroniztion of identicl Rössler prototype-4 systems with unknown prmeters. Numericl simultions re shown to vlidte nd illustrte the effectiveness of the proposed dptive control nd synchroniztion schemes for the Rössler prototype-4 system. KEYWORDS Adptive Control, Chos, Stiliztion, Synchroniztion, Rössler Prototype-4 System. 1. INTRODUCTION Chotic systems re nonliner dynmicl systems tht re highly sensitive to initil conditions. The sensitive nture of chotic systems is commonly clled s the utterfly effect [1]. Chos is n interesting nonliner phenomenon nd hs een extensively nd intensively studied in the lst two decdes [1-30]. Chos theory hs een pplied in mny scientific disciplines such s Mthemtics, Computer Science, Microiology, Biology, Ecology, Economics, Popultion Dynmics nd Rootics. Thus, the ehviour of chotic system is highly unpredictle even if the system mthemticl description is deterministic. The first three-dimensionl chotic system ws discovered y Lorenz in 1963 [], when he ws studying wether models. From then on, mny Lorenz-like chotic systems such s Rössler system [3], Chen system [4], Lü system [5] nd Liu system [6] were reported nd nlyzed. The control of chotic systems is to design stte feedck control lws tht stilizes the chotic systems round the unstle equilirium points. Active control technique is used when the system prmeters re known nd dptive control technique is used when the system prmeters re unknown [7-10]. Synchroniztion of chotic systems is phenomenon tht my occur when two or more chotic oscilltors re coupled or when chotic oscilltor drives nother chotic oscilltor. Becuse of DOI : /ijit

2 the utterfly 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 prolem in the chos literture [11-38]. In 1990, Pecor nd Crroll [11] introduced method to synchronize two identicl chotic systems nd showed tht it ws possile for some chotic systems to e completely synchronized. From then on, chos synchroniztion hs een widely explored in vriety of fields including physicl systems [1], chemicl systems [13], ecologicl systems [14], secure communictions [15-17], etc. In most of the chos synchroniztion pproches, the mster-slve or drive-response formlism hs een used. If prticulr chotic system is clled the mster or drive system nd nother 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 y Pecor nd Crroll [11], vriety of impressive pproches hve een proposed for the synchroniztion of chotic systems such s the OGY method [18], ctive control method [19-4], dptive control method [5-30], smpled-dt feedck synchroniztion method [31], time-dely feedck method [3], ckstepping method [33], sliding mode control method [34-38], etc. In this pper, we discuss the dptive control nd synchroniztion of the Rössler prototype-4 system (Rössler, [39], 1979) with unknown prmeters. In the first prt of the pper, we devise stte feedck control scheme for stilizing the uncertin Rössler prototype-4 system out its unstle equilirium t the origin. The stility results for dptive control nd prmeter estimtion re estlished using Lypunov stility theory. In the second prt of the pper, we devise stte feedck control scheme for synchronizing two identicl uncertin Rössler prototype-4 systems. The stility results for dptive synchroniztion nd prmeter estimtion re estlished using Lypunov stility theory. This pper is orgnized s follows. In Section, we derive results for the dptive control of Rössler prototype-4 system with unknown prmeters. In Section 3, we derive results for the dptive synchroniztion of Rössler prototype-4 systems with unknown prmeters. Section 4 contins summry of the min results derived in this pper.. ADAPTIVE CONTROL OF RÖSSLER PROTOTYPE-4 CHAOTIC SYSTEM.1 Theoreticl Results The Rössler prototype-4 system (Rössler, 1979) is one of the clssicl three-dimensionl chotic systems. The Rössler prototype-4 system dynmics is descried y = x x 1 3 = x 1 = ( x x ) x 3 3 where xi, ( i = 1,,3) re the stte vriles nd, re positive constnts. The Rössler prototype-4 system (1) is chotic when the prmeter vlues re tken s (1) = 0.5 nd = 0.5 () 1

3 The stte orits of the Rössler prototype-4 system (1) is descried in Figure 1. Figure 1. Stte Orits of the Rössler Prototype-4 Chotic System When the prmeter vlues re tken s in (), the system (1) is chotic nd the system lineriztion mtrix t the equilirium point E 0 = (0,0,0) is given y A = which hs the eigenvlues λ = i, λ = i nd λ 3 = Since λ 1 nd λ re eigenvlues with positive rel prt, it is immedite from Lypunov stility theory [40] tht the system (1) is unstle t the equilirium point E 0 = (0,0,0). In this section, we design dptive control lw for glolly stilizing the Rössler prototype-4 chotic system (1) when the prmeter vlues re unknown. Thus, we consider the controlled Rössler prototype-4 system given y the dynmics = x x + u = x + u 1 = ( x x ) x + u (3) where u1, u nd u3 re feedck controllers to e designed using the sttes nd estimtes of the unknown prmeters of the system. 13

4 In order to ensure tht the controlled system (3) glolly converges to the origin symptoticlly, we consider the following dptive control functions u = x + x k x u = x k x 1 u = ˆ( x x ) + x ˆ k x where â nd ˆ re estimtes of the prmeters nd, respectively, nd ki,( i = 1,,3) re positive constnts. Sustituting the control lw (4) into the chotic Rössler prototype-4 system (3), we otin = k x = k x = ( ˆ )( x x ) ( ˆ ) x k x Let us now define the prmeter errors s e = ˆ nd e = ˆ. (6) Using (6), the closed-loop dynmics (5) cn e written compctly s = k x = k x = e ( x x ) e x k x For the derivtion of the updte lw for djusting the prmeter estimtes pproch is used. Consider the qudrtic Lypunov function (4) (5) (7) ˆ, ˆ, the Lypunov 1 V ( x1, x, x3, e, e ) = ( x1 + x + x3 + e + e ) (8) which is positive definite function on Note lso tht R 5. = ˆ & nd = ˆ. & (9) Differentiting V long the trjectories of (7) nd using (9), we otin V& = k1 x1 k ˆ ˆ x k3 x3 + e x3( x x ) & e & + x3 (10) In view of Eq. (10), the estimted prmeters re updted y the following lw: ˆ & = x3( x x ) + k4e ˆ & = x + k e 3 5 where k4 nd k5 re positive constnts. Sustituting (11) into (10), we get (11) 14

5 V& = k x k x k x k e k e (1) which is negtive definite function on R 5. Thus, y Lypunov stility theory [40], we otin the following result. Theorem 1. The Rössler prototype-4 chotic system (3) with unknown prmeters is glolly nd 3 exponentilly stilized for ll initil conditions x(0) R y the dptive control lw (4), where the updte lw for the prmeters is given y (11) nd ki, ( i = 1, K,5) re positive constnts.. Numericl Results For the numericl simultions, the fourth order Runge-Kutt method is used to solve the chotic system (3) with the dptive control lw (4) nd the prmeter updte lw (11). The prmeters of the Rössler prototype-4 chotic system re selected s = = 0.5. For the dptive nd updte lws, we tke k = 4, ( i = 1,, K,5). Suppose tht the initil vlues of the estimted prmeters re ˆ(0) = 4 nd ˆ(0) = 9. The initil vlues of the Rössler prototype-4 system re tken s x (0) = (8,9,1). i When the dptive control lw (4) nd the prmeter updte lw (11) re used, the controlled Rössler prototype-4 system converges to the equilirium E 0 = (0,0,0) exponentilly s shown in Figure. The prmeter estimtes ˆ( t ) nd ˆ( t) re shown in Figure 3. Figure. Time Responses of the Controlled Rössler Prototype-4 System 15

6 Figure 3. Prmeter Estimtes ˆ( t), ˆ ( t ) 3. ADAPTIVE SYNCHRONIZATION OF IDENTICAL RÖSSLER PROTOTYPE-4 CHAOTIC SYSTEMS 3.1 Theoreticl Results In this section, we discuss the dptive synchroniztion of identicl Rössler prototype-4 chotic systems (Rössler, 1979) with unknown prmeters. As the mster system, we consider the Rössler prototype-4 dynmics descried y = x x 1 3 = x 1 = ( x x ) x 3 3 where xi, ( i = 1,,3) re the stte vriles nd, re unknown system prmeters. The system (13) is chotic when the prmeter vlues re tken s = 0.5 nd = 0.5. As the slve system, we consider the controlled Rössler prototype-4 dynmics descried y y& = y y + u y& = y + u 1 y& = ( y y ) y + u where yi, ( i = 1,,3) re the stte vriles nd ui, ( i = 1,,3) re the nonliner controllers to e designed. The synchroniztion error is defined y (13) (14) 16

7 e = y x e = y x e = y x Then the error dynmics is otined s = e e + u = e + u 1 = ( e y + x ) e + u Let us now define the dptive control functions u1 ( t), u( t), u3( t) s u = e + e k e u = e k e 1 u = ˆ( e y + x ) + e ˆ k e (15) (16) (17) where â nd ˆ re estimtes of the prmeters nd, respectively, nd ki,( i = 1,,3) re positive constnts. Sustituting the control lw (17) into (16), we otin the error dynmics s = k e = k e = ( ˆ )( e y + x ) ( ˆ ) e k e Let us now define the prmeter errors s e = ˆ nd e = ˆ. (19) Sustituting (19) into (18), the error dynmics simplifies to = k e = k e = e ( e y + x ) e e k e For the derivtion of the updte lw for djusting the estimtes of the prmeters, the Lypunov pproch is used. Consider the qudrtic Lypunov function 1 V ( e1, e, e3, e, e ) = ( e1 + e + e3 + e + e ), (1) which is positive definite function on Note lso tht R 5. = ˆ & nd = ˆ & () Differentiting V long the trjectories of (0) nd using (), we otin (18) (0) 17

8 V& = k1e1 k ˆ ˆ e k3e3 + e e3 ( e y + x ) & e & + e3 (3) In view of Eq. (3), the estimted prmeters re updted y the following lw: ˆ & = e3 ( e y + x ) + k4e ˆ & = e + k e 3 5 where k4 nd k5 re positive constnts. Sustituting (4) into (3), we get , V& = k e k e k e k e k e (5) which is negtive definite function on R 5. Thus, y Lypunov stility theory [40], 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. The identicl Rössler prototype-4 chotic systems (13) nd (14) with unknown prmeters re glolly nd exponentilly synchronized for ll initil conditions y the dptive control lw (17), where the updte lw for prmeters is given y (4) nd ki,( i = 1, K,5) re positive constnts. 3. Numericl Results For the numericl simultions, the fourth order Runge-Kutt method is used to solve the two systems of differentil equtions (13) nd (14) with the dptive control lw (17) nd the prmeter updte lw (4). The prmeter vlues of the Rössler prototype-4 chotic systems re tken s = 0.5 nd = 0.5 We tke the positive constnts ki, ( i = 1, K,5) s k = 4 for i = 1,, K,5. i Suppose tht the initil vlues of the estimted prmeters re ˆ(0) = 8 nd ˆ(0) = 5. We tke the initil vlues of the mster system (13) s x (0) = 4, x (0) = 3, x (0) = We tke the initil vlues of the slve system (14) s y (0) = 1, y (0) = 9, y (0) =. 1 3 Figure 4 shows the dptive chos synchroniztion of the identicl Rössler prototype-4 chotic systems (13) nd (14). Figure 5 shows tht the estimted vlues of the prmeters â nd ˆd converge to the system prmeters = 0.5 nd = 0.5. (4) 18

9 Figure 4. Adptive Synchroniztion of Identicl Rössler Prototype-4 Chotic Systems Figure 5. Prmeter Estimtes ˆ( t), ˆ ( t ) 4. CONCLUSIONS In this pper, we pplied dptive control theory for the stiliztion nd synchroniztion of the Rössler prototype-4 chotic system (Rössler, 1979) with unknown system prmeters. First, we designed dptive control lws to stilize the Rössler prototype-4 chotic system to its unstle equilirium point t the origin sed on the dptive control theory nd Lypunov stility theory. Then we derived dptive synchroniztion scheme nd updte lw for the estimtion of system prmeters for identicl Rössler prototype-4 chotic systems with unknown prmeters. Our synchroniztion schemes were estlished using dptive control theory nd Lypunov stility 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 Rössler prototype-4 chotic system. Numericl simultions re shown to demonstrte the effectiveness of the proposed dptive stiliztion nd synchroniztion schemes for the Rössler prototype-4 chotic system. 19

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11 [7] Sundrpndin, V. (011) Adptive control nd synchroniztion of hyperchotic Newton-Leipnik system, Interntionl Journl of Advnced Informtion Technology, Vol. 1, No. 3, pp -33. [8] Sundrpndin, V. (011) Adptive synchroniztion of hyperchotic Lorenz nd hyperchotic Lü systems, Interntionl Journl of Instrumenttion nd Control Systems, Vol. 1, No. 1, pp [9] Sundrpndin, V. (011) Adptive control nd synchroniztion of hyperchotic Ci system, Interntionl Journl of Control Theory nd Computer Modeling, Vol. 1, No. 1, pp [30] Sundrpndin, V. (011) Adptive control nd synchroniztion of the uncertin Sprott J system, Interntionl Journl of Mthemtics nd Scientific Computing, Vol. 1, No. 1, pp [31] Yng, T. & Chu, L.O. (1999) Control of chos using smpled-dt feedck control, Internt. J. Bifurct. Chos, Vol. 9, pp [3] Prk, J.H. & Kwon, O.M. (003) A novel criterion for delyed feedck control of time-dely chotic systems, Chos, Solitons nd Frctls, Vol. 17, pp [33] Yu, Y.G. & Zhng, S.C. (006) Adptive ckstepping synchroniztion of uncertin chotic systems, Chos, Solitons nd Frctls, Vol. 7, pp [34] Konishi, K.., Hiri, M. & Kokme, H. (1998) Sliding mode control for clss of chotic systems, Phys. Lett. A, Vol. 45, pp [35] Sundrpndin, V. nd S. Sivperuml (011) Anti-synchroniztion of hyperchotic Lorenz systems y sliding mode control, Interntionl Journl on Computer Science nd Engineering, Vol. 3, No. 6, pp [36] Sundrpndin, V. (011) Sliding mode controller design for synchroniztion of Shimizu-Moriok chotic systems, Interntionl Journl of Informtion Sciences nd Techniques, Vol. 1, No. 1, pp 0-9. [37] Sundrpndin, V. (011) Glol chos synchroniztion of four-wing chotic systems y sliding mode control, Interntionl Journl of Control Theory nd Computer Modeling, Vol. 1, No. 1, pp [38] Sundrpndin, V. (011) Glol chos synchroniztion of hyperchotic Newton-Leipnik systems y sliding mode control, Interntionl Journl of Informtion Technology, Convergence nd Services, Vol. 1, No. 4, pp [39] Rössler, O.E. (1979) Continuous chos four prototype equtions, Annls of the New York Acdemy of Sciences, Vol. 316, pp [40] Hhn, W. (1967) The Stility of Motion, Springer, New York, U.S.A. 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 Stility Theory, Soft Computing, Opertions Reserch, Numericl Anlysis nd Scientific Computing, Popultion Biology, etc. He hs pulished over 180 reserch rticles in interntionl journls nd two text-ooks with Prentice-Hll of Indi, New Delhi, Indi. He hs pulished over 50 ppers in Interntionl Conferences nd 100 ppers in Ntionl Conferences. He is the Editor-in-Chief of the AIRCC control journls Interntionl Journl of Instrumenttion nd Control Systems, Interntionl Journl of Control Theory nd Computer Modeling, nd Interntionl Journl of Informtion Technology, Control nd Automtion. He is n Associte Editor of the journls Interntionl Journl of Informtion Sciences nd Techniques, Interntionl Journl of Control Theory nd Applictions, Interntionl Journl of Computer Informtion Systems, Interntionl Journl of Advnces in Science nd Technology. He hs delivered severl Key Note Lectures on Control Systems, Chos Theory, Scientific Computing, MATLAB, SCILAB, etc. 1