THE DESIGN OF ADAPTIVE CONTROLLER AND SYNCHRONIZER FOR QI-CHEN SYSTEM WITH UNKNOWN PARAMETERS

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1 THE DESIGN OF ADAPTIVE CONTROLLER AND SYNCHRONIZER FOR QI-CHEN SYSTEM WITH UNKNOWN PARAMETERS Sundrpndin Vidynthn 1 1 Reserh nd Development Centre, Vel Teh Dr. RR & Dr. SR Tehnil University Avdi, Chenni , Tmil Ndu, INDIA sundrvtu@gmil.om ABSTRACT This pper investigtes the design problem of dptive ontroller nd synhronizer for the Qi-Chen system (2005), when the system prmeters re unknown. First, we build n dptive ontroller to stbilize the Qi- Chen hoti system to its unstble equilibrium t the origin. Then we build n dptive synhronizer to hieve globl hos synhroniztion of the identil Qi-Chen hoti systems with unknown prmeters. The results derived for dptive stbiliztion nd dptive synhroniztion for the Qi-Chen hoti system re estblished using dptive ontrol theory nd Lypunov stbility theory. Numeril simultions hve been shown to demonstrte the effetiveness of the dptive ontrol nd synhroniztion shemes derived in this pper for the Qi-Chen hoti system. KEYWORDS Adptive Control, Chos, Choti Systems, Synhroniztion, Qi-Chen System. 1. INTRODUCTION Choti systems re nonliner dynmil systems with the following hrteristis: extreme sensitivity to hnges in initil onditions, rndom-like behviour, deterministi motion, trjetories of hoti systems pss through ny point n infinite number of times. Experimentlly, hos ws first disovered by Lorenz ([1], 1963) while he ws simulting wether models. A hoti system simpler thn the Lorenz system ws proposed by Rössler ([2], 1976). The theoretil equtions of the Rössler system were lter found to be useful in modelling equilibrium in hemil retions. The ontrol of hoti systems is to design stte feedbk ontrol lws tht stbilizes the hoti systems round the unstble equilibrium points. Ative ontrol tehnique is used when the system prmeters re known nd dptive ontrol tehnique is used when the system prmeters re unknown [3-4]. Synhroniztion of hoti systems is phenomenon tht my our when two or more hoti ttrtors re oupled or when hyperhoti ttrtor drives nother hyperhoti ttrtor. In the lst two dedes, there hs been signifint interest in the literture on the synhroniztion of hoti nd hyperhoti systems [5-16]. DOI : /ijse

2 In 1990, Peor nd Crroll [5] introdued method to synhronize two identil hoti systems nd showed tht it ws possible for some hoti systems to be ompletely synhronized. From then on, hos synhroniztion hs been widely explored in vriety of fields inluding physil systems [6], hemil systems [7], eologil systems [8], seure ommunitions [9-10], et. In most of the hos synhroniztion pprohes, the mster-slve or drive-response formlism hs been used. If prtiulr hoti system is lled the mster or drive system nd nother hoti system is lled the slve or response system, then the ide of synhroniztion is to use the output of the mster system to ontrol the slve system so tht the output of the slve system trks the output of the mster system symptotilly. The seminl work by Peor nd Crroll (1990) hs been followed by vriety of impressive pprohes in the literture suh s the smpled-dt feedbk method [11], OGY method [12], time-dely feedbk method [13], bkstepping method [14], tive ontrol method [15-20], dptive ontrol method [21-25], sliding mode ontrol method [26-28], et. This pper is orgnized s follows. In Setion 2, we give desription of the Qi-Chen hoti system (Qi, Chen, Du, Chen nd Yun, [29], 2005). In Setion 3, we derive results for the dptive ontrol of Qi-Chen hoti system with unknown prmeters. In Setion 4, we derive results for the dptive synhroniztion of the identil Qi-Chen hoti systems with unknown prmeters. Setion 5 ontins summry of the min results derived in this pper. 2. SYSTEM DESCRIPTION The Qi-Chen system ([29], 2005) is desribed by the 3D dynmis = ( x x ) + x x = x x x x = x x bx (1) where x 1, x 2, x 3 re the stte vribles of the system nd, b, re onstnt, positive prmeters of the system. The system (1) is hoti when the prmeter vlues re tken s = 35, b = 8 / 3 nd = 80 (2) Figure 1 desribes the strnge ttrtor of the Qi-Chen hoti system (1). When the prmeter vlues re tken s in (2) for the Qi-Chen hoti system (1), the system lineriztion mtrix t the equilibrium point E 0 = (0,0,0) is given by A = / 3 whih hs the eigenvlues λ = , λ = , λ =

3 Sine λ3is positive eigenvlues of A, it follows from Lypunov stbility theory [30] tht the Qi-Chen system (1) is unstble t the equilibrium point E 0 = (0,0,0). Figure 1. The Strnge Attrtor of the Qi-Chen Choti System 3. ADAPTIVE CONTROL OF THE QI-CHEN CHAOTIC SYSTEM 3.1 Theoretil Results In this setion, we design dptive ontrol lw for globlly stbilizing the Qi-Chen system (2005), when the prmeter vlues re unknown. Thus, we onsider the ontrolled Qi-Chen system, whih is desribed by the 3D dynmis = ( x x ) + x x + u = x x x x + u = x x bx + u (3) 83

4 where u1, u2 nd u3re feedbk ontrollers to be designed using the sttes x1, x2, x3 nd estimtes ˆ, bˆ, ˆ of the unknown prmeters, b, of the system. In order to ensure tht the ontrolled system (3) globlly onverges to the origin symptotilly, we onsider the following dptive ontrol funtions u = ˆ( x x ) x x k x u = x ˆ + x + x x k x u = x x + bx ˆ k x (4) where ˆ, b ˆ nd ĉ re estimtes of the prmeters, b nd, respetively, nd ki,( i = 1, 2,3) re positive onstnts. Substituting the ontrol lw (4) into the ontrolled Qi-Chen dynmis (3), we obtin = ( ˆ ) ( x x ) k x = ( ˆ ) x k x = ( b bˆ ) x k x (5) Let us now define the prmeter errors s e = ˆ, e = b bˆ nd e = ˆ (6) b Using (6), the losed-loop dynmis (5) n be written omptly s = e ( x x ) k x = e x k x = e x k x 3 b For the derivtion of the updte lw for djusting the prmeter estimtes Lypunov pproh is used. (7) ˆ, bˆ nd ˆ, the Consider the qudrti Lypunov funtion V ( x1, x2, x3, e, eb, e ) = ( x1 + x2 + x3 + e + eb + e ) (8) 2 whih is positive definite funtion on Note lso tht R 6. ˆ ˆ& = &, = b, = ˆ & (9) b Differentiting V long the trjetories of (7) nd using (9), we obtin 84

5 ˆ& ( ) ( ˆ& ) V& = k1x1 k2x ˆ 2 k3x3 + e x1 ( x2 x1 ) & + eb x3 b + e x1 x2 (10) In view of Eq. (10), the estimted prmeters re updted by the following lw: ˆ & = x1 ( x2 x1 ) + k4e ˆ& 2 b = x3 + k5eb ˆ & = x x + k e (11) where k4, k5 nd k6 re positive onstnts. Substituting (11) into (10), we get V& = k x k x k x k e k e k e (12) b 6 whih is negtive definite funtion on R 6. Thus, by Lypunov stbility theory [30], we obtin the following result. Theorem 1. The ontrolled Qi-Chen system (1) with unknown prmeters is globlly nd 3 exponentilly stbilized for ll initil onditions x(0) R by the dptive ontrol lw (4), where the updte lw for the prmeters is given by (11) nd ki, ( i = 1, K,6) re positive onstnts. 3.2 Numeril Results For the numeril simultions, the fourth order Runge-Kutt method is used to solve the hoti system (3) with the dptive ontrol lw (4) nd the prmeter updte lw (11). The prmeters of the Qi-Chen system (3) re seleted s = 35, b = 8 / 3 nd = 80. For the dptive nd updte lws, we tke k = 5, ( i = 1,2, K,6). Suppose tht the initil vlues of the estimted prmeters re ˆ (0) = 6, bˆ (0) = 18, ˆ (0) = 5 The initil stte of the ontrolled Qi-Chen system (3) is tken s x (0) = 21, x (0) = 16, x (0) = i When the dptive ontrol lw (4) nd the prmeter updte lw (11) re used, the ontrolled modified hyperhoti Lü system onverges to the equilibrium E 0 = (0, 0, 0) exponentilly s shown in Figure 2. 85

6 The time-history of the prmeter estimtes is shown in Figure 3. The time-history of the prmeter estimtion errors is shown in Figure 4. Figure 2. Time Responses of the Controlled Qi-Chen System Figure 3. Time-History of the Prmeter Estimtes ˆ ( t), bˆ ( t), ˆ ( t ) 86

7 Figure 4. Time-History of the Prmeter Estimtes e, eb, e 4. ADAPTIVE SYNCHRONIZATION OF IDENTICAL QI-CHEN CHAOTIC SYSTEMS 4.1 Theoretil Results In this setion, we disuss the dptive synhroniztion of identil Qi-Chen hoti systems (2005) with unknown prmeters. As the mster system, we onsider the Qi-Chen dynmis desribed by = ( x x ) + x x = x x x x = x x bx (13) where xi, ( i = 1, 2,3) re the stte vribles nd, b, re unknown system prmeters. The system (13) is hoti when the prmeter vlues re tken s = 35, b = 8 / 3 nd =

8 As the slve system, we onsider the modified hyperhoti Lü dynmis desribed by y& = ( y y ) + y y + u y& = y y y y + u y& = y y by + u (14) where yi, ( i = 1, 2,3) re the stte vribles nd ui, ( i = 1, 2,3) re the nonliner ontrollers to be designed. The synhroniztion error is defined by e = y x, ( i = 1, 2,3) (15) i i i Then the error dynmis is obtined s = ( e e ) + y y x x + u = e e y y + x x + u = be + y y x x + u (16) Let us now define the dptive ontrol funtions u1( t), u2( t), u3( t) s u = ˆ( e e ) y y + x x k e u = e ˆ + e + y y x x k e u = be ˆ y y + x x k e (17) where ˆ, bˆ nd ĉ re estimtes of the prmeters, b nd, respetively, nd ki,( i = 1, 2,3) re positive onstnts. Substituting the ontrol lw (17) into (16), we obtin the error dynmis s = ( ˆ )( e e ) k e = ( ˆ ) e k e = ( b bˆ ) e k e (18) Let us now define the prmeter errors s e = ˆ e = b bˆ b e = ˆ (19) 88

9 Substituting (19) into (18), the error dynmis simplifies to = e ( ) e e k e = e e k e = e e k e 3 b (20) For the derivtion of the updte lw for djusting the estimtes of the prmeters, the Lypunov pproh is used. Consider the qudrti Lypunov funtion V ( e1, e2, e3, e, eb, e ) = ( e1 + e2 + e3 + e + eb + e ) (21) 2 whih is positive definite funtion on Note lso tht R 6. ˆ = & e ˆ& & b = b = ˆ & (22) Differentiting V long the trjetories of (20) nd using (22), we obtin V& = k1e1 k2e ˆ ˆ ˆ 2 k3e3 + e e1 ( e2 e1 ) & e & + b e3 b + e e1e 2 & (23) In view of Eq. (23), the estimted prmeters re updted by the following lw: ˆ & = e1 ( e2 e1 ) + k4e ˆ& 2 b = e3 + k5eb & ˆ = e e + k e (24) where k4, k5 nd k6 re positive onstnts. Substituting (24) into (23), we get V& = k e k e k e k e k e k e (25) b 6 From (25), we find tht V & is negtive definite funtion on Thus, by Lypunov stbility theory [30], it is immedite tht the synhroniztion error nd the prmeter error dey to zero exponentilly with time for ll initil onditions. Hene, we hve proved the following result. R 6. 89

10 Theorem 2. The identil Qi-Chen systems (13) nd (14) with unknown prmeters re globlly nd exponentilly synhronized for ll initil onditions by the dptive ontrol lw (17), where the updte lw for prmeters is given by (24) nd ki,( i = 1, K,6) re positive onstnts. 4.2 Numeril Results For the numeril simultions, the fourth order Runge-Kutt method is used to solve the two systems of differentil equtions (13) nd (14) with the dptive ontrol lw (17) nd the prmeter updte lw (24). We tke the prmeter vlues s in the hoti se, viz. = 35, b = 8 / 3, = 80 We tke the positive onstnts ki, ( i = 1, K,8) s k = 5 for i = 1,2, K,6. i Suppose tht the initil vlues of the estimted prmeters re ˆ (0) = 12, bˆ (0) = 9, ˆ (0) = 26 We tke the initil vlues of the mster system (13) s x (0) = 7, x (0) = 5, x (0) = We tke the initil vlues of the slve system (14) s y (0) = 33, y (0) = 18, y (0) = Figure 5 shows the dptive hos synhroniztion of the identil Qi-Chen systems. Figure 6 shows the time-history of the synhroniztion error e1, e2, e 3. Figure 7 shows the time-history of the prmeter estimtes ˆ ( t), bˆ ( t), ˆ ( t ). From this figure, it is ler tht the prmeter estimtes onverge to the originl vlues = 35, b = 8 / 3, = 80, respetively. Figure 8 shows the time-history of the prmeter estimtion errors e, eb, e. 90

11 Figure 5. Adptive Synhroniztion of the Qi-Chen Systems Figure 6. Time-History of the Synhroniztion Error e1, e2, e 3 91

12 Figure 7. Time-History of the Prmeter Estimtes ˆ ( t), bˆ ( t), ˆ ( t ) Figure 8. Time-History of the Prmeter Estimtion Error e, eb, e 92

13 5. CONCLUSIONS In this pper, we pplied dptive ontrol theory for the stbiliztion nd synhroniztion of the Qi-Chen system (2005) with unknown system prmeters. First, we designed dptive ontrol lws to stbilize the Qi-Chen system to its unstble equilibrium point t the origin bsed on the dptive ontrol theory nd Lypunov stbility theory. Then we derived dptive synhroniztion sheme nd updte lw for the estimtion of system prmeters for the identil Qi-Chen systems with unknown prmeters. Our synhroniztion shemes were estblished using Lypunov stbility theory. Sine the Lypunov exponents re not required for these lultions, the proposed dptive ontrol method is very effetive nd onvenient to hieve hos ontrol nd synhroniztion of the Qi-Chen hoti system. Numeril simultions re shown to demonstrte the effetiveness of the proposed dptive stbiliztion nd synhroniztion shemes. REFERENCES [1] Lorenz, E.N. (1963) Deterministi nonperiodi flow, J. Atmos. Phys. Vol. 20, pp [2] Rössler, O.E. (1976) An eqution for ontinuous hos, Physis Letters A, Vol. 57, pp [3] Wng, X., Tin, L. & Yu, L. (2006) Adptive ontrol nd slow mnifold nlysis of new hoti system, Internt. J. Nonliner Siene, Vol. 21, pp [4] Sun, M., Tin, L., Jing, S. & Xun, J. (2007) Feedbk ontrol nd dptive ontrol of the energy resoure hoti system, Chos, Solitons & Frtls, Vol. 32, pp [5] Peor, L.M. & Crroll, T.L. (1990) Synhroniztion in hoti systems, Phys. Rev. Lett., Vol. 64, pp [6] Lkshmnn, M. & Murli, K. (1996) Nonliner Osilltors: Controlling nd Synhroniztion, World Sientifi, Singpore. [7] Hn, S.K., Kerrer, C. & Kurmoto, Y. (1995) Dephsing nd bursting in oupled neurl osilltors, Phys. Rev. Lett., Vol. 75, pp [8] Blsius, B., Huppert, A. & Stone, L. (1999) Complex dynmis nd phse synhroniztion in sptilly extended eologil system, Nture, Vol. 399, pp [9] Feki, M. (2003) An dptive hos synhroniztion sheme pplied to seure ommunition, Chos, Solitons nd Frtls, Vol. 18, pp [10] Murli, K. & Lkshmnn, M. (1998) Seure ommunition using ompound signl from generlized synhronizble hoti systems, Phys. Rev. Lett. A, Vol. 241, pp [11] Yng, T. & Chu, L.O. (1999) Control of hos using smpled-dt feedbk ontrol, Internt. J. Bifurt. Chos, Vol. 9, pp [12] Ott, E., Grebogi, C. & Yorke, J.A. (1990) Controlling hos, Phys. Rev. Lett., Vol. 64, pp [13] Prk, J.H. & Kwon, O.M. (2003) A novel riterion for delyed feedbk ontrol of time-dely hoti systems, Chos, Solitons nd Frtls, Vol. 17, pp [14] Yu, Y.G. & Zhng, S.C. (2006) Adptive bkstepping synhroniztion of unertin hoti systems, Chos, Solitons nd Frtls, Vol. 27, pp [15] Ho, M.C. & Hung, Y.C. (2002) Synhroniztion of two different hoti systems by using generlized tive ontrol, Physis Letters A, Vol. 301, pp [16] Hung, L., Feng, R. & Wng, M. (2004) Synhroniztion of hoti systems vi nonliner ontrol, Physis Letters A, Vol. 320, pp [17] Tng, R.A., Liu, Y.L. & Xue, J.K. (2009) An extended tive ontrol for hos synhroniztion, Physis Letters A, Vol. 373, No. 6, pp [18] Sundrpndin, V. (2011) Output regultion of the Sprott-G hoti system by stte feedbk ontrol, Interntionl Journl of Instrumenttion nd Control Systems, Vol. 1, No. 1, pp [19] Sundrpndin, V. (2011) Globl hos synhroniztion of Lorenz nd Pehlivn hoti systems by nonliner ontrol, Interntionl Journl of Advnes in Siene nd Tehnology, Vol. 2, No. 3, pp [20] Lei, Y., Xu, W. & Zheng, H. (2005) Synhroniztion of two hoti nonliner gyros using tive ontrol, Physis Letters A, Vol. 343, No. 1, pp

14 [21] Lio, T.L. & Tsi, S.H. (2000) Adptive synhroniztion of hoti systems nd its pplitions to seure ommunitions, Chos, Solitons nd Frtls, Vol. 11, pp [22] Prk, J.H., Lee, S.M. & Kwon, O.M. (2007) Adptive synhroniztion of Genesio-Tesi hoti system vi novel feedbk ontrol, Physis Letters A, Vol. 371, pp [23] Sundrpndin, V. (2011) Adptive ontrol nd synhroniztion of hyperhoti Chen system, Interntionl Journl of Informtion Tehnology Convergene nd Servies, Vol. 1, No. 3, pp [24] Sundrpndin, V. (2011) Adptive synhroniztion of hyperhoti Lorenz nd hyperhoti Lü systems, Interntionl Journl of Instrumenttion nd Control Systems, Vol. 1, No. 1, pp [25] Sundrpndin, V. (2011) Adptive ontrol nd synhroniztion of hyperhoti Ci system, Interntionl Journl of Control Theory nd Computer Modeling, Vol. 1, No. 1, pp [26] Konishi, K., Hiri, M. & Kokme, H. (1998) Sliding mode ontrol for lss of hoti systems, Physis Letters A, Vol. 245, pp [27] Sundrpndin, V. (2011) Sliding mode ontroller design for synhroniztion of Shimizu-Moriok hoti system, Interntionl Journl of Informtion Sienes nd Tehniques, Vol. 1, No. 1, pp [28] Sundrpndin, V. (2011) Globl hos synhroniztion of four-wing hoti systems by sliding mode ontrol, Interntionl Journl of Control Theory nd Computer Modeling, Vol., 1 No. 1, pp [29] Qi, G., Chen, G., Du, S., Chen, Z. nd Yun, Z. (2005) Anlysis of new hoti system, Physi A, Vol. 352, pp [30] Hhn, W. (1967) The Stbility of Motion, Springer, New York. Author Dr. V. Sundrpndin obtined his Dotor of Siene degree in Eletril nd Systems Engineering from Wshington University, St. Louis, USA in My He is Professor t the R & D Centre t Vel Teh Dr. RR & Dr. SR Tehnil University, Chenni, Tmil Ndu, Indi. He hs published over 230 refereed interntionl publitions. He hs published over 150 ppers in Ntionl nd Interntionl Conferenes. He is the Editor-in- Chief of the AIRCC Journls - Interntionl Journl of Instrumenttion nd Control Systems, Interntionl Journl of Control Systems nd Computer Modelling, nd Interntionl Journl of Informtion Tehnology, Control nd Automtion. His reserh interests re Liner nd Nonliner Control Systems, Chos Theory nd Control, Soft Computing, Optiml Control, Opertions Reserh, Mthemtil Modelling nd Sientifi Computing. He hs delivered severl Key Note Letures on Control Systems, Chos Theory nd Control, Sientifi Computing using MATLAB/SCILAB, et. 94