SLIDING CONTROLLER DESIGN FOR THE HYBRID CHAOS SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC XU SYSTEMS Sudarapadia Vaidyaatha 1 1 Research ad Developmet Cetre, Vel Tech Dr. RR & Dr. SR Techical Uiversity Avadi, Cheai-600 062, Tamil Nadu, INDIA sudarvtu@gmail.com ABSTRACT This paper derives ew results for the slidig cotroller desig for the hybrid sychroizatio of idetical hyperchaotic Xu systems (Xu, Cai ad Zheg, 2009). I hybrid sychroizatio of a pair of chaotic systems cosistig of master ad slave systems, the odd states of the systems are completely sychroized, while the eve states are ati-sychroized so that complete sychroizatio (CS) ad ati-sychroizatio (AS) coexist i the chaos sychroizatio. The SMC-based stability results derived i this paper for the hybrid sychroizatio of idetical hyperchaotic Xu systems are established usig Lyapuov stability theory. Numerical simulatios are depicted to illustrate ad validate the hybrid sychroizatio schemes for the idetical hyperchaotic Xu systems. KEYWORDS Slidig Mode Cotrol, Hybrid Sychroizatio, Hyperchaotic Systems, Hyperchaotic Xu System. 1. INTRODUCTION Chaotic systems are oliear dyamical systems that are highly sesitive to iitial coditios ad exhibit radom behaviour i determiistic motio. The sesitive ature of chaotic systems is commoly called as the butterfly effect [1]. Sychroizatio of chaotic systems is a pheomeo which may occur whe two or more chaotic oscillators are coupled or whe a chaotic oscillator drives aother chaotic oscillator. Because of the butterfly effect which causes the expoetial divergece of the trajectories of two idetical chaotic systems havig early the same iitial coditios, sychroizig two chaotic systems is ideed a very challegig problem i the literature. Hyperchaotic system is usually defied as a chaotic system with more tha oe positive Lyapuov expoet. The first hyperchaotic system was discovered by O.E. Rössler (1979). Sice hyperchaotic system has the characteristics of high capacity, high security ad high efficiecy, it has the potetial of broad applicatios i oliear circuits, secure commuicatios, lasers, eural etworks, biological systems ad so o. Thus, the studies o hyperchaotic systems, viz. cotrol, sychroizatio ad circuit implemetatio are very challegig problems i the chaos literature. DOI : 10.5121/ijics.2012.2406 61
I most of the chaos sychroizatio approaches, the master-slave or drive-respose formalism is used. If a particular chaotic system is called the master or drive system ad aother chaotic system is called the slave or respose system, the the idea of the ati-sychroizatio is to use the output of the master system to cotrol the slave system so that the states of the slave system have the same amplitude but opposite sigs as the states of the master system asymptotically. Sice the pioeerig work by Pecora ad Carroll ([2], 1990), chaos sychroizatio problem has bee studied extesively ad itesively i the literature [2-17]. Chaos theory has bee applied to a variety of fields such as physical systems [3], chemical systems [4], ecological systems [5], secure commuicatios [6-8], etc. I the last two decades, various schemes have bee successfully applied for chaos sychroizatio such as PC method [2], OGY method [9], active cotrol method [10-14], adaptive cotrol method [15-20], time-delay feedback method [21], backsteppig desig method [22], sampled-data feedback method [23], etc. I hybrid sychroizatio of master ad slave systems, the odd states are completely sychroized (CS) ad the eve states are ati-sychroized (AS). The co-existece of CS ad AS i the sychroizatio ehaces the security of secure commuicatio systems. I this paper, we derive ew results based o the slidig mode cotrol [24-26] for the hybrid sychroizatio of idetical hyperchaotic Xu systems ([27], Xu, Cai ad Zheg, 2009). I robust cotrol systems, the slidig mode cotrol method is ofte adopted due to its iheret advatages of easy realizatio, fast respose ad good trasiet performace as well as its isesitivity to parameter ucertaities ad exteral disturbaces. This paper has bee orgaized as follows. I Sectio 2, we describe the problem statemet ad our methodology usig slidig mode cotrol (SMC). I Sectio 3, we discuss the hybrid sychroizatio of idetical hyperchaotic Xu systems. I Sectio 4, we summarize the mai results obtaied i this paper. 2. PROBLEM STATEMENT AND OUR METHODOLOGY USING SMC Cosider the chaotic system described by x& = Ax + f ( x) (1) where x R is the state of the system, A is the matrix of the system parameters ad f : R R is the oliear part of the system. We cosider the system (1) as the master or drive system. As the slave or respose system, we cosider the followig chaotic system described by the dyamics y& = Ay + f ( y) + u (2) m where y R is the state of the system ad u R is the cotroller to be desiged. We defie the hybrid sychroizatio error as 62
yi xi if i is odd ei = yi + xi if i is eve (3) The the error dyamics is obtaied as e& = Ae + η( x, y) + u (4) The objective of the ati-sychroizatio problem is to fid a cotroller u such that lim e( t) = 0 for all e(0) R. (5) t To solve this problem, we first defie the cotrol u as u = η( x, y) + Bv (6) where B is a costat gai vector selected such that ( A, B ) is cotrollable. Substitutig (6) ito (4), the error dyamics simplifies to e& = Ae + Bv (7) which is a liear time-ivariat cotrol system with sigle iput v. Thus, the origial ati-sychroizatio problem ca be replaced by a equivalet problem of stabilizig the zero solutio e = 0 of the system (7) by a suitable choice of the slidig mode cotrol. I the slidig mode cotrol, we defie the variable s( e) = Ce = c e + c e + L + ce (8) 1 1 2 2 where C [ c c c ] = L is a costat vector to be determied. 1 2 I the slidig mode cotrol, we costrai the motio of the system (7) to the slidig maifold defied by { R ( ) 0} S = e s e = which is required to be ivariat uder the flow of the error dyamics (7). Whe i slidig maifold S, the system (7) satisfies the followig coditios: s( e ) = 0 (9) which is the defiig equatio for the maifold S ad s& ( e ) = 0 (10) which is the ecessary coditio for the state trajectory e( t ) of (7) to stay o the slidig maifold S. 63
Usig (7) ad (8), the equatio (10) ca be rewritte as [ ] s& ( e) = C Ae + Bv = 0 (11) Solvig (11) for v, we obtai the equivalet cotrol law v t = CB CA e t (12) ( ) ( ) 1 ( ) eq where C is chose such that CB 0. Substitutig (12) ito the error dyamics (7), we obtai the closed-loop dyamics as & = ( ) (13) 1 e I B CB C Ae The row vector C is selected such that the system matrix of the cotrolled dyamics 1 I B( CB) C A is Hurwitz, i.e. it has all eigevalues with egative real parts. The the cotrolled system (13) is globally asymptotically stable. To desig the slidig mode cotroller for (7), we apply the costat plus proportioal rate reachig law s& = q sg( s) k s (14) where sg( ) deotes the sig fuctio ad the gais q > 0, k > 0 are determied such that the slidig coditio is satisfied ad slidig motio will occur. From equatios (11) ad (14), we ca obtai the cotrol v( t) as v( t) = ( CB) C( ki + A) e + q sg( s) (15) which yields 1 [ ] [ ] [ ] v( t) = + + > CB C ki + A e q s e < 1 ( CB) C( ki A) e q, if s( e) 0 1 ( ) ( ), if ( ) 0 (16) Theorem 2.1. The master system (1) ad the slave system (2) are globally ad asymptotically hybrid sychroized for all iitial coditios x(0), y(0) R by the feedback cotrol law u( t) = η( x, y) + Bv( t) (17) where v( t) is defied by (15) ad B is a colum vector such that ( A, B) is cotrollable. Also, the slidig mode gais k, q are positive. Proof. First, we ote that substitutig (17) ad (15) ito the error dyamics (4), we obtai the closed-loop error dyamics as e& = Ae B( CB) C( ki + A) e + qsg( s) (18) 1 [ ] 64
To prove that the error dyamics (18) is globally asymptotically stable, we cosider the cadidate Lyapuov fuctio defied by the equatio 1 2 V ( e) = s ( e) (19) 2 which is a positive defiite fuctio o R. Differetiatig V alog the trajectories of (18) or the equivalet dyamics (14), we get & & (20) 2 V ( e) = s( e) s( e) = ks qsg( s) s which is a egative defiite fuctio o R. This calculatio shows that V is a globally defied, positive defiite, Lyapuov fuctio for the error dyamics (18), which has a globally defied, egative defiite time derivative V &. Thus, by Lyapuov stability theory [28], it is immediate that the error dyamics (18) is globally asymptotically stable for all iitial coditios e(0) R. This meas that for all iitial coditios e(0) R, we have lim e( t) = 0 t Hece, it follows that the master system (1) ad the slave system (2) are globally ad asymptotically hybrid sychroized for all iitial coditios x(0), y(0) R. 3. HYBRID SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC XU SYSTEMS VIA SLIDING MODE CONTROL 3.1 Theoretical Results I this sectio, we apply the slidig mode cotrol results derived i Sectio 2 for the hybrid sychroizatio of idetical hyperchaotic Xu systems ([27], Xu et al. 2009). Thus, the master system is described by the Xu dyamics x& = a( x x ) + x 1 2 1 4 x& = bx + fx x 2 1 1 3 x& = cx ε x x 3 3 1 2 x& = dx + x x 4 4 1 3 where x1, x2, x3, x 4 are state variables ad a, b, c, d, ε, f system. (21) are positive, costat parameters of the The Xu system (21) is hyperchaotic whe the parameters are chose as 65
a = 10, b = 40, c = 2.5, d = 2, ε = 1 ad f = 16 Figure 1 illustrates the phase portrait of the hyperchaotic Xu system. Figure 1. Phase Portrait of the Hyperchaotic Xu System The slave system is described by the cotrolled hyperchaotic Xu dyamics y& = a( y y ) + y + u 1 2 1 4 1 y& = by + fy y + u 2 1 1 3 2 y& = cy ε y y + u 3 3 1 2 3 y& = dy + y y + u 4 4 1 3 4 (22) where y1, y2, y3, y4 are state variables ad u1, u2, u3, u4 are the cotrollers to be desiged. 66
The hybrid sychroizatio error is defied by e e e e = y x 1 1 1 = y + x 2 2 2 = y x 3 3 3 = y + x 4 4 4 (23) The error dyamics is easily obtaied as e& = a( e e ) + e 2ax 2x + u 1 2 1 4 2 4 1 e& = be + 2 bx + f ( y y + x x ) + u 2 1 1 1 3 1 3 2 e& = ce ε y y + ε x x + u 3 3 1 2 1 2 3 e& = de + y y + x x + u 4 4 1 3 1 3 4 (24) We write the error dyamics (24) i the matrix otatio as e& = Ae + η( x, y) + u (25) where a a 0 1 b 0 0 0 A =, 0 0 c 0 0 0 0 d η( x, y) 2ax2 2x4 2 bx + f ( y y + x x ) 1 1 3 1 3 = ε ( y1 y2 x1x2 ) y y + x x 1 3 1 3 ad u1 u 2 u = u3 u 4 (26) The slidig mode cotroller desig is carried out as detailed i Sectio 2. First, we set u as u = η( x, y) + Bv (27) where B is chose such that ( A, B) is cotrollable. We take B as 1 1 B = 1 1 (28) I the hyperchaotic case, the parameter values are take as a = 10, b = 40, c = 2.5, d = 2, ε = 1 ad f = 16 67
The slidig mode variable is selected as [ 9 1 1 9] 9 1 2 3 9 4 s = Ce = e = e + e + e e (29) which makes the slidig mode state equatio asymptotically stable. We choose the slidig mode gais as k = 6 ad q = 0.2. We ote that a large value of k ca cause chatterig ad a appropriate value of q is chose to speed up the time take to reach the slidig maifold as well as to reduce the system chatterig. From Eq. (15), we ca obtai v( t) as v( t) = 2e 48e 1.75e + 13.5e 0.1sg( s) (30) 1 2 3 4 Thus, the required slidig mode cotroller is obtaied as u = η( x, y) + Bv (31) where η( x, y), B ad v( t) are defied as i the equatios (26), (28) ad (30). By Theorem 2.1, we obtai the followig result. Theorem 3.1. The idetical hyperchaotic Xu systems (21) ad (22) are globally ad asymptotically hybrid sychroized for all iitial coditios with the slidig mode cotroller u defied by (31). 3.2 Numerical Results I this sectio For the umerical simulatios, the fourth-order Ruge-Kutta method with timestep h = 10 is used to solve the hyperchaotic Xu systems (21) ad (22) with the slidig mode 6 cotroller u give by (31) usig MATLAB. I the hyperchaotic case, the parameter values are give by a = 10, b = 40, c = 2.5, d = 2, ε = 1 ad f = 16 The slidig mode gais are chose as k = 6 ad q = 0.2. The iitial values of the master system (21) are take as x (0) = 14, x (0) = 9, x (0) = 27, x (0) = 10 1 2 3 4 The iitial values of the slave system (22) are take as y (0) = 4, y (0) = 12, y (0) = 6, y (0) = 22 1 2 3 4 Figure 2 illustrates the hybrid sychroizatio of the hyperchaotic Xu systems (21) ad (22). 68
Figure 3 illustrates the time-history of the sychroizatio errors e1, e2, e3, e 4. Figure 2. Hybrid Sychroizatio of Idetical Hyperchaotic Xu Systems Figure 3. Time-History of the Hybrid Sychroizatio Error 69
4. CONCLUSIONS I this paper, we have derived slidig mode cotrollers to achieve hybrid sychroizatio for the idetical hyperchaotic Xu systems (2009). Our hybrid sychroizatio results for the idetical hyperchaotic Xu systems have bee proved usig Lyapuov stability theory. Sice the Lyapuov expoets are ot required for these calculatios, the slidig mode cotrol method is very effective ad coveiet to achieve hybrid sychroizatio for the idetical hyperchaotic Xu systems. Numerical simulatios are also show to illustrate the effectiveess of the hybrid sychroizatio results derived i this paper usig the slidig mode cotrol. REFERENCES [1] Alligood, K.T., Sauer, T. & Yorke, J.A. (1997) Chaos: A Itroductio to Dyamical Systems, Spriger, New York. [2] Pecora, L.M. & Carroll, T.L. (1990) Sychroizatio i chaotic systems, Phys. Rev. Lett., Vol. 64, pp 821-824. [3] Lakshmaa, M. & Murali, K. (1996) Noliear Oscillators: Cotrollig ad Sychroizatio, World Scietific, Sigapore. [4] Ha, S.K., Kerrer, C. & Kuramoto, Y. (1995) Dephasig ad burstlig i coupled eural oscillators, Phys. Rev. Lett., Vol. 75, pp 3190-3193. [5] Blasius, B., Huppert, A. & Stoe, L. (1999) Complex dyamics ad phase sychroizatio i spatially exteded ecological system, Nature, Vol. 399, pp 354-359. [6] Cuomo, K.M. & Oppeheim, A.V. (1993) Circuit implemetatio of sychroized chaos with applicatios to commuicatios, Physical Review Letters, Vol. 71, pp 65-68. [7] Kocarev, L. & Parlitz, U. (1995) Geeral approach for chaotic sychroizatio with applicatios to commuicatio, Physical Review Letters, Vol. 74, pp 5028-5030. [8] Tao, Y. (1999) Chaotic secure commuicatio systems history ad ew results, Telecommu. Review, Vol. 9, pp 597-634. [9] Ott, E., Grebogi, C. & Yorke, J.A. (1990) Cotrollig chaos, Phys. Rev. Lett., Vol. 64, pp 1196-1199. [10] Ho, M.C. & Hug, Y.C. (2002) Sychroizatio of two differet chaotic systems usig geeralized active etwork, Physics Letters A, Vol. 301, pp 424-428. [11] Huag, L., Feg, R. & Wag, M. (2005) Sychroizatio of chaotic systems via oliear cotrol, Physical Letters A, Vol. 320, pp 271-275. [12] Che, H.K. (2005) Global chaos sychroizatio of ew chaotic systems via oliear cotrol, Chaos, Solitos & Fractals, Vol. 23, pp 1245-1251. [13] Sudarapadia, V. (2011) Global chaos sychroizatio of four-scroll ad four-wig chaotic attractors by active oliear cotrol, Iteratioal Joural o Computer Sciece ad Egieerig, Vol. 3, No. 5, pp. 2145-2155. [14] Sudarapadia, V. (2011) Global chaos sychroizatio of Li ad Liu-Che-Liu chaotic systems by active oliear cotrol, Iteratioal Joural of Advaces i Sciece ad Techology, Vol. 3, No. 1, pp. 1-12. [15] Lu, J., Wu, X., Ha, X. & Lü, J. (2004) Adaptive feedback sychroizatio of a uified chaotic system, Physics Letters A, Vol. 329, pp 327-333. [16] Che, S.H. & Lü, J. (2002) Sychroizatio of a ucertai uified system via adaptive cotrol, Chaos, Solitos & Fractals, Vol. 14, pp 643-647. [17] Sudarapadia, V. (2011) Adaptive cotrol ad sychroizatio of hyperchaotic Cai system, Iteratioal Joural of Cotrol Theory ad Computer Modellig, Vol. 1, No. 1, pp 1-13. [18] Sudarapadia, V. (2011) Adaptive sychroizatio of hyperchaotic Lorez ad hyperchaotic Liu systems, Iteratioal Joural of Istrumetatio ad Cotrol Systems, Vol. 1, No. 1, pp 1-18. [19] Sudarapadia, V. (2011) Adaptive cotrol ad sychroizatio of Liu s four-wig chaotic system with cubic oliearity, Iteratioal Joural of Computer Sciece, Egieerig ad Applicatios, Vol. 1, No. 4, pp 127-138. 70
[20] Sudarapadia, V. & Karthikeya, R. (2011) Global chaos sychroizatio of Pa ad Lü chaotic systems via adaptive cotrol, Iteratioal Joural of Iformatio Techology, Covergece ad Services, Vol. 1, No. 5, pp. 49-66. [21] Park, J.H. & Kwo, O.M. (2003) A ovel criterio for delayed feedback cotrol of time-delay chaotic systems, Chaos, Solitos & Fractals, Vol. 17, pp 709-716. [22] Wu, X. & Lü, J. (2003) Parameter idetificatio ad backsteppig cotrol of ucertai Lü system, Chaos, Solitos & Fractals, Vol. 18, pp 721-729. [23] Zhao, J. & J. Lu (2006) Usig sampled-data feedback cotrol ad liear feedback sychroizatio i a ew hyperchaotic system, Chaos, Solitos & Fractals, Vol. 35, pp 376-382. [24] Slotie, J.E. & Sastry, S.S. (1983) Trackig cotrol of oliear systems usig slidig surface with applicatio to robotic maipulators, Iteratioal Joural of Cotrol, Vol. 38, pp 465-492. [25] Utki, V.I. (1993) Slidig mode cotrol desig priciples ad applicatios to electric drives, IEEE Tras. Idustrial Electroics, Vol. 40, pp 23-36, 1993. [26] Sudarapadia, V. (2011) Global chaos sychroizatio of four-wig chaotic systems by slidig mode cotrol, Iteratioal Joural of Cotrol Theory ad Computer Modellig, Vol. 1, No. 1, pp 15-31. [27] Xu, J., Cai, G. & Zheg, Sog. (2009) A ovel hyperchaotic system ad its cotrol, Joural of Ucertai Systems, Vol. 3, No. 2, pp. 137-144. [28] Hah, W. (1967) The Stability of Motio, Spriger, New York. AUTHOR Dr. V. Sudarapadia obtaied his Doctor of Sciece degree i Electrical ad Systems Egieerig from Washigto Uiversity, St. Louis, USA i May 1996. He is Professor ad Dea of the Research ad Devleopmet Cetre at Vel Tech Dr. RR & Dr. SR Techical Uiversity, Cheai, Tamil Nadu, Idia. He has published over 280 refereed iteratioal joural publicatios. He has published over 170 papers i Natioal ad Iteratioal Cofereces. He is the Editor-i-Chief of the AIRCC Jourals - Iteratioal Joural of Istrumetatio ad Cotrol Systems, Iteratioal Joural of Cotrol Systems ad Computer Modellig, Iteratioal Joural of Iformatio Techology, Cotrol ad Automatio, Iteratioal Joural of Chaos, Cotrol, Modelig ad Simulatio, ad Iteratioal Joural of Iformatio Techology, Modelig ad Computig. His research iterests are Liear ad Noliear Cotrol Systems, Chaos Theory ad Cotrol, Soft Computig, Optimal Cotrol, Operatios Research, Mathematical Modellig ad Scietific Computig. He has delivered several Key Note Lectures o Cotrol Systems, Chaos Theory, Scietific Computig, Mathematical Modellig, MATLAB ad SCILAB. 71