Anti-Synchronization of two Different Chaotic Systems via Optimal Control with Fully Unknown Parameters *

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1 ISSN , England, UK Journal of Informaton and Computng Scence Vol. 5, No., 00, pp Ant-Synchronzaton of two Dfferent Chaotc Systems va Optmal Control wth Fully Unknown Parameters * Honglan Zhu + Huayn Insttute of Technology, Huaan, 3003, Chna, (Receved March 5, 009, accepted August 6, 009) Abstract. Ths paper presents ant-synchronzaton of two dfferent chaotc systems usng optmal control method. The proposed technque s appled to acheve chaos ant-synchronzaton for the new fourdmensonal and hyperchaotc Lü systems wth fully unknown parameters. Numercal smulatons results are presented to demonstrate the effectveness of the method. Key Words: Ant-synchronzaton;Optmal; Adaptve.. Introducton Snce the dea of synchronzng chaotc systems was ntroduced by Pecora and Carroll [] n 990, chaos synchronzaton has receved ncreasng attenton due to ts theoretcal challenge and ts great potental applcatons n secure communcaton, chemcal reacton and bologcal systems []. Recently several dfferent types of synchronzatons have been proposed n the lterature, for example, generalzed synchronzaton [3,,5,6], phase synchronzaton[7,8], lag synchronzaton[9,0,], antsynchronzaton[,3,] and so on. The ant-synchronzaton (AS) method s that the state vectors of synchronzed systems have the same absolute values but opposte sgns,.e. the sum of the output sgnals of two systems can converge to zero. Recently many methods have been used to acheve ant-synchronzaton between chaotc systems see [5, 6, 7]. There are a few papers that studed the problem of optmal controllng chaos and optmal synchronzaton of the chaotc systems wth unknown system parameters see [8, 9, 0]. Most of the methods mentoned above synchronze two dfferent chaotc systems usng adaptve methods. However, these methods of the synchronzaton are far from optmal synchronzaton. In general, the adaptve synchronzaton s not necessary satsfy the optmalty condtons. Our attenton n the present paper s to study optmal synchronzaton of two dfferent systems wth complete uncertan system parameters. Ths paper s organzed as follows: Secton explan the prncple of optmal control. In Secton 3 descrbes the new four-dmensonal system and hyperchaotc Lü system. In Secton,the antsynchronzaton of the new four-dmensonal system and hyperchaotc Lü system wth completely uncertan system parameters s acheved wth the optmal and adaptve control technque. In Secton 5 numercal results demonstrate the effectveness of the proposed control technque are presented. Fnally, Secton 6 contans a summarzed concluson of the results.. Prncple of Optmal Control Nonlnear chaotc system x f( t x) y gty ( ) utxy ( ) n n n where xy R, f and g are RR R and dfferentable. The frst equaton of the system () s master system and the second s slave system. where utxy) ( s control functon. Let et () xt () yt (), our goal s to desgn the controller and satsfy: + Correspondng author. E-mal address: zhuhonglan@gmal.com. () Publshed by World Academc Press, World Academc Unon

2 Honglan Zhu, et al: Ant-Synchronzaton of two Dfferent Chaotc Systems E mn [ q( x) u T Ru] dt () t0 Fgure. chaotc attractors for the new four-dmensonal system when a 35b0cd 0. (a) chaotc attractor n ( x x) space; (b) chaotc attractor n ( x x3 x) space. Fgure. chaotc attractors for the new four-dmensonal system when a 30b0c37d 0. (a) chaotc ( x x ) ( x x x ) space. and then attractor n space; (b) chaotc attractor n 3 lm et ( ) 0 t where qx ( ) s a functon whch s contnuous,dfferentable and postve.accordng to Dynamc Programmng the optmal control s based on Hamlton-Jacob-Bellman equaton: mn[ E w] [ E w] 0 u u (3) T where wq( x) u Ru and u s optmal controller. 3. Systems Descrpton Recently a new Four-Dmensonal Chaotc System s studed [].The system can be descrbed by: x ax ( x) xxx 3 x bx ( x) xxx 3 () x 3 cx3 xx x x dx xx x3 where x ( 3) are the state varables of the system and abcd are all postve real constant JIC emal for contrbuton: edtor@jc.org.uk

3 Journal of Informaton and Computng Scence, Vol. 5 (00) No., pp parameters. When a35b0c and d (088] the system () s chaotc wth a postve Lyapunov exponent (for example, wth d 0, the chaotc attractor s shown n Fg.). When a30b0c37 and d 0 the system () s also chaotc.the chaotc attractor as shown n Fg.. Lü system as a typcal transacton system, found by Lü and Chen, whch connects the Lorenz and Chen attractors and represents the transton from one to the other.recently, Chen A-mn others proposed hyperchaotc Lü system[].the hyperchaotc Lü system s descrbed by: y ay ( y) y y cy yy3 (5) y 3 yy by3 y yy 3dy The system s based on Lü system, the frst equaton wth a controller, the second and three equaton non-controller. Then fourth equaton contans an addtonal controller. Whch abc s admsson to the system parameters. Gan control parameter d s to be determned. By analyss of the dynamcs of the system, ncludng the bfurcaton dagram, Lyapunov exponent spectrum Poncare mappng, the system and crcut smulaton experments confrm hyperchaotc. When a36b3c0 and d takes other dfferent values, system performance of the dfferent dynamcs: when 03d 06, system s a perodc orbt; when 06 d 035,system s chaotc attractor; and when 035 d 30,there are two ndex greater than zero system s super chaotc attractor. In ths paper, we consder the system s hyperchaotc. Selectng parameters a36b3c0 and 035 d 30.When a 36b3c 0 and d, hyperchaotc attractor as shown n Fg.3. Fgure 3. Hyperchaotc attractors for the Lü system when a 36b3c0 and d. (a) hyperchaotc attractor ( x x ) ( x x x ) space. n space; (b) hyperchaotc attractor n 3. Ant-Synchronzaton Va Optmal Control The purpose of ths secton s to ntroduce a development optmal control law for resolvng the optmal ant-synchronzaton between the new Four-Dmensonal system and hyperchaotc Lü system wth completely unknown system parameters. All dervatons of the optmal controllers are based on the Lyapunov Bellman technque[3]. In order to observe the ant-synchronzaton behavor n the new Four-Dmensonal system and hyperchaotc Lü system we assume that Four-Dmensonal system drves the hyperchaotc Lü system. Therefore, we defne the master system and slave systems as follows JIC emal for subscrpton: publshng@wau.org.uk

4 Honglan Zhu, et al: Ant-Synchronzaton of two Dfferent Chaotc Systems x ax ( x) xxx 3 x bx ( x) xxx 3 (6) x 3 cx3 xx x x dx xx x3 and y a ( y y) y vu y cy yy 3v u (7) y 3 yy by 3 v3 u3 y yy 3dy v u where the controllers are consst of two parts,one s nonlnear feedback control v ( 3) and the other s optmal control u ( 3).In order to determne the control functons to realze ant-synchronzaton between systems (6) and (7).Let us defne the state errors between the response and derve system whch s to be controlled and the controllng derve system as e x ye x ye3 e3 y3e x y (8) Add (6) to (7) and usng the notaton (8) yelds e ae ( a a)( yy) ae xx3x vu e be ( e y y) xxx 3 cy v u (9) e 3 ce3 cy3 yy xx x by 3 v3 u3 e de ( d d) y xx x3yy 3v u and then we let v ae ˆ ( aˆaˆ)( y y) xx3x v ( y y ee) bˆ x ˆ x3x cy yy3 (0) v ˆ 3 cˆ( e3 y3) by3 xxx yy v ˆ ˆ d( e y) dy xxx3 yy3 where abcda ˆ ˆˆ ˆ ˆ ˆ ˆ b ˆ cd are estmates of abc d a bc d respectvely.then wth controllers (0) we get the error dynamcal system (9) can be descrbed by e ae ( aa )( y y) u e be ( e yy) u () e 3 ce ( 3 y3) by 3 u3 e de ( dd ) y u where a aˆab bˆbc cˆcd dˆda ˆ ˆ ˆ aab bbc cc and d d ˆ d. Clearly, the optmal synchronzaton problem s now replaced by the equvalent problem of optmal stablzng the error system () usng a sutable choce of the control law u u u 3 u wth mnmum cost. Let us now formulate the followng theorem. Theorem. Wth the nonlnear feedback controllers u ( c n) e( 3 ) () and the system parameters nonlnear updatng rule JIC emal for contrbuton: edtor@jc.org.uk

5 Journal of Informaton and Computng Scence, Vol. 5 (00) No., pp a ˆ ( k)[ ce c( y y) e a ] bˆ ( cek)( ee y yb ) c ˆ ( c3e3k3)( e3 y33c ) dˆ ( cek)( e yd ) a ˆ ( ce k5)( yy 5a ) bˆ ( c3y3 k6)( e3 6b ) c ˆ ( cy k7)( e 7c ) dˆ ( cy k8)( e 8d ) where, n and c satsfy the condtons c n ( 3) () Then the new four-dmensonal system (6) and hyperchaotc Lü system (7) are optmally asymptotcally ant-synchronzed along the optmal trajectores specfed by the ntegral performance ndex I [ e nu ] ( a) ( b) ( c) ( d) ( a ) ( b) ( c ) ( d ) dt(5) t where t 0 s a fxed tme moment and ( 3 8) are non-negatve constants. Proof. The proof of ths theorem depends upon the choce of the performance measure as gven by (5) and verfyng that both of the optmal controllers () and updatng rules (3) mnmze ths performance measure and asymptotcally stablzes the equlbrum state ().Assume that the functon E represents the mnmum value of the performance measure and the fst we let w{ [ e nu ] a c a b 3c d 5 6b 7 8d consequently E mn wdt u (6) t 0 where u ( uuu3u), the functon E s referred to as the value functon. and t satsfes the Bellman- Hamlton-Jacob equaton(3).accordng to the error system() we consder a Lyapunov functon as follows: E { ce d ka kb k3c kd k5a k6b k7c k8 (7) Assume u ( 3) are needed optmal controller and accordng to the prncple of the optmal control and ths leads to mn[ E w] c e k aa k bb k cc k dd k a a k b b k c c k d d w (8) u substtutng(),(3)and()nto(8),we can get mn[ E w] 0 (9) u It mples that the controller u satsfy the Bellman-Hamlton-Jacob equaton(3) and the controller () s optmal. Next we wll prove that wth the optmal controller () and the system parameters updatng rule (3) two systems acheve ant-synchronzaton. Lyapunov functon n (7) along the optmal trajectores of the closed-loop system, we get (3) JIC emal for subscrpton: publshng@wau.org.uk

6 6 Honglan Zhu, et al: Ant-Synchronzaton of two Dfferent Chaotc Systems E { ce k k k k k k k k substtutng (),(3),()nto(0),we can get a b 3c d 5a 6b 7c 8 E e( a b 3c d 5a 6b 7c 8d ) 0 () snce E s a postve decreasng functon and E s negatve semdefnte,t follows that the equlbrum ( e 0e 0 0e3 0e 0a 0b 0c d 0a 0b 0c 0d 0) of the system (8) s unformly stable,.e. e and ee3e L ab cda b cd L.From Eq.(), we can easly show that the squares of e ee3 and e are ntegrable wth respect to tme t,.e. eee3e L. Next by Barbalat s Lemma, for any ntal condton, the system (8) mples that e e e 3 e L,whch n turn ples ( ee e e ) (0000) as t. Thus n the closed-loop system m 3 y x y x y x y x as t. 3 3 d Consequently, wth the optmal feedback control law () and updatng rule (3) the optmal synchronzaton of both derve and response systems s acheved wth complete unknown system parameters, whch completes the proof. 5. Smulaton In ths secton, we wll show a seres of numercal smulatons to demonstrate the effectveness of the proposed control scheme. All smulaton procedures are coded and executed usng the Matlab software. Fourth order Runge-Kutta ntegraton method s used to solve two systems of dfferental equatons (6) and (7).In addton, a tme step sze 000 s employed. 3 We assume that ( 8), n ( 3 ), k ( 8). We wll select the parameters of the new four-dmensonal system a 30b 0c37d 0 and the parameters of hyperchaotc Lü system a 36b3c0d.Therefore, both the new four-dmensonal system and hyperchaotc lü system exhbt chaotc behavor. The ntal values of the master and slave systems are ( x(0), x(0), x3( 0), x(0)) ( 0,0,0, 0) and ( y(0), y(0), y3(0), y(0)) ( 0.3,.5,3.,0.). So the ntal values of the error system s ( e (0), e (0), e (0), e (0)) ( 9.7,.5, 3., 0.).The antsynchronzaton of systems (6) and (7) va optmal control law () and parameters update rule (3) are shown n Fg. and Fg.5. (0) Fgure. Dynamcs of ant-synchronzaton errors 3 hyperchaotc Lü system wth tme. (ee e e ) between the new four-dmensonal system and t JIC emal for contrbuton: edtor@jc.org.uk

7 Journal of Informaton and Computng Scence, Vol. 5 (00) No., pp Fgure 5. Adaptve parameters estmaton errors:(a) abcdˆ ˆ ˆˆ ;(b) aˆ bˆ ˆ ˆ c d 6. Concluson In ths paper, chaos ant-synchronzaton between two dfferent chaotc systems wth dfferent structures and parameters va optmal and adaptve control s presented. The new four-dmensonal system and the hyperchaotc Lü system are taken as an llustratve example to verfy the effectveness of the proposed method. 7. References [] L.M. Pecora, T.L. Carroll. Phys. Rev. Lett. 990, 6: 8. [] Chen G, Dong X. Form chaos to order: methodologes, perspectves and applcatons. Sngapore: World Scentfc [3] Y. W. Wang, Z. H. Guan. Generalzed synchronzaton of contnuous chaotc system. Chaos, Soltons and Fractals. 006, 7: 97. [] K. Mural, M. Lakshmanan. Secure communcaton usng a compound sgnal from generalzed synchronzable chaotc systems. Physcs Letters. 998, (A): 303. [5] S. S. Yang, C. K. Duan. Generalzed synchronzaton n chaotc systems. Chaos, Soltons and Fractals. 998, 0: 703. [6] L. Kocarev, U. Parltz. Gneralzed synchronzaton, predctablty, and equvalence of undrectonally coupled dynamcal systems. Phys.Rev.Lett. 996, 76: 86. [7] G. Santobon, A. Y. Pogromsky, H. Njmejer. An observer for phase synchronzaton of chaos. Physcs Letters. 00, 9(A): 65. [8] G. R. Mchael, S. P. Arkady, K. Jrgen. Phase synchronzaton of chaotc oscllators. Phys.Rev.Lett. 996, 76: 80. [9] C. L, X. Lao, K. Wong. Lag synchronzaton of hyperchaos wth applcaton to secure communcatons. Chaos, Soltons and Fractals. 005, 3: 83. [0] Y. Chen, X. Chen and S. Gu. Lag synchronzaton of structurally nonequvalent chaotc systems wth tme delays. Nonlnear Analyss [] I. S. Taheron, Y. C. La. Observablty of lag synchronzaton of coupled chaotc oscllators. Phys.Rev.E. 999, 59: 67. [] G. H. L, S. P. Zhou. An observer-based ant-synchronzaton. Chaos, Soltons and Fractals. 006, 9: 95. [3] G. H. L. Synchronzaton and ant-synchronzaton of Colptts oscllators usng actve control. Chaos, Soltons and Fractals. 005, 6: 87. [] J. Hu, S. Chen, L. Chen. Adaptve control for ant-synchronzaton of Chua s chaotc system. Physcs Letters. 005, 339(A): 55. [5] Guo-Hu L, Sh-Png Zhou. Ant-synchronzaton n dfferent chaotc systems. Chaos, Soltons and Fractals. 006, 3: 56. [6] T.-Y. Chang, J.-S. Ln, T.-L. Lao, J.-J.Yan. Ant-synchronzaton of uncertan unfed chaotc systems wth deadzone nonlnearty. Nonlnear Analyss. 007,do:0.06/j.na [7] Amr Abbas Emadzadeh, Mohammad Haer. Ant- Synchronzaton of two Dfferent Chaotc Systems va Actve Control. Transacton on engneerng, Computng and technology. 005, 6: 6. [8] El-Gohary A, Bukhar F. Optmal control of Lorenz system durng dfferent tme ntervals. Appl Math Comput. 003, : 337. [9] Pecora LM, Carroll TL. Synchronzng a chaotc systems. IEEE Trans Crc Syst. 99, 38: [0] El-Gohary A. Optmal Synchronzaton of Rössler System wth complete uncertan parameters. Chaos, Soltons and Fractals. 006, 7(): JIC emal for subscrpton: publshng@wau.org.uk

8 8 Honglan Zhu, et al: Ant-Synchronzaton of two Dfferent Chaotc Systems [] Q G Y, Du S Z., Chen G R, et al. On a fourdmensonal chaotc system. Chaos Soltons and Fractals. 005, 3: 67. [] Chen A M, Lu J A, Ln J H, et a. Generatng hyperehaotc Lt attractor va state feedback control. Physcs. 006, 36(A): 03. [3] Awad EL-Gohary. synchronzaton of Rössler system wth complete uncertan parameters. Chaos, Soltons and Fractals. 006, 7: 35. JIC emal for contrbuton: edtor@jc.org.uk