Adaptive Synchronization of Rössler System Based on. Dynamic Surface ControlPF

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1 FP Mar. 6 Journal of Electronc Scence an Technology of Chna Vol. No. Aaptve Synchronzaton of Rössler System Base on * Dynamc Surface ControlPF LIU Tong-shuan, XU Hao, GUAN Xn-png Insttute of Electrcal Engneerng, Yanshan Unversty Qnhuangao Hebe 66 Chna Abstract A synchronzaton scheme for Rössler system base on Dynamc Surface Control (DSC) s propose n ths paper. The DSC metho s a recursve esgn proceure lke conventonal backsteppng methos. Dfferent from the backsteppng esgn, a frst-orer flter s ntrouce n every DSC esgn step. For ths ntrouce flter, the ervatve of the selecte vrtual control s avoe an then the rawback of exploson of complexty exstng n backsteppng esgn s overcome. Moreover, aaptve metho s use for controller esgn when the system parameters are unknown. Fnally, a numercal example s gven to llustrate the effectveness an performance of the propose metho. Key wors aaptve synchronzaton; Rössler system; ynamc surface control Snce the poneerng works of Pecora an [] CarrollP P, the synchronzaton of chaotc systems has been extensvely stue n the last two ecaes ue to ts possble applcaton to secure communcaton an [-] bologcal systemsp P. Recently, a great eal of schemes have been propose to eal wth ths problem, [5] such as observer-base methop P, mpulsve control [6] [7] methop P, aaptve methop P, ntellgent control lke [8-9] fuzzy an neural network methosp P, just to menton a few. In the last several years, backstepptng metho has been successfully use to eal wth chaos [] synchronzaton problemp P. Ths approach s a step by step recursve esgn proceure together wth selecton of Lyapunov functon canate for synthetc control nput. In every esgn step, a vrtual controller s erve to make the selecte Lyapunov functon negatve efnte. However, the vrtual controllers tme ervatves are necessarly neee to synthesze the fnal control nput n the backsteppng esgn. Therefore, the esgner has to face the fact that the complexty of the esgne controller grows rastcally as the orer of the system, n, ncreases for the repeate fferentatons of some nonlnear functons. Ths s the so-calle problem of exploson of complexty. Dynamc surface control (DSC) s propose to overcome the problem of exploson of complexty [-] recentlyp P. In DSC, frst-orer flter s ntrouce n every esgn step to avo the tme ervatve of the vrtual control. Consequently, the requrement on the smoothness of the nonlnear terms n the system s relaxe an the problem of the exploson of complexty s elmnate n ths metho. Motvate by ths, we propose a chaos synchronzaton scheme base on DSC technque to smplfy the control nput erve from backsteppng methos n the exste contrbutons. Moreover, aaptve metho s use n the esgn proceure on the conton that the system parameters are unknown. Prelmnary To clearly llustrate the propose scheme, we frstly recall the esgn proceure of the DSC. Conser the system wth strct-feeback form as follows x = x f( x) x = x f( x, x, L, x), n M () xn = u fn( x, xl, xn) y = x n where x R, u R an y R are the system state, nput an output, respectvely. f s smooth functon satsfyng f () =. The objectve s to esgn a control law u to make the system output y track a gven sgnal x step by step. Receve * Supporte by Natonal Nature Scence Founaton of Chna (No. 67, 6)

2 7 Step : for x x f ( x ) = n Eq.(), efne the frst error surface as S = x x. The tme ervatve of S s S = x f x Select the vrtual control x = f x ks >. Then let the vrtual control x pass through a frst orer flter to ntrouce a new varable x, x x = x x() = x() where > s the tme constant. Step : n ths step, efne the secon error surface S = x x, then as S = x f x Choose the vrtual control x = f x k S >. Let x pass through the flter, x x = x x() = x() where > s the tme constant. Step : for x = x (,,, ) f x x L x t the th error surface S s efne as S = x x, then S = x f x the vrtual control s selecte as x = f x ks >. Introuce a new varable x ( ) by lettng x pass through the followng flter, x( ) x( ) = x x( ) () = x () Step n: n the last step, efne the error surface Sn = u fn xn then the real control law s obtane as follows u = fn xn knsn Journal of Electronc Scence an Technology of Chna Vol. The escrpton above s the etale esgn proceure of DSC technque. In the followng secton, we wll propose a scheme for synchronzaton objectve of Rössler system base on ths technque. Aaptve Synchronzaton Rössler System va DSC The famous Rössler system takes the form as follows, x= y z y = x ay () z = xz bz c where a, b an c are the unknown parameters. In our esgn, we rearrange the orer of the equatons. Let x = y, x = x, x = z, system Eq.() can be rewrtten nto x = x ax x = x x () x = xx bx c It s to see that the system of Eq.() s n the strct-feeback form. We take Eq.() as the master system an the slave system s y = y ay y = y y () y = yy by c u where y s the state of slave system, u s the control nput whch wll be esgne, the system parameters a, b an c are unknown but entcal wth those n Eq.(). Defne synchronzaton error e = y x, then we can obtan the error system from Eqs.() an (). e = e ae e = e e (5) e= be ee xe xe u Now, the synchronzaton problem has been change nto esgnng a control nput u to make the error

3 No. LIU Tong-shuan, et al: Aaptve Synchronzaton of Rössler System Base on Dynamc Surface Control 7 system of Eq.(5) stable at the orgn. Assste by DSC technque an aaptve metho, we wll synthesze the controller u step by step. Step : for the frst equaton n Eq.(5), e = e ae, efne the error surface as: S = e. The tme ervatve of S s S = e = e ae (6) Choose the vrtual control n ths step e = k S ae ˆ s a postve constant, â s the estmate of a whch satsfes the followng upate law ˆ ˆ a = λ t S e λσ a (7) where λ, σ are ajustable postve constants. Let e pass through a frst-orer flter to ntrouce a new varable: e e = e e() = e() where >, e s the flter output. Step : n ths step, efnng S = e e, we can be obtan S = e e e (8) Then the vrtual control s selecte as e = ks e e >. Introuce a new varable by lettng e pass through a frst-orer flter e e = e e() = e() where >, e s the flter output. Step : n the last step, the error surface s S = e e an the tme ervatve of t s S= be ee xe xe u e (9) Then the fnal control nput s conclue as u = ks be ˆ ee xe xe e () >, ˆb s the estmate of b an satsfes the upate law ˆ ˆ b = λ t S e λσ b () where λ, σ are ajustable postve constants. For Rössler system, for a an b there exst respectvely a scope n whch the system show chaotc behavor. So t s reasonable for us to have the followng nequaltes for a an b: a a b b max Base on the aforementone analyss, we can conclue the followng theorem for the synchronzaton problem. Theorem Gven any postve scalar γ, the followng nequalty = S ε a% b% γ hols for all ntal contons, there exst k,, λ, δ whch ensure the soluton of the error system s unformly ultmately boune n any small bouneness wth the esgne control law of Eq.() together wth the conton of Eq.() an the upate laws of Eqs.(7) an (), where ε = e e s the flter error, a% = a aˆ an b % = b bˆ are the parameter estmate errors. Proof In every esgn step, substtute the selecte vrtual control ntro S / an conser ε = e e, then we can rewrte Eqs(6), (8) an (9) as S = ks S ε ae % S = ks S ε () S = ks be % From e e = e we can fn e = ε. Takng tme ervatve of ε = e e an substtutng e = ε nto t, we obtan that max ε ε = k S aˆ e ˆ e a ε ε = k S e e ˆ () S t,e t,s t, e t an a all epen on the term S, ε, aˆ an b, so we enote w( S ˆ ˆ ˆ, ε, a, b) = k S a e e ˆ w ( S, ε, aˆ, b) = k S e e Snce M: = { = S = ε a% b% γ} s compact 7 set n space R, there exst maxmums η an η for w an w on M, respectvely.

4 xbb xbb xbb 7 Journal of Electronc Scence an Technology of Chna Vol. By choosng the Lyapunov functon canate as V = S a b% ε λ % λ = = the tme ervatve of V along Eqs(7), (), () an () s V = = S S = ε ε a% a% b% b% = S( ks S ε ae % ) S( ks be % ) S( ks S ε) ε( ε w) ε( ε w) λ a% a% λ b% b% Conserng the upate law an the fact we have ± xy x y V ( k ) S ( k ) S ( k ) S ( ) ε ( ) ε εw ε w σaa % ˆ σbb % ˆ For any postve scalar μ, we have ε w μ μ εw. Furthermore, nequaltes aa % ˆ a a% an bb % ˆ b b% always hol. Then V ( k ) S ( k ) S ( k ) S η η ( ) ε ( ) ε μ μ w ηε w ηε ( ) ( ) μ η μ η μ σ σ ( a a% ) ( b b % ) Choosng a postve scalar α satsfe the followng nequaltes k η μ k (5) η k μ we have σ σ V αv μ amax bmax Let α > ( μ σamax σbmax ) γ, then V / < when V = γ. So V γ s an nvarant set. Therefore We conclue that Vt ( ) s boune by ( μ σamax σbmax ) α. Furthermore, one can select k, an σ to make the boun any small. Therefore, all sgnals S, ε, a% an b% are unformly ultmately boune wth any small boun. Then the bouneness of e s obtane from e = S. e epens only on S an e, so the selecte vrtual control e s boune wth any small boun. Consequently, e, the output after e passng through a frst-orer flter s boune. From ths an S = e e, we can conclue the bouneness of e. Smlarly, the same result for e can be obtane. Ths completes the proof of the theorem. The theorem ensures that the synchronzaton errors convergent to an arbtrarly small value, so one can conclue that two chaotc systems wll be synchronze wth the esgne controller ae to the slave system. xbb:ybb/v xbb:ybb/v xbb:ybb/v B B 5 5 (a) Waveforms of xb an yb B B 5 5 (b)waveforms of xb an yb 5 :ybb :ybb :ybb B B 5 5 (c) Waveforms of xb an yb Fg. Waveforms of x an y

5 B B No. LIU Tong-shuan, et al: Aaptve Synchronzaton of Rössler System Base on Dynamc Surface Control 75 Numercal Smulaton In ths secton, a numercal example s gven below. The system of Eq.() s taken as the master system an the unknown system parameters are value as a = c=., b = 5.7. The entcal controlle system of Eq.() s taken as the slave system. The ntal contons of these two systems are x () = (,, ) an y () = (.5,.5,.5), respectvely. When the esgne controller s not ae to the slave system, the tme responses of the states of the two systems are splaye n Fg.. It s obvous that the two systems are not synchronze ebb (a) Synchronzaton error eb 8 6 k= k = k =, = =.5, λ = λ =, σ = When the esgne control law s ae to the slave system, the synchronzaton errors wll change nto zero. Therefore, the synchronzaton between the master an slave system s acheve. Fg. shows the convergence of the synchronzaton errors. In the smulaton, the spee of convergence of the synchronzaton errors can be ncrease by ncreasng the gan k, k an k. On the other han, the bg k wll make the states of the slave system oversze nstantaneously. Hence, the choce of the parameters k, k an k s a compromse of the two ses. Conclusons An aaptve synchronzaton scheme for Rössler system base on DSC technque has propose n ths paper. One can synthesze the fnal control law followng the propose esgn proceure to mplement synchronzaton objectve between two Rössler systems. Furthermore, snce a frst-orer flter s ntrouce n every esgn step, ths metho overcomes the rawback of exploson of complexty exstng n conventonal backsteppng methos. Consequently, the controller s much smpler compare wth that obtane from backsteppng methos. ebb ebb (b) Synchronzaton error eb 5 5 (c) Synchronzaton error eb B Fg. Synchronzaton errors e, e, e Then the control law, Eq.(), s esgne, n whch aˆ an b ˆ satsfy the upate laws n Eqs.(7) an (). The parameters of the controller s selecte as References [] Carroll L, Pecora L M. Synchronzng chaotc crcuts[j]. IEEE Trans Crc. Syst. I 99, 8(): [] L Z; Xu D. A secure communcaton scheme usng projectve chaos synchronzaton[j]. Chaos, Soltons an Fractals,, (): [] L S, Alvarez G, Chen G. Breakng a chaos-base secure communcaton scheme esgne by an mprove moulaton metho[j]. Chaos, Soltons an Fractals, 5, 5(): 9-. [] L Y, L X, Ouyang G, et al. Strength an Drecton of Phase Synchronzaton of Neural Networks[Z]. Internatonal Symposum on Neural Networks, Chongqng, Chna, [5] Hua C, Guan X. Synchronzaton of chaotc systems base on aaptve observer esgn[j]. Chnese Physcs,, (9): [6] Sun J, Zhang Y. Impulsve control of Rossler systems[j]. Physcs Letters A,, 6(5-6): 6-. [7] Hua C, Guan X, Sh P. Aaptve feeback control for a class of chaotc systems[j]. Chaos, Soltons an Fractals, 5, (): [8] Wang Y, Guan Z, Wang H O. LMI-base fuzzy stablty an synchronzaton of Chen s system[j]. Physcs Letters A,

6 T 76 Journal of Electronc Scence an Technology of Chna Vol., (-): Bref Introucton to Author(s) [9] Guan X.; Tang Y, Fan Z, et al. Neural network base robust LIU Tong-shuan ( 刘同栓 ) receve hs B.S. egree from aaptve synchronzaton of a chaotc system[j]. Acta Yanshan Unversty n. He s now an M.S. canate wth Physca Snca., 5(): -5. (n Chnese) the Insttute of Electrcal Engneerng, Yanshang Unversty. Hs [] Yu Y, Zhang S. Aaptve backsteppng synchronzaton of research nterests nclue ntellgent control an nonlnear control. uncertan chaotc system[j]. Chaos, Soltons an Fractals, XU Hao ( 许皓 ) receve hs B.S. egree from Yanshan, (): Unversty n. He s now an M.S. canate wth Yanshan [] Swaroop D, Geres J C, Yp P P, et al. Dynamc Surface Unversty. Hs research nterests nclue chaos control an Control of Nonlnear Systems[Z]. In Proceengs of the robust control. Amercan Control Conference, Albuquerque, New Mexco, GUAN Xn-png ( 关新平 ) was born n 96. He s 997. currently the ean of the Insttute of Electrcal Engneerng, [] Swaroop D, Herck J K, Yp P P, et al. Dynamc surface Yanshang Unversty. Hs research nterests nclue nonlnear control for a class of nonlnear systems[j]. IEEE Trans on control, robust control, network control an wreless network. Auto. Control,, 5(): He s presently a senor member of IEEE (Contnue from page 6) [6] Pang S, Huang J. SOPC Practce Technology[M]. Bejng: Publshng House of Tsnghua Unversty, 5. [7] Xu Y. USB Interface Program Desgn Base on 85 Sngle-Chp[M]. Bejng: Publshng House of Behuang Unversty,. [8] Art B, Lozano J. The Wnows Devce Drver Book[M]. Inanapols: Prentce Hall PTR,. [9] Walter O. Programmng the Wnows Drver Moel[M]. Remon: Mcrosoft Press, 999. Bref Introucton to Author(s) YAN Yong-mng ( 鄢永明 ) was born n Hunan, Chna, n 968. He grauate from Bejng Insttute of Image Processng n 99. As a grauate canate, from October to September, he was engage n the fel of new electrc evces an ts applcaton n Hunan Unversty..From October to November 5, he jone the Crcut an System Research Lab, Department of Electroncs, Pekng Unversty as a representatve of project cooperaton. ZENG Yun ( 曾云 )T was born n Hunan, Chna, n 956. He receve hs M.S. egree n Hunan Unversty n 988. As a faculty member of Department of Apple Physcs, Hunan Unversty, hs research actvtes have been concerne wth the new electrc evces an ts applcaton. He s currently a professor wth Hunan Unversty an a Senor Member of Chnese Insttute of Electroncs.