國立交通大學機械工程學系碩士論文 研究生 : 江振賓 指導教授 : 戈正銘教授 中華民國九十九年六月. 雙 Ge-Ku 系統,Sprott 4 系統及 Rossler 系統的 渾沌與渾沌同步. Chaos and Chaos Synchronization of Double Ge-Ku

Transcription

1 國立交通大學機械工程學系碩士論文 雙 Ge-Ku 系統,Sprott 4 系統及 Rossler 系統的 渾沌與渾沌同步 Chaos and Chaos Synchronzaton of Double Ge-Ku System, Sprott 4 System and Rossler System 研究生 : 江振賓 指導教授 : 戈正銘教授 中華民國九十九年六月

2 雙 Ge-Ku 系統,Sprott 4 系統及 Rossler 系統的 渾沌與渾沌同步 Chaos and Chaos Synchronzaton of Double Ge-Ku System,Sprott 4 Systems and Rossler System 研究生 : 江振賓 指導教授 : 戈正銘 Student: Chen- Bn Chang Advsor: Zheng-Mng Ge 國立交通大學機械工程學系碩士論文 A Thess Submtted to Department of Mechancal Engneerng College of Engneerng Natonal Chao Tung Unversty In Partal Fulfllment of the Requrement For the Degree of master of scence In Mechancal Engneerng June 00 Hsnchu, Tawan, Republc of Chna 中華民國九十九年六月

3 雙 Ge-Ku 系統, Sprott 4 系統及 Rossler 系統的渾沌與渾沌同步 學生 : 江振賓 指導教授 : 戈正銘 國立交通大學 機械工程學系 摘要 本篇論文以相圖 龐卡萊映射圖 李亞普洛夫指數以及分歧圖等數值方法研究新 Double Ge-Ku 系統的渾沌現象 應用主動控制獲得雙重及多重渾沌交織同步 對此系統應用部分區域穩定性理論和實用漸進穩定理論來達成廣義同步 更進一步使用新模糊模型來研究 Sprott 4 系統以及 Rossler 系統的模糊模型和渾沌同步 此外, 將探討新模糊邏輯常數控制器應用在投影同步及含有不確定度的渾沌系統 在以上研究中, 皆可由相圖和時間歷程圖得到驗證

4 Chaos and Chaos Synchronzaton of Double Ge-Ku System, Sprott 4 System and Rossler System Student:Chen- Bn Chang Advsor:Zheng-Mng Ge Department of Mechancal Engneerng, Natonal Chao Tung Unversty Abstract In ths thess, the chaotc behavor of new Double Ge-Ku system s studed by phase portrats, tme hstores, Poncaré maps, Lyapunov exponents and bfurcaton dagrams. New type for chaotc synchronzaton, double and multple symplectc synchronzaton, are obtaned by actve control. A new knd of chaotc generalzed synchronzaton, dfferent translaton pragmatcal generalzed synchronzaton, s obtaned by pragmatcal asymptotcal stablty theorem and partal regon stablty theory. A new method, usng new fuzzy model, s studed for fuzzy modelng and synchronzaton of Sprott 4 system and Rossler system. Moreover, the new fuzzy logc constant controller s studed for projectve synchronzaton and chaotc system wth uncertanty. Numercal analyses, such as phase portrats and tme hstores are provded to verfy the effectveness of all above studes.

5 誌謝 此篇論文及碩士學業之完成, 首先必須感謝指導教授 戈正銘老師的 耐心指導與教誨 老師在專業領域上的成就以及對於文學和史學上的熱情, 都令學生印象深刻且受益匪淺 戈老師在學生的碩士生涯中扮演了很重要的角色 在研究上, 戈老師給了許多意見指引學生正確的研究方向, 在研究過程中, 也讓學生學習到當發現問題時, 解決問題的能力 在研究所的後段時間, 小宇學長更是不辭辛勞教導我們研究新的題目, 並且與老師共同傳授做研究的技巧, 著實讓我上了一課 經由這兩年的相處, 讓學生發現在學術研究之餘也能體會到古典文學之美, 這都使學生開拓了不一樣的視野 在這段研究的日子當中, 承蒙張晉銘 李仕宇 徐瑜韓 張育銘 陳志銘 陳聰文學長的熱心指導, 同時也感謝泳厚 尚恩 翔平同學的相互勉勵及幫忙, 使得本篇論文能夠順利完成 最後感謝我的家人, 讓我可以不必擔心課業以外的事物, 無後顧之憂的完成學業 最後, 僅以此論文獻給你們

6 CONTENTS CHINESE ABSTRACT.. ABSTRACT Acknowledgment.. CONTENTS..v LIST OF FIGURES...v Chapter Introducton Chapter Double Symplectc Synchronzaton for Double Ge-Ku System. Prelmnary.6. Double Symplectc Synchronzaton Scheme 6. Synchronzaton of Two Dfferent New Chaotc Systems...8 Chapter Multple Symplectc Synchronzaton for Double Ge-Ku System. Prelmnary.... Multple Symplectc Synchronzaton Scheme.... Synchronzaton of Three Dfferent Chaotc Systems... Chapter 4 Usng Partal Regon Stablty Theory for Dfferent Translaton Pragmatcal Generalzed Synchronzaton 4. Prelmnary The Scheme of Dfferent Translaton Pragmatc Generalzed by Partal Regon Theory 7 4. Dfferent Translaton Pragmatcal Synchronzaton of new DoubleGe-Ku Chaotc System 40 Chapter 5 Robust Projectve Ant-Synchronzaton of Non-autonomous and Chaotc Uncertan Stochastc Systems Va Fuzzy Logc Constant Controller 5. Prelmnary Projectve Ant-Synchronzaton by FLCC Scheme Smulaton Results 6 Chapter 6 Fuzzy Modelng and Synchronzaton of Chaotc Systems by a New Fuzzy Model 6. Prelmnary...79 v

7 6. New Fuzzy Model Theory New Fuzzy Model of Chaotc Systems Fuzzy Synchronzaton Scheme Smulaton Results 89 Chapter 7 Conclusons...98 Appendx A GYC Partal Regon Stablty Theory...00 Appendx B Pragmatcal Asymptotcal Stablty Theory.08 References v

8 LIST OF FIGURES Fg.. The chaotc attractor of a new Ge-Ku-Duffng system..4 Fg.. The chaotc attractor of uncontrolled new Double Ge-Ku system.4 Fg.. The bfurcaton dagram of uncontrolled new Double Ge-Ku system 5 Fg..4.The Lyapunov exponents of uncontrolled new Double Ge-Ku system..5 Fg..5 The phase portrat of the controlled new Double Ge-Ku system for CASE. 6 Fg..6 Tme hstores of the state errors for CASE.6 Fg..7 Tme hstores of x y and x y for CASE.7 Fg..8 The chaotc attractor of a new Ge-Ku-van der Pol system.7 Fg..9 The phase portrat of the controlled new Double Ge-Ku system for CASE 8 Fg..0 Tme hstores of the state errors for CASE 8 Fg.. Tme hstores of x y and x y for CASE...9 Fg.. The chaotc attractor of a new Ge-Ku Matheu system 9 Fg.. The phase portrat of the controlled new Double Ge-Ku system for CASE...0 Fg..4 Tme hstores of the state errors for CASE 0 Fg..5 Tme hstores of x y and x y for CASE Fg.. The chaotc attractor of the Lorenz system.0 Fg.. The chaotc attractor of the Chen system...0 Fg.. The phase portrat of the controlled DGK system Fg..4 Tme hstores of the state errors for Case Fg..5 Tme hstores of G( xy,, z, t) and F( x, y, zt, ) for Case v

9 Fg..6 The chaotc attractor of the Rossler system Fg..7 The phase portrat of the controlled DGK system... Fg..8 Tme hstores of the state errors for Case... Fg..9 Tme hstores of G( xy,, z, t) and F( x, y, zt, ) for Case..4 Fg..0 The chaotc attractor of a Sprott system 4 Fg.. The phase portrat of the controlled DGK system...5 Fg.. Tme hstores of the state errors for Case 5 Fg.. Tme hstores of G( xy,, z, t) and F( x, y, zt, ) for Case 6 Fg. 4. Coordnate translaton of state x 49 Fg. 4. Coordnate translaton of state y.49 Fg. 4. Phase portrat of the error dynamcs for Case 50 Fg. 4.4 Tme hstores of errors for Case..50 Fg. 4.5 Tme hstores of x, y for Case..5 Fg. 4.6 Tme hstores of parameter errors for Case.5 Fg. 4.7 Phase portrat of the error dynamc for Case 5 Fg. 4.8 Tme hstores of errors for Case..5 Fg. 4.9 Tme hstores of x, y for Case 5 Fg. 4.0 Tme hstores of parameter errors for Case 5 Fg. 4. Phase portrat of the error dynamc for Case 54 Fg. 4. The chaotc attractor of the Ge-Ku-van der Pol system...54 Fg. 4. Tme hstores of errors for Case 55 Fg. 4.4 Tme hstores of x, y for Case 55 Fg. 4.5 Tme hstores of parameter errors for Case...56 Fg. 5. The confguraton of fuzzy logc controller 70 Fg. 5. Membershp functon..70 Fg. 5. The chaotc behavor of Sprott 4 system [50].7 v

10 Fg. 5.4 Projectons of phase portrat of Eq (5.9)...7 Fg. 5.5 Tme hstores of error dervatves for master and slave chaotc systems wthout controllers 7 Fg. 5.6 Tme hstores of states for Example the FLCC s comng nto after 0s 7 Fg. 5.7 Tme hstores of errors for Example the FLCC s comng nto after 0s 7 Fg. 5.8 Tme hstores of states for the tradtonal controller s comng nto after 0s 7 Fg. 5.9 Tme hstores of states for the tradtonal controller s comng nto after 0s 74 Fg. 5.0 The Raylegh nose used...74 Fg. 5. Projecton of phase portrat of Eq (5.8)..75 Fg. 5. Projectons of phase portrat of chaotc DGK systems.75 Fg. 5. Tme hstores of error dervatves for master and slave chaotc uncertan stochastc systems wthout controllers.76 Fg. 5.4 Tme hstores of states for Example the FLCC s comng nto after 0s 76 Fg. 5.5 Tme hstores of errors for Example the FLCC s comng nto after 0s 77 Fg. 5.6 Tme hstores of states for the tradtonal controller s comng nto after 0s 77 Fg. 5.7 Tme hstores of states for the tradtonal controller s comng nto after 0s 78 Fg. 6. Chaotc behavor of Sprott 4 system...9 Fg. 6. Chaotc behavor of Sprott 4 system wth uncertanty 9 v

11 Fg. 6. Chaotc behavor of new fuzzy Sprott 4 system wth uncertanty..94 Fg. 6.4 Chaotc behavor of Rossler system 94 Fg. 6.5 Chaotc behavor of Rossler system wth uncertanty.95 Fg. 6.6 The Raylegh nose used 95 Fg. 6.7 Chaotc behavor of new fuzzy Rossler system wth uncertanty 96 Fg. 6.8 Tme hstores of errors for Example 96 Fg. 6.9 Tme hstores of errors for Example...97 x

12 Chapter Introducton Chaos s a very nterestng nonlnear phenomenon. Chaos as a modern theory s dscovered by the age of computer n the mddle of the past century, whch has an epoch-makng sgnfcance n exact scence. Generally speakng, desgnng a system to mmc the behavor of another chaotc system s called synchronzaton. Synchronzaton of chaotc systems has receved a sgnfcant attenton, snce Pecora and Carroll presented the chaos synchronzaton method to synchronze two dentcal chaotc systems wth dfferent ntal values n 990 []. In recent years, synchronzaton n chaotc dynamc system s a very nterestng problem and has been wdely studed. Chaos synchronzaton has been appled n bologcal systems [,], secure communcaton [4,5], and many other dscplnes. By lnear and nonlnear control [6-] dfferent types of synchronzaton for nteractng chaotc systems, such as complete synchronzaton [], phase synchronzaton [], lag synchronzaton [4], generalzed synchronzaton [5-0], antcpatng synchronzaton can be obtaned. Varous approaches for achevng chaos synchronzaton usng fuzzy systems have been proposed []. Fuzzy set theory was frst presented by Zadeh [], whle fuzzy logc control (FLC) schemes have been wdely developed and successfully deployed n many applcatons []. Furthermore, adaptve fuzzy controllers have been used to control and synchronze chaotc systems [4,5]. Uncertantes assocated wth the mathematcal characterzaton of a system can lead to unrelable damage [6]. Probablstc analyss provdes a tool for ncorporatng randoms n the analyss of frequency response by generally descrbng the randoms as

13 randoms varables. Whle the studes dscussed above addressed measurement nose, some researchers have started lookng at the effect of nose present n the model. Collns et al. [7] proposed a statstcal dentfcaton procedure by treatng the ntal parameters as normally dstrbuted random varables. In recent years, some chaos synchronzatons based on fuzzy systems have been proposed snce the fuzzy set theory was ntated by Zadeh [], such as fuzzy sldng mode controllng technque [8,9], LMI-based synchronzaton [0] and extended backsteppng sldng mode controllng technque []. The fuzzy logc control (FLC) scheme have been wdely developed and have been successfully appled to many applcatons []. Recently, Yau and Sheh [] proposed a new dea n desgnng fuzzy logc controllers - constructng fuzzy rules subject to a common Lyapunov functon such that the master-slave chaotc systems satsfy stablty condton n the Lyapunov sense. In [], there are two man controllers n ther slave system. One s used n elmnaton of nonlnear terms and the other s bult by fuzzy rules subject to a common Lyapunov functon. Therefore, the resultng controllers are n nonlnear form. In [], the regular form s necessary. In order to carry out the new method, the orgnal system must to be transformed nto ther regular form. In recent years, fuzzy logc proposed [4] has receved much attenton as a powerful tool for the nonlnear control. Among varous knds of fuzzy methods, Takag-Sugeno fuzzy (T-S fuzzy) system s wdely accepted as a useful tool for desgn and analyss of fuzzy control system [5-40]. Currently, some chaos control and synchronzaton based on T-S fuzzy systems have been proposed, such as fuzzy sldng mode controllng technque, LMI-based synchronzaton [4-4] and robust control [44]. Furthermore, two dfferent nonlnear systems may have dfferent numbers of nonlnear terms. It causes dfferent numbers of lnear subsystems. For synchronzaton of two dfferent nonlnear systems, the tradtonal method usng the

14 dea of PDC to desgn the fuzzy control law for stablzaton of the error dynamcs can not be used here, snce the number of subsystems becomes very large. Ge and Ku [45] gave a chaotc system formed by a smple pendulum wth ts pvot rotatng about an axs as Fg... Ths chaotc system s x x x ax sn x[ b( c cos x ) d sn wt] where a, b, c, d are parameters. After smplfcaton, sn x x, addton of couplng terms, we get the Double Ge-Ku system x x x fx x g h x kx x fx x g h x lx (.) x cos x and (.) where a, b, c, d, g are parameters. In Chapter, a new type of synchronzaton, double symplectc synchronzaton, G( x, y, t) F( x, y, t) s studed. Tradtonal generalzed synchronzaton and symplectc synchronzaton are specal cases of the double symplectc synchronzaton. Snce the symplectc functons are presented at both the rght hand sde and the left hand sde of the equalty. The double symplectc synchronzaton may be appled to ncrease the securty of secret communcaton due to the complexty of ts synchronzaton form. By usng the Barbalat s lemma[46], the double symplectc synchronzaton can be acheved. Fnally, the effectveness and feasblty of our proposed scheme are verfed by numercal smulatons. In Chapter, a new type of synchronzaton, multple symplectc synchronzaton s studed. When the double symplectc functons are extended to a more general form, G(x,y,z w,t) = F(x,y,z w,t), multple symplectc synchronzaton s acheved. The multple symplectc synchronzaton may be appled to ncrease the securty of

15 secret communcaton more effectvely than double symplectc synchronzaton due to more complexty of ts synchronzaton form. In Chapter 4, a new chaos generalzed synchronzaton strategy of dfferent translaton pragmatcal synchronzaton by stablty theory of partal regon [47-49] s proposed. By usng the stablty theory of partal regon, the Lyapunov functon s a smple lnear homogeneous functon of error states, the controllers are more smple snce they are n lower degree than that of tradtonal controllers, whle the tradtonal Lyapunov functon s a quadratc form of error states. In Chapter 5, we propose a new strategy whch s also constructng fuzzy rules subject to a Lyapunov drect method. Error dervatves are used to be upper bound and lower bound. Through ths new approach, a smplest controller,.e. constant controller, can be obtaned and the dffculty n realzaton of complcated controllers n chaos synchronzaton by Lyapunov drect method can be also overcome. Unlke conventonal approaches, the resultng control law has less maxmum magntude of the nstantaneous control command and t can reduce the actuator saturaton phenomenon n real physcal system. Some computer smulaton examples are gven n ths Chapter. In Chapter 6, the new fuzzy model s proposed. It gves a new way to lnearze complcated nonlnear system and only two subsystems are concluded. In smulaton examples, Sprott 4 system [50] and Rossler system are used. Fnally, conclusons are presented n Chapter 7. 4

16 Fg. The pendulum on rotatng arm. 5

17 Chapter Double Symplectc Synchronzaton for Double Ge-Ku System. Prelmnary In ths Chapter, a new type of synchronzaton, double symplectc synchronzaton, G( x, y, t) F( x, y, t) s proposed. Double symplectc synchronzaton s an extenson of symplectc synchronzaton, y F( x, y, t). Because the symplectc functons are presented at both the rght hand sde and the left hand sde of the equalty, t s called double symplectc synchronzaton. Snce the synchronzaton form s more complex than generalzed and symplectc synchronzatons, so we usually use t on the purpose of secret communcaton. The scheme s effectve as shown by three examples as follows.. Double Symplectc Synchronzaton Scheme Consder two dfferent nonlnear chaotc systems, Partner A and Partner B, descrbed by x f( x, t), (.) y C( t) y + g( u, t) u, (.) where x [,,, ] T n x x xn R and y [,,, ] are the state vectors of T n y y yn R Partner A (.) and Partner B (.), n n C R s the gven matrx, f and g are contnuous nonlnear vector functons, and u s the controller. Our goal s to desgn the controller u such that G( x, y, t) asymptotcally approaches F( x, y, t).for smplcty take G( x, y, t) x y and F( x, y, t) s a contnuous nonlnear vector functon. 6

18 Property [5]: An m n matrx A of real elements defnes a lnear mappng y Ax from n R nto m R, and the nduced p-norm of A for p,, and s gven by m n T max j, max ( ), max j. j (.) j A a A A A A a The useful property of nduced matrx norms for real matrx A s as follow: A A A. (.4) Theorem: For chaotc systems Partner A (.) and Partner B (.), f the controller u s desgned as u I D F D Ff x D F C y g y D F f x g y ( y ) [ x (, t) y ( ( t) (, t)) t (, t) (, t) C( t)( x F) K( x y F)], (.5) where DF x, DF y, DF t are the Jacoban matrces of F( x, y, t), K dag(,,, ), and satsfes k k k m mn( k ), (.6) C() t then the double symplectc synchronzaton wll be acheved. Proof: Defne the error vectors as e x y F( x, y, t), (.7) then the followng error dynamcs can be obtaned by ntroducng the desgned controller de e x y DxFx DyFy DtF dt f ( x, t) C( t) y g( y, t) D Ff ( x, t) D F( C( t) y g( y, t)) D F ( I D F) u ( C( t) K) e. y x y t (.8) Choose a non-negatve Lyapunov functon of the form 7

19 T Vt () ee. (.9) Takng the tme dervatve of Vt () along the trajectory of Eq. (.8), we have Vt () T ee T T e C() t e e Ke C( t) e mn( k ) e ( C( t) mn( k )) e. (.0) Snce M mn( k ) C ( t) 0, then V ( t) M e MV ( t). Therefore, t can be obtaned that V t ( ) V (0)e Mt (.) and t lm V( ) d t s bounded. Besdes, () 0 Vt s unformly contnuous. Accordng to Barbalat s lemma [46], the concluson can be drawn that lm Vt ( ) 0,.e. t lm e ( t) 0 t. Thus, the double symplectc synchronzaton can be acheved asymptotcally.. Synchronzaton of Two Dfferent New Chaotc Systems CASE Consder a new Ge-Ku-Duffng(GKD) system as Partner A descrbed by x x x hx x l p x kx x x x fx gx (.) where h=0.; l=; p=40; k=54; f=6; g=0; and the ntal condtons are x (0), x (0).4, x (0) 5. Eq. (.) can be rewrtten n the form of Eq. (.), 8

20 x t hx x l p x kx x x fx gx where fx (, ). The chaotc attractor of the new GKD system s shown n Fg... The controlled new Double Ge-Ku (DGK) system s consdered as Partner B descrbed by y y u y ay y bc y dy u y ay y bc y ey u where a=-0.5; b=-.4; c=.9; d=-4.5; e=6.;, u, u, u T (.) u s the controller, and the ntal condton s y (0), y (0).4, y (0) 5. The chaotc attractor of uncontrolled new DGK system s shown n Fg.., Bfurcaton dagram n Fg.. and Lyapunov exponents n Fg..4. Eq. (.) can be rewrtten n the form of Eq. (.), where 0 0 C ( t) bc a 0 and 0 0 a bc 0 gy (, t) by dyy. By applyng Property, t s derved that C() t a bc by ey y C () t a bc C ( t) ( a bc) Then C ( t). s estmated. Defne x y F( x, y, t) x y, and our goal s to acheve the double symplectc x y synchronzaton x y F( x, y, t). Accordng to Theorem, the nequalty mn( ) C() t must be satsfed. It can be obtaned that mn( k ). f we choose 9 k

21 k K 0 k and desgn the controller as 0 0 k u x y x x y x y x y x y u x y { hx x [ l( p x ) kx ]} x { ay y [ b( c y ) dy ]} hx x [ l( p x ) kx ] ay y [ b( c y ) dy ] x y x y u x y ( x x fx gx ) x { ay y [ b( c y ) ey ]} x x fx gx ay y [ b( c y ) ey ] x y x y The Theorem s satsfed and the double symplectc synchronzaton s acheved. The phase portrat of the controlled DGK system and the tme hstores of the state errors are shown n Fg..5 and Fg..6 and Fg..7, respectvely. CASE Consder a new Ge-Ku-van der Pol(GKv) system as Partner A descrbed by x x x fx x p k x mx x gx h x x lx (.4) where f=0.08; p=-0.5; k=00.56; m=-000.0; g=0.6; h=0.08; l=0.0; and the ntal condtons are x (0) 0.0, x(0) 0.0, x(0) 0.0. Eq. (.4) can be rewrtten n the form of Eq. (.), where x fx (, t) fx x[ p( k x ) mx. gx h( x ) x lx The chaotc attractor of the GKv system s shown n Fg..8. The controlled DGK system s consdered as Partner B descrbed by y y u y ay y bc y dy u y ay y bc y ey u 0 (.5)

22 where a=-0.5; b=-.4; c=.9; d=-4.5; e=6.;, u, u, u T u s the vector controller, and the ntal condtons are y (0) 0.0, y (0) 0.0, y (0) 0.0. Eq. (.5) can be rewrtten n the form of Eq. (.), where 0 0 C ( t) bc a 0 0 l g and 0 gy (, t) by dyy by ey y. By applyng Property, t s derved that C ( t) a bc, C () t a bc, and C ( t) ( a bc) Then C ( t). s estmated. Defne x y F( x, y, t) x y, and our goal s to acheve the double symplectc x y synchronzaton x y F( x, y, t). Accordng to Theorem, the nequalty mn( k ) C() t must be satsfed. It can be obtaned that mn( k ).. Thua we k choose K 0 k and desgn the controller as 0 0 k u x y x x y x y x y x y u x y { fx x [ p( k x ) mx ]} x { ay y [ b( c y ) dy ]} fx x [ p( k x ) mx ] ay y [ b( c y ) dy ] x y x y u x y [ gx h( x ) x lx ] x { ay y [ b( c y ) ey ]} gx h( x ) x lx ay y [ b( c y ) ey ] x y x y The Theorem s satsfed and the double symplectc synchronzaton s acheved.

23 The phase portrat of the controlled DGK system and the tme hstores of the errors are shown n Fg..9, Fg..0 and Fg.., respectvely. CASE Consder the Ge-Ku Matheu(GKM) system as Partner A descrbed by x x x kx x f m x nxx x ( g hx ) x lx pxx (.6) where k=-0.6; f=5; m=; n=0.; g=8; h=0; l=0.5; p=0.; and the ntal condtons are x (0) 0.0, x(0) 0.0, x(0) 0.0 Eq. (.6) can be rewrtten n x t kx x f m x nx x ( g hx ) x lx px x the form of Eq. (.), where fx (, ) attractor of the GKM system s shown n Fg... The controlled DGK system s consdered as Partner B descrbed by.the chaotc y y u y ay y bc y dy u y ay y bc y ey u (.7) where a=-0.5; b=-.4; c=.9; d=-4.5; e=6.;, u, u, u T u s the vector controller, and the ntal condtons are y (0) 0.0, y (0) 0.0, y (0) 0.0. Eq. (.7) can be rewrtten n the form of Eq. (.), where 0 0 C ( t) bc a 0 0 l g and 0 gy (, t) by dyy by ey y. By applyng Property, t s derved that

24 C ( t) a bc, C () t a bc, and C ( t) ( a bc) Then C( t). s estmated. Defne x y F( x, y, t) x y, and our goal s to acheve the x y double symplectc synchronzaton x y F( x, y, t). Accordng to Theorem, the nequalty mn( k ) C() t must be satsfed. It can be obtaned that mn( k ).. k Thus we choose K 0 k and desgn the controller as 0 0 k u x y x x y x y x y x y u x y { kx x [ f ( m x ) nx x ]} x { ay y [ b( c y ) dy ]} kx x [ f ( m x ) nx x ] ay y [ b( c y ) dy ] x y x y u x y [ ( g hx ) x lx px x ] x { ay y [ b( c y ) ey ]} ( g hx ) x lx px x ay y [ b( c y ) ey ] x y x y The Theorem s satsfed and the double symplectc synchronzaton s acheved. The phase portrat of the controlled DGK system and the tme hstores of the state errors are shown n Fg., Fg..4 and Fg..5, respectvely.

25 Fg.. The chaotc attractor of a new Ge-Ku-Duffng system. Fg.. The chaotc attractor of uncontrolled new Double Ge-Ku system. 4

26 b Fg.. The bfurcaton dagram of uncontrolled new Double Ge-Ku system. b Fg..4.The Lyapunov exponents of uncontrolled new Double Ge-Ku system. 5

27 Fg..5 The phase portrat of the controlled new Double Ge-Ku system for CASE. Fg..6 Tme hstores of x y and x y for CASE. 6

28 Fg..7 Tme hstores of the state errors for CASE. Fg..8 The chaotc attractor of a new Ge-Ku-van der Pol system. 7

29 Fg..9 The phase portrat of the controlled new Double Ge-Ku system for CASE. Fg..0 Tme hstores of x y and x y for CASE. 8

30 Fg.. Tme hstores of the state errors for CASE. Fg.. The chaotc attractor of a new Ge-Ku Matheu system. 9

31 Fg.. The phase portrat of the controlled new Double Ge-Ku system for CASE. Fg..4 Tme hstores of x y and x y for CASE. 0

32 Fg..5 Tme hstores of the state errors for CASE.

33 Chapter Multple Symplectc Synchronzaton for Double Ge-Ku System. Prelmnary In ths Chapter, a new type of synchronzaton, multple symplectc synchronzaton s studed. Symplectc synchronzaton and double symplectc synchronzaton are specal cases of the multple symplectc synchronzaton. When the double symplectc functons s extended to a more general form, G(x,y,z w,t) = F(x,y,z w,t), t s called multple symplectc synchronzaton. The multple symplectc synchronzaton may be appled to ncrease the securty of secret communcaton due to the complexty of ts synchronzaton form.. Multple Symplectc Synchronzaton Scheme Chaos synchronzaton, frst proposed by Fujsaka and Yamada n 98, dd not receved great attenton untl 990. Among many knds of synchronzatons, the generalzed synchronzaton s nvestgated. It means there exst a functonal relatonshp between the states of the master and those of the slave. Symplectc synchronzaton s defned as y H( x, y, t), where x, y are the state vectors of the master and of the slave, respectvely. The fnal desred state y of the slave not only depends upon the master state x but also depends upon the slave state y tself. Therefore the slave s not a tradtonal pure slave obeyng the master completely but plays a role to determne the fnal desred state of the slave system. We call ths knd of synchronzaton, symplectc synchronzaton, and call the master system Partner A, the slave system Partner B. Snce the symplectc functons are presented at both the rght hand sde and the

34 left hand sde of the equalty, t s called double symplectc synchronzaton, G( x, y, t) F( x, y, t), where x, y are state vectors of Partner A and Partner B, respectvely, G( x, y, t) and F( x, y, t) are gven vector functons of x, y and tme. When the double symplectc functons s extended to a more general form, G(x,y,z w,t) = F(x,y,z w,t), s called multple symplectc synchronzaton, where xy,, z,, ware state vectors of Partner A and Partner B, respectvely, G(x,y,z w,t)and F(x,y,z w,t)are gven vector functons of xy,, z,, w and tme.. Synchronzaton of Three Dfferent Chaotc Systems CASE. Defne x y z G( x, y, z, t) x y z, x y z sn sn sn sn sn sn x y z x y z x y z F( x, y, z, t) x sn y z x sn y z x sn y z, x y z x y z x y z and we wsh to acheve the multple symplectc synchronzaton G(x,y,z,t) = F(x,y,z,t). Let e = G(x,y,z,t) - F(x,y,z,t) Consder a Lorenz system s descrbed by x w x w x x w x x x x x w x xx (.) 8 where w 0, w, w 8, and the ntal condton s x (0) 0.0, x (0) 0.0, x (0) 0.0. The chaotc attractor of the Lorenz system s shown n Fg... The Chen system s descrbed by z h ( z z) z h h z z z h z z zz h z

35 (.) where h 5, h 7., h, and the ntal condton s z (0) 0.5, z (0) 0.6,z (0) 0.5. The chaotc attractor of the Chen system s shown n Fg... The controlled Double Ge-Ku (DGK) system s descrbed by y y u y ay y bc y dy u y ay y bc y ey u (.) where a=-0.5; b=-.4; c=.9; d=-4.5; e=6., s the controller parameters, and the ntal condton s y (0) 0.0, y (0) 0.0,y (0) 0.0,. Thus we desgn the controller as sn cos sn sn x ay y bc y dy cos y z x sn y h z z sn cos x sn y h z z u w x w x y h z z w x w x y z x y y z x y h z z w x x x x y z w x x x y z x ay y b c y ey y z u w x x x x ay y b c y dy sn cos sn sn cos sn sn cos sn h h z z z h z w x w x y z x y y z x y h h z z z h z w x x x x y z x ay y b c y dy y z x y h h z zz hz w x x x y z x ay y b c y ey y z x y h h z zz h z 4

36 u w x xx ay y b c y ey zz h z w x w x sn y z x y cos y z x sn y zz h z w x x xx sn y z x ay y bc y dy cos y z x sn y zz hz w x xx sn y z x ay y bc y ey cos y x sn y z z h z The Theorem n Chapter s satsfed and the multple symplectc synchronzaton s acheved. The phase portrat of the controlled DGK system and the tme hstores of the state errors and the tme hstores of G( xy,, z, t) and F( x, y, zt, ) are shown n Fg.. and Fg..4 and Fg..5, respectvely. z CASE. Defne x y z cos x yz cos x yz cos x yz G( x, y, z, t) x y z, F( x, y, z, t), cos x yz cos x yz cos x yz x y z cos x yz cos x yz cos x yz and we wsh to acheve the multple symplectc synchronzaton G(x,y,z,t) = F(x,y,z,t). Let e = G(x,y,z,t) - F(x,y,z,t) Consder a Rossler system s descrbed by x x x x x w x x w x x w x (.4) where w 0.5, w 0., w 0, and the ntal condton s x (0), x (0).4, x (0) 5,. The chaotc attractor of the Rossler system s shown n Fg..6. 5

37 The Chen system s descrbed z h ( z z) z h h z z z h z z zz h z (.5) where h 5, h 7., h, and the ntal condton s z (0) 0.5, z (0) 0.6,z (0) 0.5. The controlled DGK system s descrbed by (.6) where a=-0.5; b=-.4; c=.9; d=-4.5; e=6., s the controller parameters, and the ntal condton s y (0) 0.0, y (0) 0.0,y (0) 0.0,. Thus we desgn the controller as sn cos cos sn u x x y h z z x x x y z x y z x y h z z x w x x y z cos sn cos cos x ay y b c y dy z x y h z z w x x w x x y z x ay y b c y ey z y y u y ay y bc y dy u y ay y bc y ey u cos x y h z z u x w x ay y b c y dy h h z zz h z sn cos cos x x x yz x yz x y h h z zz h z sn cos x w x x y z x ay y b c y dy z cos x y h h z zz h z w xx w x sn x cos cos y z x ay y b c y ey z x y h h z zz hz 6

38 u w x x w x ay y b c y ey z z h z x x sn x y z cos x y z cos x y z z h z sn cos x w x x y z x ay y b c y dy z cos sn x y z z h z w x x w x x y z cos x ay y b c y ey z cos x y z z h z The Theorem n Chapter s satsfed and the multple symplectc synchronzaton s acheved. The phase portrat of the controlled DGK system and the tme hstores of the state errors and the tme hstores of G( xy,, z, t) and F( x, y, zt, ) are shown n Fg..7 and Fg..8 and Fg..9, respectvely. CASE. Defne x y z x yz x yz x yz G( x, y, z, t) x y z, F( x, y, z, t) x yz x yz x yz, x y z x yz x yz x yz and we wsh to acheve the multple symplectc synchronzaton. G(x,y,z,t) = F(x,y,z,t). Let e = G(x,y,z,t) - F(x,y,z,t) Consder a Sprott E system s descrbed by x xx x x x x w x (.7) where w 4, and the ntal condton s x (0), x(0), x(0),. The chaotc attractor of the Sprott E system s shown n Fg..0. The Chen system s descrbed 7

39 z h ( z z) z h h z z z h z z zz h z (.8) where h 5, h 7., h, and the ntal condton s z (0) 0.5, z (0) 0.6, z (0) 0.5. The controlled DGK system s descrbed by y y u y ay y bc y dy u y ay y bc y ey u (.9) where a=-0.5; b=-.4; c=.9; d=-4.5; e=6., are the controller parameters, and the ntal condton s y (0) 0.0, y (0) 0.0,y (0) 0.0,. Thus we desgn the controller as u xx y h z z xx yz x yz x y h z z x x y z x ay y b c y dy z x y h z z w x y z x ay y b c y ey z x y h z z w x yz x ay y bc y ey z x y u x x ay y b c y dy h h z zz hz x x y z x y z x y h h z z z h z x x y z x ay y b c y dy z x y h h z zz hz u w x ay y b c y ey z z h z x x y z x y z x y z z h z x x y z h h z z z h z w x yz x ay y b c y ey z x y zz h z x ay y b c y dy z x y z z h z The Theorem n Chapter s satsfed and the multple symplectc synchronzaton 8

40 s acheved. The phase portrat of the controlled DGK system and the tme hstores of the state errors and the tme hstores of G( xy,, z, t) and F( x, y, zt, ) are shown n Fg.. and Fg.. and Fg.., respectvely. 9

41 Fg.. The chaotc attractor of the Lorenz system. Fg.. The chaotc attractor of the Chen system. 0

42 Fg.. The phase portrat of the controlled DGK system. Fg..4 Tme hstores of G( xy,, z, t) and F( x, y, zt, ) for Case.

43 Fg..5 Tme hstores of the errors for Case. Fg..6 The chaotc attractor of the Rossler system.

44 Fg..7 The phase portrat of the controlled DGK system. Fg..8 Tme hstores of G( xy,, z, t) and F( x, y, zt, ) for Case.

45 Fg..9 Tme hstores of the errors for Case. Fg..0 The chaotc attractor of a Sprott E system. 4

46 Fg.. The phase portrat of the controlled DGK system. Fg.. Tme hstores of G( xy,, z, t) and F( x, y, zt, ) for Case. 5

47 Fg.. Tme hstores of the errors for Case. 6

48 Chapter 4 Dfferent Translaton Pragmatcal Generalzed Synchronzaton by Stablty Theory of Partal Regon for Double Ge-Ku System 4. Prelmnary In ths Chapter, a new strategy to acheve dfferent translaton generalzed synchronzaton by stablty theory of partal regon by whch the Lyapunov functon s smple lnear homogeneous functon of error states, the controllers are more smple snce they are n lower degree than that of tradtonal controllers. By a pragmatcal theorem of asymptotcal stablty based on an assumpton of equal probablty of ntal pont, an adaptve control law s derved such that t can be proved strctly that the common zero soluton of error dynamcs and of parameter dynamcs s asymptotcally stable. Numercal smulatons of a new Double Ge-Ku(DGK) system are gven to show the effectveness of the proposed scheme. 4. The Scheme of Dfferent Translaton Pragmatcal Generalzed Synchronzaton by Stablty Theory of Partal Regon There are two dentcal nonlnear dynamcal systems, and the master system synchronzes the slave system. The master system s gven by x Ax ( f, x ) B (4.) The master system after the orgn of x-coordnate system s translated to ( k, k,, k ) s x Ax ( f, x ) B (4.) n x [ x, x,, x ] x k [ x k, x k,, x k ] R denotes a state T where n n 7

49 vector where k [ k, k,, k] s a constant vector wth postve coeffcent k as shown n Fg.4.. A s an n n uncertan constant coeffcents matrx, f s a nonlnear vector functon, and B s a vector of uncertan constant coeffcents n f. The slave system s gven by y Ay ( f, y ) B ( u) t (4.) The slave system after the orgn of y-coordnate system s translated to ( k, k,, k ) s y Ay ( f, y ) B ( u) t (4.) n y [ y, y,, y ] y k [ y k, y k,, y k ] R denotes a state T where n n vector where k s a constant vector wth postve coeffcent k as shown n Fg.4.. A s an n n estmated coeffcent matrx, B s a vector of estmated coeffcents n f, and u( t) [ u( t), u( t),, u ( )] T n n t R s a control nput vector. Our goal s to desgn a controller ut () so that the state vector of the translated slave system (4.) asymptotcally approaches the state vector of the translated master system (4.) plus a gven nonchaotc or chaotc vector functon F( t) [ F ( t), F ( t),, F ( t)] T : n y G( x) x F(.) t (4.) Ths generalzed synchronzaton can be accomplshed when t, the lmt of the error vector e( t) [ e, e,, e ] T n approaches zero: lme 0 (4.4) t where e x F( ) t y x y ( F. ) t (4.5) From Eq. (4.5) we have 8

50 e xy F() t (4.6) e Ax Ay ( f, x) B ( f, y) B ( F) t. ( u) t (4.7) where k and k are chosen to guarantee that the error dynamcs always occurs n the frst quadrant of e coordnate system. A Lyapunov functon V ( e, Ac, Bc) s chosen as a postve defnte functon n frst quadrant of e coordnate system by stablty theory n partal regon as shown n Appendx A: V( e, Ac, Bc ) e ca cb (4.8) where Ac A A, Bc B B, A c and B c are two column matrces whose elements are all the elements of matrx A c and of matrx B c, respectvely. Its dervatve along any soluton of the dfferental equaton system consstng of Eq. (4.7) and update parameter dfferental equatons for A c and B c s V ( e, A B, ) Ax Ay f x( B, f) y B(, F ) t u ( t ) A ( B) (4.9) c c c c where ut (), A c, and B c are chosen so that V Ce, C s a dagonal negatve defnte matrx, and V s a negatve sem-defnte functon of e and parameter dfferences A c and B c. By pragmatcal asymptotcally stablty theorem n Appendx B, the Lyapunov functon used s a smple lnear homogeneous functon of states and the controllers are smpler because they n lower order than that of tradtonal controllers. In many papers [5-55], tradtonal Lyapunov stablty theorem and Babalat lemma are used to prove the error vector approaches zero, as tme approaches nfnty. But the queston, why the estmated parameters also approach to the uncertan parameters, remans no answer. By pragmatcal asymptotcal stablty theorem, the queston can be answered strctly. 9

51 4. Dfferent Translaton Pragmatcal Generalzed Synchronzaton of New Double Ge-Ku Chaotc System Case The followng chaotc systems are two translated DGK systems of whch the old orgn s translated to ( x, x, x) (50,50,50), ( y, y, y) (50,50,50) to guarantee the error dynamcs always happens n the frst quadrant of e coordnate system. x x 50 x a( x 50) ( x 50){ b[ c ( x 50) ] d( x 50)} x a( x 50) ( x 50){ b[ c ( x 50) ] g( x 50)} y y 50 u y a( y 50) ( y 50){ b[ c ( y 50) ] d( y 50)} u y a( y 50) ( y 50){ b[ c ( y 50) ] g( y 50)} u (4.0) (4.) Let ntal states be ( x, x, x) (50.0,50.0,50.0), ( y, y, y ) (50.0,50.0,50.0) and system parameters a 0.5, b.4, c.9, d 4.5, g 6.. The state error s e x y F() t x y e cos t where F() t cost e s a nonchaotc gven functon of tme. We fnd that the error dynamc wthout controller always exsts n frst quadrant as shown n Fg.4..,,, (4.) cos t lme lm( x y e ) t t Our am s lm e 0. We obtan the error dynamcs: t 40

52 cost e x 50 y 50 u (sn t) e e a( x 50) ( x 50){ b[ c ( x 50) ] d( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] d( y 50)} u (sn t) e e a( x 50) ( x 50){ b[ c ( x 50) ] g( x 50)} ay ( 50) ( y 50){ b[ c ( y 50) ] g( y 50)} u (sn t) e cost cost (4.) where a a aˆ, b b bˆ, c c cˆ, d d dˆ, g g gˆ, and a, b, c, d, g are estmates of uncertan parameters a, b, c, d and g respectvely. Usng dfferent translaton pragmatcal synchronzaton by stablty theory of partal regon, we can choose a Lyapunov functon n the form of a postve defnte functon n frst quadrant: V e e e a b c d g (4.4) Its tme dervatve s V e e e a b c d g cost ( x 50 y 50 u (sn t) e ) a( x 50) ( x 50){ b[ c ( x 50) ] d( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] d( y 50)} u (sn t) e cost a( x 50) ( x 50){ b[ c ( x 50) ] g( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] g( y 50)} u (sn t) e cost wz a b c d g (4.5) Choose a aˆ ae b bˆ be c cˆ ce d dˆ de g gˆ ge (4.6) 4

53 cost u x 50 y 50 (sn t) e e u a( x 50) ( x 50){ b[ c ( x 50) ] d( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] d( y 50)} (sn t) e e ae be ce u a( x 50) ( x 50){ b[ c ( x 50) ] gx ( 50)} a( y 50) ( y 50){ b[ c ( y 50) ] g( y 50)} (sn t) e e de ge We obtan cost cost (4.7) V e e e 0 (4.8) whch s a negatve sem-defnte functon of e, e, e, a, b, c, d, g, n the frst quadrant. The Lyapunov asymptotcal stablty theorem s not satsfed. We can not obtan that common orgn of error dynamcs () and parameter dynamcs (6) s asymptotcally stable. By pragmatcal asymptotcally stablty theorem, D s a 8-manfold, n=8 and the number of error state varables p=. When e e e 0 and a, b, c, d, g take arbtrary values, V 0, so X s of dmensons, m=n-p=8-=5, m+<n s satsfed. Accordng to the pragmatcal asymptotcally stablty theorem, error vector e approaches zero and the estmated parameters also approach the uncertan parameters. The equlbrum pont s pragmatcally asymptotcally stable. Under the assumpton of equal probablty, t s actually asymptotcally stable. The smulaton results are shown n Fgs Case The followng chaotc systems are two translated DGK systems of whch the old orgn s translated to ( x, x, x) (50,50,50), ( y, y, y) (50,50,50) to guarantee that the error dynamcs always happens n the frst quadrant of e coordnate system. 4

54 x x 50 x a( x 50) ( x 50){ b[ c ( x 50) ] d( x 50)} x a( x 50) ( x 50){ b[ c ( x 50) ] g( x 50)} y y 50 u y a( y 50) ( y 50){ b[ c ( y 50) ] d( y 50)} u y a( y 50) ( y 50){ b[ c ( y 50) ] g( y 50)} u (4.9) (4.0) Let ntal states be ( x, x, x) (50.0,50.0,50.0), ( y, y, y ) (50.0,50.0,50.0) and system parameters a 0.5, b.4, c.9, d 4.5, g 6.. The state error s e x y F() t, where Ft () z [ z, z, z] s the state vector of Chen system: zh( z z ) z ( w h) z z z wz z z z lz (4.) Let ntal states be z [ z, z, z] [0.5,0.6,0.5] and system parameters h 5, l, w 7.. The Chen system s chaotc, Ft () z [ z, z, z] s a gven chaotc vector functon of tme. Defne e x y z We fnd that the error dynamcs wthout controller always exsts n frst quadrant as shown n Fg.4.7. lm e lm[ x y z ],,, (4.) t t Our am s lm e 0. t We obtan the error dynamcs: 4

55 e x 50 y 50 u h( z z) e a( x 50) ( x 50){ b[ c ( x 50) ] d( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] d( y 50)} u ( w h) z zz wz e a( x 50) ( x 50){ b[ c ( x 50) ] g( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] g( y 50)} u zz lz (4.) where a a aˆ, b b bˆ, c c cˆ, d d dˆ, g g gˆ, and a, b, c, d, g, are estmates of uncertan parameters a, b, c, d and g respectvely. Usng dfferent translaton pragmatcal synchronzaton by stablty theory of partal regon, we can choose a Lyapunov functon n the form of a postve defnte functon n frst quadrant: V e e e a b c d g (4.4) Its tme dervatve s V e e e a b c d g ( x 50 y 50 u h( z z )) a( x 50) ( x 50){ b[ c ( x 50) ] d( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] d( y 50)} u ( w h) z z z wz a( x 50) ( x 50) { b[ c ( x 50) ] g( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] g( y 50)} u z z lz a b c d g (4.5) Choose a aˆ ae b bˆ be c cˆ ce d dˆ de g gˆ ge (4.6) 44

56 u x 50 y 50 h( z z) e u a( x 50) ( x 50){ b[ c ( x 50) ] d( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] d( y 50)} ( w h) z zz wz e ae be ce u a( x 50) ( x 50){ b[ c ( x 50) ] gx ( 50)} a( y 50) ( y 50){ b[ c ( y 50) ] g( y 50)} zz lz e de ge (4.7) We obtan V e e e 0 (4.8) whch s a negatve sem-defnte functon of e, e, e, a, b, c, d, g n the frst quadrant. The Lyapunov asymptotcal stablty theorem s not satsfed. We can not obtan that common orgn of error dynamcs () and parameter dynamcs (6) s asymptotcally stable. By pragmatcal asymptotcally stablty theorem, D s a 8-manfold, n=8 and the number of error state varables p=. When e e e 0 and a, b, c, d, g, take arbtrary values, V 0, so X s of dmensons, m=n-p=8-=5, m+<n s satsfed. Accordng to the pragmatcal asymptotcally stablty theorem, error vector e approaches zero and the estmated parameters also approach the uncertan parameters. The equlbrum pont s pragmatcally asymptotcally stable. Under the assumpton of equal probablty, t s actually asymptotcally stable. The smulaton results are shown n Fgs Case The followng chaotc systems are the two translated DGK systems of whch the old orgn s translated to ( x, x, x) (50,50,50), ( y, y, y) (50,50,50) to guarantee error dynamcs always happens n the frst quadrant of e coordnate system. 45

57 x x 50 x a( x 50) ( x 50){ b[ c ( x 50) ] d( x 50)} x a( x 50) ( x 50){ b[ c ( x 50) ] g( x 50)} y y 50 u y a( y 50) ( y 50){ b[ c ( y 50) ] d( y 50)} u y a( y 50) ( y 50){ b[ c ( y 50) ] g( y 50)} u (4.9) (4.0) Let ntal states be ( x, x, x) (50.0,50.0,50.0), ( y, y, y ) (50.0,50.0,50.0) and system parameters a 0.5, b.4, c.9, d 4.5, g 6.. The state error s e x y F() t, where Ft () z [ z, z, z] s state vector the new Ge-Ku-van der Pol(GKv) system: z z z fz z[ f( f z ) f4z] z gz h( z ) z lz (4.) Let ntal states be ( z, z, z) (0.0,0.0,0.0) and system parameters f 0.08, f 0.5, f 00.56, f , h 0.6, l 0.08, w 0.0, the Ge-Ku-van der Pol system [56] s chaotc n Fg.4., Ft () z [ z, z, z] s a gven chaotc vector functon of tme. Defne e x y z We fnd that the error dynamc wthout controller always exsts n frst quadrant as shown n Fg. 4.. lm e lm[ x y z ],,, (4.) t t Our am s lm e 0. t We obtan the error dynamcs: 46

58 e x 50 y 50 u z e a( x 50) ( x 50){ b[ c ( x 50) ] d( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] d( y 50)} u fz z[ f( f z ) f4z] e a( x 50) ( x 50){ b[ c ( x 50) ] g( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] g( y 50)} u hz l( z ) z wz (4.) where a a aˆ, b b bˆ, c c cˆ, d d dˆ, g g gˆ, and a, b, c, d, g, are estmates of uncertan parameters a, b, c, d and g respectvely. Usng dfferent translaton pragmatcal synchronzaton by stablty theory of partal regon, we can choose a Lyapunov functon n the form of a postve defnte functon n frst quadrant: V e e e a b c d g (4.4) Its tme dervatve s V e e e a b c d g ( x 50 y 50 u z ) a( x 50) ( x 50){ b[ c ( x 50) ] d( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] d( y 50)} u f z z [ f ( f z ) f z ] 4 a( x 50) ( x 50){ b[ c ( x 50) ] g( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] g( y 50)} u hz l( z ) z wz a b c d g (4.5) Choose 47

59 a aˆ ae b bˆ be c cˆ ce d dˆ de g gˆ ge u x 50 y 50 z e u a( x 50) ( x 50){ b[ c ( x 50) ] d( x 50)} a( y 50) ( y 50){ b[ c ( y 50) ] d( y 50)} fz z[ f( f z ) f4z] e ae be ce u a( x 50) ( x 50){ b[ c ( x 50) ] gx ( 50)} a( y 50) ( y 50){ b[ c ( y 50) ] g( y 50)} hz l( z ) z wz e de ge We obtan (4.6) (4.7) V e e e 0 (4.8) whch s a negatve sem-defnte functon of e, e, e, a, b, c, d, g, n the frst quadrant. The Lyapunov asymptotcal stablty theorem s not satsfed. We can not obtan that common orgn of error dynamcs () and parameter dynamcs (6) s asymptotcally stable. By pragmatcal asymptotcally stablty theorem, D s a 8-manfold, n=8 and the number of error state varables p=. When e e e 0 and a, b, c, d, g, take arbtrary values, V 0, so X s of dmensons, m=n-p=8-=5, m+<n s satsfed. Accordng to the pragmatcal asymptotcally stablty theorem, error vector e approaches zero and the estmated parameters also approach the uncertan parameters. The equlbrum pont s pragmatcally asymptotcally stable. Under the assumpton of equal probablty, t s actually asymptotcally stable. The smulaton results are shown n Fgs

60 Fg. 4. Coordnate translaton of state x. Fg. 4. Coordnate translaton of state y. 49

61 Fg. 4. Phase portrat of the error dynamcs for Case. Fg. 4.4 Tme hstores of x, y for Case. 50

62 Fg. 4.5 Tme hstores of errors for Case. Fg. 4.6 Tme hstores of parameter errors for Case. 5

63 Fg. 4.7 Phase portrat of the error dynamc for Case. Fg. 4.8 Tme hstores of x, y for Case. 5

64 Fg. 4.9 Tme hstores of errors for Case. Fg. 4.0 Tme hstores of parameter errors for Case. 5

65 Fg. 4. The chaotc attractor of the Ge-Ku-van der Pol system. Fg. 4. Phase portrat of the error dynamc for Case. 54

66 Fg. 4. Tme hstores of x, y for Case. Fg. 4.4 Tme hstores of errors for Case. 55

67 Fg. 4.5 Tme hstores of parameter errors for Case. 56

68 Chapter 5 Robust Projectve Ant-Synchronzaton of Non-autonomous and Chaotc Uncertan Stochastc Systems Va Fuzzy Logc Constant Controller 5. Prelmnary In ths Chapter, the smplest controller fuzzy logc constant controller (FLCC) whch s derved by fuzzy logc desgn and Lyapunov drect method s ntroduced. Controllers n tradtonal method are always complcated or nonlnear. FLCC proposed are such smple controllers whch are constructed by constant number decded by the values of the upper and lower bound of the error dervatves. Therefore ths tool s further used n projectve ant-synchronzaton of chaotc uncertan and stochastc systems wth to show the robustness and effectveness of FLCC. There are two cases llustrated n smulaton results to show the feasblty of the FLCC Sprott 4 system and Double Ge-Ku (DGK) system. Comparson of the effcency and complexty for the FLCC for tradtonal nonlnear controllers are gven n tables and fgures. 5. Projectve Ant-Synchronzaton by FLCC Scheme Consder the followng non-autonomous or uncertan chaotc system x Ax f(x) (5.) where x [ x, x, x ] T n n R denotes a state vector, A s an n n constant coeffcent matrx and f s a nonlnear vector functon, s uncertan term. The goal system whch can be ether chaotc or regular, s y By g(y) u (5.) 57

69 where y [ y, y, y ] T n n R denotes a state vector, B s an n n constant coeffcent matrx, g s a nonlnear vector functon, and u [ u, u, u ] T n n R s the fuzzy logc controller needed to be desgned. In order to make the chaotc state x approachng the goal state y, defne e xy as the state error, here s projectve constant. The projectve chaos ant-synchronzaton s accomplshed n the sense that [57]: where lme lm( x y ) 0 (5.) t t e xy (5.4) From Eq. (-4) we have the followng error dynamcs: e x y [ A x f (x) ] B[ y g (y) u] (5.5) Accordng to Lyapunov drect method, we have the followng Lyapunov functon to derve the fuzzy logc controller for ant-synchronzaton: V f ( e,... em,... en ) ( e... em... en ) 0 (5.6) The dervatve of the Lyapunov functon n Eq. (5.5) s: V e e... me me... nene (5.7) If the controllers ncluded n e... em... en can be sutably desgned to acheve the target: V 0, then the two chaotc systems are asymptotcally stable. The desgn process of FLCC s ntroduced n the followng secton. We use one sgnal, error dervatves e ( t) e, e e, e m n T, as the antecedent part of the proposed FLCC to desgn the control nput u that wll be used n the consequent part of the proposed FLCC as follows: T, u um, un u u (5.8) 58

70 where u s a constant column vector and the FLCC accomplshes the objectve to stablze the error dynamcs (5.5). The strategy of the FLCC desgnng s proposed as follow and the confguraton of the strategy s shown n Fg. 5.. Assume the upper bound and lower bound of e m are Z m and Z m, then the FLCC can be desgn step by step as follow: () If e m s detected as postve ( em 0 ), we have to desgn a controller for e 0, thenv e e 0can be acheved. Therefore we have the followng th f then m fuzzy rule as: m m Rule : IF Rule : IF em s M THEN u m = -Z m (5.9) em s M THEN u m = -Z m (5.0) Rule : IF em s M THEN u m = e m (5.) () If e m s detected as negatve ( em 0 ), we have to desgn a controller for e 0, thenv e e 0can be acheved. Therefore we have the followng th f then m fuzzy rule as: m m Rule : IF Rule : IF em s M THEN u m = Z m (5.) em s M THEN u m = Z m (5.) Rule :IF em s M THEN u m = e m (5.4) () If em approaches to zero, then the ant-synchronzaton s nearly acheved. Therefore we have the followng th f then fuzzy rule as: Rule :IF Rule : IF e s M THEN u m = e 0 (5.5) m m m e s M THEN u m = e 0 (5.6) m Rule : IF e s M THEN u m = e 0 (5.7) m em em Zm em em Zm where M, M and M sgn( ) sgn( ), M, M and M Z Z Z Z m m refer to the membershp functons of postve (P), negatve (N) and zero (Z) separately 59 m m m

71 whch are presented n Fg. 5. For each case, u m, = ~ s the -rd output of e m whch s a constant controller. The centrod defuzzfer evaluates the output of all rules as follows: M um u m (5.8) M The fuzzy rule base s lsted n Table 5., n whch the nput varables n the antecedent part of the rules are e m and the output varable n the consequent part s u m. Table 5. Rule-table of FLCC Rule Antecedent Consequent Part e m u m Negatve (N) u m Postve (P) u m Zero (Z) u m After desgnng approprate fuzzy logc constant controllers and beng substtuted nto Eq. (5.7), a negatve defnte of dervatves of Lyapunov functon V can be obtaned and the asymptotcally stablty of Lyapunov theorem can be acheved. Consequently, the processes of FLCC desgnng to control a system followng the trajectory of a goal system s - gettng the upper bound and lower bound of the error dervatves of the goal and control systems wthout any controller,.e. Z e Z. Through the fuzzy logc system whch follows the rules of Eq. (5.9) m m m ~ Eq. (5.7), a negatve defnte of dervatves of Lyapunov functon V can be obtaned and the asymptotcally stablty of Lyapunov theorem can be acheved. 5. Smulaton Results 60

72 There are two cases n ths secton. Each example s dvded nto two parts projecton ant-synchronzaton by FLCC and tradtonal method. In the end of each example, we further conclude the smulaton results of the two controllers and lst tables and fgures to show the effectveness and robustness of our method. 5.. Example. Projectve Ant-synchronzaton of Non-autonomous Chaotc Systems by FLCC The master Sprott 4 system [50] wth uncertanty s : dx () t x dt dx () t x dt dx () t ax x x x dt (5.9) For ntal condton x, x, ) = (0,, ), s Bsn(wt), =Csn(wt) and ( 0 0 x0 parameters a=-0.5, B=0.0, C=.0, w=0.0, chaos of the Sprott 4 system [50] appears. The chaotc behavor of Sprott 4 system [50] s shown n Fg 5.. And Eq. (5.9) s shown n Fg 5.4. The dentcal slave DGK system s: dy () t y u dt dy() t b y y [ b ( b y ) b y ] u dt dy() t b y y [ b ( b y ) b y ] u dt 4 5 (5.0) For ntal condton y, y, ) = (0.0, 0.0, 0.0) and parameters b = -0.5, ( 0 0 y0 b = -.4, b =.9, b 4 = -4.5, b 5 =6., chaos of the slave DGK system appears as well. u, u and u are FLCC to ant-synchronze the slave DGK system to master one,.e., the projecton of phase portrats of system (5.0) wth chaotc behavors s shown 6

73 n Fg. 5.. lm e 0 t (5.) where the error vector e ( t) x ( t) y( t) e e ( t) A x ( t) y ( t) e ( t) x( t) y( t) (5.) Let A. From Eq. (5.), we have the followng error dynamcs: e A[ x ] ( y u) e A[ x ] ( b y y[ b ( b y ) b4 y] u) e A[ ax x x x ] ( b y y[ b ( b y ) b5 y ] u) (5.) Choosng Lyapunov functon as: V ( e e e ) (5.4) Its tme dervatve s: V e e e e e e e { A[ x ] ( y u )} e { A[ x ] ( b y y [ b ( b y ) b y ] u )} 4 e { A[ a x x x x ] ( b y y [ b ( b y ) b y ] u )} 5 (5.5) In order to desgn FLCC, we dvde Eq. (5.5) nto three parts as follows: Assume V ( e e e ) V V V, then V ee ee ee V V V, where V e. Part : V ee e{ A[ x ] ( y u)} V e, V e and Part : V e e e { A[ x ] ( b y y [ b ( b y ) b y ] u )} 4 6

74 Part : V e e e A( a x x x x ) ( b y y [ b ( b y ) b y ] u ) 5 FLCC n Part, and can be obtaned va the fuzzy rules n Table. The maxmum value and mnmum value wthout any controller can be observed n tme hstores of error dervatves shown n Fg 5.5: Z, Z, Z 6. The ant-synchronzaton scheme s proposed n Part, and and let e e 0, e e 0 and e e 0. Hence we have V V V V V V V 0. It s clear that all of the rules n FLCC can lead the Lyapunov functon satsfes the asymptotcally stable theorem. The smulaton results are shown n Fg. 5.6 and Fg Projectve Ant-synchronzaton of Sprott 4 System [50]and DGK System by Tradtonal Method In order to lead the dervatve of Lyapunov functon n Eq. (5.5) to negatve defnte, we choose tradtonal nonlnear controller as: u [ A( x ) y e ] u { A( x ) b y y[ b ( b y ) b4 y] e} u { A( 0.5 x x x x ) b y y[ b ( b y ) b5 y ] e} (5.6) then we can obtan V e e e e e e0 (5.7) The dervatve of Lyapunov functon s negatve defnte and the error dynamcs n Eq. (5.) are gong to acheve asymptotcally stable. The smulaton results are shown n Fg. 5.8 and Fg Robust FLCC Compared wth Tradtonal Method In ths secton, we compare the numercal smulaton results n 5.. and 5.. 6

75 whch are lsted n Tables 5. and 5.. Comparng the two smulaton results n Tables 5. and 5., t s clear to fnd out that: () The performance of the error convergence of states by FLCC s much better than by tradtonal method; () The controller n FLCC desgnng are much smpler than tradtonal ones. Consequently, even the system contans parameter uncertanty, the FLCC can stll reman the hgh performance to ant-synchronze the two chaotc systems wth uncertanty and exactly and effcently. Table 5.. Controllers are compared between the tradtonal method and FLCC method. Controller FLCC Tradtonal u u u Z, e 0 Z, e 0 A( x ) y e { A( x ) b y y [ b ( b y ) b y ] e } 4 Z 6, e 0 { A[ a( ) x bx x x ] cy y sn y e } Table 5.. FLCC s compared to tradtonal method between errors data. Tme(s) FLCC Tradtonal 64

76 e e e e e e Example. Chaos Projectve Ant-synchronzaton of Uncertan Stochastc Sprott 4 System [50] and DGK System The uncertan stochastc Sprott 4 system [50] wth nose s: dx () t x dt dx() t x dt dx () t a x ( ) x x x dt (5.8) For ntal condton x, x, ) = (0,, ), the uncertan term s Dsn(wt), ( 0 0 x0 the stochastc term s Raylegh nose and parameters a=-0.5, D=0., w=0.5, chaos of the uncertan stochastc Sprott 4 system appears. Raylegh nose s shown n Fg 65

77 5.0. The chaotc behavor of Eq. (5.8) s shown n Fg 5.. The dentcal slave DGK system s: dy () t y u dt dy() t b y y [ b ( b y ) b y ] u dt dy() t b y y [ b ( b y ) b y ] u dt 4 5 (5.9) When ntal condton y, y, ) = (0.0, 0.0, 0.0) and parameters b = -0.5, ( 0 0 y0 b = -.4, b =.9, b 4 = -4.5, b 5 =6., chaos of the slave DGK system appears as well. u, u and u are FLCC to ant-synchronze the slave DGK system to the master one,.e., the projecton of phase portrats of system (-) wth chaotc behavors s shown n Fg. 5.. lm e 0 t (5.0) where the error vector e ( t) x ( t) y( t) e e ( t) A x ( t) y ( t) e ( t) x( t) y( t) (5.) Let A. From Eq. (5.), we have the followng error dynamcs: e A[ x ( y u)] e A[ x ( b y y[ b ( b y ) b4 y] u)] e A[ ax ( ) x x x ( b y y[ b ( b y ) b5 y] u)] (5.) Choosng Lyapunov functon as: V ( e e e) (5.) Its tme dervatve s: 66

78 V e e e e e e e { A[ x ( y u )]} e { A[ x ( b y y [ b ( b y ) b y ] u )]} 4 e { A[ a x ( ) x x x ( b y y [ b ( b y ) b y ] u )]} 5 (5.4) In order to desgn FLCC, we dvde Eq. (5.4) nto three parts as follows: AssumeV ( e e e ) V V V, then V e e ee ee V V V Part : V ee e A[ x ( y u)], wherev e, V e andv e. Part : Part : V e e e A[ x ( b y y [ b ( b y ) b y ] u )] 4 V e e e A[ a x ( ) x x x ( b y y [ b ( b y ) b y ] u )] 5 FLCC n Part, and can be obtaned va the fuzzy rules n Table 5.. The maxmum value and mnmum value wthout any controller can be observed n tme hstores of error dervatves shown n Fg 5.: Z, Z, Z 5. FLCC are proposed n Part, and and make e e 0, e e 0 and V V e e 0. Hence, we have V V V 0. It s clear that all of the rules n V V our FLCC can lead the Lyapunov functon satsfes the asymptotcally stable theorem. The smulaton results are shown n Fg. 5.4 and Chaos Projectve Ant-synchronzaton of Uncertanty Stochastc Sprott 4 System [50] and DGK System by Tradtonal Method Accordng to Eq. (5.4), we desgn complcated controllers to ant-synchronze chaotc system wth uncertanty by tradtonal method. We choose the controllers as follows 67

79 u [ A( x ) y e ] u { Ax b y y[ b ( b y ) b4 y] e} u { A( 0.5 x ( ) x x x ) b y y[ b ( b y ) b5 y] e} (5.5) then we can obtan V e e e e e e0 (5.6) The dervatve of Lyapunov functon s negatve defnte and the error dynamcs n Eq. (5.) are gong to acheve asymptotcally stable. The smulaton results are shown n Fg. 5.6 and Fg Robust FLCC Compared to Tradtonal Method In ths case, we compare the numercal smulaton results n 5..4 and 5..5 whch are lsted n Table 5.4 and 5.5. The two man superortes are stll exsted () The performance of the error convergence of states by FLCC are much better than by tradtonal method; () The controllers n FLCC desgnng are much smpler than tradtonal ones Table 5.4. Controllers are compared between the tradtonal method and FLCC method. Controller FLCC Tradtonal u u Z 4, e 0 A( x ) y e u Z, e 0 Z 5, e 0 { Ax b y y [ b ( b y ) b y ] e } 4 { A( 0.5 x ( ) x x x ) b y y [ b ( b y ) b y ] e } 5 68

80 Table 5.5. FLCC s compared to tradtonal method between errors data. Tme(s) FLCC Tradtonal e e e e 69

81 e e

82 Fg. 5. The confguraton of fuzzy logc controller. Fg. 5. Membershp functon. 7

83 Fg. 5. The chaotc behavor of Sprott 4 system [50] Fg. 5.4 Projectons of phase portrat of Eq (5.9). 7

84 Fg. 5.5 Tme hstores of error dervatves for master and slave chaotc systems wthout controllers. Fg. 5.6 Tme hstores of states for Example the FLCC s comng nto after 0s. 7

85 Fg. 5.7 Tme hstores of errors for Example the FLCC s comng nto after 0s. Fg. 5.8 Tme hstores of states for the tradtonal controller s comng nto after 0s. 74

86 Fg. 5.9 Tme hstores of states for the tradtonal controller s comng nto after 0s. Fg. 5.0 The Raylegh nose used. 75

87 Fg. 5. Projecton of phase portrat of Eq (5.8). Fg. 5. Projectons of phase portrat of chaotc DGK systems. 76

88 Fg. 5. Tme hstores of error dervatves for master and slave chaotc uncertan stochastc systems wthout controllers. Fg. 5.4 Tme hstores of states for Example the FLCC s comng nto after 0s. 77

89 Fg. 5.5 Tme hstores of errors for Example the FLCC s comng nto after 0s. Fg. 5.6 Tme hstores of states for the tradtonal controller s comng nto after 0s. 78

90 Fg. 5.7 Tme hstores of states for the tradtonal controller s comng nto after 0s. 79

91 Chapter 6 Fuzzy Modelng and Synchronzaton of Chaotc Systems by a New Fuzzy Model 6. Prelmnary In ths paper, a new fuzzy model [58] s used to smulate and synchronze two totally dfferent chaotc systems, Sprott systems and Rossler system. Va the new fuzzy model, a complex nonlnear system s lnearzed to a smple form. Only two lnear subsystems are needed and the numbers of fuzzy rules can be decreased from N to N. Sprott systems and Rossler system are used for examples n numercal smulaton results to show the effectveness and feasblty of our new model. 6. New Fuzzy Model Theory In system analyss and desgn, t s mportant to select an approprate model representng a real system. As an expresson model of a real plant, the fuzzy mplcatons and the fuzzy reasonng method suggested by Takag and Sugeno are tradtonally used. The new fuzzy model s also descrbed by fuzzy IF-THEN rules. The core of the new fuzzy model s that we express each nonlnear equaton nto two lnear sub-equatons by fuzzy IF-THEN rules and take all the frst lnear sub-equatons to form one lnear subsystem and all the second lnear sub-equatons to form another lnear subsystem. The overall fuzzy model of the system s acheved by fuzzy blendng of ths two lnear subsystem models. Consder a contnuous-tme nonlnear dynamc system as follows: Equaton : 80

92 rule : rule : where IF z (t) s M THEN (t) A x(t) B u(t), x IF z (t) s M THEN (t) A x(t) B u(t), (6.) x T (t), x (t),..., x n(t)] x(t) [x, T (t), u (t),..., u n(t)] u(t) [u,,...n, where n s the number of nonlnear terms. z (t) s nonlnear term, M, M are fuzzy sets,, B A are column vectors and x (t) A x(t) B u(t), j j j,, s the output from the frst and the second IF-THEN rules. ut () s vector controller. Gven a par of ( x ( t),u( t)) and take all the frst lnear sub-equatons to form one lnear subsystem and all the second lnear sub-equatons to form another lnear subsystem, the fnal output of the fuzzy system s nferred as follows: Ax(t) Bu(t) Ax(t) Bu(t) Ax(t) Bu(t) Ax(t) Bu(t) x(t) M M (6.) Ax(t) Bu(t) Ax(t) Bu(t) where M and M are dagonal matrces as followng: da ( M ) M M..., da ( M ) M M... M Note that for each equaton : j M j(z(t)), M j (z (t)) 0, =,,, n and j=,. 8 M

93 Va the new fuzzy model, the fnal form of the fuzzy model becomes very smple. The new model provdes a much more convenent approach for fuzzy model research and fuzzy applcaton. The smulaton results of chaotc systems are dscussed n next Secton. 6. New Fuzzy Model of Chaotc Systems In ths Secton, the new fuzzy models of two dfferent chaotc systems, Sprott 4 system [50] and Rossler system, are gven for Model and Model. Model : New fuzzy model of Sprott 4 system wth uncertanty The Sprott 4 system wth uncertanty s: x x x x x 0.5x x x x (6.) wth ntal states (0.0,, 0.0). Uncertan terms are = 0.sn(50t) and = 0.0cos(50t). The chaotc attractor of the Sprott 4 system s shown n Fg. 6. and the chaotc attractor of Eq.(6.) s shown n Fg. 6.. If T-S fuzzy model s used for representng local lnear models of Sprott 4 system, N, N 8, 8 fuzzy rules and 8 lnear subsystems are need. The process of modelng s shown as follows: T-S fuzzy model: Assume that: () Z Z [, ] and Z 0 () Z Z [, ] and Z 0 () x Z Z [, ] and Z 0 8

94 Then we have the followng T-S fuzzy rules: Rule : IF s M, s M and x s M THEN X A X, Rule : IF s M, s M and x s M THEN X A X, Rule : IF s M, s M and x s M THEN X A X, Rule 4: IF s M, s M and x s M THEN X A4 X, Rule 5: IF s M, s M and x s M THEN X A5 X, Rule 6: IF s M, s M and x s M THEN X A6 X, Rule 7: IF s M, s M and x s M THEN X A7 X, Rule 8: IF s M, s M and x s M THEN X A8 X, Then the fnal output of the Sprott 4 system can be composed by fuzzy lnear subsystems mentoned above. It s obvously an neffcent and complcated work. New fuzzy model: By usng the new fuzzy model, Sprott 4 system can be lnearzed as smple lnear equatons. The steps of fuzzy modelng are shown as follows: Steps of fuzzy modelng: Step : Assume that [ Z, Z] and Z 0, then the frst equaton of (-) can be represented by new fuzzy model as followng: Rule: IF s M, THEN x x Z, (6.4) where R u l e : I F s M, T H E N x x Z, ( 6. 5 ) M ( ), M ( ), Z Z 8

95 and Z 0.. M and M are fuzzy sets of the frst equaton of (-) and M M. Step : Assume that [ Z, Z] and Z 0, then the second equaton of (-) can be exactly represented by new fuzzy model as followng: where Rule : IF s M, THEN x x Z, (6.6) Rule : IF s M, THEN x x Z (6.7) M ( ), M ( ), Z Z and Z 0.0. M and M are fuzzy sets of the second equaton of (6.) and M M Step :. Assume that x [ Z, Z] and Z 0, then the thrd equaton of (6.) can be exactly represented by new fuzzy model as followng: where Rule : IF x s M, THEN x 0.5x x x xz, (6.8) Rule : IF x s M, THEN x 0.5x x x x Z (6.9) M x ( x ), M ( ), Z Z and Z.5. M and M are fuzzy sets of the thrd equaton of (6.) and M M. Here, we call Eqs.(6.4), (6.6) and (6.8) the frst lnear subsystem under the fuzzy rules, and Eqs.(6.5), (6.7) and (6.9) the second lnear subsystem under the fuzzy rules. The frst lnear subsystem s 84

96 x x Z x x Z x 0.5x x x x Z (6.0) The second lnear subsystem s x x Z x x Z x 0.5x x x x Z (6.) The fnal output of the fuzzy Sprott 4 system s nferred as follows and the chaotc behavor of fuzzy system s shown n Fg. 6.. Now we have: x M 0 0 x Z x 0 M 0 x Z x 0 0 M 0.5x x x xz M 0 0 x Z 0 M 0 x Z 0 0 M 0.5x x x xz (6.) Eq. (6.) can be rewrtten as a smple mathematcal expresson: ~ X (t) (A X(t) b ) (6.) where are dagonal matrces as follows: M M, da( M M M da( M ) ) 0 0 A 0 0, Z 0.5 Z b Z 0 A , Z 0.5 b Z Z 0 Va new fuzzy model, the number of fuzzy rules can be greatly reduced. Only two lnear subsystems are enough to express such complex chaotc behavors. The smulaton results are smlar the orgnal chaotc behavor of the Sprott 4 system as 85

97 show n Fg. 6.. Model : New fuzzy model of Rossler system wth uncertanty The Rossler system wth uncertanty s: y y y 0. y y 0.5y4 y 0. y y 9y (6.4) where uncertanty terms s whte nose (PSD=0.), 4 s Raylegh nose and ntal condtons are chosen as (,.4, 5). The chaotc attractor of Rossler system s shown n Fg. 6.4, the chaotc attractor of Eq.(6.4) s shown n Fg. 6.5 and the Raylegh nose s shown n Fg New fuzzy model: Assume that: () Z4 Z4 0. [, ] and Z 4 0, () 4 Z5 Z5 [, ] and Z 5 0, () y Z6 Z6 [, ] and Z 6 0, then we have the followng new fuzzy rules: Rule : IF 0. s N, THEN y y y Z4, (6.5) Rule : IF 0. s N, THEN y y y Z4 (6.6) where N 0. ( ), Z 4 N 0. Z ( ) 4 and Rule : IF 4 s N,THEN y y 0.5yZ5, (6.7) Rule : IF 4 s N,THEN y y 0.5yZ5, (6.8) 86

98 where N ( 4 ), Z 5 N ( 4 ). Z 5 and Rule : IF x s N,THEN y 0. yz 6 9y, (6.9) Rule : IF x s N,THEN y 0. yz 6 9y, (6.0) where N x ( ), Z 6 N x ( ). Z 6 n Eqs. (6.5)~(6.0), Z4 0.4, Z5 4 and Z N, N, N, N, N and N are fuzzy sets of Eq.(6.4) and N N, N N and N N Here, we call (6.5),(6.7) and (6.9) the frst lner subsystem under the fuzzy rules and (6.6), (6.8) and (6.0) the second lner subsystem under the fuzzy rules. The frst lnear subsystem s y y y Z4 y y 0.5yZ5 y 0. y Z 9y 6 (6.) The second lnear subsystem s y y y Z4 y y 0.5yZ5 y 0. y Z 9y 6 (6.) The fnal output of the fuzzy Rossler system s nferred as follows and the chaotc behavor of fuzzy system s shown n Fg

99 T y N 0 0 y y Z4 y 0 N 0 y 0.5yZ 5 y 0 0 N 0. y Z 9x 6 T N 0 0 y y Z4 0 N 0 y 0.5yZ N 0. y Z 9x 6 (6.) where Eq. (6.) can be rewrtten as a smple mathematcal expresson: Y ( t) ( C Y ( t) c~ ) (6.4) N N, da( N N N da( N ) ) 0 C 0.5Z5 0, Z Z4 c 0 0. C 0 0.5Z 0, Z c Z Va new fuzzy model, two lnear subsystems are enough to express such complex chaotc behavors. The smulaton results are smlar the orgnal chaotc behavor of the Rossler system. 6.4 Fuzzy Synchronzaton Scheme In ths Secton, we derve the new fuzzy synchronzaton scheme based on our new fuzzy model to synchronze two totally dfferent fuzzy chaotc systems. The followng fuzzy systems as the master and slave systems are gven: master system: ~ X (t) (A X(t) b ) (6.5) slave system: 88

100 Y (t) (C Y(t) ~ c ) BU(t) (6.6) Eq. (6.5) and Eq. (6.6) represent the two dfferent chaotc systems, and n Eq. (6.6) there s control nput U(t). Defne the error sgnal as e(t) X(t) Y(t), we have: where ~ e (t) X(t) Y(t) (A X(t) b ) (C Y(t) ~ c ) BU(t) (6.7) 89 The fuzzy controllers are desgned as follows: U (t) u (t) u (t) (6.8) u (t) FX(t) PY(t ), ~ u (t) b ~ c such that e(t) 0 as t. Our desgn s to determne the feedback gans F and P. By substtutng U(t) nto Eq.(6.7), we obtan: (A BF )X(t) (C BP )Y(t e (t) ) (6.9) Theorem : The error system n Eq. (6.9) s asymptotcally stable and the slave system n Eq. (6.6) can synchronze the master system n Eq. (6.5) under the fuzzy controller n Eq. (6.8) f the followng condtons below can be satsfed: Proof: G (A BF ) (A BF ) (C BP ) 0, =~. (6.0) The errors n Eq. (6.9) can be exactly lnearzed va the fuzzy controllers n Eq. (6.8) f there exst the feedback gans F such that ( A BF ) (A BF ) (C BP ) (C BP ) 0. (6.) Then the overall control system s lnearzed as e (t) Ge (t), (6.) where (A BF ) (A BF ) (C BP ) (C BP ) 0. G As a consequence, the zero soluton of the error system (6.) lnearzed va the

101 90 fuzzy controller (6.8) s asymptotcally stable. 6.5 Smulaton Results There are two examples n ths Secton to nvestgate the effectveness and feasblty of our new fuzzy model. Example : Synchronzaton of Master and Slave Sprott 4 system The fuzzy Sprott 4 system n Eq. (6.) s chosen as the master system and the fuzzy slave Sprott 4 system, wth fuzzy controllers s as follows: ) ( ~ ) ) ( ( ) ( t BU c t C Y t Y (6.) where are dagonal matrces ) ( N N N da, ) ( N N N da and C Z, 0 Z c Z C Z, 0 Z c Z. Therefore, the error and error dynamcs are: y x y x y x e e e, ) ( ~ ) ) ( ( ) ( ) ~ ) ( ( t BU c t C Y t Y b t A X y x y x y x e e e (6.4) B s chosen as an dentty matrx and the fuzzy controllers n Eq. (6.8) are used:

102 e x x e A BF x A BF x e x x y y C BP y C BP y y y (6.5) Accordng to Eq. (6.0), we have G A BF A BF C BP C BP. G s chosen as: G 0 0 (6.6) 0 0 Thus, the feedback gans F, F, P and P can be determned by the followng equatons: 0 F B A G F B A G (6.7) 0 P B C G P B C G The synchronzaton errors are shown n Fg

103 9 Example : Synchronzaton of Sprott 4 system and Rossler system. The fuzzy Sprott 4 system n Eq. (6.) s chosen as the master system and the fuzzy slave Rossler system, wth fuzzy controllers s as follows: ) ( ~ ) ) ( ( ) ( t BU c t C Y t Y (6.8) where are dagonal matrces, and C Z Z, Z c C Z Z, Z c Therefore, the error and error dynamcs are:, (6.9) B s chosen as an dentty matrx and the fuzzy controllers n Eq. (6.8) are used: y y C BP y C BP y y y (6.40) ) ( N N N da ) ( N N N da y x y x y x e e e ) ( ~ ) ) ( ( ) ( ) ~ ) ( ( t BU c t C Y t Y b t A X y x y x y x e e e x x x BF A x x x BF A e e e

104 Accordng to Eq. (6.0), we have G A BF A BF C BP C. G s chosen as: BP 0 G (6.4) Thus, the feedback gans F, F, P and P can be determned by the followng equaton: 0 F B A G F B A G (6.4) P B C G P B C G The synchronzaton errors are shown n Fg

105 Fg. 6. Chaotc behavor of Sprott 4 system. Fg. 6. Chaotc behavor of Sprott 4 system wth uncertanty. 94

106 Fg. 6. Chaotc behavor of new fuzzy Sprott 4 system wth uncertanty. Fg. 6.4 Chaotc behavor of Rossler system. 95

107 Fg. 6.5 Chaotc behavor of Rossler system wth uncertanty. Fg. 6.6 The Raylegh nose used. 96

108 Fg. 6.7 Chaotc behavor of new fuzzy Rossler system wth uncertanty. Fg. 6.8 Tme hstores of errors for Example. 97