An open-plus-closed-loop control for chaotic Mathieu-Duffing oscillator

Transcription

1 Appl. Mah. Mech. -Engl. Ed. 3(1), 19 7 (9) DOI: 1.17/s z c Shanghai Universiy and Springer-Verlag 9 Applied Mahemaics and Mechanics (English Ediion) An open-plus-closed-loop conrol for chaoic Mahieu-Duffing oscillaor Jian-he SHEN ( ) 1,, Shu-hui CHEN (íï) 1 (1. Deparmen of Applied Mechanics and Engineering, Sun Ya-sen Universiy, Guangzhou 5175, P. R. China;. School of Mahemaics and Compuer Science, Fujian Normal Universiy, Fuzhou 357, P. R. China) (Communicaed by Li-qun CHEN) Absrac By using he idea of open-plus-closed-loop(opcl) conrol, a conroller composed of an exernal exciaion and a linear feedback is designed o enrain chaoic rajecories of Mahieu-Duffing oscillaor o is periodic and higher periodic orbis. The global basin of enrainmen of his open-plus-closed-loop conrol is proved by combining he Lyapunov sabiliy heory wih a comparaive heorem of iniial value problems for second-order ordinary differenial equaions. Numerical simulaions are performed o verify he heoreical resuls. Mahieu-Duffing oscillaor, chaos conrol, OPCL conrol, global enrain- Key words men Chinese Library Classificaion O3, O31 Mahemaics Subjec Classificaion 3H5, 7Q5, 7H5 Inroducion Since chaos was discovered, many researchers in he scienific as well as in he engineering communiy have begun o doub wheher chaos is conrollable owing o is special dynamic properies such as sensiiviy o iniial condiions and ergodiciy, ec. This doub was no removed unil 199 when O, Grebogi and Yorke s pioneer paper [1] came ou. In Ref. [1], by aking he ergodiciy of chaos ino accoun, he auhors demonsraed ha he chaoic rajecories could be conrolled o he periodic orbis embedded in he chaoic aracor by iny variaions of he sysem parameers. Since hen, he conrol of chaos had araced inensive researches from mahemaical, mechanical, physical, chemical and even social communiies, ec. Many effecive mehods were developed and numerous poenial applicaions were found. Abou he heory, mehods and engineering applicaions of chaos conrol, one can refer o he excellenly reviewed papers [-] and monograph [5]. Alhough here are quie a few mehods relaing o chaos conrol, one disincion can be made beween hose ha use he closed-loop conrol and hose ha use he open-loop conrol. These wo mehods have heir advanages and disadvanages, respecively []. In 1995, Jackson Received Jul. 19, 8 / Revised Nov. 5, 8 Projec suppored by he Naional Naural Science Foundaion of China (No ) Corresponding auhor Shu-hui CHEN, Professor, sscsh@mail.sysu.edu.cn

2 Jian-he SHEN and Shu-hui CHEN and Grosu [] proposed an open-plus-closed-loop (OPCL) conrol of complex dynamic sysems by combining he advanages of he open-loop conrol wih he closed-loop conrol. This mehod is robus o limied accuracy of daa and he effecs of noise [7]. I has been applied and developed for some relaed problems of conrol. Wheeler and Schieve [8] applied he OPCL conrol o enrain sable periodic orbis from a chaoic aracor in a noise neural sysem. Chen and Liu adoped his sraegy o conrol discree chaos and hyperchaos [9] and synchronize chaoic and hyperchaoic maps [1]. Moreover, Chen and Liu developed a modified OPCL approach o conrol chaos in nonlinear oscillaions [11] and a parameric OPCL approach o conrol chaoic maps [1]. Tian e al. [13] proposed a nonlinear OPCL conrol in which he conrolled sysem is expanded o higher order a he goal dynamics. However, a necessary sep in he OPCL conrol and is variaions is he Taylor expansion a he goal dynamics, which essenially means he local validiy of his kind of mehod. Thus, he esimaion of he basin of enrainmen in he OPCL conrol becomes imporan and someimes difficul in pracical applicaions. Jackson and Grosu [] explored he global basin of enrainmen in he auonomous Duffing oscillaor, Lorenz sysem and Rössler sysem by assuming ha he goal dynamics is exponenially bounded; hey also explored he local basin of enrainmen in he Chua s sysem. Chen and Liu [9-1] also gave he esimaions of he basin of enrainmen for some discreely dynamical sysems. In his paper, for he Mahieu-Duffing oscillaor which is a non-auonomous sysem, an OPCL conrol law is designed o enrain he periodic and higher-periodic orbis from he chaoic aracor. The global basin of enrainmen of his OPCL conrol law is proved by combining he Lyapunov sabiliy heory [1] wih a comparaive heorem of he iniial value problem for second-order ordinary differenial equaions. The paper is srucured as follows. In he nex secion, bifurcaions and roue o chaos of he Mahieu-Duffing oscillaor are reviewed briefly. In Secion, an OPCL conrol law is developed o enrain he periodic and higher-periodic orbis from he chaoic aracor. Secion 3 is devoed o proving he global basin of enrainmen of he OPCL conrol. Numerical simulaions are carried ou in Secion o demonsrae heoreical resuls. In Secion 5, several remarks are furher discussed. Finally, he conclusion of his paper is given. 1 Bifurcaions and roue o chaos of he Mahieu-Duffing oscillaor Consider he Mahieu-Duffing oscillaor described by he following non-auonomous secondorder ordinary differenial equaion: ẍ + ξẋ (α + β sin(ω))x + γx 3 =, (1) in which he do represens he derivaive wih respec o ime. Equaion (1) can be used o model some mechanical and engineering problems such as he one-mode ransverse vibraion of an axially moving beam wih harmonic flucuaion speed [15]. By increasing he ampliude of he parameric exciaion and fixing he oher parameers as ξ =.15, α = 1., Ω =., γ = 1., () Eq. (1) displays an infinie period-doubling bifurcaion roue o chaos [1]. According o Ref. [1], he dynamics of he Mahieu-Duffing oscillaor can be summarized as follows: a) For β (3.,.85 77), Eq. (1) exhibis he sable period-1 moions, where period-1 means T = π/ω = π. b) For β (.85 77, ), Eq. (1) exhibis he sable period- moions. c) For β ( , 5.5 5), Eq. (1) exhibis he sable period- moions. d) For β (5.5 5, 5. 5), Eq. (1) exhibis he sable period-8 moions. e) For β (5. 5, 5.7 1), Eq. (1) exhibis he sable period-1 moions, and so on.

3 An open-plus-closed-loop conrol for chaoic Mahieu-Duffing oscillaor 1 f) When he conrol parameer reaches β 5.3, he Mahieu-Duffing oscillaor eners ino chaos, which can be seen from Figs. 1 and. x() x() x() Fig. 1 The ime hisory of he chaoic rajecory of Eq. (1) Fig. The phase porrai of he chaoic rajecory of Eq. (1) Also, he analyical approximaions for he periodic and higher-periodic orbis under he cerain parameer values were explicily calculaed in Ref. [1]. For example, when β =., he analyical approximaion for he period-1 orbi is given by x() = cos(Ω) cos(Ω) cos(3Ω) cos(Ω) cos(5ω) cos(Ω) cos(7Ω) cos(8ω) sin(Ω) sin(Ω) sin(3ω) sin(Ω) sin(5ω). 58 1sin(Ω) sin(7Ω) sin(8ω), (3) in which Ω =.. An OPCL conrol for chaoic Mahieu-Duffing oscillaor Supposing ha (x(), ẋ()) is an arbirary periodic orbi (including he higher-periodic orbi) and (y(), ẏ()) is an arbirary chaoic rajecory, our aim is o have lim y() x() = lim ẏ() ẋ() =. () When β = β p, he periodic orbi x() saisfies and when β = β c, he chaoic orbi y() saisfies ẍ + ξẋ (α + β p sin(ω))x + γx 3 =, (5) ÿ + ξẏ (α + β c sin(ω))y + γy 3 =. () Adding he acion of he conroller u() o Eq. () yields ÿ + ξẏ (α + β c sin(ω))y + γy 3 = u(). (7) By inroducing an error variable e = y x, consequenly, Eq. (7) becomes (ë + ẍ) + ξ(ė + ẋ) (α + β c sin(ω))(e + x) + γ(e + x) 3 = u(). (8)

4 Jian-he SHEN and Shu-hui CHEN Simplifying Eq. (8) yields ë + ξė (α + β c sin(ω) 3γx )e + 3γxe + γe 3 + β sin(ω)x = u(), (9) in which β = β p β c. If he conroller is designed o be in which and u() = u o () + u c (), (1) u o () = β sin(ω)x (11) u c () = [k 1 (α + β c sin(ω)) + 3γx ](y x) + (k + ξ)(ẏ ẋ) (1) can be regarded as he open-loop and closed-loop conrol pars, respecively, and k i (i = 1, ) are he feedback coefficiens, Eq. (9) urns ou o be ë k ė k 1 e + 3γxe + γe 3 =. (13) Obviously, e = is he equilibrium of Eq. (13). Because of he boundedness of he periodic orbi x(), he local sabiliy of Eq. (13) a he equilibrium e = is deermined by is linearized equaion ë k ė k 1 e =. (1) According o Hurwiz crierion, if he feedback coefficiens are boh negaive, Eq. (1) is globally asympoically sable which means ha Eq. (13) is locally asympoically sable. Therefore, by he ergodiciy of chaos, i can be concluded ha he conrol goal () is sponaneously saisfied if he chaoic rajecory nears he arge periodic orbi. In fac, in he nex secion, he global asympoic sabiliy of Eq. (13) will be proved which indicaes ha any chaoic rajecories in he phase space can be conrolled o he goal periodic orbis under he acion of he OPCL conroller (1). 3 Global basin of enrainmen Firs, a comparaive heorem of iniial value problems for he second-order ordinary differenial equaions is given. Lemma 1 Consider he following iniial value problems for he second-order ordinary differenial equaions: { z1 = f(, z 1, ż 1 ), z 1 ( ) = z, ż 1 ( ) = z, (15) and { z = F(, z, ż ), z ( ) = z, ż ( ) = z, in which he do denoes he derivaive wih respec o ime. For any (, z i, ż i ) R 3, i = 1,, if he righ-hand side vecor fields of Eqs. (15) and (1) saisfy (1) f(, z 1, ż 1 ) F(, z, ż ), (17) he soluions z 1 () and z () of iniial value problems (15) and (1) saisfy, respecively, z 1 () z (),.

5 Thus, An open-plus-closed-loop conrol for chaoic Mahieu-Duffing oscillaor 3 Proof Define a funcion φ() = z 1 () z (), and i can be known from Eq. (17) ha and furhermore, which finally indicaes ha φ() = z 1 () z () = f(, z 1, ż 1 ) F(, z, ż ). (18) φ() φ( ) =,, (19) φ() φ( ) =,, () z 1 () z (),. (1) This ends he proof of Lemma 1. Equaion (13) is a non-auonomous differenial equaion, and is global asympoic sabiliy is difficul o deermine. For his reason, he global asympoic sabiliy of he following wo comparaive equaions of Eq.(13) which are auonomous are analyzed: ë k ė k 1 e + 3γM 1 e + γe 3 =, () ë k ė k 1 e + 3γM e + γe 3 =, (3) in which M 1 = min x(), M = max x(), T is he period of x(), and γ >. [, T] [, T] If he global asympoic sabiliy of Eqs. () and (3) a he equilibrium e = is proved, he global asympoic sabiliy of Eq. (13) a he equilibrium e = can be obained by aking Lemma 1 ino accoun. In his manner, he global basin of enrainmen of he OPCL conrol for he chaoic Mahieu-Duffing oscillaor is demonsraed. Lemma Equaion () is globally asympoically sable a he equilibrium e = if he feedback coefficiens saisfy k 1 γm 1 <, k <. Proof Consruc he following Lyapunov funcion: V (e, ė) = ė k 1e + γm 1 e 3 + γe = ė + e (γ (e + M 1) k 1 γm1 ). () The Lyapunov funcion () is posiive definie if he feedback coefficien saisfies Furhermore, k 1 γm 1 >. dv (e, ė) Eq. () = k ė d is negaive definie if k <. Consequenly, he conclusion in Lemma can be drawn according o he Lyapunov sabiliy heory [1]. Similarly, he following conclusion can be drawn.

6 Jian-he SHEN and Shu-hui CHEN Lemma 3 Equaion (3) is globally asympoically sable a he equilibrium e = if he feedback coefficiens saisfy k 1 γm <, k <. From Lemmas 1 3, one can obain he following conclusion direcly. Theorem 1 If he feedback coefficiens saisfy k 1 min{ γm, γm 1 } <, k <, (5) he basin of he OPCL conrol for he chaoic Mahieu-Duffing oscillaor is global. Numerical simulaions.1 Conrolling he chaoic rajecories o a period-1 orbi Wih he parameer β = 5.3, he soluion rajecory of Eq. (1) is chaoic, and wih he parameer β =., Eq. (1) exhibis a period-1 soluion. We firs wan o conrol he chaoic rajecory o his period-1 orbi which is currenly unsable and embedded in he chaoic aracor. Subsiuing he parameer values in Eq. () and β =.7 ino Eq. (1), i urns ou ha u() =.7xsin + [(k sin + 3x )(y x) + (k +.5)(ẏ ẋ)]. () I can be noed from Eq. () ha he arge periodic orbis are involved in he conroller. Since he exac and analyical expressions for hese orbis canno be obained easily, hey are usually calculaed by numerical inegraions or analyical approximaions. In his paper, we choose he laer. From he view of pracical conrol, he advanage of using he analyical approximaions is ha hey are more convenien o implemen. Chen and Dong [17] and Li e al. [18] applied he harmonic balance mehod o obain he firs-order approximaion for he arge periodic orbis. However, since he accuracy of he firs-order approximaion is somewha low, he performances of heir conrol are no good enough. Here, we uilize he higher-order approximaion obained in our previous paper [1] by he incremenal balance mehod. For insance, we can use Eq. (3) as he analyical approximaion for he arge period-1 orbi of he Mahieu-Duffing oscillaor wih he parameer β =.. The performance of conrolling he chaoic rajecory o he arge period-1 orbi is illusraed in Figs. 3 and. I is shown in he figures ha he originally chaoic rajecory can be driven acually and quickly o he arge period-1 orbi under he acion of he OPCL conroller (). In he simulaions, we se M 1 and M o be M = M 1 = 3. respecively from Eq. (3), and choose he feedback coefficiens as k 1 =. and k =. such ha he condiion (5) in Theorem 1 is saisfied. Thus, according o Theorem 1, he basin of he enrainmen of he OPCL conrol is global. Hence, he iniial condiion of he chaoic rajecory can be chosen arbirarily, for example (y(), ẏ()) = (5., 5.).. Conrolling he chaoic rajecories o a higher-periodic orbi Similarly, he chaoic rajecories of he Mahieu-Duffing oscillaor can be conrolled o he higher-periodic orbis we are ineresed in. Choosing he same feedback coefficiens and he same iniial condiions of he chaoic rajecory wih hose in Figs. 3 and, he simulaions of conrolling he chaoic rajecory o he arge period- and period- orbis of he Mahieu- Duffing oscillaor wih he parameers β =.8 and β = 5.1 are shown in Figs. 5 and Figs. 7 8, respecively. The conrol of he chaoic rajecory o he oher arge orbis wih higher periodiciy can be carried ou in he same way and hence is omied here.

7 An open-plus-closed-loop conrol for chaoic Mahieu-Duffing oscillaor 5 y() y() y() Fig. 3 The ime hisories of he conrolled chaoic rajecory ( ) and he numerical arge period-1 orbi ( ) (k 1 =. and k =.) Fig. The phase porrais of he conrolled chaoic rajecory ( ) and he numerical arge period-1 orbi ( ) (k 1 =. and k =.) y() y() y() Fig. 5 The ime hisories of he conrolled chaoic rajecory ( ) and he numerical arge period- orbi ( ) (k 1 =. and k =.) Fig. The phase porrais of he conrolled chaoic rajecory ( ) and he numerical arge period- orbi ( ) (k 1 =. and k =.) 3 y() 1 y() y() Fig. 7 The ime hisories of he conrolled chaoic rajecory ( ) and he numerical arge period- orbi ( ) (k 1 =. and k =.) Fig. 8 The phase porrais of he conrolled chaoic rajecory ( ) and he numerical arge period- orbi ( ) (k 1 =. and k =.) 5 Furher discussions Remark 1 Theorem 1 ells us ha any chaoic rajecories saring from he basin of aracion can be conrolled o he arge periodic orbis. Acually, no only he chaoic responses,

8 Jian-he SHEN and Shu-hui CHEN bu also he non-chaoic responses can each be conrolled o he arge periodic responses under he acion of he presen OPCL conroller. Remark I can be known from he above discussions ha k 1c = min{ γm 1, γm } and k c = are he criical feedback coefficiens under which he chaoic Mahieu-Duffing oscillaor is globally OPCL conrollable, while k 1c = and k c = are he criical ones under which he chaoic Mahieu-Duffing oscillaor is locally OPCL conrollable only. Remark 3 I can be concluded from Remark ha he presen OPCL conroller canno fulfil he conrol of he chaoic rajecory o he arge periodic orbis if he feedback coefficiens in Eq. (1) are chosen o be posiive. Numerical simulaion demonsraes his conclusion. For example, replacing he feedback coefficiens in Figs. 3 and by k 1 =.5 and k =.3, he simulaing resul of enraining he chaoic rajecory o he period-1 orbi is illusraed in Figs. 9 and 1. From hese figures, i can be seen ha he conrolled chaoic rajecory does no converge o he arge period-1 orbi. I diverges o infiniy as ime increases. Remark Numerical simulaions show ha he linear sae error feedback conroller u() = k(y x) (7) is sufficien o enrain he chaoic rajecory o he arge periodic orbis if he feedback gain k is chosen o be relaively large (in he sense of absolue value in his paper). For example, choosing k = in Eq. (7), he conrolling of he chaoic rajecory o he period-1 orbi is simulaed and given in Fig. 11 which demonsraes he above claim. However, i is difficul o prove his fac heoreically. y() y() y() Fig. 9 The ime hisories of he conrolled chaoic rajecory ( ) and he numerical arge period-1 orbi ( ) (k 1 =.5 and k =.3) Fig. 1 The phase porrais of he conrolled chaoic rajecory ( ) and he numerical arge period-1 orbi ( ) (k 1 =.5 and k =.3) y() y() Fig. 11 Linear sae error feedback conrol wih k =. ( he phase porrai of he conrolled chaoic rajecory; he phase porrai of he numerical arge period-1 orbi)

9 An open-plus-closed-loop conrol for chaoic Mahieu-Duffing oscillaor 7 Remark 5 Alhough he paper discussed only he Mahieu-Duffing oscillaor, i is believed ha he idea of his paper can be exended o perform he esimaions of basin of enrainmen in oher nonlinear oscillaors. Conclusion An OPCL conrol law for enraining he chaoic rajecories of he Mahieu-Duffing oscillaor o is arbirarily arge periodic and higher-periodic orbis is designed. The global basin of enrainmen of he OPCL conrol is explored by combining he Lyapunov sabiliy heory wih a comparaive heorem of iniial value problems for second-order ordinary differenial equaions. The criical feedback coefficiens under which he chaoic Mahieu-Duffing oscillaor is globally and locally OPCL conrollable respecively are obained heoreically and demonsraed numerically. References [1] O, E., Grebogi, C., and Yorke, J. Conrolling chaoss. Physical Review Leers (11), (199) [] Andrievskii, B. R. and Fradkov, A. L. Conrol of chaos: mehods and applicaions. I. mehod. Auomaion and Remoe Conrol (5), (3) [3] Fradkov, A. L. and Evans, R. J. Conrol of chaos: mehods and applicaions in engineering. Annual Reviews in Conrol 9, 33 5 (5) [] Boccalei, S., Grebogi, C., Lai, Y. C., Mancini, H., and Maza, D. The conrol of chaos: heory and applicaions. Physics Repors 39, () [5] Chen, G. R. and Dong, X. N. From Chaos o Order: Perspecives, Mehodologies and Applicaions, World Scienific, Singapore (1998) [] Jackson, E. A. and Grosu, I. An open-plus-closed-loop conrol of complex dynamics sysems. Physica D 85, 1 9 (1995) [7] Jackson, E. A. The OPCL conrol for enrainmen, model-resonance and migraion acions on muli-aracor sysems. Chaos 7(), (1997) [8] Wheeler, D. W. and Schieve, W. C. Enrainmen conrol in a noisy neural sysem. Physical Review E 7(19), 1 (3) [9] Chen, L. Q. An open-plus-closed-loop conrol for discree chaos and hyperchaos. Physics Leers A 81, (1) [1] Chen, L. Q. and Liu, Y. Z. An open-plus-closed-loop approach o synchronizaion of chaoic and hyperchaoic maps. Inernaional Journal of Bifurcaion and Chaos 1(5), () [11] Chen, L. Q. and Liu, Y. Z. A parameric open-plus-closed-loop approach o conrol chaos in nonlinear oscillaions. Physics Leers A, (1999) [1] Chen, L. Q. and Liu, Y. Z. The parameric open-plus-closed-loop conrol of chaoic maps and is robusness. Chaos, Solions and Fracals 1, () [13] Tian, Y. C., Tadé, M. O., and Tang, J. Y. Nonlinear open-plus-closed-loop conrol of dynamics sysems. Chaos, Solions and Fracals 11, () [1] Sloine, J. J. E. and Li, W. P. Applied Nonlinear Conrol, China Machine Press, Beijing () [15] Ravindra, B. and Zhu, W. D. Low-dimensional chaoic response of axially acceleraing coninuum in he supercriical regime. Archive of Applied Mechanics 8, (1998) [1] Shen, J. H., Lin, K. C., Chen, S. H., and Sze, K. Y. Bifurcaion and roue-o-chaos analyses for Mahieu-Duffing oscillaor by he incremenal harmonic balance mehod. Nonlinear Dynamics 5(), 3 1 (8) [17] Chen, G. R. and Dong, X. N. On feedback conrol of chaoic coninuous-ime sysems. IEEE Transacions on Circuis and Sysems-1 : Fundamenal Theory and Applicaions (9), (1993) [18] Li, R. H., Xu, W., and Li, S. Chaos conrol and synchronizaion of he Φ - van der Pol sysem driven by exernal and parameric exciaions. Nonlinear Dynamics 53, 1 71 (8)