Synchronization of Fractional Chaotic Systems via Fractional-Order Adaptive Controller

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1 Synchonizaion of Facional Chaoic Sysems via Facional-Ode Adapive Conolle S.H. Hosseinnia*, R. Ghadei*, A. Ranjba N.*, J. Sadai*, S. Momani** * Noshivani Univesiy of Technology, Faculy of Elecical Compue Engineeing, P.O. Box , Babol, Ian, (a.anjba@ni.ac.i),(h.hoseinnia@su.ni.ac.i) ** epamen of Mahemaics, Muah Univesiy, P.O. Box: 7, Al-Kaak, Jodan Absac: In his pape, an adapive facional conolle has been designed o conol chaoic sysems. In fac, his conolle is a facional conolle, which he coefficiens will be uned accoding o a pope adapaion mechanism. The adapaion law will be consuced fom a sliding suface via gadien mehod. The adapive facional conolle is implemened on a gyo sysem o signify he pefomance of he poposed echniue. Keywods: Facional adapive conolle, Sliding-Mode Conol, Facional-ode conolle, gyo sysem, Chaos. INTROUCTION Facional calculus is an old mahemaical opic since 7h cenuy. Alhough i has a long hisoy, is applicaions o physics engineeing ae jus a ecen focus of inees. Many sysems ae known o display facional ode dynamics, such as eahuake oscillaion (He, 998), Riccai (Odiba, Momani, 8; Cang e. al,. 7), wave euaion (Jafai Momani, 7), chaoic euaions in conol engineeing (Ge Ou, 8). Thee is a new opic o invesigae he conol dynamics of facional ode dynamical sysem. The behaviou of nonlinea chaoic sysems when hei models become facional have widely been invesigaed (Li Chen, 4; Ahmad Hab Ahmad, ; Ahmad e. al., 7; Ahmad, 5; Nimmo Evans, 999). Sensiive dependence on iniial condiions is an impoan chaaceisic of chaoic sysems. Theefoe, chaoic sysems ae difficul o be synchonized o conolled. A chaeing-fee fuzzy sliding-mode conol (FSMC) saegy fo unceain chaoic sysems has been poposed in (Yau Chen, 6). In (Zhang e. al., 4) he auhos poposed an acive sliding mode conol mehod fo synchonizing wo chaoic sysems peubed by paameic unceainy. An algoihm o deemine paamees of he acive sliding mode conolle in synchonizing diffeen chaoic sysems has been sudied by (Tavazoei Haei, 7). In (Yau, 4) an adapive sliding mode conolle is pesened fo a class of mase slave chaoic synchonizaion sysems wih unceainies. In (Wang Ge, ) backsepping conol has been poposed o synchonize he chaoic sysems. Even hough, synchonizaion has been implemened in many chaoic sysems wih inege deivaives, bu a few woks ae epoed on facional ode chaoic sysem. I is because; poof of sabiliy of he facional ode is moe complex han he sysem wih inege ode. In his pape, an adapive facional conolle has been poposed as a novel idea o conol sysems wih facional ode dynamic. This conolle is in essence a conolle bu facional chaaceisics. coefficiens K P, K I K will be updaed accoding o a pope gadien-based adapaion mechanism. This pape is oganized as follows: Pimaily, he poposed facional conolle will be pesened in secion, o conol such simila sysems. The pefomance of he conolle will be invesigaed when i is used o synchonize a gyo dynamic. Ulimaely, he wok will be concluded a secion 4.. FRACTIONAL AAPTIVE CONTROLLER ESIGN The following model epesens a chaoic sysem wih facional ode dynamic: x = x x = f( X, ) ()

2 whee, < X = [ x, x ] is he sae veco. T Conside he model in () as a mase. A seconday goal is o synchonize a usually simple dynamic, called slave, o follow a known sysem, called Mase. Fom poin of view of he slave, funcion f (.) in () is an unknown nonlinea funcion. A facional dynamic of slave can be geneally epesened as: y = y y = f( Y, ) +Δ f( Y, ) + d( ) + u( ),< whee, T () Y = [ y, y ] is sae of he slave dynamic, Δf (.) ss fo unceainy, dis () disubance u () is he conol signal o synchonize he slave wih he mase. I is suggesed ha o synchonize via an adapive facional conolle. The synchonizaion eo is defined by ei = xi y whee i =,. Schemaic diagam of he i closed loop sysem ogehe wih he poposed adapive facional conolle is shown in Fig.(). Facional Conolle Slave Sysem Mase Sysem Adapaion Law Fig.: Schemaic diagam of a synchonizaion mechanism Supposed ha conolle is of he following fom: α α α u () = K () P e() + K I e() + K e() I should be noed ha he conolle would be of he classic one ifα =, α =, α =. The eason behind he selecion is ha his kind of conolle is mos popula in he lieaue. Fuhemoe, he facional conolle povides he sabiliy wih moe degee of feedom (Tavazoei Haei, 8). To have a facional ode, paamees ae chosen as α <, α < α <. Paamees of conolle, i.e. K, P K I K will be updaed via a pope gadien- based adapaion mechanism o povide a obus synchonizing conolle (Chang, 5). The following facional ode diffeenial euaion descibes a followe dummy oupu sae y by: + - y (4) = x + ke + ke The sliding suface will also be defined as he eo beween wo oupus, which is as follows: S = y (5) y When he sliding mode is acivaed i.e. S =, heefoe we have: y (6) = y Since e = x y e = e&, eplacing euaion (6) in (4) immediaely esuls as: e (7) + k e+ ke = O in a sae space foma: e = e E= AE (8) e = ke ke whee, A = k k is he gain maix e E = e is he eo veco. If gains k k ae popely chosen such ha he sabiliy condiion in ineualiy ag( eig( A)) > π / holds, heefoe he eo e () asympoically ends o zeo when. Le us cidae he following funcion as a Lyapunov funcion in em of he sliding suface: V = S (9) The sliding condiion will be of he fom: V& = SS& < () When euaion () is me, unboundedness of he sliding suface will be guaaneed when ime ends o infiniy. This means S () when. The gadien seach algoihm is calculaed in he diecion opposie o he enegy flow. Moeove, i is uie inuiive o choose SS & as an eo funcion. Fom euaion (5) using euaion (), we have: S& = y& y& = ( ( y)) y& = () ( f( Y, ) +Δ f( Y, ) + d( ) + u ) y&. Pe muliplying boh sides of euaion () by S yields: SS& = S[ ( f ( Y, ) +Δ f ( Y, ) + d( ) + u ) y& ]. () Le us define he following euaion: U = ( u ) () coefficiens will be obained if one uses he gadien of he adapaion law (Chang, 5), which ae as follows: (4) α KP = γ = γ = γs ( e ( )) KP U KP

3 ( ( )) (5) α KI = γ = γ = γs e KI U KI (6) α K = γ = γ = γs ( e ( )) K U K Subsiuion ofα =, α = α = in euaions (4) (6) esuls he following fom: K& () P = γse (7) K & I = γs e () τ dτ (8) d (9) K& = γs e () d whee, γ > is he leaning ae. I should be noed ha γ K, K P I K should be caefully seleced o mainain he convegence (Chang, 5). The poposed conolle is applied on a facional gyo dynamic in a synchonizaion ask o show he pefomance of he echniue.. SYNCHRONIZATION OF UNCERTAIN FRACTIONAL GYRO SYSTEM. Sysem descipion Accoding o he sudy by Chen (Chen, ), dynamics of a symmeical gyo wih linea-pluscubic damping of he angle θ can be expessed as (Yau, 8): ( cos θ) θ& + a bsinθ + c & θ + c & θ () βsin( ω)sinθ =, whee, βsin( ω) epesens a paameic exciaion, cθ & c & θ ae linea nonlinea damping ems, ( cos θ ) a bsinθ is a especively, nonlinea esilience foce. Given he saes x = θ, x = & θ, ( cos θ ) f( θθ, & ) = a c & θ c & θ + ( b+ βsin( ω))sinθ, his sysem can be ansfomed ino he following nominal sae fom: x& = x () x& = f( x, x) This gyo sysem demonsaes complex dynamics. The behavio has been sudied by Chen (Chen, ) fo vaiey of β in he ange < β < 6 consan values of a =, b =, c =.5, c =.5 ω =. Le us conside he facion gyo dynamic in he following sae space foma: x = x () x = f( x, x) Fig.() shows he phase poai of gyo chaoic sysem wih facional deivaive in pesence of β = 5.5, iniial condiions of ( x, x ) = (, ) =.97. To show he effeciveness of he poposed conolle, he pocedue is implemened on facional gyo dynamic.. Implemenaion Conside sysem () as a mase, which is peubed wih such an unceainy. A slave sysem may be defined as he following euaion: y = y () y = f( y, y) +Δ f( y, y) + d ( ) + u ( ) Iniial condiions of mase slave saes ae inenionally defined diffeenly as x () =, x () =, y () =.6 y () =.8, especively. In ode o chose an unceainy disubance, Δ f( y, y) =.sin( y) d () =.cos( π ) ae assigned, especively. Pimaily seing of coefficiens ae chosen eual o k p () =, k = K = he leaning ae has been seleced as γ =. Fuhemoe k k in () ae seleced as.5, especively. Simulaion esuls have shown in Figs. () o (6). In Fig. (), synchonizaion of x, y x, y ae made pefec. The sliding suface he conol inpu ae shown in Fig. (4) (5) especively, wheeas Fig. (6) shows he synchonizaion eo. I should be noed ha he conol signal, u () has been acivaed in = s. x x Fig.: Phase poai of facional gyo chaoic sysem 4. CONCLUSION: An adapive facional conolle is poposed o synchonize a chaoic sysem. Coefficiens paamees of he conolle ae updaed using a gadien-based adapaion mechanism. The conolle has successfully been applied on he dynamic of facional gyo sysem. The simulaion esuls veify he significance of he poposed conolle.

4 .5 x,y x,y Fig. : Synchonizaion of Facional gyo s sysem 4 S() Fig. 4: Sliding suface 5 u() Fig. 5: The conol signal

5 .5 e e Fig. 6: Eo of synchonizaion REFRENCES: Ahmad W. M, Hab Ahmad M.(), On nonlinea conol design fo auonomous chaoic sysems of inege facional odes, Chaos, Solions & Facals, 8, Ahmad W M, Reyad El-Khazali, Yousef Al-Assaf. (4) Sabilizaion of genealized facional ode chaoic sysems using sae feedback conol, Chaos, Solions & Facals,,4 5. Ahmad WM. (5), Hypechaos in facional ode nonlinea sysems. Chaos, Solions & Facals, 6, Aena. P, R. Caponeo, L. Founa,. Poo (997), Chaos in a facional ode uffing sysem, Poceedings ECCT Budapes, Cang, J., Yue Tan, Hang Xu, Shi-Jun Liao (7), Seies soluions of non-linea Riccai diffeenial euaions wih facional ode, Chaos Solions Facals, In Pess. Chang, W- J-J Yan (5), Adapive obus conolle design based on a sliding mode fo unceain chaoic sysems,chaos, Solions & Facals, 6(), Chen, H.K. (), Chaos chaos synchonizaion of a symmeic gyo wih linea-plus-cubic damping, Jounal of Sound Vibaion, 55, Ge, Z-M, Chan-Yi Ou (8), Chaos synchonizaion of facional ode modified duffing sysems wih paamees excied by a chaoic signal, Chaos, Solions & Facals, 5(4), He, JH.(998), Nonlinea oscillaion wih facional deivaive is applicaions, Inenaional Confeence on Vibaing Engineeing 98, alian, China, Jafai, H., Shahe Momani (7), Solving facional diffusion wave euaions by modified homoopy peubaion mehod, Physics Lee A, 7( 5-6), Li C, Chen G. (4), Chaos in he facional ode Chen sysem is conol, Chaos, Solions & Facals,, Nimmo S, Evans AK. (999), The effecs of coninuously vaying he facional diffeenial ode of chaoic nonlinea sysems. Chaos, Solions & Facals,, 8. Odiba, Z., S. Momani (8), Modified homoopy peubaion mehod: Applicaion o uadaic Riccai diffeenial euaion of facional ode, Chaos, Solions Facals, 6(), Tavazoei M. S. M. Haei (7), eeminaion of acive sliding mode conolle paamees in synchonizing diffeen chaoic sysems, Chaos, Solions Facals, Tavazoei M. S. M. Haei (8). Chaos conol via a simple facional-ode conolle,physics Lees A, 7(6), Wang C, Ge SS (), Adapive synchonizaion of unceain chaoic sysems via backsepping design. Chaos, Solions & Facals,,99 6. Yau, H-T (4), esign of adapive sliding mode conolle fo chaos synchonizaion wih unceainies, Chaos, Solions Facals,,4 47 Yau, H-T, Chieh-Li Chen (6),Chaeing-fee fuzzy sliding-mode conol saegy fo unceain chaoic sysems, Chaos, Solions Facals, Yau, H-T (8), Chaos synchonizaion of wo unceain chaoic nonlinea gyos using fuzzy sliding mode conol, Mechanical Sysems Signal Pocessing, (), Zhang H, Xi-Kui Ma a,b, Wei-Zeng Liu (4), Synchonizaion of chaoic sysems wih paameic unceainy using acive sliding mode conol, Chaos, Solions Facals, 49 5.

On Control Problem Described by Infinite System of First-Order Differential Equations

On Control Problem Described by Infinite System of First-Order Differential Equations Ausalian Jounal of Basic and Applied Sciences 5(): 736-74 ISS 99-878 On Conol Poblem Descibed by Infinie Sysem of Fis-Ode Diffeenial Equaions Gafujan Ibagimov and Abbas Badaaya J'afau Insiue fo Mahemaical