Control and Synchronization of Chaotic Fractional-Order Coullet System via Active Controller

Transcription

1 Control and Synchronzaton of Chaotc Fractonal-Order Coullet System va Actve Controller M. Shahr T.*, A. Ranjbar N.*, R. Ghader*, S. H. Hossenna*, S. Moman** * Noushrvan Unversty of Technology, Faculty of Electrcal and Computer Engneerng, P.O. Box , Babol, Iran, (a.ranjbar@nt.ac.r),(h.hosenna@stu.nt.ac.r) ** Department of Mathematcs, Mutah Unversty, P.O. Box: 7, Al-Karak, Jordan Abstract: In ths paper, fractonal order Coullet system s studed. An actve control technue s appled to control ths chaotc system. Ths type of controller s also appled to synchronze chaotc fractonal-order systems n master slave structure. The synchronzaton procedure s shown va smulaton. The boundary of stablty s obtaned by both of theoretcal analyss and smulaton result. The numercal smulatons show the effectveness of the proposed controller. Keywords: Fractonal-order Dfferental Euatons (FDEs), Chaos, Coullet system, Synchronzaton, Nonlnear control, Actve control.. INTRODUCTION Fractonal calculus has -year hstory. However, applcatons of fractonal calculus n physcs and engneerng have just begun (Hlfer, ). Many systems are known to dsplay fractonal order dynamcs, such as vscoelastc systems (Bagley and Calco, 99), delectrc polarzaton, and electromagnetc waves. In the recent years, emergence of effectve methods n dfferentaton and ntegraton of non-nteger order euatons makes fractonal-order systems more and more attractve for the systems control communty. The TID controller (Lune, 994.), the fractonal PID controller (Podlubny, 999), the CRONE controllers (Oustaloup, 999,996, 995) and the fractonal lead-lag compensator (Raynaud and Zerga, ; Monje and Llle, 4.) and Smple Fractonal Controller (Tavazoe and Haer, 8), are some of well-known fractonal-order controllers. More recently, there has been a new trend to nvestgate the control and dynamcs of fractonal order chaotc systems (Ahmad, a, b, c ; L, 4, ). It has been shown (Ahmad, b) that nonlnear chaotc systems may keep ther chaotc behavor when ther models become fractonal. In (Ahmad, a), chaos control was successfully nvestgated for fractonal chaotc systems, where controllers have been desgned usng a backsteppng technue. It was demonstrated that nonlnear controllers, whch s desgned to stablze the ntegral chaotc model, mght stll stablze the fractonal order model (Ahmad, 4). It s smultaneously shown that chaos exsts n the fractonal order Chen system when the order s less than. A Lnearzng feedback technue has successfully been appled on ths chaotc system. In (L et. al, ), synchronzaton of fractonal order chaotc systems has been studed. Utlzng an approxmaton approach of fractonal operator, the system behaves chaotc, when the order s less than number of state.e.. Chaos control and synchronzaton have wdely been studed for an ntegral order system (Ge and et. al., 5a, b, 4a, b, c, d). The work wll be generalzed for a fractonal order Coullet system. Synchronzaton of Coullet system s reported n (Jan-Bng et. al., 8) usng backsteppng method. There s stll lack of report to control and synchronze a fractonal-order Coullet system. In ths paper, the am s to control and synchronze two chaotc fractonal-order Coullet systems, usng an Actve Control. Ths paper s organzed as follows: Coullet system wll be descrbed n secton. Actve controller wll be presented n secton. Ths controller s appled to synchronze two dentcal fractonal-order Coullet systems n secton 4. Ultmately, the work wll be concluded at secton 5.. SYSTEM DISCRIPTION Some of nonlnear systems represent a chaotc behavour. These systems are very senstve to ntal condtons. Ths means two dentcal dstnct systems but wth a mnor devaton n ther ntal condton may result completely dfferent. Ths means havng known bounded ntal condtons, there s less chance for the dynamcs to predct the behavour. Ths determnstc treatment s called chaos. Consder the followng Arnéodo Coullet dynamcs euaton (Arnéodo et. al., 98):

2 x& + ax& + bx& + cx + dx = () =.97 whch exhbts chaotc dynamcs for varous values of four parameters (Coullet et. al., 979; Arnéodo et. al.,.5 98). The system s smulated for the followng set {a=.8, b=-., c= -.45,d=-}. x -.5 The system n () wll be wrtten n the state space format as: x& x& () x& = cx + bx + ax+ dx A phase portrat of the system n Fgure () represents a chaotc behavour. - x = x.5.5 x x x x x = x.5 Fg.. Phase portrat of Chaotc Coullet system x Although some works have reported on chaotc Coullet system, there s stll lack of report on fractonal order Coullet system. In ths paper, an actve controller as a new control system s appled on fractonal order chaotc system. To defne the fractonal order Coullet chaotc system the euaton wll be defned as (), n whch wll changed n accordance to fractonal order of defned system. It can be shown that for some range of, the fractonal order Coullet system s un (Tavazoe and Haer, 8). A resonance property of fractonal order Coullet chaotc system s shown n Fgure () for dfferent values of =.97,.95, and.9. The numercal smulatons have carred out usng the SIMULINK based on the freuency doman approxmaton. & = & & D x x D x D x = cx + bx + ax+ dx + u() t () x.5 - Fg.. Phase portrat of chaotc Coullet system for dfferent values of the fracton parameter. In contnung the system wll be stablzed usng an actve control.. ACTIVE CONTROL OF COULLET SYSTEM In the actve control method, the control sgnals are drectly added to the fractonal-order system dynamcs as: Dx Dx Dx = f( x) + u( t) (4) The control sgnals u constructed from two parts. The frst part s consdered to elmnate the nonlnear part of euaton (4). The second part vt () acts as external nput n (4) whch s desgned to stablze the system: ut () = - f( x) + vt () (5) where, v = k x k x k x (6) -.5 x.5

3 where, k s chosen accordng to the desgner. The desgn procedure conssts of spottng gans k to stablze dynamcs n (4). In the followng, relevant matrces and relatons are gven for Control of Coullet chaotc system. Now, regardng the fractonal order Coullet system n euaton (4), the control nput s determned as: ut () = cx bx ax dx + v (7) Dynamcs n (7) can be represented by a fractonal state space dynamcs: = = D x x D x D x v (8) A proper adjustment of the gan parameter locates un egenvalue to a poston. The euaton wll be rewrtten n the followng form: D X = AX, A =.... k k... kn (9) It s evdent that egenvalues of matrx A spots the stablty performance. The stablty wll be shown n the followng for fractonal system.. Stablty analyss for Fractonal order systems A fractonal order lnear nvarant (LTIFO) system may be defned n the followng state-space format: α D x = Ax + Bu y = Cx n r () where, x R, u R and y R denote states, nput and output vectors of the system wll be shown n n by A R n r, B R p n and C R respectvely, and s the fractonal commensurate order. Fractonal order dfferental euatons are at least as as ther nteger orders counterparts, because systems wth memory are typcally more than ther memory-less alternatves (Ahmed. E. et al, 7; Hosen na et. al, 8; Ranjbar et. al, 8). It has been shown that the autonomous dynamcs D x = Ax, x() s asymptotcally f the followng condton s met (Matgnon and Llle, 996): arg( eg( A)) > π / () p where, < < and eg(a) represents the egenvalues of matrx A. In addton, ths system s f arg( eg ( A)) π / and those crtcal egenvalues whch satsfy arg( eg ( A)) = π / have geometrc multplcty of. The stablty regon for < α < s shown n Fgure. ω απ / απ / Un Un Fg.. Stablty regon of the LTIFO system wth fractonal order, < α < Agan the gan K wll be chosen accordng to euaton (9). The stablty regon wll be obtaned by euaton () to stablze the system.. applyng ths controller on fractonal-order Coullet systems Let us consder the fractonal order Coullet chaotc system regardng the parameters a=.8, b= -., c= -.45,d=-, =.97 and controller u (t): D x D x D x =.45x.x +.8 x x + u( t) σ () From euaton (7) the control nput of system () s determned as: u =+.45x+.x.8x+ x + v () Intal condtons are chosen as: x () =, x () = and x () =. Parameters of the controllers are also selected as k =, k = 5and k = 4.The correspondng egenvalues are obtaned as -.9 and.6 ±.6894, whch satsfy the stablty condton of arg( eg ( A)) > π /. Ths means, the lnearzng state feedback and together wth proper parameter adjustment stablzes un system. In ths secton, numercal smulatons have carred out usng the SIMULINK based on proper solver. The employed step sze n ths smulaton s..

4 =.9 x - - Eulbrum pont x - - =.95 x x - x Eulbrum pont - - x = x - x Eulbrum pont x Fg. 4. Phase portrat of controlled system (left) and the control sgnal (rght) for =.97,.95,.9 The phase portrat of the controlled system (left) and the approprate control sgnal (rght) are shown n Fgure (4), consderng dfferent (=.97,.95,.9). As t can be seen, the system s approached to a steady state. Note that the control s actvated n t= Seconds. 4. SYNCHRONIZATION OF FRACTIONAL- ORDER COULLET SYSTEM Bascally, chaos synchronzaton problem means makng two systems oscllate n a same way. Let us call a partcular dynamcal system master and a dfferent dynamcal system a slave. The goal s to synchronze the slave system wth the master usng an actve controller. In order to acheve ths synchronzaton, a nonlnear control system that obtans sgnals from the master system and controls the slave system should be desgned. Let us consder two fractonal order Coullet systems wth dfferent ntal condton as master and slave systems respectvely: D x master D x D x cx bx ax dx ( x, x, x ) = ( a, b, c ) = D y = y slave D y = y D y = cy + by + ay + dy + u ( y, y, y ) = ( a, b, c ) (4) 5) ( As t has already been mentoned; the control sgnal should be desgned n a way whch the slave follows the master. The error s defned as the dscrepancy of the relevant states.e. e = y x for =,,.

5 (a) x,y x,y x,y (b) x,y x,y (c) x,y x,y x,y x,y Fgure 5: The control sgnal and synchronzaton of states x, x, x for a: =.97,b: =.95,c: =.9 Deducng the master dynamcs from the slave one, approaches us to: De = e De = e (6) De = ce + be + ae+ dx ( y) u e = y x (7) ut () = ce be ae+ d( x y) + v (8) By choosng the proper value for control parameters (K ), lm e = therefore x = y and then synchronzaton t between master and slave wll be happened. A lnearzng state-nput feedback usng MATLAB 7.4s used to synchronze a fractonal Coullet chaotc system. Intal condtons of the master and slave are chosen

6 as x () =, x () =, x () = and y () =.5 y () =.5 and y () = respectvely. The control law usng parameters as a=.8, b= -. and c= -.45 are obtaned consderng euaton (5) as follows: ut () =.45e (9).e +.8e x + y + v It should be noted that proper selecton of gan parameters k causes the error tends to zero, and therefore the slave follows the Master. Smulaton result for k = 4, k = 5and k = 8s shown n Fgure (5). The control sgnal and the synchronzaton of the states x, xand x are shown f Fgure (5) for =.97, =.95and =.9 n (a), (b) and (c) respectvely. When the control s actvated nt = s, those two systems are mmedately synchronzed. The smulaton results verfy the sgnfcance of the proposed controller on the fractonal system. 4. CONCLUSIONS In ths paper, control and synchronzaton of chaotc Coullet system wth fractonal orders s nvestgated. Actve Control method has been chosen to control ths chaotc system. It has been shown that by proper selecton of the control parameters (K ), the master and slave systems are synchronzed. Numercal smulatons show the effcency of the proposed controller n the synchronze task. REFERENCES: Ahmed, E.,A.M.A. El-Sayed and H.A.A. El-Saka (7). Eulbrum ponts, stablty and numercal solutons of fractonal order predator prey and rabes models. Journal of Mathematcs Analyss, 5 (), Ahmad, WM (4). Stablzaton of generalzed fractonal order chaotc systems usng state feedback control. Chaos. Soltons & Fractals,,4 5. Ahmad, WM and Harba AM (a). On nonlnear control desgn for autonomous chaotc systems of nteger and fractonal orders. Chaos, Soltons & Fractals, 8, Ahmad, W. and Sprott JC (b). Chaos n fractonal order system autonomous nonlnear systems. Chaos, Soltons & Fractals, 6, 9 5. Arnéodo, A., P. Coullet and C. Tresser (98). Possble new strange attractors wth spral structure. Communcaton n Mathematcal Physcs. 79(4), Bagley, RL. and Calco RA (99). Fractonal order state euatons for the control of vscoelastcally damped structures. Journal of Gude Control Dynamc,4, 4. Coullet, P., C. Tresser and A. Arnéodo (979). Transton to stochastcty for a class of forced oscllators. Physcs. Letter A, 7, 68. Ge, Z-M and Chen Y-S (5a). Adaptve synchronzaton of undrectonal and mutual coupled chaotc systems. Chaos, Soltons & Fractals, 6(), Ge, Z-M and Cheng J-W (5b). Chaos synchronzaton and parameter dentfcaton of three scales brushless DC motor system. Chaos, Soltons & Fractals, 4(), Ge, Z-M and Leu W-Y (4a). Chaos synchronzaton and parameter dentfcaton for loudspeaker systems. Chaos, Soltons & Fractals, (5), 47. Ge, Z-M and Chen Y-S(4b). Synchronzaton of undrectonal coupled chaotc systems va partal stablty. Chaos, Soltons & Fractals, (),. Ge, Z-M and Chang C-M (4c). Chaos synchronzaton and parameters dentfcaton of sngle scale brushless DC motors. Chaos, Soltons & Fractals,(4), Ge, Z-M and Chen C-C (4d). Phase synchronzaton of coupled chaotc multple scales systems. Chaos, Soltons & Fractals, (): Hlfer, R., (). Applcaton of fractonal calculus n physcs. New Jersey: World Scentfc. Hosen Na, S.H., A.N. Ranjbar and D.D. Ganj, H. Soltan, J. Ghasem (8). A Soluton of Rccat Nonlnear Dfferental Euaton Usng Enhanced Homotopy Perturbaton method(ehpm), Internatonal Journal of Engneerng, (), 7-8. Jan-Bng, Hu, Yan Han and Lng-Dong Zhao (8). Synchronzaton n the Geneso Tes and Coullet systems usng the Backsteppng Approach, Journal of Physcs, 96, -6. L, C and Chen G (4). Chaos n the fractonal order Chen system and ts control. Chaos, Soltons & Fractals,, L, C., Lao X and Yu J (). Synchronzaton of fractonal order chaotc systems. Physcs Revew E, 68, 67. Lune, B.J. (994). Three-parameter tuneable tlt-ntegral dervatve (TID) controller, US Patent US Matgnon, D., Llle (996). Stablty result on fractonal dfferental euatons wth applcatons to control, processng, n: IMACS-SMC Proceedngs, Llle, France, Monje, C.A. and V. Felu (4). The fractonal-order lead compensator, n: IEEE Internatonal Conference on Computatonal Cybernetcs,Venna, Austra, August September. Oustaloup, A., J. Sabater and P. Lanusse (999). Fractonal Calculus Appl. Anal, (),. Oustaloup, A., X. Moreau and M. Noullant (996), Control Eng. Practce, 4 (8),-8. Oustaloup, A., B. Matheu and P. Lanusse (995). The CRONE control of resonant plants: Applcaton to a flexble transmsson, European Journal of Control, (),. Podlubny, I. (999). Fractonal-Order Systems and PI λ D µ - Controllers, IEEE Transacton on Automatc Control, 44 (), 8. Raynaud, H.F. and A. Zerga Inoh (). State-space representaton for fractonal order controllers, Automatca, 6,7-. Tavazoe, M. S. and M. Haer (8). Chaos control va a smple fractonal-order controller, Physcs Letters A, 7 (6), Ranjbar, N. A.,Hosen Na S.H. and D.D. Ganj, H. Soltan, J. Ghasem (8). Mantanng the Stablty of Nonlnear Dfferental Euatons by the Enhancement of HPM, Physcs Letters A,