Global Chaos Synchronization of the Pehlivan Systems by Sliding Mode Control

Transcription

1 Global Chaos Sychroizatio of the Pehliva Systems by Slidig Mode Cotrol Dr. V. Sudarapadia Professor (Systems & Cotrol Eg.), Research ad Developmet Cetre Vel Tech Dr. RR & Dr. SR Techical Uiversity Avadi, Cheai , Tamil Nadu, INDIA Abstract This paper ivestigates the problem of global chaos sychroizatio of idetical Pehliva chaotic systems (Pehliva ad Uyaroglu, 200) by slidig mode cotrol. The stability results derived i this paper for the sychroizatio of idetical Pehliva systems are established usig Lyapuov stability theory. Sice the Lyapuov expoets are ot required for these calculatios, the slidig mode cotrol method is very effective ad coveiet to achieve global chaos sychroizatio of the idetical Pehliva chaotic systems. Numerical simulatios are show to illustrate the effectiveess of the sychroizatio schemes derived i this paper for the idetical Pehliva chaotic systems. Keywords- oliear cotrol systems; chaos; slidig-mode cotrol; Pehliva system. I. INTRODUCTION Chaotic systems are dyamical systems that are highly sesitive to iitial coditios. This sesitivity is popularly kow as the butterfly effect []. Sice the semial work by Pecora ad Carroll ([2], 990), chaos sychroizatio problem has bee studied extesively ad itesively i the literature [2-7]. Chaos theory has bee applied to a variety of fields such as physical systems [3], chemical systems [4], ecological systems [5], secure commuicatios [6-8] etc. I the last two decades, various schemes have bee successfully applied for chaos sychroizatio such as PC method [2], OGY method [9], active cotrol method [0-2], adaptive cotrol method [3-4], time-delay feedback method [5], backsteppig desig method [6], sampled-data feedback sychroizatio method [7], etc. I most of the chaos sychroizatio approaches, the master-slave or drive-respose formalism is used. If a particular chaotic system is called the master or drive system ad aother chaotic system is called the slave or respose system, the the goal of the global chaos sychroizatio is to use the output of the master system to cotrol the slave system so that the states of the slave system track the states of the master system asymptotically. I other words, global chaos sychroizatio is achieved whe the differece of the states of the master ad slave systems coverge to zero asymptotically with time. I this paper, we derive ew results based o the slidig mode cotrol [8-20] for the global chaos sychroizatio of idetical Pehliva chaotic systems (Pehliva ad Uyaroglu, 200). I robust cotrol systems, slidig mode cotrol is ofte adopted due to its iheret advatages of easy realizatio, fast respose ad good trasiet performace as well as its isesitivity to parameter ucertaities ad exteral disturbaces. This paper has bee orgaized as follows. I Sectio II, we describe the problem statemet ad our methodology usig slidig mode cotrol. I Sectio III, we discuss the global chaos sychroizatio of idetical Pehliva systems ([2], 200). I Sectio IV, we summarize the mai results obtaied i this paper. II. PROBLEM STATEMENT AND OUR METHODOLOGY USING SLIDING MODE CONTROL I this sectio, we describe the problem statemet for the global chaos sychroizatio for idetical chaotic systems ad our methodology usig slidig mode cotrol. Cosider the chaotic system described by () x Ax f ( x) x R is the state of the system, A is the matrix of the system parameters ad f : R the oliear part of the system. We cosider the system () as the master or drive system. R is ISSN : Vol. 3 No. 5 May

2 As the slave or respose system, we cosider the followig chaotic system described by the dyamics (2) y Ay f( y) u m y R is the state of the system ad u R is the cotroller to be desiged. If we defie the sychroizatio error as e y x, (3) the the error dyamics is obtaied as (4) e Ae( x, y) u, ( x, y) f( y) f( x) (5) The objective of the global chaos sychroizatio problem is to fid a cotroller u such that lim et ( ) 0 for all e(0) R. t To solve this problem, we first defie the cotrol u as u ( x, y) Bv (6) B is a costat gai vector selected such that ( A, B ) is cotrollable. Substitutig (5) ito (4), the error dyamics simplifies to e AeBv (7) which is a liear time-ivariat cotrol system with sigle iput v. Thus, the origial global chaos sychroizatio problem ca be replaced by a equivalet problem of stabilizig the zero solutio e 0 of the system (7) by a suitable choice of the slidig mode cotrol. I the slidig mode cotrol, we defie the variable se Cecece ce (8) () 2 2 C c c c is a costat vector to be determied. 2 I the slidig mode cotrol, we costrai the motio of the system (7) to the slidig maifold defied by S xr s( e) 0 which is required to be ivariat uder the flow of the error dyamics (7). Whe i slidig maifold S, the system (7) satisfies the followig coditios: se () 0 (9) which is the defiig equatio for the maifold S ad se () 0 (0) which is the ecessary coditio for the state trajectory et () of (7) to stay o the slidig maifold S. Usig (7) ad (8), the equatio (0) ca be rewritte as se () C AeBv 0 () Solvig () for v, we obtai the equivalet cotrol law v t CB CA e t (2) eq () ( ) () C is chose such that CB 0. ISSN : Vol. 3 No. 5 May

3 Substitutig (2) ito the error dyamics (7), we obtai the closed-loop dyamics as (3) e I B( CB) C Ae The row vector C is selected such that the system matrix of the cotrolled dyamics I BCB ( ) C Ais Hurwitz, i.e. it has all eigevalues with egative real parts. The the cotrolled system (3) is globally asymptotically stable. To desig the slidig mode cotroller for (7), we apply the costat plus proportioal rate reachig law (4) s qsg( s) k s sg( ) deotes the sig fuctio ad the gais q 0, k 0 coditio is satisfied ad slidig motio will occur. From equatios () ad (4), we ca obtai the cotrol vt () as which yields are determied such that the slidig vt ( ) ( CB) CkI ( Ae ) qsg( s) (5) vt () ( CB) C( ki A) e q, if s( e) 0 ( CB) C( ki A) e q, if s( e) 0 Theorem. The master system () ad the slave system (2) are globally ad asymptotically sychroized for all iitial coditios x(0), y(0) R by the feedback cotrol law ut () ( xy, ) Bvt () (7) vt () is defied by (5) ad B is a colum vector such that ( AB, ) is cotrollable. Also, the slidig mode gais kqare, positive. Proof. First, we ote that substitutig (7) ad (5) ito the error dyamics (4), we obtai the closed-loop error dyamics as e AeB( CB) C( ki A) eqsg( s) (8) To prove that the error dyamics (8) is globally asymptotically stable, we cosider the cadidate Lyapuov fuctio defied by the equatio Ve s e 2 2 () () (9) which is a positive defiite fuctio o R. Differetiatig V alog the trajectories of (8) or the equivalet dyamics (4), we get (20) 2 Ve () sese ()() ks qsg() ss which is a egative defiite fuctio o R. This calculatio shows that V is a globally defied, positive defiite, Lyapuov fuctio for the error dyamics (8), which has a globally defied, egative defiite time derivative V. Thus, by Lyapuov stability theory [22], it is immediate that the error dyamics (8) is globally asymptotically stable for all iitial coditios e(0) R. This meas that for all iitial coditios e(0) R, lim et ( ) 0 t we have (6) ISSN : Vol. 3 No. 5 May

4 Hece, it follows that the master system () ad the slave system (2) are globally ad asymptotically sychroized for all iitial coditios x(0), y(0) R. This completes the proof. III. GLOBAL CHAOS SYNCHRONIZATION OF THE IDENTICAL PEHLIVAN CHAOTIC SYSTEMS A. Theoretical Results I this sectio, we apply the slidig mode cotrol results derived i Sectio II for the global chaos sychroizatio of idetical Pehliva chaotic systems (Pehliva ad Uyaroglu, [2], 200). Thus, the master system is described by the Pehliva dyamics x x2 x x ax x x x bx x x, x, x are state variables ad, 2 3 abare positive, costat parameters of the system. The slave system is also described by the Pehliva dyamics y y2 yu y ay y y u y by y u y, y2, y3are state variables ad u, u2, u3are the cotrollers to be desiged. The Pehliva systems (2) ad (22) are chaotic whe a 0.5 ad b 0.5. Fig. illustrates the chaotic portrait of the Pehliva chaotic system (2). (2) (22) The chaos sychroizatio error is defied by Figure. Chaotic Portrait of the Pehliva Chaotic System e y x, ( i,2,3) (23) i i i ISSN : Vol. 3 No. 5 May

5 The error dyamics is easily obtaied as e e e e e u 2 ae y y x x u yy xx u We write the error dyamics (24) i the matrix otatio as (25) e Ae( x, y) u (24) 0 0 u A 0 a 0, ( x, y) yy3xx 3 ad u u 2. (26) yy2 xx 2 u 3 The slidig mode cotroller desig is carried out as detailed i Sectio II. First, we set u as u ( x, y) Bv (27) B is chose such that ( A, B) is cotrollable. We take B as B. I the chaotic case, the parameter values are a 0.5 ad b 0.5. The slidig mode variable is selected as s Ce 6 5 ee 6e 5e (29) 2 3 which makes the slidig mode state equatio asymptotically stable. We choose the slidig mode gais as k 6 ad q 0.. We ote that a large value of k ca cause chatterig ad a appropriate value of q is chose to speed up the time take to reach the slidig maifold as well as to reduce the system chatterig. From Eq. (5), we ca obtai vt () as v( t) 2.5 e20 e2 5 e sg( s) (30) Thus, the required slidig mode cotroller is obtaied as u ( x, y) Bv (3) ( x, y), B ad vt () are defied as i the equatios (26), (28) ad (30). By Theorem, we obtai the followig result. Theorem 2. The idetical Pehliva chaotic systems (2) ad (22) are globally ad asymptotically sychroized for all iitial coditios with the slidig mode cotroller u defied by (3). B. Numerical Results 6 For the umerical simulatios, the fourth-order Ruge-Kutta method with time-step h 0 is used to solve the Liu-Che four-scroll chaotic systems (2) ad (22) with the slidig mode cotroller u give by (3) usig MATLAB. (28) ISSN : Vol. 3 No. 5 May

6 I the chaotic case, the parameter values are a 0.5 ad b 0.5. The slidig mode gais are chose as k 6 ad q 0.. The iitial values of the master system (2) are take as x (0) 0, x (0) 6, x (0) ad the iitial values of the slave system (22) are take as y (0) 2, y (0) 9, y (0) Fig. 2 illustrates the complete sychroizatio of the idetical Pehliva chaotic systems (2) ad (22). Figure 2. Sychroizatio of Idetical Pehliva Chaotic Systems IV. CONCLUSIONS I this paper, we have deployed slidig mode cotrol (SMC) to achieve global chaos sychroizatio for the idetical Pehliva chaotic systems (Pehliva ad Uyaroglu, 200). Our sychroizatio results for the idetical Pehliva chaotic systems have bee established usig Lyapuov stability theory. Sice the Lyapuov expoets are ot required for these calculatios, the slidig mode cotrol method is very effective ad coveiet to achieve global chaos sychroizatio for the idetical Pehliva chaotic systems. Numerical simulatios are also show to illustrate the effectiveess of the sychroizatio results derived i this paper usig slidig mode cotrol for the idetical Pehliva chaotic systems. REFERENCES [] K.T. Alligood, T. Sauer ad J.A. Yorke, Chaos: A Itroductio to Dyamical Systems, New York: Spriger, 997. [2] L.M. Pecora ad T.L. Carroll, Sychroizatio i chaotic systems, Physical Review Letters, vol. 64, pp , 990. [3] M. Lakshmaa ad K. Murali, Chaos i Noliear Oscillators: Cotrollig ad Sychroizatio, Sigapore: World Scietific, 996. [4] S.K. Ha, C. Kerrer ad Y. Kuramoto, Dephasig ad burstig i coupled eural oscillators, Physical Review Letters, vol. 75, pp , 995. [5] B. Blasius, A. Huppert ad L. Stoe, Complex dyamics ad phase sychroizatio i spatially exteded ecological system, Nature, vol. 3999, pp , 999. [6] K.M. Cuomo ad A.V. Oppeheim, Circuit implemetatio of sychroized chaos with applicatios to commuicatios, Physical Review Letters, vol. 7, pp , 993. [7] L. Kocarev ad U. Parlitz, Geeral approach for chaotic sychroizatio with applicatios to commuicatio, Physical Review Letters, vol. 74, pp , 995. [8] Y. Tao, Chaotic secure commuicatio systems history ad ew results, Telecommu. Rev. vol. 9, pp , 999. [9] E. Ott, C. Grebogi ad J.A. Yorke, Cotrollig chaos, Physical Review Letters, vol. 64, pp , 990. ISSN : Vol. 3 No. 5 May

7 [0] M.C. Ho ad Y.C. Hug, Sychroizatio of two differet chaotic systems usig geeralized active etwork, Physics Letters A, vol. 30, pp , [] L. Huag, R. Feg ad M. Wag, Sychroizatio of chaotic systems via oliear cotrol, Physical Letters A, vol. 320, pp , [2] H.K. Che, Global chaos sychroizatio of ew chaotic systems via oliear cotrol, Chaos, Solitos ad Fractals, vol. 23, pp , [3] J. Lu, X. Wu, X. Ha ad J. Lü, Adaptive feedback sychroizatio of a uified chaotic system, Physics Letters A, vol. 329, pp , [4] S.H. Che ad J. Lü, Sychroizatio of a ucertai uified system via adaptive cotrol, Chaos, Solitos ad Fractals, vol. 4, pp , [5] J.H. Park ad O.M. Kwo, A ovel criterio for delayed feedback cotrol of time-delay chaotic systems, Chaos, Solitos ad Fractals, vol. 7, pp , [6] X. Wu ad J. Lü, Parameter idetificatio ad backsteppig cotrol of ucertai Lü system, Chaos, Solitos ad Fractals, vol. 8, pp , [7] K. Murali ad M. Lakshmaa, Secure commuicatio usig a compoud sigal usig sampled-data feedback, Applied Mathematics ad Mechaics, vol., pp , [8] J.E. Slotie ad S.S. Sastry, Trackig cotrol of oliear systems usig slidig surface with applicatio to robotic maipulators, Iterat. J. Cotrol, vol. 38, pp , 983. [9] V.I. Utki, Slidig mode cotrol desig priciples ad applicatios to electric drives, IEEE Trasactios o Idustrial Electroics, vol. 40, pp , 993. [20] R. Saravaakumar, K. Vioth Kumar ad K.K. Ray, Slidig mode cotrol of iductio motor usig simulatio approach, Iterat. J. Cotrol of Computer Sciece ad Network Security, vol. 9, pp , [2] I. Pehliva ad Y. Uyaroglu, A ew chaotic attractor from geeral Lorez family ad its electroic experimetal implemetatio, Turk. J. Elec. Eg. & Comp. Sciece, vol. 8, pp. 7-84, 200. [22] K. Ogata, Moder Cotrol Egieerig, 3rd ed., New Jersey: Pretice Hall, 997. AUTHORS PROFILE Dr. V. Sudarapadia is curretly Professor (Systems ad Cotrol Egieerig), Research ad Developmet Cetre at Vel Tech Dr. RR & Dr. SR Techical Uiversity, Cheai, Tamil Nadu, Idia. He has published two books titled Numerical Liear Algebra ad Probability, Statistics ad Queueig Theory (New Delhi: Pretice Hall of Idia). He has published 90 papers i Natioal Cofereces ad 42 papers i Iteratioal Cofereces. He has published over 00 refereed joural publicatios. He is the Editor-i-Chief of Iteratioal Joural of Mathematics ad Scietific Computig ad Iteratioal Joural of Mathematical Scieces ad Applicatios. He is the Associate Editor of Iteratioal Joural o Cotrol Theory ad Applicatios, Joural of Mathematics ad Statistics, Iteratioal Joural of Advaced Research i Computer Sciece, Joural of Electroics ad Electrical Egieerig, Joural of Statistics ad Mathematics, etc. He is a Editorial Board Member of the Jourals Scietific Research ad Essays, Iteratioal Joural of Soft Computig ad Bioiformatics, ISST Joural of Mathematics ad Statistics, Iteratioal Joural of Egieerig, Sciece ad Techology, etc. His curret research iterests are Liear ad Noliear Cotrol Systems, Noliear Dyamical Systems ad Bifurcatios, Chaos ad Cotrol, Optimal Cotrol, Soft Computig, Operatios Research, Mathematical Modellig, Stochastic Modellig, Scietific Computig with MATLAB/SCILAB, etc. He has delivered several Key Note Lectures i Moder Cotrol Systems, Chaos ad Cotrol, Mathematical Modellig, Scietific Computig, etc. ISSN : Vol. 3 No. 5 May