SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEMS

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1 SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC SYSTEMS Sudarapadia Vaidyaatha 1 1 Research ad Developmet Cetre, Vel Tech Dr. RR & Dr. SR Techical Uiversity Avadi, Cheai , Tamil Nadu, INDIA sudarvtu@gmail.com ABSTRACT This paper ivestigates the global chaos sychroizatio of idetical Shimizhu-Morioka chaotic systems (Shimizu ad Morioka, 1980) by slidig mode cotrol. The stability results derived i this paper for the complete sychroizatio of idetical Shimizu-Morioka chaotic 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 Shimizu-Morioka chaotic systems. Numerical simulatios are show to illustrate ad validate the sychroizatio schemes derived i this paper for the idetical Shimizu-Morioka systems. KEYWORDS Slidig Mode Cotrol, Chaos Sychroizatio, Chaotic Systems, Shimizu-Morioka Systems. 1. INTRODUCTION Chaotic systems are dyamical systems that are highly sesitive to iitial coditios. The sesitive ature of chaotic systems is commoly called as the butterfly effect [1]. Sychroizatio of chaotic systems is a pheomeo which may occur whe two or more chaotic oscillators are coupled or whe a chaotic oscillator drives aother chaotic oscillator. Because of the butterfly effect which causes the expoetial divergece of the trajectories of two idetical chaotic systems started with early the same iitial coditios, sychroizig two chaotic systems is seemigly a very challegig problem. 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 idea of the sychroizatio is to use the output of the master system to cotrol the slave system so that the output of the slave system tracks the output of the master system asymptotically. Sice the pioeerig work by Pecora ad Carroll ([2], 1990), chaos sychroizatio problem has bee studied extesively ad itesively i the literature [2-17]. 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 [10-12], adaptive cotrol method [13-14], time-delay feedback method [15], backsteppig desig method [16], sampled-data feedback method [17], etc. 20

2 I this paper, we derive ew results based o the slidig mode cotrol [18-20] for the global chaos sychroizatio of idetical Shimizu-Morioka systems ([21], Shimizu ad Morioka, 1980). I robust cotrol systems, the slidig mode cotrol method 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 2, we describe the problem statemet ad our methodology usig slidig mode cotrol (SMC). I Sectio 3, we discuss the global chaos sychroizatio of idetical Shimizu-Morioka chaotic systems. I Sectio 4, we summarize the mai results obtaied i this paper. 2. PROBLEM STATEMENT AND OUR METHODOLOGY USING SMC I this sectio, we describe the problem statemet for the global chaos sychroizatio for idetical chaotic systems ad our methodology usig slidig mode cotrol (SMC). Cosider the chaotic system described by x& = Ax + f ( x) (1) where x R is the state of the system, A is the matrix of the system parameters ad f : R R is the oliear part of the system. We cosider the system (1) as the master or drive system. As the slave or respose system, we cosider the followig chaotic system described by the dyamics where y& = Ay + f ( y) + u (2) y R is the state of the system ad If we defie the sychroizatio error as m u R is the cotroller to be desiged. e = y x, (3) the the error dyamics is obtaied as where e& = Ae + η( x, y) + u, (4) η ( x, y) = f ( y) f ( x) (5) The objective of the global chaos sychroizatio problem is to fid a cotroller u such that lim e( t) = 0 for all e(0) R. t 21

3 To solve this problem, we first defie the cotrol u as u = η( x, y) + Bv (6) where B is a costat gai vector selected such that ( A, B ) is cotrollable. Substitutig (5) ito (4), the error dyamics simplifies to e& = Ae + Bv (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 s( e) = Ce = c e + c e + L + ce (8) where C [ c c c ] = L is a costat vector to be determied. 1 2 I the slidig mode cotrol, we costrai the motio of the system (7) to the slidig maifold defied by { R ( ) 0} S = x s e = 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: s( e ) = 0 (9) which is the defiig equatio for the maifold S ad s& ( e ) = 0 (10) which is the ecessary coditio for the state trajectory e( t ) of (7) to stay o the slidig maifold S. Usig (7) ad (8), the equatio (10) ca be rewritte as [ ] s& ( e) = C Ae + Bv = 0 (11) Solvig (11) for v, we obtai the equivalet cotrol law v t = CB CA e t (12) ( ) ( ) 1 ( ) eq where C is chose such that CB 0. Substitutig (12) ito the error dyamics (7), we obtai the closed-loop dyamics as 22

4 & ( ) (13) 1 e = I B CB C Ae The row vector C is selected such that the system matrix of the cotrolled dyamics 1 I B( CB) C A is Hurwitz, i.e. it has all eigevalues with egative real parts. The the cotrolled system (13) is globally asymptotically stable. To desig the slidig mode cotroller for (7), we apply the costat plus proportioal rate reachig law s& = q sg( s) k s (14) where sg( ) deotes the sig fuctio ad the gais q > 0, k > 0 are determied such that the slidig coditio is satisfied ad slidig motio will occur. From equatios (11) ad (14), we ca obtai the cotrol v( t) as v( t) = ( CB) C( ki + A) e + qsg( s) (15) which yields 1 [ ] [ ] [ ] + + > v( t) = + < 1 ( CB) C( ki A) e q, if s( e) 0 1 ( CB) C( ki A) e q, if s( e) 0 (16) Theorem 1. The master system (1) ad the slave system (2) are globally ad asymptotically sychroized for all iitial coditios x(0), y(0) R by the feedback cotrol law u( t) = η( x, y) + Bv( t) (17) where v( t) is defied by (15) ad B is a colum vector such that ( A, B) is cotrollable. Also, the slidig mode gais k, q are positive. Proof. First, we ote that substitutig (17) ad (15) ito the error dyamics (4), we obtai the closed-loop error dyamics as e& = Ae B( CB) C( ki + A) e + q sg( s) (18) 1 [ ] To prove that the error dyamics (18) is globally asymptotically stable, we cosider the cadidate Lyapuov fuctio defied by the equatio 1 2 V ( e) = s ( e) (19) 2 which is a positive defiite fuctio o R. 23

5 Differetiatig V alog the trajectories of (18) or the equivalet dyamics (14), we get & & (20) 2 V ( e) = s( e) s( e) = ks q sg( s) s 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 (18), which has a globally defied, egative defiite time derivative V &. Thus, by Lyapuov stability theory [22], it is immediate that the error dyamics (18) is globally asymptotically stable for all iitial coditios e(0) R. This meas that for all iitial coditios e(0) R, we have lim e( t) = 0 t Hece, it follows that the master system (1) ad the slave system (2) are globally ad asymptotically sychroized for all iitial coditios x(0), y(0) R. This completes the proof. 3. GLOBAL SYNCHRONIZATION OF IDENTICAL SHIMIZU-MORIOKA CHAOTIC SYSTEMS USING SLIDING MODE CONTROL 3.1 Theoretical Results I this sectio, we apply the slidig mode cotrol results derived i Sectio 2 for the global chaos sychroizatio of idetical Shimizu-Morioka (SM) chaotic systems ([21], Shimizu ad Morioka, 1980). Thus, the master system is described by the SM dyamics x& = x 1 2 x& = x λx x x x& = x ξ x (21) where x1, x2, x 3 are state variables ad α, β, γ, ω are positive, costat parameters of the system. The slave system is described by the cotrolled SM dyamics y& = y + u y& = y λ y y y + u y& = y ξ y + u (22) where y1, y2, y3 are state variables ad u1, u2, u3 are the cotrollers to be desiged. The Shimizu-Morioka systems (21) ad (22) are chaotic whe λ = 0.75 ad ξ =

6 Figure 1 illustrates the double-wig butterfly-shaped attractor of the SM system (21). Figure 1. State Orbits of the Shimizu-Morioka Chaotic System The chaos sychroizatio error is defied by e = y x, ( i = 1,2,3) (23) i i i The error dyamics is easily obtaied as e& = e + u e& = e λe y y + x x + u e& = ξe + y x + u (24) We write the error dyamics (24) i the matrix otatio as e& = Ae + η( x, y) + u (25) where A = 1 λ 0, 0 0 ξ 0 η( x, y) = y1 y3 x1x y1 x 1 ad u u 1 = u 2 u 3. (26) The slidig mode cotroller desig is carried out as detailed i Sectio 2. First, we set u as 25

7 u = η( x, y) + Bv (27) where B is chose such that ( A, B) is cotrollable. We take B as 1 B = 1. 1 (28) I the chaotic case, the parameter values are λ = 0.75 ad ξ = The slidig mode variable is selected as [ 2 1 2] s = Ce = e = e + e e (29) which makes the slidig mode state equatio asymptotically stable. We choose the slidig mode gais as k = 5 ad q = 0.1. 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. (15), we ca obtai v( t) as v( t) = 11 e 6.25e e 0.1 sg( s) (30) Thus, the required slidig mode cotroller is obtaied as u = η( x, y) + Bv (31) where η( x, y), B ad v( t) are defied as i the equatios (26), (28) ad (30). By Theorem 1, we obtai the followig result. Theorem 2. The idetical Shimizu-Morioka chaotic systems (21) ad (22) are globally ad asymptotically sychroized for all iitial coditios with the slidig mode cotroller u defied by (31). 3.2 Numerical Results I this sectio For the umerical simulatios, the fourth-order Ruge-Kutta method with timestep h = 10 is used to solve the Shimizu-Morioka chaotic systems (21) ad (22) with the slidig 6 mode cotroller u give by (31) usig MATLAB. I the chaotic case, the parameter values are 26

8 λ = 0.75 ad ξ = The slidig mode gais are chose as k = 5 ad q = 0.1. The iitial values of the master system (21) are take as x (0) = 30, x (0) = 25, x (0) = ad the iitial values of the slave system (22) are take as y (0) = 4, y (0) = 18, y (0) = Figure 2 illustrates the complete sychroizatio of the idetical Shimizu-Morioka chaotic systems (21) ad (22). Figure 2. Sychroizatio of Idetical Shimizu-Morioka Chaotic Systems 27

9 4. CONCLUSIONS I this paper, we have deployed slidig mode cotrol (SMC) to achieve global chaos sychroizatio for the idetical Shimizu-Morioka chaotic systems (1980). Our sychroizatio results for the idetical Shimizu-Morioka chaotic systems have bee proved 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 Shimizu-Morioka chaotic systems. Numerical simulatios are also show to illustrate the effectiveess of the sychroizatio results derived i this paper usig the slidig mode cotrol. REFERENCES [1] Alligood, K.T., Sauer, T. & Yorke, J.A. (1997) Chaos: A Itroductio to Dyamical Systems, Spriger, New York. [2] Pecora, L.M. & Carroll, T.L. (1990) Sychroizatio i chaotic systems, Phys. Rev. Lett., Vol. 64, pp [3] Lakshmaa, M. & Murali, K. (1996) Noliear Oscillators: Cotrollig ad Sychroizatio, World Scietific, Sigapore. [4] Ha, S.K., Kerrer, C. & Kuramoto, Y. (1995) Dephasig ad burstlig i coupled eural oscillators, Phys. Rev. Lett., Vol. 75, pp [5] Blasius, B., Huppert, A. & Stoe, L. (1999) Complex dyamics ad phase sychroizatio i spatially exteded ecological system, Nature, Vol. 399, pp [6] Cuomo, K.M. & Oppeheim, A.V. (1993) Circuit implemetatio of sychroized chaos with applicatios to commuicatios, Physical Review Letters, Vol. 71, pp [7] Kocarev, L. & Parlitz, U. (1995) Geeral approach for chaotic sychroizatio with applicatios to commuicatio, Physical Review Letters, Vol. 74, pp [8] Tao, Y. (1999) Chaotic secure commuicatio systems history ad ew results, Telecommu. Review, Vol. 9, pp [9] Ott, E., Grebogi, C. & Yorke, J.A. (1990) Cotrollig chaos, Phys. Rev. Lett., Vol. 64, pp [10] Ho, M.C. & Hug, Y.C. (2002) Sychroizatio of two differet chaotic systems usig geeralized active etwork, Physics Letters A, Vol. 301, pp [11] Huag, L., Feg, R. & Wag, M. (2005) Sychroizatio of chaotic systems via oliear cotrol, Physical Letters A, Vol. 320, pp [12] Che, H.K. (2005) Global chaos sychroizatio of ew chaotic systems via oliear cotrol, Chaos, Solitos & Fractals, Vol. 23, pp [13] Lu, J., Wu, X., Ha, X. & Lü, J. (2004) Adaptive feedback sychroizatio of a uified chaotic system, Physics Letters A, Vol. 329, pp [14] Che, S.H. & Lü, J. (2002) Sychroizatio of a ucertai uified system via adaptive cotrol, Chaos, Solitos & Fractals, Vol. 14, pp [15] Park, J.H. & Kwo, O.M. (2003) A ovel criterio for delayed feedback cotrol of time-delay chaotic systems, Chaos, Solitos & Fractals, Vol. 17, pp [16] Wu, X. & Lü, J. (2003) Parameter idetificatio ad backsteppig cotrol of ucertai Lü system, Chaos, Solitos & Fractals, Vol. 18, pp [17] Zhao, J. & J. Lu (2006) Usig sampled-data feedback cotrol ad liear feedback sychroizatio i a ew hyperchaotic system, Chaos, Solitos & Fractals, Vol. 35, pp

10 [18] Slotie, J.E. & Sastry, S.S. (1983) Trackig cotrol of oliear systems usig slidig surface with applicatio to robotic maipulators, Iterat. J. Cotrol, Vol. 38, pp [19] Utki, V.I. (1993) Slidig mode cotrol desig priciples ad applicatios to electric drives, IEEE Tras. Idustrial Electroics, Vol. 40, pp 23-36, [20] Saravaakumar, R., Vioth Kumar, K. & Ray, K.K. (2009) Slidig mode cotrol of iductio motor usig simulatio approach, Iterat. J. Cotrol of Comp. Sci. Network Security, Vol. 9, pp [21] Shimizu, T. & Morioka, N. (1980) O the bifurcatio of a symmetric limit cycle to a asymmetric oe i a simple model, Physics Letters A, Vol. 76, pp [22] Hah, W. (1967) The Stability of Motio, Spriger, New York. Author Dr. V. Sudarapadia is a Professor (Systems & Cotrol Egieerig) i the R & D Cetre at Vel Tech Dr. RR & Dr. SR Techical Uiversity, Cheai, Idia. His curret research areas are Liear ad Noliear Cotrol Systems, Chaos Theory, Dyamical Systems, Soft Computig, etc. He has published over 150 refereed iteratioal joural publicatios. He has published over 45 papers i Iteratioal Cofereces ad over 90 papers i Natioal Cofereces. He has published two text-books titled Numerical Liear Algebra ad Probability, Statistics ad Queueig Theory with Pretice Hall of Idia, New Delhi, Idia. He is a Seior Member of AIRCC ad the Editor-i-Chief of the AIRCC iteratioal jourals Iteratioal Joural of Istrumetatio ad Cotrol Systems ad Iteratioal Joural of Cotrol Theory ad Computer Modelig. He is the Associate Editor of may Cotrol Jourals like Iteratioal Joural of Cotrol Theory ad Applicatios, Iteratioal Joural of Computer Iformatio ad Scieces, ad Iteratioal Joural of Advaces i Sciece ad Techology. He has delivered may Key Note Lectures o Noliear Cotrol Systems, Moder Cotrol Systems, Chaos ad Cotrol, Mathematical Modellig, Scietific Computig with SCILAB/MATLAB, etc. 29