ANTI-SYNCHRONIZING SLIDING CONTROLLER DESIGN FOR IDENTICAL PAN SYSTEMS

Transcription

1 ANTI-SYNCHRONIZING SLIDING CONTROLLER DESIGN FOR IDENTICAL PAN SYSTEMS Sudarapadia Vaidyaatha Research ad Developmet Cetre, Vel Tech Dr. RR & Dr. SR Techical Uiversity Avadi, Cheai , Tamil Nadu, INDIA ABSTRACT The ati-sychroizatio of idetical Pa systems (Pa, Xu ad Zhou, 200) has bee ivestigated i this paper ad ew results have bee derived by slidig mode cotrol. I this paper, we first derive a geeral result for the ati-sychroizatio of idetical chaotic systems. As a applicatio, we derive atisychroizig slidig cotroller for idetical Pa systems. A umerical example usig MATLAB has bee preseted so as to demostrate the ati-sychroizig slidig cotroller for the Pa systems. KEYWORDS Pa System, Ati-Sychroizatio, Slidig Cotrol, Chaos.. INTRODUCTION Chaos theory is a active research area that deals with oliear dyamical systems which are sesitive to iitial coditios []. This area has diverse applicatios like Ecoomics [2], Physics [3], Biology [4], Chemistry [5], Image Processig [6] ad Secure Commuicatios [7]. The chaos sychroizatio problem, which deals with the sychroizatio of a pair of chaotic systems called the master ad slave systems, has bee studied rigorously i the literature. Some miscellaeous methods of sychroizatio ca be listed as PC method [8], OGY method [9], active cotrol method [0-], adaptive cotrol method [2], backsteppig method [3], sampled-data feedback method [4], etc. This paper deals with the desig of slidig cotroller desig for the ati-sychroizatio of idetical Pa systems ([5], 200). First, a geeral result is derived for ati-sychroizatio of chaotic systems. Next, as a applicatio, ew results are derived for the ati-sychroizatio of idetical Pa systems. Numerical simulatios usig MATLAB are show to illustrate the mai results of this paper. 2. PROBLEM STATEMENT AND OUR METHODOLOGY USING SMC We cosider the chaotic system described by x = Ax + f () x () where f : R x R is the state of the system, A is the R is the oliear part of the system. We take the system () as the master system. matrix of the system parameters ad

2 Next, we cosider the followig chaotic system as the slave system described by y = Ay + f () y + u (2) where y R is the state of the system ad If we defie the ati-sychroizatio error as m u R is the cotroller to be desiged. e = y + x, (3) the the error dyamics is obtaied as = Ae + ( x,) y +, u (4) where ( x,)()() y = f y + f x (5) The desig goal is to fid a cotroller u such that lim() e t 0= t for all e(0) R. Our methodology is detailed ext. We defie the cotrol u as u = ( x,) y + Bv (6) I Eq. (6), B is a matrix carefully selected such that ( A,) B is cotrollable. Next, we substitute the defiitio of u from (6) ito (4). The the error dyamics becomes = Ae + Bv (7) It is oted that Eq. (7) is a liear sigle-iput, time-ivariat cotrol system. I SMC, we defie the slidig variable as s e Ce c e c e c e (8) () = = I SMC, we costrai the motio of the system (7) to the slidig maifold defied by { R :() 0 } S = e s e = such that the motio is ivariat uder the flow of the error dyamics (7). 2

3 Whe i slidig maifold S, the system (7) satisfies the followig coditios: s() e = 0 (9) ad s () e = 0 (0) With the help of equatios (7) ad (8), the equatio (0) ca be rearraged as s () e = C Ae + Bv = 0 () Solvig () for v, we obtai the equivalet cotrol law v()() eq t () CB CA e t = (2) where C is chose such that CB 0. Next, we substitute (2) ito the error dyamics (7). The we obtai the closed-loop error dyamics as e = I B() CB C Ae (3) The row vector C is chose so that the system liearizatio matrix of the cotrolled dyamics I B() CB C A is Hurwitz, i.e. it has all eigevalues with egative real parts. The, by Lyapuov stability theory, the cotrolled system (3) is globally asymptotically stable. A slidig mode cotroller for (7) is desiged as follows. Our strategy is to apply the costat plus proportioal rate reachig law give by s = qsg() s k s (4) where the gais qad k are positive. We obtai the cotrol v() t from equatios () ad (4) as follows: which yields v()()() t = CBsg() C ki + A e + q s (5) v() t = ()() CB C, kiif + () A e0+ q s e > ()() CB C, kiif + () A e0 q s e < (6) 3

4 Theorem 2.. Cosider the master system () ad the slave system (2). These two chaotic systems are globally ad asymptotically ati-sychroized for all iitial coditios x(0),(0) y R by the feedback cotrol law u()( t =,)() x y + Bv t (7) where v() t is defied by (5) ad B is a colum vector such that ( A,) B is cotrollable. Also, the slidig mode gais k, q are positive. Proof. To start with, we substitute (7) ad (5) ito the error dyamics (4). The we obtai the closed-loop error dyamics as = Ae B()() CB Csg() ki + A e + q s (8) We wish to prove that the error dyamics (8) is globally asymptotically stable. For this purpose, we cosider the cadidate Lyapuov fuctio defied by 2 2 V ()() e = s e (9) which is a positive defiite fuctio o R. We differetiate V alog the trajectories of (8) or the equivalet dyamics (4). This yields (20) 2 V ()()() e = s e s e = sg() ks q s s which is a egative defiite fuctio o R. Thus, the dyamics (8) is globally asymptotically stable by Lyapuov stability theory [6]. This completes the proof. 3. ANTI-SYNCHRONIZATION OF IDENTICAL PAN SYSTEMS USING SLIDING MODE CONTROL 3. Theoretical Results We take the Pa system as the master system give by x = a() x x 2 x = cx x x 2 3 x = bx + x x (2) where the costats a, b, c are positive. 4

5 We take the cotrolled Pa system as the slave system give by y = a() y y + u 2 y = cy y y + u y = by + y y + u (22) where u, u 2, u 3 are the cotrollers to be desiged. The Pa system is chaotic whe a = 0, b = 8 / 3 ad c = 6. Figure illustrates the strage attractor of the chaotic Pa system (2). Figure. Strage Attractor of the Pa System The chaos ati-sychroizatio error is defied by e = y + x, ( i =,2,3) (23) i i i The error dyamics is easily obtaied as = a() e e + u 2 = ce y y x x + u = be + y y + x x + u (24) 5

6 The error dyamics (24) ca be compactly writte i matrix form as = Ae + ( x,) y + u (25) where a a 0 A = c 0 0, 0 0 b 0 ( x,) y = y y3 xx 3 y y2 + xx 2 ad u u = u 2 u 3. (26) Usig the results outlied i Sectio 2, we desig the ati-sychroizig slidig mode cotroller for idetical Pa systems as follows. First, we set u as u = ( x,) y + Bv (27) where B is take so that ( A,) B is cotrollable. We choose B as B = (28) I the chaotic case, the parameter values are take as a = 0, b = 8 / 3 ad c = 6. The slidig mode variable is selected as [ 6 6 ] s = Ce = e = e + e + e (29) which reders the slidig dyamics asymptotically stable. We choose the slidig mode gais as k = 6 ad q = 0.. From Eq. (5), we ca obtai v() t as v() t = e e e sg() s (30) 2 3 Thus, we have derived the required ati-sychroizig cotroller as u = ( x,) y + Bv (3) 6

7 where ( x,), y B ad v() t are give by the equatios (26), (28) ad (30). By Theorem 2., we arrive at the followig result. Theorem 3.. The idetical Pa systems (2) ad (22) are globally ad asymptotically atisychroized for all iitial coditios with the slidig mode cotroller u defied by (3). 3.2 Numerical Results I this sectio, we show the umerical simulatios obtaied via the fourth-order Ruge-Kutta 6 method with time-step h = 0 usig MATLAB for solvig the Pa systems (2) ad (22). Note that the slidig mode cotroller u has bee give by the formula (3). The parameter values of the Pa system are take as i the chaotic case, viz. a = 0, b = 8 / 3 ad c = 6. The slidig mode gais are chose as k = 6 ad q = 0.. The iitial values of the Pa system (2) are take as x (0) = 30, x(0) = 5, (0) x 4= 2 3 The iitial values of the cotrolled Pa system (22) are take as y (0) = 4, (0) y 0, = (0) y 26 = 2 3 Figure 2 illustrates the ati-sychroizatio of the idetical Pa systems (2) ad (22). Figure 3 illustrates the time-history of the ati-sychroizatio errors e, e2, e3. Figure 2. Ati-Sychroizatio of Idetical Pa Systems 7

8 4. CONCLUSIONS Figure 3. Time-History of the Ati-Sychroizatio Error This paper derived ew results for the ati-sychroizatio of chaotic systems via slidig mode cotrol (SMC). As a applicatio, ew results were derived usig slidig cotrol to achieve ati-sychroizatio for the idetical Pa systems (20 0). Our ati-sychroizatio results for the idetical Pa systems have bee proved usig Lyapuov stability theory. REFERENCES [] Sprott, J.C. (200) Elegat Chaos, World Scietific, Sigapore. [2] Feichtiger, G. (200) Chaos theory i operatios research, Iteratioal Trasactios i Operatioal Research, Vol. 9, No. 8, pp [3] Efetov, K. (995) Supersymmetry i quatum chaos ad mesoscopic physics, Physica D: Noliear Pheomea, Vol. 83, pp [4] Cross, S.S. & Cotto, D.W.K. (994) Chaos ad atichaos i pathology, Huma Pathology, Vol. 25, No. 7, pp [5] Femat, R. (2000) Chaos i a class of reactig systems iduced by robust asymptotic feedback, Physica D: Noliear Pheomea, Vol. 36, pp [6] Rhouma, R. & Belghith, S. (2008) Cryptoaalysis of a ew image ecryptio algorithm based o hyperchaos, Physics Letters A, Vol. 372, No. 38, pp [7] Li, Y., Wag, Y. & Wag, A. (2008) Message filterig characteristics of semicoductor laser as receiver i optical chaos commuicatio, Optics Commu., Vol. 28, No. 9, pp [8] Pecora, L.M. & Carroll, T.L. (990) Sychroizatio i chaotic systems, Physical Review Letters, Vol. 64, pp

9 [9] Ott, E., Grebogi, C. & Yorke, J.A. (990) Cotrollig chaos, Phys. Rev. Lett., Vol. 64, pp [0] Ucar, A., Logre, K.E. & Bai, E.W. (2007) Chaos sychroizatio i RCL -shuted Josephso juctio via active cotrol, Chaos, Solitos & Fractals, Vol. 3, No., pp 05-. [] Jia, Q. (2008) Chaos cotrol ad sychroizatio of the Newto -Leipik chaotic system, Chaos, Solitos & Fractals, Vol. 35, No. 4, pp [2] Salarieh, H. & Alasty, A. (2008) Adaptive chaos sychroizatio i Chua s systems with oisy parameters, Mathematics ad Computers i Simulatio, Vol. 79, No. 3, pp [3] Zhag, J., Li, C., Zhag, H. & Yu, J. (2004) Chaos sychroizatio usig sigle variable feedback based o backsteppig method, Chaos, Solitos & Fractals, Vol. 2, No. 5, pp [4] Yag, T., Yag, L.B. & Yag, C.M. (998) Theory of cotrol of chaos usig sampled data, Physics Letters A, Vol. 246, pp [5] Pa, L., Xu, D. & Zhou, W. (200) Cotrollig a ovel chaotic attractor usig liear feedback, J. Iform. Comput. Sciece, Vol. 5, pp [6] Hah, W. (967) The Stability of Motio, Spriger, New York. Author Dr. V. Sudarapadia eared his Doctor of Sciece degree i Electrical ad Systems Egieerig from Washigto Uiversity, St. Louis, USA i May 996. Curretly, he is workig as Professor ad Dea at the Research ad Developmet Cetre at Vel Tech Dr. RR & Dr. SR Techical Uiversity, Cheai, Tamil Nadu, Idia. Dr. Sudarapadia has published a total of 300 papers i refereed iteratioal jourals ad 90 papers i Natioal ad Iteratioal Cofereces. Dr. Sudarapadia is the Editor-i-Chief of the AIRCC Cotrol Jourals - Iteratioal Joural of Istrumetatio ad Cotrol Systems, Iteratioal Joural of Cotrol Systems ad Computer Modellig, Iteratioal Joural of Iformatio Techology, Cotrol ad Automatio, Iteratioal Joural of Chaos, Cotrol, Modellig ad Simulatio, ad Iteratioal Joural of Iformatio Techology, Modelig ad Computig. His research iterests are Chaos Theory, Cotrol Systems, Optimal Cotrol, Operatios Research, Itelliget Computig, Mathematical Modellig ad Scietific Computig. He has delivered several Key Note Lectures o Cotrol Systems, Chaos Theory, Scietific Computig ad Mathematical Modellig. He has coducted several workshops o Mathematical Modellig usig MATLAB ad SCILAB. 9