Global Chaos Synchronization of the Hyperchaotic Qi Systems by Sliding Mode Control

Transcription

1 Dr. V. Sudarapadia t al. / Itratioal Joural o Computr Scic ad Egirig (IJCSE) Global Chaos Sychroizatio of th Hyprchaotic Qi Systms by Slidig Mod Cotrol Dr. V. Sudarapadia Profssor, Rsarch ad Dvlopmt Ctr Vl Tch Dr. RR & Dr. SR Tchical Uivrsity, Chai , INDIA sudarvtu@gmail.com S. Sivaprumal Rsarch Scholar, School of Elctroics ad Elctrical Egirig, Sighaia Uivrsity, Jhujhuu, Rajastha , INDIA ad Assistat Profssor, Dpartmt of Elctroics ad Commuicatio Egirig Vl Tch Dr. RR & Dr. SR Tchical Uivrsity, Avadi, Chai , INDIA sivaprumasl@gmail.com Abstract This papr ivstigats th problm of global chaos sychroizatio of idtical hyprchaotic Qi systms (008) by slidig mod cotrol. Th stability rsults drivd i this papr for th sychroizatio of idtical hyprchaotic Qi systms ar stablishd usig Lyapuov stability thory. Sic th Lyapuov xpots ar ot rquird for ths calculatios, th slidig mod cotrol mthod is vry ffctiv ad covit to achiv global chaos sychroizatio of th idtical hyprchaotic Qi systms. Numrical simulatios ar show to illustrat th ffctivss of th sychroizatio schms drivd i this papr. Kywords-oliar cotrol systms; chaos; sychroizatio; slidig mod cotrol; hyprchaotic Qi systm. I. INTRODUCTION Chaotic systms ar dyamical systms that ar highly ssitiv to iitial coditios. Th ssitiv atur of chaotic systms is commoly calld as th buttrfly ffct []. Sychroizatio of chaotic systms is a phomo which may occur wh two or mor chaotic oscillators ar coupld or wh a chaotic oscillator drivs aothr chaotic oscillator. Bcaus of th buttrfly ffct which causs th xpotial divrgc of th trajctoris of two idtical chaotic systms startd with arly th sam iitial coditios, sychroizig two chaotic systms is smigly a vry challgig problm. I most of th chaos sychroizatio approachs, th mastr-slav or driv-rspos formalism is usd. If a particular chaotic systm is calld th mastr or driv systm ad aothr chaotic systm is calld th slav or rspos systm, th th ida of th sychroizatio is to us th output of th mastr systm to cotrol th slav systm so that th output of th slav systm tracks th output of th mastr systm asymptotically. Sic th piorig work by Pcora ad Carroll ([], 990), chaos sychroizatio problm has b studid xtsivly ad itsivly i th litratur [-7]. Chaos thory has b applid to a varity of filds such as physical systms [3], chmical systms [4], cological systms [5], scur commuicatios [6-8], tc. I th last two dcads, various schms hav b succssfully applid for chaos sychroizatio such as PC mthod [], OGY mthod [9], activ cotrol mthod [0-], adaptiv cotrol mthod [3-4], tim-dlay fdback mthod [5], backstppig dsig mthod [6], sampld-data fdback mthod [7], tc. I this papr, w driv w rsults basd o th slidig cotrol [8-0] for th global chaos sychroizatio of idtical hyprchaotic Qi systms ([], 008). I robust cotrol systms, th slidig cotrol mthod is oft adoptd du to its ihrt advatags of asy ralizatio, fast rspos ad good trasit prformac as wll as its issitivity to paramtr ucrtaitis ad xtral disturbacs. ISSN : Vol. 3 No. 6 Ju 0 430

2 Dr. V. Sudarapadia t al. / Itratioal Joural o Computr Scic ad Egirig (IJCSE) This papr has b orgaizd as follows. I Sctio II, w dscrib th problm statmt ad our mthodology usig slidig mod cotrol. I Sctio III, w discuss th global chaos sychroizatio of idtical hyprchaotic Qi systms. I Sctio IV, w summariz th mai rsults obtaid i this papr. II. PROBLEM STATEMENT AND OUR METHODOLOGY USING SLIDING MODE CONTROL I this sctio, w dscrib th problm statmt for th global chaos sychroizatio for idtical chaotic systms ad our mthodology usig slidig cotrol. Cosidr th chaotic systm dscribd by () x = Ax + f ( x) x R is th stat of th systm, A is th whr matrix of th systm paramtrs ad f : R R is th oliar part of th systm. W cosidr th systm () as th mastr or driv systm. As th slav or rspos systm, w cosidr th followig chaotic systm dscribd by th dyamics () whr y = Ay+ f( y) + u m y R is th stat of th systm ad u R is th cotrollr to b dsigd. If w dfi th sychroizatio rror as = y x, (3) th th rror dyamics is obtaid as = A+ η( x, y) + u, (4) whr η ( x, y) = f( y) f( x) (5) Th objctiv of th global chaos sychroizatio problm is to fid a cotrollr u such that lim t ( ) = 0 for all (0) R. t To solv this problm, w first dfi th cotrol u as u = η( x, y) + Bv (6) whr B is a costat gai vctor slctd such that ( A, B ) is cotrollabl. Substitutig (5) ito (4), th rror dyamics simplifis to = A+ Bv (7) which is a liar tim-ivariat cotrol systm with sigl iput v. Thus, th origial global chaos sychroizatio problm ca b rplacd by a quivalt problm of stabilizig th zro solutio = 0 of th systm (7) by a suitabl choic of th slidig cotrol. I th slidig cotrol, w dfi th variabl s = C= c+ c + + c (8) () I th slidig cotrol, w costrai th motio of th systm (7) to th slidig maifold dfid by { R ( ) 0} S = x s = which is rquird to b ivariat udr th flow of th rror dyamics (7). Wh i slidig maifold S, th systm (7) satisfis th followig coditios: s () = 0 (9) ISSN : Vol. 3 No. 6 Ju 0 43

3 Dr. V. Sudarapadia t al. / Itratioal Joural o Computr Scic ad Egirig (IJCSE) which is th dfiig quatio for th maifold S ad s () = 0 (0) which is th cssary coditio for th stat trajctory t () of (7) to stay o th slidig maifold S. Usig (7) ad (8), th quatio (0) ca b rwritt as s () = C A+ Bv = 0 () Solvig () for v, w obtai th quivalt cotrol law v t = CB CA t () q () ( ) () whr C is chos such that CB 0. Substitutig () ito th rror dyamics (7), w obtai th closd-loop dyamics as = (3) I B( CB) C A Th row vctor C is slctd such that th systm matrix of th cotrolld dyamics I BCB ( ) C Ais Hurwitz. Th th systm (3) is globally asymptotically stabl. To dsig th slidig cotrollr for (7), w apply th costat plus proportioal rat rachig law (4) s = qsg( s) k s whr sg( ) dots th sig fuctio ad th gais q > 0, k > 0 ar dtrmid such that th slidig coditio is satisfid ad slidig motio will occur. From quatios () ad (4), w ca obtai th cotrol vt () as which yilds vt ( ) = ( CB) CkI ( + A ) + qsg( s) (5) CB C ki A q s vt () = + + > ( CB) C( ki + A) q, if s( ) < 0 ( ) ( ), if ( ) 0 Thorm. Th mastr systm () ad th slav systm () ar globally ad asymptotically sychroizd for all iitial coditios x(0), y(0) R by th fdback cotrol law ut () = η( xy, ) + Bvt () (7) vt is dfid by (5) ad B is a colum vctor such that ( A, B) is cotrollabl. Also, th slidig mod whr () gais kqar, positiv. Proof. First, w ot that substitutig (7) ad (5) ito th rror dyamics (4), w obtai th closd-loop rror dyamics as = A B( CB) C( ki + A) + qsg( s) (8) To prov that th rror dyamics (8) is globally asymptotically stabl, w cosidr th cadidat Lyapuov fuctio dfid by th quatio V () = s() (9) which is a positiv dfiit fuctio o R. Diffrtiatig V alog th trajctoris of (8) or th quivalt dyamics (4), w gt (6) ISSN : Vol. 3 No. 6 Ju 0 43

4 Dr. V. Sudarapadia t al. / Itratioal Joural o Computr Scic ad Egirig (IJCSE) (0) V () = ss ()() = ks qsg() ss which is a gativ dfiit fuctio o R. This calculatio shows that V is a globally dfid, positiv dfiit, Lyapuov fuctio for th rror dyamics (8), which has a globally dfid, gativ dfiit tim drivativ V. Thus, by Lyapuov stability thory [], it is immdiat that th rror dyamics (8) is globally asymptotically stabl for all iitial coditios (0) R. This mas that for all iitial coditios (0) R, w hav lim t ( ) = 0 t Hc, it follows that th mastr systm () ad th slav systm () ar globally ad asymptotically sychroizd for all iitial coditios x(0), y(0) R. This complts th proof. III. GLOBAL CHAOS SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC QI SYSTEMS A. Thortical Rsults I this sctio, w apply th slidig mod cotrol rsults drivd i Sctio for th global chaos sychroizatio of idtical hyprchaotic Qi systms ([], 008). Th hyprchaotic Qi systm is o of th paradigms of th four-dimsioal hyprchaotic systms discovrd by G. Qi, M.A. Wyk, B.J. Wyk ad G. Ch (008). Thus, th mastr systm is dscribd by th hyprchaotic Qi dyamics x = ax ( x) + xx 3 x = bx ( + x) xx 3 x = cx ε x + x x x = dx + fx + x x x x x x ar stat variabls of th systm ad abcd,,,, ε, f whr,, 3, 4 ar positiv, costat paramtrs of th systm. Th slav systm is dscribd by th cotrolld hyprchaotic Qi dyamics y = ay ( y) + yy 3+ u y = by ( + y) yy 3 + u y = cy ε y + y y + u y = dx + fy + y y + u whr y, y, y3, y4ar stat variabls ad u, u, u3, u4ar th cotrollrs to b dsigd. Th Qi systm () is hyprchaotic wh th paramtr valus ar tak as a = 50, b = 4, c = 3, d = 8, ε = 33 ad f = 30. Th hyprchaotic portrait of th Qi systm () is illustratd i Fig.. Th sychroizatio rror is dfid by () () ISSN : Vol. 3 No. 6 Ju 0 433

5 Dr. V. Sudarapadia t al. / Itratioal Joural o Computr Scic ad Egirig (IJCSE) = y x = y x = y x = y x (3) Th rror dyamics is asily obtaid as = a( ) + y y x x + u 3 3 = b( + ) y y + x x + u 3 3 = c ε + y y x x + u = d + f + y y x x + u Figur. Stat Portrait of th Hyprchaotic Qi Systm W writ th rror dyamics (4) i th matrix otatio as = A+ η( x, y) + u (5) whr (4) a a 0 0 b b 0 0 A =, 0 0 c ε 0 0 f d η( xy, ) yy3 xx3 y y + x x 3 3 = yy xx y y x x ad u u u =. u3 u4 (6) ISSN : Vol. 3 No. 6 Ju 0 434

6 Dr. V. Sudarapadia t al. / Itratioal Joural o Computr Scic ad Egirig (IJCSE) Th slidig mod cotrollr dsig is carrid out as dtaild i Sctio. First, w st u as u = η( x, y) + Bv (7) whr B is chos such that ( AB, ) is cotrollabl. W tak B as B =. I th hyprchaotic cas, th paramtr valus ar a= 50, b= 4, c= 3, d = 8, ε = 33 ad f = 30. (8) Th slidig mod variabl is slctd as [ 8 ] s = C= = (9) which maks th slidig mod stat quatio asymptotically stabl. W choos th slidig mod gais as k = 4 ad q = 0.. W ot that a larg valu of k ca caus chattrig ad a appropriat valu of q is chos to spd up th tim tak to rach th slidig maifold as wll as to rduc th systm chattrig. From Eq. (5), w ca obtai vt () as vt ( ) = sg( s) (30) 3 4 Thus, th rquird slidig mod cotrollr is obtaid as u = η( x, y) + Bv (3) whr η( x, y), B ad vt () ar dfid as i th quatios (6), (8) ad (30). By Thorm, w obtai th followig rsult. Thorm. Th idtical hyprchaotic Qi systms () ad () ar globally ad asymptotically sychroizd for all iitial coditios with th slidig cotrollr u dfid by (3). B. Numrical Rsults 6 For th umrical simulatios, th fourth-ordr Rug-Kutta mthod with tim-stp h = 0 is usd to solv th hyprchaotic Qi chaotic systms () ad () with th slidig cotrollr u giv by (3) usig MATLAB. I th hyprchaotic cas, th paramtr valus ar a= 50, b= 4, c= 3, d = 8, ε = 33 ad f = 30. Th slidig mod gais ar chos as k = 4 ad q = 0.. Th iitial valus of th mastr systm () ar tak as x (0) = 6, x (0) = 6, x (0) =, x (0) = ad th iitial valus of th slav systm () ar tak as ISSN : Vol. 3 No. 6 Ju 0 435

7 Dr. V. Sudarapadia t al. / Itratioal Joural o Computr Scic ad Egirig (IJCSE) y (0) = 4, y (0) = 5, y (0) = 4, y (0) = Fig. illustrats th complt sychroizatio of th idtical hyprchaotic Qi systms () ad (). Figur. Sychroizatio of Idtical Hyprchaotic Qi Systms IV. CONCLUSIONS I this papr, w hav dployd slidig cotrol to achiv global chaos sychroizatio for th idtical hyprchaotic Qi systms (008). Our sychroizatio rsults for th idtical hyprchaotic Qi systms hav b provd usig Lyapuov stability thory. Sic th Lyapuov xpots ar ot rquird for ths calculatios, th slidig cotrol mthod is vry ffctiv ad covit to achiv global chaos sychroizatio for th idtical hyprchaotic Qi systms. Numrical simulatios ar also show to illustrat th ffctivss of th sychroizatio rsults drivd i this papr usig th slidig mod cotrol. REFERENCES [] K.T. Alligood, T. Saur ad J.A. York, Chaos: A Itroductio to Dyamical Systms, Sprigr, Nw York, 997. [] L.M. Pcora ad T.L. Carroll, Sychroizatio i chaotic systms, Physical Rviw Lttrs, vol. 64, pp. 8-84, 990. [3] M. Lakshmaa ad K. Murali, Noliar Oscillators: Cotrollig ad Sychroizatio, World Scitific, Sigapor, 996. [4] S.K. Ha, C. Krrr ad Y. Kuramoto, Dphasig ad burstig i coupld ural oscillators, Physical Rviw Lttrs, vol. 75, pp , 995. [5] B. Blasius, A. Hupprt ad L. Sto, Complx dyamics ad phas sychroizatio i spatially xtdd cological systm, Natur, vol. 399, pp , 999. [6] K.M. Cuomo ad A.V. Opphim, Circuit implmtatio of sychroizd chaos with applicatios to commuicatios, Physical Rviw Lttrs, vol. 7, pp , 993. [7] L. Kocarv ad U. Parlitz, Gral approach for chaotic sychroizatio with applicatios to commuicatio, Physical Rviw Lttrs, vol. 74, pp , 995. [8] Y. Tao, Chaotic scur commuicatio systms history ad w rsults, Tlcommuicatio Rviw, vol. 9, pp , 999. [9] E. Ott, C. Grbogi ad J.A. York, Cotrollig chaos, Physical Rviw Lttrs, vol. 64, pp , 990. ISSN : Vol. 3 No. 6 Ju 0 436

8 Dr. V. Sudarapadia t al. / Itratioal Joural o Computr Scic ad Egirig (IJCSE) [0] M.C. Ho ad Y.C. Hug, Sychroizatio of two diffrt chaotic systms usig gralizd activ cotrol, Physics Lttrs A, vol. 30, pp , 00. [] L. Huag, R. Fg ad M. Wag, Sychroizatio of chaotic systms via oliar cotrol, Physics Lttrs A, vol. 30, pp. 7-75, 005. [] H.K. Ch, Global chaos sychroizatio of w chaotic systms via oliar cotrol, Chaos, Solitos ad Fractals, vol. 3, pp. 45-5, 005. [3] J. Lu, X. Wu, X. Ha ad J. Lü, Adaptiv fdback sychroizatio of a uifid chaotic systm, Physics Lttrs A, vol. 39, pp , 004. [4] S.H. Ch ad J. Lü, Sychroizatio of a ucrtai uifid systm via adaptiv cotrol, Chaos, Solitos ad Fractals, vol. 4, pp , 00. [5] J.H. Park ad O.M. Kwo, A ovl critrio for dlayd fdback cotrol of tim-dlay chaotic systms, Chaos, Solitos ad Fractals, vol. 7, pp , 003. [6] X. Wu ad J. Lü, Paramtr idtificatio ad backstppig cotrol of ucrtai Lü systm, Chaos, Solitos ad Fractals, vol. 8, pp. 7-79, 003. [7] J. Zhao ad J. Lü, Usig sampld-data fdback cotrol ad liar fdback sychroizatio i a w hyprchaotic systm, Chaos, Solitos ad Fractals, vol. 35, pp , 006. [8] J.E. Sloti ad S.S. Sastry, Trackig cotrol of oliar systms usig slidig surfac with applicatio to robotic maipulators, Itratioal Joural of Cotrol, vol. 38, pp , 983. [9] V.I. Utki, Slidig mod cotrol dsig pricipls ad applicatios to lctric drivs, IEEE Trasactios o Idustrial Elctroics, vol. 40, pp. 3-36, 993. [0] R. Saravaakumar, K. Vioth Kumar ad K.K. Ray, Slidig mod cotrol of iductio motor usig simulatio approach, Itratioal Joural of Computr Scic ad Ntwork Scurity, vol. 9, pp , 009. [] G. Qi, M.A. Wyk, B.J. Wyk ad G. Ch, O a w hyprchaotic systm, Physics Lttrs A, vol. 37, pp. 4-36, 008. [] W. Hah, Th Stability of Motio, Sprigr, Nw York, 967. AUTHORS PROFILE Dr. V. Sudarapadia was bor o July 5, 967 at Uttamapalayam, Thi district, Tamil Nadu, Idia. H obtaid his D.Sc. dgr i Elctrical ad Systms Egirig from Washigto Uivrsity, USA i 996 H is workig as Profssor (Systms ad Cotrol Egirig), Rsarch ad Dvlopmt Ctr at Vl Tch Dr. RR & Dr. SR Tchical Uivrsity, Chai, Tamil Nadu, Idia. H has publishd graduat-lvl books titld, Numrical Liar Algbra ad Probability, Statistics ad Quuig Thory with PHI Larig Privat Limitd, Idia. H has publishd ovr 30 rfrd itrtaioal joural publicatios. H has publishd 90 paprs i Natioal Cofrcs ad 45 paprs i Itratioal Cofrcs. H is th Editor-i-Chif of Itratioal Joural of Mathmatics ad Scitific Computig ad Itratioal Joural of Mathmatical Scics ad Applicatios. H is a Associat Editor of Itratioal Joural o Cotrol Thory ad Applicatios, Itratioal Joural of Advacs i Scic ad Tchology, Itratioal Joural of Computr Iformatio Systms, Joural of Elctroics ad Elctrical Egirig, tc. His rsarch itrsts ar i th aras of Liar ad Noliar Cotrol Systms, Chaos Thory, Dyamical Systms ad Stability Thory, Optimal Cotrol, Opratios Rsarch, Soft Computig, Modllig ad Scitific Computig, Numrical Mthods, tc. H has dlivrd svral Ky Not Lcturs o Noliar Cotrol Systms, Chaos ad Cotrol, Scitific Modllig ad Computig with SCILAB, tc. Mr. S.Sivaprumal was bor o July 09, 983 at Thirukkoyilur, Tamil Nadu, Idia. H is workig as Assistat Profssor i Vl Tch Dr.RR & Dr.SR Tchical Uivrsity, Chai ad pursuig Ph.D. i th School of Elctroics ad Elctrical Egirig, Sighaia Uivrsity, Rajastha, Idia. H obtaid M.E. dgr i VLSI Dsig ad B.E. dgr i Elctroics ad Commuicatios Egirig from Aa Uivrsity, Chai, Idia i th yar 007 ad 005 rspctivly. H is workig as a Assistat Profssor i th Dpartmt of Elctroics ad Commuicatio Egirig at Vl Tch Dr. RR & Dr. SR Tchical Uivristy, Chai, Tamil Nadu, Idia. H has publishd two paprs i rfrd Itratioal Jourals. H has also publishd paprs o VLSI ad Cotrol Systms i Natioal ad Itratioal Cofrcs. His currt rsarch itrsts ar Robotics, Commuicatios, Cotrol Systms ad VLSI Dsig. ISSN : Vol. 3 No. 6 Ju 0 437