A NEW CRITERION FOR SYNCHRONIZATION OF COUPLED CHAOTIC OSCILLATORS WITH APPLICATION TO CHUA S CIRCUITS

Transcription

1 Internatonal Journal of Bfurcaton and Chaos, Vol. 9, No. 6 (1999) c World Scentfc Publshng Company A NEW CRITERION FOR SYNCHRONIZATION OF COUPLED CHAOTIC OSCILLATORS WITH APPLICATION TO CHUA S CIRCUITS XIAO FAN WANG and ZHI QUAN WANG Department of Automatc Control, Nanjng Unversty of Scence & Technology, Nanjng, Chna GUANRONG CHEN Department of Electrcal and Computer Engneerng, Unversty of Houston, Houston, Texas , USA Receved October 16, 1998; Revsed November 17, 1998 A new crteron s gven for choosng the couplng constant n a system of coupled chaotc oscllators to guarantee ther synchronzaton. The crteron s derved from a new observer desgn methodology based on Lyapunov stablty theory. As an example and applcaton, we prove the conjecture that synchronzaton of two chaotc Chua crcuts can be acheved wth the second state as the couplng varable provded that the couplng constant s sutably chosen accordng to the new crteron. 1. Introducton Recently, there has been consderable nterest n chaos synchronzaton and ts applcatons n communcatons [Ogorzalek, 1993; Hasler, 1997; Chen, 1998; Chen & Dong, 1998]. In the studes of varous chaos synchronzaton problems, senstvty ssues wth respect to parameter msmatch and nose dsturbances are of great mportance. One of the wdely used crteron for chaotc synchronzaton s that all condtonal Lyapunov exponents of the coupled system are negatve. However, recent research has shown that ths s nether a necessary nor a practcal suffcent condton: Chaos synchronzaton s stll possble wth the exstence of a postve condtonal Lyapunov exponent; and the presence of extremely small dsturbances (from system nose or parameter msmatch) can cause chaos synchronzaton to break down. Two typcal engneerng approaches to understandng and achevng chaos synchronzaton are nonlnear observer desgn [Njmejer & Mareels, 1997; Fradkov et al., 1999] and Lyapunov stablty methods [Wu & Chua, 1994; d Bernardo, 1996]. Although the Lyapunov stablty theoretc approach s mathematcally rgorous and can sometmes provde robust and global results, there s generally no routne procedure for constructng a sutable Lyapunov functon for a gven par of coupled chaotc systems. On the contrary, there are some standard methods for observer desgn for a general system; so ths approach can be advantageous n some applcatons. In ths paper, by applyng a new observer desgn methodology developed for general nonlnear systems [Rajaman, 1998; Rajaman & Cho, 1998], we present a new crteron for choosng the couplng constant n a coupled system of chaotc oscllators, so as to guarantee ther synchronzaton. Ths new observer desgn method s based on the Lyapunov stablty theory; therefore, n a way t combnes the aforementoned two successful approaches for chaos E-mal: wangxf@mal.njust.edu.cn 1169

2 1170 X. F. Wang et al. synchronzaton. Wu and Chua [1994] have conjectured that synchronzaton of two chaotc Chua crcuts can be acheved wth the second state as the couplng varable when the couplng constant s large enough. In ths paper, as an example and applcaton, we confrm ths conjecture by means of showng that synchronzaton can be acheved when the couplng constant s sutably chosen from among a certan nterval of real values. The concluson that we have reached s consstent wth the well-known observaton that, for two Chua s crcuts coupled by the second-state varable, too large a couplng effect would actually decrease the synchronzaton speed. Throughout the paper, denotes the Eucldean norm, cond(t) = T T 1 the condtonal number of matrx T, σ mn (M) the smallest sngular values of matrx M [Golub & van Loan, 1983], Re{a} the real parts of a complex number a, and j = A New Observer Desgn Methodology Consder a nonlnear system descrbed by ẋ = Ax +Φ(x) (1a) y = Cx (1b) where x R n and y R m are state and output of the system, respectvely, A R n n and C R m n are constant matrces, Φ(x) R n s a nonlnear functon satsfyng the Lpschtz condton Φ(x) Φ(ˆx) ρ x ˆx (1c) for a constant ρ (called the Lpschtz constant). The observer s assumed to be of the form ˆx = Aˆx +Φ(ˆx)+L(y Cˆx), (2) where L s a constant gan matrx. It s clear that the estmaton error e = x ˆx satsfes ẋ =(A LC)e +[Φ(x) Φ(ˆx)]. (3) It mght be easly speculatng that stablty can be guaranteed by smply placng the egenvalues of (A LC) on the left-half complex plane, far away from the magnary axs. However, ths ntuton s not always correct due to the nonlnearty exstng n the system. Therefore, more rgorous crtera are needed for the desgn of the observer. Theorem 1 [Rajaman, 1998]. Suppose that n system (1), the par (A, C) s observable. If L can be chosen to ensure that () (A LC) s stable n the sense that the real parts of ts egenvalues are all negatve; () the followng sngular value condton s satsfed: mn σ mn(a LC jωi) >ρ, (4) ω R + then there exsts a postve defnte matrx P such that V = e T Pe s a Lyapunov functon for the error dynamcal system (3). Consequently, ths error system wth the observer gven by (2) s asymptotcally stable n the sense that lm e(t) =0. t Remark. The par (A, C) s sad to be observable f all dstnct states of the correspondng lnear system ẋ = Ax, y = Cx are dstngushable from the output y(t). It s well known n lnear system theory that the par (A, C) s observable f and only f the followng rank condton s satsfed [Kalath, 1980]: C CẠ rank. = n, CA n 1 or equvalently, for any complex number λ, ( ) λi A rank = n. C Theorem 2 [Rajaman, 1998]. Let {λ } n =1 and T be the egenvalues and the matrx consstng of the correspondng egenvectors of (A LC), respectvely, namely, A LC = TΛT 1, Λ=dag(λ 1,..., λ n ). If mn Re{ λ } >ρ cond(t), (5) then nequalty (4) holds. Remark. Roughly speakng, by the well-known Bauer Fke Theorem n lnear algebra [Gloub & van

3 New Crteron for Synchronzaton of Coupled Chaotc Oscllators 1171 Loan, 1983], mn Re{ λ } ρ cond(t) characterzes the convergence speed of the error e(t) tendng to zero. The above results present a suffcent condton for system stablty n terms of the egenvalues and egenvectors of (A LC) aswellasthelpschtz constant ρ: The egenvalues have to be suffcently negatve and the egenvectors have to be suffcently well-condtoned, so as to domnate the Lpschtz nonlnearty of the system. Generally, the problem of choosng the gan matrx L so that (5) holds s a complex nonlnear optmzaton problem. Nevertheless, an analytcal soluton for a stable observer gan can be found f A s stable and the dstance to unobservablty of the par (A, C) exceeds the Lpschtz constant ρ [Rajaman & Cho, 1998]. In the next secton, we gve a new and easly verfed crteron for synchronzaton of coupled chaotc oscllators based on the above exstng results. 3. A New Crteron for Synchronzaton of Coupled Chaotc Oscllators Consder two x -coupled chaotc systems descrbed by ẋ = Ax +Φ(x) (6) and ˆx = Aˆx +Φ(ˆx) ck(ˆx x) (7) where x and ˆx are n-dmensonal states of the master system (6) and the slave system (7), respectvely, c(> 0) s the couplng constant, and k 1 k 2 K =... Rn n, k n k =1; k j =0 forj. The slave system (7) can be vewed as an observer for the master system (6), where x s the scalar output of (6) and C =[c 1,c 2,..., c n ], c =1; c j =0 forj, L=[l 1,l 2,..., l n ] T, l = c; l j =0 forj, ck=lc. Theorems 1 and 2 together mply the followng result: Theorem 3. Let {λ } n =1 and T be the egenvalues and the matrx consstng of the correspondng egenvectors of (A ck), that s, A ck = TΛT 1, Λ=dag(λ 1,..., λ n ). (8) For a gven couplng constant c, f mn Re{ λ } >ρ cond(t), (9) then the master system (6) and the slave system (7) are synchronzed n the sense that the synchronzaton error e = x ˆx tends to zero asymptotcally: lm e(t) =0. t Clearly, just lke many other Lyapunov functon based crtera, Theorem 3 only provdes a suffcent condton; namely, f we cannot fnd a constant c such that condton (9) s satsfed, t does not mean synchronzaton between (6) and (7) cannot be acheved. Note also that t s sometmes possble to use a coordnate transform to reduce the value of the term on the rght-hand sde of (9). Consder the followng nonsngular coordnate transform: z = Dx, ẑ = Dˆx (10) where D R n n s a nonsngular matrx. The dynamcs for z and ẑ are then gven by where ż = Az + Φ(z), ẑ = Aẑ + Φ(z) ck(ẑ z), (11a) (11b) A=DAD 1, K = DKD 1, Φ(z,=DΦ(D 1 z). Therefore, A ck = D(A ck)d 1. We denote by ρ D the new Lpschtz constant; that s, Φ(z) Φ(ẑ) ρ D z ẑ. (12) It s easy to see that ρ D ρ cond(d). (13)

4 1172 X. F. Wang et al. Clearly, x and ˆx are synchronzed f and only f z and ẑ are synchronzed. Note that the egenvalues and the matrx of the egenvectors of A ck are {λ } n =1 and DT, respectvely. We have mmedately the followng smple generalzaton of Theorem 3: Theorem 4. For a gven couplng constant c, f mn Re{ λ } > ρ D cond(dt) (14) then the master system (6) and the slave system (7) are synchronzed n the sense specfed above. Now, the problem becomes how to choose a sutable transform matrx D to mnmze the value of ρ D cond(dt). There does not exst a general method for choosng an optmal transform matrx; and yet a natural choce of the transform matrx D would smply be one that mnmzes the value of (cond(dt)). Snce the condton number of any matrx s greater than or equal to one, we can acheve ths as follows: Let T T = QR be the Q R decomposton of T T [Golub & van Loan, 1983], where Q s an orthogonal matrx and R s an upper trangular matrx. Choosng D = R T, we have DT = Q T,so cond(dt) =cond(q)=1. In ths case, condton (14) becomes mn Re{ λ } > ρ R T. (15) Moreover, snce orthogonal transformaton does not change the condton number of a matrx, we have ρ R T ρ cond(r) ρ cond(t), whch mples that the above coordnate transformaton cannot ncrease the product of the Lpschtz constant and the condton number of the matrx of the egenvectors. In case studes, t s possble to have a better estmate of the value of ρ R T. For example, consder the specal case where the nonlnear part Φ(x) conssts of only one nonlnear element n the form Φ(x) =[ϕ(x 1 ),0,..., 0] T, (16) where ϕ(x 1 ) ϕ(ˆx 1 ) ρ x 1 ˆx 1. (17) In ths case, we have Φ(z) Φ(ẑ) = R T (Φ(R T z) Φ(R T ẑ))} = R 1 (ϕ(r 11 z 1 ) ϕ(r 11 ẑ 1 )) ρ r 11 R 1 z 1 ẑ 1 ρ R T z 1 ẑ 1 (18) where r 11 s the element of the (1, 1) entry of R and R 1 s the frst column of R T. 4. Applcaton to Coupled Chua s Crcuts A wdely used system for generatng chaotc sgnals s Chua s crcut [Chua et al., 1993a]. Synchronzaton of two coupled Chua s crcuts have been studed through lnear couplng or feedback [Chua et al., 1993b, 1996; Wu & Chua, 1994; Schezer, 1997; Wang & Wang, 1998], among many other approaches. In partcular, t was proven n [Wu & Chua, 1994] that synchronzaton of two Chua s crcuts can be acheved wth the frst state as the couplng varable when the couplng constant s large enough. The key here s that the unque nonlnearty n the crcut s a pecewse lnear resstor whch depends only on the frst state. If ths state s taken as the drvng sgnal, communcaton between the two crcuts s not very secure snce one could easly fnd out, at any tme, n whch lnear regon the drve system s operatng. As a result, ths would enable an unauthorzed recever to estmate the system parameters by usng conventonal lnear short-term system dentfcaton methods. It was conjectured, based on smulatons, that synchronzaton of two Chua s crcuts can be acheved wth the second state as the couplng varable when the couplng constant s large enough [Wu & Chua, 1994], whch would sgnfcantly ncrease the securty of communcatons between two coupled Chua s crcuts. However, no rgorous argument has been gven so far to our knowledge. Based on the new synchronzaton crteron establshed above, we herewth prove that synchronzaton of two Chua s crcuts can be acheved wth the second state as the couplng varable when the couplng constant belongs to some nterval of real values.

5 New Crteron for Synchronzaton of Coupled Chaotc Oscllators 1173 In the dmensonless form, two x 2 -coupled Chua s crcuts are descrbed by ẋ 1 α(x 2 x 1 + f(x 1 )) ẋ 2 = ẋ 3 x 1 x 2 + x 3 βx 2 γx 3 (17a) ˆx 1 α(ˆx 2 ˆx 1 + f(ˆx 1 )) ˆx 2 = ˆx 1 ˆx 2 +ˆx 3 c(ˆx 2 x 2 ) ˆx 3 βˆx 2 γˆx 3 (17b) mn Re { λ } ρ cond( T) c Fg. 1. The value of mn Re{ λ } ρ cond(t) for two x 2-coupled Chua s crcuts. where f( ) s a pecewse-lnear functon: bx 1 a + b x 1 > 1 f(x 1 )= ax 1 x 1 1 bx 1 + a b x 1 < 1 (17c) mn Re { λ} ρ T R where α>0, β>0, γ>0anda<b<0. Usng the above-defned notaton, we have α α 0 A ck = 1 1 c 1, 0 β γ The Lpschtz constant for f s ρ = a,.e. Fg. 2. The value of mn Re{ λ } ρ R T for two x 2-coupled Chua s crcuts. x xˆ c f(x 1 ) f(ˆx 1 ) a x 1 ˆx 1 a x ˆx. (18) In ths study, the system parameters are chosen to be α = , β = , γ =0.0385, a = , b = (19) Wthout coordnate transforms, we cannot fnd a couplng constant so that condton (9) s satsfed (Fg. 1). However, by usng the aforementoned coordnate transform wth D = R T, we can fnd a range of values for the couplng constant such that condton (15) s satsfed (Fg. 2): mn Re{ λ } >ρ R T ρ r 11 R 1, for any c (2.7, 8.3), whch mples that two x 2 -coupled Chua s crcuts are ndeed synchronzed wth the system parameters as gven by (19) and wth any couplng constant c (2.7, 8.3). We have found that the maxmum value of mn Re{ λ } ρ R T s attaned at c =7.4. Thus, we have shown that two x 2 -coupled Chua s crcuts can synchronze as long as a sutable couplng constant s used. Although we dd Fg. 3. Synchronzaton of two x 2-coupled Chua s crcuts performed wth c = 7.4. not exactly fnd a unversal threshold c > 0such that condton (15) holds for any couplng constant c>c [Wu & Chua, 1994], we have found a convergence range for the couplng constant to be c (2.7, 8.3), and more mportantly, we have found that, n contrast to ntuton, ncreasng the couplng constant n two x 2 -coupled Chua s crcuts may actually decrease the convergence speed of the synchronzaton error. Ths has been confrmed by smulaton results performed wth c = 7.4, 74 and 740, respectvely, as shown n Fgs The new crteron serves as a gudelne for choosng a sutable couplng constant n two x 2 - coupled Chua s crcuts to guarantee synchronzaton and to obtan a fast convergence speed. t

6 1174 X. F. Wang et al. x xˆ Fg. 4. Synchronzaton of two x 2-coupled Chua s crcuts performed wth c = 74. x xˆ Fg. 5. Synchronzaton of two x 2-coupled Chua s crcuts performed wth c = Conclusons In ths paper, we have derved a new suffcent crteron for synchronzaton of coupled chaotc oscllators. As an applcaton example of ths crteron, we have proven the conjecture that synchronzaton of two x 2 -coupled Chua s crcuts can be acheved provded that the couplng constant s sutably chosen accordng to the new crteron. Ths suggests that the second, nstead of the frst state of the crcut can be rgorously used for synchronzaton, so as to ncrease the communcaton securty between two coupled Chua s crcuts. Acknowledgments Ths work s supported n part by the Open Research Laboratory of Computer Network & Supportng Technology for Informaton Integraton (State Educaton Commttee of the Southeast Unversty, P. R. Chna). References Chen, G. [1998] Control and synchronzaton of chaotc systems (a bblography), ECE Department, Unversty of Houston, TX avalable from anonymous ftp: ftp.egr.uh.edu/pub/tex/chaos.tex Chen, G. & Dong, X. [1998] From Chaos to Order t t Methodologes, Perspectves and Applcatons (World Scentfc, Sngapore). Chua, L. O., Wu, C. W., Huang, A. & Zhong, G. Q. [1993a] A unversal crcut for studyng and generatng chaos, Part I+II, IEEE Trans. Crcuts Syst. 40(10), Chua, L. O., Itoh, M., Kocarev, L. & Eckert, K. [1993b] Chaos synchronzaton n Chua s crcut, J. Crcuts Syst. Comput. 3(1), Chua, L. O., Yang, T., Zhong, G. Q. & Wu, C. W. [1996] Adaptve synchronzaton of Chua s oscllators, Int. J. Bfurcaton and Chaos 6(1), Dedeu, H., Kennedy, M. P. & Hasler, M. [1993] Chaos shft keyng: Modulaton and demodulaton of a chaotc carrer usng self-synchronzng Chua s crcuts, IEEE Trans. Crcuts Syst. 40(10), d Bernardo, M. [1996] An adaptve approach to the control and synchronzaton of contnuous-tme chaotc systems, Int. J. Bfurcaton and Chaos 6, Fradkov, A. L., Njmejer, H. & Pogromsky, A. Yu. [1999] Adaptve observer-based synchronzaton, n Controllng Chaos and Bfurcatons n Engneerng Systems, ed.chen,g.(crcpress,bocaraton,usa), pp Golub, G. H. & van Loan, C. F. [1983] Matrx Computatons (Johns Hopkns Unversty Press). Hasler, M. [1997] An ntroducton to the synchronzaton of chaotc systems: Coupled skew tent maps, IEEE Trans. Crcuts Syst. 44(10), Kalath, T. [1980] Lnear Systems (Prentce-Hall, Englewood Clffs). Hjmejer, H. & Mareels, I. [1997] An observer looks at synchronzaton, IEEE Trans. Crcuts Syst. I44, Ogorzalek, M. [1993] Tamng chaos Part I: Synchronzaton, IEEE Trans. Crcuts Syst. I40(10), Pecora, L. M. & Carroll, T. L. [1991] Drvng systems wth chaotc sgnals, Phys. Rev. Lett. A44, Rajaman, R. [1998] Observers for Lpschtz nonlnear systems, IEEE Trans. Automat. Contr. 43(3), Rajaman, R. & Cho, Y. M. [1998] Exstence and desgn of observers for nonlnear systems: Relaton to dstance to unobservablty, Int. J. Contr. 69(5), Schwezer, J., Kennedy, M. P., Hasler, M. & Dedeu, H. [1995] Synchronzaton theorem for a chaotc system, Int. J. Bfurcaton and Chaos 5(1), Wang, X. F. & Wang, Z. Q. [1998] Synchronzaton of Chua s oscllators wth the thrd state as the drvng sgnal, Int. J. Bfurcaton and Chaos 8(7), Wu, C. W. & Chua, L. O. [1994] A unfed framework for synchronzaton and control of dynamcal systems, Int. J. Bfurcaton and Chaos 4(4),