Fuzzy Modeling and H Synchronization of Different Hyperchaotic Systems via T S Models

Transcription

1 Appl. Math. Inf. Sci. 7, No. L, Applied Mathematics & Infomation Sciences An Intenational Jounal c 23 NS Natual Sciences ublishing Co. Fuzzy Modeling and H Synchonization of Diffeent Hypechaotic Systems via T S Models Yan Zhao,2, Tieyan Zhang, Dongsheng Yang 2 and Xiaohui Zhang Depatment of Automatic Contol Engineeing, Shenyang Institute of Engineeing, Shenyang 36, China 2 College of Infomation Science and Engineeing, Notheasten Univesity, Shenyang 89, China Received: 4 Sep. 22, Revised: 22 Nov. 22, Accepted: 2 Dec. 22 ublished online: Feb. 23 Abstact: In this pape, a unified fuzzy H contol appoach is poposed to deal with the poblem of synchonization of two diffeent hypechaotic systems. The T S fuzzy models with a small numbe of fuzzy IF-THEN ules ae employed to epesent many typical hypechaotic systems exactly. Based on the T S fuzzy hypechaotic models, a fuzzy H synchonization contolle is designed via paallel distibuted compensation DC techniques. The sufficient condition fo the H synchonization of two diffeent hypechaotic systems is deived by applying Lyapunov stability theoy. The esults ae expessed in tems of LMIs, and theefoe it is vey convenient to check in pactice. This method is a univesal one fo synchonization of two diffeent hypechaotic systems. Numeical examples ae given to demonstate the validity of the poposed fuzzy modeling and hypechaotic synchonization scheme. Keywods: hypechaotic systems, fuzzy modeling, T S fuzzy model, H synchonization, linea matix inequalitylmi. Intoduction Since the pioneeing wok of Caoll and ecoa [], synchonization of two chaotic dynamical systems has been paid moe and moe attention due to its potential applications in many fields. Recently, many new hypechaotic systems have been found o ceated by eseaches continually [2-5]. Moeove, synchonization of two hypechaotic systems has been studied extensively in the last few yeas. Vaious contol methods have been applied to synchonize two hypechaotic o chaotic systems, such as obust contol [6], adaptive contol [7-], linea feedback contol [2], delayed feedback contol [3], impulsive contol [4-9], fuzzy contol [2-2], etc. Among these synchonization methods, some only focus on the synchonization of two identical hypechaotic systems. Othes only dealt with one o two kinds of specified hypechaotic systems. Thee ae few unified methods suitable fo synchonization of two diffeent hypechaotic systems. At pesent, the Takagi Sugeno T S fuzzy model poposed in [22] is widely applied to many fields because of its simple stuctue with local dynamics [23-27]. Tanaka et al [28] established the accuate T S fuzzy epesentation fo many kinds of typical chaotic systems and Lian et al [29] pesented a synthesis appoach fo synchonization of chaotic systems based on T S fuzzy models. But fo hypechaotic systems, to the best knowledge of authos, thee ae few esults about this wok. Theefoe, in this pape, we will give a systematic scheme to epesent many classes of well-known hypechaotic systems by the T S fuzzy models exactly and then we design a geneal fuzzy contolle to ealize the H synchonization of two diffeent hypechaotic systems by paallel distibuted compensation DC and LMI techniques. 2. T S fuzzy modeling of hypechaotic systems 2.. T S fuzzy model Conside the continuous-time T S fuzzy ule base descibed as follows: R i : IF p t is M i,..., and p q t is M iq, THEN ẋt = A i xt + B i ut + b, whee i =,2,..., is the numbe of fuzzy ules, xt R n and ut R m denote the state vecto and the Coesponding autho zhaoyan@sie.edu.cn c 23 NS Natual Sciences ublishing Co.

2 94 Y. Zhao, T. Zhang, D. Yang, X. Zhang: Fuzzy Modeling and H Synchonization... input vecto, espectively, A i R n n and B i R n m ae known system matix and input matix with appopiate dimensions, espectively, p t, p 2 t,..., p q t ae pemise vaiables, M i j is a fuzzy set j =,2,...,q, b is a constant vecto. By taking a standad fuzzy infeence stategy, i.e., using a singleton fuzzifie, poduct fuzzy infeence and cente aveage defuzzifie, the final continuous-time fuzzy T S system is infeed as follows: whee ẋt = ω ipta i xt + B i ut + b ω, ipt ω i pt = q j= M i j p j t, in which M i j p j t is the degee of membeship of p j t in M i j, with ω i pt >, ω i pt, i =,2,...,. By using µ i pt= ω i pt / ω ipt instead of ω i pt, Eq. is ewitten as: Note that ẋt = µ i pta i xt + B i ut + b. 2 µ i pt =, µ i pt, i =,2,...,, in which µ i pt can be egaded as the fiing stength of the ith IF-THEN ules Fuzzy modeling of hypechaotic systems In this subsection, we pesent a systematic T S fuzzy modeling method fo many typical hypechaotic systems. Note that most of hypechaotic systems can be expessed as follows: ẋt = Ãxt + gxt + b, 3 whee xt = x t,x 2 t,x 3 t,x 4 t T, the linea tem Ãxt and the nonlinea tem gxt epesent the linea pat and the nonlinea pat of the hypechaotic systems espectively, b is the constant vecto of the hypechaotic systems. Accoding to the boundedness of hypechaotic systems, one can assume that x i t [ c i d i, c i + d i ], whee d i >. Next, Accoding to vaious foms of the nonlinea pat gxt, the following T S fuzzy modeling method fo hypechaotic systems is given below. Fist, we assume that thee is a common facto in gxt of hypechaotic systems. In this situation, one can choose the common facto occued in gxt as the pemise vaiable of T S fuzzy ules. Fo example, in the following hypechaotic system [25], the nonlinea tems include x tx 2 t and x tx 3 t. Among these two nonlinea tems, the common facto is x t. Theefoe we can choose x t as the pemise vaiable. ẋ t = ax 2 t x t, ẋ 2 t = dx t + cx 2 t x tx 3 t x 4 t, 4 ẋ 3 t = x tx 2 t lx 3 t, ẋ 4 t = x t + k. The membeship functions ae chosen as F x t = c x t, 2 F 2 x t = 2 d + c x t d. And the fuzzy IF-THEN ules ae defined as: R : IF x t is F, THEN ẋt = A xt + b, R 2 : IF x t is F 2, THEN ẋt = A 2 xt + b, whee xt = x t,x 2 t,x 3 t,x 4 t T. Fo the above fuzzy ule base, by taking standad fuzzy infeence stategy, i.e., using a singleton fuzzifie, poduct fuzzy infeence and cente aveage defuzzifie, we can get the following T S fuzzy system: ẋt = 2 F ix ta i xt + b 2 F. 5 ix t The est wok is to detemine A, A 2 and b such that Eqs. 5 and 3 ae equivalent completely. Fo the above hypechaotic system 4, each pat of 3 is pesented below: a a d c à = l x gxt = tx 3 t x tx 2 t b =. k it is easy to see that Eq. 5 is equal to 3 as long as A, A 2 t and b ae chosen as follows: a a d c c A = + d c + d l a a d c c A 2 = d c d l c 23 NS Natual Sciences ublishing Co.

3 Appl. Math. Inf. Sci. 7, No. L, / 95 b = k T. Consequently, the exact T S fuzzy models of the above hypechaotic system is obtained as 5. If thee is no common facto in the nonlinea pat gxt, we need to choose any two state vaiables occuing in gxt as the pemise vaiables. The est modeling pocess is simila to the above, and theefoe is omitted. To summaize, most existing hypechaotic systems can be epesented by the following T S fuzzy models exactly: R i : IF p t is F i and... and p q t is F iq, THEN ẋt = A i xt + b, whee i =,2,...,, is the numbe of fuzzy IF-THEN ules, the pemise vaiable p j t is a pope state vaiable and b is a constant vecto. Then, applying the poduct-infeence ule, singleton fuzzifie, and the cente of gavity defuzzifie to the above fuzzy ule base, the oveall fuzzy hypechaotic system can be infeed as: ẋt = whee µ i pt = q F i j p j t j= µ i pta i xt + b, 6 / q F i j p j t. j= 3. Fuzzy H synchonization of two diffeent hypechaotic systems In this section, we will investigate the synchonization of two diffeent hypechaotic systems based on the T S fuzzy models. Fistly, an assumption is given as follows. Assumption 3.: The numbes of fuzzy ules of two diffeent fuzzy hypechaotic systems to synchonize ae identical. Conside the fuzzy hypechaotic system 6 as the dive system, and the following fuzzy hypechaotic system as the esponse system: ˆxt = ˆµ i ˆpt  i ˆxt + ˆb + ut. 7 Based on DC technique, a fuzzy contolle can be designed to ealize the synchonization as follows: Subcontolle u d t: R i : IF p t is F i and... and p q t is F iq, THEN u d t = K i xt, i =,2,...,. Subcontolle u t: R i : IF ˆp t is ˆF i and... and ˆp q t is ˆF iq, THEN u t = ˆK i ˆxt, i =,2,...,. The oveall fuzzy contolle is constucted by the paallel connection, i.e., ut =u d t + u t = µ i ptk i xt ˆµ i ˆpt ˆK i ˆxt. 8 Denote the eo signal as et = xt ˆxt. Substituting 8 into 7, we can deive the closed-loop synchonization eo system as follows: whee ėt = µ i pta i K i et+ ˆµ i ˆpt 9  i ˆK i et + ϖ, ϖt = ϖ t + ϖ 2 t + b ˆb, ϖ t = ϖ 2 t = µ i pta i K i ˆxt, ˆµ i ˆpt  i ˆK i xt. Fo the synchonization eo system 9, conside the following H pefomance index: e T tetdt < 2 et e+ γ 2 + γ2 2 2 ϖ T tϖtdt, whee γ and γ 2 ae pescibed attenuation level, t f is teminal time. Theoem 3.: Consideing the fuzzy eo system 9, fo the given constants γ and γ 2, if thee exist a matix = T >, and eal matices M i, N i with appopiate dimensions, such that the following LMIs ae satisfied: Ai + A T i MT i M i + I 2 2 γ2 I <, and T  i +  i N T i N i + I 2 2 γ2 2I <, 2 whee i =,2,...,. Then we can choose K i = M i, ˆK i = N i in 8, such that the obust pefomance is guaanteed. oof: Now, we select the Lyapunov function as: V t = e T tet, whee is a positive definite matix. The time deivative of V t along the tajectoy of 9 is given by V t =ė T tet + e T tėt =e T tx +Y T et + e T tx +Y et + ϖ T tet + e T tϖt T et et = Ψ, ϖt ϖt c 23 NS Natual Sciences ublishing Co.

4 96 Y. Zhao, T. Zhang, D. Yang, X. Zhang: Fuzzy Modeling and H Synchonization... whee Ψ = X T + X +Y T + Y, X = Y = Based on, define J = And we have J = µ i pta i K i, ˆµ i ˆpt  i ˆK i. 2e T tet γ 2 + γ 2 2 ϖ T tϖt + V t dt V tdt. 2e T tet γ 2 + γ 2 2 ϖ T tϖt + V t dt V t f +V { µ i pt + ˆµ i ˆpt T et et Ξ ϖt ϖt T } et et Γ dt ϖt ϖt +V, G whee Ξ = 2 H 2 γ2 I, Γ = 2 2 γ2 2 I, G = A i K i T +A i K i +I, H = T  i ˆK i + Âi ˆK i +I. It is easy to see that if the following inequalities hold: G 2 2 γ2 I <, 3 and H 2 2 γ2 2 I then we have J = V. <, 4 2e T tet γ 2 + γ 2 2 ϖ T tϖt dt 5 Fom 5, it is obvious that the H pefomance index is guaanteed. Denoting M i = K i, N i = ˆK i, then 3 and 4 ae equivalent to the LMIs and 2. This completes the poof. Remak 3.: If the numbes of fuzzy ules of two diffeent hypechaotic systems to synchonize ae not identical, thee is anothe method we can adopt in [23] to solve the synchonization poblem. The authos in [23] developed a switching contolle to synchonize the two diffeent fuzzy chaotic systems with the non-identical numbes of fuzzy ules. 4. Simulation study To visualize the effectiveness of the theoetical analysis and design, some examples ae included fo illustation. Fistly, we investigate the T S fuzzy modeling of Liu hypechaotic system and Loenz hypechaotic system. Liu hypechaotic system [2]: ẋ t = ax 2 t x t, ẋ 2 t = lx t kx tx 3 t + x 4 t, ẋ 3 t = cx 3 t + gx 2t, 6 ẋ 4 t = dx t, whee a =, l = 4, c = 2.5, d =.6, k = and g = 4. Hee, Liu hypechaotic system has two positive Lyapunov exponents λ =.49 and λ 2 = The attacto of Liu hypechaotic system is shown in Fig. and Fig. 2. The initial state vecto is chosen as x =,,, T. Fom the simulation we can get x [ c d, c + d ], whee c =.2347, d = Obviously, the nonlinea tems ae kx tx 3 t and gx 2 t in 6. thee exists a common facto x t in the nonlinea tems. Then we can deive the T S fuzzy models of Liu hypechaotic system as follows: R : IF x t is F, THEN ẋt = A xt + b, R 2 : IF x t is F 2, THEN ẋt = A 2 xt + b 2, whee xt = x t,x 2 t,x 3 t,x 4 t T, A = A 2 = b = b 2 = T. The membeship functions ae chosen as: F x t =.2347 x t, F 2 x t = x t Loenz hypechaotic system [3]: ˆx t = a ˆx t ˆx 2 t + ˆx 4 t, ˆx 2 t = l ˆx t ˆx 2 t ˆx t ˆx 3 t, ˆx 3 t = ˆx t ˆx 2 t k ˆx 3 t, ˆx 4 t = ˆx t ˆx 3 t + d ˆx 4 t, 7 whee a =, l = 28, k = 8/3 and d =.3. Hee, Loenz hypechaotic system has two positive Lyapunov exponents λ = and λ 2 = The attacto of Loenz hypechaotic system is shown in Fig. 3 and Fig. 4. The initial state vecto is chosen as ˆx =,2,,2 T. Fom the simulation we can get ˆx [ c d, c + d ], whee c 23 NS Natual Sciences ublishing Co.

5 Appl. Math. Inf. Sci. 7, No. L, / x 3 t 5 x 4 t x 2 t x t x 2 t x t 2 Figue ojection of Liu hypechaotic attacto on x x 2 x 3 Figue 2 ojection of Liu hypechaotic attacto on x x 2 x 4 c =.4366, d = Then we can deive the T S fuzzy models of Loenz hypechaotic system as follows: R : IF ˆx t is ˆF, THEN ˆxt = Â ˆxt + ˆb, R 2 : IF ˆx t is ˆF 2, THEN ˆxt = Â 2 ˆxt + ˆb 2, whee ˆxt = ˆx t, ˆx 2 t, ˆx 3 t, ˆx 4 t T, Â = / Â 2 = / ˆb = ˆb 2 = T. x 3 t x 2 t 2 Figue 3 ojection of Loenz hypechaotic attacto on x x 2 x 3 2 x t 2 The membeship functions ae chosen as: ˆF ˆx t =.4366 ˆx t, ˆF 2 ˆx t = ˆx t In the following, based on the above T S fuzzy hypechaotic models, we investigate the synchonization between Liu hypechaotic system and Loenz hypechaotic system. We conside Liu hypechaotic system as the dive system, and Loenz hypechaotic system as the esponse system. The initial conditions fo the dive system and the esponse system ae x = 5,25,, T, 3 2 x 4 t 2 2 x 2 t 2 2 x t 2 ˆx = 25, 2,3, T. Accoding to Theoem, choosing γ = γ 2 =., we can get the positive definite matix and feedback gains K i, Figue 4 ojection of Loenz hypechaotic attacto on x x 2 x 4 c 23 NS Natual Sciences ublishing Co.

6 98 Y. Zhao, T. Zhang, D. Yang, X. Zhang: Fuzzy Modeling and H Synchonization... et e2t e3t e4t tsec 5. Conclusions In this wok, a T S-fuzzy-model-based synchonization method fo two diffeent hypechaotic systems is poposed. Based on the exact fuzzy hypechaotic models, a univesal fuzzy H synchonization contolle is designed via the Lyapunov stability theoy and DC technique. The elevant esults ae pesented in the fom of LMIs. Numeical simulation esults ae given to demonstate the effectiveness of the poposed scheme. This implies that fuzzy-model-based synchonization appoach fo two diffeent hypechaotic systems is vey flexible and useful in pactical applications. Figue 5 Synchonization eo et between Liu hypechaotic system and Loenz hypechaotic system ˆK i i =,2 computed by LMI toolbox in Matlab 7. as follows: = K = K 2 = ˆK = ˆK 2 = The synchonization contol signals ae imposed when t = 2sec. The synchonization eo cuves ae pesented in Fig. 5. Fom Fig. 5, we can see that H synchonization between the above two diffeent hypechaotic systems is deived apidly. Remak 4.: If a pai of smalle γ and γ 2 ae chosen, then we can deive moe satisfactoy synchonization esults. But the highe gains of the contolle ae needed at the same time. Acknowledgement The authos acknowledge the financial suppot of the National Natue Science Foundation, poject No , , 694, 62386, the Key oject of Chinese Ministy of Education, poject No. 2233, the Key Technologies R & D ogam of Liaoning ovince, poject No , 2273, the ogam fo Liaoning Excellent Talents in Univesity, poject No. LJQ237, the ogam fo Liaoning Innovative Reseach Team in Univesity, poject No. LT29 and the Science and Technology ogam of Shenyang, poject No. F , F-44-2-, the Key Laboatoy oject of Liaoning ovincial Education Depatment, poject No. 28S58. The authos ae gateful to the anonymous efeee fo a caeful checking of the details and fo helpful comments that impoved this pape. Refeences [] L.M. ecoa, T.L. Caoll. Synchonization in chaotic systems, hysics Review Lette 64, [2] C.X. Liu, T.Liu, L.Liu, et al, A new chaotic attacto, Chaos, Solitons & Factals 22, [3] Q. Jia, Hypechaos geneated fom the Loenz chaotic system and its contol, hysics Lette A, 366, [4] S. Nikolov, S. Clodong, Hypechaos-chaos-hypechaos tansition in modified Rössle systems, Chaos, Solitons Factals 28, [5] L. Shan, J. Li, Z.Q. Wang, Fuzzy feedback contol of a new Liu chaotic system with paametic uncetainty, Jounal of Systems Engineeing and Electonics 29, in Chinese [6] L.Q. Shen, M. Wang, Robust synchonization on a class of uncetain chaotic systems, Jounal of Systems Engineeing and Electonics 3, in Chinese [7] H.G. Zhang, Y.H. Xie, Z.L. Wang, et al, Adaptive synchonization between two diffeent chaotic neual netwoks with time delay, IEEE Tansaction on Neual Netwoks 8, c 23 NS Natual Sciences ublishing Co.

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8 2 Y. Zhao, T. Zhang, D. Yang, X. Zhang: Fuzzy Modeling and H Synchonization... Xianghui Zhang is pesently employed as Associate ofesso at SIE, China. He eceived his Maste degee in electic electonics and electic tansmission fom Notheasten Univesity in 29. His cuent eseach inteests ae fuzzy contol. c 23 NS Natual Sciences ublishing Co.