Simple Observer Based Synchronization of Lorenz System with Parametric Uncertainty

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1 IOSR Journal of Electrical and Electronic Engineering (IOSR-JEEE) ISSN: Volume, Iue 6 (Nov. - Dec. 0), PP 4-0 Simple Oberver Baed Synchronization of Lorenz Sytem with Parametric Uncertainty Manih Kumar Vaihnav, Dr. B.B. Sharma, (EED, National Intitute of Technology, Hamirpur (H.P), India) Abtract: In thi paper, oberver baed ynchronization of chaotic Lorenz ytem i preented. The virtual ytem concept i ued for ynchronization of oberver tate to actual ytem tate. Firt, ynchronization i preented without any uncertainty in ytem parameter and further ynchronization i achieved for the cae with uncertainty in ytem parameter. Adaptation law for uncertain parameter are developed a a conequence of tability analyi. Convergence of uncertain parameter are alo analyzed. Simulation reult are preented in the end to verify the ynchronization of oberver tate to actual ytem tate. Keyword:Contraction theory, Exponential convergence, Oberver, Parametric uncertainty, Synchronization, Virtual ytem. I. Introduction Since lat few decade, ynchronization & tability of chaotic ytem have been explored by many reearcher [-4]. Synchronization of two chaotic ytem mean that ytem trajectorie of both the ytem are varying in ynchronized way keeping in view the ignificant application of chaotic ytem in different field, many chaotic ytem are tudied and analyzed. In thi paper, oberver baed ynchronization i preented for chaotic Lorenz ytem. Chaotic Lorenz ytem i decribed by et of differential equation propoed by meteorologit/mathematician Edward N. Lorenz, who wa tudying thermal variation in an air cell underneath a thunderhead. Since it inception, the Lorenz ytem ha become one of the mot widely tudied ytem of ODE becaue of it wide range of behaviour. In the preent paper contraction theory reult are utilized to addre ynchronization of Lorenz ytem. Contraction theory i recently developed tool for analyzing the tability and ynchronization of nonlinear control ytem. Some of important reult of contraction theory can be found in literature [5-0] & reference therein. In oberver baed ynchronization approach, the actual ytem tate are recontructed by utilizing the information available at the actual ytem output. The output may be function of one or more tate or the combination of tate. More information related to uch approache can be found in literature [-4]. In thi paper, it i aumed that the output i the function of only one tate of actual ytem. In mot of the cae Lyapunov baed analyi i ued for oberver baed ynchronization of chaotic ytem. Here, Contraction theory i ued for the ynchronization of the Lorenz ytem. The advantage of thi method i that, there i no iue of election of energy function and it handle the ynchronization problem effectively. Adaptation law for uncertain parameter are developed in term of etimated tate. Here adaptation law are obtained while etablihing tability uing virtual ytem concept. Thi paper i outlined a follow: In ection-, preliminarie of contraction theory and virtual ytem are preented. In ection-, problem are formulated for ynchronization of chaotic Lorenz ytem for both parametric certainty and parametric uncertainty cae. In ection-4, imulation reult are preented to verify the ynchronization of oberver tate to actual ytem tate. Finally, concluion i preented in ection-5 of paper. Conider a nonlinear ytem x = f( x, t) () II. Preliminarie m Where x and fx (, t) i continuou differentiable function uch that, f : ( m) vector function. The main definition and theorem of contraction are taken from [5] and reproduced here for the clarity purpoe. Let x be the virtual diplacement in x, firt variation in x will be fx (, t) x = x x d ( T ) T T λ (, ) T m t dt f x x x x x x x x x () x () Now taking the time derivative of quared ditance 4 Page

2 Where f repreent the Jacobian matrix J of the ytem () and λ m( x, t) repreent the larget eigenvalue of x the ymmetric part of Jacobian matrix J. If λ m( x, t) i trictly uniformly negative definite (UND), then any infiniteimal length converge exponentially to zero. Then all trajectorie of the ytem will converge together exponentially. Definition. A region of the tate pace i called a contraction region with repect to a uniformly poitive T definite metric Mx (, t) Θ ( x, t) Θ( x, t) where Θ tand for a differential coordinate tranformation matrix, if equivalently f F Θ Θ Θ x (4) I uniformly negative definite (UND). Thi lead to the following convergence reult. Lemma.For ytem (), any trajectory which tart in a ball of contant radiu centered about a given trajectory and contained at all the time in a contraction region, remain in that ball and converge exponentially to given trajectory. Global exponential convergence to given trajectory i guaranteed if whole tate pace region i contracting. The following Lemma i ued to analyze the convergence of oberver tate to actual ytem tate and taken from [6]. Lemma.If a virtual ytem defined for actual ytem and oberver ytem along with parameter adaptationlaw, then incremental behaviour in term of virtual diplacement could be repreented in matrix form a Φ J : ( ˆ Q y, t) Φ (5) T θ Q ( yˆ, t) : 0 θ where Qy ( ˆ, t) i a nonlinear regreion matrix function of oberver output ŷ, Φ and θ repreent tate vector and uncertain parameter vector of virtual ytem, repectively, then ytem in (5) would be emi contracting in nature ubject to following condition: (i) Jacobian of virtual ytem dynamic w.r.t. it tate Φ i.e. ubmatrix J i uniformly negative definite (UND). (ii) All the variable involved in (5) are bounded. Semi-contracting nature of virtual ytem implie that particular olution of the virtual ytem converge to xa time t and etimate each other aymptotically with parameter etimate remaining bounded i.e. xˆ vector ˆθ remain bounded. II. Virtual Sytem Theory Let the dynamic of a nonlinear ytem i given a x ( x, θ, t) x (6) m Where x, θ i parameter of the ytem and function ( x, θ, t) I 0. Now virtual ytem i defined a y ( x, θ, t) y (7) In differential frame work y ( x, θ, t) y One can ee that the virtual ytem i contracting w.r.t. y uing the baic reult of contraction. Since actual ytem i a particular olution of the virtual ytem, o the actual ytem tate will alo converge together, exponentially. In cae of oberver deign, virtual ytem i defined uch a both actual ytem and oberver are particular olution of the virtual ytem. For actual ytem a x xu (8) Let oberver be defined a 5 Page

3 xˆ xˆ u L( x xˆ ) (9) Where u i control input and L i gain of the oberver, elected a L. Let the virtual ytem i defined a z z u L( x z) (0) The value of L i elected uch that the Jacobian of virtual ytem i uniformly negative definite (UND). It will etablih contracting nature of virtual ytem and further it will lead to exponential convergence of oberver tate to actual ytem tate. III. The Invetigation Of Chaotic Lorenz Sytem Synchronization Baed On Contraction Theory The equation of Lorenz ytem i decribed a follow: x ax ax x rx xx x x xx bx () Where a, r and b are real contant parameter and x, x and x are variable. Let the meaurable output of Lorenz ytem be given a y cx=[ 0 0] x ; () The ytem i chaotic when a 0, r=0 and b 8/. Initial condition for the ytem are taken a [- -7 5] T and imulation i run for 80 econd with tep ize of 0.0.The behaviour of the ytem i hown in Fig.(). Three dimenional phae portrait i hown in Fig. (a). The open loop repone of the ytem to initial condition i hown in Fig. (b)-(d). Fig. Lorenz ytem without controller: (a) Phae portrait in three dimenion and (b) - (d) time variation of tate trajectorie III. Oberver deign for chaotic Lorenz ytem for without parametric uncertainty The tructure of the oberver for the ytem () can be taken a 6 Page

4 xˆ axˆ axˆ L ( y yˆ ) xˆ ˆ ˆ ˆ ˆ ˆ rx x x x L ( y y) xˆ ˆ ˆ ˆ ˆ x x bx L ( y y) () And the output of the oberver i given by yˆ Cxˆ [ 0 0] x ˆ (4) Where L, L and L repreent the gain of full order oberver. Select gain of oberver uch that oberver tate converge exponentially to actual ytem tate. Let the virtual ytem for actual ytem () and oberver ytem () be elected a Φ aφ aφ L ( y yφ) Φ rφ Φ ΦΦ L ( y yφ) Φ ΦΦ bφ L ( y yφ) (5) And the output of the virtual ytem i given by yφ CΦ [ 0 0] Φ (6) Now uing the definition of y and Φ Φ JΦ (7) Where J repreent the Jacobian matrix of virtual ytem and given a y in virtual ytem (5) and defining the virtual diplacement Φ a a L a 0 r Φ L Φ Φ L b T J J J can be repreented a a r Φ L Φ L al a r Φ L 0 Φ L 0 b J (8) The ymmetric part of Jacobian J (9) The gain L i ; i=,, are elected uitably that enure J to be UND in nature, o that contracting nature of virtual ytem i etablihed. Since actual ytem and oberver are particular olution of virtual ytem, o thee will converge to each other, exponentially. To enure UND nature of J following condition hould be atified. (i) L (ii) (iii) a ( a r Φ L) ( al ) 4 Φ L a r Φ L a L b b Page

5 Fig. Convergence of behaviour for Lorenz ytemwith certain parameter: (a) (c) time variation of tate trajectorie and (d) variation of tate etimation error. III. Oberver deign for chaotic Lorenz ytem for parametric uncertainty Here, it i aumed that parameter a of Lorenz ytem i uncertain and it i aumed that uncertain parameter appear in tate equation correponding to meaurable variable in the tate dynamic. So modified oberver dynamic for uncertain parameter cae i preented a xˆ ax ˆˆ ax ˆˆ L ( y yˆ ) xˆ ˆ ˆ ˆ ˆ ˆ rx x x x L ( y y) xˆ ˆ ˆ ˆ ˆ x x bx L ( y y) (0) Where â repreent the etimated value of parameter a and parametric error i given by definition, the oberver dynamic i modified a a aˆ a.uing thi xˆ axˆ axˆ ax ˆ ax ˆ ˆ L ( y y) xˆ ˆ ˆ ˆ ˆ ˆ rx x xx L ( y y) xˆ ˆ ˆ ˆ ˆ x x bx L ( y y) () Let the adaptation law for uncertain parameter be aˆ ( xˆ xˆ )( x xˆ ) Defining a new virtual ytem a () Φ aφ aφ a xˆ a ˆ x L ( y yφ) Φ rφ Φ ΦΦ L ( y yφ) Φ ΦΦ bφ L ( y yφ) () 8 Page

6 And the adaptation law a a ( xˆ xˆ )( x Φ) (4) Where a i the parameter of virtual ytem and parametric error i defined a a a a.the Jacobian matrix in thi cae will be a L ˆ ˆ a 0 ( x x) r Φ L Φ 0 J Φ L Φ b 0 (5) ( xˆ ˆ x) T J J J can be repreented a a r Φ L Φ L The ymmetric part of Jacobian J al 0 a r Φ L 0 0 (6) Φ L 0 b By electing the uitable value of gain L i ; i=,,, UND nature of J ytem can be enure. So by uing Lemma, it can be aid that oberver tate will converge to actual ytem tate. It i not aid confidently about the convergence of etimate of uncertain parameter to it actual value. Although uncertain parameter i bounded in our cae. IV. Simulation To verify the ynchronization of the oberver tate to actual ytem tate for the cae without parametric uncertainty, initial condition for the propoed oberver i taken a[ ] T.The value of gain L ; i=,,, are choen o that J i UND in nature. The deired value of oberver gain are L 40, L and L 8. Simulation i run for 40 econd with tep ize of 0.0. Initially oberver gain are kept zero and witched on at t= econd. The convergence of oberver tate ˆx to actual tate x i hown in Fig. (). In parametric uncertainty cae, the initial condition for oberver are taken a [ ] T.The imulation i run for 40 econd with tep ize of 0.0 econd. Initially, parametric etimate for uncertain parameter a i taken a zero. Here, oberver gain are witched on at t=5 econd. The convergence behaviour i hown in Fig. (). The time variation of tate trajectorie of oberver and actual Lorenz ytem i preented in Fig. (a)-(c), variation of tate etimation error and convergence behaviour of etimate of uncertain parameter are preented in Fig. (d) & (e), repectively. i 9 Page

7 Fig. Convergence behaviour of oberver for Lorenz ytem with uncertain parameter: (a) (c) time variation of tate trajectorie of oberver and actual Lorenz ytem, (d) variation of tate etimation error and (e) convergence behaviour of etimate of uncertain parameter. V. Concluion Synchronization cheme for chaotic Lorenz ytem i preented uing oberver baed methodology. The approach utilize virtual ytem theory concept. Contraction theory provide very effective way of etablihing ynchronization. Parameter adaptation law i derived in term of etimated tate. Finally, etablihing imulation reult are preented to illutrate the aymptotic converge of oberver tate to the actual ytem tate. Parametric etimate remain bounded while etablihing thee reult. Reference [] Jayaram A., Tadi M.: Synchronization of chaotic ytem baed on SDRE method. Chao Soliton Fractal vol.8, Nr., pp (006). [] Carroll T.L. and Pecora L.M.: Synchronizing chaotic circuit. IEEE Tran. vol.8, pp (99). [] CarrollT.L. and Pecora L.M.: Cacading ynchronized chaotic ytem. Phyica D 67, 6-40 (99). [4] Carroll T.L. and Pecora L.M.: Synchronizing non-autonomou chaotic circuit. IEEE Tran. Circuit Syt. 40, 646 (995). [5] Lohmiller W., and Slotine J.J.E.: Control ytem deign for mechanical ytem uing Contraction theory.ieee Tran. Automatic Control, vol.45, No.5 (000). [6] Jouffroy J. and Slotine J.J.E.: Methodological remark on contraction theory. IEEE Conference on Deciion and Control Atlanti, Paradie Iland, Bahama, pp (004). [7] Jouffroy J., Some ancetor of contraction analyi, Proc. of IEEE Conference on Deciion and Control, pp , Seville, Spain (005). [8] B. B. Sharma and I. N. Kar: Deign of aymptotically convergent frequency etimator uing contraction theory. IEEE Tran. on Automatic Control, vol.5, No 8, pp.9-97 (008). [9] B.B.Sharma and I.N.Kar: Contraction theory baed recurive deign of tabilizing controller for a cla of nonlinear ytem. IET control theory & application, pp (00). [0] B.B.Sharma and I. N.Kar: Contraction baed adaptive control of a cla of nonlinear ytem. Proc. American control conference, pp (009). [] Liao T.L., Huang N.S.: An oberver baed approach for chaotic ynchronization and ecure communication. IEEE Tran. Circuit Syt.I 46(9), (999). [] Nijmeijer H., MareelI.M.: An oberver look at ynchronization. IEEE Tran. Circuit Syt.I 44(0), (997). [] Boutayeb M., Darouach M., Rafaralahy H.: Generalized tate-pace oberver for chaotic ynchronization with application to ecure communication. IEEE Tran. Circuit Syt.I 49(), (00). [4] Cherrier E., Boutayeb M., Ragot J.: Oberver-baed ynchronization and input recovery for a cla of nonlinear chaotic model. IEEE Tran. Circuit Syt.I 5(9), (006). [5] LohmillerW. and SlotineJ.J.E.: On contraction analyi for nonlinear ytem. Automatica, vol.4, No.6, pp (998). [6] B.B.Sharma and I. N.kar: Oberver-baed ynchronization cheme for a cla of chaotic ytem uing contraction theory.nonlinear Dynamic, vol.6, No., pp (0). 0 Page