基于状态观测器的广义投影同步

Generalized projective synchronization of a class of hyperchaoticsystems based on state observerWang Xing-Yuan ⇑,Fan BingSchool of Electronic &Information Engineering,Dalian University of Technology,Dalian 116024,Chinaa r t i c l e i n f o Article history:Received 7May 2010Received in revised form 29May 2011Accepted 12June 2011Available online 21June 2011Keywords:Hyperchaotic systemGeneralized projective synchronizationState observera b s t r a c tIn this paper,the generalized projective synchronization of a class of hyperchaotic systemsis studied.On the basis of the state observer,it is not necessary to calculate the Lyapunovexponents,which makes this scheme simpler.Hyperchaotic Lüsystem and hyperchaoticRössler systems are used as examples to validate the effectiveness of the proposed method.Ó2011Elsevier B.V.All rights reserved.1.IntroductionSince the synchronization of chaotic systems has been proposed by Pecora and Carroll in 1990[1,2],chaos synchroniza-tion has received considerable attention [3,4].In practice,due to the parameter mismatch problem in complete synchroni-zation,many scholars have proposed and implemented a number of different types of chaos synchronization methods [5–9],such as lag synchronization [5],phase and anti-phase synchronization [6],generalized synchronization [7]and generalized projective synchronization [8,9].Projective synchronization means that the driving-response systems can be synchronized up to a scaling factor a (a proportional relation).Complete synchronization and anti-synchronization are the special cases of projective synchronization when a =1and a =À1,respectively [10].The proportionality between the synchronized dynam-ical states is of great use in practice.In the application of secure communication,this feature can be used to extend binary digital to M-nary digital communication for achieving fast communication [10–13].The early studies of projective synchro-nization reported that the projective synchronization was usually observable only in a class of systems with partial-linearity[10,11].But recently some researchers have achieved control of the projective synchronization in a class of general chaotic systems,and named this new synchronization as ‘‘generalized projective synchronization’’(GPS)[9,10].Due to higher confidentiality,higher efficiency and other advantages,hyperchaotic systems have wide applications in the non-linear circuits,security communication,neural network,etc.Therefore,in recent years it has attracted much attention[14,15].However the computation of Lyapunov exponents in hyperchaotic systems is still a difficult issue [16].To avoid the complexity of calculation,state observer was adopted by some researchers.One of the features of this method is that it is not necessary to calculate the Lyapunov exponents [16,17].Based on the above studies,this paper presents a generalized pro-jective synchronization (GPS)scheme for a class of hyperchaotic systems using state observer.The numerical simulations of hyperchaotic Lüsystem and hyperchaotic Rössler system further validate the effectiveness of this method.1007-5704/$-see front matter Ó2011Elsevier B.V.All rights reserved.⇑Corresponding author.Tel.:+86041187090067.E-mail addresses:wangxy@ (X.-Y.Wang),winterfice@ (B.Fan).The rest of this paper is organized as follows.In Section 1,a GPS scheme based on state observer is presented.The two hyperchaotic systems which are used as examples are briefly described in Section 2.Sections 4.1and 4.2give the simulation results of hyperchaotic Lüsystem and hyperchaotic Rössler system,respectively.The last part is the conclusion of the paper.2.GPS scheme designConsider the following chaotic system_x¼Ax þB F ðx ÞþG ;w ¼Dx þF ðx Þ; ð1Þwhere x and w is the state vector and output of the driving system,respectively.B F (x )is the nonlinear part of the system,G is the constant of the system.A and B are known constant matrices,D is an undetermined gain matrix.In order to achieve GPS,we need to construct the corresponding response system of Eq.(1).The state observer is designed as_y¼Ay þB F ðy ÞþB ða w Àv ÞþKG ;v ¼Dy þF ðy Þ; ð2Þwhere y and v is the state vector and output of the observer,respectively.K is the unknown constant matrix,a is the scaling factor.So complete synchronization and anti-synchronization are two special cases of projective synchronization when a =1and a =À1,respectively.Now define the error vector as e =y Àa x ,so the error system is determined as follows,_e¼_y Àa _x ¼Ay þa B F ðx ÞþBD ða x Ày ÞÀa Ax Àa B F ðx ÞþðK Àa I ÞG ¼ðA ÀBD Þðy Àa x ÞþðK Àa I ÞG ¼ðA ÀBD Þe þðK Àa I ÞG :ð3ÞIt can be seen that if Eq.(3)satisfies the following equation ðK Àa I Þ¼0;ð4Þi.e.K =a I ,Eq.(3)can be rewritten as _e¼ðA ÀBD Þe :By using state observer,it is not necessary to calculate the Lyapunov exponents [16,17].As long as all the eigenvalueswith negative real parts,the error system satisfies _e!0and system (1)and system (2)can achieve GPS.It can be seen that if the gain matrix D is selected appropriately to guarantee all the eigenvalues of A ÀBD with negative real parts,the error system will be stable and the GPS between the two systems will be achieved.3.System descriptionThe hyperchaotic Lüsystem [18]can be described as follows,_x 1¼a ðx 2Àx 1Þþx 4;_x 2¼Àx 1x 3þcx 2;_x 3¼x 1x 2Àbx 3;_x 4¼x 1x 3þrx 4;8>>><>>>:ð5Þwhere a ,b ,c and r are real constants.When a =36,b =3,c =20,À1.036r 6À0.46,system (5)has periodic orbit,when a =36,b =3,c =20,À0.46<r 6À0.35,system (5)has chaotic attractor,when a =36,b =3,c =20,À0.35<r 61.3,system (5)has hyperchaotic attractor [18].The hyperchaotic Rössler system [19]can be described as follows,_x 1¼bx 2þcx 1;_x 2¼3þx 2x 3;_x 3¼Àx 2Àx 4;_x 4¼x 1þx 3þax 4;8>>><>>>:ð6Þ954X.-Y.Wang,B.Fan /Commun Nonlinear Sci Numer Simulat 17(2012)953–9634.Numerical simulation4.1.Hyperchaotic LüsystemRewrite Eq.(5)as_x 1_x 2_x 3_x 42666437775¼Àa a 010c 0000Àb 0000r 2666437775x 1x 2x 3x 42666437775þB x 1x 2x 1x 3 ;w ¼D x 1x 2x 3x 42666437775þx 1x 2x 1x 3 :8>>>>>>>>>>>>><>>>>>>>>>>>>>:ð7ÞThen,we have A ¼Àa a 010c 0000Àb 0000r 2666437775;G ¼0;F ðx Þ¼x 1x 2x 1x 3;where D is an undetermined gain matrix.Here,pole placement method is used to construct matrix D to guarantee all the eigenvalues of A ÀBD with negative real parts.LetB ¼000À110012666437775and select the eigenvalues of A ÀBD as (À1,À2,À3,À4),then we can getD ¼001:0000À30:51769:471400:7714 :Substituting A ,B and D for Eq.(2),the response system,i.e.,state observer of the driving system (7)is described by_y 1_y 2_y 3_y 42666437775¼Àa a 010c 0000Àb 0000r 2666437775y 1y 2y 3y 42666437775þ000À110012666437775y 1y 2y 1y 3 þ000À110012666437775ða w Àv Þ;v ¼001:0000À30:51769:471400:7714 y 1y 2y 3y 42666437775þy 1y 2y 1y 3 :8>>>>>>>>>>>>><>>>>>>>>>>>>>:ð8ÞWe select parameters as a =36,b =20,c =3and r =1.3to ensure system (5)is hyperchaotic.The initial values of the dynam-ical system (7)and the response system (8)are:(x 1(0),x 2(0),x 3(0),x 4(0))=(3,4,5,6)and (y 1(0),y 2(0),y 3(0),y 4(0))=(1,2,8,9).Firstly,when a =1,the simulation results of GPS between drive system (7)and response system (8)are shown in Figs.1and 2.From Fig.1(the red 1line stands for the drive system,blue line represents the response system),it can be seen that the state variables of the system (7)and system (8)tend to evolve in the same direction all the time and after a finite period of time,tend to close and get to synchronization.Fig.2shows the trajectories of e 1(t ),e 2(t ),e 3(t )and e 4(t )of the error system quickly approach to zero,respectively,which implies that state vectors tend to be synchronized.Then we set the scaling factor as a =À1.From Fig.3,we can see that the state trajectories of system (7)and system (8)tend to evolve in the opposite direction.The same results are shown in Fig.5with a =À2.It is clear that the state variables of two systems tend to be proportionally synchronized in the opposite direction.Figs.4and 6show the trajectories of e 1(t ),e 2(t ),e 3(t )and e 4(t )in the error system with a =À1and a =À2,respectively.It can be seen that after a finite period of time,the state errors converge to zero and the ratio of the amplitudes of the two systems tends to a constant scaling factor,which X.-Y.Wang,B.Fan /Commun Nonlinear Sci Numer Simulat 17(2012)953–963955Fig.1.State variables of system(7)and system(8)with a=1.Fig.2.The time evolution of errors with a=1.also implies that state vectors tend to be synchronized proportionally.In other words,the GPS between master and slaveFig.3.State variables of system(7)and system(8)with a=À1.Fig.5.State variables of system(7)and system(8)with a=À2.4.2.Hyperchaotic Rössler systemSimilar to the above example,rewrite Eq.(6)as follows_x 1_x 2_x 3_x 42666437775¼c b 0000000À10À1101a 2666437775x 1x 2x 3x 42666437775þB x 2x 3½ þ03002666437775;w ¼D x 1x 2x 3x 42666437775þ½x 2x 3 ;8>>>>>>>>>>>>><>>>>>>>>>>>>>:ð9Þwhere A ¼c b 0000000À10À1101a 2666437775;G ¼03002666437775;F ðx Þ¼½x 2x 3 :D is an undetermined gain matrix.Pole placement method is used to construct matrix D to guarantee all the eigenvalues of A ÀBD with negative real parts.LetB ¼01002666437775and select the eigenvalues of A ÀBD as (À1,À2,À3,À4),then we can getD ¼½À59:471910:3000À7:3416À32:8385 :From Eq.(4)we getK ¼a 0000a 0000a 0000a2666437775:Substituting A ,B ,D ,K and G in Eq.(2),the response system,i.e.,state observer of the driving system (9)can be obtained as _y 1_y 2_y 3_y 426666643777775¼c b 0000000À10À1101a 26666643777775y 1y 2y 3y 426666643777775þ010026666643777775y 2y 3½ þ010026666643777775ða w Àv Þþa 0000a 0000a 0000a 26666643777775030026666643777775;v ¼½À59:471910:3000À7:3416À32:8385 y 1y 2y 3y 426666643777775þ½y 2y 3 :8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:ð10ÞSystem parameters are chosen as a =0.25,b =À0.5,c =0.05to ensure system (6)is hyperchaotic.The initial values of the dynamical system (9)and the response system (10)are:(x 1(0),x 2(0),x 3(0),x 4(0))=(15,À10,À10,10)and (y 1(0),y 2(0),y 3(0),y 4(4))=(À16,À8,À2,9).Similar to the above example,we studied the GPS of drive system (9)and response system (10)with a =1,a =À1and a =À2.Figs.7–12are the corresponding simulation results.Figs.7,9and 11show the state vectors of drive system (9)and re-sponse system (10)tend to be proportionally synchronized and the ratio of the amplitudes of the two systems tends to a constant scaling factor,respectively.From Figs.8,10and 12,it can be seen that the trajectories of e 1(t ),e 2(t ),e 3(t )and e 4(t )of the error system approach to zero after an infinite period of time,respectively,which implies that drive systemX.-Y.Wang,B.Fan /Commun Nonlinear Sci Numer Simulat 17(2012)953–963959Fig.7.State variables of system(9)and system(10)with a=1.Fig.9.State variables of system(9)and system(10)with a=À1.Fig.11.State variables of system(9)and system(10)with a=À2.X.-Y.Wang,B.Fan/Commun Nonlinear Sci Numer Simulat17(2012)953–963963 5.ConclusionIn this paper,a generalized projective synchronization scheme for a class of hyperchaotic systems has been proposed. Based on state observer,there is no need to calculate the Lyapunov exponents.Furthermore,hyperchaotic Lüsystem and hyperchaotic Rössler system are chosen to validate the effectiveness of the proposed scheme.The proposed GPS scheme has good application prospects in the secure communicationfield.AcknowledgementsThe research is supported by the National Natural Science Foundation of China(No.60573172,60973152),the Superior University Doctor Subject Special Scientific Research Foundation of China(No.20070141014)and the Natural Science Foun-dation of Liaoning province(No.20082165).References[1]Pecora LM,Carroll TL.Synchronization of chaotic systems.Phys Rev Lett1990;64(8):821–4.[2]Cheng GR,LüJH.Dynamics analyse,control and synchronization in lorenz systems.Beijing:Science Press;2003.p.185–258.[3]Guan XP,Fan ZP,Chen CL,Hua CC.Chaotic control and its application on secure communication.Beijing:National Defence Industry Press;2002.p.168–225.[4]Wang XY.Chaos in the complex nonlinearity system.Beijing:Electronics Industry Press;2003.p.91–113.[5]Li ChD,Liao XF,Wong g synchronization of hyperchaos with application to secure communications.Chaos Solitons Fract2005;23(1):183–93.[6]Ho MC,Hung YC,Chou CH.Phase and anti-phase synchronization of two chaotic systems by using active control.Phys Lett A2002;296(1):43–8.[7]Vasegh N,Khellat F.Projective synchronization of chaotic time-delayed systems via sliding mode controller.Chaos Solitons Fract2009;42(2):1054–61.[8]Li GH.Generalized projective synchronization between Lorenz system and Chen’s system.Chaos Solitons Fract2007;32(4):1454–8.[9]Fenga CF,Zhang Y,Suna JT,Qia W,Wanga YH.Generalized projective synchronization in time-delayed chaotic systems.Chaos Solitons Fract2008;38(3):743–7.[10]Wen G,Xu D.Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems.Chaos Solitons Fract2005;26(1):71–7.[11]H Li G.Generalized projective synchronization of two chaotic systems by using active control.Chaos Solitons Fract2006;30(1):77–82.[12]Chee CY,Xu D.Secure digital communication using controlled projective synchronisation of chaos.Chaos,Solitons Fract2005;23(3):1063–70.[13]Wang ZL.Projective synchronization of hyperchaotic Lüsystem and Liu system.Nonlinear Dynam2010;59(3):455–62.[14]Tama LM,Chenb JH,Chenc HK,Toua WMS.Generation of hyperchaos from the Chen–Lee system via sinusoidal perturbation.Chaos Solitons Fract2008;38(3):826–39.[15]Grassi GO.Observer-based hyperchaos synchronization in cascaded discrete-time systems.Chaos Solitons Fract2009;40(2):1029–39.[16]Zhang ZW,Li H.The Chen’s synchronization based on state put Simul2005;26(4):198–201.[17]Zhou LJ,Duan YB.The hyperchaotic synchronous secure communication based on state observer.Sci Technol Eng2010;10(23):5648–51.[18]Chen A,Lu JA,LüJH,Yu S.Generating hyperchaotic Lüattractor via state feedback control.Physica A2006;364(15):103–10.[19]Rössler OE.An equation for hyperchaos.Phys Lett A1979;71(2–3):155–7.。