Synchronizing Chaotic Systems Based on Tridiagonal Structure

Transcription

1 Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-, 008 Synchronizing Chaotic Systems Based on Tridiagonal Structure Bin Liu, Min Jiang Zengke Zhang Department of Automation,Tsinghua University, China Cisdi Company, Chongqing,China Abstract: The design approach based on tridiagonal structure combines the structure analysis with the design of stabilizing controller. During the design procedure, the original nonlinear affine s is transformed into a stable with special tridiagonal structure. In this study, the method is proposed for synchronizing chaotic s. There are several advantages in this method for synchronizing chaotic s: a) it presents a atic procedure for construct a proper controller in chaos synchronization; b) it can be applied to a variety of chaotic s with lower triangular structure. Examples of Lorenz, Chua s circuit and Duffing are presented.. INTRODUCTION Synchronizing chaotic s and circuits has received great interest in recent years. Generally the two chaotic s in synchronization are called drive and response respectively. The idea of synchronization is to use the output of the drive to control the response, and make the output of the response follow the output of the drive. Many approaches have been proposed for chaos synchronization Ott [990],Carroll [99],Bai [997] and Liao [000]. Backstepping method Sepulchre [997] has become one of the most important approaches for synchronizing chaotic s Li [006],Zhang [004] in recent years. The main advantage of this method is the atic construction of a Lyapunov function for the nonlinear s, and control goal can be achieved with reduced control effort. Notice the original s is transformed into the with special tridiagonal structure using backstepping. There are a class of design methods proposed in Liu [007] including direct design method and recursive design method based on tridiagonal structurem, which is to transform the original s into the with stable tridiagonal structure by inputs and coordinate change. The recursive design method is similar to backstepping and can design the controllers with more parameters than backstepping for nonlinear s. This paper focuses on showing the effectiveness of the recursive design method based on tridiagonal structure to a wider variety of chaotic s. The paper is organized as follows: In Section the class of chaotic s considered in this work and the problem formulation are presented. In Section 3, two theorems about s with tridiagonal structure are given and the recursive design method based on special tridiagonal structure is given. In Section 4, the method is utilized for several s such as Lorenz, Chua s circuit and Duffing. Numerical simulations are carried out to confirm the validity of the proposed theoretical approach. In Section 5 conclusion is presented.. PROBLEM FORMULATION In general, typical dynamics of chaotic s such as Lorenz, Chua s circuit and Duffing all belong to the as following: ẋ fx) ) x [x,, x n ] T R n are state variables. Assume that drive is expressed as Eq. ). Then response which is coupled with ) by u is as following: ẏ f y) + g y)u ) y [y,, y n ] T R n are state variables and u R m are inputs. Let us define the state errors between the response and the drive as e y x, e y x e n y n x n 3) e [e,, e n ] T 4) Subtract ) from ). Notice Eqs.3) and 4), finally error can be derived as ė f y) f x) + g y)u 5) The problem to realize the synchronization between two chaotic s now is transformed into another problem on how to choose control law u to make e converge to zero with time increasing. 3. CONTROL BASED ON TRIDIAGONAL STRUCTURE In this section, nonlinear control methods based on tridiagonal structure are introduced. First of all, two theorems about s with specially tridiagonal structure are introduced /08/$ IFAC / KR

2 7th IFAC World Congress IFAC'08) Seoul, Korea, July 6-, 008 Theorem. Consider the s with state-dependent coefficients as follows: ẋ Ax)x 6) x [x,, x n ] T, A x) is called coefficient matrix and has a class of tridialognal structure. If the A x) [a ij ] R n n, i, j,, n satisfies the following conditions: ) i j >, a ij 0; ) i j, a ij /a ji const R ; 3) i j 0, a ij < 0; the 6) is asymptotically stable. Proof. Let k i aij a ji, j i ), and take the matrix Λ as Λ diag{d, d,, d n } 7) d, d k,, d i d i ki. Choose the Lyapunov function candidate as V xt Λ x 8) The derivative of V is given by V x T P x 9) matrix P diag{ a,, a nn }is a positive matrix. Therefore, the equilibrium x 0 is globally stable. We know the stable s with special structure. Then, the stabilization of nonlinear affine s can be transformed into the construction of the from the original nonlinear affine s. There exist a method based on tridiagonal structure for the s with lower triangular structure. The method is suitable for the s with lower triangular structure. Consider the with lower triangular ẋ f x ) + g x ) x 0) ẋ f x, x ) + g x, x ) u x [x, x ] T R n, u R. The objective of control based on tridiagonal structure is to transform the s into a that has a special tridiagonal structure and satisfies the conditions of theorem. The design procedure is as follows: ) Take the variable x as virtual input, and design the controller of the s as follows: ẋ f x ) + g x ) x ) The virtual controller can be got as x α x ) g x ) f x ) k x ) ) The variable x is not the practical controller, and the error between x and α x ) is z x α x ). The derivative of z is as follows: ż f x ) + g x ) u α x ) ) Take the practical controller as u g x, x ) l g x ) f x ) k z + α x )) We can get the new s as follows: [ ] [ ] [ ż k g z ) ż ż l g z ) k ż ] 3) 4) z x. The asymptotical stability of 4) can be guaranteed by Theorem. The similar procedure can be used for the with lower triangular structure as follows: ẋ f x ) + g x )x ẋ f x, x ) + g x, x )x 3 5). ẋ n f n x,, x n ) + g n x,, x n )u g i ) 0. The above can be transformed into the as follows: k g z). l ż g z) k z 6) gn z) l n g n z) k n k i > 0, z R n. The objective of the design method is to transform the 5) into the s 6) by using coordinate changes and inputs. In the next section, the design method is used to synchronize chaotic s. 4. SYNCHRONIZATION VIA THE DESIGN BASED ON TRIDIAGONAL STRUCTURE In this section, Lorenz, Chua s circuit and Duffing are presented for synchronizing by the design method based on tridialognal structure. 4. Lorenz In Lorenz, external excitation does not exit. Drive Lorenz and response Lorenz can be described respectively as 7) and 8) ẋ σ y x ) ẏ ρx y x z 7) ż βz + x y ẋ σ y x ) ẏ ρx y x z ż βz + x y + u 8) σ, ρ, β > 0. Let e x x x, e y y y, e z z z 9) Subtract Eq. 7) from Eq. 8), consider Eq. 9), and obtain ė x σ e y e x ) ė y ρe x e y e x e z e x z x e z ė z βe z + e x e y + e x y + x e y + u 0) The problem of synchronization between drive and response can be transformed into a problem on how to realize the asymptotical stabilization of 0). Now the objective is to find a control law u for transforming the 0) into a with tridiagonal structure. First we consider the ) ė x σ e y e x ) ) Take e y as a virtual control, we can get e y α e x ) 0, k > 0 ) Define a new variable e e y α e x ), and obtain the ė x σe x + σz ė f e x, e y, z, ė x ) + g e x, e y, z ) e z 3) 538

3 7th IFAC World Congress IFAC'08) Seoul, Korea, July 6-, 008 Fig.. Synchronized states x, x of modified Lorenz f e x, e y, z, ė x ) ρe x e y e x z 4) g e x, e y, z ) e x x Suppose g ) 0, and take e y as a virtual control, we can get Fig.. Synchronized states y, y of modified Lorenz e z α e x, e y, z ) g e x, e y, z ) f e x, e y, z, ė x ) l σe x k e ) 5) Define a new variable e 3 e z α e x, e y, z ), and obtain the ė x σ k ) e x σz ė l σe x k e + g e x, e y, z ) e 3 6) ė 3 f 3 e x, e y, e z, z ) + g 3 e x, e y, e z, z ) u f 3 e x, e y, e z, z, α ) βe z + e x e y + e x y + x e y α e x, e y, z ) 7) g 3 e x, e y, e z, z ) 8) Let the controller as u g 3 e x, e y, e z, z ) f 3 e x, e y, e z, z, α ) k 3 e 3 l g e ) 9) Substitute 9) into 6), and obtain [ ėx ė ė3 ] [ ] [ ] k σ 0 ex l σ k g ) e 0 l g ) k 3 e 3 30) The globally asymptotical stability of 30) can be assured according to theorem, that is e x, e, e 3 0. From the definition of e, e 3, we can get e x, e y, e z coverage to zero with time increasing. The parameters are selected as σ 0, β 8/3, ρ 8, l l k k k 3 and initial condition is taken as x 0) 0, y 0) 5, z 0) 0, x 0) 4, y 0) 0, z 0) 8. With the control law 9), as we can see from Fig. -Fig. 3, the slave is driven to chaotic state gradually, which synchronizes with the master. Fig. 3. Synchronized states z, z of modified Lorenz 4. Chua s circuit In order to further test the effectiveness of the method, Chua s circuit, which was the first physical dynamical capable of generating chaotic phenomena in the laboratory, is proposed for synchronizing. The circuit considered here contains a cubic nonlinearity and the drive 3) and response 3) are described by the following set of differential equations: ẋ α y x 3 ) cx ẏ x y + z 3) ż βy ẋ α y x 3 ) cx ẏ x y + z 3) ż βy Subtract 3) from 33), and obtain the error s as follows: ė x αe y αe x e x + 3x e x + 3x ) αcex + u ė y e x e y + e z 33) ė z βe y e x x x, e y y y, e z z z. Now the objective is to find a control law u for transforming the 33) into a with tridiagonal structure. First we consider the 34) ė z βe y 34) Take e y as a virtual control, we can get e y α e z ) k e z, k > 0 35) 539

4 7th IFAC World Congress IFAC'08) Seoul, Korea, July 6-, 008 Take a new variable e e y α e z ), and obtain the as follows: ė z k βe z βz 36) ė f e z, e y ) + g e z, e y ) e x f e z, e y ) e y + e z α e z ) g e z, e y ) 37) Suppose g ) 0, and take e y as a virtual control, we can get e x α e x, e y, z ) g e x, e y, z ) f e x, e y, z, ė x ) +l βe z k e ) 38) Take a new variable e 3 e x α e z, e y, z ), and obtain the ė z k βe z βe ė l βe z k e + g e z, e y ) e 3 39) ė 3 f 3 e x, e y, e z, z ) + g 3 e x, e y, e z, z ) u f 3 e x, e y, e z, x ) αe y αe x e x + 3x e x + 3x ) αcex α e x, e y, z ) g 3 e x, e y, e z, z ) 40) Let the controller as u g 3 e x, e y, e z, z ) f 3 e x, e y, e z, z, α ) k 3 e 3 l g ) 4) Substitute 4) into 4), we can obtain [ ] [ ] [ ] ėz k β β 0 ex ė l β k g ) e 4) ė3 0 l g ) k 3 e 3 The globally asymptotical stability of 4) can be assured according to theorem, that is e z, e, e 3 0. From the definition of e, e 3, we can get e x, e y, e z coverage to zero with time increasing. Choose α 0, β 6, c 0.43, l l k k k 3 and take initial condition as x 0), y 0), z 0),x 0) 0, y 0) 5, z 0) 5. With the control law 4), as we can see from Fig. 4-Fig. 6, the slave is driven to chaotic state gradually, which synchronizes with the master. Fig. 4. Synchronized states x, x of modified Chua s circuit Fig. 5. Synchronized states y, y of modified Chua s circuit 4.3 Duffing Lorenz and Chua s circuit discussed above can generate chaotic phenomena under no external excitation condition while Duffing can generate chaotic phenomena only under external excitation. So here we classify Duffing to another chaotic and make Duffing an example to illustrate how to use this method to synchronize chaotic s with external excitation. The following set of differential equations formulates two Duffing s. The first is drive and the second response ẋ y ẏ αx + by x 3 43) + c cos0.4t) Fig. 6. Synchronized states z, z of modified Chua s circuit ẋ y ẏ αx + by x 3 + c cost) a > 0, b < 0, c are known parameters. 44) Subtract 43) from 44), and obtain the error as follows: 540

5 7th IFAC World Congress IFAC'08) Seoul, Korea, July 6-, 008 Fig. 7. Synchronized states x, x of modified Duffing ė x e y ė y ae x + be y e x e x + 3x e x + 3x ) +c [cos t cos 0.4t)] + u 45) e x x x, e y y y. Now the objective is to find a control law u for transforming the 33) into a with tridiagonal structure. First we consider the 46) ė x e y 46) Take e y as a virtual control, we can get α e x ) k e x, k > 0 47) Take a new variable e e y α e x ), and obtain the ė x k e x + z ė f e x, e y, t) + g e x, e y ) u 48) f e x, e y, t) ae x + be y e x e x + 3x e x + 3x ) +c [cos t cos 0.4t)] α 49) g 50) Take the controller as u g e x, e y, t) f e x, e y, t) k e l g e x ) 5) Substitute 4) into 4), we can obtain [ ] [ ] [ ] ėx k ex 5) ė l k e The globally asymptotical stability of 4) can be assured according to theorem, that is e x, e 0. From the definition of e, we can get e x, e y coverage to zero with time increasing. Take the parameters as a.8, β 0., c., l k k, and select initial condition as x 0), y 0), x 0), y 0). With the control law 5), as we can see from Fig. 7 and Fig. 8, the slave is driven to chaotic state gradually, which synchronizes with the master. 5. CONCLUSION In this paper, the design method based on tridiagonal structure has been proposed and used to synchronize Fig. 8. Synchronized states y, y of modified Duffing chaotic s. The advantages of this method can be summarized as follows: a) it presents a atic procedure for selecting proper controllers in chaos synchronization; b) it can be applied to a variety of chaotic s with lower triangular structure. The technique has been successfully applied to the Lorenz, Chua s circuit and Duffing. Numerical simulations have verified the effectiveness of the method. REFERENCES E. Ott, C. Grebogi, and J.A. Yorke. Controlling chaos. Phys Rev Lett, 64:96-99, 990. T.L. Carroll, and L.M. Pecora. Synchronizing chaotic circuits. IEEE Trans Circ Syst I,38: , 99. E.W. Bai, E.E. Lonngran, and J.A.Yorke. Synchronization of two Lorenz s using active control. Chaos, Solitons & Fractals, 8:5-58, 997. T.L. Liao. Adaptive synchronization of two Lorenz s. Chaos, Solitons & Fractals, 9:555-56, 998. K.M. Cuomo, A.V. Oppenheim, and S.H. Strogatz. Synchronization of Lorenz-based chaotic circuits with applications to communications. IEEE Trans Circ Syst I, 40:66-633, 993. T.L. Liao, and S.H. Tsai. Adaptive synchronization of chaotic s and its application to secure communications. Chaos, Solitons & Fractals, :387-96, 000. E.W. Bai, and K.E. Lonngren. Synchronization and control of chaotic s. Chaos, Solitons & Fractals, 0:57-575, 999. H. Mohammad, and K. Behzad. Comparison between different synchronization methods of identical chaotic s. Chaos, Solitons & Fractals, 4:00-0, 006. R. Sepulchre, M. Iankovic, and P.V. Kokotovic. Constructive Nonlinear Control. Springer, 997. G.H. Li. Projective synchronization of chaotic using backstepping control. Chaos, Solitons & Fractals, : , 006. J.H. Park. Chaos, Solitons & Fractals, 5: , 990. J. Zhang, C.G. Li, H.B. Zhang, and J.B. Yu. Chaos synchronization using single variable feedback based on backstepping method. Chaos, Solitons & Fractals, 5: 83-93, 004. M.Y. Chen, D.H. Zhou, and Y. Shang. Nonlinear feedback control of Lorenz. Chaos, Solitons & Fractals, : ,

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