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Synchronizing chaotic systems using control based on tridiagonal structure
Chaos, Solitons and Fractals 39 (2009) 2274–2281 www.elsevier.com/locate/chaos
Synchronizing chaotic systems using control based on tridiagonal structure Bin Liu *, Yiming Zhou, Min Jiang, Zengke Zhang Department of Automation, Tsinghua University, Beijing 100084, China Accepted 26 June 2007
Abstract The direct design approach based on tridiagonal structure combines the structure analysis with the design of stabilizing controller and the original nonlinear affine systems is transformed into a stable system with special tridiagonal structure using the method. In this study, the direct method is proposed for synchronizing chaotic systems. There are several advantages in this method for synchronizing chaotic systems: (a) it presents an easy procedure for selecting proper controllers in chaos synchronization; (b) it constructs simple controllers easy to implement. Examples of Lorenz system, Chua’s circuit and Duffing system are presented. 2007 Elsevier Ltd. All rights reserved.
1. Introduction Synchronizing chaotic systems and circuits has received great interest in recent years due to its potential application to secure communications. Generally the two chaotic systems in synchronization are called drive system and response system, respectively. The idea of synchronization is to use the output of the drive system to control the response system, and make the output of the response system follow the output of the drive system. Many approaches have been proposed for chaos synchronization [1–8]. Backstepping method [9] has become one of the most important approaches for synchronizing chaotic systems [10–16] in recent years. The main advantage of this method is the systematic construction of a Lyapunov function for the nonlinear systems, and control goal can be achieved with reduced control effort. The disadvantage of this method is that the design procedure is complex and the controller obtained by backstepping is not simple enough for many systems. The direct design approaches transform the original systems into the system with stable tridiagonal structure by inputs. The direct design method can design the controllers with simpler structure than backstepping for nonlinear systems with low dimension. This paper focuses on increasing the effectiveness of the direct method based on tridiagonal structure to a wider variety of chaotic systems. The paper is organized as follows: In Section 2, the class of chaotic systems considered in this work and the problem formulation are presented. In Section 3, two theorems about systems with tridiagonal structure are given and the direct method based on special tridiagonal structure is given. In Section 4, the direct method is utilized *
Corresponding author. E-mail address: [email protected] (B. Liu).
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.099
B. Liu et al. / Chaos, Solitons and Fractals 39 (2009) 2274–2281
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for several systems such as Lorenz system, Chua’s circuit and Duffing system. Numerical simulations are carried out to confirm the validity of the proposed theoretical approach. In Section 5 conclusion is presented.
2. Problem formulation In general, typical dynamics of chaotic systems such as Lorenz system, Chua’s circuit and Duffing system all belong to the system as following: x_ ¼ f ðxÞ;
ð1Þ T
n
where x = [x1, . . ., xn] 2 R are state variables. Assume that drive system is expressed as Eq. (1). Then response system which is coupled with system (1) by u is as following: y_ ¼ f1 ðyÞ þ g1 ðyÞu;
ð2Þ
where y = [y1, . . ., yn]T 2 Rn are state variables and u 2 Rm are inputs. Let us define the state errors between the response system and the drive system as e 1 ¼ y 1 x1 ; e 2 ¼ y 2 x2 . . . en ¼ y n xn T
e ¼ ½e1 ; . . . ; en :
ð3Þ ð4Þ
Subtract (1) from (2). Notice Eqs. (3) and (4), finally error system can be derived as e_ ¼ f1 ðyÞ f ðxÞ þ g1 ðyÞu:
ð5Þ
The problem to realize the synchronization between two chaotic systems now is transformed into another problem on how to choose control law u to make e converge to zero with time increasing.
3. Preliminaries In this section, nonlinear control methods based on tridiagonal structure are introduced. There are two methods based on tridiagonal structure. One is the indirect method, and another is the direct method. First of all, two theorems about systems with specially tridiagonal structure are introduced. 3.1. Stability theorems about the systems with tridiagonal structure Theorem 1. Consider the systems with state-dependent coefficients as follows: x_ ¼ AðxÞx;
ð6Þ T
where x = [x1, . . ., xn] , A(x) is called coefficient matrix and has a class of tridiagonal structure. If the A(x) = [aij] 2 Rn·n,i,j = 1, . . ., n satisfies the following conditions: (1) ji jj > 1, aij = 0; (2) ji jj = 1, aij/aji = const 2 R; (3) ji jj = 0, aij < 0; the system (6) is asymptotically stable. a
Proof. Let k i ¼ aijji ; ðj i ¼ 1Þ, and take the matrix K as K ¼ diagfd 1 ; d 2 ; . . . ; d n g;
ð7Þ
pffiffiffiffi where d 1 ¼ 1; d 2 ¼ k 1 ; . . . ; d i ¼ d i1 k i . Take the Lyapunov function candidate as 1 V ¼ xT K2 x; 2
ð8Þ
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then the derivative of V is given by V_ ¼ xT P x;
ð9Þ
where matrix P = diag{a11, . . ., ann} is a positive matrix. Therefore, the equilibrium x = 0 is globally stable.
h
Theorem 2. Consider the systems with state-dependent coefficients as (6), if the coefficient matrix A(x) = [aij],i, j = 1, . . ., n satisfies the following assumptions: 1. 2. 3. 4. 5.
ji jj > 1, aij = 0; ji jj = 1, aij/aji = const 2 R; ji jj = 0, aij < 0, i, j 6 l; ji jj = 0, aij = 0, i, j > l; The invariant set of the system x_ ¼ AðxÞjx1 ¼x2 ¼...¼xl ¼0 x only includes the origin.
Then the system (6) is asymptotically stable. a
Proof. Let k i ¼ aijji ;
ðj i ¼ 1Þ, and take the matrix K as
K ¼ diagfd 1 ; d 2 ; . . . ; d n g; pffiffiffiffi where d 1 ¼ 1; d 2 ¼ k 1 ; . . . ; d i ¼ d i1 k i . Take the Lyapunov function candidate as 1 V ¼ xT K2 x; 2 then its time derivative is as follows: V_ ¼ xT P x;
ð10Þ
ð11Þ
ð12Þ
where the matrix P = diag{a11, . . ., all,0, . . ., 0} is a semi-positive matrix. Therefore, the invariant set of (6) is the invariant set about x_ ¼ AðxÞjx1 ¼x2 ¼...¼xl ¼0 x. From the assumption 5, the equilibrium x = 0 is globally stable. We know the stable systems with special structure. Then, the stabilization of nonlinear affine systems can be transformed into the construction of the system from the original nonlinear affine systems. There exist an indirect method and a direct method based on tridiagonal structure. The direct method will be introduced in the next subsection. h 3.2. Direct method based on tridiagonal structure Consider the system x_ ¼ f ðxÞ þ gðxÞu; n
ð13Þ
m
where x 2 R , u 2 R . The objective of control based on tridiagonal structure is to transform the systems into a system that has a special tridiagonal structure and satisfies the conditions of Theorem 2. The objective of the direct design method is to transform the system (1) into the systems (6) directly according to the original systems, and does not use coordinate changes. There is a theorem as follows: Theorem 3. Suppose for a stable tridiagonal matrix A(x) satisfying AðxÞx f ðxÞ 2 spanfgðxÞg;
ð14Þ
then we can construct controllers u to stabilize the systems. Proof. From (14), there exists a function a(x) satisfying AðxÞx f ðxÞ ¼ gðxÞaðxÞ:
ð15Þ
Substitute u = a(x) into (13), we can get the equation as (6). We can get that the system is asymptotically stable by the Theorem 1. The selection of the matrix with state-dependent coefficients is an important problem. For simplicity, the matrix is selected according to the coefficient matrix of the new system that is transformed from the original system. In the next section, the direct design method will be used to control chaotic systems. h
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4. Synchronization via the direct design method based on tridiagonal structure In this section, Lorenz system, Chua’s circuit and Duffing system are presented for synchronizing by the direct design method based on tridialognal structure. 4.1. Lorenz system In Lorenz system, external excitation does not exit. Drive Lorenz system and response Lorenz system can be described, respectively as (16) and (17) x_ 1 ¼ rðy 1 x1 Þ y_ 1 ¼ qx1 y 1 x1 z1 z_ 1 ¼ bz1 þ x1 y 1 x_ 2 ¼ rðy 2 x2 Þ þ u1 y_ 2 ¼ qx2 y 2 x2 z2
ð16Þ
ð17Þ
z_ 2 ¼ bz2 þ x2 y 2 þ u2 ; where r, q, b > 0. Let ex ¼ x2 x1 ; ey ¼ y 2 y 1 ; ez ¼ z2 z1 :
ð18Þ
Subtract Eq. (16) from Eq. (17) and obtain e_ x ¼ rðey ex Þ þ u1 e_ y ¼ qex ey ex ez ex z1 x1 ez e_ z ¼ bez þ ex ey þ ex y 1 þ x1 ey þ u2
ð19Þ
The problem of synchronization between drive and response system can be transformed into a problem on how to realize the asymptotical stabilization of system (19). Now, the objective is to find a control law u = [u1, u2] for transforming the system (19) into a system with tridiagonal structure. The system (19) can be rewritten as 32 3 2 3 2 3 2 ex r r 0 u1 e_ x 76 7 6 7 6 7 6 ð20Þ 1 ex x1 54 ey 5 þ 4 0 5: 4 e_ y 5 ¼ 4 q z1 e_ z y1 ex þ x1 b ez u2 According to (20), we can take the state-dependent coefficient matrix as 3 2 0 r k 1 ðq z1 Þ 7 6 Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ ¼ 4 q z1 1 ex x1 5; 0 ex þ x1 b we solve the equation as follows: 2 3 2 r ex 6 7 6 Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ4 ey 5 4 q z1 ez y1
r 1 ex þ x1
32 3 2 3 ex 0 u1 76 7 6 7 ex x1 54 ey 5 ¼ 4 0 5; u2 b ez
ð21Þ
ð22Þ
and obtain the controllers as u1 ¼ rey k 1 ðq z1 Þey u2 ¼ y 1 ex Substituting (23) into (19), the origin system is transformed into the system as following: 2 3 2 3 e_ x ex 6 7 6 7 4 e_ y 5 ¼ Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ4 ey 5 e_ z ez The globally asymptotical stability of (24) can be assured according to Theorem 1.
ð23Þ
ð24Þ
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The parameters are selected as r = 10, b = 8/3, q = 28 and initial condition is taken as x1(0) = 20, y1(0) = 5, z1(0) = 2, x2(0) = 24, y2(0) = 20, z2(0) = 28. With the control law (23), as we can see from Fig. 1, the slave system is driven to chaotic state gradually, which synchronizes with the master system. 4.2. Chua’s circuit In order to further test the effectiveness of the method, Chua’s circuit, which was the first physical dynamical system capable of generating chaotic phenomena in the laboratory, is proposed for synchronizing. The circuit considered here contains a cubic nonlinearity and the drive system (25) and response system (26) are described by the following set of differential equations: x_ 1 ¼ aðy 1 x31 cx1 Þ y_ 1 ¼ x1 y 1 þ z1 z_ 1 ¼ by 1
ð25Þ
x_ 2 ¼ aðy 2 x32 cx2 Þ y_ 2 ¼ x2 y 2 þ z2 z_ 2 ¼ by 2 :
ð26Þ
Subtract (25) from (26), and the error systems can be written as e_ x ¼ aey aex ðe2x þ 3x1 ex þ 3x21 Þ acex þ u e_ y ¼ ex ey þ ez e_ z ¼ bey
ð27Þ
where ex = x2 x1, ey = y2 y1, ez = z2 z1. Now the objective is to find a control law u for transforming the system (27) into a system with tridiagonal structure. The system (27) can be rewritten as follows: 32 3 2 3 2 3 2 ex 1 e_ x ac aðe2x þ 3x1 ex þ 3x21 Þ a 0 76 7 6 7 6 7 6 ð28Þ 1 1 1 54 ey 5 þ 4 0 5u: 4 e_ y 5 ¼ 4 0 e_ z ez 0 1 0
Fig. 1. Synchronized states of modified Lorenz system and error trajectories with the direct method.
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Notice e2x þ 3x1 ex þ 3x21 ¼
2 3 3 ex þ x1 þ x21 P 0: 2 4
According to (28), we select a state-dependent coefficient matrix as 3 2 k 1 aðe2x þ 3x1 ex þ 3x21 Þ a 0 7 6 Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ ¼ 4 1 1 1 5; 0 1 0 From Eq. (14), we can solve the equation as 2 3 2 ex ac aðe2x þ 3x1 ex þ 3x21 Þ a 6 7 6 Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ4 ey 5 4 1 1 ez 0 1
ð29Þ
k 1 > 0:
32 3 2 3 ex u 0 76 7 6 7 1 5 4 ey 5 ¼ 4 0 5: 0 ez 0
ð30Þ
ð31Þ
The controller u is obtained as u ¼ k 1 ex þ acex : Substitute (32) into (27), we can obtain the following system 2 3 2 3 e_ x ex 6 7 6 7 4 e_ y 5 ¼ Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ4 ey 5: e_ z ez
ð32Þ
ð33Þ
It is stable from Theorem 2. Choose a = 10, b = 16, c = 0.143 and take initial condition as x1(0) = 1, y1(0) = 2, z1(0) = 1, x2(0) = 10, y2(0) = 5, z2(0) = 5. With the control law (32), as we can see from Fig. 2, the slave system is driven to chaotic state gradually, which synchronizes with the master system. 4.3. Duffing system Lorenz system and Chua’s circuit discussed above can generate chaotic phenomena under no external excitation condition while Duffing system can generate chaotic phenomena only under external excitation. So, here we classify
Fig. 2. Synchronized states of modified Chua’s circuit and error trajectories with the direct method.
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Fig. 3. Synchronized states of modified Duffing system and error trajectories with the direct method.
Duffing system to another chaotic system and make Duffing system an example to illustrate how to use this method to synchronize chaotic systems with external excitation. The following set of differential equations formulates two Duffing systems. The first is drive system and the second response system x_ 1 ¼ y 1
ð34Þ
y_ 1 ¼ ax1 þ by 1 x31 þ c cos ð0:4tÞ x_ 2 ¼ y 2 y_ 2 ¼ ax2 þ by 2 x32 þ c cos ðtÞ;
ð35Þ
where a > 0, b < 0, c are known parameters. Subtract (35) from (36), the error system can be written as e_ x ¼ ey e_ y ¼ aex þ bey ex ðe2x þ 3x1 ex þ 3x21 Þ þ c½cos t cosð0:4tÞ þ u where ex = x2 x1,ey = y2 y1. Now, system with tridiagonal structure. The 0 e_ x ¼ e_ y a ðe2x þ 3x1 ex þ 3x21 Þ
ð36Þ
the objective is to find a control law u for transforming the system (36) into a system (36) can be rewritten as follows: 1 ex 0 0 þ þ u: ð37Þ b ey c cos t c cosð0:4tÞ 1
According to (37), we select a state-dependent coefficient matrix as 0 1 Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ ¼ ; k 1 > 0: k 1 b From Eq. (14), we can solve the equation for u. ex 0 1 ex 0 Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ ¼ : ey k 1 b ey u
ð38Þ
ð39Þ
The solution u is obtained as u ¼ k 1 ex ða ðe2x þ 3x1 ex þ 3x21 ÞÞex c½cos t cosð0:4tÞ: Substitute (40) into (37), and obtain the following system e_ x 0 1 ex ¼ : e_ y k 1 b ey
ð40Þ
ð41Þ
It is stable according to Theorem 2. Take the parameters as a = 1.8, b = 0.1, c = 1.1, and select initial condition as x1(0) = 1, y1(0) = 1, x2(0) = 2, y2(0) = 2. With the control law (40), as we can see from Fig. 3, the slave system is driven to chaotic state gradually, which synchronizes with the master system.
5. Conclusion In this paper, the direct design method based on tridiagonal structure has been used to synchronize chaotic systems. The advantages of this method can be summarized as follows: (a) it presents an easy procedure for selecting proper
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controllers in chaos synchronization; (b) it constructs simple controllers easy to implement. The technique has been successfully applied to the Lorenz system, Chua’s circuit and Duffing system. Numerical simulations have verified the effectiveness of the method.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. Carroll TL, Pecora LM. Synchronizing chaotic circuits. IEEE Trans Circ Syst I 1991;38:453–6. Bai EW, Lonngran EE. Synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 1997;8:51–8. Liao TL. Adaptive synchronization of two Lorenz systems. Chaos, Solitons & Fractals 1998;9:1555–61. Cuomo KM, Oppenheim AV, Strogatz SH. Synchronization of Lorenz-based chaotic circuits with applications to communications. IEEE Trans Circ Syst I 1993;40:626–33. Liao T-L, Tsai S-H. Adaptive synchronization of chaotic systems and its application to secure communications. Chaos, Solitons & Fractals 2000;11:1387–96. Bai E-W, Lonngren KE. Synchronization and control of chaotic systems. Chaos, Solitons & Fractals 1999;10:1571–5. Mohammad H, Behzad K. Comparison between different synchronization methods of identical chaotic systems. Chaos, Solitons & Fractals 2006;4:1002–22. Krstic M, Kanellakopoulos I, Kokotovic P. Nonlinear and adaptive control design. New York: John Wiley; 1995. Li G-H. Projective synchronization of chaotic system using backstepping control. Chaos, Solitons & Fractals 2006;2:490–4. Park JH. Synchronization of Genesio chaotic system via backstepping approach. Chaos, Solitons & Fractals 2006;5:1369–75. Zhang J, Li CG, Zhang HB, Yu JB. Chaos synchronization using single variable feedback based on backstepping method. Chaos, Solitons & Fractals 2004;5:1183–93. Chen M-Y, Zhou D-H, Shang Y. Nonlinear feedback control of Lorenz system. Chaos, Solitons & Fractals 2004;2:295–304. Yu Y-G, Zhang S-C. Controlling uncertain Lu¨ system using backstepping design. Chaos, Solitons & Fractals 2003;5:897–902. Mascolo S, Grassi G. Controlling chaotic dynamics using backstepping design with application to the Lorenz system and Chua’s circuit. Int J Bifur Chaos 1999;9:1425–34. Rodrigues HM, Alberto LFC, Bretas NG. On the invariance principle: generalizations and applications to synchronization. IEEE Trans Circ Syst – I: Fundamental Theory Appl 2000;47:730–9.
Synchronizing chaotic systems using control based on tridiagonal structure Bin Liu *, Yiming Zhou, Min Jiang, Zengke Zhang Department of Automation, Tsinghua University, Beijing 100084, China Accepted 26 June 2007
Abstract The direct design approach based on tridiagonal structure combines the structure analysis with the design of stabilizing controller and the original nonlinear affine systems is transformed into a stable system with special tridiagonal structure using the method. In this study, the direct method is proposed for synchronizing chaotic systems. There are several advantages in this method for synchronizing chaotic systems: (a) it presents an easy procedure for selecting proper controllers in chaos synchronization; (b) it constructs simple controllers easy to implement. Examples of Lorenz system, Chua’s circuit and Duffing system are presented. 2007 Elsevier Ltd. All rights reserved.
1. Introduction Synchronizing chaotic systems and circuits has received great interest in recent years due to its potential application to secure communications. Generally the two chaotic systems in synchronization are called drive system and response system, respectively. The idea of synchronization is to use the output of the drive system to control the response system, and make the output of the response system follow the output of the drive system. Many approaches have been proposed for chaos synchronization [1–8]. Backstepping method [9] has become one of the most important approaches for synchronizing chaotic systems [10–16] in recent years. The main advantage of this method is the systematic construction of a Lyapunov function for the nonlinear systems, and control goal can be achieved with reduced control effort. The disadvantage of this method is that the design procedure is complex and the controller obtained by backstepping is not simple enough for many systems. The direct design approaches transform the original systems into the system with stable tridiagonal structure by inputs. The direct design method can design the controllers with simpler structure than backstepping for nonlinear systems with low dimension. This paper focuses on increasing the effectiveness of the direct method based on tridiagonal structure to a wider variety of chaotic systems. The paper is organized as follows: In Section 2, the class of chaotic systems considered in this work and the problem formulation are presented. In Section 3, two theorems about systems with tridiagonal structure are given and the direct method based on special tridiagonal structure is given. In Section 4, the direct method is utilized *
Corresponding author. E-mail address: [email protected] (B. Liu).
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.099
B. Liu et al. / Chaos, Solitons and Fractals 39 (2009) 2274–2281
2275
for several systems such as Lorenz system, Chua’s circuit and Duffing system. Numerical simulations are carried out to confirm the validity of the proposed theoretical approach. In Section 5 conclusion is presented.
2. Problem formulation In general, typical dynamics of chaotic systems such as Lorenz system, Chua’s circuit and Duffing system all belong to the system as following: x_ ¼ f ðxÞ;
ð1Þ T
n
where x = [x1, . . ., xn] 2 R are state variables. Assume that drive system is expressed as Eq. (1). Then response system which is coupled with system (1) by u is as following: y_ ¼ f1 ðyÞ þ g1 ðyÞu;
ð2Þ
where y = [y1, . . ., yn]T 2 Rn are state variables and u 2 Rm are inputs. Let us define the state errors between the response system and the drive system as e 1 ¼ y 1 x1 ; e 2 ¼ y 2 x2 . . . en ¼ y n xn T
e ¼ ½e1 ; . . . ; en :
ð3Þ ð4Þ
Subtract (1) from (2). Notice Eqs. (3) and (4), finally error system can be derived as e_ ¼ f1 ðyÞ f ðxÞ þ g1 ðyÞu:
ð5Þ
The problem to realize the synchronization between two chaotic systems now is transformed into another problem on how to choose control law u to make e converge to zero with time increasing.
3. Preliminaries In this section, nonlinear control methods based on tridiagonal structure are introduced. There are two methods based on tridiagonal structure. One is the indirect method, and another is the direct method. First of all, two theorems about systems with specially tridiagonal structure are introduced. 3.1. Stability theorems about the systems with tridiagonal structure Theorem 1. Consider the systems with state-dependent coefficients as follows: x_ ¼ AðxÞx;
ð6Þ T
where x = [x1, . . ., xn] , A(x) is called coefficient matrix and has a class of tridiagonal structure. If the A(x) = [aij] 2 Rn·n,i,j = 1, . . ., n satisfies the following conditions: (1) ji jj > 1, aij = 0; (2) ji jj = 1, aij/aji = const 2 R; (3) ji jj = 0, aij < 0; the system (6) is asymptotically stable. a
Proof. Let k i ¼ aijji ; ðj i ¼ 1Þ, and take the matrix K as K ¼ diagfd 1 ; d 2 ; . . . ; d n g;
ð7Þ
pffiffiffiffi where d 1 ¼ 1; d 2 ¼ k 1 ; . . . ; d i ¼ d i1 k i . Take the Lyapunov function candidate as 1 V ¼ xT K2 x; 2
ð8Þ
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then the derivative of V is given by V_ ¼ xT P x;
ð9Þ
where matrix P = diag{a11, . . ., ann} is a positive matrix. Therefore, the equilibrium x = 0 is globally stable.
h
Theorem 2. Consider the systems with state-dependent coefficients as (6), if the coefficient matrix A(x) = [aij],i, j = 1, . . ., n satisfies the following assumptions: 1. 2. 3. 4. 5.
ji jj > 1, aij = 0; ji jj = 1, aij/aji = const 2 R; ji jj = 0, aij < 0, i, j 6 l; ji jj = 0, aij = 0, i, j > l; The invariant set of the system x_ ¼ AðxÞjx1 ¼x2 ¼...¼xl ¼0 x only includes the origin.
Then the system (6) is asymptotically stable. a
Proof. Let k i ¼ aijji ;
ðj i ¼ 1Þ, and take the matrix K as
K ¼ diagfd 1 ; d 2 ; . . . ; d n g; pffiffiffiffi where d 1 ¼ 1; d 2 ¼ k 1 ; . . . ; d i ¼ d i1 k i . Take the Lyapunov function candidate as 1 V ¼ xT K2 x; 2 then its time derivative is as follows: V_ ¼ xT P x;
ð10Þ
ð11Þ
ð12Þ
where the matrix P = diag{a11, . . ., all,0, . . ., 0} is a semi-positive matrix. Therefore, the invariant set of (6) is the invariant set about x_ ¼ AðxÞjx1 ¼x2 ¼...¼xl ¼0 x. From the assumption 5, the equilibrium x = 0 is globally stable. We know the stable systems with special structure. Then, the stabilization of nonlinear affine systems can be transformed into the construction of the system from the original nonlinear affine systems. There exist an indirect method and a direct method based on tridiagonal structure. The direct method will be introduced in the next subsection. h 3.2. Direct method based on tridiagonal structure Consider the system x_ ¼ f ðxÞ þ gðxÞu; n
ð13Þ
m
where x 2 R , u 2 R . The objective of control based on tridiagonal structure is to transform the systems into a system that has a special tridiagonal structure and satisfies the conditions of Theorem 2. The objective of the direct design method is to transform the system (1) into the systems (6) directly according to the original systems, and does not use coordinate changes. There is a theorem as follows: Theorem 3. Suppose for a stable tridiagonal matrix A(x) satisfying AðxÞx f ðxÞ 2 spanfgðxÞg;
ð14Þ
then we can construct controllers u to stabilize the systems. Proof. From (14), there exists a function a(x) satisfying AðxÞx f ðxÞ ¼ gðxÞaðxÞ:
ð15Þ
Substitute u = a(x) into (13), we can get the equation as (6). We can get that the system is asymptotically stable by the Theorem 1. The selection of the matrix with state-dependent coefficients is an important problem. For simplicity, the matrix is selected according to the coefficient matrix of the new system that is transformed from the original system. In the next section, the direct design method will be used to control chaotic systems. h
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4. Synchronization via the direct design method based on tridiagonal structure In this section, Lorenz system, Chua’s circuit and Duffing system are presented for synchronizing by the direct design method based on tridialognal structure. 4.1. Lorenz system In Lorenz system, external excitation does not exit. Drive Lorenz system and response Lorenz system can be described, respectively as (16) and (17) x_ 1 ¼ rðy 1 x1 Þ y_ 1 ¼ qx1 y 1 x1 z1 z_ 1 ¼ bz1 þ x1 y 1 x_ 2 ¼ rðy 2 x2 Þ þ u1 y_ 2 ¼ qx2 y 2 x2 z2
ð16Þ
ð17Þ
z_ 2 ¼ bz2 þ x2 y 2 þ u2 ; where r, q, b > 0. Let ex ¼ x2 x1 ; ey ¼ y 2 y 1 ; ez ¼ z2 z1 :
ð18Þ
Subtract Eq. (16) from Eq. (17) and obtain e_ x ¼ rðey ex Þ þ u1 e_ y ¼ qex ey ex ez ex z1 x1 ez e_ z ¼ bez þ ex ey þ ex y 1 þ x1 ey þ u2
ð19Þ
The problem of synchronization between drive and response system can be transformed into a problem on how to realize the asymptotical stabilization of system (19). Now, the objective is to find a control law u = [u1, u2] for transforming the system (19) into a system with tridiagonal structure. The system (19) can be rewritten as 32 3 2 3 2 3 2 ex r r 0 u1 e_ x 76 7 6 7 6 7 6 ð20Þ 1 ex x1 54 ey 5 þ 4 0 5: 4 e_ y 5 ¼ 4 q z1 e_ z y1 ex þ x1 b ez u2 According to (20), we can take the state-dependent coefficient matrix as 3 2 0 r k 1 ðq z1 Þ 7 6 Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ ¼ 4 q z1 1 ex x1 5; 0 ex þ x1 b we solve the equation as follows: 2 3 2 r ex 6 7 6 Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ4 ey 5 4 q z1 ez y1
r 1 ex þ x1
32 3 2 3 ex 0 u1 76 7 6 7 ex x1 54 ey 5 ¼ 4 0 5; u2 b ez
ð21Þ
ð22Þ
and obtain the controllers as u1 ¼ rey k 1 ðq z1 Þey u2 ¼ y 1 ex Substituting (23) into (19), the origin system is transformed into the system as following: 2 3 2 3 e_ x ex 6 7 6 7 4 e_ y 5 ¼ Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ4 ey 5 e_ z ez The globally asymptotical stability of (24) can be assured according to Theorem 1.
ð23Þ
ð24Þ
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The parameters are selected as r = 10, b = 8/3, q = 28 and initial condition is taken as x1(0) = 20, y1(0) = 5, z1(0) = 2, x2(0) = 24, y2(0) = 20, z2(0) = 28. With the control law (23), as we can see from Fig. 1, the slave system is driven to chaotic state gradually, which synchronizes with the master system. 4.2. Chua’s circuit In order to further test the effectiveness of the method, Chua’s circuit, which was the first physical dynamical system capable of generating chaotic phenomena in the laboratory, is proposed for synchronizing. The circuit considered here contains a cubic nonlinearity and the drive system (25) and response system (26) are described by the following set of differential equations: x_ 1 ¼ aðy 1 x31 cx1 Þ y_ 1 ¼ x1 y 1 þ z1 z_ 1 ¼ by 1
ð25Þ
x_ 2 ¼ aðy 2 x32 cx2 Þ y_ 2 ¼ x2 y 2 þ z2 z_ 2 ¼ by 2 :
ð26Þ
Subtract (25) from (26), and the error systems can be written as e_ x ¼ aey aex ðe2x þ 3x1 ex þ 3x21 Þ acex þ u e_ y ¼ ex ey þ ez e_ z ¼ bey
ð27Þ
where ex = x2 x1, ey = y2 y1, ez = z2 z1. Now the objective is to find a control law u for transforming the system (27) into a system with tridiagonal structure. The system (27) can be rewritten as follows: 32 3 2 3 2 3 2 ex 1 e_ x ac aðe2x þ 3x1 ex þ 3x21 Þ a 0 76 7 6 7 6 7 6 ð28Þ 1 1 1 54 ey 5 þ 4 0 5u: 4 e_ y 5 ¼ 4 0 e_ z ez 0 1 0
Fig. 1. Synchronized states of modified Lorenz system and error trajectories with the direct method.
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Notice e2x þ 3x1 ex þ 3x21 ¼
2 3 3 ex þ x1 þ x21 P 0: 2 4
According to (28), we select a state-dependent coefficient matrix as 3 2 k 1 aðe2x þ 3x1 ex þ 3x21 Þ a 0 7 6 Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ ¼ 4 1 1 1 5; 0 1 0 From Eq. (14), we can solve the equation as 2 3 2 ex ac aðe2x þ 3x1 ex þ 3x21 Þ a 6 7 6 Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ4 ey 5 4 1 1 ez 0 1
ð29Þ
k 1 > 0:
32 3 2 3 ex u 0 76 7 6 7 1 5 4 ey 5 ¼ 4 0 5: 0 ez 0
ð30Þ
ð31Þ
The controller u is obtained as u ¼ k 1 ex þ acex : Substitute (32) into (27), we can obtain the following system 2 3 2 3 e_ x ex 6 7 6 7 4 e_ y 5 ¼ Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ4 ey 5: e_ z ez
ð32Þ
ð33Þ
It is stable from Theorem 2. Choose a = 10, b = 16, c = 0.143 and take initial condition as x1(0) = 1, y1(0) = 2, z1(0) = 1, x2(0) = 10, y2(0) = 5, z2(0) = 5. With the control law (32), as we can see from Fig. 2, the slave system is driven to chaotic state gradually, which synchronizes with the master system. 4.3. Duffing system Lorenz system and Chua’s circuit discussed above can generate chaotic phenomena under no external excitation condition while Duffing system can generate chaotic phenomena only under external excitation. So, here we classify
Fig. 2. Synchronized states of modified Chua’s circuit and error trajectories with the direct method.
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Fig. 3. Synchronized states of modified Duffing system and error trajectories with the direct method.
Duffing system to another chaotic system and make Duffing system an example to illustrate how to use this method to synchronize chaotic systems with external excitation. The following set of differential equations formulates two Duffing systems. The first is drive system and the second response system x_ 1 ¼ y 1
ð34Þ
y_ 1 ¼ ax1 þ by 1 x31 þ c cos ð0:4tÞ x_ 2 ¼ y 2 y_ 2 ¼ ax2 þ by 2 x32 þ c cos ðtÞ;
ð35Þ
where a > 0, b < 0, c are known parameters. Subtract (35) from (36), the error system can be written as e_ x ¼ ey e_ y ¼ aex þ bey ex ðe2x þ 3x1 ex þ 3x21 Þ þ c½cos t cosð0:4tÞ þ u where ex = x2 x1,ey = y2 y1. Now, system with tridiagonal structure. The 0 e_ x ¼ e_ y a ðe2x þ 3x1 ex þ 3x21 Þ
ð36Þ
the objective is to find a control law u for transforming the system (36) into a system (36) can be rewritten as follows: 1 ex 0 0 þ þ u: ð37Þ b ey c cos t c cosð0:4tÞ 1
According to (37), we select a state-dependent coefficient matrix as 0 1 Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ ¼ ; k 1 > 0: k 1 b From Eq. (14), we can solve the equation for u. ex 0 1 ex 0 Aðex ; ey ; ez ; x1 ; y 1 ; z1 Þ ¼ : ey k 1 b ey u
ð38Þ
ð39Þ
The solution u is obtained as u ¼ k 1 ex ða ðe2x þ 3x1 ex þ 3x21 ÞÞex c½cos t cosð0:4tÞ: Substitute (40) into (37), and obtain the following system e_ x 0 1 ex ¼ : e_ y k 1 b ey
ð40Þ
ð41Þ
It is stable according to Theorem 2. Take the parameters as a = 1.8, b = 0.1, c = 1.1, and select initial condition as x1(0) = 1, y1(0) = 1, x2(0) = 2, y2(0) = 2. With the control law (40), as we can see from Fig. 3, the slave system is driven to chaotic state gradually, which synchronizes with the master system.
5. Conclusion In this paper, the direct design method based on tridiagonal structure has been used to synchronize chaotic systems. The advantages of this method can be summarized as follows: (a) it presents an easy procedure for selecting proper
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controllers in chaos synchronization; (b) it constructs simple controllers easy to implement. The technique has been successfully applied to the Lorenz system, Chua’s circuit and Duffing system. Numerical simulations have verified the effectiveness of the method.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. Carroll TL, Pecora LM. Synchronizing chaotic circuits. IEEE Trans Circ Syst I 1991;38:453–6. Bai EW, Lonngran EE. Synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 1997;8:51–8. Liao TL. Adaptive synchronization of two Lorenz systems. Chaos, Solitons & Fractals 1998;9:1555–61. Cuomo KM, Oppenheim AV, Strogatz SH. Synchronization of Lorenz-based chaotic circuits with applications to communications. IEEE Trans Circ Syst I 1993;40:626–33. Liao T-L, Tsai S-H. Adaptive synchronization of chaotic systems and its application to secure communications. Chaos, Solitons & Fractals 2000;11:1387–96. Bai E-W, Lonngren KE. Synchronization and control of chaotic systems. Chaos, Solitons & Fractals 1999;10:1571–5. Mohammad H, Behzad K. Comparison between different synchronization methods of identical chaotic systems. Chaos, Solitons & Fractals 2006;4:1002–22. Krstic M, Kanellakopoulos I, Kokotovic P. Nonlinear and adaptive control design. New York: John Wiley; 1995. Li G-H. Projective synchronization of chaotic system using backstepping control. Chaos, Solitons & Fractals 2006;2:490–4. Park JH. Synchronization of Genesio chaotic system via backstepping approach. Chaos, Solitons & Fractals 2006;5:1369–75. Zhang J, Li CG, Zhang HB, Yu JB. Chaos synchronization using single variable feedback based on backstepping method. Chaos, Solitons & Fractals 2004;5:1183–93. Chen M-Y, Zhou D-H, Shang Y. Nonlinear feedback control of Lorenz system. Chaos, Solitons & Fractals 2004;2:295–304. Yu Y-G, Zhang S-C. Controlling uncertain Lu¨ system using backstepping design. Chaos, Solitons & Fractals 2003;5:897–902. Mascolo S, Grassi G. Controlling chaotic dynamics using backstepping design with application to the Lorenz system and Chua’s circuit. Int J Bifur Chaos 1999;9:1425–34. Rodrigues HM, Alberto LFC, Bretas NG. On the invariance principle: generalizations and applications to synchronization. IEEE Trans Circ Syst – I: Fundamental Theory Appl 2000;47:730–9.