- Email: [email protected]
Synchronization of uncertain chaotic systems with parameters perturbation via active control
Chaos, Solitons and Fractals 21 (2004) 39–47 www.elsevier.com/locate/chaos
Synchronization of uncertain chaotic systems with parameters perturbation via active control Hao Zhang *, Xi-kui Ma School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China Accepted 19 September 2003
Abstract This paper presents an active control method for Synchronizing two uncertain chaotic systems with parameters perturbation and generalizes it to general uncertain chaotic systems. And a sufficient condition is drawn for the stability of the error dynamics, and is applied to guiding the design of the controllers. Finally, numerical simulations are used to show the robustness and effectiveness of the proposed control strategy. 2003 Elsevier Ltd. All rights reserved.
1. Introduction Since Pecora and Carroll proposed a method to synchronize chaotic systems and realized it in electronic circuits [1], chaos synchronization has attracted the interest of many researchers as a key technique of secret communication, and various effective control methods [2–11] have been proposed to achieve chaos synchronization. Recently, Bai and Lognngren [12] used active control techniques to synchronize two Lorenz systems. Moreover, researchers examined these techniques further and applied them to other systems [13–15]. The aforementioned active control methods are proposed on the assumption that all the parameters of the chaotic systems are invariant and determinate, but in fact system parameters always fluctuate within some scope in engineering application because they are inevitably perturbed by external inartificial factors such as environment temperature, voltage fluctuation, mutual interfere among components, etc. In this case, the applications of the above methods are limited in Ref. [12–15], and it is very important for real chaotic systems to implement chaos synchronization instead. In this paper, we will take the fluctuation of the system parameters into full account for the synchronization problem of two uncertain chaotic systems. Active control method is proposed to achieve the synchronization of two general uncertain chaotic systems with parameters perturbation. And, a sufficient condition is drawn for the stability of the error dynamics. Finally, the effectiveness and feasibility of the proposed control technique is numerically verified. The paper is organized as follows. In Section 2, we study the synchronization of two Lorenz chaotic systems with parameter perturbation via active control. Section 3 extends this control method to general uncertain chaotic systems with parameters perturbation, and gives a sufficient condition for the stability of the error dynamics and the controller design. Moreover, numerical simulations are given for illustration and verification. Finally some concluding remarks are given. 2. Active control method In Ref. [12], all the parameters of two Lorenz systems are not perturbed by external environment, i.e., they are determinate, while here we assume that there are two Lorenz systems with parameter perturbation Db ¼ edðtÞ, where *
Corresponding author. E-mail address: [email protected] (H. Zhang).
0960-0779/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.09.014
40
H. Zhang, X.-k. Ma / Chaos, Solitons and Fractals 21 (2004) 39–47
e ¼ 0:05 is the amplitude value of the stochastic noise edðtÞ, and that the master system is to control the salve system. The systems are 8 < x_ 1 ¼ rðx2 x1 Þ x_ ¼ rx1 x2 x1 x3 ð1Þ : 2 x_ 3 ¼ x1 x2 ðb þ DbÞx3 and
8 < y_ 1 ¼ rðy2 y1 Þ þ u1 ðtÞ y_ ¼ ry1 y2 y1 y3 þ u2 ðtÞ : 2 y_ 3 ¼ y1 y2 ðb þ DbÞy3 þ u3 ðtÞ
ð2Þ
where r ¼ 10, r ¼ 28 and b ¼ 83. They are rewritten by x_ ¼ ðA þ DAÞx þ f ðxÞ
ð3Þ
y_ ¼ ðA þ DAÞy þ f ðyÞ þ uðtÞ
ð4Þ
and
where
0
1 r r 0 A ¼ @ r 1 0 A; 0 0 b
0
0 0 DA ¼ e@ 0 0 0 0
1 0 0 A; dðtÞ
0
1 0 f ðxÞ ¼ @ x1 x3 A; x1 x2
0
1 0 f ðyÞ ¼ @ y1 y3 A y1 y2
and uðtÞ ¼ ðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞT . In Eq. (4), we have introduced three control functions u1 ðtÞ, u2 ðtÞ and u3 ðtÞ. These functions are to be determined. In order to ascertain the control functions, we subtract Eq. (3) from Eq. (4). It is convenient to define the differences between the Lorenz system that is to be controlled and the controlling system using e 1 ¼ y1 x 1 ;
e2 ¼ y2 x2 ;
e 3 ¼ y3 x 3
ð5Þ
Using this notation, we obtain e_ ¼ ðA þ DAÞe þ F ðx; yÞ þ uðtÞ
ð6Þ
where e ¼ ðe1 ; e2 ; e3 ÞT and 0 1 0 F ðx; yÞ ¼ @ y1 y3 þ x1 x3 A y1 y2 x1 x2 We define the active control functions u1 ðtÞ, u2 ðtÞ and u3 ðtÞ as uðtÞ ¼ V ðtÞ F ðx; yÞ
ð7Þ
This leads to e_ ¼ ðA þ DAÞe þ V ðtÞ
ð8Þ
Eq. (8) describes the error dynamics and can be considered in terms of a control problem where the system to be controlled is a linear system with a control input vector V ðtÞ ¼ ðv1 ðtÞ; v2 ðtÞ; v3 ðtÞÞT as functions of the error vector e ¼ ðe1 ; e2 ; e3 ÞT . As long as these feedbacks stabilize the system, the error vector e converges to zero as time t goes to infinity. This implies that the master system (1) and the salve system (2) is synchronized finally. There are many possible choices for the control V ðtÞ. Without loss of generality, we choose V ðtÞ ¼ Ke where K is a 3 · 3 constant matrix. Let us choose a particular form of the matrix K that is given by 0 1 0 0 0 K ¼ @ 0 60 0 A 0 0 0
ð9Þ
H. Zhang, X.-k. Ma / Chaos, Solitons and Fractals 21 (2004) 39–47
41
0.2
noise
0.1 0
-0.1 -0.2
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60
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t 100
0 0
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60
70
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t 100
20 10 e1
0
-10 -20 -30
20
e2
0 -20 -40
40
e3
20 0 -20 -40
50
||e||
40 30 20 10 0
800
||u||
600 400 200
Fig. 1. Synchronization of two Lorenz systems with parameter perturbation (e ¼ 0:05). (a) Noise, (b) error e1 , (c) error e2 , (d) error e3 , (e) error vector e and (f) controller output u.
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H. Zhang, X.-k. Ma / Chaos, Solitons and Fractals 21 (2004) 39–47
For this particular choice, the closed loop system has all eigenvalues that are found to have negative real parts according to the following theorem and corollary. This choice will lead to a stable system and as we will observe in a numerical investigation, lead to the synchronization of two Lorenz systems (see Fig. 1). The numerical experiment is as follows. The four-order Runge–Kutta algorithm is used to all of our simulations with time step being equal to 0.001. The initial values are given arbitrarily. After 20 s, the motion trajectories have entered into the chaotic attractor. From then on, we begin to control the chaotic system by using the above active controller. The simulation results are illustrated in Fig. 1. As expected, one can observe that the error vector e converges to zero finally after the controller is activated in Fig. 1(b)–(e). It is evident that the trajectories of the salve system asymptotically approach the ones of the master system. And when these uncertain chaotic systems are synchronized, the controller output asymptotically approaches zero, i.e., the proposed control method has a low control price (see Fig. 1(f)).
3. Generalization to general uncertain chaotic systems 3.1. Problem formulation Assume that there are two identical uncertain nonlinear dynamical systems with parameters perturbation and that the master system is to control the salve system. The master system is given by x_ ¼ ðA þ DAÞx þ f ðxÞ
ð10Þ
T
where x ¼ ðx1 ; x2 ; . . . ; xn Þ 2 Rn denotes the state vector, A is a n n system matrix, DA ¼ eA2 is a n n perturbation matrix, perturbation coefficient e > 0 2 R is the maximal amplitude value of all noise, A2 is a n n noise matrix, and the nonlinear vector-function f : Rn ! Rn is continuously differentiable and satisfies the following global Lipschitz condition, i.e., kuðx1 Þ uðx2 Þk 6 ckx1 x2 k, 8x1 ; x2 2 Rn for some c > 0. And the salve system is given by y_ ¼ ðA þ DAÞy þ f ðyÞ þ uðtÞ
ð11Þ
where y ¼ ðy1 ; y2 ; . . . ; yn ÞT 2 Rn denotes the state vector and uðtÞ ¼ ðu1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞÞT 2 Rn is the control input vector. Thus, our goal is to design an appropriate active controller uðxÞ such that the trajectory of the salve system (11) asymptotically approaches the master system (10) and finally implement synchronization, in the sense that lim ky xk ¼ 0
t!1
ð12Þ
where k k is the Euclidean norm. 3.2. Active controller design and analysis Using active control techniques, Eq. (10) is substracted from Eq. (11), and we obtain y_ x_ ¼ ðA þ DAÞðy xÞ þ f ðyÞ f ðxÞ þ uðtÞ
ð13Þ
Let e ¼ y x be error vector. Then, Eq. (13) can be rewritten as e_ ¼ ðA þ DAÞe þ F ðx; yÞ þ uðtÞ
ð14Þ
where F ðx; yÞ ¼ f ðyÞ f ðxÞ. According to the clue of active control [12–15], we can use the control input vector-function uðtÞ to eliminate all items that cannot be shown in the form of the error vector e. By this way, the vector-function uðtÞ can be determined. uðtÞ ¼ V ðtÞ F ðx; yÞ
ð15Þ
And Eq. (14) is rewritten as e_ ¼ ðA þ DAÞe þ V ðtÞ
ð16Þ
Eq. (16) describes the error dynamics and can be considered in terms of a control problem where the system to be controlled is a linear system with a control input V ðtÞ as functions of the error vector e. As long as these feedbacks
H. Zhang, X.-k. Ma / Chaos, Solitons and Fractals 21 (2004) 39–47
43
stabilize the system, the error vector e converges to zero as time t goes to infinity. This implies that the master system (10) and the salve system (11) are synchronized finally. There are many possible choices for the control V ðtÞ. Without loss of generality, we choose V ðtÞ ¼ Ke
ð17Þ
where K is a n n constant matrix. This leads to e_ ¼ ðA1 þ eA2 Þe
ð18Þ
where A1 ¼ A þ K. Definition. Let all the elementary factors of a matrix B1 be linear, then there exist the right eigenvector xðiÞ and left eigenvector y ðiÞ , ði ¼ 1; 2; . . . ; nÞ, which satisfy T
y ðiÞ xðjÞ ¼ 0;
i 6¼ j and
kxðiÞ k ¼ ky ðjÞ k ¼ 1
ð19Þ
such that T
si ¼ xðiÞ y ðiÞ
and
T
jsi j ¼ jxðiÞ y ðjÞ j 6 kxðiÞ kky ðiÞ k ¼ 1
ð20Þ
And we define T
bij ¼ y ðiÞ B2 xðjÞ
ði ¼ 1; 2; 3; . . . ; n and j ¼ 1; 2; 3; . . . ; nÞ
ð21Þ
where B2 is a n n matrix. Lemma [16]. Assume that B1 is a n n matrix, then the eigenvalue ki ðeÞ of the matrix B ¼ B1 þ eB2 is located in the union set of the circular disk 9 8 > > > > n = < X jbij =si j ; where i ¼ 1; 2; 3; . . . ; n di ¼ zkz ki ebii =si k 6 e > > > > ; : j¼1 j6¼i
Theorem. Suppose that all the elementary factors of a matrix B1 are linear and all the eigenvalues of B1 have negative real parts. Then all the eigenvalues of the matrix B ¼ B1 þ eB2 have negative real parts if Reðki þ ebii =si Þ > e
n X
jbij =si j
ð22Þ
j¼1 j6¼i
holds, where e is a small positive number. Proof. Since all the elementary factors of matrix B1 are linear, there exists a nonsingular matrix P and a diagonal matrix D, which make B1 ¼ PDP 1 ¼ P diagðk1 ; k2 ; . . . ; kn ÞP 1 hold. Thus, according to the above definition, every column of P is in parallel with the right eigenvector xðiÞ and every row of P is in parallel with left eigenvector y ðiÞ . xðiÞ and y ðiÞ are T normalized and satisfy the expression (19), where i ¼ 1; 2; 3; . . . ; n. And let xðiÞ be the ith column of P and y ðiÞ =si be the ith row of P , then we deduce the conclusion that PP 1 ¼ I holds from the expression (20). Thus, using the expression (21) and the definitions of xðiÞ and y ðiÞ , we have 2
b11 =s1 6 b =s 6 21 2 P 1 ðB1 þ eB2 ÞP ¼ D þ eP 1 B2 P ¼ diagðki Þ þ e6 6 .. 4 . bn1 =sn
b12 =s1 b22 =s2 .. . bn2 =sn
3 b1n =s1 b2n =s2 7 7 7 ... 7 5 bnn =sn
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According to the above Lemma, the eigenvalue ki ðeÞ of the matrix B ¼ B1 þ eB2 is located in the union set of the circular disk 9 8 > > > > n = < X di ¼ zkz ki ebii =si k 6 e jbij =si j ; where i ¼ 1; 2; 3; . . . ; n > > > > ; : j¼1 j6¼i
Since all the eigenvalues of B1 have negative real part, and Reðki þ ebii =si Þ > e
n X
jbij =si j P 0
j¼1 j6¼i
i.e., Reðki þ ebii =si Þ < 0 holds, the centers of all circular disks are located in the left half plane. Since jReðki þ ebii =si Þj ¼ Reðki þ ebii =si Þ > e
n X
jbij =si j
j¼1 j6¼i
holds, where ði ¼ 1; 2; . . . ; nÞ, the distance between the center of every circular disk and the imaginary axis is larger than its radius. Thus, all circular disks are located in the left half plane, i.e., all the eigenvalues of B have negative real parts. h Corollary. Assume that all the elementary factors of matrix A1 are linear and all the eigenvalues of A1 have negative real parts. Then the error dynamics (18) is asymptotically stable if Reðki þ ebii =si Þ > e
n X
jbij =si j
j¼1 j6¼i
holds. Remarks. If the elements of the matrix K is chosen properly, the above corollary holds. This implies that the closed loop system will be stable. 3.3. Numerical results and some discussions We have introduced active control approach for the synchronization of two uncertain chaotic systems with parameters perturbation. In what follows, let us now investigate the applications of the active control method to synchronize two identical Chen’s system systems with parameter perturbation through the numerical experiments. Recently, in a research on anticontrolling chaos, Chen [17,18] has developed a new chaotic system, bearing the name of the Chen’s system, which is more complex than Lorenz system and R€ ossler system in topological structure. It is given by 8 < x_ 1 ¼ aðx2 x1 Þ x_ ¼ ðc aÞx1 x1 x3 þ cx2 : 2 x_ 3 ¼ x1 x2 bx3
ð23Þ
where a, b and c are three real positive parameters. When a ¼ 35, b ¼ 3 and c ¼ 28, the Chen’s system behaves chaotically (see Fig. 2). Assume that the system parameter b is perturbed by the stochastic noise edðtÞ with the amplitude value of e ¼ 0:1, as is shown in Fig. 3(a). According to the foregoing design method, we define the control functions as uðtÞ ¼ V ðtÞ F ðx; yÞ
ð24Þ
H. Zhang, X.-k. Ma / Chaos, Solitons and Fractals 21 (2004) 39–47
45
Fig. 2. Phase portrait for Chen’s system.
where K is a 3 · 3 gain matrix, 0 1 0 1 V1 ðtÞ e1 @ V2 ðtÞ A ¼ K @ e2 A and V3 ðtÞ e3
0
1 0 F ðx; yÞ ¼ @ y1 y3 þ x1 x3 A y1 y2 x 1 x 3
Thus, the above controlled system can be given by e_ ¼ ðA þ eA2 þ KÞe
ð25Þ
where 0
a A ¼ @c a 0
1 a 0 c 0 A 0 b
0
0 and A2 ¼ @ 0 0
1 0 0 0 0 A 0 dðtÞ
Referring to the above corollary, let us choose a particular form of the matrix K that is given by 0 1 0 0 0 K ¼ @ 0 30 0 A 0 0 0 For this particular choice, all the eigenvalues of the closed loop system satisfy the above theorem through our investigation and analysis. This choice will lead to a stable system, and we will observe in the following numerical results. Similarly, the initial values are given arbitrarily. After 20 s, the motion trajectories have entered into the chaotic attractor. From then on, the above active controller is activated. The numerical results are illustrated in Fig. 3. As expected, one can observe that the trajectories of the salve system asymptotically approach the ones of the master system in Fig. 3(b)–(d). Fig. 3(e) shows that the error vector e eventually converges to zero after the controller is activated. And, Fig. 3(f) displays when these uncertain chaotic systems are synchronized, the controller output asymptotically approaches zero. This implies the proposed control method has low control cost. Thus, the proposed control approach is robust and has the capability to resist external perturbation.
4. Conclusions In this paper, we have presented an active control method for the synchronization of two uncertain chaotic systems with parameters perturbation, and have generalized it to general uncertain chaotic systems. And we have given a sufficient condition for the stability of the error dynamics, and have suggested the design method for the active controller. Finally, numerical results verify the effectiveness and robustness of the proposed control approach.
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noise
0.5
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-0.5 0
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e1
20 0
-20 -40
0
40
e2
20 0
-20 -40
60 40 e3
20 0
-20 -40
80
||e||
60 40 20 0
300
||u||
200 100 0
Fig. 3. Synchronization of two Chen’s systems with parameter perturbation (e ¼ 0:1). (a) Noise, (b) error e1 , (c) error e2 , (d) error e3 , (e) error vector e and (f) controller output u.
H. Zhang, X.-k. Ma / Chaos, Solitons and Fractals 21 (2004) 39–47
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In conclusion, the proposed control strategy can be easily extended to chaos control and is particularly suited for experimental situations. Moreover, applying the control approach to synchronizing two different uncertain chaotic systems with parameters perturbation remains a topic for further research.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Pecora LM, Carroll TL. Phys Rev Lett 1990;64:821–4. Closson TLL, Roussel Marc R. Phys Rev Lett 2000;85:3974–7. Murakami Atsushi, Ohtsubo Junji. Phys Rev E 2001;63:066203. Pyragas K. Phys Lett A 1992;170:421–8. Yang T, Chua LO. IEEE Trans Circuits Syst I 1997;44:976–88. Arecchi FT et al. Int J Bifurcat Chaos 1998;8:1643–55. Itoh M et al. Int J Bifurcat Chaos 2001;11:551–60. Sarasola C et al. Int J Bifurcat Chaos 2003;13:177–91. Chen G. Controlling chaos and bifurcations in engineering. Boca Raton: CRC Press; 1999. Heagy JF, Carroll TL. Chaos 1994;4:385–90. Chen G. Chaos, Solitons & Fractals 1997;8:1461–70. Bai EW et al. Chaos, Solitons & Fractals 1997;8:51–8. Bai EW et al. Chaos, Solitons & Fractals 2000;11:1041–4. Agiza HN, Yassen MT. Phys Lett A 2001;278:191–7. Ho MC et al. Phys Lett A 2002;301:424–8. Xu CX et al. Matrix analysis. Xi’an: Northwest Polytechnical University Press; 1991. Chen G, Ueta T. Int J Bifurcat Chaos 1999;9:1465–6. Ueta T, Chen G. Int J Bifurcat Chaos 2000;10:1917–31.
Synchronization of uncertain chaotic systems with parameters perturbation via active control Hao Zhang *, Xi-kui Ma School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China Accepted 19 September 2003
Abstract This paper presents an active control method for Synchronizing two uncertain chaotic systems with parameters perturbation and generalizes it to general uncertain chaotic systems. And a sufficient condition is drawn for the stability of the error dynamics, and is applied to guiding the design of the controllers. Finally, numerical simulations are used to show the robustness and effectiveness of the proposed control strategy. 2003 Elsevier Ltd. All rights reserved.
1. Introduction Since Pecora and Carroll proposed a method to synchronize chaotic systems and realized it in electronic circuits [1], chaos synchronization has attracted the interest of many researchers as a key technique of secret communication, and various effective control methods [2–11] have been proposed to achieve chaos synchronization. Recently, Bai and Lognngren [12] used active control techniques to synchronize two Lorenz systems. Moreover, researchers examined these techniques further and applied them to other systems [13–15]. The aforementioned active control methods are proposed on the assumption that all the parameters of the chaotic systems are invariant and determinate, but in fact system parameters always fluctuate within some scope in engineering application because they are inevitably perturbed by external inartificial factors such as environment temperature, voltage fluctuation, mutual interfere among components, etc. In this case, the applications of the above methods are limited in Ref. [12–15], and it is very important for real chaotic systems to implement chaos synchronization instead. In this paper, we will take the fluctuation of the system parameters into full account for the synchronization problem of two uncertain chaotic systems. Active control method is proposed to achieve the synchronization of two general uncertain chaotic systems with parameters perturbation. And, a sufficient condition is drawn for the stability of the error dynamics. Finally, the effectiveness and feasibility of the proposed control technique is numerically verified. The paper is organized as follows. In Section 2, we study the synchronization of two Lorenz chaotic systems with parameter perturbation via active control. Section 3 extends this control method to general uncertain chaotic systems with parameters perturbation, and gives a sufficient condition for the stability of the error dynamics and the controller design. Moreover, numerical simulations are given for illustration and verification. Finally some concluding remarks are given. 2. Active control method In Ref. [12], all the parameters of two Lorenz systems are not perturbed by external environment, i.e., they are determinate, while here we assume that there are two Lorenz systems with parameter perturbation Db ¼ edðtÞ, where *
Corresponding author. E-mail address: [email protected] (H. Zhang).
0960-0779/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.09.014
40
H. Zhang, X.-k. Ma / Chaos, Solitons and Fractals 21 (2004) 39–47
e ¼ 0:05 is the amplitude value of the stochastic noise edðtÞ, and that the master system is to control the salve system. The systems are 8 < x_ 1 ¼ rðx2 x1 Þ x_ ¼ rx1 x2 x1 x3 ð1Þ : 2 x_ 3 ¼ x1 x2 ðb þ DbÞx3 and
8 < y_ 1 ¼ rðy2 y1 Þ þ u1 ðtÞ y_ ¼ ry1 y2 y1 y3 þ u2 ðtÞ : 2 y_ 3 ¼ y1 y2 ðb þ DbÞy3 þ u3 ðtÞ
ð2Þ
where r ¼ 10, r ¼ 28 and b ¼ 83. They are rewritten by x_ ¼ ðA þ DAÞx þ f ðxÞ
ð3Þ
y_ ¼ ðA þ DAÞy þ f ðyÞ þ uðtÞ
ð4Þ
and
where
0
1 r r 0 A ¼ @ r 1 0 A; 0 0 b
0
0 0 DA ¼ e@ 0 0 0 0
1 0 0 A; dðtÞ
0
1 0 f ðxÞ ¼ @ x1 x3 A; x1 x2
0
1 0 f ðyÞ ¼ @ y1 y3 A y1 y2
and uðtÞ ¼ ðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞT . In Eq. (4), we have introduced three control functions u1 ðtÞ, u2 ðtÞ and u3 ðtÞ. These functions are to be determined. In order to ascertain the control functions, we subtract Eq. (3) from Eq. (4). It is convenient to define the differences between the Lorenz system that is to be controlled and the controlling system using e 1 ¼ y1 x 1 ;
e2 ¼ y2 x2 ;
e 3 ¼ y3 x 3
ð5Þ
Using this notation, we obtain e_ ¼ ðA þ DAÞe þ F ðx; yÞ þ uðtÞ
ð6Þ
where e ¼ ðe1 ; e2 ; e3 ÞT and 0 1 0 F ðx; yÞ ¼ @ y1 y3 þ x1 x3 A y1 y2 x1 x2 We define the active control functions u1 ðtÞ, u2 ðtÞ and u3 ðtÞ as uðtÞ ¼ V ðtÞ F ðx; yÞ
ð7Þ
This leads to e_ ¼ ðA þ DAÞe þ V ðtÞ
ð8Þ
Eq. (8) describes the error dynamics and can be considered in terms of a control problem where the system to be controlled is a linear system with a control input vector V ðtÞ ¼ ðv1 ðtÞ; v2 ðtÞ; v3 ðtÞÞT as functions of the error vector e ¼ ðe1 ; e2 ; e3 ÞT . As long as these feedbacks stabilize the system, the error vector e converges to zero as time t goes to infinity. This implies that the master system (1) and the salve system (2) is synchronized finally. There are many possible choices for the control V ðtÞ. Without loss of generality, we choose V ðtÞ ¼ Ke where K is a 3 · 3 constant matrix. Let us choose a particular form of the matrix K that is given by 0 1 0 0 0 K ¼ @ 0 60 0 A 0 0 0
ð9Þ
H. Zhang, X.-k. Ma / Chaos, Solitons and Fractals 21 (2004) 39–47
41
0.2
noise
0.1 0
-0.1 -0.2
0
10
20
30
40
50 (a)
60
70
80
90
t 100
0
10
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0 0
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50 (f)
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90
t 100
20 10 e1
0
-10 -20 -30
20
e2
0 -20 -40
40
e3
20 0 -20 -40
50
||e||
40 30 20 10 0
800
||u||
600 400 200
Fig. 1. Synchronization of two Lorenz systems with parameter perturbation (e ¼ 0:05). (a) Noise, (b) error e1 , (c) error e2 , (d) error e3 , (e) error vector e and (f) controller output u.
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H. Zhang, X.-k. Ma / Chaos, Solitons and Fractals 21 (2004) 39–47
For this particular choice, the closed loop system has all eigenvalues that are found to have negative real parts according to the following theorem and corollary. This choice will lead to a stable system and as we will observe in a numerical investigation, lead to the synchronization of two Lorenz systems (see Fig. 1). The numerical experiment is as follows. The four-order Runge–Kutta algorithm is used to all of our simulations with time step being equal to 0.001. The initial values are given arbitrarily. After 20 s, the motion trajectories have entered into the chaotic attractor. From then on, we begin to control the chaotic system by using the above active controller. The simulation results are illustrated in Fig. 1. As expected, one can observe that the error vector e converges to zero finally after the controller is activated in Fig. 1(b)–(e). It is evident that the trajectories of the salve system asymptotically approach the ones of the master system. And when these uncertain chaotic systems are synchronized, the controller output asymptotically approaches zero, i.e., the proposed control method has a low control price (see Fig. 1(f)).
3. Generalization to general uncertain chaotic systems 3.1. Problem formulation Assume that there are two identical uncertain nonlinear dynamical systems with parameters perturbation and that the master system is to control the salve system. The master system is given by x_ ¼ ðA þ DAÞx þ f ðxÞ
ð10Þ
T
where x ¼ ðx1 ; x2 ; . . . ; xn Þ 2 Rn denotes the state vector, A is a n n system matrix, DA ¼ eA2 is a n n perturbation matrix, perturbation coefficient e > 0 2 R is the maximal amplitude value of all noise, A2 is a n n noise matrix, and the nonlinear vector-function f : Rn ! Rn is continuously differentiable and satisfies the following global Lipschitz condition, i.e., kuðx1 Þ uðx2 Þk 6 ckx1 x2 k, 8x1 ; x2 2 Rn for some c > 0. And the salve system is given by y_ ¼ ðA þ DAÞy þ f ðyÞ þ uðtÞ
ð11Þ
where y ¼ ðy1 ; y2 ; . . . ; yn ÞT 2 Rn denotes the state vector and uðtÞ ¼ ðu1 ðtÞ; u2 ðtÞ; . . . ; un ðtÞÞT 2 Rn is the control input vector. Thus, our goal is to design an appropriate active controller uðxÞ such that the trajectory of the salve system (11) asymptotically approaches the master system (10) and finally implement synchronization, in the sense that lim ky xk ¼ 0
t!1
ð12Þ
where k k is the Euclidean norm. 3.2. Active controller design and analysis Using active control techniques, Eq. (10) is substracted from Eq. (11), and we obtain y_ x_ ¼ ðA þ DAÞðy xÞ þ f ðyÞ f ðxÞ þ uðtÞ
ð13Þ
Let e ¼ y x be error vector. Then, Eq. (13) can be rewritten as e_ ¼ ðA þ DAÞe þ F ðx; yÞ þ uðtÞ
ð14Þ
where F ðx; yÞ ¼ f ðyÞ f ðxÞ. According to the clue of active control [12–15], we can use the control input vector-function uðtÞ to eliminate all items that cannot be shown in the form of the error vector e. By this way, the vector-function uðtÞ can be determined. uðtÞ ¼ V ðtÞ F ðx; yÞ
ð15Þ
And Eq. (14) is rewritten as e_ ¼ ðA þ DAÞe þ V ðtÞ
ð16Þ
Eq. (16) describes the error dynamics and can be considered in terms of a control problem where the system to be controlled is a linear system with a control input V ðtÞ as functions of the error vector e. As long as these feedbacks
H. Zhang, X.-k. Ma / Chaos, Solitons and Fractals 21 (2004) 39–47
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stabilize the system, the error vector e converges to zero as time t goes to infinity. This implies that the master system (10) and the salve system (11) are synchronized finally. There are many possible choices for the control V ðtÞ. Without loss of generality, we choose V ðtÞ ¼ Ke
ð17Þ
where K is a n n constant matrix. This leads to e_ ¼ ðA1 þ eA2 Þe
ð18Þ
where A1 ¼ A þ K. Definition. Let all the elementary factors of a matrix B1 be linear, then there exist the right eigenvector xðiÞ and left eigenvector y ðiÞ , ði ¼ 1; 2; . . . ; nÞ, which satisfy T
y ðiÞ xðjÞ ¼ 0;
i 6¼ j and
kxðiÞ k ¼ ky ðjÞ k ¼ 1
ð19Þ
such that T
si ¼ xðiÞ y ðiÞ
and
T
jsi j ¼ jxðiÞ y ðjÞ j 6 kxðiÞ kky ðiÞ k ¼ 1
ð20Þ
And we define T
bij ¼ y ðiÞ B2 xðjÞ
ði ¼ 1; 2; 3; . . . ; n and j ¼ 1; 2; 3; . . . ; nÞ
ð21Þ
where B2 is a n n matrix. Lemma [16]. Assume that B1 is a n n matrix, then the eigenvalue ki ðeÞ of the matrix B ¼ B1 þ eB2 is located in the union set of the circular disk 9 8 > > > > n = < X jbij =si j ; where i ¼ 1; 2; 3; . . . ; n di ¼ zkz ki ebii =si k 6 e > > > > ; : j¼1 j6¼i
Theorem. Suppose that all the elementary factors of a matrix B1 are linear and all the eigenvalues of B1 have negative real parts. Then all the eigenvalues of the matrix B ¼ B1 þ eB2 have negative real parts if Reðki þ ebii =si Þ > e
n X
jbij =si j
ð22Þ
j¼1 j6¼i
holds, where e is a small positive number. Proof. Since all the elementary factors of matrix B1 are linear, there exists a nonsingular matrix P and a diagonal matrix D, which make B1 ¼ PDP 1 ¼ P diagðk1 ; k2 ; . . . ; kn ÞP 1 hold. Thus, according to the above definition, every column of P is in parallel with the right eigenvector xðiÞ and every row of P is in parallel with left eigenvector y ðiÞ . xðiÞ and y ðiÞ are T normalized and satisfy the expression (19), where i ¼ 1; 2; 3; . . . ; n. And let xðiÞ be the ith column of P and y ðiÞ =si be the ith row of P , then we deduce the conclusion that PP 1 ¼ I holds from the expression (20). Thus, using the expression (21) and the definitions of xðiÞ and y ðiÞ , we have 2
b11 =s1 6 b =s 6 21 2 P 1 ðB1 þ eB2 ÞP ¼ D þ eP 1 B2 P ¼ diagðki Þ þ e6 6 .. 4 . bn1 =sn
b12 =s1 b22 =s2 .. . bn2 =sn
3 b1n =s1 b2n =s2 7 7 7 ... 7 5 bnn =sn
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According to the above Lemma, the eigenvalue ki ðeÞ of the matrix B ¼ B1 þ eB2 is located in the union set of the circular disk 9 8 > > > > n = < X di ¼ zkz ki ebii =si k 6 e jbij =si j ; where i ¼ 1; 2; 3; . . . ; n > > > > ; : j¼1 j6¼i
Since all the eigenvalues of B1 have negative real part, and Reðki þ ebii =si Þ > e
n X
jbij =si j P 0
j¼1 j6¼i
i.e., Reðki þ ebii =si Þ < 0 holds, the centers of all circular disks are located in the left half plane. Since jReðki þ ebii =si Þj ¼ Reðki þ ebii =si Þ > e
n X
jbij =si j
j¼1 j6¼i
holds, where ði ¼ 1; 2; . . . ; nÞ, the distance between the center of every circular disk and the imaginary axis is larger than its radius. Thus, all circular disks are located in the left half plane, i.e., all the eigenvalues of B have negative real parts. h Corollary. Assume that all the elementary factors of matrix A1 are linear and all the eigenvalues of A1 have negative real parts. Then the error dynamics (18) is asymptotically stable if Reðki þ ebii =si Þ > e
n X
jbij =si j
j¼1 j6¼i
holds. Remarks. If the elements of the matrix K is chosen properly, the above corollary holds. This implies that the closed loop system will be stable. 3.3. Numerical results and some discussions We have introduced active control approach for the synchronization of two uncertain chaotic systems with parameters perturbation. In what follows, let us now investigate the applications of the active control method to synchronize two identical Chen’s system systems with parameter perturbation through the numerical experiments. Recently, in a research on anticontrolling chaos, Chen [17,18] has developed a new chaotic system, bearing the name of the Chen’s system, which is more complex than Lorenz system and R€ ossler system in topological structure. It is given by 8 < x_ 1 ¼ aðx2 x1 Þ x_ ¼ ðc aÞx1 x1 x3 þ cx2 : 2 x_ 3 ¼ x1 x2 bx3
ð23Þ
where a, b and c are three real positive parameters. When a ¼ 35, b ¼ 3 and c ¼ 28, the Chen’s system behaves chaotically (see Fig. 2). Assume that the system parameter b is perturbed by the stochastic noise edðtÞ with the amplitude value of e ¼ 0:1, as is shown in Fig. 3(a). According to the foregoing design method, we define the control functions as uðtÞ ¼ V ðtÞ F ðx; yÞ
ð24Þ
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Fig. 2. Phase portrait for Chen’s system.
where K is a 3 · 3 gain matrix, 0 1 0 1 V1 ðtÞ e1 @ V2 ðtÞ A ¼ K @ e2 A and V3 ðtÞ e3
0
1 0 F ðx; yÞ ¼ @ y1 y3 þ x1 x3 A y1 y2 x 1 x 3
Thus, the above controlled system can be given by e_ ¼ ðA þ eA2 þ KÞe
ð25Þ
where 0
a A ¼ @c a 0
1 a 0 c 0 A 0 b
0
0 and A2 ¼ @ 0 0
1 0 0 0 0 A 0 dðtÞ
Referring to the above corollary, let us choose a particular form of the matrix K that is given by 0 1 0 0 0 K ¼ @ 0 30 0 A 0 0 0 For this particular choice, all the eigenvalues of the closed loop system satisfy the above theorem through our investigation and analysis. This choice will lead to a stable system, and we will observe in the following numerical results. Similarly, the initial values are given arbitrarily. After 20 s, the motion trajectories have entered into the chaotic attractor. From then on, the above active controller is activated. The numerical results are illustrated in Fig. 3. As expected, one can observe that the trajectories of the salve system asymptotically approach the ones of the master system in Fig. 3(b)–(d). Fig. 3(e) shows that the error vector e eventually converges to zero after the controller is activated. And, Fig. 3(f) displays when these uncertain chaotic systems are synchronized, the controller output asymptotically approaches zero. This implies the proposed control method has low control cost. Thus, the proposed control approach is robust and has the capability to resist external perturbation.
4. Conclusions In this paper, we have presented an active control method for the synchronization of two uncertain chaotic systems with parameters perturbation, and have generalized it to general uncertain chaotic systems. And we have given a sufficient condition for the stability of the error dynamics, and have suggested the design method for the active controller. Finally, numerical results verify the effectiveness and robustness of the proposed control approach.
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noise
0.5
0
-0.5 0
10
20
30
40
50 (a)
60
70
80
90
t 100
10
20
30
40
50 (b)
60
70
80
90
t 100
0
10
20
30
40
50 (c)
60
70
80
90
t 100
0
10
20
30
40
50 ( d)
60
70
80
90
t 100
0
10
20
30
40
50 ( e)
60
70
80
90
t 100
0
10
20
30
40
50 (f)
60
70
80
90
t 100
40
e1
20 0
-20 -40
0
40
e2
20 0
-20 -40
60 40 e3
20 0
-20 -40
80
||e||
60 40 20 0
300
||u||
200 100 0
Fig. 3. Synchronization of two Chen’s systems with parameter perturbation (e ¼ 0:1). (a) Noise, (b) error e1 , (c) error e2 , (d) error e3 , (e) error vector e and (f) controller output u.
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In conclusion, the proposed control strategy can be easily extended to chaos control and is particularly suited for experimental situations. Moreover, applying the control approach to synchronizing two different uncertain chaotic systems with parameters perturbation remains a topic for further research.
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