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Adaptive synchronization of Lü system with uncertain parameters
Chaos, Solitons and Fractals 21 (2004) 657–667 www.elsevier.com/locate/chaos
Adaptive synchronization of L€ u system with uncertain parameters E.M. Elabbasy *, H.N. Agiza, M.M. El-Dessoky Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Accepted 11 December 2003
Abstract This paper addresses the synchronization problem of two L€ u systems in the presence of unknown system parameters. Based on Lyapunov stability theory an adaptive control law is derived to make the states of two identical L€ u systems with unknown system parameters asymptotically synchronized. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization schemes. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Chaos has been developed and thoroughly studied over the past two decades. A chaotic system is a nonlinear deterministic system that displays complex and unpredictable behavior. The sensitive dependence on the initial conditions and on the system’s parameter variation is a prominent characteristic of chaotic behavior. Research efforts have investigated chaos control and chaos synchronization problems in many physical chaotic systems. Synchronization in chaotic dynamical systems has been a theme in nonlinear sciences and received considerable attention. Recently, synchronization in chaotic dynamical systems has been extensively investigated in the past few years [1–25], and many possible applications, especially to secret communication, have been discussed by computer simulation and even realized in laboratory condition [10–12]. The idea of synchronization is to use the output of the drive system to control the response system so that the output of the response system follows the output of the drive system asymptotically. Synchronization via adaptive control has been successfully tested in a variety of nonlinear dynamical systems, including Lorenz equations, the Rossler system, Chua’s circuit and Chen system [12–21]. Recently, a coupled of Lorenz system is synchronized by using active control [15–19]. The object of this work is to study chaos synchronization of two identical L€ u systems with unknown parameters. The key idea is that we introduce the parameters update law into the design of the adaptive synchronization controller based on Lyapunov stability theorem. In 1963, Lorenz found the first canonical chaotic attractor [27], which has just been mathematically confirmed to exist [28]. In 1999, Chen found another similar but topologically not equivalent chaotic attractor [29], as the dual of the Lorenz system, in a sense defined by Vanecek and Celikovsky [30]: The Lorenz system satisfies the condition a12 a21 > 0 while Chen system satisfies a12 a21 < 0, where a12 , a21 are the corresponding elements in the constant matrix A ¼ ðaij Þ33 for the linear part of the system. Very recently, L€ u and Chen found a new chaotic system [24–26], bearing the name of
*
Corresponding author. E-mail addresses: [email protected] (E.M. Elabbasy), [email protected] (H.N. Agiza), [email protected] (M.M. El-Dessoky). 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.028
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Fig. 1. Shows the chaotic attractor of L€ u system at a ¼ 36, b ¼ 3 and c ¼ 20.
the L€ u system, which satisfies the condition a12 a21 ¼ 0, thereby bridging the gap between the Lorenz and Chen attractors [25,26]. Therefore L€ u attractor is a new chaotic attractor, which connects the Lorenz attractor and Chen attractor and represents the transition from one to the other and is described by x_ ¼ aðy xÞ y_ ¼ xz þ cy
ð1Þ
z_ ¼ xy bz where a, b and c are three unknown uncertain parameters. This new system exhibits a chaotic attractor at the parameter values a ¼ 36, b ¼ 3 and c ¼ 20 (see Fig. 1).
2. Adaptive synchronization of L€u system In order to observe the synchronization behavior in the L€ u system, we have two L€ u systems where the drive system with three state variables denoted by the subscript 1 drives the response system having identical equations denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system. The two L€ u systems are described, respectively, by the following equations: x_ 1 ¼ aðy1 x1 Þ y_ 1 ¼ x1 z1 þ cy1 z_ 1 ¼ x1 y1 bz1
ð2Þ
x_ 2 ¼ aðy2 x2 Þ þ u1 ðtÞ y_ 2 ¼ x2 z2 þ cy2 þ u2 ðtÞ z_ 2 ¼ x2 y2 bz2 þ u3 ðtÞ
ð3Þ
and
where U ¼ ½u1 u2 u3 T is the controller functions. The controller U is to be determined for the purpose of synchronizing the two identical L€ u systems with same but unknown parameters a, b and c in spite of the differences in initial conditions. Subtracting Eq. (2) from Eq. (3), yields the error dynamical system between e_ x ¼ aðey ex Þ þ u1 ðtÞ e_ y ¼ x2 ez z1 ex þ cey þ u2 ðtÞ e_ z ¼ x2 ey þ y1 ex bez þ u3 ðtÞ
ð4Þ
where ex ¼ x2 x1 , ey ¼ y2 y1 and ez ¼ z2 z1 . The goal of the control is to find a controller U ¼ ½u1 u2 u3 T and a parameter estimation update law for Eq. (4) such that the states of response system (3) and the states of drive system (2) are globally synchronized asymptotically, i.e.
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lim keðtÞk ¼ 0 for all a; b and c 2 R
t!1
where eðtÞ ¼ ½ex ey ez T . Let the unknown uncertain parameter b be a positive number. Now, we can pick a Lyapunov function for the system (4) 1 V ðex ; ey ; ez ; ~aÞ ¼ ðe2x þ e2y þ e2z þ ~a2 Þ 2
ð5Þ
where ~a ¼ a a1 , a1 is an estimate value of the unknown parameter a, and require that its derivative along the solutions of the system (4) satisfy dV ðex ; ey ; ez ; ~aÞ 6 ðe2x þ e2y þ be2z Þ dt
ð6Þ
We therefore need to find a controller U and a parameter estimation update law a_ 1 to guarantee that for all eðtÞ 2 R3 , the inequality (6) holds. We have the time derivative of V ðex ; ey ; ez ; ~aÞ along the solutions of the system (4) dV ðex ; ey ; ez ; ~aÞ ¼ ex e_ x þ ey e_ y þ ez e_ z þ ~a~a_ dt ¼ aex ðey ex þ u1 ðtÞÞ þ ey ðx2 ez z1 ex þ cey þ u2 ðtÞÞ þ ez ðx2 ey þ y1 ex bez þ u3 ðtÞÞ ~ aa_ 1 ¼ aex ðey ex þ u1 ðtÞÞ þ ey ðcey z1 ex þ u2 ðtÞÞ þ ez ðy1 ex bez þ u3 ðtÞÞ ~ aa_ 1 There are many possible choices for the controller U and the parameter estimation update law a_ 1 . We choose u1 ðtÞ ¼ aey þ ða1 1Þex u2 ðtÞ ¼ z1 ex ð1 þ cÞey u3 ðtÞ ¼ y1 ex
ð7Þ
a_ 1 ¼ e2x under with this choice dV ðex ; ey ; ez ; ~aÞ ¼ ð~a þ 1Þe2x e2y be2z þ ~ae2x ¼ ðe2x þ e2y þ be2z Þ dt
ð8Þ
This leads to lim keðtÞk ¼ 0 for all a; c 2 R and b > 0
t!1
Hence, the synchronization of two L€ u systems is achieved under the controller and a parameter estimation update law Eq. (7). It is clear that the controller in Eq. (7) is independent of unknown uncertain parameter b, if b > 0. If the condition of unknown uncertain parameter b > 0 is cancelled, we take a Lyapunov function for Eq. (4) as follows 1 V ðex ; ey ; ez ; ~a; ~bÞ ¼ ðe2x þ e2y þ e2z þ ~a2 þ ~b2 Þ 2 where ~a ¼ a a1 , ~b ¼ b b1 , ~a and ~b are estimate values of the unknown parameter a and b, respectively.
ð9Þ
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We have the time derivative of V ðex ; ey ; ez ; ~a; ~bÞ along the solutions of Eq. (4) dV ðex ; ey ; ez ; ~a; ~ bÞ ¼ ex e_ x þ ey e_ y þ ez e_ z þ ~a~a_ þ ~b~b_ dt ¼ aex ðey ex þ u1 ðtÞÞ þ ey ðx2 ez z1 ex þ cey þ u2 ðtÞÞ þ ez ðx2 ey þ y1 ex bez þ u3 ðtÞÞ ~ aa_ 1 ~ bb_ 1 ¼ aex ðey ex þ u1 ðtÞÞ þ ey ðcey z1 ex þ u2 ðtÞÞ þ ez ðy1 ex bez þ u3 ðtÞÞ ~ aa_ 1 ~ bb_ 1 There are many possible choices for the controller U and the parameter estimation update laws a_ 1 and b_ 1 as follows: u1 ðtÞ ¼ aey þ ða1 1Þex u2 ðtÞ ¼ z1 ex ð1 þ cÞey u3 ðtÞ ¼ y1 ex þ ðb1 1Þez
ð10Þ
a_ 1 ¼ e2x b_ 1 ¼ e2z under this choice (10), we have dV ðex ; ey ; ez ; ~a; ~ bÞ ¼ ð~a þ 1Þe2x e2y ð~b þ 1Þe2z þ ~ ae2x þ ~ be2z ¼ ðe2x þ e2y þ e2z Þ dt
ð11Þ
This leads to lim keðtÞk ¼ 0
t!1
for all a; b and c 2 R
We consider now the case where the three parameters are unknown. Let the error system (4) be rewritten in the following form e_ x ¼ aðey ex Þ þ u1 ðtÞ e_ y ¼ x2 z2 þ x1 z1 þ cey þ u2 ðtÞ e_ z ¼ x2 y2 x1 y1 bez þ u3 ðtÞ
ð12Þ
By following the active control approach of Bai and Lonngren [18,19], we have introduced three control functions u1 ðtÞ, u2 ðtÞ and u3 ðtÞ in (12). These functions are to be determined for the purpose of synchronizing the two L€ u systems with same but unknown parameters a, b and c in spite of the differences in initial conditions. Clearly, the original synchronization problem can be replaced by the equivalent problem of stabilizing the zero solution of the system (12) by a suitable choice of the active functions u1 ðtÞ, u2 ðtÞ and u3 ðtÞ. Let us now define the active control functions u1 ðtÞ, u2 ðtÞ and u3 ðtÞ as u1 ðtÞ ¼ a1 ðey ex Þ ex u2 ðtÞ ¼ x2 z2 x1 z1 ðc1 þ 1Þey
ð13Þ
u3 ðtÞ ¼ x1 y1 x2 y2 þ ðb1 1Þez Where a1 , b1 and c1 are estimates of a, b and c respectively. Then the error dynamical system (12) can be expressed by e_ x ¼ aðey ex Þ a1 ðey ex Þ ex e_ y ¼ ðc c1 Þey ey e_ z ¼ ðb1 bÞez ez Let us introduce the parameter error ~a ¼ a a1 ; ~b ¼ b b1 and ~c ¼ c c1
ð14Þ
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then the error system can be described by e_ x ¼ a~ðey ex Þ ex e_ y ¼ ~cey ey e_ z ¼ ~bez ez
ð15Þ
For the derivation of the update law for adjusting the parameters a1 , b1 , and c1 , the Lyapunov approach is used. Consider a quadratic function 1 V ðex ; ey ; ez ; ~a; ~b; ~cÞ ¼ ðe2x þ e2y þ e2z þ ~a2 þ ~b2 þ ~c2 Þ 2
ð16Þ
Differentiating (16) using (15) gives ~ d~a d~b d~c d~ a db d~c b þ ~c aðey ex Þex þ ~ce2y ~ be2z þ ~ a þ~ V_ ¼ ex e_ x þ ey e_ y þ ez e_ z þ ~a þ ~b þ ~c ¼ e2x e2y e2z þ ~ dt dt dt dt dt dt
ð17Þ
Fig. 2. Display the solution of the L€ u system (2) and (3) where the control functions are deactivated: (a) signals x1 and x2 , (b) signals y1 and y2 and (c) signals z1 and z2 (x1 , y1 and z1 - - -; x2 , y2 and z2 – – –).
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In view of (17), the estimated parameters are updated by the following law da1 ¼ ðey ex Þex dt db1 ¼ e2z dt dc1 ¼ e2y dt d~a da1 d~b db1 d~c dc1 , and , using (18) then we get from (17) Since ¼ ¼ ¼ dt dt dt dt dt dt _V ¼ ðe2 þ e2 þ e2 Þ 6 0 x y z
ð18Þ
Since V is a positive decreasing function and V_ is negative semidefinite, it follows that the equilibrium (ex ¼ 0, ey ¼ 0, ~ ~c 2 L1 . ez ¼ 0, ~a ¼ 0, ~b ¼ 0, ~c ¼ 0) of the system (14) and (18) is uniformly stable, i.e. ex ðtÞ; ey ðtÞ; ez ðtÞ 2 L1 and ~ a; b;
Fig. 3. Shows that the time response of states for drive system ðx1 ; y1 ; z1 Þ and response system ðx2 ; y2 ; z2 Þ with the control a_ 1 ¼ e2x : (a) signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 and (d) show that the error system tends to zero (x1 , y1 and z1 - - -; x2 , y2 and z2 – – –).
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663
From Eq. (18), we can easily show that the squares of ex ðtÞ, ey ðtÞ and ez ðtÞ are integrable with respect to time t. i.e. ex ðtÞ; ey ðtÞ; ez ðtÞ 2 L2 . Next by Barbalat’s Lemma, for any initial condition, the system (14) implies that e_ x ðtÞ; e_ y ðtÞ; e_ z ðtÞ 2 L1 , which in turn implies ðex ðtÞ; ey ðtÞ; ez ðtÞÞ ! ð0; 0; 0Þ as t ! 1. Thus, in the closed-loop system x2 ðtÞ ! x1 ðtÞ, y2 ðtÞ ! y1 ðtÞ, z2 ðtÞ ! z1 ðtÞ as t ! 1. This implies that the two L€ u systems have synchronized with adaptive control.
3. Numerical results Fourth-order Runge–Kutta method is used to solve the systems of differential equations (2), (3) and (10). In addition, a time step of size 0.001 is employed. The parameters are chosen to be a ¼ 36, b ¼ 3 and c ¼ 20 in all simulations so that the L€ u system exhibits a chaotic behavior if no control is applied. The initial states of the drive system are x1 ð0Þ ¼ 5, y1 ð0Þ ¼ 8 and z1 ð0Þ ¼ 10 and initial states of the response system are x2 ð0Þ ¼ 40, y2 ð0Þ ¼ 4 and z2 ð0Þ ¼ 5, hence the error system has the initial values ex ð0Þ ¼ 45, ey ð0Þ ¼ 12 and ez ð0Þ ¼ 15. The results of the
Fig. 4. Shows that the time response of states for drive system ðx1 ; y1 ; z1 Þ and response system ðx2 ; y2 ; z2 Þ with the control b_ ¼ e2z : (a) signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 and (d) show that the error system tends to zero (x1 , y1 , and z1 - - -; x2 , y2 and z2 – – –).
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80.00
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(a)
(b)
60.00 40.00
x2
40.00
20.00
y1
20.00
0.00 0.00
x1 y2 -20.00
t 0.00
4.00
8.00
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-20.00
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0.00
4.00
8.00
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4.00
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(c)
(d) 40.00
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0.00
ey 0.00 -20.00
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-20.00 0.00
4.00
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-40.00
t 0.00
Fig. 5. Shows that the time response of states for drive system ðx1 ; y1 ; z1 Þ and response system ðx2 ; y2 ; z2 Þ with the control c_ ¼ e2y : (a) signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 and (d) show that the error system tends to zero (x1 , y1 , and z1 - - -; x2 , y2 and z2 – – –).
simulation of the two identical L€ u systems without active control are shown in Fig. 2 displays the time response of the states: (a) x1 and x2 , (b) y1 and y2 , and (c) z1 and z2 . Synchronization of the systems (2) and (3) via adaptive control and the adaptation law (7) with the initial estimated parameter a1 ð0Þ ¼ 0:1, are shown in Fig. 3(a)–(d): (a) displays the trajectories x1 ðtÞ and x2 ðtÞ, (b) displays the trajectories y1 ðtÞ and y2 ðtÞ, (c) displays the trajectories z1 ðtÞ and z2 ðtÞ and (d) shows that the trajectories of ex , ey and ez tended to zero after t P 6. Synchronization of the systems (2) and (3) for the case of the estimate unknown parameter b with the initial estimated parameter b1 ð0Þ ¼ 0:2, are shown in Fig. 4(a)–(d): (a) displays the trajectories x1 ðtÞ and x2 ðtÞ, (b) displays the trajectories y1 ðtÞ and y2 ðtÞ, (c) displays the trajectories z1 ðtÞ and z2 ðtÞ and (d) shows that the trajectories of ex , ey and ez tended to zero after t P 10. Synchronization of the systems (2) and (3) for the case of the estimate unknown parameter c with the initial estimated parameters c1 ð0Þ ¼ 1, are displayed in Fig. 5(a)–(d): (a) displays the trajectories x1 ðtÞ and x2 ðtÞ, (b) displays the
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Fig. 6. Shows that the time response of states for drive system ðx1 ; y1 ; z1 Þ and response system ðx2 ; y2 ; z2 Þ with the control law (10): (a) signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 and (d) show that the error system tends to zero (x1 , y1 , and z1 - - -; x2 , y2 and z2 – – –).
trajectories y1 ðtÞ and y2 ðtÞ, (c) displays the trajectories z1 ðtÞ and z2 ðtÞ and (d) shows that the trajectories of ex , ey and ez tended to zero after t P 10. Fig. 6 shows the simulation results of chaos synchronization of systems (2) and (3) by using the adaptive control and the adaptation law (10) for the case of the estimate unknown parameter a and b with the initial estimated parameters a1 ð0Þ ¼ 0:1, b1 ð0Þ ¼ 0:2. Fig. 6(a) displays the trajectories x1 ðtÞ and x2 ðtÞ, Fig. 6(b) displays the trajectories y1 ðtÞ and y2 ðtÞ, Fig. 6(c) displays the trajectories z1 ðtÞ and z2 ðtÞ and Fig. 6(d) shows that the trajectories of ex , ey and ez tended to zero after t P 3. Finally we assume that all the system parameters a, b and c are unknown and the initial estimated parameters a1 ð0Þ ¼ 0:1, b1 ð0Þ ¼ 0:2, c1 ð0Þ ¼ 1. In this case the numerical solutions of the systems (2), (3) and (12) together with (13) and the adaptation law (18) are displayed in Fig. 7; Fig. 7(a) displays the trajectories x1 ðtÞ and x2 ðtÞ, Fig. 7(b) displays the trajectories y1 ðtÞ and y2 ðtÞ, Fig. 7(c) displays the trajectories z1 ðtÞ and z2 ðtÞ and Fig. 3(d) shows that the trajectories of ex , ey and ez tended to zero after t P 2.
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Fig. 7. Shows that the time response of states for drive system ðx1 ; y1 ; z1 Þ and response system ðx2 ; y2 ; z2 Þ with the control laws (13) and (18): (a) signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 and (d) show that the error system tends to zero (x1 , y1 , and z1 - - -; x2 , y2 and z2 – – –).
4. Conclusion This paper has addressed the adaptive synchronization problem of two identical L€ u systems. Based on Lyapunov stability theory the parameter update law introduced into the design of the adaptive synchronization controller can overcome the limitation of active control scheme. All results are proved by using Lyapunov stability theorem. Numerical simulations are used to verify the effectiveness of the proposed synchronization techniques.
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Adaptive synchronization of L€ u system with uncertain parameters E.M. Elabbasy *, H.N. Agiza, M.M. El-Dessoky Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Accepted 11 December 2003
Abstract This paper addresses the synchronization problem of two L€ u systems in the presence of unknown system parameters. Based on Lyapunov stability theory an adaptive control law is derived to make the states of two identical L€ u systems with unknown system parameters asymptotically synchronized. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization schemes. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Chaos has been developed and thoroughly studied over the past two decades. A chaotic system is a nonlinear deterministic system that displays complex and unpredictable behavior. The sensitive dependence on the initial conditions and on the system’s parameter variation is a prominent characteristic of chaotic behavior. Research efforts have investigated chaos control and chaos synchronization problems in many physical chaotic systems. Synchronization in chaotic dynamical systems has been a theme in nonlinear sciences and received considerable attention. Recently, synchronization in chaotic dynamical systems has been extensively investigated in the past few years [1–25], and many possible applications, especially to secret communication, have been discussed by computer simulation and even realized in laboratory condition [10–12]. The idea of synchronization is to use the output of the drive system to control the response system so that the output of the response system follows the output of the drive system asymptotically. Synchronization via adaptive control has been successfully tested in a variety of nonlinear dynamical systems, including Lorenz equations, the Rossler system, Chua’s circuit and Chen system [12–21]. Recently, a coupled of Lorenz system is synchronized by using active control [15–19]. The object of this work is to study chaos synchronization of two identical L€ u systems with unknown parameters. The key idea is that we introduce the parameters update law into the design of the adaptive synchronization controller based on Lyapunov stability theorem. In 1963, Lorenz found the first canonical chaotic attractor [27], which has just been mathematically confirmed to exist [28]. In 1999, Chen found another similar but topologically not equivalent chaotic attractor [29], as the dual of the Lorenz system, in a sense defined by Vanecek and Celikovsky [30]: The Lorenz system satisfies the condition a12 a21 > 0 while Chen system satisfies a12 a21 < 0, where a12 , a21 are the corresponding elements in the constant matrix A ¼ ðaij Þ33 for the linear part of the system. Very recently, L€ u and Chen found a new chaotic system [24–26], bearing the name of
*
Corresponding author. E-mail addresses: [email protected] (E.M. Elabbasy), [email protected] (H.N. Agiza), [email protected] (M.M. El-Dessoky). 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.028
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Fig. 1. Shows the chaotic attractor of L€ u system at a ¼ 36, b ¼ 3 and c ¼ 20.
the L€ u system, which satisfies the condition a12 a21 ¼ 0, thereby bridging the gap between the Lorenz and Chen attractors [25,26]. Therefore L€ u attractor is a new chaotic attractor, which connects the Lorenz attractor and Chen attractor and represents the transition from one to the other and is described by x_ ¼ aðy xÞ y_ ¼ xz þ cy
ð1Þ
z_ ¼ xy bz where a, b and c are three unknown uncertain parameters. This new system exhibits a chaotic attractor at the parameter values a ¼ 36, b ¼ 3 and c ¼ 20 (see Fig. 1).
2. Adaptive synchronization of L€u system In order to observe the synchronization behavior in the L€ u system, we have two L€ u systems where the drive system with three state variables denoted by the subscript 1 drives the response system having identical equations denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system. The two L€ u systems are described, respectively, by the following equations: x_ 1 ¼ aðy1 x1 Þ y_ 1 ¼ x1 z1 þ cy1 z_ 1 ¼ x1 y1 bz1
ð2Þ
x_ 2 ¼ aðy2 x2 Þ þ u1 ðtÞ y_ 2 ¼ x2 z2 þ cy2 þ u2 ðtÞ z_ 2 ¼ x2 y2 bz2 þ u3 ðtÞ
ð3Þ
and
where U ¼ ½u1 u2 u3 T is the controller functions. The controller U is to be determined for the purpose of synchronizing the two identical L€ u systems with same but unknown parameters a, b and c in spite of the differences in initial conditions. Subtracting Eq. (2) from Eq. (3), yields the error dynamical system between e_ x ¼ aðey ex Þ þ u1 ðtÞ e_ y ¼ x2 ez z1 ex þ cey þ u2 ðtÞ e_ z ¼ x2 ey þ y1 ex bez þ u3 ðtÞ
ð4Þ
where ex ¼ x2 x1 , ey ¼ y2 y1 and ez ¼ z2 z1 . The goal of the control is to find a controller U ¼ ½u1 u2 u3 T and a parameter estimation update law for Eq. (4) such that the states of response system (3) and the states of drive system (2) are globally synchronized asymptotically, i.e.
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lim keðtÞk ¼ 0 for all a; b and c 2 R
t!1
where eðtÞ ¼ ½ex ey ez T . Let the unknown uncertain parameter b be a positive number. Now, we can pick a Lyapunov function for the system (4) 1 V ðex ; ey ; ez ; ~aÞ ¼ ðe2x þ e2y þ e2z þ ~a2 Þ 2
ð5Þ
where ~a ¼ a a1 , a1 is an estimate value of the unknown parameter a, and require that its derivative along the solutions of the system (4) satisfy dV ðex ; ey ; ez ; ~aÞ 6 ðe2x þ e2y þ be2z Þ dt
ð6Þ
We therefore need to find a controller U and a parameter estimation update law a_ 1 to guarantee that for all eðtÞ 2 R3 , the inequality (6) holds. We have the time derivative of V ðex ; ey ; ez ; ~aÞ along the solutions of the system (4) dV ðex ; ey ; ez ; ~aÞ ¼ ex e_ x þ ey e_ y þ ez e_ z þ ~a~a_ dt ¼ aex ðey ex þ u1 ðtÞÞ þ ey ðx2 ez z1 ex þ cey þ u2 ðtÞÞ þ ez ðx2 ey þ y1 ex bez þ u3 ðtÞÞ ~ aa_ 1 ¼ aex ðey ex þ u1 ðtÞÞ þ ey ðcey z1 ex þ u2 ðtÞÞ þ ez ðy1 ex bez þ u3 ðtÞÞ ~ aa_ 1 There are many possible choices for the controller U and the parameter estimation update law a_ 1 . We choose u1 ðtÞ ¼ aey þ ða1 1Þex u2 ðtÞ ¼ z1 ex ð1 þ cÞey u3 ðtÞ ¼ y1 ex
ð7Þ
a_ 1 ¼ e2x under with this choice dV ðex ; ey ; ez ; ~aÞ ¼ ð~a þ 1Þe2x e2y be2z þ ~ae2x ¼ ðe2x þ e2y þ be2z Þ dt
ð8Þ
This leads to lim keðtÞk ¼ 0 for all a; c 2 R and b > 0
t!1
Hence, the synchronization of two L€ u systems is achieved under the controller and a parameter estimation update law Eq. (7). It is clear that the controller in Eq. (7) is independent of unknown uncertain parameter b, if b > 0. If the condition of unknown uncertain parameter b > 0 is cancelled, we take a Lyapunov function for Eq. (4) as follows 1 V ðex ; ey ; ez ; ~a; ~bÞ ¼ ðe2x þ e2y þ e2z þ ~a2 þ ~b2 Þ 2 where ~a ¼ a a1 , ~b ¼ b b1 , ~a and ~b are estimate values of the unknown parameter a and b, respectively.
ð9Þ
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We have the time derivative of V ðex ; ey ; ez ; ~a; ~bÞ along the solutions of Eq. (4) dV ðex ; ey ; ez ; ~a; ~ bÞ ¼ ex e_ x þ ey e_ y þ ez e_ z þ ~a~a_ þ ~b~b_ dt ¼ aex ðey ex þ u1 ðtÞÞ þ ey ðx2 ez z1 ex þ cey þ u2 ðtÞÞ þ ez ðx2 ey þ y1 ex bez þ u3 ðtÞÞ ~ aa_ 1 ~ bb_ 1 ¼ aex ðey ex þ u1 ðtÞÞ þ ey ðcey z1 ex þ u2 ðtÞÞ þ ez ðy1 ex bez þ u3 ðtÞÞ ~ aa_ 1 ~ bb_ 1 There are many possible choices for the controller U and the parameter estimation update laws a_ 1 and b_ 1 as follows: u1 ðtÞ ¼ aey þ ða1 1Þex u2 ðtÞ ¼ z1 ex ð1 þ cÞey u3 ðtÞ ¼ y1 ex þ ðb1 1Þez
ð10Þ
a_ 1 ¼ e2x b_ 1 ¼ e2z under this choice (10), we have dV ðex ; ey ; ez ; ~a; ~ bÞ ¼ ð~a þ 1Þe2x e2y ð~b þ 1Þe2z þ ~ ae2x þ ~ be2z ¼ ðe2x þ e2y þ e2z Þ dt
ð11Þ
This leads to lim keðtÞk ¼ 0
t!1
for all a; b and c 2 R
We consider now the case where the three parameters are unknown. Let the error system (4) be rewritten in the following form e_ x ¼ aðey ex Þ þ u1 ðtÞ e_ y ¼ x2 z2 þ x1 z1 þ cey þ u2 ðtÞ e_ z ¼ x2 y2 x1 y1 bez þ u3 ðtÞ
ð12Þ
By following the active control approach of Bai and Lonngren [18,19], we have introduced three control functions u1 ðtÞ, u2 ðtÞ and u3 ðtÞ in (12). These functions are to be determined for the purpose of synchronizing the two L€ u systems with same but unknown parameters a, b and c in spite of the differences in initial conditions. Clearly, the original synchronization problem can be replaced by the equivalent problem of stabilizing the zero solution of the system (12) by a suitable choice of the active functions u1 ðtÞ, u2 ðtÞ and u3 ðtÞ. Let us now define the active control functions u1 ðtÞ, u2 ðtÞ and u3 ðtÞ as u1 ðtÞ ¼ a1 ðey ex Þ ex u2 ðtÞ ¼ x2 z2 x1 z1 ðc1 þ 1Þey
ð13Þ
u3 ðtÞ ¼ x1 y1 x2 y2 þ ðb1 1Þez Where a1 , b1 and c1 are estimates of a, b and c respectively. Then the error dynamical system (12) can be expressed by e_ x ¼ aðey ex Þ a1 ðey ex Þ ex e_ y ¼ ðc c1 Þey ey e_ z ¼ ðb1 bÞez ez Let us introduce the parameter error ~a ¼ a a1 ; ~b ¼ b b1 and ~c ¼ c c1
ð14Þ
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then the error system can be described by e_ x ¼ a~ðey ex Þ ex e_ y ¼ ~cey ey e_ z ¼ ~bez ez
ð15Þ
For the derivation of the update law for adjusting the parameters a1 , b1 , and c1 , the Lyapunov approach is used. Consider a quadratic function 1 V ðex ; ey ; ez ; ~a; ~b; ~cÞ ¼ ðe2x þ e2y þ e2z þ ~a2 þ ~b2 þ ~c2 Þ 2
ð16Þ
Differentiating (16) using (15) gives ~ d~a d~b d~c d~ a db d~c b þ ~c aðey ex Þex þ ~ce2y ~ be2z þ ~ a þ~ V_ ¼ ex e_ x þ ey e_ y þ ez e_ z þ ~a þ ~b þ ~c ¼ e2x e2y e2z þ ~ dt dt dt dt dt dt
ð17Þ
Fig. 2. Display the solution of the L€ u system (2) and (3) where the control functions are deactivated: (a) signals x1 and x2 , (b) signals y1 and y2 and (c) signals z1 and z2 (x1 , y1 and z1 - - -; x2 , y2 and z2 – – –).
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In view of (17), the estimated parameters are updated by the following law da1 ¼ ðey ex Þex dt db1 ¼ e2z dt dc1 ¼ e2y dt d~a da1 d~b db1 d~c dc1 , and , using (18) then we get from (17) Since ¼ ¼ ¼ dt dt dt dt dt dt _V ¼ ðe2 þ e2 þ e2 Þ 6 0 x y z
ð18Þ
Since V is a positive decreasing function and V_ is negative semidefinite, it follows that the equilibrium (ex ¼ 0, ey ¼ 0, ~ ~c 2 L1 . ez ¼ 0, ~a ¼ 0, ~b ¼ 0, ~c ¼ 0) of the system (14) and (18) is uniformly stable, i.e. ex ðtÞ; ey ðtÞ; ez ðtÞ 2 L1 and ~ a; b;
Fig. 3. Shows that the time response of states for drive system ðx1 ; y1 ; z1 Þ and response system ðx2 ; y2 ; z2 Þ with the control a_ 1 ¼ e2x : (a) signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 and (d) show that the error system tends to zero (x1 , y1 and z1 - - -; x2 , y2 and z2 – – –).
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From Eq. (18), we can easily show that the squares of ex ðtÞ, ey ðtÞ and ez ðtÞ are integrable with respect to time t. i.e. ex ðtÞ; ey ðtÞ; ez ðtÞ 2 L2 . Next by Barbalat’s Lemma, for any initial condition, the system (14) implies that e_ x ðtÞ; e_ y ðtÞ; e_ z ðtÞ 2 L1 , which in turn implies ðex ðtÞ; ey ðtÞ; ez ðtÞÞ ! ð0; 0; 0Þ as t ! 1. Thus, in the closed-loop system x2 ðtÞ ! x1 ðtÞ, y2 ðtÞ ! y1 ðtÞ, z2 ðtÞ ! z1 ðtÞ as t ! 1. This implies that the two L€ u systems have synchronized with adaptive control.
3. Numerical results Fourth-order Runge–Kutta method is used to solve the systems of differential equations (2), (3) and (10). In addition, a time step of size 0.001 is employed. The parameters are chosen to be a ¼ 36, b ¼ 3 and c ¼ 20 in all simulations so that the L€ u system exhibits a chaotic behavior if no control is applied. The initial states of the drive system are x1 ð0Þ ¼ 5, y1 ð0Þ ¼ 8 and z1 ð0Þ ¼ 10 and initial states of the response system are x2 ð0Þ ¼ 40, y2 ð0Þ ¼ 4 and z2 ð0Þ ¼ 5, hence the error system has the initial values ex ð0Þ ¼ 45, ey ð0Þ ¼ 12 and ez ð0Þ ¼ 15. The results of the
Fig. 4. Shows that the time response of states for drive system ðx1 ; y1 ; z1 Þ and response system ðx2 ; y2 ; z2 Þ with the control b_ ¼ e2z : (a) signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 and (d) show that the error system tends to zero (x1 , y1 , and z1 - - -; x2 , y2 and z2 – – –).
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80.00
60.00
(a)
(b)
60.00 40.00
x2
40.00
20.00
y1
20.00
0.00 0.00
x1 y2 -20.00
t 0.00
4.00
8.00
12.00
t
-20.00
16.00
0.00
4.00
8.00
12.00
16.00
4.00
8.00
12.00
16.00
60.00
60.00
(c)
(d) 40.00
z2
40.00
ex 20.00
ez
20.00
z1
0.00
ey 0.00 -20.00
t
-20.00 0.00
4.00
8.00
12.00
16.00
-40.00
t 0.00
Fig. 5. Shows that the time response of states for drive system ðx1 ; y1 ; z1 Þ and response system ðx2 ; y2 ; z2 Þ with the control c_ ¼ e2y : (a) signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 and (d) show that the error system tends to zero (x1 , y1 , and z1 - - -; x2 , y2 and z2 – – –).
simulation of the two identical L€ u systems without active control are shown in Fig. 2 displays the time response of the states: (a) x1 and x2 , (b) y1 and y2 , and (c) z1 and z2 . Synchronization of the systems (2) and (3) via adaptive control and the adaptation law (7) with the initial estimated parameter a1 ð0Þ ¼ 0:1, are shown in Fig. 3(a)–(d): (a) displays the trajectories x1 ðtÞ and x2 ðtÞ, (b) displays the trajectories y1 ðtÞ and y2 ðtÞ, (c) displays the trajectories z1 ðtÞ and z2 ðtÞ and (d) shows that the trajectories of ex , ey and ez tended to zero after t P 6. Synchronization of the systems (2) and (3) for the case of the estimate unknown parameter b with the initial estimated parameter b1 ð0Þ ¼ 0:2, are shown in Fig. 4(a)–(d): (a) displays the trajectories x1 ðtÞ and x2 ðtÞ, (b) displays the trajectories y1 ðtÞ and y2 ðtÞ, (c) displays the trajectories z1 ðtÞ and z2 ðtÞ and (d) shows that the trajectories of ex , ey and ez tended to zero after t P 10. Synchronization of the systems (2) and (3) for the case of the estimate unknown parameter c with the initial estimated parameters c1 ð0Þ ¼ 1, are displayed in Fig. 5(a)–(d): (a) displays the trajectories x1 ðtÞ and x2 ðtÞ, (b) displays the
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Fig. 6. Shows that the time response of states for drive system ðx1 ; y1 ; z1 Þ and response system ðx2 ; y2 ; z2 Þ with the control law (10): (a) signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 and (d) show that the error system tends to zero (x1 , y1 , and z1 - - -; x2 , y2 and z2 – – –).
trajectories y1 ðtÞ and y2 ðtÞ, (c) displays the trajectories z1 ðtÞ and z2 ðtÞ and (d) shows that the trajectories of ex , ey and ez tended to zero after t P 10. Fig. 6 shows the simulation results of chaos synchronization of systems (2) and (3) by using the adaptive control and the adaptation law (10) for the case of the estimate unknown parameter a and b with the initial estimated parameters a1 ð0Þ ¼ 0:1, b1 ð0Þ ¼ 0:2. Fig. 6(a) displays the trajectories x1 ðtÞ and x2 ðtÞ, Fig. 6(b) displays the trajectories y1 ðtÞ and y2 ðtÞ, Fig. 6(c) displays the trajectories z1 ðtÞ and z2 ðtÞ and Fig. 6(d) shows that the trajectories of ex , ey and ez tended to zero after t P 3. Finally we assume that all the system parameters a, b and c are unknown and the initial estimated parameters a1 ð0Þ ¼ 0:1, b1 ð0Þ ¼ 0:2, c1 ð0Þ ¼ 1. In this case the numerical solutions of the systems (2), (3) and (12) together with (13) and the adaptation law (18) are displayed in Fig. 7; Fig. 7(a) displays the trajectories x1 ðtÞ and x2 ðtÞ, Fig. 7(b) displays the trajectories y1 ðtÞ and y2 ðtÞ, Fig. 7(c) displays the trajectories z1 ðtÞ and z2 ðtÞ and Fig. 3(d) shows that the trajectories of ex , ey and ez tended to zero after t P 2.
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Fig. 7. Shows that the time response of states for drive system ðx1 ; y1 ; z1 Þ and response system ðx2 ; y2 ; z2 Þ with the control laws (13) and (18): (a) signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 and (d) show that the error system tends to zero (x1 , y1 , and z1 - - -; x2 , y2 and z2 – – –).
4. Conclusion This paper has addressed the adaptive synchronization problem of two identical L€ u systems. Based on Lyapunov stability theory the parameter update law introduced into the design of the adaptive synchronization controller can overcome the limitation of active control scheme. All results are proved by using Lyapunov stability theorem. Numerical simulations are used to verify the effectiveness of the proposed synchronization techniques.
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