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Observer-based control design for three well-known chaotic systems
Chaos, Solitons and Fractals 29 (2006) 381–392 www.elsevier.com/locate/chaos
Observer-based control design for three well-known chaotic systems S.H. Mahboobi a, M. Shahrokhi b
b,*
, H.N. Pishkenari
a
a Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9465, Tehran, Iran Department of Chemical and Petroleum Engineering, Sharif University of Technology, P.O. Box 11365-9465, Tehran, Iran
Accepted 16 August 2005
Abstract In this paper, a singularity-free approach is proposed for controlling three well-known chaotic systems namely Lorenz, Chen and Lu. The control design guarantees the regulation of two states and boundedness of the remaining state. The stability of the proposed scheme has been shown using the Lyapunov stability theorem. Implementation of the proposed control technique requires system states, while in most of practical applications only the system output is available. To overcome this problem, a nonlinear observer is coupled with the controller. Simulation results have illustrated the effectiveness and robustness of the proposed schemes. If the control action is applied to the second system equation, all states will be regulated. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, there has been considerable interest in the control of chaotic dynamical systems. In particular, the control of the well-known Lorenz system has been studied extensively [1,2]. Gallegos [3] applied the input-state feedback linearization to control the Lorenz system. It has been discovered that the complete input-state linearization results in controller singularity when applied to the Lorenz system [4]. Yu [5] proposed a variable structure control strategy to stabilize the Lorenz chaos. To prevent an exploding control action due to singular control law, lower and upper bounds have been considered for the control action and consequently the resulting controller is only locally stabilizing [5]. Regarding the singularity problem in controlling the Lorenz system, Lenz and Obradovic [4] and Zeng and Singh [6] have obtained some singularity free results and proposed a global approach using partial linearization. Backstepping [7] is a stepwise procedure for selecting the control Lyapunov functions (clf) systematically that allows the design of adaptive controllers for a class of nonlinear processes. In the past decade, backstepping [7] has become one of the most common design methods for adaptive nonlinear control because of its ability to guarantee global stabilities, desirable tracking and transient performance for a wide class of strict-feedback system [8]. *
Corresponding author. Tel.: +98 21 66165419; fax: +98 21 66022853. E-mail address: [email protected] (M. Shahrokhi).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.042
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Ge et al. [9] have shown that many well-known chaotic systems as paradigms in the research of chaos, including Duffing oscillator, van der Pol oscillator, Rossler system and several types of Chuas circuits, can be transformed into a class of nonlinear systems in the ‘‘strict-feedback’’ form, and adaptive backstepping can be employed and extended for controlling of these chaotic systems. However, the Lorenz system, as shown by Ge et al. [9] cannot be controlled directly using backstepping method due to its singularity problem. Wang and Ge [10] have proposed a control scheme for the Lorenz system based on the adaptive backstepping method which can overcome the singularity problem. In this paper, a singularity-free approach is proposed for controlling three well-known chaotic systems namely Lorenz, Chen and Lu. The control design guarantees the output regulation and boundedness of the states. The stability of the proposed schemes has been shown using the Lyapunov stability theorem. For most of the published works in the literature full information of states is assumed, while in most of the practical applications only the system output is available. Implementation of the proposed control technique requires system states. To overcome this limitation, a nonlinear observer can be coupled with the controller. Recently, observer design has been used extensively for synchronization of chaotic systems [11–14]. In this work the nonlinear observer proposed by Thau [15] has been used. Regarding control of the Lorenz system, the performance of the proposed scheme has been compared with the performance of a recent control design proposed in the literature. The latter control strategy requires at least two states and two control efforts, while the proposed approach has the same performance by requiring only one state (system output) and one control input. Simulation results have illustrated the effectiveness and robustness of the proposed schemes.
2. Lorenz, Chen and Lu chaotic systems The Lorenz system is an approximate solution for a partial differential equation describing turbulent fluid convection, in which a flat fluid layer is heated from below and cooled from the above. This system is one of paradigms in exploration of chaos and is described by x_ 1 ¼ p1 ðx2 x1 Þ x_ 2 ¼ p2 x1 x2 x1 x3 x_ 3 ¼ x1 x2 p3 x3 þ u
ð1Þ
where x1, x2 and x3 are system states and p1, p2 and p3 are positive constant parameters. Convection in a vertical loop is a simple physical realization of the Lorenz system. In this model, x1 is proportional to the flow velocity, x2 is proportional to the horizontal temperature difference and x3 is proportional to the vertical temperature difference p1, p2 and p3 are real positive parameters denoting the Prandtl and the Rayleigh numbers and the geometric factor, respectively [16]. The control action is denoted by u. Recently, in the attempt of anti-controlling chaos, Chen [17] has developed a new chaotic system, called the Chens chaotic system, which is derived from the classical Lorenz system. The nonlinear differential equations that describe the Chens attractor are given below: x_ 1 ¼ p1 ðx2 x1 Þ x_ 2 ¼ ðp2 p1 Þx1 þ p2 x2 x1 x3 x_ 3 ¼ x1 x2 p3 x3 þ u
ð2Þ
From the control engineering point of view, the chaotic Chens attractor is relatively difficult to control as compared to the Lorenzs system due to the prominent three-dimensional and complex topological features of its attractor, especially its rapid change in velocity in the z-direction. Lu chaotic system [18] is similar to the above systems and it is expressed with the following equations: x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ x_ 2 ¼ ð28 35aÞx1 þ ð29a 1Þx2 x1 x3 aþ8 x3 þ u x_ 3 ¼ x1 x2 3 where a = 0.8.
ð3Þ
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3. Controller design 3.1. Lorenz system For the Lorenz system, we have proposed a design procedure using a new form of backstepping method which can overcome the singularity caused by x1x3 term. The goal is designing a control effort u which can regulate states x1 and x2, and keep x3 bounded. Backstepping design is a systematic Lyapunov-based control technique, which can be applied to strict-feedback, pure feedback and block-strict-feedback systems. For the Lorenz system in its strict-feedback form, the typical backstepping method cannot be used directly for the control purpose, due to the singularity caused by x1x3 term. To overcome this problem the two first steps in backstepping method are combined and performed in one step. The procedure is described below. If we select x3 as x3 ¼ w ¼ np2 ;
n>1
1 p2
ð4Þ
then the system including first and second equations of the Lorenz system will be stable. The Lyapunov function for proving the stability of the first two linear equations is given below: x1 1 h1 h3 ¼ x21 þ h2 x1 x2 þ x22 ð5Þ V 0 ¼ ½x1 x2 H 2 2 2 x2 where H is a symmetric positive definite matrix in the following form: h1 h2 H¼ h2 h3
ð6Þ
which satisfies the Lyapunov equation: AT H þ HA ¼ Q
ð7Þ
In the above equation A is p1 p1 A¼ ð1 nÞp2 1
ð8Þ
and Q is a symmetric positive definite matrix. Since eigenvalues of matrix A are negative, there exists a positive definite matrix H which satisfies Eq. (7). For convenience we set: Q ¼ I 22
ð9Þ
which leads to h1 ¼
ðp1 1 þ p1 p2 np1 p2 p22 þ 2np22 n2 p22 Þ 2p1 ðp1 þ 1Þð1 p2 þ np2 Þ
ð10Þ
h2 ¼
ð1 p2 þ np2 Þ 2ðp1 þ 1Þð1 p2 þ np2 Þ
ð11Þ
h3 ¼
ð2p1 1 þ p2 np2 Þ 2ðp1 þ 1Þð1 p2 þ np2 Þ
ð12Þ
Now, we can use the standard form of backstepping control design [8]. Let us define f and g as below: p1 ðx2 x1 Þ 0 f ¼ ; g¼ p 2 x1 x2 x1
ð13Þ
Consider the following Lyapunov function for the Lorenz system: 1 V ¼ V 0 þ ðx3 wÞ2 2
ð14Þ
If x3 satisfies the following equation: x_ 3 ¼
ow oV 0 ðf þ gx3 Þ g lðx3 wÞ; oðx1 ; x2 Þ oðx1 ; x2 Þ
l>0
ð15Þ
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the time derivative of V becomes: x1 1 lðx3 wÞ2 V_ ¼ ½x1 x2 Q 2 x2
ð16Þ
which is negative definite and thereby guarantees the stability of the Lorenz system. Using the third equation of the system, u can be determined as: u ¼ h2 x21 þ h3 x1 x2 x1 x2 þ p3 x3 lx3 þ lnp2
ð17Þ
3.2. Chen system Similar to the proposed control technique for the Lorenz system, we have the following equations: x3 ¼ w > 2p2 p1 p1 p1 A¼ p2 p1 w p2 ð2p22 þ 4p2 p1 2p21 3wp1 þ 2wp2 w2 Þ 2p1 ðp1 p2 Þðp1 2p2 þ wÞ ðp1 þ wÞ h2 ¼ 2ðp1 p2 Þðp1 2p2 þ wÞ ð3p1 þ 2p2 wÞ h3 ¼ 2ðp1 p2 Þðp1 2p2 þ wÞ h1 ¼
u ¼ h2 x21 þ h3 x1 x2 x1 x2 þ p3 x3 lx3 þ lw
ð18Þ ð19Þ ð20Þ ð21Þ ð22Þ ð23Þ
Using the Lyapunov function given by Eq. (16) the stability of the system can be shown. The proposed control laws for these systems lead to exponential regulation of x1, x2 and boundedness of x3. In addition, the control action remains bounded and approaches to p3x3 as times approaches infinity. 3.3. Lu system Controller design for the Lu chaotic system can be fulfilled via an approach similar to the procedure mentioned above and is left to the reader. Note: As mentioned above, the proposed scheme doesnt regulate the third system state. In order to regulate all of the states we can apply the control input to the second equation of the system instead of the third one. For example, the system equations for the Lorenz system become: x_ 1 ¼ p1 ðx2 x1 Þ x_ 2 ¼ p2 x1 x2 x1 x3 þ u x_ 3 ¼ x1 x2 p3 x3
ð24Þ
If the control action is selected as given below: u ¼ x1 x3 þ k 1 x1 þ k 2 x2 the characteristic matrix for the subsystem consisted of the two first system equations will be p1 p1 A¼ p2 þ k 1 1 þ k 2
ð25Þ
ð26Þ
and its characteristic equation will be k2 þ ð1 k 2 p1 Þk þ p1 ðp2 þ k 1 þ k 2 1Þ ¼ 0
ð27Þ
In order to stabilize the aforementioned subsystem the following conditions must be satisfied: k 2 6 1 p1 ;
k 1 þ k 2 P 1 p2
ð28Þ
Under the above conditions the two first system states will be regulated. Regulation of the two first states and the stability of internal dynamics (third equation) guarantee the regulation of the third state.
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4. Thau observer overview As can be seen, implementation of the proposed control technique (see Eqs. (17) and (23)) requires all of the system states, while in practice, the only accessible state is the system output which is usually assumed to be x1. Therefore it is necessary to estimate the remaining system states from the output by using an appropriate observer. In this section the Thau observer [15] for nonlinear systems will be discussed briefly and then it will be applied for state estimation. Consider the following system: x_ ¼ Ax þ f ðxÞ þ Bu ð29Þ y ¼ Cx where A, B, C and f(Æ) are known and f is a globally Lipschitzian vector field such that f(0) = 0. If (A, C) is observable, there exists K such that the eigenvalues of A0 = A KC are in the left half-plane. Thau observer has the following form [15]: ( ^x_ ¼ A^x þ f ð^xÞ þ Bu þ Kðy ^y Þ ð30Þ ^y ¼ C^x and observer error is defined as e ¼ ^x x
ð31Þ
The dynamic of observer error is described by the following equation: e_ ¼ ðA KCÞe þ f ð^xÞ f ðxÞ ¼ A0 e þ f ð^xÞ f ðxÞ
ð32Þ
Since A0 is stable, for any given positive definite matrix Q, there exists a unique positive definite P, such that the following Lyapunov equation is satisfied: AT0 P þ PA0 ¼ 2Q
ð33Þ
If K is selected such that A0 can satisfy the Lyapunov equation and L satisfies the following inequality: L<
kmin ðQÞ kP k
ð34Þ
where L is Lipschitz constant kf ðx1 Þ f ðx2 Þk 6 Lkx1 x2 k
ð35Þ
for all x1 and x2, then the Thau nonlinear observer is asymptotically stable [15]. A simpler version of the Thau inequality is proposed by Starkov and Esquer [19] and is given below. If S matrix defined as S ¼ A0 þ AT0
ð36Þ
is a stable matrix and 2L < kmin ðSÞ
ð37Þ
then Eq. (30) is an asymptotic observer for system (29).
5. Thau observer design For the Lorenz system we have 3 3 2 2 0 0 p1 p1 7 7 6 6 0 5; f ðxÞ ¼ 4 x1 x3 5 A ¼ 4 p2 1 0 0 p3 x1 x2 We will use the common values of the parameters p1 = 10, p2 = 28 and p3 = 8/3 for simulation study.
ð38Þ
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S.H. Mahboobi et al. / Chaos, Solitons and Fractals 29 (2006) 381–392
The absorbing ball is described by [20]: B ¼ fx21 þ x22 þ ðx3 p1 p2 Þ2 6 t2 ðp1 þ p2 Þ2 g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ p3 maxðp1 t¼ 1 ; 1Þ 2 In order to compute 0 0 f ðxÞ ¼ x3 x2
the Lipschitz constant L(B) restricted on B we get: 0 0 0 x1 x1 0
ð39Þ
ð40Þ
Therefore LðBÞ2 6 kf 0 ðxÞk2B 6 ð2x21 þ x22 þ x23 Þjx2B
ð41Þ
Since x 2 B we have jx3 j 6 p1 þ p2 þ tðp1 þ p2 Þ ¼ ðt þ 1Þðp1 þ p2 Þ jx2 j 6 tðp1 þ p2 Þ jx1 j 6 tðp1 þ p2 Þ
ð42Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LðBÞ2 6 ðp1 þ p2 Þ 4t2 þ 2t þ 1
ð43Þ
By substituting the numerical values we have ð44Þ
LðBÞ 6 97:49 The S matrix is given by: 2 2ðp1 þ k 1 Þ p1 þ p2 k 2 6 S ¼ 4 p1 þ p2 k 2 2 k 3 0
k 3
3
7 0 5 2p3
ð45Þ
The eigenvalues of matrix S become negative by proper choice of the matrix K. For simplicity we set k3 = 0 and choose k1 and k2 such that: k 1 > p1 ð46Þ k 22 þ 2k 2 ðp1 þ p2 Þ ðp1 þ p2 Þ2 þ 4ðp1 þ k 1 Þ > 0 Under the above conditions S is a stable matrix. By choosing K as given below [21]: 2 3 10 6 7 K ¼ 4 28 5
ð47Þ
0 the eigenvalues of S become 2, 5.333 and 20 and we have kmin ðSÞ ¼ 2
ð48Þ
As can be seen the inequality (37) is not met and therefore we introduce a rescaling parameter [21] such that: 0
kmin ðSÞ 2 ¼ 2L 194:98
Applying the rescaling we have 8 > < w_ 1 ¼ p1 ðw2 w1 Þ w_ 2 ¼ p2 w1 w2 lw1 w3 > : w_ 3 ¼ lw1 w2 p3 w3 where x = lw. Equations for the observer are
ð49Þ
ð50Þ
S.H. Mahboobi et al. / Chaos, Solitons and Fractals 29 (2006) 381–392
8 > ^_ 1 ¼ ðp1 þ k 1 Þ^ ^ 2 þ k 1 w1 w1 þ p 2 w :_ ^ 3 ¼ l^ ^ 2 p3 w ^ 3 k3w ^ 1 þ k 3 w1 w w1 w
387
ð51Þ
We can design the observer for the Chen system using a similar approach and finally we get 8 > < w_ 1 ¼ p1 ðw2 w1 Þ w_ 2 ¼ ðp2 p1 Þw1 þ p2 w2 lw1 w3 > : w_ 3 ¼ lw1 w2 p3 w3 where x = lw. Equations for the observer are 8 > ^_ 1 ¼ ðp1 þ k 1 Þ^ ^ 2 þ k 1 w1 w1 þ p 2 w :_ ^ 3 ¼ l^ ^ 2 p3 w ^ 3 k3w ^ 1 þ k 3 w1 w w1 w
ð52Þ
ð53Þ
Using the state estimates, the control actions for Lorenz and Chen systems become u ¼ h2 x21 þ h3 x1^x2 x1^x2 þ p3^x3 l^x3 þ lnp2 u¼
h2 x21
ð54Þ
þ h3 x1^x2 x1^x2 þ p3^x3 l^x3 þ lw
ð55Þ
6. Simulation results In this section, simulations are conducted to demonstrate the effectiveness of the proposed controllers coupled with the designed observers. In all simulations for Lorenz and Chen systems, we have assumed initial conditions x1 = 10, x2 = 20, x3 = 30. The parameters of the Chen system are set to p1 = 35, p2 = 28, p3 = 3. Fig. 1 shows the capability of the control law (54), in regulation of the Lorenz system. The controller parameters are l¼2 n¼2 The initial estimates of the states are set to zero. Fig. 2 depicts the effectiveness of the observer in estimation of the Lorenz system states. Fig. 3 shows the control effort required to regulate the Lorenz system states. Fig. 4 shows the capability of the control law (55), in regulation of the Chen system. The initial state estimates and the controller parameters are set to the following values:
60 50
x1 x2 x3
System States
40 30 20 10 0 -10
0
0.5
1
1.5
2
Time (sec)
Fig. 1. States of the controlled Lorenz system.
2.5
S.H. Mahboobi et al. / Chaos, Solitons and Fractals 29 (2006) 381–392 40 e1 e2 e3
State Estimation Errors
30 20 10 0 -10 -20 -30 0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 2. State estimation errors of the Lorenz system.
200 150
Control Action
100 50 0 -50 -100 -150 -200 -250 -300
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 3. Control action for the Lorenz system.
40
x1 x2 x3
30
System States
388
20
10
0
-10
-20
0
0.5
1
1.5
2
Time (sec)
Fig. 4. States of the controlled Chen system.
2.5
S.H. Mahboobi et al. / Chaos, Solitons and Fractals 29 (2006) 381–392 40 e1 e2 e3
State Estimation Errors
30 20 10 0 -10 -20 -30 -40 -50
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 5. State estimation errors of the Chen system.
150 100
Control Action
50 0 -50 -100 -150 -200 -250 -300
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 6. Control action for the Chen system.
60 x1 x2 x3
50
System States
40 30 20 10 0 -10 -20
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 7. System responses of the controlled Lorenz system in the presence of +20% parameter uncertainties.
389
S.H. Mahboobi et al. / Chaos, Solitons and Fractals 29 (2006) 381–392 40 x1 x2 x3
35
System States
30 25 20 15 10 5 0 -5
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 8. System responses of the controlled Chen system in the presence of +20% parameter uncertainties.
30 x1 x2 x3
20
System States
10 0 -10 -20 -30 -40 -50 -60 -70
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 9. State variations vs. time for the controlled Lorenz system using the Hua controller.
30 e1 e2 e3
25
State Estimation Errors
390
20 15 10 5 0 -5 -10 -15 -20
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 10. State estimation error for the Lorenz system using the Hua controller.
S.H. Mahboobi et al. / Chaos, Solitons and Fractals 29 (2006) 381–392
^x1 ¼ 0; w ¼ 32 l¼3
^x2 ¼ 0;
391
^x3 ¼ 0
Fig. 5 depicts the effectiveness of the observer performance. Fig. 6 shows the control effort required to regulate the system states. In order to exhibit the robustness of the proposed control schemes, for Lorenz and Chen systems, we have considered conditions in which the system parameters can change from their nominal values. The parameter uncertainties are assumed to be 20%. Figs. 7 and 8 show the closed loop responses in the presence of parameter uncertainties, for Lorenz and Chen systems respectively. To demonstrate the efficiency of the designed controller, we have compared its performance with a recent controller proposed in literature. Hua et al. [22] designed a controller with two manipulated variables applied to the second and third equations to overcome the problem of chaos control for Lorenz and Chen systems. Their scheme requires two system states for estimating the third state. Figs. 9–12 show the results of the Huas controller applied to Lorenz and Chen systems. As can be seen the controller has almost the same performance as the proposed controller at the expense of one additional control input and one
100 x1 x2 x3
80
System States
60 40 20 0 -20 -40 -60 -80
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 11. State variations vs. time for the controlled Chen system using the Hua controller.
30 e1 e2 e3
State Estimation Errors
20
10
0
-10
-20
-30
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 12. State estimation errors for the Chen system using the Hua controller.
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extra measurement. It should be mentioned that the Huas controller regulates all of the states, but as mentioned earlier, this goal could be reached by applying the input to the second state equation.
7. Conclusions In this paper, a singularity-free design for controlling three well-known chaotic systems namely Lorenz, Chen and Lu has been proposed. It has been shown that the origin is globally asymptotically stable. To overcome the problem of inaccessibility of the system states, a nonlinear observer is coupled with the proposed controller. In order to demonstrate the effectiveness of the proposed approach, its performance has been compared through simulation with the performance of a recent control design proposed in the literature. The latter control strategy requires at least two states and two control efforts, while the proposed approach show the same performance by requiring only one state (system output) and one control input. Simulation results have been conducted to show the effectiveness and robustness of the proposed scheme.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
[22]
Fradkov AL, Pogromsky AYu. Introduction to control of oscillations and chaos. Singapore: World Scientific; 1998. Lorenz EN. Deterministic non-periodic flow. J Atmosph Sci 1963;20:130–41. Gallegos JA. Nonlinear regulation of a Lorenz system by feedback linearization techniques. Dyn Contr 1994;4:277–98. Lenz H, Obradovic D. Robust control of the chaotic Lorenz system. Int J Bifurc Chaos 1997;7:2847–54. Yu XH. Controlling Lorenz chaos. Int J Syst Sci 1996;27:355–9. Zeng Y, Singh SN. Adaptive control of chaos in Lorenz system. Dyn Contr 1997;7:143–54. Kanellakopoulos I, Kokotovic P, Morse A. Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans Automat Control 1991;36:1241–53. Krstic M, Kanellakopoulos I, Kokotovic P. Nonlinear and adaptive control design. New York: John Wiley; 1995. Ge SS, Wang C, Lee TH. Adaptive backstepping control of a class of chaotic systems. Int J Bifurc Chaos 2000;10:1149–56. Wang C, Ge SS. Adaptive backstepping control of uncertain Lorenz system. Int J Bifurc Chaos 2001;11:1115–9. Huijberts HJC. Nonlinear discrete-time synchronization via extended observers. Int J Bifurc Chaos 2001;11:1997–2001. Nijmeijer H. A dynamical control view on synchronization. Physica D 2001;154:219–28. Feki M. Observer-based exact synchronization of ideal and mismatched chaotic systems. Phys Lett A 2003;309:53–60. Santoboni G, Yu A, Nijmeijer H. An observer for phase synchronization of chaos. Phys Lett A 2001;291:265–73. Thau FE. Observing the states of nonlinear dynamic systems. Int J Control 1973;17:471–9. Crespo LG, Sun JQ. On the feedback linearization of the Lorenz system. J Vibr Control 2004;10:85–100. Chen G, Ueta T. Yet another chaotic attractor. Int J Bifurc Chaos 1999;9:1465–6. Lu J, Chen G, Zhang S. A new chaotic attractor coined. Int J Bifurc Chaos 2002:659–61. Starkov KE, Esquer M. Construction of observers for nonlinear systems based on the solution of Thau inequality. In: Proc. of the 14th ICSE conference, Coventry, UK, vol. 2, 2000. p. 508–12. Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–3. Starkov K, Escobar SH. Observer-based synchronization of coupled chaotic systems with applications to the Chua system, Sprott systems and the Lorenz system. In: International conference on control, virtual instrumentation, and digital systems CICINDI2002, Pachuca, Mexico, 2002. Hua C, Guan X, Li X, Shi P. Adaptive observer-based control for a class of chaotic systems. Chaos, Solitons & Fractals 2004;22:103–10.
Observer-based control design for three well-known chaotic systems S.H. Mahboobi a, M. Shahrokhi b
b,*
, H.N. Pishkenari
a
a Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9465, Tehran, Iran Department of Chemical and Petroleum Engineering, Sharif University of Technology, P.O. Box 11365-9465, Tehran, Iran
Accepted 16 August 2005
Abstract In this paper, a singularity-free approach is proposed for controlling three well-known chaotic systems namely Lorenz, Chen and Lu. The control design guarantees the regulation of two states and boundedness of the remaining state. The stability of the proposed scheme has been shown using the Lyapunov stability theorem. Implementation of the proposed control technique requires system states, while in most of practical applications only the system output is available. To overcome this problem, a nonlinear observer is coupled with the controller. Simulation results have illustrated the effectiveness and robustness of the proposed schemes. If the control action is applied to the second system equation, all states will be regulated. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, there has been considerable interest in the control of chaotic dynamical systems. In particular, the control of the well-known Lorenz system has been studied extensively [1,2]. Gallegos [3] applied the input-state feedback linearization to control the Lorenz system. It has been discovered that the complete input-state linearization results in controller singularity when applied to the Lorenz system [4]. Yu [5] proposed a variable structure control strategy to stabilize the Lorenz chaos. To prevent an exploding control action due to singular control law, lower and upper bounds have been considered for the control action and consequently the resulting controller is only locally stabilizing [5]. Regarding the singularity problem in controlling the Lorenz system, Lenz and Obradovic [4] and Zeng and Singh [6] have obtained some singularity free results and proposed a global approach using partial linearization. Backstepping [7] is a stepwise procedure for selecting the control Lyapunov functions (clf) systematically that allows the design of adaptive controllers for a class of nonlinear processes. In the past decade, backstepping [7] has become one of the most common design methods for adaptive nonlinear control because of its ability to guarantee global stabilities, desirable tracking and transient performance for a wide class of strict-feedback system [8]. *
Corresponding author. Tel.: +98 21 66165419; fax: +98 21 66022853. E-mail address: [email protected] (M. Shahrokhi).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.042
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Ge et al. [9] have shown that many well-known chaotic systems as paradigms in the research of chaos, including Duffing oscillator, van der Pol oscillator, Rossler system and several types of Chuas circuits, can be transformed into a class of nonlinear systems in the ‘‘strict-feedback’’ form, and adaptive backstepping can be employed and extended for controlling of these chaotic systems. However, the Lorenz system, as shown by Ge et al. [9] cannot be controlled directly using backstepping method due to its singularity problem. Wang and Ge [10] have proposed a control scheme for the Lorenz system based on the adaptive backstepping method which can overcome the singularity problem. In this paper, a singularity-free approach is proposed for controlling three well-known chaotic systems namely Lorenz, Chen and Lu. The control design guarantees the output regulation and boundedness of the states. The stability of the proposed schemes has been shown using the Lyapunov stability theorem. For most of the published works in the literature full information of states is assumed, while in most of the practical applications only the system output is available. Implementation of the proposed control technique requires system states. To overcome this limitation, a nonlinear observer can be coupled with the controller. Recently, observer design has been used extensively for synchronization of chaotic systems [11–14]. In this work the nonlinear observer proposed by Thau [15] has been used. Regarding control of the Lorenz system, the performance of the proposed scheme has been compared with the performance of a recent control design proposed in the literature. The latter control strategy requires at least two states and two control efforts, while the proposed approach has the same performance by requiring only one state (system output) and one control input. Simulation results have illustrated the effectiveness and robustness of the proposed schemes.
2. Lorenz, Chen and Lu chaotic systems The Lorenz system is an approximate solution for a partial differential equation describing turbulent fluid convection, in which a flat fluid layer is heated from below and cooled from the above. This system is one of paradigms in exploration of chaos and is described by x_ 1 ¼ p1 ðx2 x1 Þ x_ 2 ¼ p2 x1 x2 x1 x3 x_ 3 ¼ x1 x2 p3 x3 þ u
ð1Þ
where x1, x2 and x3 are system states and p1, p2 and p3 are positive constant parameters. Convection in a vertical loop is a simple physical realization of the Lorenz system. In this model, x1 is proportional to the flow velocity, x2 is proportional to the horizontal temperature difference and x3 is proportional to the vertical temperature difference p1, p2 and p3 are real positive parameters denoting the Prandtl and the Rayleigh numbers and the geometric factor, respectively [16]. The control action is denoted by u. Recently, in the attempt of anti-controlling chaos, Chen [17] has developed a new chaotic system, called the Chens chaotic system, which is derived from the classical Lorenz system. The nonlinear differential equations that describe the Chens attractor are given below: x_ 1 ¼ p1 ðx2 x1 Þ x_ 2 ¼ ðp2 p1 Þx1 þ p2 x2 x1 x3 x_ 3 ¼ x1 x2 p3 x3 þ u
ð2Þ
From the control engineering point of view, the chaotic Chens attractor is relatively difficult to control as compared to the Lorenzs system due to the prominent three-dimensional and complex topological features of its attractor, especially its rapid change in velocity in the z-direction. Lu chaotic system [18] is similar to the above systems and it is expressed with the following equations: x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ x_ 2 ¼ ð28 35aÞx1 þ ð29a 1Þx2 x1 x3 aþ8 x3 þ u x_ 3 ¼ x1 x2 3 where a = 0.8.
ð3Þ
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3. Controller design 3.1. Lorenz system For the Lorenz system, we have proposed a design procedure using a new form of backstepping method which can overcome the singularity caused by x1x3 term. The goal is designing a control effort u which can regulate states x1 and x2, and keep x3 bounded. Backstepping design is a systematic Lyapunov-based control technique, which can be applied to strict-feedback, pure feedback and block-strict-feedback systems. For the Lorenz system in its strict-feedback form, the typical backstepping method cannot be used directly for the control purpose, due to the singularity caused by x1x3 term. To overcome this problem the two first steps in backstepping method are combined and performed in one step. The procedure is described below. If we select x3 as x3 ¼ w ¼ np2 ;
n>1
1 p2
ð4Þ
then the system including first and second equations of the Lorenz system will be stable. The Lyapunov function for proving the stability of the first two linear equations is given below: x1 1 h1 h3 ¼ x21 þ h2 x1 x2 þ x22 ð5Þ V 0 ¼ ½x1 x2 H 2 2 2 x2 where H is a symmetric positive definite matrix in the following form: h1 h2 H¼ h2 h3
ð6Þ
which satisfies the Lyapunov equation: AT H þ HA ¼ Q
ð7Þ
In the above equation A is p1 p1 A¼ ð1 nÞp2 1
ð8Þ
and Q is a symmetric positive definite matrix. Since eigenvalues of matrix A are negative, there exists a positive definite matrix H which satisfies Eq. (7). For convenience we set: Q ¼ I 22
ð9Þ
which leads to h1 ¼
ðp1 1 þ p1 p2 np1 p2 p22 þ 2np22 n2 p22 Þ 2p1 ðp1 þ 1Þð1 p2 þ np2 Þ
ð10Þ
h2 ¼
ð1 p2 þ np2 Þ 2ðp1 þ 1Þð1 p2 þ np2 Þ
ð11Þ
h3 ¼
ð2p1 1 þ p2 np2 Þ 2ðp1 þ 1Þð1 p2 þ np2 Þ
ð12Þ
Now, we can use the standard form of backstepping control design [8]. Let us define f and g as below: p1 ðx2 x1 Þ 0 f ¼ ; g¼ p 2 x1 x2 x1
ð13Þ
Consider the following Lyapunov function for the Lorenz system: 1 V ¼ V 0 þ ðx3 wÞ2 2
ð14Þ
If x3 satisfies the following equation: x_ 3 ¼
ow oV 0 ðf þ gx3 Þ g lðx3 wÞ; oðx1 ; x2 Þ oðx1 ; x2 Þ
l>0
ð15Þ
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the time derivative of V becomes: x1 1 lðx3 wÞ2 V_ ¼ ½x1 x2 Q 2 x2
ð16Þ
which is negative definite and thereby guarantees the stability of the Lorenz system. Using the third equation of the system, u can be determined as: u ¼ h2 x21 þ h3 x1 x2 x1 x2 þ p3 x3 lx3 þ lnp2
ð17Þ
3.2. Chen system Similar to the proposed control technique for the Lorenz system, we have the following equations: x3 ¼ w > 2p2 p1 p1 p1 A¼ p2 p1 w p2 ð2p22 þ 4p2 p1 2p21 3wp1 þ 2wp2 w2 Þ 2p1 ðp1 p2 Þðp1 2p2 þ wÞ ðp1 þ wÞ h2 ¼ 2ðp1 p2 Þðp1 2p2 þ wÞ ð3p1 þ 2p2 wÞ h3 ¼ 2ðp1 p2 Þðp1 2p2 þ wÞ h1 ¼
u ¼ h2 x21 þ h3 x1 x2 x1 x2 þ p3 x3 lx3 þ lw
ð18Þ ð19Þ ð20Þ ð21Þ ð22Þ ð23Þ
Using the Lyapunov function given by Eq. (16) the stability of the system can be shown. The proposed control laws for these systems lead to exponential regulation of x1, x2 and boundedness of x3. In addition, the control action remains bounded and approaches to p3x3 as times approaches infinity. 3.3. Lu system Controller design for the Lu chaotic system can be fulfilled via an approach similar to the procedure mentioned above and is left to the reader. Note: As mentioned above, the proposed scheme doesnt regulate the third system state. In order to regulate all of the states we can apply the control input to the second equation of the system instead of the third one. For example, the system equations for the Lorenz system become: x_ 1 ¼ p1 ðx2 x1 Þ x_ 2 ¼ p2 x1 x2 x1 x3 þ u x_ 3 ¼ x1 x2 p3 x3
ð24Þ
If the control action is selected as given below: u ¼ x1 x3 þ k 1 x1 þ k 2 x2 the characteristic matrix for the subsystem consisted of the two first system equations will be p1 p1 A¼ p2 þ k 1 1 þ k 2
ð25Þ
ð26Þ
and its characteristic equation will be k2 þ ð1 k 2 p1 Þk þ p1 ðp2 þ k 1 þ k 2 1Þ ¼ 0
ð27Þ
In order to stabilize the aforementioned subsystem the following conditions must be satisfied: k 2 6 1 p1 ;
k 1 þ k 2 P 1 p2
ð28Þ
Under the above conditions the two first system states will be regulated. Regulation of the two first states and the stability of internal dynamics (third equation) guarantee the regulation of the third state.
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4. Thau observer overview As can be seen, implementation of the proposed control technique (see Eqs. (17) and (23)) requires all of the system states, while in practice, the only accessible state is the system output which is usually assumed to be x1. Therefore it is necessary to estimate the remaining system states from the output by using an appropriate observer. In this section the Thau observer [15] for nonlinear systems will be discussed briefly and then it will be applied for state estimation. Consider the following system: x_ ¼ Ax þ f ðxÞ þ Bu ð29Þ y ¼ Cx where A, B, C and f(Æ) are known and f is a globally Lipschitzian vector field such that f(0) = 0. If (A, C) is observable, there exists K such that the eigenvalues of A0 = A KC are in the left half-plane. Thau observer has the following form [15]: ( ^x_ ¼ A^x þ f ð^xÞ þ Bu þ Kðy ^y Þ ð30Þ ^y ¼ C^x and observer error is defined as e ¼ ^x x
ð31Þ
The dynamic of observer error is described by the following equation: e_ ¼ ðA KCÞe þ f ð^xÞ f ðxÞ ¼ A0 e þ f ð^xÞ f ðxÞ
ð32Þ
Since A0 is stable, for any given positive definite matrix Q, there exists a unique positive definite P, such that the following Lyapunov equation is satisfied: AT0 P þ PA0 ¼ 2Q
ð33Þ
If K is selected such that A0 can satisfy the Lyapunov equation and L satisfies the following inequality: L<
kmin ðQÞ kP k
ð34Þ
where L is Lipschitz constant kf ðx1 Þ f ðx2 Þk 6 Lkx1 x2 k
ð35Þ
for all x1 and x2, then the Thau nonlinear observer is asymptotically stable [15]. A simpler version of the Thau inequality is proposed by Starkov and Esquer [19] and is given below. If S matrix defined as S ¼ A0 þ AT0
ð36Þ
is a stable matrix and 2L < kmin ðSÞ
ð37Þ
then Eq. (30) is an asymptotic observer for system (29).
5. Thau observer design For the Lorenz system we have 3 3 2 2 0 0 p1 p1 7 7 6 6 0 5; f ðxÞ ¼ 4 x1 x3 5 A ¼ 4 p2 1 0 0 p3 x1 x2 We will use the common values of the parameters p1 = 10, p2 = 28 and p3 = 8/3 for simulation study.
ð38Þ
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The absorbing ball is described by [20]: B ¼ fx21 þ x22 þ ðx3 p1 p2 Þ2 6 t2 ðp1 þ p2 Þ2 g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ p3 maxðp1 t¼ 1 ; 1Þ 2 In order to compute 0 0 f ðxÞ ¼ x3 x2
the Lipschitz constant L(B) restricted on B we get: 0 0 0 x1 x1 0
ð39Þ
ð40Þ
Therefore LðBÞ2 6 kf 0 ðxÞk2B 6 ð2x21 þ x22 þ x23 Þjx2B
ð41Þ
Since x 2 B we have jx3 j 6 p1 þ p2 þ tðp1 þ p2 Þ ¼ ðt þ 1Þðp1 þ p2 Þ jx2 j 6 tðp1 þ p2 Þ jx1 j 6 tðp1 þ p2 Þ
ð42Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LðBÞ2 6 ðp1 þ p2 Þ 4t2 þ 2t þ 1
ð43Þ
By substituting the numerical values we have ð44Þ
LðBÞ 6 97:49 The S matrix is given by: 2 2ðp1 þ k 1 Þ p1 þ p2 k 2 6 S ¼ 4 p1 þ p2 k 2 2 k 3 0
k 3
3
7 0 5 2p3
ð45Þ
The eigenvalues of matrix S become negative by proper choice of the matrix K. For simplicity we set k3 = 0 and choose k1 and k2 such that: k 1 > p1 ð46Þ k 22 þ 2k 2 ðp1 þ p2 Þ ðp1 þ p2 Þ2 þ 4ðp1 þ k 1 Þ > 0 Under the above conditions S is a stable matrix. By choosing K as given below [21]: 2 3 10 6 7 K ¼ 4 28 5
ð47Þ
0 the eigenvalues of S become 2, 5.333 and 20 and we have kmin ðSÞ ¼ 2
ð48Þ
As can be seen the inequality (37) is not met and therefore we introduce a rescaling parameter [21] such that: 0
kmin ðSÞ 2 ¼ 2L 194:98
Applying the rescaling we have 8 > < w_ 1 ¼ p1 ðw2 w1 Þ w_ 2 ¼ p2 w1 w2 lw1 w3 > : w_ 3 ¼ lw1 w2 p3 w3 where x = lw. Equations for the observer are
ð49Þ
ð50Þ
S.H. Mahboobi et al. / Chaos, Solitons and Fractals 29 (2006) 381–392
8 > ^_ 1 ¼ ðp1 þ k 1 Þ^ ^ 2 þ k 1 w1 w1 þ p 2 w
387
ð51Þ
We can design the observer for the Chen system using a similar approach and finally we get 8 > < w_ 1 ¼ p1 ðw2 w1 Þ w_ 2 ¼ ðp2 p1 Þw1 þ p2 w2 lw1 w3 > : w_ 3 ¼ lw1 w2 p3 w3 where x = lw. Equations for the observer are 8 > ^_ 1 ¼ ðp1 þ k 1 Þ^ ^ 2 þ k 1 w1 w1 þ p 2 w
ð52Þ
ð53Þ
Using the state estimates, the control actions for Lorenz and Chen systems become u ¼ h2 x21 þ h3 x1^x2 x1^x2 þ p3^x3 l^x3 þ lnp2 u¼
h2 x21
ð54Þ
þ h3 x1^x2 x1^x2 þ p3^x3 l^x3 þ lw
ð55Þ
6. Simulation results In this section, simulations are conducted to demonstrate the effectiveness of the proposed controllers coupled with the designed observers. In all simulations for Lorenz and Chen systems, we have assumed initial conditions x1 = 10, x2 = 20, x3 = 30. The parameters of the Chen system are set to p1 = 35, p2 = 28, p3 = 3. Fig. 1 shows the capability of the control law (54), in regulation of the Lorenz system. The controller parameters are l¼2 n¼2 The initial estimates of the states are set to zero. Fig. 2 depicts the effectiveness of the observer in estimation of the Lorenz system states. Fig. 3 shows the control effort required to regulate the Lorenz system states. Fig. 4 shows the capability of the control law (55), in regulation of the Chen system. The initial state estimates and the controller parameters are set to the following values:
60 50
x1 x2 x3
System States
40 30 20 10 0 -10
0
0.5
1
1.5
2
Time (sec)
Fig. 1. States of the controlled Lorenz system.
2.5
S.H. Mahboobi et al. / Chaos, Solitons and Fractals 29 (2006) 381–392 40 e1 e2 e3
State Estimation Errors
30 20 10 0 -10 -20 -30 0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 2. State estimation errors of the Lorenz system.
200 150
Control Action
100 50 0 -50 -100 -150 -200 -250 -300
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 3. Control action for the Lorenz system.
40
x1 x2 x3
30
System States
388
20
10
0
-10
-20
0
0.5
1
1.5
2
Time (sec)
Fig. 4. States of the controlled Chen system.
2.5
S.H. Mahboobi et al. / Chaos, Solitons and Fractals 29 (2006) 381–392 40 e1 e2 e3
State Estimation Errors
30 20 10 0 -10 -20 -30 -40 -50
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 5. State estimation errors of the Chen system.
150 100
Control Action
50 0 -50 -100 -150 -200 -250 -300
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 6. Control action for the Chen system.
60 x1 x2 x3
50
System States
40 30 20 10 0 -10 -20
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 7. System responses of the controlled Lorenz system in the presence of +20% parameter uncertainties.
389
S.H. Mahboobi et al. / Chaos, Solitons and Fractals 29 (2006) 381–392 40 x1 x2 x3
35
System States
30 25 20 15 10 5 0 -5
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 8. System responses of the controlled Chen system in the presence of +20% parameter uncertainties.
30 x1 x2 x3
20
System States
10 0 -10 -20 -30 -40 -50 -60 -70
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 9. State variations vs. time for the controlled Lorenz system using the Hua controller.
30 e1 e2 e3
25
State Estimation Errors
390
20 15 10 5 0 -5 -10 -15 -20
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 10. State estimation error for the Lorenz system using the Hua controller.
S.H. Mahboobi et al. / Chaos, Solitons and Fractals 29 (2006) 381–392
^x1 ¼ 0; w ¼ 32 l¼3
^x2 ¼ 0;
391
^x3 ¼ 0
Fig. 5 depicts the effectiveness of the observer performance. Fig. 6 shows the control effort required to regulate the system states. In order to exhibit the robustness of the proposed control schemes, for Lorenz and Chen systems, we have considered conditions in which the system parameters can change from their nominal values. The parameter uncertainties are assumed to be 20%. Figs. 7 and 8 show the closed loop responses in the presence of parameter uncertainties, for Lorenz and Chen systems respectively. To demonstrate the efficiency of the designed controller, we have compared its performance with a recent controller proposed in literature. Hua et al. [22] designed a controller with two manipulated variables applied to the second and third equations to overcome the problem of chaos control for Lorenz and Chen systems. Their scheme requires two system states for estimating the third state. Figs. 9–12 show the results of the Huas controller applied to Lorenz and Chen systems. As can be seen the controller has almost the same performance as the proposed controller at the expense of one additional control input and one
100 x1 x2 x3
80
System States
60 40 20 0 -20 -40 -60 -80
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 11. State variations vs. time for the controlled Chen system using the Hua controller.
30 e1 e2 e3
State Estimation Errors
20
10
0
-10
-20
-30
0
0.5
1
1.5
2
2.5
Time (sec)
Fig. 12. State estimation errors for the Chen system using the Hua controller.
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extra measurement. It should be mentioned that the Huas controller regulates all of the states, but as mentioned earlier, this goal could be reached by applying the input to the second state equation.
7. Conclusions In this paper, a singularity-free design for controlling three well-known chaotic systems namely Lorenz, Chen and Lu has been proposed. It has been shown that the origin is globally asymptotically stable. To overcome the problem of inaccessibility of the system states, a nonlinear observer is coupled with the proposed controller. In order to demonstrate the effectiveness of the proposed approach, its performance has been compared through simulation with the performance of a recent control design proposed in the literature. The latter control strategy requires at least two states and two control efforts, while the proposed approach show the same performance by requiring only one state (system output) and one control input. Simulation results have been conducted to show the effectiveness and robustness of the proposed scheme.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
[22]
Fradkov AL, Pogromsky AYu. Introduction to control of oscillations and chaos. Singapore: World Scientific; 1998. Lorenz EN. Deterministic non-periodic flow. J Atmosph Sci 1963;20:130–41. Gallegos JA. Nonlinear regulation of a Lorenz system by feedback linearization techniques. Dyn Contr 1994;4:277–98. Lenz H, Obradovic D. Robust control of the chaotic Lorenz system. Int J Bifurc Chaos 1997;7:2847–54. Yu XH. Controlling Lorenz chaos. Int J Syst Sci 1996;27:355–9. Zeng Y, Singh SN. Adaptive control of chaos in Lorenz system. Dyn Contr 1997;7:143–54. Kanellakopoulos I, Kokotovic P, Morse A. Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans Automat Control 1991;36:1241–53. Krstic M, Kanellakopoulos I, Kokotovic P. Nonlinear and adaptive control design. New York: John Wiley; 1995. Ge SS, Wang C, Lee TH. Adaptive backstepping control of a class of chaotic systems. Int J Bifurc Chaos 2000;10:1149–56. Wang C, Ge SS. Adaptive backstepping control of uncertain Lorenz system. Int J Bifurc Chaos 2001;11:1115–9. Huijberts HJC. Nonlinear discrete-time synchronization via extended observers. Int J Bifurc Chaos 2001;11:1997–2001. Nijmeijer H. A dynamical control view on synchronization. Physica D 2001;154:219–28. Feki M. Observer-based exact synchronization of ideal and mismatched chaotic systems. Phys Lett A 2003;309:53–60. Santoboni G, Yu A, Nijmeijer H. An observer for phase synchronization of chaos. Phys Lett A 2001;291:265–73. Thau FE. Observing the states of nonlinear dynamic systems. Int J Control 1973;17:471–9. Crespo LG, Sun JQ. On the feedback linearization of the Lorenz system. J Vibr Control 2004;10:85–100. Chen G, Ueta T. Yet another chaotic attractor. Int J Bifurc Chaos 1999;9:1465–6. Lu J, Chen G, Zhang S. A new chaotic attractor coined. Int J Bifurc Chaos 2002:659–61. Starkov KE, Esquer M. Construction of observers for nonlinear systems based on the solution of Thau inequality. In: Proc. of the 14th ICSE conference, Coventry, UK, vol. 2, 2000. p. 508–12. Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–3. Starkov K, Escobar SH. Observer-based synchronization of coupled chaotic systems with applications to the Chua system, Sprott systems and the Lorenz system. In: International conference on control, virtual instrumentation, and digital systems CICINDI2002, Pachuca, Mexico, 2002. Hua C, Guan X, Li X, Shi P. Adaptive observer-based control for a class of chaotic systems. Chaos, Solitons & Fractals 2004;22:103–10.