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Chaos control of chaotic dynamical systems using backstepping design
Chaos, Solitons and Fractals 27 (2006) 537–548 www.elsevier.com/locate/chaos
Chaos control of chaotic dynamical systems using backstepping design M.T. Yassen
1
Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Accepted 31 March 2005
Abstract This work presents chaos control of chaotic dynamical systems by using backstepping design method. This technique is applied to achieve chaos control for each of the dynamical systems Lorenz, Chen and Lu¨ systems. Based on Lyapunov stability theory, control laws are derived. We used the same technique to enable stabilization of chaotic motion to a steady state as well as tracking of any desired trajectory to be achieved in a systematic way. Numerical simulations are shown to verify the results. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Dynamic chaos is a very interesting nonlinear effect which has been intensively studied during the last two decades. The effect is very common, it has been detected in a large number of dynamic systems of various physical nature. However, this effect is usually undesirable in practice, and it restricts the operating range of many electronic and mechanic devices. Recently, controlling this kind of complex dynamical systems has attracted a great deal of attention within the engineering society. Chaos control, in a broader sense, can be divided into two categories [1]: one is to suppress the chaotic dynamical behavior and the other is to generated or enhance chaos in nonlinear systems (known as chaotincation or anti-control of chaos [2,3]). Nowadays, different techniques and methods have been proposed to achieve chaos control. For instance, OGY method [4], differential geometric method [5], feedback and nonfeedback control [6–9], inverse optimal control [10], adaptive control [11,12] and backstepping design technique [13]. In 1963, Lorenz [14] found the first canonical chaotic attractor, which has just been mathematically confirmed to exist [15]. In 1999, Chen [2] found another similar but topologically not equivalent chaotic attractor, as the dual of the Lorenz system, in a sense defined ˆ elikovskyˆ [16]: the Lorenz system satisfies the condition a12a21 > 0 while Chen system satisfies by Vaneˆcˆek and C a12a21 < 0, where a12, a21 are the corresponding elements in the constant matrix A = (aij)3·3 for the linear part of the system. Very recently, Lu¨ and Chen [17–19] found a new chaotic system, bearing the name of the Lu¨ system, which satisfies the condition a12a21 = 0, thereby bridging the gap between the Lorenz and Chen attractors [18,19]. In this work, chaos in Lorenz, Chen and Lu¨ systems is controlled by using backstepping design method. At the same time we used the same method to enable stabilization of chaotic motion to a steady state as well as tracking of any
1
E-mail address: [email protected] Present address: Mathematics Department, Faculty of Science, King Khalid University, Abha, Saudi Arabia.
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.03.046
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desired trajectory to be achieved in a systematic way. Computer simulation is also given for the purpose of illustration and verification.
2. Controlling Lorenz system The Lorenz system is described by 8 x_ ¼ aðy xÞ; > > < y_ ¼ cx xz y; > > : z_ ¼ xy bz;
ð1Þ
which has a chaotic attractor when a = 10, b = 8/3, c = 28. We will use backstepping method to design a controller. In order to control Lorenz system we add a control input u1 to the third equation of system (1). Then the controlled Lorenz system is 8 x_ ¼ aðy xÞ; > > < ð2Þ y_ ¼ cx xz y; > > : z_ ¼ xy bz þ u1 . Our objective is to find a control law u1 for stabilizing the state of the controlled system (2) at a bounded point. Starting from the first equation, a stabilizing function a1(x) has to be designed for the virtual control y in order to 2 make the derivative of V 1 ðxÞ ¼ x2 , i.e., V_ 1 ðxÞ ¼ ax2 þ axy be negative definite. Assume that a1(x) = px and define an error variable y ¼ y a1 ðxÞ
ð3Þ
Then we obtained the ðx; y Þ-subsystem ( x_ ¼ ay að1 pÞx; y_ ¼ cx xz y px apy þ apð1 pÞx.
ð4Þ
We can construct a Lyapunov function as follows: 1 V 2 ðx; y Þ ¼ V 1 ðxÞ þ y 2 . 2 Calculating the time derivative of V 2 ðx; y Þ along system (4), we have V_ 2 ¼ að1 pÞx2 ð1 þ apÞy 2 xy ½z a c þ p apð1 pÞ. We can choose z ¼ a2 ðx; y Þ ¼ a þ c p þ apð1 pÞ. Apparently, V_ 2 is negative definite if
1 a
< p < 1. Similarly, let
z ¼ z a2 ðx; y Þ; then we get the following system in the ðx; y ; zÞ coordinates 8 x_ ¼ ay að1 pÞx; > > < y_ ¼ cx xz y px apy þ apð1 pÞx; > > :_ z ¼ xy þ px2 bz b½a þ c p þ apð1 pÞ þ u1 . We can construct a Lyapunov function as follows: 1 V 3 ðx; y ; zÞ ¼ V 2 ðx; y Þ þ z2 . 2 Calculating the time derivative of V 3 ðx; y ; zÞ along system (6), we have V_ 3 ¼ að1 pÞx2 ð1 þ apÞy 2 bz2 þ z½px2 bða þ c p þ apð1 pÞÞ þ u1
ð5Þ
ð6Þ
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It is clear that V_ 3 becomes negative definite by choosing the control input u1 as follows: u1 ¼ bða þ c p þ apð1 pÞÞ px2
ð7Þ
Therefore we have proved that system (6) has been stabilized at the origin point (0, 0, 0). According to (3) and (5) the system (2) has been stabilized at the point (0, 0, a2). In order to control Lorenz system to the origin point (0, 0, 0), we add a control input u2 to the second equation of system (1). Thus the controlled system becomes: 8 x_ ¼ aðy xÞ; > > < y_ ¼ cx xz y þ u2 ; ð8Þ > > : z_ ¼ xy bz; For the virtual control y we design a stabilizing function a1(x) to make the derivative of V1(x) = x2/2, i.e. V_ 1 ¼ ax2 þ axy be negative definite as y = a1(x). We can choose a1(x) = 0 and define an error variable y ¼ y a1 ðxÞ
ð9Þ
then we can obtain the following ðx; y Þ-subsystem: ( x_ ¼ ax þ ay y_ ¼ cx xz y þ u2
ð10Þ
We can construct a Lyapunov function as follows: 1 V 2 ðx; y Þ ¼ V 1 ðxÞ þ y 2 . 2 Calculating the time derivative of V 2 ðx; y Þ along system (10), we have V_ 2 ¼ ax2 y 2 þ y ðax þ cx xz þ u2 Þ
ð11Þ
In order to make (11) be negative definite, choose u2 ¼ xðz ða þ cÞÞ
ð12Þ
Therefore we have proved that in the ðx; y Þ coordinates the equilibrium (0, 0) of the subsystem (10) is asymptotically stable. According to (9), a1(x) = 0, x ! 0, y ! 0, and the third equation of system (8), we get that (x, y, z) in controlled system (8) tends to (0, 0, 0) as t ! 1 when we choose the control input u2 = x(z (a + c)).
3. Tracking any desired trajectory In this section, we will find a control law u2 so that a scaler output x(t) of Lorenz system can track any desired trajectory r(t). Let x be the deviation between the output x and the desired trajectory r(t), i.e. x ¼ x rðtÞ. Define a func2 tion U 1 ¼ x2 and calculate its time derivative along the controlled system (8). U_ 1 ¼ xx_ ¼ ðx rÞ½aðy xÞ r_ becomes negative definite by choosing the virtual control y as y ¼xþ
r_ þ r x a 2
Þ, then Let U 2 ¼ U 1 þ y2 where y ¼ y ðx þ r_ þrx a r_ þ r x aða 1Þðy xÞ þ €r þ r_ U_ 2 ¼ U_ 1 þ y y_ ¼ ðx rÞ2 þ y x cx y xz þ u2 a a becomes negative definite by choosing the control law u2 as follows: u2 ¼ xz þ ða 1Þy þ xð2 c aÞ þ
€r þ 2_r þ r x a
ð13Þ
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4. Numerical simulations In this section, numerical simulations are carried out using QBASIC. Fourth-order Runge–Kutta integration method is used to solve two systems of differential Eqs. (2) and (8). In addition, a time step size 0.01 is employed. We select the parameters of Lorenz system as a = 10, b = 8/3, c = 28 so that the Lorenz system exhibits a chaotic behavior. The initial states of the controlled Lorenz systems (2) and (8) are x0(0) = 10, y0(0) = 10, z0(0) = 10. Fig. 1 shows that the Lorenz system can be stabilized with the control law u1 (7) where p = 0.1 to the bounded point (0, 0, a2). Fig. 2 shows that the Lorenz system can be stabilized with the control law u2 (12) to the origin point (0, 0, 0). Fig. 3 shows that the scaler output x(t) can track the desired trajectory r(t) = sin(t) with the control input u2 (13). In all simulations the control input is activated at time t = 20.
Fig. 1. The time response of the states x, y, z for the controlled Lorenz system (2) where the control input is (7) and p = 0.1. The control is activated at time t = 20.
Fig. 2. The time response of the states x, y, z for the controlled Lorenz system (8) where the control input is (12). The control is activated at time t = 20.
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Fig. 3. Output x(t) of Lorenz system tracks the trajectory r(t) = sin(t) with the control (13). The control is activated at time t = 20.
5. Controlling Chen system The Chen system is described by 8 > < x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy; > : z_ ¼ xy bz;
ð14Þ
which has a chaotic attractor when a = 35, b = 3, c = 28. We will use backstepping method to design a controller. In order to control Chen system we add a control input u to the third equation of system (14). Then the controlled Chen system is 8 > < x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy; ð15Þ > : z_ ¼ xy bz þ u. Our objective is to find a control law u for stabilizing the state of the controlled system (15) at a bounded point. Starting from the first equation, a stabilizing function b1(x) has to be designed for the virtual control y in order to 2 make the derivative of V 1 ðxÞ ¼ x2 , i.e., V_ 1 ðxÞ ¼ ax2 þ axy be negative definite. Assume that b1(x) = px and define an error variable y ¼ y b1 ðxÞ Then we obtained the ðx; y Þ-subsystem x_ ¼ ay að1 pÞx; y_ ¼ xz þ ðc a þ cp þ pð1 pÞÞx ðap cÞy . We can construct a Lyapunov function as follows: 1 V 2 ðx; y Þ ¼ V 1 ðxÞ þ y 2 . 2 Calculating the time derivative of V 2 ðx; y Þ along system (17), we have V_ 2 ¼ að1 pÞx2 ðap cÞy 2 xy ½z c cp pð1 pÞ.
ð16Þ
ð17Þ
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We can choose z ¼ b2 ðx; y Þ ¼ c þ cp þ pð1 pÞ Apparently, V_ 2 is negative definite if ac < p < 1. Similarly, let z ¼ z b2 ðx; y Þ; then we get the following system in the ðx; y ; zÞ coordinates: 8 x_ ¼ ay að1 pÞx > < y_ ¼ xz þ ðc a þ cp þ pð1 pÞÞx ðap cÞy > : z_ ¼ xy þ px2 bz b½c þ cp þ pð1 pÞ þ u
ð18Þ
ð19Þ
We can construct a Lyapunov function as follows: 1 V 3 ðx; y ; zÞ ¼ V 2 ðx; y Þ þ z2 . 2 Calculating the time derivative of V 3 ðx; y ; zÞ along system (19), we have V_ 3 ¼ að1 pÞx2 ð1 þ apÞy 2 bz2 þ z½px2 bðc þ cp þ pð1 pÞÞ þ u It is clear that V_ 3 becomes negative definite by choosing the control input u as follows: u ¼ bðc þ cp þ pð1 pÞÞ px2
ð20Þ
Therefore we have proved that system (19) has been stabilized at the origin point (0, 0, 0). According to (16) and (18) the system (15) has been stabilized at the point (0, 0, b2). In order to control Chen system to the origin point (0, 0, 0), we add a control input b to the second equation of system (14). Thus the controlled system becomes: 8 x_ ¼ aðy xÞ; > > < y_ ¼ ðc aÞx xz þ cy þ b; ð21Þ > > : z_ ¼ xy bz; For the virtual control y we design a stabilizing function a1(x) to make the derivative of V1(x) = x2/2, i.e. V_ 1 ¼ ax2 þ axy be negative definite as y = a1(x). We can choose a1(x) = 0 and define an error variable y ¼ y a1 ðxÞ then we can obtain the following ðx; y Þ-subsystem: ( x_ ¼ ax þ ay y_ ¼ ðc aÞx xz þ cy þ b
ð22Þ
ð23Þ
We can construct a Lyapunov function as follows: 1 V 2 ðx; y Þ ¼ V 1 ðxÞ þ y 2 . 2 Calculating the time derivative of V 2 ðx; y Þ along system (23), we have V_ 2 ¼ ax2 þ y ðcx xz þ cy þ bÞ
ð24Þ
In order to make (24) be negative definite, choose b ¼ cx ð1 þ cÞy þ xz
ð25Þ
Therefore we have proved that in the ðx; y Þ coordinates the equilibrium (0, 0) of the subsystem (23) is asymptotically stable. According to (22), a1(x) = 0, x ! 0, y ! 0, and the third equation of system (21), we get that (x,y,z) in controlled system (21) tends to (0, 0, 0) as t ! 1 when we choose the control input b ¼ cx ð1 þ cÞy þ xz.
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6. Tracking any desired trajectory In this section, we will find a control law b so that a scaler output x(t) of Chen system can track any desired trajectory r(t). Let x be the deviation between the output x and the desired trajectory r(t), i.e. x ¼ x rðtÞ. Define a function 2 U 1 ¼ x2 and calculate its time derivative along the controlled system (21). U_ 1 ¼ xx_ ¼ ðx rÞ½aðy xÞ r_ becomes negative definite by choosing the virtual control y as y¼
a1 r þ r_ xþ a a 2
r Let U 2 ¼ U 1 þ y2 where y ¼ y ða1 x þ rþ_ Þ, then a a a1 r þ r_ r_ þ €r U_ 2 ¼ U_ 1 þ y y_ ¼ ðx rÞ2 þ y xþ ðc aÞx xz þ cy þ b ða 1Þðy xÞ a a a
becomes negative definite by choosing the control law b as follows: a1 r þ 2_r þ €r xþ . b ¼ xz ð2 a þ cÞy c 1 a a
ð26Þ
7. Numerical simulations In this section, numerical simulations are carried out using QBASIC. Fourth order Runge–Kutta integration method is used to solve two systems of differential Eqs. (15) and (21). In addition, a time step size 0.001 is employed. We select the parameters of Chen system as a = 35, b = 3, c = 28 so that the Chen system exhibits a chaotic behavior. The initial states of the controlled Chen systems (15) and (21) are x0(0) = 10, y0(0) = 10, z0(0) = 10. Fig. 4 shows that the Chen system can be stabilized with the control law u (20) where p = 0.9 to the bounded point (0, 0, b2). Fig. 5 shows that the Chen system can be stabilized with the control law b (25) to the origin point (0, 0, 0). Fig. 6 shows that the scaler output x(t) can track the desired trajectory r(t) = 5sin(t) with the control input b (26). In all simulations the control input is activated at time t = 10.
Fig. 4. The time response of the states x, y, z for the controlled Chen system (15) where the control input is (20) and p = 0.9. The control is activated at time t = 10.
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Fig. 5. The time response of the states x, y, z for the controlled Chen system (21) where the control input is (25). The control is activated at time t = 10.
Fig. 6. Output x(t) of Chen system tracks the trajectory r(t) = 5 sin(t) with the control (26). The control is activated at time t = 10.
8. Controlling Lu¨ system The Lu¨ system is described by 8 x_ ¼ aðy xÞ; > > < y_ ¼ xz þ cy; > > : z_ ¼ xy bz;
ð27Þ
which has a chaotic attractor when a = 36, b = 3, c = 20. We will use backstepping method to design a controller. In order to control Lu¨ system we add a control input d to the third equation of system (27). Then the controlled Lu¨ system is
M.T. Yassen / Chaos, Solitons and Fractals 27 (2006) 537–548
8 x_ ¼ aðy xÞ; > > < y_ ¼ xz þ cy; > > : z_ ¼ xy bz þ d.
545
ð28Þ
Our objective is to find a control law d for stabilizing the state of the controlled system (28) at a bounded point. Starting from the first equation, a stabilizing function c1(x) has to be designed for the virtual control y in order to 2 make the derivative of V 1 ðxÞ ¼ x2 , i.e., V_ 1 ðxÞ ¼ ax2 þ axy be negative definite. Assume that c1(x) = px and define an error variable y ¼ y c1 ðxÞ. Then we obtained the ðx; y Þ-subsystem ( x_ ¼ ay að1 pÞx; y_ ¼ xz ðap cÞy þ pxða ap þ cÞ.
ð29Þ
ð30Þ
We can construct a Lyapunov function as follows: 1 V 2 ðx; y Þ ¼ V 1 ðxÞ þ y 2 . 2 Calculating the time derivative of V 2 ðx; y Þ along system (30), we have V_ 2 ¼ að1 pÞx2 ðap cÞy 2 xy ½z pa þ ap2 cp a. We can choose z ¼ c2 ðx; y Þ ¼ apð1 pÞ þ cp þ a. Apparently, V_ 2 is negative definite if ac < p < 1. Similarly, let z ¼ z c2 ðx; y Þ; then we get the following system in the ðx; y ; zÞ coordinates: 8 x_ ¼ ay að1 pÞx; > > < y_ ¼ xz ðap cÞy þ pxða ap þ cÞ; > > : z_ ¼ xy þ px2 bz b½a þ cp þ apð1 pÞ þ d.
ð31Þ
ð32Þ
We can construct a Lyapunov function as follows: 1 V 3 ðx; y ; zÞ ¼ V 2 ðx; y Þ þ z2 . 2 Calculating the time derivative of V 3 ðx; y ; zÞ along system (32), we have V_ 3 ¼ að1 pÞx2 ðap cÞy 2 bz2 þ z½px2 abpð1 pÞ bcp ba þ d. It is clear that V_ 3 becomes negative definite by choosing the control input d as follows: d ¼ bða þ cp þ apð1 pÞÞ px2 .
ð33Þ
Therefore we have proved that system (32) has been stabilized at the origin point (0, 0, 0). According to (29) and (31) the system (28) has been stabilized at the point (0, 0, c2). In order to control Lu¨ system to the origin point (0, 0, 0), we add a control input d1 to the second equation of system (27). Thus the controlled system becomes: 8 x_ ¼ aðy xÞ; > > < y_ ¼ xz þ cy þ d1 ; ð34Þ > > : z_ ¼ xy bz; For the virtual control y we design a stabilizing function a1(x) to make the derivative of V1(x) = x2/2, i.e. V_ 1 ¼ ax2 þ axy be negative definite as y = a1(x). We can choose a1(x) = 0 and define an error variable y ¼ y a1 ðxÞ
ð35Þ
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then we can obtain the following ðx; y Þ-subsystems: x_ ¼ ax þ ay ; y_ ¼ xz þ cy þ d1 .
ð36Þ
We can construct a Lyapunov function as follows: 1 V 2 ðx; y Þ ¼ V 1 ðxÞ þ y 2 . 2 Calculating the time derivative of V 2 ðx; y Þ along system (36), we have V_ 2 ¼ ax2 þ y ðax xz þ cy þ d1 Þ.
ð37Þ
In order to make (37) be negative definite, choose d1 ¼ ð1 þ cÞy þ xðz aÞ.
ð38Þ
Therefore we have proved that in the ðx; y Þ coordinates the equilibrium (0, 0) of the subsystem (36) is asymptotically stable. According to (35), a1(x) = 0, x ! 0, y ! 0, and the third equation of system (34), we get that (x,y,z) in controlled system (34) tends to (0, 0, 0) as t ! 1 when we choose the control input d1 ¼ ð1 þ cÞy þ xðz aÞ.
9. Tracking any desired trajectory In this section, we will find a control law d1 so that a scaler output x(t) of Lu¨ system can track any desired trajectory r(t). Let x be the deviation between the output x and the desired trajectory r(t), i.e. x ¼ x rðtÞ. Define a function 2 U 1 ¼ x2 and calculate its time derivative along the controlled system (34). U_ 1 ¼ xx_ ¼ ðx rÞ½aðy xÞ r_ becomes negative definite by choosing the virtual control y as y¼
a1 r þ r_ xþ . a a 2
r Let U 2 ¼ U 1 þ y2 where y ¼ y ða1 x þ rþ_ Þ, then a a r_ þ €r a1 r þ r_ xþ xz þ cy þ d1 ða 1Þðy xÞ U_ 2 ¼ U_ 1 þ y y_ ¼ ðx rÞ2 þ y a a a
Fig. 7. The time response of the states x, y, z for the controlled Lu¨ system (28) where the control input is (33) and p = 0.8. The control is activated at time t = 10.
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becomes negative definite by choosing the control law d1 as follows: d1 ¼ xz ð2 a þ cÞy
ða 1Þ2 r þ 2_r þ €r xþ . a a
ð39Þ
10. Numerical simulations In this section, numerical simulations are carried out using QBASIC. Fourth order Runge–Kutta integration method is used to solve two systems of differential Eqs. (28) and (34). In addition, a time step size 0.001 is employed. We select the parameters of Lu¨ system as a = 36, b = 3, c = 20 so that Lu¨ system exhibits a chaotic behavior. The initial states of the controlled Lu¨ systems (28) and (34) are x0(0) = 10, y0(0) = 10, z0(0) = 10. Fig. 7 shows that the Lu¨ system can be stabilized with the control law d (33) where p = 0.8 to the bounded point (0, 0, c2). Fig. 8 shows that the Lu¨
Fig. 8. The time response of the states x, y, z for the controlled Lu¨ system (34) where the control input is (38). The control is activated at time t = 10.
Fig. 9. Output x(t) of Lu¨ system tracks the trajectory rðtÞ ¼ 5 sinðtÞ with the control (39). The control is activated at time t = 10.
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system can be stabilized with the control law d1 (38) to the origin point (0, 0, 0). Fig. 9 shows that the scaler output x(t) can track the desired trajectory r(t) = 5sin(t) with the control input d1 (39). In all simulations the control input is activated at time t = 10.
11. Conclusion This work demonstrates that chaos control of chaotic dynamical systems using backstepping design is achieved. At the same time we used the same method to enable stabilization of chaotic motion to a steady state as well as tracking of any desired trajectory to be achieved in a systematic way. The chaotic motion of Lorenz, Chen and Lu¨ systems is stabilized to a bounded point and to the origin point. Numerical simulations are used to verify the effectiveness of the proposed chaos control technique.
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Chaos control of chaotic dynamical systems using backstepping design M.T. Yassen
1
Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Accepted 31 March 2005
Abstract This work presents chaos control of chaotic dynamical systems by using backstepping design method. This technique is applied to achieve chaos control for each of the dynamical systems Lorenz, Chen and Lu¨ systems. Based on Lyapunov stability theory, control laws are derived. We used the same technique to enable stabilization of chaotic motion to a steady state as well as tracking of any desired trajectory to be achieved in a systematic way. Numerical simulations are shown to verify the results. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Dynamic chaos is a very interesting nonlinear effect which has been intensively studied during the last two decades. The effect is very common, it has been detected in a large number of dynamic systems of various physical nature. However, this effect is usually undesirable in practice, and it restricts the operating range of many electronic and mechanic devices. Recently, controlling this kind of complex dynamical systems has attracted a great deal of attention within the engineering society. Chaos control, in a broader sense, can be divided into two categories [1]: one is to suppress the chaotic dynamical behavior and the other is to generated or enhance chaos in nonlinear systems (known as chaotincation or anti-control of chaos [2,3]). Nowadays, different techniques and methods have been proposed to achieve chaos control. For instance, OGY method [4], differential geometric method [5], feedback and nonfeedback control [6–9], inverse optimal control [10], adaptive control [11,12] and backstepping design technique [13]. In 1963, Lorenz [14] found the first canonical chaotic attractor, which has just been mathematically confirmed to exist [15]. In 1999, Chen [2] found another similar but topologically not equivalent chaotic attractor, as the dual of the Lorenz system, in a sense defined ˆ elikovskyˆ [16]: the Lorenz system satisfies the condition a12a21 > 0 while Chen system satisfies by Vaneˆcˆek and C a12a21 < 0, where a12, a21 are the corresponding elements in the constant matrix A = (aij)3·3 for the linear part of the system. Very recently, Lu¨ and Chen [17–19] found a new chaotic system, bearing the name of the Lu¨ system, which satisfies the condition a12a21 = 0, thereby bridging the gap between the Lorenz and Chen attractors [18,19]. In this work, chaos in Lorenz, Chen and Lu¨ systems is controlled by using backstepping design method. At the same time we used the same method to enable stabilization of chaotic motion to a steady state as well as tracking of any
1
E-mail address: [email protected] Present address: Mathematics Department, Faculty of Science, King Khalid University, Abha, Saudi Arabia.
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.03.046
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desired trajectory to be achieved in a systematic way. Computer simulation is also given for the purpose of illustration and verification.
2. Controlling Lorenz system The Lorenz system is described by 8 x_ ¼ aðy xÞ; > > < y_ ¼ cx xz y; > > : z_ ¼ xy bz;
ð1Þ
which has a chaotic attractor when a = 10, b = 8/3, c = 28. We will use backstepping method to design a controller. In order to control Lorenz system we add a control input u1 to the third equation of system (1). Then the controlled Lorenz system is 8 x_ ¼ aðy xÞ; > > < ð2Þ y_ ¼ cx xz y; > > : z_ ¼ xy bz þ u1 . Our objective is to find a control law u1 for stabilizing the state of the controlled system (2) at a bounded point. Starting from the first equation, a stabilizing function a1(x) has to be designed for the virtual control y in order to 2 make the derivative of V 1 ðxÞ ¼ x2 , i.e., V_ 1 ðxÞ ¼ ax2 þ axy be negative definite. Assume that a1(x) = px and define an error variable y ¼ y a1 ðxÞ
ð3Þ
Then we obtained the ðx; y Þ-subsystem ( x_ ¼ ay að1 pÞx; y_ ¼ cx xz y px apy þ apð1 pÞx.
ð4Þ
We can construct a Lyapunov function as follows: 1 V 2 ðx; y Þ ¼ V 1 ðxÞ þ y 2 . 2 Calculating the time derivative of V 2 ðx; y Þ along system (4), we have V_ 2 ¼ að1 pÞx2 ð1 þ apÞy 2 xy ½z a c þ p apð1 pÞ. We can choose z ¼ a2 ðx; y Þ ¼ a þ c p þ apð1 pÞ. Apparently, V_ 2 is negative definite if
1 a
< p < 1. Similarly, let
z ¼ z a2 ðx; y Þ; then we get the following system in the ðx; y ; zÞ coordinates 8 x_ ¼ ay að1 pÞx; > > < y_ ¼ cx xz y px apy þ apð1 pÞx; > > :_ z ¼ xy þ px2 bz b½a þ c p þ apð1 pÞ þ u1 . We can construct a Lyapunov function as follows: 1 V 3 ðx; y ; zÞ ¼ V 2 ðx; y Þ þ z2 . 2 Calculating the time derivative of V 3 ðx; y ; zÞ along system (6), we have V_ 3 ¼ að1 pÞx2 ð1 þ apÞy 2 bz2 þ z½px2 bða þ c p þ apð1 pÞÞ þ u1
ð5Þ
ð6Þ
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It is clear that V_ 3 becomes negative definite by choosing the control input u1 as follows: u1 ¼ bða þ c p þ apð1 pÞÞ px2
ð7Þ
Therefore we have proved that system (6) has been stabilized at the origin point (0, 0, 0). According to (3) and (5) the system (2) has been stabilized at the point (0, 0, a2). In order to control Lorenz system to the origin point (0, 0, 0), we add a control input u2 to the second equation of system (1). Thus the controlled system becomes: 8 x_ ¼ aðy xÞ; > > < y_ ¼ cx xz y þ u2 ; ð8Þ > > : z_ ¼ xy bz; For the virtual control y we design a stabilizing function a1(x) to make the derivative of V1(x) = x2/2, i.e. V_ 1 ¼ ax2 þ axy be negative definite as y = a1(x). We can choose a1(x) = 0 and define an error variable y ¼ y a1 ðxÞ
ð9Þ
then we can obtain the following ðx; y Þ-subsystem: ( x_ ¼ ax þ ay y_ ¼ cx xz y þ u2
ð10Þ
We can construct a Lyapunov function as follows: 1 V 2 ðx; y Þ ¼ V 1 ðxÞ þ y 2 . 2 Calculating the time derivative of V 2 ðx; y Þ along system (10), we have V_ 2 ¼ ax2 y 2 þ y ðax þ cx xz þ u2 Þ
ð11Þ
In order to make (11) be negative definite, choose u2 ¼ xðz ða þ cÞÞ
ð12Þ
Therefore we have proved that in the ðx; y Þ coordinates the equilibrium (0, 0) of the subsystem (10) is asymptotically stable. According to (9), a1(x) = 0, x ! 0, y ! 0, and the third equation of system (8), we get that (x, y, z) in controlled system (8) tends to (0, 0, 0) as t ! 1 when we choose the control input u2 = x(z (a + c)).
3. Tracking any desired trajectory In this section, we will find a control law u2 so that a scaler output x(t) of Lorenz system can track any desired trajectory r(t). Let x be the deviation between the output x and the desired trajectory r(t), i.e. x ¼ x rðtÞ. Define a func2 tion U 1 ¼ x2 and calculate its time derivative along the controlled system (8). U_ 1 ¼ xx_ ¼ ðx rÞ½aðy xÞ r_ becomes negative definite by choosing the virtual control y as y ¼xþ
r_ þ r x a 2
Þ, then Let U 2 ¼ U 1 þ y2 where y ¼ y ðx þ r_ þrx a r_ þ r x aða 1Þðy xÞ þ €r þ r_ U_ 2 ¼ U_ 1 þ y y_ ¼ ðx rÞ2 þ y x cx y xz þ u2 a a becomes negative definite by choosing the control law u2 as follows: u2 ¼ xz þ ða 1Þy þ xð2 c aÞ þ
€r þ 2_r þ r x a
ð13Þ
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4. Numerical simulations In this section, numerical simulations are carried out using QBASIC. Fourth-order Runge–Kutta integration method is used to solve two systems of differential Eqs. (2) and (8). In addition, a time step size 0.01 is employed. We select the parameters of Lorenz system as a = 10, b = 8/3, c = 28 so that the Lorenz system exhibits a chaotic behavior. The initial states of the controlled Lorenz systems (2) and (8) are x0(0) = 10, y0(0) = 10, z0(0) = 10. Fig. 1 shows that the Lorenz system can be stabilized with the control law u1 (7) where p = 0.1 to the bounded point (0, 0, a2). Fig. 2 shows that the Lorenz system can be stabilized with the control law u2 (12) to the origin point (0, 0, 0). Fig. 3 shows that the scaler output x(t) can track the desired trajectory r(t) = sin(t) with the control input u2 (13). In all simulations the control input is activated at time t = 20.
Fig. 1. The time response of the states x, y, z for the controlled Lorenz system (2) where the control input is (7) and p = 0.1. The control is activated at time t = 20.
Fig. 2. The time response of the states x, y, z for the controlled Lorenz system (8) where the control input is (12). The control is activated at time t = 20.
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Fig. 3. Output x(t) of Lorenz system tracks the trajectory r(t) = sin(t) with the control (13). The control is activated at time t = 20.
5. Controlling Chen system The Chen system is described by 8 > < x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy; > : z_ ¼ xy bz;
ð14Þ
which has a chaotic attractor when a = 35, b = 3, c = 28. We will use backstepping method to design a controller. In order to control Chen system we add a control input u to the third equation of system (14). Then the controlled Chen system is 8 > < x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy; ð15Þ > : z_ ¼ xy bz þ u. Our objective is to find a control law u for stabilizing the state of the controlled system (15) at a bounded point. Starting from the first equation, a stabilizing function b1(x) has to be designed for the virtual control y in order to 2 make the derivative of V 1 ðxÞ ¼ x2 , i.e., V_ 1 ðxÞ ¼ ax2 þ axy be negative definite. Assume that b1(x) = px and define an error variable y ¼ y b1 ðxÞ Then we obtained the ðx; y Þ-subsystem x_ ¼ ay að1 pÞx; y_ ¼ xz þ ðc a þ cp þ pð1 pÞÞx ðap cÞy . We can construct a Lyapunov function as follows: 1 V 2 ðx; y Þ ¼ V 1 ðxÞ þ y 2 . 2 Calculating the time derivative of V 2 ðx; y Þ along system (17), we have V_ 2 ¼ að1 pÞx2 ðap cÞy 2 xy ½z c cp pð1 pÞ.
ð16Þ
ð17Þ
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We can choose z ¼ b2 ðx; y Þ ¼ c þ cp þ pð1 pÞ Apparently, V_ 2 is negative definite if ac < p < 1. Similarly, let z ¼ z b2 ðx; y Þ; then we get the following system in the ðx; y ; zÞ coordinates: 8 x_ ¼ ay að1 pÞx > < y_ ¼ xz þ ðc a þ cp þ pð1 pÞÞx ðap cÞy > : z_ ¼ xy þ px2 bz b½c þ cp þ pð1 pÞ þ u
ð18Þ
ð19Þ
We can construct a Lyapunov function as follows: 1 V 3 ðx; y ; zÞ ¼ V 2 ðx; y Þ þ z2 . 2 Calculating the time derivative of V 3 ðx; y ; zÞ along system (19), we have V_ 3 ¼ að1 pÞx2 ð1 þ apÞy 2 bz2 þ z½px2 bðc þ cp þ pð1 pÞÞ þ u It is clear that V_ 3 becomes negative definite by choosing the control input u as follows: u ¼ bðc þ cp þ pð1 pÞÞ px2
ð20Þ
Therefore we have proved that system (19) has been stabilized at the origin point (0, 0, 0). According to (16) and (18) the system (15) has been stabilized at the point (0, 0, b2). In order to control Chen system to the origin point (0, 0, 0), we add a control input b to the second equation of system (14). Thus the controlled system becomes: 8 x_ ¼ aðy xÞ; > > < y_ ¼ ðc aÞx xz þ cy þ b; ð21Þ > > : z_ ¼ xy bz; For the virtual control y we design a stabilizing function a1(x) to make the derivative of V1(x) = x2/2, i.e. V_ 1 ¼ ax2 þ axy be negative definite as y = a1(x). We can choose a1(x) = 0 and define an error variable y ¼ y a1 ðxÞ then we can obtain the following ðx; y Þ-subsystem: ( x_ ¼ ax þ ay y_ ¼ ðc aÞx xz þ cy þ b
ð22Þ
ð23Þ
We can construct a Lyapunov function as follows: 1 V 2 ðx; y Þ ¼ V 1 ðxÞ þ y 2 . 2 Calculating the time derivative of V 2 ðx; y Þ along system (23), we have V_ 2 ¼ ax2 þ y ðcx xz þ cy þ bÞ
ð24Þ
In order to make (24) be negative definite, choose b ¼ cx ð1 þ cÞy þ xz
ð25Þ
Therefore we have proved that in the ðx; y Þ coordinates the equilibrium (0, 0) of the subsystem (23) is asymptotically stable. According to (22), a1(x) = 0, x ! 0, y ! 0, and the third equation of system (21), we get that (x,y,z) in controlled system (21) tends to (0, 0, 0) as t ! 1 when we choose the control input b ¼ cx ð1 þ cÞy þ xz.
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6. Tracking any desired trajectory In this section, we will find a control law b so that a scaler output x(t) of Chen system can track any desired trajectory r(t). Let x be the deviation between the output x and the desired trajectory r(t), i.e. x ¼ x rðtÞ. Define a function 2 U 1 ¼ x2 and calculate its time derivative along the controlled system (21). U_ 1 ¼ xx_ ¼ ðx rÞ½aðy xÞ r_ becomes negative definite by choosing the virtual control y as y¼
a1 r þ r_ xþ a a 2
r Let U 2 ¼ U 1 þ y2 where y ¼ y ða1 x þ rþ_ Þ, then a a a1 r þ r_ r_ þ €r U_ 2 ¼ U_ 1 þ y y_ ¼ ðx rÞ2 þ y xþ ðc aÞx xz þ cy þ b ða 1Þðy xÞ a a a
becomes negative definite by choosing the control law b as follows: a1 r þ 2_r þ €r xþ . b ¼ xz ð2 a þ cÞy c 1 a a
ð26Þ
7. Numerical simulations In this section, numerical simulations are carried out using QBASIC. Fourth order Runge–Kutta integration method is used to solve two systems of differential Eqs. (15) and (21). In addition, a time step size 0.001 is employed. We select the parameters of Chen system as a = 35, b = 3, c = 28 so that the Chen system exhibits a chaotic behavior. The initial states of the controlled Chen systems (15) and (21) are x0(0) = 10, y0(0) = 10, z0(0) = 10. Fig. 4 shows that the Chen system can be stabilized with the control law u (20) where p = 0.9 to the bounded point (0, 0, b2). Fig. 5 shows that the Chen system can be stabilized with the control law b (25) to the origin point (0, 0, 0). Fig. 6 shows that the scaler output x(t) can track the desired trajectory r(t) = 5sin(t) with the control input b (26). In all simulations the control input is activated at time t = 10.
Fig. 4. The time response of the states x, y, z for the controlled Chen system (15) where the control input is (20) and p = 0.9. The control is activated at time t = 10.
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Fig. 5. The time response of the states x, y, z for the controlled Chen system (21) where the control input is (25). The control is activated at time t = 10.
Fig. 6. Output x(t) of Chen system tracks the trajectory r(t) = 5 sin(t) with the control (26). The control is activated at time t = 10.
8. Controlling Lu¨ system The Lu¨ system is described by 8 x_ ¼ aðy xÞ; > > < y_ ¼ xz þ cy; > > : z_ ¼ xy bz;
ð27Þ
which has a chaotic attractor when a = 36, b = 3, c = 20. We will use backstepping method to design a controller. In order to control Lu¨ system we add a control input d to the third equation of system (27). Then the controlled Lu¨ system is
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8 x_ ¼ aðy xÞ; > > < y_ ¼ xz þ cy; > > : z_ ¼ xy bz þ d.
545
ð28Þ
Our objective is to find a control law d for stabilizing the state of the controlled system (28) at a bounded point. Starting from the first equation, a stabilizing function c1(x) has to be designed for the virtual control y in order to 2 make the derivative of V 1 ðxÞ ¼ x2 , i.e., V_ 1 ðxÞ ¼ ax2 þ axy be negative definite. Assume that c1(x) = px and define an error variable y ¼ y c1 ðxÞ. Then we obtained the ðx; y Þ-subsystem ( x_ ¼ ay að1 pÞx; y_ ¼ xz ðap cÞy þ pxða ap þ cÞ.
ð29Þ
ð30Þ
We can construct a Lyapunov function as follows: 1 V 2 ðx; y Þ ¼ V 1 ðxÞ þ y 2 . 2 Calculating the time derivative of V 2 ðx; y Þ along system (30), we have V_ 2 ¼ að1 pÞx2 ðap cÞy 2 xy ½z pa þ ap2 cp a. We can choose z ¼ c2 ðx; y Þ ¼ apð1 pÞ þ cp þ a. Apparently, V_ 2 is negative definite if ac < p < 1. Similarly, let z ¼ z c2 ðx; y Þ; then we get the following system in the ðx; y ; zÞ coordinates: 8 x_ ¼ ay að1 pÞx; > > < y_ ¼ xz ðap cÞy þ pxða ap þ cÞ; > > : z_ ¼ xy þ px2 bz b½a þ cp þ apð1 pÞ þ d.
ð31Þ
ð32Þ
We can construct a Lyapunov function as follows: 1 V 3 ðx; y ; zÞ ¼ V 2 ðx; y Þ þ z2 . 2 Calculating the time derivative of V 3 ðx; y ; zÞ along system (32), we have V_ 3 ¼ að1 pÞx2 ðap cÞy 2 bz2 þ z½px2 abpð1 pÞ bcp ba þ d. It is clear that V_ 3 becomes negative definite by choosing the control input d as follows: d ¼ bða þ cp þ apð1 pÞÞ px2 .
ð33Þ
Therefore we have proved that system (32) has been stabilized at the origin point (0, 0, 0). According to (29) and (31) the system (28) has been stabilized at the point (0, 0, c2). In order to control Lu¨ system to the origin point (0, 0, 0), we add a control input d1 to the second equation of system (27). Thus the controlled system becomes: 8 x_ ¼ aðy xÞ; > > < y_ ¼ xz þ cy þ d1 ; ð34Þ > > : z_ ¼ xy bz; For the virtual control y we design a stabilizing function a1(x) to make the derivative of V1(x) = x2/2, i.e. V_ 1 ¼ ax2 þ axy be negative definite as y = a1(x). We can choose a1(x) = 0 and define an error variable y ¼ y a1 ðxÞ
ð35Þ
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then we can obtain the following ðx; y Þ-subsystems: x_ ¼ ax þ ay ; y_ ¼ xz þ cy þ d1 .
ð36Þ
We can construct a Lyapunov function as follows: 1 V 2 ðx; y Þ ¼ V 1 ðxÞ þ y 2 . 2 Calculating the time derivative of V 2 ðx; y Þ along system (36), we have V_ 2 ¼ ax2 þ y ðax xz þ cy þ d1 Þ.
ð37Þ
In order to make (37) be negative definite, choose d1 ¼ ð1 þ cÞy þ xðz aÞ.
ð38Þ
Therefore we have proved that in the ðx; y Þ coordinates the equilibrium (0, 0) of the subsystem (36) is asymptotically stable. According to (35), a1(x) = 0, x ! 0, y ! 0, and the third equation of system (34), we get that (x,y,z) in controlled system (34) tends to (0, 0, 0) as t ! 1 when we choose the control input d1 ¼ ð1 þ cÞy þ xðz aÞ.
9. Tracking any desired trajectory In this section, we will find a control law d1 so that a scaler output x(t) of Lu¨ system can track any desired trajectory r(t). Let x be the deviation between the output x and the desired trajectory r(t), i.e. x ¼ x rðtÞ. Define a function 2 U 1 ¼ x2 and calculate its time derivative along the controlled system (34). U_ 1 ¼ xx_ ¼ ðx rÞ½aðy xÞ r_ becomes negative definite by choosing the virtual control y as y¼
a1 r þ r_ xþ . a a 2
r Let U 2 ¼ U 1 þ y2 where y ¼ y ða1 x þ rþ_ Þ, then a a r_ þ €r a1 r þ r_ xþ xz þ cy þ d1 ða 1Þðy xÞ U_ 2 ¼ U_ 1 þ y y_ ¼ ðx rÞ2 þ y a a a
Fig. 7. The time response of the states x, y, z for the controlled Lu¨ system (28) where the control input is (33) and p = 0.8. The control is activated at time t = 10.
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becomes negative definite by choosing the control law d1 as follows: d1 ¼ xz ð2 a þ cÞy
ða 1Þ2 r þ 2_r þ €r xþ . a a
ð39Þ
10. Numerical simulations In this section, numerical simulations are carried out using QBASIC. Fourth order Runge–Kutta integration method is used to solve two systems of differential Eqs. (28) and (34). In addition, a time step size 0.001 is employed. We select the parameters of Lu¨ system as a = 36, b = 3, c = 20 so that Lu¨ system exhibits a chaotic behavior. The initial states of the controlled Lu¨ systems (28) and (34) are x0(0) = 10, y0(0) = 10, z0(0) = 10. Fig. 7 shows that the Lu¨ system can be stabilized with the control law d (33) where p = 0.8 to the bounded point (0, 0, c2). Fig. 8 shows that the Lu¨
Fig. 8. The time response of the states x, y, z for the controlled Lu¨ system (34) where the control input is (38). The control is activated at time t = 10.
Fig. 9. Output x(t) of Lu¨ system tracks the trajectory rðtÞ ¼ 5 sinðtÞ with the control (39). The control is activated at time t = 10.
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system can be stabilized with the control law d1 (38) to the origin point (0, 0, 0). Fig. 9 shows that the scaler output x(t) can track the desired trajectory r(t) = 5sin(t) with the control input d1 (39). In all simulations the control input is activated at time t = 10.
11. Conclusion This work demonstrates that chaos control of chaotic dynamical systems using backstepping design is achieved. At the same time we used the same method to enable stabilization of chaotic motion to a steady state as well as tracking of any desired trajectory to be achieved in a systematic way. The chaotic motion of Lorenz, Chen and Lu¨ systems is stabilized to a bounded point and to the origin point. Numerical simulations are used to verify the effectiveness of the proposed chaos control technique.
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