Nonlinear Analysis: Real World Applications 11 (2010) 3336–3343
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Chaos synchronization of an energy resource system based on linear control Zuolei Wang ∗ School of Mathematical Sciences, Yancheng Teachers University, Yancheng, 224002, China
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Article history: Received 22 September 2009 Accepted 12 November 2009 Keywords: Synchronization Energy resource system Linear control Chaos
abstract Synchronization of an energy resource system is investigated. Three linear control schemes are proposed to synchronize a chaotic energy resource system via the backstepping method. This can be viewed as an improvement to the existing results of Tian et al. (2006) [14]. Because we use simpler controllers to realize a global asymptotical synchronization. In the first two schemes, the sufficient conditions for achieving synchronization of two identical energy resource systems using linear feedback control are derived by using Lyapunov stability theorem. In the third scheme, the synchronization condition is obtained by numerical method, in which only one state variable controller is contained. Finally, three numerical simulation examples are performed to verify these results. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Chaos is a very interesting nonlinear phenomenon, and chaos synchronization has been intensively studied since 1990 [1–4]. In the past decade, many techniques for chaos control and synchronization have been developed, such as the feedback method, adaptive technique, time delay feedback approach, and back-stepping method [5–8]. Furthermore, chaos synchronization has many potential applications in chemical reactors, secure communication, biomedics, and so on [9–13]. Recently, the problem of control and synchronization for a chaotic energy resource system has attracted increasing attention because of its potential importance in actual applications [14–18]. An energy resource system belongs to the class of complex nonlinear systems. With the continuous development of the economy, the issue of energy supply and demand has been paid more and more attention in recent years. So in 2005, a new dynamical system called an energy resource demand–supply system was proposed by Sun et al. [19]. It is a three-dimensional autonomous system exhibiting very complex dynamical behaviors. According to Shilnikov theory as well as detailed simulation, such an energy resource system exhibits small horseshoes as well the horseshoe chaos. Achievements have already been obtained from a chaotic energy resource system, for example, active synchronization [14], non-autonomous feedback control [15], adaptive control [16,17] and time-delay feedback control [18]. All the controllers derived from the above-mentioned methods are nonlinear. In a real industry process, because linear feedback controllers are economic and easy to implement, they possess a high value in applications. The linear feedback control technique has been used to suppress and synchronize chaos by various authors [20,21]. Due to the simplicity in configuration and implementation, the linear controller is especially attractive and it has been commonly adopted for practical implementations. We remark that we avoid the use of a nonlinear feedback control, which can cancel the nonlinear part of the error system. The design of the linear controller is achieved through an application of the optimal control and Lyapunov stability theories, which guarantee the global stability of the nonlinear error system.
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Z. Wang / Nonlinear Analysis: Real World Applications 11 (2010) 3336–3343
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Fig. 1. Views of the chaotic attractor of system (1): (a) (x, y); (b) (x, z); (c) (y, z).
In this paper, the synchronization of an energy resource system is addressed by employing a back-stepping procedure. Based on the stability theory of a cascade-connected system [22,23], we propose a linear control for chaos control and synchronization for unified chaotic systems. It seems that the controller is simple and can be viewed as an improvement to the existing results of [14]. The detailed arrangement of this paper is as follows. In Section 2, the dynamical system is introduced. In Section 3, linear controllers are used to synchronize two identical chaotic systems and numerical simulations are shown to verify the results. Section 4 draws some conclusions. 2. Systems description An energy resource system can be expressed by the following equations (see [19]): x˙ = a1 x(1 − x/M ) − a2 (y + z ), y˙ = −b1 y − b2 z + b3 x[N − (x − z )], z˙ = c1 z (c2 x − c3 ),
(1)
where x(t ) is the energy resource shortage in region A, y(t ) is the energy resource supply increment in region B, z (t ) is the energy resource import in region A; M , N , ai , bj , cj (i = 1, 2; j = 1, 2, 3) are parameters that are all positive real. When a1 = 0.09, a2 = 0.15, b1 = 0.06, b2 = 0.083, b3 = 0.07, c1 = 0.2, c2 = 0.5, c3 = 0.4, M = 1.8, N = 1.0, this system exhibits chaotic behavior (see Fig. 1). 3. Synchronization of energy system In order to obtain synchronization of the energy resource system (1), the drive system with subscript 1 is written as x˙ 1 = a1 x1 − a2 (y1 + z1 ) − a3 x21 , y˙ 1 = −b1 y1 − b2 z1 − b3 x1 (x1 − z1 ) + b4 x1 , z˙1 = d1 x1 z1 − d2 z1 ,
(2)
where a3 = a1 /M , b4 = b3 N , d1 = c1 c2 , d2 = c1 c3 . The controlled response system with subscript 2 can be expressed as x˙ 2 = a1 x2 − a2 (y2 + z2 ) − a3 x22 + u1 , y˙ 2 = −b1 y2 − b2 z2 − b3 x2 (x2 − z2 ) + b4 x2 + u2 , z˙2 = d1 x2 z2 − d2 z2 + u3 ,
(3)
where u1 , u2 , u3 are controllers. Let the error variables be e 1 = x2 − x1 ,
e2 = y2 − y1 ,
e3 = z2 − z1 .
(4)
The error system of (2) and (3) can be derived as e˙ 1 = a1 e1 − a2 (e2 + e3 ) + a3 x22 − a3 x21 + u1 , e˙ 2 = −b1 e2 − b2 e3 − b3 x2 (x2 − z2 ) + b3 x1 (x1 − z1 ) + b4 e1 + u2 , e˙ 3 = d1 x2 z2 − d1 x1 z1 − d2 e3 + u3 .
(5)
For two identical chaotic systems without controllers u1 , u2 and u3 , the trajectories of the two identical systems will quickly separate and can not synchronize on the condition that the initial values (x1 (0), y1 (0), z1 (0)) 6= (x2 (0), y2 (0), z2 (0)). However, with appropriate control schemes, the two systems will approach synchronization for any initial value.
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3.1. Three linear controllers In this subsection, we consider that three simple linear controllers are needed to synchronize the two identical energy resource systems. For this, we have the following theorem. Theorem 1. Systems (2) and (3) will approach global and exponential asymptotical synchronization for any initial condition with the following control law: u1 = −ge1 ,
u2 = −ge2 ,
u3 = −ge3 ,
(6)
where g is a sufficiently large feedback gain. Proof. Substituting (6) into (5), one can obtain e˙ 1 = a1 e1 − a2 (e2 + e3 ) − a3 x22 + a3 x21 − ge1 , e˙ 2 = −b1 e2 − b2 e3 − b3 x2 (x2 − z2 ) + b3 x1 (x1 − z1 ) + b4 e1 − ge2 , e˙ 3 = d1 x2 z2 − d1 x1 z1 − d2 e3 − ge3 .
(7)
Consider the following Lyapunov function: V =
1 2
(e21 + e22 + e23 ).
(8)
With (6), the time derivative of the Lyapunov function V along the trajectories of system (7) is V˙ = e1 e˙ 1 + e2 e˙ 2 + e3 e˙ 3
= e1 [a1 e1 − a2 (e2 + e3 ) − a3 (x2 + x1 )e1 − ge1 ] + e2 [−b1 e2 − b2 e3 − b3 (x1 + x2 )e1 + b3 z2 e1 + b3 x1 e3 + b4 e1 − ge2 ] + e3 (d1 z2 e1 + d1 x1 e3 − d2 e3 − ge3 ).
(9)
Since a chaotic system has bounded trajectories, there exists a positive constant L, such that |x| < L, |y| < L and |z | < L; thus V˙ ≤ (a1 + 2La3 − g )e21 − (b1 + g )e22 − (d2 + g − d1 L)e23
+(a2 + 3Lb3 + b4 ) |e1 | |e2 | + (d1 L + a2 ) |e1 | |e3 | + (b2 + b3 L) |e2 | |e3 | = −(|e1 | , |e2 | , |e3 |)P (|e1 | , |e2 | , |e3 |)T ,
(10)
where
1
1 − (d1 L + a2 ) 2 1 − ( b 2 + b 3 L) . 2 g + d2 − d1 L
− (a2 + 3Lb3 + b4 )
g − a1 − 2La3
2
1 P = − (a2 + 3Lb3 + b4 ) 2 1 − (d1 L + a2 ) 2
g + b1 1
− (b2 + b3 L) 2
(11)
Obviously, to ensure that the origin of error system (7) is asymptotically stable, the matrix P should be positive definite. This is the case if the following three inequalities hold: g − a11 > 0,
(12)
g + (b1 − a11 )g − b1 a11 − 2
a212
> 0,
(13)
g + (b1 + a33 − a11 )g + (a33 b1 − a11 b1 − 3
2
+ 2a12 a13 a23 −
a212 a33
− b1 a11 a33 +
a11 a223
a212
a223
−
−
b1 a213
−
a213
− a11 a33 )g
> 0,
(14)
where a11 = a1 + 2La3 , a12 = − 12 (a2 + 3Lb3 + b4 ), a13 = − 21 (d1 L + a2 ), a23 = − 12 (b2 + b3 L), a33 = d2 − d1 L. Denote f11 (g ) = g − a11 ,
(15)
f12 (g ) = g + (b1 − a11 )g − b1 a11 − 2
a212
,
f13 (g ) = g + (b1 + a33 − a11 )g + (a33 b1 − a11 b1 − 3
2
(16) a212
a223
− + 2a12 a13 a23 − a212 a33 − b1 a11 a33 + a11 a223 − b1 a213 .
−
a213
− a11 a33 )g (17)
It is easy to observe that, for a sufficiently large positive feedback gain g, f11 (g ), f12 (g ) and f13 (g ) will be positive. Hence, the matrix P is positive definite, and V˙ is negative definite. So, based on Lyapunov stability theory, we know that the origin of
Z. Wang / Nonlinear Analysis: Real World Applications 11 (2010) 3336–3343
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Fig. 2. State trajectories of the drive system (the solid line) and the response system (the dashed–dotted line): (a) (x1 , x2 ); (b) (y1 , y2 ); (c) (z1 , z2 ).
Fig. 3. Synchronization error in two chaotic systems with g = 0.03: (a) e1 ; (b) e2 ; (c) e3 .
error system (7) is asymptotically stable. It follows that the states x2 , y2 and z2 of response system (3) and the states x1 , y1 and z1 of drive system (2) are ultimately synchronized asymptotically. Therefore, the response system (3) is synchronous with the drive system (2). This completes the proof. In the numerical simulation, the fourth-order Runge–Kutta integration method is employed to solve the systems of differential equations with time step size 0.001. The parameters of the energy system are selected as in Section 2. The initial values of drive system and the response system are chosen as (x1 (0), y1 (0), z1 (0)) = (0.82, 0.29, 0.48) and (x2 (0), y2 (0), z2 (0)) = (0.78, 0.3, 0.25), respectively. Fig. 2 shows the time evolution curves of the drive system and the response system. Fig. 3 displays the synchronization errors between systems (2) and (3) (where g = 0.03). From Figs. 2 and 3, it is obvious that the synchronization errors converge asymptotically to zero and two different systems are indeed achieved with synchronization. From the theoretical analysis, it is known that conditions for synchronization obtained analytically in Theorem 1 are only the sufficient conditions. In order to explore the synchronization behavior of the scheme, g is taken as a bifurcation parameter to see how the solution of y2 − y1 evolves with variation of the parameter. From the bifurcation diagram plotted in Fig. 4, it can be observed that y2 − y1 converges to zero at about g = 0.0096, which means that synchronization can be achieved when g > 0.0096. 3.2. Two linear controllers In this subsection, we use two simple linear controllers to realize a global asymptotical synchronization. The chaotic response energy system with two linear controllers that we consider can be expressed as x˙ 2 = a1 x2 − a2 (y2 + z2 ) − a3 x22 + u1 , y˙ 2 = −b1 y2 − b2 z2 − b3 x2 (x2 − z2 ) + b4 x2 , z˙2 = d1 x2 z2 − d2 z2 + u3 , where u1 = −ge1 , u3 = −ge3 , respectively.
(18)
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Fig. 4. Bifurcation diagram of y2 − y1 versus the parameter g at x1 = 0.2.
Then we have the following theorem. Theorem 2. Systems (2) and (18) will approach global and exponential asymptotical synchronization for any initial condition with the following control law: u1 = −ge1 ,
u3 = −ge3 ,
(19)
where g is a sufficiently large feedback gain. Proof. The error system of systems (2) and (18) can obtained as e˙ 1 = a1 e1 − a2 (e2 + e3 ) − a3 x22 + a3 x21 − ge1 , e˙ 2 = −b1 e2 − b2 e3 − b3 x2 (x2 − z2 ) + b3 x1 (x1 − z1 ) + b4 e1 , e˙ 3 = d1 x2 z2 − d1 x1 z1 − d2 e3 − ge3 .
(20)
Consider the following Lyapunov function: 1
V =
2
(e21 + e22 + e23 ).
(21)
The time derivative of the Lyapunov function V along the trajectories of system (20) is V˙ = e1 e˙ 1 + e2 e˙ 2 + e3 e˙ 3
= e1 [a1 e1 − a2 (e2 + e3 ) − a3 (x2 + x1 )e1 − ge1 ] + e2 [−b1 e2 − b2 e3 − b3 (x1 + x2 )e1 + b3 z2 e1 + b3 x1 e3 + b4 e1 ] + e3 (d1 z2 e1 + d1 x1 e3 − d2 e3 − ge3 ).
(22)
Since a chaotic system has bounded trajectories, there exists a positive constant L, such that |x| < L, |y| < L and |z | < L; thus V˙ ≤ (a1 + 2La3 − g )e21 − b1 e22 − (g + d2 − d1 L)e23
+ (a2 + 3Lb3 + b4 ) |e1 | |e2 | + (d1 L + a2 ) |e1 | |e3 | + (b2 + b3 L) |e2 | |e3 | = −(|e1 | , |e2 | , |e3 |)P (|e1 | , |e2 | , |e3 |)T ,
(23)
where
1
− (a2 + 3Lb3 + b4 )
g − a1 − 2La3
2
1 P = − (a2 + 3Lb3 + b4 ) 2 1 − (d1 L + a2 )
b1 1
− (b2 + b3 L)
2
2
1 − (d1 L + a2 ) 2 1 − ( b 2 + b 3 L) . 2 g + d2 − d1 L
(24)
Obviously, to ensure that the origin of error system (20) is asymptotically stable, the matrix P should be positive definite. This is the case if the following three inequalities hold: g − a11 > 0,
(g − a11 )b1 −
(25) a212
> 0,
b1 g + (b1 a33 − b1 a11 − 2
(26) a223
−
a212
)g + 2a12 a13 a23 −
a212 a33
− b1 a11 a33 +
a11 a223
−
b1 a213
> 0,
(27)
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Fig. 5. State trajectories of the drive system (the solid line) and the response system (the dashed–dotted line): (a) (x1 , x2 ); (b) (y1 , y2 ); (c) (z1 , z2 ).
Fig. 6. Synchronization error in two chaotic systems with g = 0.03: (a) e1 ; (b) e2 ; (c) e3 .
where 1 a12 = − (a2 + 3Lb3 + b4 ), 2 1 1 = − (d1 L + a2 ), a23 = − (b2 + b3 L), 2 2
a11 = a1 + 2La3 , a13
a33 = d2 − d1 L.
(28)
Denote f21 (g ) = g − a11 , f22 (g ) = (g − a11 )b1 −
(29) a212
,
f23 (g ) = b1 g + (b1 a33 − b1 a11 − 2
(30) a223
−
a212
)g + 2a12 a13 a23 −
a212 a33
− b1 a11 a33 +
a11 a223
−
b1 a213
.
(31)
It is easy to observe that, for a sufficiently large positive feedback gain g, f21 (g ), f22 (g ) and f23 (g ) (where b1 is a positive constant) will be positive. Hence, the matrix P is positive definite, and V˙ is negative definite. Therefore, based on Lyapunov stability theory, we know that the origin of error system (20) is asymptotically stable and the controlled system (18) is synchronous with system (2). This completes the proof. In the numerical simulation, the values of the parameters and the initial values of the drive system and the response system are identical to those above. Fig. 5 shows the time evolution curves of the drive system and the response system. Fig. 6 displays the synchronization errors between systems (2) and (18) (where g = 0.03). From Figs. 5 and 6, it is obvious that the synchronization errors converge asymptotically to zero and two systems with different initial values are indeed achieved with synchronization. From the bifurcation diagram plotted in Fig. 7, it can be observed that y2 − y1 converges to zero at about g = 0.0273, which means that synchronization can be achieved when g > 0.0273.
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Fig. 7. Bifurcation diagram of y2 − y1 versus the parameter g at x1 = 0.2.
Fig. 8. Bifurcation diagram of y2 − y1 versus the parameter g at x1 = 0.2.
3.3. Single linear controller In this subsection, we consider the case when the synchronization of the two identical energy systems can be obtained via a single linear controller. Consider the following response system with a single linear controller: x˙ 2 = a1 x2 − a2 (y2 + z2 ) − a3 x22 + u1 , y˙ 2 = −b1 y2 − b2 z2 − b3 x2 (x2 − z2 ) + b4 x2 , z˙2 = d1 x2 z2 − d2 z2 ,
(32)
where u1 = −ge1 (g > 0) is the controller. Employing numerical stimulation, it can be obtained that system (2) will synchronize with system (32) for certain values of g. The initial values of the two systems and the values of parameters are identical to those above. The bifurcation diagram is plotted in Fig. 8, from which it can be observed that y2 − y1 converges to zero at about g = 0.0246, which means that synchronization can be achieved when g > 0.0246. Fig. 9 shows the time evolution curves of the drive system and the response system. Fig. 10 displays the synchronization errors between systems (2) and (32) (where g = 0.03). From Figs. 9 and 10, it is obvious that the synchronization errors converge asymptotically to zero and two systems are indeed achieved with synchronization. 4. Conclusion We have studied the synchronization of an energy resource system when it exhibits chaotic behavior, and proposed three schemes with only the simplest linear controllers to synchronize two identical energy resource systems. Based on the Lyapunov stability theory, the sufficient conditions for synchronization are obtained analytically in the first two schemes. By a numerical method, the synchronization condition can be obtained in the third scheme, in which only one state variable controller is contained, which is of important significance in synchronization. Furthermore, three numerical simulation examples are provided to show the effectiveness of the developed methods.
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Fig. 9. State trajectories of the drive system (the solid line) and the response system (the dashed–dotted line): (a) (x1 , x2 ); (b) (y1 , y2 ); (c) (z1 , z2 ).
Fig. 10. Synchronization error in two chaotic systems with g = 0.03: (a) e1 ; (b) e2 ; (c) e3 .
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