Control of the Fractional-Order Chen Chaotic System via Fractional-Order Scalar Controller and Its Circuit Implementation

A fractional-order scalar controller which involves only one state variable is proposed. By this fractional-order scalar controller, the unstable equilibrium points in the fractional-order Chen chaotic system can be asymptotically stable. The present control strategy is theoretically rigorous. Some circuits are designed to realize these control schemes. The outputs of circuit agree with the results of theoretical results.


Introduction
In the last few decades, chaotic behaviors have been discovered in many areas of science and engineering such as mathematics, physics, chemistry, electronics, medicine, economy, biological science, and social science.In 1990, Ott et al. presented the OGY method of chaotic control [1].After that, chaos control has attracted increasing attention among scientists in various fields.Many control schemes [1,2] have been presented, such as feedback control, parametric perturbation control, adaptive control, and fuzzy control.On the other hand, the chaotic or hyperchaos behaviors have been found in many fractional-order dynamical systems.Many fractional-order chaotic systems have been presented, the fractional-order Chua's chaotic circuit [3], the fractional-order Duffing chaotic system [4], the fractionalorder memristor-based chaotic system [5], the fractionalorder Lorenz chaotic system [6], the fractional-order Chen chaotic system [7], and so forth [8,9].Moreover, control and synchronization of fractional-order chaotic systems have attracted much attention in the recent years [10][11][12][13][14][15][16].
Compared to the traditional controller (integer-order controller), the fractional-order controller has many advantages, such as less sensitivity to parameter variations and better disturbance rejection ratios [17].It is possible that traditional controller (integer-order controller) will be replaced by fractional-order controller in the future.Recently, a fractional-order vector controller is addressed to stabilize the unstable equilibrium points for integer-order chaotic systems by Tavazoei and Haeri [17].Zhou and Kuang have presented another fractional-order vector controller to stabilize the nonequilibrium points for integer-order chaotic systems [18].However, only integer-order chaotic systems are discussed in [17,18], and only fractional-order vector controller is investigated.
Up to now, to the best of our knowledge, very few results on chaotic control are reported by fractional-order scalar controller.Motivated by the above-mentioned discussions, some fractional-order scalar controllers are presented to control the fractional-order Chen chaotic systems in this paper.Only one system state variable is used in the fractionalorder scalar controller.The control scheme is simple and theoretical.Moreover, some circuits are designed to realize these control schemes, and the circuit results agree with the theoretical results.
The outline of this paper is as follows.In Section 2, some mathematical preliminaries are addressed for the fractionalorder system.In Section 3, some fractional-order scalar controller are proposed to stabilize the unstable equilibrium points in the fractional-order Chen chaotic system.In Section 4, some circuits are designed to realize the control schemes.The conclusion is finally drawn in Section 5.

Control of the Unstable Equilibrium Points for the Fractional-Order Chen Chaotic System via a Fractional-Order Scalar Controller
In this section, some fractional-order scalar controllers which involve only one state variable are addressed.The unstable equilibrium points of the fractional-order Chen chaotic system can be asymptotically stable by these fractional-order scalar controllers.
In 1963, E. N. Lorenz reported the first chaotic model that revealed the complex and fundamental behaviors of the nonlinear dynamical systems.In 1999, Chen found another chaotic model in a simple three-dimensional autonomous system, which nevertheless is not topologically equivalent to the Lorenz chaotic model.The fractional-order Chen chaotic model is described as where 0 <  < 1 is the fractional order.The fractional-order Chen chaotic system has chaotic attractor for  ≥ 0.83 [19].The fractional-order Chen chaotic attractor with  = 0.9 is shown as in Figure 1. Figure 1: The fractional-order Chen chaotic attractor with  = 0.9.
There are three unstable equilibrium points in the above fractional-order Chen chaotic system.The unstable equilibrium points are  0 = (0, 0, 0) and  ± = (± √ 63, ± √ 63, 21), respectively.Our goal is how to control the unstable equilibrium points via a fractional-order scalar controller.

Case 1: Control of the Unstable Equilibrium
Theorem 1.Let the controlled system be where  1    2 + 2  2 is the scalar fractional-order controller and  1 and  2 are feedback coefficients.If  1 > −1 and  2 > 105, then the controlled system (8) will be asymptotically converged to the equilibrium point  0 = (0, 0, 0).
Proof.The unstable equilibrium point  0 = (0, 0, 0) in the fractional-order Chen chaotic system is also the equilibrium point in the controlled system (8).The Jacobi matrix of the controlled system at equilibrium point  0 = (0, 0, 0) is The eigenvalues are because
Therefore, all eigenvalues of the Jacobi matrix at equilibrium point  0 = (0, 0, 0) in the controlled system (8) have negative real part.This result implies that the controlled system will be asymptotically converged to the equilibrium point  0 = (0, 0, 0).The proof is completed.
Proof.It is easily to obtain that the unstable equilibrium point  0 = (0, 0, 0) in the fractional-order Chen chaotic system is also the equilibrium point in the controlled system (13).
The Jacobi matrix of the controlled system (13) at equilibrium point  0 = (0, 0, 0) is The eigenvalues are because
Therefore, all eigenvalues of the Jacobi matrix at equilibrium point  0 = (0, 0, 0) in the controlled system (13) have negative real part.This result indicates that the controlled system (13) will be asymptotically converged to the equilibrium point  0 = (0, 0, 0).The proof is completed.
Similarly, we can easily control the fractional-order Chen chaotic system that will be asymptotically converged to the unstable equilibrium point  − = (− √ 63, − √ 63, 21).Remark 4. In this section, we only discuss that all eigenvalues of the Jacobi matrix at equilibrium point in the controlled system have negative real part.Recently, Li and Ma [25] reported the more rigorous result on the local asymptotical stability of the nonlinear fractional differential system.Their result also can be applied to control the unstable equilibrium point in the fractional-order Chen chaotic system.Remark 5.Only one system state variable and its fractionalorder derivative are used in our fractional-order scalar controller.This is the main contribution in our work.

Circuit Implementation of the Control Scheme for the Fractional-Order Chen Chaotic System
In this subsection, some circuits are designed to realize these control schemes for the fractional-order Chen chaotic system, and the circuit results fit the theoretical results mentioned in Section 3. Now, many references on the guidelines to design circuits for the fractional-order chaotic systems are reported.By the circuit design methods [9,[26][27][28][29], the circuits are designed as mentioned below to realize the fractional-order chaotic system ( 8), (13), and ( 18), and the circuit experiments are obtained.(8).Now, let  1 = 1 and  2 = 200 in the controlled system (8).According to Theorem 1, the controlled system (8) will be asymptotically converged to the unstable equilibrium point  0 = (0, 0, 0).By the circuit design method [9,27,28], the circuit diagram designed to realize the controlled system ( 8) is presented as shown in Figures 2 and 3.

Case 1: Realize Physically the Controlled Fractional-Order Chen Chaotic System
The first equation, the second equation, and the third equation in controlled system (8) are realized by Figures 2(a), 2(b), and 2(c), respectively.The operator   /  is realized by Figure 3.
According to the circuit design methods, the resistors in Figure 2 are chosen as  1 = 100 kΩ,  2 = 2.86 kΩ,  3 = 3.57 kΩ,  4 = 14.3 kΩ,  5 = 33.3kΩ,  6 = 100 kΩ, and  7 = 0.5 kΩ, respectively.Here and later, the capacitors and resistors in Figure 3  By choosing the circuit output  1 in Figure 2(a) as the vertical axis input, Figure 4(a) shows the circuit experiment displayed on the oscilloscope.Similarly, Figure 4(b) shows the circuit experiment displayed on the oscilloscope with the circuit outputs  2 in Figure 2(b) and Figure 4(c) shows the circuit experiment displayed on the oscilloscope with the circuit outputs  3 in Figure 2(c).In this paper, the vertical coordinate unit is V (volt) and the horizontal coordinate unit is second (s).
According to Figure 4, the circuit results fit the theoretical results mentioned in Theorem 1.
Here, the first equation, the second equation, and the third equation in controlled system (13) are realized by Figures 5(a), 5(b), and 5(c), respectively.The operator   /  is realized by Figure 3.
Similarly, by choosing the circuit output  1 in Figure 5(a) as the vertical axis input, Figure 6(a) shows the circuit experiment displayed on the oscilloscope.Similarly, Figure 6(b) shows the circuit experiment displayed on the oscilloscope with the circuit outputs  2 in Figure 5(b) and Figure 6(c) shows the circuit experiment displayed on the oscilloscope with the circuit outputs  3 in Figure 5(c).
According to Figure 6, the circuit results agree with the theoretical results mentioned in Theorem 2. (18).Now, let  5 = −1 in the controlled system (18).According to Theorem 3, the controlled system (18) will be asymptotically converged to the unstable equilibrium point  + = ( √ 63, √ 63, 21).Similarly, the circuit diagram designed to realize the controlled system (18) is displayed as shown in Figure 7.

Case 3: Realize Physically the Controlled Fractional-Order Chen Chaotic System
Similarly, the first equation, the second equation, and the third equation in controlled system (18) are realized by Figures 7(a), 7(b), and 7(c), respectively.The operator   /  is realized by Figure 3.The resistors and capacitors in Figure 7 are chosen as Case 1 and Case 2.
By choosing the circuit output  1 in Figure 7(a) as the vertical axis input, Figure 8(a) shows the circuit experiment displayed on the oscilloscope.Similarly, Figure 8(b) shows the circuit experiment displayed on the oscilloscope with the circuit outputs  2 in Figure 7(b) and Figure 8(c) shows the circuit experiment displayed on the oscilloscope with the circuit outputs  3 in Figure 7(c).
According to Figure 8, the circuit results agree with the theoretical results mentioned in Theorem 3.

Conclusions
In order to control of the unstable equilibrium points for the fractional-order Chen chaotic system, some fractionalorder scalar controllers are proposed, and only one state variable is used in the fractional-order scalar controller.The control scheme is theoretically rigorous.Moreover, three fractional-order chaotic circuits are designed to realize the control strategy, and the circuit experiments are obtained.The experiment results agree with the theoretical results.Furthermore, some results [30][31][32][33] on the effect of noises or disturbances in control or synchronization problems of chaotic systems have been proposed.The anticontrol or  antisynchronization problems for fractional chaotic systems with disturbances or noises have been also discussed in [34].So, the effect of noises or disturbances for our control scheme is our further work.

Figure 4 :
Figure 4: The circuit experiment displayed on the oscilloscope.