Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 698507, 9 pages

http://dx.doi.org/10.1155/2014/698507

Research Article

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

Qiong Huang, Chunyang Dong, and Qianbin Chen

Key Lab of Mobile Communication Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received 20 March 2014; Revised 26 May 2014; Accepted 4 June 2014; Published 22 June 2014

Academic Editor: Chuandong Li

Copyright © 2014 Qiong Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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 fractional-order memristor-based chaotic system [5], the fractional-order 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 [10–16].

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 fractional-order 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 fractional-order 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.

2. Mathematical Preliminaries

In this paper, we use the Caputo definition of fractional derivative, which is

where denoted the Caputo operator, is the first integer which is not less than , and is the l-order derivative for ; that is, .

Consider the following nonlinear fractional-order system:

where are continuous function, are fractional order, and are state vectors.

First, we recall the stability results of nonlinear fractional-order systems [19–24]. Let the equilibrium point of system (2) be and let the Jacobian matrix be . are the eigenvalues of the Jacobian matrix . If are satisfied, then the equilibrium point is asymptotically stable [19–24].

Second, we recall the improved version of Adams-Bashforth-Moulton algorithm [14] for the fractional-order systems. Consider the following two-dimensional nonlinear fractional-order system:

with initial condition . Let and let . Then, the two-dimensional fractional-order system can be discretized as follows

where

The error of this algorithm is

3. 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 is the fractional order. The fractional-order Chen chaotic system has chaotic attractor for [19]. The fractional-order Chen chaotic attractor with is shown as in Figure 1.

(a)

(b)

Figure 1: The fractional-order Chen chaotic attractor with .

There are three unstable equilibrium points in the above fractional-order Chen chaotic system. The unstable equilibrium points are and , respectively. Our goal is how to control the unstable equilibrium points via a fractional-order scalar controller.

3.1. Case  1: Control of the Unstable Equilibrium Point

Theorem 1. Let the controlled system be

where is the scalar fractional-order controller and and are feedback coefficients. If and , then the controlled system (8) will be asymptotically converged to the equilibrium point .

Proof. The unstable equilibrium point 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 is

The eigenvalues are

because

So

Therefore, all eigenvalues of the Jacobi matrix at equilibrium point in the controlled system (8) have negative real part. This result implies that the controlled system will be asymptotically converged to the equilibrium point . The proof is completed.

Theorem 2. Consider the controlled system is as follows:

where is a fractional-order scalar controller and and are feedback coefficients. If and , then the controlled system (13) will be asymptotically converged to the equilibrium point .

Proof. It is easily to obtain that the unstable equilibrium point 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 is

The eigenvalues are

because

So

Therefore, all eigenvalues of the Jacobi matrix at equilibrium point 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 . The proof is completed.

3.2. Case  2: Control of the Unstable Equilibrium Points

Theorem 3. Consider the controlled system is

where is the scalar fractional-order controller and is feedback coefficient. If , then the controlled system (18) will be asymptotically converged to the equilibrium point .

Proof. The Jacobian matrix at the equilibrium point in the controlled system (18) is

Its characteristic equation is

where , , and .

Because , the following yields

This result indicates that all eigenvalues of the Jacobi matrix at equilibrium point in the controlled system (18) have negative real part. So, the controlled system (18) will be asymptotically converged to the equilibrium point . 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 .

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 fractional-order derivative are used in our fractional-order scalar controller. This is the main contribution in our work.

4. 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, 26–29], the circuits are designed as mentioned below to realize the fractional-order chaotic system (8), (13), and (18), and the circuit experiments are obtained.

4.1. Case  1: Realize Physically the Controlled Fractional-Order Chen Chaotic System (8)

Now, let and in the controlled system (8). According to Theorem 1, the controlled system (8) will be asymptotically converged to the unstable equilibrium point . 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.

(a)

(b)

(c)

Figure 2: The circuit diagram designed to realize the fractional-order controlled system (8) for .

Figure 3: Circuit diagram for box .

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 , , , , , , and , respectively. Here and later, the capacitors and resistors in Figure 3 are chosen as , , , , , and . The operational amplifiers are of the type of LF353N, the multipliers are of the type of AD633, and the power is supplied by .

By choosing the circuit output 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 in Figure 2(b) and Figure 4(c) shows the circuit experiment displayed on the oscilloscope with the circuit outputs in Figure 2(c). In this paper, the vertical coordinate unit is V (volt) and the horizontal coordinate unit is second (s).

(a)

(b)

(c)

Figure 4: The circuit experiment displayed on the oscilloscope.

According to Figure 4, the circuit results fit the theoretical results mentioned in Theorem 1.

4.2. Case 2: Realize Physically the Controlled Fractional-Order Chen Chaotic System (13)

Now, let and in the controlled system (13). According to Theorem 2, the controlled system (8) will be asymptotically converged to the unstable equilibrium point . Similarly, the circuit diagram designed to realize the controlled system (13) is as shown in Figure 5.

(a)

(b)

(c)

Figure 5: The circuit diagram designed to realize the fractional-order controlled system (13) for .

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.

According to the circuit design methods, the resistors in Figure 5 are chosen as , and , respectively. The resistors   are the same as in Figure 2.

Similarly, by choosing the circuit output 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 in Figure 5(b) and Figure 6(c) shows the circuit experiment displayed on the oscilloscope with the circuit outputs in Figure 5(c).

(a)

(b)

(c)

Figure 6: The circuit experiment displayed on the oscilloscope.

According to Figure 6, the circuit results agree with the theoretical results mentioned in Theorem 2.

4.3. Case  3: Realize Physically the Controlled Fractional-Order Chen Chaotic System (18)

Now, let in the controlled system (18). According to Theorem 3, the controlled system (18) will be asymptotically converged to the unstable equilibrium point . Similarly, the circuit diagram designed to realize the controlled system (18) is displayed as shown in Figure 7.

(a)

(b)

(c)

Figure 7: The circuit diagram designed to realize the fractional-order controlled system (18) for .

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 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 in Figure 7(b) and Figure 8(c) shows the circuit experiment displayed on the oscilloscope with the circuit outputs in Figure 7(c).

(a)

(b)

(c)

Figure 8: The circuit experiment displayed on the oscilloscope.

According to Figure 8, the circuit results agree with the theoretical results mentioned in Theorem 3.

5. Conclusions

In order to control of the unstable equilibrium points for the fractional-order Chen chaotic system, some fractional-order 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 [30–33] 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Science and Technology Specific Project of China (2012YQ20022404), National Natural Science Foundation of China (no. 60972070), Program for Changjiang Scholars and Innovative Research Team in University (IRT1299), and the special fund of Chongqing key laboratory (CSTC).

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