Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 217843, 8 pages
http://dx.doi.org/10.1155/2011/217843
Research Article

A Novel Control Method for Integer Orders Chaos Systems via Fractional-Order Derivative

1Key Laboratory of Network control and Intelligent Instrument of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2Institute of Applied Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received 30 August 2010; Accepted 3 March 2011

Academic Editor: Antonia Vecchio

Copyright © 2011 Ping Zhou and Fei Kuang. 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 control method is obtained to stabilize the point in chaos attractor of integer orders chaos systems. The control law has simple structure and is designed easily. Two examples are also given to illustrate the effectiveness of the theoretical result.

1. Introduction

Chaotic phenomena has been observed in many areas of science and engineering such as mechanics, electronics, physics, medicine, ecology, biology, and economy. To avoid troubles arising from unusual behaviors of a chaotic system, chaos control has received a great deal of interest among scientists from various research fields in the past few decades [18]. On the other hand, derivatives and integrals of fractional order have been found in many applications in recent years in physics and engineering. Many systems [413] are known to display fractional-order dynamics, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, and electromagnetic waves.

In the recent years, emergence of effective methods in the differentiation and integration of non integer order equations makes fractional-order systems more and more attractive for the systems control community. It is verified that the fractional-order controllers can have better disturbance rejection ratios and less sensitivity to plant parameter variations compared to the traditional controllers [14]. However, few results on control the saddle point of index 2 in integer orders chaos system are presented via fractional-order derivative. A fractional-order control method [14] is presented to stabilize the unstable equilibrium of integer orders chaos systems, but the method cannot apply to the desired point in chaotic attractor. Motivated by that, in this paper, a very simple control method is presented for a class of integer orders chaotic systems via fractional-order derivative. The order of fractional-order derivative is only determined by the eigenvalues at the desired point in integer orders chaotic attractor. The control technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous.

2. Control Chaos via Fractional-Order Derivative

Consider the following 3D integer orders chaos system: where are real state variables and is differentiable function.

Let be the any point in the chaos attractor of integer orders system (2.1), the Jacobian matrix of at point , and , the eigenvalues of , respectively. In this paper, we always assume that , , and , especially   is called the saddle point of index  2 if it also is the equilibrium point of system (2.1). In order to stabilize the point   via fractional-order derivative, we introduce the following controller: where is the fractional order and will be determined later. Then, we can construct the control system for (2.1) as follows:

Hence, the problem of stabilization of in chaos attractor of system (2.1) is shifted into the stability of in system (2.3). So, the stabilization problem of in system (2.1) can be solved if is suitably designed such that in system (2.3) is asymptotically stable.

Now, we state the main result in this paper as follows.

Theorem 2.1. If   in , then the point is asymptotically stable in control system (2.3); that is, the point in chaos system (2.1) can be stabilized via fractional-order derivative.

Proof. According to system (2.3) and controller (2.2), we can obtain that
It can be seen that the point   is one equilibrium point of system (2.4) and the Jacobian matrix at this point for system (2.4) is the same as the Jacobian matrix at this point for system (2.1); therefore, the eigenvalues of the Jacobian matrix of (2.4) at point are , .
For , we have For , we also derive that In term of (2.5) and (2.6), for any eigenvalue of , we obtain that which implies that the equilibrium point of system (2.4) is asymptotically stable [15, 16]; that is, the point in the chaos attractor of integer orders chaos system (2.1) can be stabilized via fractional-order derivative; the proof is completed.

Remark 2.2. Especially, if is the saddle point of index 2 of system (2.1), then the control law obtained in this paper is similar with the controller obtained in [14]. So, Theorem 2.1 proposed by this paper extends the controller proposed by [14].

Remark 2.3. By Theorem 2.1 proposed in this paper, we know that it is possible to stabilize the closed-loop system without applying any changes on the places of the poles.

Remark 2.4. If is the same saddle point for hyperchaotic systems, which implies that the eigenvalues of are , , and , respectively, then this point in hyperchaotic systems can be stabilized via the proposed controller.

3. Applications

Now, we take Lü chaos system [17] and hyperchaotic Chen system [18] for numerical simulation, respectively. All the numerical simulation of fractional-order system in this paper is based on [19]. It is a direct time-domain approximation numerical simulation. Reference [20] has shown that using frequency-domain approximation in the numerical simulations of fractional systems may result in wrong consequences. This mistake has occurred in the recent literature that found the lowest-order chaotic systems among fractional-order systems.

We introduce the numerical solution of fractional differential equations in [19]. There have been several definitions of fractional derivatives. In the following, we introduce the most common one of them where m is the first integer which is not less than , is the m-order derivative in the usual sense, and is the q-order Riemann-Liouville integral operator with the expression Here, stands for gamma function, and is generally called “q-order Caputo differential operator.”

Consider the following fractional-order system: with initial condition . Now, set , . The above system can be discretized as follows: where and for ,

The error of this approximation is described as follows:

Now, we choose Lü chaos system for numerical simulation. The Lü chaos system [17] is

Now, we choose in chaos attractor of chaos system (3.9). Since the eigenvalues of the Jacobian matrix at for system (3.9) are , , we have . According to Theorem 2.1, in order to stabilize this point, we choose for controller (2.2).

For example, the simulation results are shown in Figure 1 for , and the initial conditions are , where , , and .

fig1
Figure 1: The control simulation result for (10, 9, 15) when .

Finally, we choose hyperchaotic Chen system for numerical simulation. The hyperchaotic Chen system [18] is

Now, we choose in chaos attractor of the hyperchaotic Chen system (3.10). Since the eigenvalues of the Jacobian matrix at for system (3.10) are ,, and , respectively, we have . According to Theorem 2.1, in order to stabilize this point, we choose for controller (2.2).

For example, the simulation results are shown in Figure 2 for and the initial conditions are , where , , , and .

fig2
Figure 2: The control simulation result for (0, 0, 19, 0) when .

4. Conclusion

We construct a novel control law for integer orders chaos system via fractional-order derivative. Any desired point in chaos attractor of integer orders chaos system can be stabilized via the fractional-order derivative. The order of fractional-order derivative of chaos system is only determined by the eigenvalues at the point. By comparison with the traditional controllers, we know that this controller without causing any changes in the eigenvalues of the system at the desired fixed points. The proposed controller is employed to control Lü chaotic system and hyperchaotic Chen system, and some numerical simulation results are obtained. Theoretical analysis and simulation results show that the control method in this paper is effective.

Acknowledgments

The work is supported jointly by National Natural Science Foundation of China under Grant no. 61004042 and Natural Science Foundation Project of CQ CSTC 2009BB2417.

References

  1. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, no. 11, pp. 1196–1199, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. G. Chen and X. Yu, Chaos Control: Theory and Applications, vol. 292 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2003. View at Zentralblatt MATH
  4. W. Deng, “Generating 3-D scroll grid attractors of fractional differential systems via stair function,” International Journal of Bifurcation and Chaos, vol. 17, no. 11, pp. 3965–3983, 2007. View at Publisher · View at Google Scholar
  5. W. Deng and J. Lü, “Generating multi-directional multi-scroll chaotic attractors via a fractional differential hysteresis system,” Physics Letters A, vol. 369, no. 5-6, pp. 438–443, 2007. View at Publisher · View at Google Scholar
  6. W. M. Ahmad and J. C. Sprott, “Chaos in fractional-order autonomous nonlinear systems,” Chaos, Solitons and Fractals, vol. 16, no. 2, pp. 339–351, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. D. A. Tikhonov and H. Malchow, “Chaos and fractals in fish school motion, II,” Chaos, Solitons and Fractals, vol. 16, no. 2, pp. 287–289, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. C. Li and G. Chen, “Chaos in the fractional order Chen system and its control,” Chaos, Solitons and Fractals, vol. 22, no. 3, pp. 549–554, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. C. G. Li, X. F. Liao, and J . B. Yu, “Synchronization of fractional order chaotic systems,” Physical Review E, vol. 68, no. 6, Article ID 067203, 3 pages, 2003. View at Publisher · View at Google Scholar
  10. X. Gao and J. Yu, “Synchronization of two coupled fractional-order chaotic oscillators,” Chaos, Solitons and Fractals, vol. 26, no. 1, pp. 141–145, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. W. H. Deng and C. P. Li, “Chaos synchronization of the fractional Lü system,” Physica A, vol. 353, no. 1–4, pp. 61–72, 2005. View at Publisher · View at Google Scholar
  12. J. G. Lu, “Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal,” Chaos, Solitons and Fractals, vol. 27, no. 2, pp. 519–525, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. J. P. Yan and C. P. Li, “On chaos synchronization of fractional differential equations,” Chaos, Solitons and Fractals, vol. 32, no. 2, pp. 725–735, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. M. S. Tavazoei and M. Haeri, “Chaos control via a simple fractional-order controller,” Physics Letters A, vol. 372, no. 6, pp. 798–807, 2008. View at Publisher · View at Google Scholar
  15. D. Matignon, “Stability results of fractional differential equations with applications to control processing,” in Proceedings of the IMACS-IEEE Multiconference on Computational Engineering in Systems Applications (CESA '96), vol. 963, Lille, France, July 1996.
  16. K. Ogata, Modern Control Engineering, Prentice-Hall, Englewood Cliffs, NJ, USA, 1990.
  17. J. Lü, G. Chen, and S. Zhang, “Dynamical analysis of a new chaotic attractor,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 5, pp. 1001–1015, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. Y. Li, W. K. S. Tang, and G. Chen, “Generating hyperchaotic attractor via state feedback control,” The International Journal of Bifurcation and Chaos, vol. 15, no. 10, pp. 3367–3375, 2005. View at Publisher · View at Google Scholar
  19. C. P. Li and G. J. Peng, “Chaos in Chen's system with a fractional order,” Chaos, Solitons and Fractals, vol. 22, no. 2, pp. 443–450, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. M. S. Tavazoei and M. Haeri, “A necessary condition for double scroll attractor existence in fractional-order systems,” Physics Letters A, vol. 367, no. 1-2, pp. 102–113, 2007. View at Publisher · View at Google Scholar