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## Synchronization of fractional-order chaotic systems based on adaptive fuzzy control

Chen Ye, Li Sheng-Gang, Liu Heng
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• #### 摘要

本文主要研究了带有未知外界扰动的分数阶混沌系统的同步问题. 基于分数阶Lyapunov稳定性理论，构造了分数阶的参数自适应规则以及模糊自适应同步控制器. 在稳定性分析中主要使用了平方Lyapunov函数. 该控制方法可以实现两分数阶混沌系统的同步，使得同步误差渐近趋于0. 最后，数值仿真结果验证了本文方法的有效性.

#### Abstract

In this paper the synchronization problem for fractional-order chaotic system with unknown external disturbance is investigated by adaptive fuzzy control. Based on the fractional Lyapunov stability theorem, an adaptive fuzzy controller, which is accompanied with fractional adaptation law, is established. Fuzzy logic system is used to approximate an unknown nonlinear function. The fuzzy approximation error can be canceled by the proposed fractional adaptation law. Just like the stability analysis in an integer-order chaotic system, the quadratic Lyapunov function is used to analyze the stability of the fractional-order closed-loop system. The control method can realize good synchronization performances between two fractional-order chaotic systems, and the synchronization error tends to zero asymptotically. Besides, the proposed controller can also guarantee the boundedness of all signals in the closed-loop system. Finally, the numerical simulation results illustrate the effectiveness of the proposed control method for fractional-order chaotic system in the presence of the external disturbances.

#### 参考文献

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#### 施引文献

•  [1] Li Y, Chen Y, Podlubny I 2009 Automatica 45 3690 [2] Podlubny I 1999 Fractional Differential Equations (New York: Academic Press) [3] Yuan L G, Yang Q G 2012 Commun. Nonlinear Sci. Numer. Simul 17 305 [4] Wen B, Cao M S, Hou Z L, Song W L, Zhang L, Lu M M, Jin H B, Fang X Y, Wang W Z, Yuan J 2013 Carbon 65 124 [5] Zhang R, Tian G, Yang S, Cao H 2015 ISA Trans. 56 102 [6] Yin C, Cheng Y, Chen Y, Stark B, Zhong S 2015 Nonlinear Dyn. 82 39 [7] Huang S, Zhang R, Chen D 2016 J. Computat. Nonlinear Dyn. 11 031007 [8] Pecora M L, Carroll T L 1990 Phys. Rev. Lett. 64 821 [9] Yu H J, Liu Y Z 2005 Acta Phys. Sin. 54 3029 (in Chinese) [于洪洁, 刘延柱 2005 物理学报 54 3029] [10] Wang H, Han Z Z, Xie Q Y, Zhang W 2009 Commun. Nonlinear Sci. Numer. Simul. 14 2239 [11] Yu N, Ding Q, Chen H 2007 J. Commun. 28 73 (in Chinese) [于娜, 丁群, 陈红 2007 通信学报 28 73] [12] Marino R, Tomei P 1996 Nonlinear control design: geometric, adaptive and robust. Prentice Hall International (UK) Ltd. [13] Yue Q, Yang J, Li G H, Li G D, Xu W, Chen J S, Wang S N 2005 Inorg. Chem. 44 5241 [14] Li R, Zhang G J, Yao H, Zhu T, Zhang Z H 2014 Acta Phys. Sin. 63 230501 (in Chinese) [李睿, 张广军, 姚宏, 朱涛, 张志浩 2014 物理学报 63 230501] [15] Kim S, Park P, Jeong C 2010 IET Control Theory Appl. 4 1828 [16] Becker R, Rannacher R 1996 A Feed-back Approach to Error Control in Finite Element Methods: Basic Analysis and Examples [17] Boulkroune A, Bouzeriba A, Bouden T 2016 Neurocomputing 173 606 [18] Mathiyalagan K, Park J H, Sakthivel R 2015 Complexity 21 114 [19] Liu H, Li S G, Sun Y G, Wang H X 2015 Chin. Phys. B 24 090505 [20] Liu H, Li S, Wang H, Huo Y, Luo J 2015 Entropy 17 7185 [21] Boulkroune A, Tadjine M, M'Saad M, Farza M 2010 Fuzzy Sets and Systems 161 797 [22] Tong S, Wang T, Tang J T 2000 Fuzzy Sets and Syst. 111 169 [23] Liu H, Li S G, Sun Y G, Wang H X 2015 Acta Phys. Sin. 64 070503 (in Chinese) [刘恒, 李生刚, 孙业国, 王宏兴 2014 物理学报 64 070503]
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##### 出版历程
• 收稿日期:  2016-04-13
• 修回日期:  2016-05-27
• 刊出日期:  2016-09-05

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