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## Adaptive fuzzy synchronization for uncertain fractional-order chaotic systems with unknown non-symmetrical control gain

Liu Heng, Li Sheng-Gang, Sun Ye-Guo, Wang Hong-Xing
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• #### 摘要

针对带有非对称控制增益的不确定分数阶混沌系统的同步问题设计了模糊自适应控制器. 模糊逻辑系统用来逼近未知的非线性函数, 非对称的控制增益矩阵被分解为一个未知的正定矩阵、一个对角线上元素为+1或-1的已知对角矩阵和 一个未知的上三角矩阵的乘积. 基于分数阶Lyapunov稳定性理论构造了模糊控制器以及分数阶的参数自适应律, 在保证所有变量有界的情况下实现驱动系统和响应系统的同步. 在分数阶系统稳定性分析中给出了一种平方Lyapunov函数的使用方法, 根据此方法很多针对整数阶系统的控制方法可以推广到分数阶系统中. 最后数值仿真结果验证了所提控制方法的可行性.

#### Abstract

In this paper the synchronization problem for the uncertain fractional-order chaotic systems with unknown non-symmetrical control gain matrices is investigated by means of adaptive fuzzy control. Fuzzy logic systems are employed to approximate the unknown nonlinear functions. We decompose the control gain matrix into a positive definite matrix, a unity upper triangular matrix, and a diagonal matrix with diagonal entries +1 or -1. The positive matrix is used to construct the Lyapunov function; the diagonal matrix is employed to design the controller. Based on the fractional Lyapunov stability theorem, an adaptive fuzzy controller, which is accompanied by fractional adaptation laws, is established. The proposed methods can guarantee the boundedness of the involved signals as well as the asymptotical convergence of the synchronization errors. It should be pointed out that the methods for using quadratic Lyapunov function in the stability analysis of the fractional-order chaotic systems are developed in this paper. Based on the results of this paper, many control methods which are valid for integer-order nonlinear systems can be extended to control fractional-order nonlinear systems. Finally, the effectiveness of the proposed methods is shown by simulation studies.

#### 作者及机构信息

###### 1. 陕西师范大学数学与信息科学学院, 西安 710119; 2. 淮南师范学院数学与计算科学系, 淮南 232038
• 基金项目: 国家自然科学基金(批准号: 11401243, 61403157)和中央高校基本科研业务费专项资金(批准号: GK201504002)资助的课题.

#### Authors and contacts

###### 1. College of Mathematics and Information Science, Shaanxi Normal Universtiy, Xi'an 710119, China; 2. Department of Mathematics and Computational Science, Huainan Normal University, Huainan 232038, China
• Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11401243, 61403157), and the Fundamental Research Funds for the Central Universities (Grant No. GK201504002).

#### 参考文献

 [1] Podlubny I 1999 Fractional differential Equations (New York: Academic Press) [2] Li Y, Chen Y Q, Podlubny I 2009 Automatica 45 3690 [3] Wang D F, Zhang J Y, Wang X Y 2013 Chin. Phys. B 22 100504 [4] Yuan L G, Yang Q G 2012 Commun. Nonlinear Sci. Numer. Simul. 17 305 [5] 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] [6] Aghababa M P 2012 Commun. Nonlinear Sci. Numer. Simul. 17 2670 [7] Mahmoud G M, Mahmoud E E 2012 Nonlinear Dyn. 67 1613 [8] Kim S H, Park P, Jeong C 2010 IET Control Theory Appl. 4 1828 [9] Kiani B A, Fallahi K, Pariz N, Leung H 2009 Commun. Nonlinear Sci. Numer. Simul. 14 863 [10] Ma S Q, Lu Q S, Feng Z S 2010 Int. J. Nonlinear Mech. 45 659 [11] Li Z J, Zeng Y C 2013 Chin. Phys. B 22 040502 [12] Zhou P, Ding R, Cao Y X 2012 Nonlinear Dyn. 70 1263 [13] Huang L L, Qi X 2013 Acta Phys. Sin. 62 080507 (in Chinese) [黄丽莲, 齐雪 2013 物理学报 62 080507] [14] Zhang R X, Yang S P 2010 Chin. Phys. B 19 020510 [15] Yang L X, Jiang J 2014 Commun. Nonlinear Sci. Numer. Simulat. 19 1496 [16] Liu J G, Huang L H, Meng Y M 2013 Int. J. Adapt. Control Signal Process. 27 1086 [17] Hosseinnia S H, Ghaderi R, Ranjbar A N, Mahmoudiana M, and Momanic S 2010 Computers and Mathematics with Applications 59 1637 [18] Yin C, Dadras S, and Zhong S M 2012 Journal of the Franklin Institute 349 2078 [19] Pan L, Zhou W N, Fang J A, Li D Q 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 3754 [20] Senol B, Ates A, Alagoz B B, Yeroglu C 2014 ISA Transactions 53 189 [21] Essounbouli N, Hamzaoui A, Zaytoon J 2006 Control Intell. Systems 34 12 [22] Tong S C, Tang J, Wang T 2000 Fuzzy Sets and Systems 111 153 [23] Tong S C, He X L, Zhang H G 2009 IEEE Trans. Fuzzy Syst. 17 1059 [24] Boulkroune A, Tadjine M, M’Saad M, Farza M 2010 Fuzzy Sets and Systems 161 797 [25] Tong S C, Liu C L, Li Y M 2010 IEEE Trans. Fuzzy Syst. 18 845 [26] Liu H, Yu H J, Xiang W 2012 Acta Phys. Sin. 61 180503 (in Chinese) [刘恒, 余海军, 向伟 2012 物理学报 61 180503] [27] Tong S C, Li Y M 2012 IEEE Trans. Fuzzy Syst. 20 168 [28] Liu H, Yu H J, Xiang W 2012 Chin. Phys. B 21 120505 [29] Pan Y P, Er M J 2013 IEEE Trans. Fuzzy Syst. 21 1123 [30] Tong S C, Li Y M 2013 IEEE Trans. Fuzzy Syst. 21 134 [31] Yang Y, Hua C C, Guan X P 2014 IEEE Trans. Fuzzy Syst. 22 631 [32] Wang L X 1994 Adaptive Fuzzy Systems and Control: Design and Stability Analysis (Englewood Cliffs: Prentice Hall) pp120-144 [33] Trigeassou J C, Maamri N, Sabatier J, Oustaloup A 2011 Signal Processing 91 437 [34] Shen J, Lam J 2014 Automatica 50 547 [35] Lin T C, Kuo C H 2011 ISA Transactions 50 548 [36] Lin T C, Lee T Y, Balas V E 2011 Chaos, Solitons & Fractals 44 791 [37] Aguila-Camacho N, Duarte-Mermoud M A, Gallegos J A 2014 Commun. Nonlinear Sci. Numer. Simulat. 19 2951 [38] Costa P R, Hsu L, Imai A K, Kokotovic P 2003 Automatica 39 1251 [39] Ahmed E, El-Sayed A M A, El-Saka H A A 2007 J. Math. Anal. Appl. 325 542 [40] L J H, Chen G R, Yu X H, Leung H 2004 IEEE Trans. Circuits Syst. I 51 2476 [41] L J H, Yu S M, Leung H, Chen G R 2006 IEEE Trans. Circuits Syst. I 53 149

#### 施引文献

•  [1] Podlubny I 1999 Fractional differential Equations (New York: Academic Press) [2] Li Y, Chen Y Q, Podlubny I 2009 Automatica 45 3690 [3] Wang D F, Zhang J Y, Wang X Y 2013 Chin. Phys. B 22 100504 [4] Yuan L G, Yang Q G 2012 Commun. Nonlinear Sci. Numer. Simul. 17 305 [5] 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] [6] Aghababa M P 2012 Commun. Nonlinear Sci. Numer. Simul. 17 2670 [7] Mahmoud G M, Mahmoud E E 2012 Nonlinear Dyn. 67 1613 [8] Kim S H, Park P, Jeong C 2010 IET Control Theory Appl. 4 1828 [9] Kiani B A, Fallahi K, Pariz N, Leung H 2009 Commun. Nonlinear Sci. Numer. Simul. 14 863 [10] Ma S Q, Lu Q S, Feng Z S 2010 Int. J. Nonlinear Mech. 45 659 [11] Li Z J, Zeng Y C 2013 Chin. Phys. B 22 040502 [12] Zhou P, Ding R, Cao Y X 2012 Nonlinear Dyn. 70 1263 [13] Huang L L, Qi X 2013 Acta Phys. Sin. 62 080507 (in Chinese) [黄丽莲, 齐雪 2013 物理学报 62 080507] [14] Zhang R X, Yang S P 2010 Chin. Phys. B 19 020510 [15] Yang L X, Jiang J 2014 Commun. Nonlinear Sci. Numer. Simulat. 19 1496 [16] Liu J G, Huang L H, Meng Y M 2013 Int. J. Adapt. Control Signal Process. 27 1086 [17] Hosseinnia S H, Ghaderi R, Ranjbar A N, Mahmoudiana M, and Momanic S 2010 Computers and Mathematics with Applications 59 1637 [18] Yin C, Dadras S, and Zhong S M 2012 Journal of the Franklin Institute 349 2078 [19] Pan L, Zhou W N, Fang J A, Li D Q 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 3754 [20] Senol B, Ates A, Alagoz B B, Yeroglu C 2014 ISA Transactions 53 189 [21] Essounbouli N, Hamzaoui A, Zaytoon J 2006 Control Intell. Systems 34 12 [22] Tong S C, Tang J, Wang T 2000 Fuzzy Sets and Systems 111 153 [23] Tong S C, He X L, Zhang H G 2009 IEEE Trans. Fuzzy Syst. 17 1059 [24] Boulkroune A, Tadjine M, M’Saad M, Farza M 2010 Fuzzy Sets and Systems 161 797 [25] Tong S C, Liu C L, Li Y M 2010 IEEE Trans. Fuzzy Syst. 18 845 [26] Liu H, Yu H J, Xiang W 2012 Acta Phys. Sin. 61 180503 (in Chinese) [刘恒, 余海军, 向伟 2012 物理学报 61 180503] [27] Tong S C, Li Y M 2012 IEEE Trans. Fuzzy Syst. 20 168 [28] Liu H, Yu H J, Xiang W 2012 Chin. Phys. B 21 120505 [29] Pan Y P, Er M J 2013 IEEE Trans. Fuzzy Syst. 21 1123 [30] Tong S C, Li Y M 2013 IEEE Trans. Fuzzy Syst. 21 134 [31] Yang Y, Hua C C, Guan X P 2014 IEEE Trans. Fuzzy Syst. 22 631 [32] Wang L X 1994 Adaptive Fuzzy Systems and Control: Design and Stability Analysis (Englewood Cliffs: Prentice Hall) pp120-144 [33] Trigeassou J C, Maamri N, Sabatier J, Oustaloup A 2011 Signal Processing 91 437 [34] Shen J, Lam J 2014 Automatica 50 547 [35] Lin T C, Kuo C H 2011 ISA Transactions 50 548 [36] Lin T C, Lee T Y, Balas V E 2011 Chaos, Solitons & Fractals 44 791 [37] Aguila-Camacho N, Duarte-Mermoud M A, Gallegos J A 2014 Commun. Nonlinear Sci. Numer. Simulat. 19 2951 [38] Costa P R, Hsu L, Imai A K, Kokotovic P 2003 Automatica 39 1251 [39] Ahmed E, El-Sayed A M A, El-Saka H A A 2007 J. Math. Anal. Appl. 325 542 [40] L J H, Chen G R, Yu X H, Leung H 2004 IEEE Trans. Circuits Syst. I 51 2476 [41] L J H, Yu S M, Leung H, Chen G R 2006 IEEE Trans. Circuits Syst. I 53 149
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##### 出版历程
• 收稿日期:  2014-10-21
• 修回日期:  2014-11-05
• 刊出日期:  2015-04-05

## 带有未知非对称控制增益的不确定分数阶混沌系统自适应模糊同步控制

• 1. 陕西师范大学数学与信息科学学院, 西安 710119;
• 2. 淮南师范学院数学与计算科学系, 淮南 232038
基金项目:

国家自然科学基金(批准号: 11401243, 61403157)和中央高校基本科研业务费专项资金(批准号: GK201504002)资助的课题.

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