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

doi:10.7498/aps.64.070503

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.

Keywords:

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).

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[3]	Wang D F, Zhang J Y, Wang X Y 2013 Chin. Phys. B 22 100504
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[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
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[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 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

Cited By

[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 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 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 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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Vol. 64, Issue 7 - 2015-04-05

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