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A lot of studies of control highlight fractional calculus in modeling systems and designing controllers have been carried out. More recently, a lot of chaotic behaviors have been found in fractional-order systems. Then, controlling the fractional-order systems, especially controlling nonlinear fractional-order systems has become a hot research subject. The design of state estimators is one of the essential points in control theory. Time delays are often considered as the sources of complex behaviors in dynamical systems. A lot progress has been made in the research of time delay systems with real variables. In recent years, fractional-order time-delay chaotic synchronization and chaotic secure communication have received ever-increasing attention. In this paper we focus our study on the synchronization of fractional-order time-delay chaotic systems and its application in secure communication. Firstly, based on the Lipschitz condition, the nonlinear fractional-order time-delay system is proposed. Secondly, the fractional-order time-delay observer for the system is constructed. The necessary and sufficient conditions for the existence of the fractional-order observer are given by some lemmas. Thirdly, the synchronous controller is designed based on the state observer and the stability theory of fractional-order system. Instead of the state variables, the output variables of drive system and response system are used to design the synchronous controller, which makes the design much more simple and practical. With the Lyapunov stability theory and fractional order matrix inequalities, the method of how to obtain the parameters of the controller is presented. The sufficient conditions for asymptotical stability of the state error dynamical system are derived. After that, with the Chen fractional-order time-delay chaotic system, the synchronous controller is designed to make the system run synchronously. Finally, the proposed approach is then applied to secure communications, where the information signal is injected into the transmitter and simultaneously transmitted to the receiver. With the observer design technique, a chaotic receiver is then derived to recover the information signal at the receiving end of the communication. In the conventional chaotic masking method, the receiver is driven by the sum of the information signal and the output of the transmitter, whose dynamics is autonomous. The simulation results show that the design of the synchronous controller works effectively and efficiently, which implies that the proposed fractional order time-delay observer in this paper runs effectively. The proposed method is able to be applied to other fractional order time-delay chaos systems, and also to chaotic secure communication system.
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Keywords:
- chaotic synchronization /
- fractional-order time-delay chaotic system /
- fractional state observe controller /
- linear matrix inequality
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[1] Xu Z, Liu C X, Yang T 2010 Acta Phys. Sin. 59 1524 (in Chinese)[许喆, 刘崇新, 杨韬2010 59 1524]
[2] Ardashir M, Sehraneh G, Okyay K, Sohrab K 2016 Appl. Soft. Comput. 49 544
[3] Liang Y, Wang X Y 2013 Acta Phys. Sin. 62 018901 (in Chinese)[梁义, 王兴元2013 62 018901]
[4] Ou Y C, Lin W T, Cheng R J, Mo J Q 2013 Acta Phys. Sin. 62 060201 (in Chinese)[欧阳成, 林万涛, 程荣军, 莫嘉琪2013 62 060201]
[5] Faieghi M R, Delavari H 2012 Commu. Non. Sci. Num. Simu. 17731
[6] Xu J, Pei L J 2006 Adv. Mech. 36 17 (in Chinese)[徐鉴, 裴利军2006力学进展36 17]
[7] Cai Q Q 2015 Acta Phys. Sin. 64 240506 (in Chinese)[柴琴琴2015 64 240506]
[8] Huang L L, Ma N 2012 Acta Phys. Sin. 61 160510 (in Chinese)[黄丽莲, 马楠2012 61 160510]
[9] Niu H, Zhang G S 2013 Acta Phys. Sin. 62 130502 (in Chinese)[牛弘, 张国山2013 62 130502]
[10] Gamal M M, Emad E M, Ayman A A 2015 Nonlin. Dyn. 80 855
[11] Velmurugan G, Rakkiyappan R 2016 J. Comput. Nonlin. Dyn. 11 031016
[12] Song Z 2016 Complexity 21 131
[13] Song X N, Song S A, Li B 2016 Optik 127 11860
[14] Jiang G P, Zheng W X, Tang W K, Chen G R 2006 IEEE Trans. Circuits-Ⅱ 53 110
[15] Jiang G P, Chen G R, Tang W K 2004 IEEE Trans. Circuits-Ⅱ 51 281
[16] He S B, Sun K H, Wang H H 2014 Acta Phys. Sin. 63 030502(in Chinese)[贺少波, 孙克辉, 王会海2013 62 030502]
[17] Deng W 2014 Adv. Comput. Math. 40 174
[18] Petráš I 2010 Commu. Non. Sci. Num. Simu. 15 384
[19] Ibrahima N D, Holger V, Mohamed D 2013 IEEE J. Emer. Sele. Top. Circ. Syst. 3 442
[20] Li Y, Chen Y, Podlubny I 2010 Comput. Math. Appl. 59 1810
[21] Mahdi P, Elham A B 2016 Circ. Syst. Signal Pr. 35 1855
[22] Darouach M, Zasadzinski M, Xu S 1994 IEEE Trans. Automat. Contr. 39 606
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