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在混沌系统的同步控制中, 由于混沌系统对初始状态的敏感性, 一旦两个混沌系统的状态初值偏差大, 其状态同步往往需要高幅值的控制律来达到, 这给同步控制实现带来了困难, 并且在同步控制中, 两个混沌系统的初始值通常是未知的. 本文考虑控制输入受限情况下的混沌同步控制问题, 基于符号函数的近似表示式, 将受限的控制输入建模为连续可微的光滑函数, 在每一个采样点将同步控制误差系统近似为局部最优线性模型并设计连续型线性二次型调节器(LQR)最优控制律. 为降低混沌同步控制律的幅值和维持同步系统采样时刻之间的动态, 设计了等价的离散最优控制律, 并通过调整LQR性能加权矩阵值, 确保同步控制信号不会超出其受限的上界. 最后对统一混沌模型下的三种不同混沌系统同步控制进行了仿真研究. 仿真结果验证了方法的有效性.It is well known that the dynamics of the chaotic system is very sensitive to the initial conditions of the state, and the synchronization of two identical chaotic systems is only obtained, in general, with the high gain control law once their initial conditions are in a certain large deviation. Furthermore, the initial conditions are commonly unknown in practice, which causes difficulty in synchronizing two chaotic systems. This paper deals with the synchronization of two unified chaotic systems with input constraint. First, the scalar sign function is utilized to approximate the constrained non-smooth input function so that a continuous smooth nonlinear input function and an approximated nonlinear synchronized error system are obtained. Then, an optimal linear quadratic regulator (LQR) continuous-time control law is designed based on the optimal linear model, which is constructed at the sampled operating point of the afore-mentioned approximated nonlinear synchronized error system. To reduce the high magnitude of the obtained control law, the continuous-time control law is digitally redesigned for the implementation and an iterative procedure is proposed to adjust the weighting matrices in the LQR performance index so as to avoid input saturation occurs. Finally, three illustrative examples of the Lorenz, the Chen and the L chaotic systems decomposed from the unified chaotic system are given to demonstrate the effectiveness of the proposed method.
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Keywords:
- unified chaotic system /
- sign function /
- input constraint /
- synchronization control
[1] Zribi M, Smaoui N, Salim H 2009 Chaos Soliton. Fract. 42 3197
[2] Huang L L, Qi X 2013 Acta Phys. Sin. 62 080507 (in Chinese) [黄丽莲, 齐雪 2013 62 080507]
[3] Qi D L, Wang Q, Yang J 2011 Chin. Phys. B 20 100505
[4] Chen Z W, Wang J, Pang S J 2012 Acta Phys. Sin. 61 220505 (in Chinese) [陈志旺, 王敬, 庞双杰 2012 61 220505]
[5] Che Y Q, Wang J, Chan W L, Tsang K M 2010 Nonlinear Dyn. 61 847
[6] Zang H Y, Min L Q, Zhao Q, Chen G R 2013 Chin. Phys. Lett. 30 040502
[7] Fu S H, Lu Q S, Du Y 2012 Chin. Phys. B 6 060507
[8] Li H Y, Hu Y A, Ren J C, Zhu M, Liu L 2012 Acta Phys. Sin. 61 140502 (in Chinese) [李海燕, 胡云安, 任建存, 朱敏, 刘亮 2012 61 140502]
[9] Shan L, Li J, Wang Z Q 2006 Acta Phys. Sin. 55 3950 (in Chinese) [单梁, 李军, 王执铨 2006 55 3950]
[10] Li C B, Chen S, Zhu H Q 2009 Acta Phys. Sin. 58 2255 (in Chinese) [李春彪, 陈谡, 朱焕强 2009 58 2255]
[11] Li S H, Cai H X 2004 Acta Phys. Sin. 53 1687 (in Chinese) [李世华, 蔡海兴 2004 53 1687]
[12] Ott E, Grebogi C, Yorke J A 1990 Phys. Rev. Lett. 64 1196
[13] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[14] Guo S M, Shieh L S, Lin C F, Chandra J 2001 Int. J. Bifurcat. Chaos 11 1079
[15] Azzaz M S, Tanougast C, Sadoudi S, Bouridance A 2013 Commun. Nonlinear Sci. Numer. Simul. 18 2035
[16] Ali S Z, Islam M K, Zafrullah M 2012 Optimal Rev. 19 320
[17] Wei Y, Fan L, Xia G Q, Chen Y L, Wu Z M 2012 Acta Phys. Sin. 61 224203 (in Chinese) [魏月, 樊利, 夏光琼, 陈于淋, 吴正茂 2012 61 224203]
[18] Luo C, Wang X Y 2013 Int. J. Mod. Phys. C 24 1350025
[19] Wang X Y, Zhang N, Ren X L, Zhang Y L 2011 Chin. Phys. B 20 020507
[20] Zhu F L 2009 Chaos Soliton. Fract. 40 2384
[21] Bouraoui H, Kemih K 2013 Acta Phys. Pol. A 123 259
[22] Shieh L S, Tsay Y T, Yates R 1983 IEEE Proc. Cont. Th. App. Part D 130 111
[23] Chen Y S, Tsai J S H, Shieh L S, Kung F C 2002 IEEE Trans. Circ. Syst. I 49 1860
[24] Xie L B, Ozkul S, Sawant M, Shieh L S, Tsai J S H 2012 Int. J. Syst. Sci. 752546
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[1] Zribi M, Smaoui N, Salim H 2009 Chaos Soliton. Fract. 42 3197
[2] Huang L L, Qi X 2013 Acta Phys. Sin. 62 080507 (in Chinese) [黄丽莲, 齐雪 2013 62 080507]
[3] Qi D L, Wang Q, Yang J 2011 Chin. Phys. B 20 100505
[4] Chen Z W, Wang J, Pang S J 2012 Acta Phys. Sin. 61 220505 (in Chinese) [陈志旺, 王敬, 庞双杰 2012 61 220505]
[5] Che Y Q, Wang J, Chan W L, Tsang K M 2010 Nonlinear Dyn. 61 847
[6] Zang H Y, Min L Q, Zhao Q, Chen G R 2013 Chin. Phys. Lett. 30 040502
[7] Fu S H, Lu Q S, Du Y 2012 Chin. Phys. B 6 060507
[8] Li H Y, Hu Y A, Ren J C, Zhu M, Liu L 2012 Acta Phys. Sin. 61 140502 (in Chinese) [李海燕, 胡云安, 任建存, 朱敏, 刘亮 2012 61 140502]
[9] Shan L, Li J, Wang Z Q 2006 Acta Phys. Sin. 55 3950 (in Chinese) [单梁, 李军, 王执铨 2006 55 3950]
[10] Li C B, Chen S, Zhu H Q 2009 Acta Phys. Sin. 58 2255 (in Chinese) [李春彪, 陈谡, 朱焕强 2009 58 2255]
[11] Li S H, Cai H X 2004 Acta Phys. Sin. 53 1687 (in Chinese) [李世华, 蔡海兴 2004 53 1687]
[12] Ott E, Grebogi C, Yorke J A 1990 Phys. Rev. Lett. 64 1196
[13] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[14] Guo S M, Shieh L S, Lin C F, Chandra J 2001 Int. J. Bifurcat. Chaos 11 1079
[15] Azzaz M S, Tanougast C, Sadoudi S, Bouridance A 2013 Commun. Nonlinear Sci. Numer. Simul. 18 2035
[16] Ali S Z, Islam M K, Zafrullah M 2012 Optimal Rev. 19 320
[17] Wei Y, Fan L, Xia G Q, Chen Y L, Wu Z M 2012 Acta Phys. Sin. 61 224203 (in Chinese) [魏月, 樊利, 夏光琼, 陈于淋, 吴正茂 2012 61 224203]
[18] Luo C, Wang X Y 2013 Int. J. Mod. Phys. C 24 1350025
[19] Wang X Y, Zhang N, Ren X L, Zhang Y L 2011 Chin. Phys. B 20 020507
[20] Zhu F L 2009 Chaos Soliton. Fract. 40 2384
[21] Bouraoui H, Kemih K 2013 Acta Phys. Pol. A 123 259
[22] Shieh L S, Tsay Y T, Yates R 1983 IEEE Proc. Cont. Th. App. Part D 130 111
[23] Chen Y S, Tsai J S H, Shieh L S, Kung F C 2002 IEEE Trans. Circ. Syst. I 49 1860
[24] Xie L B, Ozkul S, Sawant M, Shieh L S, Tsai J S H 2012 Int. J. Syst. Sci. 752546
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