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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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