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系统地研究了Langford 系统Hopf分岔极限环幅值非线性反馈控制问题.根据中心流形定理和规范型降维理论, 推导含控制增益项的非线性曲率系数控制公式和幅值近似解,通过数值仿真验证并绘制极限环幅控关系曲线. 所推导的控制公式为Langford 系统极限环幅值控制提供了方便有效的控制方法.
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关键词:
- 幅值控制 /
- Langford系统 /
- 极限环 /
- Hopf分岔
The control of amplitude of limit cycle emerging from the Hopf bifurcation in Langford system under a nonlinear feedback controller is investigate in this paper. Explicit nonlinear control formulae and amplitude approximations in terms of control gains are derived by utilizing the center manifold theory and normal form reduction. Gain-amplitude curves for controlled systems are drawn and verified by numerical simulations. The formulae and expressions for the Langford system present a convenient approach to obtaining an effective analytical control in this system.-
Keywords:
- amplitude control /
- Langford system /
- limit cycle /
- Hopf bifurcation
[1] Berns D W, Moiola J L, Chen G R 1998 Automatica 34 1567
[2] Oueini S S Nayfeh A H 1999 J. Sound and Vibration 224 33
[3] Berns D W, Moiola J L, Chen G R 2000 Int. J. Bifur. Chaos 10 2815
[4] Angulo F, Bernardo M, Fossas E 2005 IEEE Trans. Circ. Sys. 52 366
[5] Tang J S, Chen Z L 2006 Int. J. Bifur. Chaos 16 1
[6] Liu X D, Huang W H 1999 J. Vibrat. Eng. 12 21 (in Chinese) [刘向东, 黄文虎 1999 振动工程学报 12 21]
[7] Qian C Z, Fu W B 2005 J. Dyn. Control 3 7 (in Chinese) [钱长照, 符文彬 2005 动力学与控制学报 3 7]
[8] Tang J S, Xiao H 2007 Acta Phys. Sin. 56 101 (in Chinese) [唐驾时, 萧寒 2005 物理学报 56 101]
[9] Liu S H, Tang J S 2008 Acta Phys. Sin. 57 6162 (in Chinese) [刘素华, 唐驾时 2008 物理学报 57 6162]
[10] Liu S H, Zhao C G 2008 J. Dyn. Control 6 141 (in Chinese) [刘素华, 赵成刚 2008 动力学与控制学报 6 141]
[11] Liu S H, Tang J S 2007 Acta Phys. Sin 56 3145 (in Chinese) [刘素华, 唐驾时 2007 物理学报 56 3145]
[12] Liu S H, Wang S G 2008 J. Hunan Univ. 35 55 (in Chinese) [刘素华, 王曙光 2008 湖南大学学报 35 55]
[13] Liu S H, Tang J S 2008 Chin. Phys. B 17 1691
[14] Liu S H 2008 Ph. D. Dissertation (Changsha: Hunan University) (in Chinese) [刘素华 2008 博士学位论文 (长沙:湖南大学)]
[15] Hassard B D, Kazarinoff N D, Wan Y 1981 Theory and Applications of Hopf Bifurcation (London: Cambridge Univ.) p35
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[1] Berns D W, Moiola J L, Chen G R 1998 Automatica 34 1567
[2] Oueini S S Nayfeh A H 1999 J. Sound and Vibration 224 33
[3] Berns D W, Moiola J L, Chen G R 2000 Int. J. Bifur. Chaos 10 2815
[4] Angulo F, Bernardo M, Fossas E 2005 IEEE Trans. Circ. Sys. 52 366
[5] Tang J S, Chen Z L 2006 Int. J. Bifur. Chaos 16 1
[6] Liu X D, Huang W H 1999 J. Vibrat. Eng. 12 21 (in Chinese) [刘向东, 黄文虎 1999 振动工程学报 12 21]
[7] Qian C Z, Fu W B 2005 J. Dyn. Control 3 7 (in Chinese) [钱长照, 符文彬 2005 动力学与控制学报 3 7]
[8] Tang J S, Xiao H 2007 Acta Phys. Sin. 56 101 (in Chinese) [唐驾时, 萧寒 2005 物理学报 56 101]
[9] Liu S H, Tang J S 2008 Acta Phys. Sin. 57 6162 (in Chinese) [刘素华, 唐驾时 2008 物理学报 57 6162]
[10] Liu S H, Zhao C G 2008 J. Dyn. Control 6 141 (in Chinese) [刘素华, 赵成刚 2008 动力学与控制学报 6 141]
[11] Liu S H, Tang J S 2007 Acta Phys. Sin 56 3145 (in Chinese) [刘素华, 唐驾时 2007 物理学报 56 3145]
[12] Liu S H, Wang S G 2008 J. Hunan Univ. 35 55 (in Chinese) [刘素华, 王曙光 2008 湖南大学学报 35 55]
[13] Liu S H, Tang J S 2008 Chin. Phys. B 17 1691
[14] Liu S H 2008 Ph. D. Dissertation (Changsha: Hunan University) (in Chinese) [刘素华 2008 博士学位论文 (长沙:湖南大学)]
[15] Hassard B D, Kazarinoff N D, Wan Y 1981 Theory and Applications of Hopf Bifurcation (London: Cambridge Univ.) p35
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