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针对Rössler系统平衡点的Hopf分岔,以Washout滤波器为控制器,详细讨论了控制器参数对Hopf分岔点位置、分岔类型以及周期解振幅的控制问题.首先根据Routh-Hurwitz判据计算了受控系统的参数空间稳定域,找出了对应的Hopf分岔边界,并由此分析了滤波器时间常数、线性控制增益对分岔点位置的影响.然后,引入Normal Form直接法方便地求出系统Hopf分岔Normal Form系数,由此确定出改变分岔类型和周期解振幅的控制器非线性增益选择原则.最后用数值计算验证了本文的结论.
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关键词:
- Rö /
- ssler系统 /
- Washout滤波器 /
- Hopf分岔 /
- Normal Form
The Washout filter is adopted to control Hopf bifurcation in Rössler system. The effect of its parameters on the position of bifurcation point, the bifurcation type, and the amplitude of periodic solution is discussed in detail. Based on Routh criterion, the stability region in parameter space is found out firstly. Then the influence of the filter’s time constant and the linear gain on the location of bifurcation point is analyzed. By using the Direct Normal Form method, the normal form of the controlled Rössler system is deduced and the coefficient of resonant term is expressed as a function of the nonlinear gain. It is revealed that, by changing the sign and the absolute value of the real part of the coefficient, the nonlinear gain can modify the periodic solution’s amplitude and Hopf bifurcation type. All the results obtained are verified by numerical simulation.-
Keywords:
- Rö /
- ssler system /
- Washout filter /
- Hopf bifurcation /
- Normal Form
[1] Wu Z, Yu P 2006 IEEE Trans on Automatic Control 51 1019
[2] Wang Y,Murray R M 2002 Automatica 38 611
[3] Goman M G, Khramtsovsky A V 1998 Philosophical Transactions: Mathematical, Physical and Engineering Sciences 356 2277
[4] Wang Q H,Zhou S X 2003 Proceeding of the CSEE 23 18(in Chinese)[王庆红、周双喜 2003 中国电机工程学报 23 18]
[5] Venkatasubramanian V,Sebattler H,Zaborszky J 1995 IEEE Trans on Automatic Control 40 1992
[6] Yu P, Chen G R 2004 International Journal of Bifurcation and Chaos 14 1683
[7] Rössler O E 1976 Phys. Lett. A 35397
[8] Chen L,Wang D S 2007 Acta Phys. Sin. 56 5661(in Chinese)[谌 龙、王德石 2007 物理学报 56 5661]
[9] Wu Z Q 1996 Ph. D. Thesis (Tianjin University) (in Chinese)[吴志强 1996 博士学位论文(天津大学)]
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[1] Wu Z, Yu P 2006 IEEE Trans on Automatic Control 51 1019
[2] Wang Y,Murray R M 2002 Automatica 38 611
[3] Goman M G, Khramtsovsky A V 1998 Philosophical Transactions: Mathematical, Physical and Engineering Sciences 356 2277
[4] Wang Q H,Zhou S X 2003 Proceeding of the CSEE 23 18(in Chinese)[王庆红、周双喜 2003 中国电机工程学报 23 18]
[5] Venkatasubramanian V,Sebattler H,Zaborszky J 1995 IEEE Trans on Automatic Control 40 1992
[6] Yu P, Chen G R 2004 International Journal of Bifurcation and Chaos 14 1683
[7] Rössler O E 1976 Phys. Lett. A 35397
[8] Chen L,Wang D S 2007 Acta Phys. Sin. 56 5661(in Chinese)[谌 龙、王德石 2007 物理学报 56 5661]
[9] Wu Z Q 1996 Ph. D. Thesis (Tianjin University) (in Chinese)[吴志强 1996 博士学位论文(天津大学)]
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