规范形Qi系统的Hopf分岔分析及控制

文章研究了一个Qi系统的Hopf分岔控制问题.根据计算的极限环曲率系数,判定原系统的Hopf分岔类型,并采用washout滤波器控制该系统的分岔行为.首先讨论了控制器的线性增益对Hopf分岔点位置的影响,然后引入规范形计算方法,求出受控系统的Hopf分岔规范形.分析了规范形中系数对控制参数的选择原则所产生之影响,以及对Hopf分岔类型及极限环幅值的影响.理论和仿真结果表明,控制器的线性增益能使原系统的Hopf分岔点延迟或消失,而非线性增益能则改变极限环的稳定性和极限环幅值的大小.最后把washout滤波器和线性控制器的控制效果作了对比,发现washout滤波器比之线性控制器具有一定的优势.

Bifurcation analysis and control of a Qi system based on normal form

This paper considers the Hopf bifurcation control for a particular dynamical system (Qi system). According to the limit cycle curvature coefficient, we determine the Hopf bifurcation type for the original system, and then the washout filter is adopted to control the Hopf bifurcation. Firstly, the influence of linear gain on the bifurcation point is analyzed; and then, the normal form theory is applied to develop the Hopf bifurcation normal form of the controlled system. We investigate the effects from the coefficients of the normal form on the parameter-selection criteria and the amplitude of the periodic solution as well as the type of Hopf bifurcation of the controlled system. Theoretical and simulation results show that the linear gain of the control function will delay or even eliminate the bifurcation point; while the nonlinear gain can change the stability of limit cycles or the amplitudes of oscillations. Finally, a comparison between the linear controller and the washout filter shows that the washout filter is superior to the linear control.