一类非自治位置时滞反馈控制系统的亚谐共振响应

研究了弹性轨道条件下，控制回路中位置反馈信号存在时滞的磁浮系统在亚谐轨道激励作用下的响应问题. 将动力学模型在平衡点处线性化，以时滞为分岔参数，得到了系统出现Hopf分岔的条件. 用中心流形约化方法得到了包含轨道扰动系统的Poincaré规范型. 用多尺度法从理论上推导了时滞磁浮系统的亚谐共振周期解，得到了自由振动的分岔响应方程，分析了周期解中自由振动项的存在条件，研究了控制参数和激励参数与周期解的关系. 最后用数值仿真的方法分析了时滞参数、控制参数对系统响应的影响，分析结果指出，使系统保持稳定的亚谐响应的时滞边界小于无扰动时的时滞边界，时滞参数不但可以抑制亚谐响应，还能够控制混沌的产生，而控制参数可以控制系统响应中自由振动项的出现和受迫振动的幅值，适当选择这些参数可以有效抑制亚谐振动响应.关键词：亚谐共振响应位置时滞反馈控制非自治磁浮系统分岔

Sub-harmonic resonances of the non-autonomous system with delayed position feedback control

The response of the Maglev system with delayed position feedback control under the sub-harmonic excitation of the flexible guideway is investigated. The dynamical model is linearized at the equilibrium. Employing time delay as its bifurcation parameter, the condition under which the Hopf bifurcation may occur is investigated. Center manifold reduction is applied to get the Poincar~ normal form of the nonlinear system with guideway disturbance so that we can study the relation between periodic solution and system parameter. The sub-harmonic resonant periodic solution of the normal form is calculated based on the method of multiple scales, and we get the bifurcation equation of the free oscillation. The existence condition of the free oscillation in the solution is analyzed. Relationship between periodic solution and control and excitation parameters is also investigated. Finally numerical method is applied to study how system and excitation parameters affect the system response. It was shown that the critical time delay to keep the response of the system stable is less than that without perturbation. Time delay can not only suppress sub-harmonic resonance, but also control the appearance of the chaos. Control parameter can govern the emergence of the free oscillation and affect the amplitude of the forced oscillation. So carefully selecting the system parameters can restrain the oscillation effectively.