时滞反馈对三稳态van der Pol系统稳态概率密度的影响

Effects of time-delay feedback on steady state probability density of tri-stable van der Pol system

利用时滞反馈调节系统随机响应特征是随机动力学与控制重要的课题之一。为解决多稳态随机系统的控制器参数设计问题,针对加性噪声激励下的三稳态van der Pol系统,研究了时滞差分反馈对系统稳态概率密度的影响。利用随机平均法得到系统幅值稳态概率密度函数的解析表达式,随后分别讨论了给定噪声强度情况下时滞和反馈强度对系统稳态概率密度的影响。结果表明,反馈强度和时滞变化均能使系统稳态概率密度曲线的拓扑结构发生改变,从而调节系统稳态响应的幅值分布;转迁集计算的结果对控制参数选择有直接指导作用。

It is an important topicin stochastic dynamics and control to adjustsystem’s stochastic response characteristics using time-delay feedback.Here,to solve the design problem of controller parameters for multi-stable stochastic systems,effects of time-delay difference feedback on steady state probability density of a tri-stable van der Pol system under additive noise excitation were studied.The stochastic averaging method wasused to derive the analytical expression for the steady state probability density function of the system’s vibration amplitude,and then effects of time delay and feedback intensity on the system’s steady state probability density under given noise intensity were discussed,respectively.Results showedthat both feedback intensity and time delay can change topological structure of the system’s steady state probability density curve toadjust amplitude distribution of the system’s steady state response;the results of transition set calculation provide a direct guidance for choosing control parameters.