位移反馈分数阶PID控制对单自由度线性振子的影响
Effect of fractional-order PID controller on the dynamical response of linear single degree-of-freedom oscillator with displacement feedback

研究了基于位移反馈分数阶PID控制的单自由度线性振子的自由振动, 通过平均法得到了系统的一阶近似 解析解. 发现分数阶PID控制器的比例环节以等效线性刚度的形式影响系统的动力学特性, 积分环节以等效线性负 阻尼和等效线性刚度的形式影响系统的动力学特性, 微分环节以等效线性阻尼和等效线性刚度的形式影响系统的 动力学特性. 对近似解析解和数值解进行了比较, 二者吻合良好, 验证了近似解析解的正确性. 从近似解析解和分 数阶系统的特征方程两个角度对系统的稳定性进行了分析. 最后利用系统的时间响应性能指标分析了位移反馈分 数阶PID控制器的系数和分数阶阶次变化时, 对单自由度线性振子控制性能的影响.
The free vibration of a linear single degree-of-freedom (SDOF) oscillator with fractional-order proportionalintegral- derivative (PID) controller based on displacement feedback is investigated by the averaging method, and the approximate analytical solution is obtained. The effects of the parameters in fractional-order PID controller on the dynamical properties are characterized, where the proportional component is characterized in the form of equivalent linear stiffness, the integral component is characterized in the form of equivalent linear negative damping and equivalent linear stiffness, and the differential component is characterized in the form of equivalent linear damping and equivalent linear stiffness. Those equivalent parameters could distinctly illustrate the effects of the parameters in fractional-order PID controller on the dynamical response. A comparison of the approximate analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximate results. The system stability is analyzed based on the approximate analytical solution and the characteristic equation of the fractional-order system. Finally, the effects on system control performance of fractional-order PID controller for linear SDOF oscillator with displacement feedback are analyzed by the time response performance metrics parameters，when the coefficients and orders of fractional-order PID controller are changed.