分数阶 PID 控制对单自由度线性振子的影响

摘要

The free vibration of a linear single degree-of-freedom (SDOF)oscillator with a fractional-order PID controller of velocity feedback was investigated by the averaging method,and the approximately analytical solution was obtained.The results indicate that the effects of the parameters in the fractional-order PID controller on the dynamical properties are characterized.The proportional component of the fractional-order PID controller is characterized in the form of equivalent linear damping.The integral component of the fractional-order PID controller is characterized in the form of equivalent linear damping and equivalent linear stiffness.The differential component of the fractional-order PID controller is characterized in the form of equivalent linear damping and equivalent linear negative stiffness.Those equivalent parameters could distinctly illustrate the effects of the parameters in the fractional-order PID controller on the dynamical response.A comparison of the analytical solution with numerical results is made,and their agreement verifies the correctness of the approximately analytical results.The effects on system control performance of the coefficients and the orders in the fractional-order PID controller were analyzed by time response performance metrics parameters.Finally,the parameters tuning of the fractional-order PID controller based on velocity feedback was demonstrated through controlling a SDOF quarter vehicle suspension model.%研究了基于速度反馈分数阶 PID 控制的单自由度线性振子的自由振动，利用平均法得到了系统的近似解析解。研究发现分数阶 PID 控制的比例环节以等效线性阻尼的形式影响系统的振幅，积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，微分环节以等效线性阻尼和等效线性负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，验证了求解过程和近似解析解的正确性。通过系统响应的性能指标分析了分数阶 PID 控制的比例系数、积分环节系数、微分环节系数以及分数阶阶次变化时，对系统控制性能的影响。最后，通过单自由度1／4车辆悬架模型的控制实例，说明了分数阶 PID 的参数整定过程。