分数阶PID控制对单自由度线性振子的影响

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摘要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，证实了基于速度反馈分数阶PID控制的优越性。

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关键词 ：分数阶PID控制, 平均法, 近似解析解

Abstract：The free vibration of a linear single degree-of-freedom (SDOF) oscillator with fractional-order PID controller of velocity feedback is investigated by the averaging method, and the approximately analytical solution is obtained. The effects of the parameters in fractional PID controller on the dynamical properties are characterized. The proportional component of fractional-order PID controller is characterized in the form of equivalent linear damping. The integral component of fractional-order PID controller is characterized in the form of equivalent linear damping and the equivalent linear stiffness. The differential component of fractional-order PID controller is characterized in the form of equivalent linear damping and the equivalent linear negative stiffness. Those equivalent parameters could distinctly illustrate the effects of the parameters in fractional PID controller on the dynamical response. A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximately analytical results. The effects on system control performance of the coefficients and the orders in fractional-order PID controller are analyzed by time response performance metrics parameters. Finally, the superiority of the fractional-order PID controller based on velocity feedback is demonstrated through the control of a SDOF quarter vehicle suspension model.

Key words： fractional-order PID control averaging method approximately analytical solution

收稿日期: 2015-06-23 出版日期: 2016-12-15

引用本文:

牛江川，申永军，杨绍普，李素娟. 分数阶PID控制对单自由度线性振子的影响[J]. 振动与冲击, 2016, 35(24): 88-95.
NIU Jiang-Chuan SHEN Yong-Jun YANG Shao-Pu LI Su-Juan. Effect of fractional-order PID Controller on the dynamical response of linear single degree-of-freedom oscillator. JOURNAL OF VIBRATION AND SHOCK, 2016, 35(24): 88-95.

链接本文:

http://jvs.sjtu.edu.cn/CN/ 或 http://jvs.sjtu.edu.cn/CN/Y2016/V35/I24/88

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