基于速度反馈分数阶PID控制的达芬振子的主共振

与传统整数阶比例-积分-微分(PID)控制器相比,分数阶比例-积分-微分控制器由于增加了两个控制参数,因此能够更灵活地控制受控对象.研究了基于速度反馈分数阶比例-积分-微分控制的达芬振子的主共振,利用平均法获得了系统的近似解析解.研究发现分数阶比例-积分-微分控制器的比例环节以等效线性阻尼的形式影响系统的共振振幅,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质量的形式影响系统的动力学特性.建立了主共振幅频响应方程的解析表达式和稳定性判断准则,并对主共振幅频响应的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.分析了分数阶比例-积分-微分控制器的比例环节系数、积分环节系数、微分环节系数以及分数阶阶次变化时,对系统主共振幅频响应的影响.对分数阶比例-积分-微分控制器与传统整数阶比例-积分-微分控制器的控制效果进行了对比,发现当控制器各个环节的系数相同时,基于速度反馈的分数阶比例-积分-微分控制对达芬振子主共振的控制效果要优于传统整数阶比例-积分-微分控制.

PRIMARY RESONANCE OF DUFFING OSCILLATOR WITH FRACTIONAL-ORDER PID CONTROLLER BASED ON VELOCITY FEEDBACK

Compared with the traditional integer-order PID controller, the fractional-order PID(FOPID) controller may present much better control performance due to its two surplus adjustable parameters.The primary resonance of Duffing oscillator with FOPID controller of velocity feedback is investigated by the averaging method, and the approximately analytical solution is obtained.The effects of the parameters in FOPID controller on the dynamical properties are studied and characterized by some equivalent parameters.The proportional component of FOPID controller is characterized in the form of equivalent linear damping.The integral component of FOPID controller is characterized in the form of the equivalent linear damping and equivalent linear stiffness.The differential component of FOPID controller is characterized in the form of the equivalent linear damping and equivalent mass.Those equivalent parameters could distinctly illustrate the effects of the parameters in FOPID controller on the dynamical response.The amplitude-frequency equation for steady-state solution and associated with the stability condition are also presented.A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximately analytical results.Finally, the effects on the amplitude-frequency performance of the coefficients and the orders in FOPID controller are analyzed, and the control performances of fractional-order and integer-order PID controller are compared. The results show that the control performance of FOPID controller is better than the traditional integer-order counterpart for controlling the vibration of the primary resonance of Duffing oscillator, when the coefficients of the two controllers are the same.