Abstract: In order to solve the problem of stability control of the linear inverted pendulum, the fractional order proportional integral (FOPI and FO[PI]) controller is designed to correct the system. Firstly, the mathematical model of inverted pendulum is established according to the Newton mechanics method. Then, using the gain robustness fractional order controller parameter simplified algorithm based on the vector designed the fractional order proportional integral controller. Finally, the validity of the fractional order proportional integral controller parameter tuning method is verified and fractional order proportional integral controller and integer order PID (IOPID) controller are used to carry out stability control simulation experiment and then compared and analyzed the angle response curves of the pendulum .The results show that the fractional order proportional integral controller has better stability control effect than the IOPID controller for the inverted pendulum systems, and in the fractional order proportional integral controller, the FO[PI] controller has better stability control effect for the inverted pendulum systems, with robustness, faster response time and less oscillation amplitude.

Key words: Control theory, linear inverted pendulum, gain robust method, fractional order controller, vector method

摘要： 针对直线一级倒立摆的稳定控制问题，设计了分数阶比例积分(FOPI和FO[PI])控制器。首先，根据Newton力学方法建立了倒立摆系统的数学模型。然后，采用基于向量的增益鲁棒性分数阶控制器参数求解简化算法，设计了分数阶比例积分控制器。最后，在MATLAB环境下进行了分数阶比例积分控制器参数整定方法的有效性验证，并且对倒立摆系统分别采用分数阶比例积分控制器和整数阶PID（IOPID）控制器进行了稳定控制仿真实验，并将得到的摆杆角度响应曲线进行了对比分析。结果表明：分数阶比例积分控制器对系统的稳定控制效果优于IOPID控制器，且在分数阶比例积分控制器中，FO[PI]控制器对系统稳定控制最好，响应时间较快、振荡幅值较小且具有鲁棒性。

关键词: 控制理论, 倒立摆, 增益鲁棒法, 分数阶控制器, 向量法