matlab大作业平衡杆小球位置控制系统华中科技大学.docVIP

MATLAB语言、控制系统分析与设计 大作业 平衡杆小球位置控制系统 设计与仿真 专 业：电气工程及其自动化 班 级： 设 计 者： 学 号： 华中科技大学电气与电子工程学院 2008年1月 平衡杆小球位置控制系统设计与仿真 一、问题描述与实验要求 A ball is placed on a beam, see figure below, where it is allowed to roll with 1 degree of freedom along the length of the beam. A lever arm is attached to the beam at one end and a servo gear at the other. As the servo gear turns by an angle theta, the lever changes the angle of the beam by alpha. When the angle is changed from the vertical position, gravity causes the ball to roll along the beam. A controller will be designed for this system so that the balls position can be manipulated. For this problem, we will assume that the ball rolls without slipping and friction between the beam and ball is negligible. The constants and variables for this example are defined as follows: M mass of the ball 0.11 kg R radius of the ball 0.015 m D lever arm offset 0.03 m G gravitational acceleration 9.8 m/s^2 L length of the beam 1.0 m J balls moment of inertia 9.99e-6 kgm^2 R ball position coordinate Alpha beam angle coordinate Theta servo gear angle System Equations The Lagrangian equation of motion for the ball is given by the following: Linearization of this equation about the beam angle, alpha = 0, gives us the following linear approximation of the system: The equation which relates the beam angle to the angle of the gear can be approximated as linear by the equation below: Substituting this into the previous equation, we get: Design requirements The design criteria for this problem are: ? Settling time less than 3 seconds ? Overshoot less than 5% 1．数学模型的建立 将上面推导的简化式做拉普拉斯变换，得到： 变化后得到： 化简为： 2．设计目标 希望能精确小球的位置，即要求小球的稳态误差为零，同时希望因扰动引起的稳态误差也能为零。对于动态性能的要求，希望小球能较快且平稳，期望调节时间Ts为3s，超调量小于5%。根据时域和频域指标的关系，可将时域性能指标转换为频率响应的约束条件。如系统的带宽与闭环系统自然振荡频率?n和阻尼比?有关，而??n与调节时间有关。相角裕度PM和阻尼比?有关，进而与超调量相关。 （1-4） zeta = -log(.05)/sqrt(pi^2+(log(.05))^2); PM = 100*zet