摘要：倒立摆是典型的多变量、非线性、强耦合的自然不稳定系统，可以把许多抽象问题直观的表达出来，所以对倒立摆系统的研究在理论上和工程实践上均有着深远意义。
本课题首先利用牛顿力学分析的方法建立了直线一级倒立摆系统的数学模型，并在此基础上分析了该系统是不稳定的，同时又是能控和能观的。然后研究了倒立摆系统的线性二次最优控制算法，并设计了倒立摆系统的二次最优控制器，利用MATLAB仿真分析倒立摆系统的Q,R矩阵，通过仿真结果获得了实时控制效果，使得摆杆保持竖直向上平衡的同时，能跟踪小车的位置。最后，结合倒立摆的二次最优控制器设计和能量自动摆起控制，实现了直线一级倒立摆的自动摆起的LQR实时控制。可以得出，LQR法具有较小的超调量和较好的稳态效果，LQR法适合应用到对稳态性能要求较高的控制系统中。6861
关键词：倒立摆；最优控制；建模与分析；MATLAB仿真；自动摆起
Quadratic optimal control for an inverted pendulum system
Abstract：Inverted pendulum is a typical multi-variable, non-linear, strong coupling and naturally unstable system，it can express many abstract problems directly, So the Research of inverted pendulum system have far-reaching significance in theory and in practice.
In this paper, we firstly use Newtonian mechanics analysis method to establish the mathematical model of the linear 1-stage inverted pendulum system, in the mean time, the system is unstable by analyzing the model, but it is controllable and observable. Then we study on algorithm of linear quadratic optimal control for inverted pendulum system, and design a quadratic optimal controller of inverted pendulum system, Simulation and analysis the matrix Q,R of an inverted pendulum system using MATLAB. By simulation results, gained the real-time control effect, While maintaining the pendulum rod balance the vertical up, tracking the car's location. Finally, Combining the design of inverted pendulum system’s quadratic optimal controller, and the automatic swing-up control, achieving linear inverted pendulum’s LQR control of automatic swing up. It can be concluded that LQR method has smaller overshoot and better steady state results, and LQR method is suitable for applications that require higher steady state performance in the control system.
KeyWords: Inverted pendulum; Optimal control; Modeling and analysis; Matlab simulation; swing-up control
目录
1 绪论 1
1.1 倒立摆系统的研究意义 1
1.2 倒立摆系统简介 1
1.2.1 倒立摆系统工作原理 1
1.2.2 倒立摆系统的特点 3
1.3 国内外研究情况简介 3
1.3.1 倒立摆稳定控制研究 3
1.3.2 倒立摆起摆控制研究 4
1.4 倒立摆的主要控制方法 5
1.5 本课题的主要工作 5
2 倒立摆系统建模和定性分析 7
2.1 直线一级倒立摆的数学模型 7
2.1.1 倒立摆系统建模假设与符号说明 7
2.1.2 牛顿力学建模 8
2.1.3 实际系统模型 11
2.2 倒立摆系统的定性分析 12
3 最优控制基本理论 15
3.1 最优控制简介 15
3.2 线性二次最优控制问题的提出 16
3.3 线性二次最优控制基本原理 17
4 一级倒立摆的二次最优控制设计 19
4.1 二次最优控制器设计 19
4.2 仿真分析与实时控制 20
4.2.1 Matlab相关介绍 20 MATLAB一级倒立摆的二次最优控制系统设计仿真:http://www.youerw.com/zidonghua/lunwen_4603.html