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基于LQR的二级倒立摆控制系统研究

摘 要

倒立摆系统是一个典型的多变量、非线性、强耦合和快速运动的高阶不稳定系统，它是检验各种新的控制理论和方法有效性的典型理想模型。在其控制过程中，能有效地反映诸如镇定性、鲁棒性、随动性以及跟踪等许多关键问题。本文主要研究二级倒立摆LQR控制方法。首先建立了二级倒立摆的数学模型，然后对二级倒立摆的数学模型进行控制设计，应用遗传算法确定系统性能指标函数中的加权阵Q，R得到系统状态反馈控制矩阵。最后，用MATLAB进行了系统仿真。在几次凑试Q矩阵值后系统的响应结果都不尽如人意，于是采用遗传算法对Q矩阵优化。仿真结果证明：经过遗传算法优化后的系统响应能更加满足设计要求。

关键词：二级倒立摆；LQR控制；遗传算法

Research on double inverted pendulum control

system based on LQR

ABSTRACT

The inverted pendulum is a typical -linear, strong-coupling, fleet and absolutely

instable. It is representative as an ideal model to prove new control theory and techniques. During the control process, pendulum can effectively reflect many key problems such as equanimity, robust, follow-up and track, therefore. This paper studies a control method of double inverted pendulum LQR. First of all, the mathematical model of the double inverted pendulum is established, then make a control design to double inverted pendulum on the mathematical model, and determine the system performance index weight matrix Q, R by using genetic algorithm in order to attain the system state feedback control matrix. Finally, the simulation of the system is made by MATLAB. After several test matrix Q value the results are not satisfactory response, then we optimize Q matrix by using Genetic Algorithm. Simulation results show: The system response can meet the design requirements effectively after Genetic Algorithm optimization.

Key words：Double inverted pendulum; LQR control; Genetic Algorithm.

目录

摘 要 ................................................................................................................................................................... I ABSTRACT ................................................................................................................................................. II 1 绪论 ................................................................................................................................................................ 1

1.1 引言 ........................................................................................................................................................... 1 1.2 倒立摆设备简介 ................................................................................................................................... 1 1.3 倒立摆系统简介 ................................................................................................................................... 3 1.4国内外研究情况 .................................................................................................................................... 5 1.5论文的主要内容 .................................................................................................................................... 8

2数学模型的建立和分析 .................................................................................................................... 9

2.1 数学建模的方法 .................................................................................................................................... 9 2.2二级倒立摆的结构和工作原理 ....................................................................................................... 9 2.3拉格朗日运动方程 ............................................................................................................................. 10 2.4推导建立数学模型 ............................................................................................................................. 11 2.5 二级倒立摆系统性能分析 .............................................................................................................. 17 2.5.1 稳定性分析 ................................................................................................................................... 17 2.5.2 能控性能观性分析 .................................................................................................................... 17

3 线性二次最优控制算法简介 ....................................................................................................... 19

3.1线性二次型最优调节器原理 .......................................................................................................... 19 3.2 加权阵Q、R的选择 ....................................................................................................................... 21

4 遗传算法 ................................................................................................................................................... 22