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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。
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本课题以固高公司直线倒立摆为研究对象，对直线二级倒立摆系统进行研究分析，运用Matlab实时控制软件对二级倒立摆系统进行二次最优控制算法的设计，解决摆杆摆角的有关控制问题。首先利用牛顿动力学方法建立了二级倒立摆系统的数学模型，利用Matlab仿真分析其开环系统的不稳定性、能控性和能观性。其次以最优控制理论为原理，设计出二级倒立摆的LQR控制器，课题研究了二次最优控制中加权矩阵Q和R难以确定的问题，通过找出二级倒立摆系统的动态响应与Q和R阵遵循的基本规律，研究和设计了连续二次最优控制策略的算法，得到最优控制器对应的反馈增益矩阵。最后，通过Simulink进行了倒立摆系统的LQR实时控制，取得较好的控制效果。9099
关键词：二级倒立摆；LQR；MATLAB仿真；实时控制
Design of Quadratic Optimal Control of a Double Inverted pendulum
ABSTRACT: The inverted pendulum is a typical high order system, with multi- variable, non-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, the inverted pendulum can effectively reflect many key problems such as equanimity, robust, follow-up and track, and so on.
This dissertation, based on the linear inverted Pendulum of Googol Company, has analyzed the double inverted Pendulum system and solved control issues of the pendulum swing angle, with the help of Matlab real-time control software experiment platform which designed by quadratic optimal control algorithm double inverted pendulum system. Firstly, the writer established the mathematical model of the double inverted pendulum system with Lagrange equations, uses Matlab software to analyze the open-loop system instability, controllability and observability. Second by using optimization control theory, design the double inverted Pendulum LQR controller. This dissertation researches that the weight matrices Q and R of the quadratic optimal control are difficult to be determined. By finding the basic law followed by the Q and R matrix and the dynamic response of the double Inverted pendulum, the writer researches and designs the algorithms of continuous quadratic optimal control strategy, gets feedback gain matrix of the optimal controller. Finally, through Simulink conducted LQR real-time control of double inverted pendulum system, the best control result has been achieved.
Key words: Double inverted pendulum; LQR control; Matlab simulation; Real-time control
目 录
1 绪论 1
1.1 倒立摆系统研究的意义 1
1.2 倒立摆种类简介 2
1.3 倒立摆的特性 3
1.4 国内外研究情况简介 3
1.5 课题的主要内容 5
2 倒立摆系统建模与分析 7
2.1 二级倒立摆的结构和工作原理 7
2.2 数学建模的方法 8
2.3 拉格朗日运动方程 8
2.4 推导建立数学模型 9
2.5 二级倒立摆系统性能分析 14
2.5.1 倒立摆系统稳定性分析 14
2.5.2 倒立摆系统可控性分析 14
3 最优控制理论的简介 16
3.1 最优控制理论的发展历程 16
3.2 线性二次型最优调节器原理 17
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