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。在对倒立摆的控制过程中能反映控制理论中的许多关键问题,如镇定问题、非线性问题、鲁棒性问题以及跟踪问题等,所以倒立摆被广泛的用来验证各种控制理论和控制方法的有效性。对倒立摆系统的研究在理论上和工程应用上具有着深远的意义,相关的科研成果已经应用到航天科技和机器人学等诸多领域。
本文首先利用牛顿力学分析的方法建立了直线一级倒立摆实物系统的线性状态方程,并在此基础上分析了该系统是不稳定的,同时又是能控的和能观的。然后详细的介绍了直线一级倒立摆实物系统的硬件和软件构成,重点阐述了基于MATLAB/Simulink实时工具箱RTW(Real-Time Workshop)的实控软件的使用方法。
本文主要研究了倒立摆系统的PID控制算法、线性二次最优控制算法和极点配置法,并设计出了这三种算法的控制器,同时利用MATLAB/Simulink仿真分析了这三种算法的优缺点。在PID控制器的设计中,本文提出了一种基于MATLAB/Sumlink的参数整定方法,较好和较快的整定了PID的控制参数,并利用MATLAB/Sumlink仿真实验看出传统的PID算法只能较好的控制摆杆的角度这一个输出量,而对另一个输出量——小车的位移却不能有效的控制,因此传统的PID控制对直线一级倒立摆这样的单输入两输出系统是无法实现稳定控制的。在LQR最优控制器的设计中,我们通过不断的MATLAB仿真实验总结出了性能矩阵Q选择的一般原则,为LQR算法的实际应用给出了一定的理论指导,同时通过MATLAB仿真,得出LQR算法是能较好的控制倒立摆系统,并具有较好的稳态效果。在极点配置法控制器的设计中,我们同样通过MATLAB仿真验证了其对倒立摆系统具有较好的控制效果。同时将极点配置法和LQR法进行了MATLAB仿真比较,得出极点配置法的响应时间较快,鲁棒性和动态性较好,而LQR算法具有较小的超调量和较好的稳态效果。最后指出极点配置法适合应用到要求快速响应的控制系统中,而LQR法适合应用到对稳态性能要求较高的控制系统中。
Inverted pendulum is a typical multi-variable, non-linear, strong coupling and naturally unstable system. During its control process, Inverted pendulum can reflect many crucial questions in the control theory, such as calm question, non-linear problem, robust question as well as tracking question and so on. Therefore, inverted pendulum has been widely used to verify some qualities and effect of certain control theory or method. The research on inverted pendulum system has the profound significance in theory and project application. The correlative scientific research achievement has already applied to astronautics science technology and subject of robot and so on.
In this paper, we firstly use Newtonian mechanics analysis method to establish the linear state equations of the linear 1-stage inverted pendulum’s physical system. In the meantime, the system is unstable by analyzing the linear state equation, but it is also controllable and observable. And then we describe in detail the hardware and software on the physical system of the linear 1-stage inverted pendulum, focusing on the use of the real time control software based on MATLAB/Simulink real-time toolkit RTW(Real-Time Workshop).
The thesis has been mainly discussed the control methods of inverted pendulum system based on PID control algorithm, the linear quadratic optimal control algorithm and the pole assignment algorithm, and then we design the controllers of the three control algorithm. We analyze the advantages and disadvantages of the three control algorithm through MATLAB/Simulink. In the design of PID controller, this paper has put forward the parameter turning method which is based on MATLAB/Simulink. Using this method, we have turned the PID control parameters better and faster. Through MATLAB/Simulink simulation, we find out that traditional PID control algorithm can be well-placed to control the pendulum’s angle of this output, and the control for another output-car displacement’s control is not effective. So PID control for linear 1-stage inverted pendulum, which is the single-input and two-output system, is unable to achieve stability control. In the design of LQR optimal controller, we have summed up the general principle of the performance matrix Q choice through MATLAB simulation, which gives a certain theoretical guidance for the practical application of LQR algorithm. Meanwhile, through MATLAB simulation we know that LQR algorithm can effectively control inverted pendulum system, which has good steady results. In the design of the pole assignment controller, we have also proved that it can control inverted pendulum effectively through MATLAB simulation. At the same time, by comparing the pole assignment method with LQR method through MATLAB simulation, we have found that the pole assignment has the faster response time, better robustness and dynamic, but LQR method has smaller overshoot and better steady-state results. Finally we have pointed out that the pole assignment method is suitable for applications that require fast response in the control system, and LQR method is suitable for applications that require higher steady-state performance in the control system.
引文
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