模糊控制在倒立摆系统中的应用研究
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摘要
倒立摆控制系统是检验控制算法的一个典型的对象,其中倒立摆摆杆级数的增加和其控制算法的更新都是倒立摆控制系统的重要研究内容。倒立摆系统本身是一个多变量、非线性、强耦合和自然不稳定的系统。由于倒立摆系统与机器人的站立、行走和火箭的飞行等有很大的相似性,所以对倒立摆控制问题的研究具有重要的理论和实际指导意义。
本文以深圳固高公司提供的倒立摆控制系统为研究对象,在讨论直线倒立摆系统数学模型的基础上,采用了LQR(Linear Quadratic Regulation)线性二次型调节器控制算法和模糊控制算法分别实现了对一级倒立摆和二级倒立摆系统的仿真和实时控制研究。主要的研究内容包括：
讨论了一级倒立摆和二级倒立摆系统的数学模型,并分析了一级倒立摆和二级倒立摆系统的稳定性、能控性、能观性和可控性,分析表明倒立摆系统在开环状态下是不稳定的,但是在其平衡点位置附近是可控制、可观测的,而且一级倒立摆系统比二级倒立摆系统可控度更高。
进行了倒立摆系统的LQR控制算法的研究,介绍了如何利用Matlab/Simulink建立倒立摆系统的仿真模型,讨论了加权矩阵Q和R的选取方法,进行一级倒立摆和二级倒立摆LQR控制器的设计、仿真和实物的实时控制,得到稳定时倒立摆系统的响应曲线,实验表明LQR算法控制倒立摆效果较好。
对于一级倒立摆系统,进行模糊控制方法的研究,设计了一级倒立摆系统的Simulink模型和模糊控制器,并对其进行仿真研究。
对于二级倒立摆系统,进行模糊控制算法的研究,为降低模糊控制器的维数,解决“规则爆炸”问题,利用最优控制方法设计了融合函数,即把小车位移、下摆摆角和上摆摆角合成为一个变量E,而把小车速度、下摆角速度和上摆角速度合成为一个变量EC,从而设计了二维模糊控制器,提升了模糊控制器的性能。对二级倒立摆系统进行了仿真及实物的实时控制,最后对二级倒立摆系统进行了抗干扰性研究并与LQR控制算法进行了比较。研究结果表明,模糊控制倒立摆系统使其具有很好的稳定性。
Inverted pendulum system is a typical object of the verifying control algorithm. The increasement of the inverted pendulum series and updating control algorithm are the major research project of the inverted pendulum control system. Inverted pendulum system has the performances of multivariable, nonlinear, strong coupling and natural instability. Because the Inverted pendulum system is great similarity with the robot standing, walking and flying rockets, etc, it has important theoretical and practical significances to research on the inverted pendulum control problems.
The inverted pendulum system was provided by GoogolTech of Shenzhen in this paper. Based on the mathematical model of linear inverted pendulum system, LQR(Linear quadratic regulator) control algorithm and fuzzy control algorithm are adopted to achieve the simulation and real-time control of one-stage inverted pendulum and two-stage inverted pendulum system. In this paper, we focus on the following problems:
Firstly, the mathematical models of one-stage inverted pendulum and two-stage inverted pendulum system were setted up. Then, the stability, controllability and observability of the system were analyzed. The result showed that the inverted pendulum system was unstable under open-loop stage, but it can be controlled and observed near the equilibrium point, and the controllability of one stage inverted pendulum system was better than two stage inverted pendulum system.
Secondly, LQR control algorithm of inverted pendulum system was researched. The inverted pendulum system model established by matlab/simulink was introduced, the method of how to select the Q and R weighting matrix was discussed. The LQR of the one-stage inverted pendulum and two-stage inverted pendulum system were designed, simulated and real-time controlled, the response curve of the stabilized inverted pendulum system was obtained. The experimental result showed that the inverted pendulum system with LQR algorithm had better performance.
Thirdly, the fuzzy control algorithm both for the one-stage inverted pendulum and the two-stage inverted pendulum system were researched, their simulink models and fuzzy controller were designed, both the two systems were simulated and real-time controlled. In order to reduce the dimension of the fuzzy controller and resolve the "rule explosion", the fusion function was designed by adopting the optimal control method. That is to synthesize the displacement of the car, the upper swing angle and the bottom swing angle as a variable E, synthesize the speed of the car, the upper angular velocity and the bottom angular velocity as a variable EC. The two-dimensional fuzzy controller was designed by employing the two velocities, and its performance was enhanced. The anti-interference performance of the two-stage inverted pendulum system was researched. Finally, compared with the LQR algorithm, the fuzzy control inverted pendulum system had the better stability than the system with LQR algorithm.
本文以深圳固高公司提供的倒立摆控制系统为研究对象,在讨论直线倒立摆系统数学模型的基础上,采用了LQR(Linear Quadratic Regulation)线性二次型调节器控制算法和模糊控制算法分别实现了对一级倒立摆和二级倒立摆系统的仿真和实时控制研究。主要的研究内容包括：
讨论了一级倒立摆和二级倒立摆系统的数学模型,并分析了一级倒立摆和二级倒立摆系统的稳定性、能控性、能观性和可控性,分析表明倒立摆系统在开环状态下是不稳定的,但是在其平衡点位置附近是可控制、可观测的,而且一级倒立摆系统比二级倒立摆系统可控度更高。
进行了倒立摆系统的LQR控制算法的研究,介绍了如何利用Matlab/Simulink建立倒立摆系统的仿真模型,讨论了加权矩阵Q和R的选取方法,进行一级倒立摆和二级倒立摆LQR控制器的设计、仿真和实物的实时控制,得到稳定时倒立摆系统的响应曲线,实验表明LQR算法控制倒立摆效果较好。
对于一级倒立摆系统,进行模糊控制方法的研究,设计了一级倒立摆系统的Simulink模型和模糊控制器,并对其进行仿真研究。
对于二级倒立摆系统,进行模糊控制算法的研究,为降低模糊控制器的维数,解决“规则爆炸”问题,利用最优控制方法设计了融合函数,即把小车位移、下摆摆角和上摆摆角合成为一个变量E,而把小车速度、下摆角速度和上摆角速度合成为一个变量EC,从而设计了二维模糊控制器,提升了模糊控制器的性能。对二级倒立摆系统进行了仿真及实物的实时控制,最后对二级倒立摆系统进行了抗干扰性研究并与LQR控制算法进行了比较。研究结果表明,模糊控制倒立摆系统使其具有很好的稳定性。
Inverted pendulum system is a typical object of the verifying control algorithm. The increasement of the inverted pendulum series and updating control algorithm are the major research project of the inverted pendulum control system. Inverted pendulum system has the performances of multivariable, nonlinear, strong coupling and natural instability. Because the Inverted pendulum system is great similarity with the robot standing, walking and flying rockets, etc, it has important theoretical and practical significances to research on the inverted pendulum control problems.
The inverted pendulum system was provided by GoogolTech of Shenzhen in this paper. Based on the mathematical model of linear inverted pendulum system, LQR(Linear quadratic regulator) control algorithm and fuzzy control algorithm are adopted to achieve the simulation and real-time control of one-stage inverted pendulum and two-stage inverted pendulum system. In this paper, we focus on the following problems:
Firstly, the mathematical models of one-stage inverted pendulum and two-stage inverted pendulum system were setted up. Then, the stability, controllability and observability of the system were analyzed. The result showed that the inverted pendulum system was unstable under open-loop stage, but it can be controlled and observed near the equilibrium point, and the controllability of one stage inverted pendulum system was better than two stage inverted pendulum system.
Secondly, LQR control algorithm of inverted pendulum system was researched. The inverted pendulum system model established by matlab/simulink was introduced, the method of how to select the Q and R weighting matrix was discussed. The LQR of the one-stage inverted pendulum and two-stage inverted pendulum system were designed, simulated and real-time controlled, the response curve of the stabilized inverted pendulum system was obtained. The experimental result showed that the inverted pendulum system with LQR algorithm had better performance.
Thirdly, the fuzzy control algorithm both for the one-stage inverted pendulum and the two-stage inverted pendulum system were researched, their simulink models and fuzzy controller were designed, both the two systems were simulated and real-time controlled. In order to reduce the dimension of the fuzzy controller and resolve the "rule explosion", the fusion function was designed by adopting the optimal control method. That is to synthesize the displacement of the car, the upper swing angle and the bottom swing angle as a variable E, synthesize the speed of the car, the upper angular velocity and the bottom angular velocity as a variable EC. The two-dimensional fuzzy controller was designed by employing the two velocities, and its performance was enhanced. The anti-interference performance of the two-stage inverted pendulum system was researched. Finally, compared with the LQR algorithm, the fuzzy control inverted pendulum system had the better stability than the system with LQR algorithm.
引文
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[10]段广仁.线性系统理论[M].哈尔滨：哈尔滨工业出版社,2004,2：87-103
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[12]汤兵勇,路林吉,王文杰.模糊控制理论与应用技术[M].北京：清华大学出版社,2002.9：78-96
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[15]席爱民.模糊控制技术[M].西安：西安电子科技大学出版社,2008：15-22
[16]刘海龙.倒立摆试验系统的设计与研究[D].大连理工大学,2006.12：23-31
[17]范醒哲,张乃尧,阵宁.三级倒立摆的两种模糊串级控制方案[C].第三届全球智能控制与自动化大会论文集,合肥,2000：145-161
[18]李洪兴,王加银,苗志宏.模糊控制系统的建模[J].中国科学(A辑)2002.9,第32卷第9期,772-781
[19]王强.基于自适应神经模糊推理的倒立摆控制研究[D].江苏：南京理工大学,2006：35-57
[20]张乃尧.基于神经网络的模糊自适应控制研究综述[C].北京：首都高校第三届自动控制学术报告会论文集,1994.173-176.
[21]邢郡,王磊,戴冠中.用BP算法加速倒立摆的模糊控制过程[J].西安：西北工业大学学报,1996,539-544.
[22]吴受章.应用最优控制[M].西安：西安交通大学出版社,1986：78-93
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[26]曲建岭,吴文海,孙俊恩.三级倒立摆系统模糊控制器设计及仿真[J].系统仿真学报.2004.3第16卷第3期,578-581
[27]阮治明.二级倒立摆系统的最佳控制[J].兵工自动化,2006,25(1)：60-63.
[28]柴飞燕.基于T-S模型的模糊控制器设计[D].兰州理工大学,2007.4：52-55
[29]薛定宇,陈阳泉.控制数学问题的MATLAB求解[M].北京：清华大学出版社,2005：110-112
[30]瞿亮.基于MATLAB的控制系统计算机仿真[M].北京：北方交通大学出版社,1998：101-115
[31]胡寿松.自动控制原理[M].北京：科学出版社,2001：516-576
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[2]朱兴.多级倒立摆控制算法的研究与应用[D].北方工业大学,2007.5：9-21
[3]汪雪琴.倒立摆系统的模糊智能控制研究[D].北京：北京化工大学,2004：84-92
[4]汤兵勇,路林吉,王文杰.模糊控制理论与应用技术[M].北京：北京清华大学出版社,2002：96-99
[5]郭钊侠,方建安,苗清影.倒立摆系统及其智能控制研究[J].东华大学,2003,29(2)：122
[6]张静.MATLAB在控制系统中的应用[M].北京：电子工业出版社,2007,5：122-124
[7]瞿亮.基于Matlab的控制系统计算机仿真[M].北京：北方交通大学出版社,1998：56-98
[8]黄苑虹.倒立摆系统的稳定控制研究[D].广东：广东工业大学,2002：12-34
[9]张晓明.多种多级倒立摆系统控制和仿真环境的研究[D].南京航空航天大学,2003：62-65
[10]段广仁.线性系统理论[M].哈尔滨：哈尔滨工业出版社,2004,2：87-103
[11]韩峻峰,李玉慧.模糊控制技术[M].重庆：重庆大学出版社,2003.5：82-135
[12]汤兵勇,路林吉,王文杰.模糊控制理论与应用技术[M].北京：清华大学出版社,2002.9：78-96
[13]张化光,何希勤.模糊自适应控制理论及其应用[M].北京：北京航空航天大学出版社,2002：45-47
[14]章卫国,杨向忠.模糊控制理论与应用[M].西安：西北工业大学出版社,2000：36-65
[15]席爱民.模糊控制技术[M].西安：西安电子科技大学出版社,2008：15-22
[16]刘海龙.倒立摆试验系统的设计与研究[D].大连理工大学,2006.12：23-31
[17]范醒哲,张乃尧,阵宁.三级倒立摆的两种模糊串级控制方案[C].第三届全球智能控制与自动化大会论文集,合肥,2000：145-161
[18]李洪兴,王加银,苗志宏.模糊控制系统的建模[J].中国科学(A辑)2002.9,第32卷第9期,772-781
[19]王强.基于自适应神经模糊推理的倒立摆控制研究[D].江苏：南京理工大学,2006：35-57
[20]张乃尧.基于神经网络的模糊自适应控制研究综述[C].北京：首都高校第三届自动控制学术报告会论文集,1994.173-176.
[21]邢郡,王磊,戴冠中.用BP算法加速倒立摆的模糊控制过程[J].西安：西北工业大学学报,1996,539-544.
[22]吴受章.应用最优控制[M].西安：西安交通大学出版社,1986：78-93
[23]解学书.最优控制理论与应用[M].北京：清华大学出版社,1986：323-329.
[24]王仲民,孙建军,岳宏.基于LQR的倒立摆最优控制系统研究[J].工业仪表与自动化装置,2005.3：6-9
[25]宋西蒙.倒立摆系统LQR_模糊控制算法研究[D].西安：西安电子科技大学,2006.1：12-46
[26]曲建岭,吴文海,孙俊恩.三级倒立摆系统模糊控制器设计及仿真[J].系统仿真学报.2004.3第16卷第3期,578-581
[27]阮治明.二级倒立摆系统的最佳控制[J].兵工自动化,2006,25(1)：60-63.
[28]柴飞燕.基于T-S模型的模糊控制器设计[D].兰州理工大学,2007.4：52-55
[29]薛定宇,陈阳泉.控制数学问题的MATLAB求解[M].北京：清华大学出版社,2005：110-112
[30]瞿亮.基于MATLAB的控制系统计算机仿真[M].北京：北方交通大学出版社,1998：101-115
[31]胡寿松.自动控制原理[M].北京：科学出版社,2001：516-576
[32]单波,徐燕,赵建涛.预测控制算法及其在倒立摆中的应用[J].2001.4,28(2)：46-51.
[33]段旭东,许可.单级倒立摆的建模与控制仿真[M].机器人技术与应用,2002.5：75-82
[34]王彪,唐超颖,沈春林.利用遗传算法进行PID参数寻优的倒立摆控制系统[J].江苏：南京航空航天大学,2004.12：1171-1174.
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