柔性倒立摆的控制方法研究
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摘要
倒立摆系统是一个高阶的、非线性、强耦合的绝对不稳定系统,它是检验各种新的控制理论与方法的一个有效的工具。
本文以直线一级柔性倒立摆为研究对象,首先基于拉格朗日方程建立了系统的数学模型,并对其性能进行了分析。其次分别应用PD控制方法、模糊控制方法以及滑模变结构控制方法对直线一级柔性倒立摆系统进行了控制研究。具体方法如下:
(1)设计了三回路PD控制器对直线一级柔性倒立摆系统的三个变量进行了控制,并利用极点配置方法整定控制器参数,最后将该控制算法运用到实际的柔性倒立摆系统中,成功地稳定住直线一级柔性倒立摆,实验证明PD控制器也能实现对非线性、多变量不稳定系统的控制。
(2)针对直线一级柔性倒立摆系统多变量的特性,提出了基于融合函数的模糊控制器。具体过程是:利用最优控制理论计算出使系统稳定的反馈增益矩阵,由反馈增益矩阵构造融合函数,由此减少了模糊控制器的输入变量维数,解决了“规则爆炸”问题。MATLAB-SIMULINK仿真和实际实验结果证明该方案的正确性和有效性。
(3)将滑模变结构控制算法应用于直线一级柔性倒立摆系统,设计了基于指数趋近率的滑模控制器,并采用指数趋近律与饱和函数相结合的准滑模变结构控制来削弱抖振。仿真结果验证了该方案的正确性和有效性。
Inverted pendulum system is a higher-order, nonlinear, strongly coupled and absolutely instable system. It is an effective tool for validating new control theories and methods.
The paper takes the linear 1-stage flexible-joint inverted pendulum as a research object. At first, it establishes the model of the inverted pendulum system based on the Lagrange equation and analyzes its performance. Secondly, it also talks about how the PD control method, the fuzzy control method and the sliding mode control method control the flexible-joint inverted pendulum system. Specific means are as follows:
(1) The three loop PD controllers are designed to control three variables of the flexible joint inverted-pendulum system and then the pole assignment is used to design the parameters of controllers. The designed controller is applied to actual inverted pendulum system. The real-time experiment result shows that the PD controller can also control multivariable, nonlinear and absolutely instable system.
(2) Based on the linear 1-stage flexible-joint inverted pendulum system property of multivariable, a fuzzy controller based on fusion function is proposed. Specific process is as follows: the feedback gain matrix is figured out, which makes the system stable by utilizing optimal control theory, and then makes up the fusion function by utilizing the feedback gain matrix which reduces the input variable dimension of the fuzzy controller,"rule explosion" problem is solved. The simulation experiment and the real-time experiment prove its validity and the correctness.
(3) The sliding model variable structure control (SMVSC) algorithm is applied to flexible joint inverted pendulum system, and sliding mode controller based on reaching rate is designed. And by introducing saturation function to the SMVSC based on reaching law, the system’s chattering is solved. The result of simulation shows that the algorithm is feasible and efficient.
本文以直线一级柔性倒立摆为研究对象,首先基于拉格朗日方程建立了系统的数学模型,并对其性能进行了分析。其次分别应用PD控制方法、模糊控制方法以及滑模变结构控制方法对直线一级柔性倒立摆系统进行了控制研究。具体方法如下:
(1)设计了三回路PD控制器对直线一级柔性倒立摆系统的三个变量进行了控制,并利用极点配置方法整定控制器参数,最后将该控制算法运用到实际的柔性倒立摆系统中,成功地稳定住直线一级柔性倒立摆,实验证明PD控制器也能实现对非线性、多变量不稳定系统的控制。
(2)针对直线一级柔性倒立摆系统多变量的特性,提出了基于融合函数的模糊控制器。具体过程是:利用最优控制理论计算出使系统稳定的反馈增益矩阵,由反馈增益矩阵构造融合函数,由此减少了模糊控制器的输入变量维数,解决了“规则爆炸”问题。MATLAB-SIMULINK仿真和实际实验结果证明该方案的正确性和有效性。
(3)将滑模变结构控制算法应用于直线一级柔性倒立摆系统,设计了基于指数趋近率的滑模控制器,并采用指数趋近律与饱和函数相结合的准滑模变结构控制来削弱抖振。仿真结果验证了该方案的正确性和有效性。
Inverted pendulum system is a higher-order, nonlinear, strongly coupled and absolutely instable system. It is an effective tool for validating new control theories and methods.
The paper takes the linear 1-stage flexible-joint inverted pendulum as a research object. At first, it establishes the model of the inverted pendulum system based on the Lagrange equation and analyzes its performance. Secondly, it also talks about how the PD control method, the fuzzy control method and the sliding mode control method control the flexible-joint inverted pendulum system. Specific means are as follows:
(1) The three loop PD controllers are designed to control three variables of the flexible joint inverted-pendulum system and then the pole assignment is used to design the parameters of controllers. The designed controller is applied to actual inverted pendulum system. The real-time experiment result shows that the PD controller can also control multivariable, nonlinear and absolutely instable system.
(2) Based on the linear 1-stage flexible-joint inverted pendulum system property of multivariable, a fuzzy controller based on fusion function is proposed. Specific process is as follows: the feedback gain matrix is figured out, which makes the system stable by utilizing optimal control theory, and then makes up the fusion function by utilizing the feedback gain matrix which reduces the input variable dimension of the fuzzy controller,"rule explosion" problem is solved. The simulation experiment and the real-time experiment prove its validity and the correctness.
(3) The sliding model variable structure control (SMVSC) algorithm is applied to flexible joint inverted pendulum system, and sliding mode controller based on reaching rate is designed. And by introducing saturation function to the SMVSC based on reaching law, the system’s chattering is solved. The result of simulation shows that the algorithm is feasible and efficient.
引文
[1]李晓燕,谢克明.平面一级倒立摆系统的智能控制策略研究[D].太原理工大学, 2005.
[2]杨亚炜,张明廉.三级倒立摆的数控稳定[J].北京航空航天大学学报, 2000, 26(3): 311-314.
[3] Li Hong-xing, Miao Zhi-hong, Wang Jia-yin. Variable universe adaptive fuzzy control on the quadruple inverted pendulum[J]. Science in China Series E: Technological Sciences, 2002, 45(2):213-224.
[4]冯艳宾,崔红梅,李凤等.李洪兴教授领导的复杂系统智能控制实验室成功实现平面运动二级倒立摆实物系统控制[J].数学通报, 2003, (6): 1-3.
[5]陈晖,李德毅,沈程智,张飞舟.云模型在倒立摆控制中的应用[J].计算机研究与发展,1999,36(10):1-10.
[6]杨世勇,徐莉萍,王培进.单机倒立摆的PID控制研究[J].控制工程, 2007, 14(增刊): 23-25.
[7]闫勤芳,邢作常,张志勇.直线柔性二级倒立摆控制系统的频率响应法设计[J].计算机工程与设计, 2005, 26(5): 1336-1338.
[8]李岩,姚旭东.二级倒立摆控制系统分析[J].沈阳工业学院学报, 1999, 18(2):86-92.
[9]杨世勇,王培进,徐莉萍.倒立摆的一种模糊控制方法[J].控制理论与应用, 2007, 26(7):10-12.
[10]张滔,谢宗安.智能控制在倒立摆系统中的应用[D].贵州大学, 2005.
[11]丛爽,张冬军,魏衡华.单级倒立摆三种控制方法的对比研究[J].系统工程与电子技术, 2001,23(11):47-49.
[12]司昌龙,张明廉.拟人智能控制及控制律转化研究[D].北京航空航天大学, 2003.
[13]马俊,李小华.倒立摆的鲁棒控制研究[D].辽宁科技大学, 2008.
[14]张明廉,孙昌龄,杨亚炜等.拟人控制二维单倒立摆[J].控制与决策, 2002, 17(1): 53-55.
[15] Ji Guo-yuan, Chen Zeng-qiang, Yuan Zhu-zhi. Adaptive high order differential feedback control for inverted pendulum system[J]. Control theory and application 2004, 21(5):676-683.
[16]杨平,徐春梅等.直线型一级倒立摆状态反馈控制设计与实现[J].上海电力学院学报, 2007,23(1): 21-25.
[17]罗晶,陈平.一阶倒立摆的PID控制[J].实验室研究与探索, 2005, 24(11): 26-28.
[18]舒怀林.基于神经网络的倒立摆控制系统[J].机床与液压, 2008, 36(3): 141-146.
[19]范醒哲,张乃尧,陈宁.三级倒立摆的两种模糊串级控制方案[J].第三届全球智能控制与自动化大会, 2000.
[20] K.Furuta, T.Okutan, H.Sone. Computer control of a double inverted pendulum[J]. Computers and Electrical Engineering, 1978, (5):67-84.
[21]丛爽,张冬军.柔性连接倒立摆系统的控制与实现[J].控制工程, 2004, 11(6): 506-509.
[22]刘金坤.先进PID控制及其MATLAB仿真[M].第一版.北京,电子工业出版社, 2003, 1-5.
[23]杨平,徐春梅,曾婧婧等. PID控制在倒立摆实时控制系统中的应用[J].微计算机信息, 2006, 22(19): 83-85.
[24]杨平,徐春梅等.直线二级倒立摆的PID实时控制[J].上海电力学院学报, 2008, 24(3): 236-247.
[25]王社阳,柏桂珍,张卯瑞.二级倒立摆系统控制方法的研究[D].哈尔滨工业大学, 2002.
[26]洪旭,过润秋.倒立摆系统模糊控制算法研究[D].西安电子科技大学, 2005.
[27]郝月龙,张井岗,陈志梅.智能滑模控制及其应用研究[D].太原科技大学, 2007.
[28]张志强,李战明.基于模糊滑模变结构的倒立摆控制方法研究[D].兰州理工大学, 2008.
[29]庄开宇,褚健,苏宏业.变结构控制理论若干问题研究及其应用[D].浙江大学, 2002.
[30]宋西蒙,过润秋.倒立摆系统LQR-模糊控制算法研究[D].西安电子科技大学, 2006.
[31]朱俊辉,周凤岐.基于变结构控制的二级倒立摆稳定控制[D].西北工业大学, 2006.
[32] Li Ning, Zhang Naiyao, Jin Kaiyan. Structure analysis of typical fuzzy controllers with unevenly distributed Input membership function[J]. Tsinghua University, 2002, 40(1): 120-123.
[33]王声远,霍伟.一种新的自适应模糊滑模控制器设计方法[D].控制与决策. 2001, 16(1): 93-96.
[34] Chen CS, Chen WL. Robust adaptive sliding-mode control using fuzzy modeling for an inverted pendulum system[J]. IEEE Trans IE 1998, 45: 297–306.
[35] Tong S, Li HX. Fuzzy adaptive sliding-mode control for a nonlinear system[J]. IEEE Trans FS 2003, 11(3):354–60.
[36]龙利,陈今润. MATLAB环境下控制系统实时仿真实验的研究[D].重庆大学, 2005.
[37]张赟宁,陈今润. MATLAB环境下控制系统综合实验平台设计与实现[D].重庆大学, 2006.
[38]刘时鹏,陈今润. MATLAB环境下直线单级倒立摆系统实时控制实验的研究与设计[D].重庆大学, 2004.
[39]段学超,仇原鹰.平面倒立摆的建模、控制与实验研究[D].西安电子科技大学, 2006.
[40] Jin Kun-liu, Fu Chun-sun. Chattering free adaptive fuzzy terminal sliding mode control for second order nonlinear system[J]. Journal of Control Theory and Applications, 2006, 4(4): 385-391.
[41] Ma Xiao-jun, Sun Zeng-qi. Analysis and design of fuzzy controller and fuzzy observer[J]. IEEE Trans on Fuzzy Systems, 1998, 6(1): 41-55.
[42]张乃尧, C. Ebert, R. Belschner, H. Strahl.倒立摆的双闭环模糊控制[J].控制与决策. 1996, 11(1):85-88.
[43] Mori Shozo, H. Nishihara, K. Furuta. Control of unstable mechanical system: Control of pendulum [J]. Int. J. of Control 1976, 23(5): 673-692.
[44] Cheng Fu-yan, Zhong Guo-min, Li You-shan, Xu Zheng-ming. Fuzzy control of a double-inverted pendulum[J]. Fuzzy Sets and Systems, 1996, 79(3): 315-321.
[45] S.Deris, S. Omatu. Stabilization of inverted pendulum by the genetic algorithm[J]. IEEE International Conference on Systems, Man and Cybemetics, 1995, 383-388.
[46] M. Yamakita, M. Iwashiro, Y Sugahara, K. Furuta. Robust swing-up control of inverted pendulum [C]. Proc. of the American Control Conference, Seattle, Washington. 1995, 290-294.
[47]张明廉,郝健康.拟人智能控制与三级倒立摆[J].航空学报, 1995,16(6): 654-661.
[48] Chang Yeoul Cha, Jung-il Park. A study on construction of control scheme for inverted pendulum with self-learning fuzzy controller[J]. Kite Journal of electronics engineering, 1996, 7(3): 48-52.
[49] Ma Xiao-Jun, Sun Zeng-Qi. Analysis and design of fuzzy controller and fuzzy observer[J]. IEEE Trans on Fuzzy Systems, 1998, 6(1): 41-55.
[50]黄丹,周少武,吴新开,张志飞.基于LQR最优调节器的倒立摆控制系统[J].微计算机信息(测控自动化). 2004, 2(20): 37-38.
[51]李爱国,王兴成.变结构控制理论及其在二级倒立摆控制中的应用[D].大连海事大学, 2004.
[52]李铁龙,王洪斌,王洪瑞.非线性滑模变结构控制及其在倒立摆系统中的应用研究[D].燕山大学, 2005.
[2]杨亚炜,张明廉.三级倒立摆的数控稳定[J].北京航空航天大学学报, 2000, 26(3): 311-314.
[3] Li Hong-xing, Miao Zhi-hong, Wang Jia-yin. Variable universe adaptive fuzzy control on the quadruple inverted pendulum[J]. Science in China Series E: Technological Sciences, 2002, 45(2):213-224.
[4]冯艳宾,崔红梅,李凤等.李洪兴教授领导的复杂系统智能控制实验室成功实现平面运动二级倒立摆实物系统控制[J].数学通报, 2003, (6): 1-3.
[5]陈晖,李德毅,沈程智,张飞舟.云模型在倒立摆控制中的应用[J].计算机研究与发展,1999,36(10):1-10.
[6]杨世勇,徐莉萍,王培进.单机倒立摆的PID控制研究[J].控制工程, 2007, 14(增刊): 23-25.
[7]闫勤芳,邢作常,张志勇.直线柔性二级倒立摆控制系统的频率响应法设计[J].计算机工程与设计, 2005, 26(5): 1336-1338.
[8]李岩,姚旭东.二级倒立摆控制系统分析[J].沈阳工业学院学报, 1999, 18(2):86-92.
[9]杨世勇,王培进,徐莉萍.倒立摆的一种模糊控制方法[J].控制理论与应用, 2007, 26(7):10-12.
[10]张滔,谢宗安.智能控制在倒立摆系统中的应用[D].贵州大学, 2005.
[11]丛爽,张冬军,魏衡华.单级倒立摆三种控制方法的对比研究[J].系统工程与电子技术, 2001,23(11):47-49.
[12]司昌龙,张明廉.拟人智能控制及控制律转化研究[D].北京航空航天大学, 2003.
[13]马俊,李小华.倒立摆的鲁棒控制研究[D].辽宁科技大学, 2008.
[14]张明廉,孙昌龄,杨亚炜等.拟人控制二维单倒立摆[J].控制与决策, 2002, 17(1): 53-55.
[15] Ji Guo-yuan, Chen Zeng-qiang, Yuan Zhu-zhi. Adaptive high order differential feedback control for inverted pendulum system[J]. Control theory and application 2004, 21(5):676-683.
[16]杨平,徐春梅等.直线型一级倒立摆状态反馈控制设计与实现[J].上海电力学院学报, 2007,23(1): 21-25.
[17]罗晶,陈平.一阶倒立摆的PID控制[J].实验室研究与探索, 2005, 24(11): 26-28.
[18]舒怀林.基于神经网络的倒立摆控制系统[J].机床与液压, 2008, 36(3): 141-146.
[19]范醒哲,张乃尧,陈宁.三级倒立摆的两种模糊串级控制方案[J].第三届全球智能控制与自动化大会, 2000.
[20] K.Furuta, T.Okutan, H.Sone. Computer control of a double inverted pendulum[J]. Computers and Electrical Engineering, 1978, (5):67-84.
[21]丛爽,张冬军.柔性连接倒立摆系统的控制与实现[J].控制工程, 2004, 11(6): 506-509.
[22]刘金坤.先进PID控制及其MATLAB仿真[M].第一版.北京,电子工业出版社, 2003, 1-5.
[23]杨平,徐春梅,曾婧婧等. PID控制在倒立摆实时控制系统中的应用[J].微计算机信息, 2006, 22(19): 83-85.
[24]杨平,徐春梅等.直线二级倒立摆的PID实时控制[J].上海电力学院学报, 2008, 24(3): 236-247.
[25]王社阳,柏桂珍,张卯瑞.二级倒立摆系统控制方法的研究[D].哈尔滨工业大学, 2002.
[26]洪旭,过润秋.倒立摆系统模糊控制算法研究[D].西安电子科技大学, 2005.
[27]郝月龙,张井岗,陈志梅.智能滑模控制及其应用研究[D].太原科技大学, 2007.
[28]张志强,李战明.基于模糊滑模变结构的倒立摆控制方法研究[D].兰州理工大学, 2008.
[29]庄开宇,褚健,苏宏业.变结构控制理论若干问题研究及其应用[D].浙江大学, 2002.
[30]宋西蒙,过润秋.倒立摆系统LQR-模糊控制算法研究[D].西安电子科技大学, 2006.
[31]朱俊辉,周凤岐.基于变结构控制的二级倒立摆稳定控制[D].西北工业大学, 2006.
[32] Li Ning, Zhang Naiyao, Jin Kaiyan. Structure analysis of typical fuzzy controllers with unevenly distributed Input membership function[J]. Tsinghua University, 2002, 40(1): 120-123.
[33]王声远,霍伟.一种新的自适应模糊滑模控制器设计方法[D].控制与决策. 2001, 16(1): 93-96.
[34] Chen CS, Chen WL. Robust adaptive sliding-mode control using fuzzy modeling for an inverted pendulum system[J]. IEEE Trans IE 1998, 45: 297–306.
[35] Tong S, Li HX. Fuzzy adaptive sliding-mode control for a nonlinear system[J]. IEEE Trans FS 2003, 11(3):354–60.
[36]龙利,陈今润. MATLAB环境下控制系统实时仿真实验的研究[D].重庆大学, 2005.
[37]张赟宁,陈今润. MATLAB环境下控制系统综合实验平台设计与实现[D].重庆大学, 2006.
[38]刘时鹏,陈今润. MATLAB环境下直线单级倒立摆系统实时控制实验的研究与设计[D].重庆大学, 2004.
[39]段学超,仇原鹰.平面倒立摆的建模、控制与实验研究[D].西安电子科技大学, 2006.
[40] Jin Kun-liu, Fu Chun-sun. Chattering free adaptive fuzzy terminal sliding mode control for second order nonlinear system[J]. Journal of Control Theory and Applications, 2006, 4(4): 385-391.
[41] Ma Xiao-jun, Sun Zeng-qi. Analysis and design of fuzzy controller and fuzzy observer[J]. IEEE Trans on Fuzzy Systems, 1998, 6(1): 41-55.
[42]张乃尧, C. Ebert, R. Belschner, H. Strahl.倒立摆的双闭环模糊控制[J].控制与决策. 1996, 11(1):85-88.
[43] Mori Shozo, H. Nishihara, K. Furuta. Control of unstable mechanical system: Control of pendulum [J]. Int. J. of Control 1976, 23(5): 673-692.
[44] Cheng Fu-yan, Zhong Guo-min, Li You-shan, Xu Zheng-ming. Fuzzy control of a double-inverted pendulum[J]. Fuzzy Sets and Systems, 1996, 79(3): 315-321.
[45] S.Deris, S. Omatu. Stabilization of inverted pendulum by the genetic algorithm[J]. IEEE International Conference on Systems, Man and Cybemetics, 1995, 383-388.
[46] M. Yamakita, M. Iwashiro, Y Sugahara, K. Furuta. Robust swing-up control of inverted pendulum [C]. Proc. of the American Control Conference, Seattle, Washington. 1995, 290-294.
[47]张明廉,郝健康.拟人智能控制与三级倒立摆[J].航空学报, 1995,16(6): 654-661.
[48] Chang Yeoul Cha, Jung-il Park. A study on construction of control scheme for inverted pendulum with self-learning fuzzy controller[J]. Kite Journal of electronics engineering, 1996, 7(3): 48-52.
[49] Ma Xiao-Jun, Sun Zeng-Qi. Analysis and design of fuzzy controller and fuzzy observer[J]. IEEE Trans on Fuzzy Systems, 1998, 6(1): 41-55.
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