基于神经网络的倒立摆控制系统数值模拟
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摘要
倒立摆控制系统通常用来检验控制策略的效果，是自动控制、机械电子等领域中非常典型的较为理想的实验装置。倒立摆系统是一个非线性、强耦合、多变量和自然不稳定的系统，这就需要给倒立摆系统施加适当的力使倒立摆系统在一个很小的范围内移动，从而使摆的摆角在很小的范围内变化。根据神经网络的自组织、自适应和泛化学习能力，应用神经网络解决倒立摆控制问题。BP神经网络在计算的过程中，需要调节其连接权系数，一般情况下采用梯度下降法来调节其权系。
人工神经元网络为非线性系统建模提供了快速、简便的学习能力和应用性能，以及精确表达过程的非线性-、复杂性的能力。其中多层前馈神经元网络以其结构直观、算法简便和理论上能逼近任意非线性连续映射的能力而得到了最广泛的应用。但是经典的BP算法存在收敛速度慢、数值稳定性差等缺点，为此许多学者将非线性优化理论中的许多算法引入到多层前馈网络的训练学习中，起了一定的改善作用。拟牛顿算法中的BFGS优化算法在求解无约束优化问题中被认为是最好的方法。论文采用BFGS方法来训练BP神经网络的权值，从而达到控制倒立摆系统的目的。
通过对倒立摆系统和离散时间系统的模拟应用，表明所提出方法的控制效率和有效性。模拟结果表明所提出的非线性系统控制方法给出了更精确的全局最优解和更快速的收敛速率。
Inverted pendulum control system which is a kind of perfect equipment in autocontrol, mechanism and electron is used to test the result of control. Inverted pendulum system is a nonlinear, coupling, variable and erratic system. In order to ensure the swing angle and the position of dolly to chang in a small range, a proper force must be applied to the inverted pendulum system. Neural network can be used to solve the problem of inverted pendulum control because of its ability of sel-organization, sel-adaption and generalization. BP neural network can work out how much force applied to the dolly through gradient degressive algorithm to adjust the joint weights.
Artificial neural networks offer speediness, simple and convenient learing ability and application performance for nonlinear system model. Multilayer feedforward neural networks are used widely because of its intuitionistic structure, simple and convenient algorithm and the ability to approach random nonlinear continuum mapping. However, BP neural network lies the problem of slow convergence speed and the bad numerical value stability and so on. A lot of algorithms in nonlinear optimization theory are used into the learning of multilayer feedforward neural networks, and get some improvement. This article uses the BFGS optimization method which is considered to be the best optimization algorithm in solving nonrestrain question, to train the weight of BP neural network.
The control of nonlinear system is applied to some simulation examples of a discrete-time system and the inverted pendulum model system to demonstrate the performance and control efficiency of the proposed method. The simulation results show that the suggested methods give better global convergent characteristics and faster convergence.
人工神经元网络为非线性系统建模提供了快速、简便的学习能力和应用性能，以及精确表达过程的非线性-、复杂性的能力。其中多层前馈神经元网络以其结构直观、算法简便和理论上能逼近任意非线性连续映射的能力而得到了最广泛的应用。但是经典的BP算法存在收敛速度慢、数值稳定性差等缺点，为此许多学者将非线性优化理论中的许多算法引入到多层前馈网络的训练学习中，起了一定的改善作用。拟牛顿算法中的BFGS优化算法在求解无约束优化问题中被认为是最好的方法。论文采用BFGS方法来训练BP神经网络的权值，从而达到控制倒立摆系统的目的。
通过对倒立摆系统和离散时间系统的模拟应用，表明所提出方法的控制效率和有效性。模拟结果表明所提出的非线性系统控制方法给出了更精确的全局最优解和更快速的收敛速率。
Inverted pendulum control system which is a kind of perfect equipment in autocontrol, mechanism and electron is used to test the result of control. Inverted pendulum system is a nonlinear, coupling, variable and erratic system. In order to ensure the swing angle and the position of dolly to chang in a small range, a proper force must be applied to the inverted pendulum system. Neural network can be used to solve the problem of inverted pendulum control because of its ability of sel-organization, sel-adaption and generalization. BP neural network can work out how much force applied to the dolly through gradient degressive algorithm to adjust the joint weights.
Artificial neural networks offer speediness, simple and convenient learing ability and application performance for nonlinear system model. Multilayer feedforward neural networks are used widely because of its intuitionistic structure, simple and convenient algorithm and the ability to approach random nonlinear continuum mapping. However, BP neural network lies the problem of slow convergence speed and the bad numerical value stability and so on. A lot of algorithms in nonlinear optimization theory are used into the learning of multilayer feedforward neural networks, and get some improvement. This article uses the BFGS optimization method which is considered to be the best optimization algorithm in solving nonrestrain question, to train the weight of BP neural network.
The control of nonlinear system is applied to some simulation examples of a discrete-time system and the inverted pendulum model system to demonstrate the performance and control efficiency of the proposed method. The simulation results show that the suggested methods give better global convergent characteristics and faster convergence.
引文
[1]李洪兴，苗志宏，王加银．四级倒立摆的变论域自适应模糊控制．中国科学．E 辑，2002，32(1)：6s-7s．
[2]李洪兴，王加银．n级倒立摆系统建模．模糊系统与数学，2002，16：251-257．
[3]田明，戴汝为．基于动态BP神经网络得系统辨识方法．自动化学报，1993，19(4)：450-453．
[4]Gupta P.An Improved Approach for Nonlinear System Identification Using Neural Networks. Journal of the Franklin Institute, 1999, 336(4):721-734.
[5]Ni X F. A New Method for Identification and Control of Nonlinear Dynamic Systems. Engng. Applic. Artif. Intell, 1996, 9(3): 231-243.
[6]曹海云，李守巨，刘迎曦．基于神经网络的岩土力学参数反演方法研究综述．测试技术学报，2006，20(增)：9-12．
[7]赵莉．单级倒立摆的模糊控制及仿真．山东师范大学学报(自然科学版，2004，19(3)：102-104．
[8]张丹红，王勤．单级倒立摆系统的神经网络逆模控制．武汉理工大学学报(信息与管理工程版)，2006，28(5)：28-31．
[9]张飞舟．拟人智能控制三级倒立摆机理的研究．北京航空航天大学学报，1999，25(2)：151-155．
[10]张丽娟，涂亚庆．小车-二级倒立摆系统的仿人智能控制策略和算法．自动化与仪器仪表，2006，5：1-5．
[11]李守巨，刘迎曦，曹海云．基于改进PID神经网络的非线性动态系统识别方法．东北大学学报(自然科学版)，2007，28(增1)：136-139．
[12]杨慧中，张素贞．BFGS修正算法在前馈神经元网络学习中的应用．华东理工那个大学学报，2001，27(5)：459-462．
[13]徐丽娜．神经网络控制．北京：电子工业出版社，2003．
[14]Shouju Li, Yingxi Liu, He Yu. "Nonlinear Dynamic System Control with Evolutionary PID Neural Networks."Proceedings of the International Conference on Sensing. Computing and Automation, 2006: 166-171.
[15]孙慧玲．基于人工神经网络的结构损伤识别和预测方法：(硕士学位论文)．大连：大连理工大学，2004．
[16]Zaheer-uddin M,Tudoroiu N."Neuro-PID tracking control of a discharge air temperature system."Energy Conversion and Management, 2004, 45: 2405-2415.
[17] MEIER H, FARWIG Z, UNBEHAUEN H. Discrete computer control of a triple-inverted pendulum. Optimal Control Application & Methods[C], 1990, 11(2): 157-171.
[18] Mori Shozo, Nishihara H and KFuruta. Control of Unstable Mechanical System: Control of pendulum. Int. J. of Control [J], 1976, 23(5): 673-692.
[19] KFuruta, Katsuhisa, Kajiwara H and KKosuge. Digital Control of a Double Inverted Pendulum on An Inclined Rail. Int. J. of Control [J], 1980, 32(5): 907-924.
[20] KFuruta, Ochia T. Attitude Control of a Triple-Inverted Pendulum. Int. J. Control[1], 1984, 39(6): 1351-1365.
[21] Watts J. Control of An Inverted Pendulum. ASEE Annual Conference [C], 1984, 2527(2): 706-710.
[22] Furuta K, Yamakita M and Kobayashi S. Swing-up Control of Inverted Pendulum Using Pseudo-state Feedback [J]. J. of Systems and Control Engineering, 1992:206-210.
[23] Fradkov A L, Guzenko P Y, Hill D J and Pogromsky A Y. Speed Gradient Control and Passivity of Nonlinear Oscillators [C]. Proc. of IFAC Symposium on Control of Nonlinear Systems, Lake Tahoe, 1995: 655-659.
[24] Wiklund, Magnus, Anders Kristenson and Astrom K J. A New Strategy for Swing Up an Inverted Pendulum [C]. In Preprints IFAC 12th World Congress, Sydney, Australia, 1993: 151-154.
[25] Yamakita M, Iwashiro M, Sugahara Y and Furuta K. Robust Swing-up Control of Inverted Pendulum[C]. Proc. of the American Control Conference, Seattle, Washington, 1995: 290-294.
[26] Tzuu-Hseng, Li S, Chin-Yin Tsai. Parallel Fuzzy Sliding Mode Control of the Cart-pole System IEEE International Conference on Industral Electronics. Control and Instrumentation, 1995, 2: 1468-1473.
[27] 张乃尧, Ebert C, Belschner R, Strahl H. 倒立摆的双闭环模糊控制.控制与决策. 1996, 11(1): 85-88.
[28] Hung T H, Yeh M F and Lu H C. A Pi-like Fuzzy Controller Implementation for the Inverted Pendulum System. IEEE KIPS 97, Beijing, China, 1997: 218-222.
[29] DeyiLi, HuiChen, JianhuaFan, ChengzhiShen. A Novel Qualitative Control Method to Inverted Pendulum Systems. Proceedings of the 14th IFAC: 485-490.
[30]Deris S, Omatu S.Stabilization of Inverted Pendulum By the Geneticalgorithm. IEEE International Conferece on Systerms, Man and Cybernetics, 1995: 383-388.
[31]Bouslama F, Ichikawa A. Application of Neural Network to Fuzzy Control. Neural Networks,1993,6:791-799.
[32]Barto A G, Sutton R S, Anderson C W. Neuronlike Adaptive Eleements That Can Solve Difficult Learning Control Problems [J].IEEE Trans.On SMC,1983, 13(5): 834-846.
[33]Anderson C W. Learning To Control an Inverted Pendulum Using Neural Networks.IEEE Control Systems Magazine, 1989, 9(3):31-37.
[34]胡叔旖，孙增忻．二级倒立摆的基干规则控制的研究．1995年中国智能自动化学会议论文集，1995：533-540．
[35]张明廉，郝健康等．拟人智能控制与三级倒立摆．航空学报，1995，16(6)：654-661．
[36]张明廉，何卫东，沈程智．规约规则发仿人控制．第一届全球华人智能控制与自动化大会论文集，1996：291-196．
[37]李祖枢，陈庆春．倒摆系统的智能控制研究及其发展动向．第二届全球华人智能控制与智能自动化大会论文集(上卷)：14-19．
[38]曾拍．小车-二级摆摆起倒立的仿人智能控制研究：(硕士学位论文)．重庆：重庆大学，2001．
[39]Shouling He,Qiang Wu,Nariman Sepehri.Control of Base-Excited Inverted Pendulums Using a Neural Inverse Modeling Approach. Proceedings of the 14th IF-AC:409-414.
[40]李俊芳，张振东，史晓霞．基于智能方法的倒立摆系统研究．燕山大学学报，2001，8(25)：56-60．
[41]黄苑虹，梁慧冰．从倒立摆装置的控制策略看控制理论的发展和应用．广东上业大学学报，2001，18(3)：49-53．
[42]李浩．提高BP算法收敛速度的几种方法．计算机学报，1997，1．
[43]曹洪洋，李万庆，李文华等．前馈神经网络的一种新的学习算法．微机发展，2001，11(4)：50-51．
[44]粟塔山．最优化计算原理与算法程序设计．长沙：国防科技大学出版社，2001．
[45]曹海云，李守巨，刘迎曦．基于改进PID神经网络的非线性动态系统控制．控制工程，2007，14(增)：38-40．
[46]Fredric Ham M, Ivica Kostanic.Priciplees of Neurocomputing for Science & Engineering. 北京：机械工业出版社，2003．
[47]刘金琨．先进PID控制MATLAB仿真．北京：电子工业出版社，2004．
[48]郑长义，王光德，刘宏，王健．基于Riccati方程的二级倒立摆控制．吉林师范大学学报(自然科学版)，2006，2：107-108．
[49]徐丽娜．神经网络控制结构及所用得神经网络．自动化技术与应用，2004，23(1)：1-5．
[50]李宜达．控制系统设计与仿真．北京：清华大学出版社，2004．
[2]李洪兴，王加银．n级倒立摆系统建模．模糊系统与数学，2002，16：251-257．
[3]田明，戴汝为．基于动态BP神经网络得系统辨识方法．自动化学报，1993，19(4)：450-453．
[4]Gupta P.An Improved Approach for Nonlinear System Identification Using Neural Networks. Journal of the Franklin Institute, 1999, 336(4):721-734.
[5]Ni X F. A New Method for Identification and Control of Nonlinear Dynamic Systems. Engng. Applic. Artif. Intell, 1996, 9(3): 231-243.
[6]曹海云，李守巨，刘迎曦．基于神经网络的岩土力学参数反演方法研究综述．测试技术学报，2006，20(增)：9-12．
[7]赵莉．单级倒立摆的模糊控制及仿真．山东师范大学学报(自然科学版，2004，19(3)：102-104．
[8]张丹红，王勤．单级倒立摆系统的神经网络逆模控制．武汉理工大学学报(信息与管理工程版)，2006，28(5)：28-31．
[9]张飞舟．拟人智能控制三级倒立摆机理的研究．北京航空航天大学学报，1999，25(2)：151-155．
[10]张丽娟，涂亚庆．小车-二级倒立摆系统的仿人智能控制策略和算法．自动化与仪器仪表，2006，5：1-5．
[11]李守巨，刘迎曦，曹海云．基于改进PID神经网络的非线性动态系统识别方法．东北大学学报(自然科学版)，2007，28(增1)：136-139．
[12]杨慧中，张素贞．BFGS修正算法在前馈神经元网络学习中的应用．华东理工那个大学学报，2001，27(5)：459-462．
[13]徐丽娜．神经网络控制．北京：电子工业出版社，2003．
[14]Shouju Li, Yingxi Liu, He Yu. "Nonlinear Dynamic System Control with Evolutionary PID Neural Networks."Proceedings of the International Conference on Sensing. Computing and Automation, 2006: 166-171.
[15]孙慧玲．基于人工神经网络的结构损伤识别和预测方法：(硕士学位论文)．大连：大连理工大学，2004．
[16]Zaheer-uddin M,Tudoroiu N."Neuro-PID tracking control of a discharge air temperature system."Energy Conversion and Management, 2004, 45: 2405-2415.
[17] MEIER H, FARWIG Z, UNBEHAUEN H. Discrete computer control of a triple-inverted pendulum. Optimal Control Application & Methods[C], 1990, 11(2): 157-171.
[18] Mori Shozo, Nishihara H and KFuruta. Control of Unstable Mechanical System: Control of pendulum. Int. J. of Control [J], 1976, 23(5): 673-692.
[19] KFuruta, Katsuhisa, Kajiwara H and KKosuge. Digital Control of a Double Inverted Pendulum on An Inclined Rail. Int. J. of Control [J], 1980, 32(5): 907-924.
[20] KFuruta, Ochia T. Attitude Control of a Triple-Inverted Pendulum. Int. J. Control[1], 1984, 39(6): 1351-1365.
[21] Watts J. Control of An Inverted Pendulum. ASEE Annual Conference [C], 1984, 2527(2): 706-710.
[22] Furuta K, Yamakita M and Kobayashi S. Swing-up Control of Inverted Pendulum Using Pseudo-state Feedback [J]. J. of Systems and Control Engineering, 1992:206-210.
[23] Fradkov A L, Guzenko P Y, Hill D J and Pogromsky A Y. Speed Gradient Control and Passivity of Nonlinear Oscillators [C]. Proc. of IFAC Symposium on Control of Nonlinear Systems, Lake Tahoe, 1995: 655-659.
[24] Wiklund, Magnus, Anders Kristenson and Astrom K J. A New Strategy for Swing Up an Inverted Pendulum [C]. In Preprints IFAC 12th World Congress, Sydney, Australia, 1993: 151-154.
[25] Yamakita M, Iwashiro M, Sugahara Y and Furuta K. Robust Swing-up Control of Inverted Pendulum[C]. Proc. of the American Control Conference, Seattle, Washington, 1995: 290-294.
[26] Tzuu-Hseng, Li S, Chin-Yin Tsai. Parallel Fuzzy Sliding Mode Control of the Cart-pole System IEEE International Conference on Industral Electronics. Control and Instrumentation, 1995, 2: 1468-1473.
[27] 张乃尧, Ebert C, Belschner R, Strahl H. 倒立摆的双闭环模糊控制.控制与决策. 1996, 11(1): 85-88.
[28] Hung T H, Yeh M F and Lu H C. A Pi-like Fuzzy Controller Implementation for the Inverted Pendulum System. IEEE KIPS 97, Beijing, China, 1997: 218-222.
[29] DeyiLi, HuiChen, JianhuaFan, ChengzhiShen. A Novel Qualitative Control Method to Inverted Pendulum Systems. Proceedings of the 14th IFAC: 485-490.
[30]Deris S, Omatu S.Stabilization of Inverted Pendulum By the Geneticalgorithm. IEEE International Conferece on Systerms, Man and Cybernetics, 1995: 383-388.
[31]Bouslama F, Ichikawa A. Application of Neural Network to Fuzzy Control. Neural Networks,1993,6:791-799.
[32]Barto A G, Sutton R S, Anderson C W. Neuronlike Adaptive Eleements That Can Solve Difficult Learning Control Problems [J].IEEE Trans.On SMC,1983, 13(5): 834-846.
[33]Anderson C W. Learning To Control an Inverted Pendulum Using Neural Networks.IEEE Control Systems Magazine, 1989, 9(3):31-37.
[34]胡叔旖，孙增忻．二级倒立摆的基干规则控制的研究．1995年中国智能自动化学会议论文集，1995：533-540．
[35]张明廉，郝健康等．拟人智能控制与三级倒立摆．航空学报，1995，16(6)：654-661．
[36]张明廉，何卫东，沈程智．规约规则发仿人控制．第一届全球华人智能控制与自动化大会论文集，1996：291-196．
[37]李祖枢，陈庆春．倒摆系统的智能控制研究及其发展动向．第二届全球华人智能控制与智能自动化大会论文集(上卷)：14-19．
[38]曾拍．小车-二级摆摆起倒立的仿人智能控制研究：(硕士学位论文)．重庆：重庆大学，2001．
[39]Shouling He,Qiang Wu,Nariman Sepehri.Control of Base-Excited Inverted Pendulums Using a Neural Inverse Modeling Approach. Proceedings of the 14th IF-AC:409-414.
[40]李俊芳，张振东，史晓霞．基于智能方法的倒立摆系统研究．燕山大学学报，2001，8(25)：56-60．
[41]黄苑虹，梁慧冰．从倒立摆装置的控制策略看控制理论的发展和应用．广东上业大学学报，2001，18(3)：49-53．
[42]李浩．提高BP算法收敛速度的几种方法．计算机学报，1997，1．
[43]曹洪洋，李万庆，李文华等．前馈神经网络的一种新的学习算法．微机发展，2001，11(4)：50-51．
[44]粟塔山．最优化计算原理与算法程序设计．长沙：国防科技大学出版社，2001．
[45]曹海云，李守巨，刘迎曦．基于改进PID神经网络的非线性动态系统控制．控制工程，2007，14(增)：38-40．
[46]Fredric Ham M, Ivica Kostanic.Priciplees of Neurocomputing for Science & Engineering. 北京：机械工业出版社，2003．
[47]刘金琨．先进PID控制MATLAB仿真．北京：电子工业出版社，2004．
[48]郑长义，王光德，刘宏，王健．基于Riccati方程的二级倒立摆控制．吉林师范大学学报(自然科学版)，2006，2：107-108．
[49]徐丽娜．神经网络控制结构及所用得神经网络．自动化技术与应用，2004，23(1)：1-5．
[50]李宜达．控制系统设计与仿真．北京：清华大学出版社，2004．