滑模变结构控制理论及其在倒立摆系统中的应用研究
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摘要
滑模变结构控制由前苏联学者Emelyanov在上世纪五十年代提出，后经Utkin和Itkis等人的进一步发展研究。由于它所具有的独特的鲁棒性以及对匹配不确定性和外部干扰的完全自适应性等特点，到上世纪七十年代，逐渐引起了西方学者的重视。目前滑模变结构控制理论已经得到了充分的发展，成为非线性控制理论的一个重要分支，其所研究的对象涉及离散系统、分布参数系统、滞后系统等。然而滑模变结构控制带来的高频抖振是其应用到实际系统的障碍。因此许多其它的先进控制技术如自适应控制、模糊控制、神经网络控制等也被综合应用到滑模变结构控制系统设计中，以解决滑模变结构控制系统的抖振，同时还要尽量保持滑模变结构所具有的性能。
本文在掌握滑模变结构控制理论的国内外研究现状，结合实际应用对滑模变结构控制理论提出的新要求基础上，对目前滑模变结构控制理论面临的几个问题进行了深入的研究，同时理论联系实际，将滑模变结构控制理论成功地应用于倒立摆系统控制中。论文的主要内容如下：
针对一类相对于输入为非线性的系统，设计了滑模变结构控制器保证非线性系统的全局渐近稳定，在此基础上，考虑到系统运行在滑模面上时所具有的特点，提出了一种基于滑模变结构控制的参数辨识方法来辨识非线性系统的未知参数。并证明了该辨识器是渐近收敛的。
针对一类二阶不确定离散系统，利用离散变结构控制系统所具有的滑动模态区，提出了一种具有强鲁棒性的时变滑模面离散变结构控制方法。通过旋转和平移初始滑模面得到一个时变的滑模面，并证明了在此滑模面条件下滑动模态的存在性。给出了滑模面旋转和平移的具体过程，同时由于滑动模态区的宽度固定，可以定量地确定出滑模面在各个采样时刻的变化量。该控制方法使得闭环系统对于任意初始状态都处于滑动模态区内，然后通过结合旋转和平移滑模面的方法，使随后任何时刻的系统状态都处于滑动模态区内。消除了传统变结构控制方法具有的趋近模态运动，提高
摘 要
了系统的鲁棒性能。
基于传统的离散系统滑模变结构控制方法，利用饱和函数代替滑模切
换函数中的符号函数来抑制滑模变结构闭环控制系统的抖振。证明了当系
统不存在不确定性和外部干扰情况时，用饱和函数的方法可以消除系统的
抖振；而当系统存在不确定性和外部干扰时，必须通过选择适当的饱和函
数参数才能达到抑制抖振目的。同时给出了为抑制抖振所需要的饱和函数
的条件。
将一种广义变结构控制方法成功地应用于一级倒立摆的自举控制实
验。该方法由摆动控制部分和平衡控制部分组成。摆动控制使摆从垂亘向
下运动到近似垂直向上位置，采用双闭环控制策略，内环控制小车的位移，
而外环采用正反馈控制器由摆的角位移和速度来确定出小车期望的位移。
平衡控制利用滑模变结构控制理论使摆到达近似垂直向上位置后能保持
垂直向上状态。
将滑模变结构控制理论应用到二级串联、二级并行倒立摆系统的平衡
控制实验中。为了降低闭环系统的抖振，分别对二级串联、二级井行倒立
摆系统利用两种不同的方法来抑制抖振。实验结果证明了所用方法的有效
性和可行性。
三级倒立摆系统被公认为自动控制理论中的一个典型而又具有挑战
性试验设备。提出了一种基于全程滑模的变结构控制方法对具有单控制输
入的三级倒立摆系统进行平衡控制的综合设计和成功的实验研究．该方法
使得倒立摆系统从初始时刻开始就运行在滑模面上，从而使系统具有较强
的鲁棒性，取得了非常好的实验效果．
最后是全文的总结与展望。
In the 1950s, Emelyanov, a Soviet researcher, first proposed variable structure control (VSC) based on sliding mode(SM), then Utkin and Itkis et al. developed the theory. Significant interest on VSC with SM has been generated in the control research realm on account of its excellent robustness and complete adaptability to the uncertainties and external disturbance in the 1970s.
So far, VSC with SM has been deeply studied as an important branch of nonlinear control theory. It has been employed to control discrete-time systems, distributed parameters systems and time-delay systems etc.. However, the chattering with high frequency in sliding mode control systems is a barrier for the application to the practice engineering problems. Therefore, many advanced control theories, such as adaptive control, fuzzy control, NN etc. have been applied to variable structure system to reduce the chattering.
Considering the current developed VSC theory and the new requirements from the practice, some desiderated problems are studied and discussed. The single, double, parallel and triple inverted pendulums have been controlled by the sliding mode controller (SMC) successfully.
An SM based identifier is presented to deal with the parameter identification problem for a class of parameter uncertain nonlinear dynamic systems with input nonlinearity. An SMC algorithm is employed to ensure the global reaching condition of the sliding mode for the nonlinear system, and an identifier is designed to identify the uncertain parameters of the nonlinear system. The asymptotical convergence of the identifier is proved.
A discrete variable structure control algorithm with time-varying sliding surface is developed for a class of second-order uncertain discrete systems.
Ill
The time-varying sliding surface is obtained by rotating and/or shifting the initial sliding mode surface, and the existence of sliding mode with the time-varying sliding surface is proved. The rotating and/or shifting procedures are presented in detail, and the movement of the time-varying sliding surface is obtained quantitatively at every sampling time by taking advantage of the range of the sliding mode band. The states of the closed loop system moved in the sliding mode band of the time-varying sliding surface globally. Reaching mode is eliminated and robustness of the closed-loop system is enhanced.
The sign function of the switching surface function is replaced by a saturate function to reduce the chattering. The chattering can be eliminated if there do not exist hasn't parameter uncertainties and external disturbance in the controlled system. An parameter of the saturate function can be adjusted to reduce the chattering if there are some parameter uncertainties and external disturbance, and how to select this parameter to reduce the chattering are introduced.
An extended VSC is designed to make the single pendulum self erected and then keep it upright. Two loops are presented to swing the pendulum up. The outer loop, which is a positive feedback loop performs position control of the cart based on the position and rate of the pendulum. The inner loop make the cart track the position. The balance control part based on sliding mode keep the pendulum upright.
The double inverted pendulum and the parallel inverted pendulum was successfully balanced by VSC with SM. Two different methods for these two systems are employed to reduce the chattering and excellent results are obtained.
Triple inverted pendulum is a mechanical idealization which poses an interesting and difficult problem in control. In this paper, a robust control algorithm based on sliding mode is employed to control a triple inverted pendulum system with single input. The employment of such a technique
IV
appears necessary because the traditional linear design process cannot incorporate the nonlinear dynamics of such system and its physical limitations. The new controller satisfies the required dynamic characteristic of the triple inverted pendulum system and keep the system
本文在掌握滑模变结构控制理论的国内外研究现状，结合实际应用对滑模变结构控制理论提出的新要求基础上，对目前滑模变结构控制理论面临的几个问题进行了深入的研究，同时理论联系实际，将滑模变结构控制理论成功地应用于倒立摆系统控制中。论文的主要内容如下：
针对一类相对于输入为非线性的系统，设计了滑模变结构控制器保证非线性系统的全局渐近稳定，在此基础上，考虑到系统运行在滑模面上时所具有的特点，提出了一种基于滑模变结构控制的参数辨识方法来辨识非线性系统的未知参数。并证明了该辨识器是渐近收敛的。
针对一类二阶不确定离散系统，利用离散变结构控制系统所具有的滑动模态区，提出了一种具有强鲁棒性的时变滑模面离散变结构控制方法。通过旋转和平移初始滑模面得到一个时变的滑模面，并证明了在此滑模面条件下滑动模态的存在性。给出了滑模面旋转和平移的具体过程，同时由于滑动模态区的宽度固定，可以定量地确定出滑模面在各个采样时刻的变化量。该控制方法使得闭环系统对于任意初始状态都处于滑动模态区内，然后通过结合旋转和平移滑模面的方法，使随后任何时刻的系统状态都处于滑动模态区内。消除了传统变结构控制方法具有的趋近模态运动，提高
摘 要
了系统的鲁棒性能。
基于传统的离散系统滑模变结构控制方法，利用饱和函数代替滑模切
换函数中的符号函数来抑制滑模变结构闭环控制系统的抖振。证明了当系
统不存在不确定性和外部干扰情况时，用饱和函数的方法可以消除系统的
抖振；而当系统存在不确定性和外部干扰时，必须通过选择适当的饱和函
数参数才能达到抑制抖振目的。同时给出了为抑制抖振所需要的饱和函数
的条件。
将一种广义变结构控制方法成功地应用于一级倒立摆的自举控制实
验。该方法由摆动控制部分和平衡控制部分组成。摆动控制使摆从垂亘向
下运动到近似垂直向上位置，采用双闭环控制策略，内环控制小车的位移，
而外环采用正反馈控制器由摆的角位移和速度来确定出小车期望的位移。
平衡控制利用滑模变结构控制理论使摆到达近似垂直向上位置后能保持
垂直向上状态。
将滑模变结构控制理论应用到二级串联、二级并行倒立摆系统的平衡
控制实验中。为了降低闭环系统的抖振，分别对二级串联、二级井行倒立
摆系统利用两种不同的方法来抑制抖振。实验结果证明了所用方法的有效
性和可行性。
三级倒立摆系统被公认为自动控制理论中的一个典型而又具有挑战
性试验设备。提出了一种基于全程滑模的变结构控制方法对具有单控制输
入的三级倒立摆系统进行平衡控制的综合设计和成功的实验研究．该方法
使得倒立摆系统从初始时刻开始就运行在滑模面上，从而使系统具有较强
的鲁棒性，取得了非常好的实验效果．
最后是全文的总结与展望。
In the 1950s, Emelyanov, a Soviet researcher, first proposed variable structure control (VSC) based on sliding mode(SM), then Utkin and Itkis et al. developed the theory. Significant interest on VSC with SM has been generated in the control research realm on account of its excellent robustness and complete adaptability to the uncertainties and external disturbance in the 1970s.
So far, VSC with SM has been deeply studied as an important branch of nonlinear control theory. It has been employed to control discrete-time systems, distributed parameters systems and time-delay systems etc.. However, the chattering with high frequency in sliding mode control systems is a barrier for the application to the practice engineering problems. Therefore, many advanced control theories, such as adaptive control, fuzzy control, NN etc. have been applied to variable structure system to reduce the chattering.
Considering the current developed VSC theory and the new requirements from the practice, some desiderated problems are studied and discussed. The single, double, parallel and triple inverted pendulums have been controlled by the sliding mode controller (SMC) successfully.
An SM based identifier is presented to deal with the parameter identification problem for a class of parameter uncertain nonlinear dynamic systems with input nonlinearity. An SMC algorithm is employed to ensure the global reaching condition of the sliding mode for the nonlinear system, and an identifier is designed to identify the uncertain parameters of the nonlinear system. The asymptotical convergence of the identifier is proved.
A discrete variable structure control algorithm with time-varying sliding surface is developed for a class of second-order uncertain discrete systems.
Ill
The time-varying sliding surface is obtained by rotating and/or shifting the initial sliding mode surface, and the existence of sliding mode with the time-varying sliding surface is proved. The rotating and/or shifting procedures are presented in detail, and the movement of the time-varying sliding surface is obtained quantitatively at every sampling time by taking advantage of the range of the sliding mode band. The states of the closed loop system moved in the sliding mode band of the time-varying sliding surface globally. Reaching mode is eliminated and robustness of the closed-loop system is enhanced.
The sign function of the switching surface function is replaced by a saturate function to reduce the chattering. The chattering can be eliminated if there do not exist hasn't parameter uncertainties and external disturbance in the controlled system. An parameter of the saturate function can be adjusted to reduce the chattering if there are some parameter uncertainties and external disturbance, and how to select this parameter to reduce the chattering are introduced.
An extended VSC is designed to make the single pendulum self erected and then keep it upright. Two loops are presented to swing the pendulum up. The outer loop, which is a positive feedback loop performs position control of the cart based on the position and rate of the pendulum. The inner loop make the cart track the position. The balance control part based on sliding mode keep the pendulum upright.
The double inverted pendulum and the parallel inverted pendulum was successfully balanced by VSC with SM. Two different methods for these two systems are employed to reduce the chattering and excellent results are obtained.
Triple inverted pendulum is a mechanical idealization which poses an interesting and difficult problem in control. In this paper, a robust control algorithm based on sliding mode is employed to control a triple inverted pendulum system with single input. The employment of such a technique
IV
appears necessary because the traditional linear design process cannot incorporate the nonlinear dynamics of such system and its physical limitations. The new controller satisfies the required dynamic characteristic of the triple inverted pendulum system and keep the system
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Utkin V. I., VSS: present and future. Automation and Remote Control, 1983, 44:1105-1120
Utkin V. I. and Yang K. D., Methods for constructing discontinuity planes in multidimensional variable structure systems. Automation and Remote Control, 1978, 39(10): 1466-1470
Utkin V.I., "Variable Structure Systems: Presebt and Future," Automation and Remote Control, 1983, 44(9): 1105-1120,
Utkin, V, I., Variable structure systems with sliding modes. IEEE Trans.
Automatic Control, 1977, 22(2):212-222
Venkataraman S.T. and Gulati S., A variable structure model reference adaptive control for nonlinear robotic manipulators. Proceedings of American Control Conference, 1992:891-893
Wang W.J., Wu G.H., and Yang D.G., Variable Structure Control Design for Uncertain Discrete-Time Systems. IEEE Trans. Automat. Contr., 1994, 39(1): 99-102
Wensel L., Vazquez N., Nair D. et. al., Computer vision based inverted pendulum. Proceedings of the 17th IEEE Instrumentation and Measurement Technology Conference, 2000:1320-1323
Widjaja M., Yurkovich S., Intelligent control for swing up and balancing of an inverted pendulum system. Proceedings of the 4th IEEE Conference on Control Applications, 1995: 534-542
Wunch W. S., Reproduction of an arbitrary function of time by discontinuous control, Ph.D. dissertation, Stanford Univ., Stanford, CA, 1953
Xu J.X., Parameter identification methodologies based on variable structure control. Int. J. Control, 1993, 57(5): 1207-1220
Xu J.X., Self-tuning variable structure control method for a class of nonlinear systems. Int. J. Robust Nonlinear Control, 1998, 8(6): 1133-1153
Yamakita M., Iwashiro M., Sugahara Y., Furuta K., Robust swing up control of double pendulum. Proceedings of the American Control Conference,1995,1: 290 -295
Yasunobu S., Yamasaki H., Evolutionary control method and swing up and stabilization control of inverted pendulum. IFSA World Congress and 20th NAFIPS International Conference. 2001, 4:2078-2083
Yi J., Yubazakia N., Hirota K., Upswing and stabilization control of inverted pendulum system basedon the SIRMs dynamically connected fuzzy inference model. Fuzzy Sets and Systems, 2001,122:139-152
Yoo D. S. and Chung M. J., A variable structure control with simple
adaptation laws for upper bounds on the norm of the uncertainties. IEEE Trans. Automat. Control, 1992, 37 (6): 860-864
Yu X., ManZ., Model reference adaptive control systems with terminal sliding modes. Int. J. Contr., 1996, 64:1165-1176
Yu X., Man Z., Multi-input uncertain linear systems with terminal sliding mode control. Automatica, 1998, 34(3): 389-392
Yu X., Man Z., Fast TSM Control for Single Input Systems. Proc. of 2000 ASCC, Shanghai, China, July, 2000
Yu X.H., Conditions for the Existence of Discrete Time Sliding Mode. Proc. IFAC 12th World Congress, 1993, 2:215-218
Yu X., Digital Variable Structure Control with Pseudo-Sliding Mode, Variable Structure and Lyapnov Control (A.S.I. Zinober ed.), Spinger-Verlag, 1994:133-155
Zhang Keqin, Hongye Su, Kaiyu Zhuang, and Jian Chu , Comments on "Discrete-Time Variable Structure Controller With a Decoupled Disturbance Compensator and Its Application to a CNC Servomechanism". IEEE Transactions on Control Systems Technology, 2003,11(1):156-157
Zhong W., Rock H., Energy and passitive based control of the double inverted pendulum on a cart. Proc. of the IEEE International Conference on Control Applications. 2001, 896-901
Zinober A.S.I., Variable Structure and Lyapnov Control. Lecture Notes in Control and Information Sciences 193, Springer-Verlag, 1994
单波，徐燕，赵建涛，预测控制算法及其在倒立摆中的应用．华北电力大学学报，2001，28(2)：46—51
高为炳，变结构控制的理论和设计分发，科学出版社，1998
高为炳，变结构控制系统的品质控制．全国控制理论与应用年会，西安，1988，pp156-159
李洪兴，苗志红，王家银，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)：65—75
罗宁苏，冯纯伯，消除变结构系统中高频颤振的一种方法．全国控制理论与应用年会，西安，1988，pp132-139
武明，季林红，金德闻等，基于能量的人体动力学平衡评价指标的仿真研究．清华大学学报(自然科学版)，2002年第42卷第2期：168—171
肖军，张石，徐心和，四级倒立摆系统的模糊控制方法研究．系统仿真学报，2001，13(6)：752—755
张冬军，从爽等，旋转平行倒立摆的摆起及平衡控制的研究．Proc．of 4th World Congress on Intelligent Control and Automation, 2002: 2370-2374
张飞舟，范跃祖，沈程智等，利用云模型实现智能控制倒立摆．控制理论与应用，2000，17(4)：519-523