不确定非线性系统的自适应迭代学习控制研究
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摘要
不确定非线性系统普遍存在于实际工程领域，充分利用系统的非线性结构特征，有利于简化控制器设计，提高控制性能。本文讨论了两类典型参数不确定非线性系统，设计了相应的自适应迭代学习控制方案，进行了相关理论分析、算例仿真和应用研究。
针对一类带未知干扰和时变参数的非线性系统，提出了一种鲁棒自适应迭代学习控制算法。在迭代域内对未知参数进行估计，利用鲁棒控制抑制干扰，引入反馈环节进一步提高控制性能；针对一类带未知常参数和时变参数的非线性系统，提出了一种组合自适应迭代学习控制算法。通过对系统非线性结构特征的分析，该算法在时域内对未知常参数进行估计，在迭代域内对未知时变参数进行估计，充分利用了时域和迭代域信息。采用Backstepping技术将上述两种算法推广到二阶不确定非线性系统中，基于Lyapunov稳定性理论分析了这两种算法在相同初始条件和对准条件下的收敛性。仿真验证了所提控制算法的有效性。
将上述算法分别应用于带未知不可重复干扰和带未知可重复时变干扰的两种参数不确定机械臂系统。推导了二自由度机械臂系统动力学模型，根据系统特性对算法进行了改进，给出了系统所有内部信号有界性和跟踪误差收敛性的证明。与传统自适应迭代学习控制的实例仿真相对比，结果表明所提方案具有更好的控制性能，在给定区间上实现了对期望轨迹的精确跟踪。
Uncertain nonlinear systems are widely existed in practice engineering. Controllerdesign can be simplified and control performance can be improved by using the nonlinearcharacteristics fully. Two classes of typical nonlinear systems with uncertain parametersare discussed and adaptive iterative learning control (AILC) approaches are designedaccordingly. Moreover, theoretical analysis, numerical simulations and applicationresearches are given.
For a class of nonlinear systems with unknown disturbance or time-varyingparameters, a robust adaptive iterative learning control (RAILC) algorithm is proposed.The unknown parameters are estimated in time-domain and the disturbance is inhibited byrobust control. Feedback control is also introduced to improve the control performance; fora class of nonlinear systems with unknown time-invariant or time-varying parameters, acomposite adaptive iterative learning control (CAILC) algorithm is proposed. Through theanalysis of the structure properties, time-invariant and time-varying parameters areestimated in time-domain and iteration-domain respectively in the algorithm, which makesfull use of the time-domain and the iteration-domain information. Both of the algorithmsare promoted to the second order uncertain nonlinear systems by means of backsteppingtechnology. The convergence analysis of the two approaches under identical initialcondition and alignment condition are carried out based on Lyapunov stability theory. Theeffectiveness of these algorithms is demonstrated by simulations.
The applications of the above algorithms in parametric uncertain robot manipulatorswith unknown nonrepeatbal disturbance and in those with unknown repeatbal time-varyingdisturbance are discussed. The dynamic modal of the robot manipulator system is given,and the algorithms are adjusted according to the system properties. The boundedness of allinternal siginals and the convergence along the iteration horizon of output tracking errorare proved. Compared with a traditional AILC, the simulation results show that betterperformance can be obtained with the proposed methods, and the accurate tracking ofdesired trajectory on specified time interval can be achieved.
针对一类带未知干扰和时变参数的非线性系统，提出了一种鲁棒自适应迭代学习控制算法。在迭代域内对未知参数进行估计，利用鲁棒控制抑制干扰，引入反馈环节进一步提高控制性能；针对一类带未知常参数和时变参数的非线性系统，提出了一种组合自适应迭代学习控制算法。通过对系统非线性结构特征的分析，该算法在时域内对未知常参数进行估计，在迭代域内对未知时变参数进行估计，充分利用了时域和迭代域信息。采用Backstepping技术将上述两种算法推广到二阶不确定非线性系统中，基于Lyapunov稳定性理论分析了这两种算法在相同初始条件和对准条件下的收敛性。仿真验证了所提控制算法的有效性。
将上述算法分别应用于带未知不可重复干扰和带未知可重复时变干扰的两种参数不确定机械臂系统。推导了二自由度机械臂系统动力学模型，根据系统特性对算法进行了改进，给出了系统所有内部信号有界性和跟踪误差收敛性的证明。与传统自适应迭代学习控制的实例仿真相对比，结果表明所提方案具有更好的控制性能，在给定区间上实现了对期望轨迹的精确跟踪。
Uncertain nonlinear systems are widely existed in practice engineering. Controllerdesign can be simplified and control performance can be improved by using the nonlinearcharacteristics fully. Two classes of typical nonlinear systems with uncertain parametersare discussed and adaptive iterative learning control (AILC) approaches are designedaccordingly. Moreover, theoretical analysis, numerical simulations and applicationresearches are given.
For a class of nonlinear systems with unknown disturbance or time-varyingparameters, a robust adaptive iterative learning control (RAILC) algorithm is proposed.The unknown parameters are estimated in time-domain and the disturbance is inhibited byrobust control. Feedback control is also introduced to improve the control performance; fora class of nonlinear systems with unknown time-invariant or time-varying parameters, acomposite adaptive iterative learning control (CAILC) algorithm is proposed. Through theanalysis of the structure properties, time-invariant and time-varying parameters areestimated in time-domain and iteration-domain respectively in the algorithm, which makesfull use of the time-domain and the iteration-domain information. Both of the algorithmsare promoted to the second order uncertain nonlinear systems by means of backsteppingtechnology. The convergence analysis of the two approaches under identical initialcondition and alignment condition are carried out based on Lyapunov stability theory. Theeffectiveness of these algorithms is demonstrated by simulations.
The applications of the above algorithms in parametric uncertain robot manipulatorswith unknown nonrepeatbal disturbance and in those with unknown repeatbal time-varyingdisturbance are discussed. The dynamic modal of the robot manipulator system is given,and the algorithms are adjusted according to the system properties. The boundedness of allinternal siginals and the convergence along the iteration horizon of output tracking errorare proved. Compared with a traditional AILC, the simulation results show that betterperformance can be obtained with the proposed methods, and the accurate tracking ofdesired trajectory on specified time interval can be achieved.
引文
1 Park B H, Kuc T Y, Lee J S. Adaptive learning control of uncertain robotic systems [J]. International Journal of Control, 1996, 65(5): 725-744
2 YE Xu-dong, DING Zheng-tao. Robust tracking control of uncertain nonlinear systems with unknown control directions [J]. Systems and Control Letters, 2001, 42(1): 1-10
3 Jian-Xin Xu, Jing Xu, Ying Tan. Robust learning control for nonlinear uncertain systems based on composite energy function [A]. 15th Triennial World Congress of the International Federation of Automatic Control [C], Barcelona, Spain, 2002:21-26
4 Jiang Zhongping, Laurent Praly. Design of robust adaptive controller for nonlinear systems with dynamic uncertainties [J]. Automatica, 1998, 34(7): 825-840
5 Kanellakopoulos I, Kokotovic P V, Morse A S. Systematic design of adaptive controllers for feedback linearizable systems [J]. IEEE Trans on Automatic Control, 1991, 36(11): 1241-1253
6 Krstic M, Kokotovic P V. Adaptive nonlinear design with controller-identifier separation and swapping [J]. IEEE Trans on Automatic Control, 1995, 40(3): 426-440
7 Pan Z, Basar T. Adaptive controller design for tracking and disturbance attenuation in parametric strict-feedback nonlinear systems [J]. IEEE Trans on Automatic Control, 1998, 43(8): 1066-1083
8 孙明轩，黄宝健．迭代学习控制[M]．北京：国防工业出版社，1999
9 Jian-Xin Xu,, Zhihua Qu. Robust iterative learning control for a class of nonlinear systems[J]. Automatica, 1998, 34(8): 983-988
10 杨小军，李俊民．基于神经网络的鲁棒自适应滑模迭代学习控制[J]．西安电子科技大学学报，2002，29(3)：382-386
11 Krstic M, Kanellakopoulos I, Kokotovic P V. Nonlinear and Adaptive Control Design [M]. New York: John Wiley and Sons Inc, 1995
12 Ge S S. Advanced control techniques of robotic manipulators [A]. Proceedings of American Control Conference [C], New York: IEEE Press, 1998
13 王强德，井元伟，张嗣瀛．一类不确定非线性系统的鲁棒自适应ε-输出跟踪控制[J]．控制与决策，2004，19(6)：711-713
14 周绍生，费树岷，冯纯伯．不确定严格非线性系统的鲁棒控制[J]．信息与控制，2000，29(3)：193-197
15 Seto D, Annaswamy A M, Baillieul J. Adaptive control of nonlinear systems with triangular structure [J]. IEEE Trans on Automatic Control, 1994, 39(7): 1411-1428
16 ZHANG Kan-jian, FENG Chun-bo, FEI Shu-min. Robust output feedback for a class of uncertain nonlinear systems [J]. Control Theory and Applications, 2003, 20(2): 173-179
17 Efe M O, Unsal C, Kaynak O, et al. Variable structure control of a class of uncertain systems [J]. Automatic, 2004, 40(1): 59-64
18 刘晓华，陈卫田．参数化不确定非线性系统的鲁棒控制[J]．控制与决策，1997，12(1)：53-57
19 费树岷，冯纯伯，宋士吉．非线性不确定系统的鲁棒性研究[J]．自动化学报，2000，26(5)：700-703
20 柯海森，叶旭东，钱建海．一类不确定非线性系统的控制器设计[J]．浙江大学学报(工学版)，2006，40(2)：230-232
21 Jian-Xin Xu, V Badrinath, Zhi-hua Qu. Robust learning control for robotic manipulators with an extension to a class of non-linear systems [J]. International Journal of Control, 2000, 73(10): 858-870
22 Jian-Xin Xu. A new periodic adaptive control approach for time-varying parameters with known periodicity [J]. IEEE Trans on Automatic Control, 2004, 49(4): 579-583
23 Jian-Xin Xu, Ying Tan, Tong-heng Lee. Iterative learning control design based on composite energy function with input saturation [J]. Automatic, 2004, 40:1371-1377
24 French M, Rogers E. Nonlinear iterative learning by an adaptive Lyapunov technique [A]. Proceedings of the 37th IEEE Conference on Design and Control [C], Tampa, Florida, 1998:175-180
25 Dimtry Gorinevsky. Distributed system loop shaping design of iterative learning control for batch processes [A]. Proceedings of the 38th IEEE Conference on Decision and Control[C], Phoenix, AZ, USA, 1999, 1:245-250
26 Tayebi A, Zaremba M B. Exponential convergence of an iterative learning controller for time-varying nonlinear systems [A]. Proceedings of the 38th IEEE Conference on Decision and Control [C], Phoenix, AZ, USA, 1999, 2: 1593-1598
27 史忠科．连续非线性系统的迭代学习控制[J]．控制理论与应用，1997，14(6)：878-881
28 Jeong G M, Choi C H. Iterative learning control for linear discrete time nonminimum phase systems[J]. Automatica, 2002, 38(2): 287-291
29 Arimoto S, Kawamura S, Miyazaki F. Bettering operation of robots by learning [J]. Journal of Robotic Systems, 1984, 1(2): 123-140
30 Heinzinger G, Frenwick D, Paden B, et al. Robust learning control [A]. Proceedings of the 28th IEEE Conference on Decision and Control [C], Tampa, Florida, 1989: 436-440
31 Saab S S, Vogt W G, Mickle M H. Learning control algorithms for traeking"slowly" varying trajectories [J]. IEEE Trans on System, Man, and Cybernetics, 1997, 27(4).. 657-670
32 Arimoto S, Nanima T, Suzuki H. Selective learning with a forgetting factor for robot motion control [A]. Proceedings of the IEEE International Conference on Robotics and Automation [C], Sacramento, CA: IEEE Press, 1991:728-733
33 谢胜利，谢振东．终端受限机器人系统轨道跟踪的新控制算法[J]．自动化学报，2001，27(2)：152-157
34 Wang Danwei. A simple iterative learning controller for manipulators with flexible joints [J]. Automatica, 1995, 31(9): 1341-1344
35 于少娟，齐向东，吴聚华．迭代学习控制理论及应用[M]．北京：机械工业出版社，2005
36 Jian-Xin Xu, Tong-heng Lee. Analysis and comparison of iterative learning control schemes [J]. Engineering Application of Artificial Intelligence, 2004, 17(6): 675-686
37 姚仲舒，杨成梧，王宏飞．任意初始状态下迭代学习控制的频域分析[J]．系统工程与电子技术，2003，25(2)：213-215
38 Geng Zheng, Carroll R, Xie Jahong. Two-dimensional model and algorithm analysis for a class of iterative learning control systems [J]. International Journal of Control, 1990, 52(4): 833-862
39 Kuc T Y, Nam K, Lee J S. An iterative learning control of robot manipulators [J]. IEEE Journal of Robotics and Automation, 1991, 7(6): 835-841
40 任雪梅，高为炳．任意初始状态下的学习控制[J]．自动化学报，1994，20(1)：74-79
41 Arimoto S, Nanima T, Suzuki H. Robustness of P-type learning with a forgetting factor for robot motion [A]. Proceedings of the 29th IEEE Conference on Decision and Control [C], Honolulu, Hawil, USA, 1990:2640-2645
42 Jian-Xin Xu. Recent advances in iterative learning control [J]. Automatica, 2005, 31(1): 132-142
43 Seo W G, Park B H, Lee J S. Adaptive fuzzy learning control for a class of nonlinear dynamic systems [J]. International Journal of Intelligent Systems, 2000, 15: 1157-1175
44 Chien C J, Fu L C. An iterative learning control of nonlinear systems using neural network design [J]. Special Issue on Iterative Learning Control, Asian Journal of Control, 2002, 4(1): 21-29
45 柯海森，叶旭东，钱建海．一类不确定非线性系统的控制器设计[J]．浙江大学学报(工学版)，2006，40(2)：230-232
46 Choi J Y, Lee J S. Adaptive learning control of uncertain robotic systems [J]. IEE Proceedings Control Theory and Applications, 2000, 147(2): 217-223
47 Jian-Xin Xu, Rui Yan. Iterative learning control design without a prior knowledge of the control direction [J]. Automatica, 2004, 40(10): 1803-1809
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50 Jian-Xin Xu, Ying Tan. A composite energy function-based learning control approach for nonlinear systems with time-varying parametric uncertainties [J]. IEEE Trans on Automatic Control, 2002, 47(11): 1940-1945
51 Tayebi A. Adaptive iterative learning control for robot manipulators [J]. Automatica, 2004, 40(7): 1195-1203
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59 朱永红，占俐琳，周健儿等．自适应鲁棒迭代学习控制在陶瓷机械手中的应用[J]．陶瓷学报，2004，25(3)：197-202
60 Jang T J, Choi C H, Ahn H S. Iterative learning control in feedback systems [J]. Automatica, 1995, 31(2): 243-248
61 Jian-Xin Xu, Rui Yan. On Initial conditions in iterative learning control [J]. IEEE Trans on Automatic Control, 2005, 50(9): 1349-1354
62 French M, Rogers E. Nonlinear iterative learning by an adaptive Lyapunov technique [J]. International Journal of Control, 2000, 73(10): 840-850
63 Chun-Te Hsu, Chiang-Ju Chien, Chia-Yu Yao. A new algorithm of adaptive iterative learning control for uncertain robotic systems [A]. Proceedings of 2003 IEEE International Conference on Robotics and Automation [C]. Taibei, 2003:4130-4135
64 Seo, W G, Park B H, Lee J S. Intelligent learning control for a class of nonlinear dynamics system [J]. IEE Proceedings Control Theory and Applications, 1999, 146(2): 165-170
65 杨胜跃，罗安，樊晓平．不确定性机器人系统自适应鲁棒迭代学习控制[J]．控制理论与应用，2003，20(5)：707-712
66 霍伟．机器人动力学与控制[M]．北京：高等教育出版社，2005：55-61
2 YE Xu-dong, DING Zheng-tao. Robust tracking control of uncertain nonlinear systems with unknown control directions [J]. Systems and Control Letters, 2001, 42(1): 1-10
3 Jian-Xin Xu, Jing Xu, Ying Tan. Robust learning control for nonlinear uncertain systems based on composite energy function [A]. 15th Triennial World Congress of the International Federation of Automatic Control [C], Barcelona, Spain, 2002:21-26
4 Jiang Zhongping, Laurent Praly. Design of robust adaptive controller for nonlinear systems with dynamic uncertainties [J]. Automatica, 1998, 34(7): 825-840
5 Kanellakopoulos I, Kokotovic P V, Morse A S. Systematic design of adaptive controllers for feedback linearizable systems [J]. IEEE Trans on Automatic Control, 1991, 36(11): 1241-1253
6 Krstic M, Kokotovic P V. Adaptive nonlinear design with controller-identifier separation and swapping [J]. IEEE Trans on Automatic Control, 1995, 40(3): 426-440
7 Pan Z, Basar T. Adaptive controller design for tracking and disturbance attenuation in parametric strict-feedback nonlinear systems [J]. IEEE Trans on Automatic Control, 1998, 43(8): 1066-1083
8 孙明轩，黄宝健．迭代学习控制[M]．北京：国防工业出版社，1999
9 Jian-Xin Xu,, Zhihua Qu. Robust iterative learning control for a class of nonlinear systems[J]. Automatica, 1998, 34(8): 983-988
10 杨小军，李俊民．基于神经网络的鲁棒自适应滑模迭代学习控制[J]．西安电子科技大学学报，2002，29(3)：382-386
11 Krstic M, Kanellakopoulos I, Kokotovic P V. Nonlinear and Adaptive Control Design [M]. New York: John Wiley and Sons Inc, 1995
12 Ge S S. Advanced control techniques of robotic manipulators [A]. Proceedings of American Control Conference [C], New York: IEEE Press, 1998
13 王强德，井元伟，张嗣瀛．一类不确定非线性系统的鲁棒自适应ε-输出跟踪控制[J]．控制与决策，2004，19(6)：711-713
14 周绍生，费树岷，冯纯伯．不确定严格非线性系统的鲁棒控制[J]．信息与控制，2000，29(3)：193-197
15 Seto D, Annaswamy A M, Baillieul J. Adaptive control of nonlinear systems with triangular structure [J]. IEEE Trans on Automatic Control, 1994, 39(7): 1411-1428
16 ZHANG Kan-jian, FENG Chun-bo, FEI Shu-min. Robust output feedback for a class of uncertain nonlinear systems [J]. Control Theory and Applications, 2003, 20(2): 173-179
17 Efe M O, Unsal C, Kaynak O, et al. Variable structure control of a class of uncertain systems [J]. Automatic, 2004, 40(1): 59-64
18 刘晓华，陈卫田．参数化不确定非线性系统的鲁棒控制[J]．控制与决策，1997，12(1)：53-57
19 费树岷，冯纯伯，宋士吉．非线性不确定系统的鲁棒性研究[J]．自动化学报，2000，26(5)：700-703
20 柯海森，叶旭东，钱建海．一类不确定非线性系统的控制器设计[J]．浙江大学学报(工学版)，2006，40(2)：230-232
21 Jian-Xin Xu, V Badrinath, Zhi-hua Qu. Robust learning control for robotic manipulators with an extension to a class of non-linear systems [J]. International Journal of Control, 2000, 73(10): 858-870
22 Jian-Xin Xu. A new periodic adaptive control approach for time-varying parameters with known periodicity [J]. IEEE Trans on Automatic Control, 2004, 49(4): 579-583
23 Jian-Xin Xu, Ying Tan, Tong-heng Lee. Iterative learning control design based on composite energy function with input saturation [J]. Automatic, 2004, 40:1371-1377
24 French M, Rogers E. Nonlinear iterative learning by an adaptive Lyapunov technique [A]. Proceedings of the 37th IEEE Conference on Design and Control [C], Tampa, Florida, 1998:175-180
25 Dimtry Gorinevsky. Distributed system loop shaping design of iterative learning control for batch processes [A]. Proceedings of the 38th IEEE Conference on Decision and Control[C], Phoenix, AZ, USA, 1999, 1:245-250
26 Tayebi A, Zaremba M B. Exponential convergence of an iterative learning controller for time-varying nonlinear systems [A]. Proceedings of the 38th IEEE Conference on Decision and Control [C], Phoenix, AZ, USA, 1999, 2: 1593-1598
27 史忠科．连续非线性系统的迭代学习控制[J]．控制理论与应用，1997，14(6)：878-881
28 Jeong G M, Choi C H. Iterative learning control for linear discrete time nonminimum phase systems[J]. Automatica, 2002, 38(2): 287-291
29 Arimoto S, Kawamura S, Miyazaki F. Bettering operation of robots by learning [J]. Journal of Robotic Systems, 1984, 1(2): 123-140
30 Heinzinger G, Frenwick D, Paden B, et al. Robust learning control [A]. Proceedings of the 28th IEEE Conference on Decision and Control [C], Tampa, Florida, 1989: 436-440
31 Saab S S, Vogt W G, Mickle M H. Learning control algorithms for traeking"slowly" varying trajectories [J]. IEEE Trans on System, Man, and Cybernetics, 1997, 27(4).. 657-670
32 Arimoto S, Nanima T, Suzuki H. Selective learning with a forgetting factor for robot motion control [A]. Proceedings of the IEEE International Conference on Robotics and Automation [C], Sacramento, CA: IEEE Press, 1991:728-733
33 谢胜利，谢振东．终端受限机器人系统轨道跟踪的新控制算法[J]．自动化学报，2001，27(2)：152-157
34 Wang Danwei. A simple iterative learning controller for manipulators with flexible joints [J]. Automatica, 1995, 31(9): 1341-1344
35 于少娟，齐向东，吴聚华．迭代学习控制理论及应用[M]．北京：机械工业出版社，2005
36 Jian-Xin Xu, Tong-heng Lee. Analysis and comparison of iterative learning control schemes [J]. Engineering Application of Artificial Intelligence, 2004, 17(6): 675-686
37 姚仲舒，杨成梧，王宏飞．任意初始状态下迭代学习控制的频域分析[J]．系统工程与电子技术，2003，25(2)：213-215
38 Geng Zheng, Carroll R, Xie Jahong. Two-dimensional model and algorithm analysis for a class of iterative learning control systems [J]. International Journal of Control, 1990, 52(4): 833-862
39 Kuc T Y, Nam K, Lee J S. An iterative learning control of robot manipulators [J]. IEEE Journal of Robotics and Automation, 1991, 7(6): 835-841
40 任雪梅，高为炳．任意初始状态下的学习控制[J]．自动化学报，1994，20(1)：74-79
41 Arimoto S, Nanima T, Suzuki H. Robustness of P-type learning with a forgetting factor for robot motion [A]. Proceedings of the 29th IEEE Conference on Decision and Control [C], Honolulu, Hawil, USA, 1990:2640-2645
42 Jian-Xin Xu. Recent advances in iterative learning control [J]. Automatica, 2005, 31(1): 132-142
43 Seo W G, Park B H, Lee J S. Adaptive fuzzy learning control for a class of nonlinear dynamic systems [J]. International Journal of Intelligent Systems, 2000, 15: 1157-1175
44 Chien C J, Fu L C. An iterative learning control of nonlinear systems using neural network design [J]. Special Issue on Iterative Learning Control, Asian Journal of Control, 2002, 4(1): 21-29
45 柯海森，叶旭东，钱建海．一类不确定非线性系统的控制器设计[J]．浙江大学学报(工学版)，2006，40(2)：230-232
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