基于LMI的移动机器人鲁棒控制研究
|
摘要
如果将机器人看作是一种能够扩展人类工作能力的有效工具，那么人类在认识和改造世界的过程中就不能没有机器人。移动机器人是机器人家族中的一个重要分支，也是进一步扩展机器人应用领域的重要研究发展方向，因此对移动机器人运动控制的研究，一直都得到普遍关注。
本文针对近年来在移动机器人运动控制方面的热点问题——非完整约束轮式移动机器人的运动控制问题和干扰抑制问题——进行了深入和细致的研究。前者因为移动机器人系统存在非完整约束，而使其运动控制具有极大的挑战性；后者则由于存在各种不确定性给所研究的问题带来许多困难。主要研究结果如下：
1．基于一类具有非完整约束的轮式移动机器人，建立了该种移动机器人的运动模型，并通过状态反馈线性化方法对其进行了线性化处理，此模型可用于移动机器人的跟踪控制。
2．轮式移动机器人的控制问题由于存在非完整运动约束而极富挑战性。利用鲁棒H~∞控制理论中的LMI方法，给出系统稳定性判据和输出反馈H~∞控制器存在的条件，同时给出H~∞控制器的设计方法。
3．H~∞控制器的阶次通常都比较高，在工程实践中出于对硬件实现性和软件可靠性等方面的考虑，往往希望控制器的阶次低一些。基于LMI方法给出降阶H~∞控制器存在的条件，同时给出降阶H~∞控制器的设计方法。
4．观测器型控制器由于其结构简单和物理意义明确而受到广泛重视。针对具有外部干扰的不确定性系统，提出基于LMI的观测器型H~∞控制器设计方法。
5．将以上控制器的设计方法应用于一类移动机器人控制器设计中，仿真结果表明该方法的有效性和合理性。
Since robots can be regarded as an effective extension of man's motor ability, it is sure to be indispensable in the course of recognition and exploration of the world. Due to its important role in theory and application, motion control of mobile robots have been given enough attention by researcher in the world.
In recent years, there are two main problems being focused. The one is problem of motion control for nonholonomic wheeled mobile robot, the other is anti interference in mobile robot's motion. The former has huge challenge for nonholonomic constraints in the course of motion, and the difficulty of the latter is the uncertainties. These problems are all deeply and systemically studied. The main contributions are given as follows:
1. The motion model of wheeled mobile robots with nonholonomic constraints is built, and the model has been linearilized Via state feedback exact linearization. The motion control of WMR can be studied using the model.
2. It is well known that the control problems of wheeled mobile robots are complicated due to nonholonomic constraints in the course of motion. A robust output feedback controller is designed using linear matrix inequality. At the same time, the stability of system, the existing condition of output feedback and the approach to controller design are presented.
3. The order of H?controller is usually higher, but we hope that the order of controller is lower for the realization of hardware and the reliability of software in engineering application. A reduced-order H" controller is designed according to linear matrix inequality approach.
4. Observer-based controller is widely accepted for its simple structure and explicit physical meaning. The LMI method to design H" controller under the assumption of observer-based type is studied. The controller parameters can be derived from three LMIs.
5. The simulation results show the effectiveness of the proposed control method.
本文针对近年来在移动机器人运动控制方面的热点问题——非完整约束轮式移动机器人的运动控制问题和干扰抑制问题——进行了深入和细致的研究。前者因为移动机器人系统存在非完整约束，而使其运动控制具有极大的挑战性；后者则由于存在各种不确定性给所研究的问题带来许多困难。主要研究结果如下：
1．基于一类具有非完整约束的轮式移动机器人，建立了该种移动机器人的运动模型，并通过状态反馈线性化方法对其进行了线性化处理，此模型可用于移动机器人的跟踪控制。
2．轮式移动机器人的控制问题由于存在非完整运动约束而极富挑战性。利用鲁棒H~∞控制理论中的LMI方法，给出系统稳定性判据和输出反馈H~∞控制器存在的条件，同时给出H~∞控制器的设计方法。
3．H~∞控制器的阶次通常都比较高，在工程实践中出于对硬件实现性和软件可靠性等方面的考虑，往往希望控制器的阶次低一些。基于LMI方法给出降阶H~∞控制器存在的条件，同时给出降阶H~∞控制器的设计方法。
4．观测器型控制器由于其结构简单和物理意义明确而受到广泛重视。针对具有外部干扰的不确定性系统，提出基于LMI的观测器型H~∞控制器设计方法。
5．将以上控制器的设计方法应用于一类移动机器人控制器设计中，仿真结果表明该方法的有效性和合理性。
Since robots can be regarded as an effective extension of man's motor ability, it is sure to be indispensable in the course of recognition and exploration of the world. Due to its important role in theory and application, motion control of mobile robots have been given enough attention by researcher in the world.
In recent years, there are two main problems being focused. The one is problem of motion control for nonholonomic wheeled mobile robot, the other is anti interference in mobile robot's motion. The former has huge challenge for nonholonomic constraints in the course of motion, and the difficulty of the latter is the uncertainties. These problems are all deeply and systemically studied. The main contributions are given as follows:
1. The motion model of wheeled mobile robots with nonholonomic constraints is built, and the model has been linearilized Via state feedback exact linearization. The motion control of WMR can be studied using the model.
2. It is well known that the control problems of wheeled mobile robots are complicated due to nonholonomic constraints in the course of motion. A robust output feedback controller is designed using linear matrix inequality. At the same time, the stability of system, the existing condition of output feedback and the approach to controller design are presented.
3. The order of H?controller is usually higher, but we hope that the order of controller is lower for the realization of hardware and the reliability of software in engineering application. A reduced-order H" controller is designed according to linear matrix inequality approach.
4. Observer-based controller is widely accepted for its simple structure and explicit physical meaning. The LMI method to design H" controller under the assumption of observer-based type is studied. The controller parameters can be derived from three LMIs.
5. The simulation results show the effectiveness of the proposed control method.
引文
1．蔡自兴．机器人学的发展趋势和发展战略．中南工业大学学报，2000，31：1-9．
2．刘德满，尹朝万．机器人智能控制技术．沈阳：东北大学出版社，1993．
3．蔡自兴．机器人学．北京：清华大学出版社，2000．
4．王越超，景兴建．轮式移动机器人的控制．机器人，2000，22(7)：724-729．
5. Nilanjan Sarkar etc. control of mechanical systems with rolling constraits: application to dynamic control of mobile robots.The Int. Journal of Robotics Research, Nov.13,No.1,Feb.1994.
6．吴卫国等．非完整轮式移动机器人控制研究．控制与决策年会论文集，1997：1286-1290．
7．董文杰，霍伟．受非完整约束移动机器人的跟踪控制．自动化学报，2000，26(1)：1-6．
8．池瑞楠等．非完整两轮驱动移动机器人的实验研究．机器人技术与研究，2001(1)：36-40．
9. Anthony M.Bloch etc.. Control and stabilization of nonholonomic dynamic system,.IEEE trans. Automatic Control, Vol.37, No. 11, Nov., 1992.
10. C.Canudas and O. J. Sordalen. exponential stabilization of mobile robots with nonholonomic constraints.IEEE trans. Automatic Control, Vol.37, No.11; Nov., 1992.
11．胡中须等．基于非完整移动机器人动态模型的鲁棒输出跟踪控制．控制与决策，2000，15(5)：599-601．
12. J. Guldner and V. I. Utkin. Stabilization of nonholonomic mobile robots using Lyapunov functions for navigation and slide model control. 33rd IEEE Conf. On Decision and Control, 1994:2967-2972.
13. R.W.Brockett etc. asymptotic stability and feedback stabilization. Differential geometric control theory, 1983:181-208.
14．陈启军等．全局指数收敛的机器人PD自适应轨迹跟踪．控制与决策，2000，15(6)：690-694．
15．马保离，霍伟．空间机器人系统的自适应控制．控制理论与应用．1996，13(2)：191-197．
16. K S Narendra, J Balakrishnan, Adaptive control using multiple models. IEEE, Trans. Automat. Contr., 1997,, 42 (2): 171-187.
17. Jungrnin Yang etc., sliding mode motion control of nonholonomic mobile robot, IEEE Control System, Apr., 1999.
18. Jungmin Yang etc., sliding mode control for trajectory tracking of nonholonomic mobile robot, IEEE trans. Robot Automat, 15(3), 1999.
19. M.L. Corradini, and G. Orlando, Robust tacking control of mobile robot in the presence of uncertainties in the dynamical model, Journal of robotic system , 2001, 18(6): 317-323.
20. Anthony M.Bloch etc.. Control and stabilization of nonholonomic dynamic system,.IEEE trans. Automatic Control, Vol.37, No.11, Nov., 1992.
21. J.C. Doyle, K. Glover, P. Khargonekar, and B. Francis, State-space solution to standard H_2 and H_∞ control problem, IEEE Trans. Automatic control, 1989, AC-34: 831-847.
22. K. Glover and J.C. Doyle. State-space formulas for all stabilizing controllers that satisfy an H_∞-norm bound and relation to risk sensitivity. Syst. Contr. Letters, 11,1998:167-172.
23. P. Gahinet, P. Apkarian. A linear matrix inequality approach to H_∞ control.Int. Jour. Of robust and nonlinear control, 1994, vol.4: 421-448.
24. P. Gahinet. Explicit controller formulas for LMI-based synthesis. Automatica, 1996,32(10):1201-1212.
25. T. Iwasaki, and R. E. Skelton. All controllers for the general H_∞ control problem: LMI existence condition and state-space formulas. Automatica, 1994, 30 (8): 1307-1317.
26. P. Apkarian, P. Gahinet. a convex characterization of H_∞ controller. IEEE Trans. on AC, 1995,40(5):853-863.
27. Yu. Nesterov, Nemirovsky. Interior-point polynomial methods in convex program. SIAM, Philadelphia, 1994.
28. Ilya Kolmanovsky etc., developments in nonholonomic control problems, IEEE Control System, Dec., 1995: 20-36.
29. George J.Pappas and K.J. Kyriakopoulos, Modelling and feedback control of nonholonomic mobile vehicles, IEEE Conference on Decision and Control, Dec., 1992.
30. D'Andrea-Novel, Campion G, Bastin G. Control of nonholonomic Wheeled mobile robots by state feedback linearization. Int. J. of Robotics Research,1995:543-559.
31. Jiang Z P, Nijeijer H. Tracking control of mobile robots: A case study in backstepping[J].Automatica, 1997, 33(7): 1393-1399.
32．吴卫国，陈辉堂，王月娟．移动机器人全局跟踪控制．自动化学报，2001，27(3)：326-331．
33. G.Campion etc., structural properties and classification of kinematic and dynamic models of wheeled mobile robots, IEEE Conference 1993.
34. J.C. Alexander and J.H. Maddocks, On the kinematics of wheeled mobile robots, Intemational Journal of Robotics Research, Oct., 1989.
35．程代展．非线性系统的几何理论．科学出版社，1988．
36．卢强，孙元章．电力系统非线性控制．科学出版社，1993．
37．冯纯伯，费树岷．非线性控制系统分析与设计(第二版)．北京，电子工业出版社，1998．
38．孙书利，崔平远．移动机器人的路径跟踪与一种适用的点阵定．信息与控制，2002，31 (7)：639-654．
39. Kyucheol Park, Hakyoung Chung and Jang Gyu Lee. point stabilization of mobile robots via state space exact feedback linearization. Proceedings of 1999 IEEE Int. Conf. Robot. Automat:2626-2631.
40. Y. J. Kanayama, Y. Kimura, F. Miyazaki, and T. Noguchi. A stable tracking control scheme for an autonomous mobile robot, in Proc. IEEE Int. Conf. On Robotics and Automation, 1990: 384-389.
41. D Andrea-Novel B, Campion G, Bastin G. Control ofnonholonomic wheeled mobile robot by state feedback linearization[J]. Int. J of Robotics research, 1995, 14 (6): 543-559.
42. Y. Zhang, S. Velinsky, and X. Feng. On the tracking control of differentially steered wheeled mobile robot. Journal of dynamic system, measurement, and control, 1997, Vol. 119: 455-461.
43. Y.Zhang etc.. Dynamic model based robust tracking control of a differentially steered wheeled mobile robot. Proceedings of the American Control Conference, 1998: 850-854.
44. T. Vinay, and B. Postma. Path planning and accurate tracking control of mobile robot. The fourth engineering mathematics and engineering conference, 2000.
45．胡中须等，基于非完整移动机器人动态模型的鲁棒输出跟踪控制，控制与决策，2000，15(5)：599-601．
46．王银河，戴冠中．一类不确定线性系统的鲁棒线性控制器设计，控制与决策，2001，16(5)：605-608．
47．唐朝晖，吴敏，桂卫华．基于LMI方法的机器人系统分散鲁棒控制，高技术通讯，2000．8．
48．程储旺，宋执环，孙优贤，不确定性时滞系统的鲁棒镇定控制器设计—LMI方法，控制理论与应用，Vol．16，No．5，Oct．，1999．
49．徐昕，李涛，伯晓晨等．Matlab工具箱应用指南——控制工程篇．北京：电子工业出版社，2000．
50．陈桂明，张明照，戚红雨等．应用Matlab建模与仿真．北京：科学出版社，2001．
51．孙秀霞，毛剑琴．基于凸优化和投影原理的降阶控制器设计方法研究．控制与决策，2000，15(4)：458-460．
52．胡中骥，施颂椒，翁正新．时滞系统的状态反馈和基于观测器的输出反馈控制设计．自动化学报，自动化学报，Vol．26，No．2，Mar．，2000．
53．吴旭光．基于状态观测器的H_∞控制器设计．控制与决策，14(5)，1999，9：433-437．
54．范训礼，谢振华，戴冠中等．基于LMI的不确定线性系统观测器设计．推进技术，Vol．21，No．2，Apr．，2000．
55．范训礼，戴航，张新家．基于观测器的不确定线性系统鲁棒控制器设计．第三届全球华人智能控制大会．
56．席斌，吴铁军．观测器型H_∞控制器设计的LMI方法．自动化学报，Vol．25，No．4，Jul．，1999．
2．刘德满，尹朝万．机器人智能控制技术．沈阳：东北大学出版社，1993．
3．蔡自兴．机器人学．北京：清华大学出版社，2000．
4．王越超，景兴建．轮式移动机器人的控制．机器人，2000，22(7)：724-729．
5. Nilanjan Sarkar etc. control of mechanical systems with rolling constraits: application to dynamic control of mobile robots.The Int. Journal of Robotics Research, Nov.13,No.1,Feb.1994.
6．吴卫国等．非完整轮式移动机器人控制研究．控制与决策年会论文集，1997：1286-1290．
7．董文杰，霍伟．受非完整约束移动机器人的跟踪控制．自动化学报，2000，26(1)：1-6．
8．池瑞楠等．非完整两轮驱动移动机器人的实验研究．机器人技术与研究，2001(1)：36-40．
9. Anthony M.Bloch etc.. Control and stabilization of nonholonomic dynamic system,.IEEE trans. Automatic Control, Vol.37, No. 11, Nov., 1992.
10. C.Canudas and O. J. Sordalen. exponential stabilization of mobile robots with nonholonomic constraints.IEEE trans. Automatic Control, Vol.37, No.11; Nov., 1992.
11．胡中须等．基于非完整移动机器人动态模型的鲁棒输出跟踪控制．控制与决策，2000，15(5)：599-601．
12. J. Guldner and V. I. Utkin. Stabilization of nonholonomic mobile robots using Lyapunov functions for navigation and slide model control. 33rd IEEE Conf. On Decision and Control, 1994:2967-2972.
13. R.W.Brockett etc. asymptotic stability and feedback stabilization. Differential geometric control theory, 1983:181-208.
14．陈启军等．全局指数收敛的机器人PD自适应轨迹跟踪．控制与决策，2000，15(6)：690-694．
15．马保离，霍伟．空间机器人系统的自适应控制．控制理论与应用．1996，13(2)：191-197．
16. K S Narendra, J Balakrishnan, Adaptive control using multiple models. IEEE, Trans. Automat. Contr., 1997,, 42 (2): 171-187.
17. Jungrnin Yang etc., sliding mode motion control of nonholonomic mobile robot, IEEE Control System, Apr., 1999.
18. Jungmin Yang etc., sliding mode control for trajectory tracking of nonholonomic mobile robot, IEEE trans. Robot Automat, 15(3), 1999.
19. M.L. Corradini, and G. Orlando, Robust tacking control of mobile robot in the presence of uncertainties in the dynamical model, Journal of robotic system , 2001, 18(6): 317-323.
20. Anthony M.Bloch etc.. Control and stabilization of nonholonomic dynamic system,.IEEE trans. Automatic Control, Vol.37, No.11, Nov., 1992.
21. J.C. Doyle, K. Glover, P. Khargonekar, and B. Francis, State-space solution to standard H_2 and H_∞ control problem, IEEE Trans. Automatic control, 1989, AC-34: 831-847.
22. K. Glover and J.C. Doyle. State-space formulas for all stabilizing controllers that satisfy an H_∞-norm bound and relation to risk sensitivity. Syst. Contr. Letters, 11,1998:167-172.
23. P. Gahinet, P. Apkarian. A linear matrix inequality approach to H_∞ control.Int. Jour. Of robust and nonlinear control, 1994, vol.4: 421-448.
24. P. Gahinet. Explicit controller formulas for LMI-based synthesis. Automatica, 1996,32(10):1201-1212.
25. T. Iwasaki, and R. E. Skelton. All controllers for the general H_∞ control problem: LMI existence condition and state-space formulas. Automatica, 1994, 30 (8): 1307-1317.
26. P. Apkarian, P. Gahinet. a convex characterization of H_∞ controller. IEEE Trans. on AC, 1995,40(5):853-863.
27. Yu. Nesterov, Nemirovsky. Interior-point polynomial methods in convex program. SIAM, Philadelphia, 1994.
28. Ilya Kolmanovsky etc., developments in nonholonomic control problems, IEEE Control System, Dec., 1995: 20-36.
29. George J.Pappas and K.J. Kyriakopoulos, Modelling and feedback control of nonholonomic mobile vehicles, IEEE Conference on Decision and Control, Dec., 1992.
30. D'Andrea-Novel, Campion G, Bastin G. Control of nonholonomic Wheeled mobile robots by state feedback linearization. Int. J. of Robotics Research,1995:543-559.
31. Jiang Z P, Nijeijer H. Tracking control of mobile robots: A case study in backstepping[J].Automatica, 1997, 33(7): 1393-1399.
32．吴卫国，陈辉堂，王月娟．移动机器人全局跟踪控制．自动化学报，2001，27(3)：326-331．
33. G.Campion etc., structural properties and classification of kinematic and dynamic models of wheeled mobile robots, IEEE Conference 1993.
34. J.C. Alexander and J.H. Maddocks, On the kinematics of wheeled mobile robots, Intemational Journal of Robotics Research, Oct., 1989.
35．程代展．非线性系统的几何理论．科学出版社，1988．
36．卢强，孙元章．电力系统非线性控制．科学出版社，1993．
37．冯纯伯，费树岷．非线性控制系统分析与设计(第二版)．北京，电子工业出版社，1998．
38．孙书利，崔平远．移动机器人的路径跟踪与一种适用的点阵定．信息与控制，2002，31 (7)：639-654．
39. Kyucheol Park, Hakyoung Chung and Jang Gyu Lee. point stabilization of mobile robots via state space exact feedback linearization. Proceedings of 1999 IEEE Int. Conf. Robot. Automat:2626-2631.
40. Y. J. Kanayama, Y. Kimura, F. Miyazaki, and T. Noguchi. A stable tracking control scheme for an autonomous mobile robot, in Proc. IEEE Int. Conf. On Robotics and Automation, 1990: 384-389.
41. D Andrea-Novel B, Campion G, Bastin G. Control ofnonholonomic wheeled mobile robot by state feedback linearization[J]. Int. J of Robotics research, 1995, 14 (6): 543-559.
42. Y. Zhang, S. Velinsky, and X. Feng. On the tracking control of differentially steered wheeled mobile robot. Journal of dynamic system, measurement, and control, 1997, Vol. 119: 455-461.
43. Y.Zhang etc.. Dynamic model based robust tracking control of a differentially steered wheeled mobile robot. Proceedings of the American Control Conference, 1998: 850-854.
44. T. Vinay, and B. Postma. Path planning and accurate tracking control of mobile robot. The fourth engineering mathematics and engineering conference, 2000.
45．胡中须等，基于非完整移动机器人动态模型的鲁棒输出跟踪控制，控制与决策，2000，15(5)：599-601．
46．王银河，戴冠中．一类不确定线性系统的鲁棒线性控制器设计，控制与决策，2001，16(5)：605-608．
47．唐朝晖，吴敏，桂卫华．基于LMI方法的机器人系统分散鲁棒控制，高技术通讯，2000．8．
48．程储旺，宋执环，孙优贤，不确定性时滞系统的鲁棒镇定控制器设计—LMI方法，控制理论与应用，Vol．16，No．5，Oct．，1999．
49．徐昕，李涛，伯晓晨等．Matlab工具箱应用指南——控制工程篇．北京：电子工业出版社，2000．
50．陈桂明，张明照，戚红雨等．应用Matlab建模与仿真．北京：科学出版社，2001．
51．孙秀霞，毛剑琴．基于凸优化和投影原理的降阶控制器设计方法研究．控制与决策，2000，15(4)：458-460．
52．胡中骥，施颂椒，翁正新．时滞系统的状态反馈和基于观测器的输出反馈控制设计．自动化学报，自动化学报，Vol．26，No．2，Mar．，2000．
53．吴旭光．基于状态观测器的H_∞控制器设计．控制与决策，14(5)，1999，9：433-437．
54．范训礼，谢振华，戴冠中等．基于LMI的不确定线性系统观测器设计．推进技术，Vol．21，No．2，Apr．，2000．
55．范训礼，戴航，张新家．基于观测器的不确定线性系统鲁棒控制器设计．第三届全球华人智能控制大会．
56．席斌，吴铁军．观测器型H_∞控制器设计的LMI方法．自动化学报，Vol．25，No．4，Jul．，1999．