一类多项式系统的稳定性分析和控制器综合
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摘要
多项式非线性系统在现实世界中广泛存在,如生化领域,化工过程,电子电路等系统,其中许多控制问题均可建模,转化或近似成多项式非线性系统。又因多项式非线性系统在非线性系统家族中具有普遍性,故多项式非线性系统的研究对非线性系统具有重要意义。由此可见,如何分析及综合多项式非线性系统对非线性理论发展以及工程应用是件非常有意义的工作。近年来,通过数值计算的方法对多项式非线性系统进行研究受到相当多的关注,并取得了一系列成果。这些方法在多项式非线性系统稳定性分析上取得了重大进展。
本论文研究了一类特殊的时不变多项式非线性系统,对其进行了稳定性分析和控制器综合。论文主要研究内容如下：
1.介绍了李亚普诺夫稳定性。在稳定性分析方面,李亚普诺夫稳定性判据是控制理论中的一大基石,它是Lyapunov在19世纪末提出,后经Malkin, Chetaev, Zubov, Krasovskii, Razumikhin等人的发展,成为成熟的体系。
2.介绍了Sum-of-Squares (SOS)最优化算法和Canonical Quadratic Distance Problems (CQDP)最优化算法的相关的概念,并且介绍了基于SOS估计吸引域的算法3.1并给出了在算法3.1基础上改进的算法3.2。最后给出了二维仿真实例,仿真结果表明,对于某些二维系统,用算法3.2可以得到比较好的结果。
3.介绍了基于Sum-of-Squares(SOS)最优化方法的扰动分析,并且介绍了基于Sum-of-Squares(SOS)的设计状态反馈控制器的算法4.1并且给出了在此算法基础上改进的设计控制器的算法4.2。最后给出了二维仿真实例,仿真结果表明,对于某些二维系统,用算法4.2可以得到比较好的结果。
Polynomial nonlinear systems appear widely in the real world. Many control problems in the field of biochemical, chemical process, electric circuits and so on can be modeled as, transformed into, or approximated by polynomial nonlinear systems. Researching polynomial nonlinear systems has significance for the investigation of nonlinear systems, because polynomial nonlinear systems have universality in the nonlinear systems family. Therefore, how to analyze and synthesize polynomial nonlinear systems is a promising work for nonlinear control theory development and engineering applications. In recent years, considerable attention has been devoted to the study of polynomial nonlinear systems in numerical approach. Significant progress has been made in the stability analysis of polynomial nonlinear systems by those numerical approaches.
This thesis explores of a class of special time-invariant polynomial nonlinear systems. And it researches their stability analysis and controller synthesis.The main research content are showed as follows:
1. The Lyapunov stability is introduced. In the field of stability analysis, the Lyapunov stability criterions are a major cornerstone of the Control Theory. It was put forward at the end of the 19th century by Lyapunov. And then it was developed by Malkin, Chetaev, Zubov, Krasovskii, Razumikhin and so on. After that it became a sophisticated system.
2. The related concepts of Sum-of-Squares (SOS) optimization algorithm and canonical quadratic distance problems(CQDP) optimization algorithm are introduced, and algorithm 3.1 bases on SOS to estimates the domain of attraction. And then the improved algorithm 3.2 is given at the basis of algorithm 3.1. Finally, it is illustrated with an example. The example shows that good results can be got by using algorithm 3.2 for some two-dimensional systems.
3. The disturbance analysis based on Sum-of-Squares (SOS) optimization algorithm is introduced. And then the state feedback controller in algorithm 4.1 based on Sum-of-Squares (SOS) is introduced. Furthermore, the improved algorithm 4.2 is given at the basis of algorit-hm 4.1 to design the state feedback controller. At last, it is illustrated with an case. The case denominates the better results can be obtained by using algorithm 4.2 for some two-dimensio-nal systems.
本论文研究了一类特殊的时不变多项式非线性系统,对其进行了稳定性分析和控制器综合。论文主要研究内容如下：
1.介绍了李亚普诺夫稳定性。在稳定性分析方面,李亚普诺夫稳定性判据是控制理论中的一大基石,它是Lyapunov在19世纪末提出,后经Malkin, Chetaev, Zubov, Krasovskii, Razumikhin等人的发展,成为成熟的体系。
2.介绍了Sum-of-Squares (SOS)最优化算法和Canonical Quadratic Distance Problems (CQDP)最优化算法的相关的概念,并且介绍了基于SOS估计吸引域的算法3.1并给出了在算法3.1基础上改进的算法3.2。最后给出了二维仿真实例,仿真结果表明,对于某些二维系统,用算法3.2可以得到比较好的结果。
3.介绍了基于Sum-of-Squares(SOS)最优化方法的扰动分析,并且介绍了基于Sum-of-Squares(SOS)的设计状态反馈控制器的算法4.1并且给出了在此算法基础上改进的设计控制器的算法4.2。最后给出了二维仿真实例,仿真结果表明,对于某些二维系统,用算法4.2可以得到比较好的结果。
Polynomial nonlinear systems appear widely in the real world. Many control problems in the field of biochemical, chemical process, electric circuits and so on can be modeled as, transformed into, or approximated by polynomial nonlinear systems. Researching polynomial nonlinear systems has significance for the investigation of nonlinear systems, because polynomial nonlinear systems have universality in the nonlinear systems family. Therefore, how to analyze and synthesize polynomial nonlinear systems is a promising work for nonlinear control theory development and engineering applications. In recent years, considerable attention has been devoted to the study of polynomial nonlinear systems in numerical approach. Significant progress has been made in the stability analysis of polynomial nonlinear systems by those numerical approaches.
This thesis explores of a class of special time-invariant polynomial nonlinear systems. And it researches their stability analysis and controller synthesis.The main research content are showed as follows:
1. The Lyapunov stability is introduced. In the field of stability analysis, the Lyapunov stability criterions are a major cornerstone of the Control Theory. It was put forward at the end of the 19th century by Lyapunov. And then it was developed by Malkin, Chetaev, Zubov, Krasovskii, Razumikhin and so on. After that it became a sophisticated system.
2. The related concepts of Sum-of-Squares (SOS) optimization algorithm and canonical quadratic distance problems(CQDP) optimization algorithm are introduced, and algorithm 3.1 bases on SOS to estimates the domain of attraction. And then the improved algorithm 3.2 is given at the basis of algorithm 3.1. Finally, it is illustrated with an example. The example shows that good results can be got by using algorithm 3.2 for some two-dimensional systems.
3. The disturbance analysis based on Sum-of-Squares (SOS) optimization algorithm is introduced. And then the state feedback controller in algorithm 4.1 based on Sum-of-Squares (SOS) is introduced. Furthermore, the improved algorithm 4.2 is given at the basis of algorit-hm 4.1 to design the state feedback controller. At last, it is illustrated with an case. The case denominates the better results can be obtained by using algorithm 4.2 for some two-dimensio-nal systems.
引文
1. Sontag E D. On diserete-time polynomial systems[J], Nonlinear Analysis Theory and Applicalions,1976,15(1):55-64.
2. Sontag E D. On the observability of polynomial systems[J], SIAM Journal on Control and Optimization,1979,17(3):139-151.
3. Nesic D, Mareels I M. Dead beat controllability of polynomial systems:symbolic computation approaches[J], IEEE Transactions on Automatic Control,1998,43(2):162-175.
4. Malkin I G. The theory of stability of motion[M], Moscow:Gostekhizdat,1952.
5. Chetaev N G. Stability of motion[M], Moseow:GITTL,1955.
6. Zubov V I. The method of A.M. Lyapunov and their application[M], Leningrad: Leningrad University,1957.
7. Krasovskii N N. Some problems of the stability theory[M], London:Academic Press, 1959.
8. Razumikhin B S. On the stability of systems with a delay [J], Prikl. Mat. Meh,1956,20(6) 500-512.
9. Razumikhin B S. Application of Liapunov's method to problems in the stability of systems with a delay [J], Automati Telemeh,1960,21(9):740-749.
10. Gu K, Kharitonov V L, Chen J. Stability of Time-Delay Systems[M], Boston:Birkhauser, 2003.
11.孙优贤,褚健.工业过程控制技术：方法篇[M],化学工业出版社,2006.
12.褚健,俞立,苏宏业.鲁棒控制理论及应用[M],浙江大学出版社,2000.
13.周克敏,Doyle J C, Glover K, et al.鲁棒与最优控制[M],国防工业出版社,2002.
14.俞立.鲁棒控制-线性矩阵不等式处理方法[M],清华大学出版社,2002.
15. Lasalle J, Lefschetz S. Stability by Liopunov's direct method with applications[M], London:Academic Press,1961.
16. Kalman R, Bertram J. Control system analysis and design via the second method of Lyapunov, Part I, continuous-time systems[J], Transcartons of the ASME, Series D, Journal of Basic Engineering,1960,82(9):371-393.
17. Richard J P. Time-delay systems:an overview of some recent advances and open problems[J], Automatica,2003,39(15):1667-1694.
18. Boyd S, Ghaoui L E, Feron E, et al. Linear matrix inequalities in system and control theory[M], Philadelphia:SIAM,1994.
19. Parrilo P A. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[D], U.S.A:California Institute of Technology,2000.
20. Prajna S, Papachristodoulou A, Seiler P, et al. SOSTOOLS:Sum of Squares Optimization Toolbox for MATLAB User'guide,2004.
21. Sturm J F. http://sedumi.mcmaster.ca/.
22. Sturm J F. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[J]. Optimization Methods and Software,1999,11(1):625-653.
23. Toh K C, Todd M J, Tutuncu R H. http://www.math.nus.edu.sg./-mattohkc/sdpt3.html.
24. Toh K C, Todd M J, Tutuncu R H. SDPT3-A Matlab software package for semidefinite programming, Version 1.3[J]. Optimization Methods and Software,1999,11(1):545-581.
25. J. Zaborszky, G. Huang, B. Zheng, T. C. Leung. On the phase portrait of a class of large nonlinear dynamic systems such as the power system[J], Automatic Control,1988,33(1):4-15.
26. E. D. Ferreira, B. H. Krogh. Using neural networks to estimate regions of stability[J], American Control Conference,1997,1989-1993.
27.张敏,罗安,何早红.包含静止无功发生器的电力系统吸引域[J],中国电机工程学报,2003,23(6)：25-28.
28.罗明河,牛存镇.化学反应工程原理及反应器设计[M],昆明：云南科技出版社,1997,335-348.
29.王建华.化学反应工程(下册,化学反应器设计)[M],成都：成都科技大学出版社,1989,49-59.
30. H. F. Rase. The influence of the Lewis number on the dynamics of chemically reacting systems[J], Chemical Reactor Design for Process Plants,1997,5(8):589.
31.胡亮,罗康碧,李沪萍.化学反应器的定态及其稳定性分析[J],昆明理工大学学报,2002,27(1)：106-109.
32.姚宏,郭雷.高维磁悬浮控制系统的稳定吸引域[J],中国控制与决策学术年会文集,2001,135-138.
33.傅雪东,吴国钊,裴海龙.欠驱动机器人Pendubot的上摆和平衡控制及吸引域分析[J],中国智能自动化学术会议论文集,2007,713-718.
34. V. I. Zubov. Mathematical Methods for the Study of Automatic Method[M], Israel: Jerusalem Academic Press,1962.
35. Tibken B. Estimation of the domain of attraction for polynomial systems via LMIs[J], Proceedings of the IEEE Conference on Decision and Control,2000,3860-3864.
36. Tibken B, Dilaver K F. Computation of subsets of the domain of attraction for polynomial systems[J], Proceedings of the IEEE conference on Decision and Control,2002,2651-2656.
37. Chesi G. Estimating the domain of attraction:a light LMI technique for a class of polyno-mial systems[J], IEEE Conference on Decision and Control,2003,5609-5614.
38. Chesi G, Tesi A, Vicino A, et al. An LMI approach to constrained optimization with homogeneous forms[J], Systems & Control Letters,2001,42(1):11-19.
39. Jarvis-Wloszek Z, Feeley R, Tan W, et al. Some Controls Applications of Sum of Squares Programming[J], Proceedings of IEEE Conference on Decision and Control,2003, 4676-4681.
40. Hachicho O, Tibken B. Estimating domains of attraction of a class of nonlinear dynam-ical systems with LMI methods based on the theory of moments[J], Proceedings of IEEE Co-nference on Decision and Control,2002,3150-3155.
41. R. Bakhtiari, M. J. YazdanPanah. Designing a linear controller for polynomial systems with the largest domain of attraction via LMIs[J], International Conference on Control and Automation,2005,449-453.
42. Chesi G, Garulli A, Tesi A, et al. Solving quadratic distance problems:an LMI-based approach[J], IEEE Transactions on Automatic Control,2003,48(2):200-212.
43. Z.W. Jarvis-Wloszek. Lyapunov based analysis and controller synthesis for polynomial systems using sum-of-squares optimization[D], U.S.A.:University of California, Berkele-y,2003.
44. W. Hahn. Stability of motion[M], Berlin:Springer,1967.
45. J. La. Salle, S. Lefschetz. Stability by Lyapunov's direct method[M], London:Academic Press,1961.
46. H. K. Khalil. Nonlinear systems[M], New York:MacMillan,1992.
47. P. A. Parrilo. Semidefinite programming relaxations for semialgebraic problems[J], Ma-thematical Programming Ser. B,2003,96(2):293-320.
48. T. Motzkin. The arithmetic-geometric inequality[J], In Inequalities, Proceedings of the Symposium at Wright-Patterson AFB,1965,3(2):202-224.
49. S. Prajna, A. Papachristodoulou. SOSTOOLS and its control applications [J], positive pol-ynomical in control, Springer,2005,35(1):146-155.
50. S. P. Bhattacharyya, H. Chapellat, and L. H. Keel. Robust Control:The Parametric Appr- oach[M], U.K.:Cambridge Univ. Press,1995.
51. R. Genesio, M. Tartaglia, and A. Vicino. On the estimation of asymptotic stability regi-ons:State of art and new proposals[J], IEEE Trans. Automat. Contr.,1985, AC(30):747-755.
52.李宁.基于矩量理论和Sum-of-Squares最优化理论的吸引域估计[D],沈阳：东北大学,2008.
2. Sontag E D. On the observability of polynomial systems[J], SIAM Journal on Control and Optimization,1979,17(3):139-151.
3. Nesic D, Mareels I M. Dead beat controllability of polynomial systems:symbolic computation approaches[J], IEEE Transactions on Automatic Control,1998,43(2):162-175.
4. Malkin I G. The theory of stability of motion[M], Moscow:Gostekhizdat,1952.
5. Chetaev N G. Stability of motion[M], Moseow:GITTL,1955.
6. Zubov V I. The method of A.M. Lyapunov and their application[M], Leningrad: Leningrad University,1957.
7. Krasovskii N N. Some problems of the stability theory[M], London:Academic Press, 1959.
8. Razumikhin B S. On the stability of systems with a delay [J], Prikl. Mat. Meh,1956,20(6) 500-512.
9. Razumikhin B S. Application of Liapunov's method to problems in the stability of systems with a delay [J], Automati Telemeh,1960,21(9):740-749.
10. Gu K, Kharitonov V L, Chen J. Stability of Time-Delay Systems[M], Boston:Birkhauser, 2003.
11.孙优贤,褚健.工业过程控制技术：方法篇[M],化学工业出版社,2006.
12.褚健,俞立,苏宏业.鲁棒控制理论及应用[M],浙江大学出版社,2000.
13.周克敏,Doyle J C, Glover K, et al.鲁棒与最优控制[M],国防工业出版社,2002.
14.俞立.鲁棒控制-线性矩阵不等式处理方法[M],清华大学出版社,2002.
15. Lasalle J, Lefschetz S. Stability by Liopunov's direct method with applications[M], London:Academic Press,1961.
16. Kalman R, Bertram J. Control system analysis and design via the second method of Lyapunov, Part I, continuous-time systems[J], Transcartons of the ASME, Series D, Journal of Basic Engineering,1960,82(9):371-393.
17. Richard J P. Time-delay systems:an overview of some recent advances and open problems[J], Automatica,2003,39(15):1667-1694.
18. Boyd S, Ghaoui L E, Feron E, et al. Linear matrix inequalities in system and control theory[M], Philadelphia:SIAM,1994.
19. Parrilo P A. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[D], U.S.A:California Institute of Technology,2000.
20. Prajna S, Papachristodoulou A, Seiler P, et al. SOSTOOLS:Sum of Squares Optimization Toolbox for MATLAB User'guide,2004.
21. Sturm J F. http://sedumi.mcmaster.ca/.
22. Sturm J F. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[J]. Optimization Methods and Software,1999,11(1):625-653.
23. Toh K C, Todd M J, Tutuncu R H. http://www.math.nus.edu.sg./-mattohkc/sdpt3.html.
24. Toh K C, Todd M J, Tutuncu R H. SDPT3-A Matlab software package for semidefinite programming, Version 1.3[J]. Optimization Methods and Software,1999,11(1):545-581.
25. J. Zaborszky, G. Huang, B. Zheng, T. C. Leung. On the phase portrait of a class of large nonlinear dynamic systems such as the power system[J], Automatic Control,1988,33(1):4-15.
26. E. D. Ferreira, B. H. Krogh. Using neural networks to estimate regions of stability[J], American Control Conference,1997,1989-1993.
27.张敏,罗安,何早红.包含静止无功发生器的电力系统吸引域[J],中国电机工程学报,2003,23(6)：25-28.
28.罗明河,牛存镇.化学反应工程原理及反应器设计[M],昆明：云南科技出版社,1997,335-348.
29.王建华.化学反应工程(下册,化学反应器设计)[M],成都：成都科技大学出版社,1989,49-59.
30. H. F. Rase. The influence of the Lewis number on the dynamics of chemically reacting systems[J], Chemical Reactor Design for Process Plants,1997,5(8):589.
31.胡亮,罗康碧,李沪萍.化学反应器的定态及其稳定性分析[J],昆明理工大学学报,2002,27(1)：106-109.
32.姚宏,郭雷.高维磁悬浮控制系统的稳定吸引域[J],中国控制与决策学术年会文集,2001,135-138.
33.傅雪东,吴国钊,裴海龙.欠驱动机器人Pendubot的上摆和平衡控制及吸引域分析[J],中国智能自动化学术会议论文集,2007,713-718.
34. V. I. Zubov. Mathematical Methods for the Study of Automatic Method[M], Israel: Jerusalem Academic Press,1962.
35. Tibken B. Estimation of the domain of attraction for polynomial systems via LMIs[J], Proceedings of the IEEE Conference on Decision and Control,2000,3860-3864.
36. Tibken B, Dilaver K F. Computation of subsets of the domain of attraction for polynomial systems[J], Proceedings of the IEEE conference on Decision and Control,2002,2651-2656.
37. Chesi G. Estimating the domain of attraction:a light LMI technique for a class of polyno-mial systems[J], IEEE Conference on Decision and Control,2003,5609-5614.
38. Chesi G, Tesi A, Vicino A, et al. An LMI approach to constrained optimization with homogeneous forms[J], Systems & Control Letters,2001,42(1):11-19.
39. Jarvis-Wloszek Z, Feeley R, Tan W, et al. Some Controls Applications of Sum of Squares Programming[J], Proceedings of IEEE Conference on Decision and Control,2003, 4676-4681.
40. Hachicho O, Tibken B. Estimating domains of attraction of a class of nonlinear dynam-ical systems with LMI methods based on the theory of moments[J], Proceedings of IEEE Co-nference on Decision and Control,2002,3150-3155.
41. R. Bakhtiari, M. J. YazdanPanah. Designing a linear controller for polynomial systems with the largest domain of attraction via LMIs[J], International Conference on Control and Automation,2005,449-453.
42. Chesi G, Garulli A, Tesi A, et al. Solving quadratic distance problems:an LMI-based approach[J], IEEE Transactions on Automatic Control,2003,48(2):200-212.
43. Z.W. Jarvis-Wloszek. Lyapunov based analysis and controller synthesis for polynomial systems using sum-of-squares optimization[D], U.S.A.:University of California, Berkele-y,2003.
44. W. Hahn. Stability of motion[M], Berlin:Springer,1967.
45. J. La. Salle, S. Lefschetz. Stability by Lyapunov's direct method[M], London:Academic Press,1961.
46. H. K. Khalil. Nonlinear systems[M], New York:MacMillan,1992.
47. P. A. Parrilo. Semidefinite programming relaxations for semialgebraic problems[J], Ma-thematical Programming Ser. B,2003,96(2):293-320.
48. T. Motzkin. The arithmetic-geometric inequality[J], In Inequalities, Proceedings of the Symposium at Wright-Patterson AFB,1965,3(2):202-224.
49. S. Prajna, A. Papachristodoulou. SOSTOOLS and its control applications [J], positive pol-ynomical in control, Springer,2005,35(1):146-155.
50. S. P. Bhattacharyya, H. Chapellat, and L. H. Keel. Robust Control:The Parametric Appr- oach[M], U.K.:Cambridge Univ. Press,1995.
51. R. Genesio, M. Tartaglia, and A. Vicino. On the estimation of asymptotic stability regi-ons:State of art and new proposals[J], IEEE Trans. Automat. Contr.,1985, AC(30):747-755.
52.李宁.基于矩量理论和Sum-of-Squares最优化理论的吸引域估计[D],沈阳：东北大学,2008.