切换系统的最优切换律研究
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摘要
本论文主要研究了切换系统的最优切换律问题。切换系统是混合动态系统中重要的一类。混合动态系统是由几个连续子系统或者离散子系统以及作用在其中的切换规则所组成的。由于切换的动作,使切换系统的动态特性变得非常复杂,在实际生产生活中,系统切换规则的设计对于切换系统的研究有着十分重要的作用。
本文以线性切换系统为例,主要分析了脉冲切换系统的最优切换律问题,研究了基于范数和基于收敛路径的脉冲切换系统的最优切换律的求取,考虑了切换系统切换规则的选择。总体而言,本文主要完成了以下几项研究工作:
(1)讨论了线性脉冲切换系统的最优切换律。首先根据系统矩阵束的约束条件,得到线性脉冲切换系统渐近稳定时切换律的取值范围。其次从每次切换后状态点的范数取值逐渐变小的角度,给出了基于范数的最优切换律的分析方法,切换系统渐近稳定的收敛过程中,采用不同的切换律,得到的系统运行结果也不同。最后,通过实例仿真,验证了设计方法的正确性。
(2)对一类周期切换线性脉冲切换系统,研究了其在渐近稳定条件下切换系统基于收敛路径的最优切换律。从渐近稳定系统收敛过程中收敛运行的路径最小的角度,给出了基于收敛路径的最优切换律的分析方法。通过仿真实例,验证了设计方法的正确性和有效性。
The optimal switching law of switched systems is researched in this article. As an important class of hybrid dynamical systems, the switched system is composed of a family of continuous-time or discrete-time subsystems and a rule orchestrating the switching among the subsystems. Due to the switching signal, its dynamical performance becomes very complicated. Hence, it is very important to choose a property switching law in the practical production.
This dissertation investigates how to choose a proper switching law from switching time for the linear switched systems. The main work incules the analysis of the optimal switching laws of impulsive swithed systems and the research on the optimal switching laws based on norm and convergence path, the choosing of switching laws. The main contributions of the research work presented in this dissertation are as follows:
(1) The optimal switching law is discussed for the linear impulsive switched systems. Firstly, asymptotically stable value range of the switching law is obtained based on the constraint condition of matrix pencil for the linear impulsive switched system. Secondly, different running results of system are got because of the choosing of switching law, in the asymptotically stable convergence process of switched system. Then, the analysis method of optimal switching law based on the norm is present in the view of the gradually decreased norm value of the state points after the switching. Finally, instance simulation verify the effectiveness of this method.
(2) The optimal switching law of switching systems based on the convergence of the optimal switching path in the asymptotic stability conditions is studied for a class of periodic switched linear impulsive switched systems. The analysis of the optimal switching based on the convergence route is present from asymptotically stable convergence process the minimum angle of path of converge operation. The simulation example, verify the correctness and validity of the design method.
本文以线性切换系统为例,主要分析了脉冲切换系统的最优切换律问题,研究了基于范数和基于收敛路径的脉冲切换系统的最优切换律的求取,考虑了切换系统切换规则的选择。总体而言,本文主要完成了以下几项研究工作:
(1)讨论了线性脉冲切换系统的最优切换律。首先根据系统矩阵束的约束条件,得到线性脉冲切换系统渐近稳定时切换律的取值范围。其次从每次切换后状态点的范数取值逐渐变小的角度,给出了基于范数的最优切换律的分析方法,切换系统渐近稳定的收敛过程中,采用不同的切换律,得到的系统运行结果也不同。最后,通过实例仿真,验证了设计方法的正确性。
(2)对一类周期切换线性脉冲切换系统,研究了其在渐近稳定条件下切换系统基于收敛路径的最优切换律。从渐近稳定系统收敛过程中收敛运行的路径最小的角度,给出了基于收敛路径的最优切换律的分析方法。通过仿真实例,验证了设计方法的正确性和有效性。
The optimal switching law of switched systems is researched in this article. As an important class of hybrid dynamical systems, the switched system is composed of a family of continuous-time or discrete-time subsystems and a rule orchestrating the switching among the subsystems. Due to the switching signal, its dynamical performance becomes very complicated. Hence, it is very important to choose a property switching law in the practical production.
This dissertation investigates how to choose a proper switching law from switching time for the linear switched systems. The main work incules the analysis of the optimal switching laws of impulsive swithed systems and the research on the optimal switching laws based on norm and convergence path, the choosing of switching laws. The main contributions of the research work presented in this dissertation are as follows:
(1) The optimal switching law is discussed for the linear impulsive switched systems. Firstly, asymptotically stable value range of the switching law is obtained based on the constraint condition of matrix pencil for the linear impulsive switched system. Secondly, different running results of system are got because of the choosing of switching law, in the asymptotically stable convergence process of switched system. Then, the analysis method of optimal switching law based on the norm is present in the view of the gradually decreased norm value of the state points after the switching. Finally, instance simulation verify the effectiveness of this method.
(2) The optimal switching law of switching systems based on the convergence of the optimal switching path in the asymptotic stability conditions is studied for a class of periodic switched linear impulsive switched systems. The analysis of the optimal switching based on the convergence route is present from asymptotically stable convergence process the minimum angle of path of converge operation. The simulation example, verify the correctness and validity of the design method.
引文
[1]郑刚,谭民,宋永华.混杂系统的研究进展[J].控制与决策, 2004,19(1):7-16
[2]牛彦杰.一类切换系统的分析与控制[J].南京工业大学,2005.5
[3]邹洪波.切换线性系统稳定性若干问题研究[D].浙江大学,2007
[4]夏君华.切换系统的稳定性分析[D].沈阳工业大学,2003.3
[5]邱秀君.天然气/汽油两用燃料发动机燃料切换的优化控制[D].重庆大学,2009.4
[6]邹洪波.切换线性系统稳定性若干问题研究[D].浙江大学,2007
[7] R.N.shorten, K.S. Narendra. On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form[J]. IEEE Transactions on Automatic Control,2003, 48(4):618-621
[8] Seidman T. I. Optimal Control for Switching Systems[J]. 21st annual Conference on Information Science and Systems, Johns Hopkins University, 1987:485-489
[9]冯伟贞.线性时变脉冲切换系统稳定性分析[J].华南师范大学学报.2006,8(3):12-18
[10]王仁明,关治洪,王燕舞,刘新芝.关于一类线性时不变切换系统的脉冲控制[J].系统工程学报, 2006,8(21):405-408
[11] Antsaklis P. Special issue on Hybrid Systems: Theory and Applications a Brief Introduction to the Theory and Applications of Hybrid Systems[J]. Proceedings of IEEE, 2000, 88(7):879-887
[12] Sira-Ramirez H. Nonlinear P-I Controller Design for Switched Mode DC-to-DC Power Converters[J]. IEEE Transactions on Circuits and Systems, 1991, 38(4):410-417
[13]翟海峰.混杂系统分析与控制设计研究[D].杭州:浙江大学,2001
[14] Boyd S, Ghaoui L.E, Feron E, etal. Linear matrix inequalities in systems and control theory[M]. Philadelphia: SIAM, USA, 1994
[15] Liu B, Liu X.Z and Liao X.X. Robust stability analysis of uncertain impulsive systems[J]. Math. Anal. Appl, 2004, 290(2):519-533
[16] Liberzon D, Morse A.S. Basic problems in stability and design of switched systems[J] . IEEE Control systems Magazine,1999,19(5): 59-70
[17]刘玉彬.脉冲切换系统的稳定性[D].华南师范大学,2007.5
[18]景丽,高立群.等时切换下线性切换系统的稳定性及鲁棒稳定性研究-LMI方法[J].计算技术与自动化.2003,22(4): 16-20.
[19] Z.Ji, L.Wang, G.Xie, F.Hao. Linear Matrix Inequality Approach to (quadratic Stabilization of Switched Systems[J]. IEE Proc. Control Theory Application, 2004,151(3): 289-294.
[20]杨磊,李俊民,曹吉利.不稳定线性切换系统的优化切换镇定[J].陕西工学院学报,2002, 18(3): 56-59.
[21] Xuping Xu, Panos J. Antsaklis. Stabilization of Second-Order LTI Switched Systems[J]. Proceedings of the 38th Corence on Decision and Control, Phoenix,Arizona USA. December 1999. 1339-1344.
[22]刘豹.现代控制理论.机械工业出版社[M]. 142-153..
[23]王美.切换系统的输出反馈控制[D].厦门大学,2009.5
[24] Mason O, Shorten R. A conjecture on the existence of common quadratic Lyapunov functions for positive linear systems [J]. Proceedings of the 2003 American Control Conference, 2003:4469-4470
[25] Mason O, Shorten R. A result on the existence of common quadratic Lyapunov functions for pairs of stable discrete-time LTI systems[J]. Proceedings of the 2003 American Control Conference, 2003:4471-4475
[26] Shorten R.N, NarendraK.S. Necessary and sufficient conditions for common quadratic Lyapunov functions for a pair of stable LTI systems whose system matrices are in companion form[J]. Proceedings of the 2003 American Control Conference, 2003:5503-5508
[27] M.S. Branicky. Multiple Lapunov functions and other analysis tools for switched and hybrid sytems[J]. IEEE Trans on Automatic Control,1998,43(4):186-200
[28] H.Ye, A.M.Michel, L.Hou. Stability analysis of systems with impulse effects[J]. IEEE Trans on Automatic Control,1998,43:1719-1723
[29] S.Pettersson, B.Lennartson. Stability and robustness for hybrid systems[J]. Proceedings of the 36th IEEE Conference on Decision and Control,1996.1202- 1207
[30] A.N.Michel. Recent trends in the stability analysis of hybrid danamical systems[J]. IEEE Trans on Circuit and Systems,1999,45(1):120一134
[31] X.Xu, P.J.Antsakis. Optimal control of switched antonomous systems[J]. Proceedings of 41st IEEE Conference on Decision and Control. Las Vegas,Nevada USA,2002.4401一4406
[32] P. Riedinger, C. Zanne, F. Kratz. Time optimal control of hybrid systems[J]. Proceedings of American Control Conference. San Diego, California USA,1997.2466- 2470
[33]侯冬梅.广义系统的次优控制[D].黑龙江大学,2008.5
[34]卢军锋.一类线性切换系统的可靠性控制研究[D].南京理工大学,2008.6
[35]王俊歌.随机脉冲系统和脉冲切换系统的稳定和控制[D].广西大学,2008.5
[36]尹玉娟,刘玉忠,赵军.具有脉冲作用的切换线性广义系统的稳定性分析[J].自动化学报,2007, 33(4): 446-448
[37]孙文安,董世山,邹其开.线性周期切换系统渐进稳定性[J].辽宁工程技术大学学报,2006,2(25): 158-160
[38]李俊圣,汪楠,罗蕾,高立群.线性切换系统基于范数的最优切换律[J].第二十三届中国控制会议论文集,2004,8:757-761
[39]毛北行.切换系统的鲁棒稳定性问题研究[D].郑州大学,2004.5
[40] BoHu, XuPing-xu,Anthony N.Michel,PanosJ.Antsaklis.Stability analysis for a class of Nonlinear Switched systems[C]. Proceedings of the38th conference on decision & conrtol.1999,9:4374-4379
[41]方晨.广义连续系统与广义离散系统的鲁棒控制[D].天津大学,2006.12
[42] Hui Ye, Michel A.N, Ling Hou. Stability theory for hybrid dynamical systems[J]. IEEE Transactions on Automatic Control, 1998,43(4):461-474
[43] Zhai G, Hu B, Yasuda K, Michal A N. Piecewise Lyapunov Functions for Switched systems with Average Dwell Time [J]. Asian Journal of Control,2000, 42(3):192-197
[44] Xinzhi Liu. Stability Analysis of Impulsive Control Systems[J]. Mathematical and Computer Modelling, 2003, 37: 1357-1370.
[45] Olsola Akinyele, John O. Adeyeye. Cone-Valued Lyapunov Functions and Stability of Hybrid Systems [J]. Dynamic of Continuous, Discrete and Impulsive Systems, 2001,8:203-214.
[46]罗定提,刘斌,刘新芝.拟线性脉冲系统的一致渐近稳定性[J].系统工程,2002,20(3): 81-86.
[47]刘斌,刘新芝,廖晓听.混合脉冲系统稳定性分析的研究进展[J].株洲工学院学报,2002, 16(1): 29-32.
[48] Bin Liu, Xinzhi Liu, Xiaoxin Liao. Stability and Robustness of quasi-Linear Impulsive Hybrid Systems[J]. J. Math. Anal. Appl.,2003, 283: 416-430.
[49] VijaySekhar Chellaboina, Sanjay P. Bhat, Wassim variance Principle for Nonlinear Hybrid and Impulsive M. Haddad[J]. An In-Dynamical Systems.Nonlinear Analysis, 2003, 53: 527-550.
[50]都强,傅希林.锥值Lyapunov函数和脉冲混合系统的稳定性[J].科学技术与工程, 2005, 5(13): 849-850.
[51] M.S. Branicky, VS. Borkar and S.K. Muter, A unified framework for hybrid control model and optimal control theory[J], IEEE Trans On Automatical Control,1998,34(1):31-45
[52] X.Xu,P.J.Antsakis. Optimal control of switched antonomous systems[J]. In Proceedings of 41St IEEE Conference on Decision and Control. Las Vegas, Nevada USA, 2002.4401-4406
[53] H.J. Sussman. A maximum principle for hybrid optimal control problems[J]. Proc. Of the CDC, 1999:425-430
[54] P.Riedinger, C.Zanne, F.Kratz. Time optimal control of hybrid systems[J]. Proceedings of American Control Conference. San Diego, California USA,1997.2466-2470
[55] X.Xu, P.J. Antsakis. An approach of solving general switched linear quadratic optimal control problems[J]. Proceedings of 40th IEEE Conference on Decision and Control.Orlando, Florida USA, 2001.2478-2483
[56] E. Skafidas, R J. Evans, I. Mareels. Optimal controllers switching for stochastic systems[J]. Proc. of 36th IEEE CDC, 3950-3955
[57] H.Zhang,X.Liu. Robust control on impulsive switched systems with disturbance[J]. Control Theory and Application.2003,78-83
[58] X.Xu and Results and Panos J. Antsaklis, Optimal Control of Switched Systems New Open Problems[J], Proceedings of the American Control Conference, Chicago,Iillnois, June 2000, 1033-1041
[59]王春芳.一类线性切换系统的优化控制策略研究[D].南京理工大学,2008.6
[60] Liu X, Zhang H, Robust stability of impulsive switched systems with disturbance[J]. The ANZIAM journal, 2006,48(2): 259-270
[61]冯伟贞.线性时变脉冲切换系统稳定性分析[J].华南师范大学学报,2006,8(3):12-18
[2]牛彦杰.一类切换系统的分析与控制[J].南京工业大学,2005.5
[3]邹洪波.切换线性系统稳定性若干问题研究[D].浙江大学,2007
[4]夏君华.切换系统的稳定性分析[D].沈阳工业大学,2003.3
[5]邱秀君.天然气/汽油两用燃料发动机燃料切换的优化控制[D].重庆大学,2009.4
[6]邹洪波.切换线性系统稳定性若干问题研究[D].浙江大学,2007
[7] R.N.shorten, K.S. Narendra. On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form[J]. IEEE Transactions on Automatic Control,2003, 48(4):618-621
[8] Seidman T. I. Optimal Control for Switching Systems[J]. 21st annual Conference on Information Science and Systems, Johns Hopkins University, 1987:485-489
[9]冯伟贞.线性时变脉冲切换系统稳定性分析[J].华南师范大学学报.2006,8(3):12-18
[10]王仁明,关治洪,王燕舞,刘新芝.关于一类线性时不变切换系统的脉冲控制[J].系统工程学报, 2006,8(21):405-408
[11] Antsaklis P. Special issue on Hybrid Systems: Theory and Applications a Brief Introduction to the Theory and Applications of Hybrid Systems[J]. Proceedings of IEEE, 2000, 88(7):879-887
[12] Sira-Ramirez H. Nonlinear P-I Controller Design for Switched Mode DC-to-DC Power Converters[J]. IEEE Transactions on Circuits and Systems, 1991, 38(4):410-417
[13]翟海峰.混杂系统分析与控制设计研究[D].杭州:浙江大学,2001
[14] Boyd S, Ghaoui L.E, Feron E, etal. Linear matrix inequalities in systems and control theory[M]. Philadelphia: SIAM, USA, 1994
[15] Liu B, Liu X.Z and Liao X.X. Robust stability analysis of uncertain impulsive systems[J]. Math. Anal. Appl, 2004, 290(2):519-533
[16] Liberzon D, Morse A.S. Basic problems in stability and design of switched systems[J] . IEEE Control systems Magazine,1999,19(5): 59-70
[17]刘玉彬.脉冲切换系统的稳定性[D].华南师范大学,2007.5
[18]景丽,高立群.等时切换下线性切换系统的稳定性及鲁棒稳定性研究-LMI方法[J].计算技术与自动化.2003,22(4): 16-20.
[19] Z.Ji, L.Wang, G.Xie, F.Hao. Linear Matrix Inequality Approach to (quadratic Stabilization of Switched Systems[J]. IEE Proc. Control Theory Application, 2004,151(3): 289-294.
[20]杨磊,李俊民,曹吉利.不稳定线性切换系统的优化切换镇定[J].陕西工学院学报,2002, 18(3): 56-59.
[21] Xuping Xu, Panos J. Antsaklis. Stabilization of Second-Order LTI Switched Systems[J]. Proceedings of the 38th Corence on Decision and Control, Phoenix,Arizona USA. December 1999. 1339-1344.
[22]刘豹.现代控制理论.机械工业出版社[M]. 142-153..
[23]王美.切换系统的输出反馈控制[D].厦门大学,2009.5
[24] Mason O, Shorten R. A conjecture on the existence of common quadratic Lyapunov functions for positive linear systems [J]. Proceedings of the 2003 American Control Conference, 2003:4469-4470
[25] Mason O, Shorten R. A result on the existence of common quadratic Lyapunov functions for pairs of stable discrete-time LTI systems[J]. Proceedings of the 2003 American Control Conference, 2003:4471-4475
[26] Shorten R.N, NarendraK.S. Necessary and sufficient conditions for common quadratic Lyapunov functions for a pair of stable LTI systems whose system matrices are in companion form[J]. Proceedings of the 2003 American Control Conference, 2003:5503-5508
[27] M.S. Branicky. Multiple Lapunov functions and other analysis tools for switched and hybrid sytems[J]. IEEE Trans on Automatic Control,1998,43(4):186-200
[28] H.Ye, A.M.Michel, L.Hou. Stability analysis of systems with impulse effects[J]. IEEE Trans on Automatic Control,1998,43:1719-1723
[29] S.Pettersson, B.Lennartson. Stability and robustness for hybrid systems[J]. Proceedings of the 36th IEEE Conference on Decision and Control,1996.1202- 1207
[30] A.N.Michel. Recent trends in the stability analysis of hybrid danamical systems[J]. IEEE Trans on Circuit and Systems,1999,45(1):120一134
[31] X.Xu, P.J.Antsakis. Optimal control of switched antonomous systems[J]. Proceedings of 41st IEEE Conference on Decision and Control. Las Vegas,Nevada USA,2002.4401一4406
[32] P. Riedinger, C. Zanne, F. Kratz. Time optimal control of hybrid systems[J]. Proceedings of American Control Conference. San Diego, California USA,1997.2466- 2470
[33]侯冬梅.广义系统的次优控制[D].黑龙江大学,2008.5
[34]卢军锋.一类线性切换系统的可靠性控制研究[D].南京理工大学,2008.6
[35]王俊歌.随机脉冲系统和脉冲切换系统的稳定和控制[D].广西大学,2008.5
[36]尹玉娟,刘玉忠,赵军.具有脉冲作用的切换线性广义系统的稳定性分析[J].自动化学报,2007, 33(4): 446-448
[37]孙文安,董世山,邹其开.线性周期切换系统渐进稳定性[J].辽宁工程技术大学学报,2006,2(25): 158-160
[38]李俊圣,汪楠,罗蕾,高立群.线性切换系统基于范数的最优切换律[J].第二十三届中国控制会议论文集,2004,8:757-761
[39]毛北行.切换系统的鲁棒稳定性问题研究[D].郑州大学,2004.5
[40] BoHu, XuPing-xu,Anthony N.Michel,PanosJ.Antsaklis.Stability analysis for a class of Nonlinear Switched systems[C]. Proceedings of the38th conference on decision & conrtol.1999,9:4374-4379
[41]方晨.广义连续系统与广义离散系统的鲁棒控制[D].天津大学,2006.12
[42] Hui Ye, Michel A.N, Ling Hou. Stability theory for hybrid dynamical systems[J]. IEEE Transactions on Automatic Control, 1998,43(4):461-474
[43] Zhai G, Hu B, Yasuda K, Michal A N. Piecewise Lyapunov Functions for Switched systems with Average Dwell Time [J]. Asian Journal of Control,2000, 42(3):192-197
[44] Xinzhi Liu. Stability Analysis of Impulsive Control Systems[J]. Mathematical and Computer Modelling, 2003, 37: 1357-1370.
[45] Olsola Akinyele, John O. Adeyeye. Cone-Valued Lyapunov Functions and Stability of Hybrid Systems [J]. Dynamic of Continuous, Discrete and Impulsive Systems, 2001,8:203-214.
[46]罗定提,刘斌,刘新芝.拟线性脉冲系统的一致渐近稳定性[J].系统工程,2002,20(3): 81-86.
[47]刘斌,刘新芝,廖晓听.混合脉冲系统稳定性分析的研究进展[J].株洲工学院学报,2002, 16(1): 29-32.
[48] Bin Liu, Xinzhi Liu, Xiaoxin Liao. Stability and Robustness of quasi-Linear Impulsive Hybrid Systems[J]. J. Math. Anal. Appl.,2003, 283: 416-430.
[49] VijaySekhar Chellaboina, Sanjay P. Bhat, Wassim variance Principle for Nonlinear Hybrid and Impulsive M. Haddad[J]. An In-Dynamical Systems.Nonlinear Analysis, 2003, 53: 527-550.
[50]都强,傅希林.锥值Lyapunov函数和脉冲混合系统的稳定性[J].科学技术与工程, 2005, 5(13): 849-850.
[51] M.S. Branicky, VS. Borkar and S.K. Muter, A unified framework for hybrid control model and optimal control theory[J], IEEE Trans On Automatical Control,1998,34(1):31-45
[52] X.Xu,P.J.Antsakis. Optimal control of switched antonomous systems[J]. In Proceedings of 41St IEEE Conference on Decision and Control. Las Vegas, Nevada USA, 2002.4401-4406
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