线性奇异时滞系统的鲁棒控制研究
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摘要
在实际的工程应用问题中,由于存在着诸如由参数的测量误差、运行环境的变化等所导致的不确定性,使得我们很难得到被控系统对象的精确模型。研究奇异系统的鲁棒控制,就是研究当系统模型存在不确定性时,如何设计控制器,使得对于满足一定范围的参数不确定性,闭环系统仍能保持稳定,并使之保持一定的动态性能。而奇异系统由于比正则系统具有更广泛的形式,其研究因此也有着更为重要的理论意义和实际意义。
本论文主要研究了线性奇异时滞系统的鲁棒控制问题,以Lyapunov泛函和线性矩阵不等式为主要工具,分别对不确定连续时间奇异时滞系统和不确定离散时间奇异系统的鲁棒稳定性进行了分析。主要内容包括以下两个方面：
1.针对具有时变时滞的连续时间标称奇异系统,通过选取恰当的Lyapunov泛函给出了更简便的具有LMI线性矩阵不等式形式的鲁棒稳定的充分条件,并将结论推广到不确定情形中去。值得注意的是该条件对时滞d(t)是没有任何限制要求的。同时,给出具体算例来验证结论的有效性。
2.针对具有时变时滞的不确定离散时间奇异系统,首先给出了当时滞有界时标称系统鲁棒稳定的线性矩阵不等式条件,并将结论推广到不确定的情形中去。同时,给出具体算例来验证结论的有效性。
It makes us impossible to get the precise model of controlled objects in most practical engineering applications, due to the existence of uncertainties caused by the measurement parameter-error, the changing of running condition, etc. To study on robust control for singular systems, is to research how to design a control law to stabilize the resultant closed-loop system and keep some dynamic performances when satisfied a range of parameter uncertainties. Therefore, the study on singular systems is significant both in theory and in practice, because singular systems have more widespread form compared with regular ones.
The robust control for linear singular systems with time-delay is studied on the basis of Lyapunov function and linear matrix inequality(LMI) in this dissertation, including the robust stability of uncertain continuous-time singular systems and uncertain discrete-time singular systems with time delays. The main contents of this dissertation are outlined as follows.
1. A sufficient condition of robust stability with LMI form is presented according to nominal continuous-time singular systems with time-varying delays, by selecting the proper Lyapunov function, and the condition is delay-independent. Then the conclusion is extended to the uncertain case. It is worth nothing that the condition has no limitation requirements on the time-delay d(t), and numerical examples show the effectiveness of the conclusion.
2. A new sufficient condition of robust stability with LMI form is presented at first according to nominal discrete-time singular systems with time-varying delays when the varying time-delay is bounded. Then the conclusion is extended to the case of uncertain condition, and numerical examples show the effectiveness.
本论文主要研究了线性奇异时滞系统的鲁棒控制问题,以Lyapunov泛函和线性矩阵不等式为主要工具,分别对不确定连续时间奇异时滞系统和不确定离散时间奇异系统的鲁棒稳定性进行了分析。主要内容包括以下两个方面：
1.针对具有时变时滞的连续时间标称奇异系统,通过选取恰当的Lyapunov泛函给出了更简便的具有LMI线性矩阵不等式形式的鲁棒稳定的充分条件,并将结论推广到不确定情形中去。值得注意的是该条件对时滞d(t)是没有任何限制要求的。同时,给出具体算例来验证结论的有效性。
2.针对具有时变时滞的不确定离散时间奇异系统,首先给出了当时滞有界时标称系统鲁棒稳定的线性矩阵不等式条件,并将结论推广到不确定的情形中去。同时,给出具体算例来验证结论的有效性。
It makes us impossible to get the precise model of controlled objects in most practical engineering applications, due to the existence of uncertainties caused by the measurement parameter-error, the changing of running condition, etc. To study on robust control for singular systems, is to research how to design a control law to stabilize the resultant closed-loop system and keep some dynamic performances when satisfied a range of parameter uncertainties. Therefore, the study on singular systems is significant both in theory and in practice, because singular systems have more widespread form compared with regular ones.
The robust control for linear singular systems with time-delay is studied on the basis of Lyapunov function and linear matrix inequality(LMI) in this dissertation, including the robust stability of uncertain continuous-time singular systems and uncertain discrete-time singular systems with time delays. The main contents of this dissertation are outlined as follows.
1. A sufficient condition of robust stability with LMI form is presented according to nominal continuous-time singular systems with time-varying delays, by selecting the proper Lyapunov function, and the condition is delay-independent. Then the conclusion is extended to the uncertain case. It is worth nothing that the condition has no limitation requirements on the time-delay d(t), and numerical examples show the effectiveness of the conclusion.
2. A new sufficient condition of robust stability with LMI form is presented at first according to nominal discrete-time singular systems with time-varying delays when the varying time-delay is bounded. Then the conclusion is extended to the case of uncertain condition, and numerical examples show the effectiveness.
引文
[1]E. J. Davision, The output control of linear time invariant multivariable systems with unmeasurable arbitrary disturbance, IEEE Transactions on Automatic Control, 17:621-629,1972.
[2]K. Zhou, J.C. Doyle and K. Glover. Robust and Optimal Control. Englewood Cliffs, N. J. Prentice Hall,1995.
[3]贾英民.鲁棒H∞控制.北京：科学出版社,2007.
[4]Mori T, Kokame H. Stability of x(t)= Ax(t)+Bx(t-τ).IEEE Trans. Automat. Contr., 1989, AC-34(4):460-462.
[5]H.H. Rosenbrock. Structural Properties of Linear Dynamical Systems, Internati-onal Journal of Control,1974,20(2):191-202.
[6]D. G. Luenberger. Dynamic Systems in Descriptor Form, IEE Transactions on Automatic Control,1977,22:312-321.
[7]P. Bernhard. On singular implicit linear dynamic systems, SIAM Journal of Control Optimization,1982,20:612-633.
[8]P.B · Readdy and P. Sannuti. Optimal control of a coupled core nuclear reactor by a singular perturbation method, IEEE Transactions on Automatic Control,1975,20(5): 766-769,.
[9]A. Ailon. On the design of output feedback for finite and infinite Pole assignment in singular systems with application to the control problem of constrained robots, Circuits, Systems & Signal Processing,1994,13(5):525-544.
[10]D. C. Luenberger and A. Arbel. Singular dynamic Leontief systems, Econometric, 1977,45(4):991-995.
[11]P. Bernhard, On singular implicit linear dynamic systems, SIAM Journal of Control Optimization,20:612-633,1982.
[12]B. Dziurla and R. W. Newcomb, The Dazing inverse and semi-state equations, Procee--dings of the International Symposium on Mathematical Theory of Networks and Systems, 283-289,1979.
[13]D. G Luenberger, Dynamic systems in descriptor form, IEEE Transactions on Auto- -matic Control,22:312-321,1977.
[14]K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, Elsevier,1989.
[15]Hale J K. Theory of functional differential equation. New York:Springer-Verlag, 1977.
[16]J. D. Cobb. Fundamental properties of the manifold of singular and regular linear systems, Journal of Mathematics Analysis and Applications,1986,120:325-353.
[17]Neweomb R. W. The semi state description of nonlinear time variable circuits. IEEE Trans. Circuits and Systems.1981, CAS-28(1):62-71.
[18]胡寿松,王执拴,胡维礼.最优控制理论与系统.南京：东南大学出版社,1994.
[19]Luenberger D. G., Arbel. Singular dynamical Leontief systems. Econometric,1977, 45(4):991-995.
[20]刘华平,孙富春,何克忠,孙增沂.奇异摄动控制系统：理论与应用.控制理论与应用,2003,20(1)：1-7.
[21]Vidyasagar M. Robust stabilization relative to the unweighted norm is generica--lly unattainable in the presence of singular plant perturbations. IEEE Trans. Automat. Contr.,1987,32(1):51-53.
[22]Lewis F L, Syrmos V L. A geometric theory for derivative feedback, IEEE Trans Automat. Contr.,1991,36(9):1111-1116.
[23]Dai LY. Singular Control System, Berlin:Spring-Verlag,1989.
[24]Wohnam W. M. Linear Multivariable Control:A Geometric Approach,3rd edn. NewYork: Springer-verlag,1985.
[25]Lewis F. A tutorial on the geometric analysis of linear time-invariant implicit systems. Automatic,1992,28:119-137.
[26]Wonham W. Algebra-geometric and lie-theoretic techniques in systems theory, part A:Interdisciplinary Mathematics. IEEE Trans. Automat. Contr.,1979,24(3):519-520.
[27]W. Q. Liu, W. Y. Yan and K. L. Teo. A frequency domain approach to control of singular systems, IEEE Transactions on Automatic Control,1997,42:885-889.
[28]Verghese G. C, Levy B. C, Kailath T. A generalized state space for singular systems. IEEE Transactions on Automatic Control,1981,26:811-831.
[29]P. N. Paraskevopoulos and F. N. Koumboulis. Decoupling and pole assignment in generalized state space systems, IEE Proceedings-Control Theory and Applications, 1991,138(6):547-560.
[30]Dai L. Singular Control Systems. Berlin:Springer-Verlag,1989.
[31]Choi H. H. Riccati equation approach to the memoryless stabilization of uncertain dynamic systems with delayed control. Electron. Lett.,1995,30:1100-1101.
[32]Lewis F L. A survey of linear singular systems. Circuits, Syst. Signal Processing, 1989,8:3-36.
[33]Takaba K, Morihara N, Katayama T. A generalized Lyapunov theorem for descriptor systems. Syst. Control Lett.,1995,24:49-51.
[34]刘永清,王伟,李远清.大型动力系统的理论与应用：滞后广义系统解的基本理论与应用.广州：华南理工大学出版社,1997.
[35]Lam J, Xu S, Zhang L. Robust D-stability analysis for uncertain discrete singular systems with state delay. IEEE Trans. Circuits Syst.Ⅰ.2002,49:551-555.
[36]杨冬梅,张庆灵,姚波等.广义系统.北京：科学出版社,2005.
[37]Zhang Q L, Lam J. Robust impulse-eliminating control for descriptor systems. Dynamics of Continuous Discrete and Impulse Systems Series B:Applications and Algorithms,2002,9:13-17.
[38]Xu S, Dooren P V, Lam J. Robust stability and stabilization for singular systems with state delay and parameter methods. Appl. Numer. Math.,1997,24:247-264.
[39]Fridman E, Shaked U. A descriptor system approach to control of linear time-delay systems. IEEE Trans. Automat. Contr.,2002,47:253-279.
[40]谢湘生,刘洪伟.滞后广义系统反馈镇定控制器设计的LMI方法[JJ.系统工程与电子技术.2000,22(11)：35-38.
[41]张先明,吴敏,何勇.线性时滞广义系统的时滞相关H∞控制.控制理论与应用,2005,22(4)649-652.
[42]王惠姣,薛安克,鲁仁全等.一类不确定奇异系统的鲁棒H∞控制,自动化学报,2007,33(12)：1300-1305.
[43]Han Q L. Stability criteria for a class of linear neutral systems with time varying discrete and distributed delays. IMA J of Mathematical Control and Information,2003,20(2):371-386.
[44]Krstic M, Deng H. Stabilization of Nonlinear Uncertain Systems. London:Sprin--ger-Verlag,1998.
[45]申铁龙.H∞控制理论及应用.北京：清华大学出版社,1996.
[46]S. Xu, P. V. Dooren, R. Stefan and J. Lam, Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Transactions on Automatic Control,2002,47(7):1122-1128.
[47]P. Gahinet, A. Nemirovskii, A. J. Laub and M. Chilall. LMI Control Toolbox User's Guide, Natick, MA, The Math Works Inc.,1995.
[48]俞立.鲁棒控制——线性矩阵不等式处理方法.北京：清华大学出版社,2002.
[49]Gu K,et al. Stability of Time-Delay Systems. Boston:Birkhauser,2003.
[50]Xie L, de Souza C E. Robust H∞ control for linear time-invariant systems with norm-bounded uncertainty in the input matrix. Sys. Control Lett.,14:389-396,1990.
[51]Boyd S, Ghaohui L E, Feron E, Balakrishnan V. Linear Matrix Inequalities in System and Control Lett.,1992,19:134-149.
[52]线性矩阵不等式(LMI)的MATLAB求解,http://wenku. baidu. com/view/552a888271fe910 ef12df846. html.
[53]闫敏.一类不确定线性时滞系统鲁棒控制方法的研究： (硕士学位论文).辽宁：大连理工大学,2008.
[54]S. L. Campbell, N. K. Nichols and W. J. Terrell, Duality, observability, and contro--lability for linear time-varying descriptor systems, circuits, systems & Signal Processing,1991,10(4):455-470.
[55]Lu R Q, Su H Y, Chu J. A memoryless robust stabilizing controller for a class of uncertain linear singular time-delay systems. Int. Journal of Systems Science, 2004,36(10):973-981.
[56]王惠姣.不确定奇异系统的鲁棒控制研究：(博士学位论文).浙江：浙江大学,2008.
[57]Zhang X M, Han Q L. Delay-dependent robust H∞ filtering for uncertain discrete time systems with time-varying delay based on a finite sum inequality. IEEE Transactions on Circuits and Systems-II,2002,47:253-270.
[58]Huijiao Wang, Anke Xue, Renquan Lu and Yun Chen. Delay-dependent robust H∞ control for uncertain discrete singular time-varying delay systems based on a Finite Sum Inequality, Proceedings of the 26th Chinese Control Conference,3:595-599,2007.
[59]王惠姣,王建中,葛铭等.不确定离散奇异系统的时滞依赖鲁棒H∞控制,控制理论与应用,25(6)：1145-1150.
[60]张庆灵,戴冠中,徐心和.离散广义系统稳定性分析与控制的Lyapunov方法.自动化学报.1998,4(5)：622-629.
[2]K. Zhou, J.C. Doyle and K. Glover. Robust and Optimal Control. Englewood Cliffs, N. J. Prentice Hall,1995.
[3]贾英民.鲁棒H∞控制.北京：科学出版社,2007.
[4]Mori T, Kokame H. Stability of x(t)= Ax(t)+Bx(t-τ).IEEE Trans. Automat. Contr., 1989, AC-34(4):460-462.
[5]H.H. Rosenbrock. Structural Properties of Linear Dynamical Systems, Internati-onal Journal of Control,1974,20(2):191-202.
[6]D. G. Luenberger. Dynamic Systems in Descriptor Form, IEE Transactions on Automatic Control,1977,22:312-321.
[7]P. Bernhard. On singular implicit linear dynamic systems, SIAM Journal of Control Optimization,1982,20:612-633.
[8]P.B · Readdy and P. Sannuti. Optimal control of a coupled core nuclear reactor by a singular perturbation method, IEEE Transactions on Automatic Control,1975,20(5): 766-769,.
[9]A. Ailon. On the design of output feedback for finite and infinite Pole assignment in singular systems with application to the control problem of constrained robots, Circuits, Systems & Signal Processing,1994,13(5):525-544.
[10]D. C. Luenberger and A. Arbel. Singular dynamic Leontief systems, Econometric, 1977,45(4):991-995.
[11]P. Bernhard, On singular implicit linear dynamic systems, SIAM Journal of Control Optimization,20:612-633,1982.
[12]B. Dziurla and R. W. Newcomb, The Dazing inverse and semi-state equations, Procee--dings of the International Symposium on Mathematical Theory of Networks and Systems, 283-289,1979.
[13]D. G Luenberger, Dynamic systems in descriptor form, IEEE Transactions on Auto- -matic Control,22:312-321,1977.
[14]K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, Elsevier,1989.
[15]Hale J K. Theory of functional differential equation. New York:Springer-Verlag, 1977.
[16]J. D. Cobb. Fundamental properties of the manifold of singular and regular linear systems, Journal of Mathematics Analysis and Applications,1986,120:325-353.
[17]Neweomb R. W. The semi state description of nonlinear time variable circuits. IEEE Trans. Circuits and Systems.1981, CAS-28(1):62-71.
[18]胡寿松,王执拴,胡维礼.最优控制理论与系统.南京：东南大学出版社,1994.
[19]Luenberger D. G., Arbel. Singular dynamical Leontief systems. Econometric,1977, 45(4):991-995.
[20]刘华平,孙富春,何克忠,孙增沂.奇异摄动控制系统：理论与应用.控制理论与应用,2003,20(1)：1-7.
[21]Vidyasagar M. Robust stabilization relative to the unweighted norm is generica--lly unattainable in the presence of singular plant perturbations. IEEE Trans. Automat. Contr.,1987,32(1):51-53.
[22]Lewis F L, Syrmos V L. A geometric theory for derivative feedback, IEEE Trans Automat. Contr.,1991,36(9):1111-1116.
[23]Dai LY. Singular Control System, Berlin:Spring-Verlag,1989.
[24]Wohnam W. M. Linear Multivariable Control:A Geometric Approach,3rd edn. NewYork: Springer-verlag,1985.
[25]Lewis F. A tutorial on the geometric analysis of linear time-invariant implicit systems. Automatic,1992,28:119-137.
[26]Wonham W. Algebra-geometric and lie-theoretic techniques in systems theory, part A:Interdisciplinary Mathematics. IEEE Trans. Automat. Contr.,1979,24(3):519-520.
[27]W. Q. Liu, W. Y. Yan and K. L. Teo. A frequency domain approach to control of singular systems, IEEE Transactions on Automatic Control,1997,42:885-889.
[28]Verghese G. C, Levy B. C, Kailath T. A generalized state space for singular systems. IEEE Transactions on Automatic Control,1981,26:811-831.
[29]P. N. Paraskevopoulos and F. N. Koumboulis. Decoupling and pole assignment in generalized state space systems, IEE Proceedings-Control Theory and Applications, 1991,138(6):547-560.
[30]Dai L. Singular Control Systems. Berlin:Springer-Verlag,1989.
[31]Choi H. H. Riccati equation approach to the memoryless stabilization of uncertain dynamic systems with delayed control. Electron. Lett.,1995,30:1100-1101.
[32]Lewis F L. A survey of linear singular systems. Circuits, Syst. Signal Processing, 1989,8:3-36.
[33]Takaba K, Morihara N, Katayama T. A generalized Lyapunov theorem for descriptor systems. Syst. Control Lett.,1995,24:49-51.
[34]刘永清,王伟,李远清.大型动力系统的理论与应用：滞后广义系统解的基本理论与应用.广州：华南理工大学出版社,1997.
[35]Lam J, Xu S, Zhang L. Robust D-stability analysis for uncertain discrete singular systems with state delay. IEEE Trans. Circuits Syst.Ⅰ.2002,49:551-555.
[36]杨冬梅,张庆灵,姚波等.广义系统.北京：科学出版社,2005.
[37]Zhang Q L, Lam J. Robust impulse-eliminating control for descriptor systems. Dynamics of Continuous Discrete and Impulse Systems Series B:Applications and Algorithms,2002,9:13-17.
[38]Xu S, Dooren P V, Lam J. Robust stability and stabilization for singular systems with state delay and parameter methods. Appl. Numer. Math.,1997,24:247-264.
[39]Fridman E, Shaked U. A descriptor system approach to control of linear time-delay systems. IEEE Trans. Automat. Contr.,2002,47:253-279.
[40]谢湘生,刘洪伟.滞后广义系统反馈镇定控制器设计的LMI方法[JJ.系统工程与电子技术.2000,22(11)：35-38.
[41]张先明,吴敏,何勇.线性时滞广义系统的时滞相关H∞控制.控制理论与应用,2005,22(4)649-652.
[42]王惠姣,薛安克,鲁仁全等.一类不确定奇异系统的鲁棒H∞控制,自动化学报,2007,33(12)：1300-1305.
[43]Han Q L. Stability criteria for a class of linear neutral systems with time varying discrete and distributed delays. IMA J of Mathematical Control and Information,2003,20(2):371-386.
[44]Krstic M, Deng H. Stabilization of Nonlinear Uncertain Systems. London:Sprin--ger-Verlag,1998.
[45]申铁龙.H∞控制理论及应用.北京：清华大学出版社,1996.
[46]S. Xu, P. V. Dooren, R. Stefan and J. Lam, Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Transactions on Automatic Control,2002,47(7):1122-1128.
[47]P. Gahinet, A. Nemirovskii, A. J. Laub and M. Chilall. LMI Control Toolbox User's Guide, Natick, MA, The Math Works Inc.,1995.
[48]俞立.鲁棒控制——线性矩阵不等式处理方法.北京：清华大学出版社,2002.
[49]Gu K,et al. Stability of Time-Delay Systems. Boston:Birkhauser,2003.
[50]Xie L, de Souza C E. Robust H∞ control for linear time-invariant systems with norm-bounded uncertainty in the input matrix. Sys. Control Lett.,14:389-396,1990.
[51]Boyd S, Ghaohui L E, Feron E, Balakrishnan V. Linear Matrix Inequalities in System and Control Lett.,1992,19:134-149.
[52]线性矩阵不等式(LMI)的MATLAB求解,http://wenku. baidu. com/view/552a888271fe910 ef12df846. html.
[53]闫敏.一类不确定线性时滞系统鲁棒控制方法的研究： (硕士学位论文).辽宁：大连理工大学,2008.
[54]S. L. Campbell, N. K. Nichols and W. J. Terrell, Duality, observability, and contro--lability for linear time-varying descriptor systems, circuits, systems & Signal Processing,1991,10(4):455-470.
[55]Lu R Q, Su H Y, Chu J. A memoryless robust stabilizing controller for a class of uncertain linear singular time-delay systems. Int. Journal of Systems Science, 2004,36(10):973-981.
[56]王惠姣.不确定奇异系统的鲁棒控制研究：(博士学位论文).浙江：浙江大学,2008.
[57]Zhang X M, Han Q L. Delay-dependent robust H∞ filtering for uncertain discrete time systems with time-varying delay based on a finite sum inequality. IEEE Transactions on Circuits and Systems-II,2002,47:253-270.
[58]Huijiao Wang, Anke Xue, Renquan Lu and Yun Chen. Delay-dependent robust H∞ control for uncertain discrete singular time-varying delay systems based on a Finite Sum Inequality, Proceedings of the 26th Chinese Control Conference,3:595-599,2007.
[59]王惠姣,王建中,葛铭等.不确定离散奇异系统的时滞依赖鲁棒H∞控制,控制理论与应用,25(6)：1145-1150.
[60]张庆灵,戴冠中,徐心和.离散广义系统稳定性分析与控制的Lyapunov方法.自动化学报.1998,4(5)：622-629.