关于时滞微分系统的稳定性研究
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摘要
稳定性是系统的一个基本结构特性,是控制系统能够正常运行的前提.稳定性问题历来是研究系统控制理论的一个重要课题.时滞现象是自然界中极其普遍的现象,它的存在往往是系统不稳定和系统性能变差的根源.因此分析时滞系统的稳定性有着重要的理论和实际意义.中立型时滞系统是一类既可以描述状态滞后又可以描述状态微分滞后的系统,在诸多系统的研究中都有重要的应用.另一方面,实际的系统都受到各种外界扰动因素的影响,因此对包含非线性扰动的时滞系统的鲁棒稳定性研究就显得尤为重要.退化系统是包含正常系统的更具一般性的动力学系统,具有许多正常系统所不能描述的复杂动态行为和静态特性.人们熟知的具有非线性负载的电力系统模型、Hopfiel礻神经网络模型、Leontief动态投入产出模型等都是退化系统.现实情况中存在大量的分数阶现象,有些系统用分数阶微分方程来描述比用整数阶微分方程来描述更加精确,分数阶微分方程比整数阶微分方程更具有普遍性而迅速成为研究的热点.
本文的主要工作是研究这几类微分系统的稳定性或滑模控制问题,分为四章.
第一章主要介绍这篇论文研究的实际背景、主要工作及所需的预备知识.
第二章讨论了一类具有非线性扰动的中立型不确定时滞系统的鲁棒稳定性问题.基于增广的Lyapunov-Krasovskii泛函和自由权矩阵的方法,推导出以线性矩阵不等式形式给出的时滞相关稳定性的充分条件.
第三章讨论了一类多时滞线性退化系统时滞依赖稳定性问题.通过系统分解的方法,推导出一个以线性矩阵不等式给出的时滞依赖稳定性的充分条件.
第四章讨论了含有状态时滞的线性分数阶系统的滑模控制设计,并说明滑模的存在性和控制设计的稳定性.
Time delays frequently appear in many practical systems, they are regarded as the major source of instability and poor performance. Therefore, a great amount of effort has been devoted to the stability analysis for the time-delay systems. There are many control systems having delay not only in the state but also in the state derivative, such systems are called neutral systems. Recently, there has been rapidly growing interest in the problem of the stability analysis for neutral delayed systems. The singular system model is a natural presentation of dynamic systems and can describe a larger class of systems than regular ones. Over the past decades, much attention has been focused on stability analysis for singular linear systems with time delay. Fractional differential equations are introduced in order to describe and imitate the practical phenomena more exactly. Thus there are much theoretical value and practical meaning of the research on fractional differential equations and systems.
In this paper, we are concerned with the robust stability problem for several class of differential systems. And a sliding mode control design for fractional systems is also given in it. There are four chapters in this paper.
In chapter1, some background knowledge of neutral differential system, singular system and fractional differential system are introduced, and the preliminary knowledge which is necessary in the paper is given.
In chapter2, we discuss the robust stability criteria for a class of neutral uncertain-ty delayed systems with nonlinear perturbation. Based on the augmented Lyapunov-Krasovskii functional and the free-weighting matrix method, sufficient conditions of delay-dependent stability expressed in terms of linear matrix inequalities are obtained.
In chapter3, the dependent-delay stability analysis for singular linear systems with multiple delays is studied. By decomposing the singular linear systems with multiple de-lays into slow and fast subsystems, LMI-based delay-dependent stability and stabilisation conditions are established.
In chapter4, we deal with a sliding mode control design for a class of linear fractional systems with state delay, and the existence of the sliding mode and the stability of the proposed control design are discussed.
本文的主要工作是研究这几类微分系统的稳定性或滑模控制问题,分为四章.
第一章主要介绍这篇论文研究的实际背景、主要工作及所需的预备知识.
第二章讨论了一类具有非线性扰动的中立型不确定时滞系统的鲁棒稳定性问题.基于增广的Lyapunov-Krasovskii泛函和自由权矩阵的方法,推导出以线性矩阵不等式形式给出的时滞相关稳定性的充分条件.
第三章讨论了一类多时滞线性退化系统时滞依赖稳定性问题.通过系统分解的方法,推导出一个以线性矩阵不等式给出的时滞依赖稳定性的充分条件.
第四章讨论了含有状态时滞的线性分数阶系统的滑模控制设计,并说明滑模的存在性和控制设计的稳定性.
Time delays frequently appear in many practical systems, they are regarded as the major source of instability and poor performance. Therefore, a great amount of effort has been devoted to the stability analysis for the time-delay systems. There are many control systems having delay not only in the state but also in the state derivative, such systems are called neutral systems. Recently, there has been rapidly growing interest in the problem of the stability analysis for neutral delayed systems. The singular system model is a natural presentation of dynamic systems and can describe a larger class of systems than regular ones. Over the past decades, much attention has been focused on stability analysis for singular linear systems with time delay. Fractional differential equations are introduced in order to describe and imitate the practical phenomena more exactly. Thus there are much theoretical value and practical meaning of the research on fractional differential equations and systems.
In this paper, we are concerned with the robust stability problem for several class of differential systems. And a sliding mode control design for fractional systems is also given in it. There are four chapters in this paper.
In chapter1, some background knowledge of neutral differential system, singular system and fractional differential system are introduced, and the preliminary knowledge which is necessary in the paper is given.
In chapter2, we discuss the robust stability criteria for a class of neutral uncertain-ty delayed systems with nonlinear perturbation. Based on the augmented Lyapunov-Krasovskii functional and the free-weighting matrix method, sufficient conditions of delay-dependent stability expressed in terms of linear matrix inequalities are obtained.
In chapter3, the dependent-delay stability analysis for singular linear systems with multiple delays is studied. By decomposing the singular linear systems with multiple de-lays into slow and fast subsystems, LMI-based delay-dependent stability and stabilisation conditions are established.
In chapter4, we deal with a sliding mode control design for a class of linear fractional systems with state delay, and the existence of the sliding mode and the stability of the proposed control design are discussed.
引文
[1]Jack Hale. Introduction to Functional Differential Equations [M]. New York:Springer Verlag. 1993.
[2]郑祖庥.泛函微分方程理论[M].合肥：安徽教育出版社,1994.
[3]Gao H. Lam J, Wang C, Wang Y. Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay [J]. IEE Proceedings Control Theory and Applica-tions,2004,151(6),691-698.
[4]Jinde Cao, Jun Wang. Delay-dependent robust stability of uncertain nonlinear systems with time delay [J]. Applied Mathematics and Computation,2004,154(1),289-297.
[5]Wang B, Liu X, Zhong S. New stability analysis for uncertain neutral system with time-varying delay [J]. Applied Mathematics and Computation,2008,197(1),457-465.
[6]张先明，吴敏,何勇.中立型线性时滞系统的时滞相关稳定性[J].自动化学报,2004,30(4),624-628.
[7]Zhang J H, Shi P, Qiu J Q. Robust stability criteria for uncertain neutral system with time delay and nonlinear uncertainties [J]. Chaos, Solitons and Fractals,2008,38(1),160-167.
[8]Zuo Z, Wang Y. New stability criteria for a class of linear systems with time-varying delay and nonlinear perturbations [J]. IEE Proceedings Control Theory and Applications,2006,153(5), 623-626.
[9]俞立.鲁棒控制一线性矩阵不等式处理方法[M].北京：清华大学出版社,2002.
[10]Gu K. An integral inequality in the stability problem of time-delay systems [J]. Decision and Control, Proceedings of the 39th IEEE Conference,2000, (3),2805-2810.
[11]何舒平,刘飞.基于观测器的非线性不确定时滞跳变系统无源控制[J].数学物理学报,2009,29A(2),334-343.
[12]Cheng Wang, Yi Shen. Improved delay-dependent robust stability criteria for uncertain time delay systems [J]. Applied Mathematics and Computation,2011,218(6),2880-2888.
[13]E.Fridman, U.Shake. An improved stabilization method for linear time-delay systems [J]. IEEE Transactions Automatic Control,2002,47(11),1931-1937.
[14]Q.L.Han. On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty [J]. Automatica,2004,40(6),1087-1092.
[15]瞿少成.不确定系统的滑模控制理论及应用研究[M].武汉：华中师范大学出版社,2008.
[16]蒋威.退化时滞微分系统[M].合肥：安徽大学出版社,1998.
[17]苏宏业.不确定时滞系统的鲁棒控制理论[M].北京：科学出版社,2007.
[18]L Y Dai. Singular Control Systems [M]. Berlin:Springer-Verlag,1989.
[19]C.J.Chang, K.F.R.Liu, K.Yeh, C.W.Chen, P.Y.Chung. Delay independent criterion for mul-tiple time-delay systems [J]. Physics Procedia,2012,25:270-277.
[20]P.Balasubramaniam, R.Krishnasamy, R.Rakkiyappan. Delay-dependent stability criterion for a class of non-linear singular Markovian jump systems with mode-dependent interval time-varying delays [J]. Communications in Nonlinear Science and Numerical Simulation,2012,17(9): 3612-3627.
[21]E K Boukas, Z K Liu. Delay-dependent stability analysis of singular linear continuous-time system [J]. The fourth International Conference on Control and Automation,10-12 June 2003, Montreal, Canada.
[22]Cheng Wang, Yi Shen. Improved delay-dependent robust stability criteria for uncertain time delay systems [J]. Applied Mathematics and Computation,2011,218(6):2880-2888.
[23]Mei Fang. Delay-dependent stability analysis for discrete singular systems with time-varying delays [J]. Acta Automatica Sinica,2010,36(5):751-755.
[24]Kwan Ho Lee, Joon Hwa Lee, Wook Hyun Kwon. Sufficient LMI conditions for H∞ output feedback stabilization of linear discrete-time systems [J]. IEEE Transactions on Automatica Control,2006,51(4):675-680.
[25]Z G Wu, H Y Su, J Chu. Improved results on delay-dependent H∞, control for singular time-delay systems [J]. Acta Automatica Sinica,2009,35(8):1101-1106.
[26]F.Gouaisbaut, M.Dambrine, J.P.Richard. Robust control of delay systems:a sliding mode control design via LMI [J]. Systems Control Letters,2002,46(4):219-230.
[27]Wei Jiang. Eigenvalue and stability of singular differential delay systems [J]. Journal of Math-ematical Analysis and Applications,2004,297(1):305-316.
[28]Piyapong Niamsup. Stability of time-varying switched systems with time-varying delay [J]. Nonlinear Analysis:Hybrid Systems,2009,3(4):631-639.
[29]廖晓听.稳定性的理论、方法和应用[M].武汉：华中科技大学出版社,2002.
[30]时宝,张德存,盖明久.微分方程理论及其应用[M].北京：国防工业出版社,2005.
[31]郭大钧.非线性泛函分析M.济南：山东科学技术出版社,2001.
[32]Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations [M]. The Netherlands:North Holland Mathematics Studies, Elsevier,2006.
[33]I.Podlubny. Fractional differential Equations [M]. San Diego:San Diego Academic Press,1999.
[34]王振滨,曹广益,朱新坚.分数阶线性定常系统的稳定性及其判据[J].控制理论与应用,2004,21(6)：922-926.
[35]卜春霞,张应奇,毛北行.变时滞控制系统的变结构控制[J].郑州大学学报(理学版),2003,35(4)：23-26.
[36]El-Khezali R, Ahmad WH. Variable structure control of fractional time-delay systems [J]. In: Proceedings of the 2nd IFAC, Workshop on fractional differentiation and its applications, Porto, Portugal, July 19-21,2006.
[37]Lee,J.H., S.W.Kim, W.H.Kwon. Memoryless controllers for state delayed systems [J]. IEEE, Trans Automat. Control,39,159-162,1994.
[38]Yuanqing Xia, Jingqing Han, Yingmin Jia. A sliding mode control for linear systems with input and state delays [J]. In:Proceedings of the 41sl IEEE, Conference on Decision and Control, Las Vepas, Nevada USA, December 2002.
[2]郑祖庥.泛函微分方程理论[M].合肥：安徽教育出版社,1994.
[3]Gao H. Lam J, Wang C, Wang Y. Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay [J]. IEE Proceedings Control Theory and Applica-tions,2004,151(6),691-698.
[4]Jinde Cao, Jun Wang. Delay-dependent robust stability of uncertain nonlinear systems with time delay [J]. Applied Mathematics and Computation,2004,154(1),289-297.
[5]Wang B, Liu X, Zhong S. New stability analysis for uncertain neutral system with time-varying delay [J]. Applied Mathematics and Computation,2008,197(1),457-465.
[6]张先明，吴敏,何勇.中立型线性时滞系统的时滞相关稳定性[J].自动化学报,2004,30(4),624-628.
[7]Zhang J H, Shi P, Qiu J Q. Robust stability criteria for uncertain neutral system with time delay and nonlinear uncertainties [J]. Chaos, Solitons and Fractals,2008,38(1),160-167.
[8]Zuo Z, Wang Y. New stability criteria for a class of linear systems with time-varying delay and nonlinear perturbations [J]. IEE Proceedings Control Theory and Applications,2006,153(5), 623-626.
[9]俞立.鲁棒控制一线性矩阵不等式处理方法[M].北京：清华大学出版社,2002.
[10]Gu K. An integral inequality in the stability problem of time-delay systems [J]. Decision and Control, Proceedings of the 39th IEEE Conference,2000, (3),2805-2810.
[11]何舒平,刘飞.基于观测器的非线性不确定时滞跳变系统无源控制[J].数学物理学报,2009,29A(2),334-343.
[12]Cheng Wang, Yi Shen. Improved delay-dependent robust stability criteria for uncertain time delay systems [J]. Applied Mathematics and Computation,2011,218(6),2880-2888.
[13]E.Fridman, U.Shake. An improved stabilization method for linear time-delay systems [J]. IEEE Transactions Automatic Control,2002,47(11),1931-1937.
[14]Q.L.Han. On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty [J]. Automatica,2004,40(6),1087-1092.
[15]瞿少成.不确定系统的滑模控制理论及应用研究[M].武汉：华中师范大学出版社,2008.
[16]蒋威.退化时滞微分系统[M].合肥：安徽大学出版社,1998.
[17]苏宏业.不确定时滞系统的鲁棒控制理论[M].北京：科学出版社,2007.
[18]L Y Dai. Singular Control Systems [M]. Berlin:Springer-Verlag,1989.
[19]C.J.Chang, K.F.R.Liu, K.Yeh, C.W.Chen, P.Y.Chung. Delay independent criterion for mul-tiple time-delay systems [J]. Physics Procedia,2012,25:270-277.
[20]P.Balasubramaniam, R.Krishnasamy, R.Rakkiyappan. Delay-dependent stability criterion for a class of non-linear singular Markovian jump systems with mode-dependent interval time-varying delays [J]. Communications in Nonlinear Science and Numerical Simulation,2012,17(9): 3612-3627.
[21]E K Boukas, Z K Liu. Delay-dependent stability analysis of singular linear continuous-time system [J]. The fourth International Conference on Control and Automation,10-12 June 2003, Montreal, Canada.
[22]Cheng Wang, Yi Shen. Improved delay-dependent robust stability criteria for uncertain time delay systems [J]. Applied Mathematics and Computation,2011,218(6):2880-2888.
[23]Mei Fang. Delay-dependent stability analysis for discrete singular systems with time-varying delays [J]. Acta Automatica Sinica,2010,36(5):751-755.
[24]Kwan Ho Lee, Joon Hwa Lee, Wook Hyun Kwon. Sufficient LMI conditions for H∞ output feedback stabilization of linear discrete-time systems [J]. IEEE Transactions on Automatica Control,2006,51(4):675-680.
[25]Z G Wu, H Y Su, J Chu. Improved results on delay-dependent H∞, control for singular time-delay systems [J]. Acta Automatica Sinica,2009,35(8):1101-1106.
[26]F.Gouaisbaut, M.Dambrine, J.P.Richard. Robust control of delay systems:a sliding mode control design via LMI [J]. Systems Control Letters,2002,46(4):219-230.
[27]Wei Jiang. Eigenvalue and stability of singular differential delay systems [J]. Journal of Math-ematical Analysis and Applications,2004,297(1):305-316.
[28]Piyapong Niamsup. Stability of time-varying switched systems with time-varying delay [J]. Nonlinear Analysis:Hybrid Systems,2009,3(4):631-639.
[29]廖晓听.稳定性的理论、方法和应用[M].武汉：华中科技大学出版社,2002.
[30]时宝,张德存,盖明久.微分方程理论及其应用[M].北京：国防工业出版社,2005.
[31]郭大钧.非线性泛函分析M.济南：山东科学技术出版社,2001.
[32]Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations [M]. The Netherlands:North Holland Mathematics Studies, Elsevier,2006.
[33]I.Podlubny. Fractional differential Equations [M]. San Diego:San Diego Academic Press,1999.
[34]王振滨,曹广益,朱新坚.分数阶线性定常系统的稳定性及其判据[J].控制理论与应用,2004,21(6)：922-926.
[35]卜春霞,张应奇,毛北行.变时滞控制系统的变结构控制[J].郑州大学学报(理学版),2003,35(4)：23-26.
[36]El-Khezali R, Ahmad WH. Variable structure control of fractional time-delay systems [J]. In: Proceedings of the 2nd IFAC, Workshop on fractional differentiation and its applications, Porto, Portugal, July 19-21,2006.
[37]Lee,J.H., S.W.Kim, W.H.Kwon. Memoryless controllers for state delayed systems [J]. IEEE, Trans Automat. Control,39,159-162,1994.
[38]Yuanqing Xia, Jingqing Han, Yingmin Jia. A sliding mode control for linear systems with input and state delays [J]. In:Proceedings of the 41sl IEEE, Conference on Decision and Control, Las Vepas, Nevada USA, December 2002.