退化时滞微分方程的解、稳定性及控制问题
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摘要
在经济、电力、控制工程等很多实际系统中,退化现象是普遍存在的,这引起学者们的广泛重视并得出了很多成果.而时滞又是客观世界与工程实际中普遍存在的现象.在许多实际系统中,要对其准确的描述,就必须同时考虑退化和时滞的影响.因此研究退化时滞微分方程解的性态具有重要的现实意义.
本文就退化时滞微分方程的解、稳定性及控制问题作了一些研究.本博士论文由四章组成,主要讨论了分数阶退化、时滞微分方程解的存在性与通解,退化时滞微分方程稳定性以及退化时滞微分系统保成本控制问题.
首先,我们叙述了问题产生的背景与意义及本文所做的主要工作.
其次,基于Banach不动点定理、Schauder不动点定理和逐步逼近法获得分数阶泛函微分方程解的存在性结果,该结果推广了整数阶常微分方程和泛函微分方程的相应结论.并且通过定义可解阵对,获得分数阶一般退化微分方程的通解表达式.该结果推广了整数阶退化微分方程和分数阶常微分方程相应结论.
再者,我们给出了退化时滞微分方程全时滞稳定性新判据,运用所得结果可以克服求解超越方程根的困难,同时便于检验全时滞稳定性.并且提出了一般退化时滞微分系统的V泛函方法,基于V泛函方法和分析技巧给出了具有多变时滞的退化微分非线性系统的渐近稳定性判据,稳定性判据被描述为矩阵等式或者矩阵不等式.所得结果在计算上是可行和有效的.
最后,我们讨论了具状态与输入时滞连续时间不确定退化微分系统保成本控制问题,利用线性矩阵不等式(LMI)方法获得了系统保成本控制器存在的充分条件,根据线性矩阵不等式的可行解给出保成本控制律的参数特征并且该保成本控制器的保成本函数存在上界.
In many practical systems, people have found that there were many degen-erate phenomena existing extensively, such as economic systems, power systems, engineering systems and so on. And this phenomenon has attracted many scholars' attention and lots of available and important results have been obtained. However, time delay is a common phenomenon in the objective world and engineering fields. In order to describe the system more accurately in many practical systems, we must take the influence of degradation and delay into consideration altogether. So, it has a practical significance to study the solution and its characteristics of degenerate differential systems with delay.
In this dissertation, solution, stability and control problems of degenerate dif-ferential equations with delay are discussed. This doctoral dissertation is composed of four chapters, which mainly studies existence conditions of solution of fractional order functional differential equations and general solution of fractional order de-generate differential equations, and stability and guaranteed cost control problem of degenerate differential equations with delay.
Firstly, the backgrounds and significance of the problems are given. The main work done in this dissertation is introduced.
Secondly, the existence results of solution of fractional order functional differ-ential equation are derived based on Banach fixed point theorem, Schauder fixed point theorem and successive approximations technique, respectively. These results extend the corresponding ones of ordinary differential equations and functional dif-ferential equations of integer order. The expressions of the general solution for fractional order ordinary differential equations are obtained by defining the solv-able matrix pair, and these results extend the corresponding ones of integer order degenerate differential equations and fractional order linear ordinary differential equations.
Further, some new stability criteria of degenerate differential equations with delay are obtained, and we can overcome the difficulties of solving roots of tran-scendental equation by applying these criteria, which are more convenient to check all-delay stability. The V-functional method for general singular differential de-lay system is proposed. The asymptotic stability criteria for degenerate differ-ential nonlinear systems with multiple time-varying delays are derived based on V-functional method and some analytical techniques, which are described as ma-trix equations or matrix inequalities. The results obtained are computationally flexible and efficient.
Finally, we discuss guaranteed cost control problem of continuous-time uncer-tain degenerate system with state and control delays. Sufficient conditions for the existence of the guaranteed cost controller are derived based on the linear inequal-ity (LMI) approach, a parameterized characterization of the guaranteed cost laws is given in term of the feasible solutions to a certain LMI, and the cost function of guaranteed cost controller exists the upper bound.
本文就退化时滞微分方程的解、稳定性及控制问题作了一些研究.本博士论文由四章组成,主要讨论了分数阶退化、时滞微分方程解的存在性与通解,退化时滞微分方程稳定性以及退化时滞微分系统保成本控制问题.
首先,我们叙述了问题产生的背景与意义及本文所做的主要工作.
其次,基于Banach不动点定理、Schauder不动点定理和逐步逼近法获得分数阶泛函微分方程解的存在性结果,该结果推广了整数阶常微分方程和泛函微分方程的相应结论.并且通过定义可解阵对,获得分数阶一般退化微分方程的通解表达式.该结果推广了整数阶退化微分方程和分数阶常微分方程相应结论.
再者,我们给出了退化时滞微分方程全时滞稳定性新判据,运用所得结果可以克服求解超越方程根的困难,同时便于检验全时滞稳定性.并且提出了一般退化时滞微分系统的V泛函方法,基于V泛函方法和分析技巧给出了具有多变时滞的退化微分非线性系统的渐近稳定性判据,稳定性判据被描述为矩阵等式或者矩阵不等式.所得结果在计算上是可行和有效的.
最后,我们讨论了具状态与输入时滞连续时间不确定退化微分系统保成本控制问题,利用线性矩阵不等式(LMI)方法获得了系统保成本控制器存在的充分条件,根据线性矩阵不等式的可行解给出保成本控制律的参数特征并且该保成本控制器的保成本函数存在上界.
In many practical systems, people have found that there were many degen-erate phenomena existing extensively, such as economic systems, power systems, engineering systems and so on. And this phenomenon has attracted many scholars' attention and lots of available and important results have been obtained. However, time delay is a common phenomenon in the objective world and engineering fields. In order to describe the system more accurately in many practical systems, we must take the influence of degradation and delay into consideration altogether. So, it has a practical significance to study the solution and its characteristics of degenerate differential systems with delay.
In this dissertation, solution, stability and control problems of degenerate dif-ferential equations with delay are discussed. This doctoral dissertation is composed of four chapters, which mainly studies existence conditions of solution of fractional order functional differential equations and general solution of fractional order de-generate differential equations, and stability and guaranteed cost control problem of degenerate differential equations with delay.
Firstly, the backgrounds and significance of the problems are given. The main work done in this dissertation is introduced.
Secondly, the existence results of solution of fractional order functional differ-ential equation are derived based on Banach fixed point theorem, Schauder fixed point theorem and successive approximations technique, respectively. These results extend the corresponding ones of ordinary differential equations and functional dif-ferential equations of integer order. The expressions of the general solution for fractional order ordinary differential equations are obtained by defining the solv-able matrix pair, and these results extend the corresponding ones of integer order degenerate differential equations and fractional order linear ordinary differential equations.
Further, some new stability criteria of degenerate differential equations with delay are obtained, and we can overcome the difficulties of solving roots of tran-scendental equation by applying these criteria, which are more convenient to check all-delay stability. The V-functional method for general singular differential de-lay system is proposed. The asymptotic stability criteria for degenerate differ-ential nonlinear systems with multiple time-varying delays are derived based on V-functional method and some analytical techniques, which are described as ma-trix equations or matrix inequalities. The results obtained are computationally flexible and efficient.
Finally, we discuss guaranteed cost control problem of continuous-time uncer-tain degenerate system with state and control delays. Sufficient conditions for the existence of the guaranteed cost controller are derived based on the linear inequal-ity (LMI) approach, a parameterized characterization of the guaranteed cost laws is given in term of the feasible solutions to a certain LMI, and the cost function of guaranteed cost controller exists the upper bound.
引文
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[2]郑祖庥.泛函微分方程理论[M].合肥：安徽教育出版社,1994.
[3]Peter Kunkel, Volker Mehrmann. Differential-Algebraic Equations[M]. Zurich: European Mathematical Society,2006.
[4]Rosenbrock H.H. Structural properties of linear dynamical systems[J]. Int. J. Contr.,1974,20(2):191-202.
[5]Luenberger D.G. Dynamical equation in descriptor form[J]. IEEE Transactions on Automatic Control,1977,22(3):312-321.
[6]Dai L. Singular control systems[M]. Berlin, New York:Springer-Verlag,1989.
[7]Campbell S. L. Singular systems of differential equations(II)[M]. Sanfrancisco, London, Melbourne:Pitman,1982.
[8]蒋威.退化时滞微分系统[M].合肥：安徽大学出版社,1998.
[9]杨冬梅等.广义系统[M].北京：科学出版社,2004.
[10]王文涛.非线性奇异控制系统[M].北京：冶金工业出版社,2005.
[11]李远清,刘永清.广义泛函微分方程解的稳定性[J].应用数学学报,1999,22(1)：130-138.
[12]刘永清,王伟,李远清.滞后广义系统解的基本理论与应用[M].广州：华南理工大学出版社,1997.
[13]Li H.. Li H. B. Zhong S. M. Stability of neutral type descriptor system with mixed delays [J]. Chaos, Solitons and Fractals,2007, (33):1796-1800.
[14]Xu S.Y., Dooren P.V. Robust stability and stabilization for singular systems with state delay and parameter uncertainty [J]. IEEE Trans. Automat. Control, 2002,47(7):1122-1126.
[15]Jiang Wei, Song Wenzhong. Controllability of singular systems with control delay[J]. Automatica,2001,37:1873-1877.
[16]Jiang Wei. Function-controllability of nonlinear singular delay differential con-trol Systems [J]. Acta Mathematica Sinica,2006,49:1153-1162.
[17]徐远通,郭志明.一种几何指标理论在泛函微分方程中的应用[M].数学学报,2001,44(6)：1027-1036.
[18]庚建设,郭志明,离散广义Emden-Fower方程的边值问题,中国科学A辑数学,2006,36(7)：721-732.
[19]唐先华,庾建设.超线性时滞微分系统的振动性[M].应用数学学报,2003,(26)(2)：328-336.
[20]郭志明,庚建设,雷光龙,具有时滞的利率—流通量方程的建立及其应用,应用数学学报,2004 Vol.27,No.4,pp.702-709.
[21]Zhang K. Y., Cao D. Q. Further results on asymptotic stability of linear neutral systems with multiple delays [J]. J. Franklin Inst.,2007,344:858-866.
[22]Cao D. Q., He P. Stability criteria of linear neutral systems with a single delay [J]. Appl. Math. Comput.,2004,148:135-143.
[23]Tan M. C., Asymptotic stability of nonlinear systems with unbounded de-lays[J]. J. Math. Anal. Appl.,2008,337:1010-1021.
[24]Park J. H., KWON O. Novel stability criterion of time delay systems with nonlinear uncertainties[J]. Appl. Math. Lett.,2005,18:683-688.
[25]Darouach M. Linear functional observers for systems with delays in state vari-ables[J]. IEEE Trans. Auto. Cont.,2001,46:491-496.
[26]Dugard L., Verriest EI. Stability and control of time-delay systems[M]. London: Springer-Verlag,1997.
[27]Fridman E. New Lyapunov-Krasovskii functionals for stability of linear re-tarded and neutral type systems [J]. Sys. Control Lett.,2001,43:309-319.
[28]Fridman E., Shaked U. Delay-dependent stability and H∞ control:constant and time-varying delays[J]. Int, J, Cont,2003,76:48-60.
[29]Chang H. L. Delay-dependent and delay-independent guaranteed cost control for uncertain neutral systems with time-varying delays via LMI approach[J]. Chaos, Solitons and Fractals,2007,33:1017-1027.
[30]Nian X., Feng J. Guaranteed-cost control of a linear uncertain system with multiple time-varying delays:an LMI approach[J]. IEE Proc. Control Theory Appl.,2003,150:17-22.
[31]Park J. H. Robust guaranteed cost control for uncertain linear differential systems of neutral type[J]. Appl. Math. Comput.,2003,140:523-535.
[32]Yu L., Gao F. Optimal guaranteed cost control of discrete-time uncertain systems with both state and input delays[J]. J. Franklin Inst.,2001,338:101-110.
[33]Shi P., Boukas E. Optimal guaranteed cost control of uncertain discrete time-delay systems [J]. J. Comput. Appl. Math.,2003,157:435-451.
[34]Chang S.S.L. Peng T.K.C., Adaptive guaranteed cost control of systems with uncertain parameters [J]. IEEE Trans. Automat. Control,1972,17:(4):474-483.
[35]Xie L. Output feedback H∞ control of systems with parameter uncertainty [J]. Int. J. Control,1996,63(4):741-750.
[36]Barmish B.R. Necessary and sufficient conditions for quadratic stabilizability of uncertain system[J]. J. Optim. Theory Appl.,1985,46(4):399-408.
[37]Leitmann G. On one approach to the control of uncertain systems[J]. ASME J. Dynamic Systems, Measurement, Control,1993,115(4):373-380.
[38]Zhang Hai, Li Xiaoyan, Jiang wei. The all-delay stability of degenerate differ-ential systems with delay, Annals of Differential Equations[J].2008,24(1):100-104.
[39]Zhang Hai, Jiang Wei, Du Jun. Guaranteed cost control of continuous-time uncertain singular system with both state selay and input delays, Annals of Differential Equations[J].2009,25(4):483-489.
[40]张海.退化时滞微分系统的周期解与稳定性问题[D].合肥：安徽大学,2007.
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