【作者】 李旭；

【导师】 李旭光；

【作者基本信息】 东北大学 ， 控制理论与控制工程， 2014， 硕士

【摘要】 尽管线性时不变时滞系统是一个古老陈旧的课题并且得到了广泛的关注,但是其研究是非常困难复杂的并且许多主要问题都没有被完全探究。与有限维系统不同,时滞系统是一个具有无穷多个特征根的无限维系统,而且这些系统的特征根的分布还不能被现有的数学方法完全研究。更加不同的是,时滞系统的一个临界虚根对应无穷多个临界时滞。这使得研究时滞系统变得非常困难,因此时滞系统的完全稳定性至今没有被彻底解决。据目前所知,解决完全稳定性的关键性质就是系统临界虚根的一致性,一致性仅仅在一些特殊的系统中被探讨过,目前还没有一般性的证明。包括一致性证明在内,本文的主要工作和结论如下：基于解析曲线框架,本文给出了Retarded时滞系统临界虚根一致性的完全证明。将临界虚根的渐近行为同Puiseux级数联系起来,同时将频域扫曲线的渐近行为同对偶Puiseux级数联系起来,通过探讨Puiseux级数对之间的内在性质来证明一致性定理。由于非退化情况的一致性证明非常简单,在本文中主要讨论退化情况的一致性。退化情况的证明非常复杂且需要大量的工作,因此将退化情况分为三种子情况并且给出逐一证明。值得一提的是,本文中的一致性证明适用于所有Retarded系统,无论其临界虚根是单根或者重根。类似于Retarded系统的结论,本文将Neutral算子的稳定性(Neutral系统稳定的必要条件)嵌入到频域扫方法中并且将临界虚根的一致性推广到Neutral系统中。可以通过观察频率足够大时的频域扫曲线来判断Neutral算子的稳定性,从而不再需要将Neutral系统从标量形式变换回矩阵形式。至于Neutral系统临界虚根的一致性,本文采用与Retarded系统相同的方法来证明。利用数值分析软件Matlab实现频域扫方法,同时利用DDEBIF-TOOL工具箱完成时滞系统临界虚根渐近行为的数值仿真。更多还原

【Abstract】 Although the stability of linear time-invariant time-delay systems (TDSs) is an old topic and has received a lot of attention, its research is full of complexity and some main problems have not been fully investigated. Unlike a finite-dimensional linear system, a time-delay system is an infinite-dimensional system involving infinitely many characteristic roots, the distribution of which cannot be fully detected by existing mathematical tools. More differently, the critical imaginary roots (CIRs) of a time-delay system correspond to infinitely many critical delays (CDs). With these intricacies the complete stability of linear time-delay systems has remained unsolved so far. The key property for solving complete stability is invariance property, which is proved only for special cases without general confirmation. Including the proof of invariance property, the main research work and conclusions in this thesis is as follows:Based on analytic curves framework, the full proof of invariance property for time-delay systems of Retarded type is finished. By connecting the asymptotic behavior of the critical roots with Puiseux series and connecting the asymptotic behavior of the frequency-sweeping curves with dual Puiseux series similarly, some internal relation between Puiseux series pairs is presented to confirm invariance property. Due to the simplicity of the proof for non-degenerate case, this thesis mainly focuses on the proof for degenerate case via dividing it into three sub-cases, which gives rise to great complexity and tremendous work. It is worth mentioning that the proof in this thesis holds for general case regardless of the multiplicity of the critical roots.Inspired by some homologous results for Retarded time-delay systems, the stability condition of the Neutral operator(as a necessary stability condition additionally required by Neutral systems) is embedded into frequency-sweeping approach and invariance property is generalized to time-delay systems of Neutral type. By observing the frequency-sweeping curves with sufficiently large frequency, the stability of the Neutral operator can be directly examined without transforming the systems in scalar form back to the matrix form. As for the confirmation of invariance property, the same methods used for Retarded type is employed.Using the numerical analysis software Matlab, the frequency-sweeping approach is implemented and the simulation for the asymptotic behavior of critical imaginary roots is finished.更多还原