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时滞微分系统的性质分析与算法研究
其他题名Research on Properties and Algorithms of Delay Differential Systems
魏萍
学位类型工学博士
导师郁文生
2006-05-26
学位授予单位中国科学院研究生院
学位授予地点中国科学院自动化研究所
学位专业控制理论与控制工程
关键词时滞微分系统 中立型微分系统 时滞无关稳定 时滞相关稳定 代数判定条件 全局双曲性 Hopf分岔 多项式完全判别系统 Delay Differential System Neutral Differential System Delay-independent Stability Delay-dependent Stability Algebraic Criterion Global Hyperbolicity Hopf Bifurcation Complete Discrimination System For Polynomials
摘要具有时滞的微分系统广泛存在于控制、物理、化工及生态等模型中,研究此类系统的性质,并给出可以检验的代数判定条件具有重要的现实意义。本文首先针对一类中立型时滞微分系统,分别讨论了时滞无关稳定性和时滞相关稳定性问题,给出代数判定条件及算法步骤。然后又讨论了一类时滞微分系统的Hopf分岔问题,给出了判定其产生Hopf分岔现象的代数条件及算法步骤。在本文中,主要的工作和贡献有: 1,考虑了一类形式较为一般的中立型线性多时滞微分系统的时滞无关稳定性问题。通过讨论相关差分方程的全局双曲性,利用多项式完全判别系统,给出该系统时滞无关稳定的代数判定条件及算法步骤。推广和统一了某些文献中的相关结论,尤其这些结论可以完全应用到相关的一类滞后型线性多时滞微分系统,最后给出例子说明方法的有效性。 2,在上面提到的工作基础上,考虑了一类形式较为一般的中立型线性多时滞微分系统的时滞相关稳定性。也就是,如果此类系统经以上提到的代数判定条件判定不是时滞无关稳定时,给出能保持其稳定性的时滞存在区间。相应的代数判定条件与算法步骤依次给出。相关结果也可以完全应用到对应的一类滞后型线性多时滞微分系统,最后通过例子说明该方法的具体实现。 3,对于文献中采用的判定时滞微分系统模型出现Hopf分岔现象的一般方法进行了总结与推广。在通常文献中,研究对象多为不超过三维的单时滞微分系统模型,尤其文献[16]对这一类工作进行了总结,并最后归结为判定一个一元多项式的实根问题。本文在上述基础上,面向一类高维的多时滞微分系统模型,沿用判定时滞微分系统模型出现Hopf分岔现象的一般方法,其中采用了不同的变量转换,将这类工作进行总结,并最后归结为判定一组二元多项式的实根问题。
其他摘要Delay differential system model can often be found in control process, physics,chemical engineering and ecology. It is more significant to study the properties of the system and establish easily testable algebraic criteria. At first, the delay-independent stability and the delay-dependent stability of a class of neutral delay differential systems are considered respectively, and the corresponding algebraic criteria and algorithms are presented in this paper. Afterwards, the Hopf bifurcation of a class of delay differential systems is studied, and the corresponding algebraic criteria and algorithms are presented. The main contributions can be summarized as follows. 1, The delay-independent stability of a class of neutral multi-delay differential systems is analyzed. By using the "Complete Discrimination System for Polynomials", easily testable algebraic criteria and algorithms for delay-independent stability of the systems are established after considering the global hyperbolicity of the related difference systems. The results generalize and unify the relevant methods in the literature. Especially, the results can be applied to the related class of retarded differential systems completely. Some numerical examples are provided to illustrate the effectiveness of the results. 2, Based on the work mentioned above, the delay-dependent stability of the class of neutral multi-delay differential systems is discussed. That is, the delay intervals for stability of the systems can be computed if the systems are not delay-independent stable tested by the algebraic criteria mentioned above. The corresponding algebraic criteria and algorithms are presented, and the results can be applied to the related class of retarded differential systems completely. A numerical example is provided to illustrate the results. 3, The general method, used in the most literature for determining the occurrence of Hopf bifurcation for delay differential systems, is summarized and generalized. The literature has just considered about Hopf bifurcation for the class of lower-dimensional one-delay differential systems. Especially, the paper[16] summarized the method used in the literature, and turned the problem into determining the real roots of a one-variable polynomial equation. In this paper, the general method is summarized again for a class of higher-dimensional multi-delay differential systems by using a different transformation, and turn the bifurcation analysis into determining the real roots of a pair of two-variable polynomial equations.
馆藏号XWLW1034
其他标识符200318014602986
语种中文
文献类型学位论文
条目标识符http://ir.ia.ac.cn/handle/173211/5902
专题毕业生_博士学位论文
推荐引用方式
GB/T 7714
魏萍. 时滞微分系统的性质分析与算法研究[D]. 中国科学院自动化研究所. 中国科学院研究生院,2006.
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