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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2022, Vol. 39 ›› Issue (3): 439-450.doi: 10.3969/j.issn.1005-3085.2022.03.008

• • 上一篇    下一篇

单摆系统的多时滞控制与特征根分析

赵东霞,   范东霞,   王婷婷,   毛  莉   

  1. 中北大学理学院，太原 030051
  • 出版日期:2022-06-15 发布日期:2022-08-15
  • 基金资助:
    国家自然科学基金 (61603351)；山西省自然科学基金 (201801D121027)；山西省基础研究计划资助项目 (20210302123046)；中北大学第十七届研究生科技立项 (20201747).

The Control and Eigenvalue Analysis for a Pendulum System with Multiple Time-delay

ZHAO Dongxia,   FAN Dongxia,   WANG Tingting,   MAO Li   

  1. School of Science, North University of China, Taiyuan 030051
  • Online:2022-06-15 Published:2022-08-15
  • Supported by:
    The National Natural Science Foundation of China (61603351); the Natural Science Foundation of Shanxi Province (201801D121027); the Fundamental Research Program of Shanxi Province (20210302123046); the 17$^{\rm th}$ Postgraduate Science and Technology Project of North University of China (20201747).

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摘要：

对单摆系统在位置反馈和时滞位置反馈下的镇定问题进行了研究。考虑到控制器本身存在滞后现象这一实际因素，构建了含有两个时滞的二阶时滞微分方程模型；建立了系统的特征方程，采用特征根分析方法以及解析函数的零点重数之和关于参数的连续依赖性，得到了系统参数及各个时滞值与系统稳定性之间的关系，给出了与时滞相关的稳定性条件及与时滞无关的稳定性条件；结合代数方程求根技巧，分析了特征根的重数，得到了特征根的重数至多为 4 的参数条件，并证明了三重特征根至多有两个；在系统参数满足稳定性条件的前提下，证明了当特征根的模趋于无穷大时，必有特征根的实部趋于负无穷大；利用渐近分析技巧，计算出了当特征根的实部趋于负无穷大时特征根的渐近表达式。最后，通过 Matlab 数值仿真验证了结论的有效性，这一方法可进一步推广到含有多个时滞的 $n$ 阶微分方程系统。

关键词: 多时滞, 稳定性, 特征根, 渐近分析, 指数型多项式

Abstract:

The stabilization of a pendulum system with position feedback and delayed position feedback controller is studied. Considering the fact that the controller itself has delay, a second-order differential equation with  two delays is established. We present the characteristic equation, obtain the relation between the system parameters and stability, and give the results for delay-dependent and delay-independent stability. By the technique for finding roots of algebra equation, the multiplicity of characteristic roots is analyzed, the parameter condition that the multiplicity is at most 4 is obtained, and it is proved that there are at most two triple characteristic roots. When the modulus of the characteristic root tends to infinity, we prove that the real part must tend to negative infinity. By using the asymptotic analysis technique, the asymptotic expression of the characteristic root is calculated. The validity of the conclusion is verified by Matlab numerical simulations. This method can be extended to $n$-order differential equations with multiple time-delays.

Key words: multiple time delay, stability, eigenvalue, asymptotic analysis, the exponential polynomial

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