非线性系统运动稳定性的一种判据
A Criterion for the Stability of Motion of Nonlinear Systems
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摘要: 对于自治的非线性系统来说,只要其线性部分系数矩阵的特征值不属于临界情形,其无扰运动在其足够小的邻域内的稳定性完全可以由其线性部分的特征值确定.关于线性系统的稳定性,已有不少简单易行的判别方法,而关于非线性系统的稳定性,很多数学家和力学家作了大量的研究工作;但大都是针对特殊类型的非线性系统解决了一些问题,直到现在为止,还没有普遍适用于任何的非线性系统的简单易行的判别方法.本文所给的是判别非线性系统稳定性的充要条件,常用的克拉索夫斯基方法只是这一方法的一个特例[1],[2].Abstract: As to an autonomous nonlinear system, the stability of the equilibrium slate in a sufficiently small neighborhood of the equilibrium state can be determined by eigen values of the linear pan of the nonlinear system provided that the eigenvalues are not in a critical case.Many methods may be used to detect the stability for a linear system.A lot of researches for determining the stability of a nonlinear system are completed by mathematicians and mechanicians but most of them are methods for the special forms of nonlinear systems.Till now.none of these methods can be conveniently applied to all nonlinear systems.The method introduced by this paper gives the necessary and sufficient conditions of the stability of a nonlinear system.The familiar Krasoyski's method is a special case of this method[1],[2].
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[1] 秦元勋、王慕秋、王联,《运动稳定性理论与应用》,科学出版社(1981),234-352 [2] 绪方胜彦,《现代控制工程》,科学出版社(1978),550-570. [3] 廿特马赫尔Ф.P.,《矩阵论》(上),高等教育出版ft(1957),293-336. -
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