主次元分析与稳定性分析

【作者】 张春凤；

【导师】 钟守铭；

【作者基本信息】 电子科技大学 ， 运筹学与控制论， 2004， 硕士

【摘要】 本论文的研究目的是寻找使网络系统达到稳定的条件,并且这些条件对网络自身的限制比较弱。从而使网络系统的设计更加容易,反过来也可以运用这些条件验证一个网络系统是否稳定。通俗的说法就是对于用一般或特殊的微分方程所描述的系统建立判别方法,以判断哪些系统受干扰或不受干扰的运动状态相差甚微,哪些则相反,同时为稳定系统的设计,提供了更好的理论和方法。本文主要运用数学工具对网络系统的几种变换形式进行了研究讨论。例如:常数变易法,不等式分析技巧,Gronwall-Bellman不等式以及经典的Liapunov函数法,Dini导数等。通过求解简单的一维微分方程,得到了仅由网络连接权矩阵的特征值和特征向量表示的解的解析表达式,根据解析式的形式分析了网络的最终输出状态,从而确定了网络达到稳定收敛的充分条件。有时在实际问题中建立的微分方程形式的模型往往很复杂,无法求出其解的解析表达式,就需要从方程本身运用一些直接方法来判断系统的渐近性态。零解的稳定性。对变换后的网络模型,本文运用解对初值的连续依赖性和解的存在唯一性以及一些变换算法,得到了不需求网络解的解析式,直接利用网络连接权矩阵的特征值符号来判断网络的渐近行为和零解的稳定性条件。对网络模型的另一种变换形式,利用的是局部线性近似的方法,由于非线性方程与它的线性部分在局部范围内的稳定性等价,可用非线性系统局部线性化后的简单线性部分代替复杂的非线性方程。在讨论了Liapunov函数的存在性后,可构造适当的V函数直接对网络系统分析渐近性、稳定性、收敛性,对复数形式的网络也进行了探讨,得到了网络输出在满足一定条件下最终收敛于权矩阵的最大特征值对应的特征向量,实现了特征提取,变换权矩阵符号又得到了网络输出最终收敛于最小特征值对应的特征向量的结论。最后论文分析了细胞神经网络模型的稳定性,证明了细胞神经网络平衡点的存在性,通过构造适当的Liapunov函数对其沿系统求Dini导数得到了平衡点全局一致渐近稳定的几个充分条件。更多还原

【Abstract】 Our goal is to find out the conditions which guarantee the networks stability. Th-ese conditions are weaker than ever and make the design of the networks easier. In general, we can verify whether the networks system is stable through these conditions.In other words, we can establish stable system base on the stability criteria.We mainly use mathematical tools to research networks model. For example cons-tance variation, inequality analyze technique, Gronwall-Bellman inequality and clas-sical Liapunov function , Dini derivation ect. The differential equation of the networks model is solved by solving a simple one dimensional equation .The solution of the neural networks that is obtained is represen-ted by the eigenvalue and eigenvector of the weight matrix. Then the asymptotic sta-ble behavior in analyzed. Sometimes differential equation form is complex , so we can’t obtain the analytic expression of the solution. We may use simple methods from the equation itself to de-termine the asymptotic and the stability of the trivial solution. We gain some stabilityconditions base on the continuous dependence of the solution for initial-value and theunique existence of a solution. Under these conditions, we needn’t solute the differe-ntial equation and only make use of the sign of eigenvalue for weight matrix to verify the asymptotic and the stability of trivial solution. For another transformation of the network model ,we make use of local linear appr-ximation .Because nonlinear equation is stable equivalence with it’s linear equation inlocal range, we only need analyze the linear equation. After discuss the existence of the Liapunov function ,we can make up apposite V function to derivate along the mo-del. We obtain the result that the output will convergence to the eigenvector correspo-ndion to the largest eigenvalue of weight matrix. Since then realize the feature extrac-tion . Through changing the sign of weight matrix ,we obtain the result that the final output vector is the eigenvector corresponding to the minimum eigenvalue of weight matix. In the final section ,we analyze the stability of cellular neural networks ,then proof the existence of equilibrium point . By meanse of Liapunov function we obtain sever-eval sufficient conditions which guarantee the equilibrium point uniformly asymptoti-cal stability.更多还原