常系数线性微分差分方程稳定性的特征值分析法

本文首次对常系数线性微分差分方程(DDE)在某一有限区域内的稳定性提出了一种定量的特征值分析方法。该方法的主要思想是先将特征值复平面上某一有限的被研究区域划分成若干个均匀的子区域。对于每个子区域,在以子域中心为圆心并包含该子域的邻域内把DDE的特征矩阵展成泰勒级数,在满足一定精度下将其截断至一定阶数,得出相应的多项式矩阵。然后,将其线性化成复矩阵束,并用求解复广义特征根的方法求出DDE在该子区域内的特征根。通过对所有子域进行计算,便可得出DDE在研究区域内的全部特征根。应用这一方法,对计及静压传感器时滞的双反射器天线系统的稳定性以及交直流电力系统在计及换流站调节器时滞和宜流线路分布参数后的小干扰稳定性进行了分析和计算,所得结果与参考文献中应用其它方法得出的结果一致。

EIGENVALUE ANALYSIS METHOD FOR STABILITY EVALUATION OF LINEAR CONSTANT-COEFFICIENT DIFFERENTIAL-DIFFERENCE EQUATIONS

An eigenvalue analysis method for stability evaluation of linear constant-coefficient differential-difference equations (DDE) is proposed in the paper. The main idea of the method is as follows. Thestudied region on the eigenvalue plane which is interested in the stability evaluation of the DDE isequally divided into a number of subregions. For each subregion, the eigenmatrix of the DDE is ex-pended into Taylor series around the centre of the subregion. The Taylor series is truncated at a certainorder to meet required accuracy and to form a matrix polynomial. The matrix polynomial is thenlinearized into a complex matrix pencil. The eigenvalues of the DDE in the subregion are calculated bysolving the generalized eigenvalues of the complex matrix pencil. In this way, all of the eigenvalues ofthe DDE in the studied region are obtained. By using the method, this paper studies the stability of two-reflector antenna stabilization systemwith the time lags of the hydrostatic pickups, as well as the small disturbance stability of AC/DC powersystems with taking the time lags of the convertor controllers and the distributed parameters of DC lineinto consideration. The results of the two systems are consistent with those obtained by other methods