一类含幅、相等式的非线性方程组的线性化及其应用
The Linearization of Certain Class of Nonlinear Simultaneous Equation Set Containing Amplitude and Phase Frequency Characteristics and Application
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摘要: 一类有理分式复函数G(jω)=1+a1jω+a2(jω)2+…+am(jω)n/b0+b1jω+b2jω+…+bn(jω)n常被用来描述系统的性能.当其有关的幅值(模)和相角(幅角)的数据能获取时,可以对G(jω)进行综合.诸未知系数ai及bi(i=0,1,…,m,…,n)将通过解一类含幅、相等式的非线性代数方程组i i=1,2,…,r.确解的初值[9].本文通过简单的数学处理,将这类非线性方程组完全线性化为同维的线性方程组,从而得以直接解.此法还可以推广到分母含纯(jω)因子和e-jωi0因子的系统.文章最后通过例子简介了这种方法在控制工程领域的应用.Abstract: A class of complex function of rational fraction type G(jω)=1+a1jω+a2(jω)2+…+am(jω)n/b0+b1jω+b2jω+…+bn(jω)n is frequently used to describe the dyna-mical properties of systems. It is however quite difficult to establish a mathematical model of this type on the basis of amplitude and phase frequency data collected from experiments conducted on the related physical system. Since the erection of mathematical model G(;o) would involve the solution of a set of nonlinear simultaneous equations and bis(i=0, 1,…,m,…,n)in. Up to now, these nonlinear equa-tiorjs have been considered to be very difficult to solve directly. In spite of the fact there are special computer programmes in certain software packages available to tackle this problem, it is by no means an easy task due to the complex procedures involved in picking up a set of initial values that should be close enough to the exact solutions. This paper proposes a simplified method of linearizing these nonlinear equations set so that direct solution is possible. The method can also be applied to systems with factors of(jω) and e-jωra in G(jω). An illustration by a workable example is furnished at the end of this paper to show its versatility.
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