有理分式的增广图示及其在工程控制论中的应用

本文研究有理分式的增广图示,分子分母分别为n及m次多项式的有理分式,它的根轨迹方程的次数,当n+m是偶数时,是y2的(n+m)/2-1次;当n+m是奇数时,是(n+m-1)/2次.因此,n+m≤10的图示数据能用公式计算有理分式的增广图示能应用于研究反馈系统及特征方程的任一实系数作参数的图线特性.用本文理论易证倒分式定理:K₁=f(n)(s)/(F)(m)(s),与K₂=F(m)(s)/f(n)(s)二者在复数平面上的根轨迹完全相同又由图示知识发现,不论n和m多大,只要有理分式的零点和极点在实轴上相间排列,它就没有复数根轨迹,这样的系统不会发生振荡,本文对这种分式可能存在的稳定区作较全面地分析.

Extended Graphical Representation of Rational Fractions with Applications to Cybernetics

In this paper, we discuss the extended graphical frepresentation of the fraction of a complex variable s

Where K is confined to be real. Three figures of the above fraction can be used in feedback systems as well as to study the properties of figures for any one coefficient of a characteristic equation as a real parameter. It is easy to prove the following theorem:
K₁=f(n)(s)/(F)(d)(s),and K₂=F(d)(s)/f(n)(s)
have the same root locus.By this graphical theory, we find out that if the zeros and poles of a fraction are alternatively placed on the axis x, then there is no complex root locus of this fraction, therefore the state of such a system is always non-oscillatory; Using these figures of this fraction, we can discuss its stable interval systematically.