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.