Extended graphical representation of rational fractions with applications to cybernetics (Q1085571)

We discuss an extended graphical representation of the fraction of a complex variable s \(K=\sum^{n}_{i=0}a_ is^{n- i}/\sum^{m}_{j=0}b_ js^{m-j}\) 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 anyone coefficient of a characteristic equation as a real parameter. It is easy to prove the following theorem: \[ K_ 1=f^{(n)}(s)/F^{(d)}(s)\quad and\quad K_ 2=F^{(d)}(s)/f^{(n)}(s) \] 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 stability interval systematically.