Extended graphical representation of rational fractions with applications to cybernetics

Abstract

In this paper, we discuss the extended graphical representation of the fraction of a complex variables

$${{K = \sum\limits_{i = 0}^n {a_i s^{n - i} } } \mathord{\left/ {\vphantom {{K = \sum\limits_{i = 0}^n {a_i s^{n - i} } } {\sum\limits_{j = 0}^m {b_j s^{m - j} } }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{j = 0}^m {b_j s^{m - j} } }}$$

WhereK 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:

$$K_1 = f^{(n)} (s)/F^{(d)} (s){\mathbf{ }}and{\mathbf{ }}K_2 = F^{(d)} (s)/f^{(n)} (s)$$

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 axisx, 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.