The asymptotic behaviour,the angles of departure,and the angles of approach,of the characteristic frequency loci

In algebraic function theory, there is a well established method which uses ‘Newton's diagram’ to find the series expansions of an algebraic function q(x) in the neighbourhood of a point x₀ . In this paper it is shown how, for a linear, time-invariant, multi-variable feedback system, this method can be used to find :

(i) the asymptotic behaviour of the characteristic frequency loci (multivariable root loci) ;

(ii) the angles of departure of the characteristic frequency loci from the open-loop poles ; and

(iii) the angles of approach of the characteristic frequency loci to the finite zeros of such a system.