The characteristic equation of a control system is given as:

\(1+\frac{K(s + 1)}{s(s+4)(s^2+2s+2)} =0\)

For a large value of s, the root loci for K ≥ 0 are asymptotic to asymptotes, where do the asymptotes intersect on the real axis?

Concept:

All the asymptotes meet at a common point on the real axis known as centroid, it is given by,

Centroid:

\(\frac{{\sum real\;part\;of\;pole - \sum real\;part\;of\;zero}}{{No.\;\;of\;poles - No.\;\;of\;zeros}}\)

Analysis:

from characteristic equation open-loop transfer function have,

Zeros = -1

Poles = 0, -4, -1 + i, -1 - i.

Therefore centroid = -5 / 3.

Important Points

In root locus when P ≠ Z then some of the root locus branches will tend to infinity along the direction of asymptotes.

The angle of Asymptotes = \(\frac{{\left( {2k \pm 1} \right)180}}{{P - Z}}\) K = 0, 1, 2, 3, …

Remember: Centroid always lies on a real axis not compulsory to be on root locus whereas saddle point can lie anywhere but it should present on the root locus.