Chapter 10

# 10.5 Evans Root Locus Construction Rule # 3: Asymptotic Angles and Centroid

This Rule deals with the asymptotic angles and centroid location. If gain K is large enough, one can see that the branches of the RL travelling towards infinity follow a straight-line path that is asymptotic to a hypothetical line, called an asymptote, at a certain angle, called an asymptotic angle. If one extended these hypothetical lines, they would all intersect at an “anchor” point, called a centroid. Evans showed that the asymptotic angles and the centroid location can be computed as shown in this Rule.

When the test point $s^*$  is close to the open loop singularities (poles, zeros), angles for vectors drawn from the singularity towards the point $s^*$ , which are used to evaluate $G(s^*)$ function, are quite different. However, as the gain K tends to approach infinity, $K \rightarrow \infty$, which is the descriptor for asymptotic condition, point $s^*$ begins to practically lie on the asymptote, and these angles all begin to look alike and approach the asymptotic angle $\theta_i$.

Recall that the total angle of the function $G(s^*)$ is equal to:

 $\angle G(s^*)=\angle_{zeros}-\angle_{poles}$ Equation 10-7

The total angles, respectively, for all vectors associated with poles and all vectors associated with zeros, will be equal to:

 $\angle_{poles} = n\cdot\theta_i$ $\angle_{zeros}=m\cdot\theta_i$ Equation 10-8

If the test point $s^*$ is to belong to the root locus, the angle of $G(s^*)$ has to meet the angle criterion:

 $\angle G(s^*)=180^{\circ}$ $m\cdot\theta_i - n\cdot\theta_i=180^{\circ}$ Equation 10-9

In the formula below, the sign in the denominator is reversed, because $m\leq n$. This will have no effect on the formula, because $+180^{\circ}=-180^{\circ}$. The asymptotes need to be anchored on the plot. To do that, a so-called root locus centroid is defined, as a “centre of gravity” of the plot.

Rule 3: The asymptotes are centred on the Real axis at the centroid, described by this equation:

 $\sigma = \frac{\sum poles - \sum zeros}{n-m}$ Equation 10-10

The branches of Root Locus that tend to infinity converge at asymptotic angles, described by this equation:

 $\theta_i = \frac{180^{\circ}\pm k\cdot360^{\circ}}{n-m}$ $k=0,1,...,(n-m-1)$ Equation 10-11

As an example, consider RL shown in Figure 10‑4. Centroid and asymptotes are calculated as follows:

$\sigma = \frac{-10-5-2}{3-0}=-5.67$, $\theta_i = \frac{180^{\circ}\pm k\cdot360^{\circ}}{3-0}=60^{\circ},180^{\circ},-60^{\circ}$

See how this shows on the RL plot in Figure 10‑6.