Which of the following point is not on the root locus of the system with loop transfer function \(G\left( s \right)H\left( s \right) = \frac{k}{{s\left( {s + 1} \right)\left( {s + 3} \right)}}\)

Concept:

1. Every branch of a root locus diagram starts at a pole (K = 0) and terminates at a zero (K = ∞) of the open-loop transfer function.

2. Root locus diagram is symmetrical with respect to the real axis.

3. Number of branches of the root locus diagram are:

N = P if P ≥ Z

= Z, if P ≤ Z

4. Number of asymptotes in a root locus diagram = |P – Z|

5. Centroid: It is the intersection of the asymptotes and always lies on the real axis. It is denoted by σ.

\(\sigma = \frac{{\sum {P_i} - \sum {Z_i}}}{{\left| {P - Z} \right|}}\)

ΣPi is the sum of real parts of finite poles of G(s)H(s)

ΣZi is the sum of real parts of finite zeros of G(s)H(s)

6. Angle of asymptotes: 

l = 0, 1, 2, … |P – Z| – 1

7. On the real axis to the right side of any section, if the sum of a total number of poles and zeros are odd, the root locus diagram exists in that section.​

Application:

RLD exists between 0 to -1 ; and -3 to -∞.

Therefore, s = -1.5 is not on the root locus of the system.

The root locus diagram can be drawn as shown below.