How to draw Root Locus diagram?

1) There are 4 poles and no zeros. P=4 and Z=0. N=4. There will be 4 branches in root locus

2) All 4 branches will start from open loop poles and terminate at infinity.

3) The branches will terminate at infinity along straight line asymptotes whose angles are determined by

where q=0..N-1

Angle of asymptotes =45, 135,225,315

4) The asymptotes meet the real axis at Centroid

Centroid= Sum of real parts of poles-sum of real parts of Zeros/(P-Z)=0-1-2-3/4=-1.5

5) Breakaway point and Break in  point  is calculated by solving dk/ds=0

Solving this gives s=-1.5, -0.381, -2.619. (Note : -1.5 is not valid breakaway point point because it doesnt lies on root locus)

6) The value of k and the point at which the root locus branch crosses the imaginary axis is determined by applying Routh Criterion to the characteristic equation. The roots at the intersection point are imaginary.

 s^4 | 1     11          k

 s^3 | 6       6            0

 s^2 | 10         k          0   

 s^1 | (60-6k)/10   0       

 s^0 |   k                        

60-6k=0 --> kmax=10

Auxiliary  equation : 10s^2+k=0

At k=10, s=+j and s=-j                  

7) Root Locus 

  

This is the root locus diagram for the given transfer function

Summary

The system is absolutely stable for 0<k<10 and at k=10, the system is marginally stable and for k>10, system is unstable.