The characteristic equation of a control system is given by

s(s + 4) (s + 5) (s + 6) + K(s + 3) = 0

The number of asymptotes and the centroid of asymptotes of this control system are

Concept:

Number of asymptotes = |P – Z|

P = number of open-loop poles

Z = number of open-loop zeros

Centroid: 

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

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)

Calculation:

given that s(s + 4) (s + 5) (s + 6) + K(s + 3) = 0

characteristic equation 1 + G(s)H(s) = 0

from the above transfer function 

P = 4 and Z = 1

1) Number of asymptotes = |P – Z| = |4 - 1| = 3

2) Centroid

σ = -12 / 3 = -4

so number of asymptotes and centroid are -3 and (-4, 0)