The open loop poles of a third order unity feedback system

The open loop poles of a third order unity feedback system are at 0, – 1, – 2. Let the frequency corresponding to the point where the root locus of the system transits to unstable region be K. Now suppose we introduce a zero in the open loop transfer function at – 3, while keeping all the earlier open loop poles intact. Which one of the following is TRUE about the point where the root locus of the modified system transits to unstable region?

1. It corresponds to a frequency greater than K
2. It corresponds to a frequency less than K
3. It corresponds to a frequency K
4. Root locus of modified system never transits

Open loop transfer function of a third order unity feedback system

G(s) =	K
	s (s + 1)(s + 2)

Now zero is introduce at – 3.
Modified transfer function,

G(s) =	K(s + 3)
	s (s + 1)(s + 2)

Closed loop characteristic equation,
1 + G(s) = 0

1 +	K(s + 3)	= 0
	s (s + 1)(s + 2)

s(s + 1)(s + 2) + K(s + 3) = 0
s³ + 3s ² + 2s + Ks + 3K = 0
s³ + 3s² + (K + 2)s + 3K = 0
Routh array corresponding to characteristic equation,

For every value of K > 0 I st column of the array are positive.
Therefore, Root locus of modified system never transits to unstable region.