Consider a general closed-loop unity feedback control system shown in Figure 5.34, where \(K=1\) is the controller gain and \(G(s)\) is the plant Transfer Function model given by

\[G(s)=\frac{s^{2}+s+a}{s(s+1)\left(s^{2}+s+1.25ight)\left(s^{2}+s+4.25ight)}\]

(a) Plot the root locus for the plant as a varies from 1.25 to 4.25 .

(b) Find the exact value of a for which the pair of poles that asymptotically approach zero changes.

R(s) + Y(s) K G(s) Figure 5.34 Closed-loop unity feedback block diagram.