Consider the feedback control system shown below where \(K \geq 0\) (Figure 5.8).

(a) Find the root locus of the system with respect to the controller gain \(K\).

(b) For what range of \(K\) is the system stable?

(c) Find the value of the gain \(K\) that will allow the system to have a root at

\[s=-\frac{1}{2}+j \frac{\sqrt{3}}{2}\]

(d) Find the controller gain \(K\) that will be required to give the system a damping ratio of \(\xi=0.5\).

R(s) Y(s) K s(s+1) Figure 5.8 Root locus diagram.