A unity feedback system has open-loop transfer function $$G\left( s \right) = {{25} \over {s\left( {s + 6} \right)}}.$$ The peak overshoot in the step-input response of the system is approximately equal to

A unity feedback system has open loop transfer function $$G\left( s \right) = {{K\left( {s + 5} \right)} \over {s\left( {s + 2} \right)}};K \ge 0$$
(a) Draw a rough sketch of the root locus plot; given that the complex roots ofthe characteristic equation move along a circle.
(b) As K increases, does the system become less stable? Justify your answer.
(c) Find the value of $$K$$ (if it exists) so that the damping $$\xi $$ of the complex closed loop poles is $$0.3.$$

Open-loop transfer function of a unity - feedback system is $$$G\left( s \right) = {G_1}\left( s \right).{e^{ - s{\tau _D}}} = {{{e^{ - s{\tau _D}}}} \over {s\left( {s + 1} \right)\left( {s + 2} \right)}}$$$
Given : $$\,\left| {{G_1}\left( {j\omega } \right)} \right| \approx 1$$ when $$\omega = 0.446$$

(a) Determine the phase margin when $${\tau _D} = 0$$
(b) Comment in one sentence on the effect of dead time on the stability of the system.
(c) Determine the maximum value of dead time $${\tau _D}$$ for the closed-loop system to be stable.

$$D\left( s \right) = {{\left( {0.5s + 1} \right)} \over {\left( {0.05s + 1} \right)}}$$ Maximum phase lead of the compensator is