A system in unity feedback configuration, has the transmittance

\[
G(s)=\frac{\mathrm{K}}{s(s+3)(s+9)}
\]

(a) What value of \(\mathrm{K}\) will force the system to exhibit peak overshoot of \(20 \%\) to a step input?

(b) For the value of \(\mathrm{K}\) found in (a), find the settling time, \(t_{s}\) and velocity error coefficient, \(\mathrm{K}_{v}\).

(c) Design a cascade compensator such that the following specifications are achieved.

(i) Peak overshoot \(\leq 15 \%\)

(ii) Settling time \(\leq 2 / 5\) of that found in part (b)

(iii) \(\mathrm{K}_{v} \geq 20\)