If one of the two general poles of an underdamped second-order system is given by

\[s=-\sigma_{d}+j \omega_{d}\]

(a) Show that the natural frequency \(\omega_{n}\) is given by

\[\omega_{n}=\sqrt{\left(\omega_{n}^{2}+\sigma_{d}^{2}ight)}\]

(b) Also show that

\[\begin{aligned}\cos \theta & =\zeta \\t_{p} & =\frac{\pi}{\omega_{d}} \\t_{s} & =\frac{4}{\sigma_{d}}\end{aligned}\]

where \(\zeta\) is the damping ratio, \(t_{p}\) is the peak time and \(t_{s}\) is the settling time.

(c) For such a second-order system, consider the case where the plant transfer function is given by

\[G(s)=\frac{1}{s(s+6)}\]

Design the controller \(K\) so that the system responds with \(15 \%\) overshoot.