Consider the system shown in Figure $($a$).$ The damping ratio of this system is $0\mathrm{.}158$ and the undamped natural frequency is $3\mathrm{.}16ra\frac{d}{sec}.$ To improve the relative stability, we employ tachometer feedback. Figure $($b$)$ shows such a tachometer$-$feedback system. $($Note: A tachometer is a sensing device which measures speed$)$

Determine the value of $K_h$ so that the damping ratio of the system is $0\mathrm{.}5.$ Draw an approximate unit$-$step response curves of both the original and tachometer$-$feedback systems. Also draw the error$-$versus$-$time curves for the unit$-$ramp response of both systems. $($Note: you can use hand sketch, but feel free to use Matlab or Simulink to submit more accurate results$)$

$($a$)$

Figure: $($a$)$ Control system; $($b$)$ control system with tachometer feedback.