Problem $1(25\%)$

A block diagram is shown in Fig. $1.$

$($a$)$ When $r(t)$ is unit step input, determine $k$ and a to let the parameters of system response $M_o=50\%$ at peak time $t_p=0\mathrm{.}5sec.$ Determine the steady state error $e_{ss}$

$($b$)$ Using the values of $k$ and a obtinaed from part $($a$),$ determine the range of $p$ for zero overshoot response and system stability. Choose any value $p$ in the range to find the steady state error $e_{ss}$

$($c$)$ Briefly comment on the effect of the $p$ variable on the steady state error $e_{ss}.$