Problem $5.(3$ points$)$ In this problem we will use methods to design proportional $($P$),$ proportional$-$integral $($PI$),$ proportional$-$derivative $($PD$),$ and proportional$-$integral$-$derivative $($PID$)$ controllers for a vectored thrust aircraft Use the following transfer function to represent the dynamics from the lateral input to the roll angle of the aircraft:

$G(s)=\frac{r}{J{s}^2+cs+mgl}$

With the following parameters

$g=9\mathrm{.}8\frac{m}{s^2},m,=1\mathrm{.}5kg$

$l,=0\mathrm{.}05m,J=0\mathrm{.}0475kg{m}^2$

We wish to design a feedback controller that tracks a given reference input with the following specifications:

steady$-$state error of less than $3\%$;

frequency response bandwidth of $20ra\frac{d}{sec}$;

step response overshoot of less than $15\%($can show this corresponds to a phase margin of approximately $45).$