PROBLEM $2$ ;

$($a$)$ Find $G_{YD}(0)-1(0,1,$

$u)$

$($b$)$ Let $H(s)=1,C(s)=\frac{K}{s},D(s)=0($it has no effect$),$ and $G(s)=\frac{2}{8s+2}.$ Find the

value of $K$ that results in OS$\%=15\%$ assuming $R(s)$ is a unit step input.

$($c$)$ Use MATLAB to plot the response given your solution to $($b$).$ Does it appear that the

overshoot condition is met? Hint: apply step input to $G_{YR}(s)$ that uses your value of $K.$ A

MATLAB command sequence to perform step response is given below:

num$=[*$vector of coefficients of the numerator polynomial from

hightest power of $s$ to lowest$*]$;

den$=[*$vector of coefficients of the denominator polynomial from

highest power of $s$ to lowestt$]$;

Gtf$=$tf$($num$,$ den$)$;

step$($Gtf$,$ Tfinal$)$; os tfinal is the final time to simulate to

Recall that entering "doc command" on the MATLAB command line will open the help

documentation for that command. Example of plotting the step input for

$G(s)=\frac{s+5}{s^2+2s+8}$ over $2$ seconds:

num $=\left[\begin{array}{cc}1& 5\end{array}\right]$;

den$=\left[\begin{array}{ccc}1& 2& 8\end{array}\right]$;

$G=tf($num$,$ den$)$;

Tfinal$=2$;

step$($G$,$Tfinal$)$;