Question $3$:

Consider the following system: $G(s)=\frac{1}{(s+2)(s-1)}$

$1-$ Analyze the open$-$loop system

$2-$ What is the optimal controller, in terms of effective cost, that can achieve a $5\%$ of overshoot?

$3-$ Design this controller $($find the control parameters$).$

$4-$ Analyze the system in closed$-$loop $($Determine Ts$,\%$OS$,$ and $es{s}_s$

$5-$ Design a PD controller so that the settling time is reduced by a factor of $2.$

$6-$ Use MATLAB to simulate the system with the PD controller in a closed$-$loop for unit step response that achieves the desired performance.

$7-$ Realize the controller designed in question $4$ with operational amplifiers.