$1.$ Design the gain value, K$,$ for the feedback control system in the following figure so that the system responds with $10\%$ overshoot and a settling time of $2$ seconds.

$2.$ Implement the previous control system in Simulink$$ and validate the result obtained by measuring the $\%$ of overshoot and stabilization time.

Were the performance requirements stated in the first paragraph achieved?

$3.$ If the answer is NO$,$ mathematically justify why the required behavior was not achieved, and how both performance requirements could be achieved.

$4.$ Validate your justification by implementing it in Simulink