Provide MATLAB Code when needed.

Automatic ship steering is particularly useful in heavy seas when it is important to maintain the ship along an accurate path. Such a control system for a large tanker is shown with the plant transfer function relating heading changes to rudder deflection in radians.

$($a$)$ Write the differential equation that relates the heading angle to rudder angle for the ship without feedback

$($b$)$ This control system uses simple proportional feedback with the gain of unity. Is the closed$-$loop system stable as shown?

$($c$)$ Is it possible to stabilize this system by changing the proportional gain from unity to a lower value?

$($d$)$ Design a control system in MATLAB that can achieve $10\%$ max overshoot and rise time less than $20$sec$.($all options are open$)$