P$2.(20$ marks$)$ 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 in the figure at right, with the plant transfer function relating heading changes to rudder deflection in radians. The differential equation that relates the heading angle to rudder angle for the ship without feedback are given as:

a$.$ Find the transfer function from output heading $\backslash (\backslash$psi $\backslash )$ to input steering angle $\backslash (\backslash$delta $\backslash ).$

b$.$ This control system uses simple proportional feedback with the gain of unity. Is the closed$-$loop system stable? $($Hint: use Routh Hurwitz criterion$)$

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