$(+10$ points$)$ Consider a submarine which aims to maintain a constant depth, $z.$ The depth

is dependent upon the submarine pitch angle $,$ the pitch rate $q,$ the heave velocity $w,$ and

the stern hydroplane angle ${}_S.$

The equations of motion of the submarine are approximated by

$z^{}(t)=w(t)-5(t)$

${}^{}(t)=q(t)$

$w^{}(t)=q(t)-\frac{1}{2}w(t)-\frac{1}{5}u(t)$

$10q^{}(t)=-q(t)-u(t)$

$($a$)$ Find the transfer function for the system with input ${}_S(s)$ and output $Z(s).$ Presume

all initial conditions are $0.$ Be sure that your answer is simplified, i$.$e$.,$ no polynomial

fractions in either the numerator or denominator.