a$.(20$ pts$)$ Assuming second$-$order dynamics, calculate the process overshoot, decay ratio, and period

of oscillation and write an approximate transfer function in the form of

$\frac{R^{\text{'}}(s)}{P^{\text{'}}(s)}=\frac{K}{{}^2{s}^2+2s+1}$

where $R^{\text{'}}$ is the instrument output deviation $($mm$),P^{\text{'}}$ is the actual pressure deviation $($psi$),$ and find

the steady state gain $(K),$ the natural period of oscillation $(),$ and the damping factor $().$

b$.(10$ pts$)$ Write an equivalent differential equation model in terms of actual $($not deviation$)$ variables.