Consider the transfer function:

$G(s)=\frac{1}{(10s+1)(8s+1)(4s+1)}$

$($a$)$ Find an FOPTD approximation and an SOPTD approximation to $G(s),$ using the Skogestad half rule.

$($b$)$ Find and plot the response of the orginal function and both approximations, to a unit step input. When plotting, plot over the domain $t=[0,75].$

$($c$)$ Discuss the tradeoff between accuracy and simplicity of the approximations.