$12\mathrm{.}7$ A process including sensor and control valve can be modeled by a fourth$-$order transfer function:

$G(s)=\frac{1}{(s+1)(0\mathrm{.}2s+1)(0\mathrm{.}04s+1)(0\mathrm{.}008s+1)}$

$($a$)$ Design PID controllers using two design methods:

$($i$)$ A SOPTD model using the model reduction approach proposed by Skogestad $($Section $6\mathrm{.}3)$ and the IMC tuning relation in Table $12\mathrm{.}1.$

$($ii$)$ Repeat part $($i$)$ for the AMIGO method and an FOPTD model obtained by model reduction.

$($b$)$ Evaluate the two controllers by simulating the closed$-$loop responses to a unit step change in a disturbance, assuming that $G_d(s)=G(s)$