$*$ For the Simulink items below, if it is easier, I would appreciate if you could provide a block diagram with equations$/$settings of the model rather than the model in Simulink itself. I understand not everyone has access to the software, but any help to model this system is greatly appreciated. $*$

A mixing tank, which adds a buffer to neutralize an acid stream, behaves as a $2-$input$/2-$output process with the $pH(P^{\text{'}}(s)),$ and the tank level $(H^{\text{'}}(s)),$ as the controlled variables and two feed flowrates and $\{$:$W_2^{\text{'}}(s))$ as the manipulated variables. The process is described by the following transfer functions:

$\left[\begin{array}{c}{P}^{\text{'}}(s)\\ H^{\text{'}}(s)\end{array}\right]=\left[\begin{array}{cc}\frac{0\mathrm{.}4}{(1\mathrm{.}5s+1)(5s+1)}& \frac{-0\mathrm{.}55}{(1\mathrm{.}2s+1)(5s+1)}\\ \frac{0\mathrm{.}05}{5s+1}& \frac{1\mathrm{.}2}{5s+1}\end{array}\right]\left[\begin{array}{c}{W}_1^{\text{'}}(s)\\ W_2^{\text{'}}(s)\end{array}\right]$

$($a$)$ Use an RGA analysis to show that the $1-1/2-2$ pairing is best.

$($b$)$ Use the IMC method to $($ignorantly$)$ tune the two feedback loops independently $($that is$,$ assume $G_{p12}=0$ and that $\{$:$G_{p21}=0).$ Build a Simulink model showing how the two independent feedback loops would respond to a unit step set point change in $H^{\text{'}}(s),$ if there were no process interactions $($i$.$e$.,$ in your model set $G_{p12}=0$ and that $G_{p21}=0).$ Do not include any set point change for the $pH$ control loop. Choose ${}_c=1$ for both controllers; Use a stop time of $20$ minutes for the simulation.

$($c$)$ Build a second Simulink model showing what happens when the process interactions are non$-$ zero $($i$.$e$.,$ include the correct transfer functions for $G_{p12}$ and $G_{p21}),$ and the level set point is a unit step change. Do not include any set point change for the $pH$ control loop. Use the same settings for the controllers that you used in part $2b.$ Show the output of the $pH$ and the level set point is a unit step change. Do not include any set point change for the pH control loand level as functions of time. By how much does the pH deviate when the level set point is changed? What is the approximate settling time for the pH$?$

$($d$)$ Tune a static decoupling controller designed to mitigate the effects of changes in W $2($s$)$ on P $($s$).$ Add this to your Simulink model, and show the response of the pH to a unit step in change in the level. Again, do not include any set point change for the pH control loop. You do not need to include the second decoupling controller, T$21.$ By how much does the pH deviate when the level set point is changed, when the static decoupler is used? What is the approximate settling time?

$($e$)$ Replace the static decoupler with a dynamic decoupler and repeat item $($d$)$ above.

$($f$)$ Based on the gains, suggest whether you think the other decoupling controller is necessary. If you were to include this controller, what would its transfer function be$?$ Does it have a large gain or a small gain? Does it have large or small time constants?