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$$ Consider the multivariable PI controller described by the following transfer matrix:

$G_c(s)=K_c+K_l\frac{1}{s}$

where $K_r$ and $K_{\frac{?}{?}}$are full $nn$ matrices respectively for the proportional and integral parts of the controller. For the following distillation column model:

$\left[\begin{array}{c}{Y}_1(s)\\ Y_2(s)\end{array}\right]=\left[\begin{array}{cc}\frac{-0\mathrm{.}6e^{-1.ss}\frac{\%}{\%}}{4\mathrm{.}75s+1}& \frac{0\mathrm{.}28e^{-4.s}\frac{\%}{\%}}{12\mathrm{.}75s+1}\\ \frac{-0\mathrm{.}27e^{-2.s}\frac{\%}{\%}}{4\mathrm{.}25s+1}& \frac{0\mathrm{.}52\frac{\%}{\%}}{14s+1}\end{array}\right]\left[\begin{array}{c}{U}_1(s)\\ U_2(s)\end{array}\right]$

The unit of time is minutes, and the dimensionless process variables, in term of percent of range, are defined as follows:

$Y_1(s)=$ top tray temperature

$Y_2(s)=$ Bottom tray temperature

$U_1(s)=$ Reflux flow set$-$point

$U_2(s)=$ Reboiler steam valve position

Implement on this process the multivariable PI controller with the following parameters:

$K_c=\left[\begin{array}{ccc}-2\mathrm{.}8923& 1\mathrm{.}4336& 8\mathrm{.}4310\end{array}\right]$

$-3\mathrm{.}9732$

$K_l=\left[\begin{array}{ccc}-0\mathrm{.}528& 0\mathrm{.}282& 0\mathrm{.}612\end{array}\right]$

$-0\mathrm{.}276$

Using MATLAB$/$Simulink$,$ obtain the closed system response to a change of $-2\%$ in the top tray temperature.

Note that the controller is $G_c(s)$ has the following structure:

$G_c(s)=\left[\begin{array}{cc}{G}_{c11}& G_{c12}\\ G_{c21}& G_{c22}\end{array}\right]$

where every element is given as $G_{cj}=k_c+k_l,\frac{1}{s}.$