techniques.

$\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]=\left[\begin{array}{cc}\frac{Q_{11}(10\mathrm{.}6s+1)e^{-s}}{(3\mathrm{.}08s+1)(4\mathrm{.}81s+1)(34\mathrm{.}4s+1)}& \frac{0\mathrm{.}27e^{-s}*}{(1\mathrm{.}283s+1)(1\mathrm{.}25s+1)}\\ 0G_{21}& \frac{-15e^{-0\mathrm{.}42s}}{c_2(2s+1)}\end{array}\right]*\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]$

The unit of time is minutes, and the process variables are $($in termg of deviations from their respective steady$-$state values$)$:

A PID controller is designed to stabilize the system using the Cohen and Coon tuning rules with the following settings:

Loop $1.K_c=2\mathrm{.}09,{}_{11}=5\mathrm{.}48$;${}_{D1}=0\mathrm{.}82$

Loop $2$: $K_{c2}=-0\mathrm{.}385$;${}_{12}=-12\mathrm{.}3$;${}_{D2}=-0\mathrm{.}02$

$y_1=$ Percent paralysis $($a measure of muscle relaxation$)$

$y_2=$ Mean Arterial Pressure; $(mmHg)$

$u_1=$ Injection rate of Atracurium $($drug for producing muscle relaxation$)$; $(m\frac{l}{m}in)$

$u_2=$ Injection rate of llloflurane $($drug for inducing unconsciousness$)(m\frac{l}{m}in)$

a$)$ Design astatic decoupler for this process

b$)$ Design a dynamic decoupler

c$)$ Simplify your result down to a lead$/$lag element

d$)$ Implement this decoupler along with the single$-$loop controllers designed in part $($c$)$ and obtain the closed$-$loop system response to step change of $-10$inmmHg in the mean arterial pressure set$-$point.

e$)$ Does the decoupler provide an improvement in closed$-$loop performance?