$3\mathrm{.}2$: The transfer function model for a solid$-$fuel boiler has been given as:

$\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]=\left[\begin{array}{cc}\frac{-e^{-2s}}{10s+1}& \frac{-1}{10s+1}\\ 0& \frac{e^{-10s}}{60s+1}\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]$

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

$y_1=$ Normalized boiler underpressure; $y_2=$ Percent flue gas $O_2$ content

$u_1=$ Damper position $(\%)$; $u_2=$ Secondary blower motor speed $($rpm$)$

a$)$ Design a dynamic decoupler for this process

b$)$ Comment on whether or not the decoupler is realizable

$(1)$

Now augment the process with a matrix of additional delays:

$(1)$

$D(s)=\left[\begin{array}{cc}1& 0\\ 0& e^{-2s}\end{array}\right]$

c$)$ Repeat the design of the dynamic decoupler of part $($a$).$ Is this decoupler now realizable? $(2)$

$3\mathrm{.}2$: The transfer function model for a solid$-$fuel boiler has been given as:

$\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]=\left[\begin{array}{cc}\frac{-e^{-2s}}{10s+1}& \frac{-1}{10s+1}\\ 0& \frac{e^{-10s}}{60s+1}\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]$

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

$y_1=$ Normalized boiler underpressure; $y_2=$ Percent flue gas $O_2$ content

$u_1=$ Damper position $(\%)$; $u_2=$ Secondary blower motor speed $($rpm$)$

a$)$ Design a dynamic decoupler for this process

b$)$ Comment on whether or not the decoupler is realizable

Now augment the process with a matrix of additional delays:

$(1)$

$(1)$

$D(s)=\left[\begin{array}{cc}1& 0\\ 0& e^{-2s}\end{array}\right]$

c$)$ Repeat the design of the dynamic decoupler of part $($a$).$ Is this decoupler now realizable? $(2)$