Pleas show all working step by step $1)$ Convert the following Laplace transfer functions to their discrete time equivalent $($perform the hand calculations$)$ assuming that the transfer function is a model of the process dynamics and that there is a zero order hold device,

$\frac{y(s)}{u(s)}=\frac{8e^{-6s}}{30s+1}$

$\frac{y(s)}{u(s)}=\frac{2e^{-10s}}{40s+1}$

$2)$ The following continuous process,

$G_p(s)=\frac{3e^{-s}}{10s+1}$

is to be controlled with a digital proportional control law. The controller is to operate with a sample time $T_s=1$ and the z$-$transfer function for the controller is$,$

$\frac{U(z)}{E(z)}=k_c$

where $k_c$ is the controller gain. Given the simplified closed loop block diagram,a$)$ Derive the closed loop transfer function describing the response of the output

given a change in the set point

b$)$ If the controller gain

use the final value theorem to determine the numerical value of the offset given a unit step change in the set point

c$)$ Derive a transfer function describing the response of the manipulated variable

given a change in the set point

d$)$ Use the initial value theorem to determine the first $($initial$)$ value of the manipulated variable when the closed loop system has been subjected to a unit step change in set point. What happens to this value if the value of the controller gain is doubled?$3.$

The following continuous process,

G$\_($p$)($s$)=(3$e$^(-$s$))/(10$s$+1)$

is to be controlled with a digital proportional control law. The controller is to operate with a sample time T$\_$s $=2$

and the z$-$transfer function for the controller is$,$

$($U$($z$))/($E$($z$))=$k$\_($c$)$

where k$\_($c$)$ is the controller gain.

a$)$ Derive the closed loop transfer function describing the response of the output Y$($z$)$ given a change in the set point SP$($z$)$

b$)$ If the controller gain k$\_($c$)=2$ use the final value theorem to determine the numerical value of the offset given a unit step change in the set point

c$)$ Derive a transfer function describing the response of the manipulated variable U$($z$)$ given a change in the set point

d$)$ Use the initial value theorem to determine the first $($initial$)$ value of the manipulated variable when the closed loop system has been subjected to a unit step change in set point.

e$)$ Use the final value theorem to determine the final value of the manipulated variable when the closed loop system has been subjected to a unit step change in set point.