A chemical plant produces an important liquid product which is stored in a tank. The liquid level in the tank is measured, and the level transmitter output is sent to a feedback controller that controls the liquid level by adjusting volumetric flow rate. A second inlet flow rate is the disturbance variable. Assuming the following hold: The liquid density and the cross$-$sectional area of the tank A are constant; The flow$-$head relation is linear, q$3=$ h$/$R; The level transmitter, $1/$P transducer, and pneumatic control valve have negligible dynamics; An electronic controller with input and output in $\%$ is used $($full scale $=100\%).$

a$)$ Propose a control system for the storage thank using an appropriate diagram. The control objective must be specified. $[5$ marks$]$

b$)$ Develop a close loop transfer function for set point changes If an integral controller is used. $[5$ marks$]$

c$)$ Develop the closed$-$loop transfer function for disturbance changes with integral control. $[5$ marks$]$

d$)$ Propose a block diagram for the control system. $[5$ marks$]$

e$)$ Discuss the process that could lead to offsets and instability in the control system and the appropriate strategies that could be implemented to remove offsets and eliminate the instability. $[10$ marks$]$

f$)$ Based on the feedback control system for a closed$-$loop response y$($t$),$ considering a unit step with the following transfer functions:

Gc $=$ Kc ; Gv $=1/3$s$+1$ ; Gp $=$ Gd $=1/4$s$+1$ ; Gm $=1/$s$+1$

Discuss the stability status of the control system given that $=10$