dynamic process control

In the figures, both the PI diagram and the block diagram for a control loop regulating the level H in a tank by controlling the outlet flow Qu are shown. Variations in the inlet flow Qi act as disturbances to the system. The signals shown in the block diagram are deviations from steady$-$state values.

$($styrer $)$means controls

$($M$$ler $)$means measures

The tank is cylindrical with a cross$-$sectional area of $4$ m$^2$ and a total height of $5$ m$.$ The level transmitter has a measuring range of $05$ m$,$ meaning it converts H $=05$ m to PV $=0->100\%.$ The time constant of the level transmitter is $2$ seconds. The control valve converts the controller signal y $(0->100\%)$ to flow Qu $(0->0\mathrm{.}1$ m$^3/$s$).$ The time constant of the control valve is $10$ seconds. The controller is a proportional controller with gain Kc$.$ The bias adjustment is $50\%($not shown in the block diagram$).$

Answer the following questions:

A$)$ Formulate a model for the tank$\text{'}$s content and explain, based on the PI diagram at the top, why the 'Process' block $($in the bottom block diagram$)$ has the signals, units, constants, and transfer functions it has!

B$)$ How much does the level H $($m$)$ increase after a setpoint $($SP$)$ step of $1\%?$

C$)$ If the SP is increased in a step of $1\%,$ what stationary error $($offset$)$ occurs on the level H $($m$)?$

D$)$ Is the found offset $($above$)$ what you would expect? Explain!

E$)$ Write down the transfer function h$($s$)/$qi$($s$)($for fixed Qu$,$ i$.$e$.,$ qu$($s$)=0).$