A liquid level control system is shown in the diagram below $($same as Figure $11\mathrm{.}22,$ page $186$ of the Textbook$)$:

Pigure $11\mathrm{.}2$ exit line.

The time constants of the level transmitter and the control valve are negligible. The crosssectional area of the tank is $0\mathrm{.}2m^2.$ At the initial steady state $q_1=10$ liters$/$minute$,q_2=10$ liters $\frac{?}{?}$ minute, $q_3=20$ liters $\frac{?}{?}$ minute and liquid level $(h)=0\mathrm{.}4m.$ Steady state gains for the transmitter and the valve are $K_m=20m\frac{A}{m},K_v=-2$ liters $\frac{?}{mi}\frac{n}{m}A.$ Controller is proportional only with $K_c=5.$ Set point for liquid level is set at $h_{ss}=0\mathrm{.}4m$

$($a$)$ With $q_1$ as the disturbance variable, $h$ as the controlled variable and $q_3$ as the manipulated variable, draw a block diagram for the feedback control loop for this system. Show all the transfer functions.

$($b$)$ At $t=0,$ the set point for liquid level is changed by a step increase to $0\mathrm{.}5m.$ Derive an expression for the change of liquid level $(h)$ as a function of time. What is the new steady state liquid level? Is there an offset?

Also, derive an expression for the variation of $q_3$ as a function of time. What is the new steady state value of $q_3?$