$(12$ marks$)$ Consider the system of two tanks as depicted in Fig. $1.$ The flow rate $F_2$ is proportional to

the height of the liquid in the first tank $($i$.$e$.F_2=h).$ You would like to control the liquid level of tank

$2(h_2)$ using a valve, that controls the flow rate $F_3$ using a PI controller with gain $K_c$ and time constant

${}_1.$

Figure $1$: System of two tanks controlled by a valve

Draw the block diagram of the closed loop system.

Derive the close$-$loop transfer functions for changes in the load or set$-$point. Assume that the transfer

functions of the measuring devices and control valves are unity.

How will $h_2$ vary as a function of time for a step change in the set point and the load $?$ What are

the closed loop static gains $?$ Draw qualitative plots for each response.