Consider the two$-$tank system below. We want to control the liquid level $h_2$ of tank $2$ by manipulating flow rate $F_1$ through a proportional controller. Initially, the system is at steady$-$state with $F_1=1f\frac{t^3}{m}in$ and $h_1=h_2=3ft.$ The two tanks have the same cross$-$sectional areas of $5f{t}^2.$ Determine the values of the controller gain which

$($I$)$ Produce a critically damped response.

$($II$)$ Produce an underdamped response with decay ratio of $\frac{1}{4}$ for $h_2.$

$($III$)$ Describe and plot the dynamic response of liquid level $h_1$ in tank $1,$ for a unit step change in the set point of $h_2.$

Required Steps

Develop the mathematical model for the system.

Draw the block diagram of the closed loop system with $F_1$ as manipulating variable.

Derive the corresponding closed$-$loop responses to load or set point changes.

Derive a general expression for the closed$-$loop static gain.

Identify the corresponding closed$-$loop transfer functions for changes in the load $(G_{\text{l}\text{o}\text{a}\text{d}})$ or set point $($Gsp$).$

Assumptions

$(1)$ Flow rates of the free streams are proportional to the liquid levels.

$(2)$ Transfer functions of the measuring devices and control valves are equal to unity.