PLEASE SOLVE ACCORDINGLY

QUESTION

Consider the following differential equation that represents the dynamic behavior of a liquid in a cylindrical

tank$,$

$A\frac{dh}{dt}=F_i-c\sqrt[\phantom{2}]{h}$

where $h$ is the height of the liquid level $(m)$ and $F_i$ is the flow rate of the inlet stream $(\frac{m^3}{s})$ of the tank.

Design and tune a feedback automatic control system for controlling the liquid level in the tank. Assume

that;

Liquid level in the tank is your controlled variable

You have air$-$to$-$close pneumatic control valve on the inlet stream pipeline, i$.$e$.$ your manipulated

variable.

Control valve has first order dynamics with Gain $=0\mathrm{.}01\frac{\frac{m^3}{s}}{\%CO}$ and Time constant $=2s.$

Sensor$/$Transmitter couple dynamic is simply a gain

Liquid level Sensor$/$Transmitter couple range is $[0-1]m$

Tank diameter is $D=0\mathrm{.}5m$ and height is $H=1\mathrm{.}5m$

Constant $c=0\mathrm{.}01\frac{\frac{m^3}{s}}{m^{0\mathrm{.}5}}$