In the surge tank shown below:

the flow rate out of the tank is given by:

where $C_v$ is the valve's constant, and $x(t)$ is the valve's opening which is a fixed value between $0$ and $1$ for the open$-$loop case or determined by the controller for the closed$-$loop case $($level control$).$

Develop the open loop dynamic model for this process and simulate it on

a$)$ Simulink, AND

b$)$ Matlab

Draw the open$-$loop block diagram for this system and specify the input $($manipulated variable, $F(t),$ and the disturbance, $F_i(t))$ and the output variable $($controlled variable, $l(t))$ for this system. How many parameters do you need to specify to solve the problem numerically? Choose a control equation to reduce the degrees of freedom of the system to zero$)$ Choose different values for $C_v,$ cross$-$sectional area of the tank, the initial level of water in the tank, and the steady$-$state value of the valve's opening $($usually taken as $\stackrel{}{x}=0\mathrm{.}5)$ and study their effect by looking at the closed loop response.

Using $x(t)$ as the manipulated variable, simulate the closed loop system using a P and a PI controller and study the effect of the controller parameters $(K_c$ and ${}_{\text{I}}$ on the output $($controlled$)$ variable, $l(t).$

Please not that the Matlab $($levelcontrol$.$m and levelMain.m$)$ and Simulink $($PID$\_$level$\_$CL$.$slx and PID$\_$limiter$\_$level$\_$CL$.$slx$)$ files existing in the RESOURCES on OWL site may be downloaded and used for this assignment, OR you may develop your own programs.