QUESTION

Differential equation represents the dynamic behavior of a liquid level in a cylindrical tank is given below,

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

where $h$ is the height of the liquid level $(m),F_i$ is the flow rate $(\frac{L}{s})$ of liquid entering the tank on top and

$c\sqrt[\phantom{2}]{h}$ is the flow rate $(\frac{L}{s})$ of liquid leaving the tank at the bottom.

Followings are given;

Control valve has first order dynamics with $K_v=5\frac{\frac{L}{s}}{\%CO}$ and ${}_v=2s.$

Sensor$/$Transmitter couple dynamic is simply a gain

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

Cylindrical tank diameter is $D=0\mathrm{.}5m$ and height is $H=1\mathrm{.}2m,$ and $A=\frac{D^2}{4}$

Constant $c=12\frac{\frac{L}{s}}{m^{0\mathrm{.}5}}$

a$)(50$ pts$)$ Tune a PI controller, i$.$e$.$ find gain $K_c$ and integral time ${}_{\text{I}},$ for a feedback automatic control

system in order to control the liquid level in the tank.

b$)(50$ pts$)$ Tune a second PI controller; that is; find only gain $(K_c)$ of a PI controller $-$ with ${}_{\text{I}}$ equal to

value that you find in part $($a$)-$ for Phase Margin of $45.$

Optional, for self study:

Obtain closed loop response of your feedback control system for set point change in liquid level from $0\mathrm{.}5m$

to $0\mathrm{.}6m$ by using MATLAB$/$Simulink or by obtaining time domain response of liquid level, $h(t).$ Plot the

response for two PI controllers that you design in part $($a$)$ and part $($b$)$ on the same plot and discuss.

Hints:

Liquid level in the tank is your controlled variable,

You have air$-$to$-$close pneumatic control valve on the inlet stream pipe, i$.$e$.F_i$ is manipulated variable,

We do not have any disturbance in this problem,

Linearize the differential equation,

Write transfer function of process, valve and sensor transmitter couple,

Find numerical values of gain and time constant of the process, valve and sensor$-$transmitter models,

Draw closed loop block diagram,

For part $($a$)$ you can use any one of the method that is presented in the lecture,

Do not forget units of parameter values

Solve accordingly. Do not use any artifical intelligence