PROBLEM $1(20\%)$

A pipe of length L$=10$ m is feeding liquid into a tank. The speed of the liquid in the pipe is u$=1$ m$/$s$.$ The level of liquid in the tank is h$.$ The cross$-$section area of the tank is A$=1$ m$2.$

$(5\%)$ a$)$ Calculate the time delay in the pipe between the inlet to outlet for the given velocity. Assume that this delay remains constant for the rest of this problem despite changes in flow rate.

$(5\%)$ b$)$ Model the system, i$.$e$.$ formulate differential equations to calculate h$($t$)$ with respect to inlet velocity u and find the open loop transfer function between h to u$.$

$(10\%)$ c$)$ If the height h is controlled by manipulating the velocity u with a proportional controller with gain Kc$,$ find the closed loop transfer function between the set point changes in h to changes in h$.$

Problem $2(20\%)$

A process given by:

Gp $=1/($s$2$ s $2)$

is controlled by a proportional controller with gain kc$.$

$(10\%)($a$)$ Show a qualitative Nyquist plot $($show only $2-3$ key points and the general shape of the plot for this problem$)$ for k~ $=1.$ Assess the closed$-$loop stability based on the Nyquist criterion.

$(10\%)($b$)$ Based on the Nyquist criterion, compute a range of kc values to obtain closed$-$loop stability.

PROBLEM $3(20\%)$

A first$-$order process is given by

Gp $=1/($s $+5)$

This process is controlled by a PI controller given by:

Gp $=$ Kc$(1+1/$S$)$

$(10\%)$ a$)$ Compute ranges of Kc values for which the closed loop is stable.

$(10\%)$ b$)$ For a controller with gain Kc $=1$ compute the closed loop time response for a unit step change in the set point.