PROBLEM $1(20\%)$

A thermometer with a time constant of $0\mathrm{.}2$ min is immersed in a temperature bath and after the thermometer comes to equilibrium with the bath, the bath temperature is increased linearly with time at the rate of $1\backslash$deg C $/$min$.$

$10\%($a$)$ what is the difference between the indicated temperature and bath temperature $($i$)0\mathrm{.}1$ min $($ii$)10$ min after the change in temperature is applied?

$5\%($b$)$ What is the maximum deviation between the indicated temperature and bath temperature and when does it occurs?

$5\%($c$)$ Plot the forcing function and the response on the same graph. After a long enough time by how many minutes the response will lag after the input?

PROBLEM $2(20\%)$

The input $($e$)$ to a PI controller is as follows:

for $0<=$ t $<1$ e $=0\mathrm{.}5,$

for $1<=$ t $<2$ e $=0$

for $2<=$ t $<3$ e $=0\mathrm{.}5$

for t $>=3$ e$=0$

$10\%$ a$)$ Find the Laplace transform E$($s$)($transform of e$($t$)).$

$10\%$ b$)$ Find and plot the output of the controller with proportional gain Kc $=2$ and reset time

$($integration constant$)$ TT $=0\mathrm{.}5$ min.