The temperature is measured using a sensor and a transmitter, TT $22,$ that sends the

measurement to a corresponding controller, TC $22,$ that compares the actual temperature

with a desired temperature set point, SP$.$ The controller automatically opens or closes a

valve to allow or prevent the flow of steam to change the temperature in the tank. The

corresponding block diagram for this system is shown in the figure part $($b$).$

Assume the following transfer functions:

G$\_($v$)($s$)=(0\mathrm{.}02)/(4$s$+1)$

G$\_($p$)($s$)=(70)/(50$s$+1)$

H$($s$)=(1)/(12$s$+1)$

$($a$)$ Assuming Gc$\_($c$)($s$)=$K$,$ find the value of K that will result in a dominant pole with $\backslash$zeta $=0\mathrm{.}7.$

Obtain the corresponding T$\_($s$).$

$($b$)$ Design a lead compensator to obtain the same damping factor as Part $($a$)$ but with a

settling time $20\%$ smaller.

$($c$)$ Verify your results with MATLAB simulation.

Hint: the value of GH$($s$)$ to use in the rlocus command is:

GH$($s$)=(1\mathrm{.}4)/(2400$s$^(3)+848$s$^(2)+66$s$+1)$

Provide the derivation to obtain the GH$($s$)$ shown in this Hint. Problem $3($Chapter $9)$:

Lead controller:

The figure part $($a$)$ shows a heat$-$exchanger process whose purpose is to maintain the temperature of a liquid at a prescribed temperature.

$($a$)$