a$.$ Find the closed$-$loop transfer function using block diagram reduction.

b$.$ Represent each subsystem with a signal$-$flow graph and find the state$-$space representation of the closed$-$loop system from the signal$-$flow graph.

c$.$ Use the signal$-$flow graph found in $($b$)$ along with Mason's rule to find the closed$-$loop transfer function.

d$.$ Replace the power amplifier with a transfer function of unity and evaluate the closed$-$loop percent overshoot, settling time, and peak time for $\backslash ($ K$=5\backslash ).$

e$.$ For the system used for $($d$),$ derive the expression for the closed$-$loop step response.

f$.$ For the simplified model in $($d$),$ find the value of preamplifier gain, $\backslash (\backslash$boldsymbol$\{$K$\}\backslash ),$ to yield $\backslash (15\backslash \%\backslash )$ overshoot. Layout Schematic