Q$-1$ Consider the system shown in figure$-1.$ To improve the performance of the system an extra feedback loop is added to the system which resulted as the system shown in figure$-2.$

a$)$ Determine the transfer function for each system.

$[16$ pts$.]$

b$)$ The value of $\backslash ($ K $\backslash )$ is to be determined so that the damping ratio of the new system is$,\backslash (\backslash$varsigma$\_\{2\}=0\mathrm{.}4\backslash ).$

$[24$ pts$.]$

c$)$ Compare the $\backslash \%-$overshoot, rise time, peak time, $\backslash (2\backslash \%\backslash )$ settling time, and the steady$-$state value of the systems given in figures $1$ and $2.$

$[20$ pts$.]$

$[$Use the Summary and Results Table given below.$]$