Q$\mathrm{.}9.$ Consider the system shown in figure$-1.$ To improve the performance of the system a feedback loop is added to the system which resulted as the system shown in figure$-2.$

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

b$)$ The value of $K$ is to be determined so that the damping ratio of the new system is$,{}_2=0\mathrm{.}4.$

$[24$ pts$.]$

c$)$ Compare the $\%-$overshoot, rise time, peak time, $2\%$ setling time, and the steady$-$state value of the systems given in figures $1$ and $2.[$Use the Summary and Results Table given below.$]$

$[20$ pts$.]$

Figure$-1$

Figure$-2$