The transfer function of a second order system is given below:

$G(s)=\frac{K}{s^2+2{}_{n}s+{}_n^2}$

a$)$ Find a region in the complex plane for the poles of the system so that the following conditions are satisfied simultaneously.

i$)t_{p}s$

ii$)t_{2}4s$

iii$)M_{p}0\mathrm{.}2$

b$)$ Show that the system output cannot be made equal to the step input value if $K=1$ and the conditions stated in $($a$)$ are forced.

c$)$ Show that solutions satisfying the conditions in $($b$)$ are possible if $K=4.$ What are the loci of the pole positions in the complex plane in this case?

d$)$ What are the values of the poles which result in a minimum settling time, in addition to the conditions stated in $($a$)?$

e$)$ Find ${}_n,,t_p,t_s$ and $M_p$ for the poles found in $($d$).$