Suppose you are to design a unity feedback controller for a first$-$order plant depicted in Fig. $3\mathrm{.}56.($As you will learn in Chapter $4,$ the configuration shown is referred to as a proportional$-$integral controller.$)$ You are to design the controller so that the closed$-$loop poles lie within the shaded regions shown in Fig. $3\mathrm{.}57.$

Figure $3\mathrm{.}56$

Unity feedback system for Problem $3\mathrm{.}30$

Figure $3\mathrm{.}57$

Desired closed$-$loop pole locations for Problem $3\mathrm{.}30$

$($a$)$ What values of ${}_n$ and $$ correspond to the shaded regions in Fig. $3\mathrm{.}57?($A simple estimate from the figure is sufficient.$)$

$($b$)$ Let $K_{}==2.$ Find values for $K$ and $K_{\text{I}}$ so the poles of the closed$-$loop system lie within the shaded regions.

$($c$)$ Prove that no matter what the values of $K_{}$ and $$ are, the controller provides enough flexibility to place the poles anywhere in the complex $($left$-$half$)$ plane.