Problem $1$: $(20$ points$)$

Using the root locus method, design a proportional plus integral controller

$G_c(s)=K_P+\frac{K_{\text{I}}}{s}$

for the feedback system in Figure $1$ so that the closed$-$loop system meets the following design specifications.

Zero steady$-$state error to a unit$-$step input.

The peak$-$overshoot to a unit$-$step input must be less than $60\%.$

Figure $1$: Feedback control system using cascade compensation.