This is the "Case Studies" from "CONTROL SYSTEMS ENGINEERING: Seventh Edition." Please go step by step.

Antenna Control: Lag$-$Lead Compensation

For the antenna azimuth position control system case study in Chapter $8,$ we obtained a $25\%$ overshoot using a simple gain adjustment. Once this percent overshoot was obtained, the settling time was determined. If we try to improve the settling time by increasing the gain, the percent overshoot also increases. In this section, we continue with the antenna azimuth position control by designing a cascade compensator that yields $25\%$ overshoot at a reduced settling time. Further, we effect an improvement in the steady$-$state error performance of the system.

PROBLEM: Given the antenna azimuth position control system shown on the front endpapers, Configuration $1,$ design cascade compensation to meet the following requirements: $(1)25\%$ overshoot, $(2)2-$second settling time, and $(3)$ K$\_($v$)=20.$