Problem $1$ Consider the unity$-$feedback system shown in Figure $(1).$ The

plant transfer function is$,$

G$($s$)=($K$)/(($s$+1)($s$+4)($s$+6))$

Design a lag network compensator that will not change the position of the

uncompensated dominant poles that result in $10\%$ overshoot but yields a

static position error constant K$\_($p$)=20.$ Simulate the uncompensated and

compensated system to verify the effects of your compensator using

MATLAB.

Problem $2$ Consider the unity$-$feedback system shown in Figure $(1),$ with

G$($s$)=($K$)/($s$($s$+20)($s$+40))$

The system is operating at $20\%$ overshoot. Design a lead compensator to

decrease the settling time by a factor of $2,$ without affecting the percent

overshoot. Simulate the uncompensated and compensated system to verify

the effects of your compensator using MATLAB.

do all work and show matlab code as well as showing all designs that are required