SOLVE STEP BY STEP FOR ALL

Problem $1(600$ pts$)$: Consider a unity feedback control system with the following open$-$loop plant transfer function Gp$($s$)$ given as$,()=1(+2)(+5)$ The goal is to design a controller to achieve the following design specifications: $1)$ M$.$O$.(\%)$ less than equal $40\mathrm{.}0,2)$ a settling time $(2\%)$ of less than and equal $2\mathrm{.}5$ sec$,$ and $3)$ zero steady$-$state error. a$)$ For a simple proportional $($P$)$ controller with a transfer function of Gc$($s$)=$ K $=50,$ use MATLAB to plot the unit step response and determine the time$-$response characteristics $($e$.$g$.,$ overshoot, settling time, steady$-$state errors$)$ of the closed$-$loop system. b$)$ Use a proper Ziegler$-$Nichol$$s method to determine control gains of a PID controller for the closedloop system in part $($a$)$; plot the unit step response of the PID closed$-$loop system and determine the new time response characteristics. c$)$ Use the MATLAB Root$-$locus editing tools of Control System design toolbox to further tune the PID gains $($found in part $($b$))$ to achieve the design specifications.

Problem $3(200$ pts$)$

For Problem $2,$ determine parameters $\backslash ($ K $\backslash )$ and $\backslash ($ a $\backslash )$ such that the response to the unit$-$step disturbance input $\backslash ((\backslash$mathrm$\{$D$\}(\backslash$mathrm$\{$s$\})=1/\backslash$mathrm$\{$s$\}\backslash )$ while $\backslash (\backslash$mathrm$\{$R$\}(\backslash$mathrm$\{$s$\})=0\backslash ))$ satisfies the following specifications: $\backslash ($ a $\backslash )$ small steady$-$state error $($less than $3\backslash \%)$ and the response attenuate rapidly $($less than $2$ seconds$).$