Problem $\backslash (2(600\backslash$mathrm$\{$pts$\})\backslash )$ : Consider the PI control system in Figure $1.$ a$)$ if the plant transfer function is approximated by $\backslash (0\mathrm{.}4\backslash$mathrm$\{$e$\}^\{-0\mathrm{.}15\}/(0\mathrm{.}5\backslash$mathrm$\{$~s$\}+1)\backslash ),$ use the Ziegler$-$Nichols's Tuning recommendations and obtain the parameters $\backslash ($ K $\backslash )$ and $\backslash ($ a $\backslash )$; b$)$ Use an analytical approach and determine parameters $\backslash ($ K $\backslash )$ and $\backslash ($ a $\backslash )$ such that the response to the unit$-$step reference input satisfy the following specifications: exhibits a maximum overshoot of $\backslash (20\backslash \%\backslash )$ or less and a settling time of $2$ sec ; and c$)$ use MATLAB $($Live script$)$ to obtain the unitstep response of the reference inputs for part $($a$)$ and part$($b$),$ determine the time characteristics of both responses $($Overshoot and settling time$)$ and compare the results.

Problem $3(200$ pts$)$

For Problem $2,$ determine parameters $\backslash ($ K $\backslash )$ and a 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: a small steady$-$state error $($less than $3\backslash \%)$ and the response attenuate rapidly $($less than $2$ seconds$).$