Q$4.$ Consider the feedback control system topology given below

where

$\backslash [$

G$($s$)=\backslash$frac$\{1\}\{$s$^\{2\}\}\backslash$& C$($s$)=\backslash$frac$\{$T s$+1\}\{$a T s$+1\}$

$\backslash ]$

a$.$ Find$/$derive the necessary conditions on $\backslash ($ T$,$ a $\backslash ),$ and $\backslash ($ K $\backslash )$ such that closed$-$loop system is BIBO stable.

b$.$ Compute unit$-$step and unit$-$ramp steady$-$state errors of the closed$-$loop system for a $(\backslash ($ T$,$ a$,$ K $\backslash ))$ tuple that makes the closed$-$loop system stable.

c$.$ Let $\backslash ($ a$=2\backslash )$ and $\backslash ($ T$=0\mathrm{.}5\backslash ),$ draw the detailed root locus diagram of the system with respect to $\backslash ($ K $\backslash$geq $0\backslash ).($In the root$-$locus diagram you are also supposed to compute following details provided that they are applicable: centroid of the asymptotes, break away$/$in points $($and associated gains$),$ the point$($s$)$ and corresponding gains when the root locus intersects the imaginary axis$).$

d$.$ This part is composed of two sub$-$questions.

i$.$ Let $\backslash ($ a$=\backslash$frac$\{1\}\{6\}\backslash )$ and $\backslash ($ T$=\backslash$frac$\{1\}\{6\}\backslash ).$ Draw the detailed root locus diagram of the system with respect to $\backslash ($ K $\backslash$geq $\backslash )0.($In the root$-$locus diagram you are also supposed to compute following details provided that they are applicable: centroid of the asymptotes, break away$/$in points $($and associated gains$),$ the point$($s$)$ and corresponding gains when the root locus intersects the imaginary axis$).$

ii$.$ Find a $\backslash ($ K $\backslash$geq $0\backslash )$ value such that maximum percent overshoot of the closed$-$loop system is approximately equal to $\backslash ($ M$\_\{$P O$\}=4\mathrm{.}32\backslash \%\backslash )$ and compute the settling$-$time $(\backslash (\backslash \%2\backslash )$ criteria$)$ of the closed$-$loop system for this gain value.

NOTE: In this homework you may use a calculator$/$computer tool like Matlab or similar to calculate the roots of high$-$order polynomials for e$.$g$.,$ finding the exact locations of the candidate break away$/$in points in the root loci.