Please include all calculations and graphs. Do not just leave a formula or write an answer without calculations. If you are not going to do these, please do not answer this question. This question is related to the field of Control engineering

Problem$-1.$ For the control system given,

$($a$)$ Sketch, by hand, its root locus for

$G_c(s)=K.$ When sketching the

root locus, if necessary, make use of

the asymptotes finding ${}_a$ and ${}_a$

that are the intersecting point and

angles with the real axis, respectively using the following formula,

${}_a=\frac{{\displaystyle \underset{?}{\overset{?}{}}}\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{p}\text{o}\text{l}\text{e}\text{s}-{\displaystyle \underset{?}{\overset{?}{}}}\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{z}\text{e}\text{r}\text{o}\text{s}}{\text{\#}\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{p}\text{o}\text{l}\text{e}\text{s}\frac{-}{\text{\#}}\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{p}\text{z}\text{e}\text{r}\text{o}\text{s}}$ and ${}_a=\frac{(2k+1)}{\text{\#}\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{p}\text{o}\text{l}\text{e}\text{s}\frac{-}{\text{\#}}\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{p}\text{z}\text{e}\text{r}\text{o}\text{s}},$ where $k=0,+-1,+-2,$dots

$($b$)$ If the root locus intersects the $j-$axis, find the values of poles $s_{1,2}$ and gain $K$ at the crossing

points. Then write the range of gain $K$ making the system stable.

$($c$)$ Determine the type of the system $($type $0,$ type $1,$ type $2,$ etc.$).$

$($d$)$Find the steady$-$state error for this P$-$controlled system for $G_C(s)=162,$ when a unit step and

then ramp inputs are introduced.

$($e$)$ It is determined that the root locus intersects the $10\%$ overshoot line $($which corresponds to a

damping ratio of $=0\mathrm{.}59)$ for $K=162,$ at $s=-2\mathrm{.}62+j3\mathrm{.}67.$ Now, justify the second order

approximation by finding the third closed$-$loop pole at this gain, then give an estimation of the

settling time using $T_s$~$=\frac{4}{{}_n}.$

$($f$)$ Design a PID controller to substitute for $G_C(s)$ so that the system's response becomes twice as

faster with the same maximum overshoot and produces no steady$-$state error to a step input.

Hint: For a simple PID controller, design a PD controller to substitute for $G_c(s)$ so that the

settling time is reduced $2$ times while keeping the same maximum overshoot. Then add a PI

controller to meet the steady$-$state error requirement. For simplicity, you may use the same gain

you found for PD controller, also you may select the zero of PI controller at $-0\mathrm{.}1.$

$($g$)$Plot the responses of the P$,$ PD and PID controlled system on the same plane for comparison.

$($h$)$ Determine the constant of proportionality, $K_p,$ the reset $($integral$)$ time $T_i$ and the rate $($derivative$)$

time $T_d$ values of the PID controller assuming that the controller has the following transfer

function,

$G_{\frac{?}{PI}D}(s)=\frac{80(s+0\mathrm{.}4)(s+10)}{s}$

$($i$)$ Realize the same PID$-$controller given in $($h$)$

i$.$ as a block diagram where three blocks for P$,$ I, D actions are connected in parallel.

Remember to determine and write the constants of proportion, integral and derivative $,$

$K_i$ and $K_d)$

ii$.$ physically with a circuit having two OPAMPs. Select the first capacitor, i$.$e$.$ the one at the

input, as $C_1=10F,$ also select the second OPAMP's impedances as $Z_3=Z_4=10k$NOTE:NNN