Consider a control system which has an open$-$loop transfer function equal to:

$G(s)H(s)=\frac{K}{(s+2{)}^2}$

Determine the roots of the characteristic equation of the system and sketch the root locus $($indicate the roots for the gain $K=0,1,4,9).$

Is this system stable? Justify.

Determine the value of gain $K$ which will provide a damping ratio of $0\mathrm{.}5.$

Assume that the system experiences a disturbance in set point, ${}_2(t),$ at $t=0$ equal to the delta function, $(t)($defined by $(t)=0$ for $t0$ and $(t)=$ for $t=0).$ What is ${}_1(s),$ the Laplace transform of ${}_i(t)?$

What is the corresponding open$-$loop control system output, ${}_0(s)($i$.$e$.$ in transformed variable $s)?$

Calculate ${}_d(t),$ the inverse Laplace transform of ${}_c(s)$

Determine the time at which the output is maximum and draw a sketch depicting the variation of ${}_o(t)$ for $t>0$ in response to the disturbance in set point.

Bode, Nyquist & root locus plots

Consider the single loop motor driven servo of Fig. $4\mathrm{.}35$ below.

Write the open$-$loop and closed$-$loop transfer functions.

Consider the case when $K_m=1,{}_1=2$ seconds, ${}_2=4$ seconds, using MATLAB find the poles.

Using MATLAB plot the Bode.

Using MATLAB plot Nyquist.

Using MATLAB Generate the root locus for $K=0\mathrm{.}34\mathrm{.}3,4\mathrm{.}40.$ While $on$ the root locus curve, right click $to$ display window which shows gain $(K),$ poles, damping, overshoot, and frequency. Show the points $K=0\mathrm{.}3$ and $0\mathrm{.}03on$ the root locus and, again, discuss stability results.

Are the stability results the same for $4\mathrm{.}3,4\mathrm{.}4,$ and $4\mathrm{.}5?In$ all cases, make sure $to$ justify your answers

$In$ all cases, make sure $to$ justify your answers.