Problem $1$:

Consider the closed$-$loop control system shown in the figure.

Using MATLAB,

$1\mathrm{.}1.$ Derive the open$-$loop transfer function of the system.

$1\mathrm{.}2.$ Plot a root$-$locus diagram for the closed$-$loop system.

$1\mathrm{.}3.$ Determine the range of the gain K required for stability.

$1\mathrm{.}4.$ Determine the value of the open$-$loop gain K such that the damping ratio $$ of a pair of dominant complex$-$conjugate closed$-$loop poles is $0\mathrm{.}55.$

For this design point,

$2\mathrm{.}1.$ Derive the closed$-$loop transfer function of the system.

$2\mathrm{.}2.$ Determine all closed$-$loop poles.

$2\mathrm{.}3.$ Plot the unit$-$step response curve.

$2\mathrm{.}4.$ From the response curve, calculate the actual maximum percent overshoot $M_P\%$ of the system.

$2\mathrm{.}5.$ From the analytical relation between $M_P\%$ and $,$ obtain the theoretical value of maximum percent overshoot at $=0\mathrm{.}55.$

$2\mathrm{.}6.$ Do the actual and theoretical values of maximum percent overshoot are equal? If not, explain.

Using Simulink,

$3\mathrm{.}1.$ Create a Simulink block diagram representing the system.

$3\mathrm{.}2.$ Determine a new value of the gain K such that the maximum percent overshoot of the closed$-$loop system is $5\%.$

$3\mathrm{.}3.$ Obtain the response $c(t)$ to the unit step input. $($Hint: simulate for $35$ sec with sampling time $0\mathrm{.}01$ sec $)$