Problem $1$:

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

$1.$ 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 $\backslash (\backslash$zeta $\backslash )$ of a pair of dominant complex$-$conjugate closed$-$loop poles is $0\mathrm{.}55.$

$2.$ 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 $\backslash (\backslash$mathrm$\{$M$\}\_\{\backslash$mathrm$\{$P$\}\}\backslash \%\backslash )$ of the system.

$2\mathrm{.}5.$ From the analytical relation between $\backslash (\backslash$mathrm$\{$M$\}\_\{\backslash$mathrm$\{$P$\}\}\backslash \%\backslash )$ and $\backslash (\backslash$zeta $\backslash ),$ obtain the theoretical value of maximum percent overshoot at $\backslash (\backslash$zeta$=0\mathrm{.}55\backslash ).$

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

$3.$ 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 $\backslash (5\backslash \%\backslash ).$

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