A negative feedback control system is depicted in Figure below. Suppose that our design objective is to find a controller $\backslash ($ G$\_\{$c$\}($s$)\backslash )$ such that our closed$-$loop system can stably track a unit step input with a steady$-$state error of zero.

$($a$)$ As a first try, consider a simple proportional controller

$\backslash [$

G$\_\{$c$\}($s$)=$K$,$

$\backslash ]$

where $\backslash ($ K $\backslash )$ is a fixed gain. Plot the root locus showing how the closed$-$loop poles change as the value of $\backslash ($ K $\backslash )$ varies from $0$ to $\backslash (\backslash$infty $\backslash ),$ and use it to determine if the system can be stabilized with the given controller $\backslash ($ G$\_\{$c$\}\backslash ).$ If it can be stabilized, use the root locus to find the range of $\backslash ($ K $\backslash )$ that will stabilize the system. $[10$pt$]$

$($b$)$ Now consider a more complex controller

$\backslash [$

G$\_\{$c$\}($s$)=1+$K s$,$

$\backslash ]$

where $\backslash ($ K $\backslash )$ is a fixed gain. This controller is known as a proportional, derivative $($PD$)$ controller. Plot the root locus showing how the closed$-$loop poles change as the value of $\backslash ($ K $\backslash )$ varies from $0$ to $\backslash (\backslash$infty $\backslash ),$ and use it to determine if the system can be stabilized with the given controller $\backslash ($ G$\_\{$c$\}\backslash ).$ If it can be stabilized, use the root locus to find the range of $\backslash ($ K $\backslash )$ that will stabilize the system. $[15$pt$]$

$($c$)$ Solve problem $($b$)$ again using the Nyquist stability criterion. That is$,$ find the range of $\backslash ($ K $\backslash )$ for which the system is stable by sketching the Nyquist plot. $[10$pt$]$