Problem $2(25$pts$)$

Consider the system shown in Fig. $1.$

$($a$)(6$pts$)$ Use MATLAB to obtain the Bode plot for $\backslash ($ K$=1\backslash ).$ Can you use the magnitude and phase plots to determine the range of positive $\backslash ($ K $\backslash )$ values for which the closed loop system is stable? If yes, what is this range?

$($b$)(7$pts$)$ Sketch the Nyquist plot of the system, and use it to determine the positive range of $\backslash ($ K $\backslash )$ values for which the closed loop system is stable.

$($c$)(6$pts$)$ Use the Nyquist plot to find the number of unstable closed loop poles for the unstable values of $\backslash ($ K $\backslash ).$

$($d$)(6$pts$)$ Using Routh's criterion, determine the ranges of $\backslash ($ K $\backslash )($positive and negative$)$ for which the closed loop system is stable.