$6\mathrm{.}27$ Consider the system given in Fig. $6\mathrm{.}91.$

$($a$)$ Use Matlab to obtain Bode plots for $K=1,$ and use the plots to estimate the range of $K$ for which the system will be stable.

$($b$)$ Verify the stable range of $K$ by using margin to determine PM for selected values of $K.$

$($c$)$ Use rlocus to determine the values of $K$ at the stability boundaries.

$($d$)$ Sketch the Nyquist plot of the system, and use it to verify the number of unstable roots for the unstable ranges of $K.$

$($e$)$ Using Routh's criterion, determine the ranges of $K$ for closed$-$loop stability of this system.

Figure $6\mathrm{.}91$

Control system for Problem $6\mathrm{.}27$