1. Consider the feedback system below. It is desired to design the PID controller such that the dominant closed-loop poles are located at s12 = -1 1 jV3. i) Determine the zero location -b for the PID controller. ii) Draw the root locus of the closed-loop poles as K varies from 0 to oo. iii) Determine the controller gain K. iv) Determine the third closed-loop pole location s. Hint: Note that the closed loop poles are the roots of characteristic polynomial s(s2 + 1) + K(s + b) (s + 1) which must be equal to (s - st ) (s - $2 ) (s - $3 ). r(t) e(t) K(s+ b) (s+1) 1 y(t) S (s + 1 ) PID controller system to be controlled