A plant (uncontrolled system), G (s) = control gains K, and K, as shown below. (1) Find the open-loop transfer function KG (s) Hint: (a) PI controller transfer function C(s) = K + can be rewritten as C(s) = K, where K and z are in terms of K and K; (b) the closed-loop characteristic equation can be written as 1 + KG(s) = 0. (2) let z = -3, hand sketch the root locus for K going from zero to infinite. a. Show calculations for the angles and the centroid of asymptotes. b. Calculate break-in and breakaway points if necessary. c. Label the points where the root locus crosses the imaginary axis. d. Provide arrows showing increasing K and indicate the locations where K = 0 and infinity. Hint: Use Matlab to help verify your root locus sketch. (A sample script can be found on Canvas under the Module of Root Locus). U(s) + 1 (s+1)(s+5)' (3) With Z = -3, use the root locus plot from (2) to identify the lowest settling time of the system without an overshoot and oscillation when the reference u(t) is a unit step input. Report this lowest settling time and the gain values K, and K. Hint: the settling time is determined by the closed-loop pole that is closest to the imaginary axis. Kp + , is controlled by a PI controller with K S Gp(s) Y(s)