1. The controller gains K, and Ky have to be selected to ensure closed-loop stability. T(S) + K P 1 s(s+12) K L(s) y(s) a) Derive the expression of the outer loop transfer function L(s). b) Write the expression of the closed-loop transfer function in the normalized form. c) For a set of controller gains K, and Ky, the closed-loop system has three stable poles with one high-frequency real pole and two dominant complex conjugate poles such that the closed-loop transfer function can be approximated as a second-order system. Find the values of K, and K, such that i) the magnitude of the high-frequency pole is 10 times the magnitude of the real part of the complex conjugate poles and ii) for the reduced-order system, the settling time and percentage maximum overshoot are 4s and 38.8%, respectively. d) Using the same controller gains, find the outer loop transfer function and draw the Nyquist plot. From the Nyquist plot, find the gain margin.