Solved: Problem 2. Given the same plant as Problem 1, you now want to complete

Question

Problem 2. Given the same plant as Problem 1, you now want to completely eliminate the steady-state error of the closed-loop system with a PI controller. r(t) + e(t) u(t) y(t) Ge Gp 4 y(t) Ge(s) = kp + kĮ 5 kps + ki Gp(s) =- 4s + 1 2 (a) Show that the closed-loop transfer function is 2(kps +ki) Gel(s) = 5 4s2 + (1 + 2kp)s + 2kj (b) From this transfer function, find equations for the natural frequency W), and damping ratio { in terms of the controller gains. (c) Find the DC gain of the transfer function. Comment on the steady-state error of the closed-loop system. (d) Find the gains kp and kį for the closed loop transfer function to have @n =1 and { =1 (critical damping). (e) Find the poles of the closed-loop system with the controller gains from (d). (f) Use MATLAB to create a single plot with 2 step response traces: one for the open loop plant and one for the closed-loop system designed in (d).