For the open-loop plant dynamics from (2.3.2), determine the closed-loop transfer function from the reference input R(s) to the shaft angle output l(s) when using a Proportional controller C(s) = kp. Rearrange this transfer function into stan- dard second-order system form. Determine how the proportional gain kp affects the dynamic behavior and the steady-state behavior of the closed-loop system. Use the Final Value Theorem to determine the steady-state reference tracking error, ess, for a constant reference (R(s) = c1s). Summarize what can and cannot be done with proportional-only control. 2. (30 points) Now determine the closed-loop transfer function for a Proportional-Velocity (PV) controller. How does adding the velocity feedback term with gain kv affect the dynamic behavior of the closed-loop system? Determine relationships that you can use to compute the control gains kp and kv in order to achieve a desired natural frequency n and damping ratio . 3. (30 points) Instead of a constant input reference, consider a ramp reference with slope A such that r(t) = At and R(s) = A 1 . Determine the steady-state tracking error for s2 a ramp reference using a PV controller in terms of A, K, kp, and kv