1. (40 points) 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 (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) = c). 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 k, affect the dynamic behavior of the closed-loop system? Determine relationships that you can use to compute the control gains kp and k, in order to achieve a desired natural frequency wn and damping ratio C. 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. Determine the steady-state tracking error for a ramp reference using a PV controller in terms of A, K, kp, and ky. 8d(s) E(s) Derivative S Negative Feedback H Proportional Control Gain Integral Control Gain Velocity Control Gain Integral 1/s Vm(s) Plant P(s) 0,(s) Figure 2.3.2: Block diagram of a PIV controller for angular position control of the Servo Unit system.