In Part b of Problem 54 in Chapter 10, we used a proportional-plus-integral (PI) speed controller that

Question:

In Part b of Problem 54 in Chapter 10, we used a proportional-plus-integral (PI) speed controller that resulted in an overshoot of 20% and a settling time, T_s = 3:92 seconds (Preitl, 2007).

a. Now assume that the system specifications require a steady-state error of zero for a step input, a ramp input steady-state error ≤2%, a %OS ≤4.32%, and a settling time ≤4 seconds. One way to achieve these requirements is to cancel the PI-controller’s zero, Z_I, with the real pole of the uncompensated system closest to the origin (located at -0.0163). Assuming exact cancellation is possible, the plant and controller transfer function becomes

Design the system to meet the requirements. You may use the following steps:

i. Set the gain, K, to the value required by the steady state error specifications. Plot the Bode magnitude and phase diagrams.

ii. Calculate the required phase margin to meet the damping ratio or equivalently the %OS requirement, using Eq. (10.73). If the phase margin found from the Bode plot obtained in Step i is greater than the required value, simulate the system to check whether the settling time is less than 4 seconds and whether the requirement of a %OS ≤ 4.32% has been met. Redesign if the simulation shows that the %OS and/or the steady-state error requirements have not been met. If all requirements are met, you have completed the design.

b. In most cases, perfect pole-zero cancellation is not possible. Assume that you want to check what happens if the PI-controller’s zero changes by ± 20%, e.g., if Z_I moves to:

Case 1: - 0:01304

or to
Case 2: - 0:01956.
The plant and controller transfer function in these cases will be, respectively:

Set K in each case to the value required by the steady-state error specifications and plot the Bode magnitude and phase diagrams. Simulate the closed loop step response for each of the three locations of Z_I: pole/zero cancellation, Case 1, and Case 2, given in the problem.
Do the responses obtained resemble a second order overdamped, critically damped, or underdamped response? Is there a need to add a derivative mode?