Writeup

The writeup is due in class on the Thursday 15 October. Only one writeup is required per group. A formal format is not required, but be sure to address the following issues. Include all your work in the writeup and justify all your answers. Tabulate results wherever appropriate.

1. Plant Identification
   1. Given your open loop step response, calculate the gain and time constant (K and ).
   2. Write the open loop transfer function of the amp-motor-tach system.
2. Integral Control
   1. Derive the relationships for the two parts of the op-amp circuit.
   2. Show that Ki/s is the appropriate transfer function for the controller, and express Ki in terms of the resistor and capacitor values.
   3. Use root locus sketching rules to draw the root locus for this system. Compare this to the root locus you plotted in Matlab.
   4. Show your work for finding the desired pole locations for each of the three design cases.
   5. Use the magnitude criterion to find the three corresponding values of Ki given these pole locations. Compare them to the ones you found in lab using Matlab.
   6. Find the closed-loop transfer function in terms of Ki. Plug in Ki for each case and verify that the closed-loop poles are where you expected.
   7. Step Response
      1. In the overdamped case, we can approximate the second-order system as a first order system. Justify this by finding the response of the overdamped system to a unit step as a function of time. Show that only the part corresponding to the dominant (slowest) pole is significant.
      2. For each of the three cases, calculate the expected Ts, Tp, %OV, and SS error from the closed-loop transfer function.
      3. Compare these expected response parameters to the ones observed in lab for each case. Did your control system behave as expected? If not, why not?
      4. Physically, how does integral action remove steady-state error? (hint - what happens if there has been an error for a significant length of time?)
      5. For the observed response parameters, calculate the pole locations which would give you these responses. Plot these ``observed'' pole locations on the same axes as the ``desired'' pole locations. Does this provide any insight as to why your results might not have been quite as expected?
   8. Disturbance Rejection
      1. Find the transfer function from disturbance to output (assume the disturbance adds in at the control signal).
      2. Find the steady-state response to a step disturbance.
      3. How does this explain what you observed?
      4. Physically, how does integral action remove steady-state effects of a disturbance?
   9. Ramp Response
      1. Calculate the expected SS error to a unit ramp in terms of Ki.
      2. In each of the three cases, compare the expected SS ramp error to the observed. Be sure to adjust the expected error for the slope of the ramp you used in lab. A unit ramp has 1V/sec slop.

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