Consider the PCM distillation column module of Appendix H. Assume that distillate MeOH composition x_D is the CV, reflux ratio R is the MV, and they are related by the following transfer function model:

1. Design a PID controller based on the IMC tuning relations and a reasonable choice for τ_c.
2. Design a PID controller based on the relay auto-tuning feature of the PCM, and the Ziegler-Nichols settings.
3. Simulate each controller for a sudden change in an unmeasured disturbance, feed composition x_F, at t = 10 min, from 0.50 to 0.55 (mole fraction). Which controller is superior? Justify your answer.
4. Attempt to obtain improved control system performance by fine tuning your better controller for part (c). For the performance criteria, consider maximum deviation from set point and settling time.