Consider the PCM furnace module of Appendix H. Assume that hydrocarbon temperature T_HC is the CV, fuel gas flow rate F is the MV, and that 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 the response of each controller to a sudden change in the hydrocarbon flow rate F_HC from 0.035 to 0.040 m³/min at t = 10 min as an unmeasured disturbance. 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.