Consider the PCM furnace module of Appendix E with the following variables (HC denotes hydrocarbon): 

CVs: HC exit temperature T_HC and oxygen exit concentration c_O2 

MVs: fuel gas flow rate F_FG and air flow rate F_A 

DV: HC flow rate F_HC 

Do the following, using the transfer function models given below:
(a) Design an MPC system using the following design parameters: Δt = 1 min, Q = diagonal [0.1, 1], R = diagonal [0.1, 0.1], P = 20, and M = 1.
(b) Repeat part (a) for the same design parameters, but where R = diagonal [0.5, 0.5].

(c) Simulate the two MPC controllers for a step change in the c_O2 set point to 1.0143 mol/m³ at t = 10 min.
(d) Repeat part (c) for a step change in F_HC at t = 10 min to 0.035 m³/min.
(e) Based on your results for parts (c) and (d), which MPC controller is superior?

Process transfer function matrix: