A crystallize is used to separate a pharmaceutical product from the fermentation extract. The three manipulated variables are the fines dissolution rate u₁, the crystallize temperature u₂, and the flow rate in the overflow u₃. The nominal values of these three inputs are 2.25 × 10^-6m³/s, 310 K, and 1.5 × 10^-6m³/s, respectively. The three variables to be controlled are the crystal size distribution, as calculated by the fines suspension density y₁; the crystal purity, as calculated by supersaturation conditions y₂; and the product rate y₃. The nominal values of these three inputs are 0.55 K, 11.23 K, and 0.12 kg/kg H₂O, respectively. These variables have multiple interactions, and the following model has been identified from experimental data for a continuous crystallize, where time is measured in s (Rohani et al., 1999):

(a) Calculate the RGA, and determine the appropriate  airings for SISO feedback control. Comment on the role of dynamics in your decision.
(b) Using the IMC tuning rules, design three PI controllers for this process.
(c) Simulate the process response to a step set point, separately, in each of the controlled outputs [use a magnitude  of +10% (relative) change]. Next, simulate the response of the system to a simultaneous pair of step changes (again, use a magnitude of +10% relative) in each of the second (purity) and third (product rate) controlled outputs. Try to tune the controller to improve the transient response to the simultaneous step changes.

0.025082(s 20.0)(s 10.4) s2 + 10.11s + 96.57 125,000(s 1.25) s+ 0.39 -1,830, 000(s +0.089) 72,600 s+ 0.2692 -0.15095 568,000 , [U3] | 2 LV3 s+ 2.11 s+0.1338 s+ 0.43 -1870 s+0.21 -0.0071 16,875 s+ 0.235 s + 0.2696 II