1. Figure Q1 shows the Williams-Otto plant, named after T. I. Williams and R. E. Otto. Raw materials A and B are mixed in a CSTR and yield the following reactions:

A + B C C + B P + E P + C G

where P is the desired product, and E and G are undesired by-products. The three reactions have an overall exothermic effect and the CSTR is cooled with water. The effluent of the reactor is cooled with cooling water and enters a decanter, where the by-product G is removed. The remaining mixture enters a distillation column where the desired product P is removed as the top distillate stream. The bottom product contains unreacted A, B and by-products C and E. Part of it is removed while the rest is recycled back to the CSTR.

1. The composition of distillate (Y₁) and bottom (Y₂) products from the distillation column are to be regulated to their desired set-points. The available manipulated variables are the reflux flow (U₁) and steam flow rate (U₂). The following input-output relationships for the distillation column have been determined experimentally:

Process transfer function A: G₂₂ =(-19.4e^-3s)/(14.4s+1)

Process transfer function B: G₁₁ = (12.8e^-s)/(16.7s+1)

Process transfer function C: G₁₂ = (-18.9e^-3s)/(21s+1)

Process transfer function D: G₂₁ = (6.6e^-7s)/(10.9s+1)

Your task is to select the pairings between controlled outputs and manipulated variables so that the resulting loops offer minimum steady-state interaction. If the best couplings still possess significant interaction between the control loops, design two decouplers that produce two non-interacting control loops.