Figure Q$1$ shows the Williams$-$Otto plant, named after T$.$ I. Williams and R$.$ E$.$ Otto. Raw

materlals A and $B$ are mixed in a CSTR and yleld the following reactions:

$A+BC,C+BP+E,P+CG$

where $P$ is the desired product, and $E$ and $G$ are undesired by$-$products. The three reactions

have an overal 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$.$

The composition of distillate $(Y_1)$ and bottom $(Y_2)$ products from the distillation

column are to be regulated to their desired set$-$points. The available manipulated

variables are the reflux flow $(U_1)$ and steam flow rate $(U_2).$ The following input$-$

output relationships for the distillation column have been determined

experimentally:

Transfer functions

Process transfer function A: $G_{22}(s)=\frac{-19\mathrm{.}4e^{-3s}}{(14\mathrm{.}4s+1)}$

Process transfer function B: $G_{11}(s)=\frac{12\mathrm{.}8e^{-s}}{(16\mathrm{.}7s+1)}$

Process transfer function C: $G_{12}(s)=\frac{-18\mathrm{.}9e^{-3s}}{(21s+1)}$

Process transfer function D: $G_{21}(s)=\frac{6\mathrm{.}6e^{-7s}}{(10\mathrm{.}9s+1)}$

Question

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.