A schematic diagram for a pH neutralization process is shown in Fig. E18.19. The transfer function matrix and relative gain array are also shown.
(a) Suppose that a multiloop control system consisting of four PID controllers is to be designed. Recommend a pairing of controlled and manipulated variables. Briefly justify your recommendation based on steady-state, dynamic, and physical considerations.
(b) Suppose that only pH _{2} and h_{2} are to be controlled using Q_{4} and Q_{6} as the manipulated variables \left(Q_{1}\right. and Q_{3} are held constant).
(i) What is the RGA for this 2 \times 2 control problem?
(ii) What pairing of controlled and manipulated variables do you recommend? (Justify your answer.)
\begin{gathered}{\left[\begin{array}{c}h_{1} \\p H_{1} \\h_{2} \\p H_{2}\end{array}\right]=\left[\begin{array}{cccc}\frac{0.43 e^{-0.8 s}}{4.32 s+1} & \frac{0.43 e^{-0.1 s}}{3.10 s+1} & \frac{0.23 e^{-1.0 s}}{5.24 s+1} & \frac{0.22 e^{-0.5 s}}{4.42 s+1} \\\frac{-0.33 e^{-1.0 s}}{2.56 s+1} & \frac{0.32 e^{-0.5 s}}{2.58 s+1} & \frac{-0.20 e^{-1.8 s}}{2.82 s+1} & \frac{0.20 e^{-0.8 s}}{3.30 s+1} \\\frac{0.22 e^{-1.1 s}}{5.52 s+1} & \frac{0.23 e^{-0.3 s}}{4.49 s+1} & \frac{0.42 e^{-0.4 s}}{3.32 s+1} & \frac{0.41 e^{-0.1 s}}{2.07 s+1} \\\frac{-0.22 e^{-1.5 s}}{3.24 s+1} & \frac{0.22 e^{-1.2 s}}{2.65 s+1} & \frac{-0.32 e^{-0.8 s}}{2.36 s+1} & \frac{0.32 e^{-0.4 s}}{2.03 s+1}\end{array}\right]\left[\begin{array}{l}Q_{1} \\Q_{3} \\Q_{4} \\Q_{6}\end{array}\right]} \\\text { RGA }=\left[\begin{array}{rrrr}0.64 & 0.72 & -0.20 & -0.20 \\0.87 & 0.85 & -0.35 & -0.35 \\-0.18 & -0.21 & 0.70 & 0.70 \\-0.36 & -0.37 & 0.85 & 0.88\end{array}\right]\end{gathered}
