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## Q. 20.11

Consider the PCM distillation column module of Appendix E with the following variables:

CVs: Overhead Composition $\left(x_{D}\right)$ and Bottom Compo$\operatorname{sition}\left(x_{B}\right)$

MVs: Reflux Ratio $(R)$ and Vapor Flow Rate $(V)$ DV: Column Feed Flow Rate $(F)$

Do the following, using the transfer function models given below:

(a) Design an MPC system using the following design parameters: $\Delta t=60 s , Q =$ diagonal $[0.1,0.1], R =$ diagonal $[0.1,1]$ $P=40, M=1$.

(b) Repeat part (a) for the same design parameters, but where $Q =$ diagonal $[0.5,0.5]$

(c) Simulate the two MPC controllers for a step change in the set point for Overhead Composition $\left(x_{D}\right)$ from $0.85$ to $0.80$.

(d) Repeat part (c) for a step change in the set point for Bottom Composition $\left(x_{B}\right)$ from $0.15$ to $0.2$.

(e) (Optional-disturbance test) Repeat part (c) for a step change in the Column Feed Flow Rate $(F)$ from $0.025$ to $0.03$.

Process transfer function matrix (all times in minutes):

$\begin{matrix} X_{D} \\ X_{B}\end{matrix}\overset{\begin{matrix} R & V\end{matrix} }{\left|\begin{array}{cc}\frac{0.14 e^{-0.94 s}}{14 s+1} & \frac{-9.0 e^{-18.6 s}}{20 s+1} \\\frac{0.16 e^{-3.7 s}}{17 s+1} & \frac{-14 e^{-0.22 s}}{10 s+1}\end{array}\right|}$

## Verified Solution

We repeat 20.10 for $R \left[\begin{array}{ll}0.1 & 0.1\end{array}\right], Q =\left[\begin{array}{ll}0.1 & 1\end{array}\right]$ and (a) $M =1$ and (b) $M =4$

First we evaluate the controller response to a step change in the oxygen concentration setpoint from $0.922$ to $1.0143$.

Next we test a step change in the hydrocarbon flow rate from $0.035 m ^{3} / min$ to $0.038 m ^{3} / min$.