# Tuning of Multivariable Model Predictive Control for Industrial Tasks
## 4 Simulation Results  
The discussed tuning method is verified in the control system of a simulated binary distillation column, which separates a two component mixture of water and methanol. The linear approximation of the process (scaled) is described by the following transfer function model introduced by [PERSON] (1963):  
\[\left[ \begin{array}{c}Y_{1}(s)\\ Y_{2}(s)\end{array} \right]=\left[ \begin{array}{cc}\frac{12.8e^{-s}}{16.7s+1}& \frac{-18.9e^{-3s}}{21s+1}\\ \frac{6.6e^{-7s}}{10.9s+1}&\frac{-19.4e^{-2s}}{14.4s+1}\end{array} \right] \left[ \begin{array}{c}U_{1}(s)\\ U_{2}(s)\end{array} \right]+\left[ \begin{array}{c}\frac{3.8e^{-8s}}{14.9s+1}\\ \frac{4.9e^{-3s}}{13.2s+1}\end{array} \right]F(s) \tag{8}\]  
The controlled variables, \(Y_{1}\) and \(Y_{2}\), are: the top product (distillate) composition and the bottom product composition, respectively. The manipulated variables, \(U_{1}\) and \(U_{2}\), are: the reflux flow rate and the vapour flow, respectively. The uncontrolled process input (the disturbance), \(F\), is the flow rate of the input stream. All variables are deviations from a typical operating point, and all variables are dimensionless. The sampling time of MPC is 1 min. To obtain a discrete process representation that is used to determine the step-response model necessary for prediction in the DMC algorithm ([PERSON], 1963), we use the forward Euler method and the zero-order holder. As a result, the following transfer function representation of the model (8) is obtained:  
\[\left[ \begin{array}{c}Y_{1}(z)\\ Y_{2}(z)\end{array} \right]=\left[ \begin{array}{cc}\frac{0.7440z^{-2}}{1-0.9419 ^{-1}}&\frac{-0.8789z^{-4}}{1-0.9535z^{-1}}\\ \frac{0.5786z^{-8}}{1-0.9123z^{-1}}&\frac{-1.3015z^{-3}}{1-0.9329z^{-1}}\end{array}  \right]\left[ \begin{array}{c}U_{1}(z)\\ U_{2}(z)\end{array} \right]+\left[ \begin{array}{c}\frac{0.2467z^{-9}}{1-0.935 12^{-1}}\\ \frac{0.3575z^{-4}}{1-0.9270z^{-1}}\end{array} \right]F(z) \tag{9}\]  
A resulting step-response model is obtained and next used for prediction in the DMC algorithm ([PERSON], 1963). The horizons used in the DMC algorithm are: the prediction horizon \(N=30\), the control horizon \(N_{u}=5\) and the horizon of process dynamics \(D=100\) (the horizon of process dynamics is the number of step-response coefficients taken into account by the model used in the DMC algorithm) ([PERSON] et al., 2017). All values of the weights that penalise the excessive increments of the manipulated variables, i.e., \(\lambda_{p,tt}\), are equal to one. In this work, the objective of the controller is to effectively compensate the influence of three types of disturbances: changes of the flow rate of the input stream \(F\), as well as the measurement noise of the first and the second process outputs, denoted as \(\delta_{1}\) and \(\delta_{2}\), respectively. As many as 600,000 min (samples) are considered, i.e., approximately 417 days. It must be emphasised that in this work, the disturbances are not generated artificially, but real industrial disturbances recorded in a distillation process are used ([PERSON] et al., 2017). All 600,000 samples of the disturbances are shown in Figure 2.  
Figure 2: All 600,000 samples of the disturbances. All coefficients \(\lambda_{p,n}=1\) for \(p=0,\ldots,N_{\rm u}-1\), \(n=1,\ldots,n_{\rm u}\)[39]. The problem to solve is to carefully select the tuning parameters \(\psi_{p,n}\) for \(p=1,\ldots,N\), \(n=1,\ldots,n_{\rm y}\). For this purpose, the tuning procedure presented in Section 3 is used. Let us first concentrate on tuning the coefficients \(\psi_{p,1}\), i.e., for the first process output, \(p=1,\ldots,N\). Initially, the trajectory of the weights \(\psi_{p,1}\) is parameterised by means of Equation (5), which is characterised by the parameters \(m_{1}\) and \(K_{1}\).