# Tuning of Multivariable Model Predictive Control for Industrial Tasks
## 4 Simulation Results
In the first step of the tuning procedure, the influence of the coefficient \(m_{1}\) on the performance criteria \(\text{SSE}_{1}\), \(\text{SSE}_{2}\), \(\sigma_{1}\), \(\sigma_{2}\), \(\text{H}_{1}\) and \(\text{H}_{2}\) is analysed, and the results are shown in Figure 3. We take into account all three criteria for the first process output, i.e., the indices \(\text{SSE}_{1}\), \(\sigma_{1}\) and \(\text{H}_{1}\), and we choose the value \(m_{1}=2\) because it gives the best control quality. Next, for the chosen parameter \(m_{1}=2\), in the second step, we assess the influence of the parameter \(K_{1}\), and it is shown in Figure 4. It is clear that the best results are obtained for \(K_{1}=100\). Finally, the trajectory of the weights \(\psi_{p,1}\) is parameterised by means of Equation (6), which is characterised by the parameter \(a_{1}\) as well as the fixed parameters \(m_{1}\) and \(K_{1}\). Figure 5 depicts the influence of the parameter \(a_{1}\) on the considered performance indices. The best set of control quality criteria is obtained for \(a_{1}=2\). Figure 4: The second step of tuning the coefficients \(\psi_{p,1}\): the influence of the coefficient \(K_{1}\) on the performance criteria \(\text{SSE}_{1}\), \(\text{SSE}_{2}\), \(\sigma_{1}\), \(\sigma_{2}\), \(\text{H}_{1}\) and \(\text{H}_{2}\) (for all 600,000 samples). Figure 3: The first step of tuning the coefficients \(\psi_{p,1}\): the influence of the coefficient \(m_{1}\) on the performance criteria \(\text{SSE}_{1}\), \(\text{SSE}_{2}\), \(\sigma_{1}\), \(\sigma_{2}\), \(\text{H}_{1}\) and \(\text{H}_{2}\) (for all 600,000 samples). All things considered, the coefficients \(\psi_{p,1}\) are characterised by the set of parameters \(m_{1}=2\), \(K_{1}=100\) and \(a_{1}=2\), while the coefficients \(\psi_{p,2}\) by the parameters \(m_{2}=3\), \(K_{2}=100\) and \(a_{1}=1\). The actual values of the coefficients used in MPC optimisation are computed from Equation (6). To evaluate the efficiency of the chosen parameters \(\psi_{p,1}\) and \(\psi_{p,2}\), let us consider the values of all six performance criteria. We not consider only the found coefficients, but additionally, we take into account the most practically cases in which all coefficients are constant, i.e., \(\psi_{p,1}=\psi_{p,2}\). In the latter case, we consider six values of the coefficients: 0.1, 1, 10, 100, 1000 and 10,000. The values of the obtained performance criteria are given in Table 1. For almost all considered indices (the only exception is \(\sigma_{2}\), but the difference is not significant), the discussed tuning method gives the best results. For constant weights \(\psi_{p,1}=\psi_{p,2}\) greater than 10,000, the results do not change. Let us now consider histograms of the control errors for constant values of the coefficients \(\psi_{p,1}\), \(\psi_{p,2}=0.1,\ 1,\ 10\) and for the tuned ones. They are shown in Figure 9. Two observations are possible. Firstly, our tuning method gives the most narrow histograms. Secondly, they do not have long \"tails\", which means that our method eliminates large negative and positive control errors. Figure 10 depicts the first 1000 samples of the control errors for constant values of the coefficients \(\psi_{p,1}\), \(\psi_{p,2}=0.1,\ 1,\ 10\) and for the tuned ones. The obtained trends confirm the observations made on the basis of the histograms, i.e., our tuning method gives the best control quality.