## 3 Results
Figure 4: Examples of identified model predictions based on simulated validation data. (**a**) 5-steps ahead prediction, (**b**) 10-steps ahead prediction, (**c**) validation data sequence. Figure 5: Examples of model predictions based on real-time lake validation data; (**a**) 5-steps ahead prediction, (**b**) 10-steps ahead prediction, (**c**) validation data sequence. Figure 6: Examples of model predictions based on real-time lake validation data, magnified figure. (**a**) 5-steps ahead prediction, (**b**) 10-steps ahead prediction, (**c**) validation data sequence. Fifteen real-time lake experiments were conducted, in which pseudorandom pod angles of rotation sequences were pod command signals. The ship's rotational speed was recorded during the whole research experiment. Sequences of pod angles of rotation combined with the ship's rotational speed were then used as validation data. Models were identified on the basis of the simulation results and regularized, using GA, on the basis of the real-time ship trial results. Exemplary model parameters, their STD values, and regularization parameters for each coefficient (\(R_{i}\)) are compiled in the Table 3. The model was identified with \(\lambda=10.377\). It gave reliable results with STDs less than 10% of the coefficient values. Figure 7 presents GA model fit to the validation data obtained during the real-time lake trial (c). There were no high-frequency oscillations in the time trials. For 5-step ahead prediction (a) results fit the validation data and reached 86%, which gives very good prediction. For 10-step ahead (b) the fit decreased and gave a reliable model with fit equal to 68%, which can be used in the MPC controller. Prediction of horizon elongation led to decreased fit. The model that reflects ship dynamics and whose fit function is greater than 55% is good enough to be a component of the MPC. Figures 8 and 9 illustrate, respectively, the regularized GA model and conventional model 5- and 10-step ahead predictions based on real-time lake validation data. These data, combined with the averaged non-regularized model fit, are presented in Table 4. In both cases prediction of horizon elongation caused the fit to decrease. In the conventionally identified model, two high-frequency oscillations make its application to MPC impossible. In the GA model the fit was greater and the output signal was smooth. Moreover, the GA model fitness function had about 20% bigger value, which proves the legitimacy of using the GA algorithm during the identification process. In Table 5 the percent improvement of the prediction ability of the tuned model is presented, compared to the reference values, guaranteeing the performance of the MPC controller and the non-regularized model prediction ability. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **5-Step Prediction** & **10-Step Prediction** & **Oscillations** \\ \hline non-regularized & 70.67\% & 54.39\% & YES \\ GA-regularized & 89.26\% & 75.92\% & NO \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Prediction ability of the GA regularized model combined with the averaged non-regularized model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **STD** & \(R_{i}\) \\ \hline \(a_{1}\) & \(-0.2268\) & 0.0153 & 100.327 \\ \(a_{2}\) & \(1.466\) & 0.0489 & 100.327 \\ \(a_{3}\) & \(-3.202\) & 0.0531 & 100.327 \\ \(a_{4}\) & \(2.961\) & 0.0195 & 100.327 \\ \hline \(b_{1}\) & \(-7.328\times 10^{-5}\)Figure 8: Regularized GA model and conventional model 5-step ahead prediction comparison based on real-time lake validation data. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **5-Step Prediction** & **10-Step Prediction** & **Oscillations** \\ \hline reference & 80\% & 65\% & NO \\ non-regularized & \(-\)9.33 percentage points & \(-\)10.61 percentage points & does not meet \\ GA-regularized & 9.26 percentage points & 10.92 percentage points & fulfills \\ \hline \hline \end{tabular}
\end{table}