## 3 Results  
The predictive model was identified on the basis of nonlinear LNG Carrier dynamics simulation data with the use of Matlab System Identification Toolbox. A large number of simulated time series was used to identify linear state-space models. However, these models were not reliable due to high standard deviations for some parameters. In Table 1, examples of model parameters and their STD values are presented. The STDs for parameters \(b_{1}\) and \(b_{2}\) are bigger than their values. Models like this occurred frequently during the classical identification process and were rejected. In order to use the model in the MPC algorithm it should be averaged, especially if a particular identified parameter varies more than 5%. Example results of two models' parameters whose STDs were smaller than 10% of the coefficient value are presented in Table 2. In the fourth column averaged coefficients are presented. Figure 4 shows 5- (a) and 10-steps ahead (b) prediction of model 1, model 2, and the averaged model. The pseudo-random angle of pod rotation (\(\delta_{2}\)) with 5 s sampling time is an input signal (c). Quantitative analysis of the particular coefficients suggests the results are too widely spread. But when one has a look at the prediction results of the model compared against simulation validation data there are hardly any differences in the trials. Model 2 has more than 10 percentage points less fit. In the 10-step predicted response for model 2, high-frequency oscillation occurs, which should be eliminated before the model is applied to the MPC algorithm. In the second stage of model verification, real-time trials on Silm Lake were done. Pseudo-random pod angles of rotation are input signals that, combined with outputs, create validation data sets. Conventionally identified models generated oscillations during validation. This is presented in Figure 5. Model 1 and the averaged model produced high-frequency oscillations with magnitude within 10% of the output signal value. However, oscillations that occurred during model 2 validation process were bigger than the output signal amplitude multiplied by 15. This means this model is useless for all prediction  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **STD** \\ \hline \(a_{1}\) & \(-0.3145\) & \(0.0142\) \\ \(a_{2}\) & \(1.736\) & \(0.0441\) \\ \(a_{3}\) & \(-3.486\) & \(0.0468\) \\ \(a_{4}\) & \(3.062\) & \(0.0168\) \\ \hline  
horizons. Moreover, the fit of the model outputs to the validation data was less than \(17\%\) for model 1 and reached \(-40\%\) for the averaged model. Using them in the prediction process will lead to significant errors. Therefore, GA regularization was applied during the model identification process. Figure 6 presents an enlarged section (50 seconds of the trial) cut of the graph presented in Figure 5. It shows that model 1 and the averaged model were not good enough, but also were not completely useless as model 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Model 1** & **Model 2** & **Average Model** \\ \hline \(a_{1}\) & \(-0.3599\) & \(-0.5825\) & \(-0.4714\) \\ \(a_{2}\) & \(1.884\) & \(2.598\) & \(2.241\) \\ \(a_{3}\) & \(-3.649\) & \(-4.421\) & \(-4.035\) \\ \(a_{4}\) & \(3.123\) & \(3.405\) & \(3.264\) \\ \hline \(b_{1}\) & \(1.258\times 10^{-4}\) & \(-2.074\times 10^{-4}\) & \(-4.08\times 10^{-5}\) \\ \(b_{2}\) & \(1.721\times 10^{-4}\) & \(-5.04\times 10^{-4}\) & \(-1.659\times 10^{-4}\) \\ \(b_{3}\) & \(1.994\times 10^{-4}\) & \(-9.322\times 10^{-4}\) & \(-3.664\times 10^{-4}\) \\ \(b_{4}\) & \(-3.716\times 10^{-4}\) & \(-1.738\times 10^{-3}\) & \(-1.055\times 10^{-3}\) \\ \hline \(k_{1}\) & \(2.368\) & \(3.206\) & \(2.787\) \\ \(k_{2}\) & \(3.919\) & \(6.363\) & \(5.141\) \\ \(k_{3}\) & \(5.184\) & \(10.29\) & \(7.737\) \\ \(k_{4}\) & \(6.288\) & \(13.77\) & \(10.029\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Examples of identified models and the average model comparison.