## 4 Conclusions  
This study conducted an experimental verification of a nonlinear model predictive control with particle swarm optimization as the solver for an overhead crane with a data-driven overhead crane dynamic model. Grammar-guided genetic programming with \(l_{0}\) regularization was used to construct an input-output dynamic model of the overhead crane. The experimental data used for identification were obtained at 10 different operating points by varying the rope length and the payload mass. A total of 15 genetic programming  
Figure 8: Comparison of the NMPC-GP and DFC at operating points (**a**) \(l=1.1\) m, \(m=10\) kg; and (**b**) \(l=1.7\) m, \(m=10\) kg. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Rope Length** \\ \(I[\mathbf{m}]\) \\ \end{tabular} } & \multirow{2}{*}{**Controller**} & **Rise Time** & \begin{tabular}{c} **Settling** \\ **Time** \\ \end{tabular} & \
\begin{tabular}{c} **Maximum** \\ **Sway Angle** \\ \end{tabular} &
\begin{tabular}{c} **Residual** \\ **Sway** \\ \end{tabular} \\ \hline \multirow{2}{*}{1.1} & NMPC-GP & 5.7 & 7.1 & 0.063 & 0.005 \\  & DFC & 5.2 & 7.3 & 0.073 & 0.005 \\  & NMPC-GP & 5.8 & 6.8 & 0.045 & 0.003 \\  & DFC & 5.7 & 7.7 & 0.045 & 0.012 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The performances of NMPC-GP and DFC. runs were carried out and compared in terms of accuracy and complexity, and a model for the NMPC was selected from the Pareto frontier. The NMPC cost function was selected to be a quadratic function of the payload positioning error, the sway angle, and the control input, which was solved with the PSO algorithm. Due to the physical limitations of the overhead crane, constraints on the control input were implemented as box constraints in the PSO algorithm. To prove the efficacy of the proposed method, three different experimental cases were carried out. In the first case, the proposed nonlinear model predictive controller was tested at different operating points with the rope length ranging from 0.8 to 2.0 m and the payload mass ranging from 10 to 50 kg. The crane position rise time was within 5.4-5.8 s, while the settling time range was within 6.4-7.2 s. The residual oscillation was within the acceptable tolerance level of the linear payload deviation of \(\pm\)0.02 m for all operating points. In the second case, the payload was subjected to an external disturbance during positioning in the time interval 1.8-2.8 s. The proposed controller positioned the payload at the desired location with a settling time of 9 s and had residual oscillations within the acceptable tolerance level. In the third case, the nonlinear model predictive controller was compared with a digital feedback controller developed using the pole placement method with a data-driven linear output-error model. The NMPC had better oscillation suppression than the DFC in the transient state, as well as lower residual oscillations. This study shows that the NMPC with PSO used to minimize the cost function of a data-driven dynamic model of an overhead crane is a viable alternative for the control of overhead cranes. The proposed method was able to position the payload with varying rope lengths and payload masses, as well as in the presence of external disturbances. A comparative study with a digital feedback controller was also performed, in which the proposed method performed better in the transient state and was able to achieve a residual oscillation within the desired tolerance. The proposed identification method was only used to identify a NARX model; however, a NARMAX model would be able to model a larger class of systems by including lagged error terms. Additionally, the crane model was assumed to have high mechanical impedance, which in some industrial settings may no longer be valid. Therefore, in future work, we aim to expand the model structure from the NARX model to the more general NARMAX model structure and incorporate the coupling between the crane velocity and the payload sway into the model.