## 7 Conclusions  
This paper has discussed the construction of a novel identification method based on ANFM for approximating the inverse dynamics of the ship's course motion. Firstly, the simulation experiments of ship inverse dynamics have been carried out and obtained two groups of different ANFM structures by using sine wave curve and sawtooth wave curve as input learning samples of ANFM, respectively. The results have shown that the ANFM had a strong approximation capability, with its output rudder angle almost always covering the ideal rudder angle. Secondly, the trained ANFM has been used as inverse controller, and the fractional-order PI\({}^{\lambda}\)D\({}^{\mu}\) controller has been connected in parallel for course tracking control of underactuated ships. The simulation results have shown that the proposed method can effectively overcome the influence of uncertainty of ship modeling parameters, track the desired course quickly and effectively, and had a good control effect. Finally, in ship course control, the effectiveness has been further evaluated by comparative experiments of four different controllers, which are the PID controller, the PID controller based on PSO algorithm, the FO PI\({}^{\lambda}\)D\({}^{\mu}\) controller based on GA algorithm and the FO PI\({}^{\lambda}\)D\({}^{\mu}\) controller based on ANFM (proposed).In terms of future works, because the system needs to set the trend of uncertain parameters artificially, and the control conditions and training conditions should be the same, but tend to be idealized, the online control mode after offline learning shows certain limitations. This method does not guarantee good control when the uncertainties no longer vary according to the established working conditions. To solve this problem, the next step is to change to an online control model, where modeling and control are carried out simultaneously to achieve full adaptivity of the intelligent controller in order to improve the generalization capability and control efficiency of ANFM and to cope with the effects of non-normative uncertainties.  
Conceptualization, G.L.; methodology, G.L.; software, H.C.; validation, G.L. and B.C.; data curation, G.L.; writing--original draft preparation, G.L.; writing--review and editing, W.D.; project administration, B.C.; funding acquisition, G.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Science Researching Plans of Liaoning Province Education Department under Grant JDL2019003, the National Natural Science Foundation of China under Grant 61803066 and U2133205, and Research Foundation for Civil Aviation University of China (3122022 PT02).  
Not applicable.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Controller Type** & **Adjust Time \(t_{s}\)** & **Rise Time \(t_{r}\)** & **Overshoot \(M\)** & \(E_{ss}\)** \\ \hline ANFM -FOID (proposed) & 32 & 18 & 4.27\% & 0.009 \\ PSO-PID & 416 & 67 & 31.16\% & 0.024 \\ PID & 519 & 71 & 14.25\% & 0.007 \\ GA-FOID & 163 & 65 & 9.53\% & 0.086 \\ \hline \end{tabular}
\end{table}
Table 4: Performance of four different controllers.