## 4 Simulation  
In this section, some simulations are carried out using the proposed control method in two different cases to verify the effectiveness. The AUV mathematical model parameters are set as shown in Table 2.  
The control method design parameters are selected as shown in Table 3, including parameters for NESO, the command integral filter, and control law. Case 1: Smooth path tracking and the reference trajectory is chosen as \(x_{t}=-100\sin(0.01t)\) and \(y_{t}=-100\cos(0.01t)\). The system disturbances are chosen as follows:  
\[d_{1}=0.2+0.3\sin(0.05t)+0.2M_{1}(X_{u}u+a_{23}vr)\]  
\[d_{2}=0.2+0.1\sin(0.05t)+0.2M_{3}(N_{r}r+a_{23}uv),\]  
which denote the external disturbances and system uncertainties. In this part, the simulation mainly demonstrates the effectiveness of the designed controller concerning system disturbance, comparing the designed controller to the one without NESO. Simulation results are presented in Figures 3-6. Figure 3 plots the trajectory tracking results. Figure 4 plots trajectory tracking errors referring to the trajectory angle, surge speed, sway speed, and pitch rate. Figure 5 plots the results of disturbance observation, and Figure 6 plots the corresponding control input in the surge direction and in the yaw direction. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** & **Units** \\ \hline \(M_{1}\) & 0.25 & kg\({}^{-1}\) \\ \(M_{2}\) & 0.2 & kg\({}^{-1}\) \\ \(M_{3}\) & 0.2 & kg\({}^{-1}\) \\ \(X_{u}\) & 20 & kg/s \\ \(Y_{v}\) & 45 & kg/s \\ \(N_{r}\) & 30 & kg \(\cdot\) m\({}^{2}\)/s \\ \(a_{12}\) & 20 & kg \(\cdot\) m\({}^{2}\) \\ \(a_{13}\) & 40 & kg \(\cdot\) m\({}^{2}\) \\ \(a_{23}\) & 60 & kg \(\cdot\) m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: AUC mathematical model parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Description** & **Symbol** \\ \hline Parameters of the NESO & \(g=40\), \(k_{lit}=100\), \(k_{11}^{so}=100\), \(k_{12}^{so}=200\), \(k_{21}^{\text{eso}}=100\), \(k_{22}^{\text{eso}}=400\) \\ \hline Parameters of the command integral filter & \(\xi=0.9\), \(\omega_{n}=20\), \(H_{M}=1000\), \(H_{R}=500\) \\ \hline Parameters of the control law & \(c_{1}=c_{2 Figure 4: _Cont._  
Figure 3: Simulation results of the trajectory tracking in case 1 and the corresponding position errors. From the simulation results provided in Figures 3-6, it can be seen that the NESO could estimate the system uncertainties effectively, and the AUV could track the desired trajectory  
Figure 4: Simulation results of some important variables in case 1. Figure 5: Estimated values for system disturbances by NESO. Figure 6: Control signals \(T\) and \(\tau\) in case 1.  
in the presence of system uncertainties. Moreover, was found that the NESO-based adaptive controller performed better than the one without NESO in trajectory tracking. Therefore, the proposed method has good robustness in the face of system uncertainties. Case 2: Piece-wise continuous path tracking and the reference trajectory are chosen as follows:  
\[\left\{ \begin{array}{l}x_{t}(t)=t,y_{t}(t)=t+50,\,(0\leq t<50)\\ x_{t}(t)=t,y_{t}(t)=100,\,(50\leq t<100)\\ x_{t}(t)=100\sin((t-100)\pi/200)+100\,y_{t}(t)=100\cos((t-100)\pi/200)+100,\,( 100\leq t<200)\\ x_{t}(t)=50\sin((t-250)\pi/100)+250\,y_{t}(t)=-50\cos((t-250)\pi/100)+250,\,( 200\leq t<250)\\ x_{t}(t)=250+1.5(t-250)\,y_{t}(t)=-50,\,(250\leq t<300)\\ x_{t}(t)=25+t\,y_{t}(t)=t-350,\,(300\leq t<400)\end{array} \right..\]  
In this part, the simulation mainly demonstrates the effectiveness of the designed controller for nonlinear trajectory tracking with sudden changes, which compared the designed controller and the sliding mode control used in the literature [35]. The simulation results are presented in Figures 7 and 8. Figure 7 shows that the actual trajectory could follow the reference trajectory with the position error in a small range. Figure 8 plots trajectory tracking errors referring to the trajectory angle, surge speed, sway speed, and pitch rate.