## 5 Numerical Simulations  
In this section, we conduct simulation studies using the MATLAB/Simulink environment. Meanwhile, two cases are applied to evaluate the performance of the proposed strategy, and the simulation results are compared with PLOS [13], the backstepping adaptive method [39], and the PID method. In addition, the following root mean squared error (RMSE) and mean integral absolute (MIA) value describing the error \(\pi_{e}\left(\tau\right)\) are considered:  
\[RMSE=\sum_{i=1}^{N}\sqrt{\frac{\left|\pi_{e}\left(\tau\right)\right|^{2}}{N}} \tag{57}\]  
\[MIA=\frac{1}{t_{\infty}-t_{0}}\int_{t_{0}}^{t_{\infty}}\left|\pi_{e}\left(\tau \right)\right|d\tau\]  
where \(N\) is the number of simulation steps. The parameters of the USV can be obtained by [29], and the initial value is \(\left[x,y,\psi\right]^{T}=\left[0,2,0\right]^{T}\), \(\left[u,v,r\right]^{T}=\left[1,0,0\right]^{T}\). The design parameters are chosen as follows: \(Q\left(t\right)=0.5I_{4\times 4}\), \(k_{x}=3\), \(k_{y}=2\), \(k_{1}=8\), \(\gamma=5\), \(\Delta_{\max}=6\), \(\Delta_{\min}=2\), \(R_{1}=0.06\), \(h_{1}=0.02\), \(\sigma_{1}=10\), \(\kappa_{1}=2\), \(\sigma_{2}=0.001\), \(\kappa_{2}=0.02\), \(\omega_{j}=1.5\left(j=1,2\right)\), \(\sigma_{1}=10\), \(K_{\Theta}=2\), \(\chi_{1}=0.1\), \(K_{n}=3\). The desired path is chosen as \(\left[x_{d},y_{d}\right]^{T}=\left[10\sin\left(0.1\theta\right)+1.1\theta, \theta\right]^{T}\). In addition, the unmodeled dynamics and the time-varying external disturbances are selected as \(\Delta f=0.3f\left(r\right)\), \(d\left(t\right)=0.8+0.4\sin\left(0.1t\right)+0.3\cos\left(0.2t\right)\), respectively. **Remark 4**. : _It is worth noting that we can improve the control ability of the controller by choosing appropriate design parameters. For example: the bandwidth \(\omega_{j}\) of enhanced TLC is selected by the pole assignment technique; \(R_{1}\) of NTD is relatively sensitive. As \(R_{1}\) becomes larger, the tracking speed becomes faster; as \(\varpi_{1}\) of LESO increases, the error \(\bar{d}\) will become smaller. Therefore, we should choose the parameters according to the actual situation._  
**Case 1. ** To verify the performance of the FPLOS guidance law, a comparison simulation is conducted between FPLOS and PLOS under the same design parameters. Simulation results are provided in Figures 5-10, and the performance indices are reported in Table 1. Figure 5 shows that the designed FPLOS provides faster convergence speed and tracking performance than PLOS. The control inputs are given in Figure 6, and it is noted that the rudder saturation problem was effectively compensated by the auxiliary system. Figure 7 depicts that the actual heading could accurately track the desired heading using the designed FPLOS. Figure 8 depicts that the predictor was capable of identifying unknown sideslip angle. The errors \(x_{e}\) and \(y_{e}\) are shown in Figures 9 and 10. It can be seen from Figure 10 that \(y_{e}\) had fast convergence performance using the proposed FPLOS. The main reason was that the lookahead distance \(\Delta\) was optimized reasonably by the fuzzy algorithm, thus achieving better convergence quality. It is shown from Table 1 that the RMSE and MIA values of FPLOS were \(\left[0.04941,0.3473,0.06539\right]\), \(\left[0.03204,0.176,0.0139\right]\), respectively, and they were only \(\left[9.42\%,53.3\%,87\%\right]\) and \(\left[74.7\%,33.65\%,30.01\%\right]\) of PLOS. Therefore, the designed FPLOS had better performance in terms of the convergence and stability. Figure 5: Path following performance. Figure 6: Control inputs \(\mu\). Figure 7: Guidance heading \(\psi_{d}\) and heading \(\psi\). **Case 2. ** Under the same condition and design parameters as Case 1, to prove the effectiveness of the designed TLC method, we provide a comparative simulation in TLC, backstepping, and the PID method.