## 4 Simulation
### Robust Tracking Control Performances with Uncertainty  
To highlight the superiority of the proposed ADFTNs, the two NN-based observers (NNOBs) in [34] with the proposed approximate optimal control polices in (29) (denoted as ADPNNs) were employed to compare with the ADPFTNs. The uncertain parameter values were set as \(\Delta_{m}=0.2+0.05\cdot\text{rand kg}\), \(\Delta_{J_{x}}=\Delta_{J_{y}}=\Delta_{J_{z}}=0.03+0.01\cdot\text{rand kg}\cdot \text{m}^{2}\), and \(\Delta_{k_{1}}=\Delta_{k_{2}}=\Delta_{k_{3}}=\Delta_{k_{4}}=\Delta_{k_{5}}= \Delta_{k_{6}}=0.005+0.001\cdot\text{rand}\). The rand function generates a random number in the interval \([0,1]\). \(\mathbf{u}_{d}=[0.1\text{sin}(0.8t),0.1\text{sin}(0.7t),0.1\text{sin}(0.5t)]^{\text {T}}\) N and \(\mathbf{\tau}_{d}=[0.1\text{sin}(0.8t)+J_{y}\dot{\theta},0.1\text{sin}(0.7t)-J_{y} \dot{\phi},0.1\text{sin}(0.5t)]^{\text{T}}\) N \(\cdot\text{m}\) with \(J_{y}\dot{\theta}\) and \(J_{y}\dot{\phi}\) being gyroscopic moments of the quadrotor UAV. The parameters of the ADPFTNs are given in Table 1.  
Figure 3 shows the time response of the observation errors for the FTNNOBs and NNOBs, respectively. It can be observed that the observation error of the FTNNOBs stabilizes to the order of \(10^{-4}\) within 0.5 s, while that of the NNOBs stabilizes to the order of \(10^{-3}\) around 5 s. It can be concluded that the FTNNOBs have faster convergence performance and higher stabilization accuracy to attenuate external disturbances and parameter uncertainties in comparison with the NNOBs. Figure 4 shows the time response of the tracking errors for the ADPFTNs and the ADPNNs, respectively. For a fair comparison, we adjusted the convergence times of the two control schemes in the simulation to make them approximately equal (Section 4.2 performs the same setup). As shown in Figure 4, the tracking errors' convergence curve of the proposed ADPFTNs is smoother than that of the ADPNNs and has higher convergence accuracy. This means that, when there are external disturbances and parameter uncertainties in the quadrotor UAV system, the FTNNOBs can quickly and accurately estimate the unknown uncertainties and compensate for the control policies. To illustrate the tracking performance of the two control schemes more directly, the 3D trajectory tracking results are shown in Figure 5. It can be observed that both the ADPFTNs and the ADPNNs can successfully track the desired trajectory. Further, it can be observed that the red solid line (the proposed ADPFTNs in this work) is more consistently close to the black solid line (desired trajectory) compared to the red dotted line (ADPNNs in [34]). Figure 6 shows the time responses of the AC NNs' weights. One can see that the curves of the position error subsystem and attitude error subsystem stabilize after 5 s. Figure 7 shows the control inputs under the frame \(\xi_{E}\).  
\begin{table}
\begin{tabular}{c c c} \hline
**Variable** & **Position Error System (i = 1)** & **Attitude Error System (i = 2)** \\ \hline \(r_{i1}\) & \(1\times 10^{-6}\) & \(1\times 10^{-6}\) \\ \(r_{i2}\) & \(1\times 10^{-5}\) & \(1\times 10^{-5}\) \\ \(k_{i1}\) & 0.01 & 0.01 \\ \(k_{2}\) & 1.2 & 1.2 \\ \(\mathbf{Q}_{i}\) & \(4\mathbf{I}_{6}\) & \(10\mathbf{I}_{6}\) \\ \(R_{i}\) & \(2.5\mathbf{I}_{3}\) & 513 \\ \(r_{3}\) & 1.6 & 1.6 \\ \(r_{4}\) & 0.2 & 0.2 \\ \(c_{i}\) & 200 & 700 \\ \(c_{i1}\) & 0.06 & 0.06 \\ \(m_{i}\) & 97 & 99 \\ \(n_{i}\) & 99 & 101 \\ \(p_{i}\) & 105 & 105 \\ \(q_{i}\) & 99 & 101 \\ \hline \end{tabular}
\end{table}
Table 1: Parameters’ selection.  
Figure 4: _Cont._  
Figure 3: Time response of the observation errors by different observers.  
Figure 4: Time response of tracking errors under ADPFTNs and ADPNNs.  
Figure 5: Three-dimensional trajectory tracking results under different control schemes.  
Figure 6: Time responses of AC NNs’ weights of ADPFTNs.  
Figure 7: Time response of control inputs under ADPFTNs and ADPNNs.