## 5 Simulation
The parameters of the FTC controller (92) were selected as follows: \(k_{1}=0.25,\,k_{2}=0.11,\,p=4,\,q=7,\,v_{1}=\text{diag}\{1.9,\,2.5,\,4.5\},\,v_{2}= \text{diag}\{3.6,\,4.8,\,1.5\},\,\zeta=0.03,\,c_{6}=1.5,\,c_{7}=1.4\), and \(c_{8}=1.4\). Figure 2 shows the evolution of residual evaluation function \(\|r\|_{2,T}\), in which the dashed line represents the fault threshold, and the solid line represents the fault detector residual norm. It can be seen that once the actuator faults happened, the designed residual exceeded the threshold \(J_{th}\) at \(t=2\) s, which could trigger the fault estimation observer to start working. At \(t=20\) s, the system entered the variable structure mode, and Figure 2b shows that as long as no fault occurs, the residual never exceeds \(J_{th}\), and therefore, the proposed fault detector could adapt to the variable structure mode. Figure 3 is the corresponding fault estimation result, and the three different types of actuator faults could be estimated accurately using the fault estimation observer (33) and the adaptive estimation algorithms (35)-(38). For the purpose of comparison, the passive non-singular terminal sliding mode-based FTC scheme designed in [32] and the FTC scheme designed in this article are adopted to simulate the same faulty attitude systems in this section. When the considered actuator fault occurs, the corresponding semi-physical simulation results (including the attitude angle \(\sigma\), and the angular velocity \(\omega\)) using the proposed FTC scheme are depicted in Figure 4, and it is seen from Figure 4 that the simulation results using the proposed FTC scheme developed in this paper had fast and accurate tracking performance. The comparison with the scheme of [32] is in Table 2, which shows that the passive non-singular terminal sliding mode-based FTC scheme could not make the output converge, and the scheme of this study had obvious advantages. Figure 3: The actuator fault estimation results. (**a**) Estimation results of LOE fault \(e_{2}\). (**b**) Estimation results of LOE fault \(e_{4}\). (**c**) Estimation results of bias fault \(f_{3}\). Figure 2: Fault detection results. (**a**) Detection results in the case of failure. (**b**) Detection results without failure. The biggest difficulty of this study was to take into account the nominal mode and the variable structure mode. Because of the uncertainty brought by the variable structure, this study designed many harmonic functions that were nearly twice as much as the nominal control to solve this problem. Finally, the effect of overshoot was limited and the final parameter setting was successful. To further emphasize the superiority of the proposed method, the control performance comparisons using three different FTC schemes are also given in Tables 2-5. Where \"--\" indicates that the curve does not converge and therefore has no convergence time, \"\(\infty\)\" indicates that the curve diverges and thus the maximum steady precision tends to infinity as time increases. \begin{table}
\begin{tabular}{c c c c c} \hline
**FTC Scheme** & **Convergence Time: \(\sigma\)** & **Steady Precision: \(\sigma\)** & **Convergence Time: \(\omega\)** & **Steady Precision: \(\omega\)** \\ \hline FTC without variable & — & \(\infty\) & — & \(\infty\) \\ structure method & — & \(\infty\) & — & \(\infty\) \\ FTC in this paper & 5.1 & \(4\times 10^{-5}\) & 5.2 & \(6\times 10^{-5}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Comparative analysis of performance with proposed FTC without variable structure harmonic functions. Figure 4: Variable structure FTC curves in multiple flight modes. (**a**) FTC response curves of attitude angles. (**b**) FTC response curves of angular velocities.