## 5 Discussions  
To validate the efficacy and superiority of the proposed spin recovery controller, numerical simulations are conducted in this section using the F-18 HARV model. Parameter values are borrowed from [12]. The initial and final conditions for simulation are obtained from bifurcation analysis results. In Ref. [12], the spin solutions of the aircraft have been obtained by bifurcation analysis. As a result, the findings obtained from this analysis are not reiterated in the present paper. The initial and final reference points are listed in Tables 1 and 2. To evaluate the performance of the spin recovery control approach, we use the NDI and finite-time SMC spin recovery methods as comparative methods [12; 18]. The simulations are conducted under two different initial spin values (oscillatory and steady). Spin recovery control is attempted after 10 s of spin motion. During the first 10 s, the control inputs are held in the fixed position, referring to Table 2. Then, the control inputs in Equation (18) are applied at t = 10 s to converge the control variables \(\left(\phi,\theta,\dot{\psi},\dot{h}\right)\) to their desired states \(\left(\dot{\psi}_{d},\theta_{d},\dot{\psi}_{d},\dot{h}_{d}\right)\). The spin recovery simulation results are exhibited in Figures 2 and 3, while the computed control commands are shown in Figure 4.  
It is apparent from Figure 2 that the proposed spin recovery controller can provide a faster convergence rate compared to NDI and the finite-time SM spin recovery control method. It can be observed that the convergence time for two initial conditions is very close to each other, indicating that the characteristic is independent of the initial conditions. It is observed from Figure 2g-i that the response curves of \(\phi\), \(\theta\), and \(\dot{\psi}\) can effectively track the desired values. For the oscillatory spin recovery, the \(\phi\) and \(\theta\) converge to the desired states at t = 28.92 s and t = 30.37 s, respectively. For the steady spin recovery, the \(\phi\) and \(\theta\) converge to the desired states at t = 23.51 s and t = 24.8 s, respectively. In the case of the oscillatory spin recovery, the aircraft spin rotation stops after completing 1.75 left turns at t = 23.35 s, as shown in Figure 2i. For the steady spin recovery, the aircraft spin rotation stops after completing 0.75 left turns at t = 14.01 s, as shown in Figure 2i. Thus, it is evident that the convergence time of \(\dot{\psi}\) is less than that of \(\phi\) and \(\theta\), indicating that the aircraft spin rotation stops well before achieving level flight. More explicitly, \(\dot{H}\) achieves the desired altitude rate for oscillatory and steady spin recovery as shown in Figure 2i at H = 1077 m \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**State Variables**} & \multicolumn{3}{c}{**Initial Spin States**} & \multirow{2}{*}{**Final Level Flight States**} \\ \cline{2-3} \cline{5-5}  & **Oscillatory** & & **Steady** \\ \hline \(V\) (m/s) & 194.39 & 221.16 & 67.3 \\ \(\alpha\) (rad) & 1.23 & 1.07 & 0.3 \\ \(\beta\) (rad) & 0.037 & 0.02 & 0 \\ \(p\) (rad/s) & \(-\)0.48 & \(-\)0.37 & 0 \\ \(q\) (rad/s) & 0.052 & 0.13 & 0 \\ \(r\) (rad/s) & \(-\)1.36 & \(-\)0.74 & 0 \\ \(\phi\) (rad) & \(-\)0.038 & \(-\)0.17 & 0 \\ \(\dot{\theta}\) (rad) & \(-\)0.34 & \(-\)0.46 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial and final state variables. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Control Variables**} & \multicolumn{3}{c}{**Initial Spin States**} & \multirow{2}{*}{**Final Level Flight States**} \\ \cline{2-3}  & **Oscillatory** & & **Steady** \\ \hline \(\delta_{e}\) (rad) & \(-\)0.16 & \(-\)0.43 & \(-\)0.058 \\ \(\delta_{a}\) (rad) & 0 & 0 & 0.41 \\ \(\delta_{r}\) (rad) & 0 & 0 & \(-\)0.076 \\ \(\delta_{t}\) & 0.43 & 0.59 & 0.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial and final control variables. and H = 1463 m, respectively.