## 4 Simulation Results and Analysis
### Hybrid Adaptive Compensation Simulation of Sliding Mode NDI  
Set the reference signal as  
\[\left\{ \begin{array}{l}\alpha_{c}=4\times\frac{\pi}{180}rad\\ q_{c}=0 rad/s\\ \theta_{c}=3.27\times\frac{\pi}{180}rad\end{array} \right.\]  
the initial condition as \(\alpha=0\) rad, \(q=0\) rad/s, \(\theta=0\) rad. The uncertainty mainly affects the aerodynamic parameters \(M_{z}^{\alpha}\), \(M_{z}^{\delta_{c}}\), \(M_{z}^{\delta_{x}}\), \(M_{z}^{\delta}\), \(M_{z}^{\delta}\), from the state matrix \(A\), \(B\), \(C\), and \(D\) of the system, it can be known that these four aerodynamic parameters mainly affect the pitch rate channel and the system output \(\dot{q}\), thereby indirectly affecting the normal overload. Therefore, the influence of uncertainty on the system is mainly manifested in the change in pitch angle and speed. Here, the uncertainty range is enlarged to \(\pm 20\%\), and the uncertainty of the pitch moment of inertia \(I_{y}\) takes the following form:  
\[I_{y}=I_{y0}(1+\Delta I_{y}(2 rand(1,1)-1)) \tag{61}\]  
The compound disturbance settings are as follows:  
\[d=[0.05 sin(0.1t),\ 0.1 sint,\ 0.1 sin(10t)]^{T} \tag{62}\]  
Figure 6: Nominal NDI for HFV under modeling error. (**a**) Tracking curve of the normal overload \(N\)_vs_. (**b**) Tracking curve of pitch rate derivative \(\dot{q}\). (**c**) Tracking curve of the pitch derivative \(\dot{\theta}\). (**d**) Tracking curve of the attack angle derivative \(\dot{a}\). (**e**) Tracking curve of the track inclination \(\mu\). In the proposed adaptive sliding mode controller, select \(P=\text{diag}(15,\,2,\,0)\), \(R_{1}=\text{diag}(0.5,\,0.5,\,0.5)\), \(R_{2}=20\). Due to high-level military secrecy, only a part of the controller parameters is presented in this study. The simulation results of NDI control, adaptive NDI control, and adaptive sliding mode NDI control are compared for the longitudinal system with \(\pm 20\%\) moment of inertia uncertainty and are shown in Figure 7, where \"DOB\" means the disturbance observer. From Figure 7, it can be seen that when the system has inertia uncertainty and elastic modal disturbance, the adaptive NDI controller can reduce the oscillation of each channel. The improved adaptive sliding mode NDI controller can more obviously suppress the oscillation of the attitude angle and angular velocity channels, effectively compensate for the disturbance and uncertain parameters of inertia, and ensure the stability and tracking performance of the HFV. Table 3 is the tracking performance table based on the statistics of Figure 6. In view of the general definition of overshoot, only Figure 6e-g has overshoot [30]. The three controllers in the table represent the three stages of improving NDI in this paper: nominal, adaptive, and adaptive sliding mode. Figure 7: Cont. The shortest response time (\(\rm t_{s,min}\)) of the hybrid compensation NDI, that is, the adaptive sliding mode controller, is only 32% of the nominal controller, the maximum steady-state error (\(\rm e_{ss,max}\)) is only 1% of the nominal controller, the overshoot (\(\rm o_{v,max}\)) is only 24% of the nominal controller. Other indicators have also been greatly improved, so the active-passive hybrid compensation NDI is more suitable for high maneuvering HFV. \begin{table}
\begin{tabular}{c c c c} \hline  & **Nominal NDI** & **Adaptive NDI** & **Adaptive Sliding Mode NDI** \\ \hline \(\rm t_{s,max}\) & 7.95 & 7.99 & 5.28 \\ \(\rm t_{s,min}\) & 5.01 & 5.10 & 1.61 \\ \(\rm e_{ss,max}\) & 0.03 & 0.001 & 0.0004 \\ \(\rm e_{ss,min}\) & 0.0003 & 0.0002 & 0.0001 \\ \(\rm o_{v,max}\) & 30.56\% & 13.89\% & 7.50\% \\ \(\rm o_{v,min}\) & 18.18\% & 0.005\% & 0.03\% \\ \hline \end{tabular}
\end{table} 
Table 3: Comparative analysis I of performance indicators between proposed algorithms. Figure 7: Comparison of the effects of different HFV controllers under inertia uncertainty and disturbance. (**a**) Tracking curves of the normal overload _Nys_. (**b**) Tracking curves of pitch rate derivative \(\dot{\vartheta}\).