## 5 Simulation Result and Discussion
### NDI Control Simulation Based on Real-Time Aerodynamic Parameter Identification  
This section first uses the conventional NDI controller for simulation and compares the control effect between the NDI controller and the PID controller under the step pitch angle command.  
Figure 8: Real-time parameter identification results. subplot (**a**,**b**) are the derivatives of the pitch moment coefficient to the deflection angle and angle of attack of the elevator, respectively.  
After the recursive identification achieves convergence, The control switch will trigger the controller switching to NDI control mode, then use the PID controller and NDI controller to track the pitch angle step command, respectively. Since the accuracy of identification has been explored and explained in detail in the previous section, this section only shows the image of the instruction tracking part.  
In this section, the initial state of the aircraft is given as follows, and the step instruction is \(10^{\circ}\). Table 3 shows the simulation initial conditions.  
Figure 9 is the NDI controller and PID controller pitch angle control effect comparison.  
In the simulation diagram, it can be clearly seen that compared with the PID controller, the NDI controller responses faster with a smaller overshoot. From the elevator deflection angle image and the pitch angular rate image, it can be seen that the NDI controller can simultaneously achieve a larger angle of the rudder deflection and faster adjustment response when controlling the aircraft to track the step pitch angle command, so as to achieve a better control effect. In order to compare the differences between the two numerically, three indicators of the rise time, peak time, and overshoot are selected to quantitatively compare the control effects of the two.  
The definitions of the three indicators are:  
1. [label=(0)]
2. Overshoot: The ratio of the instantaneous maximum deviation (\(\theta_{emax}\)) to the steady-state value (\(\theta_{\infty}\)), expressed in a percentage.
3. Rise time: The time required for the response curve to rise from 10% of the steady-state value to 90% of the steady-state value.
4. Peak time: Time it takes for a step response curve to exceed its steady-state value to reach its first peak.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Symbol** & **Value** \\ \hline pitching angle & \(\theta\) & \(7.25^{\circ}\) \\ angle of attack & \(\alpha\) & \(7.25^{\circ}\) \\ elevator & \(\delta_{de}\) & \(-7^{\circ}\) \\ altitude & \(H\) & \(7500\) m \\ speed & \(V\) & \(150\) m/s \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulation initial state.  
Figure 9: _Cont._  
Table 4 shows the quantitative comparison of simulation results.  
According to the three indicators defined here, the control effects of different controllers can be quantitatively compared. Combined with simulation images and numerical analysis, it can be seen that compared with the traditional PID controller, the adaptive NDI controller can make the aircraft converge to the target command faster, and the response  
\begin{table}
\begin{tabular}{c c c} \hline
**Description** & **PID Controller** & **NDI Controller** \\ \hline rise time(s) & 0.39 & 0.36 \\ peak time(s) & 0.61 & 0.55 \\ overshoot (\%) & 35.21 & 26.78 \\ \hline \end{tabular}
\end{table}
Table 4: Control effect index comparison (NDI vs. PID).  
Figure 9: Comparison of NDI and PID control effects. subplot (**a–c**) are pitch angle, elevator deflection angle and pitch angle velocity, respectively.  
speed is faster, the overshoot is smaller, and the convergence speed is faster, indicating that the adaptive NDI controller has better control effect.