## 4 Simulation Verification and Discussion  
### Influence of Model Mismatches and Sensor Noises  
To assess the control performance of the proposed control structure for supersonic flight, the research aircraft is first trimmed at an altitude of 10,000 m and a speed of 1.5 Mach. The resulting lateral-directional modes of the aircraft at the trimmed point are presented in Table 2. As shown, all modes are stable. However, the Dutch-roll mode exhibits a damping ratio of only approximately 0.06, which is roughly ten times lower than the minimum damping ratio of 0.4 required for level 1, class IV flight vehicles, according to military specifications [40]. Additionally, the frequency of this mode is below the required level of 1 rad/s. Figure 9: Inner-loop control structure. The actuator is modeled as a second-order linear system with deflection angle and rate limits, as specified in Table 3. The sampling rate is set to \(\Delta t=0.005\) s, which corresponds to the operation step of the digital flight control computer. The parameter \(K_{\phi}\) in the NDI control was selected as 2 to satisfy the bank angle tracking velocity requirement, and the remaining control parameters are presented in Table 4.  
The gain of the error controller is selected as slightly greater than the reference model gain to ensure that the PCH gain is a positive number. The estimated natural frequency and damping ratio of the actuator are selected to be the same as the actuator model used for simulation. For angular acceleration estimation, the bandwidth of the ESO estimator is chosen to be 30 rad/s, which should establish a tradeoff between noise suppression and estimation accuracy. The number of control parameters that need to be adjusted is quite small, and each of them is connected to the bandwidth principle in the frequency domain, which can be easily chosen under physical restrictions. It should be noted that the onboard model calculation depicted in Figure 9 is a simplified version designed to decrease the reliance on specific models. This simplified calculation omits the consideration of damping and static aerodynamic coefficients and focuses solely on the control efficiency parameters. This approach is based on Equation (33) and results in the following simplified expression:  
\[\dot{x}_{mc}=v_{cmd} \tag{42}\]  
To explain it another way, the unaccounted-for dynamics are treated as a component of the overall disturbance that is to be estimated by the ESO. As a result, the ESO is capable of achieving a high degree of accuracy in estimating the disturbance. Figure 10 displays the results of the nominal simulation for the ESO-based INDI controller. As depicted in Figure 10a, the estimated roll angular acceleration closely tracks the actual value computed by the aircraft model's dynamic equations. Despite the significant deviation between the simplified onboard model results and the actual value, the ESO's estimates demonstrate high accuracy. As shown in Figure 10b, the estimated yaw angular \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Channel** & \(K_{r(\cdot)}\) & \(K_{e(\cdot)}\) & \(\overset{\wedge}{\omega}_{a}\) **(rad/s)** & \(\hat{\xi}_{a}\) &
\begin{tabular}{c} **ESO \(\omega_{o}\)** \\ **(rad/s)** \\ \end{tabular} \\ \hline Roll & 6 & 8 & 60 & 0.7 & 30 \\ Yaw & 3 & 5 & 60 & 0.7 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Control parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Value** \\ \hline Dutch-roll mode & \(-0.561\pm 8.76i\)* \\ Roll mode & \(-3.4\) \\ Spiral mode & \(-0.0157\) \\ Dutch-roll mode frequency (rad/s) & 8.78 \\ Dutch-roll mode damping ratio & 0.0639 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristics of the lateral–directional modes. acceleration exhibits good steady-state accuracy, although the transients are not as sound as those of the roll channel. Furthermore, the angular acceleration generated by the rudder is even opposite to the actual value because the rudder deflection generates the side force, which produces the angular acceleration.