## 4 Simulation Verification and Discussion
### Flying Quality and Stability Margin Evaluation  
To investigate the influence of the cut-off frequency \(\omega_{n}\) on flying quality, the CF-based INDI controller is analyzed using a low-order equivalent system (LOES) based on linear analysis. The evaluation of the lateral closed-loop system, following MIL-STD-1797A specifications [44], includes the equivalent gain \(n\)\(\kappa_{\Phi}\), Dutch-roll damping \(\zeta_{d}\), Dutch-roll frequency \(\omega_{d}\), roll time constant \(T_{r}\), spiral root \(T_{s}\), and equivalent time delay \(\tau_{\Phi}\).  
In Figure 15, it can be observed that the flying quality assessment results are at level 1, indicating satisfactory flight quality. Moreover, as the cut-off frequency \(\omega_{n}\) increases, the aircraft's mode frequency also increases. However, as shown in Figure 16, the phase margin first increases and then decreases with an increase in \(\omega_{n}\). This indicates that a very large value of \(\omega_{n}\) should be avoided, as it may lead to a decay in the phase margin, posing a risk to the stability of the controller.  
Evaluation indicators, such as gain and phase margin, are crucial for practical engineering applications of the ESO-based INDI controller. The B  
Figure 16: Variation of gain and phase margins at the breakpoint of the actuators with the cut-off frequency.  
Figure 15: Results of equivalent system analysis as a variation of the cut-off frequency.  
aileron and rudder input points are presented in Figure 17a,b, respectively. Remarkably, the crossover frequency remains largely unchanged, even in the presence of the parameter uncertainties listed in Table 5. Additionally, the gain curve exhibits a consistent slope around the cross point, indicating the robustness of the controller under parameter perturbations.  
Figure 18 presents the boxplots of the gain and phase margins. The lower and upper edges of the boxes correspond to the 25 th and 75 th percentiles, respectively. On the basis of the analysis of 1000 samples, it can be observed that the gain margins exceed 6 dB, and the phase margins exceed 45 degrees within the range of 25-75% of the samples. These results indicate that the gain and phase margins of the system meet the required flying qualities of 6 dB and 45 degrees, respectively [45].  
To further evaluate the impact of time delay in the angular acceleration signal on the stability margin, we add the delay to the feedback of the roll and yaw angular rate signals. The angular acceleration is estimated by ESO with gyroscope measurements. For convenience, we apply a first-order Pade approximation to the delay, which is most accurate at low frequencies [42].  
As seen in Figure 19a,b, the phase margin is more sensitive to time delay compared with the gain margin, and it decays more rapidly with an increasing delay. Even with a delay of 25 ms, the gain margin remains greater than 6 dB, while the phase margin drops below 30 degrees. As described in Section 3.4, it is essential to consider the delay in the synchronization filter design. As shown in Figure 19a,b, incorporating the delay approximation into the synchronization filter significantly improves both the gain and phase margins.  
Figure 17: Bode diagram breaks at the actuators: (**a**) Aileron and (**b**) Rudder.  
Figure 18: Gain and phase margins at the actuator: (**a**) Gain margins and (**b**) Phase margins.  
Even with a 50 ms delay, the gain and phase margins increase by approximately 4 dB and 20 degrees, respectively.