## 4 Simulation Study
### Comparison of the Robust Performance  
The response of the tailless aircraft model controlled using the NDI-PI, NDI-DO, and NDI-AW with the sensor noise is shown in Figure 7. For comparison, the nominal response of the NDI-DO was also plotted with blue dotted lines. It turns out that the noise was not blown up for the three schemes, i.e., the steady responses of \(\alpha\), \(\mu\), and \(\beta\) were remarkably influenced by the sensor noise. However, it is to be mentioned that the disturbance observers succeeded to capture the disturbances induced by the bank maneuver as shown in Figure 7d. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Distribution** & **Mean** & **Standard Deviation** \\ \hline Noise of & & & \\ \(\alpha\), \(\beta\), \(\mu\) & normal & 0 & 2.0\({}^{\circ}\) \\ Noise of & & normal & 0.4\({}^{\circ}\)/s \\ \(p,q,r\) & & & \\ Bias of \(C_{i0}\), \(\Delta c_{i}\), \(\hat{C}_{i_{k}}\) & normal & 0 & 30\% \\ Mismatch of CG & & normal & 0 & 5 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Distributions for parameters affecting the robustness. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Coefficient** & **NDI-PI** & **NDI-DO** & **NDI-AW** \\ \hline \(K_{p},K_{q},K_{r}\) & 20.0, 20.0, 10.0 & 10.0, 10.0, 5.0 & 10.0, 10.0, 5.0 \\ \(K_{p},K_{p},K_{p},K_{p}\) & 2.0, 2.0, 2.0 & 2.0, 2.0 & 2.0, 2.0 \\ \(K_{k_{1}},K_{k_{2}},K_{l_{y}}\) & 0.1, 0.1, 0.1 & 0.2, 0.2, 0.2 & 0.2, 0.2, 0.2 \\ \(l_{p},l_{q},l_{r}\) & — & 15.0, 15.0, 15.0, & 15.0, 15.0, 15.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Controller and observer gains. Figure 8 presents the robust simulation testing with the aerodynamic bias and the CG mismatch following the distributions in Table 4. Responses of five realizations in the distribution space are plotted for comparison with the nominal response of the NDI-DO controller. Figure 6: The nominal response of the tailless aircraft model controlled using NDI-proportional-integral elements (PI), nonlinear disturbance operator based NDI (NDI-DO), and NDI-anti-windup (AW). Figure 7: Response of the tailless aircraft model controlled using NDI-PI, NDI-DO, and NDI-AW with the sensor noise. It is noted that the NDI-PI controller suffers from the proposed disturbances and the tracking performance is unsatisfactory. As discussed in ref [36], it is the CG mismatch, especially along the aircraft's longitudinal axis, which will bring remarkable external disturbances to the aircraft's flight dynamics. As a result, the step responses of NDI-PI are scattered as shown in Figure 8a,b. In the case of \(\alpha\), remarkable response undershoot or overshoot is observed for each realization ranged from about \(-34\%\) to \(+68\%\) of the reference step signal, and the integrators in the outer loop controller of Equation (29) tend to eliminate the steady error gradually. In the case of \(\mu\), a significant tracking error is caused not only by the bank maneuver but also by the coupling effect of the pitch maneuver. In the case of \(\beta\), the performance is satisfactory with a slightly increased magnitude of coupling error. Compared with the NDI-PI controller, the NDI-DO and NDI-AW controllers are not sensitive to the aerodynamic biases or the CG mismatch. The \(\alpha\) and \(\mu\) responses are recovered close to that of the nominal case, except that the overshoots and damping deteriorate marginally, and the coupling influence of \(\mu\) maneuver to \(\alpha\) and \(\beta\) increases slightly. It is the disturbance observers of Equations (32) and (48) that estimate the aerodynamic biases and the CG mismatch as a set of lumped disturbances, respectively, as shown in Figure 8d. Additionally, then, the estimated lumped disturbances are substituted into Equation (33) for dynamic inversion compensation. As a result, the disturbance rejection abilities of NDI-DO and NDI-AW have been improved significantly. Besides, the notable coupling error is observed in \(\mu\) and \(\beta\) at about 6 s and 26 s, when the aircraft model nose up or nose down rapidly.