## 4 Simulation Study
### Investigation of the Anti-Windup Performance  
In this part, aggressive maneuvers for the longitudinal, lateral, and directional channels are performed with their respective square and doublet reference inputs. The am  
Figure 8: Response of the tailless aircraft model controlled using NDI-PI, NDI-DO, and NDI-AW with the aerodynamic bias and the center of gravity (CG) mismatch.  
plitude of the maneuvers is chosen through a trial-and-error approach to ensure that the problem of actuator saturation is excited. For the sake of simplicity, the disturbances listed in Table 4 were turned off for this simulation.  
The results are shown in Figure 9 with the corresponding responses of the aircraft model controlled by the three schemes. It turns out that the both the NDI-PI and NDI-DO schemes suffered from the large amplitude maneuvers, while the NDI-AW scheme shows significantly improved performance.  
The first remarkable response mismatch of the three schemes is observed during the time from about 10 to 15 s, i.e., the tracking performance of the NDI-PI scheme deteriorated in terms of furious coupling, overshoot, oscillation and steady-error. However, the tracking performances of the other two schemes were satisfactory and close to each other. Such an outcome was in agreement with the prospective effect of the disturbance observer. On one hand, the NDI-DO and the NDI-AW controllers estimated the lumped disturbances using Equations (32) and (48), respectively, which were then compensated using the inner-loop dynamic inversion controller of Equation (33). As a result, the influence of the lumped disturbances to the closed-loop system was mostly rejected by the disturbance observer augmented controllers. On the other hand, though transient saturation was observed for both the NDI-DO and the NDI-AW schemes, the problem of windup was not excited sufficiently to cause the closed-loop performance decay.  
Figure 9: Response of the tailless aircraft model controlled using NDI-PI, NDI-DO, and NDI-AW with the actuator saturation.  
During the bank and sideslip maneuver from 15 to 20 s, the performance of not only the NDI-PI but also the NDI-DO deteriorated significantly, while the response of the NDI-AW was still satisfactory with marginal overshoot in \(\mu\) and coupling error in \(\alpha\). Regarding the performance maintenance of the NDI-AW scheme in the presence of actuator saturation, the effect of the anti-windup modified observer was investigated. As soon as the deflections of the aerodynamic surfaces saturate, the anti-windup modification of Equation (48) in the NDI-AW scheme became active, which took the actuator induced delay and error into account. Hence, the deflection saturations induced nonlinear disturbance is estimated and compensated by the NDI-AW scheme, whereas this is not the fact for the NDI-DO scheme. As a result, the estimated lumped disturbance of the NDI-AW in Figure 9d was of much smaller magnitude than that of the NDI-DO. Furthermore, according to Equation (39), the estimation error of the disturbance observer was related to the time derivative of the disturbances. Since the lumped disturbance of the NDI-DO consists of the nonlinear, noncontinuous deflection saturations, the resulting estimation error blows up. This explains the performance decay of the NDI-DO scheme.