## 4 Results and Discussion
### Estimation Accuracy--Head-on Gusts
Figures 35-37 indicate that time constants of 0.8 s and 1.0 s provided the most accurate estimation results relative to the reference measurement. Analyzing the arrows labeled with numerical values for the selected rising gust speed in Figure 37 enables a computational comparison. With a time constant of 0.5 s, the peak gust value in the full estimation falls 3.94 m/s below the reference, while the low-frequency estimation shows an even larger discrepancy of 5.69 m/s. Both estimation methods exhibit a delay of 0.71 s relative to the reference. Increasing the time constant to 0.8 s decreases the peak value difference to 3.02 m/s, without affecting the delay in the full estimation. A further increase to 1.0 s reduces the discrepancy to 2.47 m/s, again with the same delay. Although a time constant of 1.5 s yields the smallest difference in the peak value, it also significantly inflates the minimum values, which is discussed below. Examining the numerical annotations on the arrows in Figure 36 allows for an accurate comparison of the minimum values in the selected weakening gust scenario. At a time constant of 0.5 s, the full estimation shows a minimal deviation of 0.05 m/s, while the low-frequency estimation differs by 0.41 m/s. The respective delays compared to the reference measurement are 0.25 s and 0.41 s. Increasing the time constant to 0.8 s results in a greater underestimation of 0.46 m/s, with a nearly unchanged delay of 0.26 s. Further increases to 1.0 and 1.5 s lead to increasingly larger underestimations. Figure 34: Comparison of the gust value with the estimation results—level flight through a gust area; single selected gust (complementary filter, (**A**) T\({}_{\text{cu}}\) = 0.5 s, (**B**) T\({}_{\text{cu}}\) = 0.8 s, (**C**) T\({}_{\text{cu}}\) = 1.0 s, (**D**) T\({}_{\text{cu}}\) = 1.5 s). In contrast to the previously analyzed flight states for the case of climb, there are differences in the most favorable results for the gust with increasing velocity (T\({}_{\rm cu}\) = 1.5 s) and the gust with decreasing velocity (T\({}_{\rm cu}\) = 0.5 s). It would be necessary to develop criteria and a methodology for accurately identifying the optimal time constant for this flight phase. The headwind estimation results were evaluated by comparing them to the low-frequency signal derived from the recorded ideal gusts measurements. Analyzing the above graphs, it can be concluded that the applied estimation method allows for a more accurate representation of the gusts shape, and therefore their velocity, particularly in their extreme values, which pose the greatest threat to the aircraft. The estimation enables a more precise and faster representation of violent gusts (where considerable gusts speed increase and decrease in a short period of time). Both the low-frequency signal and the estimation are delayed relative to the recorded headwind gusts, but the complete estimation responses slightly precede the information derived from the low-frequency signal. Table 2 summarizes the computational comparisons of the selected cases discussed above (the most favorable  
Figure 35: Comparison of the gust value with the estimation results—climb crossing a gust area; entire simulation (complementary filter, (**A**) T\({}_{\rm cu}\) = 0.5 s, (**B**) T\({}_{\rm cu}\) = 0.8 s, (**C**) T\({}_{\rm cu}\) = 1.0 s, (**D**) T\({}_{\rm cu}\) = 1.5 s). Figure 36: Comparison of the gust value with the estimation results—climb crossing a gust area; series of selected gusts (complementary filter, (**A**) T\({}_{\rm cu}\) = 0.5 s, (**B**) T\({}_{\rm cu}\) = 0.8 s, (**C**) T\({}_{\rm cu}\) = 1.0 s, (**D**) T\({}_{\rm cu}\) = 1.5 s). results are in bold). Negative values in the difference columns mean underestimation, and positive values mean overestimation.