## 3 Experiments and Discussion
### Experiments on Vertical Acceleration and EDR Estimation
#### 3.2.2 Analysis of EDR Estimation  
This section further explores the estimation accuracy by comparing the estimated EDR and input EDR value. Taking the B737-800 aircraft as an example, the EDR values under different intensities were estimated, respectively, and the results were compared with the transfer function-based algorithm. Using the stability derivatives given in the modeling data, a linear-transfer function model similar to the Model 4 in ref [12] was established for comparison, because the Model 4 followed the acceleration transients with the best accuracy [12]. The choice of input-data-length and EDR update was a compromise between having enough resolution to capture instantaneous aircraft bumping and having enough samples in the window to provide stable computational statistics. For aircraft in cruise, a nominal one-minute EDR reporting interval was used. Within the reporting window, sub-windows were chosen for the computation of EDR value. Since a 10 s sub-window was used in EDR estimation, the square root of median \(\varepsilon^{1/3}\) values were computed in one-minute period. Here, the estimated EDR values were both generated every 10 s.  
Figure 13a shows the 100 times experiment results of the two algorithms. Under the four TCs, the estimated EDR values were all neighboring to the theoretical values. However, with the increase of intensity, the variance of EDR value by transfer function-based method increased, and the number of outliers increased. Figure 13b shows the statistical results of the two algorithms. We found that the performance of the transfer function-based method based on single equilibrium point deteriorated in severe turbulence. The actual flight state seriously deviated from equilibrium point, resulting in increasing deviation in acceleration computation. On the contrary--benefitted from accurately obtained vertical acceleration--the EDR estimation accuracy was still satisfactory. Compared with the transfer function-based method, the variance of EDR value by the recommended method was smaller, and there were few outliers. Figure 12: Acceleration spectrum of different locations. With the Grid 8 refinement, the CPU time of one-time complete EDR estimation was 3.17 s under MATLAB 2018a software environment with i7-9700 processor. The CPU time of vertical acceleration computation was 2.79 s, which mainly contributed to the algorithm complexity. As a result, the algorithm was capable to output the EDR value every 10 s. If the code was converted to C/C++, it could be further optimized and embedded into a real-time system onboard the aircraft. Furthermore, it should be noted that real atmospheric turbulence conforms to von Karman model in a probabilistic sense. The proposed EDR estimation algorithm was still effective by real input of turbulence measurements. A major problem of the commonly used EDR algorithm is that the control-surface deflection has adverse effects on EDR estimation [15]. As far as the traditional acceleration-based EDR algorithm is concerned, there is no control input in transfer function. With the recommended method in this paper, the vertical acceleration induced by aircraft control or vertical turbulence can be effectively distinguished. The effect of turbulent wind on the vertical acceleration is effectively separated from the aerodynamic force \(\mathbf{F}_{i}^{0}\). Therefore, the change of incremental aerodynamic derivatives is only decided by turbulent wind. A simulation was designed in which the aircraft was approved to climb to the altitude above 11,000 m after encountering turbulence. As shown in Figure 14, during this process, the elevator deflected, and the aircraft kept climbing at a pitch angle of 5 degrees. In order to explore the dynamic process of EDR estimation, the EDR value was given per second in a sliding window of 10 s. The change of aerodynamic force and vertical acceleration caused by turbulence and elevator deflection were obtained.