## 2 Theoretical Background
### Proposed Methods for Estimating Head-On Gusts  
The wind gust velocity component along the airplane's longitudinal axis (Ug) can be determined by calculating the vector difference between the measured TAS and the ground speed component GS\({}_{\rm{x}}\) along the airplane's \(x\)-axis (27). A positive value of Ug indicates a headwind, where the wind is blowing directly toward the nose of the aircraft (Figure 2). \[U_{g}=GS_{x}-TAS \tag{27}\]  
Figure 2: Illustration of relations from Equation (27). Using the dependencies presented in Figure 2 and knowing the true heading (HGDtrue) and track (TRK) angles, it is also possible to estimate not only the Ug component but also the direction and total speed of the wind, as well as lateral component of wind gust. This estimation is effective for gradual changes in wind speed and direction. Equation (27) can be easily derived from the general form presented in references [33; 34; 35; 36]. TAS is measured on board the transport aircraft using systems such as the ADC (Air Data Computer) or ADIRU/ADIRS (Air Data Inertial Reference System). To determine the component GSx, it is necessary to measure the GS and true track (TRK), as well as the aircraft's true heading (HDGtrue). These data can be obtained through the satellite navigation system (GNSS) and/or the ADIRU/ADIRS system. TAS and GSx measurements exhibit significant inertia and time constants, which can last several seconds, depending on the system structure, sensors, and algorithms. Consequently, the estimation of Ug can be associated with the low-frequency component of headwind estimation (Ug-io), as indicated by identity (28). \[U_{g-io}\equiv U_{g} \tag{28}\]  
Dangerous wind gusts are typically marked by rapid changes, and to estimate them, it is essential to consider the fast-changing state variables. The research hypothesis of this study posits that the proposed method for estimating the headwind component, as described by Equation (27), along with the transient changes in its value (gusts) outlined in Equation (18), can be effectively used in a complementary manner. Equation (27) serves to determine the overall value of the wind speed in the form of the headwind component. The measurement characteristics of GS and TAS, particularly their significant delays, impose strict limitations on the rapid and precise detection of gusts. While Equation (18) can estimate short-term variations in the headwind component, it cannot provide its absolute value. A practical solution to address this challenge is the implementation of a complementary filtering system. The complementary filter (CF) is a cost-effective and efficient data fusion technique that integrates a low-pass filter with a high-pass filter [37]. The general structure of the CF (Figure 3) incorporates low- and high-frequency inputs from the composite signal. The transfer function of the complementary filter is expressed by Equation (29). Across the entire frequency spectrum, the linear CF maintains a unity amplitude (0 dB magnitude). Additionally, the phase shift remains linear and is equal to 0 degrees (Figure 4). \[EST=\frac{1}{T_{c}\cdot s+1}\cdot LF+\frac{T_{c}\cdot s}{T_{c}\cdot s+1}\cdot HF \tag{29}\]  
LF--low-frequency signal,  
HF--high-frequency signal,  
EST--output of CF. Figure 3: Diagram of complementary filter. On the basis of dependencies (18) and (27), the complete law of headwind estimation (including temporary gusts), can be formulated as (30) and, finally, as Equation (31). \[U_{g-est\_CF}=\frac{1}{T_{cu}\cdot s+1}\cdot U_{g-lo}+\frac{T_{cu}\cdot s}{T_{cu} \cdot s+1}\cdot u_{g-hi} \tag{30}\]  
\[ \begin{array}{l}U_{g-est\_CF}=\frac{1}{T_{cu}\cdot s+1}\cdot(GS_{x}-TAS)+\\ +\frac{T_{cu}\cdot s}{T_{cu}\cdot s+1}\cdot(u+k_{1u}\cdot\dot{q}+k_{2u}\cdot q+ k_{3u}\cdot\delta_{H})\end{array}  \tag{31}\]  
After assuming \(u=\frac{1}{s}\cdot a_{x}\) and neglecting the model parameters that depend on the flight state, this can be expressed by approximation (32).