## 2 Theoretical Background  
### Analysis of the Possibility of Head-On Gusts Estimation  
The wind gust speed component in the longitudinal axis of the airplane (\(\mathbf{U_{g}}\)) can be estimated from the vector difference in measured true airspeed (\(\mathbf{TAS}\)) and the ground speed component \(\mathbf{GS_{x}}\), measured on the \(x\)-axis of the airplane (1). Positive sign of \(\mathbf{U_{g}}\) indicates the wind blowing into the nose of the plane--headwind (Figure 1). \[\mathbf{U_{g}}=\mathbf{GS_{x}}-\mathbf{TAS} \tag{1}\]  
Using the dependencies presented in Figure 1 and knowing the true heading (\(\text{HGD}_{\text{true}}\)) and track (TRK) angles, apart from the \(U_{g}\) component, it is also possible to estimate the direction and total speed of wind as well as the lateral component of wind gust. In this paper, this issue was omitted by focusing on the wind component occurring in the longitudinal axis of the aircraft. This parameter is crucial for flight safety due to the possibility of an unexpected stall. This estimation works well for slow changes in wind speed and direction. Equation (1) can be easily derived from the general form presented in the works [1; 22; 23; 24; 25]. The TAS measurement is carried out on-board the transport aircraft directly with the use of the systems air data computer (ADC) or air data inertial reference system (ADIRU/ADIRS). Determination of the component \(GS_{x}\) requires measurement of \(GS\) and true track (TRK) as well as measuring the airplane's true heading (HDGtrue). These data can be obtained from the satellite navigation system (GNSS) and/or the ADIRU/ADIRS. Measurement of TAS and \(GS_{x}\) is characterised by considerable inertia and time constants of up to a few seconds (depending on the structure of the system, sensors, and algorithms used). Therefore, in the following part of the article, the estimation of \(U_{g}\) is identified with the low-frequency component of headwind estimation (\(U_{g\text{-}lo}\)), according to the identity (2). \[U_{g\text{-}lo}\equiv U_{g} \tag{2}\]  
Dangerous gusts of wind are usually characterised by immediate changes, and for their estimation, it is necessary to refer to the fast-changing state variables. The mathematical model of the airplane may be the equations of longitudinal and lateral motion [26]. They are usually two systems of linearised ordinary differential Equations (3) and (4). \[\frac{d}{dt}X_{lon}=A_{lon}\cdot X_{lon}+B_{lon}\cdot U_{lon}+C_{lon}\cdot Z_{lon} \tag{3}\]  
\[\frac{d}{dt}X_{lat}=A_{lat}\cdot X_{lat}+B_{lat}\cdot U_{lat}+C_{lat}\cdot Z_{ lat} \tag{4}\]  
State variables are defined as (for longitudinal and lateral motion, respectively):  
\[X_{lon}=(u,w,q,v)^{\text{T}}\;X_{lat}=(v,p,r,\varphi)^{\text{T}} \tag{5}\]  
Control vectors (\(U\)) and disturbances (\(Z\)) are instead:  
\[U_{lon}=(\delta_{H},\delta_{T})^{\text{T}}\;Z_{lon}=(w_{g},u_{g})^{\text{T}}\;U _{lat}=(\delta_{L},\delta_{K})^{\text{T}}\;Z_{lat}=(v_{g})^{\text{T}} \tag{6}\]  
The presented mathematical model of the aircraft was constructed based on the theory of flight mechanics. Matrices of coefficients (7) and (8) appearing in the equations of longitudinal motion (3) and lateral motion (4) contain the corresponding aerodynamic derivatives. \[A_{lon}= \begin{bmatrix}X_{u}&X_{w}&0&-g\cdot\cos\Theta_{0}\\ Z_{u}&Z_{w}&U_{0}&-g\cdot\sin\Theta_{0}\\ \hline M_{u}&M_{w}&M_{q}&0\\ 0&0&1&0\\ \end{bmatrix} B_{lon}= \begin{bmatrix}X_{\delta H}&X_{\delta T}\\ Z_{\delta H}&Z_{\delta T}\\ \hline M_{\delta H}&M_{\delta T}\\ 0&0\\ \end{bmatrix} C_{lon}= \begin{bmatrix}-X_{w}&-X_{u}\\ -Z_{w}&-Z_{u}\\ -M_{w}&-M_{u}\\ 0&0\\ \end{bmatrix}  \tag{7}\]  
Figure 1: Illustration of relations from Equation (1). \[A_{lat}= \begin{bmatrix}Y_{V}&Y_{p}&Y_{r}-U_{0}&g\cdot\cos\Theta_{0}\\ L^{\prime}{}_{p}&L^{\prime}{}_{p}&L^{\prime}{}_{r}&0\\ N^{\prime}{}_{p}&N^{\prime}{}_{p}&N^{\prime}{}_{r}&0\\ 0&1&tg\Theta_{0}&0\end{bmatrix} B_{lat}=