## 2 Theoretical Background  
### Analysis of the Possibilities of Wind Gusts Estimation (3D Problem)  
The airplane's mathematical model may take the form of equations describing its longitudinal and lateral motion [29,30]. Typically, these are two sets of linearized ordinary differential Equations (1) and (2). \[\frac{d}{dt}X_{lon}=A_{lon}\cdot X_{lon}+B_{lon}\cdot U_{lon}+C_{lon}\cdot Z_{lon} \tag{1}\]  
\[\frac{d}{dt}X_{lat}=A_{lat}\cdot X_{lat}+B_{lat}\cdot U_{lat}+C_{lat}\cdot Z_{lat} \tag{2}\]  
The state variables are specified as follows for longitudinal and lateral motion, respectively:  
\[X_{lon}\ =\ (u,\ w,\ q,\ \theta)^{\mathrm{T}}\ \ \ \ \ X_{lat}\ =\ (v,\ p,\ r,\ \varphi)^{ \mathrm{T}} \tag{3}\]  
The control (U) and disturbance vectors (Z) are defined as:  
\[U_{lon}\ =\ (\delta_{\mathrm{H}},\ \delta_{\mathrm{T}})^{\mathrm{T}}\ \ \ \ Z_{lon}\ =\ \left(w_{\mathrm{g}},\ u_{\mathrm{g}}\right)^{ \mathrm{T}}\ \ \ U_{lat}\ =\ (\delta_{\mathrm{a}},\ \delta_{\mathrm{f}})^{\mathrm{T}}\ \ \ \ Z_{lat}\ =\ \left(V_{\mathrm{g}}\right)^{ \mathrm{T}} \tag{4}\]  
The mathematical model of the aircraft presented here is built upon the principles of flight mechanics. The coefficient matrices (5)-(7) in the longitudinal motion, described by Equation (1), and lateral motion, described by Equation (2), include the respective aerodynamic derivatives. \[A_{lon}= \begin{bmatrix}X_{u}&X_{w}&0&-g\cdot\cos\theta_{0}\\ Z_{u}&Z_{w}&U_{0}&-g\cdot\sin\theta_{0}\\ M_{u}&M_{w}&M_{q}&0\\ 0&0&1&0\end{bmatrix} A_{lat}= \begin{bmatrix}Y_{v}&Y_{p}&Y_{r}-U_{0}&g\cdot\cos \theta_{0}\\ L^{\prime}_{v}&L^{\prime}_{p}&L^{\prime}_{r}&0\\ N^{\prime}_{v}&N^{\prime}_{p}&N^{\prime}_{r}&0\\ 0&1&\tan\theta_{0}&0\end{bmatrix}  \tag{5}\]  
\[B_{lon}= \begin{bmatrix}X_{\delta H}&X_{\delta T}\\ Z_{\delta H}&Z_{\delta T}\\ M_{\delta H}&M_{\delta T}\\ 0&0\end{bmatrix} \ \ \ \ B_{lat}= \begin{bmatrix}Y_{\delta A}&X_{\delta R}\\ L^{\prime}_{\delta A}&L^{\prime}_{\delta R}\\ N^{\prime}_{\delta A}&N^{\prime}_{\delta R}\\ 0&0\end{bmatrix}  \tag{6}\]  
\[C_{lon}= \begin{bmatrix}-X_{w}&-X_{u}\\ -Z_{w}&-Z_{u}\\ -M_{w}&-M_{u}\\ 0&0\end{bmatrix} \ \ \ \ \ C_{lat}= \begin{bmatrix}Y_{v}\\ L^{\prime}_{v}\\ N^{\prime}_{v}\\ 0\end{bmatrix}  \tag{7}\]  
The variation in gust values along the airplane's longitudinal axis (u\({}_{\mathrm{g}}\)) and vertical axis (w\({}_{\mathrm{g}}\)) is reflected in the longitudinal motion, as shown in Equation (1), while changes in gust values along the lateral axis (v\({}_{\mathrm{g}}\)) appear in the lateral motion in Equation (2). The gust parameters (u\({}_{\mathrm{g}}\), w\({}_{\mathrm{g}}\), v\({}_{\mathrm{g}}\)) are included in the disturbance vectors \(Z_{lon}\) and \(Z_{lat}\) (4), respectively. By substituting the matrices \(A_{lon}\), \(B_{lon}\), and \(C_{lon}\) and the vectors \(X_{lon}\), \(U_{lon}\), and \(Z_{lon}\) into Equation (1), we obtain four equations. The first, second, and third of these can be directly expressed as dependences (8)-(10). State variable \(\theta\), represented by the fourth of the equations, is the integral of the pitch angular velocity q. As a result, the described process is shifted in phase by 90 degrees, and consequently, there is an additional delay. For this reason, this equation will not be very useful for gust estimation tasks. \[\dot{u}=X_{u}\cdot\left(u-u_{g}\right)+X_{w}\cdot\left(w-w_{g}\right)-g\cdot \theta\cdot\cos\theta+X_{\delta H}\cdot\delta_{H}+X_{\delta T}\cdot\delta_{T} \tag{8}\]\[\ddot{w}=Z_{u}\cdot\left(u-u_{g}\right)+Z_{w}\cdot\left(w-w_{g}\right)+U_{0}\cdot q -g\cdot\phi\cdot\sin\theta+Z_{\phi H}\cdot\delta_{H}+Z_{ST}\cdot\delta_{T} \tag{9}\]  
\[\dot{q}=\overline{M_{u}}\cdot\left(u-u_{g}\right)+\overline{M_{w}}\cdot\left(w -w_{g}\right)+\overline{M_{q}}\cdot q+\overline{M_{\delta H}}\cdot\delta_{H}+ \overline{M_{ST}}\cdot\delta_{T} \tag{10}\]  
It is important to note that q, u, u, w, wg, \(\vartheta\), \(\delta_{\mathrm{H}}\), and \(\delta_{\mathrm{T}}\) represent changes in values, not absolute values.