## 2 Theoretical Background
### Analysis of the Possibilities of Wind Gusts Estimation (3D Problem)
\[\ddot{w}=Z_{u}\cdot\left(u-u_{g}\right)+Z_{w}\cdot\left(w-w_{g}\right)+U_{0}\cdot q -g\cdot\phi\cdot\sin\theta+Z_{\phi H}\cdot\delta_{H}+Z_{ST}\cdot\delta_{T} \tag{9}\]1  
Estimations of ug and wg will correspond to the high-frequency estimation of headwind (ug-hi) and vertical gust (wg-hi). This has been extensively discussed in the authors' previous work [28]. Using the state Equation (2) for lateral motion, relationships (20)-(22) can be derived, linking the lateral gust vg to the linear velocity v and the angular velocities p and r in the aircraft's body-fixed coordinate system. In these linearized state equations, the rudder deflection \(\delta_{\text{R}}\) and aileron deflection \(\delta_{\text{A}}\) are also variables. \[\ddot{w}=Z_{u}\cdot\left(u-u_{g}\right)+Z_{w}\cdot\left(w-w_{g}\right)+U_{0}\cdot q -g\cdot\phi\cdot\sin\theta+Z_{\phi H}\cdot\delta_{H}+Z_{ST}\cdot\delta_{T} \tag{9}\]2  
\[\ddot{w}=Z_{u}\cdot\left(u-u_{g}\right)+Z_{w}\cdot\left(w-w_{g}\right)+U_{0}\cdot q -g\cdot\phi\cdot\sin\theta+Z_{\phi H}\cdot\delta_{H}+Z_{ST}\cdot\delta_{T} \tag{9}\]3  
\[\ddot{w}=Z_{u}\cdot\left(u-u_{g}\right)+Z_{w}\cdot\left(w-w_{g}\right)+U_{0}\cdot q -g\cdot\phi\cdot\sin\theta+Z_{\phi H}\cdot\delta_{H}+Z_{ST}\cdot\delta_{T} \tag{9}\]4  
Equation (21) describes the aperiodic rolling motion, where the influence of the lateral gust is significantly smaller compared to its effect on the yaw rate r and the linear velocity v. For this reason, the analysis of Equation (21) was omitted, focusing instead on relationships (20) and (22). By transforming Equations (20) and (22) with respect to the lateral gust vg, we obtain Equations (23) and (24), respectively. \[\ddot{w}=Z_{u}\cdot\left(u-u_{g}\right)+Z_{w}\cdot\left(w-w_{g}\right)+U_{0}\cdot q -g\cdot\phi\cdot\sin\theta+Z_{\phi H}\cdot\delta_{H}+Z_{ST}\cdot\delta_{T} \tag{9}\]5  
\[\ddot{w}=Z_{u}\cdot\left(u-u_{g}\right)+Z_{w}\cdot\left(w-w_{g}\right)+U_{0}\cdot q -g\cdot\phi\cdot\sin\theta+Z_{\phi H}\cdot\delta_{H}+Z_{ST}\cdot\delta_{T} \tag{9}\]6  
By introducing gain coefficients in place of constant expressions in Equations (23) and (24), we obtain Equations (25) and (26). Preliminary research has shown that relationship (25) is particularly useful for estimating the fast-varying component of lateral gusts (vg-hi). \[\ddot{w}=Z_{u}\cdot\left(u-u_{g}\right)+Z_{w}\cdot\left(w-w_{g}\right)+U_{0}\cdot q -g\cdot\phi\cdot\sin\theta+Z_{\phi H}\cdot\delta_{H}+Z_{ST}\cdot\delta_{T} \tag{9}\]7  
\[\ddot{w}=Z_{u}\cdot\left(u-u_{g}\right)+Z_{w}\cdot\left(w-w_{g}\right)+U_{0}\cdot q -g\cdot\phi\cdot\sin\theta+Z_{\phi H}\cdot\delta_{H}+Z_{ST}\cdot\delta_{T} \tag{9}\]8  
An advantage of estimating wind gust velocities using state equations is the ability to directly utilize measurement results obtained in the aircraft's body-fixed coordinate system (F\({}_{\text{B}}\)). However, a drawback of this approach is the difficulty in estimating gust values in a three-dimensional framework (due to interdependencies in the algorithms) and its limitation to selected, identified flight states of the specific aircraft. These algorithms are also highly sensitive to the dynamic characteristics and errors of individual sensors [31; 32], and the high-frequency signals ug-hi, vg-hi, and wg-hi, provided as high-pass input to the complementary filter, require corrections that are challenging to determine analytically [28]. For these reasons, estimating wind gusts in a 3D framework is very difficult and problematic to achieve using only state equations. This paper proposes a complementary approach to the problem of gust estimation. It is assumed that analyzing measurements performed in three reference frames and correlating them can yield significantly better results, particularly for slow-varying signals. The chosen reference frames are the Earth-fixed frame (F\({}_{\rm{v}}\)), the airflow-fixed frame (F\({}_{\rm{w}}\)), and the aircraft-fixed frame (F\({}_{\rm{B}}\)).