## 2 Theoretical Background
### Analysis of the Possibilities of Wind Gusts Estimation (3D Problem)
This is due to the fact that Equations (1) and (2) describe an incremental model. This characteristic was considered during the modelling of the estimation system. By assuming constant thrust (\(\delta_{\mathrm{T}}=0\)), we reduce dependencies (8)-(10) to the forms (11)-(13):  
\[\dot{u}=X_{u}\cdot\left(u-u_{g}\right)+X_{w}\cdot\left(w-w_{g}\right)-g\cdot \theta\cdot\cos\theta+X_{\delta H}\cdot\delta_{H} \tag{11}\]  
\[\dot{w}=Z_{u}\cdot\left(u-u_{g}\right)+Z_{w}\cdot\left(w-w_{g}\right)+U_{0} \cdot q-g\cdot\theta\cdot\sin\theta+Z_{\delta H}\cdot\delta_{H} \tag{12}\]  
\[\dot{q}=\overline{M_{u}}\cdot\left(u-u_{g}\right)-\overline{M_{w}}\cdot\left( w-w_{g}\right)+\overline{M_{q}}\cdot q+M_{\delta H}\cdot\delta_{H} \tag{13}\]  
By transforming the dependencies (11) and (12) with respect to ug and wg, we obtain Equations (14) and (15):  
\[u_{g}=u-\frac{1}{X_{u}}\cdot\dot{u}-\frac{g}{X_{u}}\cdot\theta\cdot\cos\theta +\frac{X_{\delta H}}{X_{u}}\cdot\delta_{H}+\frac{X_{w}}{X_{u}}\cdot\left(w-w_ {g}\right) \tag{14}\]  
\[w_{g}=w-\frac{1}{Z_{w}}\cdot\dot{w}-\frac{g}{Z_{w}}\cdot\theta\cdot\sin\theta +\frac{Z_{\delta H}}{Z_{w}}\cdot\delta_{H}+\frac{Z_{u}}{Z_{w}}\cdot\left(u-u_ {g}\right) \tag{15}\]  
The first two expressions on the right-hand side of Equations (14) and (15) indicate that the gust can be determined by knowing the changes in velocities u and w, as well as their first derivatives. Considering the effect of the change in the aircraft's pitch angle and the deflection of the elevator is necessary to account for transient processes (larger time constants characterize the effect of gusts on the aircraft's pitch angle change and the interaction of the pilot or autopilot in the longitudinal channel). The last terms in the sum of Equations (14) and (15) are related to the mutual (cross) interaction of head-on and vertical gusts in the longitudinal motion model of the aircraft. Equation (13) holds great potential for developing an algorithm for estimating head-on and vertical gusts, as it relates their values to the pitch angular velocity q. This parameter is characterized by relatively small time constants in response to the occurring gusts. Equation (13) also includes the first derivative of q with respect to time, which is additionally characterized by a phase shift of \(-90\) degrees. The only slow-varying parameter in this equation is the relationship linking the gust estimate with the aircraft's response to the elevator deflection. After transformation with respect to ug and wg, Equation (13) takes the form of Equations (16) and (17). The results obtained in this work [28] suggest that Equation (16) can be used for estimating the fast-varying component of the head-on gust. \[u_{g}=u-\frac{1}{\overline{M_{u}}}\cdot\dot{q}+\frac{\overline{M_{q}}}{\overline {M_{u}}}\cdot q+\frac{M_{\delta H}}{\overline{M_{u}}}\cdot\delta_{H}+\frac{ \overline{M_{w}}}{\overline{M_{u}}}\cdot\left(w-w_{g}\right) \tag{16}\]  
\[w_{g}=w-\frac{1}{\overline{M_{w}}}\cdot\dot{q}+\frac{\overline{M_{q}}}{ \overline{M_{w}}}\cdot q+\frac{M_{\delta H}}{\overline{M_{w}}}\cdot\delta_{H}+ \frac{\overline{M_{u}}}{\overline{M_{w}}}\cdot\left(u-u_{g}\right) \tag{17}\]  
In this approach to the problem, the vertical and headwind gust estimation algorithms are mutually dependent. When attempting to determine gust values along only one of these axes, it can be simplified by assuming a one-dimensional gust. For example, in work [28], a dominant headwind gust was assumed. In this case, we can assume wg = 0(no vertical gust) and W = const (the model is incremental; so, we can also assume w = 0). By introducing gain factors, Equation (16) transforms into Equation (18). \[\ddot{w}=Z_{u}\cdot\left(u-u_{g}\right)+Z_{w}\cdot\left(w-w_{g}\right)+U_{0}\cdot q -g\cdot\phi\cdot\sin\theta+Z_{\phi H}\cdot\delta_{H}+Z_{ST}\cdot\delta_{T} \tag{9}\]0  
Similarly, assuming the absence of headwind gusts, the value of the fast-varying components of the vertical gust can be estimated using Equation (19).