## 2 Problem Formulation and Preliminaries  
The longitudinal flight dynamics of a small fixed-wing UAV in the presence of wind disturbances is described as in [27; 33] by:  
\[\dot{h}(t) =V(t)\sin\gamma(t)+w_{h}(t) \tag{1}\] \[\dot{V}(t) =\frac{T(\delta_{T},V)\cos a(t)-D(a,V)}{m}-g\sin\gamma(t)-\cos \left(\gamma(t)\right)\dot{w}_{x}(t)-\sin\left(\gamma(t)\right)\dot{w}_{h}(t)\] (2) \[\dot{\gamma}(t) =\frac{T(\delta_{T},V)\sin a(t)+L(a,V,\delta_{e})}{mV(t)}-\frac{ \left(g+\dot{w}_{h}(t)\right)\cos\gamma(t)}{V(t)}+\frac{\sin\left(\gamma(t) \right)}{V(t)}\dot{w}_{x}(t)\] (3) \[\dot{\theta}(t) =q(t)\] (4) \[\dot{q}(t) =\frac{M(a,V,\delta_{e},q)}{I_{yy}} \tag{5}\]  
where \(h(t)\) denotes the altitude; \(V(t)\) is the airspeed vehicle velocity; \(\gamma(t)\) and \(\theta(t)\) correspond to the flight-path and pitch angle, respectively; and \(q(t)\) is the pitch rate with the angle-of-attack given by \(a(t)=\theta(t)-\gamma(t)\). The acceleration values of the wind along with the horizontal and vertical axis are denoted by \(\dot{w}_{x}(t)\), \(\dot{w}_{h}(t)\), respectively. Furthermore, the external lift, drag, pitch and thrust moments are expressed as:\[L(a,V,\delta_{e}) =\frac{1}{2}\rho V^{2}S\Big{(}C_{L_{0}}+C_{L_{a}}a+C_{L_{k_{\mu}}} \delta_{e}\Big{)}\] \[D(a,V) =\frac{1}{2}\rho V^{2}S\big{(}C_{D_{0}}+C_{D_{a}}a\big{)}\] \[M(a,V,\delta_{e},q) =\frac{1}{2}\rho V^{2}S\varepsilon\Big{(}C_{M_{0}}+C_{M_{a}}a+C_{ M_{q}}\frac{\varepsilon}{2V}q+C_{M_{\mu}}\delta_{e}\Big{)}\] \[T(\delta_{T},V) =\frac{1}{2}\rho S_{prop}C_{prop}\Big{[}(k_{m}\delta_{T})^{2}-V^{ 2}\Big{]}.\]  
Assuming a flight occurring within a vertical plane above a flat Earth, we establish a fixed coordinate system on the ground as the inertial reference frame. The coordinates of the aircraft as well as the axes of the wind gusts are illustrated in Figure 1. The control input consists of the throttle setting \(\delta_{T}(t)\) and the elevator deflection \(\delta_{e}(t)\). Both control signals are amplitude constrained, i.e., \(\delta_{T}(t)\leq\delta_{T}\) for a positive constant \(\delta_{T}\leq 1\) and \(\|\delta_{e}(t)\|\leq\delta_{e}\) for a positive constant \(\delta_{e}\). Owing to the engine dynamics, the throttle command is also rate constrained, that is, \(\|\delta_{T}(t)\|\leq\bar{r}\), and thus the actual throttle setting is given by:  
\[\dot{V}(t) =\frac{T(\delta_{T},V)\cos a(t)-D(a,V)}{m}-g\sin\gamma(t)-\cos \left(\gamma(t)\right)\dot{w}_{x}(t)-\sin\left(\gamma(t)\right)\dot{w}_{h}(t)\]0  
where \(\phi(t)\) denotes a control signal to be designed. Moreover, the states of the system need to be constrained within some safety bounds. The inherent properties of fixed-wing UAVs impose limitations on angle-of-attack owing to the stall phenomenon and saturated heading rate due to pitch-rate constraints. Thus, the nonlinear system ((1)-(6)) can be written in compact form as:  
\[\dot{V}(t) =\frac{T(\delta_{T},V)\cos a(t)-D(a,V)}{m}-g\sin\gamma(t)-\cos \left(\gamma(t)\right)\dot{w}_{x}(t)-\sin\left(\gamma(t)\right)\dot{w}_{h}(t)\]1  
where \(x\coloneqq[h(t),V(t),\gamma(t),\theta(t),q(t),\delta_{T}(t)]^{T}\) denotes the state vector, \(u\coloneqq[\phi(t),\delta_{e}(t)]^{T}\) is the control input, and \(d(t)\coloneqq[w_{h}(t),\dot{w}_{x}(t),\dot{w}_{h}(t)]^{T}\) contains the disturbance components. The output of the system is chosen to be the vector \(y\coloneqq[h(t),V(t)]^{T}\). In this work, assuming no prior knowledge on the system dynamics \(f(x),\ g(x),\ z(x)\), and the external wind disturbances d(t), we aim at designing an approximation-free, state feedback controller for the constrained system (7) that meets the following properties:  
* The states of the system are constrained within a compact set. * The desired trajectory \(y_{d}\coloneqq[h_{d}(t),V_{d}(t)]^{T}\) is tracked with adaptive prescribed performance specifications. Finally, to solve the aforementioned problem, we pose the following assumptions:  
**Assumption 1**.