## 3 Controller Design
Finally the constrained elevator deflection applied to the UAV is:  
\[h_{d}(t)=V_{d}(t)\sin\gamma_{d}(t)+k_{h}\mathcal{D}\bigg{(}\frac{h(t)-h_{d}(t) }{p_{1}(t)}\bigg{)}\mathcal{T}\bigg{(}\frac{h(t)-h_{d}(t)}{p_{1}(t)}\bigg{)}\]6  
According to [33], it is customary to assign a positive sign to the elevator control signal \(\delta_{e}(t)\), representing the primary deflection of the elevator. The function \(M(a,V,\delta_{e},q)\) exhibits strictly decreasing behavior and thus, a positive deflection of the elevator generates a nose-down pitching moment, while a negative deflection elicits a nose-up pitching moment. The proposed control scheme (8)-(22) is summarized in Table 1 for readers' convenience. \begin{table}
\begin{tabular}{c} \hline \hline \multicolumn{2}{c}{**Intermediate Commands**} \\ \(\gamma_{d}(t)=\arcsin\Big{(}\sigma\Big{(}\frac{-q(t)}{V_{d}(t)},\sin(\bar{ \gamma})\Big{)}\Big{)}\) see (8) \\ \(F_{x}(t)=-\frac{k_{\gamma}}{p_{2}(t)}\mathcal{D}\Big{(}\frac{V(t)-V_{d}(t)}{p_ {2}(t)}\Big{)}\mathcal{T}\Big{(}\frac{V(t)-V_{d}(t)}{p_{2}(t)}\Big{)}\) see (10) \\ \(F_{h}(t)=-\frac{k_{\gamma}}{V(t)p_{3}(t)}\mathcal{D}\Big{(}\frac{\gamma(t)- \gamma_{d}(t)}{p_{3}(t)}\Big{)}\mathcal{T}\Big{(}\frac{\gamma(t)-\gamma_{d}( t)}{p_{3}(t)}\Big{)}\) see (12) \\ \(u_{d}(t)=\sqrt{F_{x}(t)^{2}+F_{h}(t)^{2}}\) \\ \(a_{d}(t)=\arctan\Big{(}\frac{F_{h}(t)}{F_{e}(t)}\Big{)}\) \\ \(\delta_{T}(t)=-k_{\gamma}\mathcal{D}\Big{(}\frac{\delta_{T}(t)-\sigma(u_{d}(t),\delta_{T})}{p_{4}(t)}\Big{)}\mathcal{T}\Big{(}\frac{\delta_{T}(t)-\sigma(u_ {d}(t),\delta_{T})}{p_{4}(t)}\Big{)}\) see (14) \\ \(\theta_{d}(t)=\sigma(a_{d}(t)+\gamma_{d}(t),\bar{\theta})\) see (17) \\ \(q_{d}(t)=-k_{\theta}\mathcal{D}\Big{(}\frac{\theta(t)-\theta_{d}(t)}{p_{5}(t) }\Big{)}\mathcal{T}\Big{(}\frac{\theta(t)-\theta_{d}(t)}{p_{5}(t)}\Big{)}\) see (18) \\ \(\delta_{e}(t)=k_{q}\mathcal{D}\Big{(}\frac{\dot{q}(t)-\sigma(a_{d}(t),\bar{ q})}{p_{6}(t)}\Big{)}\mathcal{T}\Big{(}\frac{\dot{q}(t)-\sigma(a_{d}(t),\bar{ q})}{p_{6}(t)}\Big{)}\) see (20) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Control Scheme. **Theorem 1**. : _Consider system (7), an output reference trajectory \(y_{d}(t)\) that obeys Assumptions 1-3 and the constrained reference state vector \(r_{d}\coloneqq[\gamma_{d}(t),\theta_{d}(t),q_{d}(t)]^{T}\) generated by (8)-(22). For sufficiently large saturation levels \(\delta_{T}\), \(\delta_{e}\), there exist upper bounds \(\hat{p}_{i},\ i=1,\ldots,6\) of the performance function \(p_{i}(t)\) such that the proposed adaptive control scheme (8)-(22) guarantees:_  
\[\|x_{i}(t)-x_{d_{i}}(t)\|<p_{i}(t)<\bar{p}_{i},\ i=1,\ldots,6\]  
_for all \(t\geq 0\) with \(x_{d}\coloneqq[y_{d}^{T},r_{d}^{T}]^{T}\)._  
Proof. : Let us first define the normalized tracking error vector:  
\[\xi\coloneqq \begin{bmatrix}\frac{h(t)-h_{d}(t)}{p_{1}(t)}\\ \frac{V(t)-V_{d}(t)}{p_{2}(t)}\\ \frac{\gamma(t)-\gamma_{d}(t)}{p_{3}(t)}\\ \frac{\gamma_{T}(t)-\sigma(u_{d}(t)\dot{x}_{T})}{p_{4}(t)}\\ \frac{\theta(t)-\theta_{d}(t)}{p_{5}(t)}\\ \frac{\tilde{q}(t)-\sigma(q_{d}(t)\dot{d})}{p_{6}(t)}\end{bmatrix}  \tag{23}\]  
Differentiating \(\xi(t)\) with respect to time we obtain the augmented closed-loop dynamical system as:  
\[\dot{\xi}\coloneqq[\dot{\xi}(t)^{T},\dot{p}_{1}(t),\dot{p}_{2}(t),\dot{p}_{3} (t),\dot{p}_{4}(t),\dot{p}_{5}(t),\dot{p}_{6}(t)]^{T}. \tag{24}\]  
Next, let us define the open set:  
\[\Omega\coloneqq\overbrace{(-1,1)\times\cdots\times(-1,1)}^{6\text{ times}}\times(0,\bar{p}_{1})\times\cdots\times(0,\bar{p}_{6})\subset\mathbb{R}^{12}.\]  
Then, the proof proceeds in three phases. First the existence of a unique maximal solution \(\zeta:[0,\tau_{\max})\to\Omega\) of (24) is ensured, i.e., \(\zeta(t)\in\Omega,\ \forall t\in[0,\tau_{\max})\). Next, we establish a sufficient condition regarding the saturation levels, such that the proposed control scheme