## 3 Controller Design
Thus, the relaxed performance function recovers its conventional form with exponential decay rate when the system operates without saturation. _Step 2._ Next, select the reference force vector along with the corresponding performance update laws as:  
\[F_{x}(t) =-\frac{k_{v}}{p_{2}(t)}\mathcal{D}\bigg{(}\frac{V(t)-V_{d}(t)}{p _{2}(t)}\bigg{)}\mathcal{T}\bigg{(}\frac{V(t)-V_{d}(t)}{p_{2}(t)}\bigg{)} \tag{10}\] \[\dot{p}_{2}(t) =-\lambda_{v}(p_{2}(t)-p_{\infty}^{2})+\frac{V(t)-V_{d}(t)}{p_{2} (t)}\big{(}\sigma(F_{x}(t),|\bar{\delta}_{T}\cos a(t)|)-F_{x}(t)\big{)}\] (11) \[F_{h}(t) =-\frac{k_{\gamma}}{V(t)p_{3}(t)}\mathcal{D}\bigg{(}\frac{\gamma (t)-\gamma_{d}(t)}{p_{3}(t)}\bigg{)}\mathcal{T}\bigg{(}\frac{\gamma(t)-\gamma _{d}(t)}{p_{3}(t)}\bigg{)}\] (12) \[\dot{p}_{3}(t) =-\lambda_{\gamma}(p_{3}(t)-p_{\infty}^{3})+\frac{\gamma(t)- \gamma_{d}(t)}{p_{3}(t)}\big{(}\sigma(F_{h}(t),|\bar{\delta}_{T}\sin a(t)|)-F_ {h}(t)\big{)}. \tag{13}\]  
with \(k_{v},k_{\gamma},\lambda_{v},\lambda_{\tau},p_{\infty}^{2},p_{\infty}^{3}>0\). Notice that \(F_{x}(t),F_{h}(t)\) act as controls in (2) and (3), regulating the horizontal and vertical motion of the system, respectively. Thus, the reference unconstrained throttle command as well as the corresponding angle-of-attack are given by \(u_{d}\coloneqq\sqrt{F_{x}(t)^{2}+F_{h}(t)^{2}}\) and \(a_{d}\coloneqq\arctan\Big{(}\frac{F_{h}(t)}{F_{x}(t)}\Big{)}\), respectively. The saturation levels incorporated into the adaptive PFs (11) and (13) take into account the feasible generated thrust depending on the actual angle-of-attack. In order to impose both input and rate constraints to the control input \(\delta_{T}(t)\) we utilize the APPC technique developed in [32] which leads us to:  
\[\bar{\delta}_{T}(t) =-k_{r}\mathcal{D}\bigg{(}\frac{\delta_{T}(t)-\sigma(u_{d}(t), \bar{\delta}_{T})}{p_{4}(t)}\bigg{)}\mathcal{T}\bigg{(}\frac{\delta_{T}(t)- \sigma(u_{d}(t),\bar{\delta}_{T})}{p_{4}(t)}\bigg{)} \tag{14}\] \[\dot{p}_{4}(t) =-\lambda_{r}(p_{4}(t)-p_{\infty}^{4})+\frac{\delta_{T}(t)- \sigma(u_{d}(t),\bar{\delta}_{T})}{p_{4}(t)}\big{(}\sigma(\bar{\delta}_{T}(t), \bar{r})-\bar{\delta}_{T}(t)\big{)} \tag{15}\]  
with \(k_{r},\lambda_{r},p_{\infty}^{4}>0\), where the input amplitude and rate limitations are regulated by \(\bar{\delta}_{T}\) and \(\bar{r}\), respectively. The dynamic throttle command is then obtained by:  
\[h_{d}(t)=V_{d}(t)\sin\gamma_{d}(t)+k_{h}\mathcal{D}\bigg{(}\frac{h(t)-h_{d}(t) }{p_{1}(t)}\bigg{)}\mathcal{T}\bigg{(}\frac{h(t)-h_{d}(t)}{p_{1}(t)}\bigg{)}\]0_Step 3._ Subsequently, consider the constrained pitch reference trajectory:  
\[h_{d}(t)=V_{d}(t)\sin\gamma_{d}(t)+k_{h}\mathcal{D}\bigg{(}\frac{h(t)-h_{d}(t) }{p_{1}(t)}\bigg{)}\mathcal{T}\bigg{(}\frac{h(t)-h_{d}(t)}{p_{1}(t)}\bigg{)}\]1  
where \(\bar{\theta}\) denotes the maximum absolute value of the reference pitch angle. Moreover, as \(\gamma_{d}(t)\) is also constrained within the compact set \([-\bar{\gamma},\bar{\gamma}]\), the reference angle-of-attack \(a_{d}(t)\) lies within the set \([-\bar{\theta}+\bar{\gamma},\bar{\theta}-\bar{\gamma}]\). Finally, utilizing once again the APPC technique we obtain the constrained elevator control command as:  
\[h_{d}(t)=V_{d}(t)\sin\gamma_{d}(t)+k_{h}\mathcal{D}\bigg{(}\frac{h(t)-h_{d}(t) }{p_{1}(t)}\bigg{)}\mathcal{T}\bigg{(}\frac{h(t)-h_{d}(t)}{p_{1}(t)}\bigg{)}\]2 \[h_{d}(t)=V_{d}(t)\sin\gamma_{d}(t)+k_{h}\mathcal{D}\bigg{(}\frac{h(t)-h_{d}(t) }{p_{1}(t)}\bigg{)}\mathcal{T}\bigg{(}\frac{h(t)-h_{d}(t)}{p_{1}(t)}\bigg{)}\]3 (19) \[h_{d}(t)=V_{d}(t)\sin\gamma_{d}(t)+k_{h}\mathcal{D}\bigg{(}\frac{h(t)-h_{d}(t) }{p_{1}(t)}\bigg{)}\mathcal{T}\bigg{(}\frac{h(t)-h_{d}(t)}{p_{1}(t)}\bigg{)}\]4 (20) \[h_{d}(t)=V_{d}(t)\sin\gamma_{d}(t)+k_{h}\mathcal{D}\bigg{(}\frac{h(t)-h_{d}(t) }{p_{1}(t)}\bigg{)}\mathcal{T}\bigg{(}\frac{h(t)-h_{d}(t)}{p_{1}(t)}\bigg{)}\]5  
with \(k_{\theta},k_{q},\lambda_{\theta},\lambda_{q},p_{\infty}^{5},p_{\infty}^{6}>0\).