## 5 Comparative Simulation Results and Discussion
### Comparison with the P-PPC Method  
In this simulation scenario, we conduct a comparative analysis between the proposed control scheme and the hybrid proportional and PPC scheme introduced in [30]. The objective is to evaluate the trajectory tracking performance of a UAV operating under varying altitude and airspeed conditions in presence of wind gusts. The altitude reference signal is defined as \(h_{d}(t)=40+20\sin(0.1t)\), and the airspeed reference signal is given by \(V_{d}(t)=45-8\cos(0.5t)\). The initial conditions of the UAV are set as \(x(0)=[0,32,0.04,0.03,0,0]^{T}\) and the initial values of the adaptive performance functions are chosen as \(p_{1}(0)=47,p_{2}(0)=5.5\), \(p_{3}(0)=0.12\), \(p_{4}(0)=0.13\), \(p_{5}(0)=0.2\), \(p_{6}(0)=1.65\). The throttle settling limit is set at \(\tilde{\delta}_{T}=1\) and the elevator deflection limit at \(\tilde{\delta}_{\epsilon}=1\). To achieve faster convergence of the states to the reference signals, we have adjusted the error convergence rate parameters, specifically setting \(\lambda_{i}=1\) for \(i\in\{h,v,\gamma,\theta,q\}\). The remaining controller parameters remain unchanged and are consistent with those presented in Table 3. The proportional gain for the P-PPC scheme has been chosen as \(k_{h}=10\), and the remaining control gains have been selected identical to the APPC controller. In order to assess the robustness of the proposed controller, we introduce wind gust disturbances into the simulation. Specifically, we induce wind gusts \(w_{x}(t)=0.3\ln(t+1)+\sin(0.3\pi t)\) and \(w_{h}(t)=0.5\ln(t+1)+\sin(0.2\pi t)\) for \(t\) in the interval \([50,80]\). These wind gusts, depicted in Figure 11, are designed to challenge the control system and evaluate its ability to maintain accurate trajectory tracking in the presence of external disturbances. The simulation period is set to 80 s, providing sufficient time to observe the controller's performance under these conditions. Figure 12a,b depict the airspeed tracking performance of the APPC and P-PPC schemes, respectively. The corresponding tracking errors are depicted along with the adaptive and conventional performance boundaries in Figure 12c,d, respectively. It can be observed that the P-PPC scheme achieves faster convergence compared to the proposed APPC scheme, which adapts its performance boundaries during the transient period. However, it is important to note that this adaptation is necessary to ensure that the constrained system tracks the reference signals with adaptive prescribed performance specifications and the boundedness of the closed-loop signals is guaranteed. Figure 11: (**a**): The velocity of the wind along with the horizontal axis. (**b**): The acceleration of the wind along with the horizontal axis. (**c**): The velocity of the wind along with the vertical axis. (**d**): The acceleration of the wind along with the vertical axis. Furthermore, the effectiveness of the proposed control scheme can be seen in Figure 13a,c, where the altitude reference signal is successfully tracked with adaptive performance. In contrast, the altitude diverges to negative infinity when applying the P-PPC approach, as depicted in Figure 13b,d as the constrained control effort can not maintain all tracking errors within the conventional performance envelopes which leads to unbounded closed-loop signals owing to the singularity phenomenon of the conventional PPC approach. This demonstrates the superiority of the proposed control scheme in ensuring stable and accurate altitude tracking. Figure 12: **(a)**: The evolution of the actual airspeed \(V(t)\) (purple line) vs. the reference airspeed \(V_{d}(t)\) (dotted line) for the proposed APPC scheme. **(b)**: The evolution of the actual airspeed \(V(t)\) (purple line) vs. the reference airspeed \(V_{d}(t)\) (dotted line) for the P-PPC scheme. **(c)**: The evolution of airspeed tracking error \(V(t)-V_{d}(t)\) (orange line) within the adaptive performance envelope delimited by \(p_{2}^{conv}(t)\) and \(-p_{2}^{conv}(t)\) (black lines) for the P-PPC scheme.