## 5 Comparative Simulation Results and Discussion
### Comparison with the Cascaded PID Method  
In this simulation scenario, we compare the performance of the proposed control scheme against the PID scheme presented in [33], for trajectory tracking of a UAV cruising at a constant altitude under varying airspeed conditions. In this particular simulation scenario, we examine the trajectory tracking performance of the UAV while maintaining a constant reference height of 80. We introduce three sinusoidal airspeed profiles with increasing frequencies, denoted as \(V_{d_{t}}(t)=55+10\cos{(f_{i}t)}\), where \(i=1,2,3\) represents different profiles and \(f=[0.05,0.5,1.5]^{T}\) represents the corresponding frequencies. The initial conditions of the UAV are set as \(x(0)=[100,77,0.04,0.03,0,0]^{T}\) and the initial values of the adaptive performance functions are chosen as \(p_{1}(0)=24,p_{2}(0)=12.5,p_{3}(0)=0.12,p_{4}(0)=0.13,p_{5}(0)=0.2,p_{6}(0)=1.65\). The throttle settling limit is set at \(\tilde{\delta}_{T}=1\) and the elevator deflection limit at \(\tilde{\delta}_{e}=0.3\). To achieve faster convergence of the output to the reference trajectory, we have changed the error convergence rate parameters for the proposed control scheme. Specifically, we chose \(\lambda_{h}=0.7\) and \(\lambda_{v}=0.7\). It is worth noting that the remaining controller parameters were intentionally kept the same as those presented in Table 3 to clarify the universality of the proposed controller. This decision was made to demonstrate that the control gains do not significantly impact the control performance. As for the PID gains, they were determined through fine-tuning,  
Figure 8: (**a**): The evolution of the flight-path angle \(\gamma(t)\) (orange line) vs. the reference, unconstrained signal \(\gamma_{u}(t)\) (black dotted line), state limit (\(\tilde{\gamma}\)) is denoted by blue dotted lines. (**b**): The evolution of the pitch angle \(\theta(t)\) (orange line) vs. the reference, unconstrained signal \(a_{d}(t)+\gamma_{u}(t)\) (black dotted line), state limit (\(\tilde{\theta}\)) is denoted by blue dotted lines. (**c**): The evolution of the pitch rate \(q(t)\) (orange line) vs. the reference, unconstrained signal \(q_{d}(t)\) (black dotted line), state limit (\(\tilde{q}\)) is denoted by blue dotted lines.  
resulting in values of \(k_{p_{h}}=2\), \(k_{i_{h}}=0.1\), \(k_{d_{h}}=0.45\) for the altitude control loop, and \(k_{p_{h}}=6\), \(k_{i_{h}}=3\), \(k_{d_{h}}=9\) for the airspeed control loop.  
By observing Figures 9 and 10, it is evident that the altitude and airspeed tracking errors, respectively, are significantly smaller for the proposed APPC scheme compared to the PID scheme. It is worth noting that as the reference frequency increases, the PID tracking error also increases, whereas the error in the proposed scheme remains close to zero. Furthermore, it is important to highlight that the error signal in the PID scheme exhibits oscillations due to input saturation, whereas in the proposed approach, the error signal is smooth without requiring an exhaustive tuning procedure.  
Figure 10: **(a)**: Airspeed error of the proposed APPC scheme for f = 0.05. **(b)**: Airspeed error of the cascaded PID scheme for f = 0.05. **(c)**: Airspeed error of the proposed APPC scheme for f = 0.5. **(d)**: Airspeed error of the cascaded PID scheme for f = 0.5. **(e)**: Airspeed error of the proposed APPC scheme for f = 1.5. **(f)**: Airspeed error of the cascaded PID scheme for f = 1.5.  
Figure 9: **(a)**: Altitude error of the proposed APPC scheme for f = 0.05. **(b)**: Altitude error of the cascaded PID scheme for f = 0.05. **(c)**: Altitude error of the proposed APPC scheme for f = 0.5. **(d)**: Altitude error of the cascaded PID scheme for f = 0.5. **(e)**: Altitude error of the proposed APPC scheme for f = 1.5. **(f)**: Altitude error of the cascaded PID scheme for f = 1.5.