## 5 Numerical Simulation
Considering that the backstepping control laws \(k_{i},i=1,\cdots,4\) are key factors to determine the convergence rate of the tracking error, we select the control gains as \(k_{1}=5\), \(k_{2}=7\), \(k_{3}=8\) and \(k_{4}=10\) to achieve good tracking performance and transition performance. Besides the proposed control scheme, the adaptive backstepping method derived from Reference [44] is also simulated for comparison purposes (hereafter referred as the compared controller) and the parameters are taken as the same as that of [44]. The compared controller is based on barrier Lyapunov function (BLF) and employs a command filter to obtain the constrained virtual controls and their time derivatives in the back-stepping design process. As the controller is essentially a kind of adaptive backstepping approach depending on nominal values of the aerodynamic coefficients, theoretically, the static uncertainties and disturbances can be handled by adaptive laws. The simulation results are shown in Figures 3-9. Figure 3 depicts the comparative curves with respect to the altitude tracking performance of the two controllers, which can guarantee the boundedness of the altitude tracking error in the presence of internal uncertainties and generalized disturbances. It is obvious that the proposed scheme has better transient and steady-state performance, as the amplitude of the tracking error is less than \(0.5m\) in the oscillating process and further falls within the range of \(\pm 0.2m\) for \(t>20\)s; while in the compared scheme, the mechanism of disturbance and uncertainty suppression is based on adaptive control rather than extended state estimation, the performance of the compared controller deteriorates significantly when the time-varying disturbances are taken into account. The noteworthy flight states in the inner loop are illustrated in Figures 4-6, which include flight path angle \(\mu\), angle of attack \(\alpha\) and pitch angular rate \(q\) and their time responses are smooth signals in a reasonable range. In particular, with the help of the FTCDs in the proposed controller, flight path angle and angle of attack can reach their steady-state values with more rapid convergence rate and mild fluctuations, even when the abrupt disturbance occurred at \(t=60\)s. When the altitude command tends to be constant, flight path angle and pitch angle rate vary slowly and converge to zero; therefore, the permissible angle of attack is guaranteed. Figure 7 exhibits the comparison curves of control inputs. Under a proposed controller, the max deflection angle is 11.1 deg. The deflection of control effector has a smoother transient process than the response in the compared controller, which can reduce the requirement of actuator bandwidth in practice. Besides, the derivatives of intermediate virtual law obtained by FTCDs are shown in Figure 8. Obviously, the FTCDs provide bounded, continuous and smooth estimations such that the problem of \"explosion of complexity\" has been eliminated. Meanwhile, the good performance of the proposed method is also attributed to the ability of the ESO in a timely accurate estimation of the compound disturbances composed with modelling uncertainties and external disturbances. As shown in Figure 9, disturbance estimations can track the actual disturbance signals in the first two seconds and have fairly small estimation error margins thereafter. Figure 4: The curves of flight path angle \(\mu\) in the comparative simulations. Figure 5: The curves of angle of attack \(\alpha\) in the comparative simulations. Figure 3: The saucer-shaped UAV altitude tracking performance: (**a**) Comparison of the response of altitude tracking and (**b**) comparison of the altitude tracking error. Figure 8: The time derivatives of the intermediate virtual control law \(x_{\mu}\), \(x_{\theta}\), and \(x_{q}\), obtained by finite-time convergent differentiators (FTCDs). Figure 6: The curves of pitch angular rate \(q\) in the comparative simulations.