## 4 Simulation Results
### Active Fault-Tolerant Control Results  
The proposed method was compared with the previously proposed robust backstepping sliding mode control (BSMC) [15] and adaptive sliding mode control [44]. In this case, the motion of the quadcopter was designed to converge to the desired roll angle \(\phi_{d}\) in the presence of an actuator fault. The desired roll angle was generated through a filter following [PERSON] et al. [24].  
\[\ddot{\phi}_{d}+3\dot{\phi}_{d}+4\phi_{d}=4\phi \tag{52}\]  
where \(\phi\) starts from \(0^{\lx@math@degree}\), reaches \(5^{\lx@math@degree}\) at \(5\) s, reduces to \(0^{\lx@math@degree}\) at \(10\) s, decreases to \(-5^{\lx@math@degree}\) at \(15\) s, and goes back to \(0^{\lx@math@degree}\) at \(20\) s, and finally, remains at \(0^{\lx@math@degree}\) for the remaining time.  
To demonstrate the effectiveness of MASMBC, a large time-varying fault was injected into actuator \(2\) with almost \(40\%\) LoCE at \(14\) s as follows:  
\[\omega_{2\_fault}=\left\{\begin{array}{ll}\omega_{2},&t<14\\ 0.6(1+0.2\sin(2\pi t/25)\omega_{2},&t\geq 14\end{array}\right. \tag{53}\]  
From the initial time to \(14\) s, Figures 8 and 9 indicate that all control methods can use the same control torque value and make a good tracking in roll angle and error. In all figures, \(\phi_{d}\), \(\phi_{P}\), \(\phi_{ASMC}\), and \(\phi_{BSMC}\) represent the desired roll angle, actual roll angle using MASMBC, actual roll angle using ASMC, and actual roll angle using BSMC, respectively. When time-varying fault is injected to almost \(40\%\) LoCE at \(14\) s (Equation (53)), the BSMC presents an oscillating performance at the beginning and a low compensation to converge to the desired roll angle. In contrast, MASMBC shows a very good tracking response after a small overshoot at the beginning. The ASMC has a smaller oscillating performance than BSMC, but this method still cannot converge to the desired value.  
Figure 7: Controller inputs: (**a**) Thrust; (**b**) Roll moment; (**c**) Pitch moment; (**d**) Yaw momentThe MASMBC uses less control effort than ASMC and BSMC when a fault occurs, and there is a similar trend for all controllers after 15 s. Figure 8a indicates that MASMBC offers a quicker compensation in tracking performance than ASMC and BSMC.  
It should be noted that by using MASMBC scheme, the tracking error in roll angle can converge quickly to zero after fault occurs (Figure 8b). MASMBC can accommodate faults automatically through fault estimation unit.  
The corresponding fault estimation in each actuator is shown in Figure 10. As can be seen, when a fault occurs in actuator 2, the paring actuator 4 is also affected. Moreover, there are some small estimation errors in actuator 1 and 3 when faults occur at 14 s. After 16 s, the fault estimation values converge to zero in actuators 1, 3, and 4.  
Figure 8: Comparison of tracking performance between modified adaptive sliding mode backstepping control (MASMBC), adaptive sliding mode control (ASMC), and backstepping sliding mode control (BSMC): (**a**) Roll angle; (**b**) Tracking error.  
Figure 9: Comparison of control torque between MASMBC, ASMC, and BSMC with 40% loss of control effectiveness (LoCE) in actuator 2.