# MRAS Using Lyapunov Theory with Sliding Modes for a Fixed-Wing MAV
## 5 Discussion  
This work has presented the results obtained after several simulations of an adaptive PD controller based on MRAS (Model Reference Adaptive System) using Lyapunov theory, and adding a sliding mode theory in the adjustment mechanism to attenuate or almost eliminate the unknown perturbations due to wind gusts.  
Considering the analysis with the \(\mathcal{L}_{2}-norm\) to know the error and control effort performance for every adjustment mechanism, a better response for altitude control was obtained with adaptive PD controller based on [PERSON] theory adding a sliding mode theory (L-SM) because a smaller error and control effort was presented to achieve the desired altitude, and the second best result in altitude was obtained with adjustment mechanisms based on [PERSON] theory without the sliding mode theory. However, it should be mentioned that the adjustment mechanism based on the second-order and high-order sliding mode significantly reduced the perturbations that are depicted in the figures called zoom-disturbances, even this perturbation reduction is depicted in the altitude, and for the yaw angle in the roll angle.  
For the yaw angle, the adjustment mechanism that presented a better performance to achieve the desired roll angle is based on the Lyapunov theory with sliding mode, although it applied more control effort than the adjustment mechanism with [PERSON] theory without sliding mode theory because this last one presented a low response in the presence of unknown perturbations as has been shown in the figure zoom-disturbances in yaw angle. The adaptive PD controller presented a better response for the roll error angle with the adjustment mechanism based on Lyapunov theory, adding a first-order sliding mode (L-SM). However, although the adjustment mechanism with Lyapunov theory without sliding mode theory was presented as a minor control effort, the response in the presence of perturbations is not acceptable in comparison with the adjustment mechanism adding the sliding mode theory to the Lyapunov theory, and this has been depicted in the figure zoom-disturbances in the roll angle.  
Thus, adding the sliding mode theory in the design of the adjustment mechanism based on the Lyapunov theory to reduce or almost eliminate the unknown perturbations by wind gusts had a significant effect.  
A PID control law was not selected to design the adaptive controller because it is necessary to design another adaptive adjustment law for the integral action, and it takes more computer cost. For future work, first, we prefer to probe the adaptive PD proposed in this work in a real flight. It should be mentioned that the presence of sensor noise may significantly influence the comparison of different control laws.  
Future work consists of developing a fixed-wing MAV testbed to analyze the adaptive control laws proposed in this work in a real flight.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app14052198/s1](https://www.mdpi.com/article/10.3390/app14052198/s1)]([https://www.mdpi.com/article/10.3390/app14052198/s1](https://www.mdpi.com/article/10.3390/app14052198/s1)).  
Conceptualization, T.E.-F. and A.S.; methodology, T.E.-F.; software, I.G.; validation, T.E.-F., A.S. and I.G.; formal analysis, W.G.; investigation, T.E.-F.; resources, I.G. and A.S.; data curation, A.S. and W.G.; writing--original draft preparation, T.E.-F.; writing-review and editing, A.S. and I.G.; visualization, A.S.; supervision, T.E.-F. and W.G.; project administration, T.E.-F.; funding acquisition, T.E.-F., A.S. and I.G. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
The authors declare no conflicts of interest.