# MRAS Using Lyapunov Theory with Sliding Modes for a Fixed-Wing MAV
## 1 Introduction  
The fixed-wing MAVs are used for civil and military applications, and a vast array of research fields in this type of aerial machines like mathematics modeling, aerodynamic design, avionics, control theory, state observers, materials design for UAVs, and other less critical research or studies of external disturbances. External disturbances are a significant natural external force over the UAVs that should be studied because every unmanned aerial vehicle (UAV) has a problem when flying in the presence of perturbations (wind gusts, wind shear, and turbulence). Then, it is not easy to achieve a stable flight or the desired performance for any unmanned aerial vehicle when the perturbations are unknown. Other authors have studied adaptive control theory to propose solutions to achieve a stable flight with fixed-wing UAVs; for example, in [1], an adaptive controller based on fuzzy logic and sliding mode control (SMC) is designed, ref. [1] used fuzzy logic to reduce the chattering in the SMC and used the SMC methodology to approximate the uncertain parameters and unknown functions caused by the external perturbations or disturbances. Two types of fuzzy adaptive SMC for different fuzzy ranges are proposed. An extended observer based on adaptive second-order SMC is presented in [2] to control attitude in a smoothly feasible route of a fixed-wing UAV. The general idea in [2] is to use the observer to estimate the unknown disturbance and bound to obtain the system stability. On the other hand, in [3], the adaptive sliding mode control is applied to the carrier landing of fixed-wing UAVs. The results are obtained via simulations to appreciate the control effectiveness; the objective of the adaptive controller in [3] is to track the desired landing path and consider a simplification of the aerodynamic model from six degrees of freedom (6-DOF) to four degrees of freedom (4-DOF) and considering the pitch and roll angles constant. In [4], an automatic tuning approach for fixed-wing UAVs is presented to obtain the controller parameters. First, specific gains and phase margins are computed or calculated. A wind tunnel is even used to validate the closed-loop response with the proposed methodology. In [5], a robust gain-schedule autopilot for a guided projectile is presented; the gain schedule is based on H-infinity baseline autopilots and proposed and evaluated as an anti-windup compensator. The gain scheduling is used to tune the gains for a proportional-integral-derivative PID applied for the attitude in a fixed-wing UAV [6]. It is subject to several operating conditions of airspeed. An automatic tuning algorithm carries it out, and it should be mentioned that it uses a family of PID cascade control. In [7], the simulation results for a model-free adaptive control where the model information for the control scheme design is not necessary to try with the uncertainties is presented, and an attitude control is applied for fixed-wing UAVs. The adaptive backstepping sliding mode is probed by numerical simulations in [8] to stabilize the altitude, attitude, and velocity for a fixed-wing UAV, and even in the controller designed, it is applied with multiple algorithm fusion. The \(\mathcal{L}_{1}\) adaptive control with linearized model dynamics is used in [9] to realize path-following with a small fixed-wing UAV subject to wind disturbances. The results are presented by numerical simulation, and even the \(\mathcal{L}_{1}\) adaptive control is presented in [10] but considering a full-order nonlinear model of the fixed-wing UAV, and the results are obtained with computer simulations. To stabilize a fixed-wing UAV in attitude and altitude, an adaptive super twisting algorithm is presented in [11], and this controller ensures finite time convergence; computer simulations obtain the results. Then, this work proposes three adjustment mechanisms based on the Lyapunov theory and sliding mode theory to obtain three robust adjustment mechanisms to auto-tune the proportional and derivative gains for an adaptive PD controller.