# Quadcopter Adaptive Trajectory Tracking Control:
## 1 Introduction
### Related Works  
It is undeniable that the conventional proportional-integral-derivative (PID) control is still widely used in a variety of many applications all over the world because it is uncomplicated to be appliedand acceptably meet given requirements in many studies [7; 8; 9; 10; 11; 12; 13; 14; 15]. The authors [16; 17; 18; 19] presented the use of a multi-loop control scheme (i.e., inner-loop and outer-loop) to control quadcopters in specific applications. Several authors improved a modified PID approach to design control laws based on nonlinear mathematical models to achieve a position tracking [20; 21]. By making use of the discrete linearisation model of tracking errors, the PID controller combined with least quadratic regulation (LQR) were presented in [22; 23] to find a discrete control input. When it comes to tracking control of mechanical systems, sliding mode control (SMC) is often employed to guarantee the stability of the systems. The authors of [24; 25] presented SMC laws combined with automatically tuning control coefficients to achieve the partial system stability. Typically, the cascade principle is also applied to design a full control law based on the Lyapunov's direct method that guarantees the asymptotic stability of each loop [25; 26; 27; 28; 29; 30; 31; 32] via the backstepping technique [33]. Model predictive control, which obtains some advantages of optimal control, was employed to carry out not only path-tracking problem, but also state constraints [34; 35]. In order to tackle external disturbances, a robust nonlinear \(\mathcal{H}_{\infty}\) controller [36] was proposed to track the reference trajectory.  
One of the obstacles of control design for quadcopter is the presence of system parameterized uncertainties in the mathematical model. The system's parameters must be updated to adjust its control gains when adding several auxiliary devices or replacing some accessories, leading to inconvenience in practice. Therefore, studies [31; 37; 38] used SMC in combination with an adaptive law based on certainty equivalence principle [33] or artificial neural networks (ANNs) [39; 40; 41] to deal with uncertainties such as inertial moments and arm length. By using Fuzzy logic, the authors [42] found a control law without any advance knowledge about mathematical model of quadcopter. When ANNs and Fuzzy logic are applied to identify the system model instead of using the explicit mathematical model, the tracking performance of the considered system has still been controversial topic.  
On the other hand, the above existing works still have some negative viewpoints in trajectory tracking problem. Many kinds of PID control (i.e., conventional form, modified form, and combination with LQR) may not clearly show the asymptotic stability. The control schemes based on SMC always use a sign function to switch between two modes, i.e., controlled variables are either outside or within the sliding surface, leading to chattering phenomenon. In other studies [17; 25; 43], the second order SMC was applied to reduce a chattering phenomenon to enhance the tracking performance. Moreover, it is a considerable drawback that SMC state variables have to reach the sliding surface in finite time to guarantee the convergence at the origin, which was ignored in the above papers. Based on the cascade principle, only a linear system which is a result of combining all linear stable subsystems can directly achieve stability. In contrast, a nonlinear model systems may not be stable because of the appearance of a finite escape time phenomenon which may cause state variables to become infinity in finite time (Chapter 3 in [44]), leading to a highly undesirable case in automatic control systems.