## 1 Introduction
Hence, a more improved observer is required to obtain fast convergence and high estimation performance than those of the previously developed observers. Meanwhile, additional attractive features of fractional-order over integer-order system in terms of stability and some functions which are not differentiable in integer sense system have been found to be differentiable in a fractional sense. Fractional calculus [38; 39] is an extension of regular integer-order calculus which makes it adequate and essential for completely characterizing many physical phenomena such as viscoelasticity, transmission lines, diffusion, and wave propagation. In the area of automatic control, fractional-order calculus has also been applied for modeling and control. The advantages of fractional-order calculus control were proved by many works, including the proportional-integral-derivative (PID) design [40; 41; 42; 43; 44; 45]. To improve the robustness of sliding mode control (SMC), SMC blended with a fractional-order sliding mode surface was developed [46; 47; 48; 49; 50; 51]. Recently, fractional-order BSC systems were studied for some chaotic systems [52; 53; 54] and servo systems [55; 56; 57]. It was confirmed that fractional-order based PID, SMC, and BSC systems showed significantly improved control performance, such as the transient, steady-state response of the closed loop system, and robustness to uncertainties for disturbance rejection compared with the controllers designed by integer-order calculus. There are some studies for quadrotor fractional-order BSC system [58; 59; 60] but there is little research on the fractional-order BSC with both a fractional-order tracking differentiator and disturbance observer. Motivated by the aforementioned works, a novel fractional-order continuous tracking differentiator and fractional-order nonlinear disturbance observer based on the sine hyperbolic function were studied to enhance the performance of a quadrotor UAV control system designed using the traditional integer-order based tracking differentiator and disturbance observer. A fractional-order filtered error surface was considered to obtain the effective control performance, and to reduce controller complexity. A proposed BSC system blended with a fractional-order tracking differentiator and a fractional-order observer was designed to obtain improved control performance despite the uncertainty over the traditional integer-order BSC system. We present the following contributions achieved in this study:  
1. The novel fractional-order BSC was designed to enforce insensitivity to variations of uncertainties of a quadrotor UAV system;
2. The proposed control strategy combined with the novel fractional-order tracking differentiator for a quadrotor UAV achieves improved settling-time convergence when compared to the traditional integer-order control system. Faster movements of a quadrotor UAV than those possible in present-day quadrotor UAVs are expected in a perturbed environment;
3. A fractional-order nonlinear disturbance observer was designed adopting the structure of the proposed tracking differentiator. Therefore, the performance of the estimation and disturbance rejection for uncertainties of the quadrotor UAV control system were improved by utilizing the proposed disturbance observer;
4. A fractional-order filtered error surface was utilized to facilitate the recursive controller design process of the BSC system and obtain simpler control system to avoid controller complexity. Finally, several comparative simulations of a nonlinear quadrotor UAV system were conducted to evaluate the performance of the proposed control scheme. The organization of remaining parts of this paper is described as follows. Section 2 presents the mathematical definitions of fractional-order calculus. The dynamic model of UAV system and fractional-order tracking differentiator are presented in Section 3.