## 1 Introduction
### Implicit NDI, INDI  
A team from Delft University in the Netherlands has conducted extensive design research on nonlinear dynamic inversion (NDI) control, a method that uses \"inversion\" of a dynamic system to design feedback linearization for general nonlinear control systems. For example,Juliana et al. [17] designed a controller using a feedback control algorithm combining classical proportional integral (PI) and proportional differential (PD) techniques through NDI to eliminate the nonlinear effects caused by uncertainty in the closed-loop system. [PERSON] et al. [18] design a transition period control method for a thrust vectoring vertical/short takeoff and landing (STOVL) UAV, that uses implicit NDI for the outer loop control circuit and improved eigenstructure assignment for the inner loop. They design guidance laws by converting the trajectory deviation into attitude control commands, and the guidance commands are tracked and the attitude is stabilized. The full six-degree-of-freedom (six-DOF) flight simulation results show that after the deceleration transition speed is lower than the minimum level flight speed, the UAV still maintains good trajectory tracking and attitude stability. Therefore, in eVTOL control design, converting the trajectory deviation into attitude control commands is also an important part, which can be accomplished by using the implicit NDI method. This is the reason why we choose implicit NDI as the outer loop control law in the transition phase control design. [PERSON] et al. [19] from Delft University proposed an improved method called incremental nonlinear dynamic inversion (INDI) control. Unlike NDI, INDI incorporates incremental control [20], and shows better robustness [21; 22] and anti-interference ability [23] in flight control. One major advantage of INDI is that it only requires a small portion of the model and does not rely on the inputs required by the control system. Additionally, INDI can handle model uncertainty better, showing robustness in the presence of inaccurate models without complex control structures or adaptive components [24]. Incremental nonlinear dynamic inversion control has been applied in eVTOL aircraft, achieving precise tracking of attitude commands [25], and its robustness has been also validated [26; 27]. [PERSON] et al. [28] simplified the design of helicopter flight controllers using a new approach based on INDI. By using feedback based on acceleration measurements, it avoided depending on information related to any changes in aerodynamic characteristics, and the control system did not require any model data that relied entirely on their state variables, thereby enhancing its robustness to inaccurate models. The test result shows that the controller stabilized the system and tracked the desired ground speed according to the specified performance characteristics. Therefore, INDI control can handle nonlinear models while providing robustness in the presence of model uncertainty without complex control structures or adaptive components. [PERSON] et al. [29] proposed a disturbance observer and demonstrated the good performance and stability of the closed-loop system. They pointed out that designing a nonlinear observer to estimate the unknown state is a means to solve the lack of sensor information. Therefore, this paper adopts an observer in the control law design to observe the information missing from the sensors. [PERSON] et al. [30] proposed a new framework for supersonic roll maneuver control that combines INDI and extended state observer (ESO). ESO reduces command oscillations and improves the servo's working life. Their proposed control achieves high-precision tracking performance during rapid roll maneuvers. Furthermore, the effectiveness and robustness of the controller were demonstrated through Monte Carlo (MC) simulation and frequency domain analysis.