## 1 Introduction
### Related Works  
In [17], an Unknown Input Observer (UIO) was proposed and used for Disturbance Accommodation Control (DAC). The external disturbance is considered an extra-state variable of the plant model. The UIO estimates the entire state, and the disturbance is cancelled with a specifically selected counteracting control signal. A Disturbance Observer (DOB) was designed in [18]. It uses the inverse of the nominal transfer function of the plant and rejects exogenous disturbances. Based on the concept of the DOB, a Multi-loop Perturbation Compensator (MPEC) was proposed in [19]. A hierarchical and recursive compensation method effectively compensates the perturbation (i.e., model uncertainty and external disturbances) to the plant. There was great improvement in the perturbation attenuation performance as the number of loops increased, but there was a reduction in the robust stability margin on the modeling error as the loop number increased. As explained in [20], the UIO offered superior application value compared to the DOB because provided an estimation of the state variables and did not only estimate the disturbance. A Nonlinear Disturbance Observer (NDO) was proposed in [21]. Such an NDO has been used in robotic manipulator systems, such as planar serial manipulators with revolute joints [22], and multi-fingered robot hands [23].  
In [24], an extended high-gain observer for nonlinear MIMO systems, in which the observer dynamics are faster than the extended system dynamics, was proposed by feeding back the output estimation error via a nonlinear function, and the nonlinear ESO (NLESO) was designed. There are two types of nonlinear functions: those that fit the rule of \"big error, small gain\", and those that fit the rule of \"small error, big gain\" [25]. the nonlinear function is generally chosen as a piecewise continuous, saturating, monotonically increasing function [26; 27; 28]. The nonlinear gains of the NLESO are designed to reduce the \"peaking phenomenon\" and to avoid large transient behaviors [29]. A nonlinear observer was also used to guarantee fast convergence and robustness concerning noise [30]. In [31; 32], a generic NLESO has been given. In [33; 34], two different nonlinear functions have been proposed. These nonlinear functions have not only a nonlinear characteristic but also have better smoothness properties. In particular, these functions can be separately adjusted to be suited to the practical application with different requirements through three design parameters. A function with two design parameters that change the shape and range of the function was proposed in [35]. A non-smooth function, which is a combination of linear and nonlinear terms, has been proposed in [25; 26; 28; 36; 37; 38] with two design parameters. Unfortunately, using this function has introduced a high-frequency chattering phenomenon [33; 35]. The work in [39] selected the hyperbolic tangent function without any parameter to ensure rapid convergence. In [40], the output estimation error was fed back using both nonlinear and switching terms to establish a novel finite-time convergent ESO. The motion control of a speed turntable with a temperature box was applied with the proposed ESO. This ESO performed better than a traditional ESO based on the experimental results. In [41], an ESO with time-varying gain was introduced to reduce drastically the peaking value. A detailed analysis with rigorous mathematical proof for the NLESO, which is the backbone of ADRC, was given in [32]. In [42], an ESO with high-order nonlinearity was studied using the Describing Function (DF) method. In addition, the nonlinear ESO and ADRC parameters were tuned using a simple and fast method.