## 3 Conventional Active Disturbance Rejection Control Problem
### Extended State Observer  
Observers acquire data about the system states from its inputs and outputs progressively. [PERSON] first recommended the rule of observers in [36], where it was concluded that the state vector of the system can be estimated by observing the input and output of the system. Subsequently, there have been numerous varieties of state observers outlined in the literature that rely upon the mathematical model of the system, including high gain observers and sliding mode observers [27]. The ESO was the first observer presented that was autonomous of the mathematical model and presented within the framework of ADRC. Furthermore, ESO has denoted estimators, which are considered a vital part of modern controls. The basic principle of the ESO is to observe the constituent parts of the generalized disturbance in real time, including model discrepancy, exogenous disturbances,  
Figure 1: Structure of conventional Active Disturbance Rejection Control (ADRC). and the unmodelled dynamics of the nonlinear system. Additionally, it compensates for unpredicted disturbances in the control signal. ESO can be classified into two types. The first is linear ESO, which is an extension of the Luenberger observer [36; 37], where the equations of the ESO contain only the linear correcting terms in order to simplify the calculations. These terms manipulate the error between the actual states of the system and the estimated states in such a way that the error approaches zero. The second type is nonlinear ESO, where the error-correcting terms include a nonlinear function of the error. These nonlinear functions have the advantage of enhancing the estimation error more rapidly and smoothly than the linear ESO. Two approaches are common for ESO tuning: the pole-placement approach, and the bandwidth-based method. If the end goal is to reduce the number of parameters of the ESO, the parameters of the ESO can be expressed as a function of the bandwidth of the ESO, allowing only a single parameter of the ESO to be chosen or tuned. Selecting a bandwidth that is too large leads to a drop in the estimation error that nevertheless remains within an acceptable bound [38]. Observer bandwidth is chosen to be sufficiently larger than the disturbance frequency and smaller than the frequency of the unmodelled dynamics [39]. However, the performance of the ESO will deteriorate if the bandwidth of the ESO is selected to be too low or too high. High values in the bandwidth of the ESO and the controller result in good tracking performance and rejection of exogenous disturbances. The side effects of adopting large values for bandwidth can thus be summarized as (1) measurement noise causing a degradation in output tracking, introducing chatter on the control signal [40]; (2) a worsening of the transient response of the ESO, as large values of bandwidth lead to what are known as high gain observers [41]; and (3) the possibility of some unmodelled high-frequency dynamics being activated beyond a certain frequency, causing inconsistency in the closed-loop system. The noise and sampling rates are considered the two main factors constraining increases in the bandwidth. Based on this, an appropriate estimator bandwidth ought to be chosen in coordination with the noise tolerance and tracking performance. The authors in [14] designed a new class of Adaptive ESO (AESO) in which the observer bandwidth varied with time to provide better performance than the LESO. The disadvantage of this method is that the parameter tuning may become more complex as AESO order increases [14]. To alleviate the peaking phenomenon caused by different initial values of the ESO, the small variable \(\varepsilon\) was designed as in [42]:  
\[\frac{1}{\varepsilon}= \begin{cases}100t^{3}&0\leq t\leq 1\\ 100&t>1\end{cases}  \tag{8}\]  
The ESO parameters are tuned using Evolutionary Algorithm (EA), optimization techniques like bacterial foraging optimization (BFO) and particle swarm optimization (PSO) rather than a manual process. Eventually, the ESO begins estimating these states.