## 5 Numerical Simulations  
To assess the effectiveness of the suggested NHOESO, the following nonlinear SISO system is considered:  
\[\left\{ \begin{array}{l}\check{\varsigma}_{1}=\xi_{2}\\ \check{\xi}_{2}=f(\xi_{1},\;\xi_{2})+w(t)+(1+\mathrm{a}_{3}\sin(t))u\\ y=x_{1}\end{array} \right. \tag{25}\]  
where \(f(\xi_{1},\xi_{2})=a_{1}\xi_{1}+a_{2}sin(\xi_{2})\), with \(a_{1}=0.2\), \(a_{2}=a_{3}=0.1\), and the external disturbances are taken as \(w(t)=\exp(-t)\cos(t)\). The reference \(r(t)\) was selected as a periodic input and was defined as \(\cos(0.5t)\) imposed at \(t=0\) s.  
The _fall-based_ control law in this simulation is  
\[u=fal(\widetilde{e}_{1},\alpha_{1},\delta_{1})+fal(\widetilde{e}_{2},\alpha_{2}, \delta_{2})-\hat{\xi}_{3} \tag{26}\]  
where  
\[fal(e,\alpha,\delta)=\left\{ \begin{array}{ll}\frac{e}{\hat{\xi}^{1-\alpha}}& \left|e\right|\leq\delta\\ \left|e\right|^{s}sgn(e)&\left|e\right|>\delta\end{array} \right.\]  
The error of the tracking is steering the control signal given by \((\widetilde{e}_{1},\,\widetilde{e}_{2})^{T}=(r_{1},\,r_{2})^{T}-\left(\hat{ \xi}_{1},\,\,\hat{\xi}_{2}\right)^{T}\). The desired transient profile vector \((r_{1},\,r_{2})^{T}\) is obtained using a conventional Tracking Differentiator (TD), given as [10]:  
\[\left\{ \begin{array}{ll}\dot{r}_{1}=r_{2},\\ \dot{r}_{2}=-Rsign\Big{(}r_{1}-r+\frac{r_{2}|r_{2}|}{2R}\Big{)}\end{array} \right. \tag{27}\]  
Different ESOs have been chosen from the literature to be compared with our proposed NHOESO. They were used to find the estimated states \(\big{(}\hat{\xi}_{1},\hat{\xi}_{2}\big{)}^{T}\) and the estimated generalized disturbance \(\hat{\xi}_{3}\). The combination of the conventional TD of (34), the _fall-based_ control law of (26), and the proposed NHOESO of (7) is called NHOESO-based ADRC (written as NADRC in the following). The dynamics of the observers that were used in these simulations are listed in Table 1. The time responses of these simulations are shown in Figures 1-4. In this work, an OPI is proposed, which is represented as  
\[OPI=w_{1}\frac{ITAE}{N_{1}}+w_{2}\frac{ISU}{N_{2}}+ \tag{28}\]  
where \(\text{ITAE}=\int_{0}^{t_{f}}t|y-r|dt\) is the integration of the time absolute error for the output signal, \(\text{ISU}=\int_{0}^{t_{f}}v^{2}\ dt\) is the integration of the square of the control signal, \(\text{IAU}=\int_{0}^{t_{f}}\lvert v\rvert dt\) is the integration of the absolute of the control signal, and \(t_{f}\) is the final simulation time. The tuning process of the ADRC units and the conventional techniques was carried out using the Genetic Algorithm (GA) in the MATLAB\({}^{\text{\textregistered}}\) environment to minimize the OPI of (28). The sample data of the tuned parameters of the ADRC units are given in Table 2. The performance indices obtained from the numerical simulations are listed in Table 3. It can be seen that the proposed observer outperformed its counterparts, where the minimum value of the ITAE of the proposed NHOESO indicates that its time-domain performance was the best among the other conventional observers found in the literature, while the minimum delivered energy of the proposed NHOESO is very clear from the value of the ISU in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Index** & **The Type of ESO** & **Dynamics** \\ \hline \multirow{3}{*}{1} & \multirow{3}{*}{LESO [44]} & \(\hat{\xi}_{1}=\hat{\xi}_{2}+\beta_{1}(y-\hat{\xi}_{1})\) \\  & & \(\hat{\xi}_{2}=\hat{\xi}_{3}+\beta_{2}(y-\hat{\xi}_{1})\) \\  & & \(\hat{\xi}_{3}=\beta_{3}(y-\hat{\xi}_{1})\) \\ \hline \multirow{3}{*}{2} & \multirow{3}{*}{NLESO type I [33]} & \(\hat{\xi}_{1}=\hat{\xi}_{2}+\beta_{1}f_{1}(y-\hat{\xi}_{1})\) \\  & & \(\hat{\xi}_{2}=\hat{\xi}_{3}+\beta_{2}f_{1}(y-\hat{\xi}_{1})\) \\ \cline{1-1}  & & \(\hat{\xi}_{3}=\beta_{3}f_{1}(y-\hat{\xi}_{1})\) \\ \cline{1-1}  & & \(\text{where}\,\,\,f_{1}(e)=\frac{b_{1}tan(b_{1}(e-b_{1})-tan(-b_{1}b_{1}))}{ \hat{\xi}-tan\tan(-b_{1}b_{1})}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The list of the ESOs used in the comparison.