## 5 Numerical Simulations
Figures 1 and 2 demonstrate precise reference signal tracking for NADRC, in contrast to ADRCs based on other verified observers in this work. The suggested NHOESO provided a much smoother control signal, as seen in Figure 3. The reason for this smoothness in the control signal was due to the nonlinear characteristic of the NHOESO, where it damped the spikes in the control signal and minimized the delivered energy to the nonlinear system. Figure 4 depicts the estimated error by the nine ESOs, including the proposed one, where the proposed NHOESO had the fewest estimated errors as compared to the other observers. The actual and estimated states of the model (32) with the NLESO type I are shown in Figure 5, and the actual and estimated states for the same model with the NHOESO are shown in Figure 6, where ITAE is the Integral Time-weighted Absolute Error (ITAE) and is defined as ITAE =\(\int_{0}^{t_{f}}t|e|\ dt\)[54], and ISU is the Integration of the Square of the controller energy and is defined as \(\text{ISU}=\int_{0}^{t_{f}}u^{2}(t)\ dt\)[55]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Index** & **The Type of ESO** & **Dynamics** \\ \hline
3 & NLESO type II [35] & \(\dot{\xi}_{1}=\dot{\xi}_{2}+\beta_{1}f_{2}(y-\dot{\xi}_{1})\) \\  & NLESO type II [35] & \(\dot{\xi}_{2}=\dot{\xi}_{3}+\beta_{2}f_{2}(y-\dot{\xi}_{1})\) \\  & NLESO type II [35] & \(\dot{\xi}_{3}=\beta_{3}f_{2}(y-\dot{\xi}_{1})\) \\  & NLESO type II [35] & \(\dot{\xi}_{1}=\dot{\xi}_{2}+\beta_{1}f_{3}(y-\dot{\xi}_{1})\) \\  & NLESO type III [35] & \(\dot{\xi}_{2}=\dot{\xi}_{3}+\beta_{2}f_{3}(y-\dot{\xi}_{1})\) \\  & NLESO type II [35] & \(\dot{\xi}_{3}=\beta_{3}f_{3}(y-\dot{\xi}_{1})\) \\  & NLESO type II [35] & \(\dot{\xi}_{3}=\dot{\xi}_{3}/\beta_{3}f_{3}(y-\dot{\xi}_{1})\) \\  & NLESO type II [35] & \(\dot{\xi}_{3}=\dot{\xi}_{1}(\frac{\dot{\xi}}{\dot{\xi}}-arc\tan(b_{2}e^{-b_{2} (e-b_{3})})\) \\ \hline
5 & NLESO type IV(a) [53] & \(\dot{\xi}_{1}=\dot{\xi}_{2}+\beta_{1}fal\left(y-\dot{\xi}_{1},\alpha,\dot{\xi}\right)\) \\  & NLESO type IV(a) [53] & \(\dot{\xi}_{2}=\dot{\xi}_{3}+\beta_{2}fal\left(y-\dot{\xi}_{1},\alpha,\dot{\xi}\right)\) \\  & NLESO type IV [36] & \(\dot{\xi}_{3}=\beta_{3}fal\left(y-\dot{\xi}_{1},\alpha,\dot{\xi}\right)\) \\
6 & NLESO type IV(b) [25] & \(\dot{\xi}_{1}=\dot{\xi}_{2}+\beta_{1}fal\left(y-\dot{\xi}_{1},\alpha_{1},\delta_ {1}\right)\) \\  & NLESO type IV(b) [25] & \(\dot{\xi}_{2}=\dot{\xi}_{3}+\beta_{2}fal\left(y-\dot{\xi}_{1},\alpha_{2},\delta_ {2}\right)\) \\  & NLESO type II [36] & \(\dot{\xi}_{3}=\beta_{3}fal\left(y-\dot{\xi}_{1},\alpha_{3},\delta_{3}\right)\) \\ \hline
7 & NLESO type V [36] & \(\dot{\xi}_{1}=\dot{\xi}_{2}+\beta_{1}tanh\left(y-\dot{\xi}_{1}\right)\) \\  & NLESO type V [36] & \(\dot{\xi}_{2}=\dot{\xi}_{3}+\beta_{2}tanh\left(y-\dot{\xi}_{1}\right)\) \\  & NLESO type II [36] & \(\dot{\xi}_{3}=\beta_{3}ftanh\left(y-\dot{\xi}_{1}\right)\) \\
8 & NLESO type VI [37] & \(\dot{\xi}_{1}=\dot{\xi}_{2}+\beta_{1}fal\left(y-\dot{\xi}_{1},\alpha,\dot{\xi} \right)+k_{1}\left(y-\dot{\xi}_{1}\right)\) \\  & NLESO type VI [37] & \(\dot{\xi}_{2}=\dot{\xi}_{3}+\beta_{2}fal\left(y-\dot{\xi}_{1},\alpha,\dot{\xi} \right)+k_{2}\left(y-\dot{\xi}_{1}\right)\) \\  & NLESO type II [36] & \(\dot{\xi}_{3}=\beta_{3}fal\left(y-\dot{\xi}_{1},\alpha,\dot{\xi}\right)+k_{3} \left(y-\dot{\xi}_{1}\right)\) \\  & NLESO type II [36] & \(\dot{\xi}_{1}=\dot{\xi}_{2}+\beta_{1}\mathcal{G}\left(y-\dot{\xi}_{1}\right)\) \\
9 & Proposed NHOESO & \(\dot{\xi}_{2}=\dot{\xi}_{3}+\beta_{2}\mathcal{G}\left(y-\dot{\xi}_{1}\right)\) \\  & NLESO type II [36] & \(\dot{\xi}_{3}=\dot{\xi}_{4}+\beta_{3}\mathcal{G}\left(y-\dot{\xi}_{1}\right)\) \\  & NLESO type II [36] & \(\dot{\xi}_{4}=\beta_{4}\mathcal{G}\left(y-\dot{\xi}_{1}\right)\) \\  & NLESO type II [36] & \(\Figure 1: The output response of the system of Equation (25). Figure 2: A magnified area of Figure 1.  
Figure 4: The estimation errors with curve indices from 1 to 9 represent LESO, NLESO types I, II, III, IV(a), IV(b), V, and VI, and NHOESO, respectively.