## 5 Simulations Results
### Hypothetical Model
In the first case, the output of the proposed system did not include any measurement noise, while in the second test case, a Gaussian noise was applied with variance (\(\sigma\)) equal to \(10^{-4}\) and the mean \(\mu=0\). The simulation results of both conventional ADRC and N-ADRC are shown in Figure 3. The estimated states \(\hat{x}_{2}\) and \(\hat{x}_{3}\) of the nonlinear system given in Equation (50) using C-ADRC scheme are shown in Figure 4. These states are also estimated using the N-ADRC scheme and are depicted in Figure 5. Moreover, the control signals for both schemes are illustrated in Figure 6.  
Figure 4: Estimated states using C-ADRC with measurement noise, (**a**) \(\hat{x}_{2}\), (**b**) \(\hat{x}_{3}\). Figure 3: The output response curves (y). (**a**) Conventional ADRC (C-ADRC) (without measurement noise), (**b**) C-ADRC (with measurement noise), (**c**) N-ADRC (without measurement noise), (**d**) N- ADRC (with measurement noise). As shown in Figure 3b, the presence of measurement noise has an adverse effect on the time response of the output for the in the case of C-ADRC, especially at the time interval [1.5,7] sec which shows large oscillations in the output response. The effect is not shown in the case of the N-ADRC (see Figure 3d). The estimated states of the C-ADRC shown in Figure 4 were highly affected by the measurement noise included in the output opposite the case of utilizing the N-ADRC, in which the noise had negligible effect at the estimated state (refer to Figure 5). This behaviour on the control signal was illustrated in Figure 6. This due to the bandwidth (\(\omega_{0}\)) of the ESO in the C-ADRC is less than the bandwidth (\(\omega_{0}^{\prime}\)) for the outer ESO. The numerical results are listed in Table 1. Adding measurement noise to the measured output significantly affected the output response--the Integral Time Absolute Error (ITAE)--and the total energy of the actuating signal (ISU) of the C-ADRC controller. In Table 1, \(\text{ITAE}=\int_{0}^{20}t|y-r|dt\) is the integration of the time absolute error for the output signal, and \(\text{ISU}=\int_{0}^{20}u^{2}\ dt\) is the integration of the square of the control signal. Figure 5: Estimated states using outer-loop LESO for the N-ADRC with measurement noise, (a) \(2_{2}\), (b) \(2_{3}\). Figure 3.0 Control signals with measurement noise: (a) C-ADRC, (b) N-ADRC. It is worth mentioning that, in our simulation, we have set the bandwidth (\(\omega_{0}\)) of the ESO in the C-ADRC to 100 rad/sec, while, for our proposed structure, a bandwidth (\(\omega_{0}\)) for the inner ESO was set to 100 rad/sec, and a bandwidth (\(\omega_{0}^{\prime}\)) for the outer ESO had a value of 22.83 rad/sec. It is clear that a big reduction in the bandwidth requirements in our proposed structure achieved a noticeable improvement in the performance in terms of both ITAE and ISU, especially in the noisy case. The estimation error of the generalized disturbances for the inner LESO is described by \(e_{3}\), which is given in Equation (12), and the generalized disturbance estimation error of the outer LESO is described by \(\zeta_{3}\), which is given in Equation (32); both of these are illustrated in Figure 3.1 The ITAE of \(e_{3}\) is 10.5769 and the ITAE of \(\zeta_{3}\) is 5.8251, displaying a percentage reduction in the ITAE equal to 45%. Figure 4 more clearly illustrates the reduction in \(\zeta_{3}\) against \(e_{3}\). As illustrated in Figure 8 the derivative of the generalized disturbance \(\Delta(t)=\frac{dL}{dt}\) is bounded during the transient period by 5.34 and at the steady state by 0.3. Assumption A1 is already satisfied. Moreover, the estimation errors of the C-ADRC with and without measurement noise are shown in Figures 9 and 10, respectively. In the same manner, the estimation errors of the N-ADRC with and without measurement noise are shown in Figures 11 and 12, respectively.