## 4 Main Results  
The innovative ADRC is constructed by adding an extra LESO, which shares an output signal with the plant to be controlled with an inner-loop LESO. The structure of the novel ADRC is presented in Figure 2. The inner LESO accomplishes the estimation of plant states and generalized disturbance. In a situation where a suitably low bandwidth \(\omega_{0}\) is selected for the inner LESO to reduce noise, the estimation of the generalized disturbance through the augmented state is associated with a relatively large estimation error, this situation is deeply considered in [43]. The outer-loop LESO will thus complete the rejection process by choosing an appropriate control law \(v\) that depends on the estimated generalized disturbance \(\hat{z}_{n+1}\). **Assumption A1**. : _The function \(L\) is continuously differentiable and there is a positive constant \(M\) such that \(\Delta(t)\) is bounded by it, i.e.,_  
\[\sup_{0\leq t\leq\infty}|\Delta(t)|\leq M\]  
This assumption represents wide range of fast and slow disturbances which exist in many real-world applications. **Assumption A2**. : _There exist constants \(\lambda_{1}\) and \(\lambda_{2}\) and positive definite, continuous differentiable functions \(V,W:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{+}\) such that_  
\[\lambda_{1}\|y\|^{2}\leq V(y)\leq\lambda_{2}\|y\|^{2} \tag{9}\]  
_Letting \(W(y)=\|y\|^{2}\), we can assume_  
\[\sum_{i=1}^{\rho}\frac{\partial V_{i}}{\partial y_{i}}(y_{i}-a_{i}y_{1})- \frac{\partial V}{\partial y_{\rho+1}}a_{\rho+1}y_{1}\leq-W(y) \tag{10}\]  
The stability of the proposed nested LESO is conducted in the following steps. Firstly, demonstrating the stability of the inner-loop LESO by deriving the error dynamics of the system in Equation (1) and proving its stability using the Lyapunov function (Theorem 1). Secondly, deriving the state-space equation of the nonlinear system combined with the inner-loop LESO (dotted square in Figure 2 given by Equation (26). Then, proving that the derivative of the generalized disturbance estimation error \(\dot{e}_{n+1}\) is upper bounded by \(M^{\prime}\), which is less than \(M\) defined in Assumption A1 (Lemma 1). Meanwhile, the stability analysis of the outer-loop LESO is demonstrated in Corollary 1 based on the results of Theorem 1. Finally, the stability analysis of the closed-loop system is given in Theorem 2. **Theorem 1**. : _Given the nonlinear plant (2) and the LESO in (3), then, the steady state estimation is given as_  
\[\lim_{t\rightarrow\infty}\lvert x_{i}(t)-\hat{x}_{i}(t)\rvert=\frac{1}{\omega_ {0}{}^{n+2-i}}\frac{2M\lambda^{2}{}_{max}(P)}{\lambda_{min}(P)} \tag{11}\]  
Figure 2: The Novel Active Disturbance Rejection Control (N-ADRC) structure with nested Linear Extended State Observers (LESOs). From Equation (15), \(\frac{d\eta_{i}(t)}{dt}=\omega_{0}{}^{n+1-i}\frac{d\omega\left(\frac{t}{a_{0}} \right)d\left(\frac{t}{a_{0}}\right)}{d\frac{t}{a_{0}}}=\omega_{i0}{}^{n-i} \frac{d\omega\left(\frac{t}{a_{0}}\right)}{d\frac{t}{a_{0}}}\) can be derived. Then,  
\[\frac{d\omega\left(\frac{t}{a_{0}}\right)}{d\frac{t}{a_{0}}}=\frac{1}{\omega_{ 0}{}^{n-i}}\frac{d\eta_{i}(t)}{dt} \tag{17}\]  
Both Equations (15) and (17) are substituted into Equation (16) and the result is  
\[\left\{ \begin{array}{rl}\frac{1}{\omega_{0}{}^{n-1}}\frac{d\eta_{1}(t)}{dt}= \frac{1}{\omega_{0}{}^{n-1}}\eta_{2}(t)-\omega_{0}a_{1}.\frac{1}{\omega_{0}{} ^{n}}\eta_{1}(t)\\ \frac{1}{\omega_{0}{}^{n-2}}\frac{d\eta_{2}(t)}{dt}=\frac{1}{\omega_{0}{}^{n- 2}}\eta_{3}(t)-\omega_{0}{}^{2}a_{2}.\frac{1}{\omega_{0}{}^{n}}\eta_{1}(t)\\ \vdots\\ \frac{d\eta_{n}(t)}{dt}=\eta_{in+1}(t)-\omega_{0}{}^{n}a_{n}.\frac{1}{\omega_{ 0}{}^{n}}\eta_{i}(t)\\ \frac{1}{\omega_{0}{}^{-1}}\frac{d\eta_{i,n+2}(t)}{dt}=\Delta-\omega_{0}{}^{n+ 1}a_{n+1}.\frac{1}{\omega_{0}{}^{n}}\eta_{1}(t)\end{array} \right. \tag{18}\]  
The time-scaled estimation error dynamics are  
\[\left\{