## 4 Main Results
\begin{subarray}{c}t\rightarrow\infty\\ a_{n+1}\to 0\end{subarray} }|\dot{e}_{n+1}|\leq|\Delta(t)|\), and \(\lim_{ \begin{subarray}{c}t\rightarrow\infty\\ a_{n+1}\to 0\end{subarray} }|\dot{e}_{n+1}|\leq M\). Consider \(M^{\prime}\) such that  
\[\lim_{ \begin{subarray}{c}t\rightarrow\infty\\ a_{n+1}\to 0\end{subarray} }|\dot{e}_{n+1}|\leq M^{\prime}\leq M \tag{31}\]  
\(\Box\)  
**Corollary 1. **_Consider the system given in Equation (27), and the linear extended state observer Equation (28). Then, \(\lim_{ \begin{subarray}{c}t\rightarrow\infty\\ t\rightarrow\infty\end{subarray} }|x_{i}(t)-\hat{z}_{i}(t)|\leq\frac{1}{\left( \omega_{0}^{\prime}\right)^{n+2-i}}\frac{2M^{\prime}\lambda^{2}_{\text{max}}(P ^{\prime})}{\lambda_{\text{min}}(P^{\prime})}\), where \(x_{i}(t)\), and \(\hat{z}_{i}(t)\), \(i\in\{1,2,\ldots,n+1\}\) denote the solutions to Equations (27) and (28) respectively, and \(\omega_{0}^{\prime}\) is the bandwidth constant of the outer LESO._  
**Proof. ** As in Theorem 1, let the estimation error of the outer-loop ESO of Equation (28) is defined as  
\[\zeta_{i}(t)=x_{i}(t)-\hat{z}_{i}(t),\ i\in\{1,2,\ldots,n+1\} \tag{32}\]  
and \(\gamma_{i}(t)=\left(\omega_{0}^{\prime}\right)^{n+1-i}\varsigma_{i}\Big{(} \frac{t}{\omega_{0}^{\prime}}\Big{)}\), \(i\in\{1,2,\ldots,n+1\}\). Consider the candidate Lyapunov function \(V^{\prime}=\langle P^{\prime}\gamma,\gamma\rangle=\gamma^{T}P^{\prime}\gamma\), where \(\gamma\in\mathbb{R}^{n+1}\) and \(P^{\prime}\) is a symmetric and positive definite matrix with \(P^{\prime}\) as a symmetric and positive definite matrix, then  
\[|x_{i}(t)-\hat{z}_{i}(t)|\leq\frac{1}{\left(\omega_{0}^{\prime}\right)^{n+1-i} }\ast\left(\frac{2M^{\prime}\lambda^{2}_{\text{max}}(P^{\prime})}{\omega_{0}^{ \prime}\lambda_{\text{min}}(P^{\prime})}\Big{(}1-e^{-\frac{t}{2\lambda_{\text{ max}}(P^{\prime})}}\Big{)}+\sqrt{\frac{V^{\prime}(\gamma(0))}{\lambda_{\text{min}}(P)}}e^{- \frac{t}{2\lambda_{\text{max}}(P^{\prime})}}\right)\]And  
\[\lim_{t\to\infty}\lvert x_{i}(t)-\hat{z}_{i}(t)\rvert\leq\frac{1}{\left(\omega_{0} ^{\prime}\right)^{n+2-i}}\frac{2M^{\prime}\lambda^{2}{}_{max}(P^{\prime})}{ \lambda_{min}(P^{\prime})} \tag{33}\]  
\(\Box\)  
**Assumption A3. **_The states \(x_{i}\ (i=1,2,\ldots,n)\) and the generalized disturbance \(f\) of a \(n\)-dimensional uncertain nonlinear SISO system (1) are estimated by a convergent outer-loop LESO which produces the estimated states \(\hat{z}_{i}\ (i=1,2,\ldots,n)\) of the plant and the estimated generalized disturbance \(\hat{z}_{n+1}\) as \(t\to\infty\) respectively, i.e.,_  
\[\lim_{t\to\infty}\lvert x_{i}-\hat{z}_{i}\rvert=0,\ i\in\{1,2,\ldots,n\} \tag{34}\]  
_and_  
\[\lim_{t\to\infty}\lvert f-\hat{z}_{n+1}\rvert=0 \tag{35}\]  
**Assumption A4. **_A Tracking Differentiator (TD) produces a trajectory \(r_{i}\,\ i\in\{1,2,\ldots,n\}\) with minimum set point change. The trajectory converges to a reference trajectory \(r^{(i-1)}\) for \(i\in\{1,2,\ldots,n\}\) as \(t\to\infty\), i.e.,_  
\[\lim_{t\to\infty}\lvert r^{(i-1)}-r_{i}\rvert=0,\ i\in\{1,2,\ldots,n\} \tag{36}\]  
The stability of the closed-loop system with the Novel Active Disturbance Rejection Control (N-ADRC) is considered in the following theorem. **Theorem 2. **_Consider an \(n\)-dimensional uncertain nonlinear SISO system given in Equation (1). The system Equation (1) is controlled by the Linearization Control Law (LCL) \(u\) given by \(u=v^{\prime}-\hat{x}_{n+1}\) where_  
\[v^{\prime}=v-\hat{z}_{n+1} \tag{37}\]  
_and \(v\) is given as,_  
\[v=\lVert_{1}(\overline{e_{1}})\overline{e_{1}}+\lVert_{2}(\overline{e_{2}}) \overline{e_{2}}+\ldots+\lVert_{n}(\overline{e_{n}})\overline{e_{n}} \tag{38}\]  
_where \(\lVert_{i}:\mathbb{R}\to\mathbb{R}^{+}\) is an even nonlinear gain function._  
_where \(\overline{e_{i}}=r_{i}-\hat{z}_{i}\,i\in\{1,2,\ldots,n\}\) is the tracking error. Assuming that Assumptions A3 and A4 hold true, then, the closed-loop system is asymptotically stable, i.e., \(\lim\limits_{t\to\infty}\lvert\overline{e_{i}}\rvert=0,\ i\in\{1,2,\ldots,n\}\)._  
**Proof.