## 4 Main Results
Thus, by using Equation (20),  
\[|x_{i}(t)-\hat{x}_{i}(t)|\leq\frac{1}{\omega_{0}^{\alpha+1-i}}\Bigg{(}\frac{2 M\lambda^{2}_{max}(P)}{\omega_{0}\lambda_{min}(P)}\Big{(}1-e^{-\frac{t}{2 \lambda_{max}(P)}}\Big{)}+\ \sqrt{\frac{V(\eta(0))}{\lambda_{min}(P)}}e^{-\frac{t}{2 \lambda_{max}(P)}}\Bigg{)}\]  
Finally,  
\[\lim_{t\rightarrow\infty}|x_{i}(t)-\hat{x}_{i}(t)|\leq\frac{1}{\omega_{0}^{ \alpha+2-i}}\frac{2M\lambda^{2}_{max}(P)}{\lambda_{min}(P)} \tag{23}\]  
It is clear that when \(\omega_{0}\rightarrow\ \infty\), \(\lim_{t\rightarrow\infty}|x_{i}(t)-\hat{x}_{i}(t)|=0\)\(\Box\)  
Based on the result of Theorem 1, a trade-off between noise tolerance and accuracy of the estimation error can be attained. Equation (23) tells us an accurate state estimation can be obtained when the bandwidth \(\omega_{0}\)  leans towards infinity, which could not be feasibly realized. Moreover, with high bandwidth \(\omega_{0}\), the LESO allows measurement noise to propagate through the system. The LESO can restrain high-frequency noises under certain conditions. Now, the estimated error of the generalized disturbance \(e_{n+1}\) will be expressed as a difference between the actual generalized disturbance \(x_{n+1}\) and the estimated one \(\hat{x}_{n+1}\). From Equation (12), setting \(i=n+1\) gives  
\[e_{n+1}=x_{n+1}-\hat{x}_{n+1}\Rightarrow x_{n+1}=e_{n+1}+\hat{x}_{n+1} \tag{24}\]  
Consider the control law \(u(t)\) described by,  
\[u(t)=\
u^{\prime}(t)-\hat{x}_{n+1}\Rightarrow\hat{x}_{n+1}=\
u^{\prime}(t)-u(t) \tag{25}\]  
Substituting Equations (24) and (25) into Equation (1) gives the original uncertain SISO system in terms of the remaining estimated generalized disturbance error \(e_{n+1}\),  
\[\left\{ \begin{array}{c}\dot{x}_{1}(t)=x_{2}(t)\\ \dot{x}_{2}(t)=x_{3}(t)\\ \vdots\\ \dot{x}_{n}(t)=e_{n+1}+\
u^{\prime}(t)\\ y(t)=x_{1}(t)\end{array} \right. \tag{26}\]  
Adding an augmented state to the resultant system Equation (26) results in  
\[\left\{ \begin{array}{c}\dot{x}_{1}(t)=x_{2}(t)\\ \dot{x}_{2}(t)=x_{3}(t)\\ \vdots\\ \dot{x}_{n}(t)=x_{n+1}+\
u^{\prime}(t)\\ \dot{x}_{n+1}=\Delta^{\prime}=\dot{e}_{n+1}\\ y(t)=x_{1}(t),\end{array} \right. \tag{27}\]The estimated generalized disturbance error \(e_{n+1}\) will be further cancelled by the outer-loop LESO expressed by,  
\[\left\{ \begin{array}{c}\dot{\hat{z}}_{1}(t)=\hat{z}_{2}(t)+l_{1}(y(t)-\hat{z}_{ 1}(t))\\ \dot{\hat{z}}_{2}(t)=\hat{z}_{3}(t)+l_{2}(y(t)-\hat{z}_{1}(t))\\ \vdots\\ \dot{\hat{z}}_{n}(t)=\hat{z}_{n+1}(t)+v^{\prime}(t)+l_{n}(y(t)-\hat{z}_{1}(t)) \\ \dot{\hat{z}}_{n+1}(t)=l_{n+1}(y(t)-\hat{z}_{1}(t))\end{array} \right. \tag{28}\]  
where \(l_{i}=\alpha_{i}\left(\omega_{0}^{\prime}\right)^{i}\), \(i\in\{1,2,\ldots,n+1\}\) are outer-loop ESO gains, \(\alpha_{i}\) are relevant constants. **Lemma 1. **_Consider the system given in Equation (2), and the linear extended state observer Equation (3). The upper bound of the derivative of the generalized disturbance estimation error is given by:_  
\[\lim_{ \begin{subarray}{c}t\rightarrow\infty\\ a_{n+1}\to 0\end{subarray} }|\dot{e}_{n+1}|\leq M^{\prime},\text{ where }M^{\prime}\leq M\]  
**Proof. ** From Equation (12), with \(i=n+1\), \(e_{n+1}=x_{n+1}-\hat{x}_{n+1}\Rightarrow\dot{e}_{n+1}=\dot{x}_{n+1}-\dot{\hat {x}}_{n+1}\). Thus, \(|\dot{e}_{n+1}|\leq|\dot{x}_{n+1}|+|\dot{\hat{x}}_{n+1}|\), and from Equations (2) and (3),  
\[|\dot{e}_{n+1}|\leq|\Delta(t)|+|\beta_{n+1}e_{1}(t)| \tag{29}\]  
From Equation (23), \(\lim_{ \begin{subarray}{c}t\rightarrow\infty\\ a_{n+1}\to 0\end{subarray} }|e_{1}(t)|\leq\frac{1}{\omega_{0}^{n+1}}\frac{2M \lambda^{2}_{\text{max}}(P)}{\lambda_{\text{min}}(P)}\). As \(\beta_{n+1}=a_{n+1}\omega_{0}^{n+1}\), \(\lim_{ \begin{subarray}{c}t\rightarrow\infty\\ t\rightarrow\infty\end{subarray} }|\beta_{n+1}e_{1}(t)|\leq a_{n+1}\frac{2M \lambda^{2}_{\text{max}}(P)}{\lambda_{\text{min}}(P)}\). Thus,  
\[\lim_{ \begin{subarray}{c}t\rightarrow\infty\\ a_{n+1}\to 0\end{subarray} }|\beta_{n+1}e_{1}(t)|=0 \tag{30}\]  
From Equations (29) and (30), \(\lim_{