## 4 The Proposed Nonlinear Higher-Order Extended State Observer (NHOESO)  
### The NHOESO Definition and Convergence Theorem  
To extend the concept of the linear extended state observer (LESO), two major improvements are included in this work. Firstly, a smooth nonlinear error function is introduced. Secondly, adding more than one augmented state to the proposed NHOESO enables it to track the generalized disturbance \(\xi_{\rho+1}\) asymptotically. Moreover, the estimation error \(e_{i}(t),\ i\in\{1,2,\ldots,\rho\}\) depends on the upper bound of the second derivative of the generalized disturbance \(\xi_{\rho+1}\). This aspect allows the NHOESO to estimate disturbances and uncertainties with higher-order derivatives. As was shown in the simulation, these improvements provided faster and more accurate estimations of the states \(\xi_{i},\ i\in\{1,2,\ldots,\rho\}\) and the generalized disturbance \(\xi_{\rho+1}\). The proposed nonlinear error function also reduced the chattering phenomenon presented in LESOs. In addition, the proposed NHOESO produced both a smooth control signal and a minimum overshoot in the output response. Starting from the nonlinear system of (3) and adding the extended states \(\xi_{\rho+1}=f+(b(x)-b_{0})u=L,\xi_{\rho+2}=\dot{L}\), System (3) can be written equivalently as  
\[\left\{ \begin{array}{l}\dot{\xi}_{i}=\xi_{i+1},\ i\in\{1,2,\ldots,\rho-1\} \\ \dot{\xi}_{\rho}=\xi_{\rho+1}+b_{0}u\\ \dot{\xi}_{\rho+1}=\xi_{\rho+2}\\ \dot{\xi}_{\rho+2}=\Delta_{h}\end{array} \right. \tag{6}\]  
where \(\Delta_{h}(t)=\ddot{L}\). The proposed NHOESO is described as  
\[\left\{ \begin{array}{l}\dot{\xi}_{i}=\xi_{i+1}+\beta_{i}\mathcal{G}(e),\ i \in\{1,2,\ldots,\rho-1\}\\ \dot{\xi}_{\rho}=\xi_{\rho+1}+\beta_{\rho}\mathcal{G}(e)+b_{0}u\\ \dot{\xi}_{\rho+1}=\dot{\xi}_{\rho+2}+\beta_{\rho+1}\mathcal{G}(e)\\ \dot{\xi}_{\rho+2}=\beta_{\rho+2}\mathcal{G}(e)\end{array} \right. \tag{7}\]  
where \(e=(y-\dot{\xi}_{1})\) is the estimation error and the parameters \(\beta_{i}\), \(i=\{1,2,\ldots,\rho+2\}\) are the gains of the estimator to be adjusted. It is supposed that \(\beta_{i}=a_{i}\omega_{0}^{i}\), where \(a_{i}\), \(i\in\{1,2,\ldots,\rho+2\}\) is the related design parameter with each \(\omega_{0}^{i}\), and \(\omega_{0}\) is the bandwidth of the NHOESO estimator. The function \(\mathcal{G}:\mathbb{R}\rightarrow\mathbb{R}\) is proposed as  
\[\mathcal{G}(e)=K_{a}|e|^{a}sign(e)+K_{\beta}|e|^{\beta}e \tag{8}\]  
where \(e=(y-\dot{\xi}_{1})\) is the estimation error, \(K_{a},K_{\beta},\alpha,\ and\ \beta\) are positive design parameters, i.e., \(\alpha,\ \beta>0\). The suggested function \(\mathcal{G}(e)\) of (8) can be expressed as  
\[\mathcal{G}(e)=\left(K_{a}\frac{|e|^{a}}{e}sign(e)+K_{\beta}|e|^{\beta}\right)e \tag{9}\]  
Since sign \((e)=e/\mid e\mid\), for \(|e|\
eq 0\), then  
\[\mathcal{G}(e)=\left\{ \begin{array}{cc}0&e=0\\ k(e)e&e\
eq 0\end{array} \right. \tag{10}\]  
Function \(k(e):\mathbb{R}/\{0\}\rightarrow\mathbb{R}^{+}\) is an even function, given as  
\[k(e)=K_{a}|e|^{a-1}+K_{\beta}|e|^{\beta} \tag{11}\]  
To prove the convergence of the NHOESO, the following assumptions are needed:  
**Assumption A3**. : \(L\) _is a continuously differentiable function._  
**Assumption A4**. : _There exists \(M_{h}\in\mathbb{R}^{+}\) such that \(sup_{t\in[0,\infty)}|\Delta_{h}(t)|=M_{h}\), where \(\Delta_{h}(t)=\tilde{L}\)._  
**Assumption A5**. : \(V:\mathbb{R}^{\rho+2}\to\mathbb{R}^{+}\) and \(W:\mathbb{R}^{\rho+2}\to\mathbb{R}^{+}\) _are continuously differentiable functions with_  
\[\lambda_{1}\|\eta\|^{2}\leq V(\eta)\leq\lambda_{2}\|\eta\|^{2},\;\;W(\eta)=\| \eta\|^{2} \tag{12}\]  
\[\sum_{i=1}^{\rho+1}\frac{\partial V(\eta)}{\eta_{i}}\bigg{(}\eta_{i+1}-a_{i}k \bigg{(}\frac{\eta_{1}}{\omega_{0}\rho}\bigg{)}\cdot\eta_{1}\bigg{)}-\frac{ \partial V(\eta)}{\partial y_{\rho+2}}a_{\rho+2}k\bigg{(}\frac{\eta_{1}}{ \omega_{0}\rho}\bigg{)}\eta_{1}\leq-W(\eta) \tag{13}\]  
**Theorem 1**.