## 4 The Proposed Nonlinear Higher-Order Extended State Observer (NHOESO)
### Mismatched Disturbances  
To satisfy the matched condition, the ESO assumes that the plant is expressed in the normal form [45; 46]. Thus, it can only be applied to systems that can be expressed in the normal form directly or by changing variables. When a system has zero dynamics, performing such a transformation can be challenging. There are also nonlinear systems with disturbances appearing in a different channel of control input, and these systems fail to satisfy the matching condition. Due to this, the ADRC is no longer able to manipulate this mismatched disturbance as before. For instance, the following nonlinear model belongs to a class of uncertain nonlinear systems in a lower triangular form with mismatched disturbance [47; 48; 49; 50; 51; 52]:  
\[\left\{\begin{array}{l}\check{\varsigma}_{i}=a_{i}\check{\varsigma}_{i+1}+ \phi_{i}(\check{\varsigma}_{1},\ldots,\check{\varsigma}_{i})+w_{i},\;i\in\{1,2, \ldots,\rho-1\}\\ \check{\varsigma}_{\rho}=\phi_{\rho}\big{(}\check{\varsigma}_{1},\check{ \varsigma}_{2},\ldots,\check{\varsigma}_{\rho}\big{)}+w_{\rho}+bu,\\ y=\check{\varsigma}_{1}.\end{array}\right. \tag{22}\]  
where \(\check{\varsigma}=\big{(}\check{\varsigma}_{1}(t),\check{\varsigma}_{2}(t), \ldots,\check{\varsigma}_{\rho}(t)\big{)}^{T}\in\mathbb{R}^{\rho}\) is the system state, \(y(t)\in\mathbb{R}\) is the measured output, \(u(t)\in\mathbb{R}\) is the control input, \(w_{i}(t)\in\mathbb{R},\;i\in\{1,2,\ldots,\rho\}\) is the unknown exogenous disturbance, and \(b\in\mathbb{R}\) is the control coefficient. The function \(\;\phi_{i}:\mathbb{R}^{i}\rightarrow\;\mathbb{R},\;i\in\{1,2,\ldots,\rho\}\,\).  
**Theorem 2**.: _A second-order nonlinear system in a lower triangular form with mismatched disturbances can be described as follows:_  
\[\left\{\begin{array}{l}\check{\varsigma}_{1}=a_{1}\check{\varsigma}_{2}+ \phi_{1}(\check{\varsigma}_{1})+w_{1}\\ \check{\varsigma}_{2}=\phi_{2}(\check{\varsigma}_{1},\check{\varsigma}_{2})+w _{2}+bu\\ y=\check{\varsigma}_{1}\end{array}\right. \tag{23}\]  
where \(\check{\varsigma}=\big{(}\check{\varsigma}_{1}(t),\check{\varsigma}_{2}(t) \big{)}^{T}\in\mathbb{R}^{2}\) is the system's state, \(y(t)\in\mathbb{R}\) is the measured output, \(u(t)\in\mathbb{R}\) is the control input, \(\;w_{i}(t)\in\mathbb{R},\;i\in\{1,2\}\) is the unknown exogenous disturbance, and \(b\in\mathbb{R}\) is the control coefficient. The function \(\;\phi_{i}:\mathbb{R}^{i}\rightarrow\;\mathbb{R},\;i\in\{1,2\}\,\). If the function \(\phi_{1}\) and the exogenous disturbance \(w_{1}\) are differentiable w.r.t \(t\), System (23) can be transformed into the following form:  
\[\left\{\begin{array}{l}\check{\tilde{\varsigma}}_{1}=\widetilde{\varsigma}_ {2},\\ \check{\tilde{\varsigma}}_{2}=f\big{(}\widetilde{\varsigma}_{1},\widetilde{ \varsigma}_{2},w_{1},\check{w}_{1},w_{2}\big{)}+\hat{b}u,\\ y=\widetilde{\varsigma}_{1}.\end{array}\right. \tag{24}\]  
where \(\;f\Big{(}\widetilde{\varsigma}_{1},\widetilde{\varsigma}_{2},w_{1},\check{w} _{1},w_{2}\Big{)}\;=\;a_{1}\phi_{2}\Big{(}\widetilde{\varsigma}_{1},\frac{ \tilde{\varsigma}_{2}-\phi_{1}(\widetilde{\varsigma}_{1})-w_{1}}{a_{1}}\Big{)} +\frac{\phi_{1}(\widetilde{\varsigma}_{1})}{\phi_{1}}\widetilde{\varsigma}_{2} +a_{1}w_{2}+\check{w}_{1}\) and \(\hat{b}=a_{1}b\).  
**Proof**: seen Appendix A.