## 4 The Proposed Nonlinear Higher-Order Extended State Observer (NHOESO)
### Justification for Adding an Additional Augmented State  
The reason to incorporate more additional states in the proposed NHOESO is explained in this subsection. The analysis of the steady-state estimation error \(e_{i}(t)\), \(i\in\{1,2,\ldots,\rho+1\}\) of the LESO can be found in [11],  
\[\lim_{t\to\infty}\big{|}\xi_{i}-\hat{\xi}_{i}\big{|}=\frac{1}{\omega_{0}\rho^{ +2-i}}\frac{2M\lambda^{2}_{max}(P)}{\lambda_{min}(P)} \tag{14}\]  
while that of the NHOESO is found in Theorem 1,  
\[\lim_{t\to\infty}\big{|}\xi_{i}-\hat{\xi}_{i}\big{|}=\frac{1}{\omega_{0}\rho^{ +3-i}}\frac{2M_{h}\lambda^{2}_{max}(P)}{\lambda_{min}(P)} \tag{15}\]  
These results illustrate that the steady-state estimation error \(\lim_{t\to\infty}\big{|}\xi_{i}-\hat{\xi}_{i}\big{|}\) of the NHOESO is more sensitive to an increase in the bandwidth \(\omega_{0}\) than that of the LESO. This is because of the presence of \(\omega_{0}\) in the denominator. Moreover, the steady-state estimation errors are proportional to \(M\) for the LESO as in (14) and to \(M_{h}\) for the NHOESO as in (15), where \(M_{h}\) and \(M\) are the limit bounds of the second and first derivatives of the generalized disturbance \(\xi_{\rho+1}\), respectively (see Assumptions A2 and A4)  
Consider a generalized disturbance \(\xi_{\rho+1}\), which is a linear function in time, i.e., \(L(t)=at\), where a is a constant. Then \(\Delta=\dot{L}=a\) and \(\Delta_{h}=\tilde{L}=0\), and based on this, the upper bound \(M\), is a non-zero constant, while \(M_{h}\) is zero. In this case, for a specific low value of \(\omega_{0}\), the steady-state estimation error of the NHOESO in (15) will be zero, while that of the LESO given in (14) has a non-negligible value. Therefore, the NHOESO is more suitable than the LESO to give an estimation of the generalized disturbance \(\xi_{\rho+1}\) of the linear type. Moreover, a generalized disturbance \(\xi_{\rho+1}\) with a higher-order derivative will worsen the steady-state estimation error for the LESO because if the generalized disturbance \(\xi_{\rho+1}\) is expressed as \(L(t)=at^{2}\), then the upper bound \(M\to\infty\), as \(t\to\infty\). The steady-state estimation error of the LESO will escape to infinity, i.e., \(\lim_{t\to\infty}\big{|}\xi_{i}-\hat{\xi}_{i}\big{|}\to\infty\), and thus, the LESO will diverge. However, \(M_{h}\) will have a constant value of \(2a\). Consequently, the NHOESO will have a small steady-state estimation error for sufficiently large NHOESO bandwidth \(\omega_{0}\). The only weak point of the NHOESO is the time required for the estimatedgeneralized disturbance \(\xi_{3}\) to be settled to its actual value \(\xi_{3}\), which is analyzed in the following example. Consider a situation where the relative degree \(\rho=2\), and the LESO is expressed as  
\[\left\{ \begin{array}{l}\dot{\xi}_{1}=\dot{\xi}_{2}+\beta_{1}\big{(}y-\dot{\xi} _{1}\big{)},\\ \dot{\xi}_{2}=\dot{\xi}_{3}+\beta_{2}\big{(}y-\dot{\xi}_{1}\big{)}+b_{0}u,\\ \dot{\xi}_{3}=\beta_{3}\big{(}y-\dot{\xi}_{1}\big{)}\end{array} \right. \tag{16}\]  
The dynamics of the NHOESO given in (7) with \(\mathcal{G}\big{(}y-\dot{\xi}_{1}\big{)}=y-\dot{\xi}_{1}\) is expressed as  
\[\left\{ \begin{array}{l}\dot{\xi}_{1}=\dot{\xi}_{2}+\beta_{1}\big{(}y-\dot{ \xi}_{1}\big{)},\\ \dot{\xi}_{2}=\dot{\xi}_{3}+\beta_{2}\big{(}y-\dot{\xi}_{1}\big{)}+b_{0}u,\\ \dot{\xi}_{3}=\dot{\xi}_{4}+\beta_{3}\big{(}y-\dot{\xi}_{1}\big{)},\\ \dot{\xi}_{4}=\beta_{4}(y-\dot{\xi}_{1})\end{array} \right. \tag{17}\]  
Given that \(\dot{\xi}_{4}=\int\beta_{4}(y-\xi_{1})\ dt\), then (17) can be expressed as  
\[\left\{ \begin{array}{l}\dot{\xi}_{1}=\dot{\xi}_{2}+\beta_{1}\big{(}y-\dot{ \xi}_{1}\big{)},\\ \dot{\xi}_{2}=\dot{\xi}_{3}+\beta_{2}\big{(}y-\dot{\xi}_{1}\big{)}+b_{0}u,\\ \dot{\xi}_{3}=\beta_{4}\int_{0}^{t}\big{(}y-\dot{\xi}_{1}\big{)}\ dt+\beta_{3} \big{(}y-\dot{\xi}_{1}\big{)}\end{array} \right.