## Appendix A Proof of Theorems 1 and 2  
**Theorem A1. (NHOESO convergence):**_Given System (6) and NHOESO (7), it follows that, under Assumptions 3, 4, and 5, for any initial conditions,_  
\[\lim_{t\to\infty}\bigl{|}\xi_{i}(t)-\hat{\xi}_{i}(t)\bigr{|}=O\biggl{(}\frac {1}{\omega_{0}\rho^{+3-i}}\biggr{)}\]  
\[\lim_{t\to\infty}\quad\bigl{|}\xi_{i}(t)-\hat{\xi}_{i}(t)\bigr{|}=0\]  
\[\omega_{0}\to\infty\]  
_where \(\xi_{i}\) and \(\hat{\xi}_{i}\) symbolize the states of (6) and (7) respectively, with \(i\in\{1,2,\ldots,\rho+2\}\)._  
**Proof of Theorem A1:** Let \(e_{i}=\xi_{i}-\hat{\xi}_{i}\), \(i\in\{1,2,\ldots,\rho+2\}\). Correspondingly, let  
\[\eta_{i}=\omega_{0}\rho^{+1-i}e_{i}\biggl{(}\frac{t}{\omega_{0}}\biggr{)},\;i \in\{1,2,\ldots,\rho+2\} \tag{11}\]  
Then, the dynamics of the estimation error can be expressed in time-scaled form, as  
\[\left\{ \begin{array}{l}\frac{d\eta_{1}}{dt}=\eta_{2}-a_{1}k\biggl{(}\frac{ \eta_{1}}{\omega_{0}\rho}\biggr{)}\eta_{1}\\ \frac{d\eta_{2}}{dt}=\eta_{3}-a_{2}k\biggl{(}\frac{\eta_{1}}{\omega_{0}\rho} \biggr{)}\eta_{1}\\ \vdots\\ \frac{d\eta_{2}}{dt}=\eta_{\rho+1}-a_{\rho}k\biggl{(}\frac{\eta_{1}}{\omega_{ 0}\rho}\biggr{)}\eta_{1}\\ \frac{d\eta_{\rho+1}}{dt}=\eta_{\rho+2}-a_{\rho+1}k\biggl{(}\frac{\eta_{1}}{ \omega_{0}\rho}\biggr{)}\eta_{1}\\ \frac{d\eta_{\rho+2}}{dt}=\frac{\Delta_{\Psi}}{\omega_{0}\rho^{-}}-a_{\rho+2} k\biggl{(}\frac{\eta_{1}}{\omega_{0}\rho}\biggr{)}\eta_{1}\end{array} \right. \tag{12}\]  
Let the candidate [PERSON] functions \(V,W:\mathbb{R}^{\rho+2}\to\mathbb{R}^{+}\) denoted by \(V(\eta)=\langle P\eta,\eta\rangle=\eta^{T}P\eta\), with \(\eta\in\mathbb{R}^{\rho+2}\) and where \(P\) is a positive definite symmetric matrix. Consider (12) of Assumption A5 with \(\lambda_{1}=\lambda_{min}(P)\) and \(\lambda_{2}=\lambda_{max}(P)\), where \(\lambda_{min}(P)\)and \(\lambda_{max}(P)\) are the minimum and maximum eigenvalues of \(P\), respectively. Finding \(\dot{V}\) w.r.t \(t\) over \(\eta\) (over the solution (A2)) is achieved in the following way:  
\[\dot{V}(\eta)\Big{|}_{along~{}(\Delta 2)}=\sum_{i=1}^{\rho+2}\frac{\partial V( \eta)}{\partial\eta_{i}}\dot{\eta}_{i}(t)\]  
Then,  
\[\dot{V}(\eta)\Big{|}_{along~{}(\Delta 2)}~{}=\sum_{i=1}^{\rho+1}\frac{ \partial V(\eta)}{\eta_{i}}\bigg{(}\eta_{i+1}(t)-a_{i}k\bigg{(}\frac{\eta_{1}( t)}{\omega_{0}{}^{p}}\bigg{)}\cdot\eta_{1}(t)\bigg{)}-\frac{\partial V(\eta)}{ \partial\eta_{\rho+2}}a_{\rho+2}k\bigg{(}\frac{\eta_{1}(t)}{\omega_{0}{}^{p}} \bigg{)}\cdot\eta_{1}(t)+\frac{\partial V(\eta)}{\partial\eta_{\rho+2}}\frac {\Delta_{h}}{\omega_{0}{}^{2}}\]  
Consider (13) of Assumption A5, then,  
\[\dot{V}(\eta)\Big{|}_{along~{}(\Delta 2)}~{}\leq-W(\eta)+\frac{\partial V( \eta)}{\partial\eta_{\rho+2}}\frac{\Delta_{h}}{\omega_{0}{}^{2}}\]  
As \(V(\eta)\leq\lambda_{max}(P)\|\eta\|^{2}\) and \(\left|\frac{\partial V(\eta)}{\partial\eta_{\rho+2}}\right|\leq\|\frac{ \partial V(\eta)}{\partial\eta}\|\), then \(\left|\frac{\partial V}{\partial\eta_{\rho+2}}\right|\leq 2\lambda_{max}(P)\eta\). As \(V(\eta)\leq\lambda_{max}(P)\|\eta\|^{2}=\lambda_{max}(P)W(\eta)\), thus, \(-W(\eta)\leq-\frac{V(\eta)}{\lambda_{max}(P)}\). Finally, because \(\lambda_{min}(P)\|\eta\|^{2}\leq V(\eta)\), this leads to \(\parallel\eta\parallel\leq\sqrt{\frac{V(\eta)}{\lambda_{min}(P)}}\).