## Appendix A Proof of Theorems 1 and 2
Based on this and given Assumption A4, \(\dot{V}(\eta)\) becomes,  
\[\dot{V}(\eta)\leq-\frac{V(\eta)}{\lambda_{max}(P)}+\frac{M_{h}}{\omega_{0}{}^ {2}}2\lambda_{max}(P)\frac{\sqrt{V(\eta)}}{\sqrt{\lambda_{min}(P)}},\]  
Since \(\frac{d}{dt}\sqrt{V(\eta)}=\frac{1}{2}\frac{1}{\sqrt{V(\eta)}}\dot{V}(\eta)\), then  
\[\frac{d}{dt}\sqrt{V(\eta)}\leq\frac{1}{2}\frac{1}{\sqrt{V(\eta)}}\bigg{(}- \frac{V(\eta)}{\lambda_{max}(P)}+\frac{M_{h}}{\omega_{0}{}^{2}}2\lambda_{max} (P)\frac{\sqrt{V(\eta)}}{\sqrt{\lambda_{min}(\eta)}}\bigg{)}\]  
which gives  
\[\frac{d}{dt}\sqrt{V(\eta)}\leq-\frac{\sqrt{V(\eta)}}{2\lambda_{max}(P)}+\frac {M_{h}}{\omega_{0}{}^{2}}\frac{\lambda_{max}(P)}{\sqrt{\lambda_{min}(P)}}\] (A3)  
Given that (A3) is an ordinary first ODE, it can be solved as  
\[\sqrt{V(\eta)}\leq\frac{2M_{h}\lambda^{2}{}_{max}(P)}{\omega_{0}{}^{2}\sqrt{ \lambda_{min}(P)}}\bigg{(}1-e^{-\frac{l}{2\lambda_{max}(P)}}\bigg{)}+\sqrt{V( \eta(0))}e^{-\frac{l}{2\lambda_{max}(P)}}\]  
From Assumption A5, we have \(\lambda_{min}(P)\|\eta\|^{2}\leq V(\eta)\). This leads to \(\parallel\eta\parallel\leq\sqrt{\frac{V(\eta)}{\lambda_{min}(P)}}\). Then,  
\[\parallel\eta\parallel\leq\sqrt{\frac{1}{\lambda_{min}(P)}}\left(\frac{2M_{h} \lambda^{2}{}_{max}(P)}{\omega_{0}{}^{2}\sqrt{\lambda_{min}(P)}}\bigg{(}1-e^{- \frac{l}{2\lambda_{max}(P)}}\bigg{)}+\sqrt{V(\eta(0))}e^{-\frac{l}{2\lambda_{ max}(P)}}\right)\]  
which gives  
\[\parallel\eta\parallel\leq\ \frac{2M_{h}\lambda^{2}{}_{max}(P)}{\omega_{0}{}^{2} \lambda_{min}(P)}\bigg{(}1-e^{-\frac{l}{2\lambda_{max}(P)}}\bigg{)}+\sqrt{ \frac{V(\eta(0))}{\lambda_{min}(P)}}e^{-\frac{l}{2\lambda_{max}(P)}}\] (A4)It results from (A1) that  
\[\big{|}\xi_{i}-\hat{\xi}_{i}\big{|}\leq\frac{1}{\omega_{0}\rho^{+1-i}}\eta(\omega_ {0}t)\]  
It follows from (A4) that  
\[\dot{V}(\eta)\Big{|}_{along~{}(\Delta 2)}~{}=\sum_{i=1}^{\rho+1}\frac{ \partial V(\eta)}{\eta_{i}}\bigg{(}\eta_{i+1}(t)-a_{i}k\bigg{(}\frac{\eta_{1}( t)}{\omega_{0}{}^{p}}\bigg{)}\cdot\eta_{1}(t)\bigg{)}-\frac{\partial V(\eta)}{ \partial\eta_{\rho+2}}a_{\rho+2}k\bigg{(}\frac{\eta_{1}(t)}{\omega_{0}{}^{p}} \bigg{)}\cdot\eta_{1}(t)+\frac{\partial V(\eta)}{\partial\eta_{\rho+2}}\frac {\Delta_{h}}{\omega_{0}{}^{2}}\]0  
Finally,  
\[\dot{V}(\eta)\Big{|}_{along~{}(\Delta 2)}~{}=\sum_{i=1}^{\rho+1}\frac{ \partial V(\eta)}{\eta_{i}}\bigg{(}\eta_{i+1}(t)-a_{i}k\bigg{(}\frac{\eta_{1}( t)}{\omega_{0}{}^{p}}\bigg{)}\cdot\eta_{1}(t)\bigg{)}-\frac{\partial V(\eta)}{ \partial\eta_{\rho+2}}a_{\rho+2}k\bigg{(}\frac{\eta_{1}(t)}{\omega_{0}{}^{p}} \bigg{)}\cdot\eta_{1}(t)+\frac{\partial V(\eta)}{\partial\eta_{\rho+2}}\frac {\Delta_{h}}{\omega_{0}{}^{2}}\]1 (A5)  
\(\square\)  
**Theorem A2. **_A second-order nonlinear system in a lower triangular form with mismatched disturbances can be described as follows:_  
\[\left\{ \begin{array}{l}\dot{\xi}_{1}=a_{1}\xi_{2}+\phi_{1}(\xi_{1})+w_{1}, \\ \dot{\xi}_{2}=\phi_{2}(\xi_{1},\tilde{\xi}_{2})+w_{2}+bu,\\ y=\tilde{\xi}_{1}\end{array} \right.\] (A6)  
where \(\tilde{\xi}=\big{(}\xi_{1}(t),\xi_{2}(t)\big{)}^{T}\in\mathbb{R}^{2}\) is the system state, \(y(t)\in\mathbb{R}\) is the measured output, \(u(t)\in\mathbb{R}\) is the control input, \(w_{i}(t)\in\mathbb{R}\), \(i\in\{1,2\}\) is the unknown exogenous disturbance, and \(b\in\mathbb{R}\) is the control coefficient. The function \(\phi_{i}:\mathbb{R}^{i}\to\mathbb{R},\ i\in\{1,2\}\). If the function \(\phi_{1}\) and the exogenous disturbance \(w_{1}\) are differentiable w.r.t \(t\), System (A6) can be transformed into the following form:  
\[\left\{ \begin{array}{l}\dot{\tilde{\xi}}_{1}=\tilde{\xi}_{2},\\ \dot{\tilde{\xi}}_{2}=f\Big{(}\tilde{\xi}_{1},\tilde{\xi}_{2},w_{1},\check{w} _{1},w_{2}\Big{)}+\hat{b}u,\\ y=\tilde{\xi}_{1}.\end{array} \right.\] (A7)  
where \(f\Big{(}\tilde{\xi}_{1},\tilde{\xi}_{2},w_{1},\check{w}_{1},w_{2}\Big{)}\)\(=\)\(a_{1}\phi_{2}\Big{(}\tilde{\xi}_{1},\frac{\tilde{\xi}_{2}-\phi_{1}(\tilde{\xi}_{1})-w_{1}}{a _{1}}\Big{)}+\frac{\partial\phi_{1}(\tilde{\xi}_{1})}{\partial\xi_{1}}\tilde{ \xi}_{2}+a_{1}w_{2}+\check{w}_{1}\) and \(\hat{b}=a_{1}b\).