## 4 Main Results
\begin{array}{rl}\frac{d\eta_{1}(t)}{dt}=\eta_{2}(t)-a_{1}\eta_{1}(t )\\ \frac{d\eta_{2}(t)}{dt}=\eta_{3}(t)-a_{2}\eta_{1}(t)\\ \vdots\\ \frac{d\eta_{n}(t)}{dt}n\eta_{n+1}(t)-a_{n}\eta_{i}(t)\\ \frac{d\eta_{n+1}(t)}{dt}=\frac{\Delta}{\omega_{0}}-a_{n+1}.\eta_{1}(t)\end{array} \right. \tag{19}\]  
Consider the candidate Lyapunov functions \(V,W:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{+}\) defined by \(V(\eta)=\langle P\eta,\eta\rangle=\eta^{T}P\eta\), where \(\eta\in\mathbb{R}^{n+1}\) and \(P\) is a symmetric and positive definite matrix. Suppose Assumption A2 (9) with \(\lambda_{1}=\lambda_{min}(P)\) and \(\lambda_{2}=\lambda_{max}(P)\), where \(\lambda_{min}(P)\) and \(\lambda_{max}(P)\) are the minimal and maximal eigenvalues of \(P\), respectively. Finding the derivative of \(V(\eta)\) w.r.t t along the solution of \(\eta\) in Equation (19), \(\dot{V}(\eta)|_{along\ (19)}=\sum_{i=1}^{n+1}\frac{\partial V(\eta)}{\partial\eta_{i}} \dot{\eta}_{i}(t)=\sum_{i=1}^{n+1}\frac{\partial V(\eta)}{\eta_{i}}(\eta_{i+1 }(t)-a_{i}.\eta_{1}(t))+\frac{\partial V(\eta)}{\partial\eta_{i+1}}\Big{(} \frac{\Delta}{\omega_{0}}-a_{n+1}.\eta_{1}(t)\Big{)}\). Then, \(\dot{V}(\eta)|_{along\ (19)}=\sum_{i=1}^{n+1}\frac{\partial V(\eta)}{\eta_{i}}(\eta_{i+1 }(t)-a_{i}.\eta_{1}(t))+\frac{\partial V(\eta)}{\partial\eta_{n+1}}\frac{ \Delta}{\omega_{0}}-\frac{\partial V(\eta)}{\partial\eta_{n+1}}a_{n+1}.\eta_{1 }(t)\). Based on Assumption A2, \(\dot{V}(\eta)|_{along\ (19)}\leq-W(\eta)+\frac{\partial V(\eta)}{\partial\eta_{n+1}} \frac{\Delta}{\omega_{0}}\). As \(V(\eta)\leq\lambda_{max}(P)\|\eta\|^{2}\) and \(|\frac{\partial V}{\partial\eta_{n+1}}|\leq\|\frac{\partial V(\eta)}{\partial \eta}\|\), then \(|\frac{\partial V}{\partial\eta_{\varphi+1}}|\leq 2\lambda_{max}(P)\|\eta\|^{2}\). 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 \(\|\eta\|\leq\sqrt{\frac{V(\eta)}{\lambda_{min}(P)}}\). From Assumption A1, \(\dot{V}_{i}(\eta_{i})\leq-\frac{V(\eta)}{\lambda_{max}(P)}+\frac{M}{\omega_{0}}2 \lambda_{max}(P)\frac{\sqrt{V_{i}(\eta)}}{\sqrt{\lambda_{min}(P)}}\). As \(\frac{d}{dt}\sqrt{V(\eta)}=\frac{1}{2}\frac{1}{\sqrt{V(\eta)}}\dot{V}(\eta)\), \(\frac{d}{dt}\sqrt{V(\eta)}\leq\frac{1}{2}\frac{1}{\sqrt{V(\eta)}}\Bigg{(}- \frac{V(\eta)}{\lambda_{max}(P)}+\frac{M}{\omega_{0}}2\lambda_{max}(P)\frac{ \sqrt{V(\eta)}}{\sqrt{\lambda_{min}(\eta)}}\Bigg{)}\). Thus,  
\[\frac{d}{dt}\sqrt{V(\eta)}\leq-\frac{\sqrt{V(\eta)}}{2\lambda_{max}(P)}+\frac{M} {\omega_{0}}\frac{\lambda_{max}(P)}{\sqrt{\lambda_{min}(P)}} \tag{20}\]  
Solving ordinary differential Equation (20) gives,  
\[\sqrt{V(\eta)}\leq\frac{2M\lambda^{2}_{max}(P)}{\omega_{0}\sqrt{\lambda_{min}(P) }}\bigg{(}1-e^{-\frac{t}{\lambda_{max}(P)}}\bigg{)}+\sqrt{V(\eta(0))}e^{-\frac{ t}{2\lambda_{max}(P)}} \tag{21}\]From Assumption A2, \(\lambda_{min}(P)\|\eta\|^{2}\leq V(\eta)\). This leads to \(\|\eta\|\leq\sqrt{\frac{V(\eta)}{\lambda_{min}(P)}}\), then Equation (21) can be described as:  
\[ \begin{array}{l}\|\eta(t)\|\leq\sqrt{\frac{1}{\lambda_{min}(P)}}\left(\frac{2 M\lambda^{2}_{max}(P)}{\omega_{0}\sqrt{\lambda_{min}(P)}}\Big{(}1-e^{-\frac{t}{2 \lambda_{max}(P)}}\Big{)}+\sqrt{V(\eta(0))}e^{-\frac{t}{2\lambda_{max}(P)}} \right)\\ \|\eta\|(t)\leq\ \frac{2M\lambda^{2}_{max}(P)}{\omega_{0}\lambda_{min}(P)}\Big{(}1-e^{- \frac{t}{2\lambda_{max}(P)}}\Big{)}+\ \sqrt{\frac{V(\eta(0))}{\lambda_{min}(P)}}e^{-\frac{t}{2\lambda_{max}(P)}} \end{array}  \tag{22}\]  
It follows from Equation (15) that \(|x_{i}(t)-\hat{x}_{i}(t)|=\frac{1}{\omega_{0}^{\alpha+1-i}}|\eta_{i}(\omega_{ 0}t)|\Rightarrow|x_{i}(t)-\hat{x}_{i}(t)|\leq\frac{1}{\omega_{0}^{\alpha+1-i }}\|\eta(t)\|\).