## Appendix A Proof of the Theorems  
In this appendix, we will present the proofs of all theorems in the main findings of the article, encompassing Theorems 1-3. Firstly, we define the following transformed error \(\varsigma(t)\):  
\[\varsigma_{i}(t)=\left[ \begin{array}{cc}\ell_{i}^{-1}&\ell_{i}^{-2}&\ell_{ i}^{-3}\end{array} \right]\left[ \begin{array}{cc}\tilde{\xi}_{i}(t)&\tilde{ \psi}_{i}(t)&\tilde{\phi}_{i}(t)\end{array} \right]^{T},i=1,2,3 \tag{1}\]  
Combining the dynamics (1) and the event-triggered extended observer (14), we obtain that  
\[ \begin{array}{rcl}\dot{\xi}_{i}(t)&=&\ell_{i}A\varsigma_{i}(t)-\ell_{i}K_{i }(\beta)\frac{r_{i}(t_{k})-\xi_{i}(t)}{\ell_{i}}+\frac{1}{\ell_{i}^{3}}B\frac {dh_{i}(t)}{dt}\\ &=&\ell_{i}A\varsigma_{i}(t)-\ell_{i}K_{i}(\beta)C\varsigma_{i}(t)+K_{i}(\beta )(r_{i}(t)-r_{i}(t_{k}))+\frac{1}{\ell_{i}^{3}}B\frac{dh_{i}(t)}{dt}\end{array}  \tag{2}\]  
where \(A=\left[ \begin{array}{cc}0&1&0\\ 0&0&1\\ 0&0&0\end{array} \right]\), \(B=\left[ \begin{array}{cc}0\\ 0\\ 1\end{array} \right]\), \(C=\left[ \begin{array}{cc}1&0&0\end{array} \right]\). And then, we consider the Lyapunov function \(V(t)=\sum\limits_{i=1}^{3}V_{i}(t)\), where \(V_{i}(t)=\varsigma_{i}(t)^{T}P_{i}\varsigma_{i}(t)\). Calculating the derivative of \(V_{1}(t)\), we have:  
\[ \begin{array}{rl}\dot{V}_{1}(t)=&\ell_{1}\varsigma_{1}(t)^{T}\Big{(}P_{1}A+ A^{T}P_{1}-P_{1}K_{1}(\beta)C-C^{T}K_{1}(\beta)^{T}P_{1}\Big{)}\varsigma_{1}(t) \\ &+2\varsigma_{1}(t)^{T}P_{1}K_{1}(\beta)(r_{1}(t)-r_{1}(t_{k}))+\frac{2}{ \ell^{3}}\varsigma_{1}(t)^{T}P_{1}B\frac{dh_{1}(x,t)}{dt}\end{array}  \tag{3}\]By choosing \(\beta_{11}\), \(\beta_{12}\) and \(\beta_{13}\), it satisfies that the roots of function  
\[s^{3}+\beta_{11}s^{2}+\beta_{12}s+\beta_{13}=0 \tag{10}\]  
have negative real parts. From (10), we show that \(A-K_{i}(\beta)C\) is Hurwitz since then there exists a positive constant \(\lambda\) such that  
\[\dot{V}_{1}(t) =-\lambda\ell_{1}\xi_{1}(t)^{\top}\xi_{1}(t)+2\xi_{1}(t)^{T}P_{1} K_{1}(\beta)(r_{1}(t)-r_{1}(t_{k}))+\frac{2}{\ell_{1}^{3}}\xi_{1}{}^{T}P_{1}B \frac{dh_{1}(t)}{dt} \tag{11}\] \[\leq-\lambda\ell_{1}\|\xi_{1}(t)\|^{2}+2\|\xi_{1}(t)\|\|P_{1}K_{1} (\beta)\|\|r_{1}(t)-r_{1}(t_{k})|+\frac{2}{\ell_{1}^{3}}\|\xi_{1}(t)\|\|P_{1} \|\left|\frac{dh_{1}(t)}{dt}\right|\]  
Combining the Assumption 1 and the driven condition (4), we have that  
\[\dot{V}_{1}\leq-\lambda\ell(1-\sigma)\|\xi_{1}\|^{2}-\lambda\sigma\ell\|\xi_{ 1}\|^{2}+2\|\xi_{1}\|\|PK(\beta)\|\Pi_{1}+\frac{2}{\ell^{3}}\|\xi_{1}\|\|P\|M_{1} \tag{12}\]  
holds for any positive constant \(\sigma\in(0,1)\), then we know that  
\[\dot{V}_{1}\leq-\lambda\ell(1-\sigma)\|\xi_{1}\|^{2},\forall\|\xi_{1}\|\geq \frac{2}{\lambda\sigma\ell}\|PK(\beta)\|\Pi_{1}+\frac{2}{\lambda\sigma\ell^{4} }\|P\|M_{1} \tag{13}\]  
And then, we obtain the inequality  
\[\|V_{1}(t)\|\leq\max\Biggl{\{}\|V(t_{0})\|e^{-\frac{\lambda(1-\sigma)\ell_{1} }{\varnothing}(t-t_{0})},\bar{\omega}\biggl{(}\frac{2}{\lambda\sigma\ell}\|PK( \beta)\|\Pi_{1}+\frac{2}{\lambda\sigma\ell^{4}}\|P\|M_{1}\biggr{)}^{2}\Biggr{\}} \tag{14}\]  
by the result of ultimate boundedness in [40]. Here, we denote the maximum and minimum eigenvalues of \(P\) as \(\bar{\omega}\) and \(\underline{\omega}\), respectively, and we have \(\underline{\omega}\|\xi_{1}(t)\|^{2}\leq V(t)\leq\bar{\omega}\|\xi_{1}(t)\|^{2}\).