## 3 Main Results
### Co-Design of the Event-Triggered and Controller Parameters  
In this section, the state controller of the proposed flexible robotic arm system using dynamic event-triggered scheme is presented in (20). **Theorem 2**. : _For given scalars \(\tau_{1}\), \(\tau_{2}\), \(\omega\), \(\sigma\in(0,1)\) and \(\varphi_{j}(j=1,2,3,4)\), if there exist matrices \(N\), and positive definite symmetric matrices \(Y_{i}\), \(\tilde{Q}_{1}\), \(\tilde{Q}_{2}\), \(\tilde{R}_{j}(j=1,2,3)\), \(W_{i}\), \(X_{i}(i=1,2,3,4)\), scalar \(\gamma>0\) satisfying the following inequalities_  
\[\Pi_{i}=\left[ \begin{array}{cccc}\Pi_{11i}&\Pi_{12i}&\Pi_{13i}&\Pi_{14i}\\ *&\Pi_{22i}&0&0\\ *&*&\Pi_{33i}&0\\ *&*&*&\Pi_{44i}\end{array} \right]<0,\;i\in\mathcal{S} \tag{31}\]  
_where_  
\[\Pi_{11i}=\left[ \begin{array}{cccc}\Gamma_{11}&B_{2}Y_{i}&\tilde{R}_{1}& \tilde{R}_{2}\\ *&\Gamma_{22}&\tilde{R}_{3}-\tilde{N}^{T}&\tilde{R}_{3}-\tilde{N}\\ *&*&\Gamma_{33}&\tilde{N}\\ *&*&*&\Gamma_{44}\end{array} \right],\;\Pi_{12i}=\left[-\,\hat{B}_{2}LY_{i}\; \;\hat{B}_{1i}\;\;X_{i}\hat{\mathcal{C}}\;\right],\]  
\[\Pi_{13i}=[\tau_{1}\mathcal{A}_{i}^{T}(t)\;\;\tau_{2}\mathcal{A}_{i}^{T}(t)\; \;\tau_{12}\mathcal{A}_{i}^{T}(t)],\;\;\Pi_{22i}=-\,\mathrm{diag}\{2\varphi_{4 }X_{i}-\varphi_{4}^{2}W,\gamma^{2}I,I\},\]  
\[\Pi_{33i}=\mathrm{diag}\Big{\{}\varphi_{1}^{2}\tilde{R}_{1}-2\varphi_{1}X_{i},\varphi_{2}^{2}\tilde{R}_{2}-2\varphi_{2}X_{i},\varphi_{3}^{2}\tilde{R}_{3} -2\varphi_{3}X_{i}\Big{\}},\]  
\[\Pi_{14i}=[\sqrt{\pi_{i,1}}X_{i}\;\;\cdots\;\sqrt{\pi_{i,i-1}}X_{i}\;\;\sqrt{ \pi_{i,i+1}}X_{i}\;\cdots\;\sqrt{\pi_{i,s}}X_{i}\;\;X_{i}],\]  
\[\Pi_{44i}=-\,\mathrm{diag}\Big{\{}X_{1},\cdots,X_{i-1},X_{i+1},\cdots,X_{s}, \varpi^{-1}I\Big{\}}\]  
\[\Gamma_{11}=X_{i}A_{i}^{T}+A_{i}X_{i}+\tilde{Q}_{1}+\tilde{Q}_{2}-\tilde{R}_{1 }-\tilde{R}_{2}+\pi_{ii}X_{i},\]  
\[\Gamma_{22}=\tilde{N}+\tilde{N}^{T}-2\tilde{R}_{3}+2\sigma\varphi_{4}X_{i}- \sigma\varphi_{4}^{2}W,\]  
\[\Gamma_{33}=-\tilde{R}_{3}-\tilde{R}_{1}-\tilde{Q}_{1},\;\Gamma_{44}=-\tilde{R }_{3}-\tilde{R}_{2}-\tilde{Q}_{2},\]  
\[\tilde{\mathcal{A}}_{i}(t)=[\;A_{r}(t)X_{i}\;\;B_{2}LY_{i}\;\;0_{n}\;\;0_{n}\;\; -\;B_{2}LY_{i}\;\;B_{1i}\;],\]  
_and \(\tilde{\mathcal{Y}}=X\mathcal{Y}X\), \(\mathcal{Y}\in\{N,N_{T},Q_{1},Q_{2},R_{i}\}(i=1,2,3)\); \(\hat{B}_{2}\), \(\hat{B}_{1i}\), \(\hat{C}\), \(\tau_{12}\) are defined in Theorem 23, and then the closed loop system in (20) with the event-triggered controller in (18) is asymptotically stable with a \(\mathcal{H}_{\infty}\) performance \(\gamma\), namely \(\|z(t)\|_{2}\leq\gamma\|\omega(t)\|_{2}\), the corresponding controller is \(K_{i}=Y_{i}X_{i}^{-1}\)._Proof. : Choose an LKF \(V(x,i,\kappa)\) as in (24). Note that when the external disturbance \(\omega(t)=0\), according to Theorem 1, we can get \(\mathcal{L}V(x,i,\kappa)<0\) naturally, this shows the asymptotic stability of the closed-loop system. \(\varsigma(\kappa(t))\) is a function of class \(K_{\infty}\), \(\varsigma(\kappa(t))=\omega\kappa(t),\omega\in\mathbb{R}^{0+}\) clearly satisfies this requirement. \[\hat{\Omega}_{i}=\left[ \begin{array}{cccc}\Omega_{1i}&\Omega_{2i}&\Omega_{3i }&I\\ *&\Omega_{4}&0&0\\ *&*&\Omega_{5}&0\\ *&*&*&\alpha^{-1}I\end{array} \right]<0,\ i\in\mathcal{S}. \tag{32}\]  
According to Theorem 1, let is assume that \(X_{i}=P_{i}^{-1},Y_{i}=K_{i}X_{i}\) (\(i\in\mathcal{S}\)). Then, multiply both sides of this inequality by \(\{X_{i},X_{i},X_{i},X_{i},X_{i},I,I,I,I,I\}\)  
\[\hat{\Omega}_{i}=\left[ \begin{array}{cccc}\hat{\Pi}_{11i}&\Pi_{12i}&\Pi_{13 i}&X_{i}\\ *&\hat{\Pi}_{22i}&0&0\\ *&*&\hat{\Pi}_{33i}&0\\ *&*&*&\alpha^{-1}I\end{array} \right]<0,\ i\in\mathcal{S}\]  
where  
\[\hat{\Pi}_{11i}=\left[ \begin{array}{cccc}\hat{\Gamma}_{11}&B_{2}\Gamma Y_{i} &\hat{R}_{1}&\tilde{R}_{2}\\ *&\Gamma_{22}&\tilde{R}_{3}-\tilde{N}^{T}&\tilde{R}_{3}-\tilde{N}\\ *&*&\Gamma_{33}&\tilde{N}\\ *&*&*&\Gamma_{44}\end{array}