## 3 Main Results
### Co-Design of the Event-Triggered and Controller Parameters
We order each component of \(\mathcal{U}\) as follows:

\[\psi((1,1))=1,\quad\psi((2,1))=2,\quad\psi((3,1))=3,\quad\psi((4,1))=4,\quad \psi((4,2))=5,\]

as a result, the infinitesimal generator of \(\psi(Z(t))\) is

\[\Xi=\left[ \begin{array}{cccccc}-\lambda_{1}&0.3\lambda_{1}&0.5\lambda_{1}& 0.2\lambda_{1}&0\\ 0.3\lambda_{2}&-\lambda_{2}&0.6\lambda_{2}&0.1\lambda_{2}&0\\ 0.3\lambda_{3}&0.2\lambda_{3}&-\lambda_{3}&0.5\lambda_{3}&0\\ 0&0&0&-\lambda_{4}&\lambda_{4}\\ 0.1\lambda_{5}&0.4\lambda_{5}&0.5\lambda_{5}&0&-\lambda_{5}\end{array} \right]. \tag{33}\]
Before presenting the result of the numerical simulation, the setup and software of the computer that we employ are given in Table 1:

Similar to [17], some appropriate physical system parameters of (5) and parameters related to the dynamic event-trigger condition in (16) are set in the following Table 2.

Let us choose \(\vartheta(t)=\left[-0.3\ 0.4\ 0.6\ -0.8\right]^{T}\), \(t\in[-\tau_{2},0]\), and select 30 s as the total simulation time, and assume that the external disturbance is \(w(t)=\exp(-t)\sin(t)\). The dy \begin{table}
\begin{tabular}{c c} \hline \hline Hardware & Type \\ \hline CPU & Intel(Core) i5-12400F \\ GPU & NVIDIA T400 4 GB GDDR6 \\ Memory stick & Kingston FURY 16 GB \\ Hard disk & Kingston 1 TB SSD \\ \hline Software & Type \\ \hline Operating system & Windows 10 \\ MATLAB & Matlab 2016b \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation environment. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Numerical Value** \\ \hline \(l_{1,\cdots,4}\) & kg \(\cdot\) m\({}^{2}\) & [1, 10] \\ \(k\) & N \(\cdot\) m \(\cdot\) rad\({}^{-1}\) & 20 \\ \(J\) & kg \(\cdot\) m\({}^{2}\) & 1 \\ \(\tau_{1}\) & s & 0.01 \\ \(M\) & kg & 5 \\ \(\tau_{2}\) & s & 0.1 \\ \(H\) & m & 1 \\ \(\sigma\) & * & 0.6 \\ \(\omega\) & * & 2 \\ \(h_{1}\) & * & 0.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the controlled object. Figure 5: Application scenarios. (**a**) Model 1; (**b**) Model 2; (**c**) Model 3; (**d**) Model 4.

namic event-triggered weighting matrix \(W_{i}\) and the corresponding controller \(K_{i}(i=1,2,3,4)\) are solved by the LMI program and given as follows:

\[ \begin{split}\text{W}_{1}&=\left[\begin{array}{ccc}32.436 &1.5572&32.527&1.7520\\ 1.5572&76.951&0.20833&-1.8370\\ 32.527&0.20833&48.101&1.6751\\ 1.7520&-1.8370&1.6751&0.35863\end{array}\right]\times 10^{-10},\;K_{1}=\left[ \begin{array}{c}0.0863\\ -0.0347\\ -0.0115\\ -1.3406\end{array}\right]^{T},\\ \text{W}_{2}&=\left[\begin{array}{ccc}34.138&0.5318&34.187&-0.0194\\ 0.5318&67.814&-0.1582&-0.3244\\ 34.187&-0.1582&50.187&0.2248\\ -0.0194&-0.3244&0.2248&0.0639\end{array}\right]\times 10^{-10},\;K_{2}=\left[ \begin{array}{c}-0.0219\\ 0.0060\\ 0.0210\\ -1.3341\end{array}\right]^{T},\\ \text{W}_{3}&=\left[\begin{array}{ccc}33.091&1.2368&33.154&1.0846\\ 1.2368&73.571&-0.1185&-1.3694\\ 33.154&-0.1185&48.938&1.1624\\ 1.0846&-1.3694&1.1624&0.3489\end{array}\right]\times 10^{-10},\;K_{3}=\left[ \begin{array}{c}0.0414\\ -0.0267\\ 0.0039\\ -1.3394\end{array}\right]^{T},\\ \text{W}_{4}&=\left[\begin{array}{ccc}36.478&-1.2711&36.377&-2.8002\\ -1.2711&55.584&-1.1178&2.6714\\ 36.377&-1.1178&52.841&-2.0251\\ -2.8002&2.6714&-2.0251&0.8078\end{array}\right]\times 10^{-10},\;K_{4}=\left[ \begin{array}{c}-0.1647\\ 0.0606\\ 0.0637\\ -1.3346\end{array}\right]^{T}.\end{split} \]

Figure 6 displays the state-response diagram of the system under semi-Markov process without controller and shows that the open loop of the flexible robotic arm in (20) is unstable. The state trajectory of the static event-triggered mechanism is shown in Figure 7. Furthermore, DoS attacks that obey the Bernoulli distribution of parameter property \(h_{1}=0.95\) operate on the input signal \(u_{t}\). Contrapositive with the time-triggered mode, the event-triggered mode achieves fewer times of sending at the expense of a part of the calming time, a total of 193 events are triggered.