## 3 Main Results
### Co-Design of the Event-Triggered and Controller Parameters
\right],\]  
\[\hat{\Gamma}_{11}=A_{i}^{T}X_{i}+X_{i}A_{i}-\tilde{R}_{1}-\tilde{R}_{2}+\tilde {Q}_{1}+\tilde{Q}_{2}+\pi_{ii}X_{i}+\sum_{i\
eq j,j=1}^{s}\pi_{ij}X_{i}X_{j}^{ -1}X_{i},\]  
\[\hat{\Pi}_{33i}=-\operatorname{diag}\Bigl{(}X_{i}(X_{i}R_{1}X_{i})^{-1}X_{i}, X_{i}(X_{i}R_{2}X_{i})^{-1}X_{i},X_{i}(X_{i}R_{3}X_{i})^{-1}X_{i}\Bigr{)}\]  
There exists \(-X_{i}\tilde{R}_{j}^{-1}X_{i}\) (\(j=1,2,3\)) nonlinear term in the equation, which is linearized by inequation \(-X_{i}\tilde{R}_{j}^{-1}X_{i}<\varphi_{j}^{2}\tilde{R}_{j}-2\varphi_{j}X_{i}\) (\(j=1,2,3\). \(i\in\mathcal{S}\)). In order to implement in the LMI toolbox of MATLAB, according to [PERSON] complement lemma \(\hat{\Omega}_{i}<0\) is further restructured to (31). Remark 5: _Under the condition of \(\varsigma(\kappa(t))=\omega\kappa(t),\omega\in\mathbb{R}^{0+}\), the inter-event intervals always have a positive lower bound, which rules out Zeno behavior. Furthermore, because the internal dynamic variable \(\kappa(t)\) is not negative, the dynamic event-triggered technique utilized might result in less data transfer, which helps conserve resources._  
The implementation procedure of the dynamic event-triggered robust controller design for flexible robotic arm systems with continuous-time phase-type semi-Markov jump process is illustrated in Algorithm 1. ```
Physical constants: \(M,g,\)\(H,\)\(k,\)\(J\); Random variables: \(I_{i}\), \(h_{1}\); Initialization: (For each state \(i\))
1. Initialize Initial state \(\vartheta(0)\);
2. Choose the lower and upper time delay in network \(\tau_{1}\) and \(\tau_{2}\);
3. Choose the event-triggered parameters \(\omega\) and \(\sigma\);
4. \(\gamma\) is the \(\mathcal{H}_{\infty}\) performance index;
```
5. Set \(P_{i}\), \(Q_{1}\)\(Q_{2}\), \(R1\), \(R2\) and \(R3\) as positive definite matrices; Iteration:
6. Let the iteration and step index \(s=0\);
7. Repeat
8. The dynamic event-triggered control \(u(t)\leftarrow(18)\)
9. IF event-triggered condition (16) is satisfied
10. not trigger, \(u(t)=x(t_{k}h)\);
11. let \(s=s+1\);
11. ELSE
12. update the signal \(u(t)=x(t_{l})\);
13. let \(s=s+1\), \(t_{k}h=t_{l}\);
14. END IF
15. IF exist DoS attacks;
16. let \(u(t)=0\);
17. ELSE
18. normal signal transmission \(u(t)\);
19. END IF
20. Go to back step (8);
21. End. ```

**Algorithm 1**_Cont._

## 4 Simulation Results

The flexible robotic arm in (20) shown grasping objects of different weights is very common in production practice. In this section, a phase-type semi-Markov chain is used to simulate the grabbing of four different objects as shown in Figure 5.

The sojourn time in the first three states is a random negative exponential distribution variable of parameter \(\lambda_{j}\)\((j=1,2,3)\). The sojourn time in the last state is split into two parts, namely two random variables with parameters \(\lambda_{4}\) and \(\lambda_{4}\) of negative exponential distribution. For example, if the process \(\hat{r}(t)\) enters a functioning condition, it must first spend some time in the first part, then stay in the second part, and then enter state 1 at the end. Without loss of generality, we assume that

\[P = \left[ \begin{array}{ccc}p_{11}&p_{12}&p_{13}&p_{14}\\ p_{21}&p_{22}&p_{23}&p_{24}\\ p_{31}&p_{32}&p_{33}&p_{34}\\ p_{41}&p_{42}&p_{43}&p_{44}\end{array} \right]=\left[ \begin{array}{ccc}0&0.3 &0.5&0.2\\ 0.3&0&0.6&0.1\\ 0.3&0.2&0&0.5\\ 0.1&0.4&0.5&0\end{array} \right],\] \[\mathbf{a^{j}} = (a^{j}_{1})=(1),\quad\mathbf{T}^{j}=(\mathbf{T}^{j}_{11})=(- \lambda_{j})(j=1,2,3),\] \[\mathbf{a^{4}} = (a^{4}_{1},a^{4}_{2})=[1,0],\quad\mathbf{T}^{4}=\left[ \begin{array} []{cc}\mathbf{T}^{4}_{11}&\mathbf{T}^{4}_{12}\\ \mathbf{T}^{4}_{21}&\mathbf{T}^{4}_{22}\end{array} \right]=\left[ \begin{array}[ ]{cc}-\lambda_{4}&\lambda_{4}\\ 0&\lambda_{5}\end{array} \right].\]

The state space of \(Z(t)=(\hat{r}(t),\chi(t))\) is clearly \(\mathcal{U}=((1,1),(2,1),(3,1),(4,1),(4,2))\).