## 3 Main Results
### Co-Design of the Event-Triggered and Controller Parameters
Compared with the time period sampling 3000 times, the data transmission rate of the event trigger is 6.43%, which greatly saves network resources. Figure 8 depicts the interval between the sampling time when the control system successfully triggered under the event triggering mechanism and the last two successful triggering moments. As can be seen from the figure, the trigger intervals under the event triggering mechanism are not equal, and all trigger intervals are greater than or equal to the sampling cycle. Note Event triggering can reduce the transmission frequency of system data and save limited resources. The system state controlled by the dynamic event triggering mechanism is shown in Figure 9, and the matching triggering interval is shown in Figure 10. It has fewer trigger times (170, 5.67%) and exhibits a good calming effect for a limited time period. This demonstrates the effectiveness of the designed dynamic event-triggering controller in saving network resources. In general, the proposed dynamic ETM can maintain the security and stability control of the flexible robotic arm systems while occupying far less network resources than the time-triggered mechanism and the static ETM. The minimum time-interval between two adjacent trigger instants is not less than the sampling period, to prevent the Zeno phenomenon. Furthermore, the threshold is dynamic, and thus the trigger condition can adaptively modify the transmission of the signal to fend off the negative effects of DoS attacks. Figure 6: System status without controller. Figure 7: Event triggering mechanism. Figure 8: Trigger time interval (193 times). Figure 9: Dynamic event-triggering mechanism. ## 5 Conclusions

We herein present a type of physical model of flexible robot arm based on a semi-Markov jump process. For the corresponding linear system, the network control method is adopted, and the stabilization effect under the time-triggered mechanism and the dynamic event-triggered mechanism is considered, respectively. In order to obtain a full-state feedback controller satisfying the attenuation index \(\gamma\) of disturbance suppression, a series of matrix inequalities are obtained by constructing LKF and using Jensen inequality, reciprocal convex lemma and Schur complement lemma. The appropriate controller \(K_{i}\) and weighted matrix \(W_{i}\) are solved using the LMI toolkit in MATLAB. Finally, an effect diagram of the numerical simulation is given. In this paper, for the first time, we take stochastic theory and LMI technologies into consideration in the control method of flexible robotic arm systems, and directly solve the controller parameters using LKF. However, this method still has some limitations. The angle \(q(\theta)\) of the robot arm is assumed to be \((-\frac{\pi}{2},\frac{\pi}{2})\), which is somewhat conservative. To extend the rotation of the arm to \((0,\ 2\pi)\), a neural-network-based approach will be considered in the future work. Conceptualization, H.Z. and W.A. ; methodology, H.Z. and Z.C. ; software, Z.C. ; validation, H.Z., W.A. and P.S. ; formal analysis, H.Z. ; investigation, P.S. ; resources, W.A. ; data curation, Z.C. ; writing--original draft preparation, Z.C. ; writing--review and editing, H.Z. ; visualization, P.S. ; supervision, P.S. ; project administration, W.A. ; funding acquisition, H.Z. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Natural Science Foundation of China (62003062, 62241201), by the National Key R&D Program of China under Grant 2022 YFE0107300, by the Science and Technology Research Project of Chongqing Municipal Education Commission (KJZD-M201900801 and KJQN201900831), and by the High-level Talents Research Project of CTBU (1953013, 1956030). Not applicable. The source codes and datasets used to support the findings of this study are available from the authors upon request via email: [EMAIL_ADDRESS]. Figure 10: Dynamic trigger time interval (170 times). ## Conflicts of Interest: The authors declare no conflict of interest.