## 1 Introduction
Compared to a static event trigger, by introducing an internal function to reduce the conservatism of the trigger condition in the form of a dynamic event trigger, this internal function can store part of the system information which allows further reduction in resource consumption under the premise of safety and control. In [23], for linear time-invariant systems, a unique dynamic event-triggered control mechanism is described. By deriving criteria for this control strategy that provide asymptotic stability and \(\mathcal{L}_{2}\) stability, the triggering parameters and feedback gain may be jointly constructed. Dynamic event-triggering mechanism is the suggested name for this class in [24] due to the introduction of an internal dynamic variable being a defining feature of it. In [25], multiple types of network attack signals are considered in multi-agent systems with multiple cyber-attacks. According to Lyapunov stability theory and the linear matrix inequalities (LMIs) method, exponentially mean-square finite-time stability is derived from these systems. The annular finite-time bounded and adaptive event-triggered control issues of networked switched systems with deception attacks are examined in [26]; however, no concrete engineering application is given in the paper. To the extent of the authors' knowledge, however, there is scarce literature on the robust properties of semi-Markov jump systems with a dynamic event-triggering mechanism. Due to the project needs and the hardware situation of the laboratory, a flexible robot arm is used as the research object. The flexible robotic arm systems proposed in this paper takes the change in moment of inertia into account for the first time. This is a necessary requirement for ensuring the safety and stability control of special robots that are applied under certain circumstances, e.g., surgical robots and assisted-living robots which have strict lightweight requirements. The main contributions of this paper are as follows: (1) We consider the \(\mathcal{H}_{\infty}\) performance of the flexible robotic arm with phase-type semi-Markov process; (2) The necessary conditions are found for robust controllers to exist by using the Lyapunov function method, furthermore, the controller and the dynamic event-triggering scheme are co-designed; (3) A simulation is conducted on the flexible robotic arm using the LMI toolbox in MATLAB, to estimate the usefulness of the designed controller gain with the comparison diagram of the time-trigger mechanism and the dynamic event-trigger mechanism. The remainder of this paper is structured as follows. Section 2 provides a description of the system and a statement of the problem, along with some fundamental ideas and lemmas. Principal conclusions from a stability analysis and time-/event-triggered robust controller for flexible robotic arm systems with semi-Markov jump process are rendered in Section 3. In Section 4, a numerical simulation method is used to demonstrate the functional operation of the suggested controller design process. Notation. There are various notations used throughout this paper. In the upper-right section of the matrix, \(-1\) and \(T\) represent the inverse and transpose of matrix, respectively, and \(*\) in the symmetric matrix denotes the omitted term, and \(\mathrm{diag}\{\cdot\}\) means a block diagonal matrix. Notation \(0_{n\times m}\) represents the zero matrix of dimension \(n\times m\). Furthermore, the \(2-\)norm of vectors in this paper are represented by \(||*||_{2}\), and \(\mathbb{R}^{0+}\) denotes the set of positive real numbers. Finally, \(\mathbb{E}\{\cdot\}\) represents the mathematical expectation. The remaining notations are less general in nature and are shown after their respective formula.