## 1 Introduction  
With the development of information technology, physical systems in practical applications are often composed of a large number of interacting elements, which are called complex systems, such as robotic arm control [1], power system [2], and multi-agent system [3, 4]. As a special class of complex systems, Markov jump systems (MJSs) are widely studied by scholars for their superior modeling capacity to deal with unexpected changes, which may include environmental disruptions, random component mistakes, or even human influences during regular operation [5, 6]. The authors in [7, 8, 9] reviewed recent advancements in modeling, stability analysis, filter design and sliding model control of MJSs. A new controller was proposed to ensure the stabilization of generalized MJSs in [10] to solve the random stabilization problem. [PERSON] et al. [11] considered both the stability analysis and stabilization of generalized Markov jump time-delay systems, with the relevant state-feedback controller derived by means of LMI. However, there exists a limitation in practical applications whereby the sojourn time of the Markov chain follows an exponential probability distribution, that is, results on MJSs are unfortunately conservative for the constant transition rates as a memoryless property of the exponential distribution. To solve this problem, a new model referred to as semi-Markov chain with less conservative sojourn time was proposed in [12; 13; 14]. Phase-type semi-Markov jump systems, which present a special class of semi-Markov jump systems, are increasingly attracting the attention of researchers, since the corresponding sojourn time obeys the characteristics of PH distribution. The problem of positive \(\mathcal{L}_{1}\) filters designed for nonlinear stochastic-switched systems, which are sensitive to the phase-type semi-Markov jump process, was explored in [15]. The work in [16] was extended to solve the problem of positive \(\mathcal{L}_{1}\) fuzzy filters designed for phase-type semi-Markov jump singular systems, which can be translated into a corresponding Markov jump singular system using an equivalent transformation [15]. Benefiting from the technology developed for the rapid growth of network communications, NCSs (network control systems) have gradually replaced physical transmission media with their unique advantages: low construction cost, small required space, low physical failure rate [17]. These advantages have recently attracted significant research interest from scholars. However, the introduction of network control systems has also created new difficulties for networked system analysis and synthesis [18]. We consider the inevitable shortcomings (packet mis-order, transmission delay, and data packet loss) of the network itself when designing a controller. On one hand, cyber-attacks will reduce the stability of networked systems. The basic goal of deception assaults is to alter the true signal while introducing fake signals. On the other hand, the occurrence of a time-delay phenomenon also causes instability or oscillation of the system, thus increasing the difficulty of system stability analysis. For a type of discrete Markovian jump system that suffers from random packet loss and time-varying delays, [PERSON] et al. addressed the finite time \(\mathcal{H}_{\infty}\) robust control issue in their work [19]. Regarding the uncertain discrete Markov jump time-varying delay polyhedron system, [PERSON] et al. [20] used Lyapunov functional theory and convex polyhedron technique, to provide the random admissibility of the generalized model error augment system. In addition, the actual system may suffer from sudden changes in internal structure or the external environment. In 1999, [PERSON] proposed the concept of an event trigger for the first time in [21], which subsequently attracted wide attention due to its superiority in network communication. In 2013, [PERSON] built an event flip-flop and NCSs into a unified time-delay system model [22].