# Fault Estimation for Semi-Markov Jump Neural Networks Based on the Extended State Method
## 1 Introduction  
With the rapid progress of science and technology, complex control systems have been widely used in computer-integrated manufacturing systems, network communication systems, agricultural machinery control systems, and modern agricultural flight control systems [1, 2, 3, 4, 5]. Furthermore, neural networks (NNs) have drawn extensive attention from both the academic field and the industrial and agricultural sectors in handling large-scale nonlinear systems. Nevertheless, these complex systems are frequently prone to unpredictable variations in their internal structure, such as equipment malfunctions, environmental alterations, disconnections of subsystems, information losses, and changes in interconnections among subsystems. Hence, how to improve the security and reliability of these systems has emerged as a crucial challenge. Fortunately, the Markov process provides an effective framework for regulating such transitions. In practical applications, system modes can switch between different states at various time points, and these state transitions can be governed by a Markov chain. This characteristic has facilitated the development of Markov jump neural networks (MJNNs), which are recognized as one of the key classes of hybrid systems. In recent years, the development of dissipation has continued to improve, particularly the dissipation rate [6]. Scholars established a more general extended dissipation rate and applied it to continuous-time neural networks [7]. They studied the estimation of a reachable set based on an incompletetransition rate in discrete-time MJNNs. The research was further expanded to address the challenge of dissipative state estimation, considering finite piecewise-homogeneous Markov chains and mixed time delays [8]. Other scholars also studied the stochastic acceptability analysis of singular Markov jump delay systems (SMJDS) and designed a dissipative state estimator to deal with imperfect communication links [9]. It is worth noting that most Markov jump systems are constrained by a constant transition rate. However, in practice scenarios, the transition rate may not be constant and can instead be described by a semi-Markov process. In practical applications, time delays are inevitable and often the primary cause of system instability, oscillations, and performance degradation. This issue is particularly pronounced in communication systems and networked control systems, where time delays are inherently random and time-varying. Consequently, time-varying delays are typically modeled using probabilistic approaches [10; 11; 12]. Nonetheless, research on discrete-time semi-Markov jump neural networks(s-MJNNs) with time delays remains relatively scarce, further motivating our exploration of this topic. In state estimation studies for neural networks, it is commonly assumed that adequate communication protocols are in place between sensors and the state estimator, ensuring synchronization and data transmission from all sensors at each sampling interval. Three communication protocols are commonly used in control systems: the Round-Robin protocol (RRP), the Weighted Try-Once-Discard protocol (WTODP), and the Stochastic Communication protocol (SCP) [13; 14; 15; 16]. Among these, RRP-based state estimation methods have been extensively studied due to their structured scheduling and efficient resource allocation. To improve communication efficiency and alleviate network congestion, RRP is adopted to schedule sensor data transmission in a cyclic and fair manner, ensuring a balanced network load and minimizing data loss. Compared to SCP and WTODP, RRP provides deterministic scheduling, preventing data starvation in critical sensors and enhancing the accuracy of real-time fault estimation. Furthermore, by integrating RRP into the observer design, the proposed method ensures a stable and systematic sensor data transmission strategy, thereby improving estimation performance under constrained communication bandwidth.