## 1 Introduction
To solve the aforementioned difficulties, considering that the communication protocol only allows one sensor node to send the latest data to the estimator during each sampling instant, we propose utilizing network protocol to partially solve the problem that the state estimation performance influenced by the MMs and bandwidth limitation for MNNs. The common communication protocols mainly include RA protocol [49; 50; 51], RR protocol [52; 53] and WTOD protocol [54; 55; 56; 57; 58; 59; 60]. Among them, the WTOD protocol is regarded as the most efficacious protocol [55], which can give the priority to sending the latest data to sensor nodes according to the principle of \"most needed\" to find the sensor node that has the maximum difference between the measured output at the present sampling instant and that of the preceding sampling instant. Therefore, this paper proposes an improved SMF algorithm based on the WTOD protocol to partially solve the influencing problem of multiple sensor-related problems and multiple network-induced phenomena on estimation performance. The difficulty in doing so is because the introduction of the WTOD protocol increases the complexity and nonlinearity of the system model, which brings significant challenge for the design and performance analysis of the estimator. In view of the preceding problem and practical requirements, this paper estimates the states of MNNs with mixed time delays, measurement saturations, MMs and gain perturbations of the estimator subject to cyber-attacks under the framework of the WTOD protocol. A non-fragile set-membership estimator is designed. The primary contributions of this paper are threefold: (1) Compared with [20], more sensor-related problems and network-induced phenomena are considered in MNNs, including cyber-attacks and MMs, which makes the established model more realistic; (2) Compared with [27], we propose that using an improved set-membership estimation algorithm based on the WTOD protocol to partially solve the difficulty of set-membership estimation in the presence of network-induced phenomena and sensor-related problems in MNNs, which can reduce both the influence of network congestion under bandwidth limitation and the MMs on state estimation; (3) Considering the introduction of stochastic variables, we improve the traditional non-fragile estimation frameworks in [19; 20] to handle the simultaneous occurrence of gain perturbation and parameter uncertainty in MNNs. The subsequent structure of this paper is organized as follows. Section 2 introduces the MNN model and designs the corresponding non-fragile estimator. The sufficient condition and proof process are provided in Section 3 for the existence of the non-fragile estimator. The parameters of the non-fragile estimator are derived through the solution of a set of LMIs. Section 4 provides the evidence to support the effectiveness and feasibility of the proposed estimation algorithm through the use of numerical examples. Section 5 gives a comprehensive overview of this paper. Notation. \(*\) represents the transposed term of symmetry in the symmetric block matrix. R\({}^{n_{x}}\) represents \(n_{x}\)-dimensional Euclidean space. \(U^{-1}\) and \(\parallel U\parallel\) represent the inverse and 2-norm of the matrix \(U\), respectively. \(I\) indicates the identity matrix with proper dimensions. \(\phi\geq Z\) (\(\phi>Z\)) means that \(\phi-Z\) is positive semi-definite (positive definite). E\(\{\,\cdot\,\}\) represents the mathematical expectation. The trace (\(N\)) is the trace of the matrix \(N\)\(\mathrm{P}\{B\}\) is the occurrence probability of the event '\(B\)'. diag\(\{\}\) indicates a block diagonal matrix. The superscript \"T\" denotes the transpose operation. \(A\in[\rho^{-},\rho^{+}]\) indicates that the value of every element in \(A\) falls within the specified interval \([\rho^{-},\rho^{+}]\), i.e., between the upper bound denoted as \(\rho^{+}\) and the lower bound denoted as \(\rho^{-}\).