## 2 Problem Formulation
.\]  
According to (8), the unknown parameter matrices \(\Delta A_{u}\), \(\Delta B_{u}\), \(\Delta D_{u}\) and \(\Delta E_{u}\) can be expressed with the following augmented form:  
\[\Delta A_{u}=T\hat{F}_{1,u}X_{1},\ \Delta B_{u}=T\hat{F}_{2,u}X_{2}, \tag{14}\] \[\Delta D_{u}=T\hat{F}_{3,u}X_{3},\ \Delta E_{u}=T\hat{F}_{4,u}X_{4},\]  
where \(T=(\tilde{T}\quad 0)^{\mathsf{T}}\), \(X_{1}=(\tilde{X}_{1}\ 0)\), \(X_{2}=(\tilde{X}_{2}\ 0)\), \(X_{3}=(\tilde{X}_{3}\ 0)\), \(X_{4}=(\tilde{X}_{4}\ 0)\) and matrix \(\hat{F}_{u}\) satisfies \(\hat{F}_{u}^{\dagger}\hat{F}_{u}\leq I\). In summary, taking into account the effects of the gain perturbation of the estimator, cyber-attacks, mixed time delays, measurement saturation and MMs in the MNNs, the following estimator is designed:  
\[K_{u,n}=K_{u}+\Delta K_{u} \tag{15}\] \[\delta_{u+1}=A\delta_{u}+BF(\delta_{u})+DG(\delta_{u-\tau_{1}})+E \sum_{i=1}^{\tau_{2}}\mu_{i}\delta_{u-i}+K_{u,n}(q_{u}-\alpha_{u}\mathcal{C}_{ 2}\delta_{u})\]  
where \(\delta_{u}\in\mathbb{R}^{n_{y}+n_{x}}\) is the estimated value of \(o_{u}\), the matrix \(K_{u}\) is the gain of estimator to be determined and \(\Delta K_{u}\) is the gain perturbation of estimator satisfying  
\[\Delta K_{u}=\hat{T}\hat{F}_{u}\hat{X}, \tag{16}\]  
where \(\hat{F}_{u}\) is an unknown time-varying matrix satisfying \(\hat{F}_{u}^{\mathsf{T}}\hat{F}_{u}\leq 1\); \(\hat{T}\) and \(\hat{X}\) are known real matrices. **Remark 2**. : _During the process of state estimation, the system may encounter some issues such as actuator degradation, numerical calculation errors and the aging of components, which can cause gain perturbations in the estimator. In engineering practice, even small gain perturbations in the estimator can also lead to decreases in estimation performance. Consequently, researchers have devised non-fragile techniques to enhance the robustness of the estimator. As we all know, non-fragile technologies are mainly implemented through LMI. Noteworthy is the fact that the structure of LMI is changed when the MNNs are subjected to the stochastic cyber-attacks, so the traditional non-fragile estimation framework based on SMF is no longer applicable. To solve this issue, this paper designs a new non-fragile set-membership estimation framework by adjusting the structure of the matrices \(\hat{T}\) and \(\hat{X}\) correspondingly, based on Lemma 3 in the following derivation._  
**Assumption 2**. : _The initial values \(\Psi_{i}\)\((i\in[-\tau,0])\) belong to the ellipsoid sets given below:_  
\[z_{i}=\{\Psi_{i}\in R^{n_{x}+n_{y}}:\Psi_{i}^{T}P_{i}^{-1}\Psi_{i}\leq 1\}, \tag{17}\]  
_where \(P_{i}\)\((i\in[-\tau,0])\) are known positive definite matrices._  
The primary aim of this paper is to design a new non-fragile set-membership estimator to estimate the state of measurement-saturated MNNs with MMs and mixed time delays subject to cyber-attacks under the framework of the WTOD protocol, which mainly includes the following two parts:  
(1) Obtain the estimator gain matrices \(K_{u}\) such that the positive definite matrices \(P_{u+1}\) and \(\delta_{u+1}\) satisfy the constraints of the ellipsoid sets as follows:  
\[ \begin{split}& z_{u+1}=\{o_{u+1}\in\mathbb{R}^{n_{x}+n_{y}}:\\ &(o_{u+1}-\delta_{u+1})^{T}P_{u+1}^{-1}\ (o_{u+1}-\delta_{u+1})\leq 1\}; \end{split}  \tag{18}\]  
(2) Design an optimization algorithm by minimizing the trace of \(P_{u+1}\) to achieve local optimal estimation performance. To facilitate subsequent derivation, some lemmas are introduced as follows. **Lemma 1** ([61]). : _Given constant matrices \(\wp_{1}\), \(\wp_{2}\) and \(\wp_{3}\) satisfying \(\wp_{1}=\wp_{1}^{T}\), \(\wp_{2}=\wp_{2}^{T}\)\(>0\), then_  
\[\wp_{1}+\wp_{3}^{T}\wp_{2}^{-1}\wp_{3}<0,\]  
_if and only if_  
\[ \begin{pmatrix}\wp_{1}&\wp_{3}^{T}\\ \wp_{3}&-\wp_{2}\end{pmatrix} <0,\ \begin{pmatrix}-\wp_{2}&\wp_{3}\\ \wp_{3}^{T}&\wp_{1}\end{pmatrix} <0.\]  
**Lemma 2** ([61]).