## 3 Main Result
### Estimator Design
Then, the subsequent inequality can be derived by combining (14) and (16):  
\[\tilde{\gamma}+\tilde{T}\tilde{F}_{u}\tilde{X}+\tilde{X}^{\mathrm{T}}\tilde{F} _{u}^{\mathrm{T}}\tilde{T}^{\mathrm{T}}<0, \tag{38}\]  
where \(\tilde{F}_{u}=\mathrm{diag}\{F_{1,u},\ F_{2,u},\ F_{3,u},\ F_{4,u},\ \hat{F},\ \hat{F}\}\), \(\tilde{T}\) and \(\tilde{X}\) are defined in (20). Based on Lemma 3, there exists a positive scalar \(\iota\) such that  
\[\tilde{\gamma}+\iota^{-1}\tilde{T}\tilde{T}^{\mathrm{T}}+\iota\tilde{X}^{ \mathrm{T}}\tilde{X}<0. \tag{39}\]  
Then, we can derive from Lemma 3 that Equation (39) is equivalent to (19). The proof of Theorem 1 is ended. \[\bar{\mathcal{C}}= \begin{pmatrix}1.5&0.8\\ 1.5&0.9\end{pmatrix} ,\tilde{H}_{\text{W}}= \begin{pmatrix}0.36\\ 0.21\end{pmatrix} ,\tilde{H}_{V}= \begin{pmatrix}1.8\\ 1.5\end{pmatrix} ,\] \[\Delta\tilde{A}_{u}= \begin{pmatrix}0.006\sin(0.6u)&0\\ 0&0.003\sin(0.6u)\end{pmatrix} ,\] \[\Delta\tilde{B}_{u}= \begin{pmatrix}0.005\sin(0.8u)&-0.005\sin(0.8u )\\ -0.001\sin(0.8u)&0.004\sin(0.8u)\end{pmatrix} ,\] \[\Delta\tilde{D}_{u}= \begin{pmatrix}0.001\cos(0.7u)&-0.002\cos(0.7u )\\ -0.005\cos(0.7u)&0.003\cos(0.7u)\end{pmatrix} ,\] \[\Delta\tilde{E}_{u}= \begin{pmatrix}0.01\sin(0.7u)&0.004\sin(0.7u )\\ 0.01\sin(0.7u)&0.004\sin(0.7u)\end{pmatrix} ,\] \[\bar{\mathcal{U}}= \begin{pmatrix}1&0\\ 0&1\end{pmatrix} ,\tilde{U}_{1}= \begin{pmatrix}0.3&0\\ 0&0.3\end{pmatrix} ,\] \[\hat{T}= \begin{pmatrix}0.05&-0.1&0&0.05\end{pmatrix} ^{\text{T}},\] \[\hat{F}_{u}\text{=}\text{sin}(rand(1)*2\pi),\ \hat{\mathcal{X}}= \begin{pmatrix}0.05&0.06\end{pmatrix} ,\] \[\tau_{1}=2,\ \tau_{2}=4.\] \[\mu_{1}=0.4,\ \mu_{2}=0.5,\ \mu_{3}=0.2,\ \mu_{4}=0.3.\]  
The nonlinear time-delay neuron activation function \(g(\tilde{\delta}_{u-\tau_{1}})\) is selected as follows:  
\[g(\bar{\delta}_{u-\tau_{1}})= \begin{pmatrix}0.1\bar{\delta}_{1,u-\tau_{1}}- \tanh(0.4\bar{\delta}_{1,u-\tau_{1}})&0.02\bar{\delta}_{2,u-\tau_{1}}+0.06 \tanh(\bar{\delta}_{2,u-\tau_{1}})\end{pmatrix} ^{\text{T}}\]  
which belongs to the sector \((\bar{S}_{1},\bar{S}_{2})\) with  
\[\bar{S}_{1}= \begin{pmatrix}-0.3&0\\ 0&-0.04\end{pmatrix} ,\bar{S}_{2}= \begin{pmatrix}0.1&0\\ 0&0.02\end{pmatrix} .\]  
The nonlinear neuron activation function \(f(\tilde{\delta}_{u-\tau_{1}})\) is selected as follows:  
\[f(\bar{\delta}_{u})= \begin{pmatrix}\tanh(0.2\bar{\delta}_{1,u})&\tanh(0.1\bar {\delta}_{2,u})\end{pmatrix} ^{\text{T}}\]  
which belongs to the sector \((\bar{f}_{1},\bar{f}_{2})\) with \(\bar{f}_{1}=\text{diag}\{0,\ 0\},\bar{f}_{2}=\text{diag}\{0.1,0.2\}\). For the UBB noises \(\tilde{W}_{u}=\tilde{\bar{V}}_{u}=7\text{sin}(u*2\pi)\) and \(R_{1,u}=R_{2,u}=49\). The parameters of bias injection attack are selected as \(\rho_{u}=20\), \(\Omega_{u}=4000\) and \(\bar{\beta}=0.05\). The initial conditions are selected as \(\Psi_{u}=\left(2\ 2\ 1-1\right)^{\text{T}}\), \(\partial_{u}=\left(0\ 0\ 0\ 0\right)^{\text{T}}\) and \(P_{u}=\text{diag}\{4,\ 4,\ 1,\ 1\}\), where \(u=-4,-3,-2,-1\). The weight coefficient matrix is \(\tilde{\omega}=\text{diag}\{1,\ 0.8\}\). The saturation level of sensors is \(\pm 1.8\). We use MATLAB R2020a for solving the optimization problem in Corollary 1. **Case 1**. : The true state and its estimation values of the MNNs without attack and MMs are shown below. Figures 3 and 4 describe the real values \(o_{u}^{(l)}\) and the estimation values \(\delta_{u}^{(l)}\ (i=1,\ 2)\) of the MNNs without MMs and cyber-attacks, respectively. From Figures 3 and 4, we can clearly see that, although there exist mixed time delays, measurement-saturation phenomena and UBB noises in the process of state estimation for MNNs, the estimation framework proposed in this paper can still accurately estimate the state of MNNs. Compared with our approach, although the approaches in [27; 55] can also estimate the state of the MNNs, the estimation accuracy is relatively low. In addition, we can see that there is a large lag and deviation between the true state and the estimated value obtained by the traditional SMF in [11]. **Case 2**.