## 2 Problem Formulation
Hence, the measurement data that are subject to cyber-attacks received by the estimator can be expressed as  
\[q_{u}=\hat{q}_{u}+\beta_{u}\Xi_{u}\rho_{u}, \tag{11}\]  
where \(\beta_{u}\in\{0,1\}\) is used to indicate whether the bias injection attack occurs at sampling instant \(u\), in which \(\beta_{u}=1\) represents the occurrence of the bias injection attack at sampling instant \(u\) and \(\beta_{u}=0\) represents the networks that are not subject to the bias injection attack at sampling instant \(u\). \(\rho_{u}\) represents the signal of bias injection attack and satisfies the following assumption:  
\[\mathrm{E}\{\rho_{u}^{\mathrm{T}}\Omega_{u}^{-1}\rho_{u}\}\leq 1, \tag{12}\]  
where \(\Omega_{u}\) is a known scaled positive definite matrix. To reflect engineering practice better, we assume that the MM is persistent, while the bias injection attack occurs stochastically and follows the Bernoulli distribution:  
\[\mathrm{P}\{\beta_{u}=1\}=\tilde{\beta},\;\mathrm{P}\{\beta_{u}=0\}=1-\tilde{ \beta},\]  
where \(\tilde{\beta}\in[0,1)\) is a known constant. Let \(o_{u}=(\tilde{\sigma}_{u}^{\mathrm{T}}\,\hat{q}_{u-1}^{\mathrm{T}})^{\mathrm{ T}}\); then, the augmented form of measurement saturated-MNNs, with MMs and mixed time delays subject to cyber-attacks under the framework of WTOD protocol, can be expressed in the following:  
\[ \begin{cases}o_{u+1}=\aleph_{u}A(o_{u})o_{u}+B(o_{u})F(o_{u})+D(o_{u})G(o_{u- \tau_{1}})+E(o_{u})\sum_{i=1}^{\tau_{2}}\mu_{i}o_{u-i}\\ \qquad+\aleph_{u}H_{1}W_{u}+\alpha_{u}I_{u}\theta(C_{1}o_{u})\\ q_{u}=\alpha_{u}C_{2}o_{u}+\alpha_{u}\Xi_{u}\theta(C_{1}o_{u})+\alpha_{u}H_{2} W_{u}+\beta_{u}\Xi_{u}\rho_{u}\\ o_{u}=\Psi_{u},\;u\in[-\tau,0]\end{cases}  \tag{13}\]  
where  
\[\aleph_{u}= \begin{pmatrix}I&0\\ 0&\alpha_{u}\end{pmatrix} ,\;A(o_{u})= \begin{pmatrix}\tilde{A}(\tilde{\varrho} _{u})&0\\ \Xi_{u}\tilde{\varrho}_{1}\tilde{C}&I-\Xi_{l_{u}}\end{pmatrix} ,\]  
\[B(o_{u})= \begin{pmatrix}\tilde{B}(\tilde{\delta}_{u})&0\\ 0&0\end{pmatrix} ,\;F(o_{u})= \begin{pmatrix}f(\tilde{\varrho}_{u})\\ 0\end{pmatrix} ,\]  
\[D(o_{u})= \begin{pmatrix}\tilde{D}(\tilde{\varrho}_{u})&0\\ 0&0\end{pmatrix} ,\;G(o_{u-\tau_{1}})= \begin{pmatrix}g(\tilde{\varrho}_{u-\tau_{ 1}})\\ 0\end{pmatrix} ,\]  
\[E(o_{u})= \begin{pmatrix}\tilde{E}(\tilde{\varrho}_{u})&0\\ 0&0\end{pmatrix} ,\;H_{1}= \begin{pmatrix}\tilde{H}_{w}&0\\ 0&\Xi_{l_{u}}\tilde{H}_{w}\end{pmatrix} ,\]\[W_{u}= \begin{pmatrix}\tilde{W}_{u}\\ \tilde{\psi}_{u}\end{pmatrix} ,\ C_{1}=(\tilde{\mathcal{C}}\quad 0),\ C_{2}=( \Xi_{u}\tilde{U}_{1}\tilde{\mathcal{C}}\quad I-\Xi_{u}),\]  
\[I_{u}=(0\quad\Xi_{l_{u}}),\ H_{2}=\Big{(}0\ \Xi_{l_{u}}\tilde{H}_{v}\Big{)}.\]  
and \(\Psi_{u}\) is the augmented initial value. By separating unknown matrices \(\Delta\tilde{A}_{u}\), \(\Delta\tilde{B}_{u}\), \(\Delta\tilde{D}_{u}\) and \(\Delta\tilde{E}_{u}\), the matrices \(A(o_{u})\), \(B(o_{u})\), \(D(o_{u})\) and \(E(o_{u})\) can be decomposed into the following expressions:  
\[A(o_{u})=A+\Delta A_{u},\ B(o_{u})=B+\Delta B_{u},\]  
\[D(o_{u})=D+\Delta D_{u},\ E(o_{u})=E+\Delta E_{u},\]  
where  
\[A= \begin{pmatrix}\vec{\tilde{A}}&0\\ \Xi_{l_{u}}\tilde{U}_{1}\tilde{\mathcal{C}}&I-\Xi_{l_{u}}\end{pmatrix} ,\ \Delta A_{u}= \begin{pmatrix}\Delta\vec{\tilde{A}}_{u}&0\\ 0&0\end{pmatrix} ,\]  
\[B= \begin{pmatrix}\vec{\tilde{B}}&0\\ 0&0\end{pmatrix} ,\ \Delta B_{u}= \begin{pmatrix}\Delta\vec{\tilde{B}}_{u}&0\\ 0&0\end{pmatrix} ,\ D= \begin{pmatrix}\vec{\tilde{D}}&0\\ 0&0\end{pmatrix} ,\]  
\[\Delta D_{u}= \begin{pmatrix}\Delta\vec{\tilde{D}}_{u}&0\\ 0&0\end{pmatrix} ,\ E= \begin{pmatrix}\vec{\tilde{E}}&0\\ 0&0\end{pmatrix} ,\ \Delta E_{u}= \begin{pmatrix}\Delta\vec{E}_{u}&0\\ 0&0\end{pmatrix}