## 3 Stability Analysis
 \tag{31}\]  
Eventually, combining (19), (29) and (31), the closed-loop NMASs can be expressed as  
\[\text{Y}(t+1)=\Lambda(\tau_{t,i})\text{Y}(t)+\varpi(\tau_{t,i})\Theta(t), \tag{32}\]  
where \(\varpi(\tau_{t,i})\Theta(t)\) is the resulting noise item, \(\text{Y}(t)=\left[ \begin{array}{c}\text{Y}_{\text{e}}(t)\\ \tilde{\text{Y}}_{\text{f}}(t)\end{array} \right]\), \(\Theta(t)=\left[ \begin{array}{c}\Theta_{1}(t)\\ \Theta_{2}(t)\end{array} \right]\), \(\Lambda(\tau_{t,i})=\left[ \begin{array}{cc}\Omega&\Phi(\tau_{t,i})\\ 0&\Gamma\end{array} \right]\), and  
\[ \begin{split}&\text{Y}_{\text{e}}(t)=\left[\begin{array}{cccc}x _{\text{e,1}}^{\text{T}}(t)&x_{\text{e,2}}^{\text{T}}(t)&\cdots&x_{\text{e,i} }^{\text{T}}(t)&\cdots&x_{\text{e,N}}^{\text{T}}(t)\end{array}\right]^{\text{ T}},\\ &\\ &\tilde{\text{Y}}_{\text{f}}(t)=\left[\begin{array}{cccc}\tilde{\text{Y}}_{ \text{f,1}}^{\text{T}}(t)&\tilde{\text{Y}}_{\text{f,2}}^{\text{T}}(t)&\cdots &\tilde{\text{Y}}_{\text{f,i}}^{\text{T}}(t)&\cdots&\tilde{\text{Y}}_{\text{f, N}}^{\text{T}}(t)\end{array}\right]^{\text{ T}},\\ &\\ &\tilde{\text{Y}}_{\text{f,i}}(t)=\left[\begin{array}{cccc}\tilde{\text{x}}_{ \text{f,i}}^{\text{T}}(t)&\tilde{\text{x}}_{\text{f,i}}^{\text{T}}(t-1)& \cdots&\tilde{\text{x}}_{\text{f,i}}^{\text{T}}(t-\tau_{t,i})\end{array} \right]^{\text{T}},\\ &\\ &\Theta_{1}(t)=\left[\begin{array}{cccc}\Theta_{\Delta v,1}^{\text{T}}(t)& \Theta_{\Delta v,2}^{\text{T}}(t)&\cdots&\Theta_{\Delta v,i}^{\text{T}}(t)&\cdots &\Theta_{\Delta v,N}^{\text{T}}(t)\\ \Theta_{\Delta v,1}^{\text{T}}(t)&\Theta_{\Delta v,2}^{\text{T}}(t)&\cdots&\Theta_ {\Delta v,j}^{\text{T}}(t)&\cdots&\Theta_{\Delta v,N}^{\text{T}}(t)\end{array} \right]^{\text{T}},\end{split} \]\[\Theta_{2}(t)=\left[ \begin{array}{cccccc}\Theta_{v,1}^{\rm T}(t)&\Theta_{v,2}^{ \rm T}(t)&\cdots&\Theta_{v,i}^{\rm T}(t)&\cdots&\Theta_{v,N}^{\rm T}(t)\\ \Theta_{\omega,1}^{\rm T}(t)&\Theta_{\omega,2}^{\rm T}(t)&\cdots&\Theta_{ \omega,j}^{\rm T}(t)&\cdots&\Theta_{\omega,N}^{\rm T}(t)\end{array} \right]^{\rm T},\]  
\[\Theta_{\Lambda\omega,j}^{\rm T}(t)=\left[ \begin{array}{cccccc}\Lambda \omega_{i}^{\rm T}(t)&\Delta\omega_{i}^{\rm T}(t-1)&\cdots&\Delta\omega_{i}^{ \rm T}(t-\tau_{t,j})\end{array} \right]^{\rm T},\]  
\[\Theta_{v,i}^{\rm T}(t)=\left[ \begin{array}{cccccc}v_{i}^{\rm T}(t+1)&v_{i} ^{\rm T}(t)&\cdots&v_{i}^{\rm T}(t-\tau_{t,j})\end{array} \right]^{\rm T},\]  
\[\Theta_{\omega,i}^{\rm T}(t)=\left[ \begin{array}{cccccc}\omega_{i}^{\rm T}( t)&\omega_{i}^{\rm T}(t-1)&\cdots&\omega_{i}^{\rm T}(t-\tau_{t,j})\end{array}  \right]^{\rm T}.\]  
It can be seen that the system matrix \(\Lambda(\tau_{t,j})\) is an upper triangular matrix, where \(\Phi(\tau_{t,j})\) is independent of the stability of the system. The noise term \(\omega(\tau_{t,j})\Theta(t)\) is bounded and has a mean of zero. According to the [PERSON] stability theorem, the stability of the system (32) depends on whether the eigenvalues of the coefficient matrices \(\Omega\) and \(\Gamma\) lie within the unit circle. Taking the expectation on both sides of Equation (32) yields:  
\[\mathbb{E}\{\mathrm{Y}(t+1)\}=\Lambda(\tau_{t,j})\mathbb{E}\{\mathrm{Y}(t)\}. \tag{33}\]  
Clearly, Theorem 1 is obtained, and the proof is finished. **Remark 2**. : _It is observed that the stability of the closed-loop system (32) is preserved regardless of RCCs and actuator faults, and Theorem 1 provides a criterion for parameter selection (i.e., \(L_{i}^{\rm a}\) and \(L_{i}^{\rm b}\)) in the proposed control protocol (17)._