## 3 Stability Analysis  
Based on the considered system model and the proposed control law in the control scheme, the following theorem is obtained. **Theorem 1**. : _The stability of the closed-loop system resulting from Equations (1), (9), (10) and (17), i.e., the control objective, is guaranteed if and only if the eigenvalues of \(\Omega\) and \(\Gamma\) are lie inside the unit circle, where the matrices \(\Omega\) and \(\Gamma\) are described by_  
\[\Omega=\left[ \begin{array}{cccc}A_{\mathrm{e,1}}-B_{\mathrm{e,1}}c_{1}&B_{ \mathrm{e,1}}d_{12}&\cdots\\ B_{\mathrm{e,2}}d_{21}+E_{\mathrm{e,2}}C_{\mathrm{e,1}}(A_{\mathrm{e,1}}-B_{ \mathrm{e,1}}c_{1})&A_{\mathrm{e,2}}-B_{\mathrm{e,2}}c_{2}&\cdots\\ \vdots&\vdots&\vdots&\ddots\\ B_{\mathrm{e,N}}d_{N1}+E_{\mathrm{e,N}}C_{\mathrm{e,1}}(A_{\mathrm{e,1}}-B_{ \mathrm{e,1}}c_{1})&B_{\mathrm{e,N}}d_{N2}+E_{\mathrm{e,N}}C_{\mathrm{e,1}}B_ {\mathrm{e,1}}d_{12}&\cdots\\ &&B_{\mathrm{e,1}}d_{1N}\\ B_{\mathrm{e,2}}d_{2N}+E_{\mathrm{e,2}}C_{\mathrm{e,1}}B_{\mathrm{e,1}}d_{1N} \\ \vdots&A_{\mathrm{e,N}}-B_{\mathrm{e,N}}c_{N}\end{array} \right],\]  
\[\Gamma=\mathrm{diag}\{\underbrace{\{I_{n1}-K_{1}C_{\mathrm{f,1}},\cdots,I_{n1 }-K_{1}C_{\mathrm{f,1}},\cdots,\underbrace{I_{n1}-K_{N}C_{\mathrm{f,N}},\cdots,I _{n1}-K_{N}C_{\mathrm{f,N}}\}}_{\tau_{\mathrm{f,N}}+1}\}.\]  
Proof. : The reference signal, without any loss of generality, is set as \(r(\cdot)=0\). The estimation error of agent \(i\) about the state and fault can be expressed as follow  
\[\bar{x}_{t,i}(t)=x_{t,i}(t)-\hat{x}_{t,i}(t|t-1). \tag{18}\]  
According to (7), (9) and (18), it can be deduced that the estimation error of agent \(i\) has the following relationship  
\[\hat{x}_{t,i}(t+1)=(I_{n1}-K_{i}C_{t,i})(A_{t,i}\hat{x}_{t,i}(t)+I_{t}\omega_{ i}(t))-K_{i}v_{i}(t+1), \tag{19}\]  
and then it can be obtained that  
\[\hat{x}_{t,i}(t+1|t)= A_{t,i}\hat{x}_{t,i}(t|t-1)+B_{t,i}u_{i}(t) \tag{20}\] \[+K_{i}C_{t,i}A_{t,i}\hat{x}_{t,i}(t)+K_{i}C_{\mathrm{f,1}}I_{t} \omega_{i}(t)+K_{i}v_{i}(t+1).\]  
Iterative computation from (4) and (12) yields the actual error broadening state and the predicted value of the error broadening state of the leader at moment \(t\)  
\[ \begin{split} x_{\mathrm{e,1}}(t)=& A_{\mathrm{e,1}}^{\tau_{ \mathrm{e,1}}^{\tau_{\mathrm{e,1}}^{\tau_{\mathrm{e,1}}^{\tau_{\mathrm{e,1}}^ {\tau_{\mathrm{e,1}}^{\tau_{\mathrm{e,1}}}}}}}}x_{\mathrm{e,1}}(t-\tau_{t,1}^{ \mathrm{e}})\\ &+\sum_{s_{1}=1}^{\tau_{\mathrm{e,1}}^{\tau_{\mathrm{e,1}^{ \tau_{\mathrm{e,1}}^{\tau_{\mathrm{e,1}}}}}}}A_{\mathrm{e,1}}^{s_{1}-1}(B_{ \mathrm{e,1}}\Delta u_{1}(t-s_{1})+F_{\mathrm{e,1}}\Delta f_{1}(t-s_{1}))\\ &+\sum_{s_{1}=1}^{\tau_{\mathrm{e,1}}^{\tau_{\mathrm{e,1}^{\tau_ {\mathrm{e,1}}^{\tau_{\mathrm{e,1}}}}}}}A_{\mathrm{e,1}}^{s_{1}-1}(I_{\mathrm{ e,1}}\Delta v_{1}(t-s_{1})-E_{\mathrm{e,1}}\Delta v_{1}(t+1-s_{1})),\end{split}  \tag{21}\]  
\[ \begin{split}\hat{x}_{\mathrm{e,1}}(t|t-\tau_{t,1}^{\mathrm{e, 1}}-1)=& A_{\mathrm{e,1}}^{\tau_{\mathrm{e,1}}^{\tau_{\mathrm{e,1} }^{\tau_{\mathrm{e,1}}^{\tau_{\mathrm{e,1}}}}}}\hat{x}_{\mathrm{e,1}}(t-\tau_{t,1}^{\mathrm{e}}|t-\tau_{t,1}^{\mathrm{e}}-1)\\ &+\sum_{s_{1}=1}^{\tau_{\mathrm{e,1}}^{\tau_{\mathrm{e,1}^{\tau_ {\mathrm{e,1}}^{\tau_{\mathrm{e,1}}}}}}}A_{\mathrm{e,1}}^{s_{1}-1}(B_{\mathrm{ e,1}}\Delta u_{1}(t-s_{1})\\ &+F_{\mathrm{e,1}}\Delta f_{1}(t-s_{1}|t-s_{1}-1)),\end{split}  \tag{22}\]  
where, according to (6), we define \(\Delta\hat{f}_{i}(t|t-1)=\Delta f_{i}(t)=I_{k}\bar{x}_{t,i}(t)\). Taking (21) minus (22) gives the result\[x_{\text{e},1}(t)-\hat{x}_{\text{e},1}(t|t-\tau_{t,1}^{\text{es}}-1)= A_{\text{e},1}^{\tau_{t,1}^{\text{es}}}\overline{I}\xi_{1}(t-\tau_{t,1}^{ \text{es}}) \tag{23}\] \[+\sum_{s_{1}=1}^{\tau_{t,1}^{\text{es}}}A_{\text{e},1}^{s_{1}-1}(I _{\text{e},1}\Delta\omega_{1}(t-s_{1})-E_{\text{e},1}\Delta v_{1}(t+1-s_{1})),\]  
where \(\overline{I}=\left[ \begin{array}{cc}I&0\end{array} \right]^{\overline{T}}\).