## 3 Stability Analysis
}^{N}l_{ij}(t)\hat{x}_{j}(t-\tau_{i,j}),\]  
where \(c_{i}=L_{i}^{\text{a}}+L_{i}^{\text{b}}\sum_{j=1\
eq i\
eq i}^{N}a_{ij}C_{ \text{e},i}\), \(d_{ij}=L_{i}^{\text{b}}a_{ij}C_{\text{e},j}\), \(g_{i}(t)=c_{i}A_{\text{e},i}^{\tau_{i,j}}\bar{\tau}\), \(l_{ij}(t)=d_{ij}A_{\text{e},i}^{\tau_{i,j}}\bar{\tau}\). Substituting (27) into (4) and (5), the actual error incremental state and incremental output of the leader at the time \(t\) can be obtained as follows  
\[x_{\text{e},1}(t+1)\] \[= (A_{\text{e},1}-B_{\text{e},1}c_{1})(x_{\text{e},1}(t)+\sum_{s_{ i}=1}^{\tau_{i1}^{\text{eqs}}}A_{\text{e},1}^{s_{i}-1}(I_{\text{e},1}\Delta \omega_{1}(t-s_{1})-E_{\text{e},1}\Delta\upsilon_{1}(t+1-s_{1})))\] \[+I_{\text{e},1}\Delta\omega_{1}(t)-E_{\text{e},1}\Delta\upsilon_ {1}(t+1)\] \[+B_{\text{e},1}\sum_{j=2}^{N}d_{1j}(x_{\text{e},j}(t)+\sum_{s_{ j}=1}^{\tau_{i1}^{\text{eqs}}}A_{\text{e},j}^{s_{j}-1}(I_{\text{e},j}\Delta \omega_{j}(t-s_{j})-E_{\text{e},i}\Delta\upsilon_{j}(t+1-s_{j}))) \tag{29}\] \[+B_{\text{e},1}(g_{1}(t)-\sum_{j=2}^{N}d_{1j}\delta_{j}(t))\bar{ x}_{1}(t-\tau_{i,1})\] \[-B_{\text{e},1}\sum_{j=2}^{N}l_{1j}(t)\hat{x}_{j}(t-\tau_{i,j}),\]  
\[\Delta y_{1}(t+1)\] \[= \text{C}_{\text{e},1}(A_{\text{e},1}-B_{\text{e},1}c_{1})(x_{ \text{e},1}(t)+\sum_{s_{1}=1}^{\tau_{i1}^{\text{eqs}}}A_{\text{e},1}^{s_{1}-1 }(I_{\text{e},1}\Delta\omega_{1}(t-s_{1})-E_{\text{e},1}\Delta\upsilon_{1}(t+1- s_{1}))) \tag{30}\] \[+\text{C}_{\text{e},1}(I_{\text{e},1}\Delta\omega_{1}(t)-E_{\text {e},1}\Delta\upsilon_{1}(t+1))\] \[+\text{C}_{\text{e},1}B_{\text{e},1}\sum_{j=2}^{N}d_{1j}(x_{ \text{e},j}(t)+\sum_{s_{j}=1}^{\tau_{i1}^{\text{eqs}}}A_{\text{e},j}^{s_{j}-1 }(I_{\text{e},j}\Delta\omega_{j}(t-s_{j})-E_{\text{e},j}\Delta\upsilon_{j}(t+1- s_{j})))\] \[= (A_{\text{e},1}-B_{\text{e},1}c_{1})(x_{\text{e},1}(t)+\sum_{s_{ i}=1}^{\tau_{i1}^{\text{eqs}}}A_{\text{e},1}^{s_{i}-1}(I_{\text{e},1}\Delta \omega_{1}(t-s_{1})-E_{\text{e},1}\Delta\upsilon_{1}(t+1-s_{1})))\]0 \[= (A_{\text{e},1}-B_{\text{e},1}c_{1})(x_{\text{e},1}(t)+\sum_{s_{ i}=1}^{\tau_{i1}^{\text{eqs}}}A_{\text{e},1}^{s_{i}-1}(I_{\text{e},1}\Delta \omega_{1}(t-s_{1})-E_{\text{e},1}\Delta\upsilon_{1}(t+1-s_{1})))\]1Then, using (28) and (30) can obtain the actual error broadening state (4) of follower \(i\) at the moment \(t+1\)  
\[ \begin{split}& x_{\text{e,i}}(t+1)\\ =&(A_{\text{e,i}}-B_{\text{e,i}}c_{i}+E_{\text{e,i}}C_{ \text{e,1}}B_{\text{e,1}}d_{1i})(x_{\text{e,i}}(t)+\sum_{s_{i}=1}^{\tau_{i,1}^ {\text{es}}}A_{\text{e,i}}^{s_{i}-1}(I_{\text{e,i}}\Delta\omega_{i}(t-s_{i}) \\ &-E_{\text{e,i}}\Delta v_{i}(t+1-s_{i}))+\sum_{s_{i}=0}^{\tau_{i, \text{es}}^{\text{min}}-1}A_{\text{e,i}}^{s_{i}}E_{\text{e,i}}\sum_{s_{1}=1}^{ \tau_{i,1}^{\text{es}}}A_{\text{e,1}}^{s_{i}-1}(I_{\text{e,1}}\Delta\omega_{1 }(t-s_{1})\\ &-E_{\text{e,i}}\Delta v_{1}(t+1-s_{1})))+(B_{\text{e,i}}d_{i1}+E_ {\text{e,i}}C_{\text{e,1}}(A_{\text{e,1}}-B_{\text{e,1}}c_{1}))(x_{\text{e,1 }}(t)\\ &+\sum_{s_{1}=1}^{\tau_{i,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} )))\\ &+\sum_{\begin{subarray}{c}j=2\\ j\
eq i\end{subarray}}^{N}(B_{\text{e,i}}d_{ij}+E_{\text{e,i}}C_{\text{e,1}}B _{\text{e,1}}d_{1j})((x_{\text{e,j}}(t)+\sum_{s_{j}=1}^{\tau_{i,j}^{\text{es} }}A_{\text{e,j}}^{s_{j}-1}(I_{\text{e,j}}\Delta\omega_{j}(t-s_{j})\\ &-E_{\text{e,j}}\Delta v_{j}(t+1-s_{j}))))+I_{\text{e,i}}\Delta\omega_{ i}(t)-E_{\text{e,i}}\Delta v_{i}(t+1)\\ &+(B_{\text{e,i}}g_{i}(t)-E_{\text{e,i}}C_{\text{e,1}}B_{\text{e,1}} I_{1i}))\tilde{x}_{i}(t-\tau_{i,j})\\ &+(B_{\text{e,i}}(c_{i}\delta_{i}(t)-\sum_{j=2}^{N}d_{ij}\delta_{j}(t )-l_{i1}(t))\\ &+E_{\text{e,i}}C_{\text{e,1}}B_{\text{e,1}}(g_{1}(t)-\sum_{j=2}^{N} d_{1j}\delta_{j}(t)))\tilde{x}_{1}(t-\tau_{i,1})\\ &-\sum_{\begin{subarray}{c}j=2\\ j\
eq i\end{subarray}}^{N}(B_{\text{e,i}}l_{ij}(t)+E_{\text{e,i}}C_{\text{e,1 }}B_{\text{e,1}}I_{1j})\tilde{x}_{j}(t-\tau_{i,j}),i\
eq 1.\end{split}