Appendix
The Appendix provides rigorous proofs of Theorem 1 and Theorem 3, which are fundamental to this paper.
1.1 A1 Proof of Theorem 1
Select the following Lyapunov function of systems (6):
$$V(k)=\sum\limits_{{i=1}}^{N} {\left( \begin{gathered} \widetilde {x}_{i}^{T}(k){P_{i1}}{\widetilde {x}_i}(k)+\widetilde {f}_{i}^{T}(k){P_{i2}}{\widetilde {f}_i}(k) \hfill \\ +\,\sum\limits_{{j=1}}^{N} {\sum\limits_{{l=1}}^{{{h_{ij}}}} {\widetilde {x}_{j}^{T}(k - l)\overline {G} _{{ij}}^{T}{R_{ij}}{{\overline {G} }_{ij}}{{\widetilde {x}}_j}(k - l} )} +\sum\limits_{{l=1}}^{{{h_{ii}}}} {\widetilde {x}_{i}^{T}(k - l){S_i}{{\widetilde {x}}_i}(k - l)} \hfill \\ \end{gathered} \right)} .$$
(26)
Take the first order difference of V(k). It follows that
$$\begin{aligned} & \Delta V(k)=V(k+1) - V(k) \\ & \quad =\sum\limits_{{i=1}}^{N} {\left( {\widetilde {x}_{i}^{T}(k+1){P_{i1}}{{\widetilde {x}}_i}(k+1) - \widetilde {x}_{i}^{T}(k){P_{i1}}{{\widetilde {x}}_i}(k)} \right.} \\ & \quad +\,\widetilde {f}_{i}^{T}(k+1){P_{i2}}{\widetilde {f}_i}(k+1) - \widetilde {f}_{i}^{T}(k){P_{i2}}{\widetilde {f}_i}(k)+\sum\limits_{{j=1}}^{N} {\widetilde {x}_{j}^{T}(k)\overline {G} _{{ij}}^{T}{R_{ij}}{{\overline {G} }_{ij}}{{\widetilde {x}}_j}(k)} \\ & \quad - \,\sum\limits_{{j=1}}^{N} {\widetilde {x}_{j}^{T}(k - {h_{ij}})\overline {G} _{{ij}}^{T}{R_{ij}}{{\overline {G} }_{ij}}{{\widetilde {x}}_j}(k - {h_{ij}})} \\ & \quad \left. {+\,\widetilde {x}_{i}^{T}(k){S_i}{{\widetilde {x}}_i}(k) - \widetilde {x}_{i}^{T}(k - {h_{ii}}){S_i}{{\widetilde {x}}_i}(k - {h_{ii}})} \right). \\ \end{aligned}$$
(27)
Due to \(||{\widetilde {g}_{ij}}||=||{g_{ij}}({x_j}(k - {h_{ij}})) - {g_{ij}}({\widehat {x}_j}(k - {h_{ij}}))|| \le ||{Z_{ij}}({\widetilde {x}_j}(k - {h_{ij}}))||\) and \(\sum\limits_{{i=1}}^{N} {\sum\limits_{{j=1}}^{N} {\widetilde {x}_{j}^{T}(k)\overline {G} _{{ij}}^{T}{R_{ij}}{{\overline {G} }_{ij}}{{\widetilde {x}}_j}(k)} } =\sum\limits_{{i=1}}^{N} {\widetilde {x}_{i}^{T}(k)\left( {\sum\limits_{{j=1}}^{N} {\overline {G} _{{ji}}^{T}{R_{ji}}{{\overline {G} }_{ji}}} } \right){{\widetilde {x}}_i}(k)}\), we can obtain
$$\begin{gathered} \Delta V(k) \le \sum\limits_{{i=1}}^{N} {\left( {\widetilde {x}_{i}^{T}(k+1){P_{i1}}{{\widetilde {x}}_i}(k+1)} \right.} +\widetilde {x}_{i}^{T}(k)\left( {\sum\limits_{{j=1}}^{N} {\overline {G} _{{ji}}^{T}{R_{ji}}{{\overline {G} }_{ji}}} - {P_{i1}}+{S_i}} \right){\widetilde {x}_i}(k) \\ +\,\widetilde {f}_{i}^{T}(k+1){P_{i2}}{\widetilde {f}_i}(k+1) - \widetilde {f}_{i}^{T}(k){P_{i2}}{\widetilde {f}_i}(k) - \widetilde {x}_{i}^{T}(k - {h_{ii}}){S_i}{\widetilde {x}_i}(k - {h_{ii}}) \\ +\,\left. {\sum\limits_{{j=1}}^{N} {\widetilde {x}_{j}^{T}(k - {h_{ij}})\left( {Z_{{ij}}^{T}{Z_{ij}} - \overline {G} _{{ij}}^{T}{R_{ij}}{{\overline {G} }_{ij}}} \right){{\widetilde {x}}_j}(k - {h_{ij}})} - \widetilde {g}_{i}^{T}{{\widetilde {g}}_i}} \right), \\ \end{gathered}$$
(28)
where, \({\widetilde {g}_i}={\left( {\begin{array}{*{20}{c}} {\widetilde {g}_{{i1}}^{T}}&{\widetilde {g}_{{i2}}^{T}}&{.....}&{\widetilde {g}_{{iN}}^{T}} \end{array}} \right)^T}\), in (28), there are
$$\begin{gathered} \widetilde {x}_{i}^{T}(k+1){P_{i1}}{\widetilde {x}_i}(k+1) \hfill \\ \quad =\,\eta _{i}^{T}(k)\overline {\varphi } _{{i1}}^{T}{P_{i1}}{\overline {\varphi } _{i1}}{\eta _i}(k) - 2\eta _{i}^{T}(k)\overline {\varphi } _{{i1}}^{T}{P_{i1}}{H_i}{C_i}{\widetilde {x}_i}(k - {h_{ii}}) \hfill \\ \quad +\,\widetilde {x}_{t}^{T}(k - {h_{ii}})C_{t}^{T}H_{i}^{T}{P_{i1}}{H_i}{C_i}{\widetilde {x}_i}(k - {h_{ii}}), \hfill \\ \end{gathered}$$
(29)
$$\begin{gathered} \widetilde {f}_{i}^{T}(k+1){P_{i2}}{\widetilde {f}_i}(k+1) \hfill \\ \quad =\,\eta _{i}^{T}(k)\overline {\varphi } _{{i2}}^{T}{P_{i2}}{\overline {\varphi } _{i2}}{\eta _i}(k)+2\eta _{i}^{T}(k)\overline {\varphi } _{{i2}}^{T}{P_{i2}}{\Gamma _i}{\Lambda _{i1}}{C_i}{H_i}{C_i}{\widetilde {x}_i}(k - {h_{ii}}) \hfill \\ \quad \quad +\,\widetilde {x}_{i}^{T}(k - {h_{ii}})C_{i}^{T}H_{i}^{T}C_{i}^{T}\Lambda _{{i1}}^{T}{\Gamma _i}{P_{i2}}{\Gamma _i}{\Lambda _{i1}}{C_i}{H_i}{C_i}{\widetilde {x}_i}(k - {h_{ii}}), \hfill \\ \end{gathered}$$
(30)
where, \({\eta _i}(k)={\left( {\begin{array}{*{20}{c}} {\widetilde {x}_{i}^{T}(k)}&{\widetilde {f}_{i}^{T}(k)}&{\widetilde {g}_{i}^{T}}&{w_{i}^{T}(k)}&{\Delta f_{i}^{T}(k)} \end{array}} \right)^T}.\)
Substitute (29), (30) into (28). Then, according to (7), and considering the zero initial condition, it follows that
$$\begin{aligned} J & =\sum\limits_{{k=0}}^{\infty } {\sum\limits_{{i=1}}^{N} {\left( {\frac{1}{{{\gamma _i}}}\widetilde {y}_{i}^{T}(k){{\widetilde {y}}_i}(k) - {\gamma _i}\mu _{i}^{T}(k){\mu _i}(k)} \right)} } \\ & \le \,\sum\limits_{{k=0}}^{\infty } {\left( {\sum\limits_{{i=1}}^{N} {\left( {\frac{1}{{{\gamma _i}}}\widetilde {y}_{i}^{T}(k){{\widetilde {y}}_i}(k) - {\gamma _i}\mu _{i}^{T}(k){\mu _i}(k)} \right)+\Delta V(k)} } \right)} \\ & =\,\sum\limits_{{k=0}}^{\infty } {\sum\limits_{{i=1}}^{N} {\eta _{i}^{T}(k)\left( {\begin{array}{*{20}{c}} {{\Phi _{i1}}}&{{\Phi _{i2}}} \\ {\Phi _{{i2}}^{T}}&{{\Phi _{i3}}} \end{array}} \right)} } {\eta _i}(k). \\ \end{aligned}$$
(31)
In (31), there are
$${\Phi _{i1}}=\overline {\varphi } _{{i1}}^{T}{P_{i1}}{\overline {\varphi } _{i1}}+\overline {\varphi } _{{i2}}^{T}{P_{i2}}{\overline {\varphi } _{i2}}+{\Phi _{i10}}+\sum\limits_{{j=1}}^{N} {\overline {G} _{{ji0}}^{T}{R_{ji}}{{\overline {G} }_{ji0}}} ,$$
(32)
$${\Phi _{i2}}=\overline {\varphi } _{{i2}}^{T}{P_{i2}}{\Gamma _i}{\Lambda _{i1}}{C_i}{H_i}{\widetilde {C}_i} - \overline {\varphi } _{{i1}}^{T}{P_{i1}}{H_i}{\widetilde {C}_i},$$
(33)
$${\Phi _{i3}}=\widetilde {Z}_{i}^{T}{\widetilde {Z}_i} - \widetilde {{\overline {G} }}_{i}^{T}{\widetilde {R}_i}{\widetilde {{\overline {G} }}_i} - {\widetilde {S}_{i0}}+\widetilde {C}_{i}^{T}H_{i}^{T}{P_{i1}}{H_i}{\widetilde {C}_i}+\widetilde {C}_{i}^{T}H_{i}^{T}C_{i}^{T}\Lambda _{{i1}}^{T}{\Gamma _i}{P_{i2}}{\Gamma _i}{\Lambda _{i1}}{C_i}{H_i}{\widetilde {C}_i},$$
(34)
where
$${\Phi _{i10}}=diag\left( {\begin{array}{*{20}{c}} { - {P_{i1}}+{S_i}+\frac{1}{{{\gamma _i}}}{C_i}^{T}{C_i}}&{ - {P_{i2}}}&{ - I}&{ - {\gamma _i}I}&{ - {\gamma _i}I} \end{array}} \right),$$
$${\overline {\Phi } _{i10}}=diag\left( {\begin{array}{*{20}{c}} { - {P_{i1}}+{S_i}+\sum\limits_{{j=1}}^{N} {\overline {G} _{{ji}}^{T}{R_{ji}}{{\overline {G} }_{ji}}} +\frac{1}{{{\gamma _i}}}{C_i}^{T}{C_i}}&{ - {P_{i2}}}&{ - I}&{ - {\gamma _i}I}&{ - {\gamma _i}I} \end{array}} \right),$$
$${\widetilde {C}_i}=\left( {\begin{array}{*{20}{c}} {0 \ldots 0}&{{C_i}}&{0 \ldots 0} \end{array}} \right).$$
For \(- \widetilde {{\overline {G} }}_{i}^{T}{\widetilde {R}_i}{\widetilde {{\overline {G} }}_i}\) in \({\Phi _{i3}}\),due to \({\widetilde {{\overline {G} }}_i}={\widetilde {G}_i}+{\widetilde {W}_i}{\widetilde {F}_i}{\widetilde {N}_i}\) and Lemma 1,we can deduce:
$$- \,\widetilde {{\overline {G} }}_{i}^{T}{\widetilde {R}_i}{\widetilde {{\overline {G} }}_i} \le - \widetilde {G}_{i}^{T}{\widetilde {R}_i}{\widetilde {G}_i}+\widetilde {G}_{i}^{T}{\widetilde {R}_i}{\widetilde {W}_i}\Gamma _{{\alpha i}}^{{ - 1}}\widetilde {W}_{i}^{T}{\widetilde {R}_i}{\widetilde {G}_i}+\widetilde {N}_{i}^{T}{\Gamma _{\beta i}}{\widetilde {N}_i}.$$
(35)
By (32–35), it follows that
$$\begin{aligned} \left( {\begin{array}{*{20}{c}} {{\Phi _{i1}}}&{{\Phi _{i2}}} \\ {\Phi _{{i2}}^{T}}&{{\Phi _{i3}}} \end{array}} \right) \le & \,\left( {\begin{array}{*{20}{c}} {{\Phi _{i10}}}&0 \\ 0&{\widetilde {Z}_{i}^{T}{{\widetilde {Z}}_i} - {{\widetilde {S}}_{i0}} - \widetilde {G}_{i}^{T}{{\widetilde {R}}_i}{{\widetilde {G}}_i}} \end{array}} \right) \\ & +\,\left( {\begin{array}{*{20}{c}} {\overline {\varphi } _{{i1}}^{T}} \\ { - \widetilde {C}_{i}^{T}H_{i}^{T}} \end{array}} \right){P_{i1}}\left( {\begin{array}{*{20}{c}} {{{\overline {\varphi } }_{i1}}}&{ - {H_i}{{\widetilde {C}}_i}} \end{array}} \right) \\ & +\,\,\left( {\begin{array}{*{20}{c}} {\overline {\varphi } _{{i2}}^{T}} \\ {\widetilde {C}_{i}^{T}H_{i}^{T}C_{i}^{T}\Lambda _{{i1}}^{T}} \end{array}} \right){P_{i2}}\left( {\begin{array}{*{20}{c}} {{{\overline {\varphi } }_{i2}}}&{{\Lambda _{i1}}{C_i}{H_i}{{\widetilde {C}}_i}} \end{array}} \right)+\sum\limits_{{j=1}}^{N} {\left( \begin{gathered} \overline {G} _{{ji0}}^{T} \hfill \\ 0 \hfill \\ \end{gathered} \right){R_{ji}}(\begin{array}{*{20}{c}} {{{\overline {G} }_{ji0}}}&0 \end{array})} \\ & +\,\left( \begin{gathered} 0 \hfill \\ \widetilde {G}_{i}^{T}{\widetilde {R}_i}{\widetilde {W}_i} \hfill \\ \end{gathered} \right)\Gamma _{{\alpha i}}^{{ - 1}}\left( {\begin{array}{*{20}{c}} 0&{\widetilde {W}_{i}^{T}{{\widetilde {R}}_i}{{\widetilde {G}}_i}} \end{array}} \right)+\left( \begin{gathered} 0 \hfill \\ \widetilde {N}_{i}^{T} \hfill \\ \end{gathered} \right){\Gamma _{\beta i}}\left( {\begin{array}{*{20}{c}} 0&{{{\widetilde {N}}_i}} \end{array}} \right). \\ \end{aligned}$$
(36)
According to the Schur complements, we can determine that
$$\left( {\begin{array}{*{20}{c}} {{\Phi _{i1}}}&{{\Phi _{i2}}} \\ {\Phi _{{i2}}^{T}}&{{\Phi _{i3}}} \end{array}} \right)<0,$$
is equivalent to
$$\begin{gathered} \left( {\begin{array}{*{20}{c}} {{\Phi _{i10}}}&0&{\overline {\varphi } _{{i1}}^{T}} \\ *&{\widetilde {Z}_{i}^{T}{{\widetilde {Z}}_i} - {{\widetilde {S}}_{i0}} - \widetilde {G}_{i}^{T}{{\widetilde {R}}_i}{{\widetilde {G}}_i}}&{ - \widetilde {C}_{i}^{T}H_{i}^{T}} \\ *&*&{ - P_{{i1}}^{{ - 1}}} \\ *&*&* \\ *&*&* \\ *&*&* \\ *&*&* \end{array}} \right.\left. {\begin{array}{*{20}{c}} {\overline {\varphi } _{{i2}}^{T}}&0&0&{\overline {G} _{{i0}}^{{'T}}} \\ 0&{\widetilde {G}_{i}^{T}{{\widetilde {R}}_i}{{\widetilde {W}}_i}}&{\widetilde {N}_{i}^{T}}&0 \\ 0&0&0&0 \\ { - P_{{i2}}^{{ - 1}}}&0&0&0 \\ *&{ - {\Gamma _{\alpha i}}}&0&0 \\ *&*&{ - \Gamma _{{\beta i}}^{{ - 1}}}&0 \\ *&*&*&{ - \widetilde {R}_{i}^{{' - 1}}} \end{array}} \right) \hfill \\ \quad +\,\,\left( \begin{gathered} 0 \hfill \\ \widetilde {C}_{i}^{T}H_{i}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right){\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\Lambda _{i1}}{C_i} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)^T}+\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\Lambda _{i1}}{C_i} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right){\left( \begin{gathered} 0 \hfill \\ \widetilde {C}_{i}^{T}H_{i}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)^T}<0. \hfill \\ \end{gathered}$$
(37)
Owing to:
$$\left( \begin{gathered} 0 \hfill \\ \widetilde {C}_{i}^{T}H_{i}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right){\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\Lambda _{i1}}{C_i} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)^T}+\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\Lambda _{i1}}{C_i} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right){\left( \begin{gathered} 0 \hfill \\ \widetilde {C}_{i}^{T}H_{i}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)^T} \le \left( \begin{gathered} 0 \hfill \\ \widetilde {C}_{i}^{T}H_{i}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)\Pi _{{i1}}^{{ - 1}}{\left( \begin{gathered} 0 \hfill \\ \widetilde {C}_{i}^{T}H_{i}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)^T}+\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\Lambda _{i1}}{C_i} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right){\Pi _{i1}}{\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\Lambda _{i1}}{C_i} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)^T},$$
(38)
by the Schur complements and robust stability theory, we can know that, if the condition in Theorem 1 holds,the matrix inequality (37) will hold. Therefore, the errors of state \({\widetilde {x}_i}(t)\) and fault estimation \({\widetilde {f}_i}(t)\), \(i=1 \ldots N\) are robust and stable, and the error systems (6) satisfy the H∞ performance \({\left\| {{y_i}(k)} \right\|_2}<{\gamma _i}{\left\| {{\mu _i}(k)} \right\|_2}\). Thus, Theorem 1 is proved.
1.2 A2 Proof of Theorem 3
We define the Lyapunov function of system (12) as follows:
$${V_c}(k)=\sum\limits_{{i=1}}^{N} {\left( {\eta _{{ci}}^{T}(k){P_{ci}}{\eta _{ci}}(k)+\sum\limits_{{j=1}}^{N} {\sum\limits_{{l=1}}^{{{h_{ij}}}} {x_{j}^{T}(k - {h_{ij}})\widetilde {{\overline {G} }}_{{ij}}^{T}{R_{cij}}{{\widetilde {{\overline {G} }}}_{ij}}{x_j}(k - {h_{ij}})} } } \right)} .$$
(39)
Taking the difference of \({V_c}(k)\) along the systems (12), it follows that
$$\begin{aligned} \Delta {V_c}(k) & ={V_c}(k+1) - {V_c}(k) \\ & \le \sum\limits_{{i=1}}^{N} {\left( {\eta _{{ci}}^{T}(k)\widetilde {{\overline {A} }}_{i}^{T}{P_{ci}}{{\widetilde {{\overline {A} }}}_i}{\eta _{ci}}(k)+g_{i}^{T}\widetilde {{\overline {G} }}_{{ci}}^{T}{P_{ci}}{{\widetilde {{\overline {G} }}}_{ci}}{g_i}} \right.} +\mu _{{ci}}^{T}(k)\widetilde {{\overline {D} }}_{i}^{T}{P_{ci}}{\widetilde {{\overline {D} }}_i}{\mu _{ci}}(k)+2\eta _{{ci}}^{T}(k)\widetilde {{\overline {A} }}_{i}^{T}{P_{ci}}{\widetilde {{\overline {G} }}_{ci}}{g_i} \\ & \quad +\,2\eta _{{ci}}^{T}(k)\widetilde {{\overline {A} }}_{i}^{T}{P_{ci}}{\widetilde {{\overline {D} }}_i}{\mu _{ci}}(k)+2g_{i}^{T}\widetilde {{\overline {G} }}_{{ci}}^{T}{P_{ci}}{\widetilde {{\overline {D} }}_i}{\mu _{ci}}(k) \\ & \quad +\,x_{i}^{T}(k)\left( {\sum\limits_{{j=1}}^{N} {\widetilde {{\overline {G} }}_{{ji}}^{T}{R_{cji}}{{\widetilde {{\overline {G} }}}_{ji}}} } \right){x_i}(k) - \sum\limits_{{j=1}}^{N} {x_{j}^{T}(k - {h_{ij}})\widetilde {{\overline {G} }}_{{ij}}^{T}{R_{cij}}{{\widetilde {{\overline {G} }}}_{ij}}{x_j}(k - {h_{ij}})} \\ & \quad +\,\sum\limits_{{j=1}}^{N} {x_{j}^{T}(k - {h_{ij}})Z_{{cij}}^{T}{Z_{cij}}{x_j}(k - {h_{ij}})} \left. { - \sum\limits_{{j=1}}^{N} {g_{{ij}}^{T}({x_j}(k - {h_{ij}})){g_{ij}}({x_j}(k - {h_{ij}}))} } \right), \\ \end{aligned}$$
(40)
where, \({g_i}={\left( {\begin{array}{*{20}{c}} {{g_{i1}}^{T}({x_1}(k - {h_{i1}}))}& \ldots &{{g_{iN}}^{T}({x_N}(k - {h_{iN}}))} \end{array}} \right)^T}.\)
According to (13), and take into account the zero initial condition, we have
$$\begin{aligned} {J_c} & \le \sum\limits_{{k=0}}^{\infty } {\left( {\sum\limits_{{i=1}}^{N} {\left( {\frac{1}{{{\gamma _{ci}}}}y_{i}^{T}(k){y_i}(k) - {\gamma _{ci}}\mu _{i}^{T}(k){\mu _i}(k)} \right)} +\Delta V(k)} \right)} \\ & =\,{\sum\limits_{{k=0}}^{\infty } {\sum\limits_{{i=1}}^{N} {\left( \begin{gathered} {\eta _{ci}}(k) \hfill \\ {g_i} \hfill \\ {\mu _i}(k) \hfill \\ {x_1}(k - {h_{i1}}) \hfill \\ . \hfill \\ . \hfill \\ {x_N}(k - {h_{iN}}) \hfill \\ \end{gathered} \right)} } ^T}\left( {\begin{array}{*{20}{c}} {{\Phi _{ci1}}}&0 \\ 0&{{\Phi _{ci2}}} \end{array}} \right)\left( \begin{gathered} {\eta _{ci}}(k) \hfill \\ {g_i} \hfill \\ {\mu _i}(k) \hfill \\ {x_1}(k - {h_{i1}}) \hfill \\ . \hfill \\ . \hfill \\ {x_N}(k - {h_{iN}}) \hfill \\ \end{gathered} \right), \\ \end{aligned}$$
(41)
where
$${\Phi _{ci1}}=\left( {\begin{array}{*{20}{c}} {\widetilde {{\overline {A} }}_{i}^{T}{P_{ci}}{{\widetilde {{\overline {A} }}}_i}+\sum\limits_{{j=1}}^{N} {\widetilde {{\overline {G} }}_{{ji0}}^{T}{R_{cji}}{{\widetilde {{\overline {G} }}}_{ji0}}} - {P_{ci}}+\frac{1}{{{\gamma _{ci}}}}C_{i}^{{'T}}C_{i}^{'}}&{\widetilde {{\overline {A} }}_{i}^{T}{P_{ci}}{{\widetilde {{\overline {G} }}}_{ci}}}&{\widetilde {{\overline {A} }}_{i}^{T}{P_{ci}}{{\widetilde {{\overline {D} }}}_i}} \\ *&{\widetilde {{\overline {G} }}_{{ci}}^{T}{P_{ci}}{{\widetilde {{\overline {G} }}}_{ci}} - I}&{\widetilde {{\overline {G} }}_{{ci}}^{T}{P_{ci}}{{\widetilde {{\overline {D} }}}_i}} \\ *&*&{\widetilde {{\overline {D} }}_{i}^{T}{P_{ci}}{{\widetilde {{\overline {D} }}}_i} - {\gamma _{ci}}I} \end{array}} \right),$$
$$\begin{aligned} {\Phi _{ci2}} & =\widetilde {Z}_{{ci}}^{T}{\widetilde {Z}_{ci}} - \widetilde {{\widetilde {{\overline {G} }}}}_{i}^{T}{\widetilde {R}_{ci}}{\widetilde {{\widetilde {{\overline {G} }}}}_i} \\ & =\,diag\left\{ {Z_{{ci1}}^{T}{Z_{ci1}} - \widetilde {{\overline {G} }}_{{i1}}^{T}{R_{ci1}}{{\widetilde {{\overline {G} }}}_{i1}}, \ldots ,Z_{{ciN}}^{T}{Z_{ciN}} - \widetilde {{\overline {G} }}_{{iN}}^{T}{R_{ciN}}{{\widetilde {{\overline {G} }}}_{iN}}} \right\}. \\ \end{aligned}$$
Due to \({\widetilde {{\widetilde {{\overline {G} }}}}_i}={\widetilde {{\widetilde {G}}}_i}+{\widetilde {{\widetilde {W}}}_i}{\widetilde {F}_i}{\widetilde {N}_i}\) and Lemma 1, we can deduce:
$${\Phi _{ci2}} \le \widetilde {Z}_{{ci}}^{T}{\widetilde {Z}_{ci}} - \widetilde {{\widetilde {G}}}_{i}^{T}{\widetilde {R}_{ci}}{\widetilde {{\widetilde {G}}}_i}+\widetilde {{\widetilde {G}}}_{i}^{T}{\widetilde {R}_{ci}}{\widetilde {{\widetilde {W}}}_i}\Gamma _{{c\alpha i}}^{{ - 1}}\widetilde {{\widetilde {W}}}_{i}^{T}{\widetilde {R}_{ci}}{\widetilde {{\widetilde {G}}}_i}+\widetilde {N}_{i}^{T}{\Gamma _{c\beta i}}{\widetilde {N}_i}.$$
(42)
So, according to the Schur complements, \({\Phi _{ci2}}<0\) will hold when (15) holds.
Note that
$$\begin{aligned} {\Phi _{ci1}}= & \left( \begin{gathered} \widetilde {{\overline {A} }}_{i}^{T} \hfill \\ \widetilde {{\overline {G} }}_{{ci}}^{T} \hfill \\ \widetilde {{\overline {D} }}_{i}^{T} \hfill \\ \end{gathered} \right){P_{ci}}\left( {\begin{array}{*{20}{c}} {{{\widetilde {{\overline {A} }}}_i}}&{{{\widetilde {{\overline {G} }}}_{ci}}}&{{{\widetilde {{\overline {D} }}}_i}} \end{array}} \right)+\left( {\begin{array}{*{20}{c}} { - {P_{ci}}+\frac{1}{{{\gamma _{ci}}}}C_{i}^{{'T}}C_{i}^{'}}&0&0 \\ 0&{ - I}&0 \\ 0&0&{ - {\gamma _{ci}}I} \end{array}} \right) \\ & +\sum\limits_{{j=1}}^{N} {\left( \begin{gathered} \widetilde {{\overline {G} }}_{{ji0}}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)} {R_{cji}}\left( {\begin{array}{*{20}{c}} {{{\widetilde {{\overline {G} }}}_{ji0}}}&0&0 \end{array}} \right). \\ \end{aligned}$$
(43)
Thus, by the Schur complements, \({\Phi _{ci1}}<0\) will hold when (44) holds
$$\left( {\begin{array}{*{20}{c}} { - {P_{ci}}+\frac{1}{{{\gamma _{ci}}}}C_{i}^{{'T}}C_{i}^{'}}&0&0&{\widetilde {{\overline {A} }}_{i}^{T}}&{\widetilde {{\overline {G} }}_{{i0}}^{{'T}}} \\ *&{ - I}&0&{\widetilde {{\overline {G} }}_{{ci}}^{T}}&0 \\ *&*&{ - {\gamma _{ci}}I}&{\widetilde {{\overline {D} }}_{i}^{T}}&0 \\ *&*&*&{ - P_{{ci}}^{{ - 1}}}&0 \\ *&*&*&*&{ - \widetilde {R}_{{ci}}^{{' - 1}}} \end{array}} \right)<0.$$
(44)
The matrix inequality (44) can be expressed as:
$$\begin{gathered} \left( {\begin{array}{*{20}{c}} { - {P_{ci}}+\frac{1}{{{\gamma _{ci}}}}\widetilde {C}_{i}^{T}{{\widetilde {C}}_i}}&0&0&{\widetilde {A}_{i}^{T}}&{\widetilde {G}_{{i0}}^{{'T}}} \\ *&{ - I}&0&{\widetilde {G}_{i}^{T}}&0 \\ *&*&{ - {\gamma _{ci}}I}&{\widetilde {D}_{i}^{T}}&0 \\ *&*&*&{ - P_{{ci}}^{{ - 1}}}&0 \\ *&*&*&*&{ - \widetilde {R}_{{ci}}^{{' - 1}}} \end{array}} \right)+\left( \begin{gathered} \widetilde {E}_{{1ci}}^{T} \hfill \\ 0 \hfill \\ \widetilde {E}_{{1ci}}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)F_{i}^{T}{\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\widetilde {M}_{1i}} \hfill \\ 0 \hfill \\ \end{gathered} \right)^T}+\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\widetilde {M}_{1i}} \hfill \\ 0 \hfill \\ \end{gathered} \right){F_i}{\left( \begin{gathered} \widetilde {E}_{{1ci}}^{T} \hfill \\ 0 \hfill \\ \widetilde {E}_{{1ci}}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)^T} \hfill \\ \quad +\,\left( \begin{gathered} 0 \hfill \\ \widetilde {N}_{i}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)\widetilde {F}_{i}^{T}{\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\widetilde {W}_{ci}} \hfill \\ 0 \hfill \\ \end{gathered} \right)^T}+\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\widetilde {W}_{ci}} \hfill \\ 0 \hfill \\ \end{gathered} \right){\widetilde {F}_i}{\left( \begin{gathered} 0 \hfill \\ \widetilde {N}_{i}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)^T}+\left( \begin{gathered} \widetilde {N}_{i}^{{'T}} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)\widetilde {F}_{i}^{{'T}}{\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\widetilde {{\widetilde {W}}}_i} \hfill \\ \end{gathered} \right)^T}+\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\widetilde {{\widetilde {W}}}_i} \hfill \\ \end{gathered} \right)\widetilde {F}_{i}^{'}{\left( \begin{gathered} \widetilde {N}_{i}^{{'T}} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)^T}<0. \hfill \\ \end{gathered}$$
(45)
By Lemma 1, we can note that (45) will hold when (46) holds:
$$\begin{gathered} \left( {\begin{array}{*{20}{c}} { - {P_{ci}}+\frac{1}{{{\gamma _{ci}}}}C_{i}^{{'T}}C_{i}^{'}}&0&0&{\widetilde {A}_{i}^{T}}&{\widetilde {G}_{{i0}}^{{'T}}} \\ *&{ - I}&0&{\widetilde {G}_{{ci}}^{T}}&0 \\ *&*&{ - {\gamma _{ci}}I}&{\widetilde {D}_{i}^{T}}&0 \\ *&*&*&{ - P_{{ci}}^{{ - 1}}}&0 \\ *&*&*&*&{ - \widetilde {R}_{{ci}}^{{' - 1}}} \end{array}} \right)+{\varepsilon _{c2i}}{\left( \begin{gathered} \widetilde {E}_{{1ci}}^{T} \hfill \\ 0 \hfill \\ \widetilde {E}_{{1ci}}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)_i}{\left( \begin{gathered} \widetilde {E}_{{1ci}}^{T} \hfill \\ 0 \hfill \\ \widetilde {E}_{{1ci}}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)^T}+\varepsilon _{{c2i}}^{{ - 1}}\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\widetilde {M}_{1i}} \hfill \\ 0 \hfill \\ \end{gathered} \right){\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\widetilde {M}_{1i}} \hfill \\ 0 \hfill \\ \end{gathered} \right)^T} \hfill \\ \quad +\,\left( \begin{gathered} 0 \hfill \\ \widetilde {N}_{i}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)\Gamma _{{cNi}}^{{ - 1}}{\left( \begin{gathered} 0 \hfill \\ \widetilde {N}_{i}^{T} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)^T}+\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\widetilde {W}_{ci}} \hfill \\ 0 \hfill \\ \end{gathered} \right){\Gamma _{cWi}}{\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\widetilde {W}_{ci}} \hfill \\ 0 \hfill \\ \end{gathered} \right)^T}+\left( \begin{gathered} \widetilde {N}_{i}^{{'T}} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)\Gamma _{{cN'i}}^{{ - 1}}{\left( \begin{gathered} \widetilde {N}_{i}^{{'T}} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)^T}+\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\widetilde {{\widetilde {W}}}_i} \hfill \\ \end{gathered} \right){\Gamma _{cW'i}}{\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {\widetilde {{\widetilde {W}}}_i} \hfill \\ \end{gathered} \right)^T}<0. \hfill \\ \end{gathered}$$
(46)
And according to the Schur complements (46), is equivalent to (14) in Theorem 3.
Thus, \({\Phi _{ci1}}<0\), \({\Phi _{ci2}}<0\) will hold when (14) and (15) hold. Then, Theorem 3 is proved.
