Appendix A Proof of Theorem 1
Proof
Let \({{\tilde{{\bar{\Xi }}} }_i} = {{{\bar{\Xi }} }_i} - \Xi _i^*\). Then choose the following Lyapunov function for systems (12):
$$\begin{aligned} \begin{array}{l} V({\varepsilon _i},{w_i},{\iota _i})\\ = \sum \limits _{i = 1}^N {{r_i}\varepsilon _i^TP_1^{ - 1}{\varepsilon _i}} + \sum \limits _{i = 1}^N {\left[ {\frac{{{{\left( {{w_i}(t) - {\bar{w}}} \right) }^2}}}{{{{{\bar{w}}}_i}}}} \right] } \\ + \sum \limits _{i = 1}^N {\left[ {{{\left( {{\iota _i}(t) - {\bar{\iota }} } \right) }^2}} \right] } + \sum \limits _{i = 1}^N {\left[ {{{\left( {{\alpha _i}(t) - {\bar{\alpha }} } \right) }^2}} \right] } \\ + \sum \limits _{i = 1}^N {tr\left( {\frac{1}{{{\tau _i}}}\tilde{\Xi }_i^T{{{{\tilde{\Xi }}}}_i}} \right) } + \sum \limits _{i = 1}^N {tr\left( {\tilde{\bar{\Xi }}_i^T{{\tilde{\bar{\Xi }}}_i}} \right) } \end{array} \end{aligned}$$
(A1)
where \({{\bar{w}}}\), \({{\bar{\iota }} }\) and \({{\bar{\alpha }} }\) are positive scalars and P is a solution of (14). \({{r_i}}\) satisfy the Lamma 3.
Then the derivative of \(V({\varepsilon _i},{w_i},{\iota _i})\) along (11) can be obtained as
$$\begin{aligned} \begin{array}{l} \dot{V}({\varepsilon _i},{w_i},{\iota _i})\\ = 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^T{P^{ - 1}}A{\varepsilon _i}} + 2\sum \limits _{i = 1}^N {\left[ {\left( {{\alpha _i}(t) - {\bar{\alpha }} } \right) {{{{\dot{\alpha }}} }_i}(t)} \right] } \\ + 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^T{P^{ - 1}}\sum \limits _{j = 1}^N {{l_{ij}}\left( {{\theta _j} + {d_j}(t) - {{\tilde{\Xi }}}_j^T{h_j}\left( {{x_j}} \right) } \right) } } \\ + 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^T{P^{ - 1}}B\sum \limits _{j = 1}^N {{l_{ij}}\left( \begin{array}{l} \kappa {w_j}(t)\Gamma {\varepsilon _j} + {g_j}\left( {\Gamma {\varepsilon _j}} \right) \\ + {\iota _j}(t){\mathop \textrm{sgn}} \left( {\Gamma {\varepsilon _j}} \right) \end{array} \right) } } \\ + 2\sum \limits _{i = 1}^N {\left[ {\frac{{\left( {{w_i}(t) - {\bar{w}}} \right) }}{{{{{\bar{w}}}_i}}}{{\dot{w}}_i}(t)} \right] } + 2\sum \limits _{i = 1}^N {\left[ {\left( {{\iota _i}(t) - {\bar{\iota }} } \right) {{{{\dot{\iota }}} }_i}(t)} \right] } \\ + 2\sum \limits _{i = 1}^N {tr\left( {\frac{1}{{{\tau _i}}}{\tau _i}{{\tilde{\Xi }}}_i^T{{\dot{{{\tilde{\Xi }}}}}_i}} \right) } + 2\sum \limits _{i = 1}^N {tr\left( {\tilde{\bar{\Xi }}_i^T{{\dot{\tilde{\bar{\Xi }}}}_i}} \right) } \end{array} \end{aligned}$$
(A2)
Since \(\Gamma = - {B^T}{P^{ - 1}}\), \(\Lambda = {P^{ -1}}B{B^T}{P^{ - 1}}\), some mathematical manipulations give that
$$\begin{aligned} \begin{array}{l} 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^T{P^{-1}}B\sum \limits _{j = 1}^N {{l_{ij}}\kappa \Gamma {\varsigma _j}} } = - 2\kappa {\varepsilon ^T}\left( {RL \otimes {P^{ - 1}}B{B^T}{P^{- 1}}} \right) \varsigma \\ \le - 2\kappa {\lambda _{2\Psi } }\left( {{I_N} \otimes {P^{-1}}B{B^T}{P^{ - 1}}} \right) \varsigma \le - 2\sum \limits _{i=1}^N {{w_i}(t)\varepsilon _i^T\Lambda {\varepsilon _i}}, \end{array} \end{aligned}$$
(A3)
$$\begin{aligned} \begin{array}{l} 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^T{P^{ -1}}B\sum \limits _{j= 1}^N {{l_{ij}}{\upsilon _j}} } = 2{\varepsilon ^T}\left( {RL \otimes {P^{ - 1}}B} \right) \upsilon \\ \le 2{\lambda _{2\Psi }}{\varepsilon ^T}\left( {{I_N} \otimes {P^{-1}}B} \right) \upsilon = - 2{\lambda _{2\Psi }}\sum \limits _{i=1}^N {{\iota _i}(t){{\left\| {{B^T}{P^{ - 1}}{\varepsilon _i}} \right\| }_1}} \end{array} \end{aligned}$$
(A4)
$$\begin{aligned} \begin{array}{l} 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^T{P^{-1}}B\sum \limits _{j= 1}^N {{l_{ij}}{g_j}\left( {\Gamma {\varepsilon _j}} \right) } } = 2{\varepsilon ^T}\left( {RL \otimes {P^{ - 1}}B} \right) g\\ \le 2{\lambda _{2\Psi }}{\varepsilon ^T}\left( {{I_N} \otimes {P^{-1}}B} \right) g = - 2{\lambda _{2\Psi }}\sum \limits _{i = 1}^N {{\alpha _i}(t)\left\| {\Gamma {\varepsilon _i}} \right\| \left\| {{x_i}} \right\| }. \end{array} \end{aligned}$$
(A5)
where \(\kappa {\lambda _{2\Psi }} \ge 1\), \({{{\bar{\alpha }} }_i} \le {\lambda _{2\Psi }}\) and \({{{\bar{\iota }} }_i} \le {\lambda _{2\Psi }}\).
Substituting \({{\dot{w}}_i}(t),{{{\dot{\iota }}} _i}(t),{{{{\dot{\alpha }}} }_i}(t),{{\dot{{{\tilde{\Xi }}}}}_i},{{\dot{\tilde{\bar{\Xi }}}}_i}\), (A3), (A4), and (A5) into (A2) gives
$$\begin{aligned} \begin{array}{l} \dot{V}({\varepsilon _i},{w_i},{\iota _i})\\ = 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^TP_1^{ - 1}{\varepsilon _i}} - 2\sum \limits _{i = 1}^N {{\bar{w}}\varepsilon _i^T\Lambda {\varepsilon _i}} \\ - 2\sum \limits _{i = 1}^N {{\bar{\iota }} {{{\bar{\iota }} }_i}{{\left\| {{\Gamma }{\varepsilon _i}} \right\| }_1}} - 2\sum \limits _{i = 1}^N {{\bar{\alpha }} \left\| {\Gamma {\varepsilon _i}} \right\| \left\| {{x_i}} \right\| } \\ + 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^T{P^{ - 1}}\sum \limits _{j = 1}^N {{l_{ij}}\left( {{\theta _j} + {d_j}(t) - {{\tilde{\Xi }}}_j^T{h_j}\left( {{x_j}} \right) } \right) } } \\ + 2\sum \limits _{i = 1}^N {tr\left( {{{{{\tilde{\Xi }}}}^T}_i\left[ { - {\sigma _i}\left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) + {h_i}\left( {{x_i}} \right) \varepsilon _i^T{P^{ - 1}}} \right] } \right) } \\ + 2\sum \limits _{i = 1}^N {tr\left( {\tilde{\bar{\Xi }}_i^T{\sigma _i}\left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) } \right) } \end{array} \end{aligned}$$
(A6)
Due to
$$\begin{aligned} \begin{array}{l} 2\sum \limits _{i = 1}^N {tr\left( {{{{{\tilde{\Xi }}}}^T}_i\left[ { - {\sigma _i}\left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) + {h_i}\left( {{x_i}} \right) \varepsilon _i^T{P^{ - 1}}} \right] } \right) } \\ + 2\sum \limits _{i = 1}^N {tr\left( {\tilde{\bar{\Xi }}_i^T{\sigma _i}\left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) } \right) } \\ = 2\sum \limits _{i = 1}^N {tr\left( {{{{{\tilde{\Xi }}}}^T}_i{h_i}\left( {{x_i}} \right) \varepsilon _i^T{P^{ - 1}}} \right) } - 2\sum \limits _{i = 1}^N {tr\left( {{\sigma _i}\left( {{\hat{\Xi }}_i^T - {\Xi }_i^{*T}} \right) \left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) } \right) } \\ + 2\sum \limits _{i = 1}^N {tr\left( {{\sigma _i}\left( {\bar{\Xi }_i^T - {\Xi }_i^{*T}} \right) \left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) } \right) } \\ = 2\sum \limits _{i = 1}^N {tr\left( {{{{{\tilde{\Xi }}}}^T}_i{h_i}\left( {{x_i}} \right) \varepsilon _i^T{P^{ - 1}}} \right) } \\ - 2\sum \limits _{i = 1}^N {tr\left( {{\sigma _i}{{\left( {{{\hat{\Xi }}_i} - {{\bar{\Xi }}_i}} \right) }^T}\left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) } \right) } \end{array} \end{aligned}$$
(A7)
So the \(\dot{V}({\varepsilon _i},{w_i},{\iota _i})\) can be rewritten as
$$\begin{aligned} \begin{array}{l} \dot{V}({\varepsilon _i},{w_i},{\iota _i})\\ = 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^TP_1^{ - 1}{\varepsilon _i}} - 2\sum \limits _{i = 1}^N {{\bar{w}}\varepsilon _i^T{\Lambda }{\varepsilon _i}} \\ - 2\sum \limits _{i = 1}^N {{\bar{\iota }} {{{\bar{\iota }} }_i}{{\left\| {{\Gamma }{\varepsilon _i}} \right\| }_1}} - 2\sum \limits _{i = 1}^N {{\bar{\alpha }} \left\| {\Gamma {\varepsilon _i}} \right\| \left\| {{x_i}} \right\| } \\ + 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^T{P^{ - 1}}\sum \limits _{j = 1}^N {{l_{ij}}\left( {{\theta _j} + {d_j}(t) - {{\tilde{\Xi }}}_j^T{h_j}\left( {{x_j}} \right) } \right) } } \\ + 2\sum \limits _{i = 1}^N {tr\left( {{{{{\tilde{\Xi }}}}^T}_i{h_i}\left( {{x_i}} \right) \varepsilon _i^T{P^{ - 1}}} \right) } \\ - 2\sum \limits _{i = 1}^N {tr\left( {{\sigma _i}{{\left( {{{\hat{\Xi }}_i} - {{\bar{\Xi }}_i}} \right) }^T}\left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) } \right) } \end{array} \end{aligned}$$
(A8)
It is well known that \(tr(W_1W_2) = tr(W_2W_1)\) holds for suitable matrices \(W_1\), \(W_2\), one has
$$\begin{aligned} \begin{array}{l} - 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^T{P^{ - 1}}\sum \limits _{j = 1}^N {{l_{ij}}{{\tilde{\Xi }}}_j^T{h_j}\left( {{x_j}} \right) } } \\ = - 2\kappa {\varepsilon ^T}\left( {RL \otimes {P^{ - 1}}} \right) {{{{\tilde{\Xi }}}}^T}h\left( x \right) \\ \le - 2\kappa {\lambda _{2\Psi }}{\varepsilon ^T}\left( {{I_N} \otimes {P^{ - 1}}} \right) {{{{\tilde{\Xi }}}}^T}h\left( x \right) \\ \le - 2\sum \limits _{i = 1}^N {tr\left( {{{\tilde{\Xi }}^T}_i{h_i}\left( {{x_i}} \right) \varepsilon _i^T{P^{ - 1}}} \right) .} \end{array} \end{aligned}$$
(A9)
Furthermore, the Hölder inequality ensures
$$\begin{aligned} \begin{array}{l} 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^T{P^{ - 1}}\sum \limits _{j = 1}^N {{l_{ij}}({\theta _j} + {d_j}(t)}) } \\ = 2{\varepsilon ^T}\left( {\mathrm{{RL}} \otimes {P^{ - 1}}} \right) \left( {\theta + d} \right) \\ \le 2{\lambda _{M\Psi }}\left( {{\theta _M} + {d_M}} \right) {\left\| {{P^{ - 1}}\varepsilon } \right\| _1} \end{array} \end{aligned}$$
(A10)
where \({\lambda _{M\Psi }}\left( {{\theta _M} + {d_M}} \right) \le {\bar{\iota }} {{{\bar{\iota }} }_i}\). Then, it follows from (A9) and (A10) that:
$$\begin{aligned} \begin{array}{l} \dot{V}({\varepsilon _i},{w_i},{\iota _i})\\ \le 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^TP_1^{ - 1}{\varepsilon _i}} - 2\sum \limits _{i = 1}^N {{\bar{w}}\varepsilon _i^T\Lambda {\varepsilon _i}} \\ - 2\sum \limits _{i = 1}^N {{\bar{\iota }} {{{\bar{\iota }} }_i}{{\left\| {{P^{ - 1}}{\varepsilon _i}} \right\| }_1}} - 2\sum \limits _{i = 1}^N {{\bar{\alpha }} \left\| {\Gamma {\varepsilon _i}} \right\| \left\| {{x_i}} \right\| } \\ + 2{\lambda _{M\Psi }}\left( {{\theta _M} + {d_M}} \right) {\left\| {{P^{ - 1}}\varepsilon } \right\| _1}\\ - 2\sum \limits _{i = 1}^N {tr\left( {{{{{\tilde{\Xi }}}}^T}_i{h_i}\left( {{x_i}} \right) \varepsilon _i^T{P^{ - 1}}} \right) } \\ + 2\sum \limits _{i = 1}^N {tr\left( {{{{{\tilde{\Xi }}}}^T}_i{h_i}\left( {{x_i}} \right) \varepsilon _i^T{P^{ - 1}}} \right) } \\ - 2\sum \limits _{i = 1}^N {tr\left( {{\sigma _i}{{\left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) }^T}\left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) } \right) } \\ \le 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^TP_1^{ - 1}A{\varepsilon _i}} - 2\sum \limits _{i = 1}^N {{\bar{w}}\varepsilon _i^T\Lambda {\varepsilon _i}} \\ - 2\sum \limits _{i = 1}^N {tr\left( {{\sigma _i}{{\left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) }^T}\left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) } \right) } \\ - 2\sum \limits _{i = 1}^N {{\bar{\alpha }} \left\| {\Gamma {\varepsilon _i}} \right\| \left\| {{x_i}} \right\| } \\ \le 2\sum \limits _{i = 1}^N {{r_i}\varepsilon _i^TP_1^{ - 1}A{\varepsilon _i}} - 2\sum \limits _{i = 1}^N {{\bar{w}}\varepsilon _i^T\Lambda {\varepsilon _i}} \\ \le - 2{\vartheta _2}\sum \limits _{i = 1}^N {\varepsilon _i^TP_1^{ - 1}{\varepsilon _i}} = - 2{\vartheta _2}{\varepsilon ^T}\left( {{I_N} \otimes {P^{ - 1}}} \right) \varepsilon \end{array} \end{aligned}$$
(A11)
where \({\vartheta _2}\) is the positive constant. Due to \(P > 0\), the \(V({\varepsilon _i},{w_i},{\iota _i})\) is non-increasing and guarantees that the signals are bounded.
Since \(V({\varepsilon _i},{w_i},{\iota _i},t) \le V(0)\) and is non-increasing, it thus has a finite limit \({V^\infty }\) as \(t \rightarrow \infty\). By noting (25), it is easy to get that
$$\begin{aligned} \int _0^{ + \infty } {{\vartheta _2}} {\varepsilon ^T}\left( {{I_N} \otimes {P^{ - 1}}} \right) \varepsilon \mathrm{{d}}t \le V(0) - {V^\infty } \end{aligned}$$
(A12)
By utilizing Lemma 2, one has \(\mathop {\lim }\limits _{t \rightarrow \infty } {\varepsilon ^T}\left( {{I_N} \otimes {P^{ - 1}}} \right) \varepsilon = \mathbf{{0}}\). So one has \(\mathop {\lim }\limits _{t \rightarrow \infty } \left\| {\varepsilon (t)} \right\| = 0\).
Define \({\hat{\Xi }}_i^* = {{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}\), then it is easy to get that:
$$\begin{aligned} \begin{array}{l} {\dot{{\hat{\Xi }}}}_i^* = {{\dot{{\hat{\Xi }}}}_i} - {{\dot{\bar{\Xi }}}_i}\\ = \left[ { - {\sigma _i}{\tau _i}\left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) + {h_i}\left( {{x_i}} \right) \varepsilon _i^T{P^{ - 1}}} \right] - {\sigma _i}\left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) \\ = - {\sigma _i}\left( {{\tau _i} + 1} \right) {\hat{\Xi }}_i^* + {h_i}\left( {{x_i}} \right) \varepsilon _i^T{P^{ - 1}} \end{array} \end{aligned}$$
(A13)
Since \(\mathop {\lim }\nolimits _{t \rightarrow \infty } \left\| {\varepsilon (t)} \right\| = 0\), \({{h_i}\left( {{x_i}} \right) }\) is uniformly bounded, \({{\sigma _i}}\) and \({{\tau _i}}\) are given positive scalars, then \(\mathop {\lim }\nolimits _{t \rightarrow \infty } {\left\| {\hat{\Xi }_i^*} \right\| _F} = 0\), i.e, \(\mathop {\lim }\nolimits _{t \rightarrow \infty } \left( {{{{\hat{\Xi }}}_i} - {{\bar{\Xi }}_i}} \right) = \mathbf{{O}}\). From (7), we can conclude that \({w_i}(t),{\iota _i}(t)\) converge to a finite value. The proof is completed.
Appendix B Proof of Theorem 2
Define \({{\tilde{{\bar{\Omega }}} }_i} = {{{\bar{\Omega }} }_i} - \Omega _i^*\). Then choose the following Lyapunov function for systems (21):
$$\begin{aligned} \begin{array}{l} V({{{{\tilde{\varepsilon }}} }_i},{{{\tilde{w}}}_i}(t),{{{{\tilde{\iota }}} }_i}(t))\\ = \sum \limits _{i = 1}^N {{r_i}{{\tilde{\varepsilon }}} _i^TQ_1^{ - 1}{{{{\tilde{\varepsilon }}} }_i}} + \sum \limits _{i = 1}^N {\left[ {\frac{{{{\left( {{{{\tilde{w}}}_i}\left( t \right) - \bar{{\tilde{w}}}} \right) }^2}}}{{{{{\tilde{w}}}_i}}}} \right] } \\ + \sum \limits _{i = 1}^N {\delta _i^T{Q_2}{\delta _i}} + \sum \limits _{i = 1}^N {\left[ {{{\left( {{{{{\tilde{\iota }}} }_i}\left( t \right) - \bar{{{\tilde{\iota }}}} } \right) }^2}} \right] } \\ + \sum \limits _{i = 1}^N {tr\left( {\frac{1}{{{{{{\tilde{\tau }}} }_i}}}{{\tilde{\Omega }}} _i^T{{{{\tilde{\Omega }}} }_i}} \right) } + \sum \limits _{i = 1}^N {tr\left( {\tilde{{\bar{\Omega }}} _i^T{{\tilde{{\bar{\Omega }}} }_i}} \right) .} \end{array} \end{aligned}$$
(B14)
The derivative of \(V({{{{\tilde{\varepsilon }}} }_i},{{\tilde{w}}_i}(t),{{{{\tilde{\iota }}} }_i}(t))\) can be obtained as
$$\begin{aligned} \begin{array}{l} \dot{V}({{{{\tilde{\varepsilon }}} }_i},{{{\tilde{w}}}_i}(t),{{{{\tilde{\iota }}} }_i}(t))\\ = 2\sum \limits _{i = 1}^N {{r_i}{{\tilde{\varepsilon }}} _i^TQ_1^{ - 1}{{\dot{{{\tilde{\varepsilon }}}} }_i}} + 2\sum \limits _{i = 1}^N {\left[ {\frac{{\left( {{{{\tilde{w}}}_i}\left( t \right) - \bar{{\tilde{w}}}} \right) }}{{{{{\tilde{w}}}_i}}}{{\dot{{\tilde{w}}}}_i}\left( t \right) } \right] } \\ + 2\sum \limits _{i = 1}^N {\delta _i^T{Q_2}{{{{\dot{\delta }}} }_i}} + 2\sum \limits _{i = 1}^N {\left[ {\left( {{{{{\tilde{\iota }}} }_i}\left( t \right) - \bar{{{\tilde{\iota }}}} } \right) {{\dot{{{\tilde{\iota }}}} }_i}\left( t \right) } \right] } \\ + 2\sum \limits _{i = 1}^N {tr\left( {\frac{1}{{{{{{\tilde{\tau }}} }_i}}}{{\tilde{\Omega }}} _i^T{{\dot{{{\tilde{\Omega }}}} }_i}} \right) } + 2\sum \limits _{i = 1}^N {tr\left( {\tilde{{\bar{\Omega }}} _i^T{{\dot{\tilde{{\bar{\Omega }}}} }_i}} \right) } \end{array} \end{aligned}$$
(B15)
where \({\bar{{\tilde{w}}}}\), \({\bar{{{\tilde{\iota }}} }}\), \({{{{{\tilde{\tau }}} }_i}}\) are positive constants.
Substituting \({{{\dot{{\tilde{w}}}}_i}\left( t \right) }\), \({{{\dot{{{\tilde{\iota }}}} }_i}\left( t \right) }\), \({{{\dot{{{\tilde{\Omega }}}} }_i}}\) and \({{{\dot{\tilde{{\bar{\Omega }} }}}_i}}\) into (B15) gives
$$\begin{aligned} \begin{array}{l} \dot{V}({{{{\tilde{\varepsilon }}} }_i},{{{\tilde{w}}}_i}(t),{{{{\tilde{\iota }}} }_i}(t))\\ = 2\sum \limits _{i = 1}^N {{r_i}{{\tilde{\varepsilon }}} _i^TQ_1^{ - 1}A{{{{\tilde{\varepsilon }}} }_i}} \\ + 2\sum \limits _{i = 1}^N {{r_i}{{\tilde{\varepsilon }}} _i^TQ_1^{ - 1}\sum \limits _{j = 1}^N {{l_{ij}}\left( \begin{array}{l} B{{\tilde{\kappa }}} {{\tilde{\Gamma }}} {{{{\tilde{\varsigma }}} }_j} + B{{{\tilde{g}}}_j}({{\tilde{\Gamma }}} {{{{\tilde{\varepsilon }}} }_j})\\ + GC{\delta _j} + B{{{{\tilde{\upsilon }}} }_j} \end{array} \right) } } \\ + 2\sum \limits _{i = 1}^N {\delta _i^T{Q_2}\left( {(A + GC){\delta _i} + {{\tilde{\Omega }}} _i^T{h_i}\left( {{{{\hat{x}}}_i}} \right) - {\theta _i} - {d_i}} \right) } \\ + 2\sum \limits _{i = 1}^N {\left[ {\left( {{{{\tilde{w}}}_i}\left( t \right) - \bar{{\tilde{w}}}} \right) {{\tilde{\varepsilon }}} _i^T{{\tilde{\Lambda }}} {{{{\tilde{\varepsilon }}} }_i}} \right] } \\ + 2\sum \limits _{i = 1}^N {\left[ {\left( {{{{{\tilde{\iota }}} }_i}\left( t \right) - \bar{{{\tilde{\iota }}}} } \right) {{\bar{{{\tilde{\iota }}}} }_i}{{\left\| {{C^{ - 1}}{{{\bar{\delta }} }_i ^T}{Q_2}} \right\| }_1}} \right] } \\ + 2\sum \limits _{i = 1}^N {tr\left( {{{\tilde{\Omega }}} _i^T\left[ { - {{{{\tilde{\sigma }}} }_i}\left( {{{{\hat{\Omega }} }_i} - {{{\bar{\Omega }} }_i}} \right) - {h_i}\left( {{{{\hat{x}}}_i}} \right) {C^{ - 1}}{\bar{\delta }} _i^T{Q_2}} \right] } \right) } \\ + 2\sum \limits _{i = 1}^N {tr\left( {\tilde{{\bar{\Omega }}} _i^T{{{{\tilde{\sigma }}} }_i}\left( {{{{\hat{\Omega }} }_i} - {{{\bar{\Omega }} }_i}} \right) } \right) } \end{array} \end{aligned}$$
(B16)
Since \({{\tilde{\Gamma }}} = - {B^T}Q_1^{ - 1}\), \({{\tilde{\Lambda }}} = Q_1^{ - 1}B{B^T}Q_1^{ - 1}\), some mathematical manipulations give that
$$\begin{aligned} \begin{array}{l} 2\sum \limits _{i = 1}^N {{r_i}{{\tilde{\varepsilon }}} _i^TQ_{^1}^{ - 1}\sum \limits _{j = 1}^N {{l_{ij}}B{{\tilde{\kappa }}} {{\tilde{\Gamma }}} {{{{\tilde{\varsigma }}} }_j}} } = - 2{{\tilde{\kappa }}} {{{{\tilde{\varepsilon }}} }^T}\left( {RL \otimes Q_1^{ - 1}B{B^T}Q_1^{ - 1}} \right) {{\tilde{\varsigma }}} \\ \le - 2{{\tilde{\kappa }}} {\lambda _{2\Psi }}\left( {{I_N} \otimes {{\tilde{\Lambda }}} } \right) {{\tilde{\varsigma }}} \le - 2\sum \limits _{i = 1}^N {{{{\tilde{w}}}_i}(t){{\tilde{\varepsilon }}} _i^T{{\tilde{\Lambda }}} {{{{\tilde{\varepsilon }}} }_i},} \end{array} \end{aligned}$$
(B17)
$$\begin{aligned} \begin{array}{l} 2\sum \limits _{i = 1}^N {{r_i}{{\tilde{\varepsilon }}} _i^TQ_{^1}^{ - 1}\sum \limits _{j = 1}^N {{l_{ij}}B{{{{\tilde{\upsilon }}} }_j}} } = 2{{{{\tilde{\varepsilon }}} }^T}\left( {RL \otimes Q_{^1}^{ - 1}B} \right) {{\tilde{\upsilon }}} \\ \le {\lambda _{2\Psi }}{{{{\tilde{\varepsilon }}} }^T}\left( {{I_N} \otimes Q_{^1}^{ - 1}B} \right) {{\tilde{\upsilon }}} = - {\lambda _{2\Psi }}2\sum \limits _{i = 1}^N {{\iota _i}(t)\left\| {Q_{^1}^{ - 1}B{{{{\tilde{\varepsilon }}} }_i}} \right\| ,} \end{array} \end{aligned}$$
(B18)
$$\begin{aligned} \begin{array}{l} 2\sum \limits _{i = 1}^N {{r_i}{{\tilde{\varepsilon }}} _i^TQ_1^{ - 1}\sum \limits _{j = 1}^N {{l_{ij}}B{{{\tilde{g}}}_j}({{{{\tilde{\varepsilon }}} }_j})} } = 2{{{{\tilde{\varepsilon }}} }^T}\left( {RL \otimes Q_{^1}^{ - 1}B} \right) {\tilde{g}}\\ \le 2{\lambda _{2\Psi }}{{{{\tilde{\varepsilon }}} }^T}\left( {{I_N} \otimes Q_{^1}^{ - 1}B} \right) {\tilde{g}} \le 2{\lambda _{2\Psi }}\sum \limits _{i = 1}^N {{{\tilde{\varepsilon }}} _i^TQ_{^1}^{ - 1}B{{{\tilde{g}}}_i}({{{{\tilde{\varepsilon }}} }_i})} \\ \le - 2{\lambda _{2\Psi }}{{\tilde{\vartheta }}} \sum \limits _{i = 1}^N {{{{{\tilde{\iota }}} }_i}(t){{\left\| {{C^{ - 1}}\left( {{{{\hat{y}}}_i} - {y_i}} \right) ^T{Q_2}} \right\| }_1},} \end{array} \end{aligned}$$
(B19)
$$\begin{aligned} \begin{array}{l} 2\sum \limits _{i = 1}^N {\delta _i^T{Q_2}\left( {{\theta _i} + {d_i}} \right) } \mathrm{{ = }}2{\delta ^T}({I_N} \otimes {Q_2})\left( {\theta + d} \right) \\ \le 2 \sum \limits _{i = 1}^N {\left( {{\theta _M} + {d_M}} \right) {\left\| {{\delta _i ^T}{Q_2}} \right\| _1}} \le 2\sum \limits _{i = 1}^N {\bar{{{\tilde{\iota }}}} {{\bar{{{\tilde{\iota }}} }}_i}{{\left\| {{C^{ - 1}}{{{\bar{\delta }} }_i ^T}{Q_2}} \right\| }_1}} \end{array} \end{aligned}$$
(B20)
where \({{\tilde{\kappa }}} {\lambda _{2\Psi }} \ge 1\), \({\lambda _{2\Psi }}{{\tilde{\vartheta }}} \ge {\bar{{{\tilde{\iota }}}} }_i\) and \(\bar{\tilde{\iota }} {{\bar{{{\tilde{\iota }}}} }_i} \ge \left( {{\theta _M} + {d_M}} \right)\).
Similar to (A7), we can derive
$$\begin{aligned} \begin{array}{l} 2\sum \limits _{i = 1}^N {tr\left( {{{\tilde{\Omega }}} _i^T\left[ { - {{{{\tilde{\sigma }}} }_i}\left( {{{{\hat{\Omega }} }_i} - {{{\bar{\Omega }} }_i}} \right) - {h_i}\left( {{{{\hat{x}}}_i}} \right) {C^{ - 1}}{\bar{\delta }} _i^T{Q_2}} \right] } \right) } \\ + 2\sum \limits _{i = 1}^N {tr\left( {\tilde{\bar{\Omega }} _i^T{{{{\tilde{\sigma }}} }_i}\left( {{{{\hat{\Omega }} }_i} - {{{\bar{\Omega }} }_i}} \right) } \right) } \\ = - 2\sum \limits _{i = 1}^N {tr\left( {{{\tilde{\Omega }}} _i^T{h_i}\left( {{{{\hat{x}}}_i}} \right) {C^{ - 1}}{\bar{\delta }} _i^T{Q_2}} \right) } \\ - 2\sum \limits _{i = 1}^N {tr\left( {{{{{\tilde{\sigma }}} }_i}{{\left( {{{{\hat{\Omega }} }_i} - {{{\bar{\Omega }} }_i}} \right) }^T}\left( {{{{\hat{\Omega }} }_i} - {{{\bar{\Omega }} }_i}} \right) } \right) } \\ \le - 2\sum \limits _{i = 1}^N {tr\left( {{{\tilde{\Omega }}} _i^T{h_i}\left( {{{{\hat{x}}}_i}} \right) {C^{ - 1}}{\bar{\delta }} _i^T{Q_2}} \right) .} \end{array} \end{aligned}$$
(B21)
It is well known that \(tr(W_1W_2) = tr(W_2W_1)\) holds for suitable matrices \(W_1\), \(W_2\), one has
$$\begin{aligned} 2\sum \limits _{i = 1}^N {\delta _i^T{Q_2}{{\tilde{\Omega }}} _i^T{h_i}\left( {{{{\hat{x}}}_i}} \right) } - 2\sum \limits _{i = 1}^N {tr\left( {{{\tilde{\Omega }}} _i^T{h_i}\left( {{{{\hat{x}}}_i}} \right) {C^{ - 1}}{\bar{\delta }} _i^T{Q_2}} \right) } \le 0. \end{aligned}$$
(B22)
Substituting (B17)-(B22) into (B16), the \(\dot{V}({{\tilde{\varepsilon }}_i},{{{\tilde{w}}}_i}(t),{{{{\tilde{\iota }}} }_i}(t))\) can be rewritten as
$$\begin{aligned} \begin{array}{l} \dot{V}({{{{\tilde{\varepsilon }}} }_i},{{{\tilde{w}}}_i}(t),{{{{\tilde{\iota }}} }_i}(t))\\ \le 2\sum \limits _{i = 1}^N {{r_i}{{\tilde{\varepsilon }}} _i^TQ_1^{ - 1}A{{{{\tilde{\varepsilon }}} }_i}} + 2\sum \limits _{i = 1}^N {{r_i}{{\tilde{\varepsilon }}} _i^TQ_1^{ - 1}\sum \limits _{j = 1}^N {{l_{ij}}GC{\delta _j}} } \\ + 2\sum \limits _{i = 1}^N {\delta _i^T{Q_2}(A + GC){\delta _i}} - 2\sum \limits _{i = 1}^N {\bar{{\tilde{w}}}{{\tilde{\varepsilon }}} _i^T\tilde{\Lambda }{{{{\tilde{\varepsilon }}} }_i}} \end{array} \end{aligned}$$
(B23)
Due to
$$\begin{aligned} \begin{array}{l} 2\sum \limits _{i = 1}^N {{r_i}{{\tilde{\varepsilon }}} _i^TQ_1^{ - 1}\sum \limits _{j = 1}^N {{l_{ij}}GC{\delta _j}} } \\ = 2{{{{\tilde{\varepsilon }}} }^T}(R \otimes Q_1^{ - 1})(L \otimes GC){{\tilde{\delta }}} \\ \le 2{{{{\tilde{\varepsilon }}} }^T}({I_N} \otimes Q_1^{ - 1})({I_N} \otimes GC){{\tilde{\delta }}} \\ \le 2{\lambda _{M\Psi }}\sum \limits _{i = 1}^N {{{\tilde{\varepsilon }}} _i^TQ_1^{ - 1}GC{\delta _i}} \end{array} \end{aligned}$$
(B24)
So the \(\dot{V}({{{{\tilde{\varepsilon }}} }_i},{{{\tilde{w}}}_i}(t),{{\tilde{\iota }}_i}(t))\) can be rewritten as
$$\begin{aligned} \begin{array}{l} \dot{V}({{{{\tilde{\varepsilon }}} }_i},{{{\tilde{w}}}_i}(t),{{{{\tilde{\iota }}} }_i}(t))\\ \le 2\sum \limits _{i = 1}^N {{r_i}{{\tilde{\varepsilon }}} _i^TQ_1^{ - 1}A{{{{\tilde{\varepsilon }}} }_i}} - 2\sum \limits _{i = 1}^N {\bar{{\tilde{w}}}{{\tilde{\varepsilon }}} _i^T{{\tilde{\Lambda }}} {{\tilde{\varepsilon }}}_i } \\ + 2{\lambda _{M\Psi }}\sum \limits _{i = 1}^N {{{\tilde{\varepsilon }}} _i^TQ_1^{ - 1}GC{\delta _i}} \\ + 2\sum \limits _{i = 1}^N {\delta _i^T{Q_2}(A + GC){\delta _i}} \end{array} \end{aligned}$$
(B25)
Define \(\xi = {\left[ {{{{{\tilde{\varepsilon }}} }^T},{\delta ^T}} \right] ^T}\), the \(\dot{V}({{{{\tilde{\varepsilon }}} }_i},{{\tilde{w}}_i}(t),{{{{\tilde{\iota }}} }_i}(t))\) can be denoted as
$$\begin{aligned} \begin{array}{l} \dot{V}({{{{\tilde{\varepsilon }}} }_i},{{{\tilde{w}}}_i}(t),{{{{\tilde{\iota }}} }_i}(t))\\ \le 2{{{{\tilde{\varepsilon }}} }^T}\left( {{I_N} \otimes \left( {{r_{\max }}Q_1^{ - 1}A - \bar{{\tilde{w}}}{{\tilde{\Lambda }}} } \right) } \right) {{\tilde{\varepsilon }}} \\ + 2{\lambda _{M\Psi }}{{{{\tilde{\varepsilon }}} }^T}\left( {{I_N} \otimes Q_1^{ - 1}GC} \right) \delta \\ + 2{\delta ^T}\left( {{I_N} \otimes {Q_2}(A + GC)} \right) \delta \\ \le {\xi ^T}\Phi \xi \end{array} \end{aligned}$$
(B26)
where
$$\begin{aligned} \Phi \mathrm{{ = }}{I_N} \otimes \left[ {\begin{array}{*{20}{c}} {{\Phi _{11}}}&{}{{\Phi _{12}}}\\ {{\Phi _{21}}}&{}{{\Phi _{22}}} \end{array}} \right] \end{aligned}$$
(B27)
and
$$\begin{aligned}\begin{array}{l} {\Phi _{11}}\mathrm{{ = }}{}Q_1^{ - 1}A + {A^T}Q_1^{ - 1} - {\varpi _1} {{\tilde{\Lambda }}},{\Phi _{12}} = \sqrt{{{{\bar{\vartheta }} }_2}{{\left( {Q_1^{ - 1}GC} \right) }^T}}, {\varpi _1} {\le {\bar{{\tilde{w}}}}}/{r_{\max }},\\ {\Phi _{21}} = \sqrt{{{{\bar{\vartheta }} }_2}{{\left( {Q_1^{ - 1}GC} \right) }^T}},{\Phi _{22}} = {Q_2}A + {A^T}{Q_2} - {C^T}C. \end{array}\end{aligned}$$
According to (25) and (26), it is not difficult to obtain \({\Phi _{11}} < - {\varpi _2}I\) and \({\Phi _{22}} < - {\varpi _3}I\). According to Schur complement, it gets that \({\Phi _{11}} < 0\), \({\Phi _{11}} - {\Phi _{12}}\Phi _{22}^{ - 1}{\Phi _{21}} < 0\) and \(\Phi < 0\) are equivalent. Since the (B26), \(\dot{V}({{\tilde{\varepsilon }}_i},{{{\tilde{w}}}_i}(t),{{{{\tilde{\iota }}} }_i}(t)) \le {\xi ^T}\Phi \xi \le 0\). Furthermore, \(V({{{{\tilde{\varepsilon }}} }_i},{{{\tilde{w}}}_i}(t),{{{{\tilde{\iota }}} }_i}(t))\) is positive definite and strictly bounded, using the LaSalle-Yoshizawa theorem, we can obtain \(\mathop {\lim }\limits _{t \rightarrow \infty } \dot{V}({\tilde{\varepsilon }_i},{{\tilde{w}}_i}(t),{{{\tilde{\iota }}} _i}(t)) = 0\). Furthermore, we can obtain \(\mathop {\lim }\limits _{t \rightarrow \infty } \xi = 0\). Thus we can conclude that MAS achieves consensus. This completes the proof.
