Adaptive NN-based distributed consensus control for nonlinear multi-agent systems under direct graphs

Abstract

This paper is mainly concerned with the leaderless consensus problem for nonlinear multi-agent systems (MAS) with unknown mismatched nonlinear dynamics and external disturbances. First, a novel adaptive controller is designed to achieve bounded consensus with a linear feedback term, the RBF neural network (RBFNN) adaptive approximation term, a discontinuous feedback term, and a state constraint term. Furthermore, on this basis, an observer-based robust uniform control scheme against disturbance is proposed to achieve the leaderless consensus of MAS with unmeasurable states under the directed graph. Compared with the previous works, the proposed controller is less conservative. Finally, two simulation-based examples are provided to verify the effectiveness of the proposed control scheme.

Appendices

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.