#### 期刊菜单

Event-Triggered Bipartite Consensus Tracking Control for Nonlinear Multi-Agent Systems
DOI: 10.12677/DSC.2022.113016, PDF, HTML, XML, 下载: 66  浏览: 128  科研立项经费支持

Abstract: This paper investigates the bipartite consensus tracking control problem of nonlinear multi-agent systems under the signed directed graph. The centralized and distributed event-triggered control protocols are presented respectively. By using matrix theory algebraic graph theory and Lyapunov stability theory, it is proved that the proposed control protocol can solve the bipartite consensus tracking control problem, and there is no Zeno behavior. The event-triggered strategy can effectively reduce the update times of control input, and achieve the purpose of saving network bandwidth and reducing communication cost. Finally, the simulation results verify the validity of the theoretical results.

1. 引言

2. 预备知识和问题描述

2.1. 图论

${\Delta }_{i}=\underset{j\in {N}_{i}}{\sum }|{a}_{ij}|$。符号图G的Laplacian矩阵L定义为 $L=\Delta -Α$。若G中所有边的权值全为正值，则称之为正符号图。

1)若 $\forall {v}_{i},{v}_{j}\in {V}_{l}\left(l\in \left\{1,2\right\}\right)$，则 ${a}_{ij}\ge 0$

2)若 $\forall {v}_{i}\in {V}_{l},{v}_{j}\in {V}_{q},l\ne q\left(l,q\in \left\{1,2\right\}\right)$，则 ${a}_{ij}\le 0$

2.2. 问题描述

${\stackrel{˙}{x}}_{i}\left(t\right)=f\left({x}_{i}\left(t\right),t\right)+{u}_{i}\left(t\right),\text{\hspace{0.17em}}i=1,2,\cdots ,n$ (1)

${\stackrel{˙}{x}}_{n+1}\left(t\right)=f\left({x}_{n+1}\left(t\right),t\right)$ (2)

$\underset{t\to +\infty }{\mathrm{lim}}|{x}_{i}\left(t\right)-{x}_{n+1}\left(t\right)|=0,\forall i\in {V}_{1}$

$\underset{t\to +\infty }{\mathrm{lim}}|{x}_{i}\left(t\right)+{x}_{n+1}\left(t\right)|=0,\forall i\in {V}_{2}$

3. 主要结果

3.1. 集中式事件触发二分一致性

${u}_{i}\left(t\right)=-\alpha {y}_{i}\left({t}_{k}\right)$ (3)

${y}_{i}\left(t\right)=\underset{j=1}{\overset{n+1}{\sum }}|{a}_{ij}|\left({x}_{i}\left(t\right)-sign\left({a}_{ij}\right){x}_{j}\left(t\right)\right),\text{\hspace{0.17em}}i=1,2,\cdots ,n$ (4)

${e}_{i}\left(t\right)=\alpha {y}_{i}\left({t}_{k}\right)-\alpha {y}_{i}\left(t\right),i=1,2,\cdots ,n$ (5)

$L=\left[\begin{array}{cc}{L}_{1}& {L}_{2}\\ 0& 0\end{array}\right]$

$y\left(t\right)={\left({y}_{1}\left(t\right),{y}_{2}\left(t\right),\cdots ,{y}_{n}\left(t\right)\right)}^{\text{T}}$$f\left(x\left(t\right),t\right)={\left(f\left({x}_{1}\left(t\right),t\right),f\left({x}_{2}\left(t\right),t\right),\cdots ,f\left({x}_{n}\left(t\right),t\right)\right)}^{\text{T}}$

$y\left(t\right)={L}_{1}x\left(t\right)+{L}_{2}{x}_{n+1}\left(t\right)$ (6)

$\stackrel{˜}{y}\left(t\right)={D}_{1}y\left(t\right)$$\stackrel{˜}{e}\left(t\right)={D}_{1}e\left(t\right)$

$\stackrel{˜}{y}\left(t\right)={D}_{1}{L}_{1}x\left(t\right)+{D}_{1}{L}_{2}{x}_{n+1}\left(t\right)$ (7)

$|f\left({x}_{1}\left(t\right),t\right)±f\left({x}_{2}\left(t\right),t\right)|\le l|{x}_{1}\left(t\right)±{x}_{2}\left(t\right)|$

$P{D}_{1}{L}_{1}{D}_{1}+{D}_{1}{L}_{1}^{\text{T}}{D}_{1}P>0$.

$f\left(e\right)=‖\stackrel{˜}{e}\left(t\right)‖-\frac{\sigma \left(\alpha {\lambda }_{0}-2l‖P‖\right)}{2‖P‖\cdot ‖{L}_{1}‖}\cdot ‖\stackrel{˜}{y}\left(t\right)‖$ (8)

$V\left(t\right)=\stackrel{˜}{y}{\left(t\right)}^{\text{T}}P\stackrel{˜}{y}\left( t \right)$

$\stackrel{˙}{V}\left(t\right)=2{\stackrel{˜}{y}}^{\text{T}}P\stackrel{˙}{\stackrel{˜}{y}}$

$\begin{array}{c}\stackrel{˜}{y}\left(t\right)={D}_{1}{L}_{1}{D}_{1}{D}_{1}^{-1}x\left(t\right)+{D}_{1}{L}_{2}{D}_{2}{D}_{2}^{-1}{x}_{n+1}\left(t\right)\\ ={D}_{1}{L}_{1}{D}_{1}\left[{D}_{1}x\left(t\right)+{\left({D}_{1}{L}_{1}{D}_{1}\right)}^{-1}\left({D}_{1}{L}_{2}{D}_{2}\right){D}_{2}^{-1}{x}_{n+1}\left(t\right)\right]\\ ={D}_{1}{L}_{1}{D}_{1}\left[{D}_{1}x\left(t\right)-{1}_{n}{D}_{2}^{-1}{x}_{n+1}\left(t\right)\right]\end{array}$ (9)

$\begin{array}{c}\stackrel{˙}{\stackrel{˜}{y}}\left(t\right)={D}_{1}{L}_{1}\stackrel{˙}{x}\left(t\right)+{D}_{1}{L}_{1}{\stackrel{˙}{x}}_{n+1}\left(t\right)\\ ={D}_{1}{L}_{1}\left[f\left(x\left(t\right),t\right)+u\left(t\right)\right]+{D}_{1}{L}_{2}f\left({x}_{n+1}\left(t\right)\right)\\ ={D}_{1}{L}_{1}\left[-\alpha y\left(t\right)-e\left(t\right)+f\left(x\left(t\right),t\right)\right]+{D}_{1}{L}_{2}{D}_{2}{D}_{2}^{-1}f\left({x}_{n+1}\left(t\right),t\right)\\ =-\alpha {D}_{1}{L}_{1}{D}_{1}\stackrel{˜}{y}\left(t\right)-{D}_{1}{L}_{1}{D}_{1}{D}_{1}e\left(t\right)+{D}_{1}{L}_{1}{D}_{1}{D}_{1}f\left(x\left(t\right),t\right)+{D}_{1}{L}_{2}{D}_{2}{D}_{2}^{-1}f\left({x}_{n+1}\left(t\right),t\right)\\ =-\alpha {D}_{1}{L}_{1}{D}_{1}\stackrel{˜}{y}\left(t\right)-{D}_{1}{L}_{1}{D}_{1}{D}_{1}e\left(t\right)+{D}_{1}{L}_{1}{D}_{1}\left[{D}_{1}f\left(x\left(t\right),t\right)-{1}_{n}{D}_{2}^{-1}f\left({x}_{n+1},t\right)\right]\end{array}$ (10)

$\begin{array}{c}\stackrel{˙}{V}\left(t\right)=2{\stackrel{˜}{y}}^{\text{T}}P\left[-\alpha {D}_{1}{L}_{1}{D}_{1}\stackrel{˜}{y}\left(t\right)-{D}_{1}{L}_{1}{D}_{1}{D}_{1}e\left(t\right)+{D}_{1}{L}_{1}{D}_{1}{D}_{1}f\left(x\left(t\right),t\right)-{1}_{n}{D}_{2}^{-1}f\left({x}_{n+1}\left(t\right),t\right)\right]\\ =-2\alpha {\stackrel{˜}{y}}^{\text{T}}P{D}_{1}{L}_{1}{D}_{1}\stackrel{˜}{y}\left(t\right)-2{\stackrel{˜}{y}}^{\text{T}}P{D}_{1}{L}_{1}{D}_{1}{D}_{1}e\left(t\right)+2{\stackrel{˜}{y}}^{\text{T}}P{D}_{1}{L}_{1}{D}_{1}\left[{D}_{1}f\left(x\left(t\right),t\right)-{1}_{n}{D}_{2}^{-1}f\left({x}_{n+1}\left(t\right),t\right)\right]\\ =-\alpha {\stackrel{˜}{y}}^{\text{T}}\left(P{D}_{1}{L}_{1}{D}_{1}+{D}_{1}{L}_{1}^{\text{T}}{D}_{1}P\right)\stackrel{˜}{y}\left(t\right)-2{\stackrel{˜}{y}}^{\text{T}}P{D}_{1}{L}_{1}{D}_{1}{D}_{1}e\left(t\right)+2{\stackrel{˜}{y}}^{\text{T}}P{D}_{1}{L}_{1}{D}_{1}{D}_{1}e\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{\stackrel{˜}{y}}^{\text{T}}P{D}_{1}{L}_{1}{D}_{1}\left[{D}_{1}f\left(x\left(t\right),t\right)-{1}_{n}{D}_{2}^{-1}f\left({x}_{n+1}\left(t\right),t\right)\right]\end{array}$

$\begin{array}{c}\stackrel{˙}{V}\left(t\right)\le -\alpha {\lambda }_{0}{‖\stackrel{˜}{y}\left(t\right)‖}^{2}+2‖\stackrel{˜}{y}\left(t\right)‖\cdot ‖P‖\cdot ‖{L}_{1}‖\cdot ‖\stackrel{˜}{e}\left(t\right)‖+2‖\stackrel{˜}{y}\left(t\right)‖\cdot ‖P‖\cdot l\cdot ‖\stackrel{˜}{y}\left(t\right)‖\\ =-\alpha {\lambda }_{0}{‖\stackrel{˜}{y}\left(t\right)‖}^{2}+2‖\stackrel{˜}{y}\left(t\right)‖\cdot ‖P‖\cdot ‖{L}_{1}‖\cdot ‖\stackrel{˜}{e}\left(t\right)‖+2l{‖\stackrel{˜}{y}\left(t\right)‖}^{2}\cdot ‖P‖\\ =-\left(\alpha {\lambda }_{0}-2l‖P‖\right){‖\stackrel{˜}{y}\left(t\right)‖}^{2}+2‖P‖\cdot ‖{L}_{1}‖\cdot ‖\stackrel{˜}{e}\left(t\right)‖\cdot ‖\stackrel{˜}{y}\left(t\right)‖\end{array}$ (11)

$‖\stackrel{˜}{e}\left(t\right)‖\le \frac{\sigma \left(\alpha {\lambda }_{0}-2l‖P‖\right)}{2‖P‖\cdot ‖{L}_{1}‖}\cdot ‖\stackrel{˜}{y}\left(t\right)‖$

$\stackrel{˙}{V}\le \left(\sigma -1\right)\left(\alpha {\lambda }_{0}-2l‖P‖\right){‖\stackrel{˜}{y}\left(t\right)‖}^{2}$

$\tau =\frac{2\alpha \sigma l‖P‖-\sigma {a}^{2}{\lambda }_{0}}{\left(2\sigma l‖P‖-\sigma \alpha {\lambda }_{0}-2\alpha ‖P‖\cdot ‖{L}_{1}‖\right)\left(\alpha l+{\alpha }^{2}‖{L}_{1}‖\right)}$

$\frac{‖\stackrel{˜}{e}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}$ 求导可得：

$\begin{array}{c}\frac{\text{d}‖\stackrel{˜}{e}\left(t\right)‖}{\text{d}t‖\stackrel{˜}{y}\left(t\right)‖}=\frac{\text{d}{\left({\stackrel{˜}{e}}^{\text{T}}\left(t\right)\stackrel{˜}{e}\left(t\right)\right)}^{1/2}}{\text{d}t{\left({\stackrel{˜}{y}}^{\text{T}}\left(t\right)\stackrel{˜}{y}\left(t\right)\right)}^{1/2}}\\ =\frac{{\stackrel{˜}{e}}^{\text{T}}\left(t\right)\stackrel{˙}{\stackrel{˜}{e}}\left(t\right)\frac{1}{‖\stackrel{˜}{e}\left(t\right)‖}‖\stackrel{˜}{y}\left(t\right)‖-{\stackrel{˜}{y}}^{\text{T}}\left(t\right)\stackrel{˙}{\stackrel{˜}{y}}\left(t\right)\frac{1}{‖\stackrel{˙}{\stackrel{˜}{y}}\left(t\right)‖}‖\stackrel{˜}{e}\left(t\right)‖}{{‖\stackrel{˜}{y}\left(t\right)‖}^{2}}\\ =\frac{{\stackrel{˜}{e}}^{\text{T}}\left(t\right)\stackrel{˙}{\stackrel{˜}{e}}\left(t\right)}{‖\stackrel{˜}{e}\left(t\right)‖\cdot ‖\stackrel{˜}{y}\left(t\right)‖}-\frac{‖\stackrel{˜}{e}\left(t\right)‖{\stackrel{˜}{y}}^{\text{T}}\left(t\right)\stackrel{˙}{\stackrel{˜}{y}}\left(t\right)}{{‖\stackrel{˜}{y}\left(t\right)‖}^{3}}=\frac{-\alpha {\stackrel{˜}{e}}^{\text{T}}\left(t\right)\stackrel{˙}{\stackrel{˜}{y}}\left(t\right)}{‖\stackrel{˜}{e}\left(t\right)‖\cdot ‖\stackrel{˜}{y}\left(t\right)‖}-\frac{‖\stackrel{˜}{e}\left(t\right)‖{\stackrel{˜}{y}}^{\text{T}}\left(t\right)\stackrel{˙}{\stackrel{˜}{y}}\left(t\right)}{{‖\stackrel{˜}{y}\left(t\right)‖}^{3}}\\ \le \frac{\alpha ‖\stackrel{˙}{\stackrel{˜}{y}}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}+\frac{‖\stackrel{˜}{e}\left(t\right)‖\cdot ‖\stackrel{˙}{\stackrel{˜}{y}}\left(t\right)‖}{{‖\stackrel{˜}{y}\left(t\right)‖}^{2}}=\left(\alpha +\frac{‖\stackrel{˜}{e}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}\right)\frac{‖\stackrel{˙}{\stackrel{˜}{y}}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}\end{array}$

$\begin{array}{c}\frac{\text{d}‖\stackrel{˜}{e}\left(t\right)‖}{\text{d}t‖\stackrel{˜}{y}\left(t\right)‖}=\left(\alpha +\frac{‖\stackrel{˜}{e}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}\right)\frac{‖\stackrel{˙}{\stackrel{˜}{y}}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}\\ \le \left(\alpha +\frac{‖\stackrel{˜}{e}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}\right)\frac{\alpha ‖{L}_{1}‖\cdot ‖\stackrel{˜}{y}\left(t\right)‖+‖{L}_{1}‖\cdot ‖\stackrel{˜}{e}\left(t\right)‖+l‖\stackrel{˜}{y}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}\\ =\left(\alpha +\frac{‖\stackrel{˜}{e}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}\right)\left(\alpha ‖{L}_{1}‖+‖{L}_{1}‖\frac{‖\stackrel{˜}{e}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}+l\right)\\ =l\left(\alpha +\frac{‖\stackrel{˜}{e}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}\right)+‖{L}_{1}‖{\left(\frac{‖\stackrel{˜}{e}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}+\alpha \right)}^{2}\\ =\alpha l\left(1+\frac{‖\stackrel{˜}{e}\left(t\right)‖}{\alpha ‖\stackrel{˜}{y}\left(t\right)‖}\right)+{\alpha }^{2}‖{L}_{1}‖{\left(\frac{‖\stackrel{˜}{e}\left(t\right)‖}{\alpha ‖\stackrel{˜}{y}\left(t\right)‖}+1\right)}^{2}\end{array}$ (12)

$\frac{\text{d}‖\stackrel{˜}{e}\left(t\right)‖}{\text{d}t‖\stackrel{˜}{y}\left(t\right)‖}\le \left(\alpha l+{\alpha }^{2}‖{L}_{1}‖\right){\left(1+\frac{‖\stackrel{˜}{e}\left(t\right)‖}{\alpha ‖\stackrel{˜}{y}\left(t\right)‖}\right)}^{2}$ (13)

$z=\frac{‖\stackrel{˜}{e}\left(t\right)‖}{‖\stackrel{˜}{y}\left(t\right)‖}$，则有：

$\stackrel{˙}{z}\le \left(\alpha l+{\alpha }^{2}‖{L}_{1}‖\right){\left(1+\frac{1}{a}z\right)}^{2}$ (14)

$\varphi \left(\tau ,0\right)=\frac{\sigma \left(\alpha {\lambda }_{0}-2l‖P‖\right)}{2‖P‖\cdot ‖{L}_{1}‖}$

$\tau =\frac{2\alpha \sigma l‖P‖-\sigma {a}^{2}{\lambda }_{0}}{\left(2\sigma l‖P‖-\sigma \alpha {\lambda }_{0}-2\alpha ‖P‖\cdot ‖{L}_{1}‖\right)\left(\alpha l+{\alpha }^{2}‖{L}_{1}‖\right)}$

3.2. 分布式事件触发二分一致性

${u}_{i}\left(t\right)=-\alpha {y}_{i}\left({t}_{k}^{i}\right)$ (15)

${y}_{i}\left(t\right)=\underset{j=1}{\overset{n+1}{\sum }}|{a}_{ij}|\left({x}_{i}\left({t}_{k}^{i}\right)-sign\left({a}_{ij}\right){x}_{j}\left({t}_{{k}^{\prime }}^{j}\right)\right),i=1,2,\cdots ,n$ (16)

${e}_{i}\left(t\right)=\alpha {y}_{i}\left({t}_{k}^{i}\right)-\alpha {y}_{i}\left(t\right),i=1,2,\cdots ,n$ (17)

${u}_{i}\left(t\right)=-\alpha {y}_{i}\left(t\right)-{e}_{i}\left(t\right)$ (18)

$\stackrel{˜}{y}\left(t\right)$ 可进一步表示为：

$\stackrel{˜}{y}\left(t\right)={D}_{1}{L}_{1}x\left(t\right)+{D}_{1}{L}_{2}{x}_{n+1}\left( t \right)$

$f\left(e\right)={\stackrel{˜}{e}}_{i}^{2}-\frac{\theta {\sigma }_{i}K}{2{\Delta }_{i}{\xi }_{\mathrm{max}}}{\stackrel{˜}{y}}_{i}^{2}$ (19)

${\lambda }_{0}={\lambda }_{\mathrm{min}}\left(P{D}_{1}{L}_{1}{D}_{1}+{D}_{1}{L}_{1}^{\text{T}}{D}_{1}P\right)$${\xi }_{\mathrm{max}}$ 为引理4中正定矩阵P的最大元素。

$V\left(t\right)=\stackrel{˜}{y}{\left(t\right)}^{\text{T}}P\stackrel{˜}{y}\left( t \right)$

$\begin{array}{c}\stackrel{˙}{V}\left(t\right)=2{\stackrel{˜}{y}}^{\text{T}}P\stackrel{˙}{\stackrel{˜}{y}}\\ =2{\stackrel{˜}{y}}^{\text{T}}P\left[-\alpha {D}_{1}{L}_{1}{D}_{1}\stackrel{˜}{y}\left(t\right)-{D}_{1}{L}_{1}{D}_{1}{D}_{1}e\left(t\right)\right]+2{\stackrel{˜}{y}}^{\text{T}}P\left[{D}_{1}{L}_{1}{D}_{1}{D}_{1}f\left(x\left(t\right),t\right)-{1}_{n}{D}_{2}^{-1}f\left({x}_{n+1}\left(t\right),t\right)\right]\\ \le -\alpha {\lambda }_{0}{‖\stackrel{˜}{y}\left(t\right)‖}^{2}+2l‖P‖\cdot {‖\stackrel{˜}{y}\left(t\right)‖}^{2}+2{\stackrel{˜}{y}}^{\text{T}}P{L}_{1}\stackrel{˜}{e}\left(t\right)\\ =-\left(\alpha {\lambda }_{0}-2l‖P‖\right){‖\stackrel{˜}{y}\left(t\right)‖}^{2}+2{\stackrel{˜}{y}}^{\text{T}}P{L}_{1}\stackrel{˜}{e}\left(t\right)\\ \le -\left(\alpha {\lambda }_{0}-2l‖P‖\right)\underset{i=1}{\sum }{\stackrel{˜}{y}}_{i}^{2}+2\underset{i=1}{\sum }\underset{j\in {N}_{i}}{\sum }{\xi }_{i}{a}_{ij}{\stackrel{˜}{y}}_{i}\left({\stackrel{˜}{e}}_{i}\left(t\right)-{\stackrel{˜}{e}}_{j}\left( t \right)\right)\end{array}$

${\stackrel{˜}{y}}_{i}{\stackrel{˜}{e}}_{i}-{\stackrel{˜}{y}}_{i}{\stackrel{˜}{e}}_{j}\le \theta {\stackrel{˜}{y}}_{i}^{2}+\frac{1}{2\theta }{\stackrel{˜}{e}}_{i}^{2}+\frac{1}{2\theta }{\stackrel{˜}{e}}_{j}^{2}$

$\begin{array}{c}\stackrel{˙}{V}\le -\left(\alpha {\lambda }_{0}-2l‖P‖\right)\underset{i=1}{\sum }{\stackrel{˜}{y}}_{i}^{2}+2\underset{i=1}{\sum }\underset{j\in {N}_{i}}{\sum }{\xi }_{i}{a}_{ij}{\stackrel{˜}{y}}_{i}\left({\stackrel{˜}{e}}_{i}\left(t\right)-{\stackrel{˜}{e}}_{j}\left(t\right)\right)\\ \le -\left(\alpha {\lambda }_{0}-2l‖P‖\right)\underset{i=1}{\sum }{\stackrel{˜}{y}}_{i}^{2}+2{\xi }_{\mathrm{max}}\underset{i=1}{\sum }\underset{j\in {N}_{i}}{\sum }{a}_{ij}\left(\theta {\stackrel{˜}{y}}_{i}^{2}+\frac{1}{2\theta }{\stackrel{˜}{e}}_{i}^{2}+\frac{1}{2\theta }{\stackrel{˜}{e}}_{j}^{2}\right)\\ =-\left(\alpha {\lambda }_{0}-2l‖P‖\right)\underset{i=1}{\sum }{\stackrel{˜}{y}}_{i}^{2}+2{\xi }_{\mathrm{max}}\underset{i=1}{\sum }{\Delta }_{i}\theta {\stackrel{˜}{y}}_{i}^{2}+2{\xi }_{\mathrm{max}}\underset{i=1}{\sum }\frac{1}{2\theta }{\Delta }_{i}{\stackrel{˜}{e}}_{i}^{2}+2{\xi }_{\mathrm{max}}\underset{i=1}{\sum }\underset{j\in {N}_{i}}{\sum }\frac{1}{2\theta }{a}_{ij}{\stackrel{˜}{e}}_{j}^{2}\\ =-\underset{i=1}{\sum }\left(\alpha {\lambda }_{0}-2l‖P‖-2{\xi }_{\mathrm{max}}{\Delta }_{i}\theta \right){\stackrel{˜}{y}}_{i}^{2}+2{\xi }_{\mathrm{max}}\underset{i=1}{\sum }\frac{1}{2\theta }{\Delta }_{i}{\stackrel{˜}{e}}_{i}^{2}+2{\xi }_{\mathrm{max}}\underset{i=1}{\sum }\frac{1}{2\theta }{\Delta }_{i}{\stackrel{˜}{e}}_{j}^{2}\end{array}$

$\stackrel{˙}{V}\left(t\right)\le \underset{i=1}{\sum }\left({\sigma }_{i}-1\right)K{\stackrel{˜}{y}}_{i}^{2}$ (20)

$\tau =\frac{\alpha \theta {\sigma }_{i}K}{\left({\alpha }^{2}‖{L}_{1}‖+\alpha l\right)\left(\theta {\sigma }_{i}K-2\alpha N{\Delta }_{i}{\xi }_{\mathrm{max}}\right)}$

$\tau =\frac{\alpha \theta {\sigma }_{i}K}{\left({\alpha }^{2}‖{L}_{1}‖+\alpha l\right)\left(\theta {\sigma }_{i}K-2\alpha N{\Delta }_{i}{\xi }_{\mathrm{max}}\right)}$

4. 数值仿真

Figure 1. Communication topology diagram

$L=\left[\begin{array}{cccc}2& -2& 0& 0\\ 0& 3& 3& 0\\ 0& 0& 6& -6\\ 0& 0& 0& 0\end{array}\right]$

Figure 2. Results of binary consistency tracking control of multi-agent system triggered by centralized events

Figure 3. Results of binary consistency tracking control of multi-agent system triggered by distributed events

5. 结论

NOTES

*第一作者。

#通讯作者。

 [1] 刘传达, 孟建良, 庞春江, 等. 多Agent在电力系统中的应用[J]. 电气时代, 2004(8): 72-74. [2] 刘佳, 陈增强, 刘忠信. 多智能体系统及其协同控制研究进展[J]. 智能系统学报, 2010, 5(1): 1-9. [3] 夏冰, 胡坚明, 张佐, 等. 基于多智能体的城市交通诱导系统可视化模拟[J]. 系统工程, 2002, 20(5): 72-78. [4] 邵晋梁, 石磊, 李彤, 张希琳. 合作竞争网络下的多智能体系统链路故障检测[EB/OL]. 中国科学: 信息科学, 2021. https://doi.org/10.1360/SSI-2021-0120, 2022-07-26. [5] 潘勇. 多智能体系统一致性与多机器人编队控制研究[D]: [硕士学位论文]. 杭州: 杭州电子科技大学, 2016. [6] Yang, L. and Jia, Y. (2012) An Iterative Learning Approach to Formation Control of Multi-Agent Systems. Systems & Control Letters, 61, 148-154. https://doi.org/10.1016/j.sysconle.2011.10.011 [7] Wei, R. (2007) Multi-Vehicle Consensus with a Time-Varying Reference State. Systems & Control Letters, 56, 474-483. https://doi.org/10.1016/j.sysconle.2007.01.002 [8] 盖彦荣, 陈阳舟, 宋学君, 等. 有领导者线性多智能体系统一致性的分析与设计[J]. 中南大学学报: 自然科学版, 2017, 48(3): 735-741. [9] Altafini, C. (2013) Consensus Problems on Networks with Antagonistic Interactions. IEEE Transactions on Automatic Control, 58, 935-946. https://doi.org/10.1109/TAC.2012.2224251 [10] 邵海滨, 潘鹿鹿, 席裕庚, 等. 符号网络下多智能体系统二分一致性的牵制控制问题[J]. 控制与决策, 2019, 34(8): 1695-1701. [11] Yu, T., Ma, L. and Zhang, H. (2018) Prescribed Performance for Bipartite Tracking Control of Nonlinear Multiagent Systems with Hysteresis Input Uncertainties. IEEE Transactions on Cybernetics, 49, 1327-1338. [12] Wen, G., Wang, H., Yu, X., et al. (2017) Bipartite Tracking Consensus of Linear Multi-Agent Systems with a Dynamic Leader. IEEE Transactions on Circuits and Systems II: Express Briefs, 65, 1204-1208. [13] Tabuada, P. (2007) Event-Triggered Real-Time Scheduling of Stabilizing Control Tasks. IEEE Transactions on Automatic Control, 52, 1680-1685. https://doi.org/10.1109/TAC.2007.904277 [14] Wei, R. (2007) Multi-Vehicle Consensus with a Time-Varying Reference State. Systems & Control Letters, 56, 474-483. https://doi.org/10.1016/j.sysconle.2007.01.002 [15] Liu, J., Zhang, Y., Yu, Y., et al. (2018) Fixed-Time Event-Triggered Consensus for Nonlinear Multiagent Systems without Continuous Communications. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49, 2221-2229. [16] Yan, H., Shen, Y., Hao, Z., et al. (2014) Decentralized Event-Triggered Consensus Control for Second-Order Multi-Agent Systems. Neurocomputing, 133, 18-24. https://doi.org/10.1016/j.neucom.2013.11.036 [17] 黄红伟, 黄天民. 事件触发机制下的多智能体领导跟随一致性[J]. 计算机工程与应用, 2017, 53(6): 29-33. [18] Hu, W., Lu, L. and Gang, F. (2015) Leader-Following Consensus of Linear Multi-Agent Systems by Distributed Event-Triggered Control. The 34th Chinese Control Conference (CCC2015), Hangzhou, 28-30 July 2015, 7050-7055. https://doi.org/10.1109/ChiCC.2015.7260754 [19] Zhong, X., et al. (2012) Event-Triggered Average-Consensus of Multi-Agent Systems with Weighted and Direct Topology. Journal of Systems Science & Complexity, 25, 845-855. [20] Liu, K.E., et al. (2016) Event-Based Broadcasting Containment Control for Multi-Agent Systems under Directed Topology. International Journal of Control, 89, 2360-2370. https://doi.org/10.1080/00207179.2016.1157899 [21] 韩文艳. 具有领导者的多智能体系统二部一致性研究[D]: [硕士学位论文]. 哈尔滨: 哈尔滨工程大学, 2021. https://doi.org/10.27060/d.cnki.ghbcu.2021.000678