﻿ 具有未知参数和时变拓扑的复杂网络同步研究 Synchronization of Complex Networks with Unknown Parameters and Time-Varying Topology

Vol. 11  No. 04 ( 2022 ), Article ID: 50870 , 11 pages
10.12677/AAM.2022.114231

Synchronization of Complex Networks with Unknown Parameters and Time-Varying Topology

Yang Liu, Anning Sun, Chuan Zhang*

School of Mathematical Science, Qufu Normal University, Qufu Shandong

Received: Mar. 26th, 2022; accepted: Apr. 21st, 2022; published: Apr. 28th, 2022

ABSTRACT

In this paper, the Lyapunov stability theory and adaptive control method are used to study the synchronization control problem of complex networks with unknown parameters and time-varying topology. Firstly, the proposed complex network model is completely new, which considers the influence of unknown parameters and time-varying topology simultaneously. Secondly, based on Lyapunov stability theory, sufficient conditions for complete synchronization and anti-synchronization of complex network are obtained by designing appropriate adaptive controller. Finally, two simulation examples are given to verify the correctness of the conclusion for the complete synchronization and anti-synchronization of adaptive control in complex networks.

Keywords:Complex Networks, Adaptive Control, Complete Synchronization, Anti-Synchronization

1. 引言

1.1. 复杂网络同步的介绍

1.2. 复杂网络同步的研究意义

1.3. 本文的主要工作和结构

2. 网络模型的建立与假设

2.1. 网络模型的建立

${\stackrel{˙}{x}}_{i}\left(t\right)={f}_{1}\left({x}_{i}\left(t\right)\right)+F\left({x}_{i}\left(t\right)\right){\theta }_{1}+c\underset{j=1;j\ne i}{\overset{N}{\sum }}{a}_{ij}\left(t\right)\left(g\left({x}_{j}\left(t\right)\right)-g\left({x}_{i}\left(t\right)\right)\right)$ (1)

$\left\{\begin{array}{l}{a}_{ii}\left(t\right)=-{\sum }_{i=1;j\ne i}^{N}{a}_{ij}\left(t\right)\\ {\sum }_{j=1}^{N}{a}_{ij}\left(t\right)=0\end{array}$

${\stackrel{˙}{x}}_{i}\left(t\right)={f}_{1}\left({x}_{i}\left(t\right)\right)+F\left({x}_{i}\left(t\right)\right){\theta }_{1}+c\underset{j=1}{\overset{N}{\sum }}{a}_{ij}\left(t\right)g\left({x}_{j}\left(t\right)\right)+{u}_{i}\left(t\right)\text{ }\text{ }\text{ }\text{ }\left(i=1,2,\cdots ,N\right)$ (2)

$s\left(t\right)$ 是同步状态，可以是一个稳定点或者混沌吸引子的周期轨道，且方程满足：

$\stackrel{˙}{s}\left(t\right)={f}_{1}\left(s\left(t\right)\right)+F\left(s\left(t\right)\right){\theta }_{2}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(i=1,2,\cdots ,n\right)$ (3)

$\left\{\begin{array}{l}{\stackrel{˙}{x}}_{i}\left(t\right)={f}_{1}\left({x}_{i}\left(t\right)\right)+F\left({x}_{i}\left(t\right)\right){\theta }_{1}+c\underset{j=1}{\overset{N}{\sum }}{a}_{ij}\left(t\right)g\left({x}_{j}\left(t\right)\right)+{u}_{i}\left(t\right)\\ \stackrel{˙}{s}\left(t\right)={f}_{1}\left(s\left(t\right)\right)+F\left(s\left(t\right)\right){\theta }_{2}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(i=1,2,\cdots ,N\right)\end{array}$ (4)

$\underset{t\to \infty }{\mathrm{lim}}‖{y}_{i}\left(t\right)-{x}_{i}\left(t\right)‖=0\text{}\left(i=1,2,\cdots ,N\right)$

$\underset{t\to \infty }{\mathrm{lim}}‖{y}_{i}\left(t\right)+{x}_{i}\left(t\right)‖=0\text{}\left(i=1,2,\cdots ,N\right)$

2.2. 相关假设与引理

$\begin{array}{l}‖f\left({y}_{i}\left(t\right)\right)-f\left({x}_{i}\left(t\right)\right)‖\le L‖{y}_{i}\left(t\right)-{x}_{i}\left(t\right)‖\\ ‖g\left({y}_{j}\left(t\right)\right)-g\left({x}_{i}\left(t\right)\right)‖\le M‖{y}_{i}\left(t\right)-{x}_{i}\left(t\right)‖,\end{array}$

$\left(\begin{array}{ll}A\left(x\right)\hfill & B\left(x\right)\hfill \\ {B}^{\text{T}}\left(x\right)\hfill & C\left(x\right)\hfill \end{array}\right)<0$

$\begin{array}{l}\left(1\right):A\left(x\right)<0,\text{ }C\left(x\right)-B{\left(x\right)}^{\text{T}}A{\left(x\right)}^{-1}B\left(x\right)<0\\ \left(2\right):C\left(x\right)<0,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }A\left(x\right)-B\left(x\right)C{\left(x\right)}^{-1}B{\left(x\right)}^{\text{T}}<0\end{array}$

$±2{x}^{\text{T}}y\le {x}^{\text{T}}Qx+{y}^{\text{T}}{Q}^{-1}y$.

1) 矩阵A的特征值为0，特征向量为 ${\left(1,1,\cdots ,1\right)}^{\text{T}}$

2) 矩阵A的特征值实部小于等于零，且所有特征值实部为零的那些特征值是实特征值0；

3) 矩阵A是不可约矩阵，则A特征值的重数为1。

3. 主要结果

3.1. 自适应控制下网络的完全同步

${e}_{i}\left(t\right)={x}_{i}\left(t\right)-s\left(t\right),\text{}i=1,2,\cdots ,N$ (5)

$\left\{\begin{array}{l}{u}_{i}\left(t\right)=-{k}_{i}\left(t\right){e}_{i}\left(t\right)\\ {\stackrel{˙}{k}}_{i}\left(t\right)={h}_{i}{e}_{i}^{\text{T}}\left(t\right){e}_{i}\left(t\right)\\ {\stackrel{˙}{\theta }}_{1}=-{\eta }_{i}{F}^{\text{T}}\left({x}_{i}\right){e}_{i}\left(t\right)\\ {\stackrel{˙}{\theta }}_{2}={\gamma }_{i}{F}^{\text{T}}\left(s\right){e}_{i}\left(t\right)\end{array}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }i=1,2,\cdots ,N$ (6)

$\begin{array}{c}{\stackrel{˙}{e}}_{i}={x}_{i}\left(t\right)-s\left(t\right)\\ ={f}_{1}\left({x}_{i}\left(t\right)\right)+F\left({x}_{i}\left(t\right)\right){\theta }_{1}\text{ }+c\underset{j=1}{\overset{N}{\sum }}{a}_{ij}\left(t\right)g\left({x}_{j}\left(t\right)\right)+{u}_{i}\left(t\right)-\left({f}_{1}\left(s\left(t\right)\right)+F\left(s\left(t\right)\right){\theta }_{2}\right)\text{ }\\ =\left({f}_{1}\left({x}_{i}\left(t\right)\right)-{f}_{1}\left(s\left(t\right)\right)\right)+F\left({x}_{i}\left(t\right)\right){\theta }_{1}-F\left(s\left(t\right)\right){\theta }_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+c\underset{j=1}{\overset{N}{\sum }}{a}_{ij}\left(t\right)\left(g\left({x}_{j}\left(t\right)\right)-g\left(s\left(t\right)\right)\right)-{k}_{i}{e}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(i=1,2,\cdots ,N\right)\end{array}$ (7)

$V\left(t\right)=\frac{1}{2}\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{e}_{i}+\frac{1}{2}\underset{i=1}{\overset{N}{\sum }}\frac{1}{{h}_{i}}{\left({k}_{i}-{k}_{i}^{*}\right)}^{2}+\frac{1}{2}\underset{i=1}{\overset{N}{\sum }}\frac{1}{{\eta }_{i}}{\theta }_{1}^{\text{T}}{\theta }_{1}+\frac{1}{2}\underset{i=1}{\overset{N}{\sum }}\frac{1}{{\gamma }_{i}}{\theta }_{2}^{\text{T}}{\theta }_{2}$

$V\left(t\right)$ 沿着误差系统(7)关于时间t求导得：

$\begin{array}{l}\stackrel{˙}{V}\left(t\right)=\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{\stackrel{˙}{e}}_{i}+\underset{i=1}{\overset{N}{\sum }}\frac{1}{{h}_{i}}\left({k}_{i}-{k}_{i}^{*}\right){\stackrel{˙}{k}}_{i}+\underset{i=1}{\overset{N}{\sum }}\frac{1}{{\eta }_{i}}{\theta }_{1}^{\text{T}}{\stackrel{˙}{\theta }}_{1}+\underset{i=1}{\overset{N}{\sum }}\frac{1}{{\gamma }_{i}}{\theta }_{2}^{\text{T}}{\stackrel{˙}{\theta }}_{2}\\ =\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}\left[\left({f}_{1}\left({x}_{i}\left(t\right)\right)-{f}_{1}\left(s\left(t\right)\right)\right)+F\left({x}_{i}\left(t\right)\right){\theta }_{1}-F\left(s\left(t\right)\right){\theta }_{2}+c\underset{j=1}{\overset{N}{\sum }}{a}_{ij}\left(t\right)\left(g\left({x}_{j}\left(t\right)\right)-g\left(s\left(t\right)\right)\right)-{k}_{i}{e}_{i}\right]\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }+\underset{i=1}{\overset{N}{\sum }}\left({k}_{i}-{k}_{i}^{*}\right){e}_{i}^{\text{T}}{e}_{i}-\underset{i=1}{\overset{N}{\sum }}{\theta }_{1}^{\text{T}}{F}^{\text{T}}\left({x}_{i}\left(t\right)\right){e}_{i}+\underset{i=1}{\overset{N}{\sum }}{\theta }_{2}^{\text{T}}{F}^{\text{T}}\left(s\left(t\right)\right){e}_{i}\\ =\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}\left({f}_{1}\left({x}_{i}\left(t\right)\right)-{f}_{1}\left(s\left(t\right)\right)\right)-{k}_{i}^{*}\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{e}_{i}+c{e}^{\text{T}}\left(t\right)\left(A\left(t\right)\otimes {I}_{N}\right)\left(g\left(x\left(t\right)\right)-g\left(s\left(t\right)\right)\right)\end{array}$

$\begin{array}{l}\le L\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{e}_{i}-{k}_{i}^{*}\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{e}_{i}+c\frac{\xi }{2}{e}^{\text{T}}\left(t\right)\left(A\left(t\right)A{\left(t\right)}^{\text{T}}\otimes {I}_{N}\right)e\left(t\right)+\frac{c{M}^{2}}{2\xi }{e}^{\text{T}}\left(t\right)e\left(t\right)\\ \le L{e}^{\text{T}}\left(t\right)e\left(t\right)-{k}^{*}{e}^{\text{T}}\left(t\right)e\left(t\right)+c\frac{\xi }{2}{\lambda }_{\mathrm{max}}\left(A\left(t\right)A{\left(t\right)}^{\text{T}}\right){e}^{\text{T}}\left(t\right)e\left(t\right)+\frac{c{M}^{2}}{2\xi }{e}^{\text{T}}\left(t\right)e\left(t\right)\\ =\left(L-{k}^{*}+c\frac{\xi }{2}{\lambda }_{\mathrm{max}}\left(A\left(t\right)A{\left(t\right)}^{\text{T}}\right)+\frac{c{M}^{2}}{2\xi }\right){e}^{\text{T}}\left(t\right)e\left( t \right)\end{array}$

$e={\left({e}_{1}^{\text{T}},{e}_{2}^{\text{T}},\cdots ,{e}_{N}^{\text{T}}\right)}^{\text{T}}$$g\left(x\left(t\right)\right)=\left(g\left({x}_{1}\left(t\right)\right),g\left({x}_{2}\left(t\right)\right),\cdots ,g\left({x}_{N}\left(t\right)\right)\right)$$g\left(s\left(t\right)\right)=\left(g\left(s\left(t\right)\right),g\left(s\left(t\right)\right)\cdots ,g\left(s\left(t\right)\right)\right)$$k=\mathrm{min}\left\{{k}_{1},{k}_{2},\cdots ,{k}_{N}\right\}$

${k}^{*}>L+\frac{c\xi }{2}{\lambda }_{\mathrm{max}}\left(A\left(t\right)A{\left(t\right)}^{\text{T}}\right)+\frac{c{M}^{2}}{2\xi }$，则 $\stackrel{˙}{V}\left(t\right)<-{e}^{\text{T}}e<0$。根据Lyapunov稳定性理论可得到复杂网络模型(4)实现完全同步且满足同步解是渐进稳定的，定理得证。

3.2. 自适应控制下网络的反同步

${e}_{i}\left(t\right)={x}_{i}\left(t\right)+s\left(t\right),\text{}i=1,2,\cdots ,N$ (8)

$\left\{\begin{array}{l}{u}_{i}\left(t\right)=-{k}_{i}\left(t\right){e}_{i}\left(t\right)\\ {\stackrel{˙}{k}}_{i}\left(t\right)={h}_{i}{e}_{i}^{\text{T}}\left(t\right){e}_{i}\left(t\right)\\ {\stackrel{˙}{\theta }}_{1}=-{\eta }_{i}{F}^{\text{T}}\left({x}_{i}\right){e}_{i}\left(t\right)\\ {\stackrel{˙}{\theta }}_{2}=-{\gamma }_{i}{F}^{\text{T}}\left(s\right){e}_{i}\left(t\right)\end{array}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }i=1,2,\cdots ,N$ (9)

$\begin{array}{c}{\stackrel{˙}{e}}_{i}={x}_{i}\left(t\right)+s\left(t\right)\\ ={f}_{1}\left({x}_{i}\left(t\right)\right)+F\left({x}_{i}\left(t\right)\right){\theta }_{1}\text{ }+c\underset{j=1;j\ne i}{\overset{N}{\sum }}{a}_{ij}\left(t\right)g\left({x}_{j}\left(t\right)\right)+{u}_{i}\left(t\right)+\left({f}_{1}\left(s\left(t\right)\right)+F\left(s\left(t\right)\right){\theta }_{2}\right)\text{ }\\ =\left({f}_{1}\left({x}_{i}\left(t\right)\right)+{f}_{1}\left(s\left(t\right)\right)\right)+F\left({x}_{i}\left(t\right)\right){\theta }_{1}+F\left(s\left(t\right)\right){\theta }_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+c\underset{j=1;j\ne i}{\overset{N}{\sum }}{a}_{ij}\left(t\right)\left(g\left({x}_{j}\left(t\right)\right)-g\left(s\left(t\right)\right)\right)-{k}_{i}{e}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(i=1,2,\cdots ,N\right)\end{array}$ (10)

$V\left(t\right)=\frac{1}{2}\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{e}_{i}+\frac{1}{2}\underset{i=1}{\overset{N}{\sum }}\frac{1}{{h}_{i}}{\left({k}_{i}-{k}_{i}^{*}\right)}^{2}+\frac{1}{2}\underset{i=1}{\overset{N}{\sum }}\frac{1}{{\eta }_{i}}{\theta }_{1}^{\text{T}}{\theta }_{1}+\frac{1}{2}\underset{i=1}{\overset{N}{\sum }}\frac{1}{{\gamma }_{i}}{\theta }_{2}^{\text{T}}{\theta }_{2}$

$V\left(t\right)$ 沿着误差系统(10)关于时间t求导得：

$\begin{array}{l}\stackrel{˙}{V}\left(t\right)=\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{\stackrel{˙}{e}}_{i}+\underset{i=1}{\overset{N}{\sum }}\frac{1}{{h}_{i}}\left({k}_{i}-{k}_{i}^{*}\right){\stackrel{˙}{k}}_{i}+\underset{i=1}{\overset{N}{\sum }}\frac{1}{{\eta }_{i}}{\theta }_{1}^{\text{T}}{\stackrel{˙}{\theta }}_{1}+\underset{i=1}{\overset{N}{\sum }}\frac{1}{{\gamma }_{i}}{\theta }_{2}^{\text{T}}{\stackrel{˙}{\theta }}_{2}\\ =\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}\left[\left({f}_{1}\left({x}_{i}\left(t\right)\right)+{f}_{1}\left(s\left(t\right)\right)\right)+F\left({x}_{i}\left(t\right)\right){\theta }_{1}+F\left(s\left(t\right)\right){\theta }_{2}+c\underset{j=1;j\ne i}{\overset{N}{\sum }}{a}_{ij}\left(t\right)\left(g\left({x}_{j}\left(t\right)\right)-g\left(s\left(t\right)\right)\right)-{k}_{i}{e}_{i}\right]\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }+\underset{i=1}{\overset{N}{\sum }}\left({k}_{i}-{k}_{i}^{*}\right){e}_{i}^{\text{T}}{e}_{i}-\underset{i=1}{\overset{N}{\sum }}{\theta }_{1}^{\text{T}}{F}^{\text{T}}\left({x}_{i}\left(t\right)\right){e}_{i}-\underset{i=1}{\overset{N}{\sum }}{\theta }_{2}^{\text{T}}{F}^{\text{T}}\left(s\left(t\right)\right){e}_{i}\\ =\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}\left({f}_{1}\left({x}_{i}\left(t\right)\right)-{f}_{1}\left(-s\left(t\right)\right)\right)-{k}_{i}^{*}\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{e}_{i}+c{e}^{\text{T}}\left(t\right)\left(A\left(t\right)\otimes {I}_{N}\right)\left(g\left(x\left(t\right)\right)-g\left(s\left(t\right)\right)\right)\end{array}$

$\begin{array}{l}\le L\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{e}_{i}-{k}_{i}^{*}\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{e}_{i}+c\frac{\xi }{2}{e}^{\text{T}}\left(t\right)\left(A\left(t\right)A{\left(t\right)}^{\text{T}}\otimes {I}_{N}\right)e\left(t\right)+\frac{c{M}^{2}}{2\xi }{e}^{\text{T}}\left(t\right)e\left(t\right)\\ \le L{e}^{\text{T}}\left(t\right)e\left(t\right)-{k}^{*}{e}^{\text{T}}\left(t\right)e\left(t\right)+c\frac{\xi }{2}{\lambda }_{\mathrm{max}}\left(A\left(t\right)A{\left(t\right)}^{\text{T}}\right){e}^{\text{T}}\left(t\right)e\left(t\right)+\frac{c{M}^{2}}{2\xi }{e}^{\text{T}}\left(t\right)e\left(t\right)\\ =\left(L-{k}^{*}+c\frac{\xi }{2}{\lambda }_{\mathrm{max}}\left(A\left(t\right)A{\left(t\right)}^{\text{T}}\right)+\frac{c{M}^{2}}{2\xi }\right){e}^{\text{T}}\left(t\right)e\left( t \right)\end{array}$

$e={\left({e}_{1}^{\text{T}},{e}_{2}^{\text{T}},\cdots ,{e}_{N}^{\text{T}}\right)}^{\text{T}}$$g\left(x\left(t\right)\right)=\left(g\left({x}_{1}\left(t\right)\right),g\left({x}_{2}\left(t\right)\right),\cdots ,g\left({x}_{N}\left(t\right)\right)\right)$$g\left(s\left(t\right)\right)=\left(g\left(s\left(t\right)\right),g\left(s\left(t\right)\right)\cdots ,g\left(s\left(t\right)\right)\right)$$k=\mathrm{min}\left\{{k}_{1},{k}_{2},\cdots ,{k}_{N}\right\}$

${k}^{*}>L+\frac{c\xi }{\text{2}}{\lambda }_{\mathrm{max}}\left(A\left(t\right)A{\left(t\right)}^{\text{T}}\right)+\frac{c{M}^{2}}{\text{2}\xi }$，则 $\stackrel{˙}{V}\left(t\right)<-{e}^{\text{T}}e<0$。根据Lyapunov稳定性理论得到复杂网络模型(4)可以实现反同步且满足同步解是渐进稳定的，定理得证。

4. 数值仿真

4.1. 完全同步的数值仿真

$\left\{\begin{array}{l}{\stackrel{˙}{x}}_{1}\left(t\right)=\alpha \left(-{x}_{1}\left(t\right)+{x}_{2}\left(t\right)-\Phi \left({x}_{1}\left(t\right)\right)\right)\hfill \\ {\stackrel{˙}{x}}_{2}\left(t\right)=\theta {x}_{1}\left(t\right)-{x}_{2}\left(t\right)+{x}_{3}\left(t\right)\hfill \\ {\stackrel{˙}{x}}_{3}\left(t\right)=-\beta {x}_{2}\left( t \right)\hfill \end{array}$

(a) 完全同步误差 ${e}_{i1}\left(t\right)$ (b) 完全同步误差 ${e}_{i2}\left( t \right)$ (c) 完全同步误差 ${e}_{i3}\left( t \right)$

Figure 1. The time evolution of complete synchronization errors of complex networks with adaptive controllers, where the adaptive controller factor ${h}_{i}=4$

(a) 完全同步误差 ${e}_{i1}\left(t\right)$ (b) 完全同步误差 ${e}_{i2}\left( t \right)$ (c) 完全同步误差 ${e}_{i3}\left( t \right)$

Figure 2. The time evolution of complete synchronization errors of complex networks with adaptive controllers, where the adaptive controller factor ${h}_{i}=12$

4.2. 反同步的数值仿真

(a) 反同步误差 ${e}_{i1}\left(t\right)$ (b) 反同步误差 ${e}_{i2}\left( t \right)$(c) 反同步误差 ${e}_{i3}\left( t \right)$

Figure 3. The time evolution of anti-synchronization errors of complex networks with adaptive controllers, where the adaptive controller factor ${h}_{i}=4$

(a) 反同步误差 ${e}_{i1}\left(t\right)$ (b) 反同步误差 ${e}_{i2}\left( t \right)$ (c) 反同步误差 ${e}_{i3}\left( t \right)$

Figure 4. The time evolution of anti-synchronization errors of complex networks with adaptive controllers, where the adaptive controller factor ${h}_{i}=12$

5. 结论

Synchronization of Complex Networks with Unknown Parameters and Time-Varying Topology[J]. 应用数学进展, 2022, 11(04): 2145-2155. https://doi.org/10.12677/AAM.2022.114231

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21. NOTES

*通讯作者。