此篇文章主要在于仿真代码，关于文献的解释请参考 【Paper】2003_Murr_Consensus Problems in Networks of Agents with Switching Topology and Time-Delays

文献原文： Consensus Problems in Networks of Agents with Switching Topology and Time-Delays

Consensus Problems in Networks of Agents with Switching Topology and Time-Delays

  • 原Paper中Figure5复现
  • 原Paper中Figure6复现
  • 原Paper中Figure6复现

三种网络：

  1. 切换拓扑无时滞
  2. 固定拓扑有时滞
  3. 离散时间最大一致性（领航者模型）

第一个问题：另一篇文章中，有提到若全连接，则系统的稳定性会被破坏。

第二个问题：第二个问题：平均一致性与系统的一致性不是一码事，平均一致性是 average-consensus,系统一致性是 general agreement。

% 此为论文中，关于Fig5(a)中a子图的复现结果。
% 输入参数包括初始状态（位置）和图的拉普拉斯矩阵
% 最后显示状态一致性收敛的踪迹和输入的变化曲线
clc;
clear;%% 输入初始化参数
X0  = [-10.2999, 0.2575, -4.4997, 3.6258, 3.0922, 9.0156, 3.5099, -2.6645, 2.4552, -4.4921]';
% 一致性决策值
AveVal = mean(X0);                                                          % 平均一致性
MaxVal = max(X0);                                                           % 最大一致性
MinVal = min(X0);                                                           % 最小一致性
n   = 10;                                                                   % 节点数量
% 度矩阵，有向拓扑结构
D =[1 0 0 0 0 0 0 0 0 0;0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0;0 0 0 1 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 0 1;];
% 邻接矩阵，有向拓扑结构
A =[0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0;0 0 0 1 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 0 1;1 0 0 0 0 0 0 0 0 0;];
% 度矩阵，有向拓扑结构
D =[1 0 0 0 0 0 0 0 0 0;0 2 0 0 0 0 0 0 0 0;0 0 2 0 0 0 0 0 0 0;0 0 0 2 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0;0 0 0 0 0 2 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 2 0 0;0 0 0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 0 2;];
% 邻接矩阵，有向拓扑结构
A =[0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 1;0 0 0 1 0 0 0 1 0 0;0 0 0 0 1 1 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 1 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 1 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 0 1;1 1 0 0 0 0 0 0 0 0;];
% 度矩阵，有向拓扑结构
D =[2 0 0 0 0 0 0 0 0 0;0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0;0 0 0 2 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 2 0 0;0 0 0 0 0 0 0 0 2 0;0 0 0 0 0 0 0 0 0 1;];
% 邻接矩阵，有向拓扑结构
A =[0 1 0 0 0 0 0 0 1 0;0 0 1 0 0 0 0 0 0 0;0 0 0 1 0 0 0 0 0 0;0 0 0 0 1 0 0 1 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 1 0 0 0 0 1 0;1 0 0 0 0 0 0 0 0 1;1 0 0 0 0 0 0 0 0 0;];
% 度矩阵，有向拓扑结构
D =[2 0 0 0 0 0 0 0 0 0;0 2 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0;0 0 0 2 0 0 0 0 0 0;0 0 0 0 2 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 2 0 0 0;0 0 0 0 0 0 0 3 0 0;0 0 0 0 0 0 0 0 2 0;0 0 0 0 0 0 0 0 0 1;];
% 邻接矩阵，有向拓扑结构
A =[0 1 0 0 0 0 0 0 1 0;0 0 1 0 0 0 0 1 0 0;0 0 0 1 0 0 0 0 0 0;0 0 0 0 1 0 0 1 0 0;0 0 0 0 0 1 1 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 1 0 0 1 0 0;0 1 0 1 0 0 0 0 1 0;1 0 0 0 0 0 0 0 0 1;1 0 0 0 0 0 0 0 0 0;];
% 拉普拉斯矩阵
L = D - A;% 矩阵的代数连通度
eig_val = eig(L);                       % L的特征值
eig_val = sort(eig_val,'ascend');       % 从小到大排序，最小特征值为0
algeb_conn = eig_val(2);                % algebraic connectivity 代数连通度：lap_matrix的第二小特征值>0，连通图%% 收敛相关参数
tBegin = 0;                                                                 % 开始时间
tEnd   = 15;                                                                % 结束时间
h      = 0.1;                                                               % 最小时间间隔
times  = (tEnd-tBegin) / h;                                                 % 迭代计算次数
X(:,1) = X0;                                                                % X的第一列等于X1
U(:,1) = -L * X0;
t(1) = tBegin;                                                              % 时间间隔记录表i = 1;
while(i <= times)t(i+1) = tBegin + i * h;                                                % 添加更新后的t值Xt = X(:,i);Ut = -L * Xt;                                                           % u = -Lx
%     Xt1 = Xt + h * Ut;                                                      % x = x + h*uXt1 = expm(-L*(tBegin+i*h)) * X0;                                                      % x = x + h*uX(:,i+1) = Xt1;                                                         % 添加更新后的Xt值U(:,i+1) = Ut;                                                          % 添加更新后的Ut值  i = i+1;
end%% 结果显示
% 绘制图像
subplot(2,1,1)
plot(t,X(1,:), t,X(2,:), t,X(3,:), t,X(4,:), t,X(5,:), t,X(6,:), t,X(7,:), t,X(8,:), t,X(9,:), t,X(10,:), 'linewidth',2.0)
% legend("x_1","x_2","x_3","x_4","x_5","x_6","x_7","x_8","x_9","x_{10}");
xlabel('Times/收敛次数');
ylabel('States/状态值');
title('States-Times');subplot(2,1,2)
plot(t,U(1,:), t,U(2,:), t,U(3,:), t,U(4,:), t,U(5,:), t,U(6,:), t,U(7,:), t,U(8,:), t,U(9,:), t,U(10,:), 'linewidth',2.0)
% legend("u_1","u_2","u_3","u_4","u_5","u_6","u_7","u_8","u_9","u_{10}");
xlabel('Times/收敛次数');
ylabel('U/输入值');
title('U-Times');

原Paper中Figure5复现

clear;
clc;
% 此为原Paper中Figure5复现代码% 状态初值如下
X0  = [-10.2999, 0.2575, -4.4997, 3.6258, 3.0922, 9.0156, 3.5099, -2.6645, 2.4552, -4.4921]';Node_Nums = 10;% a图度矩阵，邻接矩阵，拉普拉斯矩阵
D_a = [ 1 0 0 0 0 0 0 0 0 0;0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0;0 0 0 1 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 0 1;];
A_a = [ 0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0;0 0 0 1 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 0 1;1 0 0 0 0 0 0 0 0 0;];
L_a = D_a - A_a;% b图度矩阵，邻接矩阵，拉普拉斯矩阵
D_b = [ 1 0 0 0 0 0 0 0 0 0;0 2 0 0 0 0 0 0 0 0;0 0 2 0 0 0 0 0 0 0;0 0 0 2 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0;0 0 0 0 0 2 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 0 2;];
A_b = [ 0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 1;0 0 0 1 0 0 0 1 0 0;0 0 0 0 1 1 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 1 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 0 1;1 1 0 0 0 0 0 0 0 0;];
L_b = D_b - A_b;% c图度矩阵，邻接矩阵，拉普拉斯矩阵
D_c = [ 2 0 0 0 0 0 0 0 0 0;0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0;0 0 0 2 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 2 0 0;0 0 0 0 0 0 0 0 2 0;0 0 0 0 0 0 0 0 0 1;];
A_c = [ 0 1 0 0 0 0 0 0 1 0;0 0 1 0 0 0 0 0 0 0;0 0 0 1 0 0 0 0 0 0;0 0 0 0 1 0 0 1 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 1 0 0 0 0 1 0;1 0 0 0 0 0 0 0 0 1;1 0 0 0 0 0 0 0 0 0;];
L_c = D_c - A_c;% d图度矩阵，邻接矩阵，拉普拉斯矩阵
D_d = [ 2 0 0 0 0 0 0 0 0 0;0 2 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0;0 0 0 2 0 0 0 0 0 0;0 0 0 0 2 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 2 0 0 0;0 0 0 0 0 0 0 3 0 0;0 0 0 0 0 0 0 0 2 0;0 0 0 0 0 0 0 0 0 1;];
% 邻接矩阵，有向拓扑结构
A_d = [ 0 1 0 0 0 0 0 0 1 0;0 0 1 0 0 0 0 1 0 0;0 0 0 1 0 0 0 0 0 0;0 0 0 0 1 0 0 1 0 0;0 0 0 0 0 1 1 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 1 0 0 1 0 0;0 1 0 1 0 0 0 0 1 0;1 0 0 0 0 0 0 0 0 1;1 0 0 0 0 0 0 0 0 0;];
L_d = D_d - A_d;% 设置收敛相关参数
tBegin = 0;                     % 开始时间
tEnd   = 20;                    % 结束时间
dt     = 0.1;                   % 最小时间间隔
times  = (tEnd-tBegin) / dt;    % 迭代计算次数
time   = 1;
% 定义记录时间数组
t(:,1)   = tBegin;
% 定义图a数据存储数组
X_a(:,1) = X0;
U_a(:,1) = -L_a * X0;
Delta_a(:,1) = X0' * L_a * X0;
% 定义图b数据存储数组
X_b(:,1) = X0;
U_b(:,1) = -L_b * X0;
Delta_b(:,1) = X0' * L_b * X0;X_c(:,1) = X0;
U_c(:,1) = -L_c * X0;
Delta_c(:,1) = X0' * L_c * X0;X_d(:,1) = X0;
U_d(:,1) = -L_d * X0;
Delta_d(:,1) = X0' * L_d * X0;
% 开始收敛计算
while(time <= times)t(:,time+1) = tBegin + time * dt;% 图a数据迭代X_a(:,time+1) = expm(-L_a * t(:,time+1)) * X0;                          % Xt = expm(-Lx)*X0 添加更新后的Xt值U_a(:,time+1) = -L_a * X_a(:,time+1);                                   % Ut = -LXt 添加更新后的Ut值 Delta_a(:,time+1) = X_a(:,time+1)'*L_a*X_a(:,time+1);                   % Delta = X^T * L * X% 图b数据迭代X_b(:,time+1) = expm(-L_b * t(:,time+1)) * X0;U_b(:,time+1) = -L_b * X_b(:,time+1);Delta_b(:,time+1) = X_b(:,time+1)'*L_b*X_b(:,time+1);X_c(:,time+1) = expm(-L_c * t(:,time+1)) * X0;U_c(:,time+1) = -L_c * X_c(:,time+1);Delta_c(:,time+1) = X_c(:,time+1)'*L_c*X_c(:,time+1);X_d(:,time+1) = expm(-L_d * t(:,time+1)) * X0;U_d(:,time+1) = -L_d * X_d(:,time+1);Delta_d(:,time+1) = X_d(:,time+1)'*L_d*X_d(:,time+1);time = time + 1;
end% 矩阵的代数连通度
eig_val_a = eig(L_a);                       % L_a的特征值
eig_val_a = sort(eig_val_a,'ascend');       % 从小到大排序，最小特征值为0
AC_a      = real(eig_val_a(2));             % algebraic connectivity 代数连通度：lap_matrix的第二小特征值>0，连通图eig_val_b = eig(L_b);                       % L_b的特征值
eig_val_b = sort(eig_val_b,'ascend');       % 从小到大排序，最小特征值为0
AC_b      = real(eig_val_b(2));             % algebraic connectivity 代数连通度：lap_matrix的第二小特征值>0，连通图eig_val_c = eig(L_c);
eig_val_c = sort(eig_val_c,'ascend');
AC_c      = real(eig_val_c(2));eig_val_d = eig(L_d);
eig_val_d = sort(eig_val_d,'ascend');
AC_d      = real(eig_val_d(2));% 绘制图a数据
subplot(4,2,1);
plot(t,X_a(1,:), t,X_a(2,:), t,X_a(3,:), t,X_a(4,:), t,X_a(5,:), t,X_a(6,:), t,X_a(7,:), t,X_a(8,:), t,X_a(9,:), t,X_a(10,:), 'linewidth',1.5)
xlabel("time(sec)");ylabel("node value");
title("Algebraic Connectivity="+num2str(AC_a));
subplot(4,2,3);
plot(t,Delta_a, 'linewidth',1.5);
xlabel("time(sec)");ylabel("disagreement");subplot(4,2,2);
plot(t,X_b(1,:), t,X_b(2,:), t,X_b(3,:), t,X_b(4,:), t,X_b(5,:), t,X_b(6,:), t,X_b(7,:), t,X_b(8,:), t,X_b(9,:), t,X_b(10,:), 'linewidth',1.5)
xlabel("time(sec)");ylabel("node value");
title("Algebraic Connectivity="+num2str(AC_b));
subplot(4,2,4);
plot(t,Delta_b, 'linewidth',1.5);
xlabel("time(sec)");ylabel("disagreement");subplot(4,2,5);
plot(t,X_c(1,:), t,X_c(2,:), t,X_c(3,:), t,X_c(4,:), t,X_c(5,:), t,X_c(6,:), t,X_c(7,:), t,X_c(8,:), t,X_c(9,:), t,X_c(10,:), 'linewidth',1.5)
xlabel("time(sec)");ylabel("node value");
title("Algebraic Connectivity="+num2str(AC_c));
subplot(4,2,7);
plot(t,Delta_c, 'linewidth',1.5);
xlabel("time(sec)");ylabel("disagreement");subplot(4,2,6);
plot(t,X_d(1,:), t,X_d(2,:), t,X_d(3,:), t,X_d(4,:), t,X_d(5,:), t,X_d(6,:), t,X_d(7,:), t,X_d(8,:), t,X_d(9,:), t,X_d(10,:), 'linewidth',1.5)
xlabel("time(sec)");ylabel("node value");
title("Algebraic Connectivity="+num2str(AC_d));
subplot(4,2,8);
plot(t,Delta_d, 'linewidth',1.5);
xlabel("time(sec)");ylabel("disagreement");figure(2)
plot(t,X_a(1,:), t,X_a(2,:), t,X_a(3,:), t,X_a(4,:), t,X_a(5,:), t,X_a(6,:), t,X_a(7,:), t,X_a(8,:), t,X_a(9,:), t,X_a(10,:), 'linewidth',1.5)

原Paper中Figure6复现

clear;
clc;
% 此为原Paper中Figure6复现代码% 状态初值如下
X0  = [-10.2999, 0.2575, -4.4997, 3.6258, 3.0922, 9.0156, 3.5099, -2.6645, 2.4552, -4.4921]';Node_Nums = 10;% b图度矩阵，邻接矩阵，拉普拉斯矩阵
D_a = [ 1 0 0 0 0 0 0 0 0 0;0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0;0 0 0 1 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 0 1;];
A_a = [ 0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0;0 0 0 1 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 0 1;1 0 0 0 0 0 0 0 0 0;];
L_a = D_a - A_a;% b图度矩阵，邻接矩阵，拉普拉斯矩阵
D_b = [ 1 0 0 0 0 0 0 0 0 0;0 2 0 0 0 0 0 0 0 0;0 0 2 0 0 0 0 0 0 0;0 0 0 2 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0;0 0 0 0 0 2 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 0 2;];
A_b = [ 0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 1;0 0 0 1 0 0 0 1 0 0;0 0 0 0 1 1 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 1 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 0 0 1;1 1 0 0 0 0 0 0 0 0;];
L_b = D_b - A_b;% c图度矩阵，邻接矩阵，拉普拉斯矩阵
D_c = [ 2 0 0 0 0 0 0 0 0 0;0 1 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0;0 0 0 2 0 0 0 0 0 0;0 0 0 0 1 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 2 0 0;0 0 0 0 0 0 0 0 2 0;0 0 0 0 0 0 0 0 0 1;];
A_c = [ 0 1 0 0 0 0 0 0 1 0;0 0 1 0 0 0 0 0 0 0;0 0 0 1 0 0 0 0 0 0;0 0 0 0 1 0 0 1 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 0 0 0 1 0 0;0 0 0 1 0 0 0 0 1 0;1 0 0 0 0 0 0 0 0 1;1 0 0 0 0 0 0 0 0 0;];
L_c = D_c - A_c;% d图度矩阵，邻接矩阵，拉普拉斯矩阵
D_d = [ 2 0 0 0 0 0 0 0 0 0;0 2 0 0 0 0 0 0 0 0;0 0 1 0 0 0 0 0 0 0;0 0 0 2 0 0 0 0 0 0;0 0 0 0 2 0 0 0 0 0;0 0 0 0 0 1 0 0 0 0;0 0 0 0 0 0 2 0 0 0;0 0 0 0 0 0 0 3 0 0;0 0 0 0 0 0 0 0 2 0;0 0 0 0 0 0 0 0 0 1;];
% 邻接矩阵，有向拓扑结构
A_d = [ 0 1 0 0 0 0 0 0 1 0;0 0 1 0 0 0 0 1 0 0;0 0 0 1 0 0 0 0 0 0;0 0 0 0 1 0 0 1 0 0;0 0 0 0 0 1 1 0 0 0;0 0 0 0 0 0 1 0 0 0;0 0 0 0 1 0 0 1 0 0;0 1 0 1 0 0 0 0 1 0;1 0 0 0 0 0 0 0 0 1;1 0 0 0 0 0 0 0 0 0;];
L_d = D_d - A_d;% 设置收敛相关参数
tBegin = 0;                     % 开始时间
tEnd   = 15;                    % 结束时间
dt     = 0.1;                   % 最小时间间隔
times  = (tEnd-tBegin) / dt;    % 迭代计算次数
time   = 1;
% 定义记录时间数组
t(:,1)   = tBegin;
% 定义图数据存储数组
% t=0时刻，状态为b
X(:,1) = X0;
U(:,1) = -L_b * X0;
Delta(:,1) = X0' * L_b * X0;% 开始收敛计算
while(time <= times)% 迭代顺序为 b->a->d->c->bif ((time>=1 && time<=10)||(time>=41 && time<=50)||(time>=81 && time<=90)||(time>=121 && time<=130))L = L_b;elseif((time>=11 && time<=20)||(time>=51 && time<=60)||(time>=91 && time<=100)||(time>=131 && time<=140))L = L_a;elseif((time>=21 && time<=30)||(time>=61 && time<=70)||(time>=101 && time<=110)||(time>=141 && time<=150))L = L_d;elseif((time>=31 && time<=40)||(time>=71 && time<=80)||(time>=111 && time<=120))L = L_c;endt(:,time+1) = tBegin + time * dt;X(:,time+1) = expm(-L * t(:,time+1)) * X0;U(:,time+1) = -L * X(:,time+1);Delta(:,time+1) = X(:,time+1)'*L*X(:,time+1);time = time + 1;
end% 绘制图数据
subplot(2,1,1);
plot(t,X(1,:), t,X(2,:), t,X(3,:), t,X(4,:), t,X(5,:), t,X(6,:), t,X(7,:), t,X(8,:), t,X(9,:), t,X(10,:), 'linewidth',1.5)
xlabel("time(sec)");ylabel("node value");
subplot(2,1,2);
plot(t,Delta, 'linewidth',1.5);
xlabel("time(sec)");ylabel("disagreement");

原Paper中Figure6复现

clear;
clc;
% 此为原Paper中Figure6复现代码% 状态初值如下
X0  = [-10.2999, 0.2575, -4.4997, 3.6258, 3.0922, 9.0156, 3.5099, -2.6645, 2.4552, -4.4921]';Node_Nums = 10;D = [   2 0 0 0 0 0 0 0 0 0;0 3 0 0 0 0 0 0 0 0;0 0 4 0 0 0 0 0 0 0;0 0 0 4 0 0 0 0 0 0;0 0 0 0 4 0 0 0 0 0;0 0 0 0 0 4 0 0 0 0;0 0 0 0 0 0 4 0 0 0;0 0 0 0 0 0 0 4 0 0;0 0 0 0 0 0 0 0 3 0;0 0 0 0 0 0 0 0 0 2;];
A   = [ 0 1 1 0 0 0 0 0 0 0;1 0 1 1 0 0 0 0 0 0;1 1 0 1 1 0 0 0 0 0;0 1 1 0 1 1 0 0 0 0;0 0 1 1 0 1 1 0 0 0;0 0 0 1 1 0 1 1 0 0;0 0 0 0 1 1 0 1 1 0;0 0 0 0 0 1 1 0 1 1;0 0 0 0 0 0 1 1 0 1;0 0 0 0 0 0 0 1 1 0;];
L = D - A;% 设置收敛相关参数
tBegin = 0;                     % 开始时间
tEnd   = 15;                    % 结束时间
dt     = 0.01;                   % 最小时间间隔
times  = (tEnd-tBegin) / dt;    % 迭代计算次数
time   = 1;
% 定义记录时间数组
t(:,1)   = tBegin;
% 定义图数据存储数组
X1(:,1) = X0;
U1(:,1) = -L * X0;
Delta(:,1) = X0' * L * X0;X2(:,1) = X0;
U2(:,1) = -L * X0;X3(:,1) = X0;
U3(:,1) = -L * X0;X4(:,1) = X0;
U4(:,1) = -L * X0;X5(:,1) = X0;
U5(:,1) = -L * X0;X6(:,1) = X0;
U6(:,1) = -L * X0;% 开始收敛计算
while(time <= times)t(:,time+1) = tBegin + time * dt;U1(:,time) = -L*X1(:,time);X1(:,time+1) = U1(:,time)*dt + X1(:,time);
%     X(:,time+1) = expm(-L * t(:,time+1)) * X0;if time<=13
%         U2(:,time) = -L*X2(:,time);
%         X2(:,time+1) = U2(:,time)*dt + X2(:,time);X2(:,time+1) = -L*X2(:,time) * dt + X2(:,time);
%         X2(:,time+1) = expm(-L * t(:,time+1)) * X0;else
%         U2(:,time) = -L*X2(:,time-13);
%         X2(:,time+1) = U2(:,time)*dt + X2(:,time-13);X2(:,time+1) = -L*X2(:,time-13) * dt + X2(:,time);
%         X2(:,time+1) = expm(-L * t(:,time+1-13)) * X0;endif time<=18
%         U3(:,time) = -L*X3(:,time);
%         X3(:,time+1) = U3(:,time)*dt + X3(:,time);X3(:,time+1) = -L*X3(:,time) * dt + X3(:,time);
%         X3(:,time+1) = expm(-L * t(:,time+1)) * X0;else
%         U3(:,time) = -L*X3(:,time-18);
%         X3(:,time+1) = U3(:,time)*dt + X3(:,time-18);X3(:,time+1) = -L*X3(:,time-13) * dt + X3(:,time);
%         X3(:,time+1) = expm(-L * t(:,time+1-18)) * X0;endif time<=26
%         U4(:,time) = -L*X4(:,time);
%         X4(:,time+1) = U4(:,time)*dt + X4(:,time);X4(:,time+1) = -L*X4(:,time) * dt + X4(:,time);
%         X4(:,time+1) = expm(-L * t(:,time+1)) * X0;else
%         U4(:,time) = -L*X4(:,time-26);
%         X4(:,time+1) = U4(:,time)*dt + X4(:,time-26);X4(:,time+1) = -L*X4(:,time-26) * dt + X4(:,time);
%         X4(:,time+1) = expm(-L * t(:,time+1-26)) * X0;endif time<=25X5(:,time+1) = -L*X5(:,time) * dt + X5(:,time);elseX5(:,time+1) = -L*X5(:,time-25) * dt + X5(:,time);endif time<=28X6(:,time+1) = -L*X6(:,time) * dt + X6(:,time);elseX6(:,time+1) = -L*X6(:,time-28) * dt + X6(:,time);endtime = time + 1;
end% 绘制图数据
subplot(3,2,1);
plot(t,X1(1,:), t,X1(2,:), t,X1(3,:), t,X1(4,:), t,X1(5,:), t,X1(6,:), t,X1(7,:), t,X1(8,:), t,X1(9,:), t,X1(10,:), 'linewidth',1.5)
xlabel("time(sec)");ylabel("state");
title("Time-Delay=0(sec)");subplot(3,2,2);
plot(t,X2(1,:), t,X2(2,:), t,X2(3,:), t,X2(4,:), t,X2(5,:), t,X2(6,:), t,X2(7,:), t,X2(8,:), t,X2(9,:), t,X2(10,:), 'linewidth',1.5)
xlabel("time(sec)");ylabel("state");
title("Time-Delay=0.13(sec)");subplot(3,2,3);
plot(t,X3(1,:), t,X3(2,:), t,X3(3,:), t,X3(4,:), t,X3(5,:), t,X3(6,:), t,X3(7,:), t,X3(8,:), t,X3(9,:), t,X3(10,:), 'linewidth',1.5)
xlabel("time(sec)");ylabel("state");
title("Time-Delay=0.18(sec)");subplot(3,2,4);
plot(t,X4(1,:), t,X4(2,:), t,X4(3,:), t,X4(4,:), t,X4(5,:), t,X4(6,:), t,X4(7,:), t,X4(8,:), t,X4(9,:), t,X4(10,:), 'linewidth',1.5)
xlabel("time(sec)");ylabel("state");
title("Time-Delay=0.26(sec)");subplot(3,2,5);
plot(t,X5(1,:), t,X5(2,:), t,X5(3,:), t,X5(4,:), t,X5(5,:), t,X5(6,:), t,X5(7,:), t,X5(8,:), t,X5(9,:), t,X5(10,:), 'linewidth',1.5)
xlabel("time(sec)");ylabel("state");
title("Time-Delay=0.25(sec)");subplot(3,2,6);
plot(t,X6(1,:), t,X6(2,:), t,X6(3,:), t,X6(4,:), t,X6(5,:), t,X6(6,:), t,X6(7,:), t,X6(8,:), t,X6(9,:), t,X6(10,:), 'linewidth',1.5)
xlabel("time(sec)");ylabel("state");
title("Time-Delay=0.28(sec)");

参考文献：

[1] N. Biggs. Algebraic Graph Theory. Cambridge Tracks in Mathematics. Cambridge University Press, 1974.

[2] I. Daubechies and J. C. Lagarias. Sets of matrices all infinite products of which converge. Linear Algebra and Its Applications, 161:227–263, 1992.

[3] I. Daubechies and J. C. Lagarias. Corrigendum/addendum to: Sets of matrices all infinite products of which converge. Linear Algebra and Its Applications, 327:69–83, 2001.

[4] J. P. Desai, J. P. Ostrowski, and V. Kumar. Modeling and control of formations of nonholonomic mobile robots. IEEE Trans. on Robotics and Automation, 17(6), December 2002.

[5] R. Diestel. Graph Theory, volume 173 of Graduate Texts in Mathematics. SpringerVerlag, 2000.

[6] A. Fax and R. M. Murray. Graph Laplacians and stabilization of vehicle formations. The 15th IFAC World Congress, June 2002.

[7] A. Fax and R. M. Murray. Information Flow and Cooperative Control of Vehicle Formations. The 15th IFAC World Congress, June 2002.

[8] M. Fiedler. Algebraic connectivity of graphs. Czechoslovak Mathematical Journal, 23(98):298–305, 1973.

[9] M. Fiedler. A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czechoslovak Mathematical Journal, 25(100):619–633, 1975.

[10] F. R. Gantmacher. Matrix Theory, volume II. Chelsea Publishing Company, 1959.

[11] C. Godsil and G. Royle. Algebraic Graph Theory, volume 207 of Graduate Texts in Mathematics. Springer, 2001.

[12] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1987.

[13] A. Jadbabaie, J. Lin, and S. A. Morse. Coordination of groups of mobile agents using nearest neighbor rules. IEEE Trans. on Automatic Control (to appear).

[14] R. W. Lawton, J. R. T. Beard and B. J. Young. A Decentralized Approach to Formation Maneuvers. IEEE Trans. on Robotics and Automation (to appear).

[15] N. A. Lynch. Distributed Algorithms. Morgan Kaufmann Publishers, Inc., 1997.

[16] M. Marcus and H. Minc. A Survey of Matrix Theory and Matrix Inequalities. Dover Publications, 1969.

[17] R. Merris. Laplacian matrices of a graph: a survey. Linear Algebra and its Applications, 197:143–176, 1994.

[18] M. Mesbahi. On a dynamic extension of the theory of graphs. Proc. of the American Control Conference, Anchorange, AL, May 2002.

[19] M. Mesbahi and F. Y. Hadegh. Formation flying of multiple spacecraft via graphs, matrix inequalities, and switching. AIAA Journal of Guidance, Control, and Dynamics, 24(2):369–377, March 2000.

[20] R. Olfati Saber and R. M. Murray. Consensus Protocols for Networks of Dynamic Agents. Proc. of the American Control Conference, June 2003.

[21] F. Paganini, J. Doyle, and S Low. Scalable laws for stable network congestion control. Proc. of the Int. Conf. on Decision and Control, Orlando, FL, Dec. 2001.

[22] C. W. Reynolds. Flocks, herds, and schools: a distributed behavioral model. Computer Graphics (ACM SIGGRAPH ’87 Conference Proceedings), 21(4):25–34, July 1987.

[23] J. Toner and Y. Tu. Flocks, herds, and schools: A quantitative theory of flocking. Physical Review E, 58(4):4828–4858, October 1998.

[24] T. Vicsek, A. Cziro´ok, E. Ben-Jacob, and O. Cohen, I. Shochet. Novel type of phase transition in a system of self-deriven particles. Physical Review Letters, 75(6):1226–1229, August, 1995.

【Paper】2003_Consensus Problems in Networks of Agents with Switching Topology and Time-Delays相关推荐

  1. 【Paper】2021_Consensus Control of Leader-Following Multi-Agent Systems in Directed Topology

    参考文献格式: Qinglai Wei, Xin Wang, Xiangnan Zhong and Naiqi Wu, "Consensus Control of Leader-Follow ...

  2. 【Paper】2021_Distributed Consensus Tracking of Networked Agent Systems Under Denial-of-Service Attack

    Y. Wan, G. Wen, X. Yu and T. Huang, "Distributed Consensus Tracking of Networked Agent Systems ...

  3. 【Paper】2009_Controllability of Multi-Agent Systems from a Graph-Theoretic Perspective 精炼版

    详细版请参考:[Paper]2009_Controllability of Multi-Agent Systems from a Graph-Theoretic Perspective 文章目录 5. ...

  4. 【Paper】2017_水下潜航器编队海洋勘测的协调控制方法研究

    友情链接:[paper]2019_Consensus Control of Multiple AUVs Recovery System Under Switching Topologies and T ...

  5. 【Paper】2019_Consensus Control of Multiple AUVs Recovery System Under Switching Topologies and Time D

    Zhang W, Zeng J, Yan Z, et al. Consensus control of multiple AUVs recovery system under switching to ...

  6. 【Paper】2015_El H_Decentralized Control Architecture for UAV-UGV Cooperation

    Decentralized Control Architecture for UAV-UGV Cooperation 1 Introduction 2 Problem Statement and Ar ...

  7. 【论文研读】【医学图像分割】【BCRNN】Fine-grained Recurrent Neural Networks for Automatic Prostate Segmentation ...

    [论文研读][医学图像分割][BCRNN]Fine-grained Recurrent Neural Networks for Automatic Prostate Segmentation inUl ...

  8. 【Paper】2009_Controllability of Multi-Agent Systems from a Graph-Theoretic Perspective

    [Rahmani, Amirreza and Ji, Meng and Mesbahi, Mehran and Egerstedt, Magnus. Controllability of Multi- ...

  9. 【ECCV2020】Spatio-Temporal Graph Transformer Networks for Pedestrian Trajectory Prediction

    [ECCV2020]用于行人轨迹预测的时空图 Transformer 网络 摘要 了解人群运动动力学对于现实世界的应用至关重要,例如监控系统和自动驾驶.这是具有挑战性的,因为它需要对具有社会意识的人群 ...

最新文章

  1. nfs原理及安装配置
  2. 崩溃了，一个HashMap跟面试官扯了半个小时
  3. Windows Server 2012体验之卸载辅助域控制器
  4. 为Eclipse安装功能扩展插件
  5. 新浪安装weiphp2.0的方法
  6. asp.net IsPostBack属性
  7. 流程控制示例：---3个实例示例：
  8. AWS云架构策略副总裁：开源是基础，也是未来
  9. Java 线程池实际运用案例
  10. 多元函数的微分学思维导图_多元函数微分学思维导图_多元函数微分学j_多元函数微分学思维导图...
  11. 【MM32F5270开发板试用】四、SPI的驱动，先点个屏幕
  12. Vue自学之路8-vue模版语法（事件绑定)
  13. Rayman的绝顶之路——Leetcode每日一题打卡17
  14. Vue3 router 默认加载 views 文件夹下全部vue文件
  15. Strategy策略模式详解与实例
  16. SAS更新SID问题
  17. IDEA中好用的功能(不断更新中)
  18. 联想电脑计算机怎么设置十进制,联想笔记本win10老友系统如何给电池设置充电阈值【图文】...
  19. 语音视频自动生成字幕在线工具介绍
  20. Word转PDF失败，报错Adobe Acrobat没激活！！！我如此穷，肯定会换个办法解决呀！

热门文章

  1. frp 路由穿透（github开源穿透软件）
  2. 函数atof,atoi，atol，strtod，strtol，strtoul 描述
  3. Python 操作 MySQL 的正确姿势
  4. 管理员账号无权限写入注册表
  5. 最简洁粗暴版的虚拟用户配置FTP
  6. Monkey原理初步和改良优化--Android自动化测试学习历程
  7. 《BI那点儿事》Microsoft 聚类分析算法——三国人物身份划分
  8. jquery 表单重置通用方法
  9. Struts2学习入门
  10. MSSQL同时操作数据加锁问题