First-order multi-agent average consistency

mathematical expression

The kinematic equation of the first-order multi-agent can be described as
x ˙ i ( t ) = ui ( t ) , i ∈ { 1 , 2 , 3 , … , N } \dot x_i(t) = u_i(t),i \in\{1,2,3,\dots,N\}x˙i​(t)=ui​(t),i∈{ 1,2,3,…,N }
inside$x_{i}ui\_\_$ _$u_i\left(t\right)$ is the control quantity, and the final expected result is
$$\underset{t\to T}{\lim }|x_i(t)-x_j(t)|=0$$

$$|x_i(t)-x_j(t)|=0,\forall t\ge T$$

Among them, the first equation in the above formula means that the time approaches $At\; T$ , the state of the agent tends to be consistent. The second equation expresses that when time exceeds$At\; time\; T,$ the agent's time has been consistent.

Express the first-order multi-agent consensus algorithm as
$$u_i\left(t\right)=-\sum _{j=1}^N{a}_{ij}(x_i(t)-x_j(t))$$
Here is a little trick. If it is expressed in the form of a matrix, then directly use the Laplacian matrix to express
$$u(t)=-x(t)\cdot L$$

simulation

Set the initial state of the agent to
$$x=\left[\begin{array}{cccc}1& 2& 3& 4\end{array}\right]$$
The connection of the agent is

智能体1 --- 智能体2
  |          |
  |          | 
智能体4 --- 智能体3

Then the Laplacian matrix is
$$L=\left[\begin{array}{cccc}2& -1& 0& -1\\ -1& 2& -1& 0\\ 0& -1& 2& -1\\ -1& 0& -1& 2\end{array}\right]$$
The simulation code is (matlab)

clc;clear;close

x = [1 2 3 4];
u = [];
A = [0 1 0 1;
     1 0 1 0;
     0 1 0 1;
     1 0 1 0;];
B = [2 0 0 0;
     0 2 0 0;
     0 0 2 0;
     0 0 0 2;];
L = B - A;
dt = 0.001;
k = 3;
lambda = 2;
mu = 0.1;

for i = 1:4000
    u1 = -x(end,:) * L;
    x1 = x(end,:) + u1 * dt;

    u = [u;u1];
    x = [x;x1];
end

Draw the curve of the control quantity and the curve of the state quantity

Fig1 = figure(1);
plot(0.001:0.001:4.001,x,'LineWidth', 1.5);
xlabel('t');
ylabel('x');
legend('agent1','agent2','agent3','agent4');
print(Fig1,'x','-dpng','-r600')

Fig2 = figure(2);
plot(0.001:0.001:4.000,u,'LineWidth', 1.5);
xlabel('t');
ylabel('u');
legend('agent1','agent2','agent3','agent4');
print(Fig2,'u','-dpng','-r600')

The change of state variables over time is shown as

The change of the control amount over time is as follows

. It can be seen that in the end, the states of the four agents are all consistent.