An improved edge reconnection operation to enhance the convergence rate of MAS consensus

Abstract

The application discloses an improved edge reconnection operation method for improving MAS consistency convergence rate, relates to the technical field of multi-agent system communication topology optimization, and comprises the following steps: a communication topology graph is established based on a multi-agent system containing n agents, a Laplacian matrix is obtained by using an adjacency matrix and a degree matrix of the graph, a second smallest eigenvalue of the Laplacian matrix and a Fiedler vector are calculated, absolute values of differences between components of the Fiedler vector are determined to determine edges to be deleted and inserted, the Laplacian matrix eigenvalue of the topology graph after the edges are deleted and inserted is recalculated, and a graph with an increased second smallest eigenvalue of the Laplacian matrix is regarded as a communication topology structure graph of a new multi-agent system; and effectiveness of the improved edge reconnection operation method is verified. The improved edge reconnection operation increases a necessary condition for increasing algebraic connectivity, maximizes an increasing range of the algebraic connectivity, reduces the iteration number of the algorithm, and reduces consumption of computing resources.

Description

Technical Field

[0001] This invention relates to the field of communication topology optimization technology for multi-agent systems, specifically a method for improving edge reconnection operation to increase the convergence rate of MAS consensus. Background Technology

[0002] In recent years, with the continuous development of computer, communication, and control science and technology, multi-agent systems (MAS) have been widely applied in various fields such as sensor networks, smart grids, intelligent transportation, data networks, and distributed computing. In practical applications, the higher the convergence rate of a multi-agent system, the more sensitive its perception capabilities, the more efficient its execution capabilities, and the better its flexibility. For systems with high real-time requirements and fast response times, such as UAV swarm combat systems and smart grids, the consistent convergence rate of the system is crucial to the success of the mission. Therefore, improving the consistent convergence rate of multi-agent systems is of paramount importance.

[0003] The consensus convergence rate of a multi-agent system (MAS) quantifies how quickly the system reaches consensus from its initial state under a certain consensus protocol. Currently, research on improving the consensus convergence rate of MAS typically focuses on two aspects: designing and improving consensus protocols and optimizing communication topologies. Regarding the design and improvement of system consensus protocols, early consensus protocols were asymptotically convergent. To improve the consensus convergence rate of MAS, numerous researchers have conducted in-depth research and continuous improvements on consensus protocols, designing consensus protocols with finite-time convergence and specified-time convergence.

[0004] Regarding the optimization of communication topology, research shows that when a consensus protocol for a multi-agent system is given (such as continuous consensus, periodic sampling consensus, or event-triggered consensus), the convergence rate of the multi-agent system is determined by the eigenvalues ​​of its communication topology graph: the greater the algebraic connectivity, the faster the system's consensus convergence rate, thus enabling agents to reach consensus more quickly. Here, algebraic connectivity is the second smallest eigenvalue of the Laplace matrix corresponding to the communication topology graph of the multi-agent system. Currently, commonly used methods for optimizing communication topology include operations such as adding edges, rotating edges, swapping edges, and reconnecting edges. These edge operations can be used to construct greedy algorithms to iteratively optimize the communication topology structure, thereby improving algebraic connectivity. However, constructing greedy algorithms and performing iterative optimization consumes significant computational resources and time.

[0005] Based on this, the purpose of this invention is to provide an improved method for edge reconnection operation to increase the convergence rate of MAS consistency, thereby reducing the number of iterations and computational resources in communication topology optimization. Summary of the Invention

[0006] To overcome the shortcomings of existing technologies, this invention provides an improved method for edge reconnection operations to enhance the convergence rate of MAS consistency. It proposes a new approach to improved edge reconnection operations, adding a necessary condition for increasing algebraic connectivity based on the original edge reconnection operations, thereby maximizing the increase in algebraic connectivity and effectively reducing the number of algorithm iterations and lowering the consumption of computing resources.

[0007] To achieve the above objectives, the present invention provides the following technical solution:

[0008] An improved edge reconnection operation method to enhance the convergence rate of MAS (Multi-Aspect Recognition System) includes the following steps:

[0009] (1) Based on a multi-agent system containing n agents, its communication topology is modeled using a simple undirected graph in graph theory, and then the communication topology graph of the multi-agent system is constructed.

[0010] (2) Based on the communication topology of the multi-agent system established in step (1), the Laplace matrix is ​​obtained by using the adjacency matrix and degree matrix of the graph, and the second smallest eigenvalue of the Laplace matrix and its Fiedler vector are calculated.

[0011] (3) Determine the edges to be deleted and inserted in the communication topology graph based on the absolute value of the difference between the components of the Fiedler vector, recalculate the eigenvalues ​​of the Laplace matrix of the topology graph with deleted and inserted edges, and regard the graph with the second smallest eigenvalue of the Laplace matrix as the new communication topology graph of the multi-agent system.

[0012] (4) Verify the effectiveness of the improved edge reconnection operation method.

[0013] In this invention, the communication topology of the multi-agent system is constructed as follows:

[0014] Let the communication topology of the multi-agent system be G = (V(G), E(G)), where V(G) = {v1, ..., v2} n Let} denote the set of vertices in graph G, and E(G) denote the set of edges in graph G; the initial state of agent i is represented by physical quantities such as its position coordinates, velocity, and yaw angle that change with time t, denoted as vector σ. i (t), i = 1, ..., n; if we consider agent i as a moving point mass on a plane, then its state is characterized by its position coordinates, i.e., σ i (t)=[x i (t),y i (t)] T , where x i (t) represents the x-coordinate of agent i, y i (t) represents the ordinate of agent i.

[0015] In this invention, the calculation process of the Fiedler vector in step (2) is as follows:

[0016] Based on the communication topology G = (V(G), E(G)) of the multi-agent system established in step (1), its adjacency matrix A(G) = [a ij ]∈R n×n Defined as: if (v i ,v j If )∈E(G), then a ij =1, otherwise a ij =0; The degree matrix D(G) of graph G is an n-dimensional diagonal matrix whose main diagonal elements are the number of neighbors of each vertex;

[0017] The Laplace matrix of graph G is L(G) = D(G) - A(G);

[0018] Calculate the eigenvalues ​​of the Laplacian matrix L(G), and arrange the eigenvalues ​​in non-decreasing order as follows: λ1(G)≤λ2(G)≤…≤λ n The second smallest eigenvalue λ2(G) is called the algebraic connectivity; the larger λ2(G) is, the faster the system's uniform convergence rate, that is, the less time the system takes to reach uniformity.

[0019] Calculate the eigenvector corresponding to the eigenvalue λ2(G), denoted as u. (2) (G)=[u1 (2) (G),...,u n (2) (G)] T This vector is also known as the Fiedler vector.

[0020] In this invention, the steps for obtaining the communication topology diagram of the new multi-agent system in step (3) are as follows:

[0021] 1) Based on the Fiedler vector obtained in step (2), select an edge (v) to be deleted from the communication topology graph. i ,v j ), satisfying |u i (2) (G)-u j (2) (G)| has the smallest value, which is denoted as e. min If multiple options exist, choose any one of them.

[0022] 2) Find the set of non-adjacent vertex pairs in the communication topology graph, where any vertex v s and v t All satisfy e min <|u s(2) (G)-u t (2) (G)|, denote the set of this vertex pair as V(D);

[0023] 3) Delete edge (v) i ,v j The resulting diagram is denoted as Gv. i v j According to graph Gv i v j The corresponding Laplacian matrix is ​​used to calculate the second smallest eigenvalue λ2(Gv). i v j ) and its corresponding eigenvector u (2) (Gv i v j At this point, in set V(D), we search for |u k (2) (Gv i v j )-u l (2) (Gv i v j The pair of non-adjacent vertices with the largest value {v} k ,v l}, Insert edge (v k ,v l The resulting graph is notated as G'; if there are multiple edges with the largest values, choose any one of them.

[0024] 4) Calculate the eigenvalues ​​of the Laplacian matrix corresponding to graph G', where the second smallest eigenvalue λ2(G') is the new key performance indicator; if λ2(G') > λ2(G), then the edge reconnection operation maximizes the increase of the key performance indicator λ2(G), and graph G' is regarded as the new communication topology of the multi-agent system; if λ2(G') ≤ λ2(G), then repeat step 1).

[0025] In this invention, the effectiveness of the improved edge reconnection operation method is verified in step (4) as follows:

[0026] Define the system error function when the system reaches consistency.

[0027] Where, vector σ i (t) represents the state of agent i at time t, and x* represents the state of the agent when the system reaches consensus, which is usually the average value of the initial states in the system;

[0028] By plotting the curve corresponding to the system error function using Matlab, we can intuitively reflect how the system error changes over time before and after the change in the communication topology of the multi-agent system. It is a curve that is a decreasing function with time t, and the function value tends to zero. Each vertex on the curve can represent the error of the system at a certain moment, and the faster the function value tends to zero, the less time it takes for the system to reach consensus.

[0029] Compared with the prior art, the beneficial effects of the present invention are:

[0030] This invention proposes an improved edge reconnection operation, which uses graphs from graph theory for modeling. Each agent is regarded as a vertex in the graph, and the communication between agents is regarded as an edge in graph theory. Edges are deleted and added based on Fiedler vectors to increase the algebraic connectivity of the communication topology graph. Compared with the traditional edge reconnection operation method, this invention adds a necessary condition for increasing algebraic connectivity, thus maximizing the increase in algebraic connectivity and effectively reducing the number of algorithm iterations and reducing the consumption of computing resources. Attached Figure Description

[0031] Figure 1 This is a flowchart of the method of the present invention.

[0032] Figure 2 This is a schematic diagram illustrating the communication topology optimization process of the multi-agent system of the present invention.

[0033] Figure 3 This is a comparison chart of error curves for the multi-agent system of the present invention.

[0034] Figure 4 This is a flowchart of the improved edge reconnection operation of the present invention. Detailed Implementation

[0035] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0036] Combination Figure 1-4 This invention will be described in detail below. This invention proposes a method for improving the consensus convergence rate of multi-agent systems based on edge reconnection operations, such as... Figure 1 As shown, the specific steps include:

[0037] Step 1: Based on the given communication topology of the multi-agent system, model it using a simple undirected graph from graph theory, and then construct the communication topology graph of the multi-agent system.

[0038] This invention takes a multi-agent system with six agents as an example, and its communication topology is as follows: Figure 2 G in a As shown in Figure G. a vertex v in i Representing agent i, graph G a In the graph G, the undirected edges represent the communication relationships between agents. We will now model this graph using graph theory. a For G a =(V(G) a ), E(G a )), where V(G a )={v1,...,v6} represents graph G a The set of vertices, E(G) a ) represents graph G a The set of edges in the middle, i.e., E(G) a )={(v1, v2), (v1, v3), (v1, v4), (v2, v3), (v4, v5), (v4, v6)}. Figure G a Undirected edge (v) i v j An undirected edge (v2, v3) represents communication between agents i and j; for example, an undirected edge (v2, v3) represents communication between agents 2 and 3. (Graph G) a Non-adjacent edges {v s v t The edge {v3, v5} represents that agents s and t do not communicate with each other. For example, the non-adjacent edge {v3, v5} represents that agents 3 and 5 do not communicate with each other. Assume that the state of agent i (1≤i≤6) is represented by its corresponding two-dimensional coordinates. The initial states of the six agents are shown in the table below:

[0039] Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 x-axis 1 0 1 3 5 5 ordinate 2 0 -2 -2 -2 2

[0040] That is, the initial values ​​of the 6 agents in the horizontal axis direction are denoted as x(0) = (x i (0)) T =(1, 0, 1, 3, 5, 5) T Let i = 1, 2, ..., 6, and the initial value along the vertical axis be denoted as y(0) = (y i (0)) T = (2, 0, -2, -2, -2, 2) T , i = 1, ..., 6.

[0041] Step 2: Based on the communication topology graph of the multi-agent system established in Step 1, as defined by the degree matrix and adjacency matrix above, the communication topology graph G... a Corresponding D(G) a ) and A(Ga They are respectively:

[0042] Here D(G) a The element in the i-th row and i-th column of ) is related to vertex v i The number of adjacent vertices (i.e., the number of agents communicating with agent i). For example, the element in the 3rd row and 3rd column indicates that the number of vertices adjacent to v3 is 2 (i.e., the number of agents communicating with agent 3 is 2). If A(G a If the element in the i-th row and j-th column of a vertex is 1, then it represents vertex v. i and vertex v j There exists an edge between them, that is, they are adjacent; if A(G) a If the element in the i-th row and j-th column of a vertex is 0, then it represents vertex v. i and vertex v j There are no edges between them, meaning they are not adjacent. For example, if the element in the 1st row and 2nd column is 1, it means there is an edge between vertex v1 and vertex v2 (i.e., agents 1 and 2 communicate); if the element in the 1st row and 5th column is 0, it means there is no edge between vertex v1 and vertex v5 (i.e., agents 1 and 5 do not communicate). Then the communication topology graph G... a The corresponding Laplace matrix L(G) a )for:

[0043] Due to the Laplace matrix L(G) a L(G) is a positive semi-definite real symmetric matrix. The Laplace matrix L(G) is calculated using Matlab. a The eigenvalues ​​of ) are arranged in non-decreasing order as follows:

[0044] λ1(G a )=0<λ2(G a )=0.4384<λ3(G a )=1<λ4(G a )=3≤λ5(G a )=3<λ6(G a ) = 4.5616, where the second smallest eigenvalue is λ2(G a The algebraic connectivity of λ²(G) = 0.4384 is considered a key performance indicator of the system. Calculations using Matlab yield the eigenvalue λ²(G). a The corresponding feature vector is u. (2) (G a = [-0.2610, -0.4647, -0.4647, 0.2610, 0.4647, 0.4647] T Vector u (2) (G aThe components at agent i are shown in the table below:

[0045] Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 <![CDATA[u1 (2) (G a )]]> <![CDATA[u2 (2) (G a )]]> <![CDATA[u3 (2) (G a )]]> <![CDATA[u4 (2) (G a )]]> <![CDATA[u5 (2) (G a )]]> <![CDATA[u6 (2) (G a )]]> -0.2610 -0.4647 -0.4647 0.2610 0.4647 0.4647

[0046] Step 3: Based on Step 2, calculate the key performance indicator λ2(G) a ) = 0.4384. The improved edge reconnection operation alters the system's communication topology, aiming to maximize the improvement of λ²(G) = 0.4384. a This accelerates the consistency convergence rate, allowing the system to converge to consistency more quickly. The specific steps are as follows:

[0047] (1)|u i (2) (G a )-u j (2) (G a | is a vector u (2) (G a The absolute value of the difference between the components at adjacent agent i and agent j. From the communication topology graph G a edge set E(G) a In the given set {(v1, v2), (v1, v3), (v1, v4), (v2, v3), (v4, v5), (v4, v6)}, select one edge (v1, v2), v3, (v1, v3), (v1, v4), (v2, v3), (v4, v5), (v4, v6)} to be deleted. i v j ), satisfying |u i (2) (G a )-u j (2) (G a The minimum value is denoted as e. min If multiple entries exist, select any one. The following table is obtained through example calculations:

[0048] <![CDATA[(v1,v2)]]> <![CDATA[(v1,v3)]]> <![CDATA[(v1,v4)]]> <![CDATA[(v2,v3)]]> <![CDATA[(v4,v5)]]> <![CDATA[(v4,v6) <!-- 4 -->]]> <![CDATA[|u i (2) (G a )-u j (2) (G a )|]]> 0.2037 0.2037 0.5220 0 0.2037 0.2037

[0049] As can be seen from the table above, due to e min =|u2 (2) (G a )-u3 (2) (G a Since | = 0, we choose edge (v2, v3) as the edge to be deleted, that is, to disconnect the communication between agent 2 and agent 3.

[0050] (2)|u s (2) (G a )-u t (2) (G a | is a vector u (2) (G aThe absolute value of the difference between the components at non-adjacent agents s and t. Find the set of non-adjacent vertex pairs in the communication topology graph, where any vertex pair {v} s v t All satisfy e min =0<|u s (2) (G a )-u t (2) (G a Let V(D) be the set of vertices.

[0051] <![CDATA[{v1,v6}]]> <![CDATA[{v1,v5}]]> <![CDATA[{v2,v6}]]> <![CDATA[{v2,v5}]]> <![CDATA[{v2,v4}]]> <![CDATA[{v3,v4}]]> <![CDATA[{v3,v5}]]> <![CDATA[{v3,v6]]]> <![CDATA[{v5,v6}]]> <![CDATA[u s (2) (G a )-u t (2) (G a )]]> 0.725 0.7257 0.9294 0.9294 0.7257 0.7257 0.9294 0.9294 0

[0052] Based on the above calculations, e is satisfied. min =0<|u s (2) (G a )-u t (2) (G a The set of non-adjacent vertex pairs of )| is V(D)={{v1,v6},{v1,v5},{v2,v6},{v2,v5},{v2,v4},{v3,v4},{v3,v5},{v3,v6}}.

[0053] (3) The graph obtained after deleting edge (v2, v3) is notated as G. a -v2v3, even if agent 2 and agent 3 lose communication, such as Figure 2 G in a As shown in -v2v3. From the definitions of the degree matrix and adjacency matrix above, we can obtain graph G. a The degree matrix D(G) corresponding to -v2v3 a -v2v3) and adjacency matrix A(G a -v2v3) are respectively:

[0054]

[0055] So, Figure G a The Laplace matrix L(G) corresponding to -v2v3 a -v2v3) is:

[0056]

[0057] Due to the Laplace matrix L(G) a -v2v3) is a positive semi-definite real symmetric matrix. The Laplace matrix L(G) is calculated using Matlab. a The eigenvalues ​​of -v2v3) are arranged in non-decreasing order as follows:

[0058] λ1(G a -v2v3)=0<λ2(G a -v2v3)=0.4384<λ3(G a -v2v3)=1<λ4(G a -v2v3)=1<λ5(G a -v2v3)=3<λ6(G a -v2v3)=4.5616.

[0059] The second smallest eigenvalue λ2(G) can be obtained through calculation using Matlab. a The eigenvector corresponding to ) = 0.4384 is u. (2) (G a -v2v3)=[-0.2610,-0.4647,-0.4647,0.2610,0.4647,0.4647] T Vector u (2) (G a The components of -v2v3) at agent i are shown in the following table:

[0060] Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 <![CDATA[u1 (2) (G a -v2v3)]]> <![CDATA[u2 (2) (G a -v2v3)]]> <![CDATA[u3 (2) (G a -v2v3)]]> <![CDATA[u4 (2) (G a -v2v3)]]> <![CDATA[u5 (2) (G a -v2v3)]]> <![CDATA[u6 (2) (G a -v2v3)]]> -0.2610 -0.4647 -0.4647 0.2610 0.4647 0.4647

[0061] At this point, we search in set V(D) for |u k (2) (G a -v2v3)-u l (2) (G a -v2v3)| The pair of non-adjacent vertices with the largest value {v k v l}

[0062]

[0063] As shown in the table above, there exist multiple pairs of non-adjacent vertices that satisfy |u k (2) (G a -v2v3)-u l (2) (G a The maximum value is found in the following graphs: {v2, v6}, {v2, v5}, {v3, v5}, and {v3, v6}. Now, arbitrarily select non-adjacent vertex pairs {v3, v5} and connect them. This allows agents 3 and 5 to communicate with each other. The resulting graph is notated as G. b ,like Figure 2 As shown. At this point, an edge (v3, v5) that maximizes the increase in key performance indicators is inserted.

[0064] (5) According to Figure 2 G in bAnd as can be seen from the definitions of degree matrix and adjacency matrix mentioned above, graph G b The corresponding degree matrix D(G) b ) and adjacency matrix A(G b They are respectively: So, Figure G b The corresponding Laplace matrix L(G) b )for:

[0065] Due to the Laplace matrix L(G) b L(G) is a positive semi-definite real symmetric matrix. The Laplace matrix L(G) is calculated using Matlab. b The eigenvalues ​​of ) are arranged in non-decreasing order as follows:

[0066] λ1(G b )=0<λ2(G b )=0.6571<λ3(G b )=1<λ4(G b )=2.5293<λ5(G b )=3<λ6(G b =4.8136. (Figure)

[0067] G b The key indicator λ2(G) b ) = 0.6571, and λ2(G a )=0.4384<λ2(G b = 0.6571. Through the specific implementation of the improved edge reconnection operation described above, by deleting edge (v2, v3) and adding edge (v3, v5) in the communication topology graph, even if agent 2 and agent 3 disconnect from communication, agent 3 and agent 5 can still communicate with each other. This results in the largest increase in the algebraic connectivity of the multi-agent system's communication topology graph. The modified communication topology graph of the multi-agent system is shown below. Figure 2 G in b As shown.

[0068] Step 4: To verify the effectiveness of the improved edge reconnection operation method, define the system error function when the system reaches consistency. Where the vector σ i (t) represents the state of agent i at time t, and x* represents the state when the system reaches consensus, usually the average of the initial states. When the system reaches consensus, the final coordinates of all agents are (x*, y*), where,

[0069] x*=(x*,y*) T Define the systematic error function. During the process of the system reaching consensus, the error Error(t) → 0. If Error(t) = 0 when t > T, then the system reaches consensus at time T. The function graph can be plotted using Matlab, which visually reflects the change of system error over time before and after the change in the communication topology of the multi-agent system, such as... Figure 3 As shown. Note that, Figure 3 The solid red line and dashed blue line in the middle represent the communication topology diagram G. a and G b The curve showing the change in system error over time. By comparison, it can be seen that the optimized system error approaches zero faster than the unoptimized system error. In particular, when the communication duration is 3 seconds, G... b The systematic error is approximately G a of This indicates the communication topology diagram corresponding to the multi-agent system in G. b It has a higher uniformity convergence rate.

[0070] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. An improved method for improving the rate of convergence of the consensus of a Multi-Agent System (MAS) by improving the edge-reconnection operation, characterized in that, The method comprises the following steps: (1) based on a multi-agent system comprising n agents, the communication topology structure of the multi-agent system is modeled by a simple undirected graph in graph theory, and a communication topology graph of the multi-agent system is constructed; (2) according to the communication topology graph of the multi-agent system established in step (1), a Laplacian matrix is obtained by using an adjacency matrix and a degree matrix of the graph, and a second smallest eigenvalue of the Laplacian matrix and a Fiedler vector are calculated; (3) according to the absolute value of the difference between components of the Fiedler vector, edges to be deleted and inserted in the communication topology graph are determined, the eigenvalue of the Laplacian matrix of the topology graph after the edges are deleted and inserted is recalculated, and the graph with the increased second smallest eigenvalue of the Laplacian matrix is regarded as a new communication topology structure graph of the multi-agent system; (4) verifying the effectiveness of the improved edge reconnection operation method; The calculation process of the Fiedler vector in step (2) is as follows: Based on the communication topology diagram of the multi-agent system established in step (1) , Representation diagram The set of vertices, Representation diagram The set of edges in the middle, and its adjacency matrix Defined as: if ,but ,otherwise ;picture degree matrix It is A 3D diagonal matrix whose main diagonal elements are the number of neighbors of each vertex; Figure laplacian matrix of the graph ; Eigenvalues of the Laplacian matrix are computed and sorted in non-decreasing order as The second smallest eigenvalue is called algebraic connectivity; The larger the algebraic connectivity is, the faster the consensus convergence rate of the system is, i.e., the system reaches consensus in less time; The eigenvalues are computed The corresponding eigenvector, denoted by This vector is also called the Fiedler vector; The steps of obtaining the new communication topology structure graph of the multi-agent system in step (3) are as follows: 1) Select an edge to be deleted from the communication topology graph according to the Fiedler vector obtained in step (2) , satisfying value is minimum, and the minimum value is denoted as ; if there are multiple edges, one of them is selected optionally 2) find a set of vertex pairs in the communication topology graph, any vertex of which satisfies and , , and record this set of vertex pairs as ; 3) Delete edges The resulting diagram is According to the diagram Calculate the second smallest eigenvalue of the corresponding Laplacian matrix. and its corresponding eigenvectors At this time, in the set In the middle, looking for The pair of non-adjacent vertices with the largest value Insert edge The resulting diagram is If there are multiple edges with the largest value, choose any one of them. 4) according to the graph the corresponding Laplacian matrix, compute the eigenvalues of this matrix, where the second smallest eigenvalue is the new key performance indicator; if then the edge reconnection operation maximizes the increase of the key performance indicator consider the graph as the communication topology graph of the new multi-agent system; if then re-execute step 1).

2. The method of claim 1, wherein the improved edge-reconnection operation increases the rate of convergence of the MAS consensus. The communication topology graph of the multi-agent system is constructed as follows: Let the communication topology graph of the multi-agent system be where denotes the set of vertices of the graph denotes the set of edges in the graph The initial state of an agent is represented by its position coordinates, velocity, and yaw angle, denoted by the vector If agent is regarded as a point mass moving on a plane, its state is characterized by the position coordinates, i.e. where denotes the horizontal coordinate of agent denotes the vertical coordinate of agent .​​ 3. The method of claim 2, wherein the improved edge-reconnection operation increases the rate of convergence of the MAS consensus. In step (4), the effectiveness of the improved edge reconnection operation method is verified as follows: Defining a system error function when the system reaches consensus , where the vector represents the state of the agent at time In the state of the system at time represents the state of the agent at the time of consensus, which is typically the average of the initial states in the system; By plotting the curve corresponding to the system error function using Matlab, we can intuitively reflect how the system error changes over time before and after the change in the communication topology of the multi-agent system; it is a function that changes with time. The function is a decreasing function, and the curve of the function value tending towards zero. Each vertex on the curve can represent the error of the system at a certain moment. The faster the function value tends towards zero, the less time it takes for the system to reach consistency.

Images