A Second-Order Discrete Multi-Agent Matrix Scale Consistency Control Method

Abstract

The present invention discloses a second-order discrete multi-agent matrix scale consistency control method, belonging to the technical field of multi-agent, which includes the following steps: S1. Construct the communication relationship among multi-agents and describe the second-order multi-agent system with a communication topology model; S2. Construct the dynamic model of the second-order discrete multi-agent system and design the matrix scale consistency control protocol under the second-order discrete multi-agent system; S3. Characterize the relationship between the eigenvalues of two system matrices and the Laplacian matrix, and deduce the consistency condition of the directed topology; S4. Analyze the influence of the network structure, coupling gain and discrete time interval on the system matrix scale consistency. By adopting the above-mentioned second-order discrete multi-agent matrix scale consistency control method, the multi-agents can better adapt to the complex network structure in real life, realize the effective cooperation of multi-agents in the network, and at the same time can more accurately reflect the complex interaction between individual states.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-agent, and particularly to a second-order discrete multi-agent matrix scale consensus control method. Background Art

[0002] Multi-Agent Systems (MAS for short), as an important branch in the field of artificial intelligence, are composed of multiple agents that can independently execute tasks, communicate with each other, cooperate or compete, and are widely used in multiple fields such as satellites, unmanned aerial vehicles, and transportation. The consensus control of multi-agent systems is at the forefront of the current swarm intelligence technology field and has important application value in national economy and national defense construction. The multi-agent consensus control problem refers to a group of autonomous agents achieving the completion of a common task through local information sharing and distributed control strategies. In the past ten-odd years, the work on multi-agent consensus has focused on the research of traditional complete consensus, that is, all agents asymptotically reach a common state.

[0003] However, in practical applications, agents may need to compete for limited resources such as positions or communication channels. At the same time, they may also need to cooperate to achieve common goals such as completing a task or maintaining the stability of the group. This requires considering the bipartite consensus problem, that is, a consistent state where competition and cooperation coexist. In practical applications, not only cooperation and competition need to be considered, but also different collaborative tasks need to be achieved according to different task assignments and physical variables. This requires considering the group consensus problem, that is, different groups tend to different consensus states.

[0004] However, most of the analyses on group consensus are based on an assumption that the sum of the elements in each row of the Laplacian matrix of the entire communication topology is zero. Obviously, this assumption is relatively strict and it does not apply to arbitrary topologies.

[0005] Therefore, some scholars have studied scalar scale consensus, also known as proportional consensus, which means that a certain scalar attribute in the system can reach consensus according to a preset proportional relationship, not only requiring the attribute values between agents to tend to be the same, but also requiring a certain proportional relationship between these values.

[0006] So far, meaningful results have been achieved in the research on matrix-scaled consensus control. Si Han-Chen, Kai Xin-tian, Jie Mei (Matrix-scaled consensus for scecond-order uncertain multi-agent systems under a directed graph. Proceedings of the 42nd Chinese Control Conference, 2023) This article studies the matrix-scaled consensus algorithm for second-order uncertain multi-agent systems under a directed topology graph. This algorithm only requires the matrix scale to be invertible, greatly relaxing the constraints on the matrix scale.

[0007] Currently, many works on multi-agent consensus focus on the research of scalar-scaled consensus. However, in many actual multi-dimensional multi-agent systems, scalar scales often cannot accurately reflect the complex interactions between individual states. Summary of the Invention

[0008] The purpose of the present invention is to provide a second-order discrete multi-agent matrix-scaled consensus control method, enabling multi-agents to better adapt to complex network structures in real life, realizing effective cooperation among multi-agents in the network, and at the same time being able to more accurately reflect the complex interactions between individual states.

[0009] To achieve the above object, the present invention provides a second-order discrete multi-agent matrix-scaled consensus control method, including the following steps:

[0010] S1. Apply the method of graph theory to construct the communication relationship between multi-agents, and use a communication topology model to describe the second-order multi-agent system;

[0011] S2. Construct the dynamic model of the second-order discrete multi-agent system, and design a matrix-scaled consensus control protocol for the second-order discrete multi-agent system;

[0012] S3. Through variable transformation, transform the second-order discrete multi-agent system into a simplified system, characterize the relationship between the eigenvalues of two system matrices 、 and the Laplacian matrix , and use matrix theory, stability theory and analysis methods to deduce the consensus conditions of the directed topology;

[0013] S4. Analyze the influence of the network structure, coupling gain and discrete time interval on the matrix-scaled consensus of the system.

[0014] Preferably, in step S1, the communication topology model describing the second-order multi-agent system is a directed graph denotes, where is a series of node sets; is the edge set; , N is the number of all agents; is a directed graph 's adjacency matrix;

[0015] In the directed graph , defines the direction of the edge pointing from node to node ; The neighbors of the agent are defined as .

[0016] Preferably, if node can directly receive the information of node , then , otherwise ; The directed graph contains a cluster of directed spanning trees if and only if at least one node has a directed path to all other nodes; A directed path is a sequence formed by a series of edge sets, representing the connection relationship between nodes; If there is at least one agent and a directed path to any other agent in a directed network, then it is said that this directed graph has a directed spanning tree.

[0017] Preferably, the dynamic equation of the second-order discrete multi-agent system in step S2 is:

[0018] ;

[0019] where represents the position of the th multi-agent at time, , ; represents the velocity of the th multi-agent at time, ; represents the control input information of the th multi-agent at time, ; represents the discrete time interval; , respectively represent the position and velocity of the th multi-agent at time.

[0020] Preferably, the matrix-scale consistent control protocol for the second-order discrete multi-agent system in step S2 is:

[0021] ;

[0022] Among them, and represent the positive coupling gain; , , are defined as: ; represents the matrix in the state of multi-agent ,

[0023] represents the position of multi-agent , represents time, represents the position of multi-agent ; represents the communication relationship between agent and agent ; represents the speed of agent ; represents the set of agents that all agents communicate or interact directly with agent ;

[0024] Substitute the control input into the second-order discrete dynamics equation to get:

[0025] ;

[0026] Among them, the matrix is the Laplacian matrix composed of, and its definition is , .

[0027] Preferably, let , and , then the system is rewritten in matrix form:

[0028] ;

[0029] ;

[0030] Among them, represents the set of positions and speeds of all multi-agents at time, represents the identity matrix;

[0031] Let , , , and , thus, a simplified system matrix is obtained:

[0032] ;

[0033] ;

[0034] ;

[0035] wherein, denotes the set of the position differences and velocity differences between all multi - agents and the first agent at time denotes the identity matrix of

[0036] Preferably, the definition of matrix scale consistency is:

[0037] For any initial state, if and only if at time and , holds, the matrix scale consistency of the second - order discrete multi - agent system is achieved, that is, the states of all agents reach consistency; by characterizing the relationship between the eigenvalues of the two system matrices , and the Laplacian matrix , for the directed topological graph , if has zero eigenvalues and the real parts of other eigenvalues are positive, then there exists a non - negative matrix , such that , , and moreover has a rank of , where .

[0038] Preferably, based on the definition result of matrix scale consistency, the following theorem is derived:

[0039] The second - order discrete multi - agent system reaches consistency if and only if the algebraic multiplicity of the eigenvalue 1 of is and all other eigenvalues of are inside the unit circle. In addition, when the system reaches consistency, then when , , where each column of is the left eigenvector corresponding to the eigenvalue 0 of ;

[0040] Under the action of the control input, the states of each agent can reach consistency in the state space, and the states of all agents achieve matrix scale consistency.

[0041] Therefore, the present invention adopts the above-mentioned second-order discrete multi-agent matrix scale consistency control method, which has the following beneficial effects:

[0042] (1) The present invention combines graph theory with the control of multi-agents, visually showing the network topology formed between the positions and velocities of multi-agents and the control action relationship;

[0043] (2) In the design process of the present invention, matrix scale is introduced, and a positive definite or negative definite matrix scale is assigned to each agent. The agent updates its state according to the product of the relative matrix scale state and the matrix scale symbol;

[0044] (3) The matrix scale consistency research proposed by the present invention can more accurately reflect the complex interactions between individual states;

[0045] (4) The matrix scale consistency protocol proposed by the present invention can be used to handle multi-dimensional state spaces and multi-agent systems with complex dynamic characteristics, and is more applicable to actual multi-dimensional multi-agent systems.

[0046] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Description of the Drawings

[0047] Figure 1 is a step block diagram of an embodiment of a second-order discrete multi-agent matrix scale consistency control method of the present invention;

[0048] Figure 2 is a network topology structure diagram of an embodiment of a second-order discrete multi-agent matrix scale consistency control method of the present invention;

[0049] Figure 3 is the present invention in Figure 2 the position trajectory diagram of multi-agents under the topological communication graph; Figure 3 In (a) of x the position component diagram of six multi-agents; Figure 3 In (b) of y the position component diagram of six multi-agents.

[0050] Figure 4 is the velocity trajectory diagram of multi-agents of the present invention under Figure 2 the topological communication graph;

[0051] Figure 5 is the state of multi-agents at different times of the present invention under Figure 2 the topological communication graph. Detailed implementation manners

[0052] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0053] Unless otherwise defined, the technical terms or scientific terms used in the present invention shall have the ordinary meanings understood by those of ordinary skill in the field to which the present invention pertains.

[0054] Embodiment 1

[0055] As Figure 1 shown, the present invention provides a second-order discrete multi-agent matrix scale consistency control method, including the following steps:

[0056] S1. Apply the method of graph theory to construct the communication relationship between multi-agents, and use the communication topology model to describe the second-order multi-agent system; wherein the communication topology model describing the second-order multi-agent system is represented by a directed graph wherein, is a series of node sets; is the edge set; , N is the number of all agents; is the adjacency matrix of the directed graph ;

[0057] In the directed graph , defines the direction of the edge pointing from node to node ; the neighbor of the agent is defined as .

[0058] If node can directly receive the information of node , then , otherwise ; the directed graph contains a cluster of directed spanning trees if and only if at least one node has a directed path leading to all other nodes; a directed path is a sequence formed by a series of edge sets, representing the connection relationship between nodes; if there is at least one agent in a directed network and there is a directed path leading to any other agent, then it is said that this directed graph has a directed spanning tree.

[0059] S2. Construct the dynamic model of a second-order discrete multi-agent system. In an actual multi-agent system, an individual usually has multi-dimensional states. These states do not exist in isolation but are interdependent and interact with each other. Therefore, by introducing a matrix scale, a broader distribution space is given to the agents, rather than being limited to a single dimension, reflecting the complex interactions between individual states. This property is applicable to dealing with multi-agent systems with multi-dimensional state spaces and complex dynamic characteristics. Under the matrix scale consistency algorithm, each agent asymptotically converges to the final point of its respective different convergence points through the inverse of its own scale matrix, and the final state of the agent is not limited to a straight line but extends to an open subspace of the state space. The network topology structure diagram in the present invention is as shown in Figure 2 where there are agents 1 - 6, a total of six agents.

[0060] Given a dynamic equation of a second-order discrete multi-agent system:

[0061] ;

[0062] where, represents the position of the th multi-agent at time, , ; represents the velocity of the th multi-agent at time, ; represents the control input information of the th multi-agent at time, ; represents the discrete time interval; , respectively represent the position and velocity of the th multi-agent at time. In the present invention, the position trajectory diagram and velocity trajectory diagram of the six agents under the Figure 2 topological communication graph are respectively as shown in Figure 3 , Figure 4 .

[0063] , respectively represent the states of the agents at time , represents the control input of the agents at time .

[0064] Design a matrix-scale consensus control protocol for a second-order discrete multi-agent system; the matrix-scale consensus control protocol for a second-order discrete multi-agent system is as follows:

[0065] ;

[0066] where, and represent positive coupling gains; , , is defined as: ; represents the matrix in the state of multi-agent ;

[0067] represents the position of multi-agent ; represents time; represents the position of multi-agent ; represents the communication relationship between agent and agent ; represents the velocity of agent ; represents the set of agents that all agents communicate directly or interact with agent .

[0068] Substitute the control input into the second-order discrete dynamics equation to get:

[0069] ;

[0070] where, the matrix is the Laplacian matrix composed of, and its definition is , .

[0071] S3. Through variable transformation, transform the second-order discrete multi-agent system into a simplified system, characterize the relationship between the eigenvalues of the two system matrices , and the Laplacian matrix , and use matrix theory, stability theory and analysis methods to derive the consensus condition of the directed topology; let , and , then the system is rewritten in matrix form:

[0072] ;

[0073] ;

[0074] where, denote the set of positions and velocities of all multi - agents at a moment, denote the identity matrix;

[0075] Let , , , and , thus, a simplified system matrix is obtained:

[0076] ;

[0077] ;

[0078] ;

[0079] wherein, denote the set of position differences and velocity differences between all multi - agents and the first agent at a moment, denote the identity matrix of

[0080] S4. Analyze the influence of network structure, coupling gain and discrete time interval on the matrix scale consistency of the system. The definition of matrix scale consistency is:

[0081] The velocity of each multi - agent changes with time. For any initial state, when and only when at the time of and , holds, the matrix scale consistency of the second - order discrete multi - agent system is achieved, that is, the states of all agents reach consistency; by characterizing the relationship between the eigenvalues of two system matrices , and the Laplacian matrix , for the directed topological graph , if has zero eigenvalues and the real parts of other eigenvalues are positive, then there exists a non - negative matrix , , such that , , and moreover has a rank of , where .

[0082] In the present invention, the states of multi - agents at different moments under the Figure 2 topological communication graph are as Figure 5As shown. According to the definition result of matrix scale consistency, the following theorem is deduced:

[0083] The second-order discrete multi-agent system reaches consensus if and only if the algebraic multiplicity of the eigenvalue 1 of is and all other eigenvalues of are inside the unit circle. In addition, when the system reaches consensus, then when , , where each column of is the left eigenvector corresponding to the eigenvalue 0 of and satisfies ;

[0084] Under the action of the control input of each agent, the state of each agent can reach consensus in the state space, and the states of all agents achieve matrix scale consistency.

[0085] Therefore, the present invention adopts the above-mentioned second-order discrete multi-agent matrix scale consistency control method, enabling multi-agents to better adapt to complex network structures in real life, realizing effective cooperation among multi-agents in the network, and at the same time being able to more accurately reflect the complex interactions between individual states.

[0086] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that they can still modify or equivalently replace the technical solutions of the present invention, and these modifications or equivalent replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A second-order discrete multi-agent matrix scale consistency control method, characterized in that: The following steps are involved: S1. Use graph theory to construct the communication relationship between multiple agents and use the communication topology model to describe the second-order multi-agent system. S2. Construct a dynamic model of a second-order discrete multi-agent system and design a matrix-scale consistent control protocol for the second-order discrete multi-agent system; The dynamic equation of the second-order discrete multi-agent system in step S2 is: ； in, Indicates Multi-agent The location at the moment, , ; Indicates Multi-agent The speed of time, ; Indicates Multi-agent The control input at the moment, ; H represents a discrete time interval; , Respectively represent Multi-agent Position and speed at a given moment; The matrix scale consistent control protocol under the second-order discrete multi-agent system in step S2 is: ； in, and represents positive coupling gain; , , Defined as: ; Representing multi-agent The matrix under the state of ; Representing multi-agent location; Indicates time; Representing multi-agent location; Representing an Agent and Agent The communication relationship between them; Representing an Agent speed; Represents all agents and agents A collection of agents that communicate or interact directly; The control input Substituting into the second-order discrete dynamics equation, we get: ； in, L The matrix is ​​the Laplacian matrix composed of , ; S3. Through variable conversion, the second-order discrete multi-agent system is transformed into a simplified system, characterizing the two system matrices , and the Laplacian matrix The relationship between eigenvalues, using matrix theory, stability theory and analytical methods, derives the consistency conditions of directed topology; S4. Analyze the influence of network structure, coupling gain and discrete time interval on the scale consistency of system matrix.

2. A second-order discrete multi-agent matrix scale consistency control method according to claim 1, characterized in that: The communication topology model of the second-order multi-agent system described in step S1 is a directed graph It indicates that, is a set of nodes; is an edge set; , N is the number of all agents; It is a directed graph The adjacency matrix of In a directed graph middle, The definition is from the node Point to Node The direction of the edge; the agent's neighbors are defined as .

3. A second-order discrete multi-agent matrix scale consistency control method according to claim 2, characterized in that: If the node Directly received node information, then ,otherwise .

4. A second-order discrete multi-agent matrix scale consistency control method according to claim 3, characterized in that: make , and , then the system is rewritten in matrix form: ； ； in, express The set of positions and velocities of all multi-agents at the moment, express Identity matrix; make , , , and , so we get a simplified system matrix: ； ； ； in, express The set of position differences and speed differences between all multi-agents and the first agent at the moment; express The identity matrix of .

5. A second-order discrete multi-agent matrix scale consistency control method according to claim 4, characterized in that: The matrix scale consistency is defined as: For any initial state, if and only if hour and , When it is established, the matrix scale consistency of the second-order discrete multi-agent system is achieved, that is, the states of all agents are consistent; by characterizing the two system matrices , and the Laplacian matrix The relationship between eigenvalues, for directed topological graphs ,if have zero eigenvalues, and the real parts of the other eigenvalues ​​are positive, then there exists a non-negative matrix , , so that , ,and The rank of ,in .

6. A second-order discrete multi-agent matrix scale consistency control method according to claim 5, characterized in that: According to the definition of matrix scale consistency, the following theorem is derived: A second-order discrete multi-agent system reaches consensus if and only if The algebraic multiplicity of the eigenvalue 1 is and All other eigenvalues ​​of are inside the unit circle, and furthermore, the system reaches consistency, then when hour, , ,in Each column of The left eigenvector corresponding to the eigenvalue 0 of ; Under the control input, the state of each agent reaches consistency in spatial state, and the states of all agents achieve matrix scale consistency.

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