Second-order discrete multi-agent matrix scale consistency control method

Abstract

The invention discloses a second-order discrete multi-agent matrix scale consistency control method, and belongs to the technical field of multi-agents, and the method comprises the following steps: S1, constructing a communication relationship among multiple agents, and describing a second-order multi-agent system by using a communication topology model; s2, constructing a kinetic model of the second-order discrete multi-agent system, and designing a matrix scale consistency control protocol under the second-order discrete multi-agent system; s3, depicting the relationship between the characteristic values of the two system matrixes # imgabs0 # and # imgabs1 # and the characteristic values of the Laplacian matrix # imgabs2 #, and deducing the consistency condition of the directed topology; and S4, analyzing the influence of the network structure, the coupling gain and the discrete time interval on the consistency of the system matrix scale. By the adoption of the second-order discrete multi-agent matrix scale consistency control method, the multiple agents can better adapt to a complex network structure in real life, effective cooperation of the multiple agents in the network is achieved, and meanwhile complex interaction between individual states can be reflected more accurately.

Description

Technical Field

[0001] This invention relates to the field of multi-agent technology, and in particular to a method for controlling the scale consistency of a second-order discrete multi-agent matrix. Background Technology

[0002] Multi-agent systems (MAS), an important branch of artificial intelligence, consist of multiple intelligent agents capable of independently performing tasks, communicating with each other, cooperating, or competing. They are widely used in satellites, unmanned aerial vehicles, transportation, and other fields. Consistency control in MAS is at the forefront of current swarm intelligence technology and has significant application value in national economic and defense construction. The MAS consistency control problem refers to a group of autonomous intelligent agents achieving a common task through local information sharing and distributed control strategies. For the past decade or so, work on MAS consistency has focused on traditional complete consistency research, meaning that all agents asymptotically reach a common state.

[0003] However, in practical applications, agents may need to compete for limited resources, such as location or communication channels, and they may also need to cooperate to achieve common goals, such as completing a task or maintaining group stability. This necessitates considering the binary consistency problem, i.e., a consistent state where competition and cooperation coexist. In practical applications, not only cooperation and competition need to be considered, but different collaborative tasks also need to be implemented based on different task allocations and physical variables. This requires considering the group consistency problem, i.e., different groups tending towards different consistency states.

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

[0005] Therefore, some scholars have studied scalar scale consistency, also known as proportional consistency, which refers to the consistency of a certain scalar attribute of the system according to a pre-set proportional relationship. This requires not only that the attribute values ​​of the agents tend to be the same, but also that there is a certain proportional relationship between these values.

[0006] Significant progress has been made in research on matrix-scaled consensus control. The paper by Si Han-Chen, Kai Xin-tian, and Jie Mei (Matrix-scaled consensus for scecond-order uncertain multi-agent systems under a directed graph. Proceedings of the 42nd Chinese Control Conference, 2023) studies a matrix-scaled consensus algorithm for second-order uncertain multi-agent systems under a directed topological graph. This algorithm only requires the matrix scale to be invertible, greatly reducing the constraints on the matrix scale.

[0007] Many current studies on multi-agent consensus focus on scalar scale consensus. However, in many real-world multi-dimensional multi-agent systems, scalar scales often fail to accurately reflect the complex interactions between individual states. Summary of the Invention

[0008] The purpose of this invention is to provide a second-order discrete multi-agent matrix scale consistency control method, which enables multi-agents to better adapt to complex network structures in real life, realize effective cooperation among multi-agents in the network, and more accurately reflect the complex interactions between individual states.

[0009] To achieve the above objectives, this invention provides a second-order discrete multi-agent matrix scaling consistency control method, comprising the following steps: S1. Apply graph theory to construct the communication relationships between multiple agents, and use a 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. S3. By transforming variables, the second-order discrete multi-agent system is converted into a simplified system, and the two system matrices are characterized. , and Laplace matrix By studying the relationships between eigenvalues ​​and applying matrix theory, stability theory, and analytical methods, the consistency conditions of directed topologies are derived. S4. Analyze the impact of network structure, coupling gain, and discrete time interval on the system matrix scale consistency.

[0010] Preferably, the communication topology model describing the second-order multi-agent system in step S1 uses a directed graph. It means that, among them, It is a set of nodes; It is an edge set; , N The number of all agents; Directed graph The adjacency matrix; In directed graphs middle, Defined from node Pointing to node The direction of the edge; the agent's neighbors are defined as... .

[0011] Preferably, if the node Can directly receive nodes Information, then ,otherwise Directed graph A set of directed spanning trees exists if and only if There must be at least one node with a directed path leading to all other nodes; a directed path is a sequence of edges that represent the connections between nodes; if there is at least one agent and a directed path leading to any other agent in a directed network, then the directed graph is said to have a directed spanning tree.

[0012] Preferably, the dynamic equation of the second-order discrete multi-agent system in step S2 is: ; in, Indicates the first Multiple agents in Location at any given moment , ; Indicates the first Multiple agents in The speed of time ; Indicates the first Multiple agents in Real-time control input information, ; Represents discrete time intervals; , They represent the first Multiple agents in Position and velocity at any given moment.

[0013] Preferably, the matrix scaling consistency control protocol for the second-order discrete multi-agent system in step S2 is as follows: ; in, and Indicates positive coupling gain; , , Defined as: ; Representing multi-agent Matrix in the state Representing multi-agent Location Indicates time, Representing multi-agent Location; Represents intelligent agents and intelligent agents The communication relationship between them; Represents intelligent agents speed; Represents all intelligent agents and intelligent agents A collection of intelligent agents that communicate or interact directly; Control input Substituting into the second-order discrete dynamics equation, we get: ; Here, the matrix is ​​a Laplace matrix composed of elements, defined as follows: , .

[0014] Preferred, let , and The system can then be rewritten in matrix form: ; ; in, express The set of positions and velocities of all multi-agents at any given moment. express identity matrix; make , , , and Therefore, a simplified system matrix is ​​obtained: ; ; ; in, express The set of position and velocity differences between all multi-agents and the first agent at any given time; express The identity matrix.

[0015] Preferably, the definition of matrix scale consistency is: For any initial state, if and only if hour and , Upon its establishment, it achieved matrix scaling consistency in the second-order discrete multi-agent system, meaning that the states of all agents reached consistency; this was achieved by characterizing the matrices of the two systems. , and Laplace matrix The relationship between eigenvalues ​​for a directed topological graph ,if have If there are zero eigenvalues ​​and the real parts of the other eigenvalues ​​are positive, then there exists a non-negative matrix. , , making , ,and The rank is ,in .

[0016] Preferably, based on the definition of matrix scale consistency, the following theorem is derived: A second-order discrete multi-agent system achieves consistency if and only if The algebraic multiplicity of the eigenvalue 1 is and All other eigenvalues ​​lie within the unit circle. Furthermore, when the system reaches uniformity, then when hour, , ,in Each column is The left eigenvector corresponding to the eigenvalue 0, and satisfying ; Under the influence of control inputs, each agent can achieve spatial consistency in its state, and the states of all agents achieve matrix-scale consistency.

[0017] Therefore, the present invention employs the above-mentioned second-order discrete multi-agent matrix scale consistency control method, which has the following beneficial effects: (1) This invention combines graph theory with multi-agent control to vividly and intuitively represent the network topology formed between the positions and velocities of multiple agents and the control relationship; (2) In the design process, the present invention introduces a matrix scale, assigning each agent a positive definite or negative definite matrix scale. The agent updates its state according to the product of the relative matrix scale state and the matrix scale sign. (3) The matrix scale consistency study proposed in this invention can more accurately reflect the complex interactions between individual states; (4) The matrix scale consensus protocol proposed in this invention can be used to process multi-dimensional state spaces and multi-agent systems with complex dynamic characteristics, and is more suitable for practical multi-dimensional multi-agent systems.

[0018] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0019] Figure 1 This is a flowchart illustrating the steps of an embodiment of a second-order discrete multi-agent matrix scale consistency control method according to the present invention. Figure 2 This is a network topology diagram of an embodiment of a second-order discrete multi-agent matrix scale consistency control method of the present invention; Figure 3 This invention is in Figure 2 A multi-agent location trajectory diagram under a topological communication graph; Figure 3 (a) in the text represents a multi-agent system with six agents. x Location component map; Figure 3 (b) in the example represents a multi-agent system with six agents. y Location component map.

[0020] Figure 4 This invention is in Figure 2 Velocity trajectory diagram of multiple agents under topological communication graph; Figure 5 This invention is in Figure 2 The states of multiple agents at different times in a topological communication graph. Detailed Implementation

[0021] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0022] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0023] Example 1 like Figure 1 As shown, this invention provides a second-order discrete multi-agent matrix scale consistency control method, comprising the following steps: S1. Using graph theory, construct the communication relationships between multiple agents and describe the second-order multi-agent system using a communication topology model; where the communication topology model describing the second-order multi-agent system uses a directed graph. It means that, among them, It is a set of nodes; It is an edge set; , N The number of all agents; Directed graph The adjacency matrix; In directed graphs middle, Defined from node Pointing to node The direction of the edge; the agent's neighbors are defined as... .

[0024] If node Can directly receive nodes Information, then ,otherwise Directed graph A set of directed spanning trees exists if and only if There must be at least one node with a directed path leading to all other nodes; a directed path is a sequence of edges that represent the connections between nodes; if there is at least one agent and a directed path leading to any other agent in a directed network, then the directed graph is said to have a directed spanning tree.

[0025] S2. Construct a dynamic model of a second-order discrete multi-agent system. In practical multi-agent systems, individuals typically possess multi-dimensional states. These states are not isolated but interdependent and mutually influential. Therefore, by introducing a matrix scale, agents are given a broader distribution space, rather than being limited to a single dimension, reflecting the complex interactions between individual states. This characteristic is suitable for handling multi-dimensional state spaces and multi-agent systems with complex dynamic characteristics. Under the matrix scale consensus algorithm, each agent converges asymptotically to its own different convergence points through the inverse of its own scale matrix. Furthermore, the final state of an agent is not limited to a straight line but extends to an open subspace of the state space. The network topology diagram in this invention is as follows: Figure 2 As shown, there are six agents, from agent 1 to agent 6.

[0026] Given the dynamic equations of a second-order discrete multi-agent system: ; in, Indicates the first Multiple agents in Location at any given moment , ; Indicates the first Multiple agents in The speed of time ; Indicates the first Multiple agents in Real-time control input information, ; Represents discrete time intervals; , They represent the first Multiple agents in Position and velocity at any given moment. In this invention... Figure 2 The position trajectory diagrams and velocity trajectory diagrams of the six agents under the topology communication graph are as follows: Figure 3 , Figure 4 As shown.

[0027] , They represent Multiple agents at time state, express Multiple agents at time Control input.

[0028] A matrix scaling consensus control protocol for second-order discrete multi-agent systems is designed; the matrix scaling consensus control protocol for second-order discrete multi-agent systems is as follows: ; in, and Indicates positive coupling gain; , , Defined as: ; Representing multi-agent The matrix in the state; Representing multi-agent Location; Indicates time; Representing multi-agent Location; Represents intelligent agents and intelligent agents The communication relationship between them; Represents intelligent agents speed; Represents all intelligent agents and intelligent agents A collection of intelligent agents that communicate or interact directly.

[0029] Control input Substituting into the second-order discrete dynamics equation, we get: ; Here, the matrix is ​​a Laplace matrix composed of elements, defined as follows: , .

[0030] S3. By transforming variables, the second-order discrete multi-agent system is converted into a simplified system, and the two system matrices are characterized. , and Laplace matrix By examining the relationships between eigenvalues ​​and applying matrix theory, stability theory, and analytical methods, the consistency conditions of directed topologies are derived; let , and The system can then be rewritten in matrix form: ; ; in, express The set of positions and velocities of all multi-agents at any given moment. express identity matrix; make , , , and Therefore, a simplified system matrix is ​​obtained: ; ; ; in, express The set of position and velocity differences between all multi-agent agents and the first agent at any given time. express The identity matrix.

[0031] S4. Analyze the impact of network structure, coupling gain, and discrete time interval on the system matrix scaling consistency. The definition of matrix scaling consistency is: The velocity of each multi-agent is time-varying, and for any initial state, the velocity is determined if and only if hour and , Upon its establishment, it achieved matrix scaling consistency in the second-order discrete multi-agent system, meaning that the states of all agents reached consistency; this was achieved by characterizing the matrices of the two systems. , and Laplace matrix The relationship between eigenvalues ​​for a directed topological graph ,if have If there are zero eigenvalues ​​and the real parts of the other eigenvalues ​​are positive, then there exists a non-negative matrix. , , making , ,and The rank is ,in .

[0032] In this invention Figure 2 The states of multiple agents at different times in a topological communication graph are as follows: Figure 5 As shown. Based on the definition of matrix scale consistency, the following theorem is derived: A second-order discrete multi-agent system achieves consistency if and only if The algebraic multiplicity of the eigenvalue 1 is and All other eigenvalues ​​lie within the unit circle. Furthermore, when the system reaches uniformity, then when hour, , ,in Each column is The left eigenvector corresponding to the eigenvalue 0, and satisfying ; Under the influence of control inputs, each agent can achieve spatial consistency in its state, and the states of all agents achieve matrix-scale consistency.

[0033] Therefore, the present invention adopts the above-mentioned second-order discrete multi-agent matrix scale consistency control method, which enables multi-agents to better adapt to the complex network structure in real life, realize the effective cooperation of multi-agents in the network, and more accurately reflect the complex interaction between individual states.

[0034] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to 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; 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: 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, ; represents a discrete time interval; , Respectively represent Multi-agent Position and speed at a given moment.

5. A second-order discrete multi-agent matrix scale consistency control method according to claim 4, characterized in that: 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, The matrix is ​​the Laplacian matrix composed of , .

6. A second-order discrete multi-agent matrix scale consistency control method according to claim 5, 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 .

7. A second-order discrete multi-agent matrix scale consistency control method according to claim 6, 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 .

8. A second-order discrete multi-agent matrix scale consistency control method according to claim 7, 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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