Two-way Formation Control Method for Multi-Agent Systems Based on Dynamic Event Triggering

Abstract

The present invention provides a two-way formation control method for multi-agent systems based on dynamic event triggering, which relates to the technical field of cooperative control and formation of multi-agent systems. The method includes establishing a second-order nonlinear multi-agent system model containing competitive relationships; designing a distributed adaptive controller; constructing a dynamic event triggering mechanism with dynamic parameters; constructing a controlled error system, and analyzing the controlled error system to obtain the sufficient conditions and exponential convergence rate for the multi-agent system to achieve adaptive two-way formation control. In addition, the present invention proves that there is no Zeno effect in the multi-agent system through inequality derivation, and designs numerical simulation cases to verify the effectiveness of the control strategy of the present invention and the correctness of the two-way formation criterion. The present invention reduces the triggering frequency by introducing a dynamic event triggering mechanism, saves control resources, and simplifies the process of analyzing the two-way formation of the second-order nonlinear multi-agent system.

Description

Technical Field

[0001] The present invention relates to the technical field of cooperative control and formation of multi-agent systems, and particularly to a two-way formation control method for multi-agent systems based on dynamic event triggering. Background Art

[0002] A multi-agent system (abbreviated as MAS) consists of multiple independent agent nodes, and these nodes complete complex tasks through information exchange and coordinated cooperation. Due to its broad application prospects, multi-agent systems have attracted the attention of many researchers from fields such as biology, electronic information, and engineering. Among them, the formation problem, as an important research direction of multi-agent systems, shows great application potential in fields such as microgrids, robot cooperative control, and unmanned aerial vehicle cooperation.

[0003] Traditionally, the research on multi-agent formation is mostly based on the assumption of the continuity of information interaction between agent nodes. However, in practical applications, due to the constraints of actual factors such as energy limitations, communication bandwidth limitations, and external interference, long-term stable and continuous communication is often difficult to achieve. To address this challenge, event-triggered control strategies have been introduced into the research of multi-agent control.

[0004] Under the event-triggered mechanism, agent nodes only exchange information with adjacent nodes when specific conditions (i.e., event-triggering conditions) are met, and the controller also updates the control intensity only at this time. This strategy can effectively reduce unnecessary communication and control updates, thereby saving control resources.

[0005] The dynamic event-triggered mechanism is a further improvement of the static event-triggered strategy. By introducing internal dynamic parameters to adjust the triggering threshold, the dynamic event-triggered mechanism can more flexibly respond to changes in the system state, further reduce the control frequency, and optimize resource utilization. It should be noted that the static event-triggered mechanism can be regarded as a special case of the dynamic event-triggered mechanism.

[0006] Although significant achievements have been made in the field of multi-agent cooperative formation control, in some practical situations, there may be both cooperative and competitive relationships between system nodes. For the two-way formation problem of multi-agents with competitive relationships, current research is still insufficient. In addition, the application of the dynamic event-triggered mechanism in the formation analysis of multi-agent systems is also relatively rare. Summary of the Invention

[0007] Therefore, an embodiment of the present invention provides a two-way formation control method for multi-agent systems based on dynamic event triggering, which is used to solve the problem in the prior art that it is difficult to effectively balance the communication frequency and control performance under complex relationships between nodes (such as coexistence of cooperation and competition), resulting in low resource utilization rate and limited system stability.

[0008] To solve the above problems, an embodiment of the present invention provides a two-way formation control method for a multi-agent system based on dynamic event triggering. The method includes:

[0009] Establish a second-order nonlinear multi-agent system model including competitive relationships;

[0010] Based on the established second-order nonlinear multi-agent system model including competitive relationships, design a distributed adaptive controller by introducing an adaptive control gain mechanism into the controller;

[0011] Based on the designed distributed adaptive controller, construct a dynamic event triggering mechanism with dynamic parameters;

[0012] Based on the distributed adaptive controller and the dynamic event triggering mechanism, construct a controlled error system, and analyze the controlled error system to obtain the sufficient conditions and exponential convergence rate for the multi-agent system to achieve adaptive two-way formation control.

[0013] Preferably, the method further includes: proving through inequality derivation that under the action of the dynamic event triggering mechanism and the distributed adaptive controller, the multi-agent system will not exhibit the Zeno effect during the two-way formation process.

[0014] Preferably, the establishment of the second-order nonlinear multi-agent system model including competitive relationships specifically includes:

[0015] Define the dynamic equation of the second-order nonlinear multi-agent system model, including the position and velocity variables of the agents, and a nonlinear continuous function. Its mathematical expression is:

[0016]

[0017] In the formula, t represents the time; represents the derivative of x i (t), and x i (t) represents the position of the i-th agent at time t; represents the derivative of v i (t), and v i (t) ∈ R n represents the velocity of the i-th agent at time t, where i = 1, 2,..., N, and R n represents the n-dimensional Euclidean space, and N represents the number of agents; u i (t) ∈ R n represents the control action applied to the i-th node; is a nonlinear continuous function;

[0018] Define the dynamic behavior of the leader agent. Its mathematical expression is:

[0019]

[0020] wherein, represents the derivative of x0(t), and x0(t) ∈ R n represents the position of the leader agent; represents the derivative of v0(t), and v0(t) ∈ R n represents the velocity of the leader agent;

[0021] A signed directed graph is introduced to represent the communication topology of the multi-agent system, including:

[0022] Signed directed graph: The communication topology of the multi-agent system is represented by a signed directed graph G = (V, E, A), where V = {1, 2,..., N} is the set of nodes, representing each agent; is the set of edges, representing the communication connections between agents; A = [a ij ∈ R N×N is the adjacency matrix of graph G;

[0023] Adjacency matrix A: The element a ij in the matrix defines the relationship between node i and node j:

[0024] a ij > 0 indicates a cooperative relationship between node i and node j; a ij < 0 represents a competitive relationship between node i and node j; a ij = 0 indicates no communication connection between node i and node j;

[0025] Laplacian matrix L:

[0026] In the multi-agent system, all agent nodes are divided into two clusters and For each cluster, the adjacent nodes within the cluster have a cooperative relationship, while the connections between nodes in the two clusters are competitive; define the matrix i = 1, 2,..., N, if the node then q i = 1, if the node then q i = -1;

[0027] Determine the conditions for the multi-agent system to achieve two-way formation, which are expressed as:

[0028]

[0029] wherein, represents the preset formation, T represents the transpose.

[0030] Preferably, by introducing an adaptive control gain mechanism into the controller, a distributed adaptive controller is designed, specifically including:

[0031] The mathematical expression of the distributed adaptive controller designed by introducing an adaptive control gain mechanism into the controller is:

[0032]

[0033] In the formula, α represents the free weight parameter, α > 0; represents the k-th triggering time of the i-th agent, k ∈ {1, 2,...}; represents the position of the j-th agent at time represents the preset formation; δ 1i represents the feedback control gain, δ 1i ≥ 0; δ 2i represents the adaptive control gain;

[0034] By defining the error vector, the update formula of the adaptive control gain δ 2i (t) is obtained:

[0035] i = 1, 2,..., N

[0036] where

[0037]

[0038] φ vi (t) = q i v i (t) - v0(t)

[0039] In the formula, represents the derivative of δ 2i (t); p represents a constant, p > 0; φ xi (t), φ vi (t) represent the error vectors.

[0040] Preferably, a dynamic event triggering mechanism with dynamic parameters is constructed, specifically including: defining the measurement errors γ xi (t), γ vi (t) as follows:

[0041] Define the measurement errors γ xi (t), γ vi (t) as follows:

[0042]

[0043] Based on the defined measurement error γ xi(t), γ vi (t), construct a dynamic event-triggering mechanism with dynamic parameters, which is expressed as follows:

[0044]

[0045] In the formula, inf{} represents the infimum; g i (t) represents the triggering function; represents the derivative of S i (t), S i (t) represents a dynamic variable and satisfies the initial value S0(t) > 0; β i represents the event update parameter, β i > 0; c represents the weight coefficient, c > 0; λ min represents the minimum eigenvalue; is a positive definite matrix, ψ2 = ||L + diag{δ 21 (t),..., δ 2N (t)}|| 2 .

[0046] Preferably, based on the distributed adaptive controller and the dynamic event-triggering mechanism, construct a controlled error system, and analyze the controlled error system to obtain the sufficient conditions and exponential convergence rate for the multi-agent system to achieve adaptive two-way formation control, specifically including:

[0047] Based on the distributed adaptive controller and the dynamic event-triggering mechanism, construct a controlled error system;

[0048] Apply the Lyapunov stability theory, and analyze the stability and convergence of the multi-agent system by constructing a Lyapunov function and taking the derivative;

[0049] According to the derivative result of the Lyapunov function, obtain the sufficient conditions and exponential convergence rate for the multi-agent system to achieve adaptive two-way formation control.

[0050] Preferably, the controlled error system is:

[0051]

[0052] In the formula, respectively represent the derivatives of φ x (t), φ v (t), represents convolution, I n is the n-dimensional identity matrix,

[0053] Preferably, the Lyapunov function is as follows:

[0054]

[0055] wherein, V(t) represents the Lyapunov function.

[0056] Preferably, according to the derivative result of the Lyapunov function, the sufficient condition for the multi-agent system to achieve adaptive two-way formation control is:

[0057]

[0058] wherein, f represents the convergence coefficient; θ = 2 + h 2 +(c - 2α)λ min B - αλ min (L + + L +T ) + α 2 , h represents the Lipschitz coefficient, and L +T represents the transpose of L + ;

[0059] Preferably, according to the derivative result of the Lyapunov function, the exponential convergence rate for the multi-agent system to achieve adaptive two-way formation control is -f.

[0060] From the above technical solutions, it can be seen that the present invention application has the following beneficial effects:

[0061] (1) Focusing on the second-order multi-agent system under the dynamic event-triggered framework, it is committed to achieving adaptive two-way formation control with an isolated leader agent. Through a carefully designed distributed adaptive controller, it ensures that the multi-agent system can follow and maintain a two-way formation pattern with the leader. The present invention uses the dynamic event-triggered strategy and the Lyapunov stability theory to strictly deduce the sufficient condition for achieving two-way formation and its exponential convergence rate, and at the same time uses technical means such as inequality analysis to effectively avoid potential Zeno phenomena in the control process.

[0062] (2) Compared with traditional continuously acting control strategies, the event-triggered control strategy adopted by the present invention shows its unique advantages, that is, it activates the control input only when the preset dynamic trigger conditions are met, significantly improving the adaptability and resource efficiency of the system in complex and changeable engineering environments. To further optimize the performance, the present invention innovatively introduces a dynamic event-triggered mechanism, which effectively reduces the trigger frequency by dynamically adjusting the trigger threshold, thereby reducing the computational burden while maximizing the utilization of control resources.

[0063] (3) In view of the coexistence of cooperation and competition relationships in the real system environment, when constructing the communication topology model of the multi-agent system, the present invention particularly considers the existence of competition factors. In the controller design stage, an adaptive control gain is creatively integrated, which can not only automatically adjust to the optimal control parameters, but also significantly enhance the control flexibility and robustness of the system, providing an effective solution for dealing with the formation control problem of multi-agent systems with competition relationships. Description of the Drawings

[0064] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly describe the drawings required in the embodiments. By referring to the drawings, the features and advantages of the present invention will be more clearly understood. The drawings are schematic and should not be construed as imposing any limitations on the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts. Among them:

[0065] Figure 1 It is a flowchart of a two-way formation control method for a multi-agent system based on dynamic event triggering provided in the embodiment;

[0066] Figure 2 It is a schematic diagram of the communication topology of the multi-agent system in the embodiment;

[0067] Figure 3 It is a schematic diagram of the error vector trajectory in the embodiment, where (a) is the curve of the position error vector of the multi-agent system, and (b) is the curve of the velocity error vector of the multi-agent system; the abscissa of (a) and (b) represents time, and the ordinate represents the formation error;

[0068] Figure 4 It is the velocity curve of the multi-agent system in the embodiment, where the abscissa in the figure represents time and the ordinate represents the agent velocity;

[0069] Figure 5 It is a schematic diagram of the position trajectories of the multi-agent system and the leader agent in the embodiment; the three coordinates in the figure are three-dimensional space coordinates;

[0070] Figure 6 It is a time slice diagram of the formation of the multi-agent system in the embodiment, where (a) is the formation of the multi-agent system at 0 seconds, (b) is the formation of the multi-agent system at 2 seconds, (c) is the formation of the multi-agent system at 7 seconds, and (d) is the formation of the multi-agent system at 8 seconds; the three coordinates of (a), (b), (c), and (d) are three-dimensional space coordinates;

[0071] Figure 7 It is a schematic diagram of the evolution curve of the adaptive control gain in the embodiment; the abscissa in the figure represents time and the ordinate represents the control gain;

[0072] Figure 8 It is a comparison graph of trigger moments under different trigger mechanisms in the embodiment; among them, (a) is the trigger moment graph under the static event trigger mechanism, and (b) is the trigger moment graph under the dynamic event trigger mechanism; the abscissa of (a) and (b) represents time, and the ordinate represents the agent number. Specific implementation manners

[0073] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0074] Embodiment 1

[0075] To address the problem in the prior art that it is difficult to effectively balance the communication frequency and control performance under complex relationships between nodes (such as coexistence of cooperation and competition), resulting in low resource utilization and limited system stability. As Figure 1 shown, an embodiment of the present invention proposes a two-way formation control method for a multi-agent system based on dynamic event triggering. The method includes:

[0076] Step S1: Establish a second-order nonlinear multi-agent system model including competitive relationships;

[0077] Step S2: Based on the established second-order nonlinear multi-agent system model including competitive relationships, design a distributed adaptive controller by introducing an adaptive control gain mechanism into the controller;

[0078] Step S3: Based on the designed distributed adaptive controller, construct a dynamic event trigger mechanism with dynamic parameters;

[0079] Step S4: Based on the distributed adaptive controller and the dynamic event trigger mechanism, construct a controlled error system, and analyze the controlled error system to obtain the sufficient conditions and exponential convergence rate for the multi-agent system to achieve adaptive two-way formation control.

[0080] As can be seen from the above technical solution, the present invention provides a two-way formation control method for a multi-agent system based on dynamic event triggering. The present invention innovatively constructs a second-order nonlinear multi-agent system model, which incorporates the concept of signed directed communication topology to more precisely characterize the complex relationship between agents where there is both cooperation and competition. In the design of the distributed controller, the present invention introduces an adaptive control gain mechanism, which not only improves the flexibility of control but also significantly enhances the control performance. Further, in order to optimize the utilization of control resources and reduce unnecessary communication load, the present invention designs a dynamic event triggering mechanism with dynamic parameters. This mechanism can effectively reduce the triggering frequency by dynamically adjusting the triggering threshold compared with the traditional static threshold event triggering strategy, thus saving valuable control resources. In addition, the present invention also constructs a controlled error system and ingeniously applies the Lyapunov stability theory. Through this theoretical framework, the present invention derives the sufficient conditions for the second-order nonlinear multi-agent system to achieve adaptive two-way formation control and the convergence rate of the system under the dynamic event triggering mechanism, providing a solid theoretical support for subsequent engineering applications.

[0081] In step S1, a second-order nonlinear multi-agent system model including competitive relationships is established, specifically including:

[0082] The present invention first defines the dynamic equation of the second-order nonlinear multi-agent system model, including the position and velocity variables of the agents and a nonlinear continuous function, and its mathematical expression is:

[0083]

[0084] In the formula, t represents the time; represents the derivative of x i (t), x i (t) represents the position of the i-th agent at time t; represents the derivative of v i (t), v i (t) ∈ R n represents the velocity of the i-th agent at time t, where i = 1, 2,..., N, R n represents the n-dimensional Euclidean space, and N represents the number of agents; u i (t) ∈ R n represents the control action applied to the i-th node; is a nonlinear continuous function.

[0085] Then, the dynamic behavior of the leader agent is defined, and its mathematical expression is:

[0086]

[0087] In the formula, represents the derivative of x0(t), where x0(t) ∈ R n represents the position of the leader agent; represents the derivative of v0(t), where v0(t) ∈ R n represents the velocity of the leader agent.

[0088] Secondly, the present invention introduces a signed digraph to represent the communication topology of the multi-agent system, including:

[0089] Signed digraph: The communication topology of the multi-agent system is represented by a signed digraph G = (V, E, A), where V = {1, 2,..., N} is the set of nodes, representing each agent; is the set of edges, representing the communication connections between agents; A = [a ij ∈ R N×N is the adjacency matrix of graph G.

[0090] Adjacency matrix A: The element a ij in the matrix defines the relationship between node i and node j:

[0091] a ij > 0 indicates a cooperative relationship between node i and node j; a ij < 0 represents a competitive relationship between node i and node j; a ij = 0 indicates no communication connection between node i and node j.

[0092] Laplacian matrix L:

[0093] In the multi-agent system, all agent nodes are divided into two clusters and For each cluster, the adjacent nodes within are in a cooperative relationship, while the connections between nodes in the two clusters are in a competitive relationship; define the matrix i = 1, 2,..., N, if the node then q i = 1, if the node then q i = -1.

[0094] Two-way formation means that the positions of the two clusters of the multi-agent system show a reverse-symmetric formation as time progresses. The present invention designs a leader agent as the two-way formation target. When one agent cluster in the multi-agent system synchronizes with the leader agent in velocity and forms and maintains a preset formation in position, and the other cluster is reverse-symmetric to it, it is said that the multi-agent system achieves two-way formation. The conditions for the multi-agent system to achieve two-way formation are:

[0095]

[0096] In the formula, represents the preset formation, T represents the transpose.

[0097] In step S2, based on the established second-order nonlinear multi-agent system model with competition relationship, a distributed adaptive controller is designed by introducing an adaptive control gain mechanism into the controller, which improves the control performance while increasing the control flexibility.

[0098] Specifically, the mathematical expression of the distributed adaptive controller is:

[0099]

[0100] In the formula, α represents the free weight parameter, α > 0; represents the k-th triggering time of the i-th agent, k ∈ {1, 2,...}; represents the position of the j-th agent at time represents the preset formation; δ 1i represents the feedback control gain, δ 1i ≥ 0; δ 2i represents the adaptive control gain.

[0101] By defining the error vector, the update formula of the adaptive control gain δ 2i (t) is obtained:

[0102] i = 1, 2,..., N

[0103] where

[0104]

[0105] φ vi (t) = q i v i (t) - v0(t)

[0106] In the formula, represents the derivative of δ 2i (t); p represents a constant, p > 0; φ xi (t), φ vi (t) represent the error vectors.

[0107] In step S3, based on the designed distributed adaptive controller, a dynamic event-triggering mechanism with dynamic parameters is constructed, specifically including:

[0108] Define the measurement errors γ xi (t), γ vi (t) as follows:

[0109]

[0110] Based on the defined measurement error γ xi (t), γ vi (t), construct a dynamic event-triggering mechanism with dynamic parameters, which is expressed as follows:

[0111]

[0112] In the formula, inf{} represents the infimum; g i (t) represents the triggering function; represents the derivative of S i (t), S i (t) represents a dynamic variable and satisfies the initial value S0(t)>0; β i represents the event update parameter, β i >0; c represents the weight coefficient, c>0; λ min represents the minimum eigenvalue; is a positive definite matrix, ψ2 = ||L + diag{δ 21 (t),..., δ 2N (t)}|| 2 .

[0113] In step S4, based on the distributed adaptive controller and the dynamic event-triggering mechanism, construct a controlled error system, and analyze the controlled error system to obtain the sufficient conditions and exponential convergence rate for the multi-agent system to achieve adaptive two-way formation control, specifically including:

[0114] First, based on the distributed adaptive controller and the dynamic event-triggering mechanism, construct a controlled error system:

[0115]

[0116] Define the following vectors Then the controlled error system can be written in a more compact form:

[0117]

[0118] In the formula, respectively represent the derivatives of φ x (t), φ v (t), represents convolution, I n is the n-dimensional identity matrix,

[0119] Hypothesis 1: Assume the vector function satisfies the following conditions:

[0120]

[0121] where z i ∈{1, -1}, i = 1, 2; h k > 0, k = 1, 2,..., n.

[0122] Next, the conditions for the multi - agent system (Equation (1)) and the leader agent (Equation (2)) to achieve two - way formation under the action of the distributed adaptive controller (Equation (3)) and the dynamic event - triggering mechanism (Equation (4)) will be derived based on the controlled error system (Equation (5)).

[0123] Furthermore, the Lyapunov function constructed in the present invention is:

[0124]

[0125] where V(t) represents the Lyapunov function.

[0126] Define It can be obtained that V(t)=V1(t)+V2(t). From the dynamic event - triggering mechanism (Equation (4)), it can be known that Furthermore, it can be obtained that By recursive derivation for the time period, it can be obtained that:

[0127]

[0128] Therefore, it can be known that S i (t)>0, V2(t)>0. When (assuming it holds, as one of the conclusions), it is calculated that V1(t)>0. So V(t)=V1(t)+V2(t)>0.

[0129] Furthermore, by taking the derivative of V1(t), it can be obtained that:

[0130]

[0131] where

[0132]

[0133] Based on Hypothesis 1, the following inequalities can be obtained:

[0134]

[0135] Among them and Furthermore, it can be obtained that:

[0136]

[0137] Based on the Hardy inequality, it can be obtained that:

[0138]

[0139] According to formulas (6), (7), and (8), it can be obtained that:

[0140]

[0141] Furthermore, based on the dynamic event-triggering mechanism (formula (4)), it can be obtained that:

[0142]

[0143] where θ = 2 + h 2 +(c - 2α)λ min B - αλ min (L + + L +T ) + α 2 , where f represents the convergence coefficient; h represents the Lipschitz coefficient, and L +T represents the transpose of L + .

[0144] From formula (9), it can be obtained that: V(t) ≤ V(0)e ft . Therefore, if f < 0, it can be concluded that the multi-agent system (formula (1)) achieves two-way formation with the leader agent (formula (2)) under the action of the dynamic event-triggering mechanism (formula (4)) and the distributed adaptive controller (formula (3)).

[0145] In summary: If Assumption 1 holds, if the inequality holds, then the solution of the error system (formula (5)) converges exponentially to 0, that is, the multi-agent system (formula (1)) achieves two-way formation with the leader agent (formula (2)).

[0146] In addition, the present invention also includes proving through inequality derivation that there will be no Zeno effect during the process of the multi-agent system achieving two-way formation under the action of the dynamic event-triggering mechanism and the distributed adaptive controller, and verifying it by designing a numerical simulation case.

[0147] Specifically, in the time period within, for ||γ vi (t)|| and ||γ xi(t) || The derivative gives:

[0148]

[0149] Among them

[0150]

[0151] Further derivation of formula (10) gives:

[0152]

[0153] Among them, F = max{||δ 2i (t)α||}, according to the dynamic event-triggering mechanism (formula (4)), the triggering time satisfies Substituting into formula (11) gives

[0154] It can be further written as: Finally, it can be obtained:

[0155]

[0156] Therefore, always holds, which indicates that the Zeno effect will not occur in the control process.

[0157] In summary, under the action of the dynamic event-triggering mechanism (formula (4)) and the distributed adaptive controller (formula (3)), the multi-agent system (formula (1)) will not exhibit the Zeno effect during the two-way formation process.

[0158] To further illustrate the advantages of the present invention, the following will be described in combination with specific simulation cases.

[0159] 1. To simulate the actual industrial scenario as much as possible, a multi-agent system composed of eight agents moving in the X-Y-Z three-dimensional space is selected, that is, N = 8, n = 3, and the communication topology diagram is shown in Figure (2). It can be seen from Figure (2) that the eight agents can be divided into two clusters and Select x, y, z as the position coordinates of the agents, and the dynamic function of the agents is as follows:

[0160]

[0161] Among them, v x , v y , v z are the agent velocity vectors, y x , u y , uz Representing the control input, the target formation vectors are as follows:

[0162]

[0163] Based on the target formation, the multi-agent system will ultimately form two square formations.

[0164] 2. Set the controller parameters diag{δ 11 ,..., δ 1N} = diag{4, 3.5, 4, 4.5, 3.5, 4, 4, 4.5}, α = 0.95, p = 0.01, c = 0.125, β i = 0.1, i = 1, 2,..., N.

[0165] 3. According to the sufficient conditions for the multi-agent system to achieve bidirectional formation, calculate the parameters to satisfy the derived bidirectional formation criterion for the multi-agent system.

[0166] 4. The construction of the simulation model based on Simulink and the result analysis show that after implementing the control strategy of the present invention, the multi-agent system successfully achieves bidirectional formation behavior. Specifically:

[0167] Convergence analysis of speed and position errors: Figure 3 Figures (a) and (b) in

[0168] show that as time goes by, the formation errors (including speed errors and position errors) of the multi-agent system gradually approach zero, which verifies the stability and accuracy of the system's formation control. Figure 4 clearly shows that the two agent clusters respectively achieve speed synchronization with equal values but opposite directions, which meets the design requirements of bidirectional formation and reflects the efficiency of the internal cooperative control of the system.

[0169] Formation and maintenance of the formation: Through Figure 5 and Figure 6 parts (a), (b), (c), and (d) in

[0170] it can be observed that the multi-agent system can form and maintain a symmetric square formation at different time points, which proves the effective control ability of the control strategy for the formation configuration in three-dimensional space. Figure 7 reveals the variation of the optimal control gain parameters determined by the adaptive strategy over time, indicating that this strategy can dynamically adjust the control parameters to adapt to the system state and optimize the control effect.

[0171] Efficiency comparison of the event-triggered mechanism:Figure 8 Parts (a) and (b) in [reference] compare the differences in triggering frequencies between the traditional static event-triggering mechanism and the dynamic event-triggering mechanism proposed in the present invention. The results show that the dynamic event-triggering mechanism significantly reduces the triggering frequency, thereby improving the system resource utilization rate and communication efficiency.

[0172] In summary, the simulation results strongly prove that the proposed multi-agent system control strategy can effectively achieve two-way formation with the leader agent under specific conditions, while demonstrating the excellent performance of the system in aspects such as formation error control, speed synchronization, formation maintenance, control gain optimization, and improvement of event-triggering efficiency.

[0173] Those skilled in the art should understand that the embodiments of the present application can be provided as methods, systems, or computer program products. Therefore, the present application can take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk memories, CD-ROMs, optical memories, etc.) containing computer-usable program codes.

[0174] The present application is described with reference to the flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or block in the flowchart and / or block diagram, and the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, such that the instructions executed by the processor of the computer or other programmable data processing devices generate means for implementing the functions specified in Figure 1 one process or multiple processes and / or blocks Figure 1 one block or multiple blocks.

[0175] These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to work in a specific manner, such that the instructions stored in the computer-readable memory generate a manufactured article including instruction means that implement the functions specified in Figure 1 one process or multiple processes and / or blocks Figure 1 one block or multiple blocks. These computer program instructions can also be loaded onto a computer or other programmable data processing device, such that a series of operation steps are executed on the computer or other programmable device to generate a computer-implemented process, so that the instructions executed on the computer or other programmable device provide for implementing the functions in Figure 1 one process or multiple processes and / or blocksFigure 1 Steps of functions specified in one or more boxes.

[0176] Obviously, the above embodiments are only examples for clear illustration and are not limitations on the implementation manners. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to list all implementation manners here. And the obvious changes or modifications derived therefrom are still within the protection scope of the present invention.

Claims

1. A two-way formation control method for multi-agent systems based on dynamic event triggering, characterized in that Including: Establish a second-order nonlinear multi-agent system model with competitive relationships; Based on the established second-order nonlinear multi-agent system model with competitive relationships, design a distributed adaptive controller by introducing an adaptive control gain mechanism into the controller; Based on the designed distributed adaptive controller, construct a dynamic event-triggering mechanism with dynamic parameters, specifically including: Define the measurement error γ xi (t), γ vi (t) as follows: where x i (t) represents the position of the i-th agent at time t; v i (t) ∈ R n represents the velocity of the i-th agent at time t, where i = 1, 2, …, N, and R n represents the n-dimensional Euclidean space; respectively represent the k-th and (k + 1)-th triggering times of the i-th agent, k ∈ {1, 2,...}; represents the position of the i-th agent at time represents the velocity of the i-th agent at time; Based on the defined measurement error γ xi (t), γ vi (t), a dynamic event-triggering mechanism with dynamic parameters is constructed, which is expressed as follows: where, inf{} represents the infimum; g i (t) represents the triggering function; represents the derivative of S i (t), S i (t) represents a dynamic variable and satisfies the initial value S0(t)>0; β i represents the event update parameter, β i >0; c represents the weight coefficient, c>0; λ min represents the minimum eigenvalue; is a positive definite matrix, b1, b2,..., b N is a diagonal matrix the elements on the main diagonal; ψ2 = ||L + diag{δ 21 (t),..., δ 2N (t)}|| 2 , δ 2i (t) represents the adaptive control gain parameter of the i-th agent at time t; φ vi (t) represents the error vector; L represents the Laplacian matrix; T represents the transpose; Based on the distributed adaptive controller and the dynamic event-triggering mechanism, construct a controlled error system, and analyze the controlled error system to obtain the sufficient conditions and exponential convergence rate for the multi-agent system to achieve adaptive two-way formation control.

2. The two-way formation control method for multi-agent systems based on dynamic event triggering according to claim 1, characterized in that The method further includes: Proving through inequality derivation that under the action of the dynamic event-triggering mechanism and the distributed adaptive controller, the multi-agent system will not exhibit the Zeno effect during the two-way formation process.

3. The two-way formation control method for multi-agent systems based on dynamic event triggering according to claim 1, characterized in that The establishment of the second-order nonlinear multi-agent system model with competitive relationships specifically includes: Define the dynamic equation of the second-order nonlinear multi-agent system model, including the position and velocity variables of the agents, and a nonlinear continuous function, and its mathematical expression is: In the formula, \(t\) represents the moment; denotes the derivative of \(x\) i (t), where \(x\) i (t) represents the position of the \(i\)-th agent at time \(t\); denotes the derivative of \(v\) i (t), where \(v\) i (t) ∈ ℝ n represents the velocity of the \(i\)-th agent at time \(t\), where \(i = 1, 2, \ldots, N\), and ℝ n denotes the \(n\)-dimensional Euclidean space, and \(N\) represents the number of agents; \(u\) i (t) ∈ ℝ n represents the control action applied to the \(i\)-th node; is a non-linear continuous function; Define the dynamic behavior of the leader agent, and its mathematical expression is: In the formula, represents the derivative of x0(t), where x0(t) ∈ R n represents the position of the leader agent; represents the derivative of v0(t), where v0(t) ∈ R n represents the velocity of the leader agent; Introduce a signed digraph to represent the communication topology of the multi-agent system, including: Signed directed graph: The communication topology of a multi-agent system is represented by a signed directed graph \(G=(V, E, A)\), where \(V = \{1, 2,\cdots, N\}\) is the set of nodes, representing each agent; is the set of edges, representing the communication connections between agents; \(A = [a ij \in R N×N is the adjacency matrix of graph \(G\); Adjacency matrix A: The element a in the matrix ij defines the relationship between node i and node j: a ij > 0 indicates a cooperative relationship between node i and node j; a ij < 0 represents a competitive relationship between node i and node j; a ij = 0 indicates that there is no communication connection between node i and node j; Laplacian matrix L: In a multi-agent system, all agent nodes are divided into two clusters and Among the nodes within each cluster, the adjacent nodes have a cooperative relationship, while the connections between nodes in the two clusters are in a competitive relationship; define the matrix If node then q i = 1, if node then q i = -1; Determine the conditions for the multi-agent system to achieve two-way formation, which are expressed as: In the formula, represents a preset formation, T represents transpose.

4. The two-way formation control method for multi-agent systems based on dynamic event triggering according to claim 3, wherein Design a distributed adaptive controller by introducing an adaptive control gain mechanism into the controller, specifically including: The mathematical expression of the designed distributed adaptive controller by introducing an adaptive control gain mechanism into the controller is: where α represents the free weight parameter, α > 0; represents the k-th triggering time of the i-th agent, k ∈ {1, 2,...}; denotes the position of the j-th agent at time represents the preset formation; δ 1i denotes the feedback control gain, δ 1i ≥ 0; δ 2i denotes the adaptive control gain; By defining the error vector, the adaptive control gain δ 2i (t) update formula: where φ vi (t) = q i v i (t) - v0(t) In the formula, represents the derivative of δ 2i (t); p represents a constant, p > 0; φ xi (t), φ vi (t) represents the error vector.

5. The two-way formation control method for multi-agent systems based on dynamic event triggering according to claim 1, characterized in that Based on the distributed adaptive controller and the dynamic event-triggering mechanism, construct a controlled error system, and analyze the controlled error system to obtain the sufficient conditions and exponential convergence rate for the multi-agent system to achieve adaptive two-way formation control, specifically including: Based on the distributed adaptive controller and the dynamic event-triggering mechanism, construct a controlled error system; Apply the Lyapunov stability theory, analyze the stability and convergence of the multi-agent system by constructing a Lyapunov function and taking its derivative; According to the derivative result of the Lyapunov function, obtain the sufficient conditions and exponential convergence rate for the multi-agent system to achieve adaptive two-way formation control.

6. The bidirectional formation control method for multi-agent systems based on dynamic event triggering according to claim 5, characterized in that The controlled error system is: wherein, respectively represent the derivatives of φ x (t) and φ v (t), denotes convolution, I n is an n-dimensional identity matrix, 7. The two-way formation control method for multi-agent systems based on dynamic event triggering according to claim 6, characterized in that, The Lyapunov function is: In the formula, V(t) represents the Lyapunov function.

8. The two-way formation control method for multi-agent systems based on dynamic event triggering according to claim 7, characterized in that, According to the derivative result of the Lyapunov function, the sufficient condition for the multi-agent system to achieve adaptive two-way formation control is: where f represents the convergence coefficient; θ = 2 + h 2 +(c - 2α)λ min B - αλ min (L + + L +T ) + α 2 , h represents the Lipschitz coefficient, L +T represents the transpose of L + ; 9. The two-way formation control method for multi-agent systems based on dynamic event triggering according to claim 8, characterized in that According to the derivative result of the Lyapunov function, the exponential convergence rate for the multi-agent system to achieve adaptive two-way formation control is -f.

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