An event-driven control method for multi-agent systems with saturated inputs

Abstract

The present invention relates to the field of multi-agent consensus control, and specifically provides an event-driven control design method applicable to multi-agent systems with saturated inputs, which is characterized by including the following steps: for agents with saturated inputs, construct the state-space model of each agent and give the assumption conditions that the system matrix needs to satisfy; based on the multi-agent system model established in step 1, design a control law that can make the system reach consensus; design an appropriate low feedback gain to optimize the non-linear agent model into a linear model; use the Lyapunov function method to prove the consensus of the event-driven control law. The present invention can optimize it into a linear system by using the low-gain technology based on the algebraic Riccati equation, reducing the dependence; the present invention can effectively reduce the number of control law updates on the basis of completing the consensus control task, saving resources; the present invention can effectively avoid the Zeno phenomenon.

Description

Technical Field

[0001] The present invention relates to the field of multi-agent consensus control, and particularly to an event-driven control method for multi-agent systems under saturated inputs. Background Art

[0002] In recent years, multi-agent systems have attracted the attention of many researchers and become a research hotspot in the field of automatic control. The wide applications of multi-agent systems in such frontier fields as unmanned aerial vehicle formations, multi-robot formations, distributed sensor networks, etc. have continuously increased people's interest in them. In the research field of multi-agent systems, the consensus problem is of great significance, which is the basis for agents to complete cooperation. Consensus means that agents analyze and calculate the information received from neighbors, and then continuously adjust their behaviors so that each state parameter (such as position, speed, angle, etc.) of each agent reaches a certain common value over time.

[0003] Input saturation non-linearity is a common non-linear problem. Due to physical constraints and safety reasons, most actual control systems are affected by input saturation. When the system input is saturated, no matter how the input signal changes, the output of the system can no longer increase, and there will be a mismatch problem between the input and output. When the controller is difficult to track the output signal of the system, the controller will be in a completely out-of-control state, which is the problem of input saturation. Therefore, designing a system controller to solve the input saturation problem has become the focus of current research.

[0004] On the other hand, in the process of multi-agent consensus, continuous communication and update of the control rate need to be maintained among agents. The information among agents is transmitted through a communication network, and in an actual multi-agent system, the communication network bandwidth and computing resources are very limited. Therefore, how to reduce the communication pressure among agents during the formation target tracking process is an urgent problem to be solved.

[0005] To solve the above problems, the literature "Dimarogonas D V, Frazzoli E, Johansson KH. Distributed Event-Triggered Control for Multi-Agent Systems[J]. IEEE Transactions on Automatic Control, 2012, 57(5): 1291-1297" introduced event-driven control into multi-agent consensus control, alleviating the communication pressure between agents, but did not consider the existence of input saturation. The literature "Du, S.L., Dong, L., & Ho, D..(2017). Event-triggered control for output synchronization of heterogeneous network with input saturation constraint. IECON 2017-43rd Annual Conference of the IEEE Industrial Electronics Society. IEEE" effectively avoided input saturation nonlinearity by using low-gain technology, but this research was for the output synchronization problem of heterogeneous multi-agent systems, rather than the consensus problem.

[0006] Therefore, applying the event-triggered control strategy to the consensus control of multi-agent systems with input saturation constraints, the main problems to be solved are as follows:

[0007] (1) How to design the triggering condition and threshold of event-triggered control to reduce the update times of the control law and save energy while enabling the multi-agent system to reach consensus;

[0008] (2) How to avoid the "Zeno behavior" that may be caused by the event-triggered mechanism, where "Zeno behavior" refers to an infinite number of event triggers within a finite time;

[0009] (3) How to design the system controller to solve the impact of saturated input problems on system performance. Summary of the Invention

[0010] The purpose of the present invention is to provide an event-driven control method for multi-agent systems with saturated inputs, theoretically ensuring its stability and avoiding the Zeno phenomenon, being able to effectively save communication resources and complete control tasks.

[0011] To achieve the above object, the present invention adopts the following technical solutions:

[0012] An event-driven control design method for a multi-agent system with saturated input, comprising the following steps:

[0013] Step 1: For agents with saturated input, construct the state-space model of each agent, i.e., the multi-agent system model, and give the assumption conditions that the system matrix needs to satisfy;

[0014] Step 2: Based on the multi-agent system model established in Step 1, design a control law that can make the system reach consensus;

[0015] Step 3: Use the low-gain technique based on the algebraic Riccati equation to design an appropriate low feedback gain to optimize the non-linear agent model into a linear model;

[0016] Step 4: Design an event-driven mechanism that can exclude the Zeno phenomenon;

[0017] Step 5: Use the Lyapunov function method to prove the consensus of the designed event-driven control law;

[0018] In the said Step 1, the individual mathematical model in the state-space form describing the motion characteristics of the agent and the undirected communication topology describing the communication relationship adopt the following formula:

[0019]

[0020] where x i (t) ∈ R n , representing the state vector of agent i, R n being the n-dimensional Euclidean space, u i (t) ∈ R m representing the input of agent i, R m being the m-dimensional Euclidean space, σ indicating that u i (t) has a saturation non-linearity limitation. Let the undirected graph G = {v, ε, A} be used to describe the information transmission between agents; where v = {1, 2,..., N} represents N nodes, representing the set of edges; the edges of the undirected graph G are represented by e ij ; this represents that agent i and agent j can transmit information to each other, and the set of all neighbors of agent i is defined as N i = {j ∈ v: (i, j) ∈ ε, j ≠ i}; the adjacency matrix of the undirected graph G is defined as A = [a ij ∈ R N×N , where R N×N is an N×N-dimensional matrix, and its elements are defined as a ij = 0 if and only if v j and v iThere is a path, namely e ij ∈ ε, a ij = 1, the diagonal matrix D = diag{d1,…,d i …,d n} is called the degree matrix, and its elements are defined as The Laplacian matrix L of the undirected graph G is defined as L = D - A ;

[0021] The assumptions that the system matrix needs to satisfy are: (A, B) is asymptotically zero controllable under bounded control, that is, (A, B) is stabilizable; all eigenvalues of A are in the closed left half s-plane;

[0022] The control law designed in step 2 to achieve consensus in the multi-agent system is specifically:

[0023]

[0024] where K is the feedback gain matrix, x j (t) is the state of agent j;

[0025] For any bounded initial state set, the consensus of the multi-agent system state can be achieved under this control law, that is:

[0026]

[0027] The theoretical basis for designing a suitable low feedback gain in step 3 to optimize the non-linear agent model into a linear model is specifically:

[0028] Step 3-1. Solve the following algebraic Riccati equation:

[0029]

[0030] Obtain the unique positive definite P(γ), where γ ∈ (0, 1) is a given constant, I is the identity matrix with appropriate dimensions, is the transpose of the system matrix A, is the product of the control matrix and its transpose, and N is the number of agents;

[0031] Step 3-2. Define the feedback gain matrix:

[0032]

[0033] The feedback gain K designed by this method corresponds uniquely to γ, and the smaller γ is, the smaller the value of K. By adjusting γ, K can be continuously reduced so that the input of each agent remains below the saturation constraint threshold; and the designed low feedback gain only requires the number of agents in the network, rather than the topology of the entire network;

[0034] The event-driven mechanism designed in step 4 that can eliminate the Zeno phenomenon is specifically as follows:

[0035] Step 4-1. Discretize the control laws designed in step 2 and step 3 to design a discontinuous-time control law, specifically as follows:

[0036]

[0037] where represents the triggering sequence of agent i. Substituting formula (6) into formula (1), the dynamic equation of the system is obtained as:

[0038]

[0039] Step 4-2. Define variables:

[0040]

[0041] where x j (t) and x i (t) represent the states of agents j and i. It should be noted that j and i are agents connected in the topological structure and can exchange information with each other. Then define the error variable:

[0042]

[0043] We define Since the undirected graph G is undirected and connected, there exists a matrix Γ = [Γ1, Γ2, …, Γ N ∈ R NXN , where Γ1, Γ2, …, Γ N are the eigenvectors corresponding to the eigenvalues of the Laplacian matrix L; that is where λ1, …, λ N are the eigenvalues of the Laplacian matrix L; for an undirected connected graph, we can choose Obviously, its corresponding eigenvalue is λ1 = 0; then we let φ = [Γ2, …, Γ N ∈ R N×N-1 , Λ = diag{λ2, …, λ N}, and Here represents the Kronecker product of two matrices;

[0044] Step 4-3. Design the event triggering sequence of agent i, i ∈ ν as:

[0045]

[0046] where σ0∈(0,1) are two constants, and λ0 = λ N ||P(γ)BB T P(γ)||; meanwhile, this rule can effectively avoid the Zeno phenomenon in theory, that is, the lower bound of the interval between any two adjacent triggering moments is:

[0047]

[0048] where denotes the difference between any two adjacent triggering moments, and is strictly positive;

[0049] In step 5, the Lyapunov function method is used for consensus proof, specifically:

[0050] Step 5-1. Select an appropriate Lyapunov function as:

[0051]

[0052] Step 5-2. Its derivative with respect to time is:

[0053]

[0054] It is worth mentioning that the first step of simplification here uses:

[0055]

[0056] Step 5-3. To judge the sign of, we will first judge the relationship between ||z(t)||, ||e(t)||, and ; first, according to the definition of z i (t), we can easily obtain and:

[0057]

[0058] Based on the event-triggering sequence formula (10), for any there is:

[0059]

[0060] Combined with formula (14), we can obtain the relationship between ||z(t)||, ||e(t)||, and :

[0061]

[0062] Step 5-4. Determine the sign of the derivative: By combining with formula (13), we can obtain:

[0063]

[0064] The establishment of formula (17) is because

[0065] Substituting ||z(t)||, ||e(t)||, and into the inequality relationship formula (16) of the above formula, we have:

[0066]

[0067] We further define λ max = λmax(p(γ)) as the maximum eigenvalue, λ min = λmin(p(γ)) as the minimum eigenvalue, and λ represents the matrix eigenvalue. is a constant, so we have:

[0068]

[0069] Substituting into the definition formula (12) of V(t), we have:

[0070]

[0071]

[0072] Taking the square root of both sides of the above formula and taking t→∞, we can obtain

[0073]

[0074] Combined with formula (17), by the second Lyapunov method, we can conclude that the multi-agent system can achieve consensus under the given event-triggered control strategy, that is, there is

[0075] Compared with the prior art, the beneficial effects of the present invention are:

[0076] For the consensus control of multi-agent systems, the present invention proposes a multi-agent time-driven control protocol that can solve the input saturation constraint, enabling the multi-agent system to achieve the expected control goal. The main advantages are as follows:

[0077] 1) Input saturation nonlinearity is a common nonlinear problem. Due to physical constraints and safety reasons, most practical control systems are affected by input saturation. When the input of the agent is subject to saturation nonlinear constraints, the low-gain technique based on the algebraic Riccati equation can be used to optimize it into a linear system. Moreover, the design of the low feedback gain does not depend on the global topological information of the system, but only on the number of agents.

[0078] 2) Based on the continuous-time or time-driven multi-agent consensus control protocol, although the control task can be completed, it is accompanied by a large amount of unnecessary waste of communication resources and control resources. The event-driven control adopted by the present invention can effectively reduce the number of updates of the control law on the basis of completing the consensus control task, and only update the control law when necessary, saving resources.

[0079] 3) The event trigger sequence adopted by the present invention can effectively avoid the Zeno phenomenon and will not cause the system to crash due to infinite triggers within a finite time. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] Figure 1 It is a schematic diagram of the design steps of the event-driven control method for a multi-agent system with saturated input according to the present invention;

[0081] Figure 2 It is a schematic diagram of an undirected topology according to an embodiment of the present invention;

[0082] Figure 3 It is a multi-agent state diagram when γ = 0.1;

[0083] Figure 4 It is a multi-agent control input diagram when γ = 0.1;

[0084] Figure 5 It is an event trigger sequence diagram when γ = 0.1;

[0085] Figure 6 It is a multi-agent state diagram when γ = 0.2;

[0086] Figure 7 It is a multi-agent control input diagram when γ = 0.2;

[0087] Figure 8 It is an event trigger sequence diagram when γ = 0.2. DETAILED DESCRIPTION OF THE INVENTION

[0088] Next, in combination with the accompanying drawings of the present invention, the technical solutions of the present invention will be clearly and completely described. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0089] Most of the existing multi-agent consensus control technologies require that multi-agents can receive information from neighboring agents in real time and can broadcast their own information, which has relatively high requirements for the bandwidth requirements and real-time communication capabilities of the agent communication network. Periodic time-driven control can convert continuous and real-time control into control at discrete trigger moments, but a relatively conservative time period selection still causes a large amount of communication resource waste. In real-world electronic devices, due to physical or safety considerations, input saturation nonlinearity usually exists. Control signals exceeding the saturation constraint threshold will be distorted, and the controller cannot obtain a complete control signal, resulting in a significant reduction in control performance and even affecting the stability of the system, which may pose a danger.

[0090] Based on the above considerations, the present invention first uses a low-gain technique based on the algebraic Riccati equation to design a low feedback gain matrix, optimizing the nonlinear system with saturated input constraints into a linear system, and this feedback only depends on the number of agents. To further overcome the drawbacks of periodic time-driven control, an event-driven control is designed in this paper, which effectively reduces the number of triggers and saves communication resources. Then, by proving that the interval between any two trigger moments in the trigger sequence has a strictly positive lower bound, the Zeno phenomenon is excluded. Finally, by choosing an appropriate Lyapunov function, it is proved that the event-driven control scheme designed by the present invention can make the multi-agent system reach consensus.

[0091] Embodiment 1

[0092] An event-driven control method applicable to a multi-agent system with saturated input, the method comprising the following steps: Figure 1 It represents the construction of a differential equation model of the agent for multi-agents under saturated constraint conditions. The specific composition is as follows:

[0093] Step 1. For agents with saturated input, construct the state space model of each agent and give the assumption conditions that its system matrix needs to satisfy. Specifically

[0094]

[0095] Here, x i (t) ∈ R n , representing the state vector of agent i, u i (t) ∈ Rm Represents the input of agent i. And Has the limitation of saturated nonlinearity, so (1) can also be written as:

[0096]

[0097] And assume that (A, B) is asymptotically zero controllable under bounded control, that is, (A, B) is stabilizable; all eigenvalues of A are in the closed left half s-plane.

[0098] The topological structure of multi-agent can be described as:

[0099] Let the undirected graph G = {ν, ε, A} be used to describe the information transmission between agents; where ν = {1, 2,..., N} represents N nodes, Represents the set of edges; the edges of the undirected graph G are represented by e ij Represents; this means that agent i and agent j can transmit information to each other, and the set of all neighbors of agent i is defined as N i = {j ∈ v: (i, j) ∈ ε, j ≠ i}; the adjacency matrix of the undirected graph G is defined as A = [a ij ∈ R N×N , where R N×N Is an N×N dimensional matrix, and its elements are defined as a ij = 0 if and only if v j And v i Have a path, that is, when e ij ∈ ε, a ij = 1, the diagonal matrix D = diag{d1,…,d i …,d n} is called the degree matrix, and its elements are defined as The Laplacian matrix L of the undirected graph G is defined as L = D - A .

[0100] Step 2. Design a control law to make the multi-agent system achieve consensus. The specific steps are as follows:

[0101]

[0102] For any bounded initial state set Under this control law, the consensus of the multi-agent system state can be achieved, that is:

[0103]

[0104] Step 3. Design a suitable low feedback gain to optimize the nonlinear agent model into a linear model. The specific process of this step is as follows:

[0105] Step 3-1. Solve the following algebraic Riccati equation:

[0106]

[0107] Obtain the unique positive definite P(γ), where γ ∈ (0, 1) is a given constant and I is the identity matrix with appropriate dimensions; Step 3-2. Define the feedback gain matrix:

[0108]

[0109] The feedback gain K designed by this method corresponds uniquely to γ, and the smaller γ is, the smaller the value of K is. By adjusting γ, K can be continuously reduced so that the input of each agent remains below the saturation constraint threshold. Moreover, the designed low feedback gain only requires the number of agents in the network, rather than the topology of the entire network.

[0110] Step 4. Design an event-driven mechanism that can eliminate the Zeno phenomenon. The specific process of this step is as follows:

[0111] Step 4-1. Discretize the control laws designed in Steps 2 and 3 to design a discontinuous-time control law, specifically:

[0112]

[0113] where represents the triggering sequence of agent i. Substituting (6) into (1) gives the dynamic equation of the system as:

[0114]

[0115] Step 4-2. Define the variable

[0116]

[0117] where x j (t) and χ i (t) represent the states of agents j and i. It should be noted that j and i are agents connected in the topology and can exchange information with each other. Then define the error variable:

[0118]

[0119] We define Since the graph G is undirected and connected, there exists a matrix Γ = [Γ1, Γ2, …, Γ N ∈ R NXN , where Γ1, Γ2, …, Γ N are the eigenvectors corresponding to the eigenvalues of the Laplacian matrix L. That is where λ1, …, λ N are the eigenvalues of the Laplacian matrix L. For an undirected connected graph, we can choose Obviously, its corresponding eigenvalue is λ1 = 0. Then we let φ = [Γ2, …, Γ N ∈ R N×N-1 , Λ = Λ = diag{λ2, …, λ N}, and Here, denotes the Kronecker product of two matrices.

[0120] Step 4-3. Design the event-triggering sequence for agent i, i ∈ V as follows:

[0121]

[0122] where σ0 ∈ (0, 1), λ0 = λ N ||P(γ)BB T P(γ)||. At the same time, this rule can effectively avoid the Zeno phenomenon in theory, that is, the lower bound of the interval between any two adjacent triggering times is:

[0123]

[0124] where denotes the difference between any two adjacent triggering times, and is strictly positive

[0125] Step 5. Use the Lyapunov function method to prove consistency. The specific process of this step is as follows:

[0126] Step 5-1. Select an appropriate Lyapunov function as:

[0127]

[0128] Step 5-2. Its derivative with respect to time is:

[0129]

[0130] It is worth mentioning that the first step of simplification here uses

[0131]

[0132] Step 5-3. To judge the sign of this derivative, we will first judge the relationship between ||z(t)||, ||e(t)||, and . First, according to the definition of z i (t), we can easily obtain and:

[0133]

[0134] Combined with (14), we can obtain the relationships among ||z(t)||, ||e(t)||, and :

[0135]

[0136] Step 5 - 4. Determine the sign of the derivative:

[0137]

[0138] The above equation holds because

[0139] Substitute ||z(t)||, ||e(t)||, and into the above equation, and the inequality relationship is:

[0140]

[0141] We further define λ M = λmax(p(γ)), λ m = λmin(p(γ)), So we have:

[0142]

[0143] Take the square root of both sides of the above equation and let t → ∞, we can get

[0144]

[0145] According to the second Lyapunov method, we can conclude that the multi - agent system (1) can achieve consensus under the event - triggered control strategy given in the present invention, that is

[0146] The following is the simulation experiment of the multi - agent event - driven control with saturated input designed by the present invention.

[0147] This simulation experiment is a numerical simulation experiment for the multi - agent system composed of 5 agents to achieve consensus in the final state. The initial states of each agent are bounded, that is, each element in its initial state vector is a random value within [-3, 3], σ0 = 0.9, γ = 0.1 or 0.2, and the simulation time is [0, 30s].

[0148] In the simulation experiment, it is assumed that the communication topology graph between agents is an undirected graph, and agents connected by edges can transmit information to each other. The constructed multi - agent communication topology graph is asFigure 2 As shown, it is obvious that the graph satisfies the assumption that the graph is connected. Therefore, the adjacency matrix and Laplacian matrix of the graph are respectively

[0149]

[0150] In addition, set the system matrix A and input matrix B as respectively and

[0151] When setting γ = 0.1, Figure 3 it can be seen that the two-dimensional state vectors of the 5 agents gradually approach the same value with the event, achieving the task of consensus control. Figure 4 It can be seen from [reference] that the control input of the agent gradually approaches 0, that is, the state difference between each agent and its neighboring agent gradually approaches 0. Figure 5 Records the time points when events are triggered. The experimental results when setting γ = 0.2 are as Figures 6 - 8 shown.

[0152] Table 1. Record table of the trigger times of agents when γ = 0.1

[0153] Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 977 times 1210 times 899 times 887 times 933 times

[0154] Table 2. Record table of the trigger times of agents when γ = 0.2

[0155] Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 956 times 1098 times 892 times 870 times 920 times

[0156] It is not difficult to see from the above table that when γ is set from 0.1 to 0.2, the trigger times of each agent are reduced.

[0157] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments. The above embodiments and descriptions in the specification are only preferred examples of the invention and are not used to limit the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements fall within the scope of the present invention claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. An event-driven control design method for a multi-agent system with saturated inputs, characterized in that, The following steps are involved: Step 1: For the agents with saturated input, construct the state space model of each agent, that is, the multi-agent system model, and give the assumptions that its system matrix needs to meet; Step 2: Based on the multi-agent system model established in step 1, design a control law that can make the system reach consistency; Step 3: Use the low-gain technique based on the algebraic Riccati equation to design an appropriate low feedback gain to optimize the nonlinear agent model into a linear model; Step 4: Design an event-driven mechanism that can eliminate Zeno's phenomenon; Step 5: Use the Lyapunov function method to prove the consistency of the designed event-driven control law; In step 1, the individual mathematical model in the state space form describing the motion characteristics of the intelligent agent and the undirected communication topology describing the communication relationship adopt the following formula: where x i (t) ∈ R n , representing the state vector of agent i, where R n is an n-dimensional Euclidean space, and u i (t) ∈ R m represents the input of agent i, where R m is an m-dimensional Euclidean space, and σ represents that u i (t) has a saturation non-linearity constraint. Let the undirected graph G = {v, ε, A} be used to describe the information transmission between agents; where v = {1, 2,..., N} represents N nodes, represents the set of edges; the edges of the undirected graph G are denoted by e ij ; this means that agent i and agent j can transmit information to each other. The set of all neighbors of agent i is defined as N i = {j ∈ v: (i, j) ∈ ε, j ≠ i}; the adjacency matrix of the undirected graph G is defined as A = [a ij ∈ R N×N , where R N×N is an N×N matrix, and its elements are defined as a ij = 0 if and only if there is a path between v j and v i , that is, when e ij ∈ ε, a ij = 1. The diagonal matrix D = diag{d1,…,d i …,d n} is called the degree matrix, and its elements are defined as The Laplacian matrix L of the undirected graph G is defined as L = D - A ; The assumptions that the system matrix needs to satisfy are: (A, B) is asymptotically zero controllable under bounded control, that is, (A, B) is stable; all eigenvalues of A are on the closed left half s-plane; The control law designed in step 2 to make the multi-agent system reach consensus is specifically: where \(K\) is the feedback gain matrix and \(x\) j (t) is the state of agent \(j\); For any bounded initial state set, the consistency of the multi-agent system state can be achieved under this control law, that is: The theoretical basis for designing a suitable low feedback gain in step 3 to optimize the nonlinear agent model to a linear model is specifically as follows: Step 3-1. Solve the following algebraic Riccati equation: Obtain the unique positive-definite \(P(\gamma)\), where \(\gamma\in(0,1)\) is a given constant, \(I\) is the identity matrix with appropriate dimension, \(A\) T is the transpose of the system matrix \(A\), \(BB\) T is the product of the control matrix and its transpose, and \(N\) is the number of agents; Step 3-2. Define the feedback gain matrix: K = B T P(γ) (5) The feedback gain K designed by this method corresponds uniquely to γ, and the smaller γ is, the smaller the value of K is. By adjusting γ, K can be continuously reduced so that the input u' of each agent i (t) is kept below the saturation constraint threshold; moreover, the designed low feedback gain only requires the number of agents in the network, rather than the topology of the entire network; The event-driven mechanism designed in step 4 to eliminate the Zeno phenomenon is specifically: Step 4-1. Discretize the control laws designed in step 2 and step 3 to design a discontinuous time control law, specifically: Among them represents the triggering sequence of agent i. Substituting equation (6) into equation (1), the dynamic equation of the system is obtained as follows: Step 4-2. Define variables: where x j (t) and x i (t) represent the states of agents j and i. It should be noted that j and i are agents connected in the topological structure and can exchange information with each other. Then, define the error variable: We define Since the undirected graph G is undirected and connected, there exists a matrix Γ = [Γ1, Γ2, …, Γ N ∈ R NXN , where Γ1, Γ2, …, Γ N are the eigenvectors corresponding to the eigenvalues of the Laplacian matrix L; that is, Γ T LΓ = diag{λ1, …, λ N}, where λ1, …, λ N are the eigenvalues of the Laplacian matrix L; for an undirected connected graph, we can choose Obviously, its corresponding eigenvalue is λ1 = 0; then we let φ = [Γ2, …, Γ N ∈ R N×N-1 , Λ = diag{λ2, …, λ N}, and Here denotes the Kronecker product of two matrices; Step 4-3. Design the event-triggering sequence of agent \(i\), where \(i\in v\). It is as follows: where σ0 ∈ (0, 1) are two constants, λ0 = λ N ||P(γ)BB T P(γ)||; meanwhile, this rule can effectively avoid the Zeno phenomenon in theory, that is, the lower bound of the interval between any two adjacent triggering moments is: wherein represents the difference between any two adjacent trigger times, and is strictly positive; In step 5, the consistency proof is performed using the Lyapunov function method, specifically: Step 5-1. Select the appropriate Lyapunov function: Step 5-2. Its derivative with respect to time is: It is worth mentioning that the first step of simplification here uses: Step 5-3. To determine the sign of, we will first determine the relationship between ‖z(t)‖, ‖e(t)‖, and ; First, according to the definition of z i (t), we can easily obtain and also: Based on the event trigger sequence formula (10), for any there is: Combined with formula (14), we can obtain the relationship between ‖z(t)‖, ||e(t)||, and : Step 5-4. Determine the sign of the derivative: Combined with formula (13), we can get: Equation (17) holds because Substituting ||z(t)||, ||e(t)|| into the above formula, and the inequality formula (16) of has: We further define λ max = λmax(p(γ)) as the maximum eigenvalue, λ min = λmin(p(γ)) as the minimum eigenvalue, and λ represents the matrix eigenvalue. is a constant, so we have: Substituting V(t) into the definition formula (12), we get: Take the square root of both sides of the above equation and take t→∞ to get: Combined with formula (17), by the second Lyapunov method, we can conclude that the multi-agent system can achieve consensus under the given event-triggered control strategy, that is, there is

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