Output consistency control method based on one-dimensional fluctuation partial differential multi-agent system

Abstract

The invention discloses an output consistency control method based on a one-dimensional fluctuation partial differential multi-agent system, and relates to the cross technical field of multi-agent system control and partial differential equation application, and the method comprises the steps: constructing a dynamic model, an interference model and a reference signal model of a multi-agent system; constructing a directed communication topological graph including a leader and a plurality of agents; based on the directed communication topological graph and the reference signal model, respectively constructing reference signal observers of the informed intelligent agent and the uninformed intelligent agent; based on the interference model, constructing a local interference observer of the intelligent agent; constructing a conversion function, and converting the dynamical model into a dynamical model under the auxiliary system according to the conversion function; constructing a control law for the dynamical model under the auxiliary system; according to the control law, adjusting the state of each agent under the auxiliary system until the tracking error is zero; according to the invention, the output consistency of the multi-agent system can be realized in a complex environment.

Description

Technical Field

[0001] The present invention relates to the intersecting technical field of multi-agent system control and partial differential equation application, and in particular to an output consistency control method for a multi-agent system based on one-dimensional wave partial differential. Background Art

[0002] Networked control of multi-agent systems (MASs) uses distributed communication and collaborative algorithms to enable multiple autonomous agents (agents) to achieve global goals without a central control unit. Its core is to leverage information exchange between individual agents to collaboratively solve complex tasks. With the rapid development of modern technology, many application areas, such as formation control, robotic swarms, intelligent transportation, and drones, urgently require the collaborative operation of multiple agents.

[0003] Related technologies often use ordinary differential equations to model how the state of an agent changes over time. Distributed control protocols are designed to achieve consensus or coordination among the agent's outputs (such as position, velocity, and decision variables) to address the problem of coordinated output in multi-agent systems. However, ordinary differential equations primarily focus on changes in the temporal dimension and are limited in their ability to effectively capture spatial dynamics. This makes their modeling methods ineffective for systems with significant spatial distribution and temporal variations.

[0004] With the continuous progress of society, partial differential equations have gradually become a key mathematical tool in the field of multi-agent collaborative control because they can accurately capture the dynamic characteristics of the system in space and time. In related technologies, wave equations are often used to describe the dynamic process of multi-agent systems, and under boundary control conditions, the leader-follower consistency of multi-agent systems is achieved; compared with the state consistency of partial differential wave equations, the output consistency requirement is more stringent. It not only requires the system to be consistent in state when not disturbed, but also emphasizes that after different disturbances are applied to each agent, the output consistency can still be achieved. However, current research on multi-agent systems is mostly focused on finite-dimensional or homogeneous systems. For infinite-dimensional heterogeneous multi-agent systems, their agents are diverse, the dimensions are infinite, and there are complex dynamic characteristics and interactions. The output consistency problem requires that the system be consistent in state when there is no disturbance and the output is still consistent when disturbed. However, none of the above methods can cope with it. Summary of the Invention

[0005] In view of this, the present invention provides an output consistency control method for a one-dimensional wave partial differential multi-agent system to solve the technical problems in the related art.

[0006] In a first aspect, the present invention provides an output consistency control method for a multi-agent system based on one-dimensional wave partial differential, comprising:

[0007] S1. Constructing a dynamic model and an external model of a multi-agent system; the external model includes an interference model and a leader's reference signal model; the multi-agent system includes a plurality of agents, wherein all of the agents are informed agents, or some are informed agents and the rest are uninformed agents;

[0008] S2, construct a directed communication topology graph including a leader and multiple agents;

[0009] S3. When some of the multiple agents are informed agents and the other are uninformed agents, constructing a reference signal observer for each informed agent and each uninformed agent based on the directed communication topology graph and the leader's reference signal model; or, when all the multiple agents are informed agents, constructing a reference signal observer for each informed agent based on the directed communication topology graph and the leader's reference signal model;

[0010] S4. Based on the interference model, construct a local interference observer for each agent;

[0011] S5, constructing a conversion function, and converting the dynamic model into a dynamic model under the auxiliary system according to the conversion function;

[0012] S6. Constructing control laws for the dynamic model under the auxiliary system;

[0013] S7. According to the control law, adjust the state of each intelligent agent in the auxiliary system until the tracking error is zero.

[0014] In an optional embodiment, the dynamic model of the multi-agent system is:

[0015] y tt (x,t)=y xx (x,t) 0 <x<1,t> 0

[0016] y x (0,t)=0 t≥0

[0017] y x (1,t)=u(t)+d(t) t≥0

[0018] y(x,0)=y 0 (x) 0≤x≤1

[0019] y t (x,0)=y 1 (x) 0≤x≤1

[0020] Among them, y(x,t) represents the state of each agent in the multi-agent system at position x and time t; y(x,0) and y t (x,0) represents the initial offset and initial velocity of each agent; y 0 (x) and y 1 (x) are the first preset parameter and the second preset parameter respectively, u(t) is the control signal, and d(t) is the interference signal.

[0021] In an optional implementation, the leader's reference signal model is:

[0022]

[0023]

[0024] Among them, w r (t) is the state of the reference signal, r(t) is the output of the reference signal model, S r is the coefficient matrix of the reference signal state, is the coefficient vector of the reference signal, w r (0) is the initial state of the reference signal, is the third preset parameter;

[0025] The interference model is:

[0026]

[0027]

[0028] in, is the state of the interference signal acting on agent i, d i (t) is the interference signal acting on agent i, is the coefficient matrix of the state of the interference signal acting on agent i, is the coefficient vector of the interference signal of agent i, is the initial state of the interference signal acting on agent i, It is the fourth preset parameter.

[0029] In an optional implementation, the directed communication topology graph is G=(V, E, A); wherein V is a node set, V={v1, v2, ..., v N}, v i is the node corresponding to agent i; E is the edge set, E={e ij |v i , v j ∈V},e ij For slave node v i To node vj A is the adjacency matrix, A=[a ij ]∈R N×N , matrix element a ij is the communication weight from agent j to agent i, the diagonal element a ii =0; the Laplace matrix of the directed communication topology graph is L, L=Δ-A, Δ is the degree matrix of the directed communication topology graph.

[0030] In an optional implementation, the S3 includes:

[0031] S31. When some of the multiple agents are informed agents and the other are uninformed agents, a reference signal observer for each informed agent is constructed based on the directed communication topology graph and the leader's reference signal model:

[0032]

[0033] Among them, i' is an informed agent, i'∈{1,2,……,n}; is the agent i's response to the reference signal w r (t) state estimate; is the gain of the reference signal observer corresponding to the informed agent; a i'0 is the weight broadcast by the leader to agent i', and a i'0 >0; is the estimated value of the reference signal model output r(t) by agent i', t>0; the initial condition of the reference signal observer corresponding to the informed agent is is the initial state value of the reference signal observer;

[0034] S32. When some of the multiple agents are informed agents and the other are uninformed agents, a reference signal observer for each uninformed agent is constructed based on the directed communication topology graph and the leader's reference signal model:

[0035]

[0036] Where i” is an uninformed agent, i”∈{n+1,……,N}, j is another agent adjacent to agent i”, i”≠j; For agent i's response to the reference signal w r (t) state estimate; a i”j is the communication weight between agent i” and its neighboring agent j, and a i”j >0; is the gain of the reference signal observer corresponding to the uninformed agent; is the estimated value of the reference signal model output r(t) by agent i”, t>0; the initial condition of the reference signal observer corresponding to the uninformed agent is is the initial state value of the reference signal observer.

[0037] In an optional embodiment, the S4 includes:

[0038] Based on the interference model, construct a local interference observer for each agent:

[0039]

[0040]

[0041] in, is the state estimation value of agent i to the interference signal acting on agent i; y i (0, t) is the state of agent i at position 0; is the estimated state value of agent i at position 0; is the estimated state value of each agent in the multi-agent system at position x and time t; u(t) is the control signal, d(t) is the estimated value of the interference signal; B is the first gain vector; is the second gain vector; is the Hurwitz matrix; is the coefficient matrix of the state of the interference signal acting on agent i; is the coefficient vector of the interference signal of agent i; c2<0,

[0042] In an optional implementation, the S5 includes:

[0043] S51. Construct the conversion function z(x,t):

[0044]

[0045] Among them, y(x,t) is the state of each agent in the multi-agent system at position x and time t. For S r The frequency corresponding to the eigenvalue of , r(t) is the output of the reference model, 1 N =(1,1,…,1) T ;

[0046] S52. According to the conversion function, the dynamic model is converted into a dynamic model under the auxiliary system:

[0047] z tt(x,t)=z xx (x,t)

[0048] z x (0,t)=0

[0049]

[0050] z(x,0)=z 0 (x),z t (x,0)=z 1 (x)

[0051] Where z(x,t) represents the state of each agent in the multi-agent system under the auxiliary system at position x and time t; 0 (x) and z 1 (x) are the fifth preset parameter and the sixth preset parameter respectively, u(t) is the control signal, and d(t) is the interference signal.

[0052] In an optional embodiment, the S6 includes:

[0053] Construct the control law for the dynamic model under the auxiliary system:

[0054]

[0055] in, And k2>2, is the estimated value of the interference signal by each agent, is the estimated value of the reference signal by each agent.

[0056] In an optional implementation, the S7 includes:

[0057] S71. Calculate the tracking error based on the dynamic model and the reference model:

[0058] e(t)=y(1,t)-r(t)

[0059] Where y(1,t) is the state of the agent at position 1 and time t, and r(t) is the output of the reference model;

[0060] S72. Convert the tracking error into a tracking error under the auxiliary system according to the conversion model:

[0061] z(1,t)=y(1,t)-r(t)=e(t)

[0062] S73. Adjust the state of each agent in the auxiliary system according to the control law until the tracking error reaches zero;

[0063]

[0064] The present invention can solve the problem of output consistency of infinite-dimensional heterogeneous multi-agent systems. By introducing wave equations to describe the dynamics of multi-agent systems, the collaborative control is extended to the field of networked partial differential equations. Compared with the ordinary differential equations in related technologies, it can more accurately describe the spatial diffusion and interaction of agents. At the same time, by constructing specific transfer functions, targeted observers and control laws, and precise regulation of each constructed model, it can achieve consistency in the output of each agent in a complex multi-agent system, effectively overcoming the shortcomings of related technologies in dealing with such infinite-dimensional heterogeneous systems, and providing a practical solution for the stable collaborative work of multi-agent systems in complex environments. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the specific embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0066] Figure 1 1 is a flow chart of an output consistency control method for a one-dimensional wave partial differential multi-agent system according to an embodiment of the present invention;

[0067] Figure 2 A directed graph of a multi-agent system in which a portion of the agents are informed and another portion of the agents are uninformed according to an embodiment of the present invention;

[0068] Figure 3 is a directed graph of a multi-agent system in which all agents are informed agents according to an embodiment of the present invention;

[0069] Figure 4 According to an embodiment of the present invention Figure 2 A curve diagram showing the output consistency results achieved under the following circumstances;

[0070] Figure 5 According to an embodiment of the present invention Figure 3 The curve diagram of the output consistency results under the condition of . DETAILED DESCRIPTION

[0071] To make the purpose, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without making creative efforts shall fall within the scope of protection of the present invention.

[0072] like Figure 1 As shown in FIG, a process of an output consistency control method for a one-dimensional wave partial differential multi-agent system includes the following steps:

[0073] S1. Construct a dynamic model and an external model of a multi-agent system; the external model includes an interference model and a reference signal model of a leader; the multi-agent system includes multiple agents, all of which are informed agents, or some are informed agents and the other are uninformed agents.

[0074] Specifically, the multi-agent system includes N agents, i∈{1,2,…,N}, i is any agent among the N agents; among them, agent i can be further divided into informed agent i' or uninformed agent i", i'∈{1,2,…,n}, i"∈{n+1,…,N}, i' represents the informed agent, that is, the agent that communicates with the reference signal, and i" represents the uninformed agent, that is, the agent that cannot directly communicate with the reference signal. These uninformed agents need to communicate with their adjacent agents.

[0075] It should be noted that the reference signal is generated by the leader and broadcast to the informed agents through the communication channel between the leader and the informed agents; the uninformed agents indirectly obtain the reference signal through interactions with adjacent agents (adjacent agents can be informed agents or uninformed agents), thereby achieving output consistency of each agent.

[0076] For example, among multiple agents, some are informed agents and the other are uninformed agents, such as Figure 2 As shown in the figure, the total number of agents is 4, i.e., i∈{1,2,3,4}, among which agent 1 and agent 2 can communicate directly with the leader (i.e., agent 0), that is, agent 1 and agent 2 are informed agents; agent 3 and agent 4 cannot communicate directly with the reference signal (agent 0). For agent 3, it needs to communicate with the adjacent agents 1 and 4. For agent 4, it needs to communicate with the adjacent agents 2 and 3. Therefore, agent 3 and agent 4 are both uninformed agents.

[0077] For example, multiple agents are informed agents, such as Figure 3 As shown, the total number of agents is 4, i.e., i∈{1,2,3,4}, among which agents 1 to 4 can communicate directly with the leader (i.e., agent 0), that is, agents 1 to 4 are all informed agents.

[0078] In an optional embodiment, the dynamic model of the multi-agent system is:

[0079] y tt (x,t)=y xx (x,t) 0 <x<1,t> 0

[0080] y x (0,t)=0 t≥0

[0081] y x (1,t)=u(t)+d(t) t≥0

[0082] y(x,0)=y 0 (x) 0≤x≤1

[0083] y t (x,0)=y 1 (x) 0≤x≤1

[0084] Among them, y(x,t) represents the state of each agent in the multi-agent system at position x and time t; y(x,0) and y t (x,0) represents the initial offset and initial velocity of each agent; y 0 (x) and y 1 (x) are the first preset parameter and the second preset parameter respectively; u(t) is the control signal; d(t) is the interference signal.

[0085] Specifically, tt (x, t) represents the rate of change of the agent's speed over time, that is, acceleration, which is used to reflect the speed of the agent's state evolution over time; y xx (x, t) represents the rate of change of the agent's state with space, that is, the curvature, which is used to reflect the degree of curvature of the agent's state in the spatial distribution; y tt (x,t)=y xx The acceleration of the agent's state (x, t) is proportional to the curvature of space. The greater the curvature of space (the steeper the waveform), the greater the acceleration of the agent's state and the more drastic the state change; conversely, the smaller the curvature of space (the flatter the waveform), the smaller the acceleration of the agent's state and the smoother the state change. x (0, t) is the right boundary condition, which means that when the agent is at position x = 0, the spatial derivative of y(x, t) is 0, that is, there is no inflow or outflow of information in the multi-agent system at the right boundary; y x (1, t) is the left boundary condition, which means that when the agent is at position x = 1, the spatial derivative of y(x, t) is determined by the control signal u(t) and the interference signal d(t); where d(t) represents the interference acting on all agents, d(t)∈R N , u(t) is the control signal, and the control signal u(t) and the interference d(t) are in the same channel; y(x,0) and y t (x,0) represent the initial offset and initial velocity of the agent, respectively, which are determined by two preset parameters y 0 (x), y 1 (x) OK.

[0086] It should be noted that the two preset parameters y 0 (x), y 1 The value of (x) can be adaptively set according to the characteristics or design requirements of the multi-intelligence system and is not specifically limited here.

[0087] In an optional implementation, the leader's reference signal model is:

[0088]

[0089]

[0090] Among them, w r (t) is the state of the reference signal, r(t) is the output of the reference signal model, S r is the coefficient matrix of the reference signal state, is the coefficient vector of the reference signal, w r (0) is the initial state of the reference signal, is the third preset parameter.

[0091] Specifically, w r (t) is the state of the reference signal, which is a vector that changes with time t; w r The derivative of (t) with respect to time t is used to describe the dynamic characteristics of the reference signal over time in the multi-agent system. r is the coefficient matrix of the reference signal state, describing how the reference signal state evolves over time, and And there are only simple eigenvalues on the imaginary axis, indicating that the reference signal model has neutral stability. T is the coefficient vector of the reference signal, i.e., the linear mapping from the state of the reference signal to the actual output; Indicates S r is a dimension of , q is a matrix dimensional vector, one dimensional vector.

[0092] It should be noted that the coefficient matrix S r , coefficient vector q, third preset parameter All of them can be adaptively set according to the physical characteristics and design requirements of the multi-agent system, and no specific limitations are given here.

[0093] The interference model is:

[0094]

[0095] in, is the state of the interference signal acting on agent i, d i (t) is the interference signal acting on agent i, is the coefficient matrix of the state of the interference signal acting on agent i, is the coefficient vector of the interference signal of agent i, is the initial state of the interference signal acting on agent i, It is the fourth preset parameter.

[0096] Specifically, is the state of the interference signal acting on agent i, which is a vector that changes with time t; for The derivative with respect to time t is used to describe Time-varying dynamics in multi-agent systems. is the coefficient matrix of the state of the interference signal acting on agent i, describing How it evolves over time, and the Hurwitz matrix S d is the matrix The block diagonal matrix assembled along the diagonal direction is Therefore, the Hurwitz matrix S d The coefficient matrix of the state of the interference signal acting on agent i is obtained is the coefficient vector of the interference signal acting on agent i, that is, to the actual output; where, Indicates S d is a dimension of The matrix, p i For one dimensional vector, one dimensional vector.

[0097] For example, Figure 2 He Ru Figure 3 As shown, when there are 4 agents, i∈{1,2,3,4}, the matrix is the matrix The block diagonal matrix assembled along the diagonal direction is

[0098] It should be noted that the coefficient matrix Sd , coefficient vector p i and the fourth preset parameter All of them can be adaptively set according to the physical characteristics and design requirements of the multi-agent system, and no specific limitations are given here.

[0099] S2. Construct a directed communication topology graph including a leader and multiple agents.

[0100] In an optional implementation, the directed communication topology graph is G=(V, E, A); wherein V is a node set, V={v1, v2, ..., v N}, each agent i in the multi-agent system is regarded as a node v i (i.e., v i is the node corresponding to agent i), the total number of nodes is N; E is the edge set, E={e ij |v i , v j ∈V}, where e ij For slave node v i To node v j A is the adjacency matrix, which is used to quantify the communication relationship between agents. A=[a ij ]∈R N×N , matrix element a ij is the communication weight from agent j to agent i. ij >0, indicating node v i Can be sent to node v j Send a message; if a ij =0, indicating node v i and node v j There is no direct communication between them. In addition, self-loops are not allowed in the directed communication topology graph G, that is, the diagonal elements a ii =0.

[0101] The Laplace matrix of the directed communication topology graph G is L, where L=Δ-A, where Δ is the degree matrix of the directed communication topology graph G (it can be either an out-degree matrix or an in-degree matrix).

[0102] S3. When some of the multiple intelligent agents are informed intelligent agents i' and the other part are uninformed intelligent agents i", a reference signal observer is constructed for each informed intelligent agent i' and each uninformed intelligent agent i" based on the directed communication topology graph and the leader's reference signal model; or, when multiple intelligent agents are all informed intelligent agents i", a reference signal observer is constructed for each informed intelligent agent based on the directed communication topology graph and the leader's reference signal model.

[0103] In an optional embodiment, step S3 includes:

[0104] S31. When some of the multiple agents are informed agents and the other are uninformed agents, a reference signal observer for each informed agent is constructed based on the directed communication topology graph and the leader's reference signal model:

[0105]

[0106] Among them, i' is an informed agent, i'∈{1,2,……,n}; is the agent i's response to the reference signal w r (t) state estimate; is the gain of the reference signal observer corresponding to the informed agent; a i'0 is the weight broadcast by the leader to agent i', and a i'0 >0; is the estimated value of the reference signal model output r(t) by agent i', t>0; the initial condition of the reference signal observer corresponding to the informed agent is is the initial state value of the reference signal observer, which can generally be set to 0; and since the matrix S r , vector q are given coefficient matrix and coefficient vector respectively, and an observer gain l can always be found i' To ensure the asymptotic convergence of the observer, is the Hurwitz matrix.

[0107] S32. When some of the multiple agents are informed agents and the other are uninformed agents, a reference signal observer for each uninformed agent is constructed based on the directed communication topology graph and the leader's reference signal model:

[0108]

[0109] Where i” is an uninformed agent, i”∈{n+1,……,N}, j is another agent adjacent to agent i”, i”≠j; For agent i's response to the reference signal w r (t) state estimate; a i”j is the communication weight between agent i” and its neighboring agent j, and a i”j >0; is the gain of the reference signal observer corresponding to the uninformed agent; is the estimated value of the reference signal model output r(t) by agent i”, t>0; the initial condition of the reference signal observer corresponding to the uninformed agent is is the initial state value of the reference signal observer, which can generally be set to 0.

[0110] For example, Figure 2 As shown, i'∈{1,2}, i"∈{3,4}, where reference signal observers are established for informed agent 1 and informed agent 2 respectively:

[0111]

[0112] Reference signal observers are established for uninformed agent 3 and uninformed agent 4 respectively:

[0113]

[0114] In an optional implementation, step S3 further includes:

[0115] When multiple agents are informed agents, a reference signal observer for each informed agent is constructed based on the directed communication topology graph and the leader's reference signal model:

[0116]

[0117] Among them, i' is an informed agent, i'∈{1,2,……,n}; is the agent i's response to the reference signal w r (t) state estimate; is the gain of the reference signal observer corresponding to the informed agent; a i'0 is the weight broadcast by the leader to agent i', and a i'0 >0; is the estimated value of the reference signal model output r(t) by agent i', t>0; the initial condition of the reference signal observer corresponding to the informed agent is And, since the matrix S r , vector q are given coefficient matrix and coefficient vector respectively, and an observer gain l can always be found i' To ensure the asymptotic convergence of the observer, is the Hurwitz matrix.

[0118] For example, Figure 3 As shown, i'∈{1,2,3,4}, where reference signal observers are established for informed agents 1 to 4 respectively:

[0119]

[0120] S4. Based on the interference model, construct a local interference observer for each agent.

[0121] In an optional implementation, the local disturbance observer is:

[0122]

[0123]

[0124] in, is the state estimation value of agent i to the interference signal acting on agent i; y i (0, t) is the state of agent i at position 0; is the estimated state value of agent i at position 0; is the estimated state value of each agent in the multi-agent system at position x and time t; u(t) is the control signal, d(t) is the estimated value of the interference signal; B is the first gain vector; is the second gain vector; is the Hurwitz matrix; is the coefficient matrix of the state of the interference signal acting on agent i; is the coefficient vector of the interference signal of agent i; c2<0, c3>

[0125] Preferably, c2=-1, c3=2, c4=4.

[0126] S5. Constructing a conversion function, and converting the dynamic model into a dynamic model under the auxiliary system according to the conversion function.

[0127] In an optional embodiment, step S5 includes:

[0128] S51. Construct the conversion function z(x,t):

[0129]

[0130] Among them, y(x,t) is the state of each agent in the multi-agent system at position x and time t. For S r The frequency corresponding to the eigenvalue of , r(t) is the output of the reference model, 1 N =(1,1,...,1) T .

[0131] Specifically, since the reference signal model is:

[0132]

[0133] And, S r The eigenvalues of are on the imaginary axis, and its solution Among them, J is the index set of eigenvalues, ω jis the frequency corresponding to eigenvalue j, For S r The eigenvalue value corresponding to the eigenvector; Initial state In the feature vector The projection in the direction, so we can get for The form of the quantity.

[0134] Based on this, the tracking reference signal is set to:

[0135]

[0136] Based on the tracking reference signal, a conversion function is constructed to transform the output consistency problem of the multi-agent system's dynamic model into a more manageable form. The goal is to use this conversion function to map the complex relationships related to output consistency in the multi-agent system's dynamic model to an auxiliary system composed of new variables z(x, t). This makes subsequent analysis of system stability and design of control laws clearer. This conversion transforms the study of output consistency in the multi-agent system's dynamic model into a study of the stability of the new system z(x, t), thereby simplifying the problem.

[0137] S52. According to the conversion function, the dynamic model is converted into a dynamic model under the auxiliary system:

[0138] z tt (x,t)=z xx (x,t)

[0139] z x (0,t)=0

[0140]

[0141] z(x,0)=z 0 (x),z t (x,0)=z 1 (x)

[0142] Where z(x,t) represents the state of each agent in the multi-agent system under the auxiliary system at position x and time t; 0 (x) and z 1 (x) are the fifth preset parameter and the sixth preset parameter respectively, u(t) is the control signal, and d(t) is the interference signal.

[0143] It should be noted that the dynamic model under the auxiliary system is basically consistent with the original dynamic model in form, and the definitions of various parameters are also the same, which will not be repeated here.

[0144] S6. Construct the control law for the dynamic model under the auxiliary system.

[0145] In an optional implementation, the control law is:

[0146]

[0147] in, and is the estimated value of the interference signal by each agent, It is the estimated value of each agent's output of the reference signal model.

[0148] Preferably, k1=0.4 and k2=2.1.

[0149] S7. According to the control law, adjust the state of each intelligent agent under the auxiliary system until the tracking error is zero.

[0150] By constructing a control law, the embodiment of the present invention can make the dynamic model of the auxiliary system remain stable under various circumstances, thereby achieving the output consistency goal of the original dynamic model.

[0151] In an optional embodiment, step S7 includes:

[0152] S71. Calculate the tracking error based on the dynamic model and the reference model:

[0153] e(t)=y(1,t)-r(t)

[0154] Among them, y(1,t) is the state of the agent at position 1 and time t, and r(t) is the output of the reference model.

[0155] Specifically, in order to achieve output consistency, y(1,t) is set as the output, and the goal is to ensure that the tracking error is zero, that is,

[0156]

[0157] in,

[0158] To achieve this goal, appropriate control strategies and observer designs are required. By precisely adjusting system parameters and gains, tracking errors can be effectively reduced, eliminating any factors that may cause output deviations and achieving consistency among various agents.

[0159] S72. Convert the tracking error into a tracking error under the auxiliary system according to the conversion model:

[0160] z(1,t)=y(1,t)-r(t)=e(t)

[0161] This formula shows that z(1,t) in the dynamic model of the auxiliary system is directly related to the error e(t) of the original dynamic model.

[0162] S73. Adjust the state of each agent in the auxiliary system according to the control law until the tracking error reaches zero;

[0163]

[0164] The present invention can output different control laws by adjusting the parameters of each model, and then adjust the state of each intelligent agent under the auxiliary system according to the different control laws until the tracking error is zero, that is, the output stops when it is consistent.

[0165] It should be noted that the parameters for adjusting each model are shown in Table 1 and Table 2 below, which will not be repeated here.

[0166] The embodiment of the present invention determines whether the constructed Lyapunov function under the auxiliary system can be asymptotically stable by constructing the Lyapunov function.

[0167] The Lyapunov stability theorem is a crucial tool for determining system stability. It determines system stability based on changes in the system's internal energy. Assuming a dynamic system's initial state is stable, if the system can maintain a constant internal stored energy after a small perturbation, then the system is stable. If the stored energy of the system decays over time—that is, the system needs to absorb energy, but the absorbed energy decreases until it reaches a minimum—then the system is asymptotically stable. If the stored energy increases while the absorbed energy increases, the system is unstable and will eventually lose control. By constructing a suitable Lyapunov function and analyzing the sign of its derivative, it is possible to determine whether the auxiliary system is asymptotically stable.

[0168] Specifically include:

[0169] Step a: Construct Lyapunov function:

[0170] V(t)=E(t)+G1(t)+G2(t)

[0171] in,

[0172]

[0173] Among them, k1, k2, and k3 are all coefficients greater than 0, which are used to adjust the proportions of each part in the Lyapunov function.

[0174] Step b: Derivate the constructed Lyapunov function:

[0175]

[0176] Where V(t) is the Lyapunov function, and are the observation errors of the disturbance and reference signals, respectively, is the reference signal vibration frequency.

[0177] Step c: Since k2>2, and according to the young inequality, we get:

[0178]

[0179] in,

[0180] Step d: According to the Bellman-Gronwald inequality, we can get

[0181]

[0182] Where S is the integral variable;

[0183] Step e: There exists a time T1>0, then for t>T1

[0184]

[0185] in, is the observation error of the perturbation to the i-th agent, is the observation error of the reference signal of the jth agent, N is the number of agents, ε, ξ i ,ζ i are numbers greater than 0 and less than 1, respectively, and are the quantity values generated during the scaling of the inequality;

[0186] Step f, let get

[0187]

[0188] so

[0189] Where M1 represents the cumulative amount of disturbance observation error and reference signal observation error over time. and denote the disturbance observation error and the reference signal observation error respectively. T1 is the upper bound of the integration time period; represents the perturbation observation error acting on the i-th agent; represents the perturbation observation error acting on the j-th agent;

[0190] Step g: There exists a time T2>T1, then for t>T2

[0191]

[0192] so

[0193] Among them, T2 is a moment after the integral upper limit T1.

[0194] Step h: According to the limit definition, we get

[0195] The embodiment of the present invention utilizes Lyapunov functionals to construct functions based on energy perspectives. By analyzing their changes over time, the consistency of system output is judged, and the stability of the observer is judged at the same time, providing rigorous support for system stability analysis and enhancing the reliability and practicality of the results.

[0196] Therefore, the dynamic model under the auxiliary system is stable and the multi-agent system achieves output consistency.

[0197] In order to verify the correctness and effectiveness of the technical effects of the present invention, simulation experiments were carried out with the help of MATLAB.

[0198] by Figure 2 The corresponding four agents are used as an example for simulation verification, and their parameters are shown in Table 1:

[0199] Table 1

[0200]

[0201] by Figure 3 The corresponding four intelligent agents are used as an example for simulation verification, and their parameters are shown in Table 2:

[0202] Table 2

[0203]

[0204] According to the relevant information in Table 1 and Table 2, the state of the multi-agent is updated using the second-order difference method. Figure 4 and Figure 5 It can be clearly seen that Figure 2 and Figure 3 In both cases, the curves of the four agents' y(1, t) over time t essentially overlap with the curve of the reference signal model output r(t). Throughout the time interval 0 to 50, although the output states of the four agents fluctuate slightly in the initial stage, they gradually become consistent with r(t) over time. This demonstrates that the multi-agent system can achieve output consistency after state updates using the control method of the present invention.

[0205] In summary, the present invention has the following beneficial effects:

[0206] (1) The requirements for network topology are simple, highly applicable, and easy to implement.

[0207] (2) By designing observers, reference signal observers are designed for informed agents and uninformed agents respectively, and local interference observers are designed for each agent. This allows us to accurately observe the states of reference signals and interference signals based on the characteristics of different types of agents, providing a reliable basis for the subsequent adjustment of control strategies and effectively improving the system's ability to cope with complex environments.

[0208] (3) By constructing a specific conversion function, the output consistency problem of the original dynamic model is transformed into the stability problem of the new system, which greatly simplifies the difficulty of handling the problem.

[0209] Although the embodiments of the present invention have been described with reference to the accompanying drawings, those skilled in the art may make various modifications and variations without departing from the spirit and scope of the present invention. Such modifications and variations are all within the scope defined by the appended claims.

Claims

1. A method for output consistency control of a multi-agent system based on one-dimensional wave partial differential, characterized in that: include: S1. Constructing a dynamic model and an external model of a multi-agent system; the external model includes an interference model and a leader's reference signal model; the multi-agent system includes a plurality of agents, wherein all of the agents are informed agents, or some are informed agents and the rest are uninformed agents; S2, construct a directed communication topology graph including a leader and multiple agents; S3. When some of the multiple agents are informed agents and the other are uninformed agents, constructing a reference signal observer for each informed agent and each uninformed agent based on the directed communication topology graph and the leader's reference signal model; or, when all the multiple agents are informed agents, constructing a reference signal observer for each informed agent based on the directed communication topology graph and the leader's reference signal model; S4. Based on the interference model, construct a local interference observer for each agent; S5, constructing a conversion function, and converting the dynamic model into a dynamic model under the auxiliary system according to the conversion function; S6. Constructing control laws for the dynamic model under the auxiliary system; S7. According to the control law, adjust the state of each intelligent agent in the auxiliary system until the tracking error is zero.

2. The method according to claim 1, characterized in that The dynamic model of the multi-agent system is: y tt (x,t)=y xx (x,t)0<x<1,t>0 y x (0,t)=0t≥0 y x (1,t)=u(t)+d(t)t≥0 y(x,0)=y 0 (x)0≤x≤1 and t (x,0)=y 1 (x)0≤x≤1 Among them, y(x,t) represents the state of each agent in the multi-agent system at position x and time t; y(x,0) and y t (x,0) represents the initial offset and initial velocity of each agent; y 0 (x) and y 1 (x) are the first preset parameter and the second preset parameter respectively, u(t) is the control signal, and d(t) is the interference signal.

3. The method according to claim 2, characterized in that The reference signal model of the leader is: Among them, w r (t) is the state of the reference signal, r(t) is the output of the reference signal model, S r is the coefficient matrix of the reference signal state, is the coefficient vector of the reference signal, w r (0) is the initial state of the reference signal, is the third preset parameter; The interference model is: in, is the state of the interference signal acting on agent i, d i (t) is the interference signal acting on agent i, is the coefficient matrix of the state of the interference signal acting on agent i, is the coefficient vector of the interference signal of agent i, is the initial state of the interference signal acting on agent i, It is the fourth preset parameter.

4. The method according to claim 3, characterized in that The directed communication topology graph is G=(V, E, A); wherein V is a node set, V={v1, v2, ..., v N }, v i is the node corresponding to agent i; E is the edge set, E={e ij |v i , v j ∈V},e ij For slave node v i To node v j A is the adjacency matrix, A=[a ij ]∈R N×N , matrix element a ij is the communication weight from agent j to agent i, the diagonal element a ii =0; the Laplace matrix of the directed communication topology graph is L, L=Δ-A, Δ is the degree matrix of the directed communication topology graph.

5. The method according to claim 4, characterized in that The S3 includes: S31. When some of the multiple agents are informed agents and the other are uninformed agents, a reference signal observer for each informed agent is constructed based on the directed communication topology graph and the leader's reference signal model: Among them, i' is an informed agent, i'∈{1,2,……,n}; is the agent i's response to the reference signal w r (t) state estimate; is the gain of the reference signal observer corresponding to the informed agent; a i'0 is the weight broadcast by the leader to agent i', and a i'0 >0; is the estimated value of the reference signal model output r(t) by agent i', The initial condition of the reference signal observer corresponding to the informed agent is is the initial state value of the reference signal observer; S32. When some of the multiple agents are informed agents and the other are uninformed agents, a reference signal observer for each uninformed agent is constructed based on the directed communication topology graph and the leader's reference signal model: Where i” is an uninformed agent, i”∈{n+1,……,N}, j is another agent adjacent to agent i”, i”≠j; For agent i's response to the reference signal w r (t) state estimate; a i”j is the communication weight between agent i” and its neighboring agent j, and a i”h >0; is the gain of the reference signal observer corresponding to the uninformed agent; is the estimated value of the reference signal model output r(t) by agent i”, The initial condition of the reference signal observer corresponding to the uninformed agent is is the initial state value of the reference signal observer.

6. The method according to claim 5, characterized in that The S4 includes: Based on the interference model, construct a local interference observer for each agent: in, is the state estimation value of agent i to the interference signal acting on agent i; y i (0, t) is the state of agent i at position 0; is the estimated state value of agent i at position 0; is the estimated state value of each agent in the multi-agent system at position x and time t; u(t) is the control signal, d(t) is the estimated value of the interference signal; B is the first gain vector; is the second gain vector; is the Hurwitz matrix; is the coefficient matrix of the state of the interference signal acting on agent i; is the coefficient vector of the interference signal of agent i; c2<0, 7. The method according to claim 6, characterized in that The S5 includes: S51. Construct the conversion function z(x,t): Among them, y(x,t) is the state of each agent in the multi-agent system at position x and time t. For S r The frequency corresponding to the eigenvalue of , r(t) is the output of the reference model, S52. According to the conversion function, the dynamic model is converted into a dynamic model under the auxiliary system: z tt (x,t)=z xx (x,t) z x (0,t)=0 z(x,0)=z 0 (x),z t (x,0)=z 1 (x) Where z(x,t) represents the state of each agent in the multi-agent system under the auxiliary system at position x and time t; 0 (x) and z 1 (x) are the fifth preset parameter and the sixth preset parameter respectively, u(t) is the control signal, and d(t) is the interference signal.

8. The method according to claim 7, characterized in that The S6 includes: Construct the control law for the dynamic model under the auxiliary system: in, And k2>2, is the estimated value of the interference signal by each agent, is the estimated value of the reference signal by each agent.

9. The method according to claim 8, characterized in that The S7 includes: S71. Calculate the tracking error based on the dynamic model and the reference model: e(t)=y(1,t)-r(t) Where y(1,t) is the state of the agent at position 1 and time t, and r(t) is the output of the reference model; S72. Convert the tracking error into a tracking error under the auxiliary system according to the conversion model: z(1,t)=y(1,t)-r(t)=e(t) S73. Adjust the state of each agent in the auxiliary system according to the control law until the tracking error reaches zero;

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