Multi-agent variable expected trajectory iteration consistency tracking method and system

Abstract

The invention discloses a multi-agent variable expected trajectory iteration consistency tracking method and system. The method comprises the following steps: constructing a dynamic model and a network communication topological graph of a multi-agent system; determining a multi-agent tracking error, constructing a controller through a P-type open-loop learning algorithm of a high-order inner membrane, performing iterative control on a dynamic model of the multi-agent system, and outputting the multi-agent system with consistent tracking; verifying the multi-agent system with consistent tracking according to a compression mapping principle and norm properties; and based on the verified multi-agent system, tracking the multi-agent through a P-type iterative learning control algorithm based on a second-order internal model. According to the method, the corresponding iterative learning controller based on the high-order internal model can be designed for the intelligent agent on the basis of the observation error, and the consistency tracking precision of the variable expected trajectory is improved. The multi-agent variable expected trajectory iteration consistency tracking method and system can be widely applied to the technical field of intelligent control.

Description

Technical Field

[0001] The present invention relates to the technical field of intelligent control, and particularly relates to a variable desired trajectory iterative consensus tracking method and system for multi-agent systems. Background Art

[0002] With the continuous development of technologies such as artificial intelligence and intelligent manufacturing, the applications of multi-agent systems (MASs) have become increasingly extensive and in-depth, including fields such as intelligent robots, traffic control, distributed intelligent decision-making, and network control. A multi-agent system is a system composed of multiple discrete agents, which can be software programs, robots, or other autonomous entities. Each agent has its own perception, decision-making, and action capabilities, and jointly achieves goals or tasks through mutual communication, sharing of information and resources, as well as cooperation and competition. The purpose of studying multi-agent systems is to solve large-scale and complex real-world problems, and solving such problems has exceeded the capabilities of a single agent. Through coordination and cooperation, such a system can complete tasks that are difficult or impossible for a single agent to complete, and has higher flexibility and scalability. However, there are some problems in the consensus control technology of multi-agent systems in related technologies. For example, in the tracking control method of multi-agent systems based on finite time in related technologies, this control method relies on time-domain feedback control, which has extremely high requirements for the resource occupancy and communication capabilities of agents, and there are difficulties in practical applications; in addition, the target trajectory set by this method can only be fixed and unchanged, and cannot meet the requirements when the target trajectory changes. At the same time, in the consensus or synchronization control technology of multi-agent systems, there are also limitations in communication and interaction, and the information exchange and communication between multi-agent systems may be affected by problems such as network delay, packet loss, and communication bandwidth limitation. Summary of the Invention

[0003] In order to solve the above technical problems, the purpose of the present invention is to provide a variable desired trajectory iterative consensus tracking method and system for multi-agent systems, which can design a corresponding iterative learning controller based on a high-order internal model for agents based on the observation error, and improve the consensus tracking accuracy of the variable desired trajectory.

[0004] The first technical solution adopted by the present invention is: A variable desired trajectory iterative consensus tracking method for multi-agent systems, comprising the following steps:

[0005] Construct a dynamic model and a network communication topology diagram of the multi-agent system, where the multi-agent system includes a leader agent and a plurality of follower agents;

[0006] Determine the multi-agent tracking error based on the network communication topology graph, construct a controller through the P-type open-loop learning algorithm of the high-order internal model, and perform iterative control on the dynamic model of the multi-agent system to output a multi-agent system with consistent tracking;

[0007] According to the contraction mapping principle and the norm property, verify the multi-agent system with consistent tracking to obtain the verified multi-agent system;

[0008] Based on the verified multi-agent system, track the multi-agent through the P-type iterative learning control algorithm based on the second-order internal model to obtain the simulated tracking result.

[0009] Furthermore, the step of constructing the dynamic model of the multi-agent system and the network communication topology graph specifically includes:

[0010] Determine the dynamic model of the follower agent and the dynamic model of the leader agent, and construct the dynamic model of the multi-agent system;

[0011] According to the communication relationship of the multi-agents in the multi-agent system, construct a non-negative weighted graph;

[0012] Define preset conditions for the non-negative weighted graph to construct the network communication topology graph.

[0013] Furthermore, the expression of the dynamic model of the follower agent is specifically as follows:

[0014]

[0015] In the above formula, i represents the i-th follower agent, k represents the k-th iteration, t represents the time variable, x i,k (t) represents the state of the follower agent i at the k-th iteration, x i,k (t + 1) represents the state of the follower agent i at the next moment at the k-th iteration, u i,k (t) represents the control input of the follower agent i at the k-th iteration, y i,k (t) represents the control output of the follower agent i at the k-th iteration, τ represents the maximum time delay of the agent, E represents the singular matrix of the generalized system, A and D represent the system matrices, B represents the input matrix, and c represents the output matrix;

[0016] The expression of the dynamic model of the leader agent is specifically as follows:

[0017]

[0018] In the above formula, denotes the desired trajectory, \(t\) denotes the time variable, \(d\) denotes the subscript symbol of the desired trajectory, \(k\) denotes the number of iterations, \(l\) denotes the order of the high-order internal model, and \(h\) l denotes the coefficient corresponding to the high-order internal model polynomial, \(\omega\) denotes the preset mathematical operator, \(H(\omega -1 ) is used to simplify the expression.

[0019] Furthermore, the step of determining the multi-agent tracking error based on the network communication topology graph, constructing a controller through the P-type open-loop learning algorithm of the high-order internal membrane, and performing iterative control on the dynamic model of the multi-agent system to output a multi-agent system with consistent tracking specifically includes:

[0020] Determine the desired trajectory of the multi-agent and the high-order polynomial;

[0021] Construct the P-type open-loop learning algorithm of the high-order internal membrane based on the high-order polynomial and construct a controller;

[0022] Determine the system tracking error based on the dynamic model of the multi-agent system;

[0023] According to the preset conditions of the network communication topology graph, perform iterative control on the dynamic model of the multi-agent system through the P-type open-loop learning algorithm of the high-order internal membrane, so that the system tracking error meets the desired trajectory of the multi-agent, and output a multi-agent system with consistent tracking.

[0024] Furthermore, the expression of the P-type open-loop learning algorithm of the high-order internal membrane is specifically as follows:[[]]END]]

[0025] u i,k+1 (t)=h 1 u i,k (t)+h 2 u i,k-1 (t)+…+h l u i,k-l+1 (t)+Γ 1 ξ i,k (t)+Γ 2 ξ i,k-1 (t)+…+Γ l ξ i,k-l+1 (t)=H(ω -1 )u i,k (t)+Γ(ω -1 )ξ i,k (t)

[0026] In the above formula, Γ(ω -1 ) represents the gain matrix, ξ i,k (t) represents the observation error, \(t\) represents the time variable, and u i,k+1 (t) represents the control input of the follower agent \(i\) at the \((k + 1)\)-th iteration, and u i,k-1(t) represents the control input of the follower agent i at the (k - 1)-th iteration, u i,k-l+1 (t) represents the control input of the follower agent i at the (k - l + 1)-th iteration, h l represents the coefficient corresponding to the high-order internal model polynomial, Γ l represents the l-th gain coefficient, u i,k (t) represents the control input of the follower agent i at the k-th iteration, H(ω -1 ) represents a mathematical symbol used to simplify expressions, ω represents a preset mathematical operator, k represents the number of iterations, l represents the order of the high-order internal model, and i represents the i-th follower agent.

[0027] Furthermore, the step of verifying the multi-agent system with consensus tracking according to the contraction mapping principle and norm properties to obtain the verified multi-agent system specifically includes:

[0028] Simplify and decompose the dynamic model of the multi-agent system by the singular value decomposition method to obtain a restricted equivalent form of the dynamic model of the multi-agent system;

[0029] Simplify the P-type open-loop learning algorithm of the high-order internal membrane, and verify the multi-agent system according to the contraction mapping principle and norm properties, combined with the restricted equivalent form of the dynamic model of the multi-agent system, to obtain the verified multi-agent system.

[0030] Furthermore, the step of tracking the multi-agent through the P-type iterative learning control algorithm based on the second-order internal model based on the verified multi-agent system to obtain the simulation tracking result specifically includes:

[0031] Based on the verified multi-agent system, set the state space of the dynamic model of the multi-agent system and the motion trajectory of the dynamic model of the multi-agent system;

[0032] Based on the state space of the dynamic model of the multi-agent system and the motion trajectory of the dynamic model of the multi-agent system, test the performance of the controller acting on the multi-agent system to obtain the test numerical simulation example result;

[0033] If the test numerical simulation example result meets the preset requirements, construct the P-type iterative learning control algorithm based on the second-order internal model and track the multi-agent to obtain the simulation tracking result.

[0034] The second technical solution adopted by the present invention is: a variable desired trajectory iterative consensus tracking system for multi-agents, including:

[0035] The first module is used to construct the dynamic model and network communication topology diagram of the multi-agent system, where the multi-agent system includes a leader agent and several follower agents;

[0036] The second module is used to determine the multi-agent tracking error based on the network communication topology diagram, construct a controller through the P-type open-loop learning algorithm of the high-order internal model, and perform iterative control on the dynamic model of the multi-agent system, and output a multi-agent system with consistent tracking;

[0037] The third module is used to verify the multi-agent system with consistent tracking according to the principle of contraction mapping and the properties of the norm, and obtain the verified multi-agent system;

[0038] The fourth module is used to track the multi-agent based on the verified multi-agent system through the P-type iterative learning control algorithm based on the second-order internal model, and obtain the simulated tracking result.

[0039] The beneficial effects of the method and system of the present invention are as follows: By constructing the dynamic model and network communication topology diagram of the multi-agent system, and then constructing a controller through the P-type open-loop learning algorithm of the high-order internal model and performing iterative control on the dynamic model of the multi-agent system, the observation error is defined according to the network communication topology diagram between multi-agents, and a corresponding iterative learning controller based on the high-order internal model is designed for the agents on the basis of the observation error, so that the output of the generalized multi-agent system can achieve precise consistent tracking of the expected trajectory with iterative changes. Finally, the multi-agent tracking error is verified, the verified multi-agent tracking error is obtained, and a P-type iterative learning control algorithm based on the second-order internal model is constructed to track the multi-agent. The iterative learning control is introduced to use the historical information carried by the system to perform iterative tracking on the target. Applying it to the consistent control of the multi-agent system can effectively improve the tracking transient performance, reduce the occupation of resources and the dependence on communication capabilities of the multi-agent system, and improve the consistent tracking accuracy of the variable expected trajectory. Description of the Drawings

[0040] Figure 1 is the flowchart of the steps of a method for iterative consistent tracking of variable expected trajectories of multi-agents according to the present invention;

[0041] Figure 2 is the structural block diagram of a system for iterative consistent tracking of variable expected trajectories of multi-agents according to the present invention;

[0042] Figure 3 is the framework schematic diagram of iterative consistent tracking provided by a specific embodiment of the present invention;

[0043] Figure 4 is the network communication topology schematic diagram provided by a specific embodiment of the present invention;

[0044] Figure 5 It is a schematic diagram of the iterative trajectories of the follower agent and the leader agent provided by a specific embodiment of the present invention;

[0045] Figure 6 It is a schematic diagram of the tracking error between the follower agent and the leader agent provided by a specific embodiment of the present invention. Specific Embodiments

[0046] The following further elaborates on the present invention in detail in conjunction with the accompanying drawings and specific embodiments. For the step numbers in the following embodiments, they are only set for the convenience of explanation and illustration, and no limitation is imposed on the order between the steps. The execution order of each step in the embodiments can be adaptively adjusted according to the understanding of those skilled in the art.

[0047] Referring to Figure 1 , the present invention provides a variable desired trajectory iterative consensus tracking method for multi-agent systems, and the method includes the following steps:

[0048] S100. Construct a dynamic model and a network communication topology diagram of the multi-agent system, where the multi-agent system includes a leader agent and a number of follower agents;

[0049] S110. Determine the dynamic models of the follower agents and the leader agent, and construct the dynamic model of the multi-agent system;

[0050] Specifically, determine the dynamic model of the follower agent, and the dynamic model of the follower agent is expressed as:

[0051]

[0052] In the above formula, i represents the i-th follower agent, k represents the k-th iteration, t represents the time variable, x i,k (t) represents the state of the follower agent i at the k-th iteration, x i,k (t + 1) represents the state of the follower agent i at the next moment at the k-th iteration, u i,k (t) represents the control input of the follower agent i at the k-th iteration, y i,k (t) represents the control output of the follower agent i at the k-th iteration, τ represents the maximum time delay of the agent, E represents the singular matrix of the generalized system, A and D represent the system matrices, B represents the input matrix, and C represents the output matrix.

[0053] Furthermore, it should be noted that i ∈ {1, 2,..., N}, and t ∈ [0, T] is the time variable. x i,k (t) ∈ R n , u i,k (t) ∈ Rr , y i,k (t) ∈ R m are respectively the state, control input, and output of agent i at the k-th iteration; where n, r, and m are the dimensions of the vector space, and rank(E) = p < n satisfies the regularity condition, that is, there exists a complex number s such that det(sE - A) ≠ 0.

[0054] Furthermore, determine the dynamic model of the leader agent in the multi-agent system, which is expressed as follows:

[0055]

[0056] In the above formula, represents the desired trajectory, t represents the time variable, d represents the subscript symbol of the desired trajectory, k represents the iteration number, l represents the order of the high-order internal model, h l represents the coefficient corresponding to the high-order internal model polynomial, ω represents a preset mathematical operator, H(ω -1 ) is used to simplify the expression.

[0057] Among them, the dynamic model of the leader agent in the multi-agent system represents the iterative change of the desired trajectory, and the initial state of each output desired trajectory can be freely set. At the same time, regarding the desired trajectory as the leader, numbered 0, and the coefficient h j ∈ R (j = 1, 2,..., l) is the coefficient of the l-th order characteristic polynomial p(z) = z l - h 1 z l-1 - h 2 z l-2 - … - h l .

[0058] S120. According to the communication relationship of the multi-agents in the multi-agent system, construct a non-negative weighted graph;

[0059] S130. Define preset conditions for the non-negative weighted graph and construct a network communication topology graph.

[0060] Specifically, describe the communication topology of the multi-agents as a non-negative weighted graph, where represents a finite node set, N is the number of nodes; is the edge set, is the weight adjacency matrix and a ij = 1 means that node j can obtain the information of node i, that is, it represents the communication relationship between agents. The Laplacian matrix of graph ζ is L = D - A 0 , where If there is a directed path from a node in graph ζ to any other node along the direction of the edges in the graph, then this graph is said to contain a directed spanning tree.

[0061] Consider a discrete singular multi-agent system with time delay that can be run repeatedly. The dynamic model of the i-th agent is as follows:

[0062] Ex i,k (t + 1) = Ax i,k (t) + Dx i,k (t - τ) + Bu i,k (t)

[0063] y i,k (t) = Cx i,k (t)

[0064] where i ∈ {1, 2, …, N} and t ∈ [0, T] is the time variable. x i,k (t) ∈ R n , u i,k (t) ∈ R m , y i,k (t) ∈ R p are the state, control input, and output of agent i at the k-th iteration, respectively. τ ∈ N + is the known time delay, and this time delay takes the maximum time delay among the agents. And when t ∈ [-τ, 0], there is x i,k (t) = ψ i,k (t), where ψ i,k (t) is the initial state function of the system. E ∈ R n×n is a singular matrix and rank(E) = r < n. There exists a complex number s that satisfies the condition det(sE - A) ≠ 0. A, B, C, and D are all real matrices of appropriate dimensions.

[0065] Introduce a linear transformation where x i1,k (t) ∈ R r , x i2,k (t) ∈ R n-r . Then the dynamic model of the i-th agent can be written in the following form. The expression of the transformed form of the dynamic model of the agent is as follows:

[0066]

[0067] where A 11 , A 12 , A 21 , D 22 , D 11 , D 12 , D 21 , D 22 , B1 and B 2 are both proper real matrices.

[0068] Assume that A 22 is invertible. Simplify the expression of the transformed form of the dynamic model of the agent to obtain the following:

[0069]

[0070] where

[0071] Let:

[0072]

[0073] Then the expression of the simplified dynamic model of the agent can be further rewritten in the following compact form, and its expression is:

[0074]

[0075] Desired trajectory is generated by a high-order internal model, which satisfies the following relationship:

[0076]

[0077] where h j ∈R (j = 1, 2,..., l) are the coefficients of the l-th order characteristic polynomial p(z) = z -1 - h 1 z l-1 - h 2 z l-2 - … - h l . For the convenience of analysis, introduce an operator ω that satisfies the condition Regard the desired trajectory as a virtual leader numbered 0, and the remaining N multi-agents as followers. And the communication topology between the leader and the followers is represented by a directed graph where and are the corresponding edge set and adjacency matrix.

[0078] Make the following assumptions about the dynamic model of the multi-agent system:

[0079] a) The communication topology graph ζ contains a spanning tree, and the leader is the root of this spanning tree.

[0080] b) The set system matrix a 22 is invertible.

[0081] c) The initial conditions of the system satisfy the following relationship: x i1,k+1 (0) = h1 x i1,k (0) + h 2 x i1,k-1 (0) + … + h l x i1,k-l+1 In equation (0), the initial states from the 0th to the (l - 1)th iteration are arbitrary constant values, and the initial states of different agents at the (l 1 (l 1 = 0, 1, …, l 1 - 1)th iteration are also arbitrary constant values.

[0082] S200. Determine the tracking error of multi - agents based on the network communication topology graph, construct a controller through the P - type open - loop learning algorithm of high - order internal model, and perform iterative control on the dynamic model of the multi - agent system to output a multi - agent system with consistent tracking;

[0083] Specifically, based on the dynamic model of the multi - agent system and the network communication topology graph, determine the tracking error of the system, and construct a controller by combining high - order internal model and iterative learning control to achieve that the system output y k (t) can fully track the system iterative variable desired trajectory That is The output error e k (t) is defined as

[0084] S210. Determine the desired trajectory of multi - agents and high - order polynomials;

[0085] Specifically, design the desired trajectory of multi - agents and the l - order polynomial p(z), where the characteristic equation is stable, that is, the roots of p(z)=0 are all inside or on the unit circle, and the roots on the unit circle are single roots or conjugate complex roots.

[0086] It should be noted that there are three distribution cases for the roots of the characteristic equation p(z)=0: there are roots outside the unit circle, all roots are inside the unit circle, and there are roots on the unit circle and the remaining roots are inside the unit circle. When the characteristic equation has roots outside the unit circle, the desired trajectory diverges as k→∞ and has no practical significance; when all roots of the characteristic equation are inside the unit circle, the desired trajectory converges to 0 as k→∞; when the characteristic equation has roots on the unit circle and the remaining roots are inside the unit circle, the desired trajectory shows a periodic change trend and does not converge to 0.

[0087] S220. Construct the P - type open - loop learning algorithm of high - order internal model based on the high - order polynomial and construct a controller;

[0088] Specifically, a P-type open-loop learning algorithm based on the l-th order internal model is constructed, and its expression is:

[0089] u i,k+1 (t) = h 1 u i,k (t) + h 2 u i,k-1 (t) + … + h l u i,k-l+1 (t) + Γ 1 ξ i,k (t) + Γ 2 ξ i,k-1 (t) + … + Γ l ξ i,k-l+1 (t) = H(ω -1 )u i,k (t) + Γ(ω -1 )ξ i,k (t)

[0090] In the above formula, Γ(ω -1 ) represents the gain matrix, ξ i,k (t) represents the observation error, t represents the time variable, u i,k+1 (t) represents the control input of the follower agent i at the (k + 1)-th iteration, u i,k-1 (t) represents the control input of the follower agent i at the (k - 1)-th iteration, u i,k-l+1 (t) represents the control input of the follower agent i at the (k - l + 1)-th iteration, h l represents the coefficient corresponding to the high-order internal model polynomial, Γ l represents the l-th gain coefficient, u i,k (t) represents the control input of the follower agent i at the k-th iteration, H(ω -1 ) is used to simplify the expression, ω represents the preset mathematical operator, k represents the number of iterations, l represents the order of the high-order internal model, and i represents the i-th follower agent;

[0091] Among them, Γ(ω -1 ) = Γ 1 + Γ 2 ω -1 + … + Γ l ω -l+1 , Γ j ∈R r×m (j = 1, 2, …, l) is the gain matrix, and i = 1, 2, …, N.

[0092] Define the observation error

[0093] S230. Determine the system tracking error based on the dynamic model of the multi-agent system;

[0094] S240. According to the preset conditions of the network communication topology diagram, the dynamic model of the multi-agent system is iteratively controlled by the P-type open-loop learning algorithm of the high-order inner membrane, so that the system tracking error meets the expected trajectory of the multi-agent, and a multi-agent system with consistent tracking is output.

[0095] Specifically, assuming that the above conditions a, b, and c are all satisfied, the above-mentioned P-type open-loop learning algorithm based on the l-th order internal model is used to control the system. If there exists a learning gain matrix Γ j ∈R r×m (j = 1, 2,..., l) such that:

[0096]

[0097] where when α satisfies the condition there is

[0098] Then when k → ∞, the system output error converges to 0, that is That is, the follower agents achieve consistent tracking of the leader agent.

[0099] S300. According to the contraction mapping principle and the norm property, the multi-agent system with consistent tracking is verified to obtain the verified multi-agent system;

[0100] Specifically, the multi-agent system is simplified and decomposed. According to the contraction mapping principle and the (L 2 , λ) norm property, the consistent tracking of the multi-agent system for the expected output trajectory is demonstrated in the L 2 space.

[0101] S310. The dynamic model of the multi-agent system is simplified and decomposed by the singular value decomposition method to obtain a restricted equivalent form of the dynamic model of the multi-agent system;

[0102] Specifically, the dynamic model of the multi-agent system is based on the singular value decomposition theory to obtain a restricted equivalent form of the system. Further, since rank(E) = p < n satisfies the regularity condition, there exist invertible matrices M and N such that And denote:

[0103]

[0104] Introduce the linear transformation [x i1,k T (t) x i2,k T (t)] T = N -1 x i,k (t), where x i1,k (t) ∈ Rp , x i2,k (t) ∈ R n -p . Then the dynamic model of the multi-agent system can be written in the following form, and its expression is:

[0105]

[0106] Set the decomposed block matrix A 22 to be invertible, then the expression of the dynamic model of the above agent system is simplified to obtain:

[0107]

[0108] where

[0109] Define the supervector as follows:

[0110]

[0111] Then the expression of the simplified dynamic model of the agent system can be further written in the following compact form, and its expression is:

[0112]

[0113] In the above formula, x 1,k (t) represents the regular part at the k-th iteration, and x 2,k (t) represents the singular part at the k-th iteration, represents the system matrix of the regular part, represents the system matrix of the singular part, represents the input matrix of the regular part, represents the input matrix of the singular part, C 1 represents the output matrix of the regular part, and C 2 represents the output matrix of the singular part, u k (t) represents the system input, and y k (t) represents the system output.

[0114] S320. Simplify the P-type open-loop learning algorithm for the high-order inner membrane, and verify the multi-agent system according to the principle of contraction mapping and the properties of norms, combined with the restricted equivalent form of the dynamic model of the multi-agent system, to obtain the verified multi-agent system.

[0115] Specifically, according to the principle of contraction mapping and the properties of the (L 2 , λ) norm, demonstrate the uniform tracking of the expected output trajectory by the multi-agent system in the L 2 space.

[0116] The P-type learning algorithm based on the l-th order high-order inner membrane can be simplified into the following expression:

[0117]

[0118] In the above formula

[0119] Denote:

[0120]

[0121] Δu k (t) = u k+1 (t) - H(ω -1 )u k (t)

[0122] It can be seen from the expression of the dynamic model of the compact form of the agent system that:

[0123]

[0124] Then there is:

[0125]

[0126] Furthermore, there is:

[0127]

[0128] Similarly, it can be obtained that:

[0129]

[0130] Taking the α-norm of the expression of Δx 2,k (t), there is:

[0131]

[0132] Select α to satisfy the condition Then there is:

[0133] ||Δx 2,k || α ≤ c 1 ||Δx 1,k || α + c 2 ||Δu k || α

[0134] Where

[0135] Given that Δx 1,k (0) = 0, taking the α-norm of both ends of the expression of Δx 1,k (t + 1), then there is:

[0136]

[0137] where

[0138] substitute α that satisfies the condition into the expression of ||Δx 1,k || α to obtain

[0139] ||Δx 1,k || α ≤α -1 c 3 ||Δx 1,k || a +α -1 c 4 ||Δx 2,k || a +α -1 c 5 ||Δu k || a ≤α -1 (c 3 +c 1 c 4 )||Δx 1,k || a +α -1 (c 5 +c 2 c 4 )||Δu k || a

[0140] Take α that satisfies the condition α -1 (c 3 +c 1 c 4 )<1, then ||Δx 1,k || α ≤α -1 c 6 ||Δu k || α where

[0141] From the definition of the output error, it can be known that:

[0142]

[0143] Taking the norm on both sides of the expression of the output error definition, we can obtain:

[0144]

[0145] Substitute α that satisfies the condition The expression and α satisfy the condition α -1 (c 3 +c 1 c 4 ) < 1. Substituting the expression into both ends of the expression defining the output error and taking the norm, we get:

[0146]

[0147] Among them,

[0148] It is known that Therefore, a sufficiently large α can be selected such that So there is:

[0149]

[0150] From the α-norm, we know that ||e k (t)|| ≤ α T ||e j || α , so there is

[0151] That is to say, the multi-agent system with state time-delay can achieve consistent tracking of the iterative variable desired trajectory under the action of the designed controller.

[0152] S400. Based on the verified multi-agent system, the multi-agent is tracked by the P-type iterative learning control algorithm based on the second-order internal model to obtain the simulation tracking result.

[0153] Specifically, the performance of the iterative learning consistency controller for the variable desired trajectory of the multi-agent system is verified through numerical simulation.

[0154] S410. Based on the verified multi-agent system, set the state space of the dynamic model of the multi-agent system and the motion trajectory of the dynamic model of the multi-agent system;

[0155] Specifically, set the state space of the simulated dynamic model of the multi-agent system as follows:

[0156] Consider the following continuous general multi-agent system composed of four agents:

[0157]

[0158] where t ∈ [0, 10] and the time-delay τ = 1. Select the initial input vector u 0 (t) = u 1 (t) = [0 0 0 0] T , the initial state x of the 0th initial iteration1,0 x(0) = [0 0 0 0] T , the initial state x of the first initial iteration 1,1 x(0) = [1 1 1 1] T .

[0159] The system matrix is:

[0160]

[0161] It is easy to verify that the system satisfies the regularity condition, and A 22 = 1 is invertible.

[0162] Set the motion trajectory of the leader agent, that is, the target desired trajectory as follows. First, set the iterative variable desired trajectory to be generated by the following second-order internal model, and its expression is:

[0163]

[0164] And the initial desired trajectory is The characteristic equation is P(z) = z 2 - 0.2z + 1 = 0. It is easy to obtain that the characteristic equation has a pair of conjugate complex roots on the unit circle.

[0165] S420. Based on the state space of the dynamic model of the multi-agent system and the motion trajectory of the dynamic model of the multi-agent system, the performance of the controller acting on the multi-agent system is tested to obtain the results of the test numerical simulation example;

[0166] Specifically, the performance of the iterative learning controller based on the high-order internal model acting on the multi-agent system is demonstrated through numerical simulation. The network communication topology diagram of this numerical simulation is as Figure 4 shown. If the j-th agent transmits information to the i-th agent, then let the entry a ij = 1, otherwise let a ij = 0. Then it can be known that the Laplacian matrix of the simulation experiment is Weight matrix

[0167] S430. If the results of the test numerical simulation example meet the preset requirements, construct a P-type iterative learning control algorithm based on the second-order internal model and track the multi-agent to obtain the simulation tracking results.

[0168] Specifically, check whether the test numerical simulation example meets the requirements. The gain matrix is selected as Γ = [0.1, -0.4], then there is

[0169]

[0170] Therefore Meets the convergence condition.

[0171] Construct a P-type iterative learning control algorithm based on a second-order internal model:

[0172] u k+1 (t) = 0.2u k (t) - u k-1 (t) - 0.1(L + B 0 )e k (t) + 0.3(L + B 0 )e k-1 (t)

[0173] The results of the obtained simulation are as Figure 5 Figure 6 shown.

[0174] In summary, as Figure 3 shown, in the embodiment of the present invention, by providing a multi-agent variable desired trajectory iterative learning consensus control method, iterative learning control is introduced to utilize the historical information carried by the system to iteratively track the target. Applying it to the consensus control of multi-agent systems can effectively improve the tracking transient performance, reduce the resource occupancy of multi-agent systems and the dependence on communication capabilities; through the combination of the internal model principle and iterative learning control, the consensus tracking of variable desired trajectories is realized, and a corresponding controller is designed for the problem of state time delay existing between multi-agents.

[0175] The embodiment of the present invention focuses on the open-loop iterative learning consensus control problem of generalized multi-agent systems, especially those systems whose desired trajectories are constantly evolving and driven by high-order internal models. A P-type open-loop iterative learning controller based on a high-order internal model is constructed and applied to the consensus control of generalized multi-agent systems with constantly changing desired trajectories. The observation error is defined according to the network communication topology diagram between multi-agents, and a corresponding iterative learning controller based on a high-order internal model is designed for the agents on the basis of the observation error, so that the output of the generalized multi-agent system can achieve precise consensus tracking of the iteratively changing desired trajectory.

[0176] The differences between the embodiment of the present invention and the prior art are as follows:

[0177] 1) The desired trajectory given by the present invention is generated by a known high-order internal model and is iteratively changing to achieve the iterative change of the desired trajectory.

[0178] 2) The present invention has no strict requirements for the initial states of the agents, and it is relatively easy to apply in practice.

[0179] 3) The present invention applies the P-type open-loop iterative learning control algorithm based on high-order internal model to the consensus control of a generalized multi-agent system, and gives the convergence conditions and corresponding proofs. The generalized multi-agent system can achieve complete consensus tracking of the target under the action of this algorithm.

[0180] 4) The present invention designs a corresponding controller for the state time-delay problem existing in the multi-agent system.

[0181] Refer to Figure 2 , a variable desired trajectory iterative consensus tracking system for multi-agents, including:

[0182] The first module is used to construct the dynamic model and network communication topology diagram of the multi-agent system. The multi-agent system includes a leader agent and several follower agents;

[0183] The second module is used to determine the multi-agent tracking error based on the network communication topology diagram, construct a controller through the P-type open-loop learning algorithm of the high-order internal membrane, and perform iterative control on the dynamic model of the multi-agent system, and output a multi-agent system with consensus tracking;

[0184] The third module is used to verify the multi-agent system with consensus tracking according to the contraction mapping principle and the norm property, and obtain the verified multi-agent system;

[0185] The fourth module is used to track the multi-agents based on the verified multi-agent system through the P-type iterative learning control algorithm based on the second-order internal model, and obtain the simulation tracking result.

[0186] The content in the above method embodiments is applicable to the system embodiments of the present invention. The functions specifically implemented by the system embodiments of the present invention are the same as those of the above method embodiments, and the beneficial effects achieved are also the same as those of the above method embodiments.

[0187] The above is a specific description of the preferred embodiment of the present invention, but the present invention is not limited to the described embodiment. Those skilled in the art can make various equivalent deformations or substitutions without departing from the spirit of the present invention, and these equivalent deformations or substitutions are all included in the scope defined by the claims of this application.

Claims

1. A multi-agent variable expected trajectory iterative consistency tracking method, characterized in that: The following steps are involved: Constructing a dynamic model and a network communication topology diagram of a multi-agent system, wherein the multi-agent system includes a leader agent and a plurality of follower agents; Based on the network communication topology, the multi-agent tracking error is determined, and the controller is constructed through the P-type open-loop learning algorithm of the high-order inner membrane to iteratively control the dynamic model of the multi-agent system, and the multi-agent system with consistent tracking is output; According to the compression mapping principle and norm properties, the multi-agent system with consistency tracking is verified to obtain the verified multi-agent system; Based on the verified multi-agent system, the multi-agent is tracked by a P-type iterative learning control algorithm based on a second-order internal model, and the simulated tracking results are obtained.

2. According to claim 1, a multi-agent variable expected trajectory iterative consistency tracking method is characterized in that: The step of constructing the dynamic model and network communication topology diagram of the multi-agent system specifically includes: Determine the dynamic model of the follower agent and the dynamic model of the leader agent, and construct the dynamic model of the multi-agent system; Construct a non-negative weighted graph based on the communication relationship of multiple agents in the multi-agent system; Define preset conditions for non-negative weighted graphs and construct network communication topology graphs.

3. According to claim 2, a multi-agent variable expected trajectory iterative consistency tracking method is characterized in that: The dynamic model of the follower agent is a generalized discrete time-delay multi-agent dynamic model. The expression of the dynamic model of the follower agent is specifically as follows: In the above formula, i represents the i-th follower agent, k represents the k-th iteration, t represents the time variable, and x i,k (t) represents the state of follower agent i at the kth iteration, x i,k (t+1) represents the state of follower agent i at the kth iteration at the next moment, u i,k (t) represents the control input of follower agent i at the kth iteration, y i,k (t) represents the control output of follower agent i at the kth iteration, τ represents the maximum time delay of the agent, E represents the singular matrix of the generalized system, A and D represent the system matrix, B represents the input matrix, and C represents the output matrix; The expression of the dynamic model of the leader agent is specifically as follows: In the above formula, represents the expected trajectory, t represents the time variable, d represents the subscript symbol of the expected trajectory, k represents the number of iterations, l represents the order of the high-order internal model, and h l represents the coefficients corresponding to the high-order internal model polynomial, ω represents the preset mathematical operator, H(ω -1 ) represents mathematical symbols used to simplify expressions.

4. According to claim 3, a multi-agent variable expected trajectory iterative consistency tracking method is characterized in that: The step of determining the multi-agent tracking error based on the network communication topology diagram, constructing a controller through a high-order inner membrane P-type open-loop learning algorithm and iteratively controlling the dynamic model of the multi-agent system to output a multi-agent system with consistent tracking specifically includes: Determine multi-agent expected trajectories and high-order polynomials; Construct a P-type open-loop learning algorithm for high-order inner membrane based on high-order polynomials and build a controller; Determine the system tracking error based on the dynamics model of the multi-agent system; According to the preset conditions of the network communication topology, the dynamic model of the multi-agent system is iteratively controlled through a high-order inner membrane P-type open-loop learning algorithm, so that the system tracking error meets the expected trajectory of the multi-agent, and outputs a multi-agent system with consistent tracking.

5. According to claim 4, a multi-agent variable expected trajectory iterative consistency tracking method is characterized in that: The expression of the P-type open-loop learning algorithm of the high-order inner membrane is specifically as follows: you i,k+1 (t)=h1u i,k (t)+h2u i,k-1 (t)+…+h l you i,k-l+1 (t)+Γ1ξ i,k (t)+Γ2ξ i,k-1 (t)+…+C l x i,k-l+1 (t)=H(ω -1 )u i,k (t)+Γ(ω -1 )ξ i,k (t) In the above formula, Γ(ω -1 ) represents the gain matrix, ξ i,k (t) represents the observation error, t represents the time variable, u i,k+1 (t) represents the control input of follower agent i at the k+1th iteration, u i,k-1 (t) represents the control input of follower agent i at the k-1th iteration, u i,k-l+1 (t) represents the control input of follower agent i at the k-l+1th iteration, h l represents the coefficients corresponding to the higher-order internal mode polynomial, Γ l represents the lth gain coefficient, u i,k (t) represents the control input of follower agent i at the kth iteration, H(ω -1 ) is used to simplify the expression, ω represents the preset mathematical operator, k represents the number of iterations, l represents the order of the high-order internal model, and i represents the i-th follower agent.

6. According to claim 5, a multi-agent variable expected trajectory iterative consistency tracking method is characterized in that: The step of verifying the multi-agent system with consistency tracking according to the compression mapping principle and the norm property to obtain the verified multi-agent system specifically includes: The dynamic model of the multi-agent system is simplified and decomposed by the singular value decomposition method, and the restricted equivalent form of the dynamic model of the multi-agent system is obtained; The P-type open-loop learning algorithm of the high-order inner membrane is simplified, and based on the compression mapping principle and norm properties, combined with the restricted equivalent form of the multi-agent system dynamics model, the multi-agent system is verified to obtain the verified multi-agent system.

7. According to claim 6, a multi-agent variable expected trajectory iterative consistency tracking method is characterized in that: The step of tracking the multi-agent system based on the verified multi-agent system by using a P-type iterative learning control algorithm based on a second-order internal model to obtain a simulated tracking result specifically includes: Based on the verified multi-agent system, set the state space of the dynamics model of the multi-agent system and the motion trajectory of the dynamics model of the multi-agent system; Based on the state space of the dynamics model of the multi-agent system and the motion trajectory of the dynamics model of the multi-agent system, the performance of the controller acting on the multi-agent system is tested, and the results of the numerical simulation examples are obtained; If the results of the numerical simulation example meet the preset requirements, a P-type iterative learning control algorithm based on the second-order internal model is constructed and the multi-agent is tracked to obtain the simulated tracking results.

8. A multi-agent variable expected trajectory iterative consistency tracking system, characterized in that: Includes the following modules: The first module is used to construct a dynamic model and a network communication topology diagram of a multi-agent system, wherein the multi-agent system includes a leader agent and a plurality of follower agents; The second module is used to determine the multi-agent tracking error based on the network communication topology, build a controller through the P-type open-loop learning algorithm of the high-order inner membrane, and iteratively control the dynamic model of the multi-agent system to output a multi-agent system with consistent tracking; The third module is used to verify the multi-agent system with consistency tracking according to the compression mapping principle and norm properties, and obtain the verified multi-agent system; The fourth module is used to track the multi-agent system based on the verified multi-agent system through a P-type iterative learning control algorithm based on the second-order internal model to obtain simulated tracking results.

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