A binary inclusive synchronization method for multi-agent systems with multi-layered signed networks

Abstract

The present invention discloses a binary inclusive synchronization method for a multi-agent system with a multi-layer symbolic network, comprising the following steps: considering a type of multi-agent system with a multi-layer symbolic network, each layer of which includes M follower agent nodes and N-M leader agent nodes, and establishing a multi-agent system model; then designing a controller based on output measurement with a multi-layer symbolic network; redefining the state error, and obtaining a multi-agent error system with a multi-layer symbolic network; obtaining the asymptotic stability condition of the multi-agent error system, and adjusting the gain of the controller to be designed, so that the multi-agent system with a multi-layer symbolic network can achieve binary inclusive synchronization. The present invention takes into account the joint influence of the multi-layer symbolic network and multiple leader agents in the actual environment, designs the controller, constructs the state error signal, and realizes the binary inclusive synchronization of the multi-agent system with a multi-layer symbolic network in the sense of asymptotic stability of the error system.

Description

Technical Field

[0001] The present invention relates to the technical field of binary synchronization of multi-agent systems, and in particular to a binary inclusive synchronization method of a multi-agent system with a multi-layer symbolic network. Background Art

[0002] In recent years, the synchronization control of multi-agent systems has shown broad application prospects in the fields of transportation, communication, aerospace, and military. Since Altafini introduced symbolic networks into the synchronization research of multi-agent systems, he opened the precedent for the study of binary synchronization, and the binary synchronization problem of multi-agent systems has attracted the attention of more and more scholars. The binary synchronization of multi-agent systems can be divided into leaderless binary synchronization and leader-follower binary synchronization according to whether there is a leader or not; further, according to the number of leaders, the leader-follower binary synchronization of multi-agent systems can be divided into binary inclusive synchronization of multiple leaders and binary tracking synchronization of a single leader.

[0003] At present, most of the research on binary synchronization of multi-agent systems is based on single-layer or switching symbolic networks, and there are relatively few studies on binary synchronization of multi-agent systems with multi-layer symbolic networks. In fact, agent nodes may move on multi-layer networks composed of various types of connections or multiple sub-networks. The multi-layer network structure describes a more general and natural real complex system, so it is more practical to extend the symbolic network to the multi-layer symbolic network. Existing research on bipartite synchronization of multi-agent systems based on multi-layer signed networks mainly includes: research on leaderless bipartite synchronization of multi-agent systems based on signed graphs in the case of no leader agent node (e.g. Bipartite consensus of fuzzy multiagent systems on multilayer coopetition networks using output measurements and itsapplication to image communication, DOI:10.1109 / ACCESS.2023.3333692; Pinningsynchronization control for stochastic multi-layer networks with couplingdisturbance, DOI:10.1016 / j.isatra.2021.10.016, etc.); research on bipartite tracking synchronization of multi-agent systems based on signed graphs in the case of a single leader agent node (e.g. Leader-following consensus ofdelayed neural networks under multi-layer signed graphs, DOI:10.1016 / j.neucom.2021.03.009; Quantized dynamic event-triggered control for fixed / preset-time bipartite synchronization of Further research found that in multiple engineering applications such as positioning navigation, environmental perception, and human-computer interaction, there are often multiple leader agents. Therefore, the binary inclusive synchronization control problem of multi-agent systems has gradually become a research hotspot.The existing bipartite containment synchronization control of multi-agent systems is basically based on single-layer or switching signed networks (for example, Bipartite containment control of fractional multi-agent systems with input delay on switching signed directed networks, DOI: 10.1016 / j.isatra.2022.09.023; High-order bipartite containment control in multi-agent systems over time-varying cooperation-competition networks, DOI: 10.1016 / j.neucom.2019.05.090, etc.). As far as we know, there is no report on the research results of the bipartite containment synchronization problem of multi-agent systems with multi-layer signed networks. Therefore, the present invention integrates the actual situation of multi-layer signed networks and multiple leaders, and proposes a bipartite containment synchronization method for multi-agent systems with multi-layer signed networks. Summary of the invention

[0004] The purpose of the invention is to solve the problems that there may be multi-layer symbolic networks in actual networks and multiple leader agents in actual engineering. However, existing research has not combined multi-layer symbolic networks and multiple leader agents. This invention proposes a binary inclusive synchronization method for a multi-agent system with a multi-layer symbolic network.

[0005] In order to achieve the above object, the present invention adopts the following technical solution: a binary inclusion synchronization method of a multi-agent system with a multi-layer symbolic network, comprising the following steps:

[0006] Step 1: Consider a multi-agent system with a multi-layer symbolic network, where each layer of the symbolic network contains M follower agent nodes and NM leader agent nodes, and model the multi-agent system;

[0007] Let the set of follower agent nodes be The set of leader agent nodes is And NM>1, the dynamic equation of the follower agent node is:

[0008]

[0009] in denote the state vector, control input and output of the i-th follower agent node in the multilayer signed network, respectively. A, B1, B2 and C are constant matrices. f(·) is a nonlinear odd function. τ(t) is a bounded and differentiable time-delay function that satisfies 0≤τ1≤τ(t)≤τ2 and The dynamic equation of the leader agent node is:

[0010]

[0011] in They represent the state vector and output of the leader node in a multi-layer symbolic network, respectively.

[0012] The multi-layer symbol network is composed of S single-layer symbol networks, each layer is represented by a symbol graph. Symbolic graph describing, k∈{1,2,...,S}, a multi-layer symbolic network is the adjacency matrix of a structurally balanced, multi-layer signed network and the Laplace transform matrix Respectively expressed as:

[0013]

[0014] in and

[0015]

[0016] Step 2: Design an output measurement based controller with a multi-layer signed network

[0017] In order to realize the bipartite inclusive synchronization control of multi-agent systems with multi-layer signed networks, a controller based on output measurement is designed with multi-layer signed networks. i (t), is expressed as follows,

[0018]

[0019] Where η is the controller gain to be designed,

[0020] Then, the dynamic equation of the follower agent node is expressed as follows:

[0021]

[0022] Step 3: Get a multi-agent error system with a multi-layer signed network

[0023] Define a diagonal matrix when hour, when hour,

[0024] Using the canonical transformation, let The dynamic equations of the multi-agent system can be expressed as follows:

[0025]

[0026] make

[0027]

[0028]

[0029] Available,

[0030]

[0031] make

[0032]

[0033] Then the Kronecker product form of the multi-agent system dynamics equation can be expressed as follows:

[0034]

[0035] Reconstruction state error

[0036] Nonlinear function error

[0037]

[0038] The error system is as follows:

[0039]

[0040] if The bipartite inclusive synchronization problem of a multi-agent system with a multi-layered signed network under the action of a controller is transformed into the stability problem of a multi-agent error system.

[0041] Construction state error in Is the leader agent node Z R (t), when the state error e(t) approaches 0, it means that the follower agent node trajectory of each layer of the symbolic network is located at the leader agent node Z R (t) In the convex hull formed, under the superposition control of each layer of symbolic network, the binary inclusive synchronization control of the multi-agent system with multi-layer symbolic network can be realized;

[0042] Step 4: Obtain the binary inclusive synchronization condition of the multi-agent system with a multi-layer symbolic network, and adjust the gain of the controller to be designed so that the multi-agent system with a multi-layer symbolic network can achieve binary inclusive synchronization.

[0043] If there exists a positive definite matrix and constants σ1>0,σ2>0, so that the following inequality holds,

[0044]

[0045] in Ω 22 =-(1-μ)U+σ2ε 2 I n , and the constant β satisfies:

[0046]

[0047] Then the error system formula (8) can achieve asymptotic stability, and the controller gain is adjusted Formulas (1) and (2) of a multi-agent system with a multi-layer signed network achieve binary inclusive synchronization.

[0048] Furthermore, in step 1, the f(·) is a nonlinear odd function, which means that for any f(·) satisfies the following inequality,

[0049]

[0050] Among them, i∈{1,2,...,NM}, ε>0 is a known constant, ρ i ≥0,

[0051] Furthermore, in step 1, the multi-layer symbol network is composed of S single-layer symbol networks, each layer is represented by a symbol graph. Description, k∈{1,2,...,S}, means that for any layer of symbolic network, the nodes have no self-loops,

[0052] Furthermore, in step 1, the symbol graph of the multi-layer symbol network It is structurally balanced, which means two subsets of nodes in each layer of the symbol graph and The same, and the following conditions are met:

[0053]

[0054] ② When When k∈{1,2,...,S},

[0055] ③When When k∈{1,2...,S},

[0056] Furthermore, the two node subsets of each layer of the symbol graph and The same means that the role of each node in each layer of the symbol graph remains unchanged, and the information interaction between the corresponding nodes of the adjacent layer of the symbol graph is a cooperative relationship.

[0057] Furthermore, in step 1, the adjacency matrix of the multi-layer symbol network is and the Laplace transform matrix Respectively expressed as For any layer symbol map The Laplace transform matrix In terms of, satisfying: matrix is the M matrix, is a non-negative matrix whose rows sum to 1.

[0058] Furthermore, in step 4, the inequality formula (9) is established, which means that the feasibility of the inequality formula (9) is converted into a "GEVP" solution problem in Matlab using the LMI tool.

[0059]

[0060] in X = 2βPC,

[0061] Using the "GEVP" toolbox in Matlab to obtain the minimum value of β min , and then adjust the controller gain

[0062] Compared with the prior art, the present invention has the following beneficial effects:

[0063] (1) The present invention comprehensively considers the joint influence of multi-layer symbolic networks and multiple leader agents in actual networks and engineering environments, uses multi-layer symbolic graphs to describe the cooperative-competitive interaction relationship of each layer of the network, designs an output measurement-based controller with a multi-layer symbolic network, and simultaneously establishes a new state error signal with a multi-layer symbolic network and multiple leader agents to solve the binary inclusive synchronization problem of a multi-agent system with a multi-layer symbolic network.

[0064] (2) In the sense of asymptotic stability of the error system, the present invention proposes a binary inclusive synchronization method for a multi-agent system with a multi-layer symbolic network, which has greater research and application value than the existing leaderless binary synchronization / single leader binary tracking synchronization research of multi-agent systems based on multi-layer symbolic networks, and the binary inclusive synchronization research of multi-agent systems based on single-layer or switching symbolic networks. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 It is the principle diagram of the present invention;

[0066] Figure 2 A topological diagram of a multi-layer symbol network in an embodiment of the present invention;

[0067] Figure 3 It is a state trajectory cross-sectional diagram at certain moments in an embodiment of the present invention;

[0068] Figure 4 is a state error trajectory diagram in an embodiment of the present invention, where (a) is the state error e i1 (t) is the trajectory diagram, (b) is the state error e i2 (t) is the trajectory diagram, (c) is the state error e i3 Trajectory diagram of (t), i=1,…,7. DETAILED DESCRIPTION

[0069] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the accompanying drawings, so that those skilled in the art can better understand the advantages and features of the present invention, thereby making a clearer definition of the protection scope of the present invention. The embodiments described in the present invention are only a part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in the field without making creative work are within the scope of protection of the present invention.

[0070] A binary inclusive synchronization method for a multi-agent system with a multi-layered signed network includes the following steps:

[0071] Consider a multi-agent system with a multi-layer symbolic network, where each node represents an agent, and each layer of the symbolic network contains M follower nodes and NM leader nodes. Let the set of follower agent nodes be The set of leader agent nodes is And NM>1, the dynamic equation of the follower node is:

[0072]

[0073] in denote the state vector, control input and output of the i-th follower agent node in the multilayer signed network, respectively. A, B1, B2 and C are constant matrices. f(·) is a nonlinear odd function. τ(t) is a bounded and differentiable time-delay function that satisfies 0≤τ1≤τ(t)≤τ2 and The dynamic equation of the leader agent node is:

[0074]

[0075] in They represent the state vector and output of the leader node in a multi-layer symbolic network, respectively.

[0076] Assumption 1: The nonlinear function f(·) is an odd function, and for any f(·) satisfies the following inequality,

[0077]

[0078] Among them, i∈{1,2,...,NM}, ε>0 is a known constant, ρ i ≥0,

[0079] The multi-layer symbol network is composed of S single-layer symbol networks, each layer is represented by a symbol graph. Description, k∈{1,2,...,S}. The nodes of each layer of the symbolic network are the same, that is, the node set And the role of each node remains unchanged. It is generally believed that the nodes of each layer of the symbolic network have no self-loops, that is,

[0080] For a symbolic network, the structural balance of the symbolic graph is very important. However, for a multi-layer symbolic network, it is obviously not enough if only the symbolic graphs of each layer are structurally balanced. So how can we determine whether the symbolic graph of a multi-layer symbolic network is structurally balanced? Through observation, it is found that if nodes i and j are cooperative in the kth layer, but competitive in the lth layer, then the multi-layer symbolic graph is structurally unbalanced. Therefore, if and only if each layer of the symbolic graph is structurally balanced and has the same subset of nodes in each layer and A multi-layer symbol graph is structurally balanced. The structural balance definition of a multi-layer symbol graph can be described as follows:

[0081] Definition 1: If two node subsets of the symbol graph at each layer and The same, and the following conditions are met:

[0082]

[0083] ② When When k∈{1,2,...,S},

[0084] ③When When k∈{1,2...,S},

[0085] Then the multi-layer symbol map is structurally balanced; otherwise the multi-layer symbol graph The structure is unbalanced.

[0086] By symbol diagram The multi-layer symbolic network described by and the Laplacian matrix Respectively expressed as:

[0087]

[0088] in

[0089] In the Laplace matrix On the basis of in and

[0090] Assumption 2: For any k∈S, each layer of the symbol graph The structure is balanced.

[0091] Assumption 3: For each layer of the symbolic graph, there is the same subset of nodes and And in each layer there exists at least one path from the leader to every follower.

[0092] From Assumptions 2 and 3, and in comparison with Definition 1, we can see that the symbol graph corresponding to the multi-layer symbol network here is It is structurally balanced, k=1,2,...,S.

[0093] Lemma 1: Under Assumption 2, for each layer of the symbol graph In terms of matrix is the M matrix, is a non-negative matrix whose rows sum to 1.

[0094] In order to realize the bipartite inclusive synchronization control of multi-agent systems with multi-layer signed networks, the controller u i (t) is as follows,

[0095]

[0096] Where η is the controller gain to be designed, is the adjacency matrix The elements in .

[0097] Then, the dynamic equation of the follower agent node (13) can be expressed as:

[0098]

[0099] Define a diagonal matrix when hour, when hour,

[0100] Using the canonical transformation, let The dynamic equations of the multi-agent system (13) and (14) are expressed as follows:

[0101]

[0102] make

[0103]

[0104]

[0105] Available,

[0106]

[0107] make

[0108]

[0109] Then the Kronecker product form of the multi-agent system dynamics equation can be expressed as follows:

[0110]

[0111] Reconstruction state error

[0112] Nonlinear function error

[0113]

[0114] The error system is as follows:

[0115]

[0116] Definition 2: A multi-agent system with a multi-layer signed network, see formulas (13) and (14), if

[0117]

[0118] Then the multi-agent system with a multi-layer symbolic network, see formulas (13) and (14), can achieve binary inclusive synchronization.

[0119] if The bipartite inclusive synchronization problem of a multi-agent system with a multi-layered signed network under the action of a controller is transformed into the stability problem of a multi-agent error system.

[0120] The state error is in Is the leader agent node Z R (t), when the state error e(t) approaches 0, it means that the follower agent node trajectory of each layer of the symbol network is located at the leader agent node Z R In the convex hull formed by (t), under the superposition control of each layer of symbolic network, the binary inclusive synchronization control of the multi-agent system with multi-layer symbolic network can be realized.

[0121] Based on the above description, the ultimate goal is to adjust the controller gain to be designed so that the controller gain satisfies the solution of the following theorem to achieve binary inclusive synchronization control of a multi-agent system with a multi-layer symbolic network:

[0122] Theorem 1. If there exists a positive definite matrix and constants σ1>0,σ2>0, so that the following inequality holds,

[0123]

[0124] in Ω 22 =-(1-μ)U+σ2ε 2 I n , and the constant β satisfies:

[0125]

[0126] Then the error system can achieve asymptotic stability, and the controller gain is adjusted It enables a multi-agent system with a multi-layer symbolic network to achieve binary inclusive synchronization.

[0127] Using the LMI tool, the feasibility of inequality formula (24) is converted into a “GEVP” problem in Matlab.

[0128]

[0129] in X = 2βPC,

[0130] Using the "GEVP" toolbox in Matlab to obtain the minimum value of β min , and then adjust the controller gain

[0131] Example:

[0132] Consider a multi-agent system consisting of 3 leaders and 7 followers. The system consists of two layers of symbolic networks, as shown in the attached figure. Figure 2 As shown, each layer of the signed network is structurally balanced and has the same subset of bipartite nodes. and Among them, nodes 8, 9, and 10 are leaders.

[0133] The system related parameters are given as follows:

[0134]

[0135] It can be obtained that τ1=0.3, τ2=0.5, μ=0.1, ε=1,

[0136] By the attached Figure 2 get

[0137]

[0138] Using the "GEVP" toolbox in Matlab to solve, we can get

[0139]

[0140]

[0141] and σ1=0.2186, σ2=0.0773, β min =8.1710.

[0142] By calculation Choose η=10>η min , substituting into the inequality formula (10) we can get In this way, both inequality formula (9) and inequality formula (10) are satisfied.

[0143] Attached Figure 3 The states of each agent node at different times are given, where the solid triangle and the dotted triangle are the convex hull sections formed by the states of the three leader nodes and their reverse symmetric convex hull sections, ▲ represents the states of agent nodes 1 to 3, and * represents the states of agent nodes 4 to 7. Figure 4 describes the trajectory of the state error signal.

[0144] By the attached Figure 3 and 4It can be seen that the state of the follower agent is always in the convex hull composed of the state of the leader agent, and the inclusion error gradually converges to 0, which indicates that the multi-agent system with a multi-layer signed network realizes bipartite inclusion synchronization control.

[0145] In summary, the present invention takes into account the joint influence of multi-layer symbolic networks and multiple leader agents in actual environments, designs a controller, constructs a state error signal, and realizes binary inclusive synchronization of a multi-agent system with a multi-layer symbolic network in the sense of asymptotic stability of the error system.

[0146] The description and practice disclosed in the present invention are easy to think and understand for ordinary technicians in the technical field, and several improvements and modifications can be made without departing from the principles of the present invention. Therefore, modifications or improvements made without departing from the spirit of the present invention should also be regarded as the protection scope of the present invention.

Claims

1. A binary inclusive synchronization method for a multi-agent system with a multi-layered signed network, characterized in that: The steps include: Step 1: Consider a multi-agent system with a multi-layer symbolic network, where each layer of the symbolic network contains M follower agent nodes and NM leader agent nodes, and model the multi-agent system; Let the set of follower agent nodes be The set of leader agent nodes is And NM>1, the dynamic equation of the follower agent node is: in denote the state vector, control input and output of the i-th follower agent node in the multilayer signed network, respectively. A1, A2, B1, B2 and C are constant matrices. f(·) is a nonlinear odd function. τ(t) is a bounded and differentiable time-delay function that satisfies 0≤τ1≤τ(t)≤τ2 and The dynamic equation of the leader agent node is: in They represent the state vector and output of the leader node in a multi-layer symbolic network respectively; The multi-layer symbol network is composed of S single-layer symbol networks, each layer is represented by a symbol graph. Symbolic graph describing, k∈{1,2,...,S}, a multi-layer symbolic network is the adjacency matrix of a structurally balanced, multi-layer signed network and the Laplace transform matrix Respectively expressed as: in and Step 2: Design an output measurement based controller with a multi-layer signed network; In order to realize the bipartite inclusive synchronization control of multi-agent systems with multi-layer signed networks, a controller based on output measurement is designed with multi-layer signed networks. i (t), expressed as follows: Where η is the controller gain to be designed, Then, the dynamic equation of the follower agent node is expressed as follows: Step 3, obtaining a multi-agent error system with a multi-layer signed network; Define a diagonal matrix when hour, when hour, Using the canonical transformation, let The dynamic equations of the multi-agent system (1) and (2) are expressed as follows: make We can get: Let F(Z F (t)) = [f T (z1(t)), f T (z2(t)),..., f T (z M (t))] T F(Z R (t))=[f T (z M+1 (t)),f T (z M+2 (t)),...,f T (z N (t))] T Then the Kronecker product form of the multi-agent system dynamics equation can be expressed as follows: Reconstruction state error Nonlinear function error The error system is as follows: if The bipartite inclusive synchronization problem of a multi-agent system with a multi-layered signed network under the action of a controller is transformed into the stability problem of a multi-agent error system. Construction state error in Is the leader agent node Z R (t), when the state error e(t) approaches 0, it means that the follower agent node trajectory of each layer of the symbol network is located at the leader agent node Z R (t) In the convex hull formed, under the superposition control of each layer of symbolic network, the binary inclusive synchronization control of the multi-agent system with multi-layer symbolic network can be realized; Step 4: Obtain the binary inclusive synchronization condition of the multi-agent system with a multi-layer symbolic network, and adjust the gain of the controller to be designed so that the multi-agent system with a multi-layer symbolic network can achieve binary inclusive synchronization: If there exists a positive definite matrix and constants σ1>0,σ2>0, so that the following inequality holds, in Ω 22 =-(1-μ)U+σ2ε 2 I n , and the constant β satisfies: Then the error system formula (8) can achieve asymptotic stability, and the controller gain is adjusted Formulas (1) and (2) of a multi-agent system with a multi-layered symbolic network achieve binary inclusive synchronization; In step 1, the f(·) is a nonlinear odd function, which means that for any f(·) satisfies the following inequality, Among them, i∈{1,2,...,NM}, ε>0 is a known constant, ρ i ≥0, In step 1, the multi-layer symbol network is composed of S single-layer symbol networks, each layer is represented by a symbol graph. Description, k∈{1,2,...,S}, means that for any layer of symbolic network, the nodes have no self-loops, In step 1, the symbol graph of the multi-layer symbol network It is structurally balanced, which means two subsets of nodes in each layer of the symbol graph and The same, and the following conditions are met: ① ② When hour, ③When hour, The two node subsets of each layer of the symbol graph and The same means that the role of each node in each layer of the symbol graph remains unchanged, and the information interaction between the corresponding nodes of the adjacent layer of the symbol graph is a cooperative relationship.

2. A binary inclusive synchronization method for a multi-agent system with a multi-layered signed network according to claim 1, characterized in that: In step 1, the adjacency matrix of the multi-layer symbol network is and the Laplace transform matrix Respectively expressed as For any layer symbol map The Laplace transform matrix In terms of, satisfying: matrix is the M matrix, is a non-negative matrix whose rows sum to 1.

3. A binary inclusive synchronization method for a multi-agent system with a multi-layered signed network according to claim 1, characterized in that: In step 4, the inequality formula (9) is established, which means that the feasibility of the inequality formula (9) is converted into a "GEVP" solution problem in Matlab using the LMI tool. in X = 2βPC, using the "GEVP" toolbox in Matlab to obtain the minimum value of β min , and then adjust the controller gain

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