A topological design method and its application in multi-agent system consensus

Abstract

The present application proposes a topology design method and its application in the consistency of multi-agent system. The topology design includes the following steps: first, considering the undirected graph composed of three nodes, the topology is divided into two types: connected but not fully connected and fully connected; then, for the case of connected but not fully connected, the relationship between nodes is divided into three cases: node 1 and node 2 are not adjacent, node 1 and node 3 are not adjacent, and node 2 and node 3 are not adjacent; then, for each case, the corresponding coupling matrix is designed, and the second largest eigenvalue of the coupling matrix is guaranteed to be less than a given negative constant and the sum of the off-diagonal elements is minimized; finally, the topology design scheme is applied to the consistency of multi-agent system, and its effectiveness is verified by numerical simulation. The present application deeply studies the design method of coupling matrix, not only promotes the development of matrix theory, but also provides a new method for the cooperative control of multi-agent system.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of multi-agent system cooperative control, and particularly relates to a topology design method and application thereof in multi-agent system consensus. BACKGROUND

[0002] Consensus control of multi-agent system is a core problem of multi-agent cooperative control, and optimization and implementation of its communication topology is crucial to improving overall performance. The communication topology of multi-agent system can be described by coupling matrix in mathematics. A large number of existing research results show that the second largest eigenvalue of the coupling weight matrix plays a key role in whether the multi-agent system can achieve consensus.

[0003] Although the coupling matrix has an important influence on the consensus of multi-agent system, the research on the design of the coupling matrix is still blank in the existing research. As an important research direction in the future, in-depth study on the design method of the coupling matrix can not only promote the development of matrix theory and provide new methods for its research in multi-agent system cooperative control, but also lay a solid theoretical foundation for multi-agent system to better play a role in more fields, such as intelligent logistics, cooperative rescue, environmental monitoring, etc. SUMMARY

[0004] The present application aims to provide a topology design method and application thereof in multi-agent system consensus to solve the problems in the background.

[0005] The present application is implemented as follows: a topology design method, the topology design comprises:

[0006] ①Obtaining an undirected graph composed of three nodes, and dividing its topology into two types of connected but not fully connected and fully connected;

[0007] ②For the case of connected but not fully connected, the relationship between the nodes is divided into three cases: node 1 and node 2 are not adjacent, node 1 and node 3 are not adjacent, and node 2 and node 3 are not adjacent;

[0008] ③Designing a corresponding coupling matrix for each case, and ensuring that the second largest eigenvalue of the coupling matrix does not exceed a given negative constant and the sum of its off-diagonal elements is minimum;

[0009] As a further technical solution of the present application, consider an undirected graph composed of three nodes, and divide its topology into two types of connected but not fully connected and fully connected. For the case of connected but not fully connected, the relationship between the nodes is divided into three cases: ① node 1 and node 2 are not adjacent; ② node 1 and node 3 are not adjacent; and ③ node 2 and node 3 are not adjacent.

[0010] For the case that node 1 and node 2 are not adjacent, the corresponding coupling matrix is defined as:

[0011]

[0012] where γ 13 and γ 23 are positive constants, and

[0013] For any negative constant σ, the following coupling matrix can be designed:

[0014]

[0015] such that λ2(Γ)≤σ and the sum of off-diagonal elements is minimal, where λ n (Γ), n = 1, 2, 3 denote the eigenvalues of matrix Γ and 0 = λ1(Γ) > λ2(Γ) ≥ λ3(Γ).

[0016] For the case that node 1 and node 3 are not adjacent, the corresponding coupling matrix is defined as:

[0017]

[0018] where γ 12 and γ 23 are positive constants, and

[0019] For any negative constant σ, the following coupling matrix can be designed:

[0020]

[0021] such that λ2(Γ)≤σ and the sum of off-diagonal elements is minimal, where λ n (Γ), n = 1, 2, 3 denote the eigenvalues of matrix Γ and 0 = λ1(Γ) > λ2(Γ) ≥ λ3(Γ).

[0022] For the case that node 2 and node 3 are not adjacent, the corresponding coupling matrix is defined as:

[0023]

[0024] where γ 12 and γ 13 are positive constants, and

[0025] For any negative constant σ, the following coupling matrix can be designed:

[0026]

[0027] such that λ2(Γ)≤σ and the sum of off-diagonal elements is minimal, where λn (Γ),n=1,2,3 represent eigenvalues of matrix Γ and 0=λ1(Γ)>λ2(Γ)≥λ3(Γ).

[0028] As a further technical solution of the application, for the full connectivity case, the corresponding coupling matrix is defined as:

[0029]

[0030] where γ 12 ,γ 13 and γ 23 are normal numbers, and

[0031] For any negative constant σ, the following coupling matrix can be designed:

[0032]

[0033] so that λ2(Γ)≤σ and the sum of off-diagonal elements is minimum, where λ n (Γ),n=1,2,3 represent eigenvalues of matrix Γ and 0=λ1(Γ)>λ2(Γ)≥λ3(Γ).

[0034] Another object of the application is to provide a topology design method for application in consensus of multi-agent system, which applies the topology design scheme to consensus control of multi-agent system and verifies its effectiveness through numerical simulation.

[0035] The mathematical model of the multi-agent system is:

[0036]

[0037] In the above formula, represents a state vector of the agent; is a continuous vector function and satisfies the following inequality:

[0038] ||f(x1)-f(x2)||≤ζ||x1-x2||;

[0039] The mean value is defined as:

[0040]

[0041] The error is defined as: i=1,2,…,S, then the error model can be obtained as:

[0042]

[0043] The control protocol is designed as:

[0044]

[0045] In the above formula, Satisfy the following conditions:

[0046]

[0047] In the formula, V={1,2,3} and Respectively represent the node set and the undirected edge set.

[0048] The sufficient condition for the multi-agent system to achieve consensus is:

[0049]

[0050] Compared with the prior art, the beneficial effects of the present application are:

[0051] The present application proposes a topology design scheme and its application in the consensus of multi-agent systems, and deeply studies the coupling matrix design method, which not only promotes the development of matrix theory, but also provides a new method for the cooperative control of multi-agent systems. BRIEF DESCRIPTION OF DRAWINGS

[0052] Figure 1 It is a flow chart of a topology design method and its application in the consensus of multi-agent systems.

[0053] Figure 2 It is a topology structure diagram in which node 1 and node 2 are not adjacent.

[0054] Figure 3 It is a topology structure diagram in which node 1 and node 3 are not adjacent.

[0055] Figure 4 It is a topology structure diagram in which node 2 and node 3 are not adjacent.

[0056] Figure 5 It is a topology structure diagram in which three nodes are fully connected.

[0057] Figure 6 It is a first simulation result analysis diagram of the present application.

[0058] Figure 7 It is a second simulation result analysis diagram of the present application.

[0059] Figure 8 It is a third simulation result analysis diagram of the present application.

[0060] Figure 9 It is a fourth simulation result analysis diagram of the present application. DETAILED DESCRIPTION

[0061] In order to make the objects, technical solutions and advantages of the present application clearer, the following further describes the present application in detail with reference to the drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and should not be used to limit the present application.

[0062] The specific implementation of the present application is described in detail below in combination with specific embodiments.

[0063] As shown in the following table, the present embodiment provides a topology design method, which includes: Figure 1

[0064] ① Consider an undirected graph composed of three nodes, and divide its topology into two types: connected but not fully connected and fully connected;

[0065] ② For the connected but not fully connected case, divide the relationship between the nodes into three cases: node 1 and node 2 are not adjacent, node 1 and node 3 are not adjacent, and node 2 and node 3 are not adjacent;

[0066] ③ Design a corresponding coupling matrix for each case, and ensure that the second largest eigenvalue of the coupling matrix does not exceed a given negative constant and the sum of its off-diagonal elements is minimum;

[0067] Lemma 1. Assuming that is a symmetric and irreducible matrix, where

[0068]

[0069] then the following can be obtained:

[0070]

[0071] wherein and

[0072] Lemma 2. For a Hermitian matrix H∈￡ n×n , the following inequality holds

[0073]

[0074] wherein H j ∈￡ (n-1)×(n-1) represents the matrix obtained by removing the jth row and the jth column of H, b i =(b i1 ,L,b ij ,L,b in ) T ∈￡ n represents the unit eigenvector corresponding to the eigenvalue λ i (H).

[0075] ​As a preferred embodiment of the present application, consider a 3-order coupling matrix Γ = (γ ij ) 3×3 where

[0076]

[0077] The main purpose of the present application is to design appropriate γ ij (i,j = 1,2,3, i≠j) such that λ2(Γ) is not greater than any given negative constant σ, and the sum of the off-diagonal elements of Γ is minimum. Considering that in the study of the consensus of multi-agent systems, a disconnected multi-agent system can be transformed into several connected multi-agent systems for study, the present application only discusses the case of three-node connectivity. In view of this, the topological structure among the three nodes is divided into two categories: connected but not fully connected and fully connected.

[0078] 1) For the case of connected but not fully connected, the relationship between the nodes is divided into three cases: ① node 1 and node 2 are not adjacent (as shown in Figure 2 ); ② node 1 and node 3 are not adjacent (as shown in Figure 3 ); and ③ node 2 and node 3 are not adjacent (as shown in Figure 4 ).

[0079] Case ①: Node 1 and node 2 are not adjacent.

[0080] Figure 2 The topological structure among the three nodes is shown, and for this case, the corresponding coupling matrix should be defined as:

[0081]

[0082] where γ 13 and γ 23 are normal numbers, and

[0083] Equation (1):

[0084] Based on Lemma 1, it can be deduced that 0 = λ1(Γ) > λ2(Γ) ≥ λ3(Γ). Based on matrix theory and equation

[0085] (1), it can be obtained that:

[0086] Equation (2):

[0087] Furthermore, since the matrix Γ is dissipative, or is the unit eigenvector corresponding to the eigenvalue λ1(Γ). Obviously,

[0088] The eigenvalues of Γ3 are -γ13 and -γ 23 According to Lemma 2, we have

[0089] Equation (3): λ2(Γ)λ3(Γ) = 3γ 13 γ 23

[0090] According to Equation (2) and Equation (3), we have

[0091] Equation (4):

[0092] According to Equation (4), we have the following equation:

[0093] Equation (5):

[0094] To ensure that Equation (5) has a feasible solution, the following inequality needs to be satisfied

[0095] Equation (6):

[0096] Since According to Equation (6), we have λ3(Γ)≤3λ2(Γ). Taking λ2(Γ) = σ and λ3(Γ) = 3λ2(Γ), according to Equation (2), we can deduce that for any negative constant σ, the sum of the off-diagonal elements of Γ is the smallest. In this case, Equation (4) can be rewritten as:

[0097] Equation (7)

[0098] According to Equation (7), we have

[0099] Equation (8):

[0100] In summary, for any negative constant σ, we can design the following coupling matrix:

[0101]

[0102] such that λ2(Γ)≤σ and the sum of the off-diagonal elements of the matrix is the smallest.

[0103] Case 2: Node 1 and Node 3 are not adjacent.

[0104] Figure 3 The topology between three nodes is shown, and for this case, the corresponding coupling matrix should be defined as:

[0105]

[0106] where γ 12 and γ 23 are normal numbers, and

[0107] Equation (9):

[0108] Based on Lemma 1, it can be derived that 0 = λ1(Γ) > λ2(Γ) ≥ λ3(Γ). Based on matrix theory and Equation (9), it can be obtained that

[0109] Equation (10):

[0110] Furthermore, since the matrix Γ is dissipative, or is the unit eigenvector corresponding to the eigenvalue λ1(Γ). Obviously,

[0111] The eigenvalues of Γ2are -γ 13 and -γ 23 . According to Lemma 2, it can be obtained that:

[0112] Equation (11): λ2(Γ)λ3(Γ) = 3a 12 a 23 ;

[0113] According to Equation (10) and Equation (11), it can be obtained that:

[0114] Equation (12):

[0115] According to Equation (12), the following equation can be obtained:

[0116] Equation (13):

[0117] To ensure that Equation (13) has a feasible solution, the following inequality needs to be satisfied:

[0118] Equation (14):

[0119] Since According to Equation (14), it can be obtained that λ3(Γ) ≤ 3λ2(Γ). Taking λ2(Γ) = σ and λ3(Γ) = 3λ2(Γ), according to Equation (10), it can be derived that for any negative constant σ, the sum of the off-diagonal elements of Γ is minimum. In this case, Equation (12) can be rewritten as:

[0120] Equation (15):

[0121] According to Equation (15), it can be obtained that:

[0122] Equation (16):

[0123] In summary, for any negative constant σ, the coupling matrix can be designed as follows:

[0124]

[0125] The condition is such that λ2(Γ)≤σ and the sum of the off-diagonal elements of Γ is minimized.

[0126] Case 3: Node 2 and Node 3 are not adjacent.

[0127] Figure 4 The topology between the three nodes is shown. For this scenario, the corresponding coupling matrix should be defined as:

[0128]

[0129] In the formula, γ 12 and γ 23 It is a positive number, and

[0130] Equation (17):

[0131] Based on Lemma 1, it can be deduced that 0 = λ1(Γ) > λ2(Γ) ≥ λ3(Γ). Based on matrix theory and equation (17), it can be obtained that:

[0132] Equation (18):

[0133] Similarly, since matrix Γ is dissipative, or It is the unit eigenvector corresponding to the eigenvalue λ1(Γ). Clearly,

[0134] The eigenvalue of Γ1 is -γ 12 and -γ 13 According to Lemma 2, we can obtain:

[0135] Equation (19): λ2(Γ)λ3(Γ)=3a 12 a 13 ;

[0136] According to equations (18) and (19), we can obtain:

[0137] Equation (20):

[0138] According to equation (20), the following equation can be obtained:

[0139] Equation (21):

[0140] To ensure that equation (21) has a feasible solution, the following inequality must be satisfied:

[0141] Equation (22):

[0142] Since According to equation (22), λ3(Γ)≤3λ2(Γ) can be obtained. Taking λ2(Γ)=σ and λ3(Γ)=3λ2(Γ), according to equation (18), it can be deduced that for any negative constant σ, the sum of the off-diagonal elements of Γ is minimum. In this case, equation (20) can be rewritten as:

[0143] Equation (23):

[0144] According to equation (23), it can be obtained that:

[0145] Equation (24):

[0146] Therefore, for any negative constant ρ, the coupling matrix can be designed as follows:

[0147]

[0148] So that λ2(Γ)≤σ and the sum of the off-diagonal elements of Γ is minimum.

[0149] 2) In the case of full connectivity, Figure 5 The topology of three nodes is described, and the coupling matrix is given as follows:

[0150]

[0151] In the formula, γ 12 , γ 13 and γ 23 are normal numbers, and

[0152] Equation (25):

[0153] Based on Lemma 1, it can be deduced that 0=λ1(Γ)>λ2(Γ)≥λ3(Γ). Based on matrix theory and equation (25), it can be obtained that:

[0154] Equation (26): λ2(Γ)+λ3(Γ)=-2(γ 12 +γ 13 +γ 23 );

[0155] Similarly, since the matrix Γ is dissipative, or is the unit eigenvector corresponding to the eigenvalue λ1(Γ). Obviously, and

[0156] According to Lemma 2, it can be obtained that:

[0157] Formula (27): λ2(Γ)λ3(Γ) = 3(a 12 a 13 +a 12 a 23 +a 13 a 23 );

[0158] According to Formula (27), it can be obtained that:

[0159]

[0160] Since λ3(Γ)≤λ2(Γ) and λ2(Γ) is required to be no more than a non-negative constant σ, it can be obtained that:

[0161] λ2(Γ)+λ3(Γ)≤2σ;

[0162] That is, γ 12 +γ 13 +γ 23 ≥-σ;

[0163] Obviously, if Formula (26) and Formula (27) have feasible solutions γ 12 ,γ 13 ,γ 23 , when λ2(Γ) = σ and λ3(Γ) = λ2(Γ), γ 12 +γ 13 +γ 23 is the smallest. In this case, Formula (26) and Formula (27) can be rewritten as:

[0164] Formula (28): Based on Formula (28), it can be obtained that:

[0165] Formula (29): Based on Formula (29), it can be obtained that:

[0166] Formula (30): In order to ensure that Formula (30) has a feasible solution, the following inequality should be satisfied:

[0167] Formula (31): According to Formula (31), it can be obtained that Formula (30) has a feasible solution if and only if . Let in Formula (29), it can be obtained that:

[0168] Formula (32): According to Formula (32), it can be obtained that:

[0169] Formula (33): Therefore, for any negative real number σ, the following coupling matrix can be designed:

[0170]

[0171] So that λ2(Γ)≤σ and the sum of off-diagonal elements is minimum.

[0172] Another object of the present application is to provide a topology design method for the consistency of multi-agent systems, apply the topology design scheme to the consistency control of multi-agent systems, and verify its effectiveness through numerical simulation, considering the following multi-agent system:

[0173] Formula (34) is: In the above formula, x represents the state vector of the agent; is a continuous vector function; u represents the control input of the agent.

[0174] f(x i (t)) satisfies the following inequality:

[0175] Formula (35) is: ||f(x1)-f(x2)||≤ζ||x1-x2||;

[0176] The multi-agent can achieve consistency if:

[0177]

[0178] The mean value is defined as:

[0179]

[0180] The error is defined as The error model can be obtained as:

[0181] Formula (36) is:

[0182] The consistency control protocol is designed as:

[0183] Formula (37) is:

[0184] In the above formula, satisfies the following conditions:

[0185]

[0186] The system of formula (34) can achieve consistency if λ2(Γ) satisfies:

[0187] Formula (38) is: Proof: Choose the following Lyapunov functional:

[0188] Formula (39) is: Taking the derivative of the Lyapunov functional of formula (39) can obtain:

[0189] Equation (40) According to Equation (35), we have

[0190] Equation (41)

[0191] Further, since

[0192]

[0193] Equation (42) can be obtained On the other hand, based on Lemma 1, we have

[0194] Equation (43) According to Equation (40) and Equation (43), we can deduce that

[0195] Equation (44)

[0196] where ξ = 1 + δ 2 + 2kλ2(Γ)λ n (Q).

[0197] According to Equation (44), we have

[0198] Equation (45): W(t)≤e ξt W(0);

[0199] According to Equation (39) and Equation (45), we can obtain

[0200]

[0201] Therefore, the system of Equation (34) can achieve consistency under the action of the controller of Equation (37).

[0202] As an extended embodiment of the present application, an example is selected for simulation verification;

[0203] The number of agents is selected k = 2, f(x i (t)) = diag(2, 3, 4)x i (t), Q = 2I3,

[0204]

[0205] Obviously, for any ψ1 and f(·) satisfies:

[0206] ||f(ψ1)-f(ψ2)||4|ψ1-ψ2||;

[0207] Thus, it can be easily obtained that the system of equation (34) with the above parameters can achieve consistency when

[0208]

[0209] Since the system of equation (34) is connected, the topology can be divided into two categories: connected but not fully connected and fully connected. The coupling weight matrix Γ is designed such that λ2(Γ)≤-2.125 and the sum of the off-diagonal elements of Γ is minimum.

[0210] 1) Connected but not fully connected

[0211] For the case that agent 1 and 2 are not adjacent, the following coupling matrix is designed:

[0212]

[0213] where λ2(Γ)=-2.125 and the sum of the off-diagonal elements of Γ is minimum. The variation process of is shown in Figure 6 .

[0214] For the case that agent 1 and 3 are not adjacent, the following coupling matrix is designed:

[0215]

[0216] where λ2(Γ)=-2.125 and the sum of the off-diagonal elements of Γ is minimum. The variation process of is shown in Figure 7 .

[0217] For the case that agent 2 and 3 are not adjacent, the following coupling matrix is designed:

[0218]

[0219] where λ2(Γ)=-2.125 and the sum of the off-diagonal elements of Γ is minimum. The variation process of is shown in Figure 8 .

[0220] 2) Fully connected

[0221] For the case that the three agents can communicate with each other, the following coupling matrix is designed:

[0222]

[0223] where λ2(A)=-2.125 and the sum of the off-diagonal elements of A is minimum. The variation process of is shown in Figure 9 .

[0224] It should be noted that, in this text, the terms "comprising", "containing" or any other variant thereof are intended to cover non-exclusive inclusion, so that a process, method or article including a series of elements not only includes those elements, but also includes other elements not explicitly listed, or further includes elements inherent to such a process, method or article. Without more limitations, the element defined by the statement "comprising a" does not exclude the presence of other identical elements in the process, method, article or device including the element.

[0225] The above is only the preferred embodiment of the present application, and does not limit the patent scope of the present application, and any equivalent structure or equivalent process transformation using the content of the present application specification and drawings, or direct or indirect application in other related technical fields, are also included in the patent protection scope of the present application.

Claims

1. A method of topological design, characterized by The topology design comprises: 1) obtaining an undirected graph composed of three nodes, and dividing the topology into two types of connected but not fully connected and fully connected; 2) for the connected but not fully connected case, the relationship between the nodes is divided into three cases that node 1 and node 2 are not adjacent, node 1 and node 3 are not adjacent, and node 2 and node 3 are not adjacent; 3) for each case, a corresponding coupling matrix is designed, and the second largest eigenvalue of the coupling matrix is ensured to be not more than a given negative constant and the sum of the non-diagonal elements of the coupling matrix is the minimum; 4) the topology design scheme is applied to the consensus control of a multi-agent system comprising node 1, node 2 and node 3, and the effectiveness is verified through numerical simulation; For the case that node 1 and node 2 are not adjacent, the corresponding coupling matrix is defined as: ； wherein and are normal numbers, and ; For any negative constant The following coupling matrix can be designed: ； such that and the sum of off-diagonal elements is minimum, where denotes the eigenvalues of the matrix and ; For the case that node 1 and node 3 are not adjacent, the corresponding coupling matrix is defined as: ； wherein and are normal numbers, and ; For any negative constant The following coupling matrix can be designed: ； such that and the sum of the off-diagonal elements is minimal, where denotes the eigenvalues of the matrix and ; For the case that node 2 and node 3 are not adjacent, the corresponding coupling matrix is defined as: ； wherein and are normal numbers, and ; For any negative constant The following coupling matrix can be designed: ； such that and the sum of off-diagonal elements is minimal, where denotes the eigenvalues of the matrix and .

2. The method of claim 1, wherein, For the fully connected case, the corresponding coupling matrix is defined as: ； wherein and are normal numbers, and ; For any negative constant The following coupling matrix is designed: ； such that and the sum of the off-diagonal elements is minimal, where denotes the eigenvalues of the matrix and .

3. The method of claim 1, wherein, The application of the topology design method in the consensus of the multi-agent system further comprises that a mathematical model of the multi-agent system is: ； In the above formula, represents a state vector of the agent; is a continuous vector function and satisfies the following inequality: ； The mean value is defined as: ； Definition of error The error model is then given by: ； The control protocol is designed as: ； In the above formulae, satisfies the following conditions: ； wherein and respectively denote a set of nodes and a set of undirected edges; The sufficient condition for the multi-agent system to achieve consensus is: 。

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