Bipartite consistency control method for second-order multi-agent system under first-class switching topology

Abstract

The invention relates to the technical field of bipartite consistency of a multi-agent system, and discloses a bipartite consistency control method of a second-order multi-agent system under a switching topology, which aims at bipartite consistency control without time delay and with communication time delay, considers a second-order multi-agent system consisting of n agents, determining a second-order multi-agent system model and parameters; determining a topological structure of the system under the switching topology according to the second-order multi-agent system model and the parameters thereof; designing a bipartite consistency control protocol of the second-order multi-agent system considering no time delay and communication time delay constraints; and the sufficient condition that the second-order multi-agent system meets the dichotomy consistency under the condition of no time delay and communication time delay in the switching topology is obtained. Compared with the prior art, the problem of dichotomy consistency of the second-order multi-agent system under the conditions of no time delay and communication time delay is solved.

Description

Technical Field

[0001] The present invention relates to the technical field of multi - agent system bipartite consensus, and particularly relates to a bipartite consensus control method for second - order multi - agent systems under a class of switching topologies. Background Art

[0002] In recent years, the consensus problem has always been one of the hot topics in the research of multi - agent systems. It has been widely applied in many fields. Multi - agent consensus means that in a system composed of multiple agents, some state variables (such as position, speed, etc.) of all agents gradually reach the same value over time, or satisfy a certain specific common relationship, so as to improve the effectiveness and robustness of the system.

[0003] However, traditional multi - agent consensus research mainly focuses on the ideal environment with a fixed topological structure and no communication delay. In actual application scenarios, the environment in which multi - agent systems are located is extremely complex. For example, in an intelligent transportation system, vehicles, as agents, during their driving process, due to factors such as road conditions and traffic signals, the communication connections between vehicles will change frequently, resulting in continuous switching of the communication topological structure; at the same time, limited by the performance of communication devices and transmission distance, etc., there must be communication delays when information is transmitted between vehicles. Another example is in the scenario of distributed robot cooperation. When robots are performing tasks, the dynamic changes in their spatial positions will cause the instability of the communication links between them, thereby triggering communication topological switching, and signals will also face delay problems during the transmission process.

[0004] When considering these two practical factors of switching topology and communication delay, the control methods designed under ideal conditions are no longer applicable. The existence of switching topology makes the information interaction mode between agents change continuously, and communication delay further deteriorates the performance of the system, resulting in agents being difficult to accurately obtain the real - time state information of other agents, which poses a huge challenge to achieving bipartite consensus control of second - order multi - agent systems.

[0005] Therefore, the present invention proposes a class of bipartite consensus control methods for second - order multi - agent systems under switching topologies to solve the bipartite consensus problems of second - order multi - agent systems without time delay and with communication time delay respectively. Summary of the Invention

[0006] Object of the Invention: Aiming at the problems mentioned in the background art, the present invention proposes a class of bipartite consensus control methods for second - order multi - agent systems under switching topologies to solve the bipartite consensus problems of second - order multi - agent systems without time delay and with communication time delay respectively.

[0007] Technical solution: The present invention proposes a method for bipartite consensus control of a second-order multi-agent system under switched topologies, aiming at bipartite consensus control without time delay and with communication time delay, including the following steps:

[0008] Step 1: Consider a second-order multi-agent system composed of n agents, and determine the model and parameters of the second-order multi-agent system;

[0009] Step 2: Determine the topological structure of the system under switched topologies according to the model and parameters of the second-order multi-agent system;

[0010] Step 3: Design a bipartite consensus control protocol for the second-order multi-agent system considering constraints of no time delay and communication time delay, specifically as follows:

[0011] Step 3.1 The bipartite consensus control protocol for the second-order multi-agent system without time delay under switched topologies is as follows:

[0012]

[0013] where k i , γ i , i = 1, 2, …, n are control gain coefficients, x i (t), x j (t) represent the positions of the i-th and j-th agents, v i (t), v j (t) represent the velocities of the i-th and j-th agents, u i (t) represents the control input of the i-th agent, a ij represents the weight of the edge in the topological structure of the system, N i (G(t)) represents the adjacency set of the i-th node considering the time-varying signal G(t);

[0014] Step 3.2 The bipartite consensus control protocol for the second-order multi-agent system with communication time delay under switched topologies is as follows:

[0015]

[0016] where k i , γ i are control gain coefficients, i = 1, 2, …, n, τ(t) is the communication delay signal of the system, and satisfies: 0 ≤ τ(t) ≤ h, h represents the upper limit of the time delay signal, is the derivative of τ(t).

[0017] Furthermore, the model and parameters of the second-order multi-agent system in Step 1 are specifically as follows:

[0018]

[0019] where x i (t) represents the position of the i-th agent, v i (t) represents the velocity of the i-th agent, and u i (t) represents the control input of the i-th agent.

[0020] Furthermore, the topological structure of the system under the switching topology in step 2 is as follows:

[0021] The switching topology undirected graph of the multi-agent system composed of n agents is defined as G=(V,ε,A), where V={v 1 , v 2 , …, v n} represents the set composed of finite nodes, ε represents the set of edges, and A=[a ij (t)]∈R n×n represents the adjacency matrix, a ij represents the weight of the edge. If there is information interaction between agent i and agent j, the corresponding element a ij is non-zero; the set of adjacency matrices N i of node i is represented; the degree matrix C∈R n×n is defined as C = diag{c i}, The Laplacian matrix of the switching topology undirected graph G is defined as L = C - A. The element l ij of the Laplacian matrix L is defined as:

[0022]

[0023] Considering the time-varying symbol G(t), the undirected graph of the possible topological structures under the switching topology of the second-order multi-agent system is redefined as G p (t). The switching signal r(t) is expressed as: p = r(t), p={1,2,3…,N} is the index set. The corresponding Laplacian matrix, degree matrix, and adjacency matrix are respectively defined as L p , C p and A p .

[0024] Furthermore, the topological structure of the second-order multi-agent system satisfies the following conditions:

[0025] 1) The graph G is a connected, weighted signed graph;

[0026] 2) All G p (t) are structurally balanced graphs.

[0027] Furthermore, the time-delay-free second-order multi-agent bipartite consensus control protocol under the switching topology in step 3.1 introduces new variables: Among them, y i is the newly introduced variable, v i is the velocity of the i-th agent, x i is the position of the i-th agent, and k is the control gain coefficient;

[0028] Combining the Kronecker product and considering the canonical transformation, the global dynamic equation of the system is obtained:

[0029]

[0030] p = r(t)

[0031] Among them, I n is the n-order identity matrix, D is the canonical transformation matrix, represents the global dynamic equation after canonical transformation under the switching topology, represents the position state of the agent under the switching topology, represents the state of the newly introduced variable y i in the switching topology, is the control gain coefficient of γ i in the switching topology, k p is the control gain coefficient of k i in the switching topology, is the Laplacian matrix after canonical transformation in the switching topology. The bipartite consensus problem of second-order multi-agent systems without time delay in the switching topology is transformed into the stability problem of the global dynamic equation;

[0032] Furthermore, considering the second-order multi-agent bipartite consensus control protocol with communication time delay in the switching topology described in step 3.2, the second-order multi-agent system model is described as:

[0033]

[0034] p = r(t)

[0035] Among them, K = diag{k 1 , k 2 , …, k n}, is the control gain coefficient of γ i in the switching topology, C p and A p are the degree matrix and adjacency matrix in the switching topology considering the time-varying sign G(t), v(t) is the velocity state matrix of the system, x(t) is the position state matrix of the system, v(t - τ(t)) is the velocity state matrix of the system with communication time delay, x(t - τ(t)) is the position state matrix of the system with communication time delay, and the canonical transformation is introduced:

[0036]

[0037] Among them, represents the global dynamic equation after gauge transformation under the switching topology, is the system adjacency matrix after gauge transformation under the switching topology; is the position matrix after gauge transformation under the switching topology, is the velocity matrix after gauge transformation under the switching topology. The bipartite consensus problem of the second-order multi-agent system with communication delay under the switching topology is transformed into the stability problem of the above formula.

[0038] Furthermore, the conditions for the second-order multi-agent system to satisfy bipartite consensus under the switching topology without delay and with communication delay are as follows:

[0039] Condition 1: The sufficient condition for the second-order multi-intelligent system without delay under the switching topology to achieve bipartite consensus is that when and only when k p > 0 and The system can achieve bipartite consensus under the bipartite consensus control protocol of the second-order multi-agent without delay under the switching topology;

[0040] Condition 2: The sufficient condition for the second-order multi-intelligent system with communication delay under the switching topology to achieve bipartite consensus is:

[0041] There exist matrices P, Q, R, and W with appropriate dimensions that satisfy the following conditions, then the system can achieve bipartite consensus under the bipartite consensus control protocol of the second-order multi-agent with communication delay under the switching topology;

[0042]

[0043] Among them,

[0044] Furthermore, considering the undirected graph G of the switching topology structure with time-varying sign G(t) p (t) does not contain self-loops, that is, a ii = 0.

[0045] Furthermore, for the matrices P, Q, R, and W with appropriate dimensions in step 4, when and only when they are positive semi-definite matrices, the system can achieve bipartite consensus under the bipartite consensus control protocol of the second-order multi-agent with communication delay under the switching topology.

[0046] Compared with the prior art, the beneficial effects of the present invention are:

[0047] 1. The present invention comprehensively considers second - order multi - agent systems with and without communication delays. By using the eigenvalue - eigenvector method, Lyapunov - Krasovskii functional, and linear matrix inequality theory respectively, sufficient conditions for the system to achieve bipartite consensus are obtained, ensuring that the system can achieve bipartite consensus under the assumed conditions. Finally, numerical simulations verify the feasibility and effectiveness of the theoretical derivation.

[0048] 2. The present invention proposes a method for controlling bipartite consensus of second - order multi - agent systems under switching topologies. When analyzing second - order systems without delays, by introducing a new variable y i , compared with some existing technologies, the analysis process is greatly simplified and the design process is more straightforward. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 is a flowchart of a method for controlling bipartite consensus of second - order multi - agent systems under switching topologies proposed by the present invention.

[0050] Figure 2 is an 8 - agent switching topology graph for the numerical simulation used in the present invention.

[0051] Figure 3 is a simulation velocity state diagram of the signal without delay in the embodiment of the present invention.

[0052] Figure 4 is a simulation position state diagram of the signal without delay in the embodiment of the present invention.

[0053] Figure 5 is a simulation position state diagram of the signal with communication delay in the embodiment of the present invention.

[0054] Figure 6 is a simulation position state diagram of the signal without delay in the embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0055] It should be understood that the specific embodiments described herein are for explaining the present invention and not for limiting the present invention.

[0056] The embodiments of the present invention propose a method for controlling bipartite consensus of second - order multi - agent systems under switching topologies. For the bipartite consensus problems of second - order multi - agents with and without communication delays respectively, the following steps are included:

[0057] Step 1: Consider a second - order multi - agent system composed of n agents, and determine the model and parameters in the system, as shown in formula (1):

[0058]

[0059] where x i(t) represents the position of the ith agent, v i (t) represents the speed of the ith agent, u i (t) represents the control input of the ith agent.

[0060] Step 2: Determine the topological structure of the system under the switching topology based on the second-order multi-agent system model and its parameters.

[0061] The switching topology undirected graph of a multi-agent system consisting of n agents is defined as G = (V, ε, A), where V = {v 1 , v 2 , …, v n} represents a set of finite nodes, ε represents a set of edges, A=[a ij (t)]∈R n*n represents the adjacency matrix, a ij represents the weight of the edge. If there is information interaction between agents i and j, the corresponding element a ij Non-zero, degree matrix C∈R n×n Defined as C = diag{c i}, The Laplacian matrix of graph G is defined as L = CA, and its elements are defined as:

[0062]

[0063] Considering the time-varying symbol G(t), the possible topological structure under the switching topology of the second-order multi-intelligence system is defined as G p (t), the switching signal r(t) is expressed as:

[0064] p=r(t) (3)

[0065] Where p = {1, 2, 3…, N} is the index set, so the corresponding Laplacian matrix, degree matrix and adjacency matrix are defined as L p ,C p and A p .

[0066] Assumption 1: Assume that graph G is a connected, weighted symbolic graph.

[0067] Assumption 2: Assume that all G p (t) are all structural equilibrium diagrams.

[0068] Step 3: Design a bipartite consensus control protocol for second-order multi-agent systems considering delay-free and communication delay constraints.

[0069] Step 3.1 The delay-free second-order multi-agent bipartite consensus control protocol under the switching topology is as follows:

[0070]

[0071] where \(k\) i , \(\gamma\) i , \(i = 1, 2, \cdots, n\) are control gain coefficients, \(x\) i (t), \(x\) j (t) represent the positions of the \(i\)-th and \(j\)-th agents, \(v\) i (t), \(v\) j (t) represent the velocities of the \(i\)-th and \(j\)-th agents, \(u\) i (t) represents the control input of the \(i\)-th agent, \(a\) ij represents the weight of the edge in the topological structure of the system, \(N\) i (G(t)) represents the adjacent set of the \(i\)-th node considering the time-varying signal G(t).

[0072] To simplify the analysis process, new variables are introduced

[0073]

[0074] Combining the Kronecker product and considering the canonical transformation, the global dynamic equation of system (1) is obtained:

[0075]

[0076] where I n is the \(n\)-order identity matrix, \(D\) is the canonical transformation matrix, represents the global dynamic equation after canonical transformation under the switching topology, represents the position state of the agents under the switching topology, represents the state of the new variable \(y\) introduced under the switching topology i , is the control gain coefficient of \(\gamma\) under the switching topology i , \(k\) p is the control gain coefficient of \(k\) under the switching topology i , is the Laplacian matrix after canonical transformation under the switching topology.

[0077] Therefore, the bipartite consensus problem of second-order multi-agent systems without time delay under the switching topology is transformed into the stability problem of equation (6).

[0078] Step 3.2 The bipartite consensus control protocol for second-order multi-agent systems with communication delays under the switching topology is as follows:

[0079]

[0080] where \(k\) i , \(\gamma\) iTo control the gain coefficient, τ(t) is the communication delay signal of the system and satisfies:

[0081]

[0082] h represents the upper limit of the time-delay signal.

[0083] Consider the bipartite consensus control protocol (7) with time-varying time-delay signals under switching topologies, and the description of system (1) is:

[0084]

[0085] where K = diag{k 1 , k 2 , …, k n}, τ(t) is the communication time-delay signal. For the control gain coefficient of γ i under the switching topology, C p and A p are the degree matrix and the adjacency matrix under the switching topology, v(t) is the velocity state matrix of the system, x(t) is the position state matrix of the system, v(t - τ(t)) is the velocity state matrix of the system with communication time-delay, and x(t - τ(t)) is the position state matrix of the system with communication time-delay. Introduce the canonical transformation:

[0086]

[0087] where represents the global dynamic equation after the canonical transformation under the switching topology, is the system adjacency matrix after the canonical transformation under the switching topology, is the position matrix after the canonical transformation under the switching topology, is the velocity matrix after the canonical transformation under the switching topology.

[0088] Therefore, the bipartite consensus problem of the second-order multi-agent system with communication time-delay under the switching topology is converted into the stability problem of Equation (10).

[0089] Step 4: Based on the eigenvalue-eigenvector, Lyapunov-Krasovskii functional, and linear matrix inequality theories, obtain the conditions for the second-order multi-agent system under the switching topology to satisfy bipartite consensus without time-delay and with communication time-delay, respectively.

[0090] Condition 1: The sufficient condition for the second-order multi-intelligent system without time-delay under the switching topology to achieve bipartite consensus is:

[0091] Under the conditions of satisfying Assumption 1 and Assumption 2, if and only if kP > 0 and The system (6) can achieve binary consensus under the control protocol (4).

[0092] Proof: Taking the Laplace transform of the system (6) gives:

[0093]

[0094] The characteristic equation is:

[0095]

[0096] Therefore:

[0097]

[0098] where k p , are all positive numbers, and the eigenvalues of the Laplacian matrix are all non - negative.

[0099] According to the stability condition,, the stability of the system (6) is equivalent to all roots having no positive real part. Obviously:

[0100]

[0101] Therefore, all roots are in the left - half open - plane, and it can be concluded that the system (6) can achieve binary consensus.

[0102] Condition 2: A sufficient condition for a second - order multi - agent system with communication delay under switching topology to achieve binary consensus is:

[0103] Under the conditions of satisfying Assumption 1 and Assumption 2, there exist matrices P, Q, R, W with appropriate dimensions that satisfy Equation (15), then the system (10) can achieve binary consensus under the control protocol (7).

[0104]

[0105] where:

[0106] Proof: Construct the following Lyapunov - Krasovskii function:

[0107] V(t) = V 1 (t)+V 2 (t)+V 3 (t)+V 4 (t) (16)

[0108] where,

[0109] It is not difficult to see that the constructed Lyapunov-Krasovskii is positive definite, and its time derivative along the trajectory of system (10) is:

[0110]

[0111] According to Newton-Leibniz and integral inequality:

[0112]

[0113]

[0114] After arrangement, we get:

[0115]

[0116] where

[0117]

[0118] Therefore, if equation (18) holds, then is satisfied, and the system (10) can achieve bipartite consensus under the control protocol (7).

[0119] Experimental numerical simulation

[0120] As Figure 2 Consider four different-structured topological graphs, which contain 8 agents, and the state sets are {G 1 , G 2 , G 3 , G 4}, under the satisfaction of the assumption conditions, the state of system G 1 starts to switch clockwise

[0121] For the second-order multi-agent system without time delay, let k 1 = k 2 = k 3 = k 4 = k 5 = k 6 = k 7 = k 8 = 5, γ 1 = γ 2 = γ 3 = γ 4 = γ 5 = γ 6 = γ 7 = γ 8 = 0.1. Without considering communication delay, the initial states of the agents are respectively (x 1 (0), v 1(0)) = (-6.65, -8.94), (x 2 (0), v 2 (0)) = (-7.87, 4.75), (x 3 (0), v 3 (0)) = (-2.55, -4.61), (x 4 (0), v 4 (0)) = (-6.03, -1.54), (x 5 (0), v 5 (0)) = (-0.20, 0.95), (x 6 (0), v 6 (0)) = (-3.21, 8.85), (x 7 (0), v 7 (0)) = (9.03, -1.64), (x 8 (0), v 8 (0)) = (8.40, 9.66).

[0122] From Figure 3 and Figure 4 it can be seen that under the condition of satisfying Condition 1, the second-order switching topology multi-agent system achieves speed state consistency and position state bipartite consistency, and the simulation results prove the feasibility and effectiveness of the control protocol (4).

[0123] For the second-order multi-agent system with communication delay, let k 1 = k 2 = k 3 = k 4 = k 5 = k 6 = k 7 = k 8 = 5, γ 1 = γ 2 = γ 3 = γ 4 = γ 5 = γ 6 = γ 7 = γ 8 = 0.1. The upper limit of the maximum time delay h = 0.1847 s is obtained through the LMI toolbox. Therefore, the time delay signal τ = 0.1sin(t) is selected to meet the simulation requirements, and the initial states of the agents are respectively (x 1 (0), v 1 (0)) = (2.34, 8.12), (x 2 (0), v 2 (0)) = (-4.69, 7.59), (x 3 (0), v 3(0)) = (6.48, 6.35), (x 4 (0), v 4 (0)) = (9.65, -4.78), (x 5 (0), v 5 (0)) = (4.60, 1.88), (x 6 (0), v 6 (0)) = (-3.12, -9.54), (x 7 (0), v 7 (0)) = (1.68, -1.49), (x 8 (0), v 8 (0)) = (-7.84, -3.74).

[0124] It can be seen from Figure 5 and Figure 6 that under the condition of satisfying Condition 2, the second-order switching topological multi-agent system realizes velocity state consensus and position state bipartite consensus, and the simulation results prove the feasibility and effectiveness of the control protocol (7).

[0125] The above embodiments are only for illustrating the technical concept and features of the present invention. Any equivalent transformation or modification made according to the spirit and essence of the present invention should be covered within the protection scope of the present invention.

Claims

1. A bipartite consistency control method for a second-order multi-agent system under a switching topology, for bipartite consistency control without time delay and with communication time delay, characterized by: The steps include: Step 1: Consider a second-order multi-agent system consisting of n agents, and determine the model and parameters of the second-order multi-agent system; Step 2: Determine the topological structure of the system under the switching topology according to the second-order multi-agent system model and its parameters; Step 3: Design a bipartite consensus control protocol for the second-order multi-agent system considering the constraints of no delay and communication delay, as follows: Step 3.1 The delay-free second-order multi-agent bipartite consensus control protocol under the switching topology is as follows: Among them, k i , γ i , i=1,2,…n is the control gain coefficient, x i (t), x j (t) represents the position of the i-th and j-th agents, v i (t), v j (t) represents the speed of the i-th and j-th agents, u i (t) represents the control input of the ith agent, a ij Represents the weight of the edge in the topological structure of the system, N i (G(t)) represents the adjacent set of the i-th node under the time-varying signal G(t); Step 3.2 The second-order multi-agent bipartite consensus control protocol with communication delay under the switching topology is as follows: Among them, k i , γ i is the control gain coefficient, i=1,2,…n, τ(t) is the communication delay signal of the system, and satisfies: 0≤τ(t)≤h, h represents the upper limit of the time-delay signal, is the derivative of τ(t).

2. The bipartite consistency control method for a second-order multi-agent system under a switching topology according to claim 1 is characterized in that: The second-order multi-agent system model and parameters in step 1 are as follows: Among them, x i (t) represents the position of the ith agent, v i (t) represents the speed of the ith agent, u i (t) represents the control input of the ith agent.

3. The bipartite consistency control method for a second-order multi-agent system under a switching topology according to claim 1 is characterized in that: The topology of the system under the switching topology in step 2 is as follows: The switching topology undirected graph of a multi-agent system consisting of n agents is defined as G = (V, ε, A), where V = {v1, v2, ..., v n } represents a set of finite nodes, ε represents a set of edges, A=[a ij (t)]∈R n×n represents the adjacency matrix, a ij Represents the weight of the edge. If there is information interaction between agents i and j, the corresponding element a ij Non-zero; the adjacency matrix set N of node i i Represents; degree matrix C∈R n×n Defined as C = diag{c i }, i=1,2,…n, the Laplacian matrix of the switching topology undirected graph G is defined as L=CA, and the Laplacian matrix L element l ij Defined as: Considering the time-varying symbol G(t), the possible topological structure undirected graph under the switching topology of the second-order multi-intelligence system is redefined as G p (t), the switching signal r(t) is expressed as: p = r(t), p = {1, 2, 3…, N} is the index set, and the corresponding Laplacian matrix, degree matrix and adjacency matrix are defined as L p ,C p and A p .

4. The bipartite consistency control method for a second-order multi-agent system under a switching topology according to claim 3 is characterized in that: The topological structure of the second-order multi-agent system satisfies the following conditions: 1) Graph G is a connected, weighted symbolic graph; 2) All G p (t) are all structural equilibrium diagrams.

5. The bipartite consistency control method for a second-order multi-agent system under a switching topology according to claim 3 is characterized in that: The second-order multi-agent bipartite consensus control protocol without time delay under the switching topology in step 3.1 introduces a new variable: Among them, y i is the new variable introduced, v i With v i (t) has the same meaning, which is the speed of the ith agent, x i With x i (t) has the same meaning, which is the position of the i-th agent, and k is the control gain coefficient; Combining the Kronecker product and considering the gauge transformation, the global dynamic equation of the system is obtained: p=r(t) in, I n is the n-order unit matrix, D is the canonical transformation matrix, represents the global dynamic equations after gauge transformation under switching topology, represents the position state of the agent under the switching topology, Indicates the introduction of a new variable y under the switching topology i status, For the switching topology, i The control gain coefficient, k p k is the switching topology i The control gain coefficient, The Laplacian matrix is ​​the canonical transformed Laplacian matrix under the switching topology. The bipartite consistency problem of the second-order multi-agent without time delay under the switching topology is transformed into the stability problem of the global dynamic equation.

6. The bipartite consistency control method for a second-order multi-agent system under a switching topology according to claim 3 is characterized in that: Considering the second-order multi-agent bipartite consensus control protocol with communication delay under the switching topology in step 3.2, the second-order multi-agent system model is described as: p=r(t) Where K = diag{k1, k2, …, k n }, i=1,2,…n, For the switching topology, i The control gain coefficient, C p and A p In order to consider the degree matrix and adjacency matrix under the switching topology of the time-varying symbol G(t), v(t) is the velocity state matrix of the system, x(t) is the position state matrix of the system, v(t-τ(t)) is the velocity state matrix of the system with communication delay, x(t-τ(t)) is the position state matrix of the system with communication delay, and the canonical transformation is introduced: in, represents the global dynamic equations after gauge transformation under switching topology, is the system adjacency matrix after canonical transformation under switching topology; is the position matrix after canonical transformation under switching topology, is the velocity matrix after canonical transformation under the switching topology, and the bipartite consistency problem of second-order multi-agents with communication delay under the switching topology is transformed into the stability problem of the above formula.

7. The bipartite consistency control method for a second-order multi-agent system under a switching topology according to claim 5 or 6, characterized in that: The sufficient conditions for the second-order multi-agent system to satisfy bipartite consistency under the switching topology without time delay and with communication time delay are: Condition 1: The sufficient condition for a time-delay-free second-order multi-agent system to achieve bipartite consistency under a switching topology is: if and only if k p >0 and The system can achieve bipartite consistency under the second-order multi-agent bipartite consistency control protocol without time delay under the switching topology; Condition 2: The sufficient condition for a second-order multi-agent system with communication delay under switching topology to achieve bipartite consistency is: There exist matrices P, Q, R, and W of appropriate dimensions that satisfy the following conditions, then the system achieves bipartite consistency under the second-order multi-agent bipartite consistency control protocol with communication delay under the switching topology; in, 8. The bipartite consistency control method for a second-order multi-agent system under a switching topology according to claim 3 is characterized in that: Consider the undirected graph G of the switching topology of the time-varying sign G(t) p (t) do not contain self-loops, that is, a ii =0.

9. The bipartite consistency control method for a second-order multi-agent system under a switching topology according to claim 1, characterized in that: The system achieves bipartite consistency under a second-order multi-agent bipartite consistency control protocol with communication delay under a switching topology if and only if the P, Q, R, and W matrices of appropriate dimensions in step 4 are semi-positive definite matrices.

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