A disturbed nonlinear multi-agent system fixed-time time-varying formation control method

Abstract

The application provides a kind of method for realizing fixed time time-varying formation control of multi-agent system under directed network quickly, wherein the topological graph of N agents is directed and connected;There are bounded disturbance and Lipshitz nonlinearity in the system composed of N agents, the system is modeled, and the formation dynamics model is established;Determine the formation information and control target of the system;Overcome the asymmetry of the graph and the constraints existing in dynamics, determine the formation error of the system;According to the formation error, formation information and control target of the system, a control protocol with fast convergence, high precision and good robustness is designed, which can make the multi-agent system realize formation in a limited time, and the convergence time has an upper bound independent of initial value, so that the convergence time can be adjusted by adjusting the control parameters;The application also gives a strict theoretical proof of the formation fixed time convergence, and the effectiveness of the algorithm is illustrated through numerical simulation examples.

Description

Technical Field

[0001] This invention belongs to the technical field of multi-agent formation control. It provides a novel method for solving the problem of time-varying formation control in multi-agent systems that simultaneously contain nonlinear and uncertain disturbances within a fixed time period. Background Technology

[0002] In the context of rapid technological advancements, multi-agent systems (MASs) have become a research hotspot in fields such as automation and robotics. A MAS consists of multiple autonomous agents that can collaborate and exchange information to accomplish complex tasks that a single agent cannot efficiently complete. Formation control, a key cooperative control behavior in MASs, aims to ensure all agents move according to a predetermined structure. It requires all agents to maintain a pre-defined relative state with each other, allowing the system to move and operate according to predetermined formations and strategies, which is crucial for improving overall system performance. This capability is particularly important in practical applications. For example, unmanned aerial vehicle (UAV) swarms can achieve coordinated reconnaissance and attack through formation flight, improving combat effectiveness and survivability. This cooperative operation not only improves mission execution efficiency but also enhances system reliability and autonomy.

[0003] Technical challenges that may arise when modeling the dynamics of formation flying vehicles include nonlinear systems and environmental disturbances. Each vehicle in formation flying possesses complex dynamic characteristics, including rotational and translational motions. These dynamic characteristics are inherently nonlinear, leading to nonlinear behavior in the formation flying system. Formation flying is also affected by various environmental factors, such as changes in wind speed and direction; these disturbances and uncertainties can also cause system instability. Previous studies have mostly focused on one constraint in nonlinear systems or disturbances, making it difficult to achieve time-varying formation control under conditions where both nonlinearity and disturbances coexist.

[0004] A key issue in formation control is convergence speed. Early research on consistency and formation primarily focused on asymptotic convergence, meaning that the states of the multiple agents only reached consistency or formed a predetermined formation when time approached infinity, failing to achieve precise and rapid convergence. Considering that some practical problems require multi-agent systems to complete specified tasks within a finite convergence time with consistent initial values, researchers have studied finite-time convergence. Compared to asymptotic convergence, finite-time convergence formation control is faster. However, finite-time convergence formation control is dependent on the initial values; as the initial values ​​increase, the convergence time of the multi-agent system also increases. Therefore, designing a fixed-time convergence algorithm that is robust to initial values ​​is crucial. Summary of the Invention

[0005] The purpose of this invention is to solve the technical problems existing in the background art, namely, the problem of time-varying formation control of a multi-agent system with both nonlinearity and interference within a fixed time when the communication topology is a directed graph. To this end, this invention proposes a fixed-time time-varying formation control method for a nonlinear multi-agent system with interference. This includes: assuming that the topology graph composed of N agents is directed and connected; and that the system composed of these N agents simultaneously has bounded perturbations and Lipschitz nonlinearity, modeling the system and establishing a formation dynamics model; determining the formation information and control objective of the system; overcoming the asymmetry of the graph and the constraints existing in the dynamics, determining the formation error of the system; and designing a fast-converging, highly accurate, and robust formation control protocol based on the system's formation error, formation information, and control objective. The designed algorithm enables the multi-agent system to achieve formation within a finite time, and the convergence time has an upper bound independent of the initial value, thus allowing adjustment of the convergence time by adjusting the control parameters. This invention also provides a rigorous theoretical proof of the fixed-time convergence of the formation and illustrates the effectiveness of the algorithm through numerical simulation examples.

[0006] The fixed-time time-varying formation control method for nonlinear multi-agent systems with disturbances, as designed according to this invention, can be applied to scenarios where formation aircraft in directed communication networks simultaneously experience nonlinearity and disturbances. Within the framework of distributed control and fixed-time control theory and Lyapunov stability theory, a dynamic model of the disturbed nonlinear multi-agent system is established based on the actual application scenario. Formation information and control objectives are determined, overcoming the asymmetry of the communication topology and system constraints. Formation error is determined, and a fixed-time time-varying formation control protocol is designed based on the formation error, formation information, and formation control objectives. The designed control protocol enables the multi-agent system to achieve formation within a finite time, with a convergence time that is independent of the initial value's upper bound. The formation error can converge precisely to zero within a fixed time, achieving the control objectives of fast convergence, high accuracy, and good robustness. This method is applicable to the control task of fixed-time time-varying formations in nonlinear multi-agent systems with disturbances and has significant practical value.

[0007] The algorithm described above for solving the distributed fixed-time formation control problem of multi-agent systems with both nonlinearity and uncertain perturbations includes the following steps in its design process:

[0008] Step 1: Describe the communication topology diagram composed of N agents and related knowledge about graphs;

[0009] Step 2: Establish a dynamic model of the disturbed nonlinear multi-agent system;

[0010] Step 3: Determine formation information and formation control objectives;

[0011] Step 4: Overcome the asymmetry of the graph and the constraints of the system to determine the formation error;

[0012] Step 5: Based on the formation error, formation information, and formation control objectives, design a fixed-time time-varying formation control protocol;

[0013] Step 6: Prove the convergence and effectiveness of the fixed-time time-varying formation control protocol.

[0014] As an improvement to the present invention, the communication topology diagram composed of N agents and related knowledge of the diagram in step 1 are described as follows:

[0015] Assumption It is a weighted directed graph, where V = {v1, v2, ..., v3}. N} represents a set of nodes. It is a directed edge set. Let be the adjacency matrix of , where a ij ≥0. The directed edges ε in G k Represents ordered node pairs (v i ,v j )∈ε, indicating that there exists a variable from v j to v i The directed information flow, i.e., v i Can receive v j Information, in which v i and v j These are referred to as child nodes and parent nodes, respectively. Weight And assume that there are no self-loops in the directed graph. There are distinct nodes in the directed graph. Make the node v i to node v j There exists a directed path whose directed sequence is: A directed spanning tree of a directed graph G contains all nodes of G and has at least one root node, where the root node satisfies the condition that there exists a unique directed path from that node to any other node in G. The Laplace matrix L of graph G is defined as: L = (l... ij ) N×N When i≠j, l ij =-a ij When i = j, and These are the in-neighbors and out-neighbors of agent i, respectively.

[0016] This invention assumes that the topological graph of the intelligent agents is a strongly connected directed graph. The Laplace matrix of graph G satisfies L1. N =0, and assume that L has an eigenvector ξ = (ξ1, ..., ξ2). N ) T satisfy and ξ i For any ξ > 0, i = 1, ..., N, such that ξ T L = 0. Furthermore, there exists a matrix Γ = diag(ξ) such that... set up Let be any positive vector, satisfying in yes The second smallest eigenvalue.

[0017] As an improvement to the present invention, the dynamic model of the disturbed nonlinear multi-agent system established in step 2 is as follows:

[0018] Consider a first-order multi-agent system with N agents and simultaneously experiencing nonlinear and uncertain disturbances, referred to as model (1).

[0019]

[0020] in Let x = [x1, x2, ..., xn] be the state of the i-th agent. N ] T . This is the control input for the i-th agent. This indicates that Lipschitz is nonlinear. This is due to unknown external interference.

[0021] f(x) in model (1) i Let (t) and t) be Lipschitz continuous functions, and let there exist Lipschitz constants β > 0. Satisfies: |f(ω,t)-f(ω′,t)|≤β|ω-ω′|. External disturbance d i (x i (t), t) is bounded, that is, there exists a positive scalar. Make: |d i (x i (t), t)|≤ε, i=1, 2,...,N.

[0022] As an improvement to the present invention, step 3, determining the formation information and formation control target, is as follows:

[0023] First, we define a fixed-time array: Let h(t) = [h1(t), h2(t), ... h N (t)] T Describe the formation structure of the agent system in the reference coordinate system. For model (1), if for any initial values ​​x1(0), x2(0), ..., x N (0), there is a control protocol ui Let i = 1, 2, ..., N, and a bounded positive scalar T independent of initial values, such that: If true, then model (1) can achieve fixed-time formation; the vector h(t) defines the basic framework of the desired formation formed by N agents, assuming h i (t) and It is bounded, that is, it satisfies

[0024] As an improvement to the present invention, based on steps 1, 2, and 3, the formation error in step 4, which overcomes the asymmetry of the graph and the constraints of the system, is determined as follows:

[0025] Let x i (t)-h i (t)=z i Let (t), l∈1,2,...,N. Let the relative error of the i-th agent be represented, then the dynamic equation of the error can be obtained:

[0026]

[0027] Where e = [e1, e2, ..., e] N ] T z = [z1, z2, ..., z N ] T , F=[f(x1(t),t),f(x2(t),t),…,f(x N (t), t)] T , D=[d(x1(t),t),d(x2(t),t),…,d(x N (t), t)] T , remember That is

[0028] As an improvement to the present invention, based on steps 1, 2, 3, and 4, the fixed-time time-varying formation control protocol designed in step 5 based on formation error, formation information, and formation control objectives is as follows:

[0029] Design a distributed formation control protocol as shown below:

[0030]

[0031] Where μ > 1, and the control parameters k1 and k2 are both positive numbers. and This control protocol enables fixed-time time-varying formation control of disturbed multi-agent systems, giving the system advantages such as high precision, fast convergence, and good robustness.

[0032] As an improvement to the present invention, the proof of the convergence and effectiveness of the fixed-time time-varying formation control protocol in step 6 is as follows:

[0033] Let the Lyapunov function be:

[0034]

[0035] The first derivative of V with respect to time is:

[0036]

[0037] Pick Given e i [0] and e i [μ] They have the same symbol, when e i =0, there is When e i ≠0

[0038] For |(k1e [0] +k2e [μ] ) T ΓLF| is estimated as:

[0039]

[0040] in: Let θ be any positive real number, θ = [θ1, θ2, ..., θ3]. N ] T =(e [0] +e [μ] ).

[0041] Remember L T The eigenvalues ​​of L are λ1≤λ2≤...≤λ N Where λ2>0, in 1 N It is L T L corresponds to the eigenvector of λ1 = 0. Since μ > 1, |e i |≤|e i | 0 +|e i | μ That is, |e i | 2 ≤(ei [0] +e i [μ] ) 2 :

[0042]

[0043] right The estimate is:

[0044]

[0045] Where L i Let L represent the i-th row.

[0046] right have:

[0047]

[0048] so

[0049] Combining formulas (6)-(9), we have:

[0050]

[0051] make when When H > 0, let but Right now

[0052] Where ||θ|| 2 for:

[0053]

[0054] in when|e i When |≥1, we have when|e i When |≤1, we have And when p>1, And it is known We can obtain:

[0055]

[0056] Furthermore, if the formation is not achieved, then at least one e exists. i ≠0, such that |sign(e i )|=1, therefore,

[0057] on the other hand:

[0058]

[0059] Right now

[0060] Combining (11), (12), and (13) above, we can obtain:

[0061]

[0062] We can obtain:

[0063]

[0064] in According to the fixed-time stability theorem, the implementation of the fixed-time formation in model (1) can be guaranteed by the protocol (3). Its convergence time T max Depend on Provided.

[0065] Compared with the prior art, the advantages of the present invention are as follows:

[0066] 1. This invention proposes a novel distributed formation control algorithm for multi-agent formation systems that simultaneously exhibit nonlinearity and uncertain disturbances. This algorithm features high precision, fast convergence, and good robustness.

[0067] 2. The control protocol proposed in this invention designs a formation control algorithm on the basis of ensuring consistent tracking, allowing the intelligent agent system to move and operate according to a predetermined formation and strategy.

[0068] 3. This invention proposes a method for multi-agent formation within a fixed time frame under conditions of simultaneous nonlinearity and uncertain bounded disturbances. This method includes designing a distributed architecture for cooperative control and a formation control protocol. By adjusting the control gains in this protocol, the tracking error can converge to zero within a finite time, and the convergence time T is obtained based on proof of the controller's convergence. max This algorithm depends only on the magnitude of the control gain and is independent of the initial state of the system. It not only improves the efficiency of task execution but also ensures the stability, reliability, and autonomy of the formation control of the multi-agent system. It realizes the formation control objective based on consistency of this invention and meets the practical application of forming time-varying formation control of multi-agent systems under the simultaneous presence of nonlinearity and disturbance. It has extremely high practical value. Attached Figure Description

[0069] Figure 1 This is a flowchart of the control method in this invention;

[0070] Figure 2This is a communication topology diagram of the intelligent agent system in this invention;

[0071] Figure 3 This is a motion trajectory diagram of the intelligent agent of the present invention;

[0072] Figure 4 This is a diagram illustrating the formation tracking error trajectory of the present invention. Detailed Implementation

[0073] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0074] This invention proposes a method for multi-agent formation within a fixed time frame under the simultaneous presence of nonlinear and uncertain bounded disturbances. This method includes designing a distributed architecture for cooperative control and a formation control protocol. Within the framework of fixed-time stability theorems and Lyapunov stability theory, this invention designs a distributed formation control algorithm aimed at achieving consensus-based formation control. This allows each agent to achieve cooperative control of the entire system based on its own and its neighbors' state information, exchanging information through the protocol to maintain the desired formation structure.

[0075] The multi-agent formation control method of the present invention is not only theoretically innovative, but also has wide applicability and important practical value in practical applications, especially in scenarios where multiple constraints exist simultaneously and fast response and high stability are required.

[0076] The algorithm described above for solving the distributed fixed-time formation control problem of multi-agent systems with both nonlinearity and uncertain perturbations includes the following steps in its design process:

[0077] Step 1: Describe the communication topology diagram composed of N agents and related knowledge about graphs;

[0078] Step 2: Establish a dynamic model of the disturbed nonlinear multi-agent system;

[0079] Step 3: Determine formation information and formation control objectives;

[0080] Step 4: Overcome the asymmetry of the graph and the constraints of the system to determine the formation error;

[0081] Step 5: Based on the formation error, formation information, and formation control objectives, design a fixed-time time-varying formation control protocol;

[0082] Step 6: Prove the convergence and effectiveness of the fixed-time time-varying formation control protocol.

[0083] Furthermore, the communication topology diagram composed of N agents and related knowledge of the diagram in step 1 are described as follows:

[0084] Assumption It is a weighted directed graph, where V = {v1, v2, ..., v3}. N} represents a set of nodes. It is a directed edge set. Let be the adjacency matrix of , where a ij ≥0. The directed edges ε in G k Represents ordered node pairs (v i v j )∈ε, indicating that there exists a variable from v j to v i The directed information flow, i.e., v i Can receive v j Information, in which v i and v j These are referred to as child nodes and parent nodes, respectively. Weight And assume that there are no self-loops in the directed graph. There are distinct nodes in the directed graph. Make the node v i to node v j There exists a directed path whose directed sequence is: A directed spanning tree of a directed graph G contains all nodes of G and has at least one root node, where the root node satisfies the condition that there exists a unique directed path from that node to any other node in G. The Laplace matrix L of graph G is defined as: L = (l... ij ) N×N When i≠j, l ij =-a ij When i = j, and These are the in-neighbors and out-neighbors of agent i, respectively.

[0085] This invention assumes that the topological graph of the intelligent agents is a strongly connected directed graph. The Laplace matrix of graph G satisfies L1. N =0, and assume that L has an eigenvector ξ = (ξ1, ..., ξ2). N ) T satisfy and ξ i For any ξ > 0, i = 1, ..., N, such that ξ T L = 0. Furthermore, there exists a matrix Γ = diag(ξ) such that... set up Let be any positive vector, satisfying in yes The second smallest eigenvalue.

[0086] Furthermore, the dynamic model of the disturbed nonlinear multi-agent system established in step 2 is as follows:

[0087] Consider a first-order multi-agent system with N agents and simultaneously experiencing nonlinear and uncertain disturbances, referred to as model (1).

[0088]

[0089] in Let x = [x1, x2, ..., xn] be the state of the i-th agent. N ] T . This is the control input for the i-th agent. This indicates that Lipschitz is nonlinear. This is due to unknown external interference.

[0090] f(x) in model (1) i Let (t) and t) be Lipschitz continuous functions, and let there exist Lipschitz constants β > 0. Satisfies: |f(ω,t)-f(ω′,t)|≤β|ω-ω′|. External disturbance d i (x i (t), t) is bounded, that is, there exists a positive scalar. Make: |d i (x i (t), t)|≤ε, i=1, 2,...,N.

[0091] Furthermore, step 3, determining the formation information and formation control target, is as follows:

[0092] First, we define a fixed-time array: Let h(t) = [h1(t), h2(t), ... h N (t)] T Describe the formation structure of the agent system in the reference coordinate system. For model (1), if for any initial values ​​x1(0), x2(0), ..., x N (0), there is a control protocol u i Let i = 1, 2, ..., N, and a bounded positive scalar T independent of initial values, such that: If true, then model (1) can achieve fixed-time formation; the vector h(t) defines the basic framework of the desired formation formed by N agents, assuming h i (t) and It is bounded, that is, it satisfies

[0093] Furthermore, based on steps 1, 2, and 3, the formation error in step 4, which overcomes the asymmetry of the graph and the constraints of the system, is determined as follows:

[0094] Let x i (t)-h i (t)=z i Let (t), i∈1,2,...,N. Let the relative error of the i-th agent be represented, then the dynamic equation of the error can be obtained:

[0095]

[0096] Where e = [e1, e2, ..., e] N ] T z = [z1, z2, ..., z N ] T , F=[f(x1(t),t),f(x2(t),t),…,f(x N (t), t)] T , D=[d(x1(t),t),d(x2(t),t),…,d(x N (t), t)] T , remember That is

[0097] Furthermore, based on steps 1, 2, 3, and 4, the fixed-time time-varying formation control protocol designed in step 5 based on formation error, formation information, and formation control objectives is as follows:

[0098] Design a distributed formation control protocol as shown below:

[0099]

[0100] Where μ > 1, and the control parameters k1 and k2 are both positive numbers. Head This control protocol enables fixed-time time-varying formation control of disturbed multi-agent systems, giving the system advantages such as high precision, fast convergence, and good robustness.

[0101] Furthermore, the proof of the convergence and effectiveness of the fixed-time time-varying formation control protocol in step 6 is as follows:

[0102] Let the Lyapunov function be:

[0103]

[0104] The first derivative of V with respect to time is:

[0105]

[0106] Pick k =min{k1,k2}, Given e i [0] and e i [μ] They have the same symbol, when e i =0, there is When e i ≠0

[0107] Where for |(k1e [0] +k2e [μ] ) T ΓLF| is estimated as:

[0108]

[0109] in: Let θ be any positive real number, θ = [θ1, θ2, ..., θ3]. N ] T =(e [0] +e [μ] ).

[0110] Remember L T The eigenvalues ​​of L are λ1≤λ2≤...≤λ N Where λ2>0, in 1N is L T L corresponds to the eigenvector of λ1 = 0. Since μ > 1, |e i |≤|e i | 0 +|e i | μ That is, |e i | 2 ≤(e i [0] +e i [μ] ) 2 :

[0111]

[0112] right The estimate is:

[0113]

[0114] Where L i Let L represent the i-th row.

[0115] right have:

[0116]

[0117] so

[0118] Combining formulas (6)-(9), we have:

[0119]

[0120] make when When H > 0, let but Right now

[0121] Where ||θ|| 2 for:

[0122]

[0123] in when|e i When |≥1, we have when|e i When |≤1, we have And when p>1, And it is known We can obtain:

[0124]

[0125] Furthermore, if the formation is not achieved, then at least one e exists. i ≠0, such that |sign(e i )|=1, therefore,

[0126] on the other hand:

[0127]

[0128] Right now

[0129] Combining (11), (12), and (13) above, we can obtain:

[0130]

[0131] We can obtain:

[0132]

[0133] in According to the fixed-time stability theorem, the implementation of the fixed-time formation in model (1) can be guaranteed by the protocol (3) proposed in this invention. Its convergence time T max Depend on Provided.

[0134] Example 1

[0135] The technical solution of the present invention will be further described with reference to the accompanying drawings and specific numerical simulation implementations. The present invention utilizes the Matlab platform for simulation, providing a numerical example to demonstrate the performance of the proposed distributed formation control algorithm.

[0136] Consider a MASs with 6 agents, where The topology diagram of the intelligent agent system is as follows: Figure 2 As shown, its adjacency matrix is: A = [0 1 0 0 0 0; 0 0 0 1 0 0; 1 1 0 0 0 0; 0 0 0 1 0; 0 0 0 0 1; 0 01 0 0 0]. The parameters of the controller (3) are selected as: k1 = 2, k2 = 2, μ = 1.2.

[0137] Let a = 0.8, then the required dynamic grouping format is:

[0138]

[0139] The expected formation is a regular hexagon.

[0140] Under the controller (2), the motion trajectory of the multi-agent system is as follows: Figure 3 As shown. From Figure 3 It can be seen that the agents can still form a formation within a fixed time under complex constraints, and the formation is a regular hexagon that changes over time. The formation tracking error trajectory diagram is shown below. Figure 4 As shown. Figure 4 The formation tracking error can be quickly brought into agreement, meaning that the multi-agent formation is completed.

[0141] It should be noted that the above embodiments are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Equivalent substitutions or alternatives made based on the above technical solutions shall all fall within the scope of protection of the present invention.

Claims

1. A fixed-time time-varying formation control method for a disturbed nonlinear multi-agent system, characterized in that, The method includes the following steps: Step 1: Describe the communication topology diagram composed of N agents and related knowledge about graphs; Step 2: Establish a dynamic model of the disturbed nonlinear multi-agent system; The dynamic model of the disturbed nonlinear multi-agent system established in step 2 is as follows: Consider a first-order multi-agent system with N agents and simultaneously experiencing nonlinear and uncertain disturbances, referred to as model (1). ， in For the first The state of each agent, recorded , For the first Control input for an intelligent agent This indicates that Lipschitz is nonlinear. This is due to unknown external interference; In model (1) For a Lipschitz continuous function, there exists a Lipschitz constant. ,have ,satisfy: External disturbances It is bounded, meaning there exists a positive scalar. , so that: ; Step 3: Determine formation information and formation control objectives; Step 3, which determines the formation information and formation control objectives, is as follows: First, we define fixed-time grouping: Let... Describe the formation structure of the agent system in the reference coordinate system. For model (1), if for any initial value... There is a control protocol and a bounded positive scalar that does not depend on initial values , so that: If true, then model (1) can achieve fixed-time formation; vector The basic framework for the expected formation of N agents is defined, assuming... and It is bounded, that is, it satisfies ; Step 4: Overcome the asymmetry of the graph and the constraints of the system to determine the formation error; Based on steps 1, 2, and 3, the formation error determined in step 4 to overcome graph asymmetry and system constraints is as follows: make ,set up Indicates the first The relative formation error of each agent can be used to derive the dynamic equation of the formation error: ， in , , ,remember = That is, ; Step 5: Based on the formation error, formation information, and formation control objectives, design a fixed-time time-varying formation control protocol; Based on steps 1, 2, 3, and 4, the fixed-time time-varying formation control protocol designed in step 5, based on formation error, formation information, and formation control objectives, is as follows: Design the following distributed formation control protocol, referred to as protocol (3): ， in Among them, control parameters and All are positive numbers, among which , and ; Step 6: Prove the convergence and effectiveness of the fixed-time time-varying formation control protocol.

2. The fixed-time time-varying formation control method for a disturbed nonlinear multi-agent system according to claim 1, characterized in that, The communication topology diagram and related knowledge of the N agents in step 1 are described as follows: Assumption It is a weighted directed graph, in which For a set of nodes, It is a directed edge set. Let be the adjacency matrix, where , Directed edges in Represents ordered node pairs , indicating the existence of from arrive The directed flow of information, i.e. Can be received Information, including and They are referred to as child node and parent node, respectively, and their weights are... And assume that there are no self-loops in the directed graph, and that there are distinct nodes in the directed graph. This makes the node To the node There exists a directed path whose directed sequence is: Directed graph The directed spanning tree contains All nodes, and having at least one root node, and the root node satisfies the condition that from that node to... A graph is defined as a unique directed path that exists in every other node. The Laplace matrix L is: ,when hour, ; hour, , and respectively intelligent agents Incoming and outgoing neighbors; Assuming the topological graph of the agent is a strongly connected directed graph, the graph... The Laplace matrix satisfies And assume that among them There is an eigenvector satisfy and , making In addition, there exists a matrix , making ,set up Let be any positive vector, satisfying ,in yes The second smallest eigenvalue.

3. The fixed-time time-varying formation control method for a disturbed nonlinear multi-agent system according to claim 1, characterized in that, The proof of the convergence and effectiveness of the fixed-time time-varying formation control protocol in step 6 is as follows: Let the Lyapunov function be: ， The first derivative with respect to time is: ， Pick , Known and Having the same symbols, when ,have ;when have , Among them, for The estimate is: ， in: , , , , Let be any positive real number. , remember eigenvalues ,in , in , yes correspond eigenvectors, because ,so ,Right now : ， right The estimate is: ， in Represents the first of L OK; right have: ， so ; Combining formulas (6)-(9), we have: ， make ,when hour, ,make ,but ,Right now ; in for: ， in ,when Sometimes, ,when Sometimes, And when hour, And it is known We can obtain: ， Furthermore, if formation is not achieved, at least one , making ,therefore, ; on the other hand: ， Right now ; Combining (11), (12), and (13) above, we can obtain: ， We can obtain: ， in , According to the fixed-time stability theorem, the implementation of fixed-time formation in model (1) can be guaranteed by the protocol (3), and its convergence time is [not specified]. Depend on Provided.

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