Delay-tolerant formation control method and device for second-order multi-agent system

Abstract

The present invention discloses a time-delay-tolerant formation control method and device for a second-order multi-agent system, studies the formation control problem of a second-order multi-agent system with multiplicative noise and time delay, proposes a distributed control protocol under a time delay and multiplicative noise environment, and under the distributed control protocol, further transforms the formation problem of the second-order multi-agent system into a mean-square asymptotic stability problem. On this basis, a formation center function is given, the formation problem is further analyzed, and finally the feasibility conditions of the formation are given, and the appropriate control gain is solved. Finally, the required formation is realized through simulation, and the accuracy and feasibility of the theory are proved. The present invention realizes the formation control of a second-order multi-agent system under a time delay and multiplicative noise environment.

Description

Technical Field

[0001] The present invention relates to the field of intelligent agent collaborative control technology, and in particular to a delay-tolerant formation control method and device for a second-order multi-agent system. Background Art

[0002] A multi-agent system usually refers to a system in which individuals with the same or different perception and communication capabilities can generate complex cluster behaviors through simple information interactions in a complex network environment.

[0003] In recent years, with the interdisciplinary development of computer science, control science, biology, physical science, statistics, sociology and other disciplines and the widespread application of multi-agent systems in the fields of multi-robot swarm control, drone formation control, spacecraft attitude control, etc., distributed coordinated control of multi-agent systems has received widespread attention from more and more scholars.

[0004] Since communication between individual agents is required, multiplicative noise is a common interference factor. Multiplicative noise is noise caused by random changes in channel characteristics. It mainly manifests itself in radio communication transmission channels. This type of noise only appears when the signal appears in the above channel. It will not actively interfere with the signal, but can play a positive role in the stability of the system. Therefore, it is necessary to consider the series of uncertainties that multiplicative noise brings to the formation control of multi-agent systems. Due to the wide application of formation control in engineering practice, further research on it has broad prospects.

[0005] On the other hand, in practical applications, due to the selection and performance differences of communication equipment, there are generally communication delays and input delays in the communication network. In reality, time delays are inevitable. Based on the application prospects of a series of multi-agent systems such as quadcopters and swarm robots in civil or military tasks such as exploration, monitoring, and rescue, and considering some common problems in the transmission of communication information between agents, the formation control of second-order multi-agent systems with delays and multiplicative noise is studied. Summary of the invention

[0006] The present invention provides a time-delay-tolerant formation control method and device for a second-order multi-agent system, so as to solve the technical problem that the formation control of the existing second-order multi-agent system does not consider the time delay and multiplicative noise at the same time.

[0007] To solve the above problems, according to a first aspect of the present invention, a time-delay-tolerant formation control method of a second-order multi-agent system is provided, and the time-delay-tolerant formation control method comprises the following steps:

[0008] S1: Setting up a distributed control protocol for a second-order multi-agent system with time delay and multiplicative noise;

[0009] S2: Obtain the control input of a single agent in the distributed control protocol according to the information of neighbors and its own formation instructions, and given the desired formation instructions, transform the formation problem of the second-order multi-agent system into an asymptotic stability problem under mean square;

[0010] S3: Construct explicit expressions for the center function and feasibility conditions for the formation based on the asymptotic stability problem under mean square;

[0011] S4: Select appropriate control gains according to the explicit expression of the center function and the feasibility conditions of the formation to determine the formation center of the second-order multi-agent system and achieve the specified formation formation.

[0012] Furthermore, after step S4, the method further includes:

[0013] S5: The effectiveness of the formation control of the second-order multi-agent system is verified by simulation with MATLAB.

[0014] Furthermore, in step S1, the step of setting a distributed control protocol with time delay and multiplicative noise for the second-order multi-agent system specifically includes:

[0015] S11: Construct a second-order multi-agent system model containing N agents. The dynamic equation of each agent is described as:

[0016]

[0017] in, represents the position of the ith agent, i=1,2,...N, represents the speed of the ith agent, represents the n-dimensional space domain, u i (t) represents the control input of the ith agent; let: θ(t)=[x T (t),v T (t)] T ;

[0018] S12: Use vectors to describe the time-varying formation goal of the second-order multi-agent system:

[0019]

[0020] Among them, h x (t) = [h 1x (t),h 2x (t),…,h Nx (t)] T ,h v(t) = [h 1v (t),h 2v (t),…,h Nv (t)] T , represents the position formation target of the i-th agent; represents the speed formation target of the i-th agent;

[0021] S13: In order to realize the formation of the second-order multi-agent system, the following time-varying formation protocol with delay and multiplicative noise is set:

[0022]

[0023]

[0024] Among them, z 1ji (t) represents the position measurement information obtained by agent i from its neighboring agents j, z 2ji (t) represents the speed measurement information obtained by agent i from its neighboring agents j; τ1>0 and τ2>0 are the delays, ρ 1ji (t) and ρ 2ji (t) is a Gaussian white noise that is independent of the scalar, f 1ji (·) and f 2ji (·) is the noise intensity function;

[0025] S14: Considering the different certainties in the position and velocity communication of each agent, to achieve the formation of the second-order multi-agent system, the distributed protocol is set as follows:

[0026]

[0027] Where k1 = [k 11 k 12 ],k2=[k 21 k 22 ],k 11 , k 12 , k 21 , k 22 is the control gain, both are constants, k 11 and k 12 Used to set the movement mode of the formation center, k 21 , k 22 The setting is used to achieve the completion of the overall formation, j∈N i denotes j as the neighbor of the i-th agent, a ij An adjacency matrix representing the influence strength between multiple agents.

[0028] Furthermore, in step S12, if the time-varying formation target h(t)=0, the formation control problem is transformed into a consistency control problem.

[0029] Furthermore, under the distributed protocol, the dynamic equation of each agent is described as:

[0030]

[0031] in,

[0032] represents the Klaunec product, η N,i represents an N-dimensional column vector, where the i-th item is 1 and the other items are 0, ω ji (s) represents independent d-dimensional Brownian motions in the complete probability space.

[0033] Furthermore, in step S2, the expression of the asymptotic stability problem under mean square is as follows:

[0034]

[0035]

[0036] in,

[0037] σ 1ij and σ 2ij represents a non-negative constant, Q i,j =[q kl ] N×N is an N×N matrix, and q ij =a ij ,q ii =-a ij , the other elements are all 0, I N-1 represents the N-1 dimensional identity matrix, I n represents the n-dimensional identity matrix, Ψ=[ψ1,ψ]∈R N×N is an orthogonal matrix, where 1 N represents the identity column matrix, Ψ -1 =Ψ T =(ψ1,ψ) T and where Λ=diag{λ2,λ3,...,λ N} and ψ i ∈R N×1is the eigenvector corresponding to the eigenvalue λi, i=1,2,3....N.

[0038] Furthermore, in step S3, the explicit expression of the central function satisfies:

[0039]

[0040] in:

[0041]

[0042]

[0043]

[0044]

[0045] Furthermore, the feasibility conditions of the formation specifically include:

[0046] A: k 21 λ2-k 11 >0 and 1+2(k 21 λ2-k 11 )τ1 2 +(k 12 -k 22 λ2)τ1>0;

[0047]

[0048] in, When conditions A and B are met, the second-order multi-agent system realizes formation control in a time-delay and noise environment.

[0049] According to another aspect of the present invention, the present invention provides a time-delay-tolerant formation control device for a second-order multi-agent system, the time-delay-tolerant formation control device comprising the following modules:

[0050] The protocol setting module is used to set a distributed control protocol with time delay and multiplicative noise for the second-order multi-agent system;

[0051] A problem conversion module is used to obtain the control input of a single agent in the distributed control protocol according to the information of neighbors and its own formation instructions, and to convert the formation problem of the second-order multi-agent system into an asymptotic stability problem under mean square given the expected formation instructions;

[0052] The conditional construction module is used to construct explicit expressions for the central function and the feasibility conditions of the formation based on the asymptotic stability problem under mean square;

[0053] The formation control module is used to select appropriate control gains according to the explicit expression of the center function and the feasibility conditions of the formation to determine the formation center of the second-order multi-agent system and realize the specified formation formation.

[0054] Furthermore, the time-delay-tolerant formation control device further comprises:

[0055] The simulation verification module is used to verify the effectiveness of the formation control of the second-order multi-agent system through simulation using MATLAB.

[0056] Compared with the prior art, the present invention has significant advantages and beneficial effects, which are specifically embodied in the following aspects:

[0057] The present invention studies the formation control problem of a second-order multi-agent system with multiplicative noise and time delay, and proposes a distributed control protocol in a time delay and multiplicative noise environment. Under the distributed control protocol, the formation problem of the second-order multi-agent system is further transformed into a mean square asymptotic stability problem. On this basis, a formation center function is given, and the formation problem is further analyzed. Finally, the feasibility conditions of the formation are given, and the appropriate control gain is solved. Finally, the required formation shape is achieved through simulation and the required formation is completed, proving the accuracy and feasibility of the theory. The present invention realizes the formation control of a second-order multi-agent system in a time delay and multiplicative noise environment. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 It is a flow chart of a time-delay-tolerant formation control method for a second-order multi-agent system in an embodiment of the present invention;

[0059] Figure 2 The UAV in the XY plane in the embodiment of the present invention and Position trajectory curve;

[0060] Figure 3 The UAV in the XY plane in the embodiment of the present invention and The velocity trajectory curve;

[0061] Figure 4 is a position error diagram under mean square in an embodiment of the present invention;

[0062] Figure 5 is a velocity error diagram under mean square in an embodiment of the present invention;

[0063] Figure 6 It is a formation position diagram of UAVs at different times in an embodiment of the present invention;

[0064] Figure 7It is a structural diagram of a time-delay-tolerant formation control device for a second-order multi-agent system in an embodiment of the present invention. DETAILED DESCRIPTION

[0065] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, specific embodiments of the present invention are described in detail below with reference to the accompanying drawings.

[0066] At present, the formation control problem of multi-agent systems has only been considered in a time-delay environment. In practical application problems, multiplicative noise is also a very common interference factor. How to maintain formation control in a time-delay and multiplicative noise environment is more worthy of consideration. Due to the lack of a suitable analysis method, the formation control problem of a second-order multi-agent system with time delay and multiplicative noise has not been considered. In order to solve the above technical problems, the present invention provides a time-delay-tolerant formation control method and device for a second-order multi-agent system, and gives the following two specific embodiments for illustration.

[0067] Embodiment 1:

[0068] See also Figure 1 , Figure 1 The flowchart of the time-delay-tolerant formation control method of the second-order multi-agent system in the embodiment of the present invention is as follows; the embodiment of the present invention provides a time-delay-tolerant formation control method of the second-order multi-agent system, and the formation control method comprises the following steps:

[0069] Step S1: Setting a distributed control protocol with time delay and multiplicative noise for the second-order multi-agent system;

[0070] Specifically, considering the influence of time delay and multiplicative noise on the multi-agent system during the formation process, a distributed control protocol with time delay and multiplicative noise is set up based on the formation control of the second-order multi-agent system.

[0071] Consider a communication topology graph (O, E, W) consisting of N quad-rotor drones, where O is a set of vertices, each node i in the graph corresponds to a quad-rotor drone, E is a set of all edges, and W is an adjacency matrix used to represent the influence strength between quad-rotor drones. If an undirected graph G has at least one node and the node has an undirected path to all other nodes, the graph is called connected. In the present invention, it is assumed that the graph G is undirected and connected. Then, it is obvious that the Laplace matrix L is symmetric. Therefore, L contains a zero eigenvalue and N-1 positive real eigenvalues.

[0072] The specific aspects of setting a distributed control protocol with time delay and multiplicative noise include:

[0073] Step S11: Construct a second-order multi-agent system model containing N agents. The dynamic equation of each agent is described as:

[0074]

[0075] in, represents the position of the ith agent, represents the speed of the ith agent, u i (t) represents the control input that needs to be designed. θ(t)=[x T (t),v T (t)] T .

[0076] Step S12: Use vectors to describe the time-varying formation goal of the second-order multi-agent system:

[0077]

[0078] in, h x (t) = [h 1x (t),h 2x (t),L,h Nx (t)] T ,h v (t) = [h 1v (t),h 2v (t),L,h Nv (t)] T L

[0079] Typically, formation control of second-order multi-agent systems aims to achieve and maintain the desired geometry or pattern by designing appropriate formation control laws.

[0080] Step S13: In order to realize the formation of the second-order multi-agent system, the following time-varying formation protocol with delay and multiplicative noise is considered;

[0081] Taking into account the measurement noise in communication, z 1ji (t) represents the position measurement information obtained by agent i from its neighboring agents j, z 2ji (t) represents the speed measurement information obtained by agent i from its neighboring agents j, which are respectively expressed as:

[0082]

[0083]

[0084] Where: τ1>0 and τ2>0 are time delays, ρ1ji (t) and ρ 2ji (t) is a Gaussian white noise that is independent of the scalar, f 1ji (·) and f 2ji (·) is the noise intensity function.

[0085] S14: For this case, consider the uncertainty of each position and velocity in the communication. In order to realize the formation of the second-order multi-agent system, the distributed protocol is given by the following form:

[0086]

[0087] Where: k1 = [k 11 k 12 ],k2=[k 21 k 22 ],k 11 , k 12 , k 21 , k 22 are all constants, k 11 and k 12 Used to design the movement of the center of the formation, k 21 , k 22 It can be designed to form an overall formation.

[0088] If h(t)=0, the formation control problem in this embodiment can be transformed into a consistency problem.

[0089] The following formula can be further obtained in the distributed protocol:

[0090]

[0091] in,

[0092] Step S2: Obtain the control input of a single agent based on the information of its neighbors and its own formation instructions. Given the desired formation instructions, the formation problem of the second-order multi-agent system is transformed into an asymptotic stability problem under mean square.

[0093] In order to obtain the relationship between the control gains, the feasibility of delayed-admissible stable feedback formation control is analyzed, and thus the formation control problem is essentially transformed into the mean square asymptotic stability problem of the stochastic system. Therefore, a time-delay-related stochastic stability criterion is established to reveal the influence of time delay and multiplicative noise in system control on formation control.

[0094] Lemma 1: If and only if When , for any given bounded initial condition, the multi-agent system can achieve a formation h(t) in the mean square sense under the formation protocol with time delay and multiplicative noise.

[0095] It should be noted that the graph G is assumed to be undirected and connected. The symmetric Laplacian matrix of G has a zero eigenvalue, denoted by λ1, and the other eigenvalues ​​are 0<λ2≤λ3≤…≤λ N In addition, there exists a Ψ=[ψ1,ψ]∈R N×N An orthogonal matrix, where Ψ -1 =Ψ T =(ψ1,ψ) T and where Λ=diag{λ2,λ3,...,λ N} and ψ i ∈R N×1 is the eigenvalue corresponding to i eigenvector of , i=1,2,3....N.

[0096] because is an N×N matrix, and q ij =a ij ,q ii =-a ij , the other elements are all 0, and we can further get:

[0097]

[0098] in

[0099] let and The following formula can be obtained:

[0100]

[0101] in

[0102] Let ζ(t)=[ζ x (t) T , v (t) T ] T , where ε(t) = ζ1(t), Will -1 =Ψ T =(ψ1,ψ) T Substituting into the above formula, we can get:

[0103]

[0104]

[0105] in

[0106] Step S3: constructing the explicit expression of the center function and the feasibility condition of the formation according to the asymptotic stability problem under mean square;

[0107] Step S31: Provide a formation control center function.

[0108] The required formation vector and external command information are determined, an expression is formed, and a specific function expression of the formation control center function of the second-order multi-agent system is given.

[0109] Theorem 1: If a second-order multi-agent system achieves formation in the mean square sense under a formation protocol with time delay and multiplicative noise, then the formation center function satisfies

[0110]

[0111] in:

[0112]

[0113]

[0114] Proof: By Lemma 1, we can get:

[0115]

[0116] in

[0117] because and The following formula is easy to obtain:

[0118]

[0119] in Further we can get:

[0120]

[0121] in

[0122] This completes the proof. In fact, the stability of the formation center depends on the matrix L 11 If max(Re(λ(L 11 )))≥0, the formation center is unstable, otherwise the formation center is stable.

[0123] By using the above formula, the conclusion in Theorem 1 can be easily proved.

[0124] In Theorem 1, μ0(t) can be regarded as a consistency function determined by the initial state and the dynamic model, μ h (t) describes the impact of h(t) on the formation center. It can be seen that the delay and multiplicative noise have no effect on the formation control center. And in Theorem 1, k 11 , k 12 The motion pattern used to determine the formation center function.

[0125] In addition, the stability of the formation center can be determined by L 11 is determined by the characteristic value of .

[0126] If max(Re(λ(L 11 )))≥0, the formation center is unstable.

[0127] Step S32: providing feasibility conditions;

[0128] Lemma 2: Assume is a Hurwitz matrix and If there exists a matrix P>0 such that

[0129]

[0130] The random system is said to be mean-square asymptotically stable.

[0131] In order to further reveal the influence of time delay and multiplicative noise in the system on formation control, the following stochastic differential equation is considered:

[0132]

[0133] Theorem 2: If the control gains satisfy the following relationship:

[0134] A: k 21 λ2-k 11 >0 and 1+2(k 21 λ2-k 11 )τ1 2 +(k 12 -k 22 λ2)τ1>0,

[0135]

[0136] in,

[0137] It is said that the second-order multi-agent system can realize formation control in a time-delay and multiplicative noise environment.

[0138] The following is a mathematical proof of this: By proposing a more intuitive P matrix,

[0139]

[0140]

[0141] Since Φ1(t)=<M7(t),PM7(t)> (t), then

[0142]

[0143] The same can be done with M8, and the following can be obtained:

[0144]

[0145] because By further calculation,

[0146]

[0147] in

[0148]

[0149] where s 21 =2(k 11 I N-1 -k 21 Λ) 2 τ1,

[0150]

[0151]

[0152] in

[0153] s 32 =(2k 11 k 12 I N-1 -2k 11 k 22 Λ-2k 12 k 21 Λ+3k 21 k 22 Λ 2 )τ1,

[0154]

[0155] because

[0156]

[0157] We can get:

[0158]

[0159] By further calculating and sorting out the conditions, we can get the final result as shown in Theorem 2.

[0160] Step S5: Verify the effectiveness of the formation control of the second-order multi-agent system through simulation using MATLAB.

[0161] In an embodiment of the present invention, formation control of a second-order multi-agent system with time delay and multiplicative noise is investigated, and numerical simulations are used to demonstrate the effectiveness of the proposed protocol.

[0162] In this embodiment, by considering the formation control of 4 UAVs with multiplicative noise and time delay, all UAVs can move in the XY direction.

[0163] In this case, let n = 2 and

[0164]

[0165]

[0166] Where: i∈{1,2,3,4},

[0167] See also Figure 2 , Figure 2 The UAV in the XY plane in the embodiment of the present invention and The position trajectory curve of the 4 drones is as follows:

[0168]

[0169] See also Figure 3 , Figure 3 The UAV in the XY plane in the embodiment of the present invention and The velocity trajectory curve.

[0170] In this embodiment, the communication topology is G;

[0171] Where O=(1,2,3,4), E={(1,2),(1,4),(2,3),(3,4),(4,3),(3,2),(4,1),(2,1)}, W=(w ij )∈R 4×4 , a 12 =a 14 =a 23 =a 34 =a 43 =a 32 =a 41 =a 21=1 and all other elements are zero.

[0172] Then the eigenvalues ​​of L are as follows:

[0173] λ1=0, λ2=2, λ3=2, λ4=4,

[0174] The initial position of the drone can be defined as:

[0175] x1(0)=[3.0,-0.5,0.7,1.5] T ,

[0176] x2(0)=[1.0,-1.5,2.9,0.5] T ,

[0177] x3(0)=[-2.0,-1.2,2.1,-1.0] T ,

[0178] x4(0)=[-2.9,0.5,-0.6,-1.5] T ,

[0179] In this embodiment, the simulation time is set to 100s.

[0180] Considering the center of the formation, choose The noise intensity function is chosen as: f 1ji (x) = σ 1ji x = 0.2x and f 2ji (x) = σ 2ji x=0.1x. Then, by finding the feasibility conditions of the formation, we can determine

[0181] The four drones will form a square formation with a rotation radius of 5m and an angular velocity of 0.2rad / s.

[0182] See also Figure 4 and Figure 5 , we can get the simulation results of formation control, Figure 4 is a position error diagram under mean square in the embodiment of the present invention, where Figure 4 (a) shows the actual position of the formation in the X plane. and the ideal formation position The errors that exist between Figure 4 (b) shows the actual position of the formation in the Y plane and the ideal formation position There are errors between. Figure 5 is a velocity error diagram under mean square in the embodiment of the present invention, where Figure 5 (a) represents the actual speed of the formation in the X plane and the speed of the ideal formation The errors that exist between Figure 5 (b) represents the actual speed of the formation in the Y plane and the speed of the ideal formation There is an error between Figure 4 and Figure 5 As can be seen, the error is always close to zero.

[0183] See also Figure 6 , Figure 6 is a formation position diagram of UAVs at different times in an embodiment of the present invention, Figure 6 Snapshots of the trajectories of the four drones and the formation center c(t) at (a) t = 0; (b) t = 30s; (c) t = 60s; and (d) t = 100s are shown, showing the positions of each drone at different times and the shapes of the graphs formed.

[0184] The results show that the four drones can always form a formation on the XY plane, from an irregular shape at the initial moment to a regular square around the formation center at the later stage. It is obvious that the four drones can reach the expected formation.

[0185] Therefore, by studying the formation control problem of second-order multi-agent systems with multiplicative noise and time delay, a new distributed control protocol with multiplicative noise and time delay is proposed. Under the distributed protocol, the formation control problem can be transformed into a mean square asymptotic stability problem. By solving the formation control center function and proving the feasibility conditions of the formation, the UAV formation is studied in practical applications, and the simulation results show the reliability of the theoretical derivation.

[0186] Embodiment 2:

[0187] refer to Figure 7 This embodiment provides a time-delay-tolerant formation control device for a second-order multi-agent system, and the time-delay-tolerant formation control device includes the following modules:

[0188] A protocol setting module 1 is used to set a distributed control protocol with time delay and multiplicative noise for the second-order multi-agent system;

[0189] Problem conversion module 2 is used to obtain the control input of a single agent in the distributed control protocol according to the information of neighbors and its own formation command, and to convert the formation problem of the second-order multi-agent system into an asymptotic stability problem under mean square given the expected formation command;

[0190] Condition construction module 3, used to construct the explicit expression of the center function and the feasibility condition of the formation according to the asymptotic stability problem under the mean square;

[0191] The formation control module 4 is used to select a suitable control gain according to the explicit expression of the center function and the feasibility condition of the formation, so as to determine the formation center of the second-order multi-agent system and realize the specified formation formation.

[0192] The simulation verification module 5 is used to verify the effectiveness of the formation control of the second-order multi-agent system through simulation using MATLAB.

[0193] It should be noted that, in this article, the terms "include", "comprises" or any other variations thereof are intended to cover non-exclusive inclusion, so that a process, method, article or system including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or system. In the absence of further restrictions, an element defined by the sentence "comprises a ..." does not exclude the existence of other identical elements in the process, method, article or system including the element.

[0194] The serial numbers of the embodiments of the present invention are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments. In a unit claim that lists several means, several of these means may be embodied by the same hardware item. The use of the words first, second, and third, etc. does not indicate any order and these words may be interpreted as identifiers.

[0195] The above are only preferred embodiments of the present invention, and are not intended to limit the patent scope of the present invention. Any equivalent structure or equivalent process transformation made using the contents of the present invention specification and drawings, or directly or indirectly applied in other related technical fields, are also included in the patent protection scope of the present invention.

Claims

1. A time-delay-tolerant formation control method for a second-order multi-agent system, characterized in that: The time-delay tolerant formation control method comprises the following steps: S1: Setting up a distributed control protocol for a second-order multi-agent system with time delay and multiplicative noise; Specifically include: S11: Construct a second-order multi-agent system model containing N agents. The dynamic equation of each agent is described as: in, represents the position of the ith agent, i=1,2,...N, represents the speed of the ith agent, represents the n-dimensional space domain, u i (t) represents the control input of the ith agent; let: S12: Use vectors to describe the time-varying formation goal of the second-order multi-agent system: Among them, h x (t) = [h 1x (t),h 2x (t),...,h Nx (t)] T ,h v (t) = [h 1v (t),h 2v (t),...,h Nv (t)] T , represents the position formation target of the i-th agent; represents the speed formation target of the i-th agent; S13: In order to realize the formation of the second-order multi-agent system, the following time-varying formation protocol with delay and multiplicative noise is set: Among them, z 1ji (t) represents the position measurement information obtained by agent i from its neighboring agents j, z 2ji (t) represents the speed measurement information obtained by agent i from its neighboring agents j; τ1>0 and τ2>0 are the delays, ρ 1ji (t) and ρ 2ji (t) is a Gaussian white noise that is independent of the scalar, f 1ji (·) and f 2ji (·) is the noise intensity function; S14: Considering the different certainties in the position and velocity communication of each agent, to achieve the formation of the second-order multi-agent system, the distributed protocol is set as follows: Where k1 = [k 11 k 12 ],k2=[k 21 k 22 ],k 11 , k 12 , k 21 , k 22 is the control gain, both are constants, k 11 and k 12 Used to set the movement mode of the formation center, k 21 , k 22 The setting is used to achieve the completion of the overall formation, j∈N i denotes j as the neighbor of the i-th agent, a ij The adjacency matrix representing the influence strength between multiple agents; S2: Obtain the control input of a single agent in the distributed control protocol based on the information of neighbors and its own formation instructions, and given the desired formation instructions, transform the formation problem of the second-order multi-agent system into an asymptotic stability problem under mean square; The expression of the asymptotic stability problem under the mean square is as follows: in, σ 1ij and σ 2ij represents a non-negative constant, Q i,j =[q kl ] N×N is an N×N matrix, and q ij =a ij ,q ii =-a ij , the other elements are all 0, I N-1 represents the N-1 dimensional identity matrix, I n represents the n-dimensional identity matrix, Ψ=[ψ1,ψ]∈R N×N is an orthogonal matrix, where 1 N represents the identity column matrix, Ψ -1 =Ψ T =(ψ1,ψ) T and where Λ=diag{λ2,λ3,...,λ N } and ψ i ∈R N×1 is the eigenvalue corresponding to i eigenvector of , i=1,2,3....N; S3: Construct explicit expressions for the center function and feasibility conditions for the formation based on the asymptotic stability problem under mean square; The explicit expression of the center function satisfies: in: S4: Select appropriate control gains according to the explicit expression of the center function and the feasibility conditions of the formation to determine the formation center of the second-order multi-agent system and achieve the specified formation formation.

2. The delay-tolerant formation control method for a second-order multi-agent system according to claim 1, characterized in that: After step S4, the method further includes: S5: The effectiveness of the formation control of the second-order multi-agent system is verified by simulation with MATLAB.

3. The delay-tolerant formation control method for a second-order multi-agent system according to claim 1, characterized in that: In step S12, if the time-varying formation target h(t)=0, the formation control problem is transformed into a consistency control problem.

4. The delay-tolerant formation control method for a second-order multi-agent system according to claim 1, characterized in that: Under the distributed protocol, the dynamic equation of each agent is described as: in, represents the Klaunec product, η N,i represents an N-dimensional column vector, where the i-th item is 1 and the other items are 0, ω ji (s) represents independent d-dimensional Brownian motions in the complete probability space.

5. The delay-tolerant formation control method for a second-order multi-agent system according to claim 1, characterized in that: The feasibility conditions of the formation specifically include: A： B： in, When conditions A and B are met, the second-order multi-agent system realizes formation control in a time-delay and noise environment.

6. A time-delay-tolerant formation control device for a second-order multi-agent system, characterized in that: The time-delay-tolerant formation control device comprises the following modules: The protocol setting module is used to set a distributed control protocol with time delay and multiplicative noise for the second-order multi-agent system; Specifically include: S11: Construct a second-order multi-agent system model containing N agents. The dynamic equation of each agent is described as: in, represents the position of the ith agent, i=1,2,...N, represents the speed of the ith agent, represents the n-dimensional space domain, u i (t) represents the control input of the ith agent; let: S12: Use vectors to describe the time-varying formation goal of the second-order multi-agent system: Among them, h x (t) = [h 1x (t),h 2x (t),…,h Nx (t)] T ,h v (t) = [h 1v (t),h 2v (t),…,h Nv (t)] T , represents the position formation target of the i-th agent; represents the speed formation target of the i-th agent; S13: In order to realize the formation of the second-order multi-agent system, the following time-varying formation protocol with delay and multiplicative noise is set: Among them, z 1ji (t) represents the position measurement information obtained by agent i from its neighboring agents j, z 2ji (t) represents the speed measurement information obtained by agent i from its neighboring agents j; τ1>0 and τ2>0 are the delays, ρ 1ji (t) and ρ 2ji (t) is a Gaussian white noise that is independent of the scalar, f 1ji (·) and f 2ji (·) is the noise intensity function; S14: Considering the different certainties in the position and velocity communication of each agent, to achieve the formation of the second-order multi-agent system, the distributed protocol is set as follows: Where k1 = [k 11 k 12 ],k2=[k 21 k 22 ],k 11 , k 12 , k 21 , k 22 is the control gain, both are constants, k 11 and k 12 Used to set the movement mode of the formation center, k 21 , k 22 The setting is used to achieve the completion of the overall formation, j∈N i denotes j as the neighbor of the i-th agent, a ij The adjacency matrix representing the influence strength between multiple agents; A problem conversion module is used to obtain the control input of a single agent in the distributed control protocol according to the information of neighbors and its own formation instructions, and to convert the formation problem of the second-order multi-agent system into an asymptotic stability problem under mean square given the expected formation instructions; The expression of the asymptotic stability problem under the mean square is as follows: in, σ 1ij and σ 2ij represents a non-negative constant, Q i,j =[q kl ] N×N is an N×N matrix, and q ij =a ij ,q ii =-a ij , the other elements are all 0, I N-1 represents the N-1 dimensional identity matrix, I n represents the n-dimensional identity matrix, Ψ=[ψ1,ψ]∈R N×N is an orthogonal matrix, where 1 N represents the identity column matrix, Ψ -1 =Ψ T =(ψ1,ψ) T and where Λ=diag{λ2,λ3,...,λ N } and ψ i ∈R N×1 is the eigenvalue corresponding to i eigenvector of , i=1,2,3....N; The conditional construction module is used to construct explicit expressions for the central function and the feasibility conditions of the formation based on the asymptotic stability problem under mean square; The explicit expression of the center function satisfies: in: The formation control module is used to select appropriate control gains according to the explicit expression of the center function and the feasibility conditions of the formation to determine the formation center of the second-order multi-agent system and realize the specified formation formation.

7. The delay-tolerant formation control device for a second-order multi-agent system according to claim 6, characterized in that: The time-delay-tolerant formation control device further comprises: The simulation verification module is used to verify the effectiveness of the formation control of the second-order multi-agent system through simulation using MATLAB.

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