Method and device for tracking control of uncertain multi-agent system under time delay and noise

Abstract

The application discloses a kind of uncertain multi-agent system tracking control methods with time delay and multiplicative noise, comprising the following steps: under time delay and multiplicative noise environment, the multi-agent system containing uncertain parameters is set distributed control protocol;According to the information of leader, the state of each agent itself and the state information received from neighbor, combine control protocol, the tracking control problem of system under time delay and multiplicative noise environment is converted into the boundedness problem of random time delay differential equation solution;The explicit expression of the upper bound of the tracking of the multi-agent system is constructed and the feasibility condition of bounded tracking;Select the control gain of suitable control protocol, realize the bounded tracking of the follower of multi-agent system to leader.The method of the application can consider the internal and external two parts of the system uncertainty, and provides a research scheme for distributed tracking control of uncertain multi-agent system under uncertain environment.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of multi-agent, and particularly relates to a tracking control method and device for uncertain multi-agent system under time delay and noise. BACKGROUND

[0002] Consensus control of multi-agent systems has been developed for nearly two decades, and many scholars have made extensive research from different angles and achieved fruitful results. Existing consensus algorithms can be roughly divided into two categories, namely leaderless consensus and leadered consensus. The latter is also known as distributed tracking. With the application of various algorithms, it is found that there are many uncertainties in practice, such as noise in the environment where the system is located, error uncertainty in system modeling, and uncertain disturbance of the environment to the system.

[0003] In actual systems, various uncertainties are inevitable and must be solved in the transition from theory to practical application. Patent CN202211217407.0 discloses a consensus tracking control method and system for multi-agent with heterogeneous dynamic uncertainty under directed communication, and the model difference of the agent is described by dynamic uncertainty characteristics. A decentralized state observer is designed based on output information, and a distributed consensus tracking control protocol is constructed using the relative output information between adjacent multi-agents, so that the output of the follower can gradually track the output trajectory of the leader. Patent CN202111528073.4 discloses a method for solving multi-agent consensus based on intermittent random noise. The noise considered is also Gaussian white noise, and the ability of the noise to positively affect system stability is reasonably utilized to design a distributed consensus control protocol based on noise, so that the error between each follower and the leader becomes smaller and smaller, and the consensus definition is established, thereby solving the problem of multi-agent system consensus.

[0004] The uncertainty of the multi-agent system includes internal and external aspects, such as internal dynamic model uncertainty, external measurement noise, and time delay. The deficiency is that the current research on the existence of uncertainty in the multi-agent system often only focuses on one aspect of uncertainty, and the designed controller can only solve one problem of uncertainty. To further promote the combination of multi-agent control theory and practical engineering application scenarios, we need to consider the internal and external uncertainties of the system. Therefore, it is necessary to study the distributed tracking control of uncertain multi-agent systems in uncertain environments. SUMMARY

[0005] In order to solve the technical problem that the existing tracking control of uncertain multi-agent system does not consider the internal and external uncertain factors at the same time, the present application provides a tracking control method for uncertain multi-agent system under time delay and noise, comprising the following steps:

[0006] S1, setting a distributed control protocol for a multi-agent system containing uncertain parameters under time delay and multiplicative noise environment;

[0007] S2, according to the information of the leader of the multi-agent system, the state of each agent itself and the state information received from the neighbors, combining the control protocol, the tracking control problem of the uncertain multi-agent system under time delay and multiplicative noise environment is converted into the boundedness problem of the solution of the random time delay differential equation;

[0008] S3, constructing the explicit expression of the tracking upper bound of the multi-agent system and the feasibility condition of the bounded tracking according to the boundedness problem of the solution of the random time delay differential equation;

[0009] S4, selecting a suitable control gain for the explicit expression of the tracking upper bound and the feasibility condition of the bounded tracking to realize the bounded tracking of the followers of the multi-agent system to the leader.

[0010] The present application also provides a tracking control device for uncertain multi-agent system under time delay and noise, comprising:

[0011] A processor;

[0012] A memory having a computer program stored thereon and capable of running on the processor;

[0013] When the computer program is executed by the processor, the steps of the tracking control method for uncertain multi-agent system under time delay and noise are realized.

[0014] Compared with the prior art, the present application has the following advantages and beneficial effects:

[0015] The present application provides a distributed tracking control protocol with multiplicative noise and time delay, under the distributed protocol, the bounded tracking control problem can be converted into the boundedness analysis problem of the root of the time delay differential equation, and the feasibility condition of realizing the bounded tracking control and the tracking upper bound of the follower tracking the leader are obtained, the method of the present application can consider the internal and external uncertainties of the system, and provides a research scheme for the distributed tracking control of the uncertain multi-agent system under uncertain environment.

[0016] The prior art for external uncertain disturbance or uncertain parameter processing, on the basis of consistency controller, often needs to introduce adaptive controller, sliding mode controller or observer. The present application successfully realizes the tracking control problem under the influence of various uncertain factors by using the communication information between agents and the distributed tracking controller. Compared with the existing method, more kinds of uncertain factors are considered, the application range is more extensive, and the algorithm is more simple and easy to realize. BRIEF DESCRIPTION OF DRAWINGS

[0017] Figure 1 is a flow chart of the tracking control method of the uncertain multi-agent system under time delay and noise of the present application;

[0018] Figure 2 is a communication topology relationship diagram between unmanned vehicles in an embodiment of the present application;

[0019] Figure 3 is a linear two-degree-of-freedom model diagram of an unmanned vehicle in an embodiment of the present application;

[0020] Figure 4 is a motion trajectory diagram of an unmanned vehicle in an embodiment of the present application;

[0021] Figure 5 is an error diagram of the follower tracking the leader in an embodiment of the present application. DETAILED DESCRIPTION

[0022] In order to make the purpose, technical scheme and advantages of the present application clearer, the embodiments of the present application will be further described below with reference to the drawings.

[0023] The embodiment of the present application provides a tracking control method of an uncertain multi-agent system under time delay and noise, comprising the following steps:

[0024] S1, in a time delay and multiplicative noise environment, a distributed control protocol is set for a multi-agent system containing uncertain parameters.

[0025] S11, a multi-agent system model containing N+1 agents is constructed, wherein "0" represents a leader, and "1, 2,..., N" respectively represents N followers.

[0026] The dynamic equation of the follower is described as:

[0027]

[0028] wherein, and represent the position and control input of the i-th agent respectively, and and represent n-dimensional and m-dimensional Euclidean space respectively; is a known constant matrix; ΔA iFor uncertain parameter matrix, it satisfies: ΔA i = FΔ i C1, Δ i T Δ i ≤I n , I n is an n-order unit matrix, where F∈R n×r , C1∈R r×n ; here n, m, n×r, r×n are the dimensions of different parameters.

[0029] The dynamic equation of the leader is described as:

[0030]

[0031] where x is the position of the leader, is the control input of the leader.

[0032] S12, the communication relationship between the leader and the follower is uniformly represented by using the graph theory method:

[0033]

[0034] where L is the Laplacian matrix representing the communication relationship between the followers, when the agent i is connected with the agent j (i≠j), (a ij is an element in the adjacency matrix, a ij is not equal to 0, which indicates that there is information flow from j to i, otherwise, a ij=0); otherwise, the element on the diagonal g indicates the connection relationship between the follower and the leader, when the agent i is directly connected with the leader, g i ≠0; otherwise, g i =0.

[0035] S13, the goal of the bounded tracking control is that all the followers track the state of the leader within a certain range limit: lim t→∞ (x i -x0)≤C0, i=1,…,N, where x i represents the state of the i-th agent, x0 represents the state of the leader, represents the upper bound of the bounded tracking, and t represents the time of the continuous system;

[0036] S14, in order to realize the bounded tracking control of the uncertain multi-agent system in the environment with communication delay and multiplicative noise, the control coordination of the multi-agent system is set as:

[0037]

[0038] wherein denotes the measurement information, τ1,τ2≥0 are time delays; denotes that agent j is a neighbor of the i-th agent, denotes all neighbors of agent i; K is the feedback gain to be designed; f lji (·) is a noise intensity function, f lji (x) = σ lji x, σ lji ≥0 is the corresponding noise intensity, the measurement noise ξ ji (t) = (ξ 1ji (t),...,ξ dji (t)) T , is a d-dimensional Gaussian white noise, and satisfies wherein w ji (t) = (w 1ji (t),...,w dji (t)) T , w ji (t) is a set of mutually independent d-dimensional Brownian motions, i = 1,2,...,N,

[0039] S2, according to the information of the leader of the multi-agent system, the state of each agent itself and the state information received from the neighbors, combining the control protocol, the tracking control problem of the uncertain multi-agent system under time delay and multiplicative noise environment is converted into the boundedness problem of the solution of the stochastic time delay differential equation.

[0040] S21, the error of the i-th agent tracking the leader is defined as e i = x i -x0, combining (1), (2), (4), the derivative of e i = x i -x0 on both sides is obtained:

[0041]

[0042]

[0043] Let e(t) = [e1(t), e2(t),...,e N (t)] T , ΔA = [ΔA1, ΔA2,..., ΔA N ] T , the derivative of e(t) is obtained:

[0044]

[0045] where B d = [F - B], Z i,j is an N x N matrix, z ii = -a ij , z ij = a ij , and the rest of the elements are 0.

[0046] S22、 The eigenvalue decomposition of S is where Λ = diag{λ1, λ2, …, λ N} is a diagonal matrix composed of eigenvalues of S, and Ψ = [ψ1, ψ2, …, ψ N ] is a matrix composed of eigenvectors corresponding to the respective eigenvalues. Thus, we have

[0047] dζ(t) = A0ζ(t)dt + A1ζ(t - τ1)dt + dM2+ q(t)dt (5)

[0048] where, I r is an r-order identity matrix.

[0049] Because we have

[0050]

[0051] where E denotes the expected value, and ‖.‖ 2 is the square of the Euclidean norm.

[0052] Considering the random system in the mean square sense, the square of the Euclidean norm is used to ensure that it is greater than zero, and the expectation is used for evaluation.

[0053] S23、In the time delay and multiplicative noise environment, if and only if lim t→∞ E‖ζ(t)‖ 2 ≤C0, , that is, all errors can be guaranteed to converge to C0 eventually, that is, all followers can have bounded tracking of the leader. The multi-agent system constructed by formula (1) realizes bounded tracking control under the control protocol designed by formula (4), and the bounded tracking control of the multi-agent system with uncertain parameter disturbance in the time delay and noise environment is converted into the boundedness problem of the solution of the stochastic time delay differential equation (5).

[0054] ​​S3. Based on the boundedness problem of the solution of the stochastic time-delay differential equation, construct the explicit expression of the tracking upper bound of the multi-agent system and the feasibility conditions of bounded tracking.

[0055] S31. Selecting a Lyapunov functional:

[0056] V(ζ t )=V1(t)+V2(t) (7)

[0057] Where, ζ t ={ζ(t+θ), θ∈[-τ1,0]}, V1(t)=z T (t)Pz(t), z T z(t) denotes the transpose of z(t), and P denotes a positive definite matrix.

[0058] Differentiate both sides of the Lyapunov functional simultaneously: dV(ζ) t )=dV1(t)+dV2(t),

[0059] Where ζ t ={ζ(t+θ):θ∈[-τ1,0]}, V1(t)=z T (t)Pz(t)

[0060]

[0061] From the following formula:

[0062]

[0063]

[0064]

[0065] in

[0066]

[0067] From the inequality (For any μ>0) we get:

[0068]

[0069]

[0070]

[0071] Then we have:

[0072]

[0073] For V2(t), the derivative is given by the following formula:

[0074]

[0075]

[0076]

[0077] Combine dV1(t) + dV2(t),

[0078]

[0079] Through the above derivation and simplification, we finally arrive at:

[0080]

[0081] Where μ is any constant greater than zero, and in the simulation of this embodiment, μ is taken as 1.

[0082]

[0083]

[0084] Note: but:

[0085]

[0086] Substituting, we get:

[0087]

[0088] Through the above derivation and subsequent organization, we finally arrive at:

[0089] dV(ζ t )≤ζ T (t)S1ζ(t)dt+dM(t)+q T (t)Q1q(t)dt+ζ T (t-τ2)Dζ(s-τ2)dt(8)

[0090] in:

[0091] d is the dimension of the noise.

[0092] Integrating both sides of equation (8) and taking the expectation, we get:

[0093]

[0094] Among them, eγt Operators used to aid in calculations, facilitating the subsequent acquisition of exponentially stable results.

[0095] make When S3 < 0 Then there must exist a constant γ. * >0, for any γ<γ * Since S2(γ) < 0, we get:

[0096]

[0097]

[0098] Definite boundary C4=τ1 2 ||P|||A1|| 2 For any Right now get:

[0099]

[0100] Where E represents the expected value, Since Ψ is an orthogonal matrix, then ||Ψ -1 || = 1; B d =[FB],‖B d || is a constant; ||Δ i Since ||≤1, ||ε||=max[||C1x0||,||u0||]. Because both ||C1x0|| and ||u0|| are bounded, there must exist a constant C. q ,make:

[0101]

[0102]

[0103] Among them, let C5 is the upper bound for tracking.

[0104] S32. Let the unstable eigenvalue of A be... definition

[0105] S33. When max(Re(λ(A)))≥0, and (A,B) is controllable, τ1∈[0,τ * ), Where max(Re(λ(A)))≥0 represents the maximum real part of the eigenvalues ​​of A (some eigenvalues ​​of A are unstable), λ(A) represents all eigenvalues ​​of A. Controllability means that if the motion of all state variables of the system can be influenced and controlled by the input, so that it reaches the origin from any initial state, then the system is said to be controllable, or more precisely, state controllable. (A,B) controllable is equivalent to system controllability.

[0106] The multi-agent system constructed by formula (1) achieves bounded tracking control under the control protocol designed by formula (5).

[0107] in

[0108] S4. Select an appropriate control gain for the explicit expression of the upper bound of tracking and the feasibility conditions of bounded tracking, so as to realize the follower-bounded tracking of the leader in the multi-agent system.

[0109] S41. The value of the control system gain K needs to be such that S3 < 0. Setting S3 < 0, we get:

[0110]

[0111] Let K = k(I) m +B T PB) -1 B T P, we get:

[0112]

[0113] Since P>0 is A T P+PA-2αPB(I m +B T PB) -1 B T P+I n The only solution for =0 Right now Then the above formula:

[0114]

[0115] in, Let λ represent any constant greater than zero. i Let i represent the i-th eigenvalue.

[0116] To make 2α-2kλ i +(9τ1λ i +2σ 2 )λ i k 2 <0, let 2α-2λ i k+(9τ1λi +2σ 2 )λ i k 2 =0, thus obtaining the upper limit of k. and lower limit value k :

[0117]

[0118] From λ i 2 -2α(9τ1λ i +2σ 2 )λ i >0

[0119]

[0120] Therefore

[0121]

[0122] If τ1>0, then we have

[0123] The control gain K = k(I) of the control protocol designed by formula (5) is obtained. m +B T PB) -1 B T P, in Implement bounded tracking control of the multi-agent system constructed by formula (1).

[0124] This embodiment also proposes a tracking and control device for an uncertain multi-agent system under time delay and noise, the device comprising:

[0125] processor;

[0126] Memory, on which computer programs that can run on a processor are stored;

[0127] Among them, the steps of implementing a tracking control method for an uncertain multi-agent system under time delay and noise when the computer program is executed by the processor.

[0128] In an example of this invention, the effectiveness of the proposed protocol is demonstrated through numerical simulation for bounded tracking control of uncertain multi-agent systems with time delays and multiplicative noise.

[0129] This embodiment verifies the effectiveness of the bounded tracking control of the uncertain multi-agent system using MATLAB simulation.

[0130] The system consists of 5 autonomous vehicles, one of which is a leader. The communication topology between the autonomous vehicles is as follows: Figure 2As shown. Each autonomous vehicle can share information with some of its neighbors or leaders, and can move in the XY direction.

[0131] All autonomous vehicle dynamics models are considered as linear two-degree-of-freedom vehicle models, such as: Figure 3 As shown. Under the assumption that the front wheel steering angle and tire slip angle are relatively small, the dynamics of the autonomous vehicle model are described as follows:

[0132]

[0133] Where θ represents the heading angle relative to the x-axis, and its integral is s; v is the lateral velocity; u is the steady longitudinal velocity; δ is the front wheel steering angle; m is the mass; I zz denoted as z, where a and b are the moments of inertia about the z-axis; a and b are the distances from the center of mass to the front and rear axles; k1 and k2 are the lateral stiffness of the front and rear wheels, respectively. In this embodiment, k1 = -70000 ± 15% (N / rad) and k2 = -60000 ± 15% (N / rad).

[0134] Rewriting the above equation in matrix form, and considering the perturbation of uncertain parameters, we get:

[0135]

[0136] Where x i =[s i θ i v i r i ] T u i =δ i , ΔA i =FΔ i C1, Δ i =diag(σ i1 ,σ i2 ,...,σ i6 ), ||σ iq ||≤1, q=1,2,...,6. k1 and k2 are both negative values, further defined as follows:

[0137]

[0138]

[0139]

[0140] in,

[0141] In this simulation, multiplicative noise is simulated using random numbers within (0,1). All autonomous vehicles initially position themselves on the y-axis, with their specific positions randomly generated. The leader moves horizontally along the positive x-axis, as shown below. Figure 4 As shown, it can be concluded that all followers will catch up with the leader's state within a limited time.

[0142] Figure 5 The figure shows the positional error of each follower autonomous vehicle in tracking the leader. The results show that the four follower autonomous vehicles can always track the leader autonomous vehicle in the XY plane and quickly meet the maximum allowable tracking error within a finite time.

[0143] Therefore, by studying the bounded tracking control problem of uncertain multi-agent systems under time delay and multiplicative noise conditions, a novel distributed control protocol with multiplicative noise and time delay is proposed. Under this distributed protocol, the bounded tracking control problem can be transformed into a boundedness analysis problem of the roots of the time-delay differential equation. The feasibility conditions for achieving bounded tracking control and the upper bound of the follower tracking the leader are solved. The tracking control of an unmanned vehicle is studied in a practical application, and simulation results demonstrate the reliability of the theoretical derivation.

[0144] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A tracking control method for an uncertain multi-agent system under time delay and noise, characterized in that, Includes the following steps: S1. Set up a distributed control protocol for a multi-agent system with uncertain parameters under time delay and multiplicative noise conditions; S2. Based on the leader information of the multi-agent system, the state of each agent and the state information received from its neighbors, and in conjunction with the control protocol, the tracking control problem of an uncertain multi-agent system under time delay and multiplicative noise environment is transformed into a boundedness problem of the solution of stochastic time-delay differential equations. S3. Based on the boundedness problem of the solution of the stochastic time-delay differential equation, construct the explicit expression of the tracking upper bound of the multi-agent system and the feasibility conditions of bounded tracking. S4. Select an appropriate control gain for the explicit expression of the upper bound of tracking and the feasibility conditions of bounded tracking, so as to realize the follower-bounded tracking of the leader in the multi-agent system. Step S4 is as follows: S41, Order ,get: make , Representing the m-order identity matrix, we get: make To obtain the upper limit of k and lower limit value : get Protocol control gain ,exist Implementation of the construction Bounded tracking control.

2. The tracking and control method for an uncertain multi-agent system under time delay and noise as described in claim 1, characterized in that, Step S1 is as follows: S11. Construct a multi-agent system model with N+1 agents, where "0" represents the leader. "" represents N followers respectively; S12. Use graph theory to uniformly represent the communication relationship between leaders and followers: S13. The objective of bounded tracking control is set as follows: ,in, Indicates the first i The location of each agent. Indicates the position of the leader. This represents the upper bound of bounded tracking. Represents positive real numbers. t Represents time; S14. Based on the multi-agent system model, communication relationships, and bounded tracking control target obtained in the above steps, set the control protocol for the multi-agent system.

3. The tracking and control method for an uncertain multi-agent system under time delay and noise as described in claim 2, characterized in that, Step S11 is as follows: The dynamic equation of the follower is: （1） in, They represent the first i The location and control input of each agent and They represent n peacekeeping m Vioclimatic space; It is a known constant matrix; Let be an uncertain parameter matrix, satisfying: , I n for n An identity matrix of order 1, wherein , ; The dynamic equation of the leader is: （2） in It is the position of leader. It is the leader's control input.

4. The tracking and control method for an uncertain multi-agent system under time delay and noise as described in claim 3, characterized in that, Step S12 is as follows: Using graph theory to uniformly represent the communication relationship between leaders and followers: （3） in It is a Laplace matrix representing the communication relationship between followers, when the agent... With intelligent agents ( ) connected, ; otherwise, ; elements on the diagonal This represents the connection between followers and leaders, when the agent... Directly connected to leaders ; otherwise, .

5. The tracking control method for an uncertain multi-agent system under time delay and noise according to claim 4, characterized in that, In step S14, the control protocol of the multi-agent system is set as follows: （4） in Indicates measurement information, Indicates time delay; Represents intelligent agents It is the first The neighbor of an intelligent agent, Represents intelligent agents i All neighbors; The feedback gain to be designed; It is a function of noise intensity. d It is the dimension of noise. This corresponds to the noise intensity, and the noise is measured. yes d Gaussian white noise, and satisfying , .

6. The tracking and control method for an uncertain multi-agent system under time delay and noise according to claim 5, characterized in that, Step S2 is as follows: S21, Definition of the i The error of an agent tracking a leader is Combining equations (1), (2), and (4), for Differentiate both sides simultaneously, let , ,right Differentiation yields: in , , yes 3D matrix , All other elements are 0; S22, eigenvalue decomposition into ,in For matrix Diagonal matrix composed of eigenvalues For matrix Let the eigenvectors corresponding to the respective eigenvalues ​​be... ,get: （5） in, , , for r An identity matrix of order 1; （6） in, This indicates taking the expected value. The square of the Euclidean norm; S23, when When the multi-agent system constructed by formula (1) achieves bounded tracking control under the control protocol designed by formula (4), the bounded tracking control of the multi-agent system is transformed into the boundedness problem of the solution of the stochastic time-delay differential equation (5).

7. The tracking and control method for an uncertain multi-agent system under time delay and noise as described in claim 6, characterized in that, Step S3 is as follows: S31. Selecting a Lyapunov functional: （7） in, , z T ( t )express z ( t The transpose of ) P Describe a positive definite matrix. , ; Differentiate both sides of the Lyapunov functional simultaneously: Expanding, we get: Further results were obtained: （8） in: , , , , , , , μ It is a constant greater than zero; d It is the dimension of noise; Integrating both sides of equation (8) and taking the expectation, we get: （9） in , ; make ,when hour, ;when hour, , where constant ,get: （10） Definite boundary , , ,when ,get: （11） in , If it is an orthogonal matrix, then ; , It is a constant; , ,because and Both are bounded, so we get: in, Let be a constant, let , To track the upper bound; S32. Let the unstable eigenvalue of A be... , ,definition ; S33, and Controllable , ,in, express A The maximum real part of the eigenvalues. express A All eigenvalues, constructed by formula (1) Formula (5) in , , , Indicates the first i 1 eigenvalue, Represents a constant greater than zero.

8. A tracking and control device for an uncertain multi-agent system under time delay and noise, characterized in that, The device includes: processor; A memory on which computer programs that can run on the processor are stored; When the computer program is executed by the processor, it implements the steps of the tracking control method for uncertain multi-agent systems under time delay and noise as described in any one of claims 1 to 7.

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