Affine formation leader selection method of second-order multi-agent system

Abstract

The invention discloses an affine formation leader selection method for a second-order multi-agent system. The method comprises the following steps: constructing a second-order multi-agent system model; based on the second-order multi-agent system model, constructing a mathematical model of second-order multi-agent leader selection, and selecting a leader selection scheme; constructing a tracking error state equation of the whole system according to the selected leader selection scheme; based on the stability requirement of the multi-agent system, adopting an H-infinity control method to obtain a minimum performance index gamma; and obtaining an optimal leader selection scheme according to the minimum performance index gamma. The second-order multi-agent leader selection method fills the blank of the second-order multi-agent leader selection problem, can effectively obtain an expected leader selection set, discusses that a control method of H infinity is used for obtaining the minimum performance index gamma, and the minimum performance index gamma is accurately calculated, so that when the system faces external interference, the leader selection set can be effectively selected. And the optimal leader set can be selected, so that the stability of the system is ensured.

Description

Technical Field

[0001] The present invention belongs to the technical field of multi-agent control, and in particular relates to an affine formation leader selection method for a second-order multi-agent system. Background Art

[0002] Multi-agent systems are widely used in various fields such as automation, robotics, game design, traffic management, e-commerce, social simulation, etc. Multi-agent formation control is one of the focuses of multi-agent system research, and it is also an important application scenario for distributed control of multi-agent systems. The multi-agent formation control task includes two subtasks: formation control and formation maneuver control. Its purpose is to guide certain agents to form specific geometric shapes while achieving translation, scaling, rotation, and other activities. In addition, in practical applications, external interference is inevitable. Under the influence of external interference, the robustness of the formation algorithm also requires higher requirements.

[0003] In a multi-agent system, the collaboration ability between agents, the ability to complete tasks, and the system's ability to perceive changes in the external environment are all directly affected by the selection of leaders. The rationalization of leader selection can not only make the system more stable, but also effectively reduce resource waste and better optimize the system's adaptive ability. The literature (F. Xiao, Q. Yang, X. Zhao and H. Fang, "A Framework for Optimized Topology Design and Leader Selection in Affine Formation Control," in IEEE Robotics and Automation Letters, vol. 7, no. 4, pp. 8627-8634, Oct. 2022, doi: 10.1109 / LRA.2022.3188883) proposed an optimal selection method for the leader of a first-order multi-agent system considering convergence speed and control energy, which improved the performance of UAV affine formation control, but this method only considers the leader selection problem of the first-order system. Multi-agent systems are more suitable for description using second-order system models, such as smart microgrids, drones, underwater robots, etc. Therefore, the affine formation control problem of second-order multi-agent systems is also worth studying. The leader selection problem of multi-agent systems has been solved for the first-order system. The leader selection problem for disturbed second-order multi-agent systems in a two-dimensional plane has not been fully solved. Summary of the invention

[0004] Purpose of the invention: In order to overcome the deficiencies in the prior art, a method for selecting a leader for an affine formation of a second-order multi-agent system is provided.

[0005] Technical solution: To achieve the above-mentioned purpose, the present invention provides an affine formation leader selection method for a second-order multi-agent system, comprising the following steps:

[0006] S1: Construct a second-order multi-agent system model;

[0007] S2: Based on the second-order multi-agent system model, a mathematical model of second-order multi-agent leadership selection is constructed to select a leader selection scheme;

[0008] S3: Construct the tracking error state equation of the whole system according to the leader selection scheme that has been selected;

[0009] S4: Based on the stability requirements of the multi-agent system, H ∞ The control method obtains the minimum performance index γ;

[0010] S5: According to the minimum performance index γ, the best leader selection scheme is obtained.

[0011] Furthermore, the construction of the second-order multi-agent system model in step S1 includes:

[0012] In R d There is a set of points consisting of N multi-agents, where d ≥ 2 and N ≥ d + 1; the relationship between the N agents is represented by an undirected topological graph express; Contains a set of nodes A set of edges Let A={a ij |i,j∈ε}∈R N×N Represents the adjacency matrix. If ij∈ε, node i can receive information from node j, then a ij =1, otherwise a ij =0; called p i ∈R d is the position of the ith agent and The configuration and structure of all intelligent agents are Indicates that p * Represents the desired formation;

[0013] Consider the d-dimensional plane motion of a multi-agent system composed of N agents, d is a positive integer, and the dynamic model of the multi-agent system is a second-order model, that is,

[0014]

[0015]

[0016] Among them, p i (t) is the location information of the ith agent, vi (t) represents the speed information of the i-th agent, u i (t) represents the control input of the ith agent, ω n (t) represents the external interference; for the convenience of calculation, p i (t) Use p i Indicates that v i (t) Use v i Indicates that ω n (t) with ω n Indicates that u i (t)Use u i express;

[0017] For the graph For all edges, the weight {ω ij} (i,j)∈ε is a set of scalars and ω ij =ω ji ∈R; if the weight satisfies the following formula, it is called equilibrium stress

[0018] ∑ (i,j)∈ε ω ij (p i -p j )=0,i=1,…,n.

[0019] The matrix form of the above formula is expressed as make The above formula is expressed as Where Ω represents the stress matrix, I d is the d×d identity matrix, is the expansion stress matrix, where Ω∈R N×N The stress matrix associated with ω is said to satisfy:

[0020]

[0021] Furthermore, in the second-order multi-agent system model of step S1, in the affine formation, the control input

[0022]

[0023] In the formula, k p ,k v is a control gain that is always positive.

[0024] Furthermore, the necessary and sufficient conditions for establishing the stability of the multi-agent system in step S1 are: rank(Ω)=nd-1 and Ω is a semi-positive definite matrix, rank is the rank of the matrix, and Ω, as the stress matrix of the system, has the inherent property of being symmetric and having 0 rows.

[0025] Furthermore, the step S2 specifically includes:

[0026] The matrix Ω is decomposed into Ω ll ,Ω ff ,Ω fl ,Ω lf Four sub-blocks, Ω ll Contains the interactions between drones in the leader set, represented by the l1,l2,…,l d+1 The sum of rows l1,l2,…,l d+1 Column composition; Ω ff Contains the interactions between all drones in the follower set, represented by the f1,f2,…,f n-d-1 The sum of rows f1,f2,…,f n-d-1 Column composition; Ω fl Describes the influence of followers on leaders, represented by the first f1, f2, …, f n-d-1 The sum of rows l1,l2,…,l d+1 Column composition; Ω lf Describe the influence of the leader on the follower, which is represented by the l1, l2, …, l d+1 The sum of rows f1,f2,…,f n-d-1 Column composition;

[0027] There are at least d+1 affine-independent points that affinely span R d , so d+1 agents are selected as leaders; the set of agents selected as leaders is represented by L = {l1,l2,…,l d+1},l i ∈{1,2,…,N}, the set of followers selected is denoted as F=V\L, then F={f1,f2,…,f n-d-1},f i ∈{1,2,…,N};

[0028] The leader selection result is expressed as K = diag{k1, k2, …, k i}, k i =1 means agent i is the leader, k i =0 means that agent i is a follower, so the task selected by the leader is to design the K matrix.

[0029] Furthermore, the construction of the tracking error state equation of the entire system in step S3 includes:

[0030] For the selected leader, it is necessary to make the leader's formation always in the desired target formation as much as possible under external interference, correct the control target of the affine formation, and design the control rate u of the follower i , when time t→∞, it needs to satisfy When time t→∞, the error between the leader's formation and the expected formation is as small as possible, that is,

[0031]

[0032] Among them, δ pf represents the error between the leader's position and the expected position, δ vf The error between the leader's speed and the expected speed, p l is the set of vectors consisting of the positions of all leaders, v f is the set of velocity vectors of all followers, v l is the set of vectors consisting of the positions of all leaders;

[0033] Order error The tracking error state equation of the entire system is:

[0034]

[0035] in,

[0036] Furthermore, in step S4, H ∞ The control method problem set-up includes:

[0037] In order to optimize the robustness of the system and reduce the impact of external interference on the system, under the condition of zero initial value, when the external environment is disturbed, the control output u i Need for guarantee

[0038] ||T δω || ∞ <γ

[0039] Where γ>0 means that the given H ∞ Performance index, T δω represents the external disturbance ω f To the leader formation error δ f The closed-loop transfer function matrix of the controller is: γ represents the maximum transfer gain from disturbance to performance output; the purpose of controller design is to adjust u i To minimize the maximum ||T δω || ∞ , ensuring that the effects of worst-case interference are appropriately suppressed.

[0040] Furthermore, in step S5, according to H ∞ The control problem is converted into a leader selection optimization problem, and the best leader selection scheme is obtained through the minimum performance indicator γ.

[0041] Furthermore, in step S5, according to H ∞The specific methods of controlling the problem to get the leader to select the optimization problem include:

[0042] There exists a positive definite matrix P, when X = P -1 ,W=KX, the above H ∞ The control problem is transformed into the form of matrix inequality:

[0043]

[0044] Based on H ∞ Control, the leader selection optimization problem is expressed as:

[0045]

[0046]

[0047]

[0048]

[0049] When X=P -1 ,W=KX,control input u i =KX, the present invention achieves the optimization of the leader configuration and robust performance of the multi-agent system by constructing and solving the minimum performance indicator γ, and ensures that the system reaches the theoretical optimal performance state by accurately monitoring the γ value.

[0050] The method of the present invention performs leadership selection for a multi-agent system composed of three or more intelligent agents. Given a network topology and corresponding optimization indicators, the present invention can generate an optimized set of leaders and apply external control to the leader set. Under the premise that the corresponding indicators are optimized, affine formation control of the multi-agent system can be achieved, including the formation, maintenance and change of the formation.

[0051] Beneficial effects: Compared with the prior art, the present invention fills the gap in the second-order multi-agent leadership selection problem and can effectively obtain the desired leader selection set. ∞ The control method obtains the minimum performance index γ, and by accurately calculating the minimum performance index γ, the system can also select the best set of leaders when facing external interference, thereby ensuring the stability of the system. This robustness is crucial in complex environments such as drone swarms, automated warehouse management, and autonomous driving fleets. In addition, the present invention makes the leader selection process more systematic by constructing the constraint problem of the leader. It greatly optimizes resource allocation and decision-making efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 Flow chart of the method of the present invention. DETAILED DESCRIPTION

[0053] The present invention is further explained below in conjunction with the accompanying drawings and specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, various equivalent forms of modifications to the present invention by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0054] like Figure 1 As shown, the present invention provides an affine formation leader selection method for a second-order multi-agent system, comprising the following steps:

[0055] S1: Construct a second-order multi-agent system model;

[0056] In R d There is a set of points consisting of N multi-agents, where d ≥ 2 and N ≥ d + 1; the relationship between the N agents is represented by an undirected topological graph express; Contains a set of nodes A set of edges Let A={a ij |i,j∈ε}∈R N×N Represents the adjacency matrix. If ij∈ε, node i can receive information from node j, then a ij =1, otherwise a ij =0; called p i ∈R d is the position of the ith agent and The configuration and structure of all intelligent agents are Indicates that p * Represents the desired formation;

[0057] Consider the d-dimensional plane motion of a multi-agent system composed of N agents, d is a positive integer, and the dynamic model of the multi-agent system is a second-order model, that is,

[0058]

[0059]

[0060] Among them, p i (t) is the location information of the ith agent, v i (t) represents the speed information of the i-th agent, u i (t) represents the control input of the ith agent, ω n (t) represents the external interference; for the convenience of calculation, p i (t) Use p i Indicates that v i (t) Use v iIndicates that ω n (t) with ω n Indicates that u i (t)Use u i express;

[0061] For the graph For all edges, the weight {ω ij} (i,j)∈ε is a set of scalars and ω ij =ω ji ∈R; if the weight satisfies the following formula, it is called equilibrium stress

[0062] ∑ (i,j)∈ε ω ij (p i -p j )=0,i=1,…,n.

[0063] The matrix form of the above formula is expressed as make The above formula is expressed as Where Ω represents the stress matrix, I d is the d×d identity matrix, is the expansion stress matrix, where Ω∈R N×N The stress matrix associated with ω is said to satisfy:

[0064]

[0065] In the affine formation, the control input

[0066]

[0067] In the formula, k p ,k v is the control gain which is always positive;

[0068] The necessary and sufficient conditions for establishing the stability of a multi-agent system are: rank(Ω) = nd-1 and Ω is a semi-positive definite matrix, rank is the rank of the matrix, Ω is the stress matrix of the system, and its inherent properties are symmetry and the number of rows is 0. n represents the number of multi-agents, and d represents the dimension of the space.

[0069] The second order in the second-order multi-agent system model represents the motion state of a system. The first order only considers position and velocity, while the second order considers acceleration on the basis of position and velocity. It can be understood as a dynamic behavior that changes over time.

[0070] S2: Based on the second-order multi-agent system model, a mathematical model of second-order multi-agent leadership selection is constructed to select a leader selection scheme;

[0071] The matrix Ω is decomposed into Ωll ,Ω ff ,Ω fl ,Ω lf Four sub-blocks, Ω ll Contains the interactions between drones in the leader set, represented by the l1,l2,…,l d+1 The sum of rows l1,l2,…,l d+1 Column composition; Ω ff Contains the interactions between all drones in the follower set, represented by the f1,f2,…,f n-d-1 The sum of rows f1,f2,…,f n-d-1 Column composition; Ω fl Describes the influence of followers on leaders, represented by the first f1, f2, …, f n-d-1 The sum of rows l1,l2,…,l d+1 Column composition; Ω lf Describe the influence of the leader on the follower, which is represented by the l1, l2, …, l d+1 The sum of rows f1,f2,…,f n-d-1 Column composition;

[0072] There are at least d+1 affine-independent points that affinely span R d , so d+1 agents are selected as leaders; the set of agents selected as leaders is represented by L = {l1,l2,…,l d+1},l i ∈{1,2,…,N}, the set of followers selected is denoted as F=V\L, then F={f1,f2,…,f n-d-1},f i ∈{1,2,…,N};

[0073] The leader selection result is expressed as K = diag{k1, k2, …, k i}, k i =1 means agent i is the leader, k i =0 means that agent i is a follower, so the task selected by the leader is to design the K matrix.

[0074] S3: Construct the tracking error state equation of the whole system according to the leader selection scheme that has been selected;

[0075] For the selected leader, it is necessary to make the leader's formation always in the desired target formation as much as possible under external interference, correct the control target of the affine formation, and design the control rate u of the follower i , when time t→∞, it needs to satisfy When time t→∞, the error between the leader's formation and the expected formation is as small as possible, that is,

[0076]

[0077] Among them, δ pf represents the error between the leader's position and the expected position, δ vf The error between the leader's speed and the expected speed, p l is the set of vectors consisting of the positions of all leaders, v f is the set of velocity vectors of all followers, v l is the set of vectors consisting of the positions of all leaders;

[0078] Order error The tracking error state equation of the entire system is:

[0079]

[0080] in,

[0081] S4: Based on the stability requirements of the multi-agent system, H ∞ The control method obtains the minimum performance index γ;

[0082] H ∞ The control method problem set-up includes:

[0083] In order to optimize the robustness of the system and reduce the impact of external interference on the system, under the condition of zero initial value, when the external environment is disturbed, the control output u i Need for guarantee

[0084] ||T δω || ∞ <γ

[0085] Where γ>0 means that the given H ∞ Performance index, T δω represents the external disturbance ω f To the leader formation error δ f The closed-loop transfer function matrix of the controller is: γ represents the maximum transfer gain from disturbance to performance output; the purpose of controller design is to adjust u i To minimize the maximum ||T δω || ∞ , ensuring that the effects of worst-case interference are appropriately suppressed.

[0086] S5: According to H ∞ The control problem is converted into a leader selection optimization problem, and the best leader selection scheme is obtained through the minimum performance index γ;

[0087] According to H ∞ The specific methods of controlling the problem to get the leader to select the optimization problem include:

[0088] There exists a positive definite matrix P, when X = P -1 ,W=KX, the above H ∞ The control problem is transformed into the form of matrix inequality:

[0089]

[0090] Based on H ∞ Control, the leader selection optimization problem is expressed as:

[0091]

[0092]

[0093]

[0094]

[0095] When X=P -1 ,W=KX,control input u i =KX, the present invention achieves the optimization of the leader configuration and robust performance of the multi-agent system by constructing and solving the minimum performance indicator γ, and ensures that the system reaches the theoretical optimal performance state by accurately monitoring the γ value.

[0096] This embodiment also provides an affine formation leader selection system for a second-order multi-agent system, which includes a network interface, a memory and a processor; wherein the network interface is used to realize the reception and transmission of signals during the process of sending and receiving information between other external network elements; the memory is used to store computer program instructions that can be run on the processor; the processor is used to execute the steps of the above-mentioned consensus method when running the computer program instructions.

[0097] The present embodiment also provides a computer storage medium, which stores a computer program, and the method described above can be implemented when the processor executes the computer program. The computer readable medium can be considered to be tangible and non-temporary. Non-limiting examples of non-temporary tangible computer-readable media include non-volatile memory circuits (such as flash memory circuits, erasable programmable read-only memory circuits or mask read-only memory circuits), volatile memory circuits (such as static random access memory circuits or dynamic random access memory circuits), magnetic storage media (such as analog or digital tapes or hard drives) and optical storage media (such as CDs, DVDs or Blu-ray discs), etc. The computer program includes processor executable instructions stored on at least one non-temporary tangible computer-readable medium. The computer program may also include or rely on stored data. The computer program may include a basic input / output system (BIOS) that interacts with the hardware of a special-purpose computer, a device driver that interacts with a specific device of a special-purpose computer, one or more operating systems, user applications, background services, background applications, etc.

[0098] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment in combination with software and hardware. Moreover, the present application may adopt the form of a computer program product implemented in one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that include computer-usable program code.

Claims

1. An affine formation leader selection method for a second-order multi-agent system, characterized in that: The steps include: S1: Construct a second-order multi-agent system model; S2: Based on the second-order multi-agent system model, a mathematical model of second-order multi-agent leadership selection is constructed to select a leader selection scheme; S3: Construct the tracking error state equation of the whole system according to the leader selection scheme that has been selected; S4: Based on the stability requirements of the multi-agent system, H ∞ The control method obtains the minimum performance index γ; S5: According to the minimum performance index γ, the best leader selection scheme is obtained.

2. The affine formation leader selection method for a second-order multi-agent system according to claim 1, characterized in that: The construction of the second-order multi-agent system model in step S1 includes: In R d There is a set of points consisting of N multi-agents, where d ≥ 2 and N ≥ d + 1; the relationship between the N agents is represented by an undirected topological graph express; Contains a set of nodes A set of edges Let A={a ij |i,j∈ε}∈R N×N Represents the adjacency matrix. If ij∈ε, node i can receive information from node j, then a ij =1, otherwise a ij =0; called p i ∈R d is the position of the ith agent and The configuration and structure of all intelligent agents are Indicates that p * Represents the desired formation; Consider the d-dimensional plane motion of a multi-agent system composed of N agents, d is a positive integer, and the dynamic model of the multi-agent system is a second-order model, that is, Among them, p i (t) is the location information of the ith agent, v i (t) represents the speed information of the i-th agent, u i (t) represents the control input of the ith agent, ω n (t) represents the external interference; for the convenience of calculation, p i (t) Use p i Indicates that v i (t) Use v i Indicates that ω n (t) with ω n Indicates that u i (t)Use u i express; For the graph For all edges, the weight {ω ij } (i,j)∈ε is a set of scalars and ω ij =ω ji ∈R; if the weight satisfies the following formula, it is called equilibrium stress ∑ (i,j)∈ε ω ij (p i -p j )=0,i=1,…,n。 The matrix form of the above formula is expressed as make The above formula is expressed as Where Ω represents the stress matrix, I d is the d×d identity matrix, is the expansion stress matrix, where Ω∈R N×N The stress matrix associated with ω is said to satisfy:

3. The affine formation leader selection method for a second-order multi-agent system according to claim 2, characterized in that: In the second-order multi-agent system model of step S1, in the affine formation, the control input In the formula, k p ,k v is a control gain that is always positive.

4. The affine formation leader selection method for a second-order multi-agent system according to claim 2, characterized in that: The necessary and sufficient conditions for establishing the stability of the multi-agent system in step S1 are: rank(Ω)=nd-1 and Ω is a semi-positive definite matrix, rank is the rank of the matrix, and Ω, as the stress matrix of the system, has the inherent property of being symmetric and having 0 rows.

5. The affine formation leader selection method for a second-order multi-agent system according to claim 2, characterized in that: The step S2 specifically includes: The matrix Ω is decomposed into Ω ll ,Ω ff ,Ω fl ,Ω lf Four sub-blocks, Ω ll Contains the interactions between drones in the leader set, represented by the l1,l2,…,l d+1 The sum of rows l1,l2,…,l d+1 Column composition; Ω ff Contains the interactions between all drones in the follower set, represented by the f1,f2,…,f n-d-1 The sum of rows f1,f2,…,f n-d-1 Column composition; Ω fl Describes the influence of followers on leaders, represented by the first f1, f2, …, f n-d-1 The sum of rows l1,l2,…,l d+1 Column composition; Ω lf Describe the influence of the leader on the follower, which is represented by the l1, l2, …, l d+1 The sum of rows f1,f2,…,f n-d-1 Column composition; There are at least d+1 affine-independent points that affinely span R d , so d+1 agents are selected as leaders; the set of agents selected as leaders is represented by L = {l1,l2,…,l d+1 },l i ∈{1,2,…,N}, the set of followers selected is denoted as F=V\L, then F={f1,f2,…,f n-d-1 },f i ∈{1,2,…,N}; The leader selection result is expressed as K = diag{k1, k2, …, k i }, k i =1 means agent i is the leader, k i =0 means that agent i is a follower, so the task selected by the leader is to design the K matrix.

6. The affine formation leader selection method for a second-order multi-agent system according to claim 5, characterized in that: The construction of the tracking error state equation of the entire system in step S3 includes: Correct the control target of the affine formation and design the control rate u of the follower i , when time t→∞, it needs to satisfy When time t→∞, the error between the leader's formation and the expected formation is as small as possible, that is, Among them, δ pf represents the error between the leader's position and the expected position, δ vf The error between the leader's speed and the expected speed, p l is the set of vectors consisting of the positions of all leaders, v f is the set of velocity vectors of all followers, v l is the set of vectors consisting of the positions of all leaders; Order error The tracking error state equation of the entire system is: in, 7. The affine formation leader selection method for a second-order multi-agent system according to claim 6, characterized in that: In step S4, ∞ The control method problem set-up includes: Under the condition of zero initial value, when the external environment is disturbed, the control output u i Need for guarantee ||T δω || ∞ <γ Where γ>0 means that the given H ∞ Performance index, T δω represents the external disturbance ω f To the leader formation error δ f The closed-loop transfer function matrix of the controller is: γ represents the maximum transfer gain from disturbance to performance output; the purpose of controller design is to adjust u i To minimize the maximum ||T δω || ∞ .

8. The affine formation leader selection method for a second-order multi-agent system according to claim 7, characterized in that: In step S5, according to H ∞ The control problem is converted into a leader selection optimization problem, and the best leader selection scheme is obtained through the minimum performance indicator γ.

9. The affine formation leader selection method for a second-order multi-agent system according to claim 8, characterized in that: In step S5, according to H ∞ The specific methods of controlling the problem to get the leader to select the optimization problem include: There exists a positive definite matrix P, when X = P -1 ,W=KX,H ∞ The control problem is transformed into the form of matrix inequality: Based on H ∞ Control, the leader selection optimization problem is expressed as:

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