A multi-agent system control method, device and equipment

Abstract

The present invention discloses a multi-agent system control method, device and equipment, and relates to the technical field of multi-agent system control. The method comprises: constructing a dynamic model of a follower and a leader, and constructing a communication topology diagram of the multi-agent system using an undirected graph; determining the feedback gain matrix and the continuous gain matrix of each follower and the leader by solving the Riccati equation inequality, and determining the control law of each agent according to the feedback gain matrix and the continuous gain matrix; determining the adaptive weight and control parameters of the control law of each agent based on the relative error of the agent, and constructing a distributed adaptive tracking control; through the distributed adaptive tracking control, each follower uses local information to track the leader of an unknown input, so that the state of the follower can converge to the state of the leader under the fixed topology and the switching topology of the controller.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-agent system control, and in particular to a multi-agent system control method, device and equipment. Background Art

[0002] With the rapid development of science and technology and the leap of computing technology, multi-agent systems (MAS) have been widely used, and typical occasions include complex systems such as formation control of unmanned aerial vehicles, distributed sensor networks and cluster scheduling of aircraft. In these multi-agent systems, factors such as the number of agents, information interaction between them, and the topological structure of communication jointly define the complexity of the multi-agent consensus control problem.

[0003] In the research field of consensus control of multi-agent systems, the existing consensus control algorithms are mainly classified into two categories: leaderless consensus control and consensus control with a leader. Among them, leader-based consensus control is also called leader-following method or distributed tracking control, which is the key technology to achieve collaboration among agents. At present, in the field of distributed tracking control, there have been extensive studies devoted to exploring the collaborative strategies of multi-agent systems.

[0004] Adaptive distributed tracking control is designed for general linear multi-agent systems. By introducing adaptive weight factors to adjust the relative errors between agents, the global information of the entire multi-agent topology network is avoided, and distributed adaptive tracking control, also known as fully distributed tracking control, is realized. Adaptive distributed tracking control of multi-agents based on event triggering is also proposed, and adaptive distributed control is realized on the basis of solving the nonlinear dynamic problems of multi-agents. However, in the actual application of consistency control based on the pilot-follower method, the leader will face many problems. For example, in military wars, the leader will be the enemy's number one target. In drone formations, the leader will be affected by various factors in the air, and the leader's control input may be non-zero to achieve a certain goal. Therefore, it is impractical for the existing methods to not consider the leader's control input. In practice, the leader's control input may also be unavailable to any follower, so how to design the leader's control input has greater flexibility in practical applications. On the other hand, the communication link of the multi-agent system in real application scenarios is susceptible to external environmental factors and system failures, resulting in communication disconnection in harsh working environments or agent failures. In addition, the addition of new agents may also cause changes in existing connections. With the time-varying nature of the communication network structure, the control difficulty of multi-agents increases accordingly. In the design of collaborative control strategies for multi-agent systems, one of the key challenges is how to design effective control mechanisms to achieve system synchronization based on surrounding information when encountering communication interruptions, reconnections, and delays.

[0005] In summary, the control methods currently used in linear multi-agent systems do not consider the problem of switching communication topology graphs, how to solve the situation where the leader's control input is unavailable to any follower or the leader's control input is unavailable to any follower. Summary of the invention

[0006] In view of the deficiency of the prior art in that the control input of the leader is not available to any follower under the switching communication topology, the present invention proposes a multi-agent system control method, device and equipment. By designing an adaptive tracking controller with distributed discontinuity of switching topology, non-smooth analysis and Riccati inequality are used to prove that the state of the follower of the controller can converge to the state of the leader under fixed topology and switching topology, thereby solving the problems existing in the prior art.

[0007] A multi-agent system control method, wherein the multi-agent system includes a plurality of followers and a leader, comprises the following steps:

[0008] Establish a dynamic model of the follower and a dynamic model of the leader, analyze each dynamic model, and obtain the state data of the follower and the state data of the leader;

[0009] Construct a distributed adaptive tracking controller based on the follower's state data and the leader's state data;

[0010] Each follower is controlled by a distributed adaptive tracking controller to track the leader with unknown control input;

[0011] The construction process of the distributed adaptive tracking controller includes the following steps:

[0012] Use undirected graphs to build the communication topology of the multi-agent system and divide the communication topology into multiple subgraphs;

[0013] Determine the Laplacian matrix of each subgraph, and determine the adjacency matrix of the followers and the communication matrix between the leader and the followers based on the Laplacian matrix;

[0014] According to the adjacency matrix of the followers and the communication matrix between the leader and the followers, the relative errors between the followers and the relative errors between the followers and the leader are determined respectively;

[0015] Inputting the state data corresponding to the follower and the state data corresponding to the leader into the Riccati equation inequality, determining the feedback gain matrix and the continuous gain matrix of each follower and the leader; and determining the control law of each follower according to the feedback gain matrix and the continuous gain matrix;

[0016] According to the feedback gain matrix and continuous gain matrix of each follower and leader, as well as the relative errors between followers and the relative errors between leader and followers, the adaptive weights and control parameters of each follower control law are determined under the condition of unknown bounded control of the leader;

[0017] Based on the control law of each follower, the adaptive weight of the control law and its control parameters, a distributed adaptive tracking controller is constructed.

[0018] Furthermore, the dynamic models of followers and leaders are respectively expressed as:

[0019] The dynamic model of the follower is expressed as:

[0020]

[0021] Where: x i ∈R n is the status information of the follower; u i ∈R m is the control input of the follower, A and B are constant matrices with corresponding dimensions, represents the first-order derivative of the following state variable, R n , Rm Both represent constant real numbers; n and m both represent dimensions;

[0022] The leader's dynamic model is expressed as:

[0023]

[0024] Where: x0∈R n is the leader’s state information; u0∈R m Provide control input for leaders; Represents the first-order derivative of the leader's multi-state variable.

[0025] Furthermore, the communication topology diagram of the multi-agent system constructed by using an undirected graph specifically includes: defining As a set of communication topology graphs, where p and Respectively represent the graph The index and index set of all subgraphs in the graph, each subgraph is non-connected, and the joint graph of the multiple subgraphs is connected; by introducing the switching signal As time t changes, the communication topology of the multi-agent system is expressed as G σ(t) .

[0026] Furthermore, the method of determining the Laplacian matrix of each subgraph and determining the adjacency matrix of the follower and the communication matrix between the leader and the follower according to the Laplacian matrix specifically includes the following steps:

[0027] Define the adjacency matrix W of the followers, the Laplacian matrix L is obtained by subtracting the degree matrix F and W between the followers, and define the communication matrix D between the leader and the followers;

[0028] The element of the adjacency matrix W = {a ij} satisfies the edge ε ij ∈E, then a ij =1, otherwise a ij =0; i represents the number of followers; j represents the number of adjacent followers of follower i; E represents the set of all edges;

[0029] Define the in-degree matrix to be a diagonal matrix in b i 、b n are all elements in the diagonal matrix, represents a real number, and N represents the number of followers; the Laplace matrix of the graph G is defined as: L = FW; when there is at least one path between any two nodes of the graph G, the undirected graph is called a connected graph; for a multi-agent system consisting of a leader and N followers, the leader can only send information to its neighbor agents, but does not receive information from anyone;

[0030] Define the communication matrix D between the leader and the followers D = diag{d1,d2,...,d N}∈R N×N ; When the leader and follower are connected, d i =1, otherwise d i =0; define graph It is composed of the leader, the edges between the leader and followers, and the followers and the graph G.

[0031] Further, the relative errors between followers and the relative errors between followers and the leader are determined respectively according to the adjacency matrix of the followers and the communication matrix between the leader and the followers; specifically including:

[0032] The relative error between the followers is expressed as The relative error between the leaders is denoted as d i (t)(x i -x0); where a ij (t) is the (i,j)th graph The associated adjacency matrix element; d i (t) is the i-th diagonal element of the communication matrix D between the leader and the follower, x i 、x j They represent the status information of follower i and follower j respectively, and x0 represents the status information of the leader.

[0033] Furthermore, the state data corresponding to the follower and the state data corresponding to the leader are input into the Riccati equation inequality to determine the feedback gain matrix and the continuous gain matrix of each follower and the leader; and the control law of each follower is determined according to the feedback gain matrix and the continuous gain matrix; wherein the Riccati equation inequality is expressed as:

[0034] PA+A T P-2PBB T P+βI n ≤0

[0035] Where β is a positive number, A and B are constant matrices with corresponding dimensions, T is the matrix transpose, and I n is an identity matrix, n represents the dimension, and p represents the solution of the above Riccati equation;

[0036] According to the Riccati equation inequality, P is solved and the feedback gain matrix and continuous gain matrix of each follower and leader are calculated respectively, which are expressed as follows: Feedback gain matrix K = -B T P, continuous gain matrix Γ = PBB T P.

[0037] Furthermore, according to the feedback gain matrix and continuous gain matrix of each follower and leader, as well as the relative error between followers and the relative error between leader and followers, the adaptive weights and control parameters of each follower control law are determined in the case of unknown bounded control of the leader; the adaptive weights of each follower control law are It is expressed as:

[0038]

[0039] Among them, τ i is a normal number, ζ i , j They represent the relative errors between followers i, j and the leader respectively, and j represents the adjacent follower of follower i.

[0040] Furthermore, based on the control law of each follower, the adaptive weight of the control law and its control parameters, a distributed adaptive tracking controller is constructed, and the distributed adaptive tracking controller is expressed as:

[0041]

[0042] Among them, u i represents the control input of the distributed adaptive tracking controller, η i is a constant and η i >0,c i is the adaptive weight of the control law, K∈R m×n is the feedback gain matrix, and a ij (t) is the (i,j)th graph The associated adjacency matrix element, d i (t) is the i-th diagonal element of the adjacency matrix D, τ i is a positive constant, Γ∈R n×n is a continuous gain matrix.

[0043] The present invention also includes a multi-agent system control device, wherein the multi-agent system includes a plurality of followers and a leader, including:

[0044] The dynamic model building module is used to build the dynamic model of the follower and the leader, and obtain the status data of the follower and the leader;

[0045] A controller building module, used to build a distributed adaptive tracking controller based on the state data of the followers and the state data of the leader;

[0046] a control module for controlling each follower to track a leader of unknown control inputs through a distributed adaptive tracking controller;

[0047] Wherein, the controller building module includes:

[0048] A communication topology graph construction unit, used to construct a communication topology graph of a multi-agent system using an undirected graph, and divide the communication topology graph into a plurality of subgraphs; wherein each of the subgraphs is non-connected, and a joint graph of the plurality of subgraphs is connected;

[0049] A partitioning unit, for determining a Laplacian matrix of each subgraph, and determining an adjacency matrix of followers and a communication matrix between leaders and followers according to the Laplacian matrix;

[0050] A relative error determination unit, used to determine the relative errors between followers and the relative errors between followers and the leader respectively according to the adjacency matrix of the followers and the communication matrix between the leader and the followers;

[0051] A control law determination unit is used to input the state data corresponding to the follower and the state data corresponding to the leader into the Riccati equation inequality to determine the feedback gain matrix and the continuous gain matrix of each follower and the leader; and determine the control law of each follower according to the feedback gain matrix and the continuous gain matrix;

[0052] A weight determination unit, for determining the adaptive weights and control parameters of each follower control law under the condition of unknown bounded control of the leader, according to the feedback gain matrix and the continuous gain matrix of each follower and the leader, as well as the relative errors between the followers and the relative errors between the leader and the followers;

[0053] The controller construction unit is used to construct a distributed adaptive tracking controller based on the control law of each follower, the adaptive weight of the control law and its control parameters.

[0054] The present invention also includes a multi-agent system control computer device, including: a memory, a processor and a computer program stored in the memory, and the processor implements the steps of the multi-agent system control method when executing the computer program.

[0055] The present invention provides a multi-agent system control method, device and equipment, which have the following beneficial effects:

[0056] The present invention aims at the distributed tracking control problem of a general linear multi-agent system with a leader. According to the feedback gain matrix and the continuous gain matrix of each follower and the leader, as well as the relative error between the followers and the relative error between the leader and the followers, the adaptive weights and control parameters of the control laws of each follower are determined under the condition of unknown bounded control of the leader. A distributed discontinuous adaptive tracking controller with a switching topology is designed. The Riccati equation inequality is used to prove that the state of the follower can converge to the state of the leader under the fixed topology and the switching topology. At the same time, an undirected graph is used to build a communication topology graph of the multi-agent system, which can be transformed into periodic switching over time, and the fixed communication topology situation cannot be directly applied to the switching communication topology situation. It is proved that when the communication topology between the followers is an undirected graph, the leader has a directed path to reach all followers under the joint graph, and the state of the leader is not provided to the followers, the state of the followers can still approach and converge to the state of the leader. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 A schematic diagram of a communication topology that may occur between a leader and a follower in an embodiment of the present invention;

[0058] Figure 2 Schematic diagram of the change of the switching topology signal σ(t) in an embodiment of the present invention;

[0059] Figure 3 This is a schematic diagram of state 1 in which four agents follow a leader in an embodiment of the present invention;

[0060] Figure 4 This is a schematic diagram of state 2 in which four agents follow a leader in an embodiment of the present invention;

[0061] Figure 5 This is a schematic diagram of state 3 in which four agents follow a leader in an embodiment of the present invention;

[0062] Figure 6 It is a schematic diagram of the trajectory of the intelligent agent 1 and the intelligent agent 2 tracking the leader under the switching topology in an embodiment of the present invention;

[0063] Figure 7 It is a schematic diagram of the trajectory of the intelligent agent 3 and the intelligent agent 4 tracking the leader under the switching topology in an embodiment of the present invention;

[0064] Figure 8 A schematic diagram of an adaptive time-varying coupling weight in an embodiment of the present invention;

[0065] Fig. 9 Schematic diagram of tracking error between agent 1 and the leader in an embodiment of the present invention;

[0066] Fig.10Schematic diagram of tracking error between agent 2 and the leader in an embodiment of the present invention;

[0067] Fig.11 Schematic diagram of tracking error between agent 3 and the leader in an embodiment of the present invention;

[0068] Fig.12 Schematic diagram of tracking error between agent 4 and the leader in an embodiment of the present invention;

[0069] Fig.13 It is a schematic diagram of the trajectory of the leader tracked by the agent 1 and the agent 2 under interference in an embodiment of the present invention;

[0070] Fig.14 It is a schematic diagram of the trajectory of the leader tracked by the agent 3 and the agent 4 under interference in an embodiment of the present invention;

[0071] Fig.15 Schematic diagram of adaptive time-varying coupling weights under interference in an embodiment of the present invention;

[0072] Fig.16 Schematic diagram of tracking error between agent 1 and the leader under interference in an embodiment of the present invention;

[0073] Fig.17 Schematic diagram of tracking error between agent 2 and the leader under interference in an embodiment of the present invention;

[0074] Fig.18 Schematic diagram of tracking error between agent 3 and leader under interference in an embodiment of the present invention;

[0075] Fig.19 Schematic diagram of tracking error between agent 4 and the leader under interference in an embodiment of the present invention;

[0076] Fig. 20 Schematic diagram of the flow of the multi-agent system control method in an embodiment of the present invention. DETAILED DESCRIPTION

[0077] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0078] The present invention designs a distributed discontinuous adaptive tracking controller based on the Riccati inequality based on the state error between neighboring agents. The controller ensures that under the switching communication topology, the communication topology between the following agents is undirected, the leader has a directed path to reach all followers under the joint graph, and the control input of the leader is bounded, so that the state of the follower can eventually effectively approach and converge to the leader state, such as Fig. 20As shown, the specific steps include:

[0079] S1. Construct a dynamic model of followers and leaders, and determine the state data corresponding to the followers and the state data corresponding to the leaders.

[0080] The agent network topology is usually described by G = {ν, ε} as an undirected graph, where Represents the set of vertices in the graph G. represents the edge set in the graph G, W = [a ij ]∈R N×N Defined as an adjacency matrix, the elements in the weighted adjacency matrix are non-negative. ε ij =(v i ,v j ) represents the slave node v i To node v j The information flow direction of a graph G is called undirected if and only if the edge ε ij and ε ji are equivalent, which means that information can be transmitted between them; if the edge ε ij and ε ji are not equivalent, and the information flow is unidirectional, then it is called a directed graph. In the present invention, unless otherwise stated, the graph G represents an undirected graph. The element W of the adjacency matrix = {a ij} satisfies the edge ε ij ∈E, then a ij =1, otherwise a ij = 0. Define the in-degree matrix to be a diagonal matrix in N represents the number of agents, and the Laplacian matrix of graph G is defined as: L=FW.

[0081] If there is at least one path between any two nodes in the graph G, then the undirected graph is called a connected graph. For a multi-agent system consisting of a leader and N followers, the leader can only send information to its neighbor agents, but does not receive information from anyone. Define the leader adjacency matrix D = diag{d1, d2, ..., d N}∈R N×N , if the leader and follower have a connection d i =1, otherwise d i = 0, define graph It consists of a leader, edges between leaders and followers, follower agents, and a graph G.

[0082] Define a new matrix H = L + D ∈ R N×N .

[0083] Lemma1: A graph G is connected if and only if the Laplacian L of G has a simple zero eigenvalue.

[0084] Lemma2: For the leader-follower system, if the undirected graph is connected, then the matrix H = L + D is symmetric positive definite.

[0085] Nonsmooth Analysis: Consider the following differential equation with a discontinuity on the right hand side.

[0086]

[0087] f:R m ×R→R m is Lebesgue-measurable and locally intrinsically bounded. The vector function x(·) is called a Filippov solution, At [t0, t1], if x(·) is absolutely continuous on [t0, t1] and satisfies the following differential inclusion for almost all t∈[t0, t1]: in Indicates all The intersection of sets with Lebesgue measure zero, It is the convex closure E of the set, B(z,δ) means the center is at z and the opening radius is δ.

[0088] Let V:R m →R is a locally Lipschitz continuous function, and the Clarke generalized gradient V is given by Where co represents the convex hull, Ω v represents the set with Lebesgue measure 0, Indicates that it does not exist. is any set of measure zero. The Lie derivative of the set-valued V of the formula is defined as

[0089] On Lyapunov's Stability Theorem of Set-Valued Mappings As described below: Lemma3: For Let f(z,t) be locally intrinsically bounded, and in the area where r>0. In addition, let V:R m →R is a regular function that satisfies V(0,t)=0 and 0<V1(‖z‖)≤V(z,t)≤V2(‖z‖). For z≠0, in Q, for some V1 and V2 belonging to Class function, if it exists The class function ω(·) has the property For z≠0, the solution z(t)≡0 is asymptotically stable.

[0090] Consider a multi-agent system consisting of N followers and a leader, whose followers’ dynamics can be described as

[0091]

[0092] Where: x i ∈R n is the state information of the multi-agent; u i ∈R m is the control input of the multi-agent, which can only use its neighbors and its own local information. A and B are constant matrices with corresponding dimensions, R n and R m represents a collection of real matrices.

[0093] The leader’s dynamic is described as:

[0094]

[0095] Where: x0∈R n The state information of the multi-agent leader; u0∈R m is the control input of the multi-agent leader, and A and B are constant matrices with corresponding dimensions.

[0096] Assumption 1: The subgraph associated with the followers is undirected, and in the union of all subgraphs in the graph, the leader has directed paths to all followers.

[0097] Assumption 2: (A,B) is stabilizable.

[0098] Assumption 3: The leader’s control input u0 is continuous and bounded, ‖u0‖ ∞ ≤γ, γ is a positive constant.

[0099] Assumption 4: Leader-related vertex 0 is an undirected graph The globally reachable points in .

[0100] If the leader associated vertex 0 is for Under the globally reachable vertices, then the undirected graph is connected, and the matrix H is symmetric positive definite.

[0101] The purpose of this invention is to solve the distributed tracking control problem in equations (2) and (3) under fixed topology and time-varying topology, and design a controller so that each follower can use local information to track the leader with unknown input. Therefore, the leader-follower consistency control is that the closed-loop system should satisfy:

[0102]

[0103] For any initial condition x i (0), i=1,2,...,N.

[0104] S2. Use an undirected graph to construct a communication topology graph of a multi-agent system, and divide the communication topology graph into multiple subgraphs; wherein each of the subgraphs is non-connected, and the joint graph of the multiple subgraphs is connected. Specifically including: Definition As a set of communication topology graphs, where p and Respectively represent the graph The index and index set of all subgraphs in the graph, each subgraph is non-connected, and the joint graph of the multiple subgraphs is connected; by introducing the switching signal As time t changes, the communication topology of the multi-agent system is expressed as G σ(t) .

[0105] S3. Determine the Laplacian matrix of each subgraph according to the divided subgraphs; and determine the adjacency matrix of the followers and the communication matrix between the leader and the followers according to the Laplacian matrix. Define the adjacency matrix A of the followers, the Laplacian matrix L is obtained by subtracting the degree matrix F between the followers and A, and define the communication matrix D between the leader and the followers;

[0106] The element of the adjacency matrix W = {a ij} satisfies the edge ε ij ∈E, then a ij =1, otherwise a ij =0;

[0107] Define the in-degree matrix to be a diagonal matrix in N represents the number of followers, and the Laplace matrix of the graph G is defined as: L = FW; when there is at least one path between any two nodes of the graph G, the undirected graph is called a connected graph; for a multi-agent system consisting of a leader and N followers, the leader can only send information to its neighbor agents, but does not receive information from anyone;

[0108] Define the communication matrix D between the leader and the followers D = diag{d1,d2,...,d N}∈R N×N ; When the leader and follower are connected, d i =1, otherwise d i =0; define graph It is composed of the leader, the edges between the leader and followers, and the followers and the graph G.

[0109] S4. Determine the relative error between followers and the relative error between followers and leaders based on the adjacency matrix of the followers and the communication matrix between the leader and followers. The relative error between followers is expressed as The relative error between leaders is denoted as d i (t)(x i -x0); where a ij (t) is the (i,j)th graph The associated adjacency matrix element; d i (t) is the i-th diagonal element of the communication matrix D between the leader and the follower, x i 、x j They represent the status information of follower i and follower j respectively, and x0 represents the status information of the leader.

[0110] S5. Input the state data corresponding to the follower and the state data corresponding to the leader into the Riccati equation inequality to determine the feedback gain matrix and the continuous gain matrix of each follower and the leader; and determine the control law of each follower according to the feedback gain matrix and the continuous gain matrix. According to the solution of the Riccati equation inequality, the feedback gain matrix and the continuous gain matrix of each follower and the leader are calculated respectively.

[0111] S6. According to the feedback gain matrix and continuous gain matrix of each follower and leader, as well as the relative error between followers and the relative error between leader and followers, the adaptive weights and control parameters of each follower control law are determined under the condition of unknown bounded control of the leader.

[0112] S7. Based on the control law of each follower, the adaptive weight of the control law and its control parameters, a distributed adaptive tracking controller is constructed.

[0113] S8. Complete distributed adaptive tracking control of linear multi-agent systems in switching topologies and with leaders having bounded unknown inputs.

[0114] The controller design in the present invention is for a leader with a non-zero control input that is unavailable to any follower, and its input may be fixed or time-varying. If the interaction topology graph between followers is undirected, when switching the topology, the leader has a directed path to all followers under the union of all subgraphs, and the control input of the leader is bounded, then the state of the follower is able to converge to the state of the leader.

[0115] To realize the distributed tracking control of multi-agent systems, a distributed tracking controller with a fixed topology and unknown input for the leader is designed for each follower based on the relative states between adjacent agents:

[0116]

[0117] Among them, η i is a constant and η i >0,c i is the time-varying coupling weight associated with each of the following, K∈R m×n is the feedback gain matrix, which will be given later. ij is the (i,j)th adjacency matrix element associated with graph G. i is the i-th diagonal element of the leader adjacency matrix D, τ i is a positive constant, Γ∈R n×n is a continuous gain matrix.

[0118] Let i =x i -x0,ζ=(ζ1,ζ2,...,ζ N ), C=diag(c1,c2,...,c N ), H=L+D, η=(η1, η2,..., η i ), using the above protocols (4), (5) and substituting them into the state equations (2) and (3) and writing them in a compact form, we obtain:

[0119]

[0120] Where H = L + D. Obviously, the distributed tracking problem is solved by the adaptive controllers (4) and (5) if the states of ζ with respect to equations (6) and (7) converge to 0.

[0121] Under the condition that Assumption 2 holds, (A, B) is stabilizable, then there exists a solution with P>0 that follows the Riccatiinequality:

[0122] PA+A T P-2PBB T P+βI n ≤0 (8)

[0123] Among them, β>0 is an adjustable parameter, and the feedback gain matrix is ​​designed as K=-B T P, the continuous gain matrix is ​​designed as Γ=PBB T P. Since Γ ≥ 0, the adaptive time-varying coupling weight c i is non-decreasing. Obviously, when ζ converges to 0, the leader-follower consistency control problem is solved by the adaptive controllers of (4) and (5).

[0124] Theorem 1: Consider the multi-agent system described by equations (2) and (3), under assumptions 1 to 4, the distributed tracking problem can be solved by the fixed-topology distributed adaptive tracking controllers (4) and (5). P>0, K=-B T P and Γ = PBB T P is the solution of inequality (8), and each adaptive weight c i It should be asymptotically stable in finite time.

[0125] Since the right side of equation (6) is discontinuous, differential inclusion and non-smooth analysis will be used to solve the stability of equation (6). Because the sign function is measurable and the local nature is bounded, the Filippov solution of equation (6) exists. The adaptive controller of equation (6) is expressed as:

[0126]

[0127] Among them, ae means almost everywhere.

[0128] Proof, choose the following Lyapunov function:

[0129]

[0130] Among them, α is a positive constant, and α ≥ γ. For the communication topology graph G that satisfies Assumption 1-4, it can be concluded from Lemma 1-2 that H is a symmetric and positive definite matrix, and it is obvious that V1 is positive definite and continuously differentiable.

[0131] By taking the derivative of formula (10):

[0132]

[0133] Using non-smooth analysis And Lemma 1, obtain the derivative of V1 and substitute equation (6) into equation (11) to obtain:

[0134]

[0135] By substituting K = -B T P, we get:

[0136]

[0137] Using x T sgn(x)=‖x‖1 and If f is continuous, then we obtain equation (14):

[0138]

[0139] Substitute equation (7) into equation (12), and substitute the above expression into:

[0140]

[0141] Substituting equation (12)-equation (13) into equation (15), we get:

[0142]

[0143] Obviously, the set-valued Lie derivative is a single-element set. By using ‖u0‖ in Assumption 2 ∞ ≤γ and Inequality and K = -B T P, and If f is continuous, we can get from equation (16):

[0144]

[0145] Since α ≥ γ, (17) becomes:

[0146]

[0147] According to Lemma 2, we can conclude that H is a symmetric and positive definite matrix. Therefore, there must be an orthogonal matrix It turns out that:

[0148] T Τ HT=diag(λ1,λ2,λ3,…λ N )=Λ (19)

[0149] where, for any i = 1, 2, 3, ... N, λ1, λ2, λ3, ..., λ N is the eigenvalue of the matrix H. So let Substituting into (18) we get:

[0150]

[0151] As long as αλ i ≥1,i=1,2,...,N holds true. By applying equation (8), equation (20) becomes:

[0152]

[0153] Therefore, for any It means Since V2(t) is bounded, the adaptive coupling weight c i is also bounded. Therefore, system (6) is globally asymptotically stable, and the states of all followers can converge to the state of the leader.

[0154] And from the matrix Λ ≥ 0, it is easy to see that the adaptive time-varying coupling weight c i is non-decreasing, and the steady-state constant α can be large enough to satisfy αλ i ≥1,i=1,2,...,N.

[0155] In this paper, we focus on the distributed tracking problem of multi-agent systems with unknown inputs when the leader has a switching topology. Due to the switching topology, this means that the graph will change over time. Therefore, all possible connections should be considered while the system is running. Definition As a set of communication topology connected graphs, where p and Respectively represent the graph The index and index set of all possible graphs of . In order to describe the time required to switch the communication topology, by introducing the switching signal As time t changes, the multi-agent system connection communication topology is G σ(t) .

[0156] Since the communication topology between multiple agents changes over time and the leader has non-zero control input, the following distributed adaptive controller is designed:

[0157]

[0158]

[0159] Among them, u i represents the control input of the ith agent, η i is a constant and η i >0,c i is the time-varying coupling weight associated with each of the following, K∈R m×n is the feedback gain matrix, and a ij (t) is the (i,j)th graph The associated adjacency matrix element. Let d i (t) is the i-th diagonal element of the leader adjacency matrix D, τ i is a positive constant, Γ∈R n×n is a continuous gain matrix.

[0160] Note 4: Compared with the controller (4) with fixed communication topology, it is worth noting that the a between the agents described under the controller (22) ij (t) and d between the leader and the agent i (t) is due to the change of communication topology over time, resulting in a ij (t) and d i (t) changes with time.

[0161] Due to the change of communication topology, and definition D σ(t) =diag{d1(t),d2(t),…,d N (t)} is the graph G σ(t) The Laplacian matrix and the adjacency matrix of the leader at time t.

[0162] In order to analyze the time-varying topology of leader-follower multi-agent systems The following general assumptions are made:

[0163] There is a switching signal is a piecewise constant. is a finite set of all possible interconnection topologies of the multi-agent system, and t0 is the initial time. The definition represents all possible vertices {0,1,2,…,N} that may appear on the graph, and is represented by represents the set of all possible topological connection graphs, is a subgraph defined on the points {0,1,2,...,N}.

[0164] The time interval [t0,∞) consists of bounded, non-overlapping, continuous time intervals [t j ,t j+1 ) for j = 0, 1, ... and t0 = 0. For every interval [t j ,t j+1 ), there exists a non-overlapping sequence of subintervals

[0165] And meet

[0166] For some integer m k is an integer greater than 0, and a constant τ>0 is given, so that the topology structure is does not change and is represented by G σ(t) . In the time period Some or all of the communication topology diagrams and their table G kj (j=0,1,...m k -1) It is allowed to be non-connected as long as the joint communication topology satisfies the definition given below.

[0167] Definition 1: A joint graph is the union of graphs, that is, the vertex set and edge set of a graph are the union of the vertex set and edge set of a graph. If this joint graph is connected, then the joint graph is jointly connected. are jointly connected, then this set of graphs is said to be jointly connected in the time period [t, t+T], T>0.

[0168] The following assumptions are important for studying problems under switching topology:

[0169] Assumption 5: Assume that each time period [t k ,t k+1 )(k=0,1,…), the joint graph of the graphs formed by the multi-agent systems (2) and (3) is jointly connected.

[0170] Let i =x i -x0,ζ=(ζ1,ζ2,...,ζ N ), C=diag(c1,c2,...,c N ), η=(η1,η2,...,η i ), using the control rate (22) and substituting it into the state equations (2) and (3) and writing it in a compact form, we can obtain:

[0171]

[0172] In order to make the overall process clearer, and before giving the results, it is necessary to make a joint diagram The Laplacian matrix L+D is transformed as follows:

[0173] Step 1: Find graph G σ(t) All possible ingredients

[0174] Step 2: The Laplacian is written as in For ingredients The corresponding Laplacian matrix. And under the switching topology, the corresponding partitioning matrix is ​​specified as make in

[0175] Step 3: For each i∈{1,2,…,n p}, considering the corresponding components The corresponding matrix Assumptions but in and Thus there is for Matrix of order, and record its The characteristic value is Will Different symbols included in the diagram Rethinking the Marking Matrix Correspondingly,

[0176] Step 4: The matrix All corresponding eigenvalues ​​are arranged together in order.

[0177] After marking in this way, for all H p The corresponding N eigenvalues ​​are recorded as definition Then there is the following lemma.

[0178] Lemma4: If the joint graph is in the time period [t k ,t (k+1) ] is jointly connected if and only if

[0179]

[0180] Theorem 2: If the multi-agent system defined by (2) and (3) satisfies Assumption 2, and the communication switching topology described by the switching signal σ(t) satisfies Assumption 5, then there exists a symmetric positive definite matrix P that can be solved by solving the algebraic inequality Riccati equation:

[0181] PA+A T P-2PBB T P+βI n ≤0 (26)

[0182] Where β is a positive number, so that the feedback gain matrix is ​​designed K = -B T P, then the continuous gain matrix Γ=PBB T P. For any initial condition, the multi-agent system (2) is able to track (3) and implement distributed adaptive tracking control in the control protocols (22) and (23).

[0183] Choose the Lyapunov function:

[0184]

[0185] By taking the derivative of (27):

[0186]

[0187] The same method as (12)-(15) is used to obtain the following equation:

[0188]

[0189] Similarly, using the same method as (16)-(18), and α≥γ, we can obtain (30):

[0190]

[0191] Since the matrix H σ(t) is symmetric, and at any non-switching time, we can always find an orthogonal matrix T σ(t) , H σ(t) Transform into diagonal form.

[0192]

[0193] where π p is some permutation of the set {1,2,…,N}.

[0194] let After substituting, we get the following expression:

[0195]

[0196] For any A positive number α can make αλ i ≥1,i=1,2,...,N, from the above we can see and then exist.

[0197] According to the Cauchy convergence criterion, the infinite sequence V2(t i ),i=0,1,2,…,For any μ>0, there always exists a positive number M μ , so that for any m>M μ :

[0198]

[0199] Rewrite inequality (33) as the sum of multiple integrals:

[0200]

[0201] For each sub-item:

[0202]

[0203] So we have:

[0204]

[0205] Because there is a finite sequence of time intervals [t m ,t m+1 ], so the positive integer n k The value of is also finite, so for any m>M μ :

[0206]

[0207] According to Lemma 4, when the system communication topology graph is jointly connected in the time interval [t m ,t m+1 ),have

[0208] And (37) can be rewritten as:

[0209]

[0210] Among them, a 1 ,a2,…a n is a positive integer.

[0211] Therefore, there is It is not difficult to conclude This means that t→∞ ζ i (s) = 0, which means that the multi-agent system can track the leader agent and achieve the control goal by utilizing the adaptive controllers (22) and (23) under the switching topology.

[0212] Experimental proof:

[0213] Consider a multi-agent system and assume that the follower agents follow the leader and perform distributed adaptive tracking control under the control protocol (22).

[0214] Consider the case of multi-agent switching in the communication topology, assuming there are 4 agents and one leader, and the A and B coefficient matrices are given as follows:

[0215]

[0216]

[0217] It is obvious that (A, B) is stable, and the matrix A has no positive real eigenvalues, and the intelligent agent system achieves adaptive tracking control through formulas (22) and (23). Figure 1 A possible communication topology diagram is given from Figure 1 It can be seen that each possible graph is disconnected, but the joint graph of the graphs, that is, their union, is connected. According to Theorem 2, multi-agents will achieve distributed adaptive tracking control. The communication topology of the interaction between multi-agents will switch in sequence

[0218] exist Figure 1 The switching signal time of adjacent graphics is 1 / 3s, that is, the switching time cycle of the entire multi-agent system is 2s. The switching signal diagram is as follows: Figure 2 shown.

[0219] First, by solving LMI (26), the feedback gain matrix is ​​designed by calculating K = -B T P, continuous gain matrix Γ = PBB T P obtained the following data:

[0220]

[0221] Next, design the leader's control input, u0 = K2x0, where It can be concluded that the leader's state trajectory is an oscillator. And in this case, u0 is bounded. However, for ‖u0‖ ∞ ≤γ, this bounded value γ depends on the leader’s initial state x0(0), and in the case of topology switching, this state may not be known to the follower. Therefore, controllers (22) and (23) are used here to ensure that the follower can converge to the leader’s state. And the parameter c i (0) can be any value, τ i =1,η i =0.2, i=1,...4. The simulation results are as follows Figure 3-Figure 12 shown.

[0222] Depend on Figure 3-Figure 5 It can be seen that under the switching of communication topology, the states of the four agents can converge to the state of the leader over time. Figure 6-Figure 7 It can be seen that as time changes and the topology switches, the four agents follow the trajectory of the leader with unknown input respectively, and finally converge to the state of the leader.

[0223] Figure 8 It can be seen that the adaptive coupled time-varying weights of the multi-agent system composed of (2) and (3) are non-decreasing and eventually converge to a bounded value.

[0224] Depend on Figure 9-12 It can be seen that under the switching topology, the state error of the four agents tracking the leader by the control rates (22) and (23) converges to 0.

[0225] The robustness of the proposed method is verified for the multi-agent system composed of (2) and (3) with interference. Still, the data of the first simulation is used, with a leader and 4 multi-agents under the switching topology, and controllers (22) and (23) are adopted to solve the distributed adaptive tracking problem.

[0226] Add interference to each agent (2) and (3) respectively, such as And i=0,...4.

[0227] where w irepresents the state disturbance value of the ith agent and is unknown but bounded. In this case, the output disturbance value of all agents is:

[0228]

[0229] In addition, the other parameter values ​​are consistent with the above parameter values. Figure 13-Figure 19 shown.

[0230] Depend on Figure 13-14 It can be seen that multi-agents can still converge to the state of the leader in the presence of external interference. Fig.15 Shown are the adaptive time-varying coupling weights of a multi-agent system under disturbance. Figure 16-Figure 19 It can be seen that under interference conditions, the errors between the multi-agent and the leader converge to a certain range, which shows that the proposed controllers (22) and (23) are robust.

[0231] The present invention considers the distributed adaptive tracking control problem of a general linear multi-agent leader with unknown bounded input in the case of switching topology, and the control input of the leader is non-zero and unavailable to any follower. Based on the relative state error between two adjacent agents, a distributed controller with adaptive coupled time-varying weights is designed through the Riccati inequality. For the leader with non-zero control input, the distributed adaptive tracking control problem under fixed topology and switching topology is discussed respectively. Under the joint graph, the algorithm proposed in the present invention enables the state of the follower to approach the state of the leader, and the simulation verifies the effectiveness and robustness of the adaptive controller.

[0232] Based on the same inventive concept, the present invention also proposes a multi-agent system control device, comprising:

[0233] The dynamic model building module is used to build the dynamic model of the follower and the leader, and obtain the status data of the follower and the leader.

[0234] A controller building module is used to build a distributed adaptive tracking controller based on the state data of the followers and the state data of the leader.

[0235] A control module is provided for controlling each follower to track a leader with unknown control input through a distributed adaptive tracking controller.

[0236] The controller building blocks include:

[0237] A communication topology graph construction unit is used to build a communication topology graph of a multi-agent system using an undirected graph, and divide the communication topology graph into multiple subgraphs; wherein each of the subgraphs is non-connected, and the joint graph of the multiple subgraphs is connected.

[0238] A partitioning unit is used to determine the Laplacian matrix of each subgraph, and determine the adjacency matrix of the followers and the communication matrix between the leader and the followers according to the Laplacian matrix.

[0239] The relative error determination unit is used to determine the relative errors between followers and the relative errors between followers and the leader respectively according to the adjacency matrix of the followers and the communication matrix between the leader and the followers.

[0240] The control law determination unit is used to input the state data corresponding to the follower and the state data corresponding to the leader into the Riccati equation inequality to determine the feedback gain matrix and the continuous gain matrix of each follower and leader; and determine the control law of each follower according to the feedback gain matrix and the continuous gain matrix.

[0241] The weight determination unit is used to determine the adaptive weights and control parameters of each follower control law under the condition of unknown bounded control of the leader according to the feedback gain matrix and continuous gain matrix of each follower and leader, as well as the relative errors between followers and the relative errors between the leader and followers.

[0242] The controller construction unit is used to construct a distributed adaptive tracking controller based on the control law of each follower, the adaptive weight of the control law and its control parameters.

[0243] Based on the same inventive concept, the present invention also proposes a multi-agent system control computer device, including: a memory, a processor and a computer program stored in the memory, and the processor implements the steps of the multi-agent system control method when executing the computer program.

[0244] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. A method for controlling a multi-agent system, wherein the multi-agent system comprises a plurality of followers and a leader, characterized in that: The control method comprises the following steps: Establish a dynamic model of the follower and a dynamic model of the leader, analyze each dynamic model, and obtain the state data of the follower and the state data of the leader; Construct a distributed adaptive tracking controller based on the follower's state data and the leader's state data; Each follower is controlled by a distributed adaptive tracking controller to track the leader with unknown control input; The construction process of the distributed adaptive tracking controller includes the following steps: Use undirected graphs to build the communication topology of the multi-agent system and divide the communication topology into multiple subgraphs; Determine the Laplacian matrix of each subgraph, and determine the adjacency matrix of the followers and the communication matrix between the leader and the followers based on the Laplacian matrix; According to the adjacency matrix of the followers and the communication matrix between the leader and the followers, the relative errors between the followers and the relative errors between the followers and the leader are determined respectively; Inputting the state data corresponding to the follower and the state data corresponding to the leader into the Riccati equation inequality, determining the feedback gain matrix and the continuous gain matrix of each follower and the leader; and determining the control law of each follower according to the feedback gain matrix and the continuous gain matrix; According to the feedback gain matrix and continuous gain matrix of each follower and leader, as well as the relative errors between followers and the relative errors between leader and followers, the adaptive weights and control parameters of each follower control law are determined under the condition of unknown bounded control of the leader; Based on the control law of each follower, the adaptive weight of the control law and its control parameters, a distributed adaptive tracking controller is constructed.

2. A multi-agent system control method according to claim 1, characterized in that: The dynamic model of the follower is expressed as: Where: x i ∈R n is the status information of the follower; u i ∈R m is the control input of the follower, A and B are constant matrices with corresponding dimensions, Represents the first-order derivative of the following state variable; R n , R m Both represent constant real numbers; n and m both represent dimensions; The leader's dynamic model is expressed as: Where: x0∈R n is the leader’s state information; u0∈R m Provide control input for leaders; Represents the first-order derivative of the leader's multi-state variable.

3. A multi-agent system control method according to claim 1, characterized in that: The communication topology diagram of the multi-agent system constructed by using an undirected graph specifically includes: defining As a set of communication topology graphs, where p and Respectively represent the graph The index and index set of all subgraphs in the graph, each subgraph is non-connected, and the joint graph of the multiple subgraphs is connected; by introducing the switching signal As time t changes, the communication topology of the multi-agent system is expressed as G σ(t) .

4. A multi-agent system control method according to claim 3, characterized in that: Determining the Laplacian matrix of each subgraph, and determining the adjacency matrix of the followers and the communication matrix between the leader and the followers according to the Laplacian matrix; The specific steps include: Define the adjacency matrix W of the followers, the Laplacian matrix L is obtained by subtracting the degree matrix F and W between the followers, and define the communication matrix D between the leader and the followers; The element of the adjacency matrix W = {a ij } satisfies the edge ε ij ∈E, then a ij =1, otherwise a ij =0; i represents the number of followers; j represents the number of adjacent followers of follower i; E represents the set of all edges; Define the in-degree matrix to be a diagonal matrix in b i 、b n are all elements in the diagonal matrix, represents a real number, and N represents the number of followers; the Laplace matrix of an undirected graph G is defined as: L = FW; when there is at least one path between any two nodes of the graph G, the undirected graph is called a connected graph; for a multi-agent system consisting of a leader and N followers, the leader can only send information to its neighbor agents, but does not receive information from anyone; Define the communication matrix D between the leader and the followers D = diag{d1,d2,...,d N }∈R N×N ; When the leader and follower are connected, d i =1, otherwise d i =0; define graph It is composed of the leader, the edges between the leader and followers, and the followers and the graph G.

5. A multi-agent system control method according to claim 4, characterized in that: The relative error between the followers is expressed as The relative error between the leaders is denoted as d i (t)(x i -x0); where a ij (t) is the (i,j)th graph The associated adjacency matrix element; d i (t) is the i-th diagonal element of the communication matrix D between the leader and the follower, x i 、x j They represent the status information of follower i and follower j respectively, and x0 represents the status information of the leader.

6. A multi-agent system control method according to claim 5, characterized in that: The Riccati equation inequality is expressed as: PA+A T P-2PBB T P+βI n ≤0 Where β is a positive number, A and B are constant matrices with corresponding dimensions, T is the matrix transpose, and I n is an identity matrix, n represents the dimension, and p represents the solution of the above Riccati equation; According to the Riccati equation inequality, P is solved and the feedback gain matrix and continuous gain matrix of each follower and leader are calculated respectively, which are expressed as follows: Feedback gain matrix K = -B T P, continuous gain matrix Γ = PBB T P.

7. A multi-agent system control method according to claim 6, characterized in that: The adaptive weights of the control laws of each follower It is expressed as: Among them, τ i is a normal number, ζ i , j They represent the relative errors between followers i, j and the leader respectively, and j represents the adjacent follower of follower i.

8. A multi-agent system control method according to claim 7, characterized in that: The distributed adaptive tracking controller is expressed as: Among them, u i represents the control input of the distributed adaptive tracking controller, η i is a constant and η i >0,c i is the adaptive weight of the control law, K∈R m×n is the feedback gain matrix, and a ij (t) is the (i,j)th graph The associated adjacency matrix element, d i (t) is the i-th diagonal element of the adjacency matrix D, τ i is a positive constant, Γ∈R n×n is a continuous gain matrix.

9. A multi-agent system control device, the multi-agent system comprising a plurality of followers and a leader, characterized in that: include: The dynamic model building module is used to build the dynamic model of the follower and the leader, and obtain the status data of the follower and the leader; A controller building module, used to build a distributed adaptive tracking controller based on the state data of the followers and the state data of the leader; a control module for controlling each follower to track a leader of unknown control inputs through a distributed adaptive tracking controller; Wherein, the controller building module includes: A communication topology graph construction unit, used to construct a communication topology graph of a multi-agent system using an undirected graph, and divide the communication topology graph into a plurality of subgraphs; wherein each of the subgraphs is non-connected, and a joint graph of the plurality of subgraphs is connected; A partitioning unit, for determining a Laplacian matrix of each subgraph, and determining an adjacency matrix of followers and a communication matrix between leaders and followers according to the Laplacian matrix; A relative error determination unit, used to determine the relative errors between followers and the relative errors between followers and the leader respectively according to the adjacency matrix of the followers and the communication matrix between the leader and the followers; A control law determination unit is used to input the state data corresponding to the follower and the state data corresponding to the leader into the Riccati equation inequality to determine the feedback gain matrix and the continuous gain matrix of each follower and the leader; and determine the control law of each follower according to the feedback gain matrix and the continuous gain matrix; A weight determination unit, for determining the adaptive weights and control parameters of each follower control law under the condition of unknown bounded control of the leader, according to the feedback gain matrix and the continuous gain matrix of each follower and the leader, as well as the relative errors between the followers and the relative errors between the leader and the followers; The controller construction unit is used to construct a distributed adaptive tracking controller based on the control law of each follower, the adaptive weight of the control law and its control parameters.

10. A multi-agent system controlling computer equipment, characterized in that: include: A memory, a processor and a computer program stored in the memory, wherein the processor implements the steps of the multi-agent system control method according to any one of claims 1 to 8 when executing the computer program.

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