A Fixed-Time Consensus Control Method for Linear Multi-Agent Systems

Abstract

The present invention relates to a fixed-time consensus control method for a linear multi-agent system, belonging to the technical field of automatic control. It is a consensus control method that enables each follower in the linear multi-agent system to reach the state of tracking the leader and maintain the same state as the leader within a fixed time. It can enable each follower in the system to reach the same state as the leader within the limited time upper limit, ensuring a relatively fast convergence speed and accuracy, and the limited time upper limit is independent of the initial conditions, which is convenient for early estimation. Compared with other existing methods, this method can transform each linear agent with multiple inputs into a single-input system and ensure that each follower has the same motion mode as the leader.

Description

Technical Field

[0001] The present invention belongs to the technical field of automatic control, and particularly relates to a fixed-time consensus control method for a linear multi-agent system, which is a consensus control method for each follower in the linear multi-agent system to reach tracking the leader and maintaining the same state as the leader within a fixed time. Background Art

[0002] The consensus control problem is an important basic problem for realizing the cooperative control of a multi-agent system, and has extensive applications in many fields such as biological systems, sensor networks, robot teams, aircraft formations, etc. A linear protocol constructed through the local information available to each agent can make the states of each agent asymptotically converge to consensus when time approaches infinity. Compared with asymptotic consensus control, the fixed-time consensus control method exhibits a faster convergence rate, higher precision, and excellent anti-interference ability. However, the consensus establishment time of the fixed-time control method is usually related to the initial conditions and is difficult to estimate in advance. Summary of the Invention

[0003] In view of the above problems, the present invention provides a fixed-time consensus control method for a linear multi-agent system, which can enable each follower in the system to reach the same state as the leader within a limited time upper bound, ensuring a faster convergence speed and precision, and the limited time upper bound is independent of the initial conditions and is convenient for advance estimation. Compared with other existing methods, this method can transform each linear agent with multiple inputs into a single-input system and ensure that each follower has the same motion mode as the leader.

[0004] A fixed-time consensus control method for a linear multi-agent system provided by the present invention, the agents include a leader and multiple followers, and the specific steps are as follows:

[0005] Step 1, establish the state models of the followers and the leader in the linear multi-agent system, and the expressions are respectively:

[0006]

[0007]

[0008] Wherein, is the derivative of the state vector of the i-th follower at time t; is the derivative of the state vector of the leader at time t; x i (t) is the state vector of the i-th follower at time t; x 0 (t) is the state vector of the leader at time t; A is the system matrix of the agent state equation; B is the control matrix of the agent state equation; u i(t) is the controller of the i-th follower at time t; u 0 (t) is the controller of the leader at time t;

[0009] Step 2: Establish the controller models of the followers and the leader, and the expressions are:

[0010] Follower controller model:

[0011] u i (t) = Kx i (t) + hv i (t);

[0012] The leader controller model is:

[0013] u 0 (t) = Kx 0 (t) + hv 0 (t)

[0014] Among them, u i (t) is the controller of the i-th follower at time t; u 0 (t) is the controller of the leader at time t; K is the state feedback coefficient matrix; v i (t) is the control input of the i-th follower at time t; v 0 (t) is the control input of the leader at time t; h is the gain of the control input of the agent;

[0015] Step 3: Perform a coordinate transformation on the state vectors of the followers and the leader in the linear multi-agent system to obtain the transformed coordinates of the followers and the leader;

[0016] Step 4: Establish a distributed fixed-time observer to obtain the observed values of each follower for the leader's state, and the expressions are:

[0017]

[0018] Among them, represents the observed value of the j-th state component of the leader by the i-th follower at time t, j = 1, 2,..., n, and n is the total dimension of the state components of the agent; represents the observed value of the (j + 1)-th state component of the leader by the i-th follower at time t; represents the observed value of the n-th state component of the leader by the i-th follower at time t; β 1 、β 2 、β 3 are the parameter weights of the observer respectively; α j is the j-th coefficient of the characteristic polynomial of the system matrix of the agent state equation after introducing state feedback; Denote the deviation value of the j-th state component estimate of the j-th follower from its neighboring agents; Denote the deviation value of the n-th state component estimate of the i-th follower from its neighboring agents;

[0019] Step 5, Obtain the fixed-time consensus control input;

[0020] Based on the observations of each follower on the leader's state in Step 4, obtain the consensus tracking error, and the expression is:

[0021]

[0022] where, Denote the derivative of the consensus tracking error of the j-th state component of the i-th follower at time t; e i,j (t) denotes the consensus tracking error of the j-th state component of the i-th follower at time t; e i,j+1 (t) denotes the consensus tracking error of the j + 1-th state component of the i-th follower at time t; Denote the derivative of the consensus tracking error of the n-th state component of the i-th follower at time t; Denote the deviation of the j-th state component estimate of the i-th follower; Denote the deviation of the n-th state component estimate of the i-th follower;

[0023] Construct the sliding mode surface of the follower, and the expression is:

[0024]

[0025] where, s i (t) denotes the sliding mode surface of the i-th follower at time t; r j , ρ j and ρ′ j are all sliding mode surface parameters of the j-th state component; τ represents the upper limit of integration at time t;

[0026] Based on the consensus tracking error and the sliding mode surface, obtain the fixed-time consensus control input quantity of the follower;

[0027] Step 6, Use the fixed-time consensus control input quantity obtained in Step 5 to control the following actions of the corresponding follower.

[0028] Optionally, in Step 4, β 1 is greater than the maximum absolute value v of the control input quantity v 0 (t) of the leader at time t max and / or β 2 >0 and / or where, λis the lower bound of the eigenvalues of the interaction matrix (L + C), 0 ≤ λ ≤ λ min (L + C); λ min (L + C) represents the minimum eigenvalue of the interaction matrix (L + C); L is the Laplacian matrix of the followers; C is the connectivity matrix between the i followers and the leader, C = diag(c 1 , c 2 , …, c N ); is the absolute value of the term with the largest absolute value among the coefficients of the characteristic polynomial of the system matrix of the state equation of the agent after introducing state feedback.

[0029] Optionally, ε is the variable in sign(ε).

[0030] Optionally, and / or and / or r j > 0, where ρ ∈ (ω, 1) and ω is a positive number less than 1.

[0031] Optionally,

[0032] Optionally, the deviation value of the estimated value of the j-th state component of the i-th follower from its adjacent agents is expressed as:

[0033]

[0034] In the formula, a ik is the weight of the edge between the i-th follower and the k-th follower, k ∈ {1, 2, …, N}, i ≠ k; represents the observed value of the j-th state component of the leader by the k-th follower at time t; represents the state vector of the j-th state component of the leader after coordinate transformation in step 3; c i is the weight of the edge between the i-th follower and the leader.

[0035] Optionally, based on the consensus tracking error and the sliding mode surface, the fixed-time consensus control input of the follower is obtained, and the expression is:

[0036]

[0037] where v i (t) represents the control input of the i-th follower at time t; g i and μ represent control parameters.

[0038] Optionally, the state feedback coefficient matrix K satisfies that the agent system (A + BK, b) after introducing state feedback is controllable, where b ∈ Im(B) and b ≠ 0, and Im(B) is the range space of the control matrix B of the agent state equation.

[0039] Compared with the prior art, the present invention has at least the following beneficial effects:

[0040] (1) The method of the present invention adopts a distributed state observer, enabling the followers to observe the state of the leader, and having fixed-time convergence, which improves the convergence speed.

[0041] (2) The method of the present invention can ensure that the linear multi-agent system reaches consensus within a fixed time upper bound, and the time upper bound is independent of the initial state, reducing the difficulty of consensus control.

[0042] (3) The method of the present invention converts each linear agent with multiple inputs into a single-input system, and ensures that each follower and the leader have the same motion mode. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] The drawings are only for the purpose of showing specific embodiments and are not considered as a limitation to the present invention.

[0044] Figure 1 is a flowchart of the method of the present invention;

[0045] Figure 2 is a schematic diagram of a multi-agent system in the method of the present invention.

[0046] Figure 3 is a result graph obtained by conducting a simulation experiment in the method of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0047] In order to more clearly understand the above objects, features and advantages of the present invention, the present invention will be further described in detail below with reference to the drawings and specific embodiments. It should be noted that, without conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. In addition, the present invention can also be implemented in other ways different from those described herein. Therefore, the protection scope of the present invention is not limited by the specific embodiments disclosed below.

[0048] A specific embodiment of the present invention, as Figure 1-2 , discloses a fixed-time consensus control method for a linear multi-agent system. The agents include a leader and multiple followers, and the specific steps are as follows:

[0049] Step 1: Establish the state models of the followers and the leader in the linear multi-agent system;

[0050] The state model expressions of the leader and followers are as follows:

[0051]

[0052]

[0053] Among them, is the derivative of the leader's state vector at time t; is the derivative of the state vector of the i-th follower at time t; A is the system matrix of the agent state equation; x 0 (t) is the leader's state vector at time t; x i (t) is the state vector of the i-th follower at time t; B is the control matrix of the agent state equation; u i (t) is the controller of the i-th follower at time t; i ∈ {1, 2, …, N}, and N is the total number of followers. u 0 (t) is the leader's controller at time t.

[0054] It can be understood that the agent state vector described in the present invention is the motion state of the agent, such as running speed, running direction, running acceleration, etc.

[0055] Let the initial state vectors of the leader and followers be:

[0056]

[0057]

[0058] Among them, x 0 (0) is the leader's state vector at the initial time; x 0,n (0) is the initial value of the n-th dimensional state component of the leader; x i (0) is the state vector of the i-th follower at the initial time; x i,n (0) is the initial value of the n-th dimensional state component of the i-th follower; n is the total dimension of the agent's state vector; is the n-dimensional Euclidean space.

[0059] Step 2: Establish the controller models of the followers and the leader, and the expressions are:

[0060] Follower controller model:

[0061] u i (t) = Kx i (t) + hv i (t);

[0062] The leader controller model is:

[0063] u 0 (t) = Kx0 (t) + hv 0 (t);

[0064] where u i (t) is the controller of the i-th follower at time t; u 0 (t) is the controller of the leader at time t; K is the state feedback coefficient matrix; v i (t) is the control input of the i-th follower at time t; v 0 (t) is the control input of the leader at time t; h is the gain of the control input of the agent;

[0065] Optionally, the state feedback coefficient matrix K satisfies that the agent system (A + BK, b) is controllable after introducing state feedback, where b ∈ Im(B) and b ≠ 0, and Im(B) is the range space of the control matrix B of the agent state equation, that is, the linear space spanned by the column vectors in the control matrix B of the agent state equation; take K = [δ 1 δ 2 … δ n-1 0][χ 1 χ 2 … χ n -1 , which ensures that the state feedback coefficient matrix K can make the intelligent system (A + BK, b) satisfy the controllability condition after introducing state feedback; construct the feedback auxiliary vector is the set of real numbers, p is the dimension of the controller state vector, and it needs to satisfy that the vector is linearly independent, and the expressions of the vectors χ 1 , χ 2 , …, χ n are:

[0066]

[0067] n is the dimension of the state vector of the agent.

[0068] Optionally, the method for determining the feedback auxiliary vectors δ 1 , δ 2 , …, δ n-1 is as follows:

[0069] If the rank of the control matrix of the agent state equation rank(B) = r, then r ≤ min(n, p), and there are r linearly independent columns in the control matrix B = [B 1 B 2 … B p of the agent state equation. Let [B 1 B 2 … ​B r ;

[0070] Let the control input of the \(i\)-th follower at time \(t\) be \(v\) i (t) gain Obtain \(Bh = B\) 1 ;

[0071] According to linear system theory, there exists a vector group

[0072] that is linearly independent, and the exponent of the system matrix \(A\) of the agent state equation satisfies \(\xi\) 1 +\(\xi\) 2 +…+\(\xi\) r = \(n\), where \(\xi\) r represents the maximum exponent of the system matrix \(A\) of the agent state equation that makes linearly independent.

[0073] At this time, let

[0074]

[0075] Here, refers to the unit vector with the \(\eta\)-th component being 1.

[0076] Obtain:

[0077] When \(\xi\) 1 ≠ 1,

[0078]

[0079] The two sets of column vectors on both sides of the equation are linearly independent. Obtain the construction of the feedback auxiliary vector \(\delta\) 1 , \(\delta\) 2 , …, \(\delta\) n-1

[0080] When \(\xi\) 1 = 1,

[0081] The two sets of column vectors on both sides of the equation are linearly independent, and obtain the construction of the feedback auxiliary vector \(\delta\) 1 , \(\delta\) 2 , …, \(\delta\) n-1 .

[0082] Step 3: Transform the coordinates of the followers and the leader in the linear multi-agent system;

[0083] Obtain the coordinate transformation matrix \(M\), and the expression is:

[0084]

[0085] Where, The system matrix of the agent state equation after introducing state feedback b = Bh; α n is the nth coefficient of the characteristic polynomial of, and the expression is:

[0086]

[0087] where s is the Laplace operator; I is the identity matrix with the same number of rows and columns as the

[0088] Perform coordinate transformation on the coordinates of the followers and the leader, and the expression is:

[0089]

[0090]

[0091] where is the state vector of the leader after coordinate transformation at time t; is the state vector of the ith follower after coordinate transformation at time t.

[0092] Step 4, Establish a distributed fixed-time observer;

[0093] Obtain the observation value of each follower on the leader's state through the distributed fixed-time observer, and the expression is:

[0094]

[0095] where represents the observation value of the jth state component of the leader by the ith follower at time t; represents the observation value of the (j + 1)th state component of the leader by the ith follower at time t; represents the observation value of the nth state component of the leader by the ith follower at time t; β 1 , β 2 and β 2 are the parameter weights of the observer respectively. Preferably, β 1 is greater than the maximum absolute value v of the control input v 0 (t) of the leader at time t, v max , β 2 > 0, λ is the lower bound of the eigenvalues of the interaction matrix (L + C), 0 ≤ λ ≤ λ min (L + B), λ min (L + B) represents the minimum eigenvalue of the interaction matrix (L + C); L is the Laplacian matrix of the undirected graph formed by the followers; C is the connectivity matrix of the i followers and the leader, C = diag(c1 , c 2 , …, c N ), where c i is the weight of the edge between the i-th follower and the leader; ε is the variable in sign(ε); is the system matrix of the agent state equation after introducing state feedback the absolute value of the term with the largest absolute value among the coefficients of the characteristic polynomial of; α j is the system matrix of the agent state equation after introducing state feedback the j-th coefficient of the characteristic polynomial of; represents the inconsistency deviation value between the estimated value of the j-th state component of the i-th follower and its adjacent agents, and the expression is:

[0096]

[0097] In the formula, a ik is the weight of the edge between the i-th follower and the k-th follower, k ∈ {1, 2, …, N}, i ≠ k; represents the observed value of the j-th state component of the leader by the k-th follower at time t; represents the state vector of the j-th state component of the leader after coordinate transformation in step 3 at time t; represents the inconsistency deviation value between the estimated value of the n-th state component of the i-th follower and its adjacent agents.

[0098] It can be understood that at this time, the variable ε in sign(ε) is

[0099] Step 5, obtain the fixed-time consensus control input;

[0100] Obtain the consensus tracking error through the distributed fixed-time observer established in step 4, and the expression is:

[0101]

[0102] Among them, represents the derivative of the consensus tracking error of the j-th state component of the i-th follower at time t; e i，j (t) represents the consensus tracking error of the j-th state component of the i-th follower at time t; e i，j+1 (t) represents the consensus tracking error of the j + 1-th state component of the i-th follower at time t; represents the inconsistency deviation of the estimated value of the j-th state component of the i-th follower, Denote the derivative of the consensus tracking error of the nth state component of the ith follower at time t; Denote the inconsistency deviation of the estimated value of the nth state component of the ith follower.

[0103] Construct the sliding mode surface of the follower, and the expression is:

[0104]

[0105] where s i (t) represents the sliding mode surface of the ith follower at time t; r j , ρ j and ρ′ j are all sliding mode surface parameters of the jth state component. Preferably, ρ ∈ (ω, 1), r j > 0; τ represents the upper limit of integration at time t.

[0106] Obtain the fixed-time consensus control input quantity of the follower based on the consensus tracking error and the sliding mode surface, and the expression is:

[0107]

[0108] where v i (t) represents the control input quantity of the ith follower at time t; g i and μ both represent control parameters. Preferably, g i > 0, μ > 1.

[0109] Step 6, use the fixed-time consensus control input quantity obtained in Step 5 to control the following actions of the corresponding follower.

[0110] As mentioned above, it is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed by the present invention should be covered within the protection scope of the present invention.

Claims

1. A fixed-time consensus control method for a linear multi-agent system, characterized in that, the agents include a leader and multiple followers, and the specific steps are as follows: Step 1: Establish the state models of the followers and the leader in the linear multi-agent system, and the expressions are respectively: wherein, is the derivative of the state vector of the i-th follower at time t; is the derivative of the state vector of the leader at time t; x i (t) is the state vector of the i-th follower at time t; x 0 (t) is the state vector of the leader at time t; A is the system matrix of the agent state equation; B is the control matrix of the agent state equation; u i (t) is the controller of the i-th follower at time t; u 0 (t) is the controller of the leader at time t; N is the total number of followers; Step 2: Establish the controller models of the followers and the leader, and the expressions are: Follower controller model: u i u(t) = Kx i (t) + hv i (t); Leader controller model: u 0 u(t) = Kx 0 (t) + hv 0 (t) where, u i (t) is the controller of the i-th follower at time t; u 0 (t) is the controller of the leader at time t; K is the state feedback coefficient matrix; v i (t) is the control input of the i-th follower at time t; v 0 (t) is the control input of the leader at time t; h is the gain of the control input of the agent; Step 3: Perform coordinate transformation on the state vectors of the followers and the leader in the linear multi-agent system to obtain the transformed coordinates of the followers and the leader; Step 4: Establish a distributed fixed-time observer to obtain the observation values of each follower for the leader's state, and the expression is: wherein, represents the observed value of the j-th state component of the i-th follower with respect to the leader at time t, where j = 1, 2, …, n, and n is the total dimension of the state components of the agents; represents the observed value of the (j + 1)-th state component of the i-th follower with respect to the leader at time t; represents the observed value of the n-th state component of the i-th follower with respect to the leader at time t; β 1 β 2 β 3 are the parameter weights of the observer respectively; α j is the system matrix of the agent state equation after introducing state feedback which is the j-th coefficient of the characteristic polynomial of; represents the inconsistent deviation value between the estimated value of the j-th state component of the j-th follower and its adjacent agent; represents the inconsistent deviation value between the estimated value of the n-th state component of the i-th follower and its adjacent agent; Step 5: Obtain the fixed-time consensus control input; Based on the observation values of each follower for the leader's state in Step 4, obtain the consensus tracking error, and the expression is: Among them, represents the derivative of the consensus tracking error of the j-th state component of the i-th follower at time t; e i,j (t) represents the consensus tracking error of the j-th state component of the i-th follower at time t; e i,j+1 (t) represents the consensus tracking error of the (j + 1)-th state component of the i-th follower at time t; represents the derivative of the consensus tracking error of the n-th state component of the i-th follower at time t; represents the disagreement deviation of the estimated value of the j-th state component of the i-th follower; represents the disagreement deviation of the estimated value of the n-th state component of the i-th follower; Construct the sliding mode surface of the follower, and the expression is: Among them, s i (t) represents the sliding mode surface of the i-th follower at time t; r j , ρ j and ρ' j are all sliding mode surface parameters of the j-th state component; τ represents the upper limit of integration at time t; Based on the consensus tracking error and the sliding mode surface, obtain the fixed-time consensus control input quantity of the follower; Step 6: Use the fixed-time consensus control input quantity obtained in Step 5 to control the following actions of the corresponding followers.

2. The fixed-time consensus control method according to claim 1, characterized in that, In step 4, β 1 is greater than the maximum absolute value of the control input v 0 (t) of the leader at time t, v max and / or β 2 >0 and / or where λ is the lower bound of the eigenvalues of the interaction matrix (L + C), 0 ≤ λ ≤ λ min (L + C); λ min (L + C) represents the minimum eigenvalue of the interaction matrix (L + C); L is the Laplacian matrix of the followers; C is the connectivity matrix of the i followers and the leader, C = diag(c 1 , c 2 , …, c N ); is the absolute value of the term with the largest absolute value among the coefficients of the characteristic polynomial of the system matrix of the state equation of the agent after introducing state feedback.

3. The fixed-time consensus control method according to claim 1, characterized in that, 「ε」 γ = |ε| γ sign(ε), ε is the variable in sign(ε).

4. The fixed-time consensus control method according to claim 1, characterized in that, and / or and / or r j > 0, where ρ ∈ (ω, 1) and ω is a positive number less than 1.

5. The fixed-time consensus control method according to claim 4, characterized in that, 6. The fixed-time consensus control method according to claim 1, characterized in that, The inconsistency deviation value between the estimated value of the j-th state component of the i-th follower and its neighboring agents is expressed as: where a ik is the weight of the edge between the \(i\)-th follower and the \(k\)-th follower, \(k\in\{1,2,\ldots,N\}\), \(i\neq k\); denotes the observation value of the \(j\)-th state component of the leader by the \(k\)-th follower at time \(t\); denotes the state vector of the \(j\)-th state component of the leader after coordinate transformation in step 3; \(c\) i is the weight of the edge between the \(i\)-th follower and the leader.

7. The fixed-time consensus control method according to claim 1, characterized in that, Based on the consensus tracking error and the sliding mode surface, obtain the fixed-time consensus control input quantity of the follower, and the expression is: where, v i (t) represents the control input of the i-th follower at time t; g i and μ represent control parameters.

8. The fixed-time consensus control method according to claim 1, characterized in that, the state feedback coefficient matrix K satisfies that the agent system (A + BK, b) is controllable after introducing state feedback, where b ∈ Im(B) and b ≠ 0, and Im(B) is the range space of the control matrix B of the agent state equation.

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