Multi-agent system fixed time encircling control method under triggering of intermittent dynamic event

Abstract

The invention provides a multi-agent system fixed time encircling control method under intermittent dynamic event triggering, and the method comprises the following steps: S1, constructing a directed topological graph and a Laplacian matrix of a multi-agent system according to an actual task; s2, constructing a dynamic model of the multi-agent system; s3, designing a multi-agent system fixed time encircling controller triggered by an intermittent dynamic event; s4, the controller designed in the step S3 is used for achieving surrounding control within fixed time, and the upper bound of convergence time is obtained; and S5, continuously operating until the encircling control of the multi-agent system is completed.

Description

Technical Field

[0001] The present invention relates to the field of artificial intelligence and robotics technology, and in particular to a fixed-time enclosure control method for a multi-agent system triggered by intermittent dynamic events. Background Art

[0002] In recent years, with the advancement of science and technology, the cooperative control of multi-agent systems has been widely used in more and more fields, such as drone formation, robot collaboration, and satellite switching. Among the many cooperative control problems, the encirclement control problem is one of the important problems. In encirclement control, the agents in the system are divided into followers and leaders. The state of the follower agent needs to be in the convex hull formed by the state of the leader agent, so as to achieve the purpose of restricting the entire system to a certain area.

[0003] Traditional multi-agent system research usually uses a fixed time interval sampling scheme to update the control input, but this scheme can cause problems such as communication congestion. In contrast, the event-triggered control strategy presets a threshold and only updates the control input when the measurement error exceeds the threshold, thereby reducing the number of control input updates. Dynamic event triggering goes a step further, and the set threshold changes with the system state, further reducing the number of input updates and alleviating communication congestion problems.

[0004] At the same time, in order to solve the problem of excessive energy consumption caused by continuous input of control signals, and to cope with the interruption of control input caused by factors such as network attacks, intermittent control is also widely used. The control process of intermittent control is divided into a control phase and an intermittent phase. The controller only inputs in the control phase and does not input in the intermittent phase, thereby reducing system energy consumption and enhancing system robustness.

[0005] In addition, the convergence speed of the system is also an important indicator for evaluating a system. The convergence time of the finite time control strategy is generally shorter than that of the asymptotic control, but the convergence time of the finite time control is related to the initial state. While the fixed time has a finite convergence time, the upper bound of its convergence time is independent of the initial state of the system and can be applied in more scenarios.

[0006] The present application is different from the prior art as follows:

[0007] Technical comparison with patent CN118331047A "A multi-agent dynamic event triggered fixed time binary consistency control method"

[0008] Patent CN118331047A is aimed at a second-order linear multi-agent system, and its topological relationship is an undirected graph; this patent is aimed at a first-order nonlinear multi-agent system, and its topological relationship is a directed graph. There is an essential difference between the two in the model of the control object. This patent has better adaptability when dealing with disturbances in real application scenarios.

[0009] Patent CN118331047A uses a continuous control strategy to achieve the control target, and each agent always has control input; this patent uses an intermittent control strategy to achieve the control target, and by dividing the control process into a control phase and an intermittent phase, the number of control input updates is reduced, the robustness of the system is enhanced, and power consumption is reduced. There is an essential difference between the two at the control strategy level, and the control strategy of this patent is more suitable for distributed multi-agent systems with limited communication resources and energy.

[0010] The control goal of patent CN118331047A is to achieve binary consistency of multi-agent system, so that the agents in the agent group reach a consensus; the control goal of this patent is to achieve encirclement control of multi-agent system, ensuring that the status of all followers is maintained within the convex hull formed by the leader. There is an essential difference between the two in terms of control goals.

[0011] Technical comparison with patent CN118244638A "Event-triggered nonlinear multi-agent fixed-time time-varying formation control method"

[0012] Patent CN118244638A adopts a static event trigger mechanism, and its trigger threshold is relatively fixed; this patent adopts a more flexible dynamic event trigger mechanism, and dynamically adjusts the trigger threshold by designing a dynamic variable that changes with the system state, thereby further reducing the number of triggers and reducing energy consumption. There is an essential difference between the two at the trigger mechanism level, and this patent can achieve the control target with fewer triggers.

[0013] Patent CN118244638A uses a continuous control strategy to achieve the control target, and each agent always has control input; this patent uses an intermittent control strategy to achieve the control target, and by dividing the control process into a control phase and an intermittent phase, the number of control input updates is reduced, the robustness of the system is enhanced, and power consumption is reduced. There is an essential difference between the two at the control strategy level, and the control strategy of this patent is more suitable for distributed multi-agent systems with limited communication resources and energy.

[0014] The control goal of patent CN118244638A is to realize the formation control of the multi-agent system, so that the agents reach and maintain the specified state; the control goal of this patent is to realize the encirclement control of the multi-agent system, so that the states of all followers remain within the convex hull formed by the leader. There is an essential difference between the two in terms of control goals.

[0015] In summary, it has become an important issue to be solved urgently to combine the dynamic event-triggered control strategy, the intermittent control strategy and the fixed-time control strategy to form a comprehensive encirclement control scheme, so as to achieve the control objectives of low energy consumption in the control process, strong system robustness and flexible design of convergence time. Summary of the invention

[0016] In order to solve the above technical problems, the present invention proposes a fixed-time enclosure control method for a multi-agent system under the triggering of intermittent dynamic events, so that the multi-agent system can complete the enclosure control within a fixed time.

[0017] To achieve the above object, the technical solution adopted by the present invention is:

[0018] A fixed-time enclosure control method for a multi-agent system triggered by intermittent dynamic events is characterized by:

[0019] Step S1: Construct the directed topological graph and Laplace matrix of the multi-agent system according to the actual task;

[0020] Step S2: construct a dynamic model of the multi-agent system;

[0021] Step S3: Design a fixed-time enclosure controller for a multi-agent system triggered by intermittent dynamic events;

[0022] Step S4: Use the controller designed in step S3 to achieve encirclement control within a fixed time, and obtain the upper bound of the convergence time;

[0023] Step S5: Continue running until the encirclement control of the multi-agent system is completed.

[0024] As a further improvement of the present invention, the process of constructing the directed topological graph and Laplace matrix of the multi-agent system in step S1 includes:

[0025] For a multi-agent system consisting of N agents, its communication topology is a directed graph G with N nodes, where each node represents an agent. The number of follower and leader agents is n and m respectively, then nodes 1 to n represent follower agents, nodes n+1 to n+m represent leader agents, and the edge set is represented by E. If node i can receive information from node j, then (i, j)∈E. The adjacency matrix is ​​represented as If (i,j)∈E, then a ij >0, otherwise a ij =0, and a ii = 0. The in-degree matrix is ​​defined as Where diag represents a diagonal matrix, The Laplacian matrix of graph G is defined as

[0026] As a further improvement of the present invention, constructing the dynamics model of the multi-agent system in step S2 includes:

[0027] The dynamic model of the follower agent is:

[0028]

[0029] The dynamic model of the leader agent is:

[0030]

[0031] Where V F With V L Represent the follower set and the leader set, that is, V F ={1,...,n},V L ={n+1,...,m+n}, represents the state of the ith agent, represents the control input of the ith agent, Represents a nonlinear function.

[0032] As a further improvement of the present invention, the process of designing a fixed-time enclosure controller for a multi-agent system triggered by an intermittent dynamic event in step S3 includes:

[0033] S31. Design of intermittent controller:

[0034]

[0035] Among them, 0<α<1<β, η1, η2, η3 and η4 are positive real numbers, all of which are parameters that need to be designed, Ψ>>1, represents the last triggering time of agent i, Defined as

[0036] S32. Design measurement error:

[0037]

[0038] where ∈ i (t) is an auxiliary variable, which is defined as:

[0039]

[0040] S33. Design trigger functions, dynamic variables and trigger conditions;

[0041] The trigger function is designed as:

[0042]

[0043] Among them, the two parameters 0<θ<1 and H>0 need to be designed.

[0044] Dynamic variables are designed as:

[0045]

[0046] Among them, Λ i (0)>0, 0<κ<1, χ1, χ2, χ3, χ4>0 are all parameters that need to be designed. Set the trigger conditions as:

[0047]

[0048] As a further improvement of the present invention, the upper bound of the convergence time obtained in step S4 is specifically:

[0049] (1) System needs to be guaranteed (1-τ)ν-ρ3<0 and ρ4-τν<0 both hold, and the following assumptions are satisfied:

[0050] Assumption 1: Each leader is a neighbor of only some of the followers, and at least one leader is connected to all other followers. In addition, there exists a positive real number δ i , i∈V F , so that δ j a ij =δ i a ji holds, where i,j∈V F ,

[0051] At the same time, according to Assumption 1, the Laplace matrix can be written as

[0052]

[0053] in

[0054] Assumption 2: There exists a positive real number ψ such that

[0055]

[0056] Where i∈V F ∪V L , k j ≥0,

[0057] (2) The upper bound of the system’s convergence time is:

[0058]

[0059] in, 0<ν<1, 0<τ<1 is the average control rate, T τ ≥0 is the elasticity parameter, And for δ i , i∈V F .

[0060] Beneficial effects: The advantages of the present invention are mainly reflected in:

[0061] 1. The present invention uses a fixed time control strategy, the system convergence time is limited and is not affected by the initial state of the system, and can be designed according to actual conditions;

[0062] 2. The present invention uses an intermittent control strategy to improve the system's robustness in dealing with control signal interruptions caused by factors such as network attacks, and is more in line with actual usage scenario requirements;

[0063] 3. The present invention uses a dynamic event trigger control strategy, which reduces the number of input updates, reduces the use of communication resources and energy consumption. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 It is a flow chart of a fixed-time enclosure control method for a multi-agent system triggered by intermittent dynamic events;

[0065] Figure 2 It is a schematic diagram of intermittent event triggered control;

[0066] Figure 3 is a topological relationship diagram of a multi-agent system example of the present invention;

[0067] Figure 4 is a simulation diagram of agent trajectories in a multi-agent system example of the present invention;

[0068] Figure 5 is a triggering time distribution diagram of follower agents in the multi-agent system example of the present invention;

[0069] Figure 6 is a control input simulation diagram in a multi-agent system example of the present invention;

[0070] Figure 7 It is a dynamic variable simulation change diagram in the multi-agent system example of the present invention. DETAILED DESCRIPTION

[0071] The present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments:

[0072] Example 1: A fixed-time enclosure control method for a multi-agent system triggered by an intermittent dynamic event, the flow chart is as follows Figure 1 As shown, the method comprises the following steps:

[0073] Step S1: Construct the directed topological graph and Laplace matrix of the multi-agent system according to the actual task.

[0074] The directed topological graph and the Laplace matrix described in step S1 are specifically constructed as follows:

[0075] For a multi-agent system consisting of N agents, its communication topology is a directed graph G with N nodes, where each node represents an agent. Let n and m represent the number of follower and leader agents respectively, then nodes 1 to n represent follower agents, and nodes n+1 to n+m represent leader agents. The edge set is represented by E. If node i can receive information from node j, then (i, j)∈E. The adjacency matrix is ​​represented as If (i,j)∈E, then a ij >0, otherwise a ij =0, and a ii = 0. The in-degree matrix is ​​defined as Where diag represents a diagonal matrix, The Laplacian matrix of graph G is defined as

[0076] Step S2: Construct a dynamic model of the multi-agent system.

[0077] The kinetic models constructed in step S2 are described as follows by category:

[0078] The dynamic model of the follower agent is:

[0079]

[0080] The dynamic model of the leader agent is:

[0081]

[0082] Where V F With V L Represent the follower set and the leader set, that is, V F ={1,...,n},V L ={n+1,...,m+n}. represents the state of the ith agent, represents the control input of the ith agent, Represents a nonlinear function.

[0083] Step S3: Design a fixed-time enclosure controller for a multi-agent system triggered by intermittent dynamic events.

[0084] The encirclement control objective described in step S3 is: For any given initial value of the bounded multi-agent system, the following formula holds true:

[0085]

[0086] Among them, k j ≥0, j∈V L ,and

[0087] The intermittent control strategy described in step S3 is specifically described as follows:

[0088] The control process diagram of the intermittent control strategy is as follows Figure 2 As shown, the control process [0, +∞) is divided into the control stage [t k ,p k ) and intermittent phase [p k ,t k+1 ), where k is a positive integer and the event triggering occurs only in the control phase. For the intermittent control strategy, there exists τ∈(0,1) and T τ ≥0, so that the following equation holds

[0089]

[0090] Where T c (t1, t2) is the total control time in the period [t1, t2), τ represents the average control rate, T τ is the elasticity parameter.

[0091] Furthermore, the present invention is based on the following assumptions:

[0092] Assumption 1: Each leader is a neighbor of only some of the followers, and at least one leader is connected to all other followers. In addition, there exists a positive real number δ i , i∈V F , so that δ j a ij =δ i a ji holds, where i,j∈V F .

[0093] At the same time, according to Assumption 1, the Laplace matrix can be written as

[0094]

[0095] in

[0096] Assumption 2: There exists a positive real number ψ such that

[0097]

[0098] Where i∈V F ∪V L , k j ≥0, and

[0099] The specific method for designing the fixed-time enclosure control algorithm of the multi-agent system under the intermittent dynamic event triggering described in step S3 is as follows:

[0100] S31. Design of intermittent controller:

[0101]

[0102] Where 0<α<1<β, η1, η2, η3 and η4 are positive real numbers, all of which are parameters that need to be designed. Ψ>>1, represents the last triggering time of agent i, Defined as

[0103] S32. Design measurement error:

[0104]

[0105] where ∈ i (t) is an auxiliary variable, which is defined as:

[0106]

[0107] S33. Design trigger functions, dynamic variables and trigger conditions

[0108] The trigger function is designed as:

[0109]

[0110] Among them, the two parameters 0<θ<1 and H>0 need to be designed.

[0111] Dynamic variables are designed as:

[0112]

[0113] Among them, Λ i (0)>0, 0<κ<1, χ1, χ2, χ3, χ4>0 are all parameters that need to be designed. Combining equation (9) and equation (10), the trigger condition is set as:

[0114]

[0115] Because when When , the inequality Θ i (t)≤Λ i(t) is established, so Established, thus we can get in, Therefore, dynamic trigger variables can effectively reduce the number of triggers.

[0116] Step S4: Use the controller designed in step S3 to achieve encirclement control within a fixed time, and obtain the upper bound of the convergence time.

[0117] To prove that the controller designed in step S3 can achieve encirclement control within a fixed time, the following lemma is introduced:

[0118] Lemma 1: For a semi-positive Lyapunov function V(t), if it satisfies:

[0119]

[0120] Among them, ρ1, ρ2, ρ3, ρ4, Γ are positive real numbers, ι>1. At the same time, (1-τ)ν-ρ3<0 and ρ4-τν<0 are guaranteed, where ν>0, and fixed-time enclosure control can be achieved. At the same time, the Lyapunov function V(t) converges to the domain described by the following formula:

[0121]

[0122] in 0<φ<1. The upper bound of the system convergence time is:

[0123]

[0124] In the formula

[0125] Lemma 2: If x1,x2,...,x M are all non-negative numbers, then when 0<p≤1, the following inequality holds:

[0126]

[0127] When q>1, the following inequality holds:

[0128]

[0129] Lemma 3: For The following inequalities hold:

[0130]

[0131] Where Ψ>>1, g=0.2785, and ||·||1 represents the 1-norm.

[0132] Lemma 4: If there exists Then the following inequality holds:

[0133]

[0134] Lemma 5: Matrix Each entry in is non-negative and the sum of the elements in each row is 1.

[0135] Lemma 6: Let For the δ i , i∈V F . And the matrix L1Δ is a positive definite matrix.

[0136] Next, we prove that when all assumptions hold and the inequality When (1-τ)ν-ρ3<0 and ρ4-τν<0 are both true, the controller designed in step S3 can achieve encirclement control within a fixed time, where

[0137]

[0138] remember Assume the Lyapunov function is:

[0139]

[0140] In the formula, ∈ F ∈ i A column stack vector, i∈V F . From this we can get:

[0141]

[0142] Among them, x F With x L x i In i∈V F With i∈V L In the case of a stacked column vector, x d is the desired state, defined as x e is the enclosing error, defined as x e (t) = (x e1 (t),...,x en (t)) T =x F -x d .

[0143] When t∈[t k ,p k ), then:

[0144]

[0145] where e F , and tanh(∈ F ) are e i , and tanh(∈ i ) where i∈V F .

[0146] Substituting equation (19) into the derivative of the Lyapunov function, we get:

[0147]

[0148] where ||·|| represents the 2-norm.

[0149] From formula (18), we can get Combined with hypothesis 2, we can know that:

[0150]

[0151] Among them, b ij For the matrix The (i,j)th element of . From this we can further obtain:

[0152]

[0153] From formula (9) and (11), we can know that:

[0154]

[0155] Substituting equations (10), (22), and (23) into equation (20), we have:

[0156]

[0157] Since 0<κ<1, η4>0, equation (24) can be further expanded to:

[0158]

[0159] From the definition of V1, we can get Combined with Lemma 2, equation (25) can be further expanded to:

[0160]

[0161] in,

[0162] When t∈[p k ,t k+1 ), then:

[0163]

[0164] Combining equations (26) and (27), we can get the derivative of the Lyapunov function as:

[0165]

[0166] Among them, 0<α<1, β>1,

[0167] The upper bound of the convergence time is:

[0168]

[0169] in, 0<ν<1, τ and T τ are the average control rate and elastic parameter respectively, and 0<τ<1, T τ ≥0.

[0170] Next, we further prove that the controller designed in step S3 can be used in actual scenarios and will not produce Zeno behavior. F , from the Lyapunov function definition (17), we can know that:

[0171]

[0172] make c ij is the (i,j)th element of matrix L1. When , the derivative of the absolute value of the measurement error can be obtained from formula (21) and Lemma 2:

[0173]

[0174] in, Integrating both sides of equation (31), we get:

[0175]

[0176] According to the trigger conditions (11) and (32), we can know that:

[0177]

[0178] From formula (33), we can get:

[0179]

[0180] Therefore, the controller designed in step S3 will not produce the Zeno phenomenon and can be used in practical situations.

[0181] Step S5: Continue running until the encirclement control of the multi-agent system is completed.

[0182] Next, simulation is used to verify the effectiveness of the fixed-time encirclement control method for multi-agent systems triggered by intermittent dynamic events.

[0183] Consider a system such as Figure 3 As shown, it consists of 2 leaders and 5 followers, namely V F ={1,...,5},V L ={6,7}. Figure 3 It can be obtained that L1 and L2 are:

[0184]

[0185] From formula (37), let Δ = diag{0.75, 1.0, 1.0, 1.5, 0.75}, so we can get δ m =0.75,δ M =1.5,σ m (L1) = 0.1112, λ M [(L1Δ) -1 ]=9.9549. At the same time, let f(x i (t),t)=x i (t)sin(2t 2 )+cos(5t),i∈V F ∪V L , from which we can get ψ=1.

[0186] Let α = 0.9, β = 1.5, eta1 = 15, eta2 = 20, eta3 = 31, eta4 = 1, θ = 0.1, Ψ = 100, κ = 0.5, H = 0.5, ν = 4, χ1 = 1, χ2 = 1, χ3 = 1.3, χ4 = 2. Let the initial state of the system be x F (0) = [-2, 4, 5, 2, 6] T , x L (0) = [1, -1] T The control phase is shown below:

[0187]

[0188] From formula (37), we can get τ = 0.7, and (1-τ)ν-ρ3<0 and ρ4-τν<0 are both true, where Figure 4 This is a position simulation diagram of the multi-agent system. It can be seen from the figure that the system completes the encirclement control in about 0.3 seconds, indicating that the control strategy can enable the multi-agent system to achieve fixed-time encirclement control. Figure 5This is the distribution diagram of the event triggering time of the follower agent. It can be seen from the figure that the event triggering time of the multi-agent system has obvious intermittent control characteristics, that is, it is only triggered during the control period but not during the intermittent period, which means that the control method can complete the encirclement control under the condition of intermittent dynamic event triggering. Figure 6 This is the control input variation diagram. Figure 7 It is a dynamic variable change diagram. It can be seen from the figure that the dynamic variable is always greater than 0, which means that the dynamic triggering method adopted by this control method can effectively reduce the number of triggering times.

[0189] The above description is only a preferred embodiment of the present invention and does not constitute any other form of limitation to the present invention. Any modification or equivalent change made based on the technical essence of the present invention still falls within the scope of protection required by the present invention.

Claims

1. A fixed-time enclosure control method for a multi-agent system triggered by intermittent dynamic events, characterized in that: Step S1: Construct the directed topological graph and Laplace matrix of the multi-agent system according to the actual task; Step S2: construct a dynamic model of the multi-agent system; Step S3: Design a fixed-time enclosure controller for a multi-agent system triggered by intermittent dynamic events; Step S4: Use the controller designed in step S3 to achieve encirclement control within a fixed time, and obtain the upper bound of the convergence time; Step S5: Continue running until the encirclement control of the multi-agent system is completed.

2. The method for controlling a multi-agent system at a fixed time under intermittent dynamic event triggering according to claim 1 is characterized in that: The process of constructing the directed topological graph and Laplace matrix of the multi-agent system in step S1 includes: For a multi-agent system consisting of N agents, its communication topology is a directed graph G with N nodes, where each node represents an agent. The number of follower and leader agents is n and m respectively, then nodes 1 to n represent follower agents, nodes n+1 to n+m represent leader agents, and the edge set is represented by E. If node i can receive information from node j, then (i, j)∈E. The adjacency matrix is ​​represented as If (i,j)∈E, then a ij >0, otherwise a ij =0, and a ii = 0, the in-degree matrix is ​​defined as Where diag represents a diagonal matrix, The Laplacian matrix of graph G is defined as 3. The method for controlling a multi-agent system at a fixed time under intermittent dynamic event triggering according to claim 1 is characterized in that: The step S2 of constructing the dynamics model of the multi-agent system includes: The dynamic model of the follower agent is: The dynamic model of the leader agent is: Where V F With V L Represent the follower set and the leader set, that is, V F ={1,...,n},V L ={n+1,...,m+n}, represents the state of the ith agent, represents the control input of the ith agent, Represents a nonlinear function.

4. The method for controlling a multi-agent system at a fixed time under intermittent dynamic event triggering according to claim 1 is characterized in that: The process of designing a fixed-time enclosure controller for a multi-agent system triggered by an intermittent dynamic event in step S3 includes: S31. Design of intermittent controller: Among them, 0<α<1<β, η1, η2, η3 and η4 are positive real numbers, all of which are parameters that need to be designed, Ψ>>1, represents the last triggering time of agent i, Defined as S32. Design measurement error: where ∈ i (t) is an auxiliary variable, which is defined as: S33. Design trigger functions, dynamic variables and trigger conditions; The trigger function is designed as: Among them, the two parameters 0<θ<1 and H>0 need to be designed. Dynamic variables are designed as: Among them, Λ i (0)>0, 0<κ<1, χ1, χ2, χ3, χ4>0 are all parameters that need to be designed. Set the trigger conditions as:

5. The method for controlling a multi-agent system at a fixed time under intermittent dynamic event triggering according to claim 1, characterized in that: The upper bound of the convergence time obtained in step S4 is specifically: (1) System needs to be guaranteed (1-τ)ν-ρ3<0 and ρ4-τν<0 both hold, and the following assumptions are satisfied: Assumption 1: Each leader is a neighbor of only some of the followers, and at least one leader is connected to all other followers, and there exists a positive real number δ i , i∈V F , so that δ j a ij =δ i a ji holds, where i,j∈V F , At the same time, according to Assumption 1, the Laplace matrix can be written as in Assumption 2: There exists a positive real number ψ such that Where i∈V F ∪V L , k j ≥0, (2) The upper bound of the system’s convergence time is: in, 0<ν<1, 0<τ<1 is the average control rate, T τ ≥0 is the elasticity parameter, And for δ i , i∈V F .

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