Fixed-time enclosure control method for multi-agent systems triggered by intermittent dynamic events

Abstract

The application provides a multi-agent system fixed-time encirclement control method under intermittent dynamic event triggering, and the method comprises the following steps: step S1: according to an actual task, a directed topological graph and a Laplace matrix of the multi-agent system are constructed; step S2: a dynamic model of the multi-agent system is constructed; step S3: a multi-agent system fixed-time encirclement controller under intermittent dynamic event triggering is designed; step S4: the controller designed in step S3 is used to realize the encirclement control in a fixed time, and an upper bound of a convergence time is obtained; and step S5: the method is continuously run until the encirclement 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 coordinated control of multi-agent systems has been widely applied in a growing number of fields, such as drone formations, robot collaboration, and satellite handover. Among these numerous coordinated control problems, encirclement control is a key one. In encirclement control, the agents in the system are divided into followers and leaders. The states of the follower agents must lie within the convex hull of the leader agent's states, thereby confining the entire system to a certain area.

[0003] Traditional multi-agent system research typically uses fixed-interval sampling to update control inputs, but this approach can cause problems such as communication congestion. In contrast, event-triggered control strategies reduce the number of control input updates by presetting a threshold and only updating the control input when the measurement error exceeds that threshold. Dynamic event triggering goes a step further, with the threshold changing with the system state, further reducing the number of input updates and alleviating communication congestion.

[0004] Intermittent control is also widely used to address the high energy consumption caused by continuous control signal input and to mitigate interruptions in control input due to factors such as cyberattacks. The intermittent control process is divided into a control phase and an intermittent phase. The controller receives input only during the control phase and not during the intermittent phase, thereby reducing system energy consumption and enhancing system robustness.

[0005] Furthermore, the system's convergence speed is also an important metric for evaluating a system. Finite-time control strategies generally have a shorter convergence time than asymptotic control strategies, but this time is dependent on the initial state. Fixed-time control, on the other hand, has a finite convergence time, but its upper bound is independent of the system's initial state, making it applicable in a wider range of scenarios.

[0006] The present application differs from the prior art in the following ways:

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

[0008] Patent CN118331047A addresses second-order linear multi-agent systems with an undirected graph topology, while this patent addresses first-order nonlinear multi-agent systems with a directed graph topology. The two differ fundamentally in their control object models, and this patent offers greater adaptability when handling disturbances in real-world applications.

[0009] Patent CN118331047A uses a continuous control strategy to achieve control objectives, with each agent always receiving control input. This patent uses an intermittent control strategy to achieve control objectives. By dividing the control process into a control phase and an intermittent phase, this reduces the number of control input updates, enhances system robustness, and reduces power consumption. The two differ fundamentally in their control strategies, with this patent's control strategy being more suitable for distributed multi-agent systems with limited communication resources and energy.

[0010] The control objective of patent CN118331047A is to achieve bipartite consistency in a multi-agent system, ensuring that the agents within the group reach consensus. The control objective of this patent is to achieve encirclement control in a multi-agent system, ensuring that the states of all followers remain within the convex hull formed by the leader. There are essential differences between the two in their control objectives.

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

[0012] Patent CN118244638A uses a static event trigger mechanism with a relatively fixed trigger threshold. This patent adopts a more flexible dynamic event trigger mechanism. By designing a dynamic variable that changes with the system state, the trigger threshold is dynamically adjusted, further reducing the number of triggers and lowering energy consumption. The two differ fundamentally in their trigger mechanisms, and this patent can achieve control objectives with fewer triggers.

[0013] Patent CN118244638A uses a continuous control strategy to achieve control objectives, with each agent always receiving control input. This patent uses an intermittent control strategy to achieve control objectives. By dividing the control process into a control phase and an intermittent phase, this reduces the number of control input updates, enhances system robustness, and reduces power consumption. The two differ fundamentally in their control strategies, with this patent's control strategy being more suitable for distributed multi-agent systems with limited communication resources and energy.

[0014] The control objective of patent CN118244638A is to achieve formation control of a multi-agent system, ensuring that the agents reach and maintain a specified state. The control objective of this patent is to achieve encirclement control of a multi-agent system, ensuring that the states of all followers remain within the convex hull formed by the leader. There is a fundamental difference between the two in their control objectives.

[0015] In summary, the dynamic event-triggered control strategy, intermittent control strategy and fixed time control strategy are combined to form a comprehensive containment control scheme, so as to realize the control goal of low energy consumption, strong system robustness and flexible design of convergence time in the control process, which has become an important issue to be solved. SUMMARY

[0016] To solve the above technical problems, the application provides a multi-agent system fixed time containment control method under intermittent dynamic event triggering, so that the multi-agent system can complete the containment control within a fixed time.

[0017] To achieve the above object, the technical scheme adopted by the application is:

[0018] The multi-agent system fixed time containment control method under intermittent dynamic event triggering is characterized in that: step S1: according to the actual task, a directed topological graph and a Laplace matrix of the multi-agent system are constructed;

[0019] Step S2: a dynamic model of the multi-agent system is constructed;

[0020] Step S3: a multi-agent system fixed time containment controller under intermittent dynamic event triggering is designed;

[0021] Step S4: the controller designed in step S3 is used to realize containment control within a fixed time, and an upper bound of the convergence time is obtained;

[0022] Step S5: continuously running until the containment control of the multi-agent system is completed.

[0023] As a further improvement of the application, the process of constructing the directed topological graph and the Laplace matrix of the multi-agent system in step S1 includes:

[0024] For a multi-agent system composed of N agents, its communication topological graph is a directed graph G with N nodes, each node corresponds to an agent, and the number of follower and leader agents is denoted as n and m respectively, then nodes 1 to n represent follower agents, and nodes to represent leader agents, the edge set is denoted as , if node i can receive information from node j, then . The adjacency matrix is denoted as , where if , then , otherwise , and . The in-degree matrix is defined as , where diag represents a diagonal matrix, The Laplacian matrix of graph G is defined as .

[0025] As a further improvement of the present application, the step S2 of constructing the dynamics model of the multi-agent system comprises:

[0026] The dynamics model of the follower agent is:

[0027]

[0028] The dynamics model of the leader agent is:

[0029]

[0030] where V F and V L represent the set of followers and the set of leaders, respectively, i.e. , , represents the state of the i-th agent, represents the control input of the i-th agent, represents a nonlinear function.

[0031] As a further improvement of the present application, the step S3 of designing the fixed-time containment controller of the multi-agent system under intermittent dynamic event-triggered comprises:

[0032] S31. Designing the intermittent controller:

[0033]

[0034] where , , , and are positive real numbers, all of which are parameters to be designed, , represents the last triggering time of agent i, is defined as ;

[0035] S32. Designing the measurement error:

[0036]

[0037] where is an auxiliary variable, which is defined as:

[0038]

[0039] S33. Designing the triggering function, the dynamic variable and the triggering condition;

[0040] The trigger function is designed as:

[0041]

[0042] in and These two parameters need to be designed. , ;

[0043] Dynamic variables are designed as follows:

[0044]

[0045] in, , , 、 、 、 These are all parameters that need to be designed. Set the trigger conditions as:

[0046] .

[0047] As a further improvement of the present invention, the upper bound of the convergence time obtained in step S4 is specifically:

[0048] (1) System needs to be guaranteed , and All of them are valid, and the following assumptions are met:

[0049] 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 , , making Established, of which ,

[0050] At the same time, according to Assumption 1, the Laplace matrix can be written as

[0051]

[0052] in , ;

[0053] Assumption 2: There exists a positive real number , so that the following formula holds

[0054]

[0055] in, , , .

[0056] (2) The upper bound of the system’s convergence time is:

[0057]

[0058] in, , , , , , is the average control rate, is the elasticity parameter, , and for , .

[0059] Beneficial effects: The advantages of the present invention are mainly reflected in:

[0060] 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;

[0061] 2. The present invention uses an intermittent control strategy, which improves the system's robustness against control signal interruptions caused by factors such as network attacks, and is more in line with the needs of actual usage scenarios;

[0062] 3. The present invention uses a dynamic event triggering control strategy, which reduces the number of input updates, reduces the use of communication resources and reduces energy consumption. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 This is a flow chart of a fixed-time enclosure control method for a multi-agent system triggered by intermittent dynamic events;

[0064] Figure 2 It is a schematic diagram of intermittent event triggered control;

[0065] Figure 3 is a topological relationship diagram of a multi-agent system instance of the present invention;

[0066] Figure 4 is a simulation diagram of agent trajectories in an example of a multi-agent system of the present invention;

[0067] Figure 5 is a triggering time distribution diagram of follower agents in the multi-agent system example of the present invention;

[0068] Figure 6 is a control input simulation diagram in the multi-agent system example of the present invention;

[0069] Figure 7 It is a dynamic variable simulation change diagram in the multi-agent system example of the present invention. DETAILED DESCRIPTION

[0070] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments:

[0071] Example 1: A fixed-time enclosure control method for a multi-agent system triggered by intermittent dynamic events, the flow chart is as follows Figure 1 As shown, the method includes the following steps:

[0072] Step S1: Construct the directed topology graph and Laplace matrix of the multi-agent system according to the actual task.

[0073] The directed topological graph and Laplace matrix in step S1 are specifically constructed as follows:

[0074] 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 be the number of follower and leader agents respectively, then nodes 1 to n represent follower agents, and nodes arrive represents the leader agent. The edge set is Indicates that if node i can receive information from node j, then The adjacency matrix is ​​represented as , among which if ,but , otherwise ,at the same time The in-degree matrix is ​​defined as , where diag represents a diagonal matrix, . The Laplacian matrix of graph G is defined as .

[0075] Step S2: Construct a dynamic model of the multi-agent system.

[0076] The kinetic models constructed in step S2 are classified as follows:

[0077] The dynamic model of the follower agent is:

[0078] (1)

[0079] The dynamic model of the leader agent is:

[0080] (2)

[0081] Where V F With V L Represent the follower set and the leader set respectively, that is, , . represents the state of the i-th agent, represents the control input of the ith agent, represents a nonlinear function.

[0082] Step S3: Design a fixed-time enclosure controller for the multi-agent system triggered by intermittent dynamic events.

[0083] The encirclement control objective described in step S3 is: for any given initial value of the bounded multi-agent system, the following formula holds:

[0084] (3)

[0085] in, , ,and .

[0086] The intermittent control strategy described in step S3 is specifically described as follows:

[0087] The control process diagram of the intermittent control strategy is as follows Figure 2 As shown, the control process [0, +∞) is divided into control stages Interval phase , where k is a positive integer and the event triggering occurs only in the control phase. For the intermittent control strategy, there is as well as , so that the following formula holds

[0088] (4)

[0089] in for The total control time in the period, τ represents the average control rate, is the elasticity parameter.

[0090] Furthermore, the present invention is based on the following assumptions:

[0091] 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 , , making Established, of which .

[0092] At the same time, according to Assumption 1, the Laplace matrix can be written as

[0093] ()

[0094] in , .

[0095] Assumption 2: There exists a positive real number , so that the following formula holds

[0096]

[0097] in, , ,and .

[0098] The specific method for designing the fixed-time enclosure control algorithm for the multi-agent system triggered by the intermittent dynamic events described in step S3 is as follows:

[0099] S31. Design of intermittent controller:

[0100] ( )

[0101] in , , , and are positive real numbers, all of which are parameters that need to be designed. , Indicates the last triggering time of agent i, Defined as .

[0102] S32. Design measurement error:

[0103] ( )

[0104] in is an auxiliary variable defined as:

[0105] ( )

[0106] S33. Design trigger functions, dynamic variables, and trigger conditions

[0107] The trigger function is designed as:

[0108] ( )

[0109] in and These two parameters need to be designed. , .

[0110] Dynamic variables are designed as follows:

[0111] ( )

[0112] in, , , 、 、 、 These are all parameters that need to be designed. Combining equations (9) and (10), the trigger condition is set as:

[0113] ( )

[0114] Because when When Established, therefore Established, thus we can get ,in, , so dynamic trigger variables can effectively reduce the number of triggers.

[0115] Step S4: Use the controller designed in step S3 to implement closing control within a fixed time, and obtain the upper bound of the convergence time.

[0116] To prove that the controller designed in step S3 can achieve encirclement control within a fixed time, the following lemma is introduced:

[0117] Lemma 1: For a semi-positive Lyapunov function V(t), if it satisfies:

[0118] ( )

[0119] in, , , , , is a positive real number, , . At the same time, ensure and Established, of which , then fixed time encirclement control can be achieved. At the same time, Lyapunov function converges to the domain described by:

[0120]

[0121] in , , The upper bound of the system convergence time is:

[0122]

[0123] In the formula , .

[0124] Lemma 2: If are all non-negative numbers, then when Then the following inequality holds:

[0125] ( )

[0126] when Then the following inequality holds:

[0127] ( )

[0128] Lemma 3: For , the following inequalities hold:

[0129] ( )

[0130] in , g=0.2785, represents the 1-norm.

[0131] Lemma 4: If there exists , , then the following inequality holds:

[0132] ( )

[0133] Lemma 5: Matrix Every entry in is non-negative and the sum of the elements in each row is 1.

[0134] Lemma 6: Let , for which , . And the matrix is a positive definite matrix.

[0135] Next, we prove that when all the assumptions are true and the inequality , and When all are true, the controller designed in step S3 can achieve encirclement control within a fixed time, where , .

[0136] remember , , let the Lyapunov function be:

[0137] ( )

[0138] Where, for A column stack vector of . From this we can get:

[0139] ( )

[0140] in, and They are exist and In the case of stacked column vectors, is the desired state, defined as , is the enclosing error, defined as .

[0141] when When , there are:

[0142] ( )

[0143] in and They are and A stacked column vector of .

[0144] Substituting formula (19) into the derivative of the Lyapunov function, we get:

[0145] ( )

[0146] in represents the 2-norm.

[0147] From formula (18), we can get Combined with hypothesis 2, we can know that:

[0148] ( )

[0149] in, is a matrix No. elements. From this we can further obtain:

[0150] ( )

[0151] From formula (9) and (11), we can know that:

[0152] (twenty three)

[0153] Substituting equations (10), (22), and (23) into equation (20), we have:

[0154] ( )

[0155] because , , formula (24) can be further scaled to:

[0156] ( )

[0157] Depend on The definition can be obtained , combined with Lemma 2, Equation (25) can be further expanded to:

[0158] ( )

[0159] in, , , .

[0160] when When , there are:

[0161] ( )

[0162] Combining equations (26) and (27), we can get the derivative of the Lyapunov function as:

[0163] ( )

[0164] in , , , , , , .

[0165] The upper bound of the convergence time is:

[0166] ( )

[0167] in, , , , With respectively the average control rate and the elasticity parameter, and , .

[0168] Next, it is further proved that the controller designed in step S3 can be used in practical scenarios and does not produce Zeno behavior. For , from the Lyapunov function definition formula (17), we have:

[0169] ( )

[0170] Let , be the i-th element of the matrix L1. When , the derivative of the absolute value of the measurement error is obtained from formula (21) and Lemma 2 as:

[0171] ( )

[0172] where , , , . Integrating both sides of formula (31), we have:

[0173] ( )

[0174] According to the trigger condition formula (11) and formula (32), we have:

[0175] ( )

[0176] From formula (33), we have:

[0177] ( )

[0178] Therefore, the controller designed in step S3 does not produce Zeno behavior and can be used in practical situations.

[0179] Step S5: continue running until the containment control for the multi-agent system is completed.

[0180] Next, the effectiveness of the multi-agent system fixed-time containment control method under intermittent dynamic event triggering is verified using simulation.

[0181] The system considered is as follows Figure 3 ​As shown, it consists of 2 leaders and 5 followers, namely , . By Figure 3 , we can get , , respectively:

[0182] ( )

[0183] ( )

[0184] By formula (37), let , thus we can get , , , . At the same time, let , thus we can get ψ = 1.

[0185] Let α = 0.9, β = 1.5, η 1 = 15, η 2 = 20, η 3 = 31, η 4 = 1, θ = 0.1, Ψ = 100, κ = 0.5, H = 0.5, ν = 4, χ 1 = 1, χ 2 = 1, χ 3 = 1.3, χ 4 = 2. Suppose the initial state of the system is , . The control stage is shown in the following formula:

[0186] ( )

[0187] From formula (37), we can get τ = 0.7, and , and are all true, where , . Figure 4 is the position simulation diagram of the multi-agent system. From the diagram, it can be seen that the system completes the surround control at about 0.3 seconds, which shows that the control strategy can make the multi-agent system realize the fixed-time surround control. Figure 5 is the event-triggered time distribution diagram of the follower agent. From the diagram, it can be seen that the event-triggered time of the multi-agent system has obvious intermittent control characteristics, that is, it is triggered only in the control period and not triggered in the intermittent period, which shows that the control method can complete the surround control under the condition of intermittent dynamic event triggering. Figure 6 is the control input change diagram. Figure 7 is the dynamic variable change diagram. From the diagram, it can be seen that the dynamic variable is always greater than 0, which shows that the dynamic triggering method adopted by the control method can effectively reduce the number of triggers.

[0188] The above description is merely a preferred embodiment of the present invention and does not constitute any other form of limitation to the present invention. Any modification or equivalent variation based on the technical essence of the present invention shall still fall within the scope of protection claimed by the present invention.

Claims

1. A fixed-time enclosure control method for a multi-agent system triggered by intermittent dynamic events, characterized by: Step S1: Construct the directed topology 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 the multi-agent system triggered by intermittent dynamic events; The trigger function is designed as: ( ) in and These two parameters need to be designed. , ; is the measurement error, , , , and is a positive real number, is an auxiliary variable; Dynamic variables are designed as follows: ( ) in, , , 、 、 、 These are all parameters that need to be designed. Set the trigger conditions as follows: ( ), Indicates the last triggering time of agent i; Step S4: Use the controller designed in step S3 to implement 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; The upper bound of the system's convergence time is: ( ) in, , , , , , is the average control rate, is the elasticity parameter, , and for the positive number , , V F represents the follower set, i.e. .

2. The method for controlling a multi-agent system with fixed time encirclement under intermittent dynamic events according to claim 1, characterized in that: The process of constructing the directed topology 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, and nodes arrive represents the leader agent, and the edge set is Indicates that if node i can receive information from node j, then , the adjacency matrix is ​​expressed as , among which if ,but , otherwise ,at the same time , the in-degree matrix is ​​defined as , where diag represents a diagonal matrix, , and thus the Laplace matrix of graph G is defined as .

3. The method for controlling a multi-agent system with fixed time encirclement under intermittent dynamic events according to claim 2, characterized in that: The step S2 of constructing the dynamic 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 respectively, that is, , , represents the state of the i-th agent, represents the control input of the ith agent, represents a nonlinear function.

4. The method for controlling a multi-agent system with fixed time encirclement under intermittent dynamic events according to claim 3 is characterized by: The process of designing a fixed-time enclosure controller for a multi-agent system triggered by intermittent dynamic events in step S3 includes: S31. Design of intermittent controller: ( ) in , , , and are positive real numbers, all of which are parameters that need to be designed. , Defined as ; S32. Design measurement error: ( ) in is an auxiliary variable defined as: ( ) S33. Design trigger functions, dynamic variables, and trigger conditions.

5. The method for controlling a multi-agent system with fixed time encirclement triggered by intermittent dynamic events according to claim 4 is characterized in that: The upper bound of the convergence time obtained in step S4 is specifically: (1) System needs to be guaranteed , and All of them are valid, and the following assumptions are met: 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 , , making Established, of which , At the same time, according to Assumption 1, the Laplace matrix can be written as ( ) in , ; Assumption 2: There exists a positive real number , so that the following formula holds ； in, , , ; (2) Determine the upper bound of the system's convergence time.

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