Consistency control method for partial differential multi-agent system containing switching topology

Abstract

The invention discloses a switching topology-containing partial differential multi-agent system consistency control method, and relates to the field of networked systems, partial differential equations and cooperative control, and the method comprises the steps: respectively constructing a follower system dynamic model and a leader dynamic model; constructing an error system; establishing a balance interaction topology for a plurality of followers in the follower system; constructing a plurality of guidance diagrams including follower systems and leaders according to the balanced interaction topology; the plurality of induction diagrams are always or intermittently kept in weak communication; constructing a boundary control law and a control target; according to a boundary control law, the dynamic model of the follower system is adjusted in real time until a control target is met; according to the method, the leader following consistency can be realized under the condition that the induction graph always or intermittently keeps weak communication, and the leader following consistency is exponentially guaranteed.

Description

Technical Field

[0001] The present invention relates to the fields of networked systems, partial differential equations, and cooperative control, and particularly relates to a method for consensus control of a partial differential multi-agent system with switching topologies. Background Art

[0002] A multi-agent system (Multi-Agent Systems, abbreviated as MASs) is a system composed of multiple interacting, autonomous agents that can sense the environment and make decisions. These agents coordinate, cooperate, or compete through a certain mechanism to complete complex tasks or solve problems. MASs can be regarded as a category of complex systems and have strong dynamics, adaptability, and robustness.

[0003] In application scenarios such as formation control, sensor networks, and secure communication, consensus is one of the most basic behaviors in multi-agent systems, which involves reaching an agreement on variables of interest such as attitude and position. In particular, in leader-follower consensus, the controlled follower agents attempt to achieve the same performance as the designated leader, and the dynamics of the leader are determined by its own behavior. Only a part of the followers (called informed agents) can directly access the leader.

[0004] In addition, the cooperative control of multi-agent systems with distributed parameters is usually represented by networked partial differential equations (Partial Differential Equations, abbreviated as PDEs). Such methods have applications in fields such as battery management and energy-efficient building optimization. In related technologies, most of the research involves issues such as the synchronization of distributed parameter agents in an abstract environment, the state consensus in parabolic agents, the output consensus and regulation of parabolic MASs, and considering the threat of cyber attacks, the secure consensus of PDE-based multi-agent systems, and the networked cooperative control of wave-type PDE agents. Obviously, the above methods have great potential in describing the attitude cooperation of flexible spacecraft, but the network consensus based on wave equations has not been solved.

[0005] Another issue that needs attention is that in practical applications, due to sensor failures or communication limitations between agents, the interaction flow often changes, and most of the above research problems are carried out under a fixed interaction topology. Based on this, in related technologies, further research is carried out on the consensus problem of parabolic PDE agents under topology switching. However, the above research problems use undirected interactions, while in practical applications, information transmission is usually unidirectional, which is difficult to meet the needs of the actual situation. Summary of the Invention

[0006] In view of this, the present invention provides a consensus control method for a partial differential multi-agent system with switching topologies to solve the technical problems in the related art.

[0007] The present invention provides a consensus control method for a partial differential multi-agent system with switching topologies, including: S1. Respectively construct the dynamic models of the follower system and the leader; the follower system includes a plurality of followers; S2. Construct an error system according to the dynamic models of the follower system and the leader; S3. Establish a balanced interaction topology for the plurality of followers in the follower system; the balanced interaction topology includes balanced digraphs corresponding to a plurality of topology instances; the topology instances represent the communication relationships among the plurality of followers; S4. Construct a plurality of induced graphs including the follower system and the leader according to the balanced interaction topology; the target induced graph is a digraph after introducing the leader into the target balanced digraph corresponding to the target topology instance; all of the plurality of induced graphs are always or intermittently weakly connected; S5. Construct a boundary control law according to the error system and the plurality of induced graphs; S6. Construct a control target according to the dynamic models of the follower system and the leader; S7. Adjust the dynamic model of the follower system in real time according to the boundary control law until the control target is satisfied and then stop.

[0008] In an alternative embodiment, the dynamic model of the follower system is: ; wherein, represents the state vector of each follower in the follower system at position , time , and , ; , represents the state vector of follower at position , time , , , ; is the first preset parameter, and ; is the control signal; and respectively represent the initial offset and initial velocity of each follower; and are the second preset parameter and the third preset parameter respectively.

[0009] In an alternative embodiment, the dynamic model of the leader is: ; where represents the state vector of the leader at position and time , and , ; is the first preset parameter, and ; and respectively represent the initial offset and the initial velocity of the leader; and are the fourth preset parameter and the fifth preset parameter respectively.

[0010] In an alternative embodiment, the error system is: ; where represents the state vector error between each follower and the leader at position and time , , represents the state vector error between follower and the leader; is the first preset parameter, and ; is the control signal; and respectively represent the initial offset error and the initial velocity error between each follower and the leader; are the errors of the fourth preset parameter and the second preset parameter respectively; are the errors between the fifth preset parameter and the second preset parameter respectively.

[0011] In an alternative embodiment, S3 includes: S31. Establish a plurality of balanced directed graphs according to various communication relationships among a plurality of followers in the follower system; The balanced directed graph is ; where is the node set, , is the node corresponding to follower , ; is the edge set, , is the directed edge from node to node , and the total number of directed edges is ; is the adjacency matrix, , and the matrix element is the communication weight from follower j to follower . The diagonal element ; The Laplacian matrix of the balanced digraph is , , , , and when , ; S32. Establish a balanced switching graph between multiple balanced digraphs, and randomly switch between multiple balanced digraphs according to the balanced switching graph; The balanced switching graph is: ; Wherein, represents the target balanced digraph, , represents the switching signal, and .

[0012] In an alternative embodiment, the S4 includes: S41. For the target balanced digraph, add the node corresponding to the leader to the node set of the target balanced digraph; S42. Add a directed edge from the node corresponding to the leader to the nodes corresponding to all followers to the edge set; S43. Change the matrix elements corresponding to the edges from the node corresponding to the leader to each weakly connected component of the target balanced digraph in the adjacency matrix to a number greater than zero, or when the state vector error between the follower and the leader in the error system satisfies the first condition, change the matrix elements corresponding to the edges from the node corresponding to the leader to each weakly connected component of the target balanced digraph in the adjacency matrix to a number greater than zero; The first condition is: ; Wherein, B is a preset threshold; represents the state vector error between follower and the leader.

[0013] In an alternative embodiment, the boundary control law is: ; Wherein, , represents the state vector error of each follower and the leader at position 1 and time t , ; is the Laplacian matrix corresponding to the target balanced digraph, , representing the matrix elements corresponding to the edges from the leader to each follower in the target balanced digraph.

[0014] In an alternative embodiment, the control objective is: ; where represents the follower at position and time of the state vector, ; represents the leader at position and time of the state vector; , .

[0015] The present invention can achieve leader-follower consensus for a partial differential multi-agent system with switching topologies, extending the undirected topology to a more general case where there is weak connection communication between followers and the leader, and this communication can be continuous or intermittent. The present invention constructs a boundary controller according to two topological cases to achieve leader-follower consensus when the induced graph remains weakly connected all the time, and the leader-follower consensus is guaranteed exponentially; it can also achieve leader-follower consensus when the induced graph remains weakly connected in some discontinuous time intervals. At the same time, the interaction between followers can be kept balanced, ensuring the coordination of the system.

[0016] In addition, it can also reduce the number of required actuators, thereby reducing the control cost, and the input operator of the multi-agent system is unbounded. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following will briefly introduce the drawings required for the description of the specific embodiments or the prior art. Obviously, the following drawings are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0018] Figure 1 is a schematic flowchart of a method for consensus control of a partial differential multi-agent system with switching topologies according to an embodiment of the present invention; Figure 2 is a partition diagram of the time interval according to an embodiment of the present invention; Figure 3It is a display diagram of the switching between multiple balanced interaction topologies according to an embodiment of the present invention; Figure 4 It is the leader state according to an embodiment of the present invention over time and space variation curve graph; Figure 5 It is the variation curve graph of the states of each follower according to an embodiment of the present invention with respect to position and time ; among which, (a) is the variation curve graph of the state of follower 1 with respect to position and time ; (b) is the variation curve graph of the state of follower 2 with respect to position and time ; (c) is the variation curve graph of the state of follower 3 with respect to position Figure 6 and time ; (d) is the variation curve graph of the state of follower 4 with respect to Specific embodiments

[0019] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative efforts fall within the scope of protection of the present invention.

[0020] As Figure 1 shown, the process of a consistency control method for a partial differential multi-agent system with a switching topology includes the following steps: S1. Dynamical models of the follower system and the leader are constructed respectively; the follower system includes multiple followers.

[0021] Among them, the follower system is a system composed of multiple agents (i.e., multiple followers), and the total number of followers is N , and .

[0022] In an alternative embodiment, the dynamical model of the follower system is: ; wherein, Indicates the state vector (i.e., lateral displacement) of each follower (i.e., each agent) in the follower system at position , at time , and , ; , Indicates the follower at position , at time The state vector, , , ; Is the first preset parameter, and ; Is the control signal; And Respectively represent the initial offset and initial velocity of each follower; And Are the second preset parameter and the third preset parameter respectively.

[0023] It should be noted that the initial offset And the initial velocity Of each follower are respectively determined by the second preset parameter And the third preset parameter . The first preset parameter , the second preset parameter And the third preset parameter Can be adaptively set according to the characteristics of the follower system or design requirements, and are not specifically limited here.

[0024] In an alternative embodiment, the dynamic model of the leader (in the embodiments of the present invention, the leader is labeled as agent 0) is: ; Wherein, Indicates the state vector (i.e., lateral displacement) of the leader at position , at time , and , ; Is the first preset parameter, and ; And Respectively represent the initial offset and initial velocity of the leader; And Are the fourth preset parameter and the fifth preset parameter respectively.

[0025] It should be noted that the initial offset And the initial velocity Determined by the fourth preset parameter and the fifth preset parameter respectively. The fourth preset parameter and the fifth preset parameter can be adaptively set according to the characteristics or design requirements of the follower system, and no specific limitation is made here.

[0026] S2. Construct an error system according to the dynamic models of the follower system and the leader.

[0027] In an optional implementation manner, the error system is: ; wherein, represents the state vector error between each follower and the leader at position and time ; , represents the state vector error between the follower and the leader; ; is the first preset parameter, and ; is the control signal; and respectively represent the initial offset error and the initial velocity error between each follower and the leader; are respectively the errors of the fourth preset parameter and the second preset parameter; are respectively the errors between the fifth preset parameter and the second preset parameter.

[0028] S3. Establish a balanced interaction topology for multiple followers in the follower system; the balanced interaction topology includes balanced directed graphs corresponding to multiple topology instances; the topology instances represent the communication relationships between multiple followers.

[0029] In an optional implementation manner, step S3 includes: S31. Establish multiple balanced directed graphs according to various communication relationships between multiple followers in the follower system; The balanced directed graph is ; wherein, is the node set, , regarding each follower body in the follower system as a node (i.e., is the node corresponding to the agent ), and the total number of nodes is N ; is the edge set, , is the slave node To the node The directed edge, and the total number of directed edges is ; Is the adjacency matrix, used to quantify the communication relationship between agents, , and the matrix element Is the follower j To the follower The communication weight. If > 0, it means that the node Can send information to the node ; If = 0, it means that there is no direct communication between the node And the node .

[0030] In addition, self-loops are not allowed in the balanced directed graph, that is, the diagonal element .

[0031] The Laplacian matrix of the balanced directed graph is , , , , and when , . Obviously, , where Represents a Dimensional vector.

[0032] S32. Establish a balanced switching graph between multiple balanced directed graphs, and randomly switch between multiple balanced directed graphs according to the balanced switching graph; The balanced switching graph is: ; Among them, Represents the target balanced directed graph (that is, the balanced directed graph corresponding to the current moment), , Represents the switching signal, and .

[0033] Specifically, the balanced switching graph connects infinitely many followers. The balanced switching graph describes the process of continuous establishment and destruction of the communication connections between agents modeled by the network wave partial differential equation PDE, which means that every time a balanced directed graph is switched, the communication relationship between multiple followers characterized by the previous balanced directed graph at the corresponding moment is destroyed, and the communication relationship between multiple followers characterized by the current balanced directed graph at the corresponding moment is established.

[0034] It should be noted that y = Represents a switching signal that changes with time t , and its output value yis always a real number greater than or equal to 0. When y takes on a specific integer value, it corresponds to a switch in the balanced interaction topology, and the switching mode is to randomly switch among multiple balanced digraphs.

[0035] S4. Construct multiple induced graphs including follower systems and leaders according to the balanced interaction topology; the target induced graph is the digraph after introducing the leader in the target balanced digraph corresponding to the target topology instance; multiple induced graphs are always or intermittently weakly connected.

[0036] In an alternative implementation, step S4 includes: S41. For the target balanced digraph, add the node corresponding to the leader to the node set of the target balanced digraph.

[0037] Specifically, the target balanced digraph is any one of the multiple balanced digraphs and represents the balanced digraph corresponding to the current moment. The node set in the target balanced digraph adds the node corresponding to the leader. At this time, , where represents the node corresponding to the leader.

[0038] S42. Add directed edges from the node corresponding to the leader to all nodes corresponding to the followers to the edge set.

[0039] Specifically, add directed edges from the node corresponding to the leader to all nodes corresponding to the followers to the edge set E , , represents the directed edge from the node corresponding to the leader to the node corresponding to the follower . At this time, .

[0040] S43. Change the matrix elements corresponding to the edges from the node corresponding to the leader to each weakly connected component of the target balanced digraph in the adjacency matrix to numbers greater than zero, or when the state vector error between the follower and the leader in the error system satisfies the first condition, change the matrix elements corresponding to the edges from the node corresponding to the leader to each weakly connected component of the target balanced digraph in the adjacency matrix to numbers greater than zero; The first condition is: ; where B is a preset threshold; represents the follower and the state vector error between the leader.

[0041] Specifically, it can be seen from Lemma 1 that for any digraph , the induced graph is weakly connected if and only if there is at least one edge from the leader to each weakly connected component of the directed graph . Furthermore, given a balanced directed graph , the matrix is positive definite if and only if the induced graph is weakly connected, where . Therefore, when the induced graph always remains weakly connected (Assumption 1), it is necessary to ensure that if and only if there is at least one edge from the leader to each weakly connected component of the balanced directed graph , then it is only necessary to ensure that the matrix elements A corresponding to the edges from the node corresponding to the leader to each weakly connected component of the target balanced directed graph in the adjacency matrix are changed to numbers greater than zero, that is , .

[0042] Furthermore, considering the communication limitations in practical applications, the weak connectivity of the induced graph can only be guaranteed in some discontinuous time intervals.

[0043] Specifically, for each time interval Figure 2 as shown in , the induced graph only remains weakly connected within of each time interval, that is . Therefore, (Assumption 2) in the dynamic models of the follower system and the leader, when the lateral displacements and velocities of each follower, as well as the lateral displacements and velocities of the leader's model, are all constrained, that is, there exist: ; where B is a preset threshold; represents the state vector (i.e., lateral displacement) of the follower at position and time , represents the result of taking the partial derivative of with respect to t (i.e., velocity).

[0044] Under the conditions of Assumption 2, the first condition can be further obtained.

[0045] Among them, the first condition is: ; where B is a preset threshold; Represents the state vector error between the follower and the leader.

[0046] Therefore, when the induced graph intermittently remains weakly connected, the state vector error between the follower and the leader in the error system needs to satisfy the first condition, and at the same time, it also needs to satisfy the condition of weak connectivity in Lemma 1, that is, changing the matrix elements corresponding to the edges from the node corresponding to the leader to each weakly connected component of the target balanced digraph in the adjacency matrix to numbers greater than zero.

[0047] For example, Figure 3 as shown, the follower system includes 4 followers, namely Agent 1, Agent 2, Agent 3, and Agent 4, and the leader is Agent 0. Among them, different communication relationships between the 4 followers correspond to different topological instances. For example, Figure 2 in the three topological instances in, corresponding balanced digraphs are established for these three topological instances, and then respective induced graphs are established according to the three balanced digraphs. At the same time, it is necessary to satisfy that "if and only if there is at least one edge from the leader to each weakly connected component of the digraph ". For the first topological instance, Agents 1, 2, and 3 form a weakly connected component (named the first component), and Agent 4 forms a weakly connected component (named the second component). There is exactly one edge from Agent 0 to the first component and exactly one edge from Agent 0 to the second component. Therefore, the first topological instance is weakly connected. The remaining topological instances are analyzed in the same way and will not be elaborated here.

[0048] S5. Construct a boundary control law according to the error system and multiple induced graphs.

[0049] In an alternative embodiment, the boundary control law is: ; where , represents the state vector error between each follower and the leader at position 1 and time t , ; is the Laplacian matrix corresponding to the target balanced digraph, , represents the matrix elements corresponding to the edges from the leader of the target balanced digraph to each follower.

[0050] S6. Construct a control objective according to the dynamic models of the follower system and the leader.

[0051] In the embodiments of the present invention, by setting the control objective, the dynamic models of the follower system and the leader achieve leader-following consistency. ​

[0052] In an alternative embodiment, the control objective is: .

[0053] Wherein, represents the state vector of the follower at position and time . ; represents the state vector of the leader at position and time ; , .

[0054] S7. According to the boundary control law, the dynamic model of the follower system is adjusted in real time until the control objective is met and then stopped.

[0055] The embodiments of the present invention can achieve leader-follower consensus in different topological scenarios, and ensure that the state of the follower converges to the desired leader state at an exponential rate. Specifically, for the dynamic models of the follower system and the leader (i.e., PDEs), two scenarios of balanced interaction topologies are constructed, namely, time uninterrupted in the ideal scenario and time interval disconnection caused by communication limitations in the actual scenario; for the first scenario, there is a balanced interaction topology and the induced graph is always weakly connected; for the second scenario, the induced graph is only weakly connected during specific time periods, and the directed graph is still balanced. Based on the above two topological situations, the present invention can achieve leader-follower consensus with the induced graph always remaining weakly connected through constructing the boundary control rate, and the leader-follower consensus is guaranteed exponentially; it can also achieve leader-follower consensus with the induced graph remaining weakly connected during some discontinuous time intervals.

[0056] To verify the technical effects of the present invention, the leader-follower consensus for the above two topological situations is proved: Before the proof, it is necessary to introduce: Lemma 1. For any directed graph , the induced graph is weakly connected if and only if there exists at least one edge from the leader to each weakly connected component of the directed graph . In addition, given a balanced directed graph , the matrix is positive definite if and only if the induced graph is weakly connected, where .

[0057] Lemma 2: Suppose the directed graph is balanced and the induced graph is weakly connected, then there exists such that .

[0058] Lemma 3: Common Lyapunov function: ; is positive definite, where ; ; ; at and when ; where , and ; Proof 1: First, according to Lemma 3, calculate the derivative of the Lyapunov function candidate along the trajectory of the error system; Differentiate to get: (1); Differentiate and similarly to get: (2); (3); where , are all system parameters.

[0059] Apply the Young inequality to Equation (3) to get: (4); where is a constant and .

[0060] Secondly, from Lemma 1, we know that: (5); Finally, according to Lemma 2, let: ; ; ; Then we have: (6); where .

[0061] Therefore, the states of all followers are synchronized with the state of the unique leader and converge at a convergence rate .

[0062] Proof 2: The embodiments of the present invention use the same selected , and Lyapunov function, and derive the derivative along the solution of the error system, and formula (5) can be obtained.

[0063] It should be noted that in this case, , which requires different treatments for the last three terms in formula (5).

[0064] For , where , the following can be obtained .

[0065] And for , the following relationship holds , where represents the last three terms in (7) related to .

[0066] Combining the above observations, it can be obtained that ; For any , there exists such that . For the case of , the following can be obtained: ; If , then the following can be obtained: ; In addition, by referring to Assumption 2 and the mean value theorem, it can be obtained that ; where , , where .

[0067] Therefore, , indicating that the states of the followers converge to the desired leader state at an exponential rate .

[0068] In order to verify the effectiveness of the technical effects of the present invention, numerical simulation is used to verify that the partial differential multi-intelligence system with switching topology can achieve leader-follower consistency.

[0069] Specifically, the relevant preset parameters are shown in Table 1: Table 1 ;

[0070] In addition, the interactive topology is updated every 0.05 seconds. Figure 3 The three balanced directed graphs shown are randomly switched. Correspondingly, the corresponding Laplace matrices acting on each follower (that is, the Laplace matrices corresponding to the three balanced directed graphs) are: ; ; ; matrix Represents the communication relationship between the leader and the followers. Obviously, these induced graphs always maintain weak connectivity, and the directed graphs between followers are balanced.

[0071] like Figure 4 As shown, leaders No energy is dissipated during the evolution process. Figure 5 As shown, the follower 1 (such as Figure 5 (a)), follower 2 (as shown Figure 5 (b)), follower 3 (as shown Figure 5 (c)), follower 4 (as shown Figure 5 The trajectory shown in (d) shows that all followers reach consensus with the leader within 30 seconds. The evolution process in Figure 6 Detailed description is given in .

[0072] The present invention is based on the leader-follower consistency control of a multi-agent system under a balanced interactive topology based on partial differential equations (PDEs). By constructing a boundary control law, it is achieved that when the interactive topology always or intermittently maintains weak connectivity, the system can achieve the desired exponential convergence.

[0073] Although the embodiments of the present invention have been described in conjunction with the accompanying drawings, those skilled in the art may make various modifications and variations without departing from the spirit and scope of the present invention, and such modifications and variations are all within the scope defined by the appended claims.

Claims

1. A consensus control method for a partial differential multi-agent system with switched topologies, characterized in that Including: S1. Respectively construct the dynamic models of the follower system and the leader; the follower system includes a plurality of followers; S2. Construct an error system according to the dynamic models of the follower system and the leader; S3. Establish a balanced interaction topology for the multiple followers in the follower system; the balanced interaction topology includes balanced directed graphs corresponding to multiple topology instances; the topology instances characterize the communication relationships among the multiple followers; S4. Construct multiple induced graphs including the follower system and the leader according to the balanced interaction topology; the target induced graph is a directed graph after introducing the leader into the target balanced directed graph corresponding to the target topology instance; all of the multiple induced graphs are always or intermittently weakly connected; S5. Construct a boundary control law according to the error system and the multiple induced graphs; S6. Construct a control objective according to the dynamic models of the follower system and the leader; S7. Adjust the dynamic model of the follower system in real time according to the boundary control law until the control objective is satisfied and then stop.

2. The method according to claim 1, wherein The dynamic model of the follower system is: ； Among them, represents the state vectors of each follower in the follower system at the position and the time , and , ; , represents the follower at the position and the time of the state vector, , , ; is the first preset parameter, and ; is the control signal; and respectively represent the initial offsets and initial velocities of each follower; and are the second preset parameter and the third preset parameter respectively.

3. The method according to claim 2, wherein The dynamic model of the leader is: ； Among them, represents the state vector of the leader at position , time , and , ; is the first preset parameter, and ; and respectively represent the initial offset and initial velocity of the leader; and are the fourth preset parameter and the fifth preset parameter respectively.

4. The method according to claim 3, wherein The error system is: ； Among them, represents the state vector error between each follower and the leader at the position of and the time of ; , represents the state vector error between the follower and the leader; is the first preset parameter, and ; is the control signal; and respectively represent the initial offset error and the initial velocity error between each follower and the leader; are respectively the errors of the fourth preset parameter and the second preset parameter; are respectively the errors between the fifth preset parameter and the second preset parameter.

5. The method according to claim 4, wherein The S3 includes: S31. Establish multiple balanced directed graphs according to various communication relationships among the multiple followers in the follower system; The balanced digraph is ; where is the node set , is the follower corresponding node ; is the edge set , is the directed edge from node to node , and the total number of directed edges is ; is the adjacency matrix , and the matrix element is the communication weight from follower j to follower , and the diagonal element ; the Laplacian matrix of the balanced digraph is , , , , and when , ; S32. Establish a balanced switching graph among the multiple balanced directed graphs, and randomly switch among the multiple balanced directed graphs according to the balanced switching graph; The balanced switching graph is: ； Among them, represents a target balanced digraph, , represents a switching signal, and .

6. The method according to claim 5, characterized in that, The S4 includes: S41. For the target balanced directed graph, add a node corresponding to the leader to the node set of the target balanced directed graph; S42. Add directed edges from the node corresponding to the leader to the nodes corresponding to all followers to the edge set; S43. Change the matrix elements corresponding to the edges from the node corresponding to the leader to each weakly connected component of the target balanced directed graph in the adjacency matrix to numbers greater than zero, or when the state vector error between the follower and the leader in the error system satisfies the first condition, change the matrix elements corresponding to the edges from the node corresponding to the leader to each weakly connected component of the target balanced directed graph in the adjacency matrix to numbers greater than zero; The first condition is: ； Among them, B is a preset threshold; represents the follower and the state vector error between the leader.

7. The method according to claim 6, wherein The boundary control law is: ； Among them, , represents the state vector error of each follower and the leader at position 1 and time t , ; is the Laplacian matrix corresponding to the target balanced digraph, , represents the matrix element corresponding to the edge from the leader to each follower in the target balanced digraph.

8. The method according to claim 7, characterized in that, The control objective is: ； Among them, represents the follower at the position of , and the time is of the state vector, ; represents the state vector of the leader at the position of , and the time is ; , .

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