Multi-agent cooperative control based on complex environment perception

Abstract

The application relates to the technical field of control, in particular to multi-agent cooperative control based on complex environment perception. The application discloses that under the condition of a communication bandwidth-limited or high-delay unstable network, how to realize efficient information interaction between agents is a difficulty. The application discloses that under the condition of a communication bandwidth-limited or high-delay unstable network, the problem of how to realize efficient information interaction between agents is solved.

Description

Technical Field

[0001] This invention relates to the field of control technology, specifically to multi-agent collaborative control based on complex environment perception. Background Technology

[0002] Distributed cooperative control of multi-agent systems is a crucial research topic in complexity science, attracting widespread attention from researchers in physics, control science, and other fields in recent years, yielding fruitful results. Multi-agent systems can be widely applied not only in physics, military science, and social sciences, but also provide valuable mathematical models and analytical tools for biological population migration, power grid coupling analysis, and population movement network analysis. From a time-domain perspective, existing research on distributed cooperative control of multi-agent systems can be divided into multi-agent systems in the complex domain and multi-agent systems in the finite domain. Multi-agent systems in the finite domain are a special type of multi-agent system. In finite-domain networks, each agent's state is taken from a finite set of integers. The agent's state update is a weighted sum of its own state value and the state values ​​received from its neighbors, calculated based on modular arithmetic. This modular arithmetic has significant advantages in relative measurement of attitude estimation and other applications. In digital communication and wireless sensor networks, the communication bandwidth in the information transmission channel is often limited.

[0003] Multi-agent applications are used in complex environments, which typically involve numerous dynamic obstacles, significant changes in lighting, or communication blind spots. Ensuring efficient information exchange between agents in unstable networks with limited bandwidth or high latency is a significant challenge. Summary of the Invention

[0004] To overcome the above-mentioned shortcomings, the present invention provides multi-agent cooperative control based on complex environment perception.

[0005] The technical solution adopted in this invention is as follows: 1. Consistency of multi-agent systems over finite fields For a linear time-invariant (LTI) system over a finite field (1) in Given the adjacency matrix of the system, Pasqualetti et al. used the transition graph to derive the necessary and sufficient condition for consensus to be achieved in a linear time-invariant multi-agent system over a finite field, as follows. Theorem 1 Assumption In a finite field The upper system (1) can achieve consistency if and only if in Representing a finite field Upper adjacency matrix The characteristic polynomial of a graph in the complex field is given by the following formula: if a graph is strongly connected and its corresponding adjacency matrix is ​​given by the given formula, then the characteristic polynomial of the graph is given by the given formula. A If the row is random, then the corresponding linear time-invariant multi-agent system can achieve consistency. This is introduced into the model. in: express Time-based intelligent agent state, It is a time lag. It is a node The in-degree neighbor set, time-delay switching multi-agent system refers to a switching multi-agent system with time delay. If the system is attacked by external forces causing node loss and disconnection of connections, or if time delay exists during information transmission, then a time-delay switching system can be used to model it. For a switching multi-agent system with time delay, the system model is as follows: (2) Using a transition graph, the graph theory criterion for achieving consistency in system (2) is given as follows: Theorem 2 in a finite field Above, the time-delay system (2) can achieve consistency if and only if Exactly contains p There are 12 loops, and all loops revolve around the vertex. The unit loop, where In the study of time-delay switched multi-agent systems over finite domains, another form of system model is... (3) in express Time-based intelligent agent Regarding the state of consensus in general linear dynamic multi-agent systems, there is currently very little literature on this topic. For general linear dynamic multi-agent systems, the following model is proposed: (4) Using state coordinate transformation, a general linear dynamic multi-agent system with time delay is established as follows: (5) Besides the consistency problem, the synchronization problem is also an important fundamental issue in the research of distributed cooperative control of multi-agent systems. Theorem 3 Assumption in In a finite field Above, system (1) can achieve synchronization if and only if in Representing a finite field Upper adjacency matrix Characteristic polynomial, By comparing Theorem 1 and Theorem 3, it can be concluded that the consistency problem and the synchronization problem of multi-agent systems over a finite field are clearly two different problems. Adjusting the amount of information transmitted according to environmental constraints can effectively reduce interference from the external environment while ensuring synchronization. Random phenomena are widely present in power networks and communication networks, and random networks can be regarded as a special type of switching network.

[0006] 2 Controllability of Multi-Agent Systems over Finite Domains Existing studies on the controllability of multi-agent systems in complex and finite fields are all conducted under a leader-follower topology. Given a multi-agent system, the agents in the system are first divided into leaders and followers based on the presence or absence of external control input. The external control input can drive the leader's state to any state in the state space. Then, by using the control strategy given to each agent to update its own state, the state equations of the agents are written in a unified system form over a finite field. Thus, the controllability problem of this control system is regarded as the controllability problem of a multi-agent system.

[0007] 2.1 Controllability of Linear Time-Invariant Multi-Agent Systems over Finite Fields The controllability of a multi-agent system over a finite field depends on several factors: the system model, topology, leader selection, order of the field, and controllability index. A special phenomenon of multi-agent systems over finite fields is... Figure 1 As shown, Consider a multi-agent system over a finite field. (6) The topological graph in Figure 1 in a finite field The corresponding Laplacian matrices over the complex field are respectively make Calculations yield the following results. In the complex field, we can derive Figure 1 It is controllable over the complex field. Figure 1 In a finite field and finite fields The Laplacian matrices on are respectively make It can be concluded that Therefore in a finite field superior In a finite field superior This leads to the conclusion that in a finite field... Figure 2 above is controllable, but it is within a finite field. The topological graph 2 above is uncontrollable; therefore, the order of the field... q This will affect the controllability of multi-agent systems over finite fields. Consider discrete-time invariant systems over finite fields. (7) in And the matrix and Some elements in the matrix are zero elements, and the remaining elements are unrelated free parameter variables. The necessary and sufficient condition for the structure of system (7) to be controllable is that for the matrix... and If there exists a set of non-zero parameter values ​​over a finite field that can ensure the controllability matrix of the system, , The controllability that satisfies full row rank is called the structural controllability of a linear time-invariant multi-agent system over a finite field, where... µ Called the controllability subscript, Theorem 2 proves that when the order of the field... q Greater than nm +1 and the topology graph contains a generating forest ( m (where is the number of spanning trees in the spanning forest). Linear time-invariant multi-agent systems over finite fields are structurally controllable. Furthermore, Theorem 4 proves that when the topological graph is a spanning tree and the spanning tree has at most two branches, linear time-invariant multi-agent systems are structurally controllable over any finite field.

[0008] The two theorems have different focuses. The algebraic criterion of Theorem 2 has a wider range of applications, but its drawback is that it can only be applied to orders. q Greater than nmThe condition +1 holds true in a finite field, but cannot be generalized to any field. It may hold true in any field, but the required graph theory conditions are clearly more stringent. For these reasons, the controllability of linear time-invariant multi-agent systems cannot be derived into a general graph theory criterion that holds true in any field.

[0009] There are two commonly used methods for judging the controllability of linear time-invariant multi-agent systems: the controllability matrix and the PBH criterion.

[0010] Definition 1 (PBH criterion over the complex field): The system (7) is uncontrollable if and only if there exists a complex number λ Make Compared to the controllability matrix, the PBH criterion only requires finding the eigenvalues ​​of the system matrix, thus having lower computational complexity and a wider range of applications. However, the PBH criterion is only applicable to judging the controllability of systems in the complex field. Finite fields do not satisfy algebraic closure, meaning that some polynomials may not have roots in the finite field. Therefore, based on the controllability matrix, researching simple and easy-to-operate graph theory criteria has become a trend in the development of controllability of multi-agent systems in finite fields.

[0011] 2.2 Controllability of Switching Multi-Agent Systems over Finite Domains Consider discrete-time switching multi-agent systems over a finite field. (8) scalar function The switching path to be designed is considered in a discrete-time switching multi-agent system (8), even if each spanning tree is controllable and the controllability index is n Furthermore, it cannot guarantee that the entire switching system (8) is controllable. For finite fields Discrete-time switching multi-agent system (8), where n = 4. For the two different switching strategies, let A 1. A 2. A 3 corresponds to the figures respectively G 1. G 2 and G The adjacency matrix is ​​3. The switching path in Figure 2(a) is The weight of a self-loop on a node is Then the adjacency matrixA 1. A 2. A 3 can be written in the following forms: make Calculations show that in a finite field... superior In a finite field superior Change the switching strategy, let The switching path in Figure 2(b) is make In a finite field The above can be obtained through calculation. Based on the above analysis, even if each switching subsystem is in the first... n Even if every step is controllable, it cannot guarantee that the discrete-time switching multi-agent system (8) will be able to achieve the desired result in the first step. n To achieve controllability step by step, the controllability subscript becomes the fifth factor affecting controllability for discrete-time switching multi-agent systems.

[0012] 2.3 Strong target controllability of time-varying multi-agent systems over finite fields Unlike discrete-time switching multi-agent systems, time-varying multi-agent systems not only allow each switching subgraph to change arbitrarily, but also have a non-fixed sum of switching subgraphs. Therefore, studying time-varying multi-agent systems has broader theoretical significance and application value. A time-varying multi-agent system over a finite field is represented as follows: in: It is the set of target nodes. matrix Defined as Similar to discrete-time switched multi-agent systems, controllability indices are also a crucial factor affecting the controllability of time-varying multi-agent systems over finite domains. Therefore, the controllability index of time-varying multi-agent systems over finite domains is crucial in the first... rAlgebraic criteria for the controllability of the target by step strength. Lemma 1 Finite Field The multi-agent system (9) on the first The step is strong and the goal is controllable if and only if in Based on Lemma 1, and utilizing graph theory tools such as zero-driven sets, dynamic paths, and bipartite graphs, we present a method for time-varying multi-agent systems over finite fields in the 1st... The graph theory criterion for step strength target controllability is not only a necessary and sufficient condition, but it also holds true in any domain, and therefore can be applied to more general topological graphs.

[0013] The beneficial effects of this invention are: This invention provides a solution to the problem of how to conduct efficient information exchange between intelligent agents in unstable networks with limited communication bandwidth or high latency. Attached Figure Description

[0014] Figure 1 It is a leader-follower network containing 4 nodes; Figure 2(a) shows the first switching strategy; Figure 2(b) shows the second switching strategy. Detailed Implementation

[0015] See attached document Figure 1-2(b) : 1. Consistency of multi-agent systems over finite fields For a linear time-invariant (LTI) system over a finite field (1) in Given the adjacency matrix of the system, Pasqualetti et al. used the transition graph to give the necessary and sufficient conditions for a linear time-invariant multi-agent system over a finite field to achieve consensus as follows. Theorem 1 Assumption In a finite field The upper system (1) can achieve consistency if and only if in Representing a finite field Upper adjacency matrix The characteristic polynomial. Using this condition, the literature

[13] designed a relative pose estimation algorithm for monitor networks. In the complex domain, if a graph is strongly connected and the corresponding adjacency matrix is AIf the row order is random, then the corresponding linear time-invariant multi-agent system can achieve consistency. Pasqualetti et al. pointed out that the above graph theory and matrix conditions cannot guarantee that linear time-invariant multi-agent systems over finite fields will achieve consistency. Furthermore, using the matrix semi-tensor product method, the output consistency problem of linear time-invariant multi-agent systems over finite fields is transformed into a set stabilization problem. Based on the input-output proper matrix method, the output state feedback control set is obtained, along with the control algorithm for calculating this set. In practical applications, due to the complexity of network communication and the limitation of network bandwidth, there will be lags in the transmission, storage, and processing of state information between agents. The existence of time lags may cause a decrease in system performance or even lead to system instability. Therefore, it is particularly necessary to explore the impact of time lags on the achievement of system consistency. Current research on the consistency problem of time-delay multi-agent systems over finite fields focuses on linear time-invariant time-delay multi-agent systems and time-delay switching multi-agent systems. Studies have investigated the consistency problems of linear time-invariant time-delay multi-agent systems over finite fields with single, multiple, and bounded time-varying delays. Unlike time-delay multi-agent systems over complex fields, the consistency of linear time-invariant time-delay multi-agent systems over finite fields depends not only on the diagonal elements of the system matrix but also on the off-diagonal elements. During information transmission, the generation of time delays is often random, and most random time delays affect the system, causing instability. Therefore, studying the consistency problem of multi-agent systems with random time delays over finite fields is more in line with practical applications. To address this, [further research is needed]. in: express Time-based intelligent agent state, It is a time lag. It is a node For linear time-invariant multi-agent systems with random time delays, algebraic criteria often fail to intuitively reflect the connectivity relationships between nodes within the system, and are also insufficient to demonstrate the impact of network structure on system controllability. A time-delayed switching multi-agent system refers to a switching multi-agent system with time delays. If the system is attacked by external forces causing node loss or disconnection, or if time delays exist during information transmission, then a time-delayed switching system can be used for modeling. The following system model is given for a time-delayed switching multi-agent system: (2) Using a transition graph, the graph theory criterion for achieving consistency in system (2) is given as follows. Theorem 2 In a finite field Above, the time-delay system (2) can achieve consistency if and only if Exactly contains p There are 12 loops, and all loops revolve around the vertex. The unit loop, where In the study of time-delay switching multi-agent systems over finite domains, there is another form of system model: (3) in express Time-based intelligent agent Based on the idea of ​​matrix half-tensor product, an algebraic criterion for system consistency is established for time-varying and time-delay switching multi-agent systems over finite fields. If the transition graph method is used, it is difficult to derive the graph theory criterion for system consistency (3). The criterion is for first-order multi-agent systems. At present, there are few relevant literatures on the consistency of multi-agent systems based on general linear dynamics. The following model is given for general linear dynamic multi-agent systems: (4) Using state coordinate transformation, it was proved that system (4) achieves consistency if and only if It is a directed acyclic graph, and It is a zero-power matrix. Based on this, a general linear dynamic multi-agent system with time delay is established as follows: (5) Besides the consistency problem, the synchronization problem is also a crucial fundamental issue in the research of distributed cooperative control of multi-agent systems. Synchronization in multi-agent systems refers to the ability of agents to adjust and update their states through local cooperation and communication, ensuring that the states of each agent are equal within a finite time. However, in practical robot arm networks, where the arms may need to be large enough to reach the same time-varying position, requiring each robot arm to reach a fixed state within a finite time becomes impractical. Therefore, studying the synchronization problem has broader significance than the consistency problem. Based on the idea of ​​transition graphs, the necessary and sufficient conditions for synchronization in linear time-invariant multi-agent systems over finite fields are given. The main conclusions can be expressed by the following theorem. Theorem 3 Assumption in In a finite field Above, system (1) can achieve synchronization if and only if in Representing a finite field Upper adjacency matrix The characteristic polynomial.

[0016] By comparing Theorem 1 and Theorem 3, it can be concluded that the consistency problem and the synchronization problem of multi-agent systems over a finite field are clearly two different problems. Furthermore, by adjusting the amount of information transmitted according to environmental constraints, the interference from the external environment can be effectively reduced while ensuring synchronization.

[0017] Random phenomena are widespread in power and communication networks. Random networks can be viewed as a special type of switching network. The synchronization problem of random switching networks over finite fields is addressed, and algebraic conditions for achieving synchronization are given. Based on the matrix semi-tensor product method and algebraic state space representation method, the cluster synchronization problem is transformed into a set stability problem, and on this basis, multi-navigator cluster synchronization is achieved. In addition, using Smith type and eigenvectors, algebraic conditions for achieving synchronization of time-delayed multi-agent systems over finite fields are given.

[0018] 2 Controllability of Multi-Agent Systems over Finite Domains Existing studies on the controllability of multi-agent systems in complex and finite fields are all conducted within a leader-follower topology. Given a multi-agent system, the agents are first divided into leaders and followers based on the presence or absence of external control input. External control input can drive the leader's state to any state in the state space. Then, through a given control strategy for each agent to update its own state, the state equations of the agents are written in a unified system form over a finite field. Thus, the controllability problem of this control system is viewed as the controllability problem of a multi-agent system.

[0019] 2.1 Controllability of Linear Time-Invariant Multi-Agent Systems over Finite Fields The controllability of multi-agent systems over finite fields depends on several factors: the system model, topology, leader selection, field order, and controllability index. Existing system models mainly focus on time-invariant, switching, and time-varying multi-agent systems. Changes in topology and leader selection are crucial factors influencing the controllability of multi-agent systems over complex and finite fields. The field order... q This can affect the controllability of the system and is a special phenomenon in multi-agent systems over a finite domain, such as... Figure 1 As shown, Consider multi-agent systems over finite fields (6) The topology in Figure 1 is in a finite field The corresponding Laplacian matrices over the complex field are respectively make Calculations yield the following results. In the complex field, we can derive Figure 1 It is controllable over the complex field. Figure 1 In a finite field and finite fields The Laplacian matrices on are respectively make Based on row transformations, we can derive Therefore in a finite field superior In a finite field superior This leads to the conclusion that in a finite field... superior Figure 1 It is controllable, but only in a finite field. Topology Figure 1 It is uncontrollable. Therefore, the order of the field... q This affects the controllability of multi-agent systems over finite fields. Consider discrete-time invariant systems over finite fields. (7) in And the matrix and Some elements in the matrix are zero elements, and the remaining elements are unrelated free parameter variables. The necessary and sufficient condition for the system (7) to be controllable is that for the matrix... and If there exists a set of non-zero parameter values ​​over a finite field that can ensure the controllability matrix of the system, The controllability that satisfies full row rank is called the structural controllability of a linear time-invariant multi-agent system over a finite field, where... µ This is called a controllability subscript, when the order of the field is... q Greater than nm +1 and the topology graph contains a generating forest ( m(This is the number of spanning trees in the generated forest). Linear time-invariant multi-agent systems over finite fields are structurally controllable.

[0020] The two theorems have different focuses. The algebraic criterion of Theorem 2 has a wider range of applications, but its drawback is that it only applies to orders. q Greater than nm The +1 condition holds true over a finite field and cannot be generalized to any field. For these reasons, the controllability of linear time-invariant multi-agent systems cannot be derived with a general graph-theoretic criterion that holds true over any field. There are two commonly used methods for judging the controllability of linear time-invariant multi-agent systems: the controllability matrix and the PBH criterion. Definition 1 (PBH Criterion over the Complex Field) The system (7) is uncontrollable if and only if there exists a complex number λ Make Compared to the controllability matrix, the PBH criterion only requires finding the eigenvalues ​​of the system matrix, thus having lower computational complexity and a wider range of applications. However, the PBH criterion is only applicable to determining the controllability of systems in the complex field. Finite fields do not satisfy algebraic closure, meaning that some polynomials may not have roots in a finite field. Therefore, based on the controllability matrix, researching simple and easy-to-operate graph theory criteria has become a trend in the development of controllability assessment for multi-agent systems in finite fields.

[0021] 2.2 Controllability of Switching Multi-Agent Systems over Finite Domains Consider discrete-time switching multi-agent systems over a finite field. (8) scalar function The switching path to be designed is considered in a discrete-time switching multi-agent system (8), even if each spanning tree is controllable and the controllability index is n However, it is also impossible to guarantee that the entire switching system (8) is controllable.

[0022] For finite fields Discrete-time switching multi-agent system (8), where n = 4. For the two different switching strategies, let A 1. A 2. A 3 corresponds to the figures respectively G 1. G 2 and G The adjacency matrix is ​​3. The switching path in Figure 2(a) is The weight of a self-loop on a node is Then the adjacency matrix A 1. A 2. A 3 can be written in the following forms: make Calculations show that in a finite field... superior In a finite field superior Change the switching strategy, let The switching path in Figure 2(b) is make In a finite field The above can be obtained through calculation. Based on the above analysis, even if each switching subsystem is in the first... n Even if every step is controllable, it cannot guarantee that the discrete-time switching multi-agent system (8) will achieve the desired result in the first step. n To achieve controllability, the controllability subscript becomes the fifth factor affecting controllability in discrete-time switching multi-agent systems. First, the discrete-time switching multi-agent system is given... p The controllable algebraic criterion: Based on the above algebraic criterion, even if each switching subsystem is uncontrollable (i.e. each switching subgraph contains unreachable nodes), the entire system (8) can be controlled by arranging the switching strategy reasonably.

[0023] 2.3 Strong target controllability of time-varying multi-agent systems over finite fields Unlike discrete-time switching multi-agent systems, time-varying multi-agent systems not only allow each switching subgraph to change arbitrarily, but also have a non-fixed sum of switching subgraphs. Therefore, studying time-varying multi-agent systems has broader theoretical significance and application value. A time-varying multi-agent system over a finite field is represented as follows: in: It is the set of target nodes. matrix Defined as Similar to discrete-time switching multi-agent systems, controllability indices are also an important factor affecting the controllability of time-varying multi-agent systems over finite domains.

[0024] Lemma 1 Finite Field The multi-agent system (9) on the first The step is strong and controllable, if and only if in Based on Lemma 1, graph theory tools such as zero-driven sets, dynamic paths, and bipartite graphs can be used to construct graphs on any domain, and thus can be applied to more general topological graphs.

Claims

1. A multi-agent cooperative control based on complex environment perception, characterized in that: For a linear time-invariant system over a finite field （1） in Given the adjacency matrix of the system, the necessary and sufficient conditions for a linear time-invariant multi-agent system over a finite field to achieve consensus are given using the transition graph as follows: Assumption In a finite field The upper system (1) can achieve consistency if and only if in Representing a finite field Upper adjacency matrix The characteristic polynomial of the complex number system is used to determine if a graph is strongly connected and its corresponding adjacency matrix is ​​0. A If the row is random, then the corresponding linear time-invariant multi-agent system can achieve consistency. This is introduced into the model. in: express Time-based intelligent agent state, It is a time lag. It is a node For linear time-invariant multi-agent systems with random time delays, algebraic criteria often fail to intuitively reflect the connectivity relationships between nodes within the system, and are also insufficient to demonstrate the impact of network structure on system controllability. A time-delayed switching multi-agent system refers to a switching multi-agent system with time delays. If the system is attacked by external forces causing node loss or disconnection, or if time delays exist during information transmission, then a time-delayed switching system can be used for modeling. The following system model is given for a time-delayed switching multi-agent system: （2） Using a transition graph, the graph theory criterion for achieving consistency in system (2) is given as follows: In a finite field Above, the time-delay system (2) can achieve consistency if and only if Exactly contains p There are 12 loops, and all loops revolve around the vertex. The unit loop, where In the study of time-delay switching multi-agent systems over finite domains, the system model can also be expressed as: （3） in express Time-based intelligent agent Regarding the state of multi-agent systems, based on the concept of matrix semi-tensor product, there is currently very little literature on consensus research in general linear dynamic multi-agent systems. For general linear dynamic multi-agent systems, the following model is established: （4） Using state coordinate transformation, if and only if It is a directed acyclic graph, and It is a zero-power matrix. Based on this, the following general linear dynamic multi-agent system with time delay is established: （5） Assumption in In a finite field Above, system (1) can achieve synchronization if and only if in Representing a finite field Upper adjacency matrix Characteristic polynomial, Consider multi-agent systems over finite fields （6） Topological graphs in finite fields The corresponding Laplacian matrices over the complex field are respectively make Calculations yield the following results. In the complex field, we can derive It is controllable over the complex field and over the finite field. and finite fields The Laplacian matrices on are respectively make It can be concluded that Therefore in a finite field superior In a finite field superior This leads to the conclusion that in a finite field... It is controllable in the above, but in the finite field The topology is uncontrollable. Therefore, the order of the field... q This will affect the controllability of multi-agent systems over finite fields. Consider a discrete-time invariant system over a finite field. （7） in And matrix and Some elements in the matrix are zero elements, and the remaining elements are unrelated free parameter variables. The necessary and sufficient condition for the structure of system (7) to be controllable is that for the matrix... and If there exists a set of non-zero parameter values ​​over a finite field that can ensure the controllability matrix of the system, The controllability that satisfies full row rank is called the structural controllability of a linear time-invariant multi-agent system over a finite field, where... µ The controllability subscript is used to define a system (7) that is uncontrollable if and only if there exists a complex number. λ Make ， Compared to the controllability matrix, the PBH criterion only requires finding the eigenvalues ​​of the system matrix, thus having lower computational complexity and a wider range of applications. However, the PBH criterion is only applicable to judging the controllability of systems in the complex field. Finite fields do not satisfy algebraic closure, meaning that some polynomials may not have roots in a finite field. Therefore, based on the controllability matrix, researching simple and easy-to-operate graph theory criteria has become a trend in the development of controllability assessment for multi-agent systems in finite fields. Controllability of switching multi-agent systems over finite domains Consider discrete-time switching multi-agent systems over a finite field （8） scalar function The switching path to be designed is considered in a discrete-time switching multi-agent system (8), pointing out that even if each spanning tree is controllable and the controllability index is n Furthermore, it cannot guarantee that the entire switching system (8) is controllable. For finite fields Discrete-time switching multi-agent system (8), where n = 4, For the two different switching strategies, let A 1. A 2. A 3 corresponds to the figures respectively. G 1. G 2 and G The adjacency matrix of 3, the switching path is The weight of a self-loop on a node is Then the adjacency matrix A 1. A 2. A 3 can be written in the following forms: make Calculations show that in a finite field... superior, In a finite field superior, Change the switching strategy, let The switching path is make In a finite field The above can be obtained through calculation. Based on the above analysis, the entire system (8) can be kept under control by reasonably arranging the switching strategy. Strong target controllability of time-varying multi-agent systems over finite fields Unlike discrete-time switching multi-agent systems, time-varying multi-agent systems not only allow each switching subgraph to change arbitrarily, but also the sum of the switching subgraphs is not fixed. Therefore, studying time-varying multi-agent systems has broader theoretical significance and application value. A time-varying multi-agent system over a finite field is represented as follows: in: It is the set of target nodes. matrix Defined as Similar to discrete-time switched multi-agent systems, controllability indices are also a crucial factor affecting the controllability of time-varying multi-agent systems over finite domains. Therefore, the controllability index of time-varying multi-agent systems over finite domains is crucial in the first... r Algebraic criteria for the controllability of the target by step strength. Lemma 1 Finite Field The multi-agent system (9) on the first The step is strong and the goal is controllable if and only if in Using graph theory tools such as zero-driven sets, dynamic paths, and bipartite graphs, we can apply them to more general topological graphs.

Images