A multi-agent system finite time cooperative control method

Abstract

The application belongs to the field of multi-agent cooperative control, and particularly relates to a multi-agent system finite time cooperative control method; the method comprises the following steps: constructing a communication topology by taking each agent in a second-order system as a communication node; constructing a dynamic model of each agent in the second-order system according to algebraic graph theory and initializing; setting a non-continuous event-triggered impulse control protocol; all agents communicate with each other through the communication topology, and each agent continuously acquires state information of itself and neighbor agents in the communication process; updating all agents at the impulse time by using the event-triggered impulse control protocol; updating all agents at the non-impulse time according to the dynamic model; constructing an error system according to the state information of the agents, so that each agent reaches consistency in a limited time; and the application can realize finite time state consistency at a faster speed and with fewer event triggers.

Description

Technical Field

[0001] This invention belongs to the field of multi-agent cooperative control, specifically relating to a finite-time cooperative control method for multi-agent systems. Background Technology

[0002] In nature, large gatherings of individuals often create coordinated, orderly, and even awe-inspiring scenes of movement, such as massive flocks of birds soaring in the sky, schools of fish swimming in the ocean, and wolf packs cooperating to hunt on land. The characteristics of these group phenomena—distribution, coordination, self-organization, stability, and the emergence of intelligence—have attracted the research interest of biologists. Later, to meet engineering needs, Minsky of MIT proposed the concept of an intelligent agent, introducing the concept of individual social behavior from the biological world into the field of computer science. At this point, biology and computer science intersected. An intelligent agent can be a corresponding software program or a physical object such as a person, vehicle, robot, or artificial satellite.

[0003] In recent years, due to the interdisciplinary development and integration of biology, computer science, artificial intelligence, control science, sociology, and other disciplines, multi-agent systems (MAS) have increasingly attracted widespread attention from scholars and have become a research hotspot in the fields of control science and artificial intelligence. Among these, the consistency problem, as a core issue in the study of MAS, has been addressed by various control methods, including finite-time control, sliding mode control, impulse control, and event-triggered control. However, these control methods require continuous control inputs, and in network control, real-time communication is also necessary for information exchange between nodes. Therefore, there is an urgent need for a control method that can both save system resources and improve the system convergence speed.

[0004] Impulse control, as a relatively simple discontinuous control method, has been successfully applied in many disciplines. In recent years, to achieve state consensus in MASs (Multi-Action System) more quickly, state impulse strategies have been implemented to solve the state consensus problem in MASs. Many scholars have obtained consensus criteria for MASs through impulse control schemes, but there are still some aspects that need improvement. For example, since impulse control is periodic, to achieve consensus quickly, some scholars have changed the constraint of the fixed pulse instant to the interval between the center or left endpoint of the time window of adjacent pulses. Although this allows for a larger control area, the pulse occurrence time is still limited by the time window setting, which is not flexible enough. While these impulse methods are effective in solving consensus problems, they do not reduce communication and computation costs. To ensure fast convergence, the pulse frequency must be designed to be high enough. Therefore, the results tend to be conservative. Unlike the above control strategies, the triggering time of the event-triggered impulse control strategy is the pulse input time. This control strategy can effectively reduce the number of information transmissions and resource consumption. However, in real life, second-order dynamic models are mostly used to describe position and velocity information, such as torque motors and gas injectors, which require acceleration to adjust the required motion. Therefore, considering the mutual influence of the agents' positions and velocities, solving the consistency problem in second-order multi-agent systems, enabling the agents to converge to a common rule or state, is more challenging and practically significant. Moreover, in real-world applications, we've found that achieving consistency in multi-agent systems can be very time-consuming. Thus, control strategies that require extended time to achieve system consistency cannot meet the application requirements of some practical systems. Finite-time control can achieve system consistency within a finite time, thus meeting real-world needs. Furthermore, finite-time control is crucial for improving control accuracy, exhibiting better robustness, and mitigating interference.

[0005] Based on the above discussion, considering the increased communication burden caused by continuous control and the need to achieve consistency within a finite time in practice, this paper aims to design an event-triggered pulse control strategy to achieve finite time consistency in a second-order multi-agent system for the finite-time consistency problem of multi-agent systems. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention proposes a finite-time cooperative control method for multi-agent systems, which includes:

[0007] S1. Construct a communication topology by treating each agent in a second-order continuous system as a communication node;

[0008] S2. Construct and initialize the dynamic models of each agent in the second-order continuous system based on algebraic graph theory;

[0009] S3. Configure the non-continuous event trigger pulse control protocol;

[0010] S4. All agents communicate with each other through the communication topology, and each agent continuously acquires its own and its neighboring agents' state information during the communication process;

[0011] S5. At pulse moments, a discontinuous event-triggered pulse control protocol is used to update the state and velocity information of all agents; at non-pulse moments, the state and velocity information of all agents are updated according to the dynamic model.

[0012] S6. Based on the event-triggered pulse control strategy proposed above, the multi-agent system can eventually achieve consensus within a finite time.

[0013] Furthermore, step S1, constructing the communication topology, includes: constructing an undirected graph. in Representing an undirected graph The set of nodes, where N is the total number of nodes; It is an undirected graph edge set, Let a be the adjacency matrix of an undirected graph G. ij Let a represent the edge weight from communication node i to communication node j. If there is no connecting edge between communication node i and communication node j, then a ij =0; based on the degree matrix of the undirected graph G and Laplace matrix in d i Degree matrix The value of the element in the i-th row and i-th column.

[0014] Furthermore, the second-order continuous system dynamic model constructed in step S2 based on algebraic graph theory is expressed as follows:

[0015]

[0016] Where, x i (t) represents the state variable of agent i at time t, v i (t) represents the velocity variable of agent i at time t, u i (t) represents the control input of agent i at time t.

[0017] Furthermore, the discontinuous event-triggered pulse control protocol is represented as follows:

[0018]

[0019]

[0020] Among them, ui (t) represents the control input of agent i at time t, q i (t) represents the joint state value of agent i at time t, γ1 and γ2 represent the impulse gain, and δ(tt) represents the joint state value of agent i at time t. k i ) represents the Dirac function, a ij This represents the edge weight from communication node i to communication node j, where i,j = 1, 2, ..., N, and N represents the total number of agents in the second-order continuous system. i Let x represent the set of neighboring agents of agent i. i (t) represents the state variable of agent i at time t, x j (t) represents the state variable of agent i's neighbor agent j at time t; t k i Let t be the kth pulse moment of agent i at time t.

[0021] Furthermore, in step S5, the process of updating the state information of each agent using a discontinuous event-triggered pulse control protocol includes: at the pulse moment, determining whether the trigger function value of the agent is greater than 0; if so, calculating the control input of the agent according to the discontinuous event-triggered pulse control protocol, and updating the state information of the agent according to the control input; otherwise, maintaining the state information of the agent.

[0022] Furthermore, the formula for calculating the trigger function value is as follows:

[0023] f i (t)=|q i (t)|-|z i (t)|,

[0024] z i (t)=γ1q i α (t)+q i (t)+γ2v i (t)

[0025] Among them, f i z(t) represents the trigger function of agent i at time t, qi(t) represents the joint state value of agent i at time t, and z(t) represents the joint state value of agent i at time t. i (t) represents the intermediate parameters of agent i at time t, v i (t) represents the velocity of agent i at time t, γ1 and γ2 represent the pulse gain, and α represents the error coefficient.

[0026] Furthermore, in step S5, the process of updating the state information of each agent according to the dynamic model at non-pulse moments includes: at non-pulse moments, obtaining the state information of each agent at the previous moment according to the dynamic model, and updating the current state information of each agent according to its own and its neighboring agents' state information at the previous moment.

[0027] The beneficial effects of this invention are:

[0028] This invention proposes a discontinuous event-triggered pulse control protocol for achieving finite-time consensus cooperative evolution in second-order multi-agent systems. The proposed event triggering conditions determine the update time of finite-time control and pulse moments, effectively reducing the number of event triggers and the system communication burden. By introducing finite-time control, we can also derive the system convergence time bound, avoiding situations where the system requires a long time to achieve consensus, thus failing to meet the application requirements of practical systems. Furthermore, in real-world applications, second-order dynamic models are often used to describe position and velocity information, such as torque motors and gas injectors, both of which require acceleration to adjust the required motion. Therefore, considering the mutual influence between the position and velocity of agents, solving the consensus problem in second-order multi-agent systems, enabling agents to converge to a common rule or state, is both challenging and practically significant. Attached Figure Description

[0029] Figure 1 This is a flowchart of the multi-agent finite-time consistency cooperative control method based on event-triggered pulses in this invention. Detailed Implementation

[0030] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. The present invention provides a finite-time cooperative control method for a multi-agent system, such as... Figure 1 As shown, it includes the following steps:

[0031] S1. Construct a communication topology by treating each agent in the second-order continuous system as a communication node.

[0032] Specifically, step S1, constructing the communication topology, includes: constructing an undirected graph. in Representing an undirected graph The set of nodes, where N is the total number of nodes; It is an undirected graph edge set, Let a be the adjacency matrix of an undirected graph G.ij Let a represent the edge weight from communication node i to communication node j. If there is no connecting edge between communication node i and communication node j, then a ij =0; based on the degree matrix of the undirected graph G and Laplace matrix in Degree matrix The value of the element in the i-th row and i-th column. Specifically, the Laplace matrix in:

[0033]

[0034] Among them, l ij N represents the element in the i-th row and j-th column of the Laplace matrix. i Let deg(i) represent the set of neighboring agents related to agent i, and deg(i) represent the value of the element in the i-th row and i-th column of the degree matrix.

[0035] S2. Construct and initialize the dynamic models of each agent in the second-order continuous system based on algebraic graph theory.

[0036] Specifically, algebraic graph theory is an important tool for studying multi-agent consensus. The dynamic model constructed in step S2 based on algebraic graph theory is expressed as follows:

[0037]

[0038] Where, x i (t) represents the state variable of agent i at time t, v i (t) represents the velocity variable of agent i at time t, u i (t) represents the control input that agent i needs to design at time t.

[0039] S3. Configure the non-continuous event trigger pulse control protocol.

[0040] Specifically, the discontinuous event-triggered pulse control protocol is represented as follows:

[0041]

[0042]

[0043] Among them, u i (t) represents the control input of agent i at time t, q i (t) represents the joint state value of agent i at time t, γ1 and γ2 represent the impulse gain, and δ(tt) represents the impulse gain. k i ) represents the Dirac function, a ijLet N represent the edge weight from communication node i to communication node j, and let N represent the total number of communication nodes. i Let x represent the set of neighboring agents of agent i. i (t) represents the state variable of agent i at time t, x j (t) represents the state variable of agent i's neighbor agent j at time t, where t k i For the pulse time sequence triggered by agent i.

[0044] S4. All agents communicate with each other through the communication topology, and each agent continuously acquires its own and its neighboring agents' state information during the communication process.

[0045] Specifically, the intelligent agent receives state information from sensors, which includes the current position information and the previous position information.

[0046] S5. Each agent updates its own state and velocity information based on the state information of its neighbors and its own position information, as well as the position information obtained by the sensors of its neighbors. During pulse moments, it uses a discontinuous event-triggered pulse control protocol to update its own state and velocity information; during non-pulse moments, it updates its own state and velocity information based on the dynamic model.

[0047] The event-triggered pulse control strategy designed in this invention is similar to the classic event-triggered strategy. The feedback loop is activated only when an event occurs, and the system trajectory reaches the design target. At each trigger moment, the pulse generator processes the information collected from the sensor and calculates new state and speed information according to the set control protocol requirements, which is then transmitted to the actuator for execution.

[0048] Specifically, in step S5, the process of updating the state information of each agent using a discontinuous event-triggered pulse control protocol includes: at the pulse moment, determining whether the trigger function value of the agent is greater than 0; if so, calculating the control input of the agent according to the discontinuous event-triggered pulse control protocol, and updating the state information of the agent according to the control input; otherwise, maintaining the state information of the agent.

[0049] Specifically, the formula for calculating the trigger function value is as follows:

[0050] f i (t)=|q i (t)|-|z i (t)|,

[0051] z i (t)=γ1q i α (t)+q i(t)+γ2v i (t)

[0052] Among them, f i z(t) represents the trigger function value of agent i at time t, qi(t) represents the joint state value of agent i at time t, and z(t) represents the trigger function value of agent i at time t. i (t) represents the intermediate parameters of agent i at time t, v i (t) represents the velocity of agent i at time t, γ1 and γ2 represent the pulse gain, and α represents the error coefficient.

[0053] Specifically, in step S5, the process of updating the state information of each agent according to the dynamic model at non-pulse times includes: at non-pulse times, obtaining the state information of each agent at the previous time according to the dynamic model, and updating the current state information of each agent according to its own and its neighboring agents' state information at the previous time.

[0054] S6. Based on the event-triggered pulse control strategy proposed above, the multi-agent system can eventually achieve consensus within a finite time.

[0055] The innovation of this invention lies primarily in the inclusion of finite-time control. Since most consensus algorithms converge asymptotically, the control algorithm can only guarantee that the controlled agent achieves consensus within an infinite timeframe. Furthermore, in practical applications, we have found that achieving consensus in multi-agent systems can be extremely time-consuming. Therefore, control strategies that require a long time to achieve system consensus cannot meet the application requirements of some real-world systems. Finite-time control enables system consensus to be achieved within a finite time, thus meeting real-world needs. In addition, finite-time control is significant for improving control accuracy, exhibiting better robustness, and suppressing interference. Moreover, in real-world applications, second-order dynamic models are often used to describe position and velocity information, such as torque motors and gas injectors, both of which require acceleration to adjust the required motion. Therefore, considering the mutual influence between the agent's position and velocity, solving the consensus problem in second-order multi-agent systems, enabling the agents to converge to a common rule or state, is both challenging and practically significant.

[0056] In this invention, when a multi-agent system satisfies 0 < α < 1, 0 < γ1, and 0 < γ2 < 2, finite-time consistency can be achieved, and the set time of this finite time τ is:

[0057]

[0058] Where D represents a matrix, λ nax Let α represent the largest eigenvalue of matrix D, α represent the error coefficient, and γ1 and γ2 represent the pulse gain.

[0059] Lemma 1: First consider an undirected connected graph G consisting of N nodes; then choose... The Laplacian matrix related to graph G The eigenvectors corresponding to the zero eigenvalues. Let l2,...,l N It is related to the Laplace matrix The eigenvectors associated with the positive eigenvalues ​​of l1, l2, ..., l N Called R N A set of orthogonal bases. Let L = (l1, l2, ..., l N Then for any x∈R N ,have Where x = [x1,...x N ] T , Let λ2,...,λ be a diagonal matrix. N It is the Laplace matrix The eigenvalues ​​of . Where R represents the set of real numbers, Ri N This represents N-dimensional Euclidean space.

[0060] Lemma 2: Let V:R→R + V(t) is a positive definite, Lipshitz continuous function if there exist two constants c > 0 and 0 < α < 1 such that the Dini derivative of V(t) satisfies the following condition:

[0061]

[0062] Among them, t o Let V(t) represent the initial time. Then V(t) satisfies

[0063] V 1-α (t)≤V 1-α (t0)-c(1-α)(t-t0),t0≤t≤τ,

[0064] And V(t)≡0, The set time τ is given by the following formula:

[0065]

[0066] Lemma 3: Suppose there exist three constants c > 0, 0 < α < 1, 0 < δ k <1, where k = {1, 2, ..., m} is a finite set of positive integers, m is a positive integer, and there is a continuous non-negative function V(t) such that the Dini derivative of V(t) satisfies the following condition:

[0067]

[0068] Then, the following inequality holds:

[0069] V 1-α (t)≤V 1-α (t0)-c(1-α)(t-t0),t0≤t≤τ,

[0070] Where τ is a constant representing the settling time.

[0071] definition Where, x i (t) represents the state variable of agent i at time t, v i (t) represents the velocity variable of agent i at time t, u i (t) represents the control input that agent i needs to design at time t.

[0072] We can use Lyapunov's stability theorem, graph theory knowledge, and basic inequalities to verify the agent after the state update. The process includes:

[0073] Consider a suitable Lyapunov candidate function:

[0074]

[0075] For any Differentiate with respect to V

[0076]

[0077] Using event triggering condition b i (t)=|q i (t)|-|z i If (t)|≥0, we have |q i (t)|>|z i (t)|.

[0078] Specifically, the formula for calculating the trigger function value is as follows:

[0079] f i (t)=|q i (t)|-|z i (t)|,

[0080] z i (t)=γ1q i α (t)+q i (t)+γ2v i (t)

[0081] Among them, f i z(t) represents the trigger function value of agent i at time t, qi(t) represents the joint state value of agent i at time t, and z(t) represents the trigger function value of agent i at time t. i (t) represents the intermediate parameters of agent i at time t, vi (t) represents the velocity of agent i at time t, γ1 and γ2 represent the pulse gain, and α represents the error coefficient.

[0082] According to Lemma 1, we can obtain

[0083]

[0084] Where λ max (D) is the largest eigenvalue of matrix D, λ min (D) is the smallest eigenvalue of matrix D, which is defined by Lemma 1.

[0085]

[0086] According to Lemma 2, we can conclude that 0 < α < 1 and 0 < γ1 must be satisfied here.

[0087]

[0088] On the other hand, when t = t k At that time, according to the condition 0 < δ of Lemma 3 k Since γ<1, therefore 0<γ<2<2, we have

[0089]

[0090] in, This indicates taking the limit.

[0091] By Lemma 3, it is obvious that:

[0092]

[0093]

[0094] Therefore, through event-triggered pulse control, a second-order multi-agent system achieves consensus within a finite time. Once consensus is reached, the system saves its latest state information, and all agents can execute corresponding operations based on the latest control input, achieving the goal of coordinated control of the system. If the system is a multi-UAV swarm, it can also achieve coordinated control of the swarm.

[0095] In this invention, unless otherwise explicitly specified and limited, the terms "installation," "setting," "connection," "fixing," "rotation," etc., should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral part; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal connection of two components or the interaction between two components. Unless otherwise explicitly limited, those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0096] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A finite-time cooperative control method for a multi-agent system, characterized in that, Includes the following steps: S1. Construct a communication topology by treating each agent in a second-order continuous system as a communication node; S2. Construct and initialize the dynamic models of each agent in the second-order continuous system based on algebraic graph theory; S3. Configure the non-continuous event trigger pulse control protocol; The discontinuous event-triggered pulse control protocol is represented as follows: in, This represents the control input of agent i at time t. This represents the joint state value of agent i at time t. Indicates pulse gain. Let a represent the Dirac function. ij This represents the edge weight from communication node i to communication node j, where i,j=1,2,…,N, and N represents the total number of agents in the second-order continuous system. i Let i represent the set of neighboring agents of agent i. This represents the state variable of agent i at time t. Let i represent the state variable of i's neighbor i, j, at time t. Let i be the k-th pulse moment of agent i at time t; Let α represent the velocity variable of agent i at time t, and let α represent the error coefficient. S4. All agents communicate with each other through the communication topology, and each agent continuously acquires its own and its neighboring agents' state information during the communication process; S5. At the pulse moment, a discontinuous event-triggered pulse control protocol is used to update the state and velocity information of all agents; at the non-pulse moment, the state and velocity information of all agents are updated according to the dynamic model. In step S5, the state information of each agent is updated using a discontinuous event-triggered pulse control protocol. The process includes: at the pulse moment, determining whether the trigger function value of the agent is greater than 0. If so, the control input of the agent is calculated according to the discontinuous event-triggered pulse control protocol, and the state information of the agent is updated according to the control input; otherwise, the state information of the agent is maintained. The formula for calculating the trigger function value is: in, Let represent the trigger function of agent i at time t. This represents the joint state value of agent i at time t. Let represent the intermediate parameters of agent i at time t. This represents the velocity of agent i at time t. This represents the pulse gain, and α represents the error coefficient. S6. Based on the event-triggered pulse control strategy proposed above, the multi-agent system eventually reaches consensus within a finite time.

2. The finite-time cooperative control method for a multi-agent system according to claim 1, characterized in that, Step S1, constructing the communication topology, includes: constructing an undirected graph. in Representing an undirected graph The set of nodes, where N is the total number of nodes; It is an undirected graph edge set, Let a be the adjacency matrix of an undirected graph G. ij Let a represent the edge weight from communication node i to communication node j. If there is no connecting edge between communication node i and communication node j, then a ij =0; the degree matrix is ​​determined based on the undirected graph G. and Laplace matrix ,in , , d i Degree matrix The value of the element in the i-th row and i-th column.

3. The finite-time cooperative control method for a multi-agent system according to claim 1, characterized in that, The dynamic model constructed in step S2 based on algebraic graph theory is expressed as follows: in, This represents the state variable of agent i at time t. This represents the velocity variable of agent i at time t. This represents the control input of agent i at time t.

4. The finite-time cooperative control method for a multi-agent system according to claim 1, characterized in that, In step S5, the process of updating the state information of each agent according to the dynamic model at non-pulse times includes: at non-pulse times, obtaining the state information of each agent at the previous time according to the dynamic model, and updating the current state information of each agent according to its own and its neighboring agents' state information at the previous time.

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