Method for fixed-time containment control of high-order continuous multi-agent systems

Abstract

The application discloses a kind of high-order continuous multi-agent system fixed time containment control method, belong to networked multi-agent system technical field.The technical points of the application include: establishing the high-order continuous multi-agent system dynamic model and combination state error prediction model with multiple leaders, to obtain the combination state error prediction value;Design event trigger mechanism, to obtain the update rule of trigger time;Design fixed time containment control protocol based on event trigger mechanism and combination state error prediction model, to obtain the tracking error model between leader and follower;Using the tracking error model, to obtain the control gain condition and adjustment time upper bound for realizing fixed time containment control;Realize the fixed time containment control of high-order continuous multi-agent system.The application overcomes the limitation of network communication on fixed time containment consistency, with the advantages of fast reaching consistency, easy to realize and the like.

Description

Technical Field

[0001] This invention relates to the field of networked multi-agent systems, and more specifically to a method for fixed-time control of a high-order continuous multi-agent system. Background Technology

[0002] A multi-agent system is an integrated system composed of multiple subsystems capable of autonomous decision-making, with each subsystem able to exchange information with others. Multi-agent systems have a wide range of applications in practice, such as smart grids, satellite systems, multi-vehicle systems, and unmanned aerial vehicle systems. Compared to single-system control in traditional systems, multi-agent systems demonstrate higher efficiency in achieving complex control objectives due to their superior environmental perception capabilities.

[0003] As research on multi-agent systems deepens and control tasks become increasingly complex, leaderless multi-agent systems and single-leader multi-agent systems often fail to meet the needs of real-world production. For multi-leader multi-agent systems, designing control protocols that enable followers to enter the convex hull formed by the leaders within a fixed timeframe has significant theoretical and practical value.

[0004] Multi-agent system cooperative control typically operates in an ideal environment free from external influencing factors such as time delays, packet loss, and attacks. However, in real-world production and daily life, networked multi-agent systems rely on networks for information exchange, and information transmission within a single agent also depends on network communication. Under these circumstances, the communication efficiency between agents will inevitably be constrained by network communication. Therefore, researching fixed-time control of multi-agent systems with predictive mechanisms is of greater practical significance.

[0005] Unlike continuous or periodic communication, event-triggered mechanisms can analyze whether the current system state meets the triggering conditions and then determine whether the control protocol should be updated, thus enabling more flexible and consistent system control. Therefore, studying event-triggered mechanisms can solve the problem of communication resource consumption in multi-agent systems and has greater practical significance. Summary of the Invention

[0006] To address this, the present invention proposes a fixed-time inclusion control method for high-order continuous multi-agent systems. By employing a prediction mechanism, the control protocol is no longer constant between two triggers, but is updated in real time using the state prediction values ​​of each agent, thereby further accelerating the speed of achieving inclusion control and solving the fixed-time inclusion control problem in high-order continuous multi-agent systems.

[0007] According to one aspect of the present invention, a method for fixed-time control of a high-order continuous multi-agent system is proposed, the method comprising the following steps:

[0008] Step 1: Establish a dynamic model and a combined state error prediction model for a high-order continuous multi-agent system with multiple leaders, and obtain the predicted value of the combined state error;

[0009] Step 2: Based on the predicted combined state error value, design an event triggering mechanism to obtain the update pattern of the triggering time;

[0010] Step 3: Based on the event triggering mechanism and the combined state error prediction model, design a fixed-time control protocol;

[0011] Step 4: Substitute the control protocol into the dynamic model to obtain the tracking error model between the leader and the follower;

[0012] Step 5: Using the tracking error model, obtain the control gain condition and the upper bound of the adjustment time for achieving fixed-time control.

[0013] Furthermore, the dynamic model described in step one is represented as follows:

[0014]

[0015]

[0016] Where: x f ∈R n x l ∈R n r f ∈R n and k∈R n These represent the states of follower f, leader l, control input of follower f, and velocity of leader l, respectively. n is the system order, k is a constant vector, t is the time variable, and R... n Let F be an n-dimensional Euclidean space, where N is the number of followers, M is the number of leaders, F = {1, 2, ..., M} is the set of followers, and L = {1, 2, ..., M} is the set of leaders.

[0017] Let the combined state error of the model be:

[0018]

[0019] Where: ξ ij (t)=x i (t)-x j (t), x i (t), x j (t) represent the states of the i-th and j-th agents at time t, respectively. ij The Laplace matrix representing the communication topology graph The element in the i-th row and j-th column.

[0020] The follower state prediction mechanism is designed as follows:

[0021]

[0022] in:

[0023]

[0024] Here Ω ij (s) can be obtained from the state value of agent i at the last trigger time. and the state values ​​of its neighboring intelligent agents The calculation shows that this value remains a constant until the next trigger. The state difference ξ between the follower and its neighboring leader is the latest triggering time of agent i before time t. ij (t)(i∈F,j∈L), no prediction is required.

[0025] The predicted error value of the combined state of the model is:

[0026]

[0027] in, Therefore Predict the state value of the agent at time t.

[0028] Furthermore, the event triggering mechanism in step two is designed as follows:

[0029] Define the error function e i (t) is:

[0030]

[0031] Where: k1>0, k2>0 and k3>0 are the control gains to be designed, ρ>1 is the ratio of two positive odd numbers, and sign(·) is the sign function.

[0032] Trigger function h i (t) is defined as:

[0033]

[0034] Where ||·||2 represents the 2-norm of the real matrix, and γ∈(0,1) is the triggering parameter to be designed. The triggering time is determined by the following triggering mechanism:

[0035]

[0036] Where inf represents the infimum function. The initial time is defined as the first trigger time of the system.

[0037] Furthermore, the fixed-time inclusion control protocol designed in step three based on the event triggering mechanism and the combined state error prediction model is as follows:

[0038]

[0039] Furthermore, in step four, the control protocol is substituted into the dynamic model to obtain the tracking error model between the leader and the follower as follows:

[0040] Substituting the control protocol into the system dynamic model yields the following form:

[0041]

[0042] Furthermore, the tracking error model between leaders and followers can be calculated as follows:

[0043]

[0044] in:, Represents the Kronecker product of matrices. This indicates the communication relationship between followers. This indicates the communication relationship between the leader and the followers. 1 M This represents an M-dimensional column vector where all elements are 1.

[0045] Furthermore, in step five, the tracking error model is used to obtain the control gain condition and the upper bound of the adjustment time for achieving fixed-time control:

[0046] According to Lyapunov stability theory, when the control gain k1 > 0 and k3 > 0... and At this time, the tracking error will decay to zero within a fixed time, meaning the estimated state can enter the convex hull formed by the leader within a fixed time. The upper bound T of the adjustment time T0 is... max As described below:

[0047]

[0048] Where, λ min (·) represents the minimum eigenvalue of the matrix. The control gain that meets the conditions is substituted into the fixed-time containment control protocol to obtain the real-time control quantity. The controller output is then applied to the dynamic model of the multi-agent system to achieve fixed-time containment control of a high-order continuous multi-agent system.

[0049] The beneficial technical effects of this invention are:

[0050] This invention studies the fixed-time inclusion control problem in multi-agent systems based on event-triggered and prediction mechanisms. Compared with existing methods for achieving fixed-time consistency, it reduces communication congestion and resource waste caused by continuous communication between agents in practical applications. It also solves the problem that the convergence time of finite-time consistency systems tends to infinity with larger initial states, and the problem of needing to redesign parameters in the control protocol for different initial states. This invention designs event-triggered functions based on combined state error prediction values ​​and uses a combination of event-triggered and prediction mechanisms to design a fixed-time inclusion control protocol. This solves the problem that finite-time inclusion control requires resetting control protocol parameters for different initial states, and that excessively large initial states lead to excessively long consensus times. Furthermore, it provides the conditions that the controller gain constant must satisfy, thus solving the problem of fixed-time inclusion control in high-order continuous multi-agent systems. Attached Figure Description

[0051] The above and other objects, features, and advantages of exemplary embodiments of the present invention will become readily apparent from the following detailed description taken in conjunction with the accompanying drawings. Several embodiments of the invention are illustrated in the drawings by way of example and not limitation, wherein:

[0052] Figure 1 This is a flowchart of a method for fixed-time control of a high-order continuous multi-agent system according to the present invention.

[0053] Figure 2 This is a communication topology diagram of a high-order continuous multi-agent system in an embodiment of the present invention;

[0054] Figure 3 This is an example diagram illustrating the fixed-time consistency achieved by agents in a high-order continuous multi-agent system according to an embodiment of the present invention, under controller parameters k1=2, k2=5, k3=0.1, positive odd ratio ρ=1.4, and trigger parameter γ=0.35. Detailed Implementation

[0055] The principles and spirit of the invention will now be described with reference to several exemplary embodiments. It should be understood that these embodiments are given merely to enable those skilled in the art to better understand and implement the invention, and are not intended to limit the scope of the invention in any way. Rather, these embodiments are provided to make this disclosure more thorough and complete, and to fully convey the scope of this disclosure to those skilled in the art.

[0056] Those skilled in the art will recognize that embodiments of the present invention can be implemented as a system, apparatus, device, method, or computer program product. Therefore, this disclosure can be specifically implemented in the following forms: entirely hardware, entirely software (including firmware, resident software, microcode, etc.), or a combination of hardware and software. It should be understood herein that any number of elements in the accompanying drawings is for illustrative purposes only and not as a limitation, and any naming is for distinction only and has no limiting meaning.

[0057] To improve the efficiency of multi-agent systems in achieving consensus within a fixed time, this invention proposes a fixed-time control method for high-order continuous multi-agent systems. The specific steps of this method include:

[0058] Step 1: Establish a dynamic model of a high-order continuous multi-agent system with multiple leaders and a combined state error prediction model to obtain the predicted value of the combined state error;

[0059] Step 2: Based on the predicted combined state error value, design an event triggering mechanism to obtain the update pattern of the triggering time;

[0060] Step 3: Based on the event triggering mechanism and the combined state error prediction model, design a fixed-time control protocol;

[0061] Step 4: Substitute the control protocol into the dynamic model to obtain the tracking error model between the leader and the follower;

[0062] Step 5: Using the tracking error model, obtain the control gain condition and the upper bound of the adjustment time for achieving fixed-time control.

[0063] The present invention will now be described in detail.

[0064] In step one, an undirected topological graph is first used to describe the communication relationships between multiple agents, and a dynamic model of a high-order continuous multi-agent system with multiple leaders is established.

[0065] According to an embodiment of the present invention, an undirected graph is used. This represents the communication relationships between N followers and M leaders. Let F = {1, 2, ..., M} be the set of followers, and L = {1, 2, ..., M} be the set of leaders. Representation diagram The adjacency matrix, a ij A value greater than 0 indicates that agent i can communicate with agent j; otherwise, a ij =0. Hypothetical diagram No self-loop, i.e., a ii =0. Therefore, the graph... The Laplace matrix can be defined as in Assuming that leaders do not communicate with each other, and leaders only send messages to their followers but do not receive messages from them, then... It can be expressed in the following block form:

[0066]

[0067] in: This indicates the communication relationship between followers. It indicates the communication relationship between leaders and followers.

[0068]

[0069] Where: x f ∈R n x l ∈R n r f ∈R n and k∈R n These represent the states of follower f, leader l, control input of follower f, and velocity of leader l, respectively. n is the system order, k is a constant vector, t is the time variable, and R... n Let it be an n-dimensional Euclidean space.

[0070] Let the combined state error of the model be:

[0071]

[0072] Where: ξ ij (t)=x i (t)-x j (t), x i (t), x j (t) represent the states of the i-th and j-th agents at time t, respectively. ij The Laplace matrix representing the communication topology graph The element in the i-th row and j-th column.

[0073] The follower state prediction mechanism is designed as follows:

[0074]

[0075] in:

[0076]

[0077] Here Ω ij (s) can be obtained from the state value of agent i at the last trigger time. and the state values ​​of its neighboring intelligent agents The calculated value is a constant that remains unchanged until the next trigger. The state difference ξ between a follower and its neighboring leader is... ij (t)(i∈F,j∈L) does not require prediction. It is the latest trigger time of agent i before time t.

[0078] The predicted value of the combined state error of the model is:

[0079]

[0080] in, Therefore Predict the state value of the agent at time t.

[0081] In step two, the event triggering mechanism based on the predicted value of the model combination state error is designed as follows:

[0082] Error function e i (t) is defined as:

[0083]

[0084] Where: k1>0, k2>0 and k3>0 are the control gains to be designed, ρ>1 is the ratio of two positive odd numbers, and sign(·) is the sign function.

[0085] Trigger function h i (t) is defined as:

[0086]

[0087] Where ||·||2 represents the 2-norm of the real matrix, and γ∈(0,1) is the triggering parameter to be designed. The triggering time is determined by the following triggering mechanism:

[0088]

[0089] Where inf represents the infimum function. The initial time is defined as the first trigger time of the system.

[0090] Consider a multi-agent system with a leader-follower relationship (1). Fixed-time consistency means that there exists a settling time T0 such that as t→T0, all followers enter the convex hull formed by the leader. A distributed event-triggered control protocol r is designed to achieve this. i (t), making the follower's state x i (i∈F) converges to the leader's state x in a fixed time. i Within the convex hull formed by (i∈L), that is, for any initial state, we have:

[0091]

[0092] Established.

[0093] Where: x L ={x N+1 ,x N+2 ,…,x N+M Let} be the set of leader states, Co(x) L ) is x L The convex hull formed is denoted as:

[0094]

[0095] In step three, a fixed-time control protocol based on the event triggering mechanism and the combined state error prediction model is designed as follows:

[0096]

[0097] The control protocol (3) can be written in the following form using the error function (2):

[0098]

[0099] In step four, the control protocol is substituted into the dynamic model to obtain the tracking error model between the leader and the follower:

[0100] Substituting the control protocol (4) into the system dynamic equation (1), we obtain the following form:

[0101]

[0102] make

[0103]

[0104] According to equation (5), It can be described as:

[0105]

[0106] It can be described as in: Represents the Kronecker product of matrices. 1 M This represents an M-dimensional column vector where all elements are 1.

[0107] definition When x F (t)→x * When (t), the initial state of any follower satisfies Then we have:

[0108]

[0109] Therefore, it can be calculated that:

[0110]

[0111] In step five, using the tracking error model, the control gain condition and the upper bound of the adjustment time for achieving fixed-time control are obtained:

[0112] Design the Lyapunov function V(t) as follows:

[0113]

[0114] Taking the derivative of the Lyapunov function V(t), we get:

[0115]

[0116] Among them: ||·|| 1+ρ Let represent the 1+ρ norm of the real matrix. Using Lyapunov stability theory, we obtain k1>0, k3>0. and According to the theory of fixed-time consistency, the tracking error will decay to zero within a fixed time, meaning that the estimated state can enter the convex hull formed by the leader within a fixed time. The upper bound T of the adjustment time T0 is... max As described below:

[0117]

[0118] Where: λ min (·) represents the minimum eigenvalue of the matrix. The control gain that meets the conditions is substituted into the fixed-time inclusion control protocol to obtain the real-time control quantity. The controller output is then applied to the dynamic model of the multi-agent system to achieve fixed-time inclusion control of a high-order continuous multi-agent system.

[0119] The technical effects of the present invention were further verified through simulation experiments.

[0120] In the simulation experiment, it is assumed that the multi-agent system (1) contains eight agents, of which six followers are represented by 1, 2, 3, 4, 5 and 6, and two leaders are represented by 7 and 8. Its communication topology is shown in the figure. as follows Figure 2 As shown.

[0121] The dynamic model of the system is described by (1), and its Laplace matrix can be obtained from the topology diagram. for

[0122]

[0123] in: for

[0124]

[0125] From this, we can obtain

[0126] The control parameters are selected as follows:

[0127] k1=2, k2=5, k3=0.1, ρ=1.4, γ=0.35

[0128] Calculations show that the parameters satisfy the above conditions, and the upper bound T of the adjustment time is obtained. max =8.2797.

[0129] The initial system value is set as follows:

[0130] x1(0)=5, x2(0)=1.5, x3(0)=-2, x4(0)=-4,

[0131] x5(0)=3,x6(0)=-3,x7(0)=-1,x8(0)=1,k=0

[0132] Figure 3 The diagram illustrates the state trajectories of a multi-agent system based on combined state error prediction and a distributed event triggering mechanism under the action of protocol (2). As can be seen from the diagram, all followers enter the convex hull formed by the leader around t = 0.1 s, and the state errors of all followers approach steady state around t = 0.6 s, which is less than the upper bound T of the adjustment time. max .

[0133] It should be noted that although several units, modules, or sub-modules are mentioned in the detailed description above, this division is merely exemplary and not mandatory. In fact, according to embodiments of the present invention, the features and functions of two or more modules described above can be embodied in one module. Conversely, the features and functions of one module described above can be further divided and embodied by multiple modules.

[0134] Furthermore, although the operations of the method of the present invention are described in a specific order in the accompanying drawings, this does not require or imply that these operations must be performed in that specific order, or that all the operations shown must be performed to achieve the desired result. Additionally or alternatively, certain steps may be omitted, multiple steps may be combined into one step, or one step may be broken down into multiple steps.

[0135] The above content is only one embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the present invention can have various modifications and variations; any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for fixed-time control of a high-order continuous multi-agent system, characterized in that, Includes the following steps: Step 1: Establish a dynamic model of a high-order continuous multi-agent system with multiple leaders and a combined state error prediction model to obtain the predicted value of the combined state error; Step 2: Based on the predicted combined state error value, design an event triggering mechanism to obtain the update pattern of the triggering time; Step 3: Based on the event triggering mechanism and the combined state error prediction model, design a fixed-time control protocol; Step 4: Substitute the control protocol into the dynamic model to obtain the tracking error model between the leader and the follower; Step 5: Using the tracking error model, obtain the control gain condition and the upper bound of the adjustment time for achieving fixed-time control.

2. The method for fixed-time control of a high-order continuous multi-agent system according to claim 1, characterized in that, The dynamic model described in step one is represented as follows: Where: x f ∈R n x l ∈R n r f ∈R n and k∈R n These represent the states of follower f, leader l, control input of follower f, and velocity of leader l, respectively. n is the system order, k is a constant vector, t is the time variable, and R... n Let F be an n-dimensional Euclidean space, where N is the number of followers, M is the number of leaders, F = {1, 2, ..., M} is the set of followers, and L = {1, 2, ..., M} is the set of leaders. The combined state error prediction model is as follows: Where: ξ ij (t)=x i (t)-x j (t), x i (t), x j (t) represent the states of the i-th and j-th agents at time t, respectively. ij The Laplace matrix representing the communication topology graph The element in the i-th row and j-th column; The follower state prediction mechanism is designed as follows: in: Here Ω ij (s) can be obtained from the state value of agent i at the last trigger time. and the state values ​​of its neighboring intelligent agents The calculation shows that this value remains a constant until the next trigger. The state difference ξ between the follower and its neighboring leader is the latest triggering time of agent i before time t. ij (t)(i∈F,j∈L), no prediction is required; The predicted error value of the combined state of the model is: in, Therefore Predict the state value of the agent at time t.

3. The method for fixed-time control of a high-order continuous multi-agent system according to claim 2, characterized in that, The event triggering mechanism described in step two is designed as follows: Define the error function e i (t) is: Where: k1>0, k2>0 and k3>0 are the control gains to be designed, ρ>1 is the ratio of two positive odd numbers, and sign(·) is the sign function; Trigger function h i (t) is defined as: Where: ||·||2 represents the 2-norm of the real matrix, γ∈(0,1) is the triggering parameter to be designed; the triggering time is determined by the following triggering mechanism: Where: inf represents the infimum function; the initial time is defined as the first trigger time of the system.

4. The method for fixed-time control of a high-order continuous multi-agent system according to claim 3, characterized in that, The fixed-time control protocol designed in step three, based on the event triggering mechanism and the combined state error prediction model, is as follows:

5. A method for implementing inclusive control of a second-order multi-agent system under network prediction according to claim 4, characterized in that, In step four, the control protocol is substituted into the dynamic model to obtain the tracking error model between the leader and the follower: Substituting the control protocol into the system dynamic model yields the following form: Furthermore, the tracking error model between leaders and followers can be calculated as follows: in:, Represents the Kronecker product of matrices. This indicates the communication relationship between followers. This indicates the communication relationship between the leader and the followers. 1 M This represents an M-dimensional column vector where all elements are 1.

6. A method for implementing inclusive control of a second-order multi-agent system under network prediction according to claim 5, characterized in that, In step five, the tracking error model is used to obtain the control gain condition and the upper bound of the adjustment time for achieving fixed-time control: According to Lyapunov stability theory, when the control gain k1 > 0 and k3 > 0... and At this time, the tracking error will decay to zero within a fixed time, meaning the estimated state can enter the convex hull formed by the leader within a fixed time. The upper bound T of the adjustment time T0 is... max As described below: Where, λ min (·) represents the minimum eigenvalue of the matrix; the control gain that meets the conditions is substituted into the fixed-time inclusion control protocol to obtain the real-time control quantity, and the controller output is applied to the dynamic model of the multi-agent system to realize the fixed-time inclusion control of the high-order continuous multi-agent system.

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