A Pre-set Time Consistency Tracking Control Method for a Second-Order Multi-Agent System

Abstract

This invention provides a predetermined-time consistency tracking control method for a second-order multi-agent system, relating to the field of multi-agent system cooperative control technology. First, it proposes a second-order distributed observer to estimate the tracking error of each follower within a predetermined time. Second, it constructs a novel non-singular sliding surface to ensure that the system state converges along the sliding surface within the predetermined time. Finally, it designs a consistency control protocol based on terminal sliding mode to overcome the singularity problem and achieves leader-follower consistency through local information exchange within the predetermined time. Theoretical analysis proves that the follower's state can track the leader's trajectory within the predetermined time. Furthermore, the predetermined-time control of the multi-agent system can meet the desired convergence time requirements by adjusting the predetermined-time parameters, which is more in line with practical application needs compared to finite-time and fixed-time consistency control.

Description

Technical Field

[0001] This invention relates to the field of multi-agent system cooperative control technology, specifically a predetermined time consistency tracking control method for a second-order multi-agent system. Background Technology

[0002] In recent years, distributed cooperative control of multi-agent systems (MAS) has received widespread attention in fields such as satellite formation, multi-missile cooperative attack, data fusion, and multi-vehicle cooperative control. Inspired by swarming phenomena, distributed cooperative control of MAS leverages the intelligence of swarm agents to solve complex problems, accomplishing large-scale, complex, and specific tasks that a single agent cannot achieve, while exhibiting stronger robustness and fault tolerance. The consistency problem is a fundamental issue in MAS cooperative control, with the control objective being to achieve state consistency among all agent members based on local information from their neighbors.

[0003] Convergence rate is a crucial performance indicator for evaluating the quality of consensus control protocols in multi-agent systems. Early research on consensus problems in multi-agent systems primarily focused on asymptotic consensus, where the system state reaches consensus as time approaches infinity. Compared to asymptotic consensus, finite-time consensus offers faster convergence and higher control accuracy, and explicitly provides an upper bound estimate of the steady-state time. However, this control strategy suffers from a drawback: the estimated upper bound of the steady-state time heavily depends on initial conditions and relevant parameters, limiting its applicability when the initial system state cannot be obtained beforehand. Fixed-time control theory, first proposed by the Russian Academy of Sciences in 2012, guarantees fast convergence of the system state while also explicitly providing an upper bound of the convergence time independent of the initial system conditions, thus becoming a hot research topic in recent years. Consistency estimates of the fixed-time convergence upper bound can be obtained under different initial system states.

[0004] While fixed-time control offers significant advantages over finite-time control in determining the upper bound of the system's stable time, it still suffers from two drawbacks: First, the convergence time upper bound determined through Lyapunov stability analysis is highly conservative. If the estimated time is much larger than the actual stable time, calculating the estimated upper bound of the system's convergence time through the control strategy becomes meaningless. Second, fixed-time consistency makes it difficult to obtain a suitable and explicit relationship between the stable time upper bound and the control parameters, making it very challenging to adjust the control parameters based on a given convergence time upper bound in practical applications. To address these issues, Sánchez-Torres proposed a pre-defined time control concept, which provides a more advanced stability characteristic for the controlled system. The stable time upper bound of pre-defined time control is an explicit parameter that can be defined in advance, thus providing a high degree of determinism for the controlled system, including a leader scenario. Combining pre-defined time control with sliding mode control improves the robustness of first-order control systems. A distributed pre-defined time controller is designed using filtering errors and extended to second-order control systems. A novel terminal sliding surface is designed to ensure system convergence along the sliding surface within a predetermined time. Simultaneously, a terminal sliding controller is designed to achieve predetermined time inclusion control for the multi-agent system. In summary, research on the predetermined time consistency problem of second-order multi-agent systems, both domestically and internationally, is still in its early stages. How to solve the predetermined time observation problem and the singularity problem of control schemes is a topic worthy of further exploration. Based on this, this invention designs a predetermined time consistency tracking control method for second-order multi-agent systems to solve the aforementioned problems. Summary of the Invention

[0005] The purpose of this invention is to provide a predetermined time consistency tracking control method for a second-order multi-agent system to solve the problems mentioned in the background art.

[0006] To achieve the above objectives, the present invention provides the following technical solution: a method for pre-time consistency tracking control of a second-order multi-agent system, comprising the following steps:

[0007] S1: Design a second-order distributed observer for each follower;

[0008] S2: A novel non-singular sliding surface is constructed to ensure that the system state converges along the sliding surface within a predetermined time.

[0009] S3: Design a pre-defined time-consistent tracking control protocol for the i-th following agent;

[0010] S4: Deploy the predetermined time sliding mode controller and the distributed predetermined time observer to the i-th follower, so that the state of all followers can be updated within the predetermined time 2T. b +T c +T dInternal tracking of the leader's trajectory, for any time t≥2T b +T c +T d All have χ i (t)=χ0(t),ν i (t)=ν0(t) holds true.

[0011] Preferably, a multi-agent system consisting of one leader and N followers has a communication topology between its agents represented by a graph. To describe it. The dynamics of the i-th follower are defined as follows:

[0012]

[0013] Where: x i (t)∈R N and v i (t)∈R N These are the position and velocity of the i-th follower, respectively; u i (t)∈R N and d i (t)∈R N It is the control input of the i-th follower and the unknown external disturbance.

[0014] The dynamics of the leader can be described as follows:

[0015]

[0016] Where: x0(t)∈R N and v0(t)∈R N These represent the leader's position and velocity, respectively; u0(t)∈R N It is the leader's control input.

[0017] The preferred second-order distributed observer is described as follows:

[0018]

[0019]

[0020] Where: ξ i η i (i = 1, 2, ..., N) represent the estimates of the position and velocity tracking errors of the i-th follower to the leader, and ξ0 = 0, η0 = 0; α, β, γ are defined positive constants satisfying 0 < α < 1, β > 1; γ1, They respectively satisfy T is a positive constant. b This is a predetermined time constant.

[0021] Preferably, the consistent tracking position error e of the i-th follower xi and speed error e vi Defined as e respectively xi =x i -x0 and e vi =v i -v0, taking the derivative with respect to it, yields:

[0022]

[0023] Rewrite in vector form:

[0024] e x =vv i =e v

[0025]

[0026] in:

[0027] Design the pre-defined non-singular sliding surface s of the i-th follower. i As shown in the formula:

[0028]

[0029] in: μ1 and μ2 are positive constants and satisfy the following condition: μ2>1; γ2, They respectively satisfy The positive constant; in order to make the sliding surface s i and its derivative at the switching point |e xi |=ε is continuous, and the control gains l1 and l2 are respectively selected as . ε T is a small positive constant as defined; c This is a predetermined time constant for the sliding phase. When In this case, the singularity problem can be overcome by switching the sliding surface from the terminal sliding surface to the general sliding surface. Furthermore, choosing appropriate parameters μ1 and μ2 also helps to avoid the singularity problem.

[0030] Preferably, the scheduled time consistency tracking and control protocol is described as follows:

[0031]

[0032] Where: p, q, k are positive constants and satisfy 0 < p < 1, q > 1; γ3, They respectively satisfy T is a positive constant. c This is the predetermined time constant for the arrival phase.

[0033] Compared with existing technologies, the advantages of this invention are as follows: This invention solves the consistency tracking problem of second-order multi-agent systems with unknown external disturbances by proposing a predetermined time control scheme based on sliding mode technology. Compared with existing finite-time consistency protocols, the upper bound of the stable time of the control protocol proposed in this invention is independent of the initial state and depends only on adjustable parameters, thus allowing the upper bound of the stable time to be determined a priori. At the same time, the upper bound of the convergence time of fixed-time consistency is a complex function related to system parameters, and the relationship between the upper bound of the convergence time and the gain of the control scheme is not clear. However, the upper bound of the stable time of the predetermined time distributed observer and the control protocol proposed in this invention can be directly determined by the preset convergence time. Therefore, this method is more in line with the requirements of theory and practical engineering and has a wider range of applications.

[0034] Of course, any product implementing this invention does not necessarily need to achieve all of the advantages described above at the same time. Attached Figure Description

[0035] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0036] Figure 1 A multi-agent network communication topology diagram;

[0037] Figure 2 A position and state curve diagram for each agent;

[0038] Figure 3 The velocity state curves of each agent;

[0039] Figure 4 A graph showing the changes in control signals for each follower;

[0040] Figure 5 Let e ​​be the positional error of each follower. xi The response curve;

[0041] Figure 6 Let e ​​be the speed error of each follower. vi The response curve. Detailed Implementation

[0042] 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 some embodiments of the present invention, and not all embodiments. 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.

[0043] Please see Figure 1-6 This invention provides a technical solution: a method for pre-time consistency tracking control of a second-order multi-agent system, comprising the following steps:

[0044] S1: Design a second-order distributed observer for each follower;

[0045] S2: A novel non-singular sliding surface is constructed to ensure that the system state converges along the sliding surface within a predetermined time.

[0046] S3: Design a pre-defined time-consistent tracking control protocol for the i-th following agent;

[0047] S4: Deploy the predetermined time sliding mode controller and the distributed predetermined time observer to the i-th follower, so that the state of all followers can be updated within the predetermined time 2T. b +T c +T d Internal tracking of the leader's trajectory, for any time t≥2T b +T c +T d All have χ i (t)=χ0(t),ν i (t)=ν0(t) holds true.

[0048] In a multi-agent system consisting of one leader and N followers, the communication topology between the agents is represented by the graph. To describe it. The dynamics of the i-th follower are defined as follows:

[0049]

[0050] Where: x i (t)∈R N and v i (t)∈R N These are the position and velocity of the i-th follower, respectively;

[0051] u i (t)∈R N and d i (t)∈R N It is the control input of the i-th follower and the unknown external disturbance.

[0052] The dynamics of the leader can be described as follows:

[0053]

[0054] Where: x0(t)∈R N and v0(t)∈R N These represent the leader's position and velocity, respectively; u0(t)∈R N It is the leader's control input.

[0055] The second-order distributed observer is described as follows:

[0056]

[0057]

[0058] Where: ξ i η i (i = 1, 2, ..., N) represent the estimates of the position and velocity tracking errors of the i-th follower to the leader, with ξ0 = 0 and η0 = 0; α, β, and γ are defined positive constants satisfying 0 < α < 1 and β > 1; γ1, They are respectively satisfied T is a positive constant. b This is a predetermined time constant.

[0059] Wherein, the consistent tracking position error e of the i-th follower xi and speed error e vi Defined as e respectively xi =x i -x0 and e vi =v i -v0, taking the derivative with respect to it, yields:

[0060]

[0061] Rewrite in vector form:

[0062] e x =vv i =e v

[0063]

[0064] in:

[0065] Design the pre-defined non-singular sliding surface s of the i-th follower. i As shown in the formula:

[0066]

[0067] in: μ1 and μ2 are positive constants and satisfy the following condition: μ2>1; γ2, They respectively satisfy The positive constant; in order to make the sliding surface s i and its derivative at the switching point |e xi |=ε is continuous, and the control gains l1 and l2 are respectively selected as . ε is a small positive constant as defined; T c This is a predetermined time constant for the sliding phase. When In this case, the singularity problem can be overcome by switching the sliding surface from the terminal sliding surface to the general sliding surface. Furthermore, choosing appropriate parameters μ1 and μ2 also helps to avoid the singularity problem.

[0068] Preferably, the scheduled time consistency tracking and control protocol is described as follows:

[0069] Where: p, q, k are positive constants and satisfy 0 < p < 1, q > 1; γ3, They respectively satisfy T is a positive constant. c This is the predetermined time constant for the arrival phase.

[0070] One specific application of this embodiment is:

[0071] A multi-agent system consisting of 4 followers (represented by 1, 2, 3, 4) and 1 leader (represented by 0), assuming the communication structure of the multi-agent system is as follows. Figure 1 As shown, the dynamics of the i-th agent are described by x. i =[x i1 ,x i2 ] T ,i∈{0,1,2,L,N}, where χ i1 , χ i2 ν represents the position in the x and y directions respectively. i1 v i2 Let x and y represent the velocities in the x and y directions, respectively; the leader's control input is set to u0 = [0.5sin(0.5t), 0.25cos(0.7t)]T, and the initial state is set to x0 = [0, 0]. T v0 = [0,0]T. The initial positions and velocities of the four followers are set to x1 = [4,-2]. T , v1=[2,-1]T, x2=[2,5]T, v2=[-1,3]T, x3=[-3,-3] TLet v3 = [-1, 2]T, x4 = [-5, 2]T, and x4 = [1, -3]T. Define the external perturbations of the four followers as d1 = 0.1[cos(0.6t), sin(0.5t)]. T , d1=0.05[cos(0.3t),sin(0.7t)] T ,d3=0.03[cos(0.5t),sin(0.3t)] T ,d4=0.07[cos(0.4t),sin(0.6t)] T The observer and controller parameters are set as follows: α = 0.6, β = 1.4, μ1 = 0.6, μ2 = 1.4, p = 0.6, q = 1.4, k = 0.6, λ = 0.6, ε = 0.005, T b =1,T c =1,T d =1, the total scheduled time for the multi-agent system is T = 2T b +T c +T d =4s.

[0072] Figure 2 The numerical simulation results show that the positional errors between any follower and leader are bounded within a predetermined time, and consensus can be reached within 1 second, representing the positional trajectories of each agent. Figure 3 The graph shows the velocity and movement trajectories of each agent. It can be seen from the graph that all four followers can achieve agreement with the leader within one second. Figure 4 The control signal variation curves for each follower show that the control signal changes smoothly and operates relatively smoothly after the system state converges. This further demonstrates that the controller maintains good dynamic characteristics during the consensus tracking control process of the multi-agent system. Figure 5 and Figure 6 The positional error e of the follower is given. xi and speed error e vi The response curves, as shown in the figure, indicate that the steady-state accuracy of the follower's position error converges to 4 × 10⁻⁶. -4 The steady-state accuracy of the speed tracking error is 6×10. -3 That is, all followers can achieve stable control of the multi-agent system consistency within a predetermined time. As can be seen from the above analysis, under the influence of unknown disturbances, all four followers can track the dynamic leader within a predetermined time, thereby achieving consistent tracking of the multi-agent system within a predetermined time.

[0073] In the description of this specification, references to terms such as "an embodiment," "example," "specific example," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0074] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to the specific implementations described. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims

1. A method for pre-set time consistency tracking control of a second-order multi-agent system, characterized in that, Includes the following steps: S1: Design a second-order distributed observer for each follower; S2: A novel non-singular sliding surface is constructed to ensure that the system state converges along the sliding surface within a predetermined time. S3: Design a pre-defined time-consistent tracking control protocol for the i-th following agent; S4: Deploy the predetermined time sliding mode controller and the distributed predetermined time observer to the i-th follower, so that the state of all followers can be updated within the predetermined time 2T. b +T c +T d Internal tracking of the leader's trajectory, for any time t≥2T b +T c +T d All have χ i (t)=χ0(t),ν i (t)=ν0(t) holds true; A multi-agent system consisting of one leader and N followers has a communication topology between its agents represented by a graph. To describe; the dynamics of the i-th follower are defined as follows: Where: x i (t)∈R N and v i (t)∈R N These are the position and velocity of the i-th follower, respectively; u i (t)∈R N and d i (t)∈R N It consists of the control input of the i-th follower and the unknown external disturbance; The dynamics of the leader can be described as follows: Where: x0(t)∈R N and v0(t)∈R N These represent the leader's position and velocity, respectively; u0(t)∈R N It is the leader's control input; The second-order distributed observer is described as follows: Where: ξ i η i (i = 1, 2, ..., N) represent the estimates of the position and velocity tracking errors of the i-th follower to the leader, with ξ0 = 0 and η0 = 0; α, β, and γ are defined positive constants satisfying 0 < α < 1 and β > 1; γ1, They are respectively satisfied T is a positive constant. b The predetermined time constant; The consistency tracking position error of the i-th follower is e xi and speed error e vi Defined as e respectively xi =x i -x0 and e vi =v i -v0, taking the derivative with respect to it, yields: Rewrite in vector form: yes x =vv i =e v in: Design the pre-defined non-singular sliding surface s of the i-th follower. i As shown in the formula: in: μ1 and μ2 are positive constants and satisfy the following condition: μ2>1; γ2, They respectively satisfy The positive constant; in order to make the sliding surface s i and its derivative at the switching point |e xi |=ε is continuous, and the control gains l1 and l2 are respectively selected as . ε is a small positive constant as defined; T c The predetermined time constant for the sliding phase; when In this case, the singularity problem is overcome by switching the sliding surface from the terminal sliding surface to the general sliding surface; The scheduled time consistency tracking and control protocol is described as follows: Where: p, q, k are positive constants and satisfy 0 < p < 1, q > 1; γ3, They are respectively satisfied T is a positive constant. d This is the predetermined time constant for the arrival phase.

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