A multi-agent system consistency control method for predetermined time and precision

Abstract

The application discloses a kind of multi-agent system predetermined time and precision consistency control method, a new practical preset time control lemma is proposed, as long as the derivative of Lyapunov function is designed to satisfy the inequality set, then the system convergence predetermined time is T p , can directly preset convergence time T P , which not only does not depend on the initial condition of system, but also is determined by only one parameter.Using general barrier function transformation, the synchronization error can be limited in the case of symmetric preset boundary, asymmetric preset boundary and without any preset boundary. The nonlinear predetermined time filter used is based on the proposed predetermined time control stability lemma, which can guarantee that the filtering error is quickly converged to an adjustable residual set within the preset time, thereby improving the control performance.

Description

Technical Field

[0001] The present invention relates to the technical field of consistency control of multi-agent systems, and in particular to a consistency control method for a multi-agent system with predetermined time and accuracy. Background Art

[0002] In recent years, interest in multi-agent systems (MASs) has increased significantly due to widespread applications such as satellite clusters, sensor networks, and multi-vehicle collaboration. In the control field, a key goal of MASs is to implement a consensus protocol to ensure that the system's output or state converges to a consistent state. Incorporating the recursive nature of backstepping, numerous control schemes have been proposed to address the consensus problem. Furthermore, system response speed is an important metric for evaluating system tracking performance. To improve system response speed, scholars have proposed finite-time control theory. This theory is superior to traditional control methods because it guarantees that the system reaches the desired state within a finite time and offers advantages such as strong robustness and high accuracy. However, finite-time stability has an unavoidable drawback: its convergence time depends on the system's initial state. Subsequently, Andery introduced fixed-time stability, in which the convergence time does not depend on the system's initial conditions. Subsequently, various fixed-time stability criteria for MASs have been proposed. However, the convergence time of fixed-time control depends on multiple parameters and is prone to overly conservative estimates, making it difficult to determine the convergence time in advance. To stabilize control systems within a precise convergence time, researchers have developed exponential and polynomial preset time (PT) criteria. Although these studies have achieved significant results in implementing PT control, the accuracy of the tracking error remains unknown due to system uncertainty, and only convergence to near zero is guaranteed.

[0003] It should be pointed out that for practical engineering applications, system response speed is certainly important, but high-precision control is equally crucial. Finite-time control / fixed-time control and the above-mentioned (PT) control do not consider the tracking accuracy issue. Therefore, a simpler Lyapunov condition analysis method is urgently needed to ensure PT stability while considering the predetermined accuracy. The technical solution to achieve the predetermined accuracy of uncertain nonlinear MASs is to use scaling transformation functions and error coordinate transformation. However, in the existing error transformation method, the preset accuracy boundary after transformation can only be symmetrical, so considering an asymmetric preset accuracy boundary is more practical.

[0004] It is worth noting that as the order of the controlled nonlinear MASs increases, the above-mentioned PT control scheme will face the problem of "computational complexity explosion". To overcome this obstacle, existing solutions have proposed a preset performance output feedback control scheme combined with command filter technology. However, the filter used is linear, and the filter error cannot decay quickly within the preset time, which will affect the control performance. Therefore, it is of great significance to consider a nonlinear preset time filter that can ensure that the filter error converges quickly to an adjustable residual set within the preset time, thereby improving the control performance. Summary of the Invention

[0005] The purpose of the present invention is to overcome the deficiencies of the prior art and provide a method for consistent control of a multi-agent system with predetermined time and accuracy.

[0006] To achieve the above objectives, the technical solutions provided by the present invention are:

[0007] A method for consistent control of predetermined time and accuracy of a multi-agent system, comprising:

[0008] Build a multi-agent system model, including models of follower agents and leader agents;

[0009] Establish a new time-stable control lemma and define the synchronization error z of the multi-agent system i,1 , the synchronization error z i,1 is the error between the output of the ith follower agent and the leader;

[0010] Establish a universal barrier function to transform the constrained synchronization error z of the i-th agent into i,1 Convert to unconstrained variable w i,1 ;

[0011] Combined with the new scheduled time control stability lemma, a nonlinear scheduled time filter is established and a scheduled time and precision controller is designed for each follower agent, including the 1st to sth virtual controllers and the actual controller;

[0012] The consistency control of the scheduled time and accuracy of the multi-agent system is achieved by combining the new scheduled time control stability lemma, universal barrier function, nonlinear scheduled time filter and the scheduled time and accuracy controller of each follower agent.

[0013] Furthermore, the established multi-agent system model includes a leader and M follower agents, and the leader and M follower agents form a directed graph

[0014] The model of the i-th follower agent is as follows:

[0015]

[0016] where, is the state vector of the ith follower agent, x i,1 , x i，2 ,...., represents the system state, where s represents the s-th order of the ith follower agent system, n i represents the total system order of the ith follower agent; and the dot above a variable represents the derivative of that variable with respect to time t; u i ∈ R represents the control input of the ith follower agent, y i ∈ R represents the output signal of the ith follower agent; is a smooth and unknown function for the ith follower agent;

[0017] The model of the leader is as follows:

[0018]

[0019] where, y r represents the leader agent output signal, represents the derivative of the leader agent output signal with respect to time, the function is a known, smooth and bounded function.

[0020] Further, the synchronization error z i,1 of the multi-agent system is defined as follows:

[0021]

[0022] where, i and j represent the ith and jth follower agents, respectively; follower agent j is called a neighbor of follower agent i when the information of the jth follower agent is transmittable to the ith follower agent, and a i,j > 0, otherwise a i,j = 0, a i,j represents the weight of the communication between the ith follower agent and the jth follower agent; y i represents the output signal of the ith follower agent, y j represents the output signal of the jth follower agent, y r represents the leader agent output signal; b i represents the weight of the communication between the ith follower agent and the leader, when the ith follower agent establishes communication with the leader b i = 1, otherwise b i = 0.

[0023] Furthermore, a new predetermined time control stability lemma is established as follows:

[0024]

[0025] Where V is a radially unbounded positive definite function, is the derivative of V, T p is the predefined convergence time, Represents the design parameter, Δ>0 is a positive constant.

[0026] Furthermore, a universal barrier function is established to transform the constrained synchronization error z of the i-th follower agent into i,1 Convert to unconstrained variable w i,1 The formula is as follows:

[0027]

[0028] Among them, Ω H Represents the upper bound function of the constraint synchronization error, Ω L Represents the lower bound function of the constraint synchronization error, and the initial values ​​of the upper and lower bounds of the constraint function and the initial value of the synchronization error of the i-th follower agent meet the following conditions Ω L (0)<z i,1 (0)<Ω H (0), and in addition, the bounding function is constrained to satisfy Ω H >0,Ω L >0, which is arbitrarily chosen by the designer and constrains the boundary to be a symmetric or asymmetric boundary.

[0029] Furthermore, the nonlinear predetermined time filter is established as follows:

[0030]

[0031] in, represents the derivative of the r-th order filter output signal of the i-th follower agent; c i,1 , c i,2 , c i,3 is an adjustable parameter, and its specific expression is: T p >0 is the scheduled time, 0 < β i <1 / 2 is the design parameter; θ i,r ≥1,τ i,r >0 is the design parameter, and r=1,…,n i -1; represents the initial value of the r-th order output signal of the filter of the i-th follower agent; α i,r (0) represents the initial value of the r-th order virtual control input signal of the i-th follower agent;

[0032] ζ i,r Represents the error between the filter output signal and the virtual control signal, and is calculated as follows:

[0033]

[0034] is the ι-th order filter output signal of the ith follower agent, α i,ι is the ith-order virtual control signal of the ith follower agent.

[0035] Furthermore, based on the backstepping control technology, a predetermined time and precision controller is designed;

[0036] The first virtual controller designed for the i-th follower agent is expressed as:

[0037]

[0038] in, Equal to the variable w after the barrier function transformation i,1 , Ω H Represents the upper bound function of the constraint synchronization error, Ω L Represents the lower bound function of the constraint synchronization error, z i,1 is the synchronization error between the follower agent i and the leader agent’s output signal, represents the radial basis function vector, and δ i,1 >0 is an adjustable parameter; is the adaptive rate of the i-th follower agent; c i,1 , c i,2 , c i,3 It is an adjustable parameter, and its specific expression is as follows:

[0039] T p >0 is the scheduled time, 0 < β i <1 / 2 is the design parameter;

[0040] The sth virtual controller designed for the i-th follower agent is expressed as:

[0041]

[0042] Represents the s-th state x of the i-th agent i,s and the filter output signal The error, represents the radial basis function vector, and where x i,s , x j,s Respectively represent the s-th order state of the i-th follower and the j-th follower system, δ i,s >0 is an adjustable parameter.

[0043] Furthermore, the actual controller and adaptation rate of the i-th follower agent are expressed as:

[0044]

[0045] in, Represents the nth agent of the i-th agent i Stage State and the filter output signal The error, λ i,2 =2-2β i ,λ i,3 =2+2β i , 0<β i <1 / 2 is the design parameter;

[0046] represents the radial basis function vector, and in Respectively represent the nth follower of the i-th follower and the j-th follower system i Stage state, is an adjustable parameter, Represents Adaptive The derivative of .

[0047] Compared with the existing technology, the principles and advantages of this technical solution are as follows:

[0048] 1. In the existing practical finite time control, Obviously, the preset time is determined by the initial condition x(0) of the system. However, for a multi-agent system, it is difficult to obtain the initial value of each agent. Therefore, these finite-time control schemes are difficult to apply to multi-agent systems. Although the existing practical fixed-time control scheme has the set time expression as Avoids dependence on initial conditions, but requires coordination Therefore, it is very difficult to determine the ideal fixed time, and the estimated convergence time is usually too conservative. This technical solution proposes a new practical preset time control lemma. As long as the derivative of the designed Lyapunov function satisfies the following inequality The system convergence time is T p , it is obvious that the convergence time T can be directly preset PThe method is not only independent of initial conditions of a system, but also determined by only one parameter, and the practical predetermined time control is superior to the existing practical limited or fixed time control scheme.

[0049] 2. The precision constraint boundary of the existing scheme is usually symmetrical, and if the constraint boundary needs to meet asymmetry in an actual engineering system, the existing method is not applicable. The universal barrier function transformation is used in the method to limit the synchronization error in the case of symmetrical preset boundary, asymmetrical preset boundary and no preset boundary.

[0050] 3. The filter discussed in the existing scheme converges to a adjustable residual set when the filter error tends to infinity in time, and the new nonlinear predetermined time filter is established based on the proposed predetermined time control stability lemma, which can guarantee that the filter error converges to a adjustable residual set in a preset time, thereby improving the control performance. BRIEF DESCRIPTION OF DRAWINGS

[0051] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the services needed to be used in the embodiments or the prior art description will be briefly introduced. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor.

[0052] Figure 1 A principle flowchart of a predetermined time and precision consistency control method for a multi-agent system of the present application;

[0053] Figure 2 A directed graph for a leader and four follower agent systems;

[0054] Figure 3 A leader output trajectory and four follower output trajectories;

[0055] Figure 4 Synchronization error trajectories of the four follower agents and a predefined precision boundary trajectory;

[0056] Figure 5 Adaptive rate trajectories of the four follower agents;

[0057] Figure 6 Second-order state trajectory graphs of the four follower agents. DETAILED DESCRIPTION

[0058] The present application will be further described below in combination with specific embodiments:

[0059] As Figure 1As shown, the method for controlling the consistency of predetermined time and accuracy of a multi-agent system described in this embodiment includes the following steps:

[0060] S1. Establish a multi-agent system model, including models of follower agents and leader agents;

[0061] In this step,

[0062] The established multi-agent system model includes a leader (marked as 0) and M follower agents, which form a directed graph S;

[0063] The model of the i-th follower agent is as follows:

[0064]

[0065] in, is the state vector of the i-th follower agent, x i,1 , x i，2 ,...., Represents the system state, where s represents the sth order of the i-th follower agent system, n i represents the total system order of the i-th follower agent; and The upper point represents the derivative of the variable with respect to time t; u i ∈R represents the control input of the i-th follower agent, y i ∈R represents the output signal of the i-th follower agent; is a smooth and unknown function for the i-th follower agent;

[0066] The leader model is as follows:

[0067]

[0068] Among them, y r represents the output signal of the leader agent, Represents the derivative of the output signal of the leader agent with respect to time, function is a known, smooth and bounded function.

[0069] S2. Establish a new time-stable control lemma and define the synchronization error z of the multi-agent system i,1 , the synchronization error z i,1 is the error between the output of the ith follower agent and the leader;

[0070] Define the synchronization error z of the multi-agent system i,1 as follows:

[0071]

[0072] where i and j represent the ith and jth follower agents, respectively; follower agent j is called a neighbor of follower agent i when the information of the jth follower agent is transmittable to the ith follower agent, and a i,j > 0, otherwise a i,j = 0, a i,j represents the weight of the communication between the ith follower agent and the jth follower agent; y i represents the output signal of the ith follower agent, y j represents the output signal of the jth follower agent, y r represents the output signal of the leader agent; b i represents the weight of the communication between the ith follower agent and the leader, when the ith follower agent establishes communication with the leader b i = 1, otherwise b i = 0.

[0073] The new predetermined time control stability lemma is established as follows:

[0074]

[0075] where V is a positive definite function with unbounded radial, is the derivative of V, T p is a predefined convergence time, represents a design parameter, and Δ > 0 is a positive constant.

[0076] S3, a general barrier function is established, and the constrained synchronization error z i,1 of the ith agent is converted into an unconstrained variable w i,1 , which is given by

[0077]

[0078] where Ω H represents the upper bound function of the constrained synchronization error, Ω L represents the lower bound function of the constrained synchronization error, and the initial values of the upper and lower bound functions of the constraint function and the initial value of the synchronization error of the ith follower agent satisfy the following conditions Ω L (0) < z i,1 (0) < Ω H (0), in addition, the constraint boundary functions satisfy Ω H > 0, Ω L > 0, which are arbitrarily selected by the designer, and the constraint boundary is a symmetric or asymmetric boundary.

[0079] S4. Combining the new scheduled time control stability lemma, a nonlinear scheduled time filter is established and a scheduled time and precision controller is designed for each follower agent, including the 1st to sth virtual controllers and the actual controller;

[0080] The nonlinear scheduled time filter is established as follows:

[0081]

[0082] in, represents the derivative of the r-th order filter output signal of the i-th follower agent; c i,1 , c i,2 , c i,3 is an adjustable parameter, and its specific expression is: T p >0 is the scheduled time, 0<βi i <1 / 2 is the design parameter; θ i,r ≥1,τ i,r >0 is the design parameter, and r=1,…,n i -1; represents the initial value of the r-th order output signal of the filter of the i-th follower agent; α i,r (0) represents the initial value of the r-th order virtual control input signal of the i-th follower agent;

[0083] ζ i,r Represents the error between the filter output signal and the virtual control signal, and is calculated as follows:

[0084]

[0085] is the ι-th order filter output signal of the ith follower agent, α i,ι is the l-th order virtual control signal of the i-th follower agent.

[0086] Based on backstepping control technology, design the scheduled time and precision controller;

[0087] The first virtual controller designed for the i-th follower agent is expressed as:

[0088]

[0089] in, Equal to the variable w after the barrier function transformation i,1 , Ω H Represents the upper bound function of the constraint synchronization error, Ω L Represents the lower bound function of the constraint synchronization error, z i,1is the synchronization error between the follower agent i and the leader agent’s output signal, represents the radial basis function vector, and δ i,1 >0 is an adjustable parameter; is the adaptive rate of the i-th follower agent; c i,1 , c i,2 , c i,3 It is an adjustable parameter, and its specific expression is as follows:

[0090] T p >0 is the scheduled time, 0 < β i <1 / 2 is the design parameter;

[0091] The sth virtual controller designed for the i-th follower agent is expressed as:

[0092]

[0093] Represents the s-th state x of the i-th agent i,s and the filter output signal The error, represents the radial basis function vector, and where x i,s , x j,s Respectively represent the s-th order state of the i-th follower and the j-th follower system, δ i,s >0 is an adjustable parameter.

[0094] The actual controller and adaptation rate of the i-th follower agent are expressed as:

[0095]

[0096] in, Represents the nth agent of the i-th agent i Stage State and the filter output signal The error, λ i,2 =2-2β i ,λ i,3 =2+2β i , 0<β i <1 / 2 is the design parameter;

[0097] represents the radial basis function vector, and in Respectively represent the nth follower of the i-th follower and the j-th follower system i Order state, δ i,ni >0 is an adjustable parameter, Represents Adaptive The derivative of .

[0098] S5. Combine the new scheduled time control stability lemma, universal barrier function, nonlinear scheduled time filter and the scheduled time and accuracy controller of each follower agent to perform consistency control of the scheduled time and accuracy of the multi-agent system.

[0099] In order to prove the effectiveness and superiority of the method described in this embodiment, a simulation experiment is carried out below:

[0100] Suppose a multi-agent system consists of a leader and four followers, whose directed graph is as follows Figure 2 , the mathematical model of the intelligent agent is as follows:

[0101]

[0102] y i =x i,1 .

[0103] The initial state of each follower agent is: The leader's output signal trajectory is y r (t) = 0.5sin(t), predefined time T P =2s, the expected accuracy limit is Ω L =0.05+0.3exp(-3t),Ω H =0.05+0.5exp(-2t). The parameters are selected as follows: β i =0.1,τ i,1 =0.1,θ i,1 =90,δ i,1 =0.2,andδ i,2 =1for(i=1,...,4). In addition, we choose the initial value of the adaptive parameter to be The following results can be obtained from the simulation results: Figure 3 It shows that the follower agent's output signal can quickly track the leader's output signal within 2 seconds, which means that the multi-agent system achieves consistency within the predetermined time. Figure 4 It is further shown that the synchronization error is limited within a predetermined boundary, and the boundary is asymmetric, achieving a predetermined accuracy. Figure 5 Explain the adaptive law i=1,2,3,4 are bounded. In addition, Figure 6 Describes the second-order system state x of the agent i,2Trajectory, i=1,2,3,4.

[0104] The embodiments described above are only preferred embodiments of the present invention and are not intended to limit the scope of implementation of the present invention. Therefore, any changes made based on the shape and principle of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for consistent control of predetermined time and accuracy of a multi-agent system, characterized in that: include: Build a multi-agent system model, including models of follower agents and leader agents; Establish a new time-stable control lemma and define the synchronization error z of the multi-agent system i,1 , the synchronization error z i,1 is the error between the output of the ith follower agent and the leader; Establish a universal barrier function to transform the constrained synchronization error z of the i-th agent into i,1 Convert to unconstrained variable w i,1 ; Combined with the new scheduled time control stability lemma, a nonlinear scheduled time filter is established and a scheduled time and precision controller is designed for each follower agent, including the 1st to sth virtual controllers and the actual controller; Combining the new time-steering stability lemma, universal barrier function, nonlinear time-steering filter and time-steering and precision controller of each follower agent, the consistency control of time-steering and precision of multi-agent system is achieved. The nonlinear scheduled time filter is established as follows: in, represents the derivative of the r-th order filter output signal of the i-th follower agent; c i,1 , c i,2 , c i,3 is an adjustable parameter, and its specific expression is: T p >0 is the scheduled time, 0<β i <1 / 2 is the design parameter; θ i,r ≥1,τ i,r >0 is the design parameter, and r=1,...,n i -1; represents the initial value of the r-th order output signal of the filter of the i-th follower agent; α i,r (0) represents the initial value of the r-th order virtual control input signal of the i-th follower agent; ζ i,r Represents the error between the filter output signal and the virtual control signal, and is calculated as follows: is the rth order filter output signal of the i-th follower agent, α i,r is the r-th order virtual control signal of the i-th follower agent.

2. A method for controlling the consistency of predetermined time and accuracy of a multi-agent system according to claim 1, characterized in that: The established multi-agent system model includes a leader and M follower agents, wherein the leader and the M follower agents form a directed graph S; The model of the i-th follower agent is as follows: in, is the state vector of the i-th follower agent, Represents the system state, where s represents the sth order of the i-th follower agent system, n i represents the total system order of the i-th follower agent; and The upper point represents the derivative of the variable with respect to time t; u i ∈R represents the control input of the i-th follower agent, y i ∈R represents the output signal of the i-th follower agent; is a smooth and unknown function for the i-th follower agent; The leader model is as follows: Among them, y r represents the output signal of the leader agent, Represents the derivative of the output signal of the leader agent with respect to time, function is a known, smooth and bounded function.

3. A method for controlling the consistency of predetermined time and accuracy of a multi-agent system according to claim 1, characterized in that: Define the synchronization error z of the multi-agent system i,1 as follows: Where i and j represent the i-th and j-th follower agents, respectively; when the information of the j-th follower agent can be transmitted to the i-th follower agent, the follower agent j is called the neighbor of the follower agent i, and a i,j > 0, otherwise a i,j =0,a i,j represents the weight of the communication between the i-th follower agent and the j-th follower agent; y i Represents the output signal of the i-th follower agent, y j represents the output signal of the j-th follower agent, y r represents the output signal of the leader agent; b i Represents the weight of the communication between the i-th follower agent and the leader. When the i-th follower agent establishes communication with the leader b i =1, otherwise b i =0.

4. A method for controlling the consistency of predetermined time and accuracy of a multi-agent system according to claim 1, characterized in that: The new scheduled time control stability lemma is established as follows: Where V is a radially unbounded positive definite function, is the derivative of V, T p is the predefined convergence time, represents the design parameter, and Δ>0 is a positive constant.

5. The method for controlling the consistency of predetermined time and accuracy of a multi-agent system according to claim 1, characterized in that: Establish a universal barrier function to transform the constrained synchronization error z of the i-th follower agent into i,1 Convert to unconstrained variable w i,1 The formula is as follows: Among them, Ω H Represents the upper bound function of the constraint synchronization error, Ω L Represents the lower bound function of the constraint synchronization error, and the initial values ​​of the upper and lower bounds of the constraint function and the initial value of the synchronization error of the i-th follower agent meet the following conditions Ω L (0)<z i,1 (0)<Ω H (0), and in addition, the bounding function is constrained to satisfy Ω H >0,Ω L >0, which is arbitrarily chosen by the designer and constrains the boundary to be a symmetric or asymmetric boundary.

6. A method for controlling the consistency of predetermined time and accuracy of a multi-agent system according to claim 1, characterized in that: Based on backstepping control technology, design the scheduled time and precision controller; The first virtual controller designed for the i-th follower agent is expressed as: in, Equal to the variable w after the barrier function transformation i,1 , Ω H Represents the upper bound function of the constraint synchronization error, Ω L Represents the lower bound function of the constraint synchronization error, z i,1 is the synchronization error between the follower agent i and the leader agent’s output signal, a i,j represents the weight of the communication between the ith follower agent and the jth follower agent, b i represents the weight of the communication between the ith follower agent and the leader, represents the radial basis function vector, and δ i,1 >0 is an adjustable parameter; is the adaptive rate of the i-th follower agent; c i,1 , c i,2 , c i,3 It is an adjustable parameter, and its specific expression is as follows: T p >0 is the scheduled time, 0<β i <1 / 2 is the design parameter; The sth virtual controller designed for the i-th follower agent is expressed as: Represents the s-th state x of the i-th agent i,s and the filter output signal The error, represents the radial basis function vector, and x i,s , x j,s Respectively represent the s-th order state of the i-th follower and the j-th follower system, δ i,s >0 is an adjustable parameter.

7. A method for controlling the consistency of predetermined time and accuracy of a multi-agent system according to claim 6, characterized in that: The actual controller and adaptation rate of the i-th follower agent are expressed as: in, represents the nth i Stage State and the filter output signal The error, λ i,2 =2-2β i ,λ i,3 =2+2β i , 0<β i <1 / 2 is the design parameter; represents the radial basis function vector, and in Respectively represent the nth follower of the i-th follower and the j-th follower system i Stage state, is an adjustable parameter, Represents Adaptive The derivative of .

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