Multi-agent system preset time and precision consistency control method

Abstract

The invention discloses a consistency control method for preset time and precision of a multi-agent system, and provides a novel practical preset time control lemma, as long as a Lyapunov function derivative is designed to meet a set inequality, the system convergence preset time is Tp, the convergence time TP can be directly preset, and the consistency control method for the preset time and precision of the multi-agent system is realized. The method does not depend on initial conditions of a system and is determined by only one parameter. By means of general barrier function transformation, synchronization errors can be limited to symmetric preset boundaries, asymmetric preset boundaries and under the condition that no preset boundaries exist. The adopted non-linear predetermined time filter is established based on the proposed predetermined time control stability lemma, and can ensure 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 a multi-agent system, and in particular to a consistency control method for a multi-agent system with predetermined time and accuracy. Background Art

[0002] In recent years, due to the wide application of satellite clusters, sensor networks and multi-vehicle collaboration, the interest in studying multi-agent systems (MASs) has greatly increased. In the field of control, a key goal of MASs is to implement a consensus protocol to ensure that the output or state of the system converges to a consistent state. Combined with the recursive nature of the backstepping method, many control schemes have been proposed to solve the consistency problem. In addition, the system response speed is an important indicator for evaluating the tracking performance of the system. In order to improve the response speed of the system, scholars have proposed the finite time control theory. This theory is superior to traditional control methods because it can ensure that the system reaches the desired state within a finite time and has advantages such as strong robustness and high accuracy. However, finite time stability has an inevitable disadvantage: its convergence time depends on the initial state of the system. Subsequently, Andery introduced fixed time stability, in which the convergence time does not depend on the initial conditions of the system. Subsequently, various fixed time stability criteria for MASs have been proposed. However, the convergence time of fixed time control is determined by multiple parameters and is prone to overly conservative estimates, making it difficult to determine the convergence time in advance. In order to stabilize the control system within the precise convergence time, some scholars have developed exponential preset time (PT) criteria and polynomial preset time criteria. Although these studies have achieved remarkable results in realizing PT control, due to system uncertainty, the accuracy of the tracking error is still unknown and can only be guaranteed to converge to near zero.

[0003] It should be pointed out that for practical engineering applications, system response speed is important, but high-precision control is equally important. 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 a scaling transformation function and an error coordinate transformation. However, in the existing error transformation method, the preset accuracy boundary after the transformation can only be symmetrical, so it is more practical to consider an asymmetric preset accuracy boundary.

[0004] It is worth noting that as the order of the controlled nonlinear MASs increases, the above PT control scheme will face the problem of "computational complexity explosion". In order to overcome this obstacle, the existing scheme proposes 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 predetermined 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 to provide a method for consistent control of a multi-agent system with predetermined time and accuracy.

[0006] To achieve the above purpose, the technical solution provided by the present invention is:

[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 multi-agent systems 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 ith 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, wherein the leader and the M follower agents form a directed graph

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

[0015]

[0016] in, is the state vector of the ith 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, and 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 ith follower agent, y i ∈R represents the output signal of the i-th follower agent; It is a smooth and unknown function for the i-th follower agent;

[0017] The leader model is as follows:

[0018]

[0019] 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.

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

[0021]

[0022] 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 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.

[0023] Furthermore, the new scheduled 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 satisfy the following condition 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 the constraint boundary is 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 ith 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 the calculation formula is as follows:

[0033]

[0034] is the ι-th order filter output signal of the ith follower agent, α i,ι is the ι-th 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 output signal, represents the radial basis function vector, and δ i,1 >0 is an adjustable parameter; is the adaptive rate of the ith 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 ith follower agent is expressed as:

[0041]

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

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

[0044]

[0045] in, represents the nth i Stage Status 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 status, is an adjustable parameter, Represents Adaptive The derivative of .

[0047] Compared with the prior art, 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 multi-agent systems, 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 schemes have the set time expression as Avoids dependence on initial conditions, but requires coordination and parameters, so 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 Then the system convergence time is T p Obviously, the convergence time T can be directly preset P, which not only does not depend on the initial conditions of the system, but is also determined by only one parameter. The practical predetermined time control of this technical solution is superior to the existing practical limited or fixed time control solutions.

[0049] 2. The accuracy constraint boundaries of existing solutions are usually symmetrical. In actual engineering systems, if the constraint boundaries need to be asymmetric, the existing methods are no longer applicable. This technical solution uses a universal barrier function transformation to limit the synchronization error to a symmetrical preset boundary, an asymmetric preset boundary, or no preset boundary.

[0050] 3. The filter discussed in the existing scheme has a filtering error that converges to an adjustable residual set when time tends to infinity, while the new nonlinear predetermined time filter described in the present technical scheme is established based on the proposed predetermined time control stability lemma, which can ensure that the filtering error converges quickly to an adjustable residual set within a preset time, thereby improving the control performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the services required for use in the embodiments or the prior art descriptions are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0052] Figure 1 It is a principle flow chart of a consistency control method for presetting time and accuracy of a multi-agent system of the present invention;

[0053] Figure 2 A directed graph of a system with a leader and four follower agents;

[0054] Figure 3 Outputs a trajectory for the leader and four followers;

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

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

[0057] Figure 6 It is the second-order state trajectory diagram of the four follower agents. DETAILED DESCRIPTION

[0058] The present invention will be further described below in conjunction with specific embodiments:

[0059] like Figure 1As shown, the method for controlling the consistency of the 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, and the leader and the M follower agents 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 ith 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, and 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 ith follower agent, y i ∈R represents the output signal of the i-th follower agent; It 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 scheduled time control stability 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 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 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 scheduled time control stability lemma is established as follows:

[0074]

[0075] 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.

[0076] S3. 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 , the formula is as follows:

[0077]

[0078] 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 satisfy the following condition 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 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 ith 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 the calculation formula is 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 the back-stepping 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 output signal, represents the radial basis function vector, and δ i,1 >0 is an adjustable parameter; is the adaptive rate of the ith 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 ith follower agent is expressed as:

[0092]

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

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

[0095]

[0096] in, represents the nth i Stage Status 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 systemi 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 precision controller of each follower agent to perform consistency control of the scheduled time and precision 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 performed 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 output signal trajectory of the leader 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 output signal can quickly track the leader 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 its boundary is asymmetric, achieving a predetermined accuracy. Figure 5 Explain the adaptive law i=1,2,3,4 is bounded. In addition, Figure 6 Describes the second-order system state x of the agent i,2 Trajectory, 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, all changes made according to the shape and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for consistent control of scheduled 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 multi-agent systems 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 ith 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; 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.

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, which form a directed graph The model of the i-th follower agent is as follows: in, is the state vector of the ith follower agent, x i,1 , represents the system state, where s represents the sth order of the i-th follower agent system, and 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 ith follower agent, y i ∈R represents the output signal of the i-th follower agent; It 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 consistent control 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 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.

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, Δ>0 is a positive constant.

5. A 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 that transforms 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 satisfy the following condition 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 the constraint boundary is a symmetric or asymmetric boundary.

6. A method for consistent control of predetermined time and accuracy of a multi-agent system according to claim 1, characterized in that: The nonlinear scheduled time filter is established as follows: in, represents the derivative of the r-th order filter output signal of the ith 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 the calculation formula is as follows: is the first r The output signal of the filter is i,r is the first r The virtual control signal.

7. A method for controlling the consistency of predetermined time and accuracy of a multi-agent system according to claim 1, characterized in that: Design the scheduled time and precision controller based on the back-stepping control technology; 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 output signal, represents the radial basis function vector, and δ i,1 >0 is an adjustable parameter; is the adaptive rate of the ith 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 ith follower agent is expressed as: Represents the s-th state x of the ith agent i,s The filter output signal The error, represents the radial basis function vector, and x i,s , x j,s denote the s-th order state of the i-th follower and j-th follower system, δ i,s >0 is an adjustable parameter.

8. A method for consistent control of predetermined time and accuracy of a multi-agent system according to claim 7, characterized in that: The actual controller and adaptation rate of the ith follower agent are expressed as: in, represents the nth i Stage Status 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 status, is an adjustable parameter, Represents Adaptive The derivative of .

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