CN113031554A - Fixed time tracking consistency control method for second-order multi-agent system - Google Patents

Fixed time tracking consistency control method for second-order multi-agent system Download PDF

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CN113031554A
CN113031554A CN202110270752.XA CN202110270752A CN113031554A CN 113031554 A CN113031554 A CN 113031554A CN 202110270752 A CN202110270752 A CN 202110270752A CN 113031554 A CN113031554 A CN 113031554A
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张建东
史国庆
陈强
吴勇
张耀中
黄晓峰
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Northwestern Polytechnical University
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Abstract

The invention provides a fixed Time tracking consistency control method of a second-order multi-agent system, which utilizes a TimeBase Generator to design a fixed Time tracking consistency algorithm of the second-order multi-agent system to avoid overlarge control input requirements, thereby more effectively solving the problem of the fixed Time tracking consistency of the second-order multi-agent system. With this algorithm, at any initial state, multiple followers of a second-order multi-agent system are able to track the state of the top leader at a given time. The invention is not easy to cause system input saturation, is simpler and more applicable and reduces the system instability caused by parameter adjustment.

Description

Fixed time tracking consistency control method for second-order multi-agent system
Technical Field
The invention relates to the field of multi-agent cooperative control, in particular to a multi-agent system control method.
Background
In the last two decades, due to the great potential value of distributed cooperative control of multi-agent systems, scientific research has paid high attention to the distributed cooperative control, which includes distributed consistency control, surrounding control, distributed optimization, formation control and the like.
In the multi-agent cooperative control method, most of the earliest cooperative control algorithms are gradually stable, that is, when the time is approaching infinity, the system is converged to the equilibrium point. Such algorithms can be implemented theoretically, but have great limitations in the engineering field because the control system is often required to complete a given task within a limited time in reality. In order to make the system converge more quickly, some researchers have proposed a finite time stability algorithm, i.e., the system can converge to the equilibrium point in a finite time. Although the method effectively improves the convergence speed of the system, the upper limit of the stabilization time of the system cannot be determined. Since the settling time of the system is affected by the initial position of the system, the time for the initial state of the system to converge to the equilibrium point is longer as the initial state of the system is farther from the equilibrium point. In some cases, where the upper bound of the system settling time is strictly known, or the system must converge to the equilibrium position at a given time at any initial position, the finite time control method cannot meet the task requirement, and therefore, the scholars propose a fixed time settling control method.
The fixed time stability control method requires that the system be stable, i.e., the system reaches an equilibrium position and remains, at any given time, regardless of the initial position of the system. Compared with a finite time stability control method, the fixed time stability control method is not limited by the initial state of the system, so that the method has wider application space and higher research value. In the field of multi-agent cooperative control, a fixed time stability control method also occupies an important position. By combining the fixed time stable control method with the multi-agent cooperative control, the fixed time cooperative control of the multi-agent can be realized, and the fixed time consistency, the fixed time lead consistency, the fixed time formation, the fixed time distributed average tracking and the like of the multi-agent are common. Most of the existing multi-agent system fixed time cooperative control needs a large initial control input, which may cause system input saturation in some systems with general performance.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a method for controlling the fixed Time tracking consistency of a second-order multi-agent system, which utilizes a Time Base Generator to design a fixed Time tracking consistency algorithm of the second-order multi-agent system to avoid overlarge control input requirements, thereby more effectively solving the problem of the fixed Time tracking consistency of the second-order multi-agent system. In addition, the fixed Time method based on the Time Base Generator only needs to adjust one algorithm parameter when adjusting the upper bound of the stable Time, thereby avoiding other uncertainties caused by adjusting excessive parameters. With this algorithm, at any initial state, multiple followers of a second-order multi-agent system are able to track the state of the top leader at a given time.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
includes two aspects: respectively carrying out fixed Time tracking consistency calculation on a second-order multi-agent system based on the Time Base Generator under the undirected communication condition and carrying out fixed Time tracking consistency calculation on the second-order multi-agent system based on the Time Base Generator under the directed communication condition;
one) two-order multi-agent system fixed Time following consistency calculation based on Time Base Generator under undirected communication condition
A second-order integrator type multi-agent system containing n +1 agents, wherein 1 agent is a leader, the remaining n agents are followers, a communication topological graph among the n followers is a directionless connected graph, the leader only keeps communication with part of the followers, and the integrator is used for carrying out a mathematical modeling mode on the multi-agent system; the leader system model is as follows:
Figure BDA0002974280960000021
wherein x is0E.g. position with R as leader, v0E.g. R isThe speed of the leader or the speed of the leader,
Figure BDA0002974280960000022
and
Figure BDA0002974280960000023
are respectively x0And v0Derivative of u0E.g., R is the control input of the leader, and assume u0≤umax，umaxMaximum acceleration for the leader;
the follower system model is as follows:
Figure BDA0002974280960000024
wherein x isi∈R，viE.g. R and uie.R are the position, speed and control input of the follower, respectively, diIs a bounded perturbation and satisfies | di|≤dmax；
Step 1.1: constructing a distributed fixed time state observer for each agent except the leader;
designing a fixed Time distributed state observer based on a Time Base Generator for each follower to observe the relative state information of the follower and the follower;
the original Time Base Generator function ξ (t) is as follows:
Figure BDA0002974280960000025
wherein, tsIs an arbitrarily set upper bound on the stabilization time;
the Time Base Generator function was constructed as follows:
Figure BDA0002974280960000031
wherein t iss1,ts2,ts3,ts4Is a stable time upper bound and t is more than or equal to 0s1≤ts2≤ts3≤ts4；
Let function k (t) be as follows:
Figure BDA0002974280960000032
wherein, κ1,κ2,κ3,κ4∈R+，δ1,δ2,δ3,δ4∈R+Taking kappa by the convergence accuracy of eight parameter adjusting systems1,κ2,κ3,κ4≥1，0<δ1,δ2,δ3,δ4≤0.1；
Order to
Figure BDA0002974280960000037
Indicating the positional deviation of the ith agent from the leader,
Figure BDA0002974280960000038
representing the speed deviation of the ith agent and the leader, and designing a fixed time distributed state observer piAnd q isiRespectively observe
Figure BDA0002974280960000035
And
Figure BDA0002974280960000036
the concrete structure is as follows:
Figure BDA0002974280960000033
let the matrix Q be L + B, L be the laplacian matrix corresponding to the follower, B be diag { a10,...,an0If ai01, the ith agent maintains communication with the leader on behalf of the ith agent, let λ1(Q) is the minimum eigenvalue of the matrix Q, such that
Figure BDA0002974280960000034
p0＝0，q0＝0，b1∈R+，c1∈R+And satisfy b1≥1，c1＞umax+dmax(ii) a If a isij1 represents that there is communication between the ith agent and the jth agent, and a is no communicationij＝0；
Observer qiAt ts1The observation of the leader and the speed error of the leader is finished within the moment, and an observer piAt ts1+ts2The observation of the position error of the leader and the observer is finished within the moment, namely the observer is to be at t1＝ts1+ts2All observations are completed within the moment;
step 1.2: constructing a fixed time controller for each agent except the leader;
designing a controller by using a sliding mode control method; designing a sliding mode surface with fixed Time by using a Time Base Generator, sliding the system to a balance position within fixed Time when the system state is positioned on the sliding mode surface, and designing control input according to the sliding mode surface to enable the system to move from an initial position to the sliding mode surface within fixed Time;
the fixed time slip form is constructed as follows:
Figure BDA0002974280960000041
the fixed time controller is constructed as follows:
Figure BDA0002974280960000042
wherein let t1＝ts2，ρ≥dmax+umax+1；
Step 1.3: substituting the controller (8) and the observer (6) into the formula (2);
the multi-agent system will be implemented in a fixed time:
Figure BDA0002974280960000043
wherein T is Ts4For a predetermined upper limit of the settling time, i.e. the fixed time, p1,ρ2∈R+Is constant and is adjusted according to the system accuracy, p1,ρ2The smaller the required system precision and the higher the control input saturation upper limit, the design of the fixed Time following consistency algorithm of the second-order multi-agent system based on the Time Base Generator under the undirected communication condition is finished;
secondly) calculating the fixed Time following consistency of a second-order multi-agent system based on the Time Base Generator under the condition of directed communication;
a second-order integrator type multi-agent system containing n +1 agents, wherein 1 agent is a leader, the remaining n agents are followers, a directed spanning tree exists in a communication topological graph among the n followers, the leader is a root node, and an agent system model is shown as a formula (1) and a formula (2);
step 2.1: constructing a distributed fixed-time state observer for each agent except the leader:
Figure BDA0002974280960000051
let H be L + B, where L is in the middle of Rn×nIs a Laplace matrix corresponding to a directed graph formed by followers, and a non-negative diagonal matrix B ═ diag { a }10,...,an0Has at least one element greater than zero if ai01, representing that the ith intelligent agent can directly acquire leader state information; definition y ═ y1,...,yn]T＝H-T1n，Y＝diag{y1,...,yn}，
Figure BDA0002974280960000052
λ1(Q) is the minimum eigenvalue of the matrix Q,
Figure BDA0002974280960000053
Figure BDA0002974280960000054
p0＝0，q0＝0d0＝0，
Figure BDA0002974280960000055
b1∈R+and c1∈R+Is constant and satisfies
Figure BDA0002974280960000056
Step 2.2: constructing a fixed time controller for each agent except the leader;
the design of the second-order multi-agent following consistency controller under the directed communication condition is the same as that under the undirected communication condition, and the controller is shown as a formula (8);
step 2.3: substituting the controller (8) and the observer (10) into the formula (2) of the multi-agent system, the multi-agent system can realize the fixed time leading consistency of the second-order multi-agent system shown in the formula (9) in fixed time.
The invention has the beneficial effect that the problem of the consistency of the fixed Time of the second-order multi-agent system under the undirected communication condition and the directed communication condition is solved by using the Time Base Generator. The fixed Time domain consistency method based on the Time Base Generator requires relatively smaller initial control input, and system input saturation is less likely to be caused, so that the application range of the algorithm is greatly improved. When the upper bound of the stabilization time is adjusted, only one parameter of the upper bound of the stabilization time needs to be adjusted, and compared with an algorithm which needs to adjust a plurality of parameters at the same time, the method is simpler and more applicable, and the instability of the system caused by adjusting the parameters is reduced.
Drawings
FIG. 1 is a multi-agent undirected communication network topology.
Fig. 2 is a speed observer state diagram under undirected communication conditions.
FIG. 3 is a state diagram of a position observer under undirected communication conditions.
FIG. 4 is a graph of multi-agent speed status under undirected communication conditions.
FIG. 5 is a graph of multi-agent location status under undirected communication conditions.
FIG. 6 is a multi-agent directed communication network topology diagram.
FIG. 7 is a velocity observer state plot under directed communication conditions.
FIG. 8 is a position observer state graph under directed communication conditions.
FIG. 9 is a multi-agent speed state graph under directed communication conditions.
FIG. 10 is a multi-agent location state graph under directed communication conditions.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
For better illustration of the present invention, the following describes a fixed time tracking consistency algorithm under the condition of undirected communication and directed communication respectively with reference to the attached drawings and embodiments.
One) two-order multi-agent system fixed Time following consistency calculation based on Time Base Generator under undirected communication condition
A second-order integrator type multi-agent system containing n +1 agents, wherein 1 agent is a leader, the remaining n agents are followers, a communication topological graph among the n followers is a directionless connected graph, the leader only keeps communication with part of the followers, and the integrator is used for carrying out a mathematical modeling mode on the multi-agent system; the leader system model is as follows:
Figure BDA0002974280960000061
wherein x is0E.g. position with R as leader, v0E.r is the speed of the leader,
Figure BDA0002974280960000062
and
Figure BDA0002974280960000063
are respectively x0And v0Derivative of u0E.g., R is the control input of the leader, and assume u0≤umax，umaxMaximum acceleration for the leader;
the follower system model is as follows:
Figure BDA0002974280960000064
wherein x isi∈R，viE.g. R and uie.R are the position, speed and control input of the follower, respectively, diIs a bounded perturbation and satisfies | di|≤dmax；
Step 1.1: constructing a distributed fixed time state observer for each agent except the leader;
because the system communication mode is distributed, only partial followers directly acquire the relative state information of the followers and the leader, and a fixed Time distributed state observer based on a Time Base Generator is designed for each follower to observe the relative state information of the followers and the leader;
the original Time Base Generator function ξ (t) is as follows:
Figure BDA0002974280960000071
wherein, tsIs an arbitrarily set upper bound on the stabilization time;
because the multi-agent system is a second-order system, the observer is composed of a double integrator, the original Time Base Generator function cannot meet the task requirement, and the reconstructed Time Base Generator function is as follows:
Figure BDA0002974280960000072
wherein t iss1,ts2,ts3,ts4Is a stable time upper bound and t is more than or equal to 0s1≤ts2≤ts3≤ts4The specific values of the upper bound of the four stable times can be set at will without special technical requirements;
let function k (t) be as follows:
Figure BDA0002974280960000073
wherein, κ1,κ2,κ3,κ4∈R+，δ1,δ2,δ3,δ4∈R+Taking kappa by the convergence accuracy of eight parameter adjusting systems1,κ2,κ3,κ4≥1，0<δ1,δ2,δ3,δ4Less than or equal to 0.1, the selection of specific numerical value is determined according to the precision of the system, the higher the precision of the system is, the higher the kappa is1,κ2,κ3,κ4The larger the value, δ1,δ2,δ3,δ4The smaller the value is;
order to
Figure BDA0002974280960000081
Indicating the positional deviation of the ith agent from the leader,
Figure BDA0002974280960000082
representing the speed deviation of the ith agent and the leader, and designing a fixed time distributed state observer piAnd q isiRespectively observe
Figure BDA0002974280960000083
And
Figure BDA0002974280960000084
the concrete structure is as follows:
Figure BDA0002974280960000085
let matrix Q be L + B, L correspond to followerLaplacian matrix of (B ═ diag { a) }10,...,an0If ai01, the ith agent maintains communication with the leader on behalf of the ith agent, let λ1(Q) is the minimum eigenvalue of the matrix Q, such that
Figure BDA0002974280960000086
p0＝0，q0＝0，b1∈R+，c1∈R+And satisfy b1≥1，c1＞umax+dmax(ii) a If a isij1 represents that there is communication between the ith agent and the jth agent, and a is no communicationij＝0；
Observer qiAt ts1The observation of the leader and the speed error of the leader is finished within the moment, and an observer piAt ts1+ts2The observation of the position error of the leader and the observer is finished within the moment, namely the observer is to be at t1＝ts1+ts2All observations are completed within the moment;
step 1.2: constructing a fixed time controller for each agent except the leader;
designing a controller by using a sliding mode control method; designing a sliding mode surface with fixed Time by using a Time Base Generator, sliding the system to a balance position within fixed Time when the system state is positioned on the sliding mode surface, and designing control input according to the sliding mode surface to enable the system to move from an initial position to the sliding mode surface within fixed Time;
the fixed time slip form is constructed as follows:
Figure BDA0002974280960000087
the fixed time controller is constructed as follows:
Figure BDA0002974280960000088
wherein let t1＝ts2，ρ≥dmax+umax+1；
Step 1.3: substituting the controller (8) and the observer (6) into the formula (2);
the multi-agent system will be implemented in a fixed time:
Figure BDA0002974280960000091
wherein T is Ts4For a predetermined upper limit of the settling time, i.e. the fixed time, p1,ρ2∈R+Is a very small constant, adjusted according to the system accuracy, p1,ρ2The smaller the required system precision and the higher the control input saturation upper limit, the design of the fixed Time following consistency algorithm of the second-order multi-agent system based on the Time Base Generator under the undirected communication condition is finished;
secondly) calculating the fixed Time following consistency of a second-order multi-agent system based on the Time Base Generator under the condition of directed communication;
under the condition of directional communication, the communication between the intelligent agents is not bidirectional any more, namely, information flow in a specified direction exists between the two intelligent agents, and only one intelligent agent can acquire the state information of the other intelligent agent. The Laplace matrix corresponding to the communication topological graph of the directed communication multi-agent system is an asymmetric matrix, the design difficulty of a control algorithm is higher than that of undirected communication, and the algorithm is more complex than that of undirected communication. But the multi-agent system for directional communication occupies less resources and has lower requirements on physical equipment, so that the extension of the undirected communication algorithm to the directional communication has practical significance.
A second-order integrator type multi-agent system comprises n +1 agents, wherein 1 agent is a leader, the remaining n agents are followers, a directed spanning tree exists in a communication topological graph among the n followers, the leader is a root node, and an agent system model is shown in a formula (1) and a formula (2).
Step 2.1: constructing a distributed fixed-time state observer for each agent except the leader:
Figure BDA0002974280960000092
let H be L + B, where L is in the middle of Rn×nIs a Laplace matrix corresponding to a directed graph formed by followers, and a non-negative diagonal matrix B ═ diag { a }10,...,an0Has at least one element greater than zero if ai01, representing that the ith intelligent agent can directly acquire leader state information; definition y ═ y1,...,yn]T＝H -T1n，Y＝diag{y1,...,yn}，
Figure BDA0002974280960000101
λ1(Q) is the minimum eigenvalue of the matrix Q,
Figure BDA0002974280960000102
Figure BDA0002974280960000103
p0＝0，q0＝0d0＝0，
Figure BDA0002974280960000104
b1∈R+and c1∈R+Is constant and satisfies
Figure BDA0002974280960000105
Step 2.2: constructing a fixed time controller for each agent except the leader;
because the relative state information of the observer and the leader is obtained, the design of the controller does not depend on the neighbor information any more, so that the design of the second-order multi-agent tracking consistency controller under the directed communication condition is the same as that under the undirected communication condition, and the controller is shown as a formula (8);
step 2.3: substituting the controller (8) and the observer (10) into the formula (2) of the multi-agent system, the multi-agent system can realize the fixed time leading consistency of the second-order multi-agent system shown in the formula (9) in fixed time.
The examples are as follows:
1) fixed Time tracking consistency algorithm of second-order multi-agent system based on Time Base Generator under undirected communication condition
Assuming a second-order integrator type multi-agent system containing 6 agents, wherein 1 agent is a leader, the remaining 5 agents are followers, the system model is respectively shown in formula (1) and formula (2), the observer and the controller are respectively shown in formula (6) and formula (8), the system network communication topology is shown in fig. 1, and the upper limit of the stable time is set as t-12 s, wherein t iss1＝ts2＝2s，ts3＝ts44 s; system parameter is κ1＝κ2＝κ3＝κ4＝2，b1＝1，c1＝8，ρ＝8，δ1＝δ2＝δ3＝δ40.01; the initial state of the system is x1＝-200,v1＝-200，x2＝-100,v2＝-100，x3＝0,v3＝0，x4＝100,v4＝100，x5＝200,v5The leader's control input is u 2000(t) +1, initial state x0＝0,v 00, the system disturbance is set to diThe initial states of the observer are all 0. In fig. 2 and 3, fig. 2 is a speed observer state curve, fig. 3 is a position observer state curve, a dotted line is a real state error between a leader and a follower, and a solid line is an observer observation value, and by comparison, the observer can successfully observe the relative state information of the agent and the leader within a given time. In fig. 4 and 5, fig. 4 is a speed state curve of the agent, fig. 5 is a position state curve of the agent, a dotted line is leader state information, and a solid line is agent state information.
2) Second-order multi-agent system fixed Time tracking consistency algorithm based on Time Base Generator under directed communication condition
Suppose a single has 5 wisdomThe second-order integrator type multi-agent system of the energy body is characterized in that 1 is a leader, the remaining 4 are followers, the system model is respectively shown in formula (1) and formula (2), the observer and the controller are respectively shown in formula (10) and formula (8), the network communication topology of the system is shown in figure 6, the upper limit of the stable time is set to be t-12 s, wherein t is ts1＝ts2＝2s，ts3＝ts44 s; system parameter is κ1＝κ2＝1，κ3＝κ4＝2，b1＝7，c1＝127，ρ＝22，δ1＝δ2＝δ3＝δ40.01; the initial state of the system is x0＝300,v0＝300，x1＝-200,v1＝-200，x2＝-100,v2＝-100，x3＝100,v3＝100，x4＝200,v4The leader's control input is u 200018sin (10t) +2, system perturbation set to diThe initial states of the observer are all 0. In fig. 7 and 8, fig. 7 is a speed observer state curve, fig. 8 is a position observer state curve, a dotted line is a real state error between a leader and a follower, and a solid line is an observer observation value, and by comparison, the observer can successfully observe the relative state information of the agent and the leader within a given time. In fig. 9 and 10, fig. 9 is a diagram of the speed state of the agent, fig. 10 is a diagram of the position state of the agent, the dotted line is the leader state information, and the solid line is the state information of the agent, and it can be seen by comparison that the agent successfully completes the tracking within a fixed time.
It should be understood that equivalents and modifications of the technical solution and inventive concept thereof may occur to those skilled in the art, and all such modifications and alterations should fall within the scope of the appended claims.

Claims (1)

1. A second-order multi-agent system fixed Time tracking consistency control method based on a Time Base Generator is characterized by comprising the following steps:
includes two aspects: respectively carrying out fixed Time tracking consistency calculation on a second-order multi-agent system based on the Time Base Generator under the undirected communication condition and carrying out fixed Time tracking consistency calculation on the second-order multi-agent system based on the Time Base Generator under the directed communication condition;
one) two-order multi-agent system fixed Time following consistency calculation based on Time Base Generator under undirected communication condition
A second-order integrator type multi-agent system containing n +1 agents, wherein 1 agent is a leader, the remaining n agents are followers, a communication topological graph among the n followers is a directionless connected graph, the leader only keeps communication with part of the followers, and the integrator is used for carrying out a mathematical modeling mode on the multi-agent system; the leader system model is as follows:
Figure FDA0002974280950000011
wherein x is0E.g. position with R as leader, v0E.r is the speed of the leader,
Figure FDA0002974280950000012
and
Figure FDA0002974280950000013
are respectively x0And v0Derivative of u0E.g., R is the control input of the leader, and assume u0≤umax，umaxMaximum acceleration for the leader;
the follower system model is as follows:
Figure FDA0002974280950000014
wherein x isi∈R，viE.g. R and uie.R are the position, speed and control input of the follower, respectively, diIs a bounded perturbation and satisfies | di|≤dmax；
Step 1.1: constructing a distributed fixed time state observer for each agent except the leader;
designing a fixed Time distributed state observer based on a Time Base Generator for each follower to observe the relative state information of the follower and the follower;
the original Time Base Generator function ξ (t) is as follows:
Figure FDA0002974280950000015
wherein, tsIs an arbitrarily set upper bound on the stabilization time;
the Time Base Generator function was constructed as follows:
Figure FDA0002974280950000021
wherein t iss1,ts2,ts3,ts4Is a stable time upper bound and t is more than or equal to 0s1≤ts2≤ts3≤ts4；
Let function k (t) be as follows:
Figure FDA0002974280950000022
wherein, κ1,κ2,κ3,κ4∈R+，δ1,δ2,δ3,δ4∈R+Taking kappa by the convergence accuracy of eight parameter adjusting systems1,κ2,κ3,κ4≥1，0<δ1,δ2,δ3,δ4≤0.1；
Order to
Figure FDA0002974280950000023
Indicating a positional deviation of the ith agent from the leader，
Figure FDA0002974280950000024
Representing the speed deviation of the ith agent and the leader, and designing a fixed time distributed state observer piAnd q isiRespectively observe
Figure FDA0002974280950000025
And
Figure FDA0002974280950000026
the concrete structure is as follows:
Figure FDA0002974280950000027
let the matrix Q be L + B, L be the laplacian matrix corresponding to the follower, B be diag { a10,...,an0If ai01, the ith agent maintains communication with the leader on behalf of the ith agent, let λ1(Q) is the minimum eigenvalue of the matrix Q, such that
Figure FDA0002974280950000028
p0＝0，q0＝0，b1∈R+，c1∈R+And satisfy b1≥1，c1＞umax+dmax(ii) a If a isij1 represents that there is communication between the ith agent and the jth agent, and a is no communicationij＝0；
Observer qiAt ts1The observation of the leader and the speed error of the leader is finished within the moment, and an observer piAt ts1+ts2The observation of the position error of the leader and the observer is finished within the moment, namely the observer is to be at t1＝ts1+ts2All observations are completed within the moment;
step 1.2: constructing a fixed time controller for each agent except the leader;
designing a controller by using a sliding mode control method; designing a sliding mode surface with fixed Time by using a Time Base Generator, sliding the system to a balance position within fixed Time when the system state is positioned on the sliding mode surface, and designing control input according to the sliding mode surface to enable the system to move from an initial position to the sliding mode surface within fixed Time;
the fixed time slip form is constructed as follows:
Figure FDA0002974280950000031
the fixed time controller is constructed as follows:
Figure FDA0002974280950000032
wherein let t1＝ts2，ρ≥dmax+umax+1；
Step 1.3: substituting the controller (8) and the observer (6) into the formula (2);
the multi-agent system will be implemented in a fixed time:
Figure FDA0002974280950000033
wherein T is Ts4For a predetermined upper limit of the settling time, i.e. the fixed time, p1,ρ2∈R+Is constant and is adjusted according to the system accuracy, p1,ρ2The smaller the required system precision and the higher the control input saturation upper limit, the design of the fixed Time following consistency algorithm of the second-order multi-agent system based on the Time Base Generator under the undirected communication condition is finished;
secondly) calculating the fixed Time following consistency of a second-order multi-agent system based on the Time Base Generator under the condition of directed communication;
a second-order integrator type multi-agent system containing n +1 agents, wherein 1 agent is a leader, the remaining n agents are followers, a directed spanning tree exists in a communication topological graph among the n followers, the leader is a root node, and an agent system model is shown as a formula (1) and a formula (2);
step 2.1: constructing a distributed fixed-time state observer for each agent except the leader:
Figure FDA0002974280950000041
let H be L + B, where L is in the middle of Rn×nIs a Laplace matrix corresponding to a directed graph formed by followers, and a non-negative diagonal matrix B ═ diag { a }10,...,an0Has at least one element greater than zero if ai01, representing that the ith intelligent agent can directly acquire leader state information; definition y ═ y1,...,yn]T＝H-T1n，Y＝diag{y1,...,yn}，
Figure FDA0002974280950000042
λ1(Q) is the minimum eigenvalue of the matrix Q,
Figure FDA0002974280950000043
Figure FDA0002974280950000044
p0＝0，q0＝0d0＝0，
Figure FDA0002974280950000045
b1∈R+and c1∈R+Is constant and satisfies
Figure FDA0002974280950000046
Step 2.2: constructing a fixed time controller for each agent except the leader;
the design of the second-order multi-agent following consistency controller under the directed communication condition is the same as that under the undirected communication condition, and the controller is shown as a formula (8);
step 2.3: substituting the controller (8) and the observer (10) into the formula (2) of the multi-agent system, the multi-agent system can realize the fixed time leading consistency of the second-order multi-agent system shown in the formula (9) in fixed time.
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