CN114280931A - Method for solving consistency of multiple intelligent agents based on intermittent random noise - Google Patents

Method for solving consistency of multiple intelligent agents based on intermittent random noise Download PDF

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CN114280931A
CN114280931A CN202111528073.4A CN202111528073A CN114280931A CN 114280931 A CN114280931 A CN 114280931A CN 202111528073 A CN202111528073 A CN 202111528073A CN 114280931 A CN114280931 A CN 114280931A
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noise
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张波
蔡良翌
陈良
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Guangdong University of Technology
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Abstract

The invention discloses a method for solving multi-agent consistency based on intermittent random noise, which comprises the following steps: aiming at a multi-agent system, establishing a mathematical model of the system; the multi-agent system comprises a leader and a plurality of followers; establishing an error system of the multi-agent system based on the mathematical model; introducing environment noise with white noise, and constructing an error system of the environment noise with the white noise; constructing a multi-agent system consistency definition based on the system errors of the multi-agent system; and determining the noise controllers meeting the judgment method, and under the action of the noise controllers, enabling the error between each follower and the leader to be smaller and smaller, so that the consistency definition is established, thereby solving the consistency problem of the multi-agent system. The use of intermittent noise can reduce control costs compared to the use of general continuous noise; in addition, the method also has the advantages of high control performance, easy determination of control parameters and the like.

Description

Method for solving consistency of multiple intelligent agents based on intermittent random noise
Technical Field
The invention relates to the field of noise stabilization, in particular to a method for solving consistency of multiple agents based on intermittent random noise.
Background
Humans were first inspired by many natural biological coordinated flight phenomena, such as ant migration, resulting in the discovery of multi-agents. With the rapid development of computer technology, complex network technology and communication technology, the coordinated control of multi-agent system has become one of the research hotspots in the control science and control engineering fields. The multi-agent system is mainly used for traffic networks, unmanned aerial vehicle formation and communication systems. Each agent state is kept consistent while approaching the leader state by a distributed controller. The consistency of a multi-agent system means that the state between each participant and the leader remains unchanged during the exercise.
In recent years, research into deterministic multi-agent systems has been perfected. In fact, a deterministic system is added with a white gaussian noise, and the mathematical model is a model of this system
Figure BDA0003410940050000011
Random differential equation of type. In the year 1951, the number of the main chain,
Figure BDA0003410940050000012
the idea of random differential equations is introduced, which is to form a mathematical model containing random noise for the first time, and then the random analysis theory is gradually improved. Due to the fact that various types of noise exist in reality, the invention researches a multi-agent system with environmental noise. If the environmental noise is white noise, the multi-agent system model itself becomes a white noise
Figure BDA0003410940050000013
Random differential equations of the type, since the ambient noise is not controllable, the consistency of the system becomes uncontrollable.
Noise is generally considered as a disturbance in the control system, however, it has been found in some studies that sometimes certain random noises contribute to the stability of the system, such as Khasminskii once used two white noises to stabilize a particular system. The invention is inspired from the intermittent work of the air conditioner from the aspect of saving the cost, intermittence the noise for stabilization and applies to a multi-agent system which is hot in the current control field.
Disclosure of Invention
The invention provides a method for solving the consistency of multiple intelligent agents based on intermittent random noise, aiming at solving the problem of consistency of the multiple intelligent agents with ambient white noise.
In order to realize the task, the invention adopts the following technical scheme:
a method for resolving multi-agent coherence based on intermittent random noise, comprising:
aiming at a multi-agent system, establishing a mathematical model of the system; the multi-agent system comprises a leader and a plurality of followers;
establishing an error system of the multi-agent system based on the mathematical model;
introducing environment noise with white noise, and constructing an error system of the environment noise with the white noise;
constructing a multi-agent system consistency definition based on the system errors of the multi-agent system;
and determining the noise controllers meeting the judgment method, and under the action of the noise controllers, enabling the error between each follower and the leader to be smaller and smaller, so that the consistency definition is established, thereby solving the consistency problem of the multi-agent system.
Further, the mathematical model of the system is represented as:
Figure BDA0003410940050000021
wherein
Figure BDA0003410940050000022
N is the number of multi-agent following the leader, t represents time, xi(t)∈RnIs the state of the ith follower, R represents the real number set, ui(t)∈RmRepresenting noise controllers added to the system, P ∈ Rn×nIs a system matrix, aijAre the elements of adjacency matrix a; q ∈ Rn×mIs an input matrix, cijIs the coupling strength between agents i and j; r isd(t)∈RnIs the state of the leader, defines the error e between the ith agent and the leaderi(t)＝xi(t)-rd(t), systematic error e (t) of the whole system [ e ]1 T(t),e2 T(t),…,eN T(t)]T。
Further, the error system of the multi-agent system is represented as:
Figure BDA0003410940050000023
wherein the noise controller
Figure BDA0003410940050000024
K is the noise controller gain, K ═ K1,K2,…,Kd],Ki∈Rm×n(i＝1,2...,d)，ξi(t)∈RdIs white Gaussian noise, satisfies
Figure BDA0003410940050000025
Bi(t) d-dimensional Brownian motion on the ith agent, IdRepresenting a d x d identity matrix.
Further, the error system of the environment noise with white noise is expressed as follows:
Figure BDA0003410940050000031
wherein S (t) ═ S1 T(t),S2 T(t),…,SN T(t)]T，
Figure BDA0003410940050000032
INIs an N-dimensional identity matrix, g (t) ═ g1 T(t),g2 T(t),…,gN T(t)]T，
Figure BDA0003410940050000033
i＝1,2,...,N；αijIs the noise interference density coefficient, F is the noise interference matrix, αijAre the elements in the matrix F; wi(t) e R is a different from B on the ith agenti(t), B (t), W (t) are different Brownian motions across the multi-agent system.
Further, the gain of the noise controller takes the following values:
Figure BDA0003410940050000034
Figure BDA0003410940050000035
where N represents a natural number set; t > 0 is called the control period, τ denotes the noise width, and T > τ; the value of K' is determined by a discriminant method.
Further, the multi-agent system consistency is defined as follows:
if a multi-agent system solves its consistency problem, then the requirement is that its error be at any ei(t0)∈RnThe following requirements are met:
Figure BDA0003410940050000036
wherein sup represents the supremum, t represents the time0Indicating the initial time, e (t) is the system error.
Further, by
Figure BDA0003410940050000037
Calculate k1In the obtained value range, selecting a value as k1Then according to k2≤||QK′||≤k3Setting k2And k3Determining the gain K of the noise controller, finally setting the control period T, and determining the noise width so as to determine the specific form of the noise controller.
Further, the discrimination method is as follows:
for a multi-agent system, if there are fourConstant k1∈R，k2＜0，k3≥0，k4Is more than or equal to 0, and satisfies the condition when t is more than or equal to 0:
(1)
Figure BDA0003410940050000041
(2)k2≤||QK′||≤k3,
(3)
Figure BDA0003410940050000042
wherein k is4To limit the intensity of the ambient noise.
Then, one can get:
Figure BDA0003410940050000043
wherein
Figure BDA0003410940050000044
In particular, for all initial values of error e (0) ∈ RnNIf, if
Figure BDA0003410940050000045
Then there are three cases:
Figure BDA0003410940050000046
Figure BDA0003410940050000047
v is an arbitrary constant on (0, 1);
Figure BDA0003410940050000048
in all three cases, there are
Figure BDA0003410940050000049
According to the consistency definition, the consistency problem of the intelligent system is solved.
Compared with the prior art, the invention has the following technical characteristics:
the invention designs an intermittent noise controller to solve the problem of consistency of a plurality of intelligent agents with environment white noise, namely an intermittent random noise stabilizing method; the intermittent noise has a great advantage over the general continuous noise, and the intermittent control can reduce the control cost. In addition, the method also has the advantages of high control performance, easy determination of control parameters and the like.
Drawings
FIG. 1 is a multi-agent system topology diagram;
FIG. 2 is a system without ambient and control noise;
FIG. 3 is a system with only ambient noise;
fig. 4 is a system with ambient noise and intermittent control noise.
Detailed Description
The invention provides a method for solving the consistency problem of a multi-agent system with environmental noise influence, which solves the consistency problem by utilizing a Gaussian white noise stabilization theory. In general, we consider noise as a thing that suppresses system stability, however, studies have shown that a suitable noise may have a positive effect on system stability. The invention introduces the concept of intermittent noise on the basis of the noise stabilization theory, and as the name suggests, the intermittent noise is discontinuous noise, and the invention has the advantage of effectively reducing the energy consumption for generating the noise. The invention applies it to a multi-agent system with environmental noise to solve the consistency problem.
Referring to the drawings, a multi-agent system according to the present invention is a linear multi-agent system with a leader, and a consistency problem is that the final state of all followers in the multi-agent system is consistent with the leader. The process of the present invention is described in further detail below.
S1, aiming at the multi-agent system, establishing a mathematical model of the system, which is expressed as follows:
Figure BDA0003410940050000051
wherein
Figure BDA0003410940050000052
N is the number of multi-agent following the leader, t represents time, xi(t)∈RnIs the state of the ith follower, ui(t)∈RmRepresenting noise controllers added to the system, P ∈ Rn×nIs a system matrix, aijIs each element of the adjacency matrix a, consisting of the numbers 0 and 1, with 1 indicating that there is information transfer between agents i and j, and 0 indicating that there is no information transfer; q ∈ Rn×mIs an input matrix, cijIs the coupling strength between agents i and j. x (t) ═ x1 T(t),x2 T(t),…,xN T(t)]T，rd(t)∈RnIs the state of the leader, drd(t)＝rd(t) dt, defining an error e between the ith agent and the leaderi(t)＝xi(t)-rd(t), systematic error e (t) of the whole system [ e ]1 T(t),e2 T(t),…,eN T(t)]T. Wherein R represents a real number set, Rn,Rn×mRepresenting the set of n dimensional euclidean spaces and all n × m real matrices.
And S2, establishing an error system of the multi-agent system based on the mathematical model.
The consistency problem is closely related to the error system, so this system is written as a corresponding error system in the form:
Figure BDA0003410940050000061
wherein the noise controller
Figure BDA0003410940050000062
K is the noise controller gain, K ═ K1,K2,…,Kd],Ki∈Rm×n(i＝1,2...,d)，ξi(t)∈RdIs white Gaussian noise, satisfies
Figure BDA0003410940050000063
Bi(t) d-dimensional Brownian motion on the ith agent, IdRepresenting a d x d identity matrix.
S3, introducing the environment noise with white noise, and constructing an error system of the environment noise with white noise, wherein the error system is represented as follows:
Figure BDA0003410940050000064
wherein S (t) ═ S1 T(t),S2 T(t),…,SN T(t)]T，
Figure BDA0003410940050000065
INIs an N-dimensional identity matrix, g (t) ═ g1 T(t),g2 T(t),…,gN T(t)]T，
Figure BDA0003410940050000066
(i＝1,2,...,N)，αijIs the noise interference density coefficient, F is the noise interference matrix, αijAre the elements in the matrix F; wi(t) e R is a different from B on the ith agenti(t), B (t), W (t) are different Brownian motions across the multi-agent system, with the difference B (t) on the controller, and W (t) is Brownian motion in ambient noise.
Since the noise controller operates intermittently, K here takes the values:
Figure BDA0003410940050000067
Figure BDA0003410940050000068
where N represents a natural number set; t > 0 is called the control period, τ denotes the noise width, and T > τ; since the noise is intermittent, at some intervals
Figure BDA0003410940050000069
When the noise controller gain K to be designed is equal to K ', wherein the value of K' is determined by a subsequent discrimination method; at other time intervals
Figure BDA00034109400500000610
When K is 0.
S4, constructing a multi-agent system consistency definition based on the system errors of the multi-agent system.
If the above-described multi-agent system solves the consistency problem, then the requirement is that its error be at any ei(t0)∈RnThe following requirements are met:
Figure BDA0003410940050000071
wherein sup represents the supremum, t represents the time0Indicating the initial time, e (t) is the system error.
S5, determining the noise controllers satisfying the discriminant method, and enabling the error e between each follower and the leader under the positive promotion of the noise controllersi(t) becomes smaller and smaller, eventually leading to a consistency definition in case the time t tends to be infinite
Figure BDA0003410940050000072
Almost certainly, this is true, thus solving the multi-agent system consistency problem.
Wherein by
Figure BDA0003410940050000073
Calculate k1In the obtained range of values ofWithin the value range, a value is selected as k1Then according to k2≤||QK′||≤k3Setting k2And k3Determining the gain K of the noise controller, finally setting a control period T and determining the noise width; wherein, the noise width can be about one half of the control period generally; therefore, the specific form of the noise controller can be obtained according to specific parameters such as gain, control period, noise width and the like.
The discrimination method is as follows:
for the multi-agent system described above, if there are four constants k1∈R，k2＜0，k3≥0，k4Is more than or equal to 0, and satisfies the condition when t is more than or equal to 0:
(1)
Figure BDA0003410940050000074
(2)k2≤||QK′||≤k3,
(3)
Figure BDA0003410940050000075
wherein k is4To limit the intensity of the ambient noise.
Then, one can get:
Figure BDA0003410940050000076
wherein
Figure BDA0003410940050000077
In particular, for all initial values of error e (0) ∈ RnNIf, if
Figure BDA0003410940050000078
Then there are three cases:
(1)
Figure BDA0003410940050000081
(2)
Figure BDA0003410940050000082
upsilon is an arbitrary constant on (0,1),
(3)
Figure BDA0003410940050000083
in all three cases, there are
Figure BDA0003410940050000084
As described in terms of the consistency definition, the above multi-agent system consistency problem is solved.
Example (b):
as shown in FIG. 2, FIG. 2 shows an error system with two dimensions and an initial value e1(0)＝(0.7,0.2)T、e2(0)＝(0.9,0.4)T、e3(0)＝(1,0.1)T、e4＝(3,0.3)TSystem matrix
Figure BDA0003410940050000085
Coefficient of coupling cijError system diagram without ambient noise interference of 1.
FIG. 3 is a diagram of a system with an applied ambient noise density coefficient matrix of
Figure BDA0003410940050000086
The error system diagram follows.
The multi-agent topology shown in FIG. 1, a system adjacency matrix can be obtained:
Figure BDA0003410940050000087
then k is obtained according to the norm of the P system matrix and the first condition in the discriminant method1Where k is selected13. Then set k2,k3To determineDetermining the input matrix Q and the controller gain K, where K is selected2＝3.5,k3＝3.6，K′＝[1,2]，Q＝[0.9,1.3]T. Next, the noise width that can meet the requirement of consistency can be calculated by using the three parameters, and the noise width range is 1.04 < τ < 2 according to the discrimination method, where τ is 1.2 and the control period T is 2, and the graph of the calming effect is shown in fig. 4.
The result shows that the method can effectively solve the problem of consistency of the multi-agent system, the system tends to be in a stable state after two periods, and in fact, the fact that the controller does not work in the interval of 1.2 < t < 2 is easily observed, so that the control cost can be greatly reduced, and the method is also one advantage different from other methods.
The above embodiments are only used to illustrate the technical solutions of the present application, and not to limit the same; although the present application has been described in detail with reference to the foregoing embodiments, it should be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; such modifications and substitutions do not substantially depart from the spirit and scope of the embodiments of the present application and are intended to be included within the scope of the present application.

Claims (7)

1. A method for resolving multi-agent coherence based on intermittent random noise, comprising:
aiming at a multi-agent system, establishing a mathematical model of the system; the multi-agent system comprises a leader and a plurality of followers;
establishing an error system of the multi-agent system based on the mathematical model;
introducing environment noise with white noise, and constructing an error system of the environment noise with the white noise;
constructing a multi-agent system consistency definition based on the system errors of the multi-agent system;
and determining the noise controllers meeting the judgment method, and under the action of the noise controllers, enabling the error between each follower and the leader to be smaller and smaller, so that the consistency definition is established, thereby solving the consistency problem of the multi-agent system.
2. The intermittent random noise-based multi-agent coherence solving method of claim 1, wherein the mathematical model of the system is represented as:
Figure FDA0003410940040000011
wherein
Figure FDA0003410940040000012
N is the number of multi-agent following the leader, t represents time, xi(t)∈RnIs the state of the ith follower, R represents the real number set, ui(t)∈RmRepresenting noise controllers added to the system, P ∈ Rn×nIs a system matrix, aijAre the elements of adjacency matrix a; q ∈ Rn×mIs an input matrix, cijIs the coupling strength between agents i and j; r isd(t)∈RnIs the state of the leader, defines the error e between the ith agent and the leaderi(t)＝xi(t)-rd(t), systematic error e (t) of the whole system [ e ]1 T(t),e2 T(t),…,eN T(t)]T。
3. The intermittent random noise-based multi-agent coherence solving method of claim 1, wherein the error system of the multi-agent system is expressed as:
Figure FDA0003410940040000013
wherein the noise controller
Figure FDA0003410940040000014
K is the noise controller gain, K ═ K1,K2,…,Kd],,Ki∈Rm×n(i＝1,2...,d)，ξi(t)∈RdIs white Gaussian noise, satisfies
Figure FDA0003410940040000021
Bi(t) d-dimensional Brownian motion on the ith agent, IdRepresenting a d x d identity matrix.
4. The intermittent random noise-based multi-agent coherence solving method according to claim 1, wherein the error system of the environment noise with white noise is represented as follows:
Figure FDA0003410940040000022
wherein S (t) ═ S1 T(t),S2 T(t),…,SN T(t)]T，
Figure FDA0003410940040000023
INIs an N-dimensional identity matrix, g (t) ═ g1 T(t),g2 T(t),…,gN T(t)]T，
Figure FDA0003410940040000024
Figure FDA0003410940040000025
αijIs the noise interference density coefficient, F is the noise interference matrix, αijAre the elements in the matrix F; wi(t) e R is a different from B on the ith agenti(t), B (t), W (t) are different Brownian motions across the multi-agent system.
5. The intermittent random noise-based multi-agent coherence solving method of claim 1, wherein the noise controller gain is chosen as follows:
Figure FDA0003410940040000026
ΔT1l＝[lT,lT+τ)，ΔT2lt + τ, (l +1) T), l ∈ N, where N represents a natural number set; t > 0 is called the control period, τ denotes the noise width, and T > τ; the value of K' is determined by a discriminant method.
6. The intermittent random noise-based multi-agent consistency resolution method of claim 1, wherein the multi-agent system consistency is defined as follows:
if a multi-agent system solves its consistency problem, then the requirement is that its error be at any ei(t0)∈RnThe following requirements are met:
Figure FDA0003410940040000027
wherein sup represents the supremum, t represents the time0Indicating the initial time, e (t) is the system error.
The intermittent random noise-based multi-agent coherence solving method of claim 1, by
Figure FDA0003410940040000031
Calculate k1In the obtained value range, selecting a value as k1Then according to k2≤||QK′||≤k3Setting k2And k3Determining the gain K of the noise controller, finally setting the control period T, and determining the noise width so as to determine the specific form of the noise controller.
7. The intermittent random noise-based multi-agent coherence solving method according to claim 1, wherein the discriminant is as follows:
for a multi-agent system, if there are four constants k1∈R，k2＜0，k3≥0，k4Is more than or equal to 0, and satisfies the condition when t is more than or equal to 0:
(1),
Figure FDA0003410940040000032
(2),k2≤||QK′||≤k3,
(3),
Figure FDA0003410940040000033
wherein k is4To limit the intensity of the ambient noise.
Then, one can get:
Figure FDA0003410940040000034
wherein
Figure FDA0003410940040000035
In particular, for all initial values of error e (0) ∈ RnNIf, if
Figure FDA0003410940040000036
Then there are three cases:
Figure FDA0003410940040000037
Figure FDA0003410940040000038
v isAny constant on (0, 1);
Figure FDA0003410940040000039
in all three cases, there are
Figure FDA00034109400400000310
According to the consistency definition, the consistency problem of the intelligent system is solved.
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