CN114280931B - Method for solving consistency of multiple intelligent agents based on intermittent random noise - Google Patents
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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
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 systemRandom differential equation of type. In the year 1951, the number of the main chain,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. The invention researches because of various types of noise in realityWhat is needed is a multi-agent system with ambient noise. If the environmental noise is white noise, the multi-agent system model itself becomes a white noiseRandom 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:
whereinN is the number of multi-agent following the leader, t represents time, x i (t)∈R n Is the state of the ith follower, R represents the real number set, u i (t)∈R m Representing noise controllers added to the system, P ∈ R n×n Is a system matrix, a ij Are the elements of adjacency matrix a; q ∈ R n×m Is an input matrix, c ij Is the coupling strength between agents i and j; r is d (t)∈R n Is the state of the leader, defines the error e between the ith agent and the leader i (t)=x i (t)-r d (t), systematic error e (t) of the whole system [ e ] 1 T (t),e 2 T (t),…,e N T (t)] T 。
Further, the error system of the multi-agent system is represented as:
wherein the noise controllerK is the noise controller gain, K ═ K 1 ,K 2 ,…,K d ],K i ∈R m×n (i=1,2...,d),ξ i (t)∈R d Is white Gaussian noise, satisfiesB i (t) d-dimensional Brownian motion on the ith agent, I d Representing a d x d identity matrix.
Further, the error system of the environment noise with white noise is expressed as follows:
wherein S (t) ═ S 1 T (t),S 2 T (t),…,S N T (t)] T ,I N Is an N-dimensional identity matrix, g (t) ═ g 1 T (t),g 2 T (t),…,g N T (t)] T ,i=1,2,...,N;α ij Is the noise interference density coefficient, F is the noise interference matrix, α ij Are the elements in the matrix F; w i (t) e R is a different from B on the ith agent i (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:
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 e i (t 0 )∈R n The following requirements are met:
wherein sup represents the supremum, t represents the time 0 Indicating the initial time, e (t) is the system error.
Further, byCalculate k 1 In the obtained value range, selecting a value as k 1 Then according to k 2 ≤||QK′||≤k 3 Setting k 2 And k 3 Determining 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 four constants k 1 ∈R,k 2 <0,k 3 ≥0,k 4 Is more than or equal to 0, and satisfies the condition when t is more than or equal to 0:
(2)k 2 ≤||QK′||≤k 3 ,
wherein k is 4 To limit the intensity of the ambient noise.
Then, one can get:
whereinIn particular, for all initial values of error e (0) ∈ R nN If, ifThen there are three cases:
in all three cases, there areAccording 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 matter of inhibiting 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 the intermittent noise is discontinuous noise as the name suggests, so that 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:
whereinN is the number of multi-agent following the leader, t represents time, x i (t)∈R n Is the state of the ith follower, u i (t)∈R m Representing noise controllers added to the system, P ∈ R n×n Is a system matrix, a ij Is 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 ∈ R n×m Is an input matrix, c ij Is the coupling strength between agents i and j. x (t) ═ x 1 T (t),x 2 T (t),…,x N T (t)] T ,r d (t)∈R n Is the state of the leader, dr d (t)=r d (t)dt, defining an error e between the ith agent and the leader i (t)=x i (t)-r d (t), systematic error e (t) of the whole system [ e ] 1 T (t),e 2 T (t),…,e N T (t)] T . Wherein R represents a real number set, R n ,R n×m Representing 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:
wherein the noise controllerK is the noise controller gain, K ═ K 1 ,K 2 ,…,K d ],K i ∈R m×n (i=1,2...,d),ξ i (t)∈R d Is white Gaussian noise, and satisfiesB i (t) d-dimensional Brownian motion on the ith agent, I d Representing 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:
wherein S (t) ═ S 1 T (t),S 2 T (t),…,S N T (t)] T ,I N Is an N-dimensional identity matrix, g (t) ═ g 1 T (t),g 2 T (t),…,g N T (t)] T ,(i=1,2,...,N),α ij Is the noise interference density coefficient, F is the noise interference matrix, α ij Are the elements in the matrix F; w is a group of i (t) e R is a different from B on the ith agent i (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:
where N represents a natural number set; t > 0 is called a control period, τ represents the noise width, and T > τ; since the noise is intermittent, at some intervalsWhen 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 intervalsWhen 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 e i (t 0 )∈R n The following requirements are met:
wherein sup represents the supremum, t represents the time 0 Indicating 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 controllers i (t) becomes smaller and smaller, eventually leading to a consistency definition in case the time t tends to be infiniteAlmost certainly, this is true, thus solving the multi-agent system consistency problem.
Wherein byCalculate k 1 In the obtained value range, selecting a value as k 1 Then according to k 2 ≤||QK′||≤k 3 Setting k 2 And k 3 Determining 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 k 1 ∈R,k 2 <0,k 3 ≥0,k 4 Is more than or equal to 0, and satisfies the condition when t is more than or equal to 0:
(2)k 2 ≤||QK′||≤k 3 ,
wherein k is 4 To limit the intensity of the ambient noise.
Then, one can get:
whereinIn particular, for all initial values of error e (0) ∈ R nN If it is determined thatThen there are three cases:
in all three cases, there areAs 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 e 1 (0)=(0.7,0.2) T 、e 2 (0)=(0.9,0.4) T 、e 3 (0)=(1,0.1) T 、e 4 =(3,0.3) T System matrixCoefficient of coupling c ij Error 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
The error system diagram follows.
The multi-agent topology shown in FIG. 1, a system adjacency matrix can be obtained:
then k is obtained according to the norm of the P system matrix and the first condition in the discriminant method 1 Where k is selected 1 3. Then set k 2 ,k 3 To determine the input matrix Q and the controller gain K, where K is selected 2 =3.5,k 3 =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 easier to observe, 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 (4)
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; the mathematical model of the system is represented as:
whereinN is the number of multi-agent following the leader, t represents time, x i (t)∈R n Is the state of the ith follower, R represents the real number set, u i (t)∈R m Representing noise controllers added to the system, P ∈ R n×n Is a system matrix, a ij Consists 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 ∈ R n×m Is an input matrix, c ij Is the coupling strength between agents i and j; r is d (t)∈R n Is the state of the leader, defines the error e between the ith agent and the leader i (t)=x i (t)-r d (t), systematic error e (t) of the whole system [ e ] 1 T (t),e 2 T (t),…,e N T (t)] T ;
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; the error system of the environment noise with white noise is represented as follows:
wherein S (t) ═ S 1 T (t),S 2 T (t),…,S N T (t)] T ,I N Is an N-dimensional identity matrix, g (t) ═ g 1 T (t),g 2 T (t),…,g N T (t)] T ,1,2, N; f is the noise interference matrix, α ij Is each element in the matrix F, representing a noise interference density coefficient; w i (t) e R is a different from B on the ith agent i (T) brownian motion, b (T), w (T) being different brownian motion across the multi agent system, superscript T denoting transposition;
constructing a multi-agent system consistency definition based on the system errors of the multi-agent system;
determining a noise controller meeting a discrimination method, and under the action of the noise controller, 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 discrimination method is as follows:
for a multi-agent system, if there are four constants k 1 ∈R,k 2 <0,k 3 ≥0,k 4 Is more than or equal to 0, and satisfies the condition when t is more than or equal to 0:
(2)k 2 ≤||QK′||≤k 3 ,
wherein k is 4 A limit for constraining the ambient noise intensity;
then, one can get:
whereinIn particular, for all initial values of error e (0) ∈ R nN If, ifThen there are three cases:
in all three cases, there areS, according to the consistency definition, the consistency problem of the intelligent system is solved;
by passingCalculate k 1 In the value range of (a) in the obtained valueWithin the range, a value is selected as k 1 Then according to k 2 ≤||QK′||≤k 3 Setting k 2 And k 3 Determining 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.
2. 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:
wherein the noise controllerK is the noise controller gain, K ═ K 1 ,K 2 ,…,K d ],K i ∈R m×n (i=1,2...,d),ξ i (t)∈R d Is white Gaussian noise, satisfiesB i (t) d-dimensional Brownian motion on the ith agent, I d Representing a d x d identity matrix, e i (t) represents the error between the ith agent and the leader, e j (t) represents the error between the jth agent and the leader.
3. The intermittent random noise-based multi-agent coherence solving method of claim 1, wherein the noise controller gain is chosen as follows:
ΔT 1l =[lT,lT+τ),ΔT 2l t + 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.
4. 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 e i (t 0 )∈R n The following requirements are met:
wherein sup denotes a supremum, t denotes time, t0 denotes an initial time, and e (t) denotes a system error.
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