CN105634828A - Method for controlling distributed average tracking of linear differential inclusion multi-agent systems - Google Patents

Abstract

The invention relates to a method for controlling distributed average tracking of linear differential inclusion multi-agent systems, relating to multi-agent systems. The method comprises the following steps of: constructing a network structure topological graph of multiple agents to obtain an adjacent matrix, an associated matrix and a Laplacian matrix of the graph; setting an initial state and an initial reference signal of each node; constructing a linear differential inclusion system, and setting a system matrix to obtain a system parameter; setting a communication manner, such that each node is only communicated with a neighbour node, operating a distributed average tracking algorithm, and adjusting the state of each node; and designing a control law according to feedback information and the system parameter, adjusting the node state to achieve consistency, and tracking an average value of the reference signal. Requirements on the network structure are low; only when a network is connected, all node states tend to be uniform; furthermore, the average value of the reference signal is tracked; in the whole calculation process, each node only uses information of the neighbour node; the calculation amount is low; and the operation efficiency of the algorithm is increased.

Description

Linear differential comprises the control method of the distributed average tracking of multi-agent system

Technical field

The present invention relates to multi-agent system, especially relate to a kind of control method that linear differential comprises the distributed average tracking of multi-agent system.

Background technology

The set that multi-agent system is made up of the multiagent system of multiple mutual coupling, each multiagent system has certain autonomy, and by the environment around perception, carries out communication with other intelligence bodies. Along with development in recent years, the distributed collaborative control of multi-agent system has become a focus of control field research. Distributed AC servo system Relatively centralized formula controls, and has that cost is little, reliability height, handiness height, extensibility advantages of higher, has engineering background and application prospect widely. The distributed convergent central issue being multi intelligent agent and studying, its target is that design distributed AC servo system device is so that the state of all intelligence bodies finally reaches consistent. Multiple agent distributed average tracking problem be sometimes counted as be consistency problem and coordinate tracking problem extensive, its core is a kind of algorithm of design, intelligence body independently is executed the task, communications exchange information can be passed through again simultaneously, mutually coordinate, finally make the state of all intelligence bodies follow the tracks of the mean value of upper multiple reference signal. Utilizing distributed average tracking can estimate the parameter of complex system, this has significant effect in multi-core microprocessor and formation regional control.

At present, in multi-agent system, big quantifier elimination mainly concentrates on linear system, and the research of non-linear system is less. But actual system nature is non-linear system. In order to solve the problem of non-linear system, a kind of being called that the method for global linearization is suggested, namely used time modified line sexual system replaces non-linear system. The explanation of linear differential comprises when being uncertain to one modified line sexual system, it is possible to be used for describing various non-linear system. The system that introducing linear differential comprises is exactly the various character in order to set up nonlinear and time-varying system, is solve nonlinear time-varying problem to provide technical support. The difficult point of the distributed average tracking problem that process linear differential comprises is:

First: the target value of tracking be all multiple time varying reference signal mean value, its for each intelligence body be unknown;

2nd: the system that linear differential comprises is the convex combination of one group of linear system, become uncertain system when being one;

3rd: in order to ensure the distributed characteristic of algorithm, the control inputs of intelligence body only can utilize local information.

Summary of the invention

It is an object of the invention to provide the control method that a kind of linear differential of the control of the average tracking for realizing time-varying uncertain system comprises the distributed average tracking of multi-agent system.

The present invention comprises the following steps:

Step 1: the network structure topological diagram of structure multiple agent, obtains the adjacency matrix of figure, incidence matrix and Laplce's matrix;

Step 2: the original state that each node is set, and initial reference signal;

Step 3: the system that structure linear differential comprises, arranges system matrix, obtains system parameter;

Step 4: signalling methods is set, make each node can only with neighbor node communication, run distributed average tracking algorithm, adjust the state of each node;

Step 5: carry out design control law according to feedback information and system parameter, knot modification state reaches the mean value of consistence and tracking reference signal.

In step 1, the network structure topological diagram of described structure multiple agent is Connected undigraph, and number of nodes is N, and limit quantity is M;

Adjacency matrix A, the incidence matrix E of figure and Laplce's matrix L are by following formulation:

L=EE^T��R^N��N

Wherein, N is number of nodes, and M is limit quantity, R^N��NFor N �� N ties up real matrix set, R^N��MFor N �� M ties up real matrix set, A is adjacency matrix, a_ij(i, j=1,2 ..., N) and represent node i and the relation of node j, if having limit to be connected between node i with node j, then a_ij=1, otherwise a_ij=0; By giving figure fixing direction, obtaining incidence matrix E is: e_ij(i=1 ..., N, j=1 ..., M) represent node i and the relation of limit j, if node i is the starting point of limit j, then e_ij=-1, if node i is the terminal of limit j, then e_ij=1, otherwise e_ij=0; L is Laplce's matrix, it is possible to by following formula diagonalization:

L=U^T�� U, �� :=diag ([��₁,��,��_N]), U^TU=UU^T=I

Wherein, U �� R^N��NFor unitary matrix, �� is that on principal diagonal, element is the diagonal matrix of eigenwert, ��_i(i=1 ..., N) and it is the eigenwert of Laplce's matrix L, do not lose generality, order: 0=��₁�ܦ�₂�ܡ��ܦ�_N, I �� R^N��NFor unit battle array.

In step 3, described linear differential comprises the kinetic equation of node state of description and the kinetic equation of reference signal is expressed as follows:

${\stackrel{\·}{x}}_i\left(t\right)=\underset{k=1}{\overset{n_s}{\Σ}}{\mathrm{\τ}}_k\left(t\right)\[A_k{x}_i\left(t\right)+B_k{u}_i\left(t\right)\],i=1,...,N$

${\stackrel{\·}{r}}_i\left(t\right)=\underset{k=1}{\overset{n_s}{\Σ}}{\mathrm{\τ}}_k\left(t\right)\[A_k{r}_i\left(t\right)+B_k{v}_i\left(t\right)\],i=1,...,N$

Wherein, x_i(t)��R^pIt is the state of node i, u_i(t)��R^qIt is the control inputs of node i, r_i(t)��R^pIt is the reference signal of node i, v_i(t)��R^qIt is the control inputs of node i reference signal, A_k��R^p��pAnd B_k��R^p��qIt is a group system matrix, it is the known constant matrices of design, ��_k(t) (k=1 ..., n_s) it is one group of randomized number, and meet ��_k(t) >=0 andP is the dimension of each node state, and q is the dimension of control inputs, n_sFor the number of linear system, p, q, n_sIt is known constant;

Described system matrix (A_k,B_k) should meet for all i=2 ..., N, (A_k,��_iB_k) can be stable, ��_iFor the nonzero eigenvalue of Laplce's matrix L.

In step 4, the difference information �� that each node i described and its neighbor node communication obtain_i1, and the difference information �� that node i obtains according to the difference of oneself state and reference signal_i2, it is expressed as follows:

${\mathrm{\Δ}}_{i1}=\underset{j=1}{\overset{N}{\Σ}}{a}_{ij}(x_i-x_j),i=1,...,N$

��_i2=x_i-r_i, i=1 ..., N

Wherein, a_ijFor the element of adjacency matrix, x_iIt is the state of node i, r_iIt it is the reference signal of node i.

In step 1,3 and 4, comprising: there is positive constant beta > 0, one group of constant ��_jkl> 0, one group of positive definite matrixWith matrix Y_j��R^q��pMake system matrix A_k,B_k(k=1 ..., n_s) meet the Bilinear Inequalities of following formulation:

$Q_j{A_k}^T+A_k{Q}_j+{\mathrm{\λ}}_i{Y_j}^T{B_k}^T+{\mathrm{\λ}}_i{B}_k{Y}_j\≤\underset{l=1}{\overset{n_Q}{\mathrm{\Σ}}}{\mathrm{\η}}_{jkl}(Q_l-Q_j)-{\mathrm{\βQ}}_j,$

K=1 ..., n_s, j, l=1 ..., n_Q, i=2 ..., N,

Wherein, A_k,B_k(k=1 ..., n_s) it is the n designed_sOrganizing linear system matrix, �� is adjustable positive constant, ��_jklIt is one group of adjustable constant, positive definite matrix Q_jWith matrix Y_jIt is unknown quantity, obtains by solving Bilinear Inequalities, n_QFor constant, represent matrix Q_jNumber, ��_i(i=2 ..., N) and it is the nonzero eigenvalue of Laplce's matrix L.

In step 4, comprising: the character that compound Laplce's quadratic function can be used, adopt the control method of distributed average tracking, the state of described each node of adjustment, wherein, compound Laplce's quadratic function be constructed as follows shown in formula:

$V\left(x\right):=\underset{\mathrm{\γ}\∈{\mathrm{\Γ}}^{n_Q}}{min}{X}^T\[L\⊗{\left(\underset{r=1}{\overset{n_Q}{\mathrm{\Σ}}}{\mathrm{\γ}}_j{Q}_j\right)}^{-1}\]X$

${\mathrm{\Γ}}^{n_Q}:=\{\mathrm{\γ}={\[{\mathrm{\γ}}_1,...,{\mathrm{\γ}}_{n_Q}\]}^T:{\mathrm{\γ}}_j\≥0,{\mathrm{\Σ}}_{j=1}^{n_Q}{\mathrm{\γ}}_j=1\}$

Wherein:For the vector of all node states represents, x_i��R^p(i=1 ..., N) and it is the state of node i, L is Laplce's matrix of figure,It is one group of positive definite matrix, n_QFor constant, represent matrix Q_jNumber, ��_j(j=1 ..., n_Q) it is one group of randomized number, and meet For n_QIndividual ��_jVector represent,It is defined as the set of the �� satisfied condition.

In steps of 5, described carrying out design control law according to feedback information and system parameter, knot modification state reaches the mean value of consistence and tracking reference signal, comprising:

(1) tracking target of following formulation is realized:

$\underset{t\→\mathrm{\∞}}{\lim }\left|\right|x_i\left(t\right)-\frac{1}{N}\underset{j=1}{\overset{N}{\Σ}}{r}_j\left(t\right)\left|\right|=0,i=1,...,N$

Wherein, x_i(t)��R^pIt is the state of node i, r_i(t)��R^pIt is the reference signal of node i,It is the mean value of all node reference signals;

(2) realization of tracking target can be regarded as the realization of two parts of following formulation:

||x_i-x_j||₂�� 0, as t �� ��

$\underset{i=1}{\overset{N}{\Σ}}(x_i-r_i)\→0,$ As t �� ��

Wherein, x_iIt is the state of node i, r_iIt is the reference signal of node i, formula | | x_i-x_j||₂�� 0 expression realizes state consistency,Represent the state and consistent with reference signal sum of realizing, when state and with reference signal and when being tending towards equal, namely mean that state mean value will be tending towards reference signal mean value, when realizing state consistency, it is possible to obtain each node state and will be tending towards the mean value of reference signal.

Compared with prior art, the Advantageous Effects of the present invention is:

(1) distributed way is adopted, for the strong adaptability of complex system.

(2) requirement of network structure is low, it is easy to realize.

(3) calculating simply, calculated amount is little, and real-time performance is good.

(4) kinetic equation of intelligence body can be used for modeling nonlinear and time-varying system, has representative widely.

(5) between the region of convergence of the algorithm designed by bigger compared to the convergence space of the algorithm designed based on quadratic form Lyapunov function.

The system that linear differential is comprised by the present invention is incorporated into the distributed average tracking control of multiple agent, and how research realizes the control of distributed average tracking based on distributed consensus method. The method has used the character of compound Laplce's quadratic function, the object of varying reference signal mean value when utilizing distributed average tracking algorithm to realize following the tracks of. The method is little to the requirement of network structure, only needs network to be that all node states that just can realize being connected reach unanimity and the mean value of tracking reference signal. In whole computation process, each node has only used the information of neighbor node, and calculated amount is little, thus improves the operation efficiency of algorithm largely.

Embodiment

Following examples describe the present invention.

The concrete steps of the distributed average tracking control method that the linear differential of the present invention comprises multi-agent system are as follows:

Step 101, it is determined that interstitial content N, random configuration one comprises the undirected connected network topological diagram of N number of node, obtains the adjacency matrix of figure, incidence matrix and Laplce's matrix, with reference to formula (1);

Step 102, arranges the original state of each node and the initial value of reference signal, and original state and initial reference signal can be random, with reference to formula (2);

Step 103, it is to construct linear differential comprises the kinetic equation of system, arranges system matrix, solving system parameter, with reference to formula (3), (4) and (5);

Step 104, the signalling methods of node is set, make its can only with neighbor node communication, fully demonstrate distributed feature, run distributed average tracking algorithm, all nodes are made to adjust the state of oneself according to the state of current time oneself, the reference signal value of oneself and the state of neighbor node, with reference to formula (6) and (7);

Step 105, design control law is carried out according to the difference information fed back and the system parameter obtained, adjust all node states and reach consistent and the mean value of tracking reference signal, with reference to formula (8), (9) and (10).

Linear differential comprises and can be used for describing the time-varying uncertain system that is made up of multiple robot, the core of its distributed average tracking is a kind of algorithm of design, intelligence body independently is executed the task, communications exchange information can be passed through again simultaneously, mutually coordinate, finally make the state of all intelligence bodies follow the tracks of the mean value of upper multiple reference signal.

Hereinafter consider a multi-agent system with N number of intelligence body.

1st step: by abstract for the communication relation between multiple agent be a network chart, represent multiple agent with node, utilize the relevant knowledge of graph theory to obtain adjacency matrix A, incidence matrix E and Laplce's matrix L of figure:

L=EE^T��R^N��N

Wherein, N is the quantity of node, and M is the quantity on limit, R^N��NFor N �� N ties up real matrix set, R^N��MFor N �� M ties up real matrix set, A is adjacency matrix, a_ij(i, j=1,2 ..., N) and represent node i and the relation of node j, if having limit to be connected between node i with node j, then a_ij=1, otherwise a_ij=0; Giving figure fixing direction, obtaining incidence matrix E is: e_ij(i=1 ..., N, j=1 ..., M) and represent node i and the relation of limit j, if node i is the starting point of limit j, then e_ij=-1, if node i is the terminal of limit j, then e_ij=1, otherwise e_ij=0; L is Laplce's matrix, it is possible to by following formula diagonalization:

L=U^T�� U, �� :=diag ([��₁,��,��_N]), U^TU=UU^T=I

Wherein, U �� R^N��NFor unitary matrix, �� is that on principal diagonal, element is the diagonal matrix of eigenwert, ��_i(i=1 ..., N) and it is the eigenwert of Laplce's matrix L, do not lose generality, order: 0=��₁�ܦ�₂�ܡ��ܦ�_N, I �� R^N��NFor unit battle array.

2nd step: the original state of each node and the initial value of reference signal are set. Original state x_iAnd reference signal initial value r (0)_i(0) can produce by randomized number, shown in the following formula of its form:

$x_i\left(0\right)=\left[\begin{array}{c}{x}_{i1}\left(0\right)\\ x_{i2}\left(0\right)\\ .\\ .\\ .\\ x_{ip}\left(0\right)\end{array}\right]\∈R^p,r_i\left(0\right)=\left[\begin{array}{c}{r}_{i1}\left(0\right)\\ r_{i2}\left(0\right)\\ .\\ .\\ .\\ r_{ip}\left(0\right)\end{array}\right]\∈R^p,i=1,\mathrm{...},N---\left(2\right)$

Wherein, p is the dimension of each node state, and the dimension of reference signal equals the dimension of state.

3rd step: be constructed as follows the kinetic equation that the linear differential shown in formula comprises system, system matrix is set, solving system parameter:

${\stackrel{\·}{x}}_i\left(t\right)=\underset{k=1}{\overset{n_s}{\Σ}}{\mathrm{\τ}}_k\left(t\right)\[A_k{x}_i\left(t\right)+B_k{u}_i\left(t\right)\],i=1,...,N,---\left(3\right)$

${\stackrel{\·}{r}}_i\left(t\right)=\underset{k=1}{\overset{n}{\Σ}}{\mathrm{\τ}}_k\left(t\right)\[A_k{r}_i\left(t\right)+B_k{v}_i\left(t\right)\],i=1,...,N,---\left(4\right)$

Wherein, x_i(t)��R^pIt is the state of node i, u_i(t)��R^qIt is the control inputs of node i, r_i(t)��R^pIt is the reference signal of node i, v_i(t)��R^qIt is the control inputs of node i reference signal, A_k��R^p��pAnd B_k��R^p��qIt is a group system matrix, it is the known constant matrices of design, ��_k(t) (k=1 ..., n_s) it is one group of randomized number, and meet ��_k(t) >=0 andP is the dimension of each node state, and q is the dimension of control inputs, n_sFor the number of linear system, p, q, n_sIt is known constant. Although formula (3) and (4) can regard n as_sIndividual linear system (A_k,B_k) convex combination, but due to time become and unknown value ��_kT the existence of (), institute becomes unknown system when (3) and (4) are with the formula. Design lines sexual system matrix (A_k,B_k) should meet for all i=2 ..., N, (A_k,��_iB_k) can be stable. Choose suitable ��, ��_jklValue, utilize Matlab work box PENlab to solve following Bilinear Inequalities:

$Q_j{A_k}^T+A_k{Q}_j+{\mathrm{\λ}}_i{y_j}^T{B_k}^T+{\mathrm{\λ}}_i{B}_k{Y}_j\≤\underset{l=1}{\overset{n_Q}{\mathrm{\Σ}}}{\mathrm{\η}}_{jkl}(Q_l-Q_j)-{\mathrm{\βQ}}_j,---\left(5\right)$

K=1 ..., n_s, j, l=1 ..., n_Q, i=2 ..., N,

Obtain system parameterY_j��Rq��p��

4th step: the signalling methods that node is set, make its can only with neighbor node communication, fully demonstrate distributed feature, run distributed average tracking algorithm so that all nodes adjust the state of oneself according to the state of current time oneself, the reference signal value of oneself and the state of neighbor node. Herein, communicate, with formula (6) and (7) description node i, the information obtained with its neighbor node respectively and node i utilizes the information of the state of self with reference signal acquisition:

${\mathrm{\Δ}}_{il}=\underset{j=1}{\overset{N}{\Σ}}{a}_{ij}(x_i-x_j),i=1,...,N---\left(6\right)$

��_i2=x_i-r_i, i=1 ..., N (7)

a_ijIt is the element of adjacency matrix, x_iIt is the state of node i, r_iIt is the reference signal of node i, from formula (6) it may be seen that node i by communicating with its neighbor node, can obtain the state of neighbor node, thus obtain oneself state and the difference of neighbor node state, it is used as the foundation of adjustment oneself state. Each node adjusts oneself state by such mode, just can realize consistence.

5th step: according to the difference information �� fed back_i1And ��_i2, and the system parameter obtainedY_j��R^q��pCarry out design control law, adjust all node states and reach consistent and the mean value of tracking reference signal, finally realize the tracking target of following formulation:

$\underset{t\→\mathrm{\∞}}{\lim }\left|\right|x_i\left(t\right)-\frac{1}{N}\underset{j=1}{\overset{N}{\Σ}}{r}_j\left(t\right)\left|\right|=0,i=1,...,N---\left(8\right)$

Wherein, x_i(t)��R^pIt is the state of node i, r_i(t)��R^pIt is the reference signal of node i,It is the mean value of all node reference signals.

The realization of tracking target can be regarded as the realization of two parts of following formulation:

||x_i-x_j||₂�� 0, as t �� �� (9)

As t �� �� (10)

Wherein, x_iIt is the state of node i, r_iIt is the reference signal of node i, formula | | x_i-x_j||₂�� 0 expression realizes state consistency,Represent the state and consistent with reference signal sum of realizing, when state and with reference signal and when being tending towards equal, namely mean that state mean value will be tending towards reference signal mean value, when realizing state consistency, it is possible to obtain each node state and will be tending towards the mean value of reference signal.

The time-varying uncertain system that the present invention can form multiple robot comprises with linear differential and represents, solves the distributed average tracking problem of time-varying uncertain system. Compared to traditional method, the distributed average tracking control algorithm tool that the linear differential that the present invention proposes comprises has the following advantages:

(1) distributed way is adopted, for the strong adaptability of complex system.

(2) requirement of network structure is low, it is easy to realize.

(3) calculating simply, calculated amount is little, and real-time performance is good.

(4) kinetic equation of intelligence body can be used for modeling nonlinear and time-varying system, has representative widely.

(5) between the region of convergence of the algorithm designed by bigger compared to the convergence space of the algorithm designed based on quadratic form Lyapunov function.

The system that linear differential is comprised by the present invention is incorporated into the distributed average tracking control of multiple agent, and how research realizes the control of distributed average tracking based on distributed consensus method, solves the average tracking problem of time-varying uncertain system. The method has used the character of compound Laplce's quadratic function, the object of varying reference signal mean value when utilizing distributed average tracking algorithm to realize following the tracks of. The method is little to the requirement of network structure, only needs network to be that all node states that just can realize being connected reach unanimity and the mean value of tracking reference signal. In whole computation process, each node has only used the information of neighbor node, and calculated amount is little, thus improves the operation efficiency of algorithm largely.

Certainly, the present invention can also have other various embodiments, and when not deviating from technical solution of the present invention, the technician of this area can make various corresponding change.

Claims (8)

1. linear differential comprises the control method of the distributed average tracking of multi-agent system, it is characterised in that comprise the following steps:

Step 1: the network structure topological diagram of structure multiple agent, obtains the adjacency matrix of figure, incidence matrix and Laplce's matrix;

Step 2: the original state that each node is set, and initial reference signal;

Step 3: the system that structure linear differential comprises, arranges system matrix, obtains system parameter;

Step 4: signalling methods is set, make each node can only with neighbor node communication, run distributed average tracking algorithm, adjust the state of each node;

Step 5: carry out design control law according to feedback information and system parameter, knot modification state reaches the mean value of consistence and tracking reference signal.

2. linear differential comprises the control method of the distributed average tracking of multi-agent system as claimed in claim 1, it is characterised in that in step 1, and the network structure topological diagram of described structure multiple agent is Connected undigraph, and number of nodes is N, and limit quantity is M.

3. linear differential comprises the control method of the distributed average tracking of multi-agent system as claimed in claim 1, it is characterised in that in step 1, and adjacency matrix A, the incidence matrix E of figure and Laplce's matrix L are by following formulation:

L=EE^T��R^N��N

Wherein, N is number of nodes, and M is limit quantity, R^N��NFor N �� N ties up real matrix set, R^N��MFor N �� M ties up real matrix set, A is adjacency matrix, a_ij(i, j=1,2 ..., N) and represent node i and the relation of node j, if having limit to be connected between node i with node j, then a_ij=1, otherwise a_ij=0; By giving figure fixing direction, obtaining incidence matrix E is: e_ij(i=1 ..., N, j=1 ..., M) and represent node i and the relation of limit j, if node i is the starting point of limit j, then e_ij=-1, if node i is the terminal of limit j, then e_ij=1, otherwise e_ij=0; L is Laplce's matrix, it is possible to by following formula diagonalization:

L=U^T�� U, �� :=diag ([��₁,��,��_N]), U^TU=UU^T=I

Wherein, U �� R^N��NFor unitary matrix, �� is that on principal diagonal, element is the diagonal matrix of eigenwert, ��_i(i=1 ..., N) and it is the eigenwert of Laplce's matrix L, do not lose generality, order: 0=��₁�ܦ�₂�ܡ��ܦ�_N, I �� R^N��NFor unit battle array.

4. linear differential comprises the control method of the distributed average tracking of multi-agent system as claimed in claim 1, it is characterized in that in step 3, described linear differential comprises the kinetic equation of node state of description and the kinetic equation of reference signal is expressed as follows:

${\stackrel{\·}{x}}_i\left(t\right)=\underset{k=1}{\overset{n_s}{\Σ}}{\mathrm{\τ}}_k\left(t\right)\[A_k{x}_i\left(t\right)+B_k{u}_i\left(t\right)\],i=1,...,N$

${\stackrel{\·}{r}}_i\left(t\right)=\underset{k=1}{\overset{n_s}{\Σ}}{\mathrm{\τ}}_k\left(t\right)\[A_k{r}_i\left(t\right)+B_k{v}_i\left(t\right)\],i=1,...,N$

Wherein, x_i(t)��R^pIt is the state of node i, u_i(t)��R^qIt is the control inputs of node i, r_i(t)��R^pIt is the reference signal of node i, v_i(t)��R^qIt is the control inputs of node i reference signal, A_k��R^p��pAnd B_k��R^p��qIt is a group system matrix, it is the known constant matrices of design, ��_k(t) (k=1 ..., n_s) it is one group of randomized number, and meet ��_k(t) >=0 andP is the dimension of each node state, and q is the dimension of control inputs, n_sFor the number of linear system, p, q, n_sIt is known constant;

Described system matrix (A_k,B_k) should meet for all i=2 ..., N, (A_k,��_iB_k) can be stable, ��_iFor the nonzero eigenvalue of Laplce's matrix L.

5. linear differential comprises the control method of the distributed average tracking of multi-agent system as claimed in claim 1, it is characterised in that in step 4, the difference information �� that each node i described and its neighbor node communication obtain_i1, and the difference information �� that node i obtains according to the difference of oneself state and reference signal_i2, it is expressed as follows:

${\mathrm{\Δ}}_{i1}=\underset{j=1}{\overset{N}{\Σ}}{a}_{ij}(x_i-x_j),i=1,...,N$

��_i2=x_i-r_i, i=1 ..., N

Wherein, a_ijFor the element of adjacency matrix, x_iIt is the state of node i, r_iIt it is the reference signal of node i.

6. linear differential comprises the control method of the distributed average tracking of multi-agent system as claimed in claim 1, it is characterised in that in step 1,3 and 4, comprising: there is positive constant beta > 0, one group of constant ��_jkl> 0, one group of positive definite matrix Q_j=Q_j ^T��R^p��pWith matrix Y_j��R^q��pMake system matrix A_k,B_k(k=1 ..., n_s) meet the Bilinear Inequalities of following formulation:

$Q_j{A_k}^T+A_k{Q}_j+{\mathrm{\λ}}_i{Y_j}^T{B_k}^T+{\mathrm{\λ}}_i{B}_k{Y}_j\≤\underset{l=1}{\overset{n_Q}{\mathrm{\Σ}}}{\mathrm{\η}}_{jkl}(Q_l-Q_j)-{\mathrm{\βQ}}_j,$

K=1 ..., n_s, j, l=1 ..., n_Q, i=2 ..., N,

Wherein, A_k,B_k(k=1 ..., n_s) it is the n designed_sOrganizing linear system matrix, �� is adjustable positive constant, ��_jklIt is one group of adjustable constant, positive definite matrix Q_jWith matrix Y_jIt is unknown quantity, obtains by solving Bilinear Inequalities, n_QFor constant, represent matrix Q_jNumber, ��_i(i=2 ..., N) and it is the nonzero eigenvalue of Laplce's matrix L.

7. linear differential comprises the control method of the distributed average tracking of multi-agent system as claimed in claim 1, it is characterized in that in step 4, comprise: the character that compound Laplce's quadratic function can be used, adopt the control method of distributed average tracking, the state of described each node of adjustment, wherein, compound Laplce quadratic function be constructed as follows shown in formula:

$V\left(x\right):=\underset{\mathrm{\γ}\∈{\mathrm{\Γ}}^{n_Q}}{min}{X}^T\[L\⊗{\left(\underset{j=1}{\overset{n_Q}{\mathrm{\Σ}}}{\mathrm{\γ}}_j{Q}_j\right)}^{-1}\]X$

${\mathrm{\Γ}}^{n_Q}:=\{\mathrm{\γ}={\[{\mathrm{\γ}}_1,...,{\mathrm{\γ}}_{n_Q}\]}^T:{\mathrm{\γ}}_j\≥0,{\mathrm{\Σ}}_{j=1}^{n_Q}{\mathrm{\γ}}_j=1\}$

Wherein: X=[x₁ ^T,x₂ ^T,��,x_N ^T]^T��R^NpFor the vector of all node states represents, x_i��R^p(i=1 ..., N) and it is the state of node i, L is Laplce's matrix of figure, Q_j=Q_j ^T��R^p��p(j=1 ..., n_Q) it is one group of positive definite matrix, n_QFor constant, represent matrix Q_jNumber, ��_j(j=1 ..., n_Q) it is one group of randomized number, and meet For n_QIndividual ��_jVector represent,It is defined as the set of the �� satisfied condition.

8. linear differential comprises the control method of the distributed average tracking of multi-agent system as claimed in claim 1, it is characterized in that in steps of 5, described carrying out design control law according to feedback information and system parameter, knot modification state reaches the mean value of consistence and tracking reference signal, comprising:

(1) tracking target of following formulation is realized:

$\underset{t\→\mathrm{\∞}}{\lim }\left|\right|x_i\left(t\right)-\frac{1}{N}\underset{j=1}{\overset{N}{\Σ}}{r}_j\left(t\right)\left|\right|=0,i=1,...,N$

Wherein, x_i(t)��R^pIt is the state of node i, r_i(t)��R^pIt is the reference signal of node i,It is the mean value of all node reference signals;

(2) realization of tracking target can be regarded as the realization of two parts of following formulation:

||x_i-x_j||₂�� 0, as t �� ��

$\underset{i=1}{\overset{N}{\Σ}}(x_i-r_i)\→0,$ As t �� ��

Wherein, x_iIt is the state of node i, r_iIt is the reference signal of node i, formula | | x_i-x_j||₂�� 0 expression realizes state consistency,Represent the state and consistent with reference signal sum of realizing, when state and with reference signal and when being tending towards equal, namely mean that state mean value will be tending towards reference signal mean value, when realizing state consistency, obtain each node state and will be tending towards the mean value of reference signal.