CN110609469B - Consistency control method of heterogeneous time-lag multi-agent system based on PI - Google Patents

Abstract

The invention provides a consistency control method of a heterogeneous time-lag multi-agent system based on PI (proportion integration), which is used for constructing a mathematical model of the heterogeneous time-lag multi-agent system; analyzing information exchange relation among all agents in the multi-agent system, constructing a topological structure of the multi-agent system by using a directed graph, and determining a Laplace matrix of the system; constructing a PI control protocol of each agent, and converting the original heterogeneous time-lag multi-agent system into a reduced-order system through reduced-order change; and selecting controller parameters of a PI control protocol to perform stability control of the reduced-order system, thereby realizing the consistency of the heterogeneous time-lag multi-agent system. The invention considers the time-varying time lag of the heterogeneous multi-agent system and the characteristic that the topological structure is switched, and is more suitable for practical application.

Description

Consistent control method for heterogeneous time-delay multi-agent systems based on PI

technical field

The invention belongs to the field of intelligent control, in particular to a consistency control method of a PI-based heterogeneous time-delay multi-agent system.

Background technique

Heterogeneous multi-agent systems are widely used in various fields such as biology, computer communication, robot cooperative control, and intelligent transportation. Literature (Yuezu Lv, Zhongkui Li, Zhisheng Duan, Distributed PI control for consensus of heterogeneous multiagent systems over directed graphs) pointed out that applying PI control protocol to linear heterogeneous multi-agent systems to achieve consensus control, however, only for fixed topology and However, in practical applications, the topology of the multi-agent system may change with time, and due to the limitation of communication bandwidth, the communication between agents is unavoidable. There will be a time-varying time delay, and the traditional consistency control method based on the PI control protocol is no longer applicable.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a consistency control method of a PI-based heterogeneous time-delay multi-agent system.

The technical solution for realizing the purpose of the present invention is: a PI-based consistency control method for a heterogeneous time-delay multi-agent system, comprising the following steps:

Step 1. Construct the mathematical model of the heterogeneous time-delay multi-agent system;

Step 2, analyzing the information exchange relationship between the agents in the multi-agent system, using the directed graph to construct the topology structure of the multi-agent system, and determining the Laplacian matrix of the system;

Step 3. Construct the PI control protocol of each agent, and convert the original heterogeneous time-delay multi-agent system into a reduced-order system by reducing the order change;

Step 4: Select the controller parameters of the PI control protocol, and perform the stability control of the reduced-order system, so as to realize the consistency of the heterogeneous time-delay multi-agent system.

Compared with the prior art, the present invention has the significant advantages that the time-varying time delay of the qualitative multi-agent system and the characteristic of switching the topology structure are considered, and the control is more suitable for practical applications.

Description of drawings

FIG. 1 is a flow chart of the consistency control method of the PI-based heterogeneous time-delay multi-agent system of the present invention.

FIG. 2 is a switching topology diagram of the heterogeneous time-delay multi-agent system of the present invention.

FIG. 3 is a graph of the state x _i1 (t) of the heterogeneous time-delay multi-agent system of the present invention.

FIG. 4 is a graph of the state v _i1 (t) of the heterogeneous time-delay multi-agent system of the present invention.

FIG. 5 is a graph of the state x _i2 (t) of the heterogeneous time-delay multi-agent system of the present invention.

FIG. 6 is a graph of the state v _i2 (t) of the heterogeneous time-delay multi-agent system of the present invention.

Detailed ways

The solution of the present invention will be further described below with reference to the accompanying drawings and specific embodiments.

The invention proposes a PI-based consistency control method for the heterogeneous time-delay multi-agent system, which transforms the original multi-agent system into a reduced-order system through order-reduced changes, and converts the consistency problem of the original system into a reduced-order system The stability problem of , by selecting the appropriate PI control protocol parameters to make the reduced-order system stable, complete the consistency control of the heterogeneous time-delay multi-agent system, as shown in Figure 1, which is divided into the following four steps:

Step 1. Construct the mathematical model of the heterogeneous time-delay multi-agent system;

Suppose a heterogeneous multi-agent system consists of n (n≥2) agents, of which m (m≥1) are first-order integrator models, and n-m (n-m≥1) are second-order integrator models, then The dynamic characteristic model of each agent is:

Among them, I _n ={1,2,...,n}, I _m ={1,2,...,m}, I _n /I _m ={m+1,m+2,...,n} represents the agent The index set where x _i (t)∈R ^N , v _i (t)∈R ^N , and _ui (t)∈R ^N represent the position state, speed state and control protocol of the ith agent, respectively. is an N-dimensional vector.

Step 2, analyze the information exchange relationship between the agents in the multi-agent system, use the directed graph to construct the topology structure of the multi-agent system, and determine the adjacency matrix and the Laplacian matrix of the system;

表示各智能体间的通信，(a_ij)_N×N表示邻接矩阵，如果(i,j)∈E，a_ij＝1，否则，a_ij＝0。拉普拉斯矩阵定义为：当i＝j时，

当i≠j时，l_ij＝-a_ij。The topological structure of a multi-agent system is represented by a directed graph G=(V,E,A), where V={1,2,...,n} represents each agent,

represents the communication between agents, (a _ij ) _N×N represents the adjacency matrix, if (i,j)∈E, a _ij =1, otherwise, a _ij =0. The Laplace matrix is defined as: when i=j,

When i≠j, l _ij =-a _ij .

Step 3. Construct the PI control protocol of each agent, and convert the original heterogeneous time-delay multi-agent system into a reduced-order system by reducing the order change;

The PI control protocol designed for the ith agent is:

a_ij(t)是在t时刻对应的拓扑结构的邻接矩阵；Among them, α, β, k _i >0 are the design parameters of the controller, 0≤τ(t)<d is the time delay,

a _ij (t) is the adjacency matrix of the topological structure corresponding to time t;

将控制协议(2)代入多智能体系统(1)，整个异质时滞多智能体系统表示为：make

Substituting the control protocol (2) into the multi-agent system (1), the entire heterogeneous time-delay multi-agent system is expressed as:

表示在切换信号σ下的拉普拉斯矩阵，L_1σ∈R^m×n，L_2σ∈R^(n-m)×n，K＝diag{k_m+1,k_m+2,…,k_n}；where σ(t):[0,+∞)→S={1,2,…,s} is the switching signal, s is the number of all possible topologies,

represents the Laplacian matrix under the switching signal σ, L _1σ ∈ R ^m×n , L _2σ ∈ R ^(nm)×n , K=diag{km ₊₁ ,km ₊₂ ,…,k _n } ;

将原异质时滞多智能体系统转为降阶系统，原异质时滞多智能体系统的一致性问题转化为降阶系统的稳定性问题：By reducing the order changes, that is

The original heterogeneous delay multi-agent system is transformed into a reduced-order system, and the consistency problem of the original heterogeneous delay multi-agent system is transformed into the stability problem of the reduced-order system:

E＝[-1_m-1 I_m-1]，F＝[0_n-1 I_n-1]^T，m_σ＝[a_12σ,a_13σ,…,a_1nσ]^T，M_σ＝[m_σ,m_σ,…,m_σ]^T∈R^(n-m)×(n-1)。in,

E=[-1 _m-1 I _m-1 ], F=[0 _n-1 I _n-1 ] ^T , m _σ =[a _12σ ,a _13σ ,...,a _1nσ ] ^T , M _σ =[m _σ ,m _σ ,…,m _σ ] ^T ∈R ^(nm)×(n-1) .

Step 4, selecting the controller parameters of the PI control protocol, and performing the stability control of the reduced-order system to realize the consistency of the heterogeneous time-delay multi-agent system;

即实现降阶系统的稳定性；According to the Lyapunov stability theory, the Lyapunov function V(t) is constructed for the reduced-order system, satisfying

That is, to achieve the stability of the reduced-order system;

The constructed Lyapunov function is:

Among them, P, Q, R are all positive definite matrices;

Taking the derivative of V ₁ (t), V ₂ (t), and V ₃ (t) respectively, we can get:

Integrating equations (6)-(8), we can get:

Wherein, η ^T (t)=[y ^T (t) y ^T (t-τ(t))];

The conditions for achieving the stability of the reduced-order system are:

According to the matrix Schur complement lemma, the above formula is equivalent to:

满足，那么系统能够一致。To sum up, for the heterogeneous time-delay multi-agent system, under the action of the PI control protocol, if there are positive definite matrices P, Q, R, the linear matrix inequality

Satisfaction, then the system can be consistent.

Example

Consider a heterogeneous multi-agent system composed of 6 agents, in which agents 1, 2, and 3 are first-order agents, and agents 4, 5, and 6 are second-order agents. The switching topology is shown in Figure 2, and switching is performed in 4 topological structures. Starting from the topology map Ga, switch to the next topology map every 0.1s, and switch in the order of _Ga →G _b →G _c →G _d →G _{a .} Given the parameters of the control protocol as α=0.2, β=0.01, k ₁ =2, k ₂ =1, k ₃ =3, when τ(t)=0.03|cos(10t)|, the The states are all 2-dimensional, and the initial value of the state of each agent in the given system is z(0)=[3,4,2,-2,5,3,4,6,-3,2,-2 ,-4] ^T , x(0)=[-4,3,-3,4,1,-2,3,5,-3,1,5,-2] ^T ,v(0)=[2 ,3,5,-2,6,4] ^T , under the action of the control protocol (2) of the system (1), the state curves of each agent x _i1 (t), v _i1 (t) and x _i2 (t ), v _i2 (t) are shown in Figures 3 and 4, respectively. It can be seen from Fig. 3 and Fig. 4 that under the control protocol with integral term, the position and velocity state of each agent in the heterogeneous time-delay multi-agent system tend to be the same, and the velocity remains 0, that is, the asymptotic realization agreed. The final position state eliminates stabilization errors due to the integral term in the control protocol.

Claims (2)

1. A consistency control method of a heterogeneous time-lag multi-agent system based on PI is characterized by comprising the following steps:

step 1, constructing a mathematical model of a heterogeneous time-lag multi-agent system;

step 2, analyzing information exchange relation among all agents in the multi-agent system, constructing a topological structure of the multi-agent system by using a directed graph, and determining a Laplace matrix of the system;

step 3, constructing a PI control protocol of each agent, and converting the original heterogeneous time-lag multi-agent system into a reduced-order system through reduced-order change;

step 4, selecting controller parameters of a PI control protocol, performing stability control on a reduced-order system, and realizing consistency of a heterogeneous time-lag multi-agent system;

in step 1, a heterogeneous multi-agent system is set to be composed of n, n is more than or equal to 2 agents, wherein m is more than or equal to 1, and is a first-order integrator model, n-m is more than or equal to 1, and is a second-order integrator model, and then the dynamic characteristic model of each agent is as follows:

wherein, I_n＝{1,2,…,n}，I_m＝{1,2,…,m}，I_n/I_mWhere { m +1, m +2, …, n } represents an index set in which the agent is located, and x_i(t)∈R^N，v_i(t)∈R^N，u_i(t)∈R^NRespectively representing the position state, the speed state and the control protocol of the ith agent, wherein the position state, the speed state and the control protocol are N-dimensional vectors;

in step 3, the PI control protocol designed for the ith agent is:

wherein, α, β, k_iMore than 0 is the design parameter of the controller, 0 is more than or equal to tau (t) and less than d is the time delay,

a_ij(t) is the adjacency matrix of the corresponding topology at time t, x_i(t)∈R^NIndicating the position state of the ith agent;

order to

The whole heterogeneous time-lag multi-agent system is represented as:

where σ (t) [0, + ∞) → S ═ 1,2, …, S is the switching signal, S is the number of all possible topologies,

denotes the Laplace matrix, L, under the switching signal σ_1σ∈R^m×n，L_2σ∈R^(n-m)×n，K＝diag{k_m+1,k_m+2,…,k_n}；

By a reduced order change, i.e.

The original heterogeneous time-lag multi-agent system is converted into a reduced-order system, and the consistency problem of the original heterogeneous time-lag multi-agent system is converted into the stability problem of the reduced-order system:

wherein,

E＝[-1_m-1 I_m-1]，F＝[0_n-1 I_n-1]^T，m_σ＝[a_12σ,a_13σ,…,a_1nσ]^T，M_σ＝[m_σ,m_σ,…,m_σ]^T∈R^(n-m)×(n-1)；

in step 4, according to the Lyapunov stability theory, a Lyapunov function V (t) is constructed for the order-reduced system to meet the requirements

Namely, the stability of the reduced-order system is realized, and the constructed Lyapunov function is as follows:

p, Q, R are positive definite matrixes;

are respectively paired with V₁(t)，V₂(t)，V₃(t) derivation, we can obtain:

by integrating the formulae (6) to (8), it is possible to obtain:

wherein eta is^T(t)＝[y^T(t) y^T(t-τ(t))]；

The conditions for realizing the stability of the order-reducing system are as follows:

as can be seen from the supplementary theorem of matrix Schur, the above formula is equivalent to:

in summary, for the heterogeneous skewed multi-agent system, under the action of the PI control protocol, if the positive definite matrix P, Q, R exists, the linear matrix inequality is made

If so, then the system can be consistent.

2. The method for controlling consistency of a PI-based heterogeneous lag multi-agent system according to claim 1, wherein in step 2, the topology of the multi-agent system is represented by a directed graph G ═ (V, E, a), where V ═ 1,2, …, n represents each agent,

representing communication between agents, (a)_ij)_N×NRepresenting the adjacency matrix if (i, j) ∈ E, a_ij1, otherwise, a_ijThe laplacian matrix is defined as 0: when the value of i is equal to j,

when i ≠ j, l_ij＝-a_ij。

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