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

Consistency control method of heterogeneous time-lag multi-agent system based on PI Download PDF

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CN110609469B
CN110609469B CN201910581674.8A CN201910581674A CN110609469B CN 110609469 B CN110609469 B CN 110609469B CN 201910581674 A CN201910581674 A CN 201910581674A CN 110609469 B CN110609469 B CN 110609469B
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马倩
王�琦
钱康
郭飞
李恩扬
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Nanjing University of Science and Technology
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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

Consistency control method of heterogeneous time-lag multi-agent system based on PI
Technical Field
The invention belongs to the field of intelligent control, and particularly relates to a consistency control method of a heterogeneous time-lag multi-agent system based on a PI (proportional integral).
Background
The heterogeneous multi-agent system is widely applied to various fields such as biology, computer communication, robot cooperative control, intelligent transportation and the like, has important significance in researching the consistency of the heterogeneous multi-agent system, and becomes a hotspot in the field of intelligent control. The document (Yuezu Lv, Zhongkui Li, zhishangsheng Duan, Distributed PI control for controlling of heterogeneous multi-agent systems over directed graphs) indicates that the PI control protocol is applied to the linear heterogeneous multi-agent system to realize consistency control, however, only the fixed topology and the non-time-lag linear heterogeneous multi-agent system are controlled, but in practical application, the topology structure of the multi-agent system may change along with the change of time, and due to the limitation of communication bandwidth, the communication between the agents inevitably has time-varying time lag, and the traditional consistency control method based on the PI control protocol is not applicable any more.
Disclosure of Invention
The invention aims to provide a consistency control method of a heterogeneous time-lag multi-agent system based on PI.
The technical solution for realizing the purpose of the invention is as follows: a consistency control method of a heterogeneous time-lag multi-agent system based on PI comprises 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;
and 4, selecting controller parameters of a PI control protocol, performing stability control on the reduced-order system, and realizing consistency of the heterogeneous time-lag multi-agent system.
Compared with the prior art, the invention has the remarkable advantages that: the time-varying time lag of the multi-agent system and the characteristic that the topological structure is switching are considered, and the control is more suitable for practical application.
Drawings
FIG. 1 is a flow chart of a consistency control method of a PI-based heterogeneous time-lag multi-agent system according to the present invention.
FIG. 2 is a switching topology structure diagram of the heterogeneous skew multi-agent system of the present invention.
FIG. 3 shows the state x of the heterogeneous time-lag multi-agent system of the present inventioni1(t) graph.
FIG. 4 shows the state v of the heterogeneous time-lag multi-agent system of the present inventioni1(t) graph.
FIG. 5 shows a heterogeneous time-lag multi-agent system state x according to the present inventioni2(t) graph.
FIG. 6 is a heterogeneous time-lapse multi-agent system of the present inventionState vi2(t) graph.
Detailed Description
The invention is further illustrated by the following examples in conjunction with the accompanying drawings.
The invention provides a consistency control method based on PI (proportional integral) aiming at a heterogeneous time-lag multi-agent system, which converts an original multi-agent system into a reduced-order system through reduced-order change, converts the consistency problem of the original system into the stability problem of the reduced-order system, stabilizes the reduced-order system by selecting proper PI control protocol parameters, and completes the consistency control of the heterogeneous time-lag multi-agent system, as shown in figure 1, the method comprises the following four steps:
step 1, constructing a mathematical model of a heterogeneous time-lag multi-agent system;
if a heterogeneous multi-agent system is composed of n (n is more than or equal to 2) agents, wherein m (m is more than or equal to 1) agents are first-order integrator models, n-m (n-m is more than or equal to 1) agents are second-order integrator models, the dynamic characteristic model of each agent is as follows:
Figure BDA0002113338360000021
wherein, In＝{1,2,…,n}，Im＝{1,2,…,m}，In/ImWhere { m +1, m +2, …, n } represents an index set in which the agent is located, and xi(t)∈RN，vi(t)∈RN，ui(t)∈RNRespectively representing the position state, the speed state and the control protocol of the ith agent, and all the position state, the speed state and the control protocol are N-dimensional vectors.
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 an adjacency matrix and a Laplace matrix of the system;
the topology of the multi-agent system is represented by directed graph G ═ (V, E, a), where V ═ {1,2, …, n } represents each agent,
Figure BDA0002113338360000022
representing each intelligenceCommunication between bodies (a)ij)N×NRepresenting the adjacency matrix if (i, j) ∈ E, aij1, otherwise, aij0. The laplace matrix is defined as: when the value of i is equal to j,
Figure BDA0002113338360000023
when i ≠ j, lij＝-aij。
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;
the PI control protocol designed for the ith agent is:
Figure BDA0002113338360000031
wherein, α, β, kiMore 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,
Figure BDA0002113338360000032
aij(t) an adjacency matrix of the corresponding topology at time t;
order to
Figure BDA0002113338360000033
Substituting a control protocol (2) into a multi-agent system (1), the whole heterogeneous time-lag multi-agent system being represented as:
Figure BDA0002113338360000034
where σ (t) [ [0, + ∞) → S ═ 1,2, …, S } is the switching signal, S is the number of all possible topologies,
Figure BDA0002113338360000035
Figure BDA0002113338360000036
denotes the Laplace matrix, L, under the switching signal σ1σ∈Rm×n，L2σ∈R(n-m)×n，K＝diag{km+1,km+2,…,kn}；
By a reduced order change, i.e.
Figure BDA0002113338360000037
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:
Figure BDA0002113338360000038
wherein,
Figure BDA0002113338360000041
Figure BDA0002113338360000042
E＝[-1m-1 Im-1]，F＝[0n-1 In-1]T，mσ＝[a12σ,a13σ,…,a1nσ]T，Mσ＝[mσ,mσ,…,mσ]T∈R(n-m)×(n-1)。
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;
according to the Lyapunov stability theory, a Lyapunov function V (t) is constructed for a reduced-order system to meet the requirement
Figure BDA0002113338360000043
Namely, the stability of the order-reduced system is realized;
the constructed Lyapunov function is:
Figure BDA0002113338360000044
p, Q, R are positive definite matrixes;
are respectively paired with V1(t)，V2(t)，V3(t) derivation, we can obtain:
Figure BDA0002113338360000045
Figure BDA0002113338360000046
Figure BDA0002113338360000051
by integrating the formulae (6) to (8), it is possible to obtain:
Figure BDA0002113338360000052
wherein eta isT(t)＝[yT(t) yT(t-τ(t))]；
The conditions for realizing the stability of the order-reducing system are as follows:
Figure BDA0002113338360000053
as can be seen from the supplementary theorem of matrix Schur, the above formula is equivalent to:
Figure BDA0002113338360000054
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
Figure BDA0002113338360000061
If so, then the system can be consistent.
Examples
Consider a heterogeneous multi-agent system consisting of 6 agents, where agents 1,2, 3 are first-order agents and agents 4, 5, 6 are second-order agents. Handover topology as shown in fig. 2, handover is performed in 4 topologies. From topology GaInitially, switch to the next topology every 0.1s, as per Ga→Gb→Gc→Gd→GaThe order of (2) is switched. The parameters of a given control protocol are α -0.2, β -0.01, k1＝2，k2＝1，k 33, when τ (t) is 0.03| cos (10t) |, the state of each agent is 2-dimensional, and the initial value of the state of each agent in the system is given as 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]TThe system (1) is under the action of the control protocol (2) and the state curve x of each agenti1(t)，vi1(t) and xi2(t)，vi2(t) are shown in FIGS. 3 and 4, respectively. It can be seen from fig. 3 and 4 that under the action of the control protocol with the integral term, the position and speed states of each agent in the heterogeneous time-lag multiple intelligent system tend to be the same, and the speed is kept 0, i.e. the consistency is gradually realized. The final position state eliminates settling 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:
Figure FDA0003524872030000011
wherein, In＝{1,2,…,n}，Im＝{1,2,…,m}，In/ImWhere { m +1, m +2, …, n } represents an index set in which the agent is located, and xi(t)∈RN，vi(t)∈RN，ui(t)∈RNRespectively 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:
Figure FDA0003524872030000012
wherein, α, β, kiMore 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,
Figure FDA0003524872030000013
aij(t) is the adjacency matrix of the corresponding topology at time t, xi(t)∈RNIndicating the position state of the ith agent;
order to
Figure FDA0003524872030000014
The whole heterogeneous time-lag multi-agent system is represented as:
Figure FDA0003524872030000021
where σ (t) [0, + ∞) → S ═ 1,2, …, S is the switching signal, S is the number of all possible topologies,
Figure FDA0003524872030000022
Figure FDA0003524872030000023
denotes the Laplace matrix, L, under the switching signal σ1σ∈Rm×n，L2σ∈R(n-m)×n，K＝diag{km+1,km+2,…,kn}；
By a reduced order change, i.e.
Figure FDA0003524872030000024
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:
Figure FDA0003524872030000025
wherein,
Figure FDA0003524872030000026
Figure FDA0003524872030000027
E＝[-1m-1 Im-1]，F＝[0n-1 In-1]T，mσ＝[a12σ,a13σ,…,a1nσ]T，Mσ＝[mσ,mσ,…,mσ]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
Figure FDA0003524872030000028
Namely, the stability of the reduced-order system is realized, and the constructed Lyapunov function is as follows:
Figure FDA0003524872030000029
p, Q, R are positive definite matrixes;
are respectively paired with V1(t)，V2(t)，V3(t) derivation, we can obtain:
Figure FDA0003524872030000031
Figure FDA0003524872030000032
Figure FDA0003524872030000033
by integrating the formulae (6) to (8), it is possible to obtain:
Figure FDA0003524872030000034
wherein eta isT(t)＝[yT(t) yT(t-τ(t))]；
The conditions for realizing the stability of the order-reducing system are as follows:
Figure FDA0003524872030000035
as can be seen from the supplementary theorem of matrix Schur, the above formula is equivalent to:
Figure FDA0003524872030000041
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
Figure FDA0003524872030000042
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,
Figure FDA0003524872030000043
representing communication between agents, (a)ij)N×NRepresenting the adjacency matrix if (i, j) ∈ E, aij1, otherwise, aijThe laplacian matrix is defined as 0: when the value of i is equal to j,
Figure FDA0003524872030000044
when i ≠ j, lij＝-aij。
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