CN113110058B - Dichotomous practical consistency control method for multi-agent system with limited communication - Google Patents

Dichotomous practical consistency control method for multi-agent system with limited communication Download PDF

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CN113110058B
CN113110058B CN202110452318.3A CN202110452318A CN113110058B CN 113110058 B CN113110058 B CN 113110058B CN 202110452318 A CN202110452318 A CN 202110452318A CN 113110058 B CN113110058 B CN 113110058B
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陈世明
姜根兰
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East China Jiaotong University
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Abstract

A dichotomous practical consistency control method for a multi-agent system with limited communication comprises the following steps: (1) based on the structural balance topology assumption, the dichotomous practical consistency problem is converted into the general practical consistency problem by carrying out standard transformation on a class of orthogonal matrixes; (2) the method comprises the steps that a framework of a differential inclusion theory and a Phillips solution is utilized to obtain a global solution of the multi-agent system under the condition that the right end of a controller is discontinuous; (3) constructing a Lyapunov function based on a quantizer, deducing the stability condition of the multi-agent system by utilizing a Lyapunov stability theory and an algebraic graph theory, and deriving an error convergence upper bound value which is irrelevant to any global information and an initial value; (4) a right-end discontinuous control instrument of a quantizer based on fusion time lag items and rounding functions is designed, so that the position states of all the agents are converged to controllable intervals with the same mode but different symbols, and therefore the two-division practical consistency of a multi-agent system with limited communication is achieved.

Description

Dichotomous practical consistency control method for multi-agent system with limited communication
Technical Field
The invention relates to a dichotomous practical consistency control method of a multi-agent system with limited communication, belonging to the technical field of multi-agents.
Background
The multi-agent system control is developed rapidly in recent years and is widely applied to the fields of unmanned aerial vehicle formation, sensor networks, mechanical arm assembly, multi-missile combined attack and the like. Consistency is taken as the basis of cooperative control and becomes a core problem in multi-agent system research. In recent years, researchers have conducted extensive research on the problem of consistency among different types of multi-agent systems, such as perfect consistency, leader-follower consistency, group consistency, proportion consistency, and the like. The above consistency achievement mainly focuses on designing the controller to make the state error of the intelligent agent finally approach zero. However, in an actual system, due to the deviation of the actuator, the calculation error and the harsh environment, the intelligent system may have situations of communication constraints (such as communication time lag, data quantization, and the like), external interference, unknown coupling, and the like, and if the deviation between the true motion states of each intelligent body is still required to be close to zero, the situation is often difficult to implement under limited conditions. Therefore, Dong X W, Xi J X, Shi Z Y, and Zhong Y S (Practical consistency for high-order linear time-innovative systems with interaction uncertainties, time-varying delay and external dimensions, International Journal of systems Science,2013) propose a Practical consistency concept that makes the state deviation function of the smart object fluctuate within a certain bounded interval, and can be applied to more complex Practical systems.
In order to solve the consistency problem in different non-ideal network environments, students have conducted deep research on multi-agent control algorithms based on practical consistency concepts. Dong X W, Xi J X, Shi Z Y, Zhong YS (Practical consistency for high-order linear time-innovative systems with interaction uncertainties, time-varying delays and external disturbances, International Journal of systems Science,2013) studied the general high-order linear time-invariant system Practical consistency problem with external disturbances, interaction uncertainty and time-varying lag in directed communication topologies. In an actual system, when the sampling period is increased, if the state error of the intelligent object tends to zero, the initial state needs to be very close, and obviously, the effectiveness of the system is limited. Bernuau E, Moulay E, Coirault P, Lsfula F (Practical Consensis of Homogeneous samples-Data Multi systems, IEEE Transactions on Automatic Control,2019) further discusses the Practical consistency of the second-order multi-agent system under the condition of Homogeneous sampling. In order to solve the problem that the performance of the system is weakened due to overlarge initial control input quantity and input saturation, Ning B, Han Q L and Zuo Z Y (Practical fixed-time sensors for integration-type multi-agent systems, A time base generator for Automation, 2019) utilize a time base generator and provide a fixed-time Practical consistency framework of a multi-agent system. It is also difficult to achieve a final null for the state error of the agent in a system with an oscillator, and therefore, Patterey E, Loria A, Elati A (Practical dynamic sensors of Stuart-Landau oscillators over terrestrial networks. International Journal of Control,2020) investigated the Practical dynamic consistency of the nonlinear inhomogeneous oscillator Stuart-Landa on heterogeneous networks. Zhang W, Ma ZJ, Wang Y. (Practical presentations of Leader-following Multi-agent systems with Unknown Coupling weights, acta Automatica Sinica,2018) investigated the Practical consistency of Leader-following Multi-agent systems with Unknown Coupling weights. Zhai S D, Liu P, Gao H (bound biparate synchronization for coupled discrete systems under anti-inflammatory. acta automatic Sinica,2020) aims at a coupled discrete system containing countermeasure relationships and time-varying topologies, considers the situation that two subsystem members with unbalanced structures or balanced structures change with time after topology switching, and realizes Bounded bidirectional synchronization. In real systems, however, the communication topology in which cooperation and competition exist simultaneously is more common.
Altafini C. (Consenssus publications on Networks With interactive interactions, IEEE Transactions on Automatic Control,2013) sets communication topology weight as negative to represent competition among agents, proposes a structural balance diagram hypothesis, and proves that the intelligent system can realize dichotomy consistency by using a Laplacian operator, namely, the intelligent agents converge to a value With the same modulus but different symbols. Meng Z Y, Shi G D, Johansson K H, Cao M, Hong Y G. (Behavors of networks with interactive interactions and switching strategies. Automatica,2016) discuss the sufficient condition for an intelligent system to achieve dichotomy in the case of unidirectional and bidirectional communications, respectively. Meng D Y, Meng Z Y, Hong Y G. (uniformity conversion for a signed network under direct switching techniques. Automatica,2018) establishes a connection between an intelligent body steady state and a directed switching symbol graph Laplace matrix. Subsequently, another scholars introduced competition and cooperation relationships to the matrix-weighted network. Pan L, Shao H, Mesbahi M, Li D (Bipartite consensives on matrix-valued weighted networks. IEEE Transactions on Circuits and Systems II: Express Briefs,2018) introduces the concept of positive and negative spanning trees, and obtains the sufficient condition for realizing dichotomy consistency of matrix weighting network, but only applies to the structural equilibrium diagram. Su H S, Chen J H, Yang Y C (The binary Systems for Multi-Agent Systems With Matrix-Weight-Based Signal networks IEEE Transactions on Circuits and Systems II: Express Briefs,2019) studies The dichotomy of Matrix weighted networks under The topology of structural imbalance, and through Matrix coupling, obtains The algebraic conditions for realizing dichotomy. Most of the above dichotomous consistency studies consider that errors between agents can eventually go to zero, but this is often difficult to achieve under the limited conditions of non-ideal communication environments.
The concept of quantitative consistency was first proposed by Akshay K, Tamer B, R (srikant. Quantized communication. automation, 2007), and then dimalogonasa D V, Johansson K H (Stability analysis for multi-agent systems using the reliability mark: Quantized communication and format control. automation, 2010) studied the consistency problem of the multi-agent system of communication information under both consistent quantization and logarithmic quantization based on the matrix spectrum theory, respectively. A hybrid system model of a multi-agent network based on hysteresis effect quantification is constructed by Ceragioli F, Claudio D P and Paolo F (diagnostics and hysteris in qualified average consensus. Automatic,2011), and the hybrid system can effectively avoid the chattering phenomenon and further analyze the limited time convergence of the system solution. Zhu Y R, Li S L, Ma J Y, Zheng Y S (double Consendings in Networks of Agents With innovative Interactions and quantification, IEEE Transactions on circuits and systems-express bridges, 2018) consider the binary consistency of the multi-agent system under the competition relationship and the communication quantification, and consider two situations of topological structure balance and topological structure imbalance respectively. Wu J, Deng Q, Han T, Yan H C (Distributed binary transmitting systems of nonlinear multi-agent systems with quantized communication. neural generating, 2020) further considers nonlinear systems and researches the problem of Distributed dichotomy of nonlinear multi-agent systems with quantized communication constraints. The above papers have made a lot of studies on communication quantification, but none of them considers communication time lag, which is also an important factor affecting the consistency of multi-agent system.
Disclosure of Invention
The invention aims to provide a communication-limited two-dimensional practical consistency control method for a multi-agent system in order to realize the convergence of the position state of an agent to a controllable interval with the same mode and different symbols.
The technical scheme of the invention is that a communication-limited two-dimensional practical consistency control method for a multi-agent system comprises the following steps: (1) based on the structural balance topology hypothesis of the multi-agent system, the multi-agent system with the competitive relationship is converted into a system with nonnegative connection weight by carrying out standard conversion through a class of orthogonal matrixes, so that the dichotomous practical consistency problem is converted into a general practical consistency problem;
(2) setting a quantization function to obtain a Phillips solution; obtaining a global solution of the multi-agent system under the condition that the right end of the controller is discontinuous by utilizing a differential frame comprising theory and Phillips solution and through a kinetic equation of the agent i;
(3) constructing a Lyapunov function based on a quantizer, deducing the stability condition of the multi-agent system by utilizing a Lyapunov stability theory and an algebraic graph theory, and deriving an error convergence upper bound value which is irrelevant to any global information and an initial value;
(4) designing a right-end discontinuous control protocol of a quantizer based on fusion time-lag terms and rounding functions to make the position states of all intelligent positions converge to controllable intervals with the same modulus and different symbols; therefore, the dichotomous practical consistency of the multi-agent system with limited communication is realized.
The communication topology of the multi-agent system is described by directed graph G ═ (V, E, a), where V ═ 1,2, …, n is a set of points,
Figure BDA0003039273420000051
as a set of edges, having directed edges eijThe state that the agent i transmits information to the agent j is represented by (i, j) epsilon E; a ═ aij]∈Rn×nIs a communication weight matrix, the diagonal element a of whichii0; if aijIf not equal to 0, the agent j is called as a neighbor of the agent i; for any i, j has aijIf the value is more than or equal to 0, the graph G is called a non-negative graph, otherwise, the graph G is a symbolic graph; the degree matrix D belongs to Rn×nIs defined as D ═ diag { DiTherein of
Figure BDA0003039273420000052
Laplace matrix L ∈ Rn×nIs defined as L ═ D-A, where
Figure BDA0003039273420000053
And is
Figure BDA0003039273420000054
If any two nodes in the graph have a directed path, the graph G is called a directed strong-communication graph.
The kinetic equation of the agent i is as follows:
Figure BDA0003039273420000055
xi(t) E.R represents the position state variable of agent i; u. ofi(t) E.R denotes agent iA control input of (2);
setting a quantization function
Figure BDA0003039273420000056
Gamma is a quantization level parameter, and eta belongs to (0, 1)]For quantizer accuracy, τ is the communication delay that occurs when agent j communicates to agent i,
Figure BDA0003039273420000057
is a rounded down function; a local solution exists on [0, T) of the system, and according to the theory of functional differential equations, the boundedness of the local solution is obtained through a back-up method, and the global solution of the system is further obtained.
The orthogonal matrix is: c ═ diag (σ), σ ═ σ [ σ ]1,σ2,…,σn],σi-1,1} }; c satisfies CTC＝CCTIs equal to I, and C-1C; for a multi-agent system with fixed undirected network topology and zero communication delay, a control protocol is often adopted
Figure BDA0003039273420000061
And (3) carrying out standard transformation on the data:
let z be Cx, C be C, and C be-1C, x is Cz, then
Figure BDA0003039273420000062
Wherein L isDD-CAC is a normalized post-transform laplacian matrix;
Figure BDA0003039273420000063
ld,ijrepresenting the canonical transformed Laplace matrix LDThe ith row and jth column of elements, ldFinger matrix LDI and j represent the elements in the ith row and the jth column;
c is an orthogonal matrix; d is a degree matrix; i is an identity matrix; σ is a diagonal element in the orthogonal matrix C; l is a Laplace matrix; x is the state matrix X of the agent＝[x1 x2 L xn](ii) a And Z is a transition transformation matrix Z ═ CX.
And setting a metering function to obtain a Phillips solution, and obtaining a global solution of the multi-agent system under the condition that the right end of the controller is discontinuous:
when the function x (T) [ -tau, T) → RnSatisfies the following conditions:
(1) x (T) is continuous at [ - τ, T) and absolutely continuous at [0, T);
(2) presence of a measurable vector function w (t) ═ w1(t),w2(t),…,wn(t)]、
w′(t)＝[w′1(t),w′2(t),…,w′n(t)]：[-τ，T)→RnSuch that x (T) has at T ∈ [0, T):
Figure BDA0003039273420000064
wherein wi(t)∈K[sign(aij)Q(xi(t))]、w′i(t)∈K[Q(xi(t))]Is the output function for the solution x (T), then x (T) is the system's philippif solution at [0, T);
given a continuous vector function phi (t) [ -tau, 0]→RnWith a measurable vector function ψ (t) ∈ K [ Q (phi (t))]:[-τ,0]→Rn、ψ′(t)∈K[sign(aij)Q(φ(t))]:[-τ,0]→RnIf it is determined that
Figure BDA0003039273420000071
Such that x (T) is the solution of the system at [0, T) and satisfies:
Figure BDA0003039273420000072
then, [ x (t), w' (t)]:[-τ,T)→Rn×RnA solution satisfying the initial conditions (phi (t), phi' (t)) for the system;
the local solution of the system on [0, T) is easily obtained; and according to the theory of functional differential equations, obtaining the boundedness of a local solution by a back-off method, and further obtaining a global solution of the system.
The lyapunov function is constructed as follows:
Figure BDA0003039273420000073
let a1＝max1≤i≤j≤ncij，
Figure BDA0003039273420000074
Figure BDA0003039273420000075
So that
Figure BDA0003039273420000076
ρ∈[0,τ]；
Figure BDA0003039273420000077
Thus, there are:
Figure BDA0003039273420000078
the following can be obtained:
Figure BDA0003039273420000081
to pair
Figure BDA0003039273420000082
All agents in the system can converge to set E at t ",
E＝{|x(t+λ)|∈C([-τ,0)；Rn):(wi(t)-w′j(t-τ))2+(w′j(t)-wi(t-τ))2≤(γ/N+1)2h, in turn, composed of
Figure BDA0003039273420000083
The following can be obtained:
Figure BDA0003039273420000084
wherein theta is1∈Z；
Any agent i to its neighbor agents i', i*With no more than n-1 pathways, in combination with the above analysis, one can obtain:
Figure BDA0003039273420000085
thus, pair
Figure BDA0003039273420000086
Figure BDA0003039273420000087
Making state x of any one agent ii(t″)∈[(θ-0.5)γη,(θ+0.5)γη]；
To pair
Figure BDA0003039273420000088
Comprises the following steps:
dist(xi(t),[(θ-0.5)γη,(θ+0.5)γη])≤ε
this can give the state x for any agent ii(t) all converge to a set [ (theta-0.5) gamma eta, (theta +0.5) gamma eta]Performing the following steps;
this yields an upper bound on error convergence:
Figure BDA0003039273420000089
in the formula, V is a Lyapunov function; v1Is a sub-function in V; v2Is another sub-function in V; q (t) is a quantization function; c. CijIs the ith row and the jth column element in the orthogonal matrix C; t is0A constant greater than zero; rho is equal to 0, tau](ii) a Tau is from agent j to agent iCommunication delay occurring at the time of transmission of a message; epsilon is any number greater than zero,
Figure BDA0003039273420000091
e is a convergence set; λ is a constant greater than zero; gamma is a quantization level parameter; η is quantizer precision; n is the number of agents; a is1＝max1≤i≤j≤ncij(ii) a Theta is a positive integer.
The right-end discontinuous control protocol of the quantizer based on the fusion time lag term and the rounding function is as follows:
Figure BDA0003039273420000092
q (x) is a quantization function, defined
Figure BDA0003039273420000093
Wherein i is 1,2, …, n, gamma is quantization level parameter, eta is (0, 1)]For quantizer accuracy, τ is the communication delay that occurs when agent j communicates to agent i,
Figure BDA0003039273420000094
to round down the function, sign (·) is a sign function.
The invention has the advantages that the invention fully applies graph theory and collective knowledge, vividly and intuitively expresses the relation of network topology and control action formed among the multi-agent system individuals with limited communication. The influence of communication constraints such as communication time lag, quantized data and the like is considered in the multi-agent system researched by the invention, so that the multi-agent system is more consistent with general practical application, and the multi-agent system researched by the invention has a cooperation and competition relationship at the same time, so that the multi-agent system is more consistent with general practical requirements; the consistency control of the intelligent system with simultaneous existence of disturbance cooperation and competition relationship has innovative significance; the communication-limited multi-agent system provided by the invention realizes the sufficient condition of dichotomous practical consistency, and provides a judgment standard for the consistency of the system. The asymptotic consistency of the multi-agent system is popularized to practical consistency, the system error is converged in a controllable interval under the condition of limited communication, and the convergence upper bound value is irrelevant to any global information and initial value and is only relevant to the quantizer parameter.
Drawings
FIG. 1 is a diagram of an example network topology;
fig. 2 shows the location consistency of a multi-agent system with limited γ -3 and η -1 communication;
fig. 3 shows the location consistency of a multi-agent system with limited γ ═ 1 and η ═ 1 communication;
fig. 4 shows the location consistency of a multi-agent system with limited communication γ ═ 0.1 and η ═ 1;
fig. 5 is a block diagram of the steps of the method of the present invention.
Detailed Description
FIG. 5 is a flow chart showing the steps of the method of the present invention.
The embodiment of the invention provides a communication-limited two-dimensional practical consistency control method for a multi-agent system, which comprises the following steps:
(1) based on the structural balance topology assumption of the multi-agent system, the multi-agent system with the competitive relationship is converted into a system with non-negative connection weight through standard transformation of one class of orthogonal matrixes, so that the dichotomous practical consistency problem is converted into a general practical consistency problem.
(2) Setting a quantization function to obtain a Phillips solution; and obtaining a global solution of the multi-agent system under the condition that the right end of the controller is discontinuous by utilizing a differential frame containing theory and Phillips solution and a kinetic equation of the agent i.
(3) The Lyapunov function is constructed based on a quantizer, the stability condition of the multi-agent system is deduced by utilizing the Lyapunov stability theory and the algebraic graph theory, and an error convergence upper bound value which is irrelevant to any global information and initial value is deduced.
(4) Designing a right-end discontinuous control protocol of a quantizer based on fusion time-lag terms and rounding functions to make the position states of all intelligent positions converge to controllable intervals with the same modulus and different symbols; therefore, the dichotomous practical consistency of the multi-agent system with limited communication is realized.
FIG. 1 illustrates network connections and communication relationships between agents in a multi-agent system with limited communication, using a directed network as an example. The topological graph satisfies the structural balance, and two subgroups in the graph are V respectively11,2, 3 and V2And 4, 5, 6, the subgroups have competition relations, and the subgroups have cooperative relations. The adjacency matrix a is subjected to canonical transformation, and the transformation matrix is given as C ═ { C ═ diag (σ), σ ═ 111-1-1-1]}。
Fig. 2 is a change curve of the position state of each intelligent agent under the action of the controller (11), and it can be known from the graph that the error of the absolute value of the position state of each intelligent agent is converged in a controllable interval [0,3], and two subgroups with a competitive relationship can finally realize convergence, namely, the multi-intelligent-agent system with limited communication can realize dichotomy and consistency.
Fig. 3 and 4 show the respective graphs of the position state of the agent when γ is 1 and γ is 0.1, respectively, under the same initial value state, and show that the fluctuation interval of the absolute value error of the position state of the agent can be controlled by changing the γ value, and the smaller the γ value is, the smaller the fluctuation interval of the error of the absolute value of the position state of the agent is.
Various alternatives, variations and modifications may be made on the concept of the present invention and are not to be excluded from the scope of the invention.

Claims (6)

1. A communication-limited, bi-parting utility consistency control method for a multi-agent system, the method comprising the steps of:
(1) based on the structural balance topology assumption of the multi-agent system, the multi-agent system with the competitive relationship is converted into a system with non-negative connection weight by carrying out standard transformation through a class of orthogonal matrixes, so that the dichotomous practical consistency problem is converted into a general practical consistency problem;
(2) setting a quantization function to obtain a Phillips solution; obtaining a global solution of the multi-agent system under the condition that the right end of the controller is discontinuous by utilizing a differential frame comprising theory and Phillips solution and through a kinetic equation of the agent i;
(3) constructing a Lyapunov function based on a quantization function, deducing the stability condition of the multi-agent system by utilizing a Lyapunov stability theory and an algebraic graph theory, and deriving an error convergence upper bound value which is irrelevant to any global information and initial value;
(4) designing a right-end discontinuous control protocol based on a quantization function fused with a time-lag term to make the position states of all intelligent positions converge to controllable intervals with the same mode and different symbols; therefore, the dichotomous practical consistency of the multi-agent system with limited communication is realized;
the right-end discontinuous control protocol of the quantization function based on the fusion time lag term is as follows:
Figure FDA0003623051200000011
q (x) is a quantization function, defined
Figure FDA0003623051200000012
Wherein i is 1,2, …, n, gamma is quantization level parameter, eta is (0, 1)]To quantify the accuracy, τ is the communication delay that occurs when agent j communicates to agent i,
Figure FDA0003623051200000013
sign () is a sign function for the rounding down function.
2. The communication-limited multi-agent system dichotomous utility consistency control method of claim 1, wherein the communication topology of the multi-agent system is described by a directed graph G (V, E, A), where V {1,2, …, n } is a set of points,
Figure FDA0003623051200000014
as a set of edges, having directed edges eijThe intelligent agent i transmits information to the intelligent agent j by (i, j) epsilon E; a ═ aij]∈Rn×nIs a communication weight matrix, whose diagonal elements aii0; if aijIf not equal to 0, the agent j is called as a neighbor of the agent i; for any i, j has aijIf the value is more than or equal to 0, the graph G is called a non-negative graph, otherwise, the graph G is a symbolic graph; the degree matrix D belongs to Rn×nDefined as D ═ diag { D ═ DiTherein of
Figure FDA0003623051200000021
Laplace matrix L ∈ Rn×nIs defined as L ═ D-A, where
Figure FDA0003623051200000022
And is
Figure FDA0003623051200000023
If any two nodes in the graph have one directed path, the graph G is called a directed strong-connectivity graph.
3. The method as recited in claim 1, wherein the dynamic equation of agent i is
Figure FDA0003623051200000024
xi(t) E.R represents the position state variable of agent i; u. ui(t) E.R represents the control input of agent i;
setting a quantization function
Figure FDA0003623051200000025
Gamma is a quantization level parameter, and eta belongs to (0, 1)]In order to quantify the accuracy of the quantization,
Figure FDA0003623051200000026
is a floor function; a local solution exists on the system [0, T), the boundedness of the local solution is obtained through a back-up method according to the theory of functional differential equations, the global solution of the system is further obtained, and T represents any time larger than 0.
4. According to claim 1The dichotomous practical consistency control method of the multi-agent system with limited communication is characterized in that the orthogonal matrix is as follows: c ═ diag (σ), σ ═ σ [ σ ]1,σ2,…,σn],σi-1,1} }; c satisfies CTC＝CCTIs equal to I, and C-1C; for a multi-agent system with fixed undirected network topology and zero communication time delay, a control protocol X is often adoptedi(t) u (t) lx (t), to which a canonical transformation:
let Z be CX, C be C, by C-1When C and X are CZ, then
Figure FDA0003623051200000027
Wherein L isDD-CAC is a normalized post-transform laplacian matrix;
Figure FDA0003623051200000028
ld,ijrepresenting the canonical transformed Laplace matrix LDThe element in the ith row and the jth column, ldFinger matrix LDI and j represent the elements in the ith row and the jth column; c is an orthogonal matrix; d is a degree matrix; i is an identity matrix; sigma is a diagonal element in the orthogonal matrix C; l is a Laplace matrix; x is the state matrix X ═ X of the agent1 x2 L xn](ii) a And Z is a transition transformation matrix Z ═ CX.
5. The method as claimed in claim 1, wherein the quantization function is designed to obtain a philippiv solution, which results in a global solution for the multi-agent system when the controller is in discontinuity at the right end:
when the function x (T) [ -tau, T) → RnSatisfies the following conditions:
(1) x (T) is continuous at [ - τ, T) and absolutely continuous at [0, T);
(2) presence of a measurable vector function w (t) ═ w1(t),w2(t),…,wn(t)]、w′(t)＝[w′1(t),w′2(t),...,w′n(t)]：[-τ，T)→RnSuch that x (T) has at T ∈ [0, T):
Figure FDA0003623051200000031
wherein wi(t)∈K[sign(aij)Q(xi(t))]、w′i(t)∈K[Q(xi(t))]Is the output function for the solution x (T), then x (T) is the system's philippif solution at [0, T); τ is a communication delay occurring when agent j transmits a communication to agent i; t represents any time greater than 0;
given a continuous vector function phi (t) [ -tau, 0]→RnWith a measurable vector function ψ (t) ∈ K [ Q (phi (t))]:[-τ,0]→Rn、ψ′(t)∈K[sign(aij)Q(φ(t))]:[-τ,0]→RnIf, if
Figure FDA0003623051200000032
Such that x (T) is the solution of the system at [0, T) and satisfies:
Figure FDA0003623051200000033
then, [ x (t), w' (t)]:[-τ,T)→Rn×RnA solution satisfying the initial conditions (phi (t), phi' (t)) for the system;
the local solution of the system on [0, T) is easily obtained; and according to the theory of functional differential equations, obtaining the boundedness of a local solution by a back-off method, and further obtaining a global solution of the system.
6. The communication-constrained multi-agent system dichotomous utility consistency control method of claim 1, wherein the lyapunov function is constructed as follows:
Figure FDA0003623051200000041
order to
Figure FDA0003623051200000042
So that
Figure FDA0003623051200000043
Figure FDA0003623051200000044
Thus, there are:
Figure FDA0003623051200000045
the following can be obtained:
Figure FDA0003623051200000046
to pair
Figure FDA0003623051200000047
All agents in the system can converge to set E at t ",
E＝{|x(t+λ)|∈C([-τ,0)；Rn):(wi(t)-w′j(t-τ))2+(w′j(t)-wi(t-τ))2≤(γ/N+1)2is, in turn, composed of
Figure FDA0003623051200000048
The following can be obtained:
Figure FDA0003623051200000049
wherein theta is1∈Z；
Any agent i to its neighbor agents i', i*With no more than n-1 pathways, in combination with the above analysis, one can obtain:
Figure FDA0003623051200000051
thus, pair
Figure FDA0003623051200000052
Making state x of any one agent ii(t″)∈(θ-0.5)γη,(θ+0.5)γη]；
To pair
Figure FDA0003623051200000053
Comprises the following steps:
dist(xi(t),[(θ-0.5)γη,(θ+0.5)γη])≤ε
this can give the state x for any agent ii(t) all converge to a set [ (theta-0.5) gamma eta, (theta +0.5) gamma eta]Performing the following steps;
this yields an upper bound on error convergence:
Figure FDA0003623051200000054
in the formula, V is a Lyapunov function; v1Is a sub-function of V; v2Is another sub-function in V; q (t) is a quantization function; c. CijIs the ith row and the jth column element in the orthogonal matrix C; t is0A constant greater than zero; rho is equal to 0, tau](ii) a τ is a communication delay occurring when agent j transmits a communication to agent i; epsilon is any number greater than zero,
Figure FDA0003623051200000055
e is a convergence set; λ is a constant greater than zero; gamma is a quantization level parameter; eta is quantization precision; n is the number of agents; a is1＝max1≤i≤j≤ ncij；θIs a positive integer.
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