CN113515066B - Nonlinear multi-intelligent system dynamic event trigger control method - Google Patents

Nonlinear multi-intelligent system dynamic event trigger control method Download PDF

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CN113515066B
CN113515066B CN202110535278.9A CN202110535278A CN113515066B CN 113515066 B CN113515066 B CN 113515066B CN 202110535278 A CN202110535278 A CN 202110535278A CN 113515066 B CN113515066 B CN 113515066B
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CN113515066A (en
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余瑶
柴小丰
孙长银
冯涛
解乃颖
刁奇
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University of Science and Technology Beijing USTB
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    • G05B19/04—Programme control other than numerical control, i.e. in sequence controllers or logic controllers
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
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Abstract

The invention discloses a nonlinear multi-intelligent system dynamic event trigger control method, which comprises the following steps: each intelligent agent synchronously samples state information and periodically monitors an event trigger function, and each intelligent agent broadcasts own state only when the event trigger function is triggered; dynamically updating parameters in the event triggering function; and verifying the validity of the control strategy through numerical simulation. The invention researches the formation control problem of a nonlinear high-order multi-agent system, adopts a dynamic event trigger mechanism based on sampling data, converts the formation problem into the stability problem of a subsystem, then constructs the subsystem into a time delay system, obtains a sufficient condition of system stability by utilizing a linear matrix inequality, and finally realizes formation control.

Description

Nonlinear multi-intelligent system dynamic event trigger control method
Technical Field
The invention relates to the technical field of dynamic event trigger control, in particular to a nonlinear multi-agent system dynamic event trigger control method based on sampling data.
Background
In recent years, distributed control of multi-agent systems has received increasing attention due to their wide application in the fields of power grids, sensor networks, transportation and robots, etc. Distributed cooperative control includes numerous branches, such as: congestion, consistency, formation and surrounding, etc. The formation control aims to enable a group of intelligent agents to realize a specified geometric formation, and the formation control method is widely applied to the scenes such as military fields, satellite networking, unmanned aerial vehicle performance and the like.
In a real system, the controller runs on an embedded microprocessor and executes periodically. Therefore, discrete control strategies are receiving increasing attention. The y.p.gao et al studied the problem of consistency of the second order system in a time-varying directed communication topology, each agent could only obtain neighbor position and velocity information at the sampling instant, and the sampling period of each agent was different. The g.h.wen et al proposed a delay input strategy for multi-agent systems with nonlinear dynamics by which a discrete system can be converted to a nonlinear system with a time-varying delay and the maximum sampling period allowed by the system can be estimated. In addition, there are more research works on multi-agent distributed cooperative control based on sampled data, which are not described here.
Considering the limited communication and computing resources of practical systems, how to reduce the use of communication bandwidth and transmission energy is of paramount importance. The event-triggered strategy can well solve this problem. The event-triggered strategy provides a natural way to accomplish the control task, and whether the control strategy is executed depends on a pre-set trigger spring rather than periodically executing the control strategy. When the trigger condition is satisfied, an event is triggered at this time, and the controller is updated. There have been some studies on the problem of multi-agent system consistency, but few studies have been made on the use of event-triggered control based on sampled data, and the present invention has been made based on this.
Disclosure of Invention
The invention aims to provide a nonlinear multi-intelligent system dynamic event triggering control method, which adopts a dynamic event triggering mechanism based on sampling data, converts formation problems into stability problems of subsystems, constructs the subsystems into time delay systems, obtains sufficient conditions of system stability by utilizing linear matrix inequality, and finally realizes formation control.
In order to solve the technical problems, the embodiment of the invention provides the following scheme:
a nonlinear multi-intelligent system dynamic event trigger control method comprises the following steps:
s1, each intelligent agent synchronously samples state information and periodically monitors an event trigger function, and each intelligent agent broadcasts own state only when the event trigger function is triggered;
s2, dynamically updating parameters in the event triggering function;
s3, verifying the effectiveness of the control strategy through numerical simulation.
Preferably, the method further comprises, prior to step S1:
giving a theory, definitions of dynamics and formation problems of each agent, and follow-up assumptions and quotations;
the graph theory includes:
consider an undirected graph G with N nodes, denoted g= { V, E, W }, where v= { V 1 ,v 2 ,...,v N The number of nodes represents the set of nodes,
Figure BDA0003069348140000021
representing a collection of edges, w= [ W ] ij ]∈R N×N Representing an adjacency matrix; for the adjacency matrix W, if (v i ,v j ) E, w is ij > 0, where v i And v j Are neighbors of each other; otherwise w ij ＝0；N i Representing a set of neighbors of the intelligent agent i;
the laplacian matrix l= [ L ] of graph G ij ]∈R N×N Is defined as
Figure BDA0003069348140000022
When a connection path exists between any two nodes in the graph, the undirected graph G is fully connected;
the problem is described as follows:
consider a multi-agent system with a communication topology G, comprising N agents, each of which has a kinetic model described as:
Figure BDA0003069348140000023
wherein x is i (t)∈R n ,u i (t)∈R m ，f(x i (t),t)∈R n Respectively represent the state, input and nonlinear dynamics of the intelligent agent i, and has
Figure BDA0003069348140000024
Figure BDA0003069348140000025
A and B are each a component having a corresponding dimensionA constant matrix of numbers;
definition 1: the multi-agent system is said to be capable of achieving time-varying state formation h if, for any given bounded initial condition, the following requirements are met:
Figure BDA0003069348140000026
wherein h is i ∈R n Representing state x i Offset of (t) and has
Figure BDA0003069348140000027
r (t) is a formation location function;
definition 2: if a controller u is used i (t) (i=1, 2,., N.) the multi-agent system is able to achieve formation h, then it is said that formation h is viable;
the hypothesis and quotation are as follows:
suppose 1: graph G is undirected graph and fully connected;
suppose 2: for any x, y ε R n Nonlinear dynamics f (x i (t), t) satisfies the following Li Puxi z condition:
Figure BDA0003069348140000031
lemma 1: the Laplacian matrix of the undirected graph G is L epsilon R N×N Then:
l has at least one zero eigenvalue, and the eigenvalue corresponds to an eigenvector of 1 N I.e. L1 N ＝0 N ；
If G is fully connected, 0 is a single eigenvalue of L, the remaining N-1 eigenvalues all have positive real parts, i.e. 0 = λ 1 ＜λ 2 ≤...≤λ N ；
And (4) lemma 2: if the matrix R > 0, then for any matrix X and scalar ρ > 0, the following is satisfied:
-XR -1 X≤ρ 2 R-2ρX
and (3) lemma 3: for symmetric matrix A 11 ，A 12 ，A 22 The following properties are equivalent:
Figure BDA0003069348140000032
Figure BDA0003069348140000033
/>
Figure BDA0003069348140000034
preferably, the step S1 specifically includes:
designing an event triggering mechanism:
the formation error is defined as:
δ i (t)＝x i (t)-h i (i＝1,2,...,N)
and has
Figure BDA0003069348140000035
Defining the measurement error as:
Figure BDA0003069348140000036
where T is the sampling period and where,
Figure BDA0003069348140000037
is the latest trigger time of agent i and has +.>
Figure BDA0003069348140000038
Defining a combined measurement as:
Figure RE-GDA0003259653840000047
wherein the method comprises the steps of
Figure BDA00030693481400000310
Is the latest trigger time of agent j and has +.>
Figure BDA00030693481400000311
The nonlinear term f (x i (t), t), the state estimate of the obtained agent i is:
Figure BDA0003069348140000041
wherein KT is the sampling time and has
Figure BDA0003069348140000042
Estimated for the state of agent i at time KT, and there is +.>
Figure BDA0003069348140000043
Then, the state estimation error is obtained as:
Figure BDA0003069348140000044
wherein x is i (KT+T) is the sampling value of the state of the agent i at the time KT+T, and has
Figure BDA0003069348140000045
The event-triggered queuing control strategy based on the sampled data is designed as follows:
Figure BDA0003069348140000046
wherein the method comprises the steps of
Figure BDA0003069348140000047
And->
Figure BDA0003069348140000048
Dynamic event trigger function g of agent i i (t) is defined as:
Figure BDA0003069348140000049
Figure BDA00030693481400000410
wherein sigma i (KT) is a dynamic parameter, and θ > 0 is a positive constant.
Preferably, the agent checks the dynamic event trigger function g only at each sampling instant KT i (t)；
First, according to
Figure BDA00030693481400000411
Updating dynamic parameter sigma i (KT)；
Then, substituting the sampled value into
Figure BDA00030693481400000412
If and only if g i When (KT) is not less than 0, the system triggers and updates the controller u i (t)；
The trigger time is expressed as:
Figure BDA00030693481400000413
and 4, lemma: for a given initial condition sigma i (0) E [0, 1) and θ > 0, with the adoption of
Figure BDA00030693481400000414
Parameter update is performed, then sigma i (KT) satisfying the following constraint at any time:
Figure BDA00030693481400000415
preferably, the demonstration is performed using mathematical induction:
when k=1, we get:
Figure BDA0003069348140000051
indicating 0.ltoreq.sigma i (T)≤σ i (0)＜1；
For the following
Figure BDA0003069348140000052
Let 0.ltoreq.sigma i (KT)≤σ i (0) < 1, give:
Figure BDA0003069348140000053
and has
Figure BDA0003069348140000054
Thus, sigma i (KT) monotonically decreases with increasing K, proving 0.ltoreq.sigma i ((K+1)T)≤σ i (KT) < 1, thereby obtaining, for any of
Figure BDA0003069348140000055
Inequality 0 +.sigma i (KT)≤σ i (0)＜1,/>
Figure BDA0003069348140000056
This is true.
Preferably, a control strategy is employed
Figure BDA0003069348140000057
And t.epsilon.KT, (K+1) T,
Figure BDA0003069348140000058
the multi-agent system has the following form:
Figure BDA0003069348140000059
delta of the formula i (t)＝x i (t)-h i (i=1, 2,) N) substituted into the above formula, further resulting in:
Figure BDA00030693481400000510
let lambda get i (i=1, 2,., N) represents the eigenvalues of the laplace matrix L, J is about the right standard type of L, then there is present
Figure BDA00030693481400000511
And U -1 ＝U T Satisfies the following formula:
U -1 LU＝U T LU＝J＝diag{λ 1 ,λ 2 ,...,λ N }
lambda is known from the quotation 1 1 =0, and its corresponding feature vector is
Figure BDA00030693481400000512
Definition of the definition
Figure BDA00030693481400000513
And->
Figure BDA00030693481400000514
The multi-agent system writes as: />
Figure BDA00030693481400000515
Defining a time delay function tau (t) =t-KT, t epsilon [ KT ], and [ ]K+1) T), 0.ltoreq.τ (T) < T is piecewise linear and has
Figure BDA00030693481400000516
At this time, a method for solving the time delay problem is adopted to solve the formation problem;
order the
Figure BDA00030693481400000517
Furthermore, the->
Figure BDA00030693481400000518
The system
Figure BDA00030693481400000519
The writing is as follows:
Figure RE-GDA0003259653840000075
Figure RE-GDA0003259653840000076
system and method for controlling a system
Figure BDA0003069348140000061
Conversion of formation control problems into systems
Figure BDA0003069348140000062
Stability problems of (2);
and (5) lemma: for multiple intelligent systems
Figure BDA0003069348140000063
Using a controller
Figure BDA0003069348140000064
And dynamic event trigger function based on sampling data
Figure BDA0003069348140000065
Figure BDA0003069348140000066
A formation h can be implemented if and only if:
Figure BDA0003069348140000067
the proving process is as follows:
Figure BDA0003069348140000068
if and only if the following formula is satisfied:
Figure BDA0003069348140000069
the method comprises the following steps:
Figure BDA00030693481400000610
due to
Figure BDA00030693481400000611
Is nonsingular, < >>
Figure BDA00030693481400000612
Description->
Figure BDA00030693481400000613
Thus, when and only when
Figure BDA00030693481400000614
When the state formation h is realized;
and (3) lemma 6: for event triggering functions
Figure RE-GDA0003259653840000089
Figure BDA00030693481400000616
Given an initial condition σ (0) =diag { σ } 1 (0),σ 2 (0),...,σ N (0) ' and sigma N-1 (0)＝diag{σ 2 (0),...,σ N (0) } $ gives the following inequality:
Figure BDA00030693481400000617
wherein t.epsilon.KT, (K+1) T,
Figure BDA00030693481400000618
and->
Figure BDA0003069348140000071
The following was demonstrated:
the method comprises the following steps of:
Figure BDA0003069348140000072
wherein the method comprises the steps of
Figure BDA0003069348140000073
From delta i (t) and e i Definition of (t) gives:
Figure BDA0003069348140000074
will be described in
Figure BDA0003069348140000075
Substituted +.>
Figure BDA0003069348140000076
The method comprises the following steps:
Figure BDA0003069348140000077
this gives:
Figure BDA0003069348140000078
from the definition of η (t), it is known that:
Figure BDA0003069348140000079
substituting the two formulas into one
Figure BDA00030693481400000710
The method comprises the following steps:
Figure BDA0003069348140000081
and (5) finishing the verification.
Preferably, for a multiple intelligent system
Figure BDA0003069348140000082
If a controller is used
Figure BDA0003069348140000083
Based on the dynamic event trigger function of the sampled data, when hypothesis 1 and hypothesis 2 are satisfied, formation h is possible when the following conditions are satisfied:
A. for the following
Figure BDA0003069348140000084
And j epsilon N i The conditions are satisfied:
(A+BK 2 )(h i -h j )＝0
B. there is a real matrix
Figure BDA0003069348140000085
Figure BDA0003069348140000086
And sigma (sigma) i (0) E [0, 1), θ > 0, ρ > 0 satisfies the following inequality:
Figure BDA0003069348140000087
Figure BDA0003069348140000088
wherein the method comprises the steps of
Figure BDA0003069348140000089
Is a matrix of a matrix, and has +.>
Figure BDA00030693481400000810
Figure BDA00030693481400000811
Figure BDA00030693481400000812
Figure BDA00030693481400000813
Figure BDA00030693481400000814
Figure BDA00030693481400000815
Figure BDA00030693481400000816
Figure BDA00030693481400000817
Figure BDA00030693481400000818
Figure BDA00030693481400000819
Figure BDA00030693481400000820
Figure BDA00030693481400000821
Figure BDA00030693481400000822
Figure BDA00030693481400000823
Figure BDA00030693481400000824
Figure BDA0003069348140000091
Figure BDA0003069348140000092
Figure BDA0003069348140000093
In addition, the controller gain matrix K 1 And K is equal to 3 From the following components
Figure BDA0003069348140000094
And->
Figure BDA0003069348140000095
Obtained.
Preferably, the proving process is as follows:
the following lyapunov function was constructed:
Figure BDA0003069348140000096
wherein p=p T ，Q＝Q T And r=r T Is a positive definite matrix;
when the condition A is satisfied, the following steps are obtained:
Figure BDA0003069348140000097
opposite type
Figure BDA0003069348140000098
Both sides are multiplied by +.>
Figure BDA0003069348140000099
Because of->
Figure BDA00030693481400000910
The method further comprises the following steps: />
Figure BDA00030693481400000911
According to the quotation mark 1,
Figure BDA00030693481400000912
is non-singular; will->
Figure BDA00030693481400000913
Both sides are multiplied by +.>
Figure BDA00030693481400000914
The method comprises the following steps of:
Figure BDA00030693481400000915
then
Figure BDA00030693481400000916
The writing is as follows:
Figure BDA00030693481400000917
according to the above formula, the derivative of V (t) is given by:
Figure BDA0003069348140000101
applying lemma 1, obtaining:
Figure BDA0003069348140000102
wherein χ is 1 ＝ε(t-τ(t))-ε(t)，χ 2 ε (T-T) - ε (T- τ (T)) and S is a real matrix;
Figure BDA0003069348140000103
if it meets the following conditions:
Figure BDA0003069348140000104
order the
Figure BDA0003069348140000105
And->
Figure BDA0003069348140000106
Multiplying the two sides by +.>
Figure BDA0003069348140000107
The above inequality is obtained as equivalent to:
Figure BDA0003069348140000108
thus there is
Figure BDA0003069348140000109
Definition of the definition
Figure BDA00030693481400001010
Will->
Figure BDA00030693481400001011
Figure BDA00030693481400001012
Figure BDA00030693481400001013
Substituted formula
Figure BDA0003069348140000111
The derivative of V (t) is written as:
Figure BDA0003069348140000112
wherein, xi= [ xi ] p,q ] 6×6 Is a symmetrical matrix and has
Figure BDA0003069348140000113
Figure BDA0003069348140000114
Figure BDA0003069348140000115
Figure BDA0003069348140000116
Figure BDA0003069348140000117
Figure BDA0003069348140000118
Figure BDA0003069348140000119
Figure BDA00030693481400001110
Ξ 2,4 ＝Ω 2 ，
Ξ 2,5 ＝Ξ 2,6 ＝0 (N-1)n×Nn ，
Figure BDA00030693481400001111
Ξ 3,4 ＝Ξ 3,5 ＝Ξ 3,6 ＝0 (N-1)n×Nn ，
Ξ 4,4 ＝Ω 3 ，
Ξ 4,5 ＝Ξ 4,6 ＝Ξ 5,5 ＝Ξ 5,6 ＝Ξ 6,6 ＝0 Nn×Nn ，
Figure BDA00030693481400001112
The schulb's law is applied,
Figure BDA00030693481400001113
equivalent to
Figure BDA00030693481400001114
Wherein the method comprises the steps of
Figure BDA00030693481400001115
Multiplying the two sides simultaneously
Figure BDA0003069348140000121
The above inequality is obtained as equivalent to:
Figure BDA0003069348140000122
applying lemma 2 to obtain:
Figure BDA0003069348140000123
the method further comprises the following steps:
Figure BDA0003069348140000124
consider the conditions
Figure BDA0003069348140000125
And the Shu Er's theorem, get:
Figure BDA0003069348140000126
combining the above formulas yields:
Figure BDA0003069348140000127
equivalent to inequality
Figure BDA0003069348140000128
At the same time satisfy inequality
Figure BDA0003069348140000129
Thus, it is
Figure BDA00030693481400001210
Satisfy->
Figure BDA00030693481400001211
The technical scheme provided by the embodiment of the invention has the beneficial effects that at least:
compared with the prior art, the invention has the advantages that: firstly, a dynamic event triggering strategy based on sampling data is provided, and the strategy can effectively avoid continuous communication and sampling; the formation problem is converted into the stability problem of the time delay system by introducing a time delay function, and then the stability problem can be solved by a linear matrix inequality and a Leeleapunov function; then, the dynamic event triggering strategy is a distributed control strategy, and parameters in the triggering conditions are dynamically changed, so that the event triggering times can be effectively reduced; secondly, the nonlinear multi-agent system is considered, the state estimation error is used for representing the influence of nonlinear dynamics on the system, and the state estimation error is introduced into the controller, so that the influence of nonlinearity can be effectively restrained.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for the description of the embodiments will be briefly described below, and it is apparent that the drawings in the following description are only some embodiments of the present invention, and other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of a method for controlling dynamic event triggering of a nonlinear multi-intelligent system based on sampling data provided by an embodiment of the invention;
FIG. 2 is a diagram of a state x of each agent according to an embodiment of the present invention i (t) a time-dependent curve;
FIGS. 3 a-3 c illustrate the formation error delta for each agent provided by embodiments of the present invention i (t) schematic;
FIGS. 4 a-4 c illustrate combined measurement q of individual agents provided by embodiments of the present invention i (t) a profile;
FIGS. 5 a-5 c illustrate the measured error e of each agent provided by embodiments of the present invention i A variation curve of (t);
FIG. 6 is a graph showing the dynamic trigger function parameter σ of each agent according to an embodiment of the present invention i (KT) time-dependent curve.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the present invention more apparent, the embodiments of the present invention will be described in further detail with reference to the accompanying drawings.
The embodiment of the invention provides a nonlinear multi-intelligent system dynamic event trigger control method, as shown in fig. 1, comprising the following steps:
s1, each intelligent agent synchronously samples state information and periodically monitors an event trigger function, and each intelligent agent broadcasts own state only when the event trigger function is triggered;
s2, dynamically updating parameters in the event triggering function;
s3, verifying the effectiveness of the control strategy through numerical simulation.
The embodiment of the invention researches the distributed formation control problem of the multi-agent system with nonlinear dynamics, and researches a dynamic event trigger control strategy based on sampling data by considering an actual digital processor and limited network resources. Firstly, each intelligent agent synchronously samples state information and periodically monitors event triggering functions, and each intelligent agent only broadcasts own state when the functions are triggered, so that the communication times can be greatly reduced. Meanwhile, the Zeno behavior can be avoided due to the periodic sampling mechanism. Further, parameters in the event triggering function are dynamically updated, so that the balance between communication frequency and formation performance can be realized. The invention converts the formation problem into the stability problem of the time delay system. Finally, the effectiveness of the control strategy provided by the invention is verified by numerical simulation.
Specifically, before the step S1, the method further includes: definition of the theory, dynamics and formation problem of each agent, and the assumption and quotation used later are given. Wherein the following symbols are used: 0 N Matrix representing all elements 0,1 N A matrix representing all elements as 1, the superscript T representing the matrixIs to be used in the present invention,
Figure BDA0003069348140000141
representing Cronecker product, metropolyl>
Figure BDA0003069348140000142
Representing a non-negative set of integers.
The graph theory includes:
consider an undirected graph G with N nodes, which can be expressed as g= { V, E, W }, where v= { V 1 ,v 2 ,...,v N And represents a collection of nodes,
Figure BDA0003069348140000143
representing a collection of edges, w= [ W ] ij ]∈R N×N Representing the adjacency matrix. For the neighbor matrix W, if (v i ,v j ) E, w is ij > 0, where v i And v j Are neighbors of each other; otherwise w ij ＝0。N i Representing a set of neighbors of agent i.
The laplacian matrix l= [ L ] of graph G ij ]∈R N×N Is defined as
Figure BDA0003069348140000144
The undirected graph G is fully connected when there is a connection path between any two nodes in the graph.
The problem is described as follows:
considering a multi-agent system with a communication topology G, comprising N agents, the kinetic model of each agent can be described as:
Figure BDA0003069348140000145
wherein x is i (t)∈R n ,u i (t)∈R m ，f(x i (t),t)∈R n Respectively represent the state, input and nonlinear dynamics of the intelligent agent i, and has
Figure BDA0003069348140000146
Figure BDA0003069348140000147
A and B are constant matrices having corresponding dimensions.
Definition 1: the multi-intelligent system (1) is said to be able to implement a time-varying state formation h if, for any given bounded initial condition, the following requirements are met:
Figure BDA0003069348140000148
wherein h is i ∈R n Representing state x i Offset of (t) and has
Figure BDA0003069348140000149
r (t) is a formation location function.
Definition 2: if a controller u is used i (t) (i=1, 2,., n.), the multi-agent system may implement formation h, then it is possible to call formation h.
The assumptions and quotations include:
suppose 1: graph G is undirected and fully connected.
Suppose 2: for any x, y ε R n Nonlinear dynamics f (x i (t), t) satisfies the following Li Puxi z condition:
Figure BDA00030693481400001410
lemma 1: the Laplacian matrix of the undirected graph G is L epsilon R N×N Then
(1) L has at least one zero eigenvalue, and the eigenvalue corresponds to an eigenvector of 1 N I.e. L1 N ＝0 N ；
(2) If G is fully connected, 0 is a single eigenvalue of L, the remaining N-1 eigenvalues all have a positive real part, i.e. 0 = λ 1 ＜λ 2 ≤...≤λ N 。
And (4) lemma 2: if the matrix R > 0, then for any matrix X and scalar ρ > 0, the following is satisfied:
-XR -1 X≤ρ 2 R-2ρX (2)
and (3) lemma 3: for symmetric matrix A 11 ，A 12 ，A 22 The following properties are equivalent:
Figure BDA0003069348140000151
Figure BDA0003069348140000152
Figure BDA0003069348140000153
/>
furthermore, the invention designs a dynamic event trigger formation control strategy based on the sampling data and provides conditions for realizing formation control.
First, an event trigger mechanism is designed:
the formation error is defined as:
δ i (t)＝x i (t)-h i (i＝1,2,...,N) (3)
and has
Figure BDA0003069348140000154
Defining the measurement error as:
Figure BDA0003069348140000155
where T is the sampling period and where,
Figure BDA0003069348140000156
is the latest triggering moment of the intelligent agent i and has
Figure BDA0003069348140000157
Defining a combined measurement as:
Figure BDA0003069348140000158
wherein the method comprises the steps of
Figure BDA0003069348140000159
Is the latest trigger time of agent j and has +.>
Figure BDA00030693481400001510
According to equation (1), the nonlinear term f (x i (t), t) can be estimated as the state of agent i:
Figure BDA00030693481400001511
wherein KT is the sampling time and has
Figure BDA00030693481400001512
Estimated for the state of agent i at time KT, and there is +.>
Figure BDA00030693481400001513
Then, the state estimation error can be obtained as:
Figure BDA00030693481400001514
wherein x is i (KT+T) is the sampling value of the state of the agent i at the time KT+T, and has
Figure BDA0003069348140000161
Nonlinear term f (x) in system (1) i (t), t) is difficult to model or to obtain accurate values. Thus, the state estimation error
Figure BDA0003069348140000162
For representing non-linear terms f (x i (t), influence of t) on the system, and use +.>
Figure BDA0003069348140000163
To eliminate the effect of nonlinear terms, the method is inspired by a model-based event-triggered strategy.
The event-triggered queuing control strategy based on the sampled data is designed as follows:
Figure BDA0003069348140000164
wherein the method comprises the steps of
Figure BDA0003069348140000165
And->
Figure BDA0003069348140000166
Inspired by a dynamic event trigger strategy, the dynamic event trigger function g of the intelligent agent i i (t) is defined as:
Figure BDA0003069348140000167
Figure BDA0003069348140000168
wherein sigma i (KT) is a dynamic parameter, θ > 0 is a positive constant, ψ=ψ T > 0 and Φ=Φ T > 0 will be calculated below.
The agent checks the dynamic event trigger function g only at each sampling instant KT i (t). First, the dynamic parameter σ is updated according to equation (10) i (KT); then substituting the sampled value into formula (9); if and only if g i When (KT) is not less than 0, the system triggers and updates the controller u i (t). The trigger time can be expressed as：
Figure BDA0003069348140000169
The dynamic parameter in the prior art is designed as σ (KT), which requires global information. In contrast, the dynamic parameter sigma of the inventive design i (KT) is distributed, requiring only neighbor information.
And 4, lemma: for a given initial condition sigma i (0) E [0,1 ] and θ > 0, and updating parameters by using formula (10), σ i (KT) satisfying the following constraint at any time:
Figure BDA00030693481400001610
the proving process is as follows:
the mathematical induction method is adopted for proving;
when k=1, it is possible to obtain:
Figure BDA00030693481400001611
indicating 0.ltoreq.sigma i (T)≤σ i (0)＜1。
For the following
Figure BDA00030693481400001612
Let 0.ltoreq.sigma i (KT)≤σ i (0) < 1, obtainable from formula (10):
Figure BDA0003069348140000171
and has
Figure BDA0003069348140000172
Therefore, it is easy to know σ i (KT) monotonically decreases with increasing K, proving 0.ltoreq.sigma i ((K+1)T)≤σ i (KT) < 1, whereby,for arbitrary +.>
Figure BDA0003069348140000173
The inequality (12) holds.
And (5) finishing the verification.
Adopts a control strategy (8), and T is [ KT, (K+1) T),
Figure BDA0003069348140000174
the multi-agent system has the following form:
Figure BDA0003069348140000175
substituting formula (3) into formula (15) can further yield:
Figure BDA0003069348140000176
let lambda get i (i=1, 2,., N) represents the eigenvalues of the laplace matrix L, J is about the right standard type of L, then there is present
Figure BDA0003069348140000177
And U -1 ＝U T Satisfies the following formula:
U -1 LU＝U T LU＝J＝diag{λ 1 ,λ 2 ,...,λ N } (17)
lambda is known from the quotation 1 1 =0, and its corresponding feature vector is
Figure BDA0003069348140000178
Definition of the definition
Figure BDA0003069348140000179
And->
Figure BDA00030693481400001710
The multi-agent system (16) may be written as: />
Figure BDA00030693481400001711
Defining a time delay function tau (T) =t-KT, T e [ KT, (k+1) T). As can be readily seen, 0.ltoreq.τ (T) < T is piecewise linear and has
Figure BDA00030693481400001712
At this time, a method for solving the latency problem may be adopted to solve the formation problem.
Order the
Figure BDA00030693481400001713
η (t) =col { e (t), ε (t) }. In addition, it is easy to know
Figure BDA00030693481400001714
The system (18) may write as:
Figure RE-GDA0003259653840000225
Figure BDA00030693481400001716
next, the formation control problem of the system (15) is converted into a stability problem of the system (20).
And (5) lemma: for multi-agent systems (1), formation h can be achieved using a controller (8) and a dynamic event trigger function (9), (10) based on sampled data, if and only if:
Figure BDA0003069348140000181
the following was demonstrated:
Figure BDA0003069348140000182
if and only if the following formula is satisfied:
Figure BDA0003069348140000183
at this time, the expression (22) can be obtained:
Figure BDA0003069348140000184
due to
Figure BDA0003069348140000185
Is nonsingular, < >>
Figure BDA0003069348140000186
Description->
Figure BDA0003069348140000187
Thus, when and only when
Figure BDA0003069348140000188
At that time, state formation h may be implemented.
And (5) finishing the verification.
E (t) and epsilon (t) represent the formation subspace and the formation complement subspace respectively, when delta i (t) when fully in the formation subspace, formation may be achieved.
And (3) lemma 6: for event-triggered functions (9) and (10), and given an initial condition σ (0) =diag { σ 1 (0),σ 2 (0),...,σ N (0) ' and sigma N-1 (0)＝diag{σ 2 (0),...,σ N (0) The following inequality can be obtained:
Figure BDA0003069348140000189
wherein t.epsilon.KT, (K+1) T,
Figure BDA00030693481400001810
and is also provided with
Figure BDA00030693481400001811
/>
The following was demonstrated: from the event trigger functions (9) and (10), it is possible to obtain:
Figure BDA00030693481400001812
wherein the method comprises the steps of
Figure BDA00030693481400001813
From delta i (t) and e i The definition of (t) can be given by:
Figure BDA00030693481400001814
substituting formula (27) into formula (26) can give:
Figure BDA0003069348140000191
thus, it is possible to obtain:
Figure BDA0003069348140000192
from the definition of η (t), it is known that:
Figure BDA0003069348140000193
Figure BDA0003069348140000194
substituting the formula (30) and the formula (31) into the formula (29) can obtain:
Figure BDA0003069348140000195
and (5) finishing the verification.
Theorem 1: for multi-agent system (1), if a controller (8) is employed, dynamic event trigger functions (9) and (10) based on sampled data, satisfying hypothesis 1 and hypothesis 2, then formation h is possible when the following conditions are satisfied:
(1) For the following
Figure BDA0003069348140000196
And j epsilon N i The conditions are satisfied:
(A+BK 2 )(h i -h j )＝0 (33)
(2) There is a real matrix
Figure BDA0003069348140000201
Figure BDA0003069348140000202
And sigma (sigma) i (0) E [0, 1), θ > 0, ρ > 0 satisfies the following inequality:
Figure BDA0003069348140000203
Figure BDA0003069348140000204
wherein the method comprises the steps of
Figure BDA0003069348140000205
Is a matrix array and has
Figure BDA0003069348140000206
Figure BDA0003069348140000207
Figure BDA0003069348140000208
Figure BDA0003069348140000209
Figure BDA00030693481400002010
Figure BDA00030693481400002011
Figure BDA00030693481400002012
Figure BDA00030693481400002013
Figure BDA00030693481400002014
Figure BDA00030693481400002015
Figure BDA00030693481400002016
Figure BDA00030693481400002017
Figure BDA00030693481400002018
Figure BDA00030693481400002019
Figure BDA00030693481400002020
Figure BDA00030693481400002021
Figure BDA00030693481400002022
Figure BDA00030693481400002023
In addition, the controller gain matrix K 1 And K is equal to 3 Can be composed of
Figure BDA00030693481400002024
And->
Figure BDA00030693481400002025
Obtained. />
The following was demonstrated:
constructing a Lyapunov function:
Figure BDA0003069348140000211
wherein p=p T ，Q＝Q T And r=r T Is a positive definite matrix.
When the condition (1) is satisfied, it is obtained that:
Figure BDA0003069348140000212
multiplying both sides of (37) simultaneously
Figure BDA0003069348140000213
Because of->
Figure BDA0003069348140000214
It is possible to further obtain:
Figure BDA0003069348140000215
according to the quotation mark 1,
Figure BDA0003069348140000216
is non-singular. Multiplying both sides of formula (38) by +.>
Figure BDA0003069348140000217
The method can obtain the following steps:
Figure BDA0003069348140000218
then, the formula (20) can be written as:
Figure BDA0003069348140000219
from equation (40), the derivative of V (t) can be obtained in the form:
Figure BDA00030693481400002110
applying lemma 1, one can get:
Figure BDA00030693481400002111
wherein χ is 1 ＝ε(t-τ(t))-ε(t)，χ 2 ε (T-T) - ε (T- τ (T)) and S is a real matrix.
Easily-known
Figure BDA00030693481400002112
If it meets the following conditions:
Figure BDA0003069348140000221
order the
Figure BDA0003069348140000222
And->
Figure BDA0003069348140000223
Multiplying both sides of formula (43) by +.>
Figure BDA0003069348140000224
The inequality (43) can be found to be equivalent to:
Figure BDA0003069348140000225
as shown in formula (34), there are
Figure BDA0003069348140000226
Zeta (T) =col { epsilon (T), epsilon (T-tau (T)), epsilon (T-T), e (T-tau (T)),
Figure BDA0003069348140000227
substituting equations (25), (40), (42) into equation (41), the derivative of V (t) can be written as:
Figure BDA0003069348140000228
wherein, xi= [ xi ] p,q ] 6×6 Is a symmetrical matrix and has
Figure BDA0003069348140000229
Figure BDA00030693481400002210
Figure BDA00030693481400002211
Figure BDA00030693481400002212
Figure BDA00030693481400002213
Figure BDA00030693481400002214
Figure BDA00030693481400002215
Figure BDA00030693481400002216
Ξ 2,4 ＝Ω 2 ，
Ξ 2,5 ＝Ξ 2,6 ＝0 (N-1)n×Nn ，
Figure BDA00030693481400002217
Ξ 3,4 ＝Ξ 3,5 ＝Ξ 3,6 ＝0 (N-1)n×Nn ，
Ξ 4,4 ＝Ω 3 ，
Ξ 4,5 ＝Ξ 4,6 ＝Ξ 5,5 ＝Ξ 5,6 ＝Ξ 6,6 ＝0 Nn×Nn ，
Figure BDA00030693481400002218
The schulb's law is applied,
Figure BDA00030693481400002219
equivalent to
Figure BDA0003069348140000231
Wherein the method comprises the steps of
Figure BDA0003069348140000232
Multiplying both sides of formula (45) simultaneously:
Figure BDA0003069348140000233
then the inequality (45) can be found to be equivalent to: />
Figure BDA0003069348140000234
Applying lemma 2 may result in:
Figure BDA0003069348140000235
it is further possible to obtain:
Figure BDA0003069348140000236
taking into account the condition (35) and the schulb theorem, one can obtain:
Figure BDA0003069348140000237
combining formula (47) with formula (48) can obtain:
Figure BDA0003069348140000238
equivalent to inequality (46) while satisfying inequality (45), so that formula (44) satisfies
Figure BDA00030693481400002310
And (5) finishing the verification.
By fully utilizing the properties of the undirected graph, the invention obtains the lemma 6, which can effectively avoid using the pseudo-inverse of the matrix compared with the lemma 2.
The numerical simulation process of the embodiment of the invention is as follows:
the control strategy is verified by a system consisting of six agents, the laplace matrix L is as follows:
Figure BDA0003069348140000239
the kinetic parameters for each agent are shown below:
Figure BDA0003069348140000241
the formation is in the form of
Figure BDA0003069348140000242
Parameter sigma i (0) The initial value is sigma 1 (0)＝σ 2 (0)＝σ 3 (0)＝σ 4 (0)＝σ 5 (0)＝σ 6 (0) =0.99 and θ=10 -7 The sampling period is t=0.001 s. K can be selected according to equation (33) 2 ＝-B -1 A, can be obtained by: k (K) 2 ＝[-1 3 1；-1 2 0；2 -3 -1]。
Using the LMI tool in MATLAB, let ρ=0.08, the parameters in condition (2) can be calculated:
Figure BDA0003069348140000243
Figure BDA0003069348140000244
status x of each agent i (t) time-dependent curves as shown in FIG. 2, it can be seen that the system state eventually tends to a specified formation.
Formation error delta for each agent i And (t) as shown in fig. 3 a-3 c, it can be seen from the graph that the formation error of the intelligent agent tends to be consistent, and the formation definition can be seen from the formation definition, and the formation control can be effectively realized by adopting a dynamic event triggering strategy based on the sampling data.
FIGS. 4 a-4 c show the combined measurement q for each agent i (t) change curve, q i (t) converges and approaches zero over time. Due to non-linear term f i (x i (t), the presence of t), combined measurement q i (t) eventually converging into a domain around zero.
FIGS. 5 a-5 c show the measured error e of each agent i From the graph, it can be seen that e i (t) gradually converges and approaches zero.
Dynamic trigger function parameter sigma for each agent i As shown in FIG. 6, the time-dependent curve of (KT) is shown as σ i (KT) decreases monotonically with time, which will help to reduce the event trigger function g i The number of triggers of (t).
In summary, the invention researches the formation control problem of the nonlinear high-order multi-agent system, adopts a dynamic event triggering mechanism based on sampling data, converts the formation problem into the stability problem of the subsystem, then constructs the subsystem into a time delay system, obtains the sufficient condition of system stability by utilizing the inequality of a linear matrix, and finally realizes the formation control.
The foregoing description of the preferred embodiments of the invention is not intended to limit the invention to the precise form disclosed, and any such modifications, equivalents, and alternatives falling within the spirit and scope of the invention are intended to be included within the scope of the invention.

Claims (7)

1. The nonlinear multi-intelligent system dynamic event triggering control method is characterized by comprising the following steps of:
s1, each agent synchronously samples state information and periodically monitors an event triggering function, and each agent only broadcasts own state when the function is triggered;
s2, dynamically updating parameters in the event triggering function;
s3, verifying the effectiveness of the control strategy through numerical simulation;
the method further comprises, prior to step S1:
giving a theory, definitions of dynamics and formation problems of each agent, and assumptions and quotations used subsequently;
the graph theory includes:
consider an undirected graph G with N nodes, denoted g= { V, E, W }, where v= { V 1 ,v 2 ,...,v N And represents a collection of nodes,
Figure FDA0004092786440000011
representing a collection of edges, w= [ W ] ij ]∈R N×N Representing an adjacency matrix; for the adjacency matrix W, if (v i ,v j ) E, w is ij > 0, where v i And v j Are neighbors of each other; otherwise w ij ＝0；N i Representing a set of neighbors of agent i;
the laplacian matrix l= [ L ] of graph G ij ]∈R N×N Is defined as
Figure FDA0004092786440000012
When a connection path exists between any two nodes in the graph, the undirected graph G is fully connected;
the problem is described as follows:
consider a multi-agent system with a communication topology G, comprising N agents, each of which has a kinetic model described as:
Figure FDA0004092786440000013
wherein x is i (t)∈R n ,u i (t)∈R m ，f(x i (t),t)∈R n Respectively represent the state, input and nonlinear dynamics of the intelligent agent i, and has
Figure FDA0004092786440000014
Figure FDA0004092786440000015
A and B are constant matrices having corresponding dimensions;
definition 1: the multi-agent system is said to be capable of achieving time-varying state formation h if, for any given bounded initial condition, the following requirements are met:
Figure FDA0004092786440000021
wherein h is i ∈R n Representing state x i Offset of (t) and has
Figure FDA0004092786440000022
r (t) is a formation location function;
definition 2: if a controller u is used i (t) (i=1, 2,., N.) the multi-agent system is able to achieve formation h, then it is said that formation h is viable;
the hypothesis and quotation are as follows:
suppose 1: graph G is undirected graph and fully connected;
suppose 2: for any x, y ε R n Nonlinear dynamics f (x i (t), t) satisfies the following Li Puxi z condition:
Figure FDA0004092786440000023
lemma 1: the Laplacian matrix of the undirected graph G is L epsilon R N×N Then:
l has at least one zero eigenvalue, and the eigenvalue corresponds to an eigenvector of 1 N I.e. L1 N ＝0 N ；
If G is fully connected, 0 is a single eigenvalue of L, the remaining N-1 eigenvalues all have a positive real part, i.e. 0 = λ 1 ＜λ 2 ≤...≤λ N ；
And (4) lemma 2: if the matrix R > 0, then for any matrix X and scalar ρ > 0, the following is satisfied:
-XR -1 X≤ρ 2 R-2ρX
and (3) lemma 3: for symmetric matrix A 11 ，A 12 ，A 22 The following properties are equivalent:
Figure FDA0004092786440000024
A 11 ＜0，
Figure FDA0004092786440000025
A 22 ＜0，
Figure FDA0004092786440000026
2. the method for controlling dynamic event triggering of a nonlinear multi-intelligent system according to claim 1, wherein the step S1 specifically comprises:
designing an event triggering mechanism:
the formation error is defined as:
δ i (t)＝x i (t)-h i (i＝1,2,...,N)
and has
Figure FDA0004092786440000031
Defining the measurement error as:
Figure FDA0004092786440000032
where T is the sampling period and where,
Figure FDA0004092786440000033
is the latest trigger time of agent i and has +.>
Figure FDA0004092786440000034
Defining a combined measurement as:
Figure FDA0004092786440000035
wherein the method comprises the steps of
Figure FDA0004092786440000036
Is the latest trigger time of agent j and has +.>
Figure FDA0004092786440000037
The nonlinear term f (x i (t), t), the state estimate of the obtained agent i is:
Figure FDA0004092786440000038
wherein KT is the sampling time and has
Figure FDA0004092786440000039
Estimate the state of agent i at time KT and has +.>
Figure FDA00040927864400000310
Then, the state estimation error is obtained as:
Figure FDA00040927864400000311
wherein x is i (KT+T) is the sampling value of the state of the agent i at the time KT+T, and has
Figure FDA00040927864400000312
The event-triggered queuing control strategy based on the sampled data is designed as follows:
Figure FDA00040927864400000313
wherein the method comprises the steps of
Figure FDA00040927864400000314
And->
Figure FDA00040927864400000315
Dynamic event trigger function g of agent i i (t) is defined as:
Figure FDA0004092786440000041
Figure FDA0004092786440000042
wherein sigma i (KT) is a dynamic parameter, and θ > 0 is a positive constant.
3. According to claim 2The nonlinear multi-agent system dynamic event trigger control method is characterized in that an agent only checks a dynamic event trigger function g at each sampling moment KT i (t)；
First, according to
Figure FDA0004092786440000043
Updating dynamic parameter sigma i (KT)；
Then, substituting the sampled value into
Figure FDA0004092786440000044
If and only if g i When (KT) is not less than 0, the system triggers and updates the controller u i (t)；
The trigger time is expressed as:
Figure FDA0004092786440000045
and 4, lemma: for a given initial condition sigma i (0) E [0, 1) and θ > 0, with the adoption of
Figure FDA0004092786440000046
Parameter update is performed, then sigma i (KT) satisfying the following constraint at any time:
0≤σ i (KT)≤σ i (0)＜1,
Figure FDA0004092786440000047
4. the nonlinear multi-intelligent system dynamic event trigger control method according to claim 3, wherein the proving is performed by adopting a mathematical induction method:
when k=1, we get:
Figure FDA0004092786440000048
indicating 0.ltoreq.sigma i (T)≤σ i (0)＜1；
For the following
Figure FDA0004092786440000051
Let 0.ltoreq.sigma i (KT)≤σ i (0) < 1, give:
Figure FDA0004092786440000052
and has
Figure FDA0004092786440000053
Thus, sigma i (KT) monotonically decreases with increasing K, proving 0.ltoreq.sigma i ((K+1)T)≤σ i (KT) < 1, thereby obtaining, for any of
Figure FDA0004092786440000054
Inequality 0 +.sigma i (KT)≤σ i (0)＜1,/>
Figure FDA0004092786440000055
This is true.
5. The method for dynamic event triggering control of a non-linear multi-intelligent system according to claim 4, wherein a control strategy is adopted
Figure FDA0004092786440000056
And t.epsilon.KT, (K+1) T) and->
Figure FDA0004092786440000057
The multi-agent system has the following form:
Figure FDA0004092786440000058
delta of the formula i (t)＝x i (t)-h i (i=1, 2,) N) substituted into the above formula, further resulting in:
Figure FDA0004092786440000059
/>
let lambda get i (i=1, 2,., N) represents the eigenvalue of the laplace matrix L, J is about the right standard type of L, then there is
Figure FDA00040927864400000510
And U -1 ＝U T Satisfies the following formula:
U -1 LU＝U T LU＝J＝diag{λ 1 ,λ 2 ,...,λ N }
lambda is known from the quotation 1 1 =0, and its corresponding feature vector is
Figure FDA00040927864400000511
Definition of the definition
Figure FDA00040927864400000512
And->
Figure FDA00040927864400000513
The multi-agent system writes to:
Figure FDA00040927864400000514
defining a time delay function tau (T) =t-KT, t.epsilon.KT, (K+1) T, 0.ltoreq.tau (T) < T being piecewise linear, and having
Figure FDA00040927864400000515
At this time, a method for solving the time delay problem is adopted to solve the formation problem;
order the
Figure FDA0004092786440000061
η (t) =col { e (t), ε (t) }, furthermore,
Figure FDA0004092786440000062
the system
Figure FDA0004092786440000063
The writing is as follows:
Figure FDA0004092786440000064
Figure FDA0004092786440000065
system and method for controlling a system
Figure FDA0004092786440000066
Conversion of formation control problems into systems
Figure FDA0004092786440000067
Stability problems of (2);
and (5) lemma: for multiple intelligent systems
Figure FDA0004092786440000068
Using a controller
Figure FDA0004092786440000069
And dynamic event trigger function based on sampling data
Figure FDA00040927864400000610
Figure FDA00040927864400000611
A formation h can be implemented if and only if:
Figure FDA00040927864400000612
the proving process is as follows:
Figure FDA0004092786440000071
if and only if the following formula is satisfied:
Figure FDA0004092786440000072
the method comprises the following steps:
Figure FDA0004092786440000073
due to
Figure FDA0004092786440000074
Is nonsingular, < >>
Figure FDA0004092786440000075
Description->
Figure FDA0004092786440000076
Thus if and only if
Figure FDA0004092786440000077
When the state formation h is realized;
and (3) lemma 6: for event triggering functions
Figure FDA0004092786440000078
Figure FDA0004092786440000079
Given an initial condition σ (0) =diag { σ } 1 (0),σ 2 (0),...,σ N (0) ' and sigma N-1 (0)＝diag{σ 2 (0),...,σ N (0) } $ gives the following inequality:
Figure FDA00040927864400000710
wherein t.epsilon.KT, (K+1) T,
Figure FDA00040927864400000711
and->
Figure FDA00040927864400000712
The following was demonstrated:
the method comprises the following steps of:
Figure FDA00040927864400000713
wherein t.epsilon.KT, (K+1) T,
Figure FDA0004092786440000081
from delta i (t) and e i Definition of (t) gives:
Figure FDA0004092786440000082
will be described in
Figure FDA0004092786440000083
Substituted +.>
Figure FDA0004092786440000084
The method comprises the following steps:
Figure FDA0004092786440000085
this gives:
Figure FDA0004092786440000086
from the definition of η (t), it is known that:
Figure FDA0004092786440000087
Figure FDA0004092786440000088
substituting the two formulas into one
Figure FDA0004092786440000091
The method comprises the following steps:
Figure FDA0004092786440000092
and (5) finishing the verification.
6. The method for dynamic event-triggered control of a non-linear multi-intelligent system of claim 5, wherein for a multi-intelligent system
Figure FDA0004092786440000093
If a controller is used->
Figure FDA0004092786440000094
Based on the dynamic event trigger function of the sampled data, when hypothesis 1 and hypothesis 2 are satisfied, formation h is possible when the following conditions are satisfied:
A. for the following
Figure FDA0004092786440000095
And j epsilon N i The conditions are satisfied:
(A+BK 2 )(h i -h j )＝0
B. there is a real matrix
Figure FDA0004092786440000096
Figure FDA0004092786440000097
And sigma (sigma) i (0) E [0, 1), θ > 0, ρ > 0 satisfies the following inequality:
Figure FDA0004092786440000098
/>
Figure FDA0004092786440000099
wherein the method comprises the steps of
Figure FDA00040927864400000910
Is a matrix array and has
Figure FDA00040927864400000911
Figure FDA00040927864400000912
Figure FDA00040927864400000913
Figure FDA00040927864400000914
Figure FDA0004092786440000101
Figure FDA0004092786440000102
Figure FDA0004092786440000103
Figure FDA0004092786440000104
Figure FDA0004092786440000105
Figure FDA0004092786440000106
Figure FDA0004092786440000107
Figure FDA0004092786440000108
Figure FDA0004092786440000109
Figure FDA00040927864400001010
Figure FDA00040927864400001011
Figure FDA00040927864400001012
Figure FDA00040927864400001013
Figure FDA00040927864400001014
In addition, the controller gain matrix K 1 And K is equal to 3 From the following components
Figure FDA00040927864400001015
And->
Figure FDA00040927864400001016
Obtained.
7. The method for controlling dynamic event triggering of a nonlinear multi-intelligent system according to claim 6, wherein the proving process is as follows:
the following lyapunov function was constructed:
Figure FDA00040927864400001017
wherein p=p T ，Q＝Q T And r=r T Is a positive definite matrix;
when the condition A is satisfied, the following steps are obtained:
Figure FDA00040927864400001018
opposite type
Figure FDA0004092786440000111
Both sides are multiplied by +.>
Figure FDA0004092786440000112
Because of->
Figure FDA0004092786440000113
The method further comprises the following steps:
Figure FDA0004092786440000114
according to the quotation mark 1,
Figure FDA0004092786440000115
is non-singular; will->
Figure FDA0004092786440000116
Both sides are multiplied by +.>
Figure FDA0004092786440000117
The method comprises the following steps:
Figure FDA0004092786440000118
then
Figure FDA0004092786440000119
The writing is as follows:
Figure FDA00040927864400001110
according to the above formula, the derivative of V (t) is given by:
Figure FDA00040927864400001111
applying lemma 1, obtaining:
Figure FDA00040927864400001112
wherein χ is 1 ＝ε(t-τ(t))-ε(t)，χ 2 ε (T-T) - ε (T- τ (T)) and S is a real matrix;
Figure FDA00040927864400001113
if it meets the following conditions:
Figure FDA0004092786440000121
order the
Figure FDA0004092786440000122
And->
Figure FDA0004092786440000123
Multiplying the two sides by +.>
Figure FDA0004092786440000124
The above inequality is obtained as equivalent to:
Figure FDA0004092786440000125
/>
thus there is
Figure FDA0004092786440000126
Definition of the definition
Figure FDA0004092786440000127
Will be described in
Figure FDA0004092786440000128
Figure FDA0004092786440000129
Figure FDA00040927864400001210
Substituted formula
Figure FDA00040927864400001211
The derivative of V (t) is written as:
Figure FDA00040927864400001212
wherein, xi= [ xi ] p,q ] 6×6 Is a symmetrical matrix and has
Figure FDA00040927864400001213
Figure FDA0004092786440000131
Figure FDA0004092786440000132
Figure FDA0004092786440000133
Figure FDA0004092786440000134
Figure FDA0004092786440000135
Figure FDA0004092786440000136
Figure FDA0004092786440000137
Ξ 2,4 ＝Ω 2 ，
Ξ 2,5 ＝Ξ 2,6 ＝0 (N-1)n×Nn ，
Figure FDA0004092786440000138
Ξ 3,4 ＝Ξ 3,5 ＝Ξ 3,6 ＝0 (N-1)n×Nn ，
Ξ 4,4 ＝Ω 3 ，
Ξ 4,5 ＝Ξ 4,6 ＝Ξ 5,5 ＝Ξ 5,6 ＝Ξ 6,6 ＝0 Nn×Nn ，
Figure FDA0004092786440000139
The schulb's law is applied,
Figure FDA00040927864400001310
equivalent to
Figure FDA00040927864400001311
Wherein the method comprises the steps of
Figure FDA00040927864400001312
Multiplying the two sides simultaneously
Figure FDA00040927864400001313
The above inequality is obtained as equivalent to:
Figure FDA00040927864400001314
applying lemma 2 to obtain:
Figure FDA0004092786440000141
the method further comprises the following steps:
Figure FDA0004092786440000142
consider the conditions
Figure FDA0004092786440000143
And the Shu Er's theorem, get:
Figure FDA0004092786440000144
combining the above formulas yields:
Figure FDA0004092786440000145
equivalent to inequality
Figure FDA0004092786440000146
At the same time satisfy inequality
Figure FDA0004092786440000147
Thus, it is
Figure FDA0004092786440000148
Satisfy->
Figure FDA0004092786440000149
/>
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