CN112099357B - Finite time clustering synchronization and containment control method for discontinuous complex network - Google Patents

Finite time clustering synchronization and containment control method for discontinuous complex network Download PDF

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CN112099357B
CN112099357B CN202010999653.0A CN202010999653A CN112099357B CN 112099357 B CN112099357 B CN 112099357B CN 202010999653 A CN202010999653 A CN 202010999653A CN 112099357 B CN112099357 B CN 112099357B
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汤泽
高悦
王艳
纪志成
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Jiangnan University
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Abstract

A finite time clustering synchronization and containment control method for a discontinuous complex network belongs to the field of information technology containment controllers. Firstly, by introducing a Filipov differential inclusion theory and a measure selection theorem, the invention designs an effective containment feedback controller which only controls partial nodes which are directly connected with other clusters in the current cluster. In order to effectively save the control cost, the invention designs a self-adaptive updating law based on the feedback control strength to obtain the optimal control strength for realizing network synchronization. Secondly, by using the finite time stability theory and the Lyapunov stability theorem, the invention obtains the judgment condition for realizing finite time cluster synchronization by the time-varying time-lag coupling and nonlinear coupling Lur' e network, and provides the convergence time estimation for the network to reach cluster synchronization. Finally, the validity and the correctness of the control scheme and the synchronization criterion are verified through a mathematical example simulation.

Description

Finite time clustering synchronization and containment control method for discontinuous complex network
Technical Field
The invention relates to a complex network synchronization technology, and belongs to the field of information technology containment controllers.
Background
The complex network is a cross-disciplinary concept, and the source of the problem is actually various actual networks, such as a communication network, a biological neural network, a social network, a power network and the like. These actual networks can be transformed into abstract networks of points and lines by means of mathematical graph theory. By researching the commonality of the abstract network and the universal method for processing the abstract network, theoretical guidance can be provided for the analysis of the actual network. Therefore, the intensive research on the complex network has important practical significance.
In most current network synchronization studies, the time required for a network to achieve synchronization is uncertain and difficult to estimate. In practical engineering applications, in order to save time and cost, people often want to accelerate the convergence rate and achieve network synchronization within a limited time. Meanwhile, limited time synchronization means optimization of synchronization convergence time, and the method has better robustness and anti-interference capability.
To date, limited time cluster synchronization for discontinuous complex networks, especially considering the problems of time-varying time lags and non-linear coupling of the networks, has received little attention from researchers. The complexity of theory and the importance of practical applications have prompted us to do current work. The invention researches the finite time clustering synchronization problem of the non-constant discontinuous Lur' e network with multiple couplings by designing an effective containment controller.
Disclosure of Invention
The technical problem to be solved by the invention is to achieve the following aims: the invention considers the non-constant discontinuous Lur' e network with time-varying time lag and nonlinear coupling, and the network model is closer to the actual engineering application system; the traditional linearization sector condition is not applicable to the discontinuous Lur' e system, the invention introduces a Filipov differential inclusion concept, converts the discontinuous function into a set value function, selects a measurable function corresponding to the discontinuous function according to a measure selection theorem, and carries out linearization processing on the nonlinear function according to the measurable function; a small number of nodes which are directly connected with other clusters in the current cluster are controlled by skillfully designing a negative feedback containment controller, and meanwhile, the optimal control strength is obtained according to an updating law of designing adaptive control, so that the control cost is effectively reduced; according to the finite time stability theorem, the invention provides the synchronous convergence time for realizing the cluster synchronization of the network, thereby greatly saving the time cost of control.
The technical scheme of the invention is as follows:
the finite time clustering synchronization and containment control method of the discontinuous complex network comprises the following steps:
step one, establishing a complex network with a non-constant non-continuous Lur' e system and determining a synchronization target of the complex network
(1.1) establishing a controlled discontinuous non-constant Lur' e dynamic network model with time-varying time lag and nonlinear coupling:
Figure BDA0002693820920000011
wherein: n represents the number of the Lur' e systems in the network,
Figure BDA0002693820920000021
represents the state vector of the ith Lur' e system,
Figure BDA0002693820920000022
represents the nth dimension component of the ith Lur' e system state vector,
Figure BDA0002693820920000023
representation to state vector yi(t) taking the derivative of the signal,
Figure BDA0002693820920000024
and
Figure BDA0002693820920000025
is a constant matrix, m and n represent dimensions; normal number c1，c2Is the coupling strength, the coupling matrix
Figure BDA0002693820920000026
Satisfy the dissipation condition, i.e.
Figure BDA0002693820920000027
Figure BDA0002693820920000028
If the ith Lur 'e system is connected with the jth Lur' e system, i is not equal to j, then: a isij＞0，bij> 0, otherwise aij＝0，bij0. The time-varying time lag tau (t) satisfies the derivative that tau (t) is more than or equal to 0 and less than or equal to tau and the time-varying time lag
Figure BDA0002693820920000029
Constant mu satisfies
Figure BDA00026938209200000210
Figure BDA00026938209200000211
Is a continuous nonlinear coupling function and
Figure BDA00026938209200000212
Figure BDA00026938209200000213
is a function of the value of the coupling vector G (y)j(t)) of the nth dimensional component of the image,
Figure BDA00026938209200000214
is a discontinuous non-linear vector value function and satisfies the condition that the initial value f (0) is 0
Figure BDA00026938209200000215
Figure BDA00026938209200000216
Representing a non-linear vector-valued function
Figure BDA00026938209200000217
Component of the m-th dimension of (1), wherein
Figure BDA00026938209200000218
Figure BDA00026938209200000219
ui(t) is a control input, which will be designed in detail later.
(1.2) determining synchronization targets for complex networks
The following independent Lur' e system is taken as a synchronization target in the complex dynamic network (1):
Figure BDA00026938209200000220
wherein:
Figure BDA00026938209200000221
is shown asμiThe synchronization target vectors of the individual clusters are,
Figure BDA00026938209200000222
represents the μ thiThe nth dimension component of the synchronization target vector of the respective cluster,
Figure BDA00026938209200000223
representing a pair synchronization target vector
Figure BDA00026938209200000224
The derivation is carried out by the derivation,
Figure BDA00026938209200000225
Figure BDA00026938209200000226
is a discontinuous non-linear vector value function and satisfies the condition that the initial value f (0) is 0
Figure BDA00026938209200000227
Figure BDA00026938209200000228
Representing a non-linear vector-valued function
Figure BDA00026938209200000229
Component of the m-th dimension of (1), wherein
Figure BDA00026938209200000230
Figure BDA00026938209200000231
Step two, acquiring state information of each node through a sensor device and establishing an error model
By defining error vectors
Figure BDA00026938209200000232
ei n(t) represents the nth-dimension component of the error vector of the ith node. The controlled error Lur' e network can be obtained by the formulas (1) and (2):
Figure BDA00026938209200000234
where i is 1,2, …, N, a non-linear function
Figure BDA0002693820920000031
Coupling function
Figure BDA0002693820920000032
Is a continuous nonlinear coupling function and
Figure BDA0002693820920000033
Figure BDA0002693820920000034
Figure BDA0002693820920000035
is a function of the value of the coupling vector
Figure BDA0002693820920000036
The nth dimensional component of (a).
Step three, designing a containment controller
Negative feedback containment controller u designed for error Lur' e networki(t)：
Figure BDA0002693820920000037
Wherein:
Figure BDA0002693820920000038
indicates belonging to the μiA set of clusters and directly connected Lur' e systems with other classes,
Figure BDA0002693820920000039
indicates belonging to the μiA set of all the Lur' e systems of each cluster.
Figure BDA00026938209200000310
Wherein: li(t) is a time-varying control strength, constant ρ > 0, sign is a sign function and
Figure BDA00026938209200000311
Figure BDA00026938209200000312
function sign (e) representing vector valuesi(t)) of the nth dimension component, ei(S) represents the integrated error vector; control gain etai＞0，αiIs greater than 0; is provided with
Figure BDA00026938209200000313
Is 1i(t) estimation, then there are normal numbers
Figure BDA00026938209200000314
Satisfy the requirement of
Figure BDA00026938209200000315
Defining a function omega (e)i(t)) the following:
Figure BDA00026938209200000316
||ei(t)||2representing an error vector eiThe square of the 2 norm of (t). To obtain an optimal control intensity, a time-varying control intensity l is usedi(t) the following adaptive update law is designed:
Figure BDA00026938209200000317
wherein the coefficient εi＞0。
Step four, judging whether finite time clustering synchronization is realized
Condition 1: function(s)
Figure BDA00026938209200000318
Except at countable point set { ρkOuter is continuously differentiable, left limit at discontinuous point
Figure BDA00026938209200000319
And right limit
Figure BDA00026938209200000320
Is present, k is a positive integer, and, in
Figure BDA00026938209200000321
Each of the compact sections of (a) and (b),
Figure BDA00026938209200000322
there are at most a limited number of jumping discontinuities.
Condition 2: function(s)
Figure BDA00026938209200000323
Satisfy condition 1, remember Filipov collection-valued mapping
Figure BDA00026938209200000324
Figure BDA00026938209200000325
Representing collection-valued mappings
Figure BDA00026938209200000326
The m-th dimension of (1), vector s ═ s1,s2,…,sm]T，smRepresenting the mth dimension component of the vector s.
Figure BDA00026938209200000327
Satisfy the requirement of
Figure BDA00026938209200000328
Presence of normal number
Figure BDA0002693820920000041
And
Figure BDA0002693820920000042
such that for an arbitrary vector s ═ s1,s2,…,sm]T，
Figure BDA00026938209200000418
ZmRepresenting the m-th component of vector Z, the following inequality holds:
Figure BDA0002693820920000043
wherein: sup denotes the supremum, i.e. the minimum upper bound,
Figure BDA0002693820920000044
Figure BDA0002693820920000045
Figure BDA0002693820920000046
correspondingly, a linearized estimation form of the non-linear vector value function is given
Figure BDA0002693820920000047
Wherein:
Figure BDA0002693820920000048
if the conditions 1 and 2 are satisfied, and the function gk(. epsilon. NCF (xi, delta.), xi > delta > 0, k ═ 1,2, …, n. If the normal z, β is present, the following condition holds:
(1) the inequality:
Figure BDA0002693820920000049
(2) the matrix inequality:
Figure BDA00026938209200000410
(3) the inequality:
Figure BDA00026938209200000411
under the action of the holdover controller (4) and the adaptive update law (7), the discontinuous Lur 'e network (1) and the target Lur' e system (2) can realize cluster synchronization in a limited time, and the synchronous convergence time is estimated to be
Figure BDA00026938209200000412
Wherein: lambda [ alpha ]maxThe representation is taken to be the maximum characteristic value,
Figure BDA00026938209200000413
if it is
Figure BDA00026938209200000414
Figure BDA00026938209200000415
Otherwise
Figure BDA00026938209200000416
max is the maximum value, ρ > 0, θ min { θ ═1,θ2,θ3}，
Figure BDA00026938209200000417
INRepresenting an N-dimensional identity matrix; v (-) represents the Lyapunov function of the design.
The invention has the beneficial effects that: the advantages brought by the invention are the indexes achieved.
1. The invention researches a finite time clustering synchronization problem with time-varying time-lag coupling and nonlinear coupling Lur' e dynamic networks. By designing a feedback containment controller, based on a Filipov differential inclusion concept, a finite time stability theory and a Lyapunov stability theorem, a sufficient condition for ensuring that the network achieves synchronization within a finite time is obtained, and estimation of synchronization convergence time is given.
2. Due to the wide application of the discontinuous system in the actual engineering, the invention considers the complex network formed by coupling the discontinuous non-constant Lur' e system, and has more practical significance.
3. It can be seen from the design of the controller that the controller is only applied to the Lur' e system with direct connection between different clusters, and at the same time it can be seen that,
Figure BDA0002693820920000051
the items are mainly used for weakening the influence caused by the mutual connection of the Lur 'e systems among different clusters, and other items are mainly used for synchronizing all non-continuous Lur' e systems belonging to the same cluster.
4. Under the action of the controller, the Lur 'e systems in the same cluster can achieve cluster synchronization within a limited time, and the state of the Lur' e systems among different clusters has no requirement, namely the complex network can achieve cluster synchronization within the limited time. This synchronous mode will save the time cost of control significantly.
5. The invention is concerned with the discontinuous Lur' e system, so the method for nonlinear continuous function linearization cannot be applied. Therefore, the invention introduces the definition of Filippav collection-value mapping to convert discontinuous functions into Filippav collection-value mapping functions, maps a certain point into a collection, and then selects a measurable function from the collection to ensure a general solution of the discontinuous functions.
Drawings
Fig. 1 is a diagram illustrating a network synchronization process.
Fig. 2 is a topology structure diagram of a network.
FIG. 3 is a state evolution curve of a Lur 'e system in three clusters, wherein (a) is the state evolution curve of the Lur' e system in cluster 1, (b) is the state evolution curve of the Lur 'e system in cluster 2, and (c) is the state evolution curve of the Lur' e system in cluster 3.
Fig. 4 is a synchronization error curve for three clusters.
FIG. 5 shows adaptive control strength li(t) evolution curve.
Detailed Description
In the following we will perform a numerical simulation example to illustrate the effectiveness of this invention.
The invention considers a discontinuous non-constant complex network:
Figure BDA0002693820920000052
wherein:
Figure BDA0002693820920000053
a state vector representing the ith Lur' e system;
Figure BDA0002693820920000054
represents the nth dimension component of the ith Lur' e system state vector,
Figure BDA0002693820920000055
representation to state vector yi(t) taking the derivative of the signal,
Figure BDA0002693820920000056
and
Figure BDA0002693820920000057
is a constant matrix; normal number c1，c2Is the coupling strength; coupling matrix
Figure BDA0002693820920000058
Figure BDA0002693820920000059
Satisfy the dissipation condition, i.e.
Figure BDA00026938209200000510
If there is a connection between the ith and jth Lur' e systems (i ≠ j), then there is: a isij＞0，bij> 0, otherwise aij＝0，b ij0. The time-varying time lag tau (t) satisfies the derivative that tau (t) is more than or equal to 0 and less than or equal to tau and the time-varying time lag
Figure BDA0002693820920000061
Constant mu satisfies
Figure BDA0002693820920000062
Figure BDA0002693820920000063
Is a continuous nonlinear coupling function and
Figure BDA0002693820920000064
Figure BDA0002693820920000065
is a function of the value of the coupling vector G (y)j(t)) of the nth dimensional component of the image,
Figure BDA0002693820920000066
is a discontinuous non-linear vector value function and satisfies the condition that the initial value f (0) is 0
Figure BDA0002693820920000067
Wherein
Figure BDA0002693820920000068
ui(t) is a control input, which will be designed in detail later.
Synchronization is a special cluster behavior, which represents that the paces of all systems in a complex network tend to be consistent, so we can regard the solution of a certain system as a synchronization target, and when all systems in the network are synchronized with the synchronization target, i.e. the paces of all systems in the network tend to be consistent. In the present invention, we consider the following solution of the independent Lur' e system as the synchronization target:
Figure BDA0002693820920000069
wherein:
Figure BDA00026938209200000610
represents the μ thiThe synchronization target vectors of the individual clusters are,
Figure BDA00026938209200000611
represents the μ thiThe nth dimension component of the synchronization target vector of the respective cluster,
Figure BDA00026938209200000612
representing a pair synchronization target vector
Figure BDA00026938209200000613
The derivation is carried out by the derivation,
Figure BDA00026938209200000614
Figure BDA00026938209200000615
is a discontinuous non-linear vector value function and satisfies the condition that the initial value f (0) is 0
Figure BDA00026938209200000616
Representing a non-linear vector-valued function
Figure BDA00026938209200000617
Component of the m-th dimension of (1), wherein
Figure BDA00026938209200000618
Figure BDA00026938209200000619
By defining error vectors
Figure BDA00026938209200000620
The controlled error Lur' e network can be obtained by the formulas (1) and (2):
Figure BDA00026938209200000621
where i is 1,2, …, N, a non-linear function
Figure BDA00026938209200000622
Coupling function
Figure BDA00026938209200000623
Is a continuous nonlinear coupling function and
Figure BDA00026938209200000624
Figure BDA00026938209200000625
Figure BDA00026938209200000626
is a function of the value of the coupling vector
Figure BDA00026938209200000627
The nth dimensional component of (a).
In order to realize the finite time clustering synchronization between the Lur' e networks (1) and (2), the invention designs the following containment control strategy by transmitting the state information of the adjacent node and the target synchronization node to each node:
Figure BDA0002693820920000071
wherein:
Figure BDA0002693820920000072
indicates belonging to the μiA set of clusters and directly connected Lur' e systems with other classes,
Figure BDA0002693820920000073
indicates belonging to the μiA set of all the Lur' e systems of each cluster.
Figure BDA0002693820920000074
Wherein: li(t) is a time-varying control strength, constant ρ > 0, sign is a sign function and
Figure BDA0002693820920000075
control gain etai＞0，αiIs greater than 0; suppose that
Figure BDA0002693820920000076
Is 1i(t) estimation, then there are normal numbers
Figure BDA0002693820920000077
Satisfy the requirement of
Figure BDA0002693820920000078
Defining a function omega (e)i(t)) the following:
Figure BDA0002693820920000079
||ei(t)||2representing an error vector eiSquare of the 2 norm of (t), for a time-varying control intensity l in order to obtain an optimal control intensityi(t) the following adaptive update law is designed:
Figure BDA00026938209200000710
wherein the coefficient εi＞0。
From the definition of the Filipov collection-valued map:
Figure BDA00026938209200000711
wherein:
Figure BDA00026938209200000712
Figure BDA00026938209200000713
showing a Filippov collection-valued map. Define the set value function SIGN (·):
Figure BDA00026938209200000714
defining: consider including
Figure BDA00026938209200000715
The Lur' e network (1) of each cluster has a time T > 0 when mu is greater thani＝μjSatisfy the following requirements
Figure BDA00026938209200000716
And for any T > T, | | yi(t)-yj(t) | | ≡ 0, and when μ ≡ 0i≠μjWhen the temperature of the water is higher than the set temperature,
Figure BDA0002693820920000081
the Lur' e network (1) is said to be capable of finite time cluster synchronization, where T is called synchronization convergence time.
In the following, we will discuss the conditions for obtaining a limited-time cluster synchronization between the Lur 'e network (1) and the targeted Lur' e system (2) by designing the holdback controller (5).
The following Lyapunov function V (t) is chosen:
Figure BDA0002693820920000082
calculating V (t) derivative L of Lie with respect to time tf(V (t)), and selecting a function according to the Filippov collection value mapping property and measure selection theorem
Figure BDA0002693820920000083
So that
Figure BDA0002693820920000084
Can obtain the product
Figure BDA0002693820920000085
Nonlinear vector value function in pair (10) formula
Figure BDA0002693820920000086
Linear estimation is performed to obtain
Figure BDA0002693820920000087
Based on the nature of the inequality, measure selection theorem and associated lemma, we can convert equation (9) to:
Figure BDA0002693820920000088
wherein: constant z > 0, InIs an n-dimensional identity matrix, vector
Figure BDA0002693820920000089
Figure BDA00026938209200000810
Matrix array
Figure BDA00026938209200000811
Figure BDA0002693820920000091
If it is
Figure BDA0002693820920000092
Otherwise
Figure BDA0002693820920000093
Figure BDA0002693820920000094
Because of the fact that
Figure BDA0002693820920000095
Figure BDA0002693820920000096
The following can be obtained:
Figure BDA0002693820920000097
according to the finite time stability theory, the synchronization convergence time can be estimated as:
Figure BDA0002693820920000098
from the above analysis, the state trajectory of the controlled error network can be obtained
Figure BDA0002693820920000099
The clustering synchronization can be achieved by the discontinuous Lur 'e network (1) and the target Lur' e system (2) within the synchronization convergence time T under the action of the controller (4) and the adaptive updating law (6). Based on the above discussion, we have obtained a condition that a non-continuous non-constant Lur' e complex network achieves cluster synchronization in a limited time, and the certification is completed.
And (4) conclusion:
if condition 1 and condition 2 are satisfied and function gk(. epsilon. NCF (xi, delta.), xi > delta > 0, k ═ 1,2, …, n. If the normal z, β is present, the following condition holds:
(1) inequality:
Figure BDA00026938209200000910
(2) the matrix inequality:
Figure BDA00026938209200000911
(3) inequality:
Figure BDA00026938209200000912
under the action of the holdover controller (4) and the adaptive update law (5), the non-continuous Lur 'e network (1) and the target Lur' e system (2) can realize cluster synchronization in a limited time, and the synchronization convergence time can be estimated to be
Figure BDA00026938209200000913
Wherein:
Figure BDA00026938209200000914
if it is
Figure BDA00026938209200000915
Otherwise
Figure BDA00026938209200000916
Figure BDA00026938209200000917
Step 1: a complex network coupled by 10 Lur' e systems is established and divided into 3 clusters (Lambda)1＝{1,2,3},Λ2＝{4,5,6}, Λ 17,8,9, 10), wherein the dynamical equations of the Lur' e system in each cluster are as follows:
Figure BDA0002693820920000101
i 1,2, …,10, j 1,2,3, where the matrix E1＝diag{-0.99,-0.97,-0.99}，E2＝diag{-1.01,-1,-0.99}，E3＝diag{-1.2,-1.2,-1.01}，M1＝M2＝M3＝I3，
Figure BDA0002693820920000102
Figure BDA0002693820920000103
Figure BDA0002693820920000104
Can obtain the product
Figure BDA0002693820920000105
Figure BDA0002693820920000106
||D1||＝3.01,||D2||＝3.00,||D32.93, selecting a nonlinear function fj(. is) f1(v)＝f2(v)＝f2(v) 0.2v +0.09sign (v), according to hypothesis 2
Figure BDA0002693820920000107
Taking the coupling strength c1＝0.3,c2Let τ (t) be 0.01+ sin (0.1t) 0.6,
Figure BDA0002693820920000108
then g isk(. epsilon. NCF (1.5,0.3), i ═ 1,2, …,10, k ═ 1,2, 3; depending on the topology of the network, control will therefore be exerted on the Lur' e system 3, 4, 9, 10.
Step 2: and (3) determining the state model of the target Lur' e system in the three clusters as shown in (13), wherein the relevant parameters are as follows: matrix E1＝diag{-0.99,-0.97,-0.99}，E2＝diag{-1.01,-1,-0.99}，E3＝diag{-1.2,-1.2,-1.01}，M1＝M2＝M3＝I3，
Figure BDA0002693820920000109
Figure BDA00026938209200001010
Figure BDA00026938209200001011
Therefore, the goal of finite time cluster synchronization is to synchronize the Lur 'e systems in the three clusters to their corresponding target Lur' e systems within a finite time T.
And step 3: and calculating specific parameters meeting the specific model by using an LMI tool box according to three inequality conditions which need to be met when the network reaches cluster synchronization in a limited time.
And 4, step 4: and (3) building a Simulink model to obtain a simulation result, and as can be seen from the graphs in FIGS. 3-5, the finite time clustering synchronization of the node states is achieved under the proposed conditions.

Claims (1)

1. The finite time clustering synchronization and containment control method of the discontinuous complex network comprises the following steps:
step one, establishing a complex network with a non-constant non-continuous Lur' e system and determining a synchronization target of the complex network
(1.1) establishing a controlled discontinuous non-constant Lur' e dynamic network model with time-varying time lag and nonlinear coupling:
Figure FDA0003110026500000011
wherein: n represents the number of the Lur' e systems in the network,
Figure FDA0003110026500000012
represents the state vector of the ith Lur' e system,
Figure FDA0003110026500000013
represents the nth dimension component of the ith Lur' e system state vector,
Figure FDA0003110026500000014
representation to state vector yi(t) taking the derivative of the signal,
Figure FDA0003110026500000015
and
Figure FDA0003110026500000016
is a constant matrix, m and n represent dimensions; normal number c1，c2Is the coupling strength, the coupling matrix
Figure FDA0003110026500000017
Satisfy the dissipation condition, i.e.
Figure FDA0003110026500000018
Figure FDA0003110026500000019
If the ith Lur 'e system is connected with the jth Lur' e system, i is not equal to j, then: a isij>0，bij>0, otherwise aij＝0，bij0; the time-varying time lag tau (t) satisfies the derivative that tau (t) is more than or equal to 0 and less than or equal to tau and the time-varying time lag
Figure FDA00031100265000000110
Constant mu satisfies
Figure FDA00031100265000000111
Is a continuous nonlinear coupling function and
Figure FDA00031100265000000112
Figure FDA00031100265000000113
is a function of the value of the coupling vector G (y)j(t)) of the nth dimensional component of the image,
Figure FDA00031100265000000114
is a discontinuous non-linear vector value function and satisfies the condition that the initial value f (0) is 0
Figure FDA00031100265000000115
Figure FDA00031100265000000116
Representing a non-linear vector-valued function
Figure FDA00031100265000000117
Component of the m-th dimension of (1), wherein
Figure FDA00031100265000000118
Figure FDA00031100265000000119
i＝1,2,…,N,j＝1,2,…,m，ui(t) is a control input, which will be designed in detail later;
(1.2) determining synchronization targets for complex networks
Taking the following independent target Lur 'e system as a synchronous target in the controlled discontinuous non-constant Lur' e dynamic network model (1):
Figure FDA00031100265000000120
wherein:
Figure FDA00031100265000000121
represents the μ thiThe synchronization target vectors of the individual clusters are,
Figure FDA00031100265000000122
represents the μ thiThe nth dimension component of the synchronization target vector of the respective cluster,
Figure FDA00031100265000000123
representing a pair synchronization target vector
Figure FDA00031100265000000124
The derivation is carried out by the derivation,
Figure FDA00031100265000000125
Figure FDA00031100265000000126
is a discontinuous non-linear vector value function and satisfies the initial value f (0) ═0, note
Figure FDA00031100265000000127
Figure FDA00031100265000000128
Representing a non-linear vector-valued function
Figure FDA00031100265000000129
Component of the m-th dimension of (1), wherein
Figure FDA00031100265000000130
Figure FDA00031100265000000131
i＝1,2,…,N,j＝1,2,…,m；
Step two, acquiring state information of each node through a sensor device and establishing an error model
By defining error vectors
Figure FDA0003110026500000021
ei n(t) an nth-dimension component of an error vector representing an ith node; obtaining a controlled error Lur' e network from equations (1) and (2):
Figure FDA0003110026500000022
where i is 1,2, …, N, a non-linear function
Figure FDA0003110026500000023
Coupling function
Figure FDA0003110026500000024
Figure FDA0003110026500000025
Is a continuous nonlinear coupling function and
Figure FDA0003110026500000026
Figure FDA0003110026500000027
Figure FDA0003110026500000028
is a function of the value of the coupling vector
Figure FDA0003110026500000029
The nth dimensional component of (a);
step three, designing a containment controller
Negative feedback containment controller u designed for error Lur' e networki(t)：
Figure FDA00031100265000000210
Wherein:
Figure FDA00031100265000000211
indicates belonging to the μiA set of clusters and directly connected Lur' e systems with other classes,
Figure FDA00031100265000000212
indicates belonging to the μiA set of all Lur' e systems of each cluster;
Figure FDA00031100265000000213
wherein: li(t) is a time-varying control intensity, constant ρ>0, sign is a sign function
Figure FDA00031100265000000214
Figure FDA00031100265000000215
Function of representing vector values
Figure FDA00031100265000000216
The nth dimensional component of ei(S) represents the integrated error vector; control gain etai>0，αi>0; is provided with
Figure FDA00031100265000000217
Is 1i(t) estimation, then there are normal numbers
Figure FDA00031100265000000218
Satisfy the requirement of
Figure FDA00031100265000000219
Defining a function omega (e)i(t)) the following:
Figure FDA00031100265000000220
||ei(t)||2representing an error vector ei(t) square of 2 norm; to obtain an optimal control intensity, a time-varying control intensity l is usedi(t) the following adaptive update law is designed:
Figure FDA00031100265000000221
wherein the coefficient εi>0；
Step four, judging whether finite time clustering synchronization is realized
Condition 1: function(s)
Figure FDA0003110026500000031
Except at countable point set { ρkOuter is continuously differentiable, left limit at discontinuous point
Figure FDA0003110026500000032
And right limit
Figure FDA0003110026500000033
Is present, k is a positive integer, and, in
Figure FDA00031100265000000323
Each of the compact sections of (a) and (b),
Figure FDA0003110026500000034
there are at most a limited number of jumping discontinuities;
condition 2: function(s)
Figure FDA0003110026500000035
Satisfy condition 1, remember Filipov collection-valued mapping
Figure FDA0003110026500000036
Figure FDA0003110026500000037
Representing collection-valued mappings
Figure FDA0003110026500000038
The m-th dimension of (1), vector s ═ s1,s2,…,sm]T，smRepresents the m-th component of the vector s;
Figure FDA0003110026500000039
satisfy the requirement of
Figure FDA00031100265000000310
Presence of normal number
Figure FDA00031100265000000311
And
Figure FDA00031100265000000312
such that for an arbitrary vector s ═ s1,s2,…,sm]T，
Figure FDA00031100265000000313
ZmRepresenting the m-th component of vector Z, the following inequality holds:
Figure FDA00031100265000000314
wherein: sup denotes the supremum, i.e. the minimum upper bound,
Figure FDA00031100265000000315
Figure FDA00031100265000000316
1,2, …, N, j 1,2, …, m; correspondingly, a linearized estimation form of the non-linear vector value function is given
Figure FDA00031100265000000317
Wherein:
Figure FDA00031100265000000318
if the conditions 1 and 2 are satisfied, and the function gk(·)∈NCF(ξ，δ)，ξ>δ>0, k ═ 1,2, …, n; if the normal z, β is present, the following condition holds:
(1) the inequality:
Figure FDA00031100265000000319
(2) the matrix inequality:
Figure FDA00031100265000000320
(3) the inequality:
Figure FDA00031100265000000321
under the action of a containment controller (4) and an adaptive updating law (7), the controlled discontinuous non-constant Lur 'e dynamic network model (1) and the target Lur' e system (2) can realize cluster synchronization in limited time, and the synchronous convergence time is estimated to be
Figure FDA00031100265000000322
Wherein: lambda [ alpha ]maxThe representation is taken to be the maximum characteristic value,
Figure FDA0003110026500000041
if it is
Figure FDA0003110026500000042
Figure FDA0003110026500000043
Otherwise
Figure FDA0003110026500000044
Where i is 1,2, …, N, max denotes the maximum value, ρ>0，θ＝min{θ1,θ2,θ3}，
Figure FDA0003110026500000045
INRepresenting an N-dimensional identity matrix; v (-) represents the Lyapunov function of the design.
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