CN111638648A - Distributed pulse quasi-synchronization method with proportional delay complex dynamic network - Google Patents

Distributed pulse quasi-synchronization method with proportional delay complex dynamic network Download PDF

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CN111638648A
CN111638648A CN202010505988.2A CN202010505988A CN111638648A CN 111638648 A CN111638648 A CN 111638648A CN 202010505988 A CN202010505988 A CN 202010505988A CN 111638648 A CN111638648 A CN 111638648A
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lur
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synchronization
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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 distributed pulse quasi-synchronization method with a proportional delay complex dynamic network belongs to the field of pulse control. Due to heterogeneity among different Lur' e systems in the network, the present invention discusses plesiochronous rather than complete synchronization of complex networks. Unlike the typical time delay, the proportional delay considered by the present invention is an unbounded time-varying delay, which greatly increases the requirement for the network to achieve synchronization. According to different functions of a distributed pulse containment control protocol and a pulse effect, a delay pulse comparison principle is effectively combined, a generalized parameter variation formula and an average pulse interval are defined, and finally, a quasi-synchronization judgment method of a coupled non-constant Lur' e network is provided. In addition, network synchronization errors under the conditions of different functional pulse effects are reasonably estimated, and corresponding index convergence speed is given. In addition, the invention also provides a relevant numerical example to illustrate the effectiveness of the network quasi-synchronization judging method and the controller design scheme.

Description

Distributed pulse quasi-synchronization method with proportional delay complex dynamic network
Technical Field
The invention relates to a complex network synchronization technology, and belongs to the technical field of information.
Background
In recent years, research into the collective behavior of complex power networks has become increasingly significant, as it has many applications in the fields of human real life and manufacturing. Of these collective behaviors of complex networks, synchronization is absolutely one of the most important behaviors, representing the uniform behavior of some or all of the systems in the network. In a complex network, there are often some uncertainties in the transmission and exchange of information between different systems, and time delay is undoubtedly one of the most important factors affecting the performance of the system under study. Due to the advantages of controllability and predictability of the proportional delay, the proportional delay is more practical and meaningful to be considered when the complex network is modeled.
The pulse control is a discontinuous control mode, provides instant power for the network, and can greatly save control cost compared with a continuous control method. To date, many existing works have discussed pulse synchronization for complex networks with positive pulse effects, but pulse effects that play a negative role in the network should also be considered and discussed.
To date, the problem of pulse synchronization in complex dynamic networks with proportional delays 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 quasi-synchronization of the coupled complex dynamic network by designing the distributed pulse containment controller.
Disclosure of Invention
The technical problem to be solved by the invention is to achieve the following aims: because of the problem of unmatched system parameters, the invention fully discusses the quasi-synchronization of a complex dynamic network consisting of different Lur' e systems; due to the inevitable reasons of some networks themselves, or sometimes the requirement for synchronization decreases, the synchronization error may no longer go to zero as time goes to finite or infinite. This synchronization phenomenon is called plesiochronous. In view of the mismatch of system parameters in real-world systems and the control cost constraints, the present invention discusses quasi-synchronization rather than full synchronization. Different from the time-invariant delay or the time-variant delay researched in the past, the method considers the proportional delay when modeling the complex dynamic network, and the proportional delay is unbounded time-variant delay; the invention adopts a distributed pulse containment controller, and obtains the quasi-synchronization condition of the Lur' e network according to the pulse comparison principle, the parameter variation expansion formula and the average pulse interval definition; according to the positive and negative effects of the pulse effect, different synchronization errors and different exponential convergence speeds are respectively estimated by skillfully constructing some parameter functions.
The technical scheme of the invention is as follows:
a distributed pulse quasi-synchronization method with a proportional delay Lur' e network comprises the following steps:
step one, establishing a complex network with different Lur' e systems and proportional delays and determining a leader system of the complex network
(1.1) building complex networks with different Lur' e systems and proportional delays:
Figure BDA0002526552380000021
wherein the content of the first and second substances,
Figure BDA0002526552380000022
is the state vector of the ith Lur' e system, i ═ 1,2, …, N; matrix array
Figure BDA0002526552380000023
And
Figure BDA0002526552380000024
is a constant matrix; the constant c > 0 is the coupling strength,
Figure BDA0002526552380000025
is an internal coupling matrix, where riNot less than 0; is given as In；InRepresenting an n-dimensional identity matrix. Function(s)
Figure BDA0002526552380000026
Is a memory-free non-linear vector value function,in that
Figure BDA0002526552380000027
The upper part of the material is continuous and micro,
Figure BDA0002526552380000028
is an external coupling matrix determined by network topology, and G is set to satisfy the condition of zero sum row
Figure BDA0002526552380000029
When the connection exists between the ith Lur 'e system and the jth Lur' e system i ≠ j, gji＝gij> 0, otherwise gijProportional delay q ∈ (0,1) is a constant related to historical time, t is the previous instant time, q is the proportional delay, zj(qt) is the state at the history time qt, qt ═ t- τ (t), where τ (t) ≧ 0 (1-q) t, τ (t) → ∞.
Let matrix C ═ C1,c2,…,cm]TWherein
Figure BDA00025265523800000210
j — 1,2, …, m, yielding:
Czi(t)＝[c1zi(t),c2zi(t),…,cmzi(t)]T，
Figure BDA00025265523800000211
(1.2) leader System for determining Complex network
The following independent Lur' e systems are taken as leaders in the complex dynamic network (1):
Figure BDA00025265523800000212
wherein the content of the first and second substances,
Figure BDA00025265523800000213
is the state vector of the independent system. Constant matrix
Figure BDA00025265523800000214
Figure BDA00025265523800000215
Therefore, synchronization between the complex dynamic network (1) and the Lur' e system (2) is considered as a leader-follower problem;
step two, acquiring state information of each node through a sensor device and establishing an error model
By defining an error vector zi(t)＝zi(t)-z(t),
Figure BDA00025265523800000216
1,2, …, N; obtain an application controller uiError Lur' e network of (t):
Figure BDA00025265523800000217
wherein
Figure BDA00025265523800000218
i＝1,2,…,N；Yi(z (t)) represents parameter mismatch, or heterogeneity between so-called Lur' e networks (1) and (2).
Is provided with a controller ui(t) obeying the impulse perturbation with coefficient mu, obtaining a controlled error Lur' e network
Figure BDA0002526552380000031
Where μ is the pulse effect, and depending on the different values of μ, the pulses may have an effect that is beneficial or not beneficial to synchronization. Ω (-) is a dirac function, and the pulse sequence ζ ═ t is set1,t2,…，tkIs a strictly increasing sequence of instantaneous values of the pulse, which satisfies tk-1＜tkAnd limk→+∞tk＝+∞，
Figure BDA0002526552380000032
Figure BDA0002526552380000033
Representing a set of natural numbers.
Step three, designing a distributed type containment controller
The distributed containment controller consists of a distributed control item and a feedback control item, and a distributed strategy based on containment control is established:
Figure BDA0002526552380000034
wherein the content of the first and second substances,
Figure BDA0002526552380000035
represents the set of all other Lur 'e systems directly connected to the ith Lur' e system.
Figure BDA0002526552380000036
Is a controlled coupling matrix which satisfies the zero-sum row condition, that is to say
Figure BDA0002526552380000037
If the ith Lur 'e system is linked with the jth Lur' e system, i is not equal to j, then wji＝wij> 0, otherwise w ij0. Non-negative parameter k, di(i-1, 2, …, N) is a control gain, and further, at least one diAnd > 0, defining the feedback control gain matrix as D ═ diag { D1,d2,…,dN}。
Comprehensively considering the control error Lur 'e network (4) and the distributed controller (5), setting an initial value, and arranging the mathematical expression of the impulse error Lur' e dynamic network with the proportional delay as follows:
Figure BDA0002526552380000038
indicating the set error zi(t) at time t ═ tk,
Figure BDA0002526552380000039
Is right-continuous, and
Figure BDA00025265523800000310
in (1). Thus, the solution of equation (6) is t at time tk，
Figure BDA00025265523800000311
Is a piecewise right continuous function, psii(0) Indicating the initial value of the error.
Step four, judging whether to realize quasi-synchronization
Considering a pulse controlled Lur' e network (6) that satisfies the Lipschitz condition; for a pulse sequence ζ ═ t1,t2,…，tkK → ∞ and the mean pulse interval is less than a positive number NaWhen there is a matrix D > 0, W > 0, scalar ρ > 0, α > 0, β > 0, li> 0, such that the expressions (29) to (31) hold for ρ ≦ 1, where D represents the feedback gain matrix, liDenotes a constant, i ═ 1,2, …, N, which constitutes the matrix L; and the equations (29), (30) and (32) hold for rho > 1, the solution of the error Lur ' e network is exponentially stable, namely, the exponential quasi-synchronization between the coupled Lur ' e network (1) and the target Lur ' e system (2) is realized through the designed distributed containment controller (5). Among them are:
Figure BDA0002526552380000041
Figure BDA0002526552380000042
Figure BDA0002526552380000043
Figure BDA0002526552380000044
Figure BDA0002526552380000045
β＝2b，
Figure BDA0002526552380000046
L＝diag{l1,l2,…,li,…lN}，
in the formula: i isNnDenotes an Nn-dimensional identity matrix, a denotes a normal number, INDenotes an N-dimensional identity matrix, b denotes a normal number, λ denotes a normal number λ ∈ (0, -), N0Which is indicative of a normal number of the cells,
Figure BDA0002526552380000047
represents a normal number
Figure BDA0002526552380000048
h represents the time to make the non-linear function,
Figure BDA0002526552380000049
normal numbers that satisfy the Lipschitz condition.
For the case where the normal ρ ≦ 1, the trajectory of the error Lur' e network (6) is at the synchronous rate
Figure BDA00025265523800000410
The exponent converges to a compact set
Figure BDA00025265523800000411
The quasi-synchronization between the Lur 'e dynamic network (1) and the Lur' e system (2) is realized; wherein a compact set
Figure BDA00025265523800000412
Is described as:
Figure BDA00025265523800000413
wherein the content of the first and second substances,
Figure BDA0002526552380000051
it is indicated that the error in the quasi-synchronization,
Figure BDA0002526552380000052
wherein
Figure BDA0002526552380000053
sup denotes supremum, T0Indicating a particular time;
and for the case where p > 1, the trajectory of the error Lur' e network (6) is at the synchronous rate
Figure BDA0002526552380000054
The exponent converges to a compact set
Figure BDA0002526552380000055
The quasi-synchronization between the Lur 'e dynamic network (1) and the Lur' e system (2) is realized; wherein a compact set
Figure BDA0002526552380000056
Can be described as
Figure BDA0002526552380000057
Wherein
Figure BDA0002526552380000058
Indicating a plesiochronous error.
The invention has the beneficial effects that: the advantages brought by the invention are the indexes achieved.
1. The invention researches a quasi-synchronization method of a non-constant coupling Lur' e dynamic network with proportional delay and parameter mismatching. By designing a distributed pulse containment controller, based on a delay pulse comparison principle, a parameter variation expansion formula and average pulse interval definition, aiming at different pulse effects, sufficient conditions for ensuring network synchronization are obtained;
2. considering that the time evolution of different and different Lur' e networks moves proportionally with the scaling factor q in the unit qt, we constructed a novel delay impulse comparison system with respect to differential inequality and system heterogeneity, and applied the extended formula of parameter variation to the proof;
3. the pulse control is a better discontinuous control method, provides instant power for the system, and greatly saves the control cost compared with the common continuous control method. The invention skillfully designs a distributed type containment controller consisting of a distributed control item and a feedback control item, and analyzes different functions of the controller by considering different pulse effects borne by the controller;
4. from the definition of the mean pulse interval, it can be found that the degree of freedom index N is adjusted0The number of times of the pulse time within the time interval (T, T) may be counted by a positive number NaAnd the time interval (T, T). In the invention, the introduction of the average pulse interval effectively reduces the conservatism and greatly saves the control cost;
5. index synchronous convergence rate in conclusion
Figure BDA0002526552380000059
It can be found that the value q in the delay term affects the convergence rate of the final synchronization. In addition, the larger the proportional delay value q is, the convergence rate is
Figure BDA00025265523800000510
The faster.
Drawings
FIG. 1 is a schematic diagram of the effect of pulses.
Fig. 2 is a diagram illustrating a network synchronization process.
FIG. 3 is a state evolution diagram, wherein (a) is the first state of three Lur ' e systems, (b) is the second state of three Lur ' e systems, and (c) is the third state of three Lur ' e systems.
Fig. 4(a) shows that when the parameter of the Lur' e network is μ ═ 0.8, diError curve e (t) when 4.55, i is 1,2, 3.
FIG. 4(b) is a phase diagram of a Lur' e system with different system parameters.
FIG. 5 shows that when the parameter of Lur' e network is μ ═ 0.8, di＝1.6,When i is 1,2,3, the synchronization error curve e (t).
Detailed Description
In the following we will perform a numerical simulation example to illustrate the effectiveness of this invention.
As shown in FIG. 1, in order to synchronize a complex network composed of N coupling nodes, the present invention designs a distributed negative feedback controller. Consider controller ui(t) cases affected by impulse disturbances, two cases are considered in the present invention:
1. when the pulse coefficient mu is smaller, the controller u is considered to bei(t) forming a pulse controller after being disturbed, wherein the pulse signal plays a positive role in synchronization and can be regarded as compensation for the original controller, thereby forming a new combined controller;
2. when the pulse coefficient mu is larger, the controller u is considered to beiAnd (t) noise is formed after disturbance, and the pulse signal has a negative effect on synchronization, and can be regarded as extra disturbance which forms interference on the synchronization of the complex network together with the original disturbance.
The invention considers a class of complex networks with different Lur' e systems and proportional delays:
Figure BDA0002526552380000061
wherein
Figure BDA0002526552380000062
Is the state vector of the ith Lur' e system; matrix array
Figure BDA0002526552380000063
Figure BDA0002526552380000064
And
Figure BDA0002526552380000065
is a constant matrix (i ═ 1,2, …, N); the constant c > 0 is the coupling strength,
Figure BDA0002526552380000066
Figure BDA0002526552380000067
is an internal coupling matrix, where riIs more than or equal to 0. In the present invention, let us assume ═ In，InRepresenting an n-dimensional identity matrix. (ii) a Function(s)
Figure BDA0002526552380000068
Is a memoryless non-linear vector valued function, in
Figure BDA0002526552380000069
Is continuously differentiable;
Figure BDA00025265523800000610
is an outcoupling matrix determined by the network topology, assuming that it satisfies the zero-sum row condition
Figure BDA00025265523800000611
If the connection exists between the ith Lur 'e system and the jth Lur' e system i ≠ j, gji＝gij> 0, otherwise gijThe factor q ∈ (0,1) is a constant relating to the historical time, in particular in the Lur 'e network model (1), the dynamic-driven state z of the ith Lur' e system at time tj(t), j ═ 1,2, …, N, and state z at historical time qtj(qt) where qt is proportional to the current instant time t and the constant proportion q. Thus, the constant q is considered to be a proportional delay. Thus, we can get qt ═ t- τ (t), where τ (t) ═ (1-q) t ≧ 0, τ (t) → ∞. From this point of view, proportional delays can be considered as a class of unbounded time-varying delays. In the following paragraphs, the matrix C ═ C1,c2,…,cm]TWherein
Figure BDA00025265523800000612
j＝1,2,…,m。
Can obtain
Czi(t)＝[c1zi(t),c2zi(t),…,cmzi(t)]T，
Figure BDA00025265523800000613
Figure BDA00025265523800000614
Synchronization as a kind of clustering behavior, the purpose of which is to make all systems in the complex network reach the same state, so that the solution of a certain system can be regarded as a leader, and correspondingly, all Lur' e systems in the complex dynamic network (1) can be regarded as followers. In the present invention, we consider as leader the solution of the independent Lur' e system as follows:
Figure BDA0002526552380000071
wherein
Figure BDA0002526552380000072
Is the state vector of the independent system. Constant matrix
Figure BDA0002526552380000073
Figure BDA0002526552380000074
Therefore, the synchronization between the Lur 'e dynamic network (1) and the Lur' e system (2) can be considered as a leader-follower problem.
By defining error vectors
Figure BDA0002526552380000075
i is 1,2, …, N. We have obtained the application controller ui(t) error Lur' e network
Figure BDA0002526552380000076
Wherein
Figure BDA0002526552380000077
i＝1,2,…,N。Yi(z (t)) represents parameter mismatch, or heterogeneity between so-called Lur' e networks (1) and (2).
In the present invention, we consider the controller u to simulate a more realistic situationi(t) obeys an impulsive perturbation with a coefficient of μ. Therefore, we obtain a controlled error Lur' e network
Figure BDA0002526552380000078
Wherein mu is a pulse effect, and according to different values of mu, the pulse can play a role in facilitating synchronization or not facilitating synchronization. Ω (-) is a dirac function, with the pulse sequence ζ ═ t1,t2,…，tkIs a strictly increasing sequence of instantaneous values of the pulse, which satisfies tk-1＜tkAnd limk→+∞tk＝+∞，
Figure BDA0002526552380000079
Figure BDA00025265523800000710
Representing a set of natural numbers.
In order to realize quasi-synchronization between the Lur' e networks (1) and (2), the invention designs the following distributed strategy based on containment control by transmitting the state information of the adjacent node and the target synchronization node to each node
Figure BDA00025265523800000711
Wherein
Figure BDA00025265523800000712
Represents the set of all other Lur 'e systems directly connected to the ith Lur' e system.
Figure BDA00025265523800000713
Is a controlled coupling matrix which satisfies the zero-sum row condition, that is to say
Figure BDA00025265523800000714
If the ith Lur 'e system is linked with the jth Lur' e system (i is not equal to j), w isji＝wij> 0, otherwise w ij0. Non-negative parameter k, di(i-1, 2, …, N) is a control gain, and further, at least one di> 0, we also define the feedback control gain matrix as D ═ diag { D1,d2,…,dN}。
Comprehensively considering the control error Lur 'e network (4) and the distributed controller (5), and setting an initial value, we put the above impulse error Lur' e dynamic network mathematical expression with proportional delay as follows:
Figure BDA0002526552380000081
in the present invention, we assume error zi(t) at time t ═ tk,
Figure BDA0002526552380000082
Is right-continuous, and
Figure BDA0002526552380000083
Figure BDA0002526552380000084
in (1). Therefore, the solution of (6) is t at time tk，
Figure BDA0002526552380000085
Is a piecewise right continuous function. Psii(0) Indicating the initial value of the error.
Definition ofConsider a Lur 'e dynamic network (1) and a target Lur' e network (2) with proportional delay and parameter mismatch. If there is a compact set
Figure BDA0002526552380000086
So that for any initial value
Figure BDA0002526552380000087
When t → + ∞ the error vector zi(t) all converge to
Figure BDA0002526552380000088
In this way, we can say that within a given error range
Figure BDA0002526552380000089
And in addition, the exponential quasi-synchronization of the non-constant error Lur' e network (6) is realized.
In the following, we will discuss the conditions for achieving exponential quasi-synchronization between the Lur 'e network (1) and the leader Lur' e network (2) by designing the distributed containment controller (5). All mathematical expressions are based on the comparative lemma and the extended formula of the parametric variational method.
The following Lyapunov function was chosen:
Figure BDA00025265523800000810
wherein
Figure BDA00025265523800000811
By using the Lyapunov function, we discuss the errors between all nodes and the target synchronization node in the complex network, and obviously v (t) > 0, because the initial states of the nodes in the complex network are not consistent, and thus the global error of the complex network is necessarily greater than 0, so in the following discussion, we will explain that the function in the formula (7) is monotonically decreasing, that is, the error of the complex network can be continuously reduced until quasi-synchronization is achieved.
First, for t ═ tk,
Figure BDA00025265523800000812
From the impulse controlled error Lur' e network (6), we conclude that
Figure BDA00025265523800000813
Considering the above equation, partial scaling, by rewriting with the kronecker product, can be simplified as:
Figure BDA0002526552380000091
secondly, for
Figure BDA0002526552380000092
We calculate the derivative of equation (7) along the error Lur' e network (6).
Figure BDA0002526552380000093
Wherein
Figure BDA0002526552380000094
β＝2b，L＝diag{l1,l2,…,lN}，
Figure BDA0002526552380000095
Wherein
Figure BDA0002526552380000096
sup denotes supremum.
In combination with the inequalities (8) and (9), we consider the following pulse comparison system with a special solution χ (t) for any normal number.
Figure BDA0002526552380000097
Let the function χ (t) be t at time tk,
Figure BDA0002526552380000098
Is right-continuous, and
Figure BDA0002526552380000099
Figure BDA00025265523800000910
in (1). ψ (0) represents the sum of the initial values of the errors. In view of the pulseBased on the comparative principle, we can deduce that for any t > 0, V (t) ≦ χ (t). According to the expansion formula of the parameter variation, the following integral equation of the proportional time-varying delay term χ (qt) related to χ (t) can be obtained
Figure BDA00025265523800000911
Wherein phi (t, s) (t is more than or equal to s and is more than or equal to 0) is a Cauchy matrix of a linear pulse system
Figure BDA00025265523800000912
(case 1.) if 0 < ρ ≦ 1, the right side of the Cauchy matrix Φ (t, s) may be determined by considering the average pulse spacing Nζ(t, σ) is calculated as follows:
Figure BDA00025265523800000913
substituting (12) into integral equation (11) yields
Figure BDA00025265523800000914
Wherein
Figure BDA00025265523800000915
Next, based on the analysis of inequality (13), an exponential estimate of χ (t) can be obtained by mathematical inversion. For this purpose, it is necessary to have
Figure BDA0002526552380000101
We will demonstrate that for any t ≧ 0, λ is satisfied if present
Figure BDA0002526552380000102
The following holds for the chi (t) inequality
Figure BDA0002526552380000103
Especially for t ═ 0, we have
Figure BDA0002526552380000104
Next, by a mathematical proof method: the validity of (15) is demonstrated by a counter-syndrome method. If this assumption does not hold for all t > 0, i.e. there is at least one instant t*> 0 satisfy
Figure BDA0002526552380000105
But for 0 < t*And (15) is still true. Therefore, according to (13), (15) and (16), we make the following calculations, among them
Figure BDA0002526552380000106
Figure BDA0002526552380000107
According to the definition of theta we have
Figure BDA0002526552380000108
Namely, it is
Figure BDA0002526552380000109
Figure BDA00025265523800001010
Then (17) can be further estimated as
Figure BDA00025265523800001011
To show the contradiction with the above conclusion (18), we proceed with the following procedure. First, a parameter function is defined as
Figure BDA00025265523800001012
Computing
Figure BDA00025265523800001013
Derivative, we get
Figure BDA00025265523800001014
In addition to this, the present invention is,
Figure BDA00025265523800001015
when the time instant t is in (20) is
Figure BDA00025265523800001016
Wherein is obtainable according to (14)
Figure BDA00025265523800001017
When T is more than 0 and less than TsIs provided with
Figure BDA00025265523800001018
This means that at 0 < T < TsWhen the temperature of the water is higher than the set temperature,
Figure BDA00025265523800001019
is increased. On the other hand, when T > TsWhen there is
Figure BDA00025265523800001020
This means
Figure BDA00025265523800001021
At T > TsIs reduced. Further, according to (14), there are
Figure BDA0002526552380000111
From (19), there can be obtained
Figure BDA0002526552380000112
Is less than 0. Therefore, we can deduceThe output t is more than or equal to 0,
Figure BDA0002526552380000113
since + λ q < 0,
Figure BDA0002526552380000114
we can get
Figure BDA0002526552380000115
Since 0 < q < 1 and λ > 0, we can get e-λqt＞e-λt. According to the analysis in (21) above, the following inequality holds
Figure BDA0002526552380000116
Defining another parametric equation s (t) as
Figure BDA0002526552380000117
It is easy to verify that s (0) ═ 0,
Figure BDA0002526552380000118
this means that
Figure BDA0002526552380000119
It further states that s (t) is a monotonically decreasing function whose initial value s (0) is 0. Thus, for any t ≧ 0, s (t) ≦ s (0) 0, that is
Figure BDA00025265523800001110
In combination with inequalities (18) and (24), one can deduce
Figure BDA00025265523800001111
This contradicts (16). We can then conclude that assumption (15) is valid for t ≧ 0. In view of the principle of comparison, we have
Figure BDA00025265523800001112
Let → 0, further obtain
Figure BDA00025265523800001113
From the above estimates of the error vector z (t), we can see that as t approaches infinity, there is a compact set
Figure BDA00025265523800001114
Wherein
Figure BDA0002526552380000121
Is a quasi-synchronization error. Furthermore, from the above analysis it can be derived that the error Lur' e network (6) is at synchronous rate
Figure BDA0002526552380000122
The exponent converges to a compact set
Figure BDA0002526552380000123
So far, for 0 < rho ≦ 1, we demonstrate that quasi-synchronization can be achieved between the Lur' e networks (1) and (2) by introducing the distributed holdback controller (5), with the error of quasi-synchronization being
Figure BDA0002526552380000124
(case 2.) if ρ > 1, the Cauchy matrix Φ (t, s) can also be estimated by the definition of the average pulse interval.
Figure BDA0002526552380000125
By similar procedures, we can obtain correspondingly
Figure BDA0002526552380000126
Wherein
Figure BDA0002526552380000127
Based on the discussion in (case 1), we can demonstrate that for any t ≧ 0, if there is a normal number
Figure BDA0002526552380000128
Satisfies the following conditions:
Figure BDA0002526552380000129
the following equation holds
Figure BDA00025265523800001210
t＞0,
By the same procedure we have
Figure BDA00025265523800001211
Likewise, let → 0
Figure BDA00025265523800001212
There is a compact set
Figure BDA00025265523800001213
Through a similar proving process, we derive an error Lur' e network (6) at the synchronous rate
Figure BDA00025265523800001214
The exponent converges to a compact set
Figure BDA00025265523800001215
In (1). Namely, when rho is more than 1, quasi-synchronization is successfully realized between the Lur' e networks (1) and (2) by designing the distributed containment controller (5), and the quasi-synchronization error is
Figure BDA0002526552380000131
Based on the above discussion, we have obtained the synchronization condition between the follower network (1) and the leader system (2), attested to completion.
Conclusion：
Consider a pulse controlled Lur' e network (6) that satisfies the Lipschitz condition. For a pulse sequence ζ ═ t1,t2,…，tkK → ∞, assuming that the mean pulse spacing is less than NaIf there is a matrix D > 0, W > 0, the scalar ρ > 0, α > 0, β > 0, li> 0 such that equations (29) - (31) hold for ρ ≦ 1; where D represents the feedback gain matrix, liIf the constants i are 1,2, …, N and the equations (29), (30) and (32) hold for ρ > 1, the solution of the error Lur ' e network is exponentially stable, i.e. the coupled Lur ' e network (1) and the target Lur ' e system (2) are exponentially quasi-synchronized by the designed distributed holddown controller (5); among them are:
Figure BDA0002526552380000132
Figure BDA0002526552380000133
Figure BDA0002526552380000134
Figure BDA0002526552380000135
Figure BDA0002526552380000136
β＝2b,
Figure BDA0002526552380000137
L＝diag{l1,l2,…,IN},
in the formula: i isNnDenotes an Nn-dimensional identity matrix, a denotes a normal number, INRepresenting an N-dimensional identity matrix, b representing a normal, a normal lambda ∈ (0, -), N0Indicates the normal number, the normal number
Figure BDA0002526552380000138
h represents a non-linear function
Figure BDA0002526552380000139
Normal numbers that satisfy the Lipschitz condition.
For the case where the normal ρ ≦ 1, the trajectory of the error Lur' e network (6) is at the synchronous rate
Figure BDA00025265523800001310
The exponent converges to a compact set
Figure BDA00025265523800001311
Namely, the designed distributed containment controller (5) realizes the exponential quasi-synchronization between the coupled Lur 'e network (1) and the target Lur' e system (2). Wherein a compact set
Figure BDA00025265523800001312
Can be described as
Figure BDA0002526552380000141
Wherein the content of the first and second substances,
Figure BDA0002526552380000142
it is indicated that the error in the quasi-synchronization,
Figure BDA0002526552380000143
to represent
Figure BDA0002526552380000144
Wherein
Figure BDA0002526552380000145
sup denotes supremum, T0Indicating a particular time;
and for the case where p > 1, the trajectory of the error Lur' e network (6) is at the synchronous rate
Figure BDA0002526552380000146
The exponent converges to a compact set
Figure BDA0002526552380000147
In the method, the designed distributed containment controller (5) realizes the exponential quasi-synchronization between the Lur 'e dynamic network (1) and the Lur' e system (2); wherein a compact set
Figure BDA0002526552380000148
Can be described as
Figure BDA0002526552380000149
Wherein
Figure BDA00025265523800001410
Indicating a plesiochronous error.
Step 1: establishing a complex network formed by coupling N Lur' e systems, wherein the specific model is as follows:
Figure BDA00025265523800001411
Figure BDA00025265523800001412
Figure BDA00025265523800001413
C＝[1 0 0],
coupled Lur' eThe network (1) consists of three different Lur' e systems as described in (35). System parameter a1＝9.78,b1＝14.97，c1＝0，p1＝1.31，q1＝0.75；a2＝10,b2＝14.87,c2＝0，p2＝1.27,q20.68; and a3＝10,b3＝15,c3＝0.0385,p3＝1.27，q30.68. Non-linear function
Figure BDA00025265523800001414
Figure BDA00025265523800001415
(|z1(t)+1|-|z1(t) -1|), k ═ 1,2, 3. Consider the coupling matrix as G ═ 1, -1, 0; -1,2, -1; 0, -1,0]Let the coupling strength c be 0.2 and the proportional time-varying delay factor q ∈ (0,1) be 0.8, and set the average pulse interval to be not more than Na0.02, the constant N is freely adjusted01. And defining the synchronization error of three states between the coupling Lur 'e network and the target Lur' e system as
Figure BDA00025265523800001416
Step 2: the state model of the target Lur' e system is determined as shown in (35), wherein the relevant parameter is a1＝9.78,b1＝14.97,c1＝0,p1＝1.31,d10.75. Therefore, the goal of plesiochronous synchronization is to synchronize the three coupled Lur 'e systems to the target Lur' e system given the synchronization error.
And step 3: the synchronization conditions under different functions of the pulse effect are researched by discussing the value of mu, and the specific parameters meeting the specific model are calculated by using an LMI tool box;
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 states of all nodes achieve quasi-synchronization under the proposed conditions.

Claims (1)

1. A distributed pulse quasi-synchronization method with a proportional delay complex dynamic network is characterized by comprising the following steps:
step one, establishing a complex network with different Lur' e systems and proportional delays and determining a leader system of the complex network
(1.1) building complex networks with different Lur' e systems and proportional delays:
Figure FDA0002526552370000011
wherein the content of the first and second substances,
Figure FDA0002526552370000012
is the state vector of the ith Lur' e system, i ═ 1,2, …, N; matrix array
Figure FDA0002526552370000013
And
Figure FDA0002526552370000014
is a constant matrix; constant c>0 is the strength of the coupling, and,
Figure FDA0002526552370000015
is an internal coupling matrix, where riNot less than 0; is given as In；InRepresenting an n-dimensional identity matrix; function(s)
Figure FDA0002526552370000016
Is a memoryless non-linear vector valued function, in
Figure FDA0002526552370000017
The upper part of the material is continuous and micro,
Figure FDA0002526552370000018
is an external coupling matrix determined by network topology, and G is set to satisfy the condition of zero sum row
Figure FDA0002526552370000019
When the i-th Lur' e seriesIf the system is connected with the jth Lur' e system i ≠ j, gji＝gij>0, otherwise gij0, proportional delay q ∈ (0,1) is a constant related to historical time, t is the previous instant time, q is the proportional delay, zj(qt) is the state at the history time qt, qt ═ t- τ (t), where τ (t) ≧ 0, τ (t) → ∞;
let matrix C ═ C1,c2,…,cm]TWherein
Figure FDA00025265523700000110
j — 1,2, …, m, yielding:
Czi(t)＝[c1zi(t),c2zi(t),…,cmzi(t)]T，
Figure FDA00025265523700000111
(1.2) leader System for determining Complex network
The following independent Lur' e systems are taken as leaders in the complex dynamic network (1):
Figure FDA00025265523700000112
wherein the content of the first and second substances,
Figure FDA00025265523700000113
is the state vector of the independent system; constant matrix
Figure FDA00025265523700000114
Figure FDA00025265523700000115
Therefore, synchronization between the complex dynamic network (1) and the Lur' e system (2) is considered as a leader-follower problem;
step two, acquiring state information of each node through a sensor device and establishing an error model
By defining error vectors
Figure FDA00025265523700000116
1,2, …, N; obtain an application controller uiError Lur' e network of (t):
Figure FDA00025265523700000117
wherein
Figure FDA00025265523700000118
i＝1,2,…,N；Yi(z (t)) represents parameter mismatch, or heterogeneity between so-called Lur' e networks (1) and (2);
is provided with a controller ui(t) obeying the impulse perturbation with coefficient mu, obtaining a controlled error Lur' e network
Figure FDA0002526552370000021
Wherein mu is a pulse effect, and according to different values of mu, the pulse can play a role in facilitating synchronization or not facilitating synchronization; Ω (-) is a dirac function, and the pulse sequence ζ ═ t is set1,t2,…，tkIs a strictly increasing sequence of instantaneous values of the pulse, which satisfies tk-1<tkAnd limk→+∞tk＝+∞，
Figure FDA0002526552370000022
Figure FDA0002526552370000023
Representing a set of natural numbers;
step three, designing a distributed type containment controller
The distributed containment controller consists of a distributed control item and a feedback control item, and a distributed strategy based on containment control is established:
Figure FDA0002526552370000024
wherein the content of the first and second substances,
Figure FDA0002526552370000025
represents all other sets of Lur 'e systems directly connected to the ith Lur' e system;
Figure FDA0002526552370000026
is a controlled coupling matrix which satisfies the zero-sum row condition, that is to say
Figure FDA0002526552370000027
If the ith Lur 'e system is linked with the jth Lur' e system, i is not equal to j, then wji＝wij>0, otherwise wij0; non-negative parameter k, di(i-1, 2, …, N) is a control gain, and further, at least one di>0, defining the feedback control gain matrix as D ═ diag { D1,d2,…,dN}；
Comprehensively considering the control error Lur 'e network (4) and the distributed controller (5), setting an initial value, and arranging the mathematical expression of the impulse error Lur' e dynamic network with the proportional delay as follows:
Figure FDA0002526552370000028
let the error zi(t) at time t ═ tk,
Figure FDA0002526552370000029
Is right-continuous, and
Figure FDA00025265523700000210
Figure FDA00025265523700000211
of (1); accordingly, it isThe solution of equation (6) is at time t ═ tk，
Figure FDA00025265523700000212
Is a piecewise right continuous function, psii(0) An initial value representing an error;
step four, judging whether to realize quasi-synchronization
Considering a pulse controlled Lur' e network (6) that satisfies the Lipschitz condition; for a pulse sequence ζ ═ t1,t2,…，tkK → ∞ and the mean pulse interval is less than a positive number NaWhen there is a matrix D>0,W>0, scalar ρ>0，α>0，β>0,li>0 such that the expressions (29) to (31) hold for ρ ≦ 1, where D represents a feedback gain matrix, liDenotes a constant, i ═ 1,2, …, N, which constitutes the matrix L; and the expressions (29), (30), (32) are relative to rho>1, the solution of the error Lur ' e network is exponentially stable, namely, the exponential quasi-synchronization between the coupled Lur ' e network (1) and the target Lur ' e system (2) is realized through the designed distributed containment controller (5); among them are:
Figure FDA0002526552370000031
Figure FDA0002526552370000032
Figure FDA0002526552370000033
Figure FDA0002526552370000034
Figure FDA0002526552370000035
β＝2b,
Figure FDA0002526552370000036
L＝diag{l1,l2,…,li,…lN},
in the formula: i isNnDenotes an Nn-dimensional identity matrix, a denotes a normal number, INDenotes an N-dimensional identity matrix, b denotes a normal number, λ denotes a normal number λ ∈ (0, -), N0Which is indicative of a normal number of the cells,
Figure FDA0002526552370000037
represents a normal number
Figure FDA0002526552370000038
h represents the time to make the non-linear function,
Figure FDA0002526552370000039
normal numbers that meet the Lipschitz condition;
for the case where the normal ρ ≦ 1, the trajectory of the error Lur' e network (6) is at the synchronous rate
Figure FDA00025265523700000310
The exponent converges to a compact set
Figure FDA00025265523700000311
The quasi-synchronization between the Lur 'e dynamic network (1) and the Lur' e system (2) is realized; wherein a compact set
Figure FDA00025265523700000312
Is described as:
Figure FDA00025265523700000313
wherein the content of the first and second substances,
Figure FDA0002526552370000041
it is indicated that the error in the quasi-synchronization,
Figure FDA0002526552370000042
wherein
Figure FDA0002526552370000043
sup denotes supremum, T0Indicating a particular time;
and for p>1 case, trajectory of error Lur' e network (6) at synchronous rate
Figure FDA0002526552370000044
The exponent converges to a compact set
Figure FDA0002526552370000045
The quasi-synchronization between the Lur 'e dynamic network (1) and the Lur' e system (2) is realized; wherein a compact set
Figure FDA0002526552370000046
Is described as
Figure FDA0002526552370000047
Wherein
Figure FDA0002526552370000048
Indicating a plesiochronous error.
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