# Partial-Information-Based Synchronization of Complex Networks with Multiple and Event-Triggered Couplings

Authors: Chi Huang; Yuning Xiong; Wei Wang
Journal: Complexity
 (2021)

---

## Abstract

We study the synchronization of complex networks by using event-sampling information. The nodes of the network are connected with event-triggered communication via multiple couplings. The couplings are split into several channels. Not all the channels are connected. Only a part of the states of each node can be communicated by the channels. An event detector is designed for each channel to independently determine the sampling moments. The couplings of the network are partial and event-triggered. Both features make that less information can be used for synchronization. The pinning controllers are also designed based on the sampled information. By establishing a time-dependent Lyapunov functional and utilizing an efficient event condition, we derive less conservative criteria for the synchronization of complex networks. Finally, the effectiveness of our main results is verified by an illustrative example, and comparisons are also presented to show how much conservatism can be reduced.


---

## Body

## 1. Introduction

During the last decades, the research of complex networks has attracted an increasing amount of attention. Without the central control, the network can perform many collective behaviors just through the local communications among adjacent neighbors. Hence, complex networks have extensive applications in many large-scale systems, such as biology, circuit systems, robotics, and unmanned vehicles. Among all the collective behaviors, synchronization is of great significance due to its wide existence in the real world, for example, fireflies and the school of fish. Recently, the synchronization of complex networks has been intensively studied. Many excellent synchronization results have also been derived in [1–4].The key to realizing synchronization is the connections among nodes. In earlier research, more of a concern was to investigate the structural topology of complex networks [5–8]. More recently, the focus of the research has shifted to the communication environment [9]. With the rapid development and spread of digital technologies, natural signals are analog and have to be converted into digital format to be used in electronic equipment. It is referred to as the sampling issue. Initially, the time-triggered sampling protocol was widely employed [10]. The main feature of time-triggered communication is that the occurrence of sampling is determined by a predefined time sequence. The time-triggered sampling has its advantage in fault-tolerant systems since the missing data can be detected immediately. However, it is a lack of flexibility and the restrictive design process [11]. All the sampling processes have to be manipulated strictly according to the time sequence. It is hard to dynamically regulate the sampling plan based on the system’s real-time performance. Therefore, the time-triggered sampling usually leads to a conservative usage of the communication resource. Another drawback of time-triggered sampling is that all components of a system should be timely synchronous during operation to guarantee the strict timing specification of the system. Thus, time-triggered sampling is hard to be carried out in a large-scale system, especially for complex networks. To overcome the deficiencies of time-triggered sampling, event-triggered sampling has been introduced. It is kind of a real-time scheduling algorithm. An event condition is designed to decide sampling instants. When the condition cannot be satisfied, the state of the system needs to be measured. Compared with the time-triggered sampling, the event-triggered counterpart can quickly react to sudden fluctuations [12]. Hence, it is considered a more competent sampling protocol. Due to its virtue, numerous attention has been drawn to this subject, which includes the stabilizing control tasks of single systems [13, 14] and the consensus issues of networked systems, such as multiagent systems [15, 16], complex networks [17, 18], coupled neural networks [19–21], sensor networks [22], and Boolean networks [23–27].Each connection of complex networks consists of several channels to transmit distinct levels of node’s information. In most existing works, all these channels can work properly. It is not realistic for many real systems. In mammalian brain networks, neurons in the same cortical area interact more intensively. However, only a small part of neurons (about 5%) can receive excitatory synapses to neurons from connected cortical areas [28, 29]. It indicates that communication among cortical areas counts on the partial couplings which are established by a fraction of neurons. Thus, it is necessary to study the partial couplings of complex networks. As it is known, there are two types of neuronal signals: electrical or chemical signals. Some of the neurons may send electrical signals along with the fundamental chemical signals, while some others may only send one kind of signal. The couplings among complex networks should have multiple layers to communicate different signals. Furthermore, the couplings belonged distinct layers are also partially connected via various parts of a node. Thus, a more realistic communication environment is reflected by the couplings considered in this work, which are multiple partial and obeyed to the event-triggered update.Despite the theoretical and practical significance, a few results have been proposed for synchronization of complex networks with multiple partial and event-triggered couplings. The difficulties of this problem can be categorized as three aspects. Firstly, the sampling scheme and partial connection lead to less information available for communication. The mechanisms of them are different. The sampling makes node’s information available at a sequence of discrete instants. It is a “time-dependent” loss of information. On the other hand, the partial connection means that only a part of the node’s information is transmitted, which can be seemed like a “space-dependent” loss. Double reductions make that much less information can be used for synchronization. Secondly, channels of any connection sample information separately, while they need to cooperatively converge. Thus, the event detector of each channel should be designed in a distributed way to realize a global object. Furthermore, the information used for design should also be sampled and partial. Thirdly, the sampled data is in essence a delayed signal [30]. Due to the complicated behaviors of a single node, the conservatism of synchronization criteria is unavoidable. It is necessary yet difficult to reduce the conservatism to make the conditions more applicable.This work will address the partial-information-based synchronization of complex networks with multiple and event-triggered couplings. The event-triggered sampling is considered for every channel of each connection, in which an event detector is designed based on the sampled and partial information. The event detector individually determines the sampling moment of the corresponding channel. A unified framework will be built to cope with the two kinds of information loss. Channels are rearranged according to their levels of connection. Pinning controllers are also deployed for a small fraction of nodes. The sampled data is used to build the pinning controllers. Since the sampled signal has a time delay, it is hard to eliminate the conservatism of the proposed synchronization criteria. To reduce the conservatism, a delay-dependent Lyapunov functional will be constructed. Finally, a simulative example is proposed to illustrate the efficiency and effectiveness of our results.Notations. ℕ is the set of nonnegative integers. ℝn and ℝn×n represent, respectively, the n dimensional Euclidean space and the set of n×n real matrices. In, 0 denote the identity matrix of size n and the zero matrix with proper dimensions, respectively. For real symmetric matrices X and Y, the notation X<Y (respectively, X≤Y) indicates that the matrix X−Y is negative definite (respectively, seminegative definite). For any two integers a≤b, the notation a:b is the set x∈ℕ|a≤x≤b. ⋅ denotes the Euclidean norm for a real vector. The superscript “⊤” represents the transpose of a matrix. diag⋯ denotes a diagonal matrix. λminA denotes the smallest eigenvalues of a matrix A. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symmetric term in a matrix is denoted by ∗.


## 2. Problem Formulation and Preliminaries

The dynamics of a node of a complex network can be described as(1)x˙it=Cxit+BFxit+αwit+uit,where xit,wit,uit∈ℝn are the state, the local coupling, and the control input of the ith node, respectively; C=diagc1,c2,…,cn, B=bkjn×n; Fzt=F1z1t,F2z2t,…,Fnznt; and α is the coupling strength. The nonlinear function Fkt (k∈1:n) is supposed to satisfy the following Lipschitz condition:(2)Fka−Fkb2≤Lka−b2,∀a,b∈ℝ,where Lk is the Lipschitz constant of function Fk.


### 2.1. Multiple Partial Coupling

When complex networks are connected via a graph, each node of the network can communicate with its neighboring nodes. Thus, the local coupling of theith node can be constructed as(3)wit=∑j=1,j≠iNgijxjt−xit,where N is the number of nodes of the network; G=gijN×N is the Laplacian matrix representing the structural topology of the complex network (1), where gij is defined as follows: if the jth node can send information to the ith node, gij>0; otherwise, gij=0. Furthermore, the diagonal elements gii=−∑j=1,j≠iNgij. It implies that the complex network would be decoupled when the complex network realizes synchronization.The local coupling (3) assumes that the node xi can receive all levels of information of the jth node, that is, xjk (∀k∈1:n). However, it is a subtle assumption for most biological systems. A more realistic situation is that only part of xjk can transmit information to other nodes. As a result, only partial information of each node can be received. Additionally, for different nodes, the transmitting information is distinct. Note that many real networks have multiweighted connections. For example, in neural networks, neurons send both electrical signals and chemical signals; in supply chain networks, the capital chain, logistic chain, and information chain have separate routes. Thus, the couplings of these kinds of complex networks should be multiple layers to transmit distinct types of signals or chains. To study the multiple and partial features of couplings, the coupling term (3) is transformed as(4)wit=∑q=1Q∑j=1,j≠iNgijAijqxjt−xit,where Q is the number of multiple couplings of the complex network (1): Aijq=diagaijq1,aijq2,…,aijqn with aijqk=0 or 1. When aijqk=0, it indicates that the kth level of information of the ith node (xik) cannot receive information of the corresponding level of the jth nodes (xjk) from the qth layer of the coupling. Otherwise, it means that xik can be updated according to the received information xjk.


### 2.2. Event-Triggered Communication

Until now, each node of the network can send real-time information to its neighboring nodes. However, such a manner of communication will consume more resources on communication media. In particular, it is needless. Recently, the event-triggered sampling scheme has been introduced as a more efficient substitute. Many existing works designed the event detector of each node. In this work, connections are split inton channels to transmit the corresponding level of node information. The event detector should be installed for each channel. It is utilized to determine the sampling instant of the channel. Specifically, the event condition for xik is designed as follows:(5)xikTmik−xikTmik+lh≤γikxikTmik+lh−skTmik+lh,where l∈ℕ, γik>0; h is a positive constant, which denotes the sampling period; Tmik with m∈ℕ is the latest sampling instant of xik; and skt is the synchronous state, which will be defined later. Thus, the next sampling instant is the time when the event condition (5) is invalid; that is,(6)Tm+1ik=Tmik+h⋅minl∈ℕ:xikTmik−xikTmik+lh>γikxikTmik+lh−skTmik+lh.Let the first sampling instant of anyxik be 0; that is, T0ik=0 for all i∈1:N and k∈1:n. Note that the event condition (5) just uses the local information of xik. Thus, it can be conveniently deployed in distributed systems.Remark 1.
The event condition (5) is designed to determine the sequence of sampling instants. When the condition is violated, it means that the difference between the current state and the last sampled state is quite large. Thus, the sampled data is out of date and needs to be refreshed via new sampling. Such a mechanism makes the event-triggered sampling scheme self-adjusting and flexible. The parameter γik regulates the sampling frequency. When γik is increasing, it becomes much harder to violate the condition, and fewer samplings will be carried out. However, it also means that less information can be used. One aim of this paper is to choose γik as large as possible to realize the synchronization of the complex network.
Considering the event-triggering update protocol, the information of any node is invariant during its two contiguous triggered instants. The coupling term ofxik (4) becomes(7)wikt=∑q=1Q∑j=1,j≠iNgijaijqkxjkTmtjk−xikTmtik,where Tmtik (Tmtjk) is the latest triggered instant of xik (xjk) before time t. That is,(8)Tmtik=maxTmik≤t:m∈ℕ.
In this work, the complex network (1) is supposed to synchronize to a given synchronous state st=s1t,s2t,…,snt⊤∈ℝn, which satisfies the dynamics of a single node with a given initial state s0; that is,(9)s˙t=Cst+BFst.
Define an error system asyit=xit−st (i∈1:N). Let(10)eikt=yikTmtik−yikt,k∈1:n.
Fort∈Tmik+lh,Tmik+lh+h (l≥0, m≥0 and Tmik+lh<Tm+1ik), it can be obtained that(11)xjkTmtjk−xikTmtik=xjkTmTmik+lhjk−sk∗TmTmik+lhjk−xikTmik+sk∗TmTmik+lhjk=yjkTmTmik+lhjk−yjkTmik+lh−yikTmik+yikTmik+lh+yjkTmik+lh−yikTmik+lh=ejkTmik+lh−eikTmik+lh+yjkTmik+lh−yikTmik+lh,
Based on the above discussions, it can be concluded that(12)xjkTmtjk−xikTmtik=ejkmh−eikmh+yjkmh−yikmhholds for t∈mh,mh+h (m∈ℕ). Thus, the coupling term (4) with an event-triggering protocol (12) can be converted as(13)wit=∑q=1Q∑j=1,j≠iNgijAijqejmh−eimh+yjmh−yimh.


### 2.3. Pinning Control

To synchronize the complex network (1) to the state st, the pinning controllers uit will be deployed. Since the complex network is a large-scale system, it is impossible and unnecessary to control every xik. Controlling a small part of them is sufficient to stabilize the whole network. Specifically, the pinning controller uikt for xik can be designed as follows:(14)uikt=dik⋅skTmtik−xikTmtik,where dik≥0 represents the control gain. When dik=0, xik is not controlled. It can be seen from (14) that the controller can only use the sampled information.Fort∈Tmik+lh,Tmik+lh+h (l≥0, m≥0, and Tmik+lh<Tm+1ik), we can have(15)uikt=dikskTmik−xikTmik=dik−yikTmik+yikTmik+lh−yikTmik+lh=−dikeikTmik+lh+yikTmik+lh.Based on (15), the pinning controller (14) can be transformed to the form of a vector:(16)ui=−Dieimh+yimh,where t∈mh,mh+h and Di=diagdi1,di2,…,din.


### 2.4. Complex Networks with Multiple Partial and Event-Triggered Couplings

Letfyit=Fxit−Fst≜f1yi1t,f1yi2t,…,fnyint⊤, where fkyikt=Fkxikt−Fkskt, for ∀k∈1:n. The dynamics of the error system yit can be obtained from (1):(17)y˙it=Cyit+Bfyit+αwit+uit.Define matricesHijq=diaghijq1,hijq2,…,hijqn (q∈1:Q, i,j∈1:N) as(18)hijqk=gijaijqk,i≠j,−∑j=1,j≠ingijaijqk−dik,i=j.Substituting (13) and (16) into (17), the complex network with multiple partial and event-triggered couplings can be described as(19)y˙it=Cyit+Bfyit+α∑q=1Q∑j=1NHijqyjmh+ejmh,i∈1:N,where t∈mh,mh+h and m∈ℕ.The following definition will be needed for the derivation of our main results.Definition 1.
The complex network (1) with multiple partial and event-triggered couplings (13) and the pinning controllers (16) is said to be globally exponential synchronization if, for any initial condition xi0, there exist positive constants M and ϵ such that the error system (19) satisfies yit≤Me−εt for all i∈1:N.
The error system (19) is just built for investigating couplings. In this manuscript, the couplings are split into channels. Thus, we should construct a new framework of error system for channels. From (19), it yields that, for ∀k∈1:n,(20)y˙ik=ckyikt+∑j=1nbkjfjyijt+α∑q=1Q∑j=1Nhijqkyjkmh+ejkmh.
Letδykt=y1kt,y2kt,…,yNkt⊤. Thus, it follows from (20) that(21)δy.k=ckδykt+∑j=1nbkjfjδyjt+αℋkδykmh+δekmh,where fjδyjt=fjy1jt,…,fjyNjt⊤ and δekmh=e1kmh,…,eNkmh⊤.
The error system (21) rearranges channels according to their positions at each coupling, instead of the couplings they belonged to like in (19).


## 2.1. Multiple Partial Coupling

When complex networks are connected via a graph, each node of the network can communicate with its neighboring nodes. Thus, the local coupling of theith node can be constructed as(3)wit=∑j=1,j≠iNgijxjt−xit,where N is the number of nodes of the network; G=gijN×N is the Laplacian matrix representing the structural topology of the complex network (1), where gij is defined as follows: if the jth node can send information to the ith node, gij>0; otherwise, gij=0. Furthermore, the diagonal elements gii=−∑j=1,j≠iNgij. It implies that the complex network would be decoupled when the complex network realizes synchronization.The local coupling (3) assumes that the node xi can receive all levels of information of the jth node, that is, xjk (∀k∈1:n). However, it is a subtle assumption for most biological systems. A more realistic situation is that only part of xjk can transmit information to other nodes. As a result, only partial information of each node can be received. Additionally, for different nodes, the transmitting information is distinct. Note that many real networks have multiweighted connections. For example, in neural networks, neurons send both electrical signals and chemical signals; in supply chain networks, the capital chain, logistic chain, and information chain have separate routes. Thus, the couplings of these kinds of complex networks should be multiple layers to transmit distinct types of signals or chains. To study the multiple and partial features of couplings, the coupling term (3) is transformed as(4)wit=∑q=1Q∑j=1,j≠iNgijAijqxjt−xit,where Q is the number of multiple couplings of the complex network (1): Aijq=diagaijq1,aijq2,…,aijqn with aijqk=0 or 1. When aijqk=0, it indicates that the kth level of information of the ith node (xik) cannot receive information of the corresponding level of the jth nodes (xjk) from the qth layer of the coupling. Otherwise, it means that xik can be updated according to the received information xjk.


## 2.2. Event-Triggered Communication

Until now, each node of the network can send real-time information to its neighboring nodes. However, such a manner of communication will consume more resources on communication media. In particular, it is needless. Recently, the event-triggered sampling scheme has been introduced as a more efficient substitute. Many existing works designed the event detector of each node. In this work, connections are split inton channels to transmit the corresponding level of node information. The event detector should be installed for each channel. It is utilized to determine the sampling instant of the channel. Specifically, the event condition for xik is designed as follows:(5)xikTmik−xikTmik+lh≤γikxikTmik+lh−skTmik+lh,where l∈ℕ, γik>0; h is a positive constant, which denotes the sampling period; Tmik with m∈ℕ is the latest sampling instant of xik; and skt is the synchronous state, which will be defined later. Thus, the next sampling instant is the time when the event condition (5) is invalid; that is,(6)Tm+1ik=Tmik+h⋅minl∈ℕ:xikTmik−xikTmik+lh>γikxikTmik+lh−skTmik+lh.Let the first sampling instant of anyxik be 0; that is, T0ik=0 for all i∈1:N and k∈1:n. Note that the event condition (5) just uses the local information of xik. Thus, it can be conveniently deployed in distributed systems.Remark 1.
The event condition (5) is designed to determine the sequence of sampling instants. When the condition is violated, it means that the difference between the current state and the last sampled state is quite large. Thus, the sampled data is out of date and needs to be refreshed via new sampling. Such a mechanism makes the event-triggered sampling scheme self-adjusting and flexible. The parameter γik regulates the sampling frequency. When γik is increasing, it becomes much harder to violate the condition, and fewer samplings will be carried out. However, it also means that less information can be used. One aim of this paper is to choose γik as large as possible to realize the synchronization of the complex network.
Considering the event-triggering update protocol, the information of any node is invariant during its two contiguous triggered instants. The coupling term ofxik (4) becomes(7)wikt=∑q=1Q∑j=1,j≠iNgijaijqkxjkTmtjk−xikTmtik,where Tmtik (Tmtjk) is the latest triggered instant of xik (xjk) before time t. That is,(8)Tmtik=maxTmik≤t:m∈ℕ.
In this work, the complex network (1) is supposed to synchronize to a given synchronous state st=s1t,s2t,…,snt⊤∈ℝn, which satisfies the dynamics of a single node with a given initial state s0; that is,(9)s˙t=Cst+BFst.
Define an error system asyit=xit−st (i∈1:N). Let(10)eikt=yikTmtik−yikt,k∈1:n.
Fort∈Tmik+lh,Tmik+lh+h (l≥0, m≥0 and Tmik+lh<Tm+1ik), it can be obtained that(11)xjkTmtjk−xikTmtik=xjkTmTmik+lhjk−sk∗TmTmik+lhjk−xikTmik+sk∗TmTmik+lhjk=yjkTmTmik+lhjk−yjkTmik+lh−yikTmik+yikTmik+lh+yjkTmik+lh−yikTmik+lh=ejkTmik+lh−eikTmik+lh+yjkTmik+lh−yikTmik+lh,
Based on the above discussions, it can be concluded that(12)xjkTmtjk−xikTmtik=ejkmh−eikmh+yjkmh−yikmhholds for t∈mh,mh+h (m∈ℕ). Thus, the coupling term (4) with an event-triggering protocol (12) can be converted as(13)wit=∑q=1Q∑j=1,j≠iNgijAijqejmh−eimh+yjmh−yimh.


## 2.3. Pinning Control

To synchronize the complex network (1) to the state st, the pinning controllers uit will be deployed. Since the complex network is a large-scale system, it is impossible and unnecessary to control every xik. Controlling a small part of them is sufficient to stabilize the whole network. Specifically, the pinning controller uikt for xik can be designed as follows:(14)uikt=dik⋅skTmtik−xikTmtik,where dik≥0 represents the control gain. When dik=0, xik is not controlled. It can be seen from (14) that the controller can only use the sampled information.Fort∈Tmik+lh,Tmik+lh+h (l≥0, m≥0, and Tmik+lh<Tm+1ik), we can have(15)uikt=dikskTmik−xikTmik=dik−yikTmik+yikTmik+lh−yikTmik+lh=−dikeikTmik+lh+yikTmik+lh.Based on (15), the pinning controller (14) can be transformed to the form of a vector:(16)ui=−Dieimh+yimh,where t∈mh,mh+h and Di=diagdi1,di2,…,din.


## 2.4. Complex Networks with Multiple Partial and Event-Triggered Couplings

Letfyit=Fxit−Fst≜f1yi1t,f1yi2t,…,fnyint⊤, where fkyikt=Fkxikt−Fkskt, for ∀k∈1:n. The dynamics of the error system yit can be obtained from (1):(17)y˙it=Cyit+Bfyit+αwit+uit.Define matricesHijq=diaghijq1,hijq2,…,hijqn (q∈1:Q, i,j∈1:N) as(18)hijqk=gijaijqk,i≠j,−∑j=1,j≠ingijaijqk−dik,i=j.Substituting (13) and (16) into (17), the complex network with multiple partial and event-triggered couplings can be described as(19)y˙it=Cyit+Bfyit+α∑q=1Q∑j=1NHijqyjmh+ejmh,i∈1:N,where t∈mh,mh+h and m∈ℕ.The following definition will be needed for the derivation of our main results.Definition 1.
The complex network (1) with multiple partial and event-triggered couplings (13) and the pinning controllers (16) is said to be globally exponential synchronization if, for any initial condition xi0, there exist positive constants M and ϵ such that the error system (19) satisfies yit≤Me−εt for all i∈1:N.
The error system (19) is just built for investigating couplings. In this manuscript, the couplings are split into channels. Thus, we should construct a new framework of error system for channels. From (19), it yields that, for ∀k∈1:n,(20)y˙ik=ckyikt+∑j=1nbkjfjyijt+α∑q=1Q∑j=1Nhijqkyjkmh+ejkmh.
Letδykt=y1kt,y2kt,…,yNkt⊤. Thus, it follows from (20) that(21)δy.k=ckδykt+∑j=1nbkjfjδyjt+αℋkδykmh+δekmh,where fjδyjt=fjy1jt,…,fjyNjt⊤ and δekmh=e1kmh,…,eNkmh⊤.
The error system (21) rearranges channels according to their positions at each coupling, instead of the couplings they belonged to like in (19).


## 3. Main Results

The synchronization criteria will be given for complex network (1) in this section. By introducing a transformation of sampled data, a delay-dependent Lyapunov functional will be built in Theorem 1, which is helpful to reduce the conservatism of our theoretical results.Theorem 1.
Letℋk=∑q=1QhijqkN×N (k∈1:n). Under the event-triggered communication condition (5), the complex networks (1) with multiple partial and event-triggered couplings (19) and the pinning controllers (16) can be exponentially synchronized, if there exist positive scalars εk, ε1k, ε2k and matrices Pk>0, Wk=W1kW2k∗W3k>0, U1k, U2k, U3k, U4k, V1k, V2k, Xk, X1k, k∈1:n, such that the following linear matrix inequalities (LMIs) are satisfied:(22)Pk+hXk+Xk⊤2+W1k−hXk+hX1k−W1kW2k∗−hX1k−hX1k⊤+hXk+Xk⊤2+W1k−W2k∗∗W3k>0,(23)Ω1kΩ2k∗Ω3k<0,(24)Ω˜1kΩ˜2k∗Ω3k<0,where(25)Ω1k=Φ11kΦ12kΦ13kΦ14khU1k⊤∗Φ22kΦ23kΦ24khU2k⊤∗∗Φ33kΦ34khU3k⊤∗∗∗Φ44khU4k⊤∗∗∗∗−hW1k,Ω2k=Ω˜2k0,Ω˜1k=Φ11kΦ12kΠ13kΦ14k∗Π22kΠ23kΦ24k∗∗Π33kΦ34k∗∗∗Φ44k,Ω˜2k=nV1k⊤0000nV2k⊤00,Ω3k=−nε1kIN0∗−nε2kIN,Φ11k=−Xk+Xk⊤2−U1k−U1k⊤+ckV1k+ckV1k⊤+∑j=1nε1j+ε2jbjk2Lk,Φ12k=Xk−X1k−W2k+U1k⊤−U2k+αV1k⊤ℋk,Φ13k=Pk−U3k−V1k⊤+ckV2k,Φ14k=−U4k+αV1k⊤ℋk,Φ22k=X1k+X1k⊤−Xk+Xk⊤2+W2k+W2k⊤+U2k+U2k⊤+εkΓk2−hW3k,Φ23k=U3k+αℋk⊤V2k,Φ24k=U4k,Φ33k=−V2k−V2k⊤,Φ34k=αV2k⊤ℋk,Φ44k=−εkIN,Π13k=hXk+Xk⊤2+Pk−U3k−V1k⊤+ckV2k,Π22k=X1k+X1k⊤−Xk+Xk⊤2+W2k+W2k⊤+U2k+U2k⊤+εkΓk2+hW3k,Π23k=U3k+αℋk⊤V2k+hW2k⊤+X1k⊤−Xk⊤,Π33k=−V2k−V2k⊤+hW1k.Proof.
Thanks to the Schur complement [31], it follows from (23) and (24) that(26)Ψk=Ψ11kΦ12kΦ13khU1k⊤∗Φ22kΦ23khU2k⊤∗∗Ψ33khU3k⊤∗∗∗−hW1k<0,(27)Ψk′=Ψ11kΦ12kΠ13kΦ14k∗Π22kΦ23kΦ24k∗∗Ψ33k′Π34k∗∗∗Π44k<0,where Ψ11k=Φ11k+nε1k−1V1k⊤V1k, Ψ33k=Φ33k+nε2k−1V2k⊤V2k, and Ψ33k′=Π33k+nε2k−1V2k⊤V2k.
Basing on the error system (21), we can establish the Lyapunov functional Vt=∑k=1nVkt, where(28)Vkt=δyk⊤tPkδykt+ςt∫mhtδy.ksδykmh⊤Wkδy.ksδykmhds+ςtδyktδykmh⊤Xkδyktδykmh,t∈mh,m+1h,ςt=m+1h−t and Xk=Xk+Xk⊤/2−Xk+X1k∗−X1k−X1k⊤+Xk+Xk⊤/2; δykt is defined in (21).
The rest of the proof consists of three steps. The first step will indicate the Lyapunov functionalVt is well defined, that is, Vt≥0, and Vt=0 if and only if δykt=0 holds for all k∈1:n. The second step will show that Vt is exponential convergent under conditions (22)–(24). Finally, the error system is exponential stable (19), and the complex network (1) can reach synchronization according to Theorem 1.Step 1. Well-definedness ofVt.ConsideringWk>0 and the Schur complement [31], it follows that W^k≜W1k−W2kW3k−1W2k⊤>0. Thus, it follows that(29)Wk−W^k0∗0=W2kW3k−1W2k⊤W2k∗W3k≥0,which further implies(30)Wk≥W^k0∗0.Based on the above inequality, it yields that(31)δy.ksδykmh⊤Wkδy.ksδykmh≥δy.k⊤sW^kδy.ks.Thus, it yields from (28) and (31) that(32)Vkt≥δyk⊤tPkδykt+ςt∫mhtδy.k⊤sW^kδy.ksds+ςtδyktδykmh⊤Xkδyktδykmh.Resorting to the Jensen inequality [32], it can be obtained that(33)∫mhtδy.k⊤sW^kδy.ksds≥1τt∫mhtδy.k⊤sdsW^k∫mhtδy.ksds≥1hδykt−δykmh⊤W^kδykt−δykmh,whereτt=t−mh. It follows from (32) and (33) that(34)Vkt≥δyktδykmh⊤ςthPk+hXk+W^k+τthPkδyktδykmh,wherePk=Pk0∗0 and W^k=W^k−W^k∗W^k. By applying the Schur complement [31], it can be obtained from (22) that Pk+hXk+W^k>0. Hence, Vkt defined in (28) is well defined. Furthermore, there must exist a number β∈0,1 such that Pk+hXk+W^k>βPk. It follows from (34) that(35)Vkt>βδyktδykmh⊤Pkδyktδykmh=βδyk⊤tPkδykt.Due toPk>0, Vkt≥0 and the equality holds if and only if deltaykt=0. Thus, the Lyapunov functional Vt is well defined.Step 2. Exponential convergence ofVtThe derivative ofVkt is(36)V˙kt=2δy.ktPkδykt−∫mhtδy.ksδykmh⊤Wkδy.ksδykmhds+ςtδy.ktδykmh⊤Wkδy.ktδykmh−δyktδykmh⊤Xkδyktδykmh+2ςtδy.kt0⊤Xkδyktδykmh.Thanks to the Jensen inequality [32], it can be obtained that(37)∫mhtδy.ksδykmh⊤Wkδy.ksδykmhds≥τtϕk⊤tW1kϕkt+2δykt−δykmh⊤W2kδykmh+τtδyk⊤mhW3kδykmh,whereϕt=1/τt∫mhtδy.ksds. Based on the Newton–Leibnitz formula, we can have(38)0=∑k=1n2δyk⊤tU1k⊤+δyk⊤mhU2k⊤+δyk.⊤tU3k⊤+δek⊤mhU4k⊤⋅−δykt+δykmh+τtϕkt.From (21), one has that(39)0=∑k=1n2δyk⊤tV1k⊤+δy.ktV2k⊤⋅−δy.kt+ckδykt+∑j=1nbkjfjδyjt+αℋkδekmh+δykmhBased on the Lipschitz condition (2), we have that(40)2δyk⊤tV1k⊤∑j=1nbkjfjδyjt≤nε1k−1δyk⊤tV1k⊤V1kδykt+ε1k∑j=1nbkj2δyj⊤tLjδyjt.(41)2δy.k⊤tV2k⊤∑j=1nbkjfjδyjt≤nε2k−1δy.k⊤tV2k⊤V2kδy.kt+ε2k∑j=1nbkj2δyj⊤tLjδyjt.Considering the event condition (5), we can have that(42)LHS=yikTmik−yikTmik+lh=eikTmik+lh,RHS=γikyikTmik+lh.Generally, it yields for any momentt=mh (m∈ℕ) that(43)eikmh≤γikyikmhholds for anym∈ℕ, i∈1:N, and k∈1:n, which further implies that(44)εkδekmh⊤δekmh≤εkδykmh⊤Γk2δykmh,whereΓk=diagγ1k,γ2k,…,γNk and εk can be any positive number.Combining (36)–(44), we have that(45)V˙t≤∑k=1nηk⊤tςthΨ¯k+τthΨkηkt,whereηkt=δyktδykmhδy.ktδemhϕkt⊤ and Ψ¯k=Ψk′0∗0. From (26) and (27), one can always find a sufficiently small positive number ε such that V˙t+εVt<0 holds for ∀t∈mh,m+1h. Solving the differential equation, we have Vt<e−εt−mhVmh. Similarly, we can get Vmh<e−εhVmh−h for any m∈ℕ. Combining all these inequalities yields that Vt<e−εtV0, ∀t>0. Thus, the Lyapunov functional Vt is exponential convergence.Step 3. Exponential synchronization of complex network (1).Considering the error system (19), we have that(46)yit2≤∑i=1Nyi⊤tyit=∑k=1nδyk⊤tδykt≤∑k=1nλmin−1Pkδyk⊤tPkδykt<β−1maxk=1nλmin−1PkVt<β−1maxk=1nλmin−1PkV0e−ϵt.
Based on Definition1, the complex network (1) with multiple partial and event-triggered couplings (13) and the pinning controllers (16) is exponential synchronization. This completes the proof.Remark 2.
Most existing results studied the macro-level connections among nodes. In this work, only a part of the channel of a node can communicate. Thus, the micro-level connections among nodes should also be investigated. A feasible method is to build an augmented system to contain both macro and micro connections. However, it will lead to a large-dimension stabilization condition, which is hard to be verified. An alternative way is to rearrange neurons. Equation (21) collects all the channels which have the same index of each node, instead of the channels that belonged to the same node like equation (1). The complex network (13) is rearranged as n subsystems. By establishing the Lyapunov functional (21) for each subsystem, the stabilization condition with smaller dimensions can be obtained.
The time-dependent Lyapunov functional is constructed in Theorem1 to deal with the sampled information. This method was introduced in [33] and improved in [34]. These Lyapunov functionals do not grow after the sampling instants. Thus, the conservatism of the main results can be reduced further. To show this fact, the following theorem is derived, in which a time-independent Lyapunov functional is established.Theorem 2.
The complex networks (1) with multiple partial and event-triggered couplings (13) and the pinning controllers (16) can achieve exponential synchronization if there exist positive scalars εk, ε1k, ε2k and matrices Pk>0, Qk>0, Rk>0, V1k, V2k, S1k, S2k, U1k, U2k, k∈1:n, such that the following LMIs are satisfied:(47)Ω1kΩ2k∗Ω3k<0,where(48)Ω1k=Φ11kΦ12k0Φ14kαV1k⊤ℋk∗Φ22kΦ23kΦ24k0∗∗Φ33k00∗∗∗Φ44kαV2k⊤ℋk∗∗∗∗−εkIN,Ω2k=hU1k0nV1k⊤0hU2khS2k000hS1k00000nV2k⊤,Ω3k=diag−hRk,−hRk,−nε1kIN,−nε2kIN,Φ11k=Qk+U1k+U1k⊤+ckV1k+ckV1k⊤+∑j=1nε1j+ε2jbjk2Lk,Φ12k=−U1k+U2k⊤+αV1k⊤ℋk,Φ22k=−U2k−U2k⊤+S2k+S2k⊤+εkΓk2,Φ23k=S1k⊤−S2k,Φ14k=Pk−V1k⊤+ckV2k,Φ24k=αℋk⊤V2k,Φ33k=−Qk−S1k−S1k⊤,Φ44k=hRk−V2k−V2k⊤.Proof.
Consider the following Lyapunov functionalVt=∑k=1nVkt, where(49)Vkt=δyk⊤tPkδykt+∫t−htδyk⊤sQkδyksds+∫−h0∫t+θtδy.k⊤sRkδy.ksdsdθ.
The derivative ofVkt is(50)V˙kt=2δy.k⊤tPkδykt+δyk⊤tQkδykt−δyk⊤t−hQkδykt−h+hδy.k⊤tRkδy.kt−∫t−htδy.k⊤θRkδy.kθdθ.
The Newton–Leibniz formula gives that(51)2δyk⊤tU1k+δyk⊤mhU2k⋅δykt−δykmh−∫mhtδyk.sds=0,(52)2δyk⊤tS1k+δyk⊤mhS2k⋅δykmh−δykt−h−∫t−hmhδyk.sds=0,where m=maxm∈ℕ|mh<t. Adding the left side of (51) and (52) into (50) yields(53)V˙kt=2δy.k⊤tPkδykt+δyk⊤tQkδykt−δyk⊤t−hQkδykt−h+hδy.k⊤tRkδy.kt+2δyk⊤tU1k+δyk⊤mhU2kδykt−δykmh−∫mhtζk⊤t,θRk−1ζkt,θdθ+2δyk⊤t−hS1k+δyk⊤mhS2kδykmh−δykt−h−∫t−hmhηk⊤t,θRk−1ηkt,θdθ+t−mhδyk⊤tU1k+δyk⊤mhU2kRk−1U1k⊤δykt+U2k⊤δykmh+mh−t−hδyk⊤t−hS1k+δyk⊤mhS2kRk−1S1k⊤δykt−hS2k⊤δykmh≤2δy.k⊤tPkδykt+δyk⊤tQkδykt−δyk⊤t−hQkδykt−h+hδy.k⊤tRkδy.kt+2δyk⊤tU1k+δyk⊤mhU2kδykt−δykmh+2δyk⊤t−hS1k+δyk⊤mhS2kt−h+hδyk⊤tU1k+δyk⊤mhU2kRk−1U1k⊤δykt+U2k⊤δykmh+hδyk⊤t−hS1k+δyk⊤mhS2kRk−1S1k⊤δykt−h+S2k⊤δykmh,where ζkt,θ=U1k⊤δykt+U2k⊤δykmh+Rkδy.kθ and ηkt,θ=S1k⊤δykt−h+S2k⊤δykmh+Rkδy.kθ. Considering (39), the Lipschitz assumption of function f (40) and (41), and the event condition (44), it yields that(54)V˙kt≤θk⊤tΞkθkt,where θkt=δyktδykmhδykt−hδy˙ktδekmh⊤,(55)Ξk=Ξ11kΦ12k+hU1kRk−1U2k⊤0Φ14kαV1k⊤ℋk∗Ξ22k−S2k+hS2kRk−1S1k⊤Φ24k0∗∗−Qk+hS1kRk−1S1k⊤00∗∗∗Φ44k+nε2k−1V2k⊤V2kαV2k⊤ℋk∗∗∗∗−εkIN,Ξ11k=Φ11k+hU1kRk−1U1k⊤+nε1k−1V1k⊤V1k,Ξ22k=Φ22k+hU2kRk−1U2k⊤+hS2kRk−1S2k⊤.
Thanks to the Schur complement [31], the negative definiteness of Ξk can be guaranteed by (47). Thus, V˙kt<0. By using the same derivation in Theorem 1, it can be concluded that the complex network (1) with multiple partial and event-triggered couplings (13) and the pinning controllers (16) can realize exponential synchronization according to Definition 1. This completes the proof.


## 4. Simulations

In this section, a simulative example will be built to illustrate the efficiency and effectiveness of the proposed synchronization criteria. In particular, a comparison will be presented to show how much conservatism can be reduced by Theorem1.A complex network with four nodes (N=4) is considered, each of which has three levels of information (n=3). The parameters for each node are x˙t=Cxt+BFxt with F1u=F2u=F3u=u+1−u−1/2 and(56)C=−0.1000−0.1000−0.1,B=0.116−0.15−0.15−0.150.116−0.2−0.120.20.116,G=−31111−21011−31101−2.The functionFk satisfies the Lipschitz condition (2) with the Lipschitz constants Lk=1 (k∈1:3). The coupling strength of the complex network is α=0.35; G is the Laplacian matrix of the connections among nodes. The trajectory of the synchronous state st with the initial state s0=−1,0,1⊤ is depicted in Figure 1.Figure 1
The trajectory of the synchronous statest.For any pair of adjacent nodes, only a part of levels of information can communicate, which leads to partial couplings. Furthermore, the layers of couplings are duplex (Q=2) in this example. Thus, two diagonal matrices are employed for each connection, that is, gij>0. They are listed as follows:(57)A121=diag1,0,0,A122=diag1,0,1,A131=diag0,1,0,A132=diag0,0,1,A141=diag0,1,1,A142=diag0,1,1,A211=diag1,0,1,A212=diag0,0,1,A231=diag1,0,0,A232=diag1,1,0,A311=diag0,0,0,A312=diag0,0,1,A321=diag0,0,1,A322=diag1,0,1,A341=diag0,1,1,A342=diag0,0,0,A411=diag1,0,0,A412=diag1,0,0,A431=diag0,1,1,A122=diag1,0,1.For each channel, an event condition (5) should be designed to realize the event-triggered sampling protocol. The sampling period h is set as 0.2 and γik=0.17, for all channels (i∈1:4,k∈1:3). Furthermore, a portion of xik deploy the feedback controllers (14). That is, d11=2, d22=2, d23=2, d31=2, d32=2, d42=2, and d43=2.By using the MATLAB LMI toolbox, the synchronization criteria (22)–(24) have a feasible solution. Based on Theorem 1, the constructed complex network (1) with multiple partial and event-triggered couplings (13) and the pinning controllers (16) can achieve synchronization. To show this fact, a simulation is given. The initial state xi0 is randomly chosen from the interval −5,5. Figure 2 displays the trajectories of the error system (19) of the complex network. It can be found that all errors yit (i∈1:4) tend to 0, which means that the synchronization can be realized.Figure 2
All the trajectories of the error systemsyit (i∈1:4) approach 0.To show the superiority of the event-triggered communication, we compare it with the time-triggered communication. LetMikt be the number of times that the event condition (5) is violated for xik. That is, Mikt=mt, where mt is the one in Tmtik which is defined in (8). If the complex network samples information for every 0.2 interval, the total sampling number of the whole network at time t would be 12×⌊t/0.2⌋, where ⌊x⌋ is the floor function, that is, the greatest integer less than or equal to x. Let Rt be ratio between the number of the event-triggered sampling and that of the time-triggered sampling number:(58)Rt=∑i=14∑k=13Mikt12×t/0.2.Figure3 shows the trajectory of Rt for t≤20. It can be seen that Rt decreases rapidly and stabilizes around 53%. It indicates that the event-triggered sampling scheme can save near half of the communication resources. This is a great advantage to its time-triggered counterpart.Figure 3
The trajectory ofRt for t≤20.Finally, we will compare the conservatism of conditions in Theorems1 and 2. Assume that all xik have the same γ for the event condition (5), that is, γik=γ (i∈1:4,k∈1:3). By changing the sampling period h, we try to find the maximum γ which allows the existence of feasible solutions for two theorems. As discussed in Remark 1, larger γ makes the event harder to be triggered. It means that the complex network can be synchronized with less frequent sampling and then has less conservatism. Figure 4 shows the maximum γs which can be found in Theorems 1 and 2. It can be seen that, for any fixed h, γ for Theorem 1 (the solid curve) is larger than the one for Theorem 2 (the dashed curve), especially when the sampling period h is increasing. Thus, Theorem 1 has less conservatism than Theorem 2. It is the merit of the delay-dependent Lyapunov functional (28).Figure 4
The maximumγ which allows the existence of feasible solutions for Theorems 1 (the solid curve) and 2 (the dashed curve), when the sampling period h is changing from 0.01 to 0.5.


## 5. Conclusion

The pinning synchronization problem of complex networks with multiple partial and event-triggered couplings has been studied in this paper. An event detector has been designed for every level of information of each node to decide the sampling instants. Furthermore, only a part of channels of connections can communicate with neighboring nodes. Thus, partial information of each node is available. Furthermore, couplings have multiple layers to transmit various kinds of information. To synchronize the complex network, the pinning control method has been used. The control input can only utilize the sampled information. Based on a rearrangement of the channels of couplings, a delay-dependent Lyapunov functional has been constructed. The synchronization criteria have been proposed with less conservatism, which is demonstrated by a numerical example.



---

*Source: 6613435-2021-04-19.xml*