Synchronization of directed complex networks with uncertainty and time-delay

Abstract

A complex earthquake monitor network includes satellite in-orbit and ground observation and requires good synchronization in time. This article investigates synchronization of complex networks with uncertainty and time-delay. Directed graphs are used to represent the interaction topology. The uncertainty is assumed to be norm bounded. Based on Lyapunov theory, sufficient conditions are given to guarantee the synchronization of the complex networks in the presence of time-delay. Simulation results are provided to demonstrate the effectiveness of the obtained results.

Introduction

Synchronization of complex dynamical networks have attracted considerable attention from so many fields such as biology, physics, robotics, and control engineering.^1–22 This is partly due to recent technological advances in communication and computation, and important practical applications, ranging from the Internet, automated highway systems, low-orbit satellites, coordinated navigation, multi-sensor of earthquake monitoring network,and so on.^23–26 A complex network consists of a large number of interconnected nodes, in which each node is a fundamental unit with specific purpose, especially when the earthquake monitor network covers types of observation instruments. For a given network, one of the basic problems is to find conditions under which synchronization can be achieved. In the past decade, numerous studies have been conducted on this problem. For example, Wang et.al. presented a uniform complex network model and investigated its synchronization in small-world and scale-free networks.^5–7 In practical applications, some information in a complex network may not be able to be transmitted completely precisely due to the existence of various disturbances, such as uncertainty, time-delay, and the variation of the network topology,^14–19 which might lead to divergence or oscillation of the complex networks. However, few works have been conducted to consider the parameter uncertainties and time-delay simultaneously for general complex networks.

For this situation, our objective in this article is to investigate global synchronization of general complex networks with uncertainty and time-delay. In the analysis, a Lyapunov-Krasovskii function is proposed, based on which sufficient conditions are obtained to guarantee the synchronization of the complex networks. In the absence and presence of time-delay, it is shown that all nodes can reach synchronization with desired $H_{\infty }$ performance even when the external disturbances and the parameter uncertainties coexist.

Graph theory

Let $\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A})$ be a directed graph of order $n$ , where $\mathcal{V}=\{{\mathrm{v}}_1,\dots ,{\mathrm{v}}_n\}$ is the set of nodes, $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ is the set of edges, and $\mathcal{A}=[a_{\mathrm{ij}}]$ is a weighted adjacency matrix. The node indexes belong to a finite index set $\mathcal{I}=\{1,2,\dots ,n\}$ . An edge of $\mathcal{G}$ is denoted by $e_{\mathrm{ij}}=({\mathrm{v}}_i,{\mathrm{v}}_j)$ , where the first element ${\mathrm{v}}_i$ of the $e_{\mathrm{ij}}$ is said to be the tail of the edge and the other ${\mathrm{v}}_j$ to be the head. The adjacency elements associated with the edges are positive, that is, $e_{\mathrm{ij}}\in \mathcal{E}\iff a_{\mathrm{ij}}>0$ . Moreover, it is assumed that $a_{\mathrm{ii}}=0$ for all $i\in \mathcal{I}$ . Correspondingly, the Laplacian with the directed graph is defined as $L=\Delta -\mathcal{A}$ , where $\Delta =[{\Delta }_{\mathrm{ij}}]$ is a diagonal matrix with ${\Delta }_{\mathrm{ii}}=\sum _{j=1}^n{a}_{\mathrm{ij}}$ . The set of neighbors of node ${\mathrm{v}}_i$ is denoted by $N_i=\{{\mathrm{v}}_j\in \mathcal{V}:({\mathrm{v}}_i,{\mathrm{v}}_j)\in \mathcal{E}\}$ . A directed path is a sequence of ordered edges of the form $({\mathrm{v}}_{i_1},{\mathrm{v}}_{i_2}),({\mathrm{v}}_{i_2},{\mathrm{v}}_{i_3}),\dots ,$ where ${\mathrm{v}}_{i_j}\in \mathcal{V}$ in a directed graph. If a directed graph has the property that $({\mathrm{v}}_i,{\mathrm{v}}_j)\in \mathcal{E}$ for any $({\mathrm{v}}_j,{\mathrm{v}}_i)\in \mathcal{E}$ , the directed graph is called undirected. If there is a directed path from every node to every other node, the graph is said to be strongly connected (connected for undirected graph). Moreover, if there exists a node such that there is a directed path from every other node to this node, the graph is said to have a spanning tree. Obviously, the Laplacian of any undirected graph is symmetric, and any undirected graph is strongly connected if and only if it has a spanning tree.^27,28

Lemma 1

The directed graph $\mathcal{G}$ has a spanning tree if and only if Laplacian $L$ of $\mathcal{G}$ has a simple zero eigenvalue (with associated eigenvector $1_n$ ) ¹² . In addition, all the other eigenvalues have positive real-parts.

Lemma 2

Consider a directed graph $\mathcal{G}$ . ¹³ Let $D$ be the 01-matrix with rows and columns indexed by the nodes and edges of $\mathcal{G}$ , and $E$ be the 01-matrix with rows and columns indexed by the edges and nodes of $\mathcal{G}$ , such that

$$\begin{array}{c}{D}_{\mathrm{uf}}=\{\begin{array}{cc}1,\hfill & \mathrm{if\; the\; node\; u\; is\; the\; tail\; of\; the\; edge}f\hfill \\ \multicolumn{1}{c}{0,}& \mathrm{otherwise}\hfill \end{array}\hfill \\ E_{\mathrm{fu}}=\{\begin{array}{cc}1,\hfill & \mathrm{if\; the\; node\; u\; is\; the\; head\; of\; the\; edge}f\hfill \\ \multicolumn{1}{c}{0,}& \mathrm{otherwise}\hfill \end{array}\hfill \end{array}$$

Then the Laplacian of $\mathcal{G}$ can be decomposed into $L=\mathrm{DW}(D^T-E)$ , where $W=\mathrm{diag}\{w_1,w_2,\dots ,w_{|\mathcal{E}|}\}$ , $w_i$ is the weight of the $i\mathrm{th}$ edge of $\mathcal{G}$ , and $|\mathcal{E}|$ is the number of the edges.

Model

Consider a dynamic network of $N$ identical nodes with diffusive couplings such that each node is an n-dimensional system. The state equations of the network are

$$\begin{array}{c}{\stackrel{·}{x}}_i(t)=f(x_i(t),t)-\sum _{j=1}^N(L_{\mathrm{ij}}+\Delta L_{\mathrm{ij}})\Gamma x_j(t-\tau )+{\omega }_i,\hfill \\ i=1,2,\dots ,N,t\ge 0\hfill \end{array}$$

(1)

with initial condition $x_i(s)=x_i(0)$ $s\in (-\infty ,0]$ , where $x_i(t)=(x_{i1}(t),x_{i2}(t),\dots ,$ $x_{\mathrm{in}}(t){)}^T\in {\mathbb{R}}^n$ are the state vector of node $i$ , ${\omega }_i$ denotes the external disturbance, $f:{\mathbb{R}}^n\times {\mathbb{R}}^{+}\to {\mathbb{R}}^n$ is a smooth nonlinear vector-valued function, $\Gamma$ is a constant inner-coupling matrix of the nodes, $L=[L_{\mathrm{ij}}]$ is the Laplacian of the network (denotes outer-coupling interaction of the network) and $\Delta L=[\Delta L_{\mathrm{ij}}]$ is the corresponding uncertainty Laplacian of $L$ , in which the off-diagonal entries of $\Delta L$ is defined as follows

$$|\Delta L_{\mathrm{ij}}(t)|=\{\begin{array}{cc}\le {\psi }_{\mathrm{ij}},\hfill & i\ne j\mathrm{and}{L}_{\mathrm{ij}}\ne 0\hfill \\ \multicolumn{1}{c}{0,}& L_{\mathrm{ij}}=0\hfill \end{array}$$

Here, we assume that $f:{\mathbb{R}}^n\times {\mathbb{R}}^{+}\to {\mathbb{R}}^n$ satisfies

$$\begin{array}{c}\parallel f(x(t),t)-f(y(t),t)\parallel \le \beta \parallel x(t)-y(t)\parallel ,\hfill \\ \forall x(t),y(t)\in {\mathbb{R}}^n,t\ge 0\hfill \end{array}$$

(2)

where $\beta >0$ is a constant scalar.

The dynamic network equation (1) is said to achieve synchronization if $\underset{t\to +\infty }{lim}[x_i(t)-x_j(t)]=0,i\in \mathcal{I}.$

Rewrite equation (1) by using the Kronecker product in a simple compact form

$$\stackrel{·}{x}(t)=\overline{F}(x(t),t)-[(L+\Delta L)\otimes \Gamma ]x(t-\tau )+\omega$$

(3)

where $\overline{F}(x(t),t)=(f^T(x_1(t),t),\dots ,f^T(x_N(t),t){)}^T$ and $\omega =({\omega }_1^T,\dots ,{\omega }_N^T{)}^T$ . Let $\overline{x}(t)=(1/N)\sum _{i=1}^N{x}_i(t)$ and ${\delta }_i(t)=x_i(t)-\overline{x}(t),i=1,2,\dots ,N$ . Then the network dynamics can be written as

$$\begin{array}{c}{\stackrel{·}{\delta }}_i(t)=f(x_i(t),t)-f(\overline{x}(t),t)-{\displaystyle \sum _{j=1}^N(L_{ij}+\Delta L_{ij})\Gamma x_j(t-\tau )}\\ +\frac{1}{N}{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N(L_{ij}+\Delta L_{ij})\Gamma x_j(t-\tau )+\pi +{\omega }_i(t)}-\frac{1}{N}{\displaystyle \sum _{j=1}^N{\omega }_i(t)}}\end{array}$$

(4)

where $\sum _{i=1}^N{\delta }_i(t)=0,\overline{f}(x(t),t)=(1/N)\sum _{j=1}^{N}f(x_j(t),t),\pi =f(\overline{x}(t),t)-\overline{f}(x(t),t)$ .

Let

$$\delta (t)=({\delta }_1^T(t),\dots ,{\delta }_N^T(t){)}^T$$

Then the equation (4) can be transformed into the following equation

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\delta }(t)=F(x(t),t)-(U\otimes I_n)[(L+\Delta L)\otimes \Gamma ]}\\ \delta (t-\tau )+I_N\otimes \pi +(U\otimes I_n)\omega \hfill \end{array}$$

(5)

where $F(x(t),t)=(F_1^T(x_1(t),t),F_2^T(x_2(t),t),\dots ,F_N^T(x_N(t),t){)}^T$ with $F_i(x_i(t),t)=f(x_i(t),t)-f(\overline{x}(t),t)$ and

$$U=\left[\begin{array}{cccc}\frac{N-1}{N}\hfill & -\frac{1}{N}\hfill & \cdots \hfill & -\frac{1}{N}\hfill \\ \multicolumn{1}{c}{-\frac{1}{N}}& \frac{N-1}{N}\hfill & \cdots \hfill & -\frac{1}{N}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{-\frac{1}{N}}& -\frac{1}{N}\hfill & \cdots \hfill & \frac{N-1}{N}\hfill \end{array}\right]$$

Our objective is to find design rules of the Laplacian $L$ to suppress disturbances of nodes and make all nodes reach synchronization with desired $H_{\infty }$ performance, that is, $J_{\omega }={\int }_0^{\infty }z(t{)}^{T}z(t)-{\gamma }^2{\omega }^T(t)\omega (t)\mathrm{dt}<0$ , where $z(t)$ is the objective signal to be attenuated, and $\gamma$ is a given constant. Then, a natural way to combine the relative information is to define output functions $z_i(t)$ computed from an average of the relative displacements of all nodes as follows

$$z_i(t)=x_i(t)-\frac{1}{N}\sum _{i=1}^N{x}_i(t),i=1,\dots ,N$$

namely

$$z(t)=(z_1^T(t),z_2^T(t),\dots ,z_N^T(t){)}^T=\delta (t)$$

According to Lemma 2, $\Delta L$ can be decomposed into $\Delta L=E_1\Sigma (t)E_2$ , where $E_1,E_2$ are specified constant matrices and $\Sigma (t)$ is a diagonal matrix, whose diagonal elements are the uncertainties of the edges, that is, the non-zero $\Delta L_{\mathrm{ij}}(t)\mathrm{s}$ . Without loss of generality, we assume $\Sigma (t{)}^T\Sigma (t)\le I_N$

Main results

In this section, we address the global synchronization of network and give design rules of the Laplacian $L$ in the presence of time-delay and uncertainty. Before the main results, we first introduce some lemmas.

Lemma 3

(Schur Complement) For given symmetric matrix $S$ with the form $S=[S_{\mathrm{ij}}]$ , $S_{11}\in {\mathcal{R}}^{r\times r},$ $S_{12}\in {\mathcal{R}}^{r\times (n-r)},$ $S_{22}\in {\mathcal{R}}^{(n-r)\times (n-r)}$ , then, $S<0$ if and only if $S_{11}<0,S_{22}-S_{21}{S}_{11}^{-1}{S}_{12}<0$ or $S_{22}<0,S_{11}-S_{12}{S}_{22}^{-1}{S}_{21}<0$ .

Lemma 4

Consider the matrix $U$ . The following statements hold. ¹³

1. The eigenvalues of $U$ are $1$ with multiplicity $N-1$ and 0 with multiplicity $1$ . The vectors $1_N^T$ and $1_N$ are the left and the right eigenvectors of $U$ associated with the zero eigenvalue, respectively.

2. There exists an orthogonal matrix $J$ such that $J^T\mathrm{UJ}=\left[\begin{array}{cc}{I}_{N-1}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]$ and the last column is $1_N/\sqrt{N}$ . Let ${\Xi }_1\in {\mathcal{R}}^{n\times n}$ be the Laplacian of any directed graph, then $J^T{\Xi }_{1}J=\left[\begin{array}{cc}{\vartheta }_1\hfill & 0\hfill \end{array}\right],$ ${\vartheta }_1\in {\mathcal{R}}^{N\times (N-1)}$ .

For simplicity, denote

$$J=\left[\begin{array}{cc}{J}_1\hfill & {\overline{J}}_1\hfill \end{array}\right]$$

where ${\overline{J}}_1=1_N/\sqrt{N}$ is the last column of $J$ and $J_1$ is the rest part.

Theorem 1

Suppose that the network graph has a spanning tree. For the network equation (1), synchronization can be reached with desired $H_{\infty }$ performance if there exist positive definite matrices $P,Q,R\in {\mathbb{R}}^{n\times n}$ and positive scalars ${ϵ}_i,(i=1,\dots ,9)$ , satisfying

$$H=\left[\begin{array}{ccccc}{H}_{11}\hfill & 0\hfill & H_{13}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& H_{22}\hfill & H_{23}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{H_{13}^T}& H_{23}^T\hfill & H_{33}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & H_{44}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & H_{55}\hfill \end{array}\right]<0$$

where

$$\begin{array}{c}\multicolumn{1}{c}{H_{11}={\overline{H}}_1+\frac{\tau }{{ϵ}_6}(J_1^{T}U{E}_1{E}_1^{T}U{J}_1)\otimes (P\Gamma {\Gamma }^{T}P)}\\ +\tau {\beta }^2{ϵ}_9{\lambda }_{\max }(R^2)I_{\kappa -n}+I_{\kappa -n}\hfill \\ \multicolumn{1}{c}{H_{13}=(J_1^T\otimes P)(U\otimes I_n)}\\ \multicolumn{1}{c}{H_{22}={\overline{H}}_2+\frac{1}{{ϵ}_8}\tau (J_1^T{E}_2^T{E}_2{J}_1)\otimes I_n}\\ +\tau {ϵ}_7(J_1^T{E}_2^T{E}_2{J}_1)\otimes ({\Gamma }^T{R}^2\Gamma )\hfill \\ \multicolumn{1}{c}{H_{23}=-\tau (J_1^T{L}^{T}U)\otimes ({\Gamma }^{T}R)}\\ \multicolumn{1}{c}{H_{33}=-{\gamma }^2{I}_{\kappa }+\frac{\tau }{{ϵ}_9}U\otimes I_n+\tau U\otimes R}\\ +{ϵ}_8\tau (U{E}_1{E}_1^{T}U)\otimes (R\Gamma {\Gamma }^{T}R)\hfill \\ \multicolumn{1}{c}{H_{44}=-\tau I_N\otimes R+\tau {ϵ}_6(E_2^T{E}_2)\otimes I_n}\\ \multicolumn{1}{c}{H_{55}=-\tau I_N\otimes R+\tau \frac{1}{{ϵ}_7}(U{E}_1{E}_1^{T}U)\otimes I_n}\end{array}$$

with

$$\begin{array}{c}\multicolumn{1}{c}{{\overline{H}}_1={\lambda }_{\max }(P)\beta I_{\kappa -n}-(J_1^T\mathrm{UL}{J}_1)\otimes (P\Gamma )-{(J_1^T\mathrm{UL}{J}_1)}^T\otimes {(P\Gamma )}^T}\\ +\frac{1}{{ϵ}_1}(J_1^T{E}_2^T{E}_2{J}_1)\otimes ({\Gamma }^T{P}^2\Gamma )\hfill \\ +\tau [(J_1^T\mathrm{UL})\otimes P\Gamma ](I_N\otimes R^{-1})[(J_1^T\mathrm{UL})\otimes P\Gamma {]}^T\hfill \\ +{ϵ}_3\tau (J_1^{T}U{E}_1{E}_1^{T}U{J}_1)\otimes (P\Gamma R^{-2}{\Gamma }^{T}P)\hfill \\ +\frac{\tau }{{ϵ}_3}(J_1^T{\mathrm{ULE}}_2^T{E}_2{L}^{T}U{J}_1)\otimes (P\Gamma {\Gamma }^{T}P)+I_{N-1}\otimes Q+\tau {\lambda }_{\max }(R){\beta }^2{I}_{\kappa -n}\hfill \\ +{ϵ}_1(J_1^{T}U{E}_1{E}_1^{T}U{J}_1)\otimes I_n\hfill \\ +{ϵ}_5\tau {\beta }^2{\lambda }_{\max }(R^2)I_{\kappa -n}+\tau {ϵ}_2{\beta }^2{\lambda }_{\max }(R^2){\lambda }_{\max }(U{E}_1{E}_1^{T}U)I_{\kappa -n}\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{{\overline{H}}_2=-(I_{N-1}\otimes Q)}\\ +\tau {[(\mathrm{UL}{J}_1)\otimes \Gamma ]}^T(I_N\otimes R)[(\mathrm{UL}{J}_1)\otimes \Gamma ]\hfill \\ +[\frac{\tau }{{ϵ}_4}(J_1^T{E}_2^T{E}_2{J}_1)\otimes ({\Gamma }^{T}R\Gamma {\Gamma }^{T}R\Gamma )+\tau {ϵ}_4(J_1^T{L}^T\mathrm{UU}{E}_1{E}_1^T\mathrm{UUL}{J}_1)\otimes I_n]\hfill \\ +\tau [\frac{1}{{ϵ}_5}(J_1^T{L}^T\mathrm{UUL}{J}_1)\otimes {\Gamma }^T\Gamma +\frac{1}{{ϵ}_2}(J_1^T{E}_2^T{E}_2{J}_1)\otimes ({\Gamma }^T\Gamma )]\hfill \end{array}$$

Proof

First, we discuss the stability of equation (5) without external disturbances, that is, $\omega (t)\equiv 0$ . Define a Lyapunov function for the system equation (5) as follows

$$\begin{array}{c}\multicolumn{1}{c}{V(t)={\delta }^T(t)(I_N\otimes P)\delta (t)+\underset{t-\tau }{\overset{t}{\int }}{\delta }^T(s)(I_N\otimes Q)\delta (s)\mathrm{ds}}\\ +\underset{-\tau }{\overset{0}{\int }}\underset{t+\theta }{\overset{t}{\int }}{\stackrel{·}{\delta }}^T(s)(I_N\otimes R)\stackrel{·}{\delta }(s)\mathrm{dsd}\theta \hfill \end{array}$$

Calculating $\stackrel{·}{V}(t)$ , we get

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}(t)={\delta }^T(t)(I_N\otimes P)F(x(t),t)-{\delta }^T(t)(I_N\otimes P)}\\ [(\mathrm{UL}+U\Delta L)\otimes \Gamma ]\delta (t-\tau )\hfill \\ +{\delta }^T(t)(I_N\otimes P)(I_N\otimes \pi )\hfill \\ +{\delta }^T(t)(I_N\otimes Q)\delta (t)-{\delta }^T(t-\tau )(I_N\otimes Q)\delta (t-\tau )\hfill \\ +\tau {\stackrel{·}{\delta }}^T(t)(I_N\otimes R)\stackrel{·}{\delta }(t)-\underset{t-\tau }{\overset{t}{\int }}{\stackrel{·}{\delta }}^T(s)(I_N\otimes R)\stackrel{·}{\delta }(s)\mathrm{ds}\hfill \end{array}$$

By the Newton-Leibniz formula, we have

$$\delta (t-\tau )=\delta (t)-\underset{t-\tau }{\overset{t}{\int }}\stackrel{·}{\delta }(s)\mathrm{ds}$$

Then, for any $x,y\in {\mathbb{R}}^n$ and any symmetric positive definite matrix $\overline{R}\in {\mathbb{R}}^{n\times n},$ we have

$$2x^{T}y\le x^T{\overline{R}}^{-1}x+y^T\overline{R}y$$

(6)

Thus, for any ${\Sigma }^T(t)\Sigma (t)\le I_N$ and any matrices $A,B\in {\mathbb{R}}^{N\times N}$

$$A\Sigma (t)B\le {ϵ}_{0}A{A}^T+\frac{1}{{ϵ}_0}{B}^{T}B$$

(7)

where ${ϵ}_0$ is a positive scalar. For any symmetric positive definite matrix $A\in {\mathbb{R}}^{n\times n}$ , it follows from equation (2) that

$$\begin{array}{c}\sum _{i=1}^N{\delta }_i^T(t)A(f(x_i(t),t)-f(\overline{x}(t),t))\le {\lambda }_{\max }(A)\beta {\delta }^T(t)\delta (t)\hfill \\ \sum _{i=1}^N(f(x_i(t),t)-f(\overline{x}(t),t){)}^{T}A(f(x_i(t),t)\hfill \\ -f(\overline{x}(t),t))\le {\lambda }_{\max }(A){\beta }^2{\delta }^T(t)\delta (t)\hfill \end{array}$$

where $\kappa =\mathrm{Nn}$ and ${\lambda }_{\max }(·)$ denotes the largest eigenvalue of a given matrix. Also, it is evident that

$${\delta }^T(t)(I_N\otimes P)(I_N\otimes \pi )=\sum _{i=1}^N{\delta }_i^T(t)P\pi =0$$

and

$${\stackrel{·}{\delta }}^T(t)(I_N\otimes R)(I_N\otimes \pi )=\sum _{i=1}^N{\stackrel{·}{\delta }}_i^{T}R\pi =0$$

As a result, we have

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}(t)=\sum _{i=1}^N{\delta }_i^T(t)P(f(x_i(t),t)-f(\overline{x}(t),t))-{\delta }^T(t)(I_N\otimes P)[(\mathrm{UL}+U\Delta L)\otimes \Gamma ]\delta (t)}\\ +\underset{t-\tau }{\overset{t}{\int }}{\delta }^T(t)(I_N\otimes P)[(\mathrm{UL}+U\Delta L)\otimes \Gamma ]\stackrel{·}{\delta }(s)\mathrm{ds}-\underset{t-\tau }{\overset{t}{\int }}{\stackrel{·}{\delta }}^T(s)(I_N\otimes R)\stackrel{·}{\delta }(s)\mathrm{ds}\hfill \\ +{\delta }^T(t)(I_N\otimes Q)\delta (t)-{\delta }^T(t-\tau )(I_N\otimes Q)\delta (t-\tau )\hfill \\ +\tau \sum _{i=1}^N(f(x_i(t),t)-f{(\overline{x}(t),t)}^{T}R(f(x_i(t),t)-f(\overline{x}(t),t)\hfill \\ +\tau {\delta }^T(t-\tau )[(\mathrm{UL}+U\Delta L)\otimes \Gamma {]}^T(I_N\otimes R)[(\mathrm{UL}+U\Delta L)\otimes \Gamma ]\delta (t-\tau )\hfill \\ -2\tau F^T(x(t),t)(I_N\otimes R)[(\mathrm{UL}+U\Delta L)\otimes \Gamma ]\delta (t-\tau )\hfill \\ \le {\delta }^T(t)[{\lambda }_{\max }(P)\beta I_{\kappa }-(\mathrm{UL}+U\Delta L)\otimes (P\Gamma )-{(\mathrm{UL}+U\Delta L)}^T\otimes {(P\Gamma )}^T\hfill \\ +\tau [(\mathrm{UL}+U\Delta L)\otimes P\Gamma ](I_N\otimes R^{-1})[(\mathrm{UL}+U\Delta L)\otimes P\Gamma {]}^T+I_N\otimes Q\hfill \\ +\tau {\lambda }_{\max }(R){\beta }^2{I}_{\kappa }]\delta (t)\hfill \\ -{\delta }^T(t-\tau )(I_N\otimes Q)\delta (t-\tau )+\tau {\delta }^T(t-\tau )[(\mathrm{UL}+U\Delta L)\otimes \Gamma {]}^T(I_N\otimes R)\hfill \\ \times [(\mathrm{UL}+U\Delta L)\otimes \Gamma ]\delta (t-\tau )\hfill \\ +{ϵ}_5\tau \sum _{i=1}^N{(f(x_i(t),t)-f(\overline{x}(t),t))}^T{R}^2(f(x_i(t),t)-f(\overline{x}(t),t))\hfill \\ +\frac{\tau }{{ϵ}_5}{\delta }^T(t-\tau )[(L^T\mathrm{UUL})\otimes {\Gamma }^T\Gamma ]\delta (t-\tau )-2\tau F^T(x(t),t)\hfill \\ \times (I_N\otimes R)[(U\Delta L)\otimes \Gamma ]\delta (t-\tau )\hfill \\ \le {\delta }^T(t)(J\otimes I_n)(J^T\otimes I_n)[{\lambda }_{\max }(P)\beta I_{\kappa }-(\mathrm{UL})\otimes (P\Gamma )-{(\mathrm{UL})}^T\otimes {(P\Gamma )}^T\hfill \\ +\frac{1}{{ϵ}_1}(E_2^T{E}_2)\otimes ({\Gamma }^T{P}^2\Gamma )\hfill \\ +\tau [(\mathrm{UL})\otimes P\Gamma ](I_N\otimes R^{-1})[(\mathrm{UL}{)}^T\otimes P\Gamma {]}^T+\tau [(U\Delta L)\otimes P\Gamma ]\hfill \\ \times (I_N\otimes R^{-1})[(U\Delta L{)}^T\otimes P\Gamma {]}^T\hfill \\ +{ϵ}_3\tau (U{E}_1{E}_1^{T}U)\otimes (P\Gamma R^{-2}{\Gamma }^{T}P)+\frac{\tau }{{ϵ}_3}({\mathrm{ULE}}_2^T{E}_2{L}^{T}U)\otimes (P\Gamma {\Gamma }^{T}P)\hfill \\ +I_N\otimes Q+\tau {\lambda }_{\max }(R){\beta }^2{I}_{\kappa }+{ϵ}_1(U{E}_1{E}_1^{T}U)\otimes I_n](J\otimes I_n)(J^T\otimes I_n)\delta (t)\hfill \\ -{\delta }^T(t-\tau )(J\otimes I_n)(J^T\otimes I_n)(I_N\otimes Q)(J\otimes I_n)(J^T\otimes I_n)\delta (t-\tau )\hfill \\ +\tau {\delta }^T(t-\tau )(J\otimes I_n)(J^T\otimes I_n)[(\mathrm{UL})\otimes \Gamma {]}^T(I_N\otimes R)[(\mathrm{UL})\otimes \Gamma ](J\otimes I_n)(J^T\otimes I_n)\delta (t-\tau )\hfill \\ +{\delta }^T(t-\tau )(J\otimes I_n)(J^T\otimes I_n)[\frac{\tau }{{ϵ}_4}(E_2^T{E}_2)\otimes ({\Gamma }^{T}R\Gamma {\Gamma }^{T}R\Gamma )+\tau {ϵ}_4(L^T\mathrm{UU}{E}_1{E}_1^T\mathrm{UUL})\otimes I_n]\hfill \\ (J\otimes I_n)(J^T\otimes I_n)\delta (t-\tau )+\tau {\delta }^T(t-\tau )(J\otimes I_n)(J^T\otimes I_n)\hfill \\ {[(U\Delta L)\otimes \Gamma ]}^T(I_N\otimes R)[(U\Delta L)\otimes \Gamma ]\otimes (J\otimes I_n)(J^T\otimes I_n)\delta (t-\tau )\hfill \\ +\tau {\delta }^T(t-\tau )(J\otimes I_n)(J^T\otimes I_n)[\frac{1}{{\epsilon }_5}(L^T\mathrm{UUL})\otimes {\Gamma }^T\Gamma +\frac{1}{{ϵ}_2}(E_2^T{E}_2)\otimes ({\Gamma }^T\Gamma )]\hfill \\ (J\otimes I_n)(J^T\otimes I_n)\delta (t-\tau )\hfill \\ +{ϵ}_5\tau {\beta }^2{\lambda }_{\max }(R^2){\delta }^T(t)(J\otimes I_n)(J^T\otimes I_n)\delta (t)\hfill \\ +\tau {ϵ}_2{\beta }^2{\lambda }_{\max }(R^2){\lambda }_{\max }(U{E}_1{E}_1^{T}U){\delta }^T(t)(J\otimes I_n)(J^T\otimes I_n)\delta (t)\hfill \end{array}$$

Let ${\overline{\delta }}^T(t)={\delta }^T(t)J_1$ . Since ${\delta }^T(t)(1_N\otimes I_n)=0_n^T$ , then ${\delta }^T(t)(J\otimes I_n)=({\overline{\delta }}^T(t),0_n^T)$ . Therefore, it follows that

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le {\overline{\delta }}^T(t)[{\lambda }_{\max }(P)\beta I_{\kappa -n}-(J_1^T\mathrm{UL}{J}_1)\otimes (P\Gamma )-{(J_1^T\mathrm{UL}{J}_1)}^T\otimes {(P\Gamma )}^T}\\ +\frac{1}{{ϵ}_1}(J_1^T{E}_2^T{E}_2{J}_1)\otimes ({\Gamma }^T{P}^2\Gamma )\hfill \\ +\tau [(J_1^T\mathrm{UL})\otimes P\Gamma ](I_N\otimes R^{-1})[(J_1^T\mathrm{UL})\otimes P\Gamma {]}^T+\tau [(J_1^{T}U\Delta L)\otimes P\Gamma ](I_N\otimes R^{-1})\hfill \\ \times {[(J_1^{T}U\Delta L{)}^T\otimes P\Gamma ]}^T\hfill \\ +{ϵ}_3\tau (J_1^{T}U{E}_1{E}_1^{T}U{J}_1)\otimes (P\Gamma R^{-2}{\Gamma }^{T}P)+\frac{\tau }{{ϵ}_3}(J_1^T{\mathrm{ULE}}_2^T{E}_2{L}^{T}U{J}_1)\otimes (P\Gamma {\Gamma }^{T}P)\hfill \\ +I_{N-1}\otimes Q+\tau {\lambda }_{\max }(R){\beta }^2{I}_{\kappa -n}+{ϵ}_1(J_1^{T}U{E}_1{E}_1^{T}U{J}_1)\otimes I_n]\overline{\delta }(t)\hfill \\ -{\overline{\delta }}^T(t-\tau )(I_{N-1}\otimes Q)\overline{\delta }(t-\tau )\hfill \\ +\tau {\overline{\delta }}^T(t-\tau )[(\mathrm{UL}{J}_1)\otimes \Gamma {]}^T(I_N\otimes R)[(\mathrm{UL}{J}_1)\otimes \Gamma ]\overline{\delta }(t-\tau )\hfill \\ +{\overline{\delta }}^T(t-\tau )[\frac{\tau }{{ϵ}_4}(J_1^T{E}_2^T{E}_2{J}_1)\otimes ({\Gamma }^{T}R\Gamma {\Gamma }^{T}R\Gamma )+\tau {ϵ}_4(J_1^T{L}^T\mathrm{UU}{E}_1{E}_1^T\mathrm{UUL}{J}_1)\otimes I_n]\overline{\delta }(t-\tau )\hfill \\ +\tau {\overline{\delta }}^T(t-\tau )[(U\Delta L{J}_1)\otimes \Gamma {]}^T(I_N\otimes R)[(U\Delta L{J}_1)\otimes \Gamma ]\overline{\delta }(t-\tau )\hfill \\ +\tau {\overline{\delta }}^T(t-\tau )[\frac{1}{{ϵ}_5}(J_1^T{L}^T\mathrm{UUL}{J}_1)\otimes {\Gamma }^T\Gamma +\frac{1}{{ϵ}_2}(J_1^T{E}_2^T{E}_2{J}_1)\otimes ({\Gamma }^T\Gamma )]\overline{\delta }(t-\tau )\hfill \\ +{ϵ}_5\tau {\beta }^2{\lambda }_{\max }(R^2){\overline{\delta }}^T(t)\overline{\delta }(t)\hfill \\ +\tau {ϵ}_2{\beta }^2{\lambda }_{\max }(R^2){\lambda }_{\max }(U{E}_1{E}_1^{T}U){\overline{\delta }}^T(t)\overline{\delta }(t)\hfill \end{array}$$

Then, a sufficient condition for $\stackrel{·}{V}(t)<0$ is

$$\begin{array}{c}{\overline{H}}_1+\tau [(J_1^{T}U\Delta L)\otimes P\Gamma ](I_N\otimes R^{-1})[(J_1^{T}U\Delta L)\otimes P\Gamma {]}^T<0,\hfill \\ {\overline{H}}_2+\tau {[(U\Delta L{J}_1)\otimes \Gamma ]}^T(I_N\otimes R)[(U\Delta L{J}_1)\otimes \Gamma ]<0\hfill \end{array}$$

(8)

By Schur Complement Lemma, we have that ${\overline{H}}_1+\tau [(J_1^{T}U\Delta L)\otimes P\Gamma ](I_N\otimes R^{-1})[(J_1^{T}U\Delta L)\otimes P\Gamma {]}^T<0$ is equivalent to

$$\begin{array}{c}\multicolumn{1}{c}{{\varphi }_1=\left[\begin{array}{cc}{\overline{H}}_1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -\tau I_N\otimes R\hfill \end{array}\right]+\left[\begin{array}{c}\tau (J_1^{T}U{E}_1)\otimes (P\Gamma )\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]}\\ (\Sigma \otimes I_n)\left[\begin{array}{cc}0\hfill & E_2\otimes I_n\hfill \end{array}\right]\hfill \\ +\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{E_2^T\otimes I_n}\end{array}\right]({\Sigma }^T\otimes I_n){\left[\begin{array}{c}\tau (J_1^{T}U{E}_1)\otimes (P\Gamma )\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]}^T<0\hfill \end{array}$$

and ${\overline{H}}_2+\tau [(U\Delta L{J}_1)\otimes \Gamma {]}^T(I_N\otimes R)[(U\Delta L{J}_1)\otimes \Gamma ]<0$ is equivalent to

$$\begin{array}{c}\multicolumn{1}{c}{{\varphi }_2=\left[\begin{array}{cc}{\overline{H}}_2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -\tau I_N\otimes R\hfill \end{array}\right]+\left[\begin{array}{c}\tau (J_1^T{E}_2^T)\otimes ({\Gamma }^{T}R)\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]}\\ ({\Sigma }^T\otimes I_n)\left[\begin{array}{cc}0\hfill & (E_1^{T}U)\otimes I_n\hfill \end{array}\right]\hfill \\ +\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{(U{E}_1)\otimes I_n}\end{array}\right](\Sigma \otimes I_n)\left[\begin{array}{c}\tau E_2{J}_1\otimes (R\Gamma )\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]<0\hfill \end{array}$$

From equation (7), we get

$${\varphi }_1\le {\overline{\varphi }}_1=\left[\begin{array}{cc}{\overline{H}}_1+\frac{\tau }{{ϵ}_6}(J_1^{T}U{E}_1{E}_1^{T}U{J}_1)\otimes (P\Gamma {\Gamma }^{T}P)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -\tau I_N\otimes R+\tau {ϵ}_6(E_2^T{E}_2)\otimes I_n\hfill \end{array}\right]$$

$${\varphi }_2\le {\overline{\varphi }}_2=\left[\begin{array}{cc}{\overline{H}}_2+\tau {ϵ}_7(J_1^T{E}_2^T{E}_2{J}_1)\otimes ({\Gamma }^T{R}^2\Gamma )\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -\tau I_N\otimes R+\tau \frac{1}{{ϵ}_7}(U{E}_1{E}_1^{T}U)\otimes (I_n)\hfill \end{array}\right]$$

Evidently, if ${\overline{\varphi }}_1<0$ and ${\overline{\varphi }}_2<0$ hold, then the synchronization can be reached in the absence of external disturbance.

Now, we discuss the performance of the closed-loop system with disturbance $\omega (t)$ . Calculating $\stackrel{·}{V}(t)$ , it follows from equation (7) that

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}(\delta )\le {\overline{\delta }}^T(t)({\overline{H}}_1+\tau [(J_1^{T}U\Delta L)\otimes P\Gamma ](I_N\otimes R^{-1})[(J_1^{T}U\Delta L)\otimes P\Gamma {]}^T)\overline{\delta }(t)}\\ +{\overline{\delta }}^T(t-\tau )({\overline{H}}_2+\tau {[(U\Delta L{J}_1)\otimes \Gamma ]}^T(I_N\otimes R)[(U\Delta L{J}_1)\otimes \Gamma ])\overline{\delta }(t-\tau )\hfill \\ +2{\delta }^T(t)(I_N\otimes P)(U\otimes I_n)\omega (t)+2\tau F^T(x(t),t)(I_N\otimes R)(U\otimes I_n)\omega (t)\hfill \\ +\tau {\omega }^T(t)(U\otimes R)\omega (t)+2\tau (I_N\otimes {\pi }^T)(U\otimes R)\omega (t)-2{\omega }^T(t)(\mathrm{UL}\otimes R\Gamma )\delta (t-\tau )\hfill \\ +{ϵ}_8\tau {\omega }^T(t)(U{E}_1{E}_1^{T}U)\otimes (R\Gamma {\Gamma }^{T}R)\omega (t)+\frac{1}{{ϵ}_8}\tau {\delta }^T(t-\tau )(E_2^T{E}_2)\otimes I_n\delta (t-\tau )\hfill \end{array}$$

for ${ϵ}_8>0$

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}(\delta )\le {\overline{\delta }}^T(t)({\overline{H}}_1+\tau [(J_1^{T}U\Delta L)\otimes P\Gamma ](I_N\otimes R^{-1})[(J_1^{T}U\Delta L)\otimes P\Gamma {]}^T)\overline{\delta }(t)}\\ +{\overline{\delta }}^T(t-\tau )({\overline{H}}_2+\tau {[(U\Delta L{J}_1)\otimes \Gamma ]}^T(I_N\otimes R)[(U\Delta L{J}_1)\otimes \Gamma ])\overline{\delta }(t-\tau )\hfill \\ +2{\overline{\delta }}^T(t)(J_1^T\otimes P)(U\otimes I_n)\omega (t)+\tau {ϵ}_9{\beta }^2{\lambda }_{\max }(R^2){\overline{\delta }}^T(t)\overline{\delta }(t)+\frac{\tau }{{ϵ}_9}{\omega }^T(t)(U\otimes I_n)\omega (t)\hfill \\ +\tau {\omega }^T(t)(U\otimes R)\omega (t)-2\tau {\omega }^T(t)(\mathrm{UL}{J}_1)\otimes (R\Gamma )\overline{\delta }(t-\tau )\hfill \\ +{ϵ}_8\tau {\omega }^T(t)(U{E}_1{E}_1^{T}U)\otimes (R\Gamma {\Gamma }^{T}R)\omega (t)+\frac{1}{{ϵ}_8}\tau {\overline{\delta }}^T(t-\tau )(J_1^T{E}_2^T{E}_2{J}_1)\otimes I_n\overline{\delta }(t-\tau )\hfill \end{array}$$

Consider the following cost performance for any $T>0$

$$J_T=\underset{0}{\overset{T}{\int }}[z^T(t)z(t)-{\gamma }^2{\omega }^T(t)\omega (t)]\mathrm{dt}$$

For any non-zero $\omega (t)\in {\mathcal{L}}_2[0,\infty )$ , $t\ge 0$ , and zero-valued initial condition (i.e. $\delta (s)=0$ $s\in (-\infty ,0]$ ), we get

$$\begin{array}{c}\multicolumn{1}{c}{J_T=\underset{0}{\overset{T}{\int }}[z^T(t)z(t)-{\gamma }^2{\omega }^T(t)\omega (t)]\mathrm{dt}}\\ =\underset{0}{\overset{T}{\int }}[{\overline{\delta }}^T(t)\overline{\delta }(t)-{\gamma }^2{\omega }^T(t)\omega (t)+\stackrel{·}{V}(t)]\mathrm{dt}-V(T)\hfill \\ \le \underset{0}{\overset{T}{\int }}{\xi }^T(t)M\xi (t)\mathrm{dt}-V(T)\hfill \end{array}$$

where

$$M=\left[\begin{array}{ccc}{M}_{11}\hfill & 0\hfill & M_{13}\hfill \\ M_{22}\hfill & M_{23}\hfill \\ *\hfill & M_{33}\hfill \end{array}\right]$$

$$\begin{array}{c}{M}_{11}={\overline{H}}_1+\tau [(J_1^{T}U\Delta L)\otimes P\Gamma ](I_N\otimes R^{-1})[(J_1^{T}U\Delta L)\otimes P\Gamma {]}^T+\tau {ϵ}_9{\beta }^2{\lambda }_{\max }(R^2)I_{\kappa -n}+I_{\kappa -n}\hfill \\ M_{13}=(J_1^T\otimes P)(U\otimes I_n)\hfill \\ M_{23}=-\tau (J_1^T{L}^{T}U)\otimes ({\Gamma }^{T}R))\hfill \\ M_{22}={\overline{H}}_2+\tau {[(U\Delta L{J}_1)\otimes \Gamma ]}^T(I_N\otimes R)[(U\Delta L{J}_1)\otimes \Gamma ]+\frac{1}{{ϵ}_8}\tau (J_1^T{E}_2^T{E}_2{J}_1)\otimes I_n\hfill \\ M_{33}=-{\gamma }^2{I}_{\kappa }+\frac{\tau }{{ϵ}_9}U\otimes I_n+\tau U\otimes R+{ϵ}_8\tau (U{E}_1{E}_1^{T}U)\otimes (R\Gamma {\Gamma }^{T}R)\hfill \end{array}$$

$${\xi }^T(t)=\left[\begin{array}{ccc}{\overline{\delta }}^T(t)\hfill & {\overline{\delta }}^T(t-\tau )\hfill & {\omega }^T(t)\hfill \end{array}\right]$$

So, if $M<0$ holds, then $J_T<0$ , that is, ${\int }_0^T{z}^T(t)z(t)\mathrm{dt}<{\gamma }^2{\int }_0^T{\omega }^T(t)\omega (t)\mathrm{dt}$ . Let $T\to +\infty$ , we have

$$\parallel z(t){\parallel }_2^2<{\gamma }^2\parallel \omega (t){\parallel }_2^2,\mathrm{i}.\mathrm{e}.,\parallel T_{\omega z}(s){\parallel }_{\infty }<\gamma$$

By Schur Complement Lemma, it follows from equation (7) that $M<0$ if $H<0$ . Also, equation (8) holds when $H<0$ . Therefore, under the condition $H<0$ , synchronization can be reached while satisfying $\parallel T_{\omega z}(s){\parallel }_{\infty }<\gamma$ .

Remark 1

In Theorem 1, we make an assumption that the network has a spanning tree. Actually $H<0$ implies that $L$ has at most a simple zero eigenvalue, and it is easy to see that this assumption is a necessary condition for $H<0$ from Lemma 1.

Corollary 1

Suppose that the network has a spanning tree. For the network equation (1) with $\tau =0$ , the synchronization can be reached with desired $H_{\infty }$ performance if there exist a positive definite matrix $P\in {\mathbb{R}}^{n\times n}$ and a positive scalar $ϵ$ , satisfying

$$K=\left[\begin{array}{cc}{K}_{11}\hfill & (J_1^{T}U\otimes P)\hfill \\ -{\gamma }^2{I}_{\kappa }\hfill \end{array}\right]<0$$

where

$$\begin{array}{c}\multicolumn{1}{c}{K_{11}=2{\lambda }_{\max }(P)\beta I_{\kappa -n}-(J_1^T\mathrm{UL}{J}_1\otimes P\Gamma )-{(J_1^T\mathrm{UL}{J}_1\otimes P\Gamma )}^T}\\ +(J_1^{T}U{E}_1{E}_1^{T}U{J}_1)\otimes (P\Gamma {\Gamma }^{T}P)+\frac{1}{ϵ}(J_1^T{E}_2^T{E}_2{J}_1)\otimes I_n+I_{\kappa -n}\hfill \end{array}$$

Simulation

In this section, we will do simulations with four nodes to show the effectiveness of the results derived in previous sections. Suppose that $\Gamma =I$

$$\begin{array}{l}f(x_{i1}(t),t)=-x_{i1}(t)+0.1\sin x_{i1}(t)\hfill \\ f(x_{i2}(t),t)=-x_{i2}(t)+0.1\sin x_{i2}(t)\hfill \\ f(x_{i3}(t),t)=-x_{i3}(t)+0.1\sin x_{i3}(t)\hfill \end{array},\Delta L=\left[\begin{array}{cccc}0.1& -0.1& 0& 0\\ 0& 0.1& -0.1& 0\\ 0& 0& 0.1& -0.1\\ -0.1& 0& 0& 0.1\end{array}\right]$$

and the time-delay is $\tau =0.13\mathrm{s}$ . Let $\gamma =1$ . It can be solved that a feasible solution is $P=0.9I$ , $R=0.4I$ , and

$$\begin{array}{c}Q=\left[\begin{array}{ccc}2.1593\hfill & 0.4888\hfill & -0.7593\hfill \\ \multicolumn{1}{c}{0.4888}& 1.6994\hfill & -0.4819\hfill \\ \multicolumn{1}{c}{-0.7593}& -0.4819\hfill & 2.1379\hfill \end{array}\right],L=\left[\begin{array}{cccc}1.4724\hfill & -1.4724\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1.4724\hfill & -1.4724\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1.4724\hfill & -1.4724\hfill \\ \multicolumn{1}{c}{-1.4724}& 0\hfill & 0\hfill & 1.4724\hfill \end{array}\right]\hfill \end{array}$$

The state trajectories of all nodes with zero initial condition are shown in Figure 1 while the energies of the synchronization errors and the disturbances are shown in Figures 2 and 3, respectively. Clearly, all nodes asymptotically reach synchronization with desired $H_{\infty }$ performance, which is consistent with Theorem 1.

Figure 1.

State trajectories of all nodes with zero initial condition.

Figure 2.

The energy of all nodes.

Figure 3.

The energy of the external disturbances.

Conclusion

This article investigates synchronization of complex networks with uncertainty and time-delay, which could find applications in the complex earthquake monitor network. Based on Lyapunov theory, sufficient conditions are given to guarantee the synchronization of the complex networks in the presence of time-delay with desired $H_{\infty }$ performance. In the absence and presence of time-delay, it is shown that all nodes can reach synchronization with desired $H_{\infty }$ performance even when the external disturbances and the parameter uncertainties coexist.