Topology Identification of Time-Scales Complex Networks

Abstract

This paper studies a topology identification problem of complex networks with dynamics on different time scales. Using the adaptive synchronization method, some criteria for a successful estimation are obtained. In particular, by regulating the original network to synchronize with an auxiliary chaotic network, this work further explores a way to avoid the precondition of linear independence. When the adaptive controller fails to achieve the outer synchronization, an impulsive control method is used. In the end, we conclude with three numerical simulations. The results obtained in this paper generalize continuous, discrete with arbitrary time step size and mixed cases.

1. Introduction

Complex networks are becoming increasingly popular in the study of networked dynamic systems, including disease transmission networks, metabolic networks, social networks, communication networks, the Internet, and so on [1]. In these networks, topology structure describes the coupling states between nodes and plays a key role in the generation of collective dynamic behaviors [2,3,4]. However, in reality, many network topologies are sometimes unknown or uncertain. For instance, if a malfunction occurs in a communication network, power grid, or the Internet, it is very important to locate the faulty position [5]. As a result, network topology identification is worth investigating.

In the past decade, many methods have been studied for topology identification, such as the chaotic ant swarm algorithm [6], ROC curve analysis [7], and outer synchronization [8]. Among them, the synchronization-based topology identification method [5,8,9,10,11,12,13] has gained a lot of attention. In brief, this method first takes the unknown topology network as the driver network, and then constructs an auxiliary network with a coupling estimator and adaptive controllers as the response network. The topology is inferred by the coupling estimator when outer synchronization is achieved between the drive network and the response network. The synchronization-based method has been employed to investigate various network models, such as delayed networks, fractional networks, and multi-layer networks, see [14,15,16,17,18]. In [19,20,21,22,23], different forms of outer synchronization have been considered for topology identification, for example, adaptive lag synchronization, anticipatory synchronization, and generalized outer synchronization. In [24,25,26], several control methods have been exploited for network synchronization. It is worth mentioning that most references were considered under an assumption of linear independence of inner coupling function, which means the identification would fail if the partial or full inner synchronization occurs on the network, see [27]. This motivates some researchers to deal with this problem. In [18], Zhao et al. added sinusoidal interference signals to the original network to destroy the inner synchronization. In [28], Zhu et al. added a regulation mechanism to the network to be identified and constructed an auxiliary network consisting of isolated nodes of periodic nonlinear dynamical behavior. In [29], Liu et al. regulated the original network to synchronize with an auxiliary network composed of isolated chaotic systems to avoid the inner synchronization.

It should be noted that most existing relevant literature on the structure identification of complex dynamical networks is considered for continuous systems. Given that in the real world, the dynamics of networks are not only continuous, but also discrete, and sometimes even hybrid. For instance, a computer control system is a discrete system. Furthermore, the signal transmission in such networks is not continuous, but intermittent. With this motivation, the theory of time scale, which unifies continuous and discrete cases into one theoretical framework, provides a new idea for the study of this problem. Recently, this novel mathematical theory has been applied to various researches [30,31,32,33,34,35], especially in the synchronization of complex networks. Some general results about synchronization conclusions [16,32,36] and control criteria [35,37,38] have been generated.

Inspired by the identification works in [5,8,9,27,39], and the improved methods in [28,29], we investigate the topology identification of complex networks with different time scales in this paper. The main contributions are summarized as follows:

• We discuss the topology identification of complex networks on the theory of time scales, which makes the proposed criteria more general. These criteria not only applies to the continuous cases, but also to the discrete cases with arbitrary time step, and even to the intermittent cases;
• To overcome the identification failure caused by the inner synchronization of complex network, we improve the synchronization-based method on time scales by constructing a chaotic auxiliary network;
• An impulsive control method is developed ensuring that the outer synchronization is between the original network and the auxiliary network. Impulsive control criteria are offered on time scales.

The paper is organized as follows. We begin by recalling some preliminaries in Section 2. The topology identification of complex dynamical networks is discussed in Section 3. A new topology identification model and an impulsive control method are proposed in Section 4. In Section 5, numerical simulations are performed.

2. Preliminaries

In this section, some basics on time scales and notations are introduced. For a monograph on time scales, we recommend the interested reader to [40].

Let $\mathbb{R}$, $\mathbb{Z}$ and $\mathbb{N}$ be the set of the real numbers, integers, and natural numbers, respectively. A time scale $\mathbb{T}$ is an arbitrary non-empty closed subset of $\mathbb{R}$. Denote $h\mathbb{Z}:=\left\{hk:h\in \mathbb{N},k\in \mathbb{Z}\right\}$. ${\mathbb{N}}_{+}:=\mathbb{N}\setminus \{0\}$. $\mu :\mathbb{T}\to \left[0,\infty \right)$ is a graininess function which is defined by

$$\mu (t):=\sigma (t)-t,$$

(1)

where $\sigma (t):=inf\{r\in \mathbb{T}:r>t\}$ is the forward jump operator. t is right-dense if $\sigma (t)=t$ and right-scattered if $\sigma (t)>t$. $\parallel ·\parallel$ denotes the usual Euclidean norm. ⊗ is the Kronecker product which is defined by

$$A\otimes B=\left[\begin{array}{cccc}{a}_{11}B& a_{12}B& \cdots & a_{1q}B\\ a_{21}B& a_{22}B& \cdots & a_{2q}B\\ ⋮& ⋮& \ddots & ⋮\\ a_{p1}B& a_{p2}B& \cdots & a_{pq}B\end{array}\right]$$

for a matrix $A=(a_{ij})\in {\mathbb{R}}^{p\times q}$ and any matrix B. $I_q$ stands for a $q\times q$ identity matrix.

Definition 1

([40]). A function $p:\mathbb{T}\to \mathbb{R}$ is regressive provided

$$1+\mu (t)p(t)\ne 0\mathit{for}\mathit{all}t\in \mathbb{T}.$$

The set of all regressive functions is denoted by $\mathcal{R}$.

Definition 2

([40]). We define the set ${\mathcal{R}}^{+}$ of all positively regressive elements of $\mathcal{R}$ by

$${\mathcal{R}}^{+}{\mathcal{R}}^{+}(\mathbb{T},\mathbb{R})=\{p\in \mathcal{R}:1+\mu (t)p(t)>0\mathit{for}\mathit{all}t\in \mathbb{T}\}.$$

Definition 3

([40]). Let $p,q\in {\mathcal{R}}^{+}$. We define the time scale circle plus ⊕ by

$$(p\oplus q)(t):=p(t)+q(t)+\mu (t)p(t)q(t)\mathit{for}\mathit{all}t\in \mathbb{T}.$$

Definition 4

([40]). Let $x:\mathbb{T}\to {\mathbb{R}}^m$. Then, the time scale derivative $x^{\Delta }$ is defined by

$$x^{\Delta }(t)=\left\{\begin{array}{cc}\hfill \underset{r\to t,r\in \mathbb{T}}{lim}\frac{x(t)-x(r)}{t-r}& \mathit{if}t\mathit{is}\mathit{right}-\mathit{dense},\hfill \\ \hfill \frac{x(\sigma (t))-x(t)}{\mu (t)}& \mathit{if}t\mathit{is}\mathit{right}-\mathit{scattered}.\hfill \end{array}\right.$$

Definition 5

([40]). Let $f:\mathbb{T}\to \mathbb{R}$. We define the Cauchy integral by

$${\int }_s^{r}f(t)\Delta t=F(s)-F(r)\mathit{for}\mathit{all}r,s\in \mathbb{T},$$

where $F^{\Delta }(t)=f(t)$ for all $t\in [r,s]$.

Lemma 1

([40]). Assume functions $\psi ,\omega :\mathbb{T}\to {\mathbb{R}}^m$, ψ and ω are differentiable at $t\in \mathbb{T}$. Then, the following identities hold:

$$\psi (\sigma (t))=\psi (t)+\mu (t){\psi }^{\Delta }(t),$$

$${(\psi \omega )}^{\Delta }(t)={\psi }^{\Delta }(t)\omega (t)+\psi (t){\omega }^{\Delta }(t)+\mu (t){\psi }^{\Delta }(t){\omega }^{\Delta }(t).$$

Definition 6

([40]). If $p\in \mathcal{R}$ and $t,s\in \mathbb{T}$, then the only solution of the initial value problem

$$y^{\Delta }=p(t)y,y(s)=1$$

is called the time scales exponential function and denoted by $e_p(t,s)$.

Lemma 2

([40]). If $p\in {\mathcal{R}}^{+}$, then

$$y^{\Delta }\le p(t)y\mathit{for}\mathit{all}t\in \mathbb{T}$$

implies

$$y(t)\le y(t_0)e_p(t,t_0)\mathit{for}\mathit{all}t\in \mathbb{T}.$$

Lemma 3

([41]). If $p\in {\mathcal{R}}^{+}$, then

$$e_p(t,s)\le exp\left\{{\int }_s^{t}p(\tau )\Delta \tau \right\}\mathit{for}\mathit{all}t>s.$$

Lemma 4

([42]). The linear matrix inequality (LMI)

$$\left[\begin{array}{cc}Q(x)& S(x)\\ S^T(x)& R(x)\end{array}\right]>0$$

is equivalent to

$$Q(x)>0\mathit{and}R(x)-S^T(x)Q^{-1}(x)S(x)>0,$$

where $Q(x)=Q^T(x)$ and $R(x)=R^T(x)$.

3. A Complex Network Model and Its Topology Identification

Consider a general time scale dynamical complex network of N dynamical nodes, each of which is an m-dimensional dynamical system on time scale $\mathbb{T}$. The network is characterized by

$$x_i^{\Delta }(t)=f(x_i(t))+\sum _{j=1}^N{c}_{ij}h(x_j(t)),i=1,2,\cdots ,N,$$

(2)

where $t\in \mathbb{T}$, $x_i:\mathbb{T}\to {\mathbb{R}}^m$, $x_i={(x_{i1},\cdots ,x_{im})}^T$ is the state vector of the i-th node, $f:{\mathbb{R}}^m\to {\mathbb{R}}^m$ is a nonlinear vector function representing the node dynamics, $h:{\mathbb{R}}^m\to {\mathbb{R}}^m$ is the inner coupling function. The outer coupling matrix $C=(c_{ij})\in {\mathbb{R}}^{N\times N}$ is defined as $c_{ii}=-{\sum }_{j=1,j\ne i}^N{c}_{ij}$. Here, $c_{ij}=c_{ji}>0$ $(i\ne j)$ if there is a connection between node i and j. Otherwise, $c_{ij}=c_{ji}=0$ $(i\ne j)$. One can easily see that the topology identification of (2) is dominated by matrix C.

We assume that the state vector $x_i$ is observable and can be fully measured. Function f and h are determined. To recover the unknown matrix C, one can take (2) as the drive network and construct a time scale response network as

$$\left\{\begin{array}{cc}\hfill y_i^{\Delta }(t)=& f(y_i(t))+\sum _{j=1}^N{d}_{ij}(t)h(y_j(t))+{\mu }_i(t),\hfill \\ \hfill {\mu }_i(t)=& -k_i(t)\left(y_i(t)-x_i(t)\right),\hfill \\ \hfill d_{ij}^{\Delta }(t)=& -{\left(y_i(t)-x_i(t)\right)}^{T}h(x_j(t)),\hfill \end{array}\right.$$

(3)

where $i,j=1,2,\cdots ,N$, $t\in \mathbb{T}$, $y_i:\mathbb{T}\to {\mathbb{R}}^m$, $y_i={(y_{i1},\cdots ,y_{im})}^T$ is the state vector of the i-th node. The outer coupling matrix $D(t)=(d_{ij}(t))\in {\mathbb{R}}^{N\times N}$ is the estimation of the matrix C at time $t\in \mathbb{T}$, $d_{ii}=-{\sum }_{j=1,j\ne i}^N{d}_{ij}$, $d_{ij}=d_{ji}$ $(i\ne j)$. ${\mu }_i(t)$ is the i-th adaptive controller. $k_i(t)$ is a positive bounded function.

Let $e_i(t)y_i(t)-x_i(t)$. We have the error dynamical network

$$e_i^{\Delta }(t)=f(y_i(t))-f(x_i(t))+\sum _{j=1}^N\left(d_{ij}(t)h(y_j(t))-c_{ij}h(x_j(t))\right)-k_i(t)e_i(t),i=1,2,\cdots ,N.$$

(4)

Assumption 1.

Assume that there exist positive constants α and β satisfying

$$\begin{array}{cc}\hfill \parallel f(x)-f(y)\parallel & \le \alpha \parallel x-y\parallel ,\hfill \\ \hfill \parallel h(x)-h(y)\parallel & \le \beta \parallel x-y\parallel ,\hfill \end{array}$$

for any $x,y\in {\mathbb{R}}^m$.

Definition 7.

The error dynamical network (4) achieves asymptotic stability at zero solution if

$$e_i(t)\to 0\mathit{as}t\to \infty ,i=1,\cdots ,N,t\in \mathbb{T}.$$

Definition 8.

The topology of network (2) can be identified if

$$\underset{t\to \infty }{lim}\left(d_{ij}(t)-c_{ij}\right)=0,i,j=1,2,\cdots ,N,t\in \mathbb{T}.$$

Lemma 5.

If the error dynamical network (4) achieves asymptotic stability at zero solution and the vectors $h(x_j)$, $j=1,2,\cdots ,N$, are linearly independent at the zero solution, then the topology of network (2) can be identified.

Proof. 

If $\underset{t\to \infty }{lim}{e}_i(t)=0$, then from (4), we have

$$\underset{t\to \infty }{lim}\sum _{j=1}^N\left(d_{ij}(t)h(y_j(t))-c_{ij}h(x_j(t))\right)=\underset{t\to \infty }{lim}\left(\sum _{j=1}^N\left(d_{ij}(t)-c_{ij}\right)h(x_j(t))\right)=0.$$

Since $h(x_j)$ are linearly independent, $d_{ij}(t)$ converges to $c_{ij}$ as $t\to \infty$, concluding the proof. □

Theorem 1.

Let $K(t)=diag(k_1(t),k_2(t),\cdots ,k_N(t))$. Assume there exist constant $\zeta >0$ and $\theta \in {\mathcal{R}}^{+}:\mathbb{T}\to \mathbb{R}$ such that the matrix $-(2I_N-\mu K)K+\left(\zeta ({\alpha }^2+{\beta }^2)-\theta \right)I_N$ is invertible and

$${\int }_{t_0}^t\theta (\tau )\Delta \tau \to -\infty \mathrm{as}t\to \infty .$$

(5)

For all $t\in \mathbb{T}$, if we have

$$\Omega =\left(\begin{array}{cccc}-(2I_N-\mu K)K+\left(\zeta ({\alpha }^2+{\beta }^2)-\theta \right)I_N& I_N-\mu K& (I_N-\mu K)D& (I_N-\mu K)(D-C)\\ I_N-\mu K& (\mu -\zeta )I_N& \mu D& \mu (D-C)\\ D(I_N-\mu K)& \mu D& \mu D^{T}D-\zeta I_N& \mu D^T(D-C)\\ (D-C)(I_N-\mu K)& \mu (D-C)& \mu {(D-C)}^{T}D& \mu {(D-C)}^T(D-C)\end{array}\right)\le 0,$$

(6)

then the error dynamical network (4) achieve asymptotical stability at zero solution.

Proof. 

Set $V(t):={\sum }_{i=1}^N{e}_i^T(t)e_i(t)$, $t\in \mathbb{T}$. We have

$$\begin{array}{cc}\hfill V^{\Delta }=& \sum _{i=1}^N\left({\left(e_i^T\right)}^{\Delta }{e}_i+{\left(e_i^T\right)}^{\sigma }{e}_i^{\Delta }\right)\hfill \\ \hfill =& 2\sum _{i=1}^N{e}_i^T(f(y_i)-f(x_i))+2\sum _{i=1}^N{e}_i^T\sum _{j=1}^N{d}_{ij}\left(h(y_j)-h(x_j)\right)\hfill \\ \hfill & +2\sum _{i=1}^N{e}_i^T\sum _{j=1}^N(d_{ij}-c_{ij})h(x_j)-2\sum _{i=1}^N{k}_i{(y_i-x_i)}^T(y_i-x_i)\hfill \\ \hfill & +\mu \left[\sum _{i=1}^N{(f(y_i)-f(x_i))}^T(f(y_i)-f(x_i))+2\sum _{i=1}^N{(f(y_i)-f(x_i))}^T\sum _{j=1}^N{d}_{ij}\left(h(y_j)-h(x_j)\right)\right.\hfill \\ \hfill & +2\sum _{i=1}^N{(f(y_i)-f(x_i))}^T\sum _{j=1}^N\left(d_{ij}-c_{ij}\right)h(x_j)-2\sum _{i=1}^N{k}_i{(f(y_i)-f(x_i))}^T(y_i-x_i)\hfill \\ \hfill & +\sum _{i=1}^N\sum _{j=1}^N{d}_{ij}{\left(h(y_j)-h(x_j)\right)}^T\sum _{j=1}^N{d}_{ij}\left(h(y_j)-h(x_j)\right)\hfill \\ \hfill & +2\sum _{i=1}^N\sum _{j=1}^N{d}_{ij}(t){\left(h(y_j(t))-h(x_j(t))\right)}^T\sum _{j=1}^N\left(d_{ij}(t)-c_{ij}\right)h(x_j(t))\hfill \\ \hfill & +\sum _{i=1}^N\sum _{j=1}^N\left(d_{ij}-c_{ij}\right)h^T(x_j)\sum _{j=1}^N\left(d_{ij}-c_{ij}\right)h(x_j)-2\sum _{i=1}^N{k}_i\sum _{j=1}^N\left(d_{ij}-c_{ij}\right)h^T(x_j)(y_i-x_i)\hfill \\ \hfill & \left.-2\sum _{i=1}^N{k}_i\sum _{j=1}^N{d}_{ij}{\left(h(y_j)-h(x_j)\right)}^T(y_i-x_i)+\sum _{i=1}^N{k}_i^2{(y_i-x_i)}^T(y_i-x_i)\right],t\in \mathbb{T}.\hfill \end{array}$$

We introduce the notations

$$\begin{array}{cc}\hfill E(t)& ={(e_1^T(t),e_2^T(t),\cdots ,e_N^T(t))}^T,t\in \mathbb{T},\hfill \\ \hfill G_1(t)& ={(g_{11}^T(t),g_{12}^T(t),\cdots ,g_{1N}^T(t))}^T,t\in \mathbb{T},\hfill \\ \hfill G_2(t)& ={(g_{21}^T(t),g_{22}^T(t),\cdots ,g_{2N}^T(t))}^T,t\in \mathbb{T},\hfill \\ \hfill H(X(t))& ={(h^T(x_1(t)),h^T(x_2(t)),\cdots ,h^T(x_N(t)))}^T,t\in \mathbb{T},\hfill \end{array}$$

(7)

where $g_{1i}(t):=f(y_i(t))-f(x_i(t)),t\in \mathbb{T}$ and $g_{2i}(t):=h(y_i(t))-h(x_i(t)),t\in \mathbb{T}$, $i=1,2,\cdots ,N$.

From Assumption 1, we have

$$\begin{array}{cc}\hfill \zeta \left(G_1^T{G}_1+G_2^T{G}_2\right)& =\zeta \left(\sum _{i=1}^N{g}_{1i}^T{g}_{1i}+\sum _{i=1}^N{g}_{2i}^T{g}_{2i}\right)\hfill \\ \hfill & \le \zeta \left({\alpha }^2+{\beta }^2\right)E^{T}E,t\in \mathbb{T}.\hfill \end{array}$$

(8)

Then, we have

$$\begin{array}{cc}\hfill V^{\Delta }-\theta V\le & 2E^T{G}_1+2E^T(D\otimes I_m)G_2+2E^T((D-C)\otimes I_m)H(X)-2E^T((K-\theta I_N)\otimes I_m)E\hfill \\ \hfill & +\zeta \left({\alpha }^2+{\beta }^2\right)E^{T}E-\zeta \left(G_1^T{G}_1+G_2^T{G}_2\right)\hfill \\ \hfill & +\mu \left[G_1^T{G}_1+2G_1^T\left(D\otimes I_m\right)G_2+2G_1^T\left((D-C)\otimes I_m\right)H(X)\right.\hfill \\ \hfill & -2G_1^T\left(K\otimes I_m\right)E+G_2^T\left(D^{T}D\otimes I_m\right)G_2+2G_2^T\left(D^T(D-C)\otimes I_m\right)H(X)\hfill \\ \hfill & -2E^T\left(KD\otimes I_m\right)G_2+H^T(X)\left({(D-C)}^T(D-C)\otimes I_m\right)H(X)\hfill \\ \hfill & \left.-2E^T\left(K(D-C)\otimes I_m\right)H(X)+E^T\left(K^2\otimes I_m\right)E\right]\hfill \\ \hfill \le & Q^T\left(\Omega \otimes I_m\right)Q,t\in \mathbb{T},\hfill \end{array}$$

where $Q={\left(E^T,G_1^T,G_2^T,H^T(X)\right)}^T$. By (6), $V^{\Delta }-\theta V\le Q^T\left(\Omega \otimes I_m\right)Q\le 0$, i.e.,

$$V^{\Delta }\le \theta V,t\in \mathbb{T}.$$

(9)

Since $\theta \in {\mathcal{R}}^{+}$, using Lemma 2, we have

$$\begin{array}{c}\hfill 0\le V(t)\le e_{\theta }(t,t_0)V(t_0),t\in \mathbb{T},\end{array}$$

(10)

where $t>t_0$. Without loss of generality, we assume the initial value $V(t_0)\ne 0$. According to Lemma 3, it follows that

$$\begin{array}{c}\hfill 0\le e_{\theta }(t,t_0)\le exp\left\{{\int }_{t_0}^t\theta (\tau )\Delta \tau \right\},t\in \mathbb{T}.\end{array}$$

(11)

From (5) and (11), we have $e_{\theta }(t,t_0)\to 0$ as $t\to \infty$. This leads to $V(t)\to 0$ as $t\to \infty$. Since the non-negativity of $V(t)$, we have $e_i\to 0$ as $t\to \infty$. Thus, we conclude that the error systems (4) achieve asymptotic stability at their zero solution. □

Remark 1.

If the condition (6) holds, due to Lemma 4, we have

$$-(2I_N-\mu K)K+\left(\zeta ({\alpha }^2+{\beta }^2)-\theta \right)I_N<0,t\in \mathbb{T}.$$

Then,

$$\theta (t)>{\lambda }_{max}\left(-(2I_N-\mu K)K+\zeta ({\alpha }^2+{\beta }^2)I_N\right),t\in \mathbb{T},$$

(12)

where ${\lambda }_{max}(·)$ is the maximum eigenvalue function.

4. Modified Topology Identification Based on Impulsive Synchronization

We can see that from Lemma 5 it would not be convenient to reveal the topology of network (2) by using the linear independence of h. In this section, we make some improvement for Lemma 5.

Consider the same dynamical network (2). In order to identify the unknown outer coupling matrix C, we construct an auxiliary network consisting of N isolated nodes

$${\widehat{y}}_i{(t)}^{\Delta }=f\left({\widehat{y}}_i(t)\right),i=1,2,\cdots ,N,$$

where ${\widehat{y}}_i:\mathbb{T}\to {\mathbb{R}}^m$, ${\widehat{y}}_i={({\widehat{y}}_{i1},\cdots ,{\widehat{y}}_{im})}^T$ is the state vector of the i-th node. We employ a chaotic system as node dynamics so that the state of the node will be chaotic. The word “chaos” is commonly used to describe a state of disorder. However, there is no universally accepted mathematical definition of chaos. One widely used definition states that a dynamical system must exhibit the features that it is sensitive to initial conditions and topologically transitive. One knows that the state ${\widehat{y}}_i$ is bounded. We set the initial states of the nodes to be different from each other. Thus, the states ${\widehat{y}}_i$, $i=1,2,\cdots ,N$, are linearly independent. We rewrite the adaptive controller ${\mu }_i(t)$ as follows:

$${\mu }_i(t)=k_i(t)\left({\widehat{y}}_i(t)-x_i(t)\right)-\sum _{j=1}^N{\widehat{d}}_{ij}(t)h({\widehat{y}}_j(t)),$$

(13)

where ${\widehat{d}}_{ij}^{\Delta }(t)=-{\left({\widehat{y}}_i(t)-x_i(t)\right)}^{T}h(x_j(t))$. Then, we apply ${\mu }_i(t)$ on the original network, which results in

$$x_i^{\Delta }(t)=f(x_i(t))-\sum _{j=1}^N\left({\widehat{d}}_{ij}(t)h({\widehat{y}}_j(t))-c_{ij}h(x_j(t))\right)+k_i(t)\left({\widehat{y}}_i(t)-x_i(t)\right).$$

(14)

Again, let ${\widehat{e}}_i(t)={\widehat{y}}_i(t)-x_i(t)$. We obtain the error dynamical system

$${\widehat{e}}_i^{\Delta }(t)=f({\widehat{y}}_i(t))-f(x_i(t))+\sum _{j=1}^N\left({\widehat{d}}_{ij}(t)h({\widehat{y}}_j(t))-c_{ij}h(x_j(t))\right)-k_i(t){\widehat{e}}_i(t),i=1,2,\cdots ,N,$$

(15)

which has the same representation as $e_i^{\Delta }(t)$ in (4). Hence, Theorem 1 can also be used for (15). Moreover, we apply impulsive controllers onto the drive network (14), thus the controlled drive network can be expressed as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{x}_i^{\Delta }(t)=f(x_i(t))-\sum _{j=1}^N\left({\widehat{d}}_{ij}(t)h({\widehat{y}}_j(t))-c_{ij}h(x_j(t))\right)+k_i(t)\left({\widehat{y}}_i(t)-x_i(t)\right),\hfill & \hfill t\ne t_l,\\ x_i\left(t_l^{+}\right)=x_i\left(t_l^{-}\right)+B_i(t_l)\left(x_i\left(t_l^{-}\right)-{\widehat{y}}_i\left(t_l^{-}\right)\right),\hfill & \hfill t=t_l,\end{array}\right.\end{array}$$

(16)

where $l\in \mathbb{N}$, the discrete time sequence of impulses satisfies

$$t_1<t_2<\cdots <t_l<\cdots \mathrm{with}\underset{l\to \infty }{lim}{t}_l=\infty .$$

$x_i\left(t_l^{+}\right)={lim}_{t\to t_l^{+}}{x}_i(t)$, $x_i\left(t_l^{-}\right)={lim}_{t\to t_l^{-}}{x}_i(t)$. $B_i(t_l)\in {\mathbb{R}}^{m\times m}$ are impulsive gains at $t_l$, and $B_i(t)=0$ for $t\ne t_l$, $i=1,2,\cdots ,N$, $l\in \mathbb{N}$. Here, we assume ${\widehat{y}}_i(t_l)={\widehat{y}}_i\left(t_l^{+}\right)={\widehat{y}}_i\left(t_l^{-}\right)$, $x_i(t_l)=x_i\left(t_l^{-}\right)$. Then, the impulsive error system is written as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\widehat{e}}_i^{\Delta }(t)=f({\widehat{y}}_i(t))-f(x_i(t))+{\sum }_{j=1}^N\left({\widehat{d}}_{ij}(t)h({\widehat{y}}_j(t))-c_{ij}h(x_j(t))\right)-k_i(t){\widehat{e}}_i(t),\hfill & \hfill t\ne t_l,\\ {\widehat{e}}_i\left(t_l^{+}\right)=\left(I_m+B_i(t_l)\right){\widehat{e}}_i\left(t_l^{-}\right),\hfill & \hfill t=t_l,\end{array}\right.\end{array}$$

(17)

where ${\widehat{e}}_i\left(t_l^{+}\right){lim}_{t\to t_l^{+}}{\widehat{e}}_i(t)$ and ${\widehat{e}}_i\left(t_l^{-}\right){lim}_{t\to t_l^{-}}{\widehat{e}}_i(t)$.

Theorem 2.

Let ${\varphi }_i(t)={\lambda }_{max}\left({\left(I_m+B_i(t)\right)}^T\left(I_m+B_i(t)\right)\right)$ and $\varphi (t)=max({\varphi }_1(t),{\varphi }_2(t),\cdots$, ${\varphi }_N(t))$. Assume $\theta \in {\mathcal{R}}^{+}:\mathbb{T}\to \mathbb{R}$. If there exists $\delta >1$, such that

$$\delta \varphi (t_l)e_{\theta }(t_l,t_{l-1})<1\mathit{for}\mathit{all}t\in \mathbb{T},l\in \mathbb{N},$$

(18)

then the impulsive system (17) is asymptotically stable at the zero solution.

Proof. 

We consider the function $V(t):={\sum }_{i=1}^N{\widehat{e}}_i^T(t){\widehat{e}}_i(t)$. It follows from (9) and (10) that

$$V^{\Delta }(t)\le \theta (t)V(t),t\in \left[t_{l-1},t_l\right),$$

(19)

and

$$V(t)\le e_{\theta }\left(t,t_{l-1}\right)V\left(t_{l-1}^{+}\right),t\in \left[t_{l-1},t_l\right).$$

(20)

It can be obtained from the second equation of (17) that

$$\begin{array}{cc}\hfill V\left(t_{l-1}^{+}\right)& =\sum _{i=1}^N{\left[\left(I_m+B_i(t_{l-1})\right){\widehat{e}}_i\left(t_{l-1}^{-}\right)\right]}^T\left[\left(I_m+B_i(t_{l-1})\right){\widehat{e}}_i\left(t_{l-1}^{-}\right)\right]\hfill \\ \hfill & =\sum _{i=1}^N{\widehat{e}}_i^T\left(t_{l-1}^{-}\right)\left[{\left(I_m+B_i(t_{l-1})\right)}^T\left(I_m+B_i(t_{l-1})\right)\right]{\widehat{e}}_i\left(t_{l-1}^{-}\right)\hfill \\ \hfill & \le \sum _{i=1}^N{\varphi }_i(t_{l-1}){\widehat{e}}_i^T\left(t_{l-1}^{-}\right){\widehat{e}}_i\left(t_{l-1}^{-}\right)\hfill \\ \hfill & \le \varphi (t_{l-1})\sum _{i=1}^N{\widehat{e}}_i^T\left(t_{l-1}^{-}\right){\widehat{e}}_i\left(t_{l-1}^{-}\right)\hfill \\ \hfill & =\varphi (t_{l-1})V\left(t_{l-1}^{-}\right).\hfill \end{array}$$

(21)

Next, we choose $l=1$, the Equation (20) can be written as

$$V(t)\le e_{\theta }(t,t_0)V\left(t_0^{+}\right),t\in \left[t_0,t_1\right).$$

Hence, we have

$$V(t_1)\le e_{\theta }(t_1,t_0)V\left(t_0^{+}\right)$$

and

$$V\left(t_1^{+}\right)\le \varphi (t_1)V(t_1)\le \varphi (t_1)e_{\theta }(t_1,t_0)V\left(t_0^{+}\right).$$

For $t\in \left[t_1,t_2\right)$, we have

$$V(t)\le e_{\theta }(t,t_1)V\left(t_1^{+}\right)\le e_{\theta }(t,t_1)\varphi (t_1)e_{\theta }(t_1,t_0)V\left(t_0^{+}\right)$$

and

$$V\left(t_2^{+}\right)\le \varphi (t_2)V(t_2)\le \varphi (t_2)e_{\theta }(t_2,t_1)\varphi (t_1)e_{\theta }(t_1,t_0)V\left(t_0^{+}\right).$$

Consequently, one can get a more general inequality

$$V(t)\le e_{\theta }(t,t_l)\varphi (t)e_{\theta }(t_l,t_{l-1})\cdots \varphi (t_2)e_{\theta }(t_2,t_1)\varphi (t_1)e_{\theta }(t_1,t_0)V\left(t_0^{+}\right),t\in \left[t_l,t_{l+1}\right).$$

Using the condition (18), we can now have

$$V(t)\le e_{\theta }(t,t_l)\frac{1}{{\delta }^l}V\left(t_0^{+}\right).$$

(22)

Due to $\delta >1$, it leads to

$$V(t)\to 0\mathrm{as}l\to \infty .$$

Therefore, we have $\widehat{e}(t)\to 0$ as $t\to \infty$, concluding the proof. □

5. Numerical Examples

In this section, three numerical examples with different time scales are given to verify our theoretical results. Precisely, examples show how the topology of dynamical networks are identified with $\mathbb{T}=\mathbb{R}$, $\mathbb{T}=h\mathbb{Z}$ and $\mathbb{T}={\displaystyle \bigcup _{j=0}^{\infty }}\left[j(a+b),j(a+b)+a\right]$, $a,b>0$, respectively. For simplicity, we consider the networks consisting of four identical dynamical nodes. Without loss of generality, the inner coupling function is simplified as $h(x_j)=x_j$ and the coefficient of adaptive controller is set as $k_i=1$, $i=1,2,\cdots ,N$. All examples are simulated by random initial values. We can find the appropriate $\zeta$, such that $\zeta \left({\alpha }^2+{\beta }^2\right)=1$.

Example 1.

In this example, we consider the continuous case and suppose $\mathbb{T}=\mathbb{R}$, such that $\mu (t)\equiv 0$. The chaotic Chen oscillator is taken to simulate each node of the network: for $i=1,2,3,4$, $f(x_i)=\left(35(x_{i2}-x_{i1});-7x_{i1}-x_{i1}{x}_{i3}+28x_{i2};x_{i1}{x}_{i2}-3x_{i3}\right)$. The unknown outer coupling matrix C is preset to $c_{1,2}=c_{2,1}=4$, $c_{1,3}=c_{3,1}=7$, $c_{1,4}=c_{4,1}=3$, $c_{2,3}=c_{3,2}=6$, $c_{2,4}=c_{4,2}=0$, $c_{3,4}=c_{4,3}=2$. The topology structure is shown in Figure 1. According to Remark 1, we have $\theta (t)>-1$. Therefore, the conditions in Theorem 1 can be satisfied when $\theta (t)\in (-1,0)$. Figure 2 shows the results of our simulation when $\mathbb{T}=\mathbb{R}$. As can be seen, the estimated value $d_{ij}$ continuously approaches the preset value $c_{ij}$ without additional control. The network topology is successfully identified.

Example 2.

In this example, we consider the discrete case and suppose $\mathbb{T}=2\mathbb{Z}$, such that $\mu (t)\equiv 2$. The chaotic Hénon map is taken to simulate each node of the network: For $i=1,2,3,4$, $f(x_i)=\left(1+x_{i2}-1.4x_{i1}^2;0.3x_{i1}\right)$. The unknown outer coupling matrix C is preset to $c_{1,2}=c_{2,1}=2$, $c_{1,3}=c_{3,1}=6$, $c_{1,4}=c_{4,1}=2$, $c_{2,3}=c_{3,2}=7$, $c_{2,4}=c_{4,2}=6$, $c_{3,4}=c_{4,3}=1$. The topology structure is shown in Figure 3. According to Remark 1, we have $\theta (t)>1$. From (18), if we choose $B_i(t_{\ell })=diag(-0.88,-0.88)$, $i=1,2,3,4$, $\theta =2$ and $\delta =1.27$, then we have ${\Delta }_{\ell }\le 2.0008$. Without loss of generality, we choose ${\Delta }_{\ell }=2$. Figure 4 shows the results of our simulation when $\mathbb{T}=2\mathbb{Z}$. As can be seen, the estimated value $d_{ij}$ approaches the actual value $c_{ij}$ discretely with impulsive control every 2 s. The network topology is successfully identified.

Example 3.

In this example, we suppose $\mathbb{T}={\displaystyle \bigcup _{j=0}^{\infty }}\left[j(a+b),j(a+b)+a\right]$, $a,b>0$. Specifically, we let $a=2$ and $b=2$ then $\mathbb{T}={\displaystyle \bigcup _{j=0}^{\infty }}\left[4j,4j+2\right]$. The graininess function of $\mathbb{T}$ is given by

$$\mu (t)=\left\{\begin{array}{cc}\hfill 0,& \mathit{if}t\in \bigcup _{j=0}^{\infty }\left[4j,4j+2\right),\hfill \\ \hfill 2,& \mathit{if}t\in \bigcup _{j=0}^{\infty }\{4j+2\}.\hfill \end{array}\right.$$

We consider the same unknown outer coupling matrix C as indicated in Example 1. The chaotic Chen oscillator is taken to simulate each node of the network: For $i=1,2,3,4$, $f(x_i)=\left(10(x_{i2}-x_{i1});28x_{i1}-x_{i1}{x}_{i3}-x_{i2};x_{i1}{x}_{i2}-\frac{8}{3}{x}_{i3}\right)$. In this case, $\theta (t)$ can be chosen by any value between $-1$ and 0 when $\mu (t)=0$. When $\mu (t)=2$, $\theta (t)>1$. From (18), if we choose $B_i(t_{\ell })=diag(-0.88,-0.88,-0.88)$, $i=1,2,3,4$, $\theta =2$ and $\delta =1.1$, then we have ${\Delta }_{\ell }\le [2.0726]=2$, which means that impulsive control occurs at each time point of multiples of 2. Figure 5 shows the results of our simulation for this case. As can be seen, the estimated value $d_{ij}$ approaches the preset value $c_{ij}$ intermittently with impulsive control every 2 s. The network topology is successfully identified.

6. Conclusions

In this paper, a topology identification problem of complex dynamical networks on different time scales is studied. We have investigated the outer synchronization between the original network and the auxiliary network on time scales. General synchronization criteria have been proposed according to the matrix inequality and time scale regressive condition. Moreover, we have applied the adaptive controller on the original network and taken the impulsive method to synchronize with the auxiliary chaotic network. An impulsive criterion has been derived to ensure the outer synchronization. Three examples on different time scales have been given to verify our results. The proposed ideas and methods can be extended to the topology identification problem of other complex networks.