1 Introduction

Complex dynamical networks (CDNs), which contains a large amount of nodes with the communication topology, are ubiquitous in nature and human society, such as transportation networks, biological networks, power networks and so on [1,2,3,4]. Generally, the dynamics of CDNs are determined by the topology structure and dynamical properties of nodes, which indicates that the dynamical behavior of CDNs is difficult to understand when the communication topology is unknown [5,6,7]. Hence, it is significant to identify the topology of CDNs, see [8,9,10,11,12,13,14,15,16,17,18,19] and references therein.

Recently, a plenty of studies are concentrated on the topology identification of CDNs, and some results are achieved. For example, in [11], Zhu et al. researched the topology identification of CDNs via synchronization method and an auxiliary network, which composed of isolated nodes. In [12], a novel method was developed to solve topology identification for Hindmarsh-Rose neural networks with sinusoidal signals when the persistently excite conditions could not hold. In [13], the topology identification for CDNs with multiple links was considered by utilizing adaptive control method and building chaotic auxiliary networks.

Since the fractional calculus can describe the various processes with memory and hereditary characteristics more accurately than integral calculus, see [20, 21], CDNs described by fractional calculus received wide attention from scholars. More recently, many works have been put into topology identification for fractional complex dynamical networks (FCDNs) [15,16,17,18,19]. Particularly, in [18], the topology identification was discussed for FCDNs via inequality techniques and synchronization-based method . In [19], Bai et al. considered the secure synchronization and topology identification for FCDNs under Denial of Service (DoS) attacks by using feedback control method.

It should be pointed out that, in the control mechanism in [14,15,16,17,18,19], the feedback / adaptive controller is used to deal with the synchronization of CDNs. Nevertheless, the adaptive/ feedback control mechanism requires to update date continuously, which will lead to additional information communication and waste of resources. To address the drawback, the event-triggered control (ETC) determined by static rule is proposed, and some results about the synchronization of CDNs were obtained by applying static ETC . see [22,23,24,25,26]. In static ETC, information is updated only when the threshold is less than or equal to the measurement error.

In order to mitigate the communication burden more effectively, in recent works, the dynamic event-triggered control(DETC) mechanism is presented, where a dynamic variable was added to the event-triggered condition. For example, in [29], Li et al. considered the synchronization for variable-order fractional piecewise-smooth networks by DETC, and established the exponential synchronization criterion. The mean-square exponential synchronization in a stochastic complex network was investigated via intermittent DETC in [30].

It is worth noting that in the aforementioned results, for CDNs, the time-scale characteristics of all nodes are required to be the same. However, in real world, ’fast’ dynamics and ’slow’ dynamics may coexist in one system, which can be modeled as a singularly perturbation system, such as circuit systems, high-speed aircraft systems, etc. [31,32,33,34]. Recently, SPCNs have attracted considerable concern of many scholars [35,36,37,38,39,40,41,42,43,44]. Particularly, in [37], Rakkiyappan et al. explored the state estimation and synchronization for SPCNs by sampling control method. In [39], the stochastic synchronization was studied for SPCNs with impulsive effects and semi-markov jump topology. It should be pointed out that, in the aforementioned papers about SPCNs [36,37,38,39,40,41,42,43,44], the topology structure of the CNs is set to be known. As far as we know, few results are focused on the topology identification for FSPCNs. Hence, the main motivation of this paper is to design topology observer and control strategy with DETM, derive synchronization and identification conditions for FSPCNs.

Inspired by the above discussion, the objective of this article is to tackle the problems of the topology identification and synchronization for FSPCNs. In this paper, the main contributions are listed below.

  1. 1)

    In the existing results [15] and [18], the topology identification and synchronization are considered for FCDNs. The topology identification and synchronization are considered for fractional CDNs with the singularly perturbation in this work. Compared with [15] and [18], the results in this paper are more universal.

  2. 2)

    An topology observer and a DETM are designed to realize the topology identification and synchronization for FSPCNs. Compared with the control methods used in [23, 25], the control strategy adopted in this paper can save resource more efficiently.

  3. 3)

    The finite-time synchronization condition is established based on a novel \(\varepsilon \)-dependent Lyapunov function, by the matrix inequality. In addition, the topology identification is successfully realized.

The remainders of this paper are arranged as follows. In section 2, some important lemmas, definitions and properties are introduced. The system description is provided in Sect. 3. In Sect. 4, the main results of the topology identification and synchronization for FSPCNs are discussed. In section 5, an application example is presented to verify the correctness of the theoretical results. Section 6 gives some conclusions.

Notation: See Table 1.

Table 1 Notation
Full size table

2 Preliminaries

Definition 1

( [20]) For \(f(\tau )\in C^n[t_{0},\ +\infty )\). Caputo fractional derivative of \(\alpha \)-order is defined as

$$\begin{aligned} {}_{t}^{c}D^{\alpha }_{t_{0}}f(\tau )= \frac{1}{\Gamma (n-\alpha )}\int _{t_{0}}^{t}\frac{f^{(n)}(v)}{(\tau -v)^{1+n-\alpha }}dv, \end{aligned}$$

where \(\tau \ge t_{0}\), \(\Gamma (\cdot )\) is Gamma function, \(\alpha \in (n-1, n)\), \(n\in N\). Especially, if \(\alpha \in (0,1)\), \({}_{t}^{c}D^{\alpha }_{t_{0}}f(\tau )= \frac{1}{\Gamma (1-\alpha )}\int _{t_{0}}^{t}\frac{f^{'}(v)}{(\tau -v)^{\alpha }}dv\).

Definition 2

( [20]) For \(f(\tau ): [t_{0},\ +\infty )\rightarrow R\), Riemann-Liouville fractional integral of \(\alpha \)-order is defined as

$$\begin{aligned} {}_{t}^{R}I^{\alpha }_{t_{0}}f(\tau )= \frac{1}{\Gamma (\alpha )}\int _{t_{0}}^{t}\frac{f(v)}{(\tau -v)^{1-\alpha }}dv, \end{aligned}$$

where \(\alpha >0\), and \(\Gamma (\alpha )=\int _{0}^{\infty }e^{-t}\tau ^{\alpha -1}d\tau \).

For convenience, \(I^{\alpha }\) and \( D^{\alpha }\) are used to denote \({}_{t}^{c}I^{\alpha }_{t_{0}} \ and \ {}_{t}^{c}D^{\alpha }_{t_{0}}\) in the following.

Property 1. ( [20]). Properties of fractional derivatives

  1. (i)

      when \(\alpha \ge \beta >0\), \( D^{\alpha }I^{\beta }f(\tau )= D^{\alpha -\beta }f(\tau )\);

    Especially, \( D^{\alpha } I^{\beta }f(\tau )=f(\tau )\), where \(\alpha =\beta \), \(0< \alpha \le 1\).

  2. (ii)

      \( D^{\alpha } u =0\), where u is a constant.

  3. (iii)

      \( D^{\alpha } (u_{1}f(\tau )+u_{2}g(\tau ))=u_{1}D^{\alpha }f(\tau )+u_{2}D^{\alpha }g(\tau )\),

    where \(u_{1},\ u_{2}\) are constants.

  4. (iv)

      If \(f(\tau )\in C^{n}([t_{0},+\infty ),R)\), then \( I^{\alpha } D^{\alpha }f(\tau ) =f(\tau )-\sum _{k=0}^{n-1}\frac{f^{(k)}(t_{0})}{k!}(\tau -t_{0})^{k}\).

    Especially, if \(0< \alpha \le 1\)and \(0<t<T\), then\(D^{\alpha }I^{\alpha }f(\tau ) =f(\tau )\), \(I^{\alpha }D^{\alpha }f(\tau )=f(\tau )-f(t_{0})\).

Definition 3

([21]). Mittag-Leffler function \(E_{{\acute{\alpha }}, {\grave{\alpha }}}(\tau )\) is defined as

$$\begin{aligned} E_{{\acute{\alpha }}, {\grave{\alpha }}}(\tau )=\sum _{k=1}^{\infty }\frac{\tau ^{k}}{\Gamma (k{\acute{\alpha }} +{\grave{\alpha }})}, \end{aligned}$$

where \(\tau \in C\), \({\acute{\alpha }}>0\), \({\grave{\alpha }}>0\). When \({\grave{\alpha }}=1\),

$$\begin{aligned} E_{{\acute{\alpha }},1}(\tau )=E_{{\acute{\alpha }}}(\tau )=\sum _{k=1}^{\infty }\frac{\tau ^{k}}{\Gamma (k{\acute{\alpha }}+1)}. \end{aligned}$$

In particularly, when \({\grave{\alpha }}=1\), \({\acute{\alpha }}=1\), \(E_{1,1}(\tau )=e^{\tau }\).

Lemma 1

([19]). Let \( 0<\alpha \le 1 \). \(G \in R^{n\times n}\), and \(G>0\). Suppose that \(r(\tau )\in R^n\) is a function, which is continuously differentiable. Then,

$$\begin{aligned} D^{\alpha }(r^{T}(\tau )Gr(\tau ))\le 2r^{T}(\tau )GD^{\alpha }r(\tau ) \end{aligned}$$

holds, for any \(\tau \ge t_{0}\).

Lemma 2

([38]). Let \(\varepsilon \in (0,\varepsilon ^{\prime }]\), \(0<\varepsilon ^{\prime }\le 1 \). If there exist symmetric matrices \(Q_{1}\), \(Q_{3}\), and matrix \(Q_{2}\), such that \(\left( {\begin{array}{cc} Q_{1} &{}\varepsilon ^{\prime }Q^{T}_{2}\\ &{}\varepsilon ^{\prime }Q_{3}\\ \end{array} } \right) >0\), \(Q_{1}>0\), then \(E(\varepsilon )Q(\varepsilon )=Q^{T}(\varepsilon )E(\varepsilon )>0\), where \(Q(\varepsilon )=\left( {\begin{array}{cc} Q_{1} &{}\varepsilon Q^{T}_{2}\\ Q_{2} &{}Q_{3}\\ \end{array} } \right) \).

Lemma 3

([38]). Let \(\check{\alpha }\in (0,1)\). If \(\mathcal {V}(\tau )\) is continuously differentiable on \([t_{0},+\infty )\), and satisfies

$$\begin{aligned} D^{\check{\alpha }}\mathcal {V}(\tau )\le -\nu \mathcal {V}(\tau ), \ \ \ \tau \ge t_{0}, \end{aligned}$$

where \(\nu \) is a positive constant, then,

$$\begin{aligned} \mathcal {V}(\tau )\le \mathcal {V}(t_0)E_{\check{\alpha }}(-\nu (\tau -t_{0})^{\check{\alpha }}), \ \ \ \tau \ge t_{0}. \end{aligned}$$

Lemma 4

([38]). Let \(\omega \) is a positive constant. \(z_{i}\in R^{n}\), \(i=1,\ 2\). The following inequality

$$\begin{aligned} z_{1}^{T}z_{2}\le \frac{\omega }{2}z_{1}^{T}z_{1}+\frac{1}{2\omega }z_{2}^{T}z_{2}, \end{aligned}$$

holds.

Lemma 5

If there is a positive definite function \(\mathcal {V}(\tau )\in C^{1}([t_{0},+\infty ),R)\), \(\check{\alpha }\in (0,1)\) such that

$$\begin{aligned} \begin{aligned} D^{\check{\alpha }}\mathcal {V}(\tau )\le - \zeta \mathcal {V}(\tau )-\mu , \end{aligned} \end{aligned}$$
(1)

where \(\zeta >0\), and \(\mu >0\). Then, \(\lim _{\tau \rightarrow T }\mathcal {V}(\tau )=0\), and \(\mathcal {V}(\tau )\equiv 0\), for \(\tau \ge T\). T is the solution of the equation

$$\begin{aligned} \begin{aligned} pE_{\check{\alpha }}\big (-\zeta (\tau -t_{0})^{\check{\alpha }}\big )-\frac{\mu }{\zeta }=0, \end{aligned} \end{aligned}$$

where \(p=\frac{\mu }{\zeta }+\mathcal {V}(t_{0})\).

Proof

Applying Property 1, from (1), it follows that,

$$\begin{aligned} \begin{aligned} D^{\check{\alpha }}\big (\mathcal {V}(\tau )+\frac{\mu }{\zeta }\big )\le - \zeta \big (\mathcal {V}(\tau )+\frac{\mu }{\zeta }\big ). \end{aligned} \end{aligned}$$

Set \(\mathcal {V}(\tau )+\frac{\mu }{\zeta }=\tilde{\mathcal {V}}(\tau )\),

$$\begin{aligned} \begin{aligned} D^{\check{\alpha }}\tilde{\mathcal {V}}(\tau )\le -\zeta \tilde{\mathcal {V}}(\tau ). \end{aligned} \end{aligned}$$

According to Lemma 3, we have

$$\begin{aligned} \begin{aligned} \tilde{\mathcal {V}}(\tau )\le \tilde{\mathcal {V}}(t_{0})E_{\check{\alpha }} \big (-\zeta (\tau -t_{0})^{\check{\alpha }}\big ). \end{aligned} \end{aligned}$$

The above inequality can be represented as

$$\begin{aligned} \begin{aligned} \mathcal {V}(\tau )\le \big (\mathcal {V}(t_{0})+\frac{\mu }{\zeta }\big )E_{\check{\alpha }} \big (-\zeta (\tau -t_{0})^{\check{\alpha }}\big )-\frac{\mu }{\zeta }. \end{aligned} \end{aligned}$$

Since \(E_{\check{\alpha }}\big (-\zeta (\tau -t_{0})^{\check{\alpha }}\big )\) is monotonically decreasing function, there exists \( T>t_0 \), such that \(\big (\mathcal {V}(t_{0})+\frac{\mu }{\zeta }\big )E_{\check{\alpha }}\big (-\zeta (T-t_{0})^{\check{\alpha }}\big ) -\frac{\mu }{\zeta }=0\), and \(\mathcal {V}(\tau )\le 0\), for \(\tau \ge T \). Noting that \(\mathcal {V}(\tau )\ge 0\), for \(\tau \ge 0 \), this derives that \(\mathcal {V}(\tau )\equiv 0\), \(\tau \ge T\). This completes the proof. \(\square \)

3 Description of Network Model

Consider a nonlinear FSPCNs with M identical nodes described by singularly perturbation system

$$\begin{aligned} \begin{aligned} E(\varepsilon )D^{\alpha }x_{i}(t)=&\left[ {\begin{array}{c} D^{\alpha }x_{1i}(t)\\ \varepsilon D^{\alpha }x_{2i}(t)\\ \end{array} } \right] \\ =f(x_{i}&(t))+d\sum _{j=1}^{M}b_{ij}\Gamma x_{j}(t),\\ \end{aligned} \end{aligned}$$
(2)

where \( \alpha \in (0,1)\), \(t\in [t_{0},\ +\infty )\), \(\varepsilon \) is the perturbation parameter satisfying \(0<\varepsilon \le \varepsilon ^{\prime } \ll 1\); \(E(\varepsilon )=\left[ {\begin{array}{cc} I_{n_1} &{} 0 \\ 0 &{} \varepsilon I_{n_2}\\ \end{array} } \right] \); \(x_{i}(t)=\left[ {\begin{array}{cc} x_{1i}(t) \\ x_{2i}(t) \\ \end{array} } \right] \in R^{n_{1}+n_{2} }\) is state vector of i-th node at time t, \(i=1, 2, \ldots , M\), where \(x_{1i}(t)=(x_{1i1}(t), x_{1i2}(t), \ldots \), \(x_{1in_{1}}(t))^{T}\in R^{n_{1}}\), \(x_{2i}(t)=(x_{2i1}(t), x_{2i2}(t), \ldots , x_{2in_{2}}(t))^{T}\in R^{n_{2}}\); The nonlinear function \(f(x_{i}(t))=\left[ {\begin{array}{c} f^{1}(x_{1i}(t)) \\ f^{2}(x_{2i}(t)) \\ \end{array} } \right] \in R^{n_{1}+n_{2}}\), where \(f^{1}(x_{1i}(t))=(f^{1}_{1}(x_{1i1}(t)),f^{1}_{2}(x_{1i2}(t)), \ldots ,f^{1}_{n_{1}}(x_{1in_{1}}(t)))^{T}\in R^{n_{1}}\), \(f^{2}(x_{2i}(t))=(f_{1}^{2}(x_{2i1}(t)),f^{2}_{2}(x_{2i2}(t)),\ldots \), \(f^{2}_{n_{2}}(x_{2in_{2}}(t)))^{T}\in R^{n_{2}}\); d represents the coupling strength; \(\Gamma =diag(\varrho _{1}, \varrho _{2}, \ldots , \varrho _{n_{1}+n_{2}})\in R^{(n_{1}+n_{2})\times (n_{1}+n_{2})}\) is internal coupling matrix; \(B=[b_{ij}]_{M\times M}\) denotes an unknown weight construction matrix, which stands for the topology in network (2). If there is a connection between nodes j and i (\(j\ne i\)), then \(b_{ij}\ne 0\), if not, \(b_{ij}=0\) (\(j\ne i\)). In addition, \(b_{ii}=-\sum _{j=1,j\ne i}^{M}b_{ij}\).

In the following, for network (2), we design a response network described by

$$\begin{aligned} \begin{aligned} E(\varepsilon )D^{\alpha }y_{i}(t)&=\left[ {\begin{array}{c} D^{\alpha }y_{1i}(t)\\ \varepsilon D^{\alpha }y_{2i}(t)\\ \end{array} } \right] \\ =f(y_{i}&(t))+d\sum _{j=1}^{M}{\hat{b}}_{ij}(t)\Gamma y_{j}(t)+u_{i}(t), \end{aligned} \end{aligned}$$
(3)

where \(y_{i}(t)=\left[ {\begin{array}{cc} y_{1i}(t) \\ y_{2i}(t) \\ \end{array} } \right] \) is state vector of y-th node at time t, \(i=1,2,\ldots ,M\), in which \(y_{1i}(t)=(y_{1i1}(t), y_{1i2}(t), \ldots , y_{1in_{1}}(t))^{T}\in R^{n_{1}}\), \(y_{2i}(t)=(y_{2i1}(t), y_{2i2}(t), \ldots , y_{2in_{2}} (t))^{T}\in R^{n_{2}}\); The function \(f(y_{i}(t))=\left[ {\begin{array}{c} f^{1}(y_{1i}(t))\\ f^{2}(y_{2i}(t)) \end{array} } \right] \in R^{n_{1}+n_{2}}\), where \(f^{1}(y_{1i}(t))=(f^{1}_{1}(y_{1i1}(t)),f^{1}_{1}(y_{1i2}(t)),\ldots , f^{1}_{n_{1}}(y_{1i n_{1}}(t)))^{T} \in R^{n_{1}}\), \(f^{2}(y_{2i}(t))=(f_{1}^{2}(y_{2i1}(t)), f^{2}_{2}(y_{2i2} (t)), \ldots , f^{2}_{n_{2}}(y_{2in_{2}}(t)))^{T} \in R^{n_{2}}\). \({\hat{B}}(t)=[{\hat{b}}_{ij}(t)]_{M\times M}\) is a known weight construction matrix, which is used to identify unknown matrix B; \(u_{i}(t)\) is the control input. The other parameters are the same as in (2).

In order to obtain the main results of this article, the following assumptions are made for \(f(\cdot )\) and unknown weight \(b_{ij}\):

(\({\mathscr {H}}_{1}\)): For \(f_{i}(\cdot )\), \(s_{1}\ne s_{2}\), the following condition is satisfied

$$\begin{aligned}{} & {} 0\le \frac{f_{i}(s_{1})-f_{i}(s_{2})}{s_{1}-s_{2}}\le z_{i}, \end{aligned}$$

where \(z_{i}\) is a positive constant.

(\({\mathscr {H}}_{2}\)):For weight \(b_{ij}\), which is norm bounded, such that

$$\begin{aligned} |b_{ij}|\le M_{b}, i,j=1,2,\ldots ,M, \end{aligned}$$

where \(M_{b}>0\) is a constant.

The synchronization error for the i-th node is represented by \(e_{i}(t)\), and \(e_{i}(t)=y_{i}(t)-x_{i}(t)\). Together (2) with (3), the synchronization error system can be expressed as

$$\begin{aligned} \begin{aligned} E(\varepsilon )D^{\alpha }e_{i}(t)=&\breve{f}(e_{i}(t)) +d\sum _{j=1}^{M}{\widetilde{b}}_{ij}(t)\Gamma y_{j}(t)\\ +d\sum _{j=1}^{M}&b_{ij}\Gamma e_{j}(t)+u_{i}(t), \end{aligned} \end{aligned}$$
(4)

where \({\widetilde{b}}_{ij}(t)={\hat{b}}_{ij}(t)-b_{ij}\), \(\breve{f}(e_{i}(t))=f(y_{i}(t))-f(x_{i}(t))\), \(i, j=1, 2, \ldots , M\).

Definition 4

Let \(x_{i}(t)\) be the solutions of (2) with initial value \(x_{i}(t_{0})=x_{i0}\), \(y_{i}(t)\) be the solutions of (3) with initial value \(y_{i}(t_{0})=y_{i0}\).

1). If there exist \(T(e_i(t_0)) \ge t_{0}\), satisfy

$$\begin{aligned} \begin{aligned} \lim \limits _{t\rightarrow T(e_i(t_0))} \parallel e_i(t)\parallel =0, and \ \ e_i(t)\equiv 0,\ \ t\ge T(e_i(t_0)), \end{aligned} \end{aligned}$$

then system (2) and (3) are deemed to achieve synchronization in \(T(e_i(t_0))\) under the controller u(t). In addition, \(T(e_i(t_0))\) is called as setting time.

2). If \(\lim \limits _{t\rightarrow T(e_i(t_0)) }\widetilde{b}_{ij}(t)=0,\ \ j, i =1, 2, \ldots , M\), then the topology in network (2) is said to realize the identification in finite time \(T(e_i(t_0))\).

4 Main Result

In this paper, our focus is to establish the conditions that ensure the synchronization of networks (2) and (3), and to identify the topology in network (2) based on DETC. To do so, the controller \( u_{i}(t)\) is designed as

$$\begin{aligned} \begin{aligned} u_{i}(t)=-Ke_{i}(t^i_{k})- c\frac{e_{i}(t_{k}^{i})\text {sgn}(e_{i}(t_{k}^{i}))}{||e_{i}(t_{k}^{i})||^{2}}, \ t\in [t_{k}^{i},t_{k+1}^{i}), \end{aligned} \end{aligned}$$
(5)

where c is positive constant; \(\text {sgn}(\cdot )\) denotes signum function; K is the control gain matrix to be designed. \(\{t^i_{k}\}\) denotes event-triggered sequence that satisfies \(0\le t^i_{0}< t^i_{1}<t^i_{2}<\ldots<t^i_{k}<\ldots \), \(k\in N^{+}\) and \(\lim _{t\rightarrow \infty }t^i_{k}=+\infty \).

Remark 1

The signum function, as a simple nonlinear function, can be easily applied in controller (5) u(t) compare to any other functions. During the operation of the controller, there is no need for complex mathematical calculations and function approximation, which helps to reduce the computational load of controller updates. Furthermore, in the proof in Theorem 1, the signum function term \(e_{i}(t_{k}^{i})\text {sgn}(e_{i}(t_{k}^{i}))\) plays a crucial role in eliminating the terms \(\vartheta _{i} e_{i}^{T}(t_{k}^{i})e_{i}(t_{k}^{i})\) and \(\bar{e}_{i}^{T}(t)\bar{e}_{i}(t)\).

Define the measurement error \({\bar{e}}_{i}(t)\) as \(\bar{e}_{i}(t)=(y_{i}(t_{k}^{i})-x_{i}(t_{k}^{i}))-(y_{i}(t)-x_{i}(t))= e_{i}(t_{k}^{i})-e_{i}(t)\).

Design DETM as

$$\begin{aligned} \begin{aligned} t_{k+1}^{i}=\inf _{t>t^{i}_{k}}\bigg \{t: \bar{e}^{T}_{i}(t)\bar{e}_{i}(t) \ge \vartheta _{i} e^{T}_{i}(t_{k}^{i})e_{i}(t_{k}^{i})+\gamma _{i}\eta _{i}(t)\bigg \}, \end{aligned} \end{aligned}$$
(6)

where \(\vartheta _{i},\ \gamma _{i}\) are positive constants, and \(0<\vartheta _{i}<1\); \(\eta _{i}(t)\) expresses an internal dynamic variable, and designed by

$$\begin{aligned} \begin{aligned} D^{\alpha }\eta _{i}(t)=-\beta _{i}&\eta _{i}(t)-\frac{1}{2}c\frac{\gamma _{i}\eta _{i}(t)}{||e_{i}(t_{k}^{i})||^{2}}\\&+\vartheta _{i} e_{i}^{T}(t_{k}^{i})e_{i}(t_{k}^{i})-\bar{e}_{i}^{T}(t)\bar{e}_{i}(t), \end{aligned} \end{aligned}$$
(7)

where parameter \(\beta _{i}>0\). The framework of DETM (6) is depicted in Fig. 1.

Fig. 1
figure 1

Framework of DETC scheme

Full size image

Remark 2

1) In DETM (6), internal dynamic variable \(\eta _{i}(t)\) is introduced, such that the threshold function \( \vartheta _{i} e^{T}_{i}(t_{k}^{i})e_{i}(t_{k}^{i})+\gamma _{i}\eta _{i}(t)\) is adjustable. Due to \(\eta _{i}(t)>0\), the average interval time can be enlarged, and the frequency of events triggered can be reduced effectively and Zeno behavior can be avoided.

2) In [27], DETM was designed as \(t_{k+1}^{i}=\inf _{t>t^{i}_{k}}\big \{t: \bar{e}^{T}_{i}(t)\bar{e}_{i}(t)\ge \vartheta _{i} e^{T}_{i}(t)e_{i}(t)+\gamma _{i}\eta _{i}(t)\big \}\). Since the threshold function contains a continuous term \(\vartheta _{i}e^{T}_{i}(t)e_{i}(t)\), the state information is requested to be known at all time. However, in practical application, not all of state information is available, which may cause the above event-triggering mechanism to be unavailable.

In [28], DETM was designed as \(t_{k+1}=\inf _{t>t_{k}}\big \{t:\bar{e}^{T}_{i}(t)\bar{e}_{i}(t)\ge \gamma _{i}\eta _{i}(t)+\beta _{i} e^{-\alpha t}+\delta \big \}\), \(\beta _i, \delta >0\). Obviously, the threshold function is independent of system states, which may cause that the control law cannot update in time.

The designed DETM (6) is only required to know the state information at time \(t^{i}_{k}\), and instead of all. It is easy to see that, controller updates depend on both internal dynamic variable \(\eta _{i}(t)\) and state information \(\bar{e}^{T}_{i}(t^{i}_{k})\) at time \(t^{i}_{k}\). Compared with [27] and [28], the designed DETM (6) is more superiority.

Remark 3

ETC updates the controller only when the certain triggering condition (6) is met, rather than updating the controller at fixed time intervals. Applying ETC to deal with FSPCNs synchronization can reduce the demand for communication resources, and improves the overall efficiency of the system operation.

To achieve the purpose of the topology identification for network (2) via controller (5) with DETM (6), we design the update law of the topology observer \({\hat{b}}_{ij}(t)\) as

$$\begin{aligned} \begin{aligned} D^{\alpha }{\hat{b}}_{ij}(t)=&-\rho \text {sgn}(\widetilde{b}_{ij}(t))(|{\hat{b}}_{ij}(t)| +M_{b})\\&-d\xi _{ij}e^{T}_{i}(t)Q^{T}(\varepsilon )\Gamma y_{j}(t). \end{aligned} \end{aligned}$$
(8)

where \(\rho >0\), and \(\xi _{ij}>0\) are real parameter to be designed.

Substitute (5) into (4), for \(t\in [t^{i}_{k},t^{i}_{k+1})\), yields

$$\begin{aligned} \begin{aligned} E(\varepsilon )D^{\alpha }e_{i}(t)=&\breve{f}(e_{i}(t)) +d\sum _{j=1}^{M} b_{ij}\Gamma e_{j}(t)\\ +d\sum _{j=1}^{M}{\widetilde{b}}_{ij}(t)&\Gamma y_{j}(t)-Ke_{i}(t^{i}_{k})- c\frac{e_{i}(t_{k}^{i})\text {sgn}(e_{i}(t_{k}^{i}))}{||e_{i}(t_{k}^{i})||^{2}}. \ \ \ \end{aligned} \end{aligned}$$
(9)

Lemma 6

Consider the designed DETM in (6) with (7). If \(\eta _{i}(t_{0}^{i})>0\), then, \(\eta _{i}(t)>0 \), \(t\ge t^i_0\), \(i=1,2,\ldots ,M\).

Proof

For \(t\in [t_{k}^{i},\ t_{k+1}^{i})\), applying (6) yields

$$\begin{aligned} \begin{aligned} \bar{e}_{i}^{T}(t)\bar{e}_{i}(t)\le \vartheta _{i}e_{i}^{T}(t_{k}^{i})e_{i}(t_{k}^{i})+\gamma _{i}\eta _{i}(t). \end{aligned} \end{aligned}$$
(10)

By (10) and (7), it follows that

$$\begin{aligned} \begin{aligned} D^{\alpha }\eta _{i}(t)\ge -\beta _{i}\eta _{i}(t)-\gamma _{i}\eta _{i}(t)= -\pi _{i}\eta _{i}(t), \end{aligned} \end{aligned}$$
(11)

where \(\pi _{i}=\beta _{i}+\gamma _{i}\).

On the basis of Lemma 3, from (11), we get

$$\begin{aligned} \eta _{i}(t)\ge \eta _{i}(t_{k}^{i})E_{\alpha }(-\pi _{i}(t-t^{i}_{k})^{\alpha }),\ \ \ \ \ t\in [t_{k}^{i},t_{k+1}^{i}). \end{aligned}$$

\(k\in \{0\} \cup N^+\).

Hence, for \(t\in [t_{0}^{i}, t_{k+1}^{i})\), we obtain that

$$\begin{aligned} \eta _{i}(t)\ge \eta _{i}(t_{0}^{i})\bigg (\Pi _{r=1}^{k}E_{\alpha }(-\pi _{i}(t_{r}^{i} -t_{r-1}^{i})^{\alpha })\bigg )E_{\alpha }(-\pi _{i}(t-t_{k}^{i})^{\alpha }). \end{aligned}$$

According to Definition 3 and \(\eta _{i}(t_{0}^{i})>0\), from the above inequality, it can be derived that \(\eta _{i}(t)\ge 0\), for \(t\ge t_{0}\). The proof is completed.

If the dynamic variable \( \eta _{i}(t)\equiv 0 \), then DETM (6) changes to a static ETM in [27].

4.1 Synchronization and Topology Identification of FSPCNs

In this subsection, the synchronization and topology identification in finite time are realized between network (2) and (3) under controller (5) and observer (8).

Theorem 1

Let \(\alpha \in (0,1)\), and \(\ \varepsilon \in (0,\varepsilon ^{\prime })\). Under the assumptions (\({\mathscr {H}}_{1}\))-(\({\mathscr {H}}_{2}\)), if there exist scalar \(\omega >0\), matrix \(Q_{1}>0\), matrix \(Q_{2}\) and symmetric matrix \({Q_{3}}\), such that

(i) \(\left( {\begin{array}{cc} Q_{1} &{}\varepsilon ^{\prime }Q^{T}_{2}\\ &{}\varepsilon ^{\prime }Q_{3}\\ \end{array} } \right) >0\);

(ii) \(\frac{1}{2}I_{M}\otimes (Q^{T}(\varepsilon ) Q(\varepsilon ))+d(B\otimes Q^{T}(\varepsilon )\Gamma ) +\frac{\omega }{2}I_{M}\otimes (Q^{T}(\varepsilon )KK^{T} Q(\varepsilon ))-\frac{1}{2}I_{M}\otimes (Q^{T} (\varepsilon )K+K^{T}Q(\varepsilon ))\le -\frac{z}{2} I_{M(n_{1}+n_{2})}\),

hold, then system (2) and (3) can achieve the finite-time synchronization under controller (5). The setting time \(T(e_i(t_0))\) is the solution of equality

$$\begin{aligned} \begin{aligned} \bigg (\mathcal {V}(t_{0})+\frac{\mu _i}{\zeta _i}\bigg )E_{\alpha }\bigg (-\zeta _i(t-t_{0})^{\alpha }\bigg )-\frac{\mu _i}{\zeta _i} =0, \end{aligned} \end{aligned}$$

where \(\zeta _i=\min \bigg \{{\frac{2}{\lambda _{\max }(Q(\varepsilon )E(\varepsilon ))} \big ( \lambda _{\min }(\Theta )\big ), 2\rho , \beta _{i}+\gamma _{i}\bigg \}}\), \(\Theta =-\frac{z}{2}I_{M(n_{1}+n_{2})}-\frac{1}{2}I_{M}\otimes (Q^{T}(\varepsilon )Q(\varepsilon ))-d(B\otimes Q^{T}(\varepsilon )\Gamma )-\frac{\omega }{2}I_{M}\otimes (Q^{T}(\varepsilon )KK^{T} Q(\varepsilon ))+\frac{1}{2}I_{M}\otimes (Q^{T}(\varepsilon )K+K^{T}Q(\varepsilon )) \),

\(\mu _i=\frac{1}{2}c\lambda _{\min }(Q^{T}(\varepsilon ))(1-\vartheta _{i})\),

\(\mathcal {V}(t_{0})=\frac{1}{2}\sum _{i=1}^{M}(y_{i0}-x_{i0})^{T}E(\varepsilon )Q(\varepsilon ) (y_{i0}-x_{i0})+\sum _{i=1}^{M}\eta _{i}(t_{0}) +\frac{1}{2}\sum _{i=1}^{M}\sum _{j=1}^{M}\frac{1}{\xi _{ij}}({\hat{b}}_{ij}(t_{0})-b_{ij})^{2}\).

In addition, the topology of system (2) can be identified in finite time under update law of the topology observer (8) successfully.

Proof

Consider the following Lyapunov function:

$$\begin{aligned} \begin{aligned} \mathcal {V}(t)=\mathcal {V}_{1}(t)+\mathcal {V}_{2}(t)+\mathcal {V}_{3}(t), \end{aligned} \end{aligned}$$
(12)

where \(\mathcal {V}_{1}(t)=\frac{1}{2}\sum _{i=1}^{M}e_{i}^{T}(t)E(\varepsilon )Q(\varepsilon ) e_{i}(t)\), \(\mathcal {V}_{2}(t)= \sum _{i=1}^{M}\eta _{i}(t)\), and \(\mathcal {V}_{3}(t)=\frac{1}{2}\sum _{i=1}^{M}\sum _{j=1}^{M}\frac{1}{\xi _{ij}}({\hat{b}}_{ij}(t)-b_{ij})^{2}\).

For \(t\in [t_{k}^{i}, t_{k+1}^{i})\), according to Lemmas 1 and 4, calculating Caputo fractional derivative of \(\alpha \)-order for \(\mathcal {V}_{1}(t)\) along with the solution of system (9), yields

$$\begin{aligned} \begin{aligned} D^{\alpha }\mathcal {V}_{1}(t) \le&\sum _{i=1}^{M} e_{i}^{T}(t)Q^{T}(\varepsilon )E(\varepsilon ) D^{\alpha }e_{i}(t)\\ =&\sum _{i=1}^{M}e_{i}^{T}(t)Q^{T}(\varepsilon )\breve{f}(e_{i}(t))\\&+d\sum _{i=1}^{M}\sum _{j=1}^{M}b_{ij}e^{T}_{i}(t)Q^{T}(\varepsilon )\Gamma e_{j}(t)\\&+d\sum _{i=1}^{M}\sum _{j=1}^{M}\widetilde{b}_{ij}(t)e^{T}_{i}(t)Q^{T}(\varepsilon )\Gamma y_{j}(t)\\&-\sum _{i=1}^{M}e^{T}_{i}(t)Q^{T}(\varepsilon )Ke_{i}(t^{i}_{k})\\&-c\frac{|e^{T}_{i}(t_{k}^{i})-\bar{e}^{T}_{i}(t)|Q^{T}(\varepsilon )e_{i}(t_{k}^{i})}{||e_{i}(t_{k}^{i})||^{2}}\\ \le&\sum _{i=1}^{M}e_{i}^{T}(t)\bigg (\frac{1}{2}Q^{T}(\varepsilon )Q(\varepsilon ) -Q^{T}(\varepsilon )K\\&d\sum _{j=1}^{M}b_{ij}Q^{T}(\varepsilon )\Gamma +\frac{\omega }{2}Q^{T}(\varepsilon )KK^{T}Q(\varepsilon )\bigg )e_{i}(t)\\&+d\sum _{i=1}^{M}\sum _{j=1}^{M}\widetilde{b}_{ij}(t)e^{T}_{i}(t){Q^{T}(\varepsilon )\Gamma y_{j}(t)}\\&+\frac{1}{2}\sum _{i=1}^{M}\breve{f}^{T}(e_{i}(t))\breve{f}(e_{i}(t))-\sum _{i=1}^{M}\frac{1}{2\omega }{\bar{e}}^{T}_{i}(t){\bar{e}}_{i}(t)\\ \\&+c\frac{\lambda _{\min }(Q^{T}(\varepsilon ))(-||e^{T}_{i}(t_{k}^{i})||^{2}+|\bar{e}^{T}_{i}(t)|e_{i}(t_{k}^{i}))}{||e_{i}(t_{k}^{i})||^{2}}. \end{aligned} \end{aligned}$$
(13)

For \(t\in [t_{k}^{i}, t_{k+1}^{i})\), by computing Caputo fractional derivative of \(\alpha \)-order of \(\mathcal {\mathcal {V}}_{2}(t)\) along with the solution of system (7), one obtains

$$\begin{aligned} \begin{aligned} D^{\alpha }\mathcal {V}_{2}(t)=&\sum _{i=1}^{M}D^{\alpha }\eta _{i}(t) =\sum _{i=1}^{M}(-\beta _{i}\eta _{i}(t)\\&-\frac{1}{2}c\frac{\gamma _{i}\eta _{i}(t)}{||e_{i}(t_{k}^{i})||^{2}}+\vartheta _{i}e_{i}^{T}(t_{k}^{i})e_{i}(t_{k}^{i}) -\bar{e}_{i}^{T}(t)\bar{e}_{i}(t)). \end{aligned} \end{aligned}$$
(14)

For \(t\in [t_{k}^{i}, t_{k+1}^{i})\), calculating Caputo fractional derivative of \(\alpha \)-order of \(\mathcal {V}_{3}(t)\) along with the solution of system (8), it yields

$$\begin{aligned} \begin{aligned}&D^{\alpha }\mathcal {V}_{3}(t)\\&\quad =\sum _{i=1}^{M}\sum _{j=1}^{M}\frac{1}{\xi _{ij}}({\hat{b}}_{ij}(t)-b_{ij})D^{\alpha }{\hat{b}}_{ij}(t)\\&\quad =-\rho \sum _{i=1}^{M}\sum _{j=1}^{M}\frac{1}{\xi _{ij}}\widetilde{b}_{ij}(t)\text {sgn}({\widetilde{b}}_{ij}(t))(|{\hat{b}}_{ij}(t)| +M_{b})\\&\qquad -d\sum _{i=1}^{M}\sum _{j=1}^{M}\widetilde{b}_{ij}(t)e^{T}_{i}(t)Q^{T}(\varepsilon )\Gamma y_{j}(t). \end{aligned} \end{aligned}$$
(15)

From \(({\mathscr {H}}_{2})\), it can be obtained that \(|{\hat{b}}_{ij}(t)|+M_{b}\ge |{\hat{b}}_{ij}(t)-b_{ij}|=|{\widetilde{b}}_{ij}(t)|\). Hence,

$$\begin{aligned} \begin{aligned}&-\rho \sum _{i=1}^{M}\sum _{j=1}^{M}\frac{1}{\xi _{ij}}\widetilde{b}_{ij}(t)(\text {sgn}({\widetilde{b}}_{ij}(t))(|{\hat{b}}_{ij}(t)|+M_{b}))\\&\quad \le -\rho \sum _{i=1}^{M}\sum _{j=1}^{M}\frac{1}{\xi _{ij}}(\widetilde{b}_{ij}(t))^{2}. \end{aligned} \end{aligned}$$
(16)

On the basis of (12)-(16), for \(t\in [t_{k}^{i}, t_{k+1}^{i})\), one has

$$\begin{aligned} \begin{aligned} D^{\alpha }\mathcal {V}(t)\le&\sum _{i=1}^{M}e_{i}^{T}(t)\bigg (\frac{1}{2}Q^{T} (\varepsilon )Q(\varepsilon ) -Q^{T}(\varepsilon )K\\&+d\sum _{j=1}^{M}b_{ij}Q^{T}(\varepsilon )\Gamma +\frac{\omega }{2}Q^{T}(\varepsilon )KK^{T}Q(\varepsilon )\bigg )e_{i}(t)\\&-\sum _{i=1}^{M}\frac{1}{2\omega }\bar{e}^{T}_{i}(t)\bar{e}_{i}(t)-\rho \sum _{i=1}^{M}\sum _{j=1}^{M}\frac{1}{\xi _{ij}}({\widetilde{b}}_{ij}(t))^{2}\\&-\sum _{i=1}^{M}\bar{e}^{T}_{i}(t)\bar{e}_{i}(t)+\sum _{i=1}^{M}\vartheta _{i}e^{T}_{i}(t^{i}_{k})e_{i}(t^{i}_{k})\\&+\frac{1}{2}\sum _{i=1}^{M}\breve{f}^{T}(e_{i}(t))\breve{f}(e_{i}(t))-\sum _{i=1}^{M}\beta _{i}\eta _{i}(t)\\&+c\frac{\lambda _{\min }(Q^{T}(\varepsilon ))(-||e^{T}_{i}(t_{k}^{i})||^{2}+|\bar{e}^{T}_{i}(t)|e_{i}(t_{k}^{i}))}{||e_{i}(t_{k}^{i})||^{2}}. \end{aligned} \end{aligned}$$
(17)

Due to (\({\mathscr {H}}_{1}\)), we have \( \breve{f}^{T}(e_{i}(t))\breve{f}(e_{i}(t))\le ze^{T}_{i}(t)e_{i}(t), \) where \(z=\max _{1\le i\le M}\big \{z_{i}^{2}\big \}\).

Utilizing Property 1 and DETM (6), it follows from (17) that,

$$\begin{aligned} \begin{aligned} D^{\alpha }\mathcal {V}(t)\le&\sum _{i=1}^{M}e_{i}^{T}(t)\bigg (\frac{1}{2}Q^{T}(\varepsilon )Q(\varepsilon )-Q^{T}(\varepsilon )K\\&+\frac{\omega }{2}Q^{T}(\varepsilon )KK^{T}Q(\varepsilon )+d\sum _{j=1}^{M}b_{ij}Q^{T}(\varepsilon )\Gamma \\&+\frac{z}{2}I_{M(n_{1}+n_{2})}\bigg )e_{i}(t) -\rho \sum _{i=1}^{M}\sum _{j=1}^{M}\frac{1}{\xi _{ij}}({\widetilde{b}}_{ij}(t))^{2}\\&-\sum _{i=1}^{M}(\beta _{i}+\frac{1}{2}c\frac{\gamma _{i}\eta _{i}(t)}{||e_{i}(t_{k}^{i})||^{2}} +\gamma _{i})\eta _{i}(t)\\&+c\frac{\lambda _{\min }(Q^{T}(\varepsilon ))(\frac{1}{2}\bar{e}^{T}_{i}(t)\bar{e}_{i}(t)-\frac{1}{2}||e^{T}_{i}(t_{k}^{i})||^{2} )}{||e_{i}(t_{k}^{i})||^{2}}\\ \le&e^{T}(t)\bigg (\frac{1}{2}I_{M}\otimes (Q^{T}(\varepsilon ) Q(\varepsilon ))+d(B\otimes Q^{T}(\varepsilon )\Gamma )\\&+\frac{\omega }{2}I_{M}\otimes (Q^{T}(\varepsilon )KK^{T}Q(\varepsilon )) -\frac{1}{2}I_{M}\otimes (Q^{T}(\varepsilon )K\\&+K^{T}Q(\varepsilon ))+\frac{z}{2}I_{M(n_{1}+n_{2})}\bigg )e(t)-\sum _{i=1}^{M}(\beta _{i}\\&+\frac{1}{2}c\frac{\gamma _{i}\eta _{i}(t)}{||e_{i}(t_{k}^{i})||^{2}}+\gamma _{i})\eta _{i}(t)-\rho \sum _{i=1}^{M}\sum _{j=1}^{M}\frac{1}{\xi _{ij}}({\widetilde{b}}_{ij}(t))^{2}\\&-\frac{1}{2}c\lambda _{\min }Q^{T}(\varepsilon )\big (1 -\frac{{||\bar{e}(t)}||^{2}}{||e(t_{k})||^{2}}\big )\\ \le&e^{T}(t)\bigg (\frac{1}{2}I_{M}\otimes (Q^{T}(\varepsilon )Q(\varepsilon ))+d(B\otimes Q^{T}(\varepsilon )\Gamma )\\&+\frac{\omega }{2}I_{M}\otimes (Q^{T}(\varepsilon )KK^{T}Q(\varepsilon )) -\frac{1}{2}I_{M}\otimes (Q^{T}(\varepsilon )K\\&+K^{T}Q(\varepsilon ))+\frac{z}{2}I_{M(n_{1}+n_{2})}\bigg )e(t)\\&-\sum _{i=1}^{M}(\beta _{i}+\gamma _{i})\eta _{i}(t)-\rho \sum _{i=1}^{M}\sum _{j=1}^{M}\frac{1}{\xi _{ij}}({\widetilde{b}}_{ij}(t))^{2}\\&-\frac{1}{2}c\lambda _{\min }(Q^{T}(\varepsilon ))\big (1 -\vartheta _{i})\le -\zeta _i \mathcal {V}(t)-\mu _i,\\ \end{aligned} \end{aligned}$$
(18)

where \(\zeta _i=\min \{{\frac{2}{\lambda _{\max }(Q(\varepsilon )E(\varepsilon ))} \big ( \lambda _{\min }(\Theta )\big ), 2\rho , \beta _{i}+\gamma _{i}\}}\), \(\Theta =-\frac{z}{2}I_{M (n_{1}+n_{2})}-\frac{1}{2}I_{M}\otimes (Q^{T}(\varepsilon ) Q(\varepsilon ))-d(B\otimes Q^{T} (\varepsilon )\Gamma )-\frac{\omega }{2}I_{M}\otimes (Q^{T}(\varepsilon )KK^{T} Q(\varepsilon )) +\frac{1}{2}I_{M}\otimes (Q^{T}(\varepsilon )K+K^{T}Q(\varepsilon ))\); \(\mu _i=\frac{1}{2}c\lambda _{\min }(Q^{T}(\varepsilon ))\big (1-\vartheta _{i})\).

By Lemma 5, we can conclude that \( \lim \limits _{t\rightarrow T(e_i(t_0))}\mathcal {V}(t)=0\), and \(\mathcal {V}(t)\equiv 0\), for \( t\ge T(e_i(t_0))\), where \(T(e_i(t_0))\) is the solution of equality \(\bigg (\mathcal {V}(t_{0})+\frac{\mu _i}{\zeta _i}\bigg )E_{\alpha }\bigg (-\zeta _i(t-t_{0})^{\alpha }\bigg )-\frac{\mu _i}{\zeta _i} =0\).

Moreover, from the expression of \(\mathcal {V}(t)\), it can be obtained that, \( \lim \limits _{t\rightarrow T(e_i(t))}\parallel e_i(t) \parallel = \lim \limits _{t\rightarrow T(e_i(t))}\widetilde{b}_{ij}(t)= 0\), and \(\parallel e_i(t) \parallel = \widetilde{b}_{ij}(t) \equiv 0\), for \( t\ge T(e_i(t_0))\). This shows that, system (2) and (3) can achieve the synchronization in finite time under observer (8) and controller (5), and the unknown topology in network (2) can be identified in finite time. The proof is complete.

Remark 4

In [15], the topology identification and synchronization were considered for fractional CDNs by the adaptive feedback control strategy. In [44], the synchronization was studied for SPCNs by the DETC method, but the topology identification issue was not discussed.

In Theorem 1, the synchronization and topology identification in finite time are exploited for FSPCNs concurrently on the basis of DETC. Obviously, the results of this paper are an improvement and extension for that in [15] and [44].

Remark 5

In the proof of Theorem 1, an \(\varepsilon \)-dependent Lyapunov function is introduced by incorporating the parameter \(\varepsilon \) into the traditional Lyapunov function for system stability analysis. The introduction of \(\varepsilon \) can enhance the sensitivity of the Lyapunov function, and then respond to smaller changes in the system state, and has stronger analytical ability than the traditional Lyapunov function. Moreover, if the parameter \(\varepsilon =1\), the \(\varepsilon \)-dependent Lyapunov function becomes a general Lyapunov function, hence, the range of applications of the \(\varepsilon \)-dependent Lyapunov function is wider.

In Theorem 1, we have investigated \(\varepsilon \in (0,1)\) in FSPCNs (2) and (3). In the following Case 1) and 2), we discuss the synchronization and topology identification in finite time between networks (2) and (3) when \(\varepsilon =0\) and \(\varepsilon =1\), respectively.

Case 1). Systems (2) and (3) become as

$$\begin{aligned} E(0)D^{\alpha }x_{i}(t)= & {} \left( {\begin{array}{c} D^{\alpha }x_{1i}(t)\\ 0\\ \end{array} } \right) \nonumber \\= & {} f(x_{i}(t))+d\sum _{j=1}^{M}b_{ij}\Gamma x_{j}(t), \end{aligned}$$
(19)
$$\begin{aligned} E(0)D^{\alpha }y_{i}(t)= & {} \left( {\begin{array}{c} D^{\alpha }y_{1i}(t)\\ 0\\ \end{array} } \right) \nonumber \\= & {} f(y_{i}(t))+d\sum _{j=1}^{M}{\hat{b}}_{ij}(t)\Gamma y_{j}(t)+u_{i}(t), \end{aligned}$$
(20)

respectively, where \(E(0)=\left[ {\begin{array}{cc} I_{n_{1}} &{} 0 \\ 0 &{} 0 \\ \end{array} } \right] \).

Corollary 1

Let \(\alpha \in (0,1)\). Under the assumptions (\(\mathscr {H}_{1}\))-(\({\mathscr {H}}_{2}\)), if there exist scalar \(\omega >0\), matrix \(Q_{2}\) and positive definite matrices \(Q_{1},Q_{3}\), such that

(i) \(E(0)Q(0)\ge 0\).

(ii) \(\frac{1}{2}I_{M}\otimes (Q^{T}(0) Q(0))+d(B\otimes Q^{T}(0)\Gamma )+\frac{\omega }{2}I_{M}\otimes (Q^{T}(0)KK^{T} Q(0))-\frac{1}{2}I_{M}\otimes (Q^{T}(0)K+K^{T}Q(0))\le - \frac{z}{2}I_{M(n_{1}+n_{2})}\), hold, where \(Q(0)=\left( {\begin{array}{cc} Q_{1} &{} 0 \\ Q_{2} &{} Q_{3} \\ \end{array} } \right) \), then system (19) and (20) can achieve the finite time synchronization under controller (5). The setting time \(T(e_i(t_0))\) is the solution of equality

$$\begin{aligned} \begin{aligned} \bigg (\mathcal {V}(t_{0})+\frac{\mu _i}{\zeta _i}\bigg )E_{\alpha }\bigg (-\zeta _i(t-t_{0})^{\alpha }\bigg )-\frac{\mu _i}{\zeta _i} =0, \end{aligned} \end{aligned}$$

where \(\zeta _i=\min \bigg \{{\frac{2}{\lambda _{\max }(Q(0)E(0))} \big ( \lambda _{\min }(\Theta )\big ), 2\rho , \beta _{i}+\gamma _{i}\bigg \}}\), \(\Theta =-\frac{z}{2}I_{M(n_{1}+n_{2})}-\frac{1}{2}I_{M}\otimes (Q^{T}(0)Q(0))-d(B\otimes Q^{T}(0)\Gamma )-\frac{\omega }{2}I_{M}\otimes (Q^{T}(0)KK^{T} Q(0))+\frac{1}{2}I_{M}\otimes (Q^{T}(0)K+K^{T}Q(0)) \),

\(\mu _i=\frac{1}{2}c\lambda _{\min }(Q^{T}(0))(1-\vartheta _{i})\),

\(\mathcal {V}(t_{0})=\frac{1}{2}\sum _{i=1}^{M}(y_{i0}-x_{i0})^{T}E(0)Q(0) (y_{i0}-x_{i0})+\sum _{i=1}^{M}\eta _{i}(t_{0}) +\frac{1}{2}\sum _{i=1}^{M}\sum _{j=1}^{M}\frac{1}{\xi _{ij}}({\hat{b}}_{ij}(t_{0})-b_{ij})^{2}\).

In addition, the topology of system (19) can be identified in finite time under update law of the topology observer (8) successfully.

Case 2). Systems (2) and (3) change as

$$\begin{aligned} D^{\alpha }x_{i}(t)=f(x_{i}(t))+d\sum _{j=1}^{M}b_{ij}\Gamma x_{j}(t), \end{aligned}$$
(21)
$$\begin{aligned} D^{\alpha }y_{i}(t)=f(y_{i}(t))+d\sum _{j=1}^{M}{\hat{b}}_{ij}(t)\Gamma y_{j}(t)+u_{i}(t), \end{aligned}$$
(22)

respectively.

Corollary 2

Let \(\alpha \in (0,1)\). Under the assumptions (\(\mathscr {H}_{1}\))-(\({\mathscr {H}}_{2}\)), if there exist scalar \(\omega >0\), matrix \(Q>0\), such that

$$\begin{aligned} \begin{aligned}&\frac{1}{2}I_{M}\otimes (Q^{T} Q)+d(B\otimes Q^{T}\Gamma )+\frac{\omega }{2}I_{M}\otimes (Q^{T}KK^{T}Q)\\&\quad -\frac{1}{2}I_{M}\otimes (Q^{T}K+K^{T}Q)\le -\frac{z}{2}I_{M(n_{1}+n_{2})}, \end{aligned} \end{aligned}$$

hold, then system (21) and (22) can achieve the finite-time synchronization under controller (5). The setting time \(T(e_i(t_0))\) is the solution of equality

$$\begin{aligned} \begin{aligned} \bigg (\mathcal {V}(t_{0})+\frac{\mu _i}{\zeta _i}\bigg )E_{\alpha }\bigg (-\zeta _i(t-t_{0})^{\alpha }\bigg ) -\frac{\mu _i}{\zeta _i}=0, \end{aligned} \end{aligned}$$

where \(\zeta _i=\min \bigg \{{\frac{2}{\lambda _{\max }(Q)} \big ( \lambda _{\min }(\Theta )\big ), 2\rho , \beta _{i}+\gamma _{i}\bigg \}}\), \(\Theta =-\frac{1}{2}I_{M}\otimes (Q^{T}Q)-d(B\otimes Q^{T}\Gamma )- \frac{\omega }{2}I_{M}\otimes (Q^{T}KK^{T} Q)+\frac{1}{2}I_{M}\otimes (Q^{T}K+K^{T}Q) -\frac{z}{2}I_{M(n_{1}+n_{2})}\),

\(\mu _i=\frac{1}{2}c\lambda _{min}(Q)\big (1-\vartheta _{i})\),

\(\mathcal {V}(t_{0})=\frac{1}{2}\sum _{i=1}^{M}(y_{i0}-x_{i0})^{T}Q (y_{i0}-x_{i0})+\sum _{i=1}^{M}\eta _{i}(t_{0}) +\frac{1}{2}\sum _{i=1}^{M}\sum _{j=1}^{M}\frac{1}{\xi _{ij}}({\hat{b}}_{ij}(t_{0})-b_{ij})^{2}\).

In addition, the topology of system (21) can be identified in finite time under the update law of the topology observer (8) successfully.

Remark 6

When set \(\varepsilon =0\), system (19) and (20) become fractional singular CDNs, which means that network nodes are connected only in slow dynamical mode, the fast dynamical connection do not exist. When take \(\varepsilon =1\), networks (21) and (22) change as fractional CDNs without the singular perturbation, and there is only a fast dynamic connection mode between networks. Corollary 1 and 2 discuss the synchronization and topology identification of FSPCNs in the above two cases respectively.

Remark 7

Traditional synchronization methods, such as asymptotic synchronization and exponential synchronization [17, 18], generally require an infinite time for a system to achieve synchronization. Finite-time synchronization method achieves the estimability of the synchronization time. Designers can adjust control parameters based on the estimated time to meet specific synchronization requirements.

4.2 Exclusion of Zeno Phenomenon

For the designed event-triggering mechanism (6), in the following, it is proved that the Zeno behavior can be excluded.

Theorem 2

For each node, under the designed event-triggering mechanism (6), the Zeno phenomenon will not occur.

Proof

From the proof process of Theorem 1, we can conclude that, there exists a positive constant \(\hbar \), such that \(||Q^{T}(\varepsilon )E(\varepsilon ) D^{\alpha }e_{i}(t)||\le \hbar \), \(i=1, 2, \cdots , M\). Noting that \(\bar{e}_{i}(t)=e_{i}(t^{i}_{k})-e_{i}(t)\), we have \(\bar{e}_{i}(t^{i}_{k})=0\). It derives that, \(E(\varepsilon )D^{\alpha } \bar{e}_{i}(t)=-E(\varepsilon )D^{\alpha }e_{i}(t)\). Hence,

$$\begin{aligned} \begin{aligned}&||\bar{e}_{i}(t)|| =|| e_{i}(t^{i}_{k})- e_{i}(t)||\\&\quad =||E^{-1}(\varepsilon )Q^{-T}(\varepsilon )D^{-\alpha }D^{\alpha }Q^{T}(\varepsilon ) E(\varepsilon )(e_{i}(t^{i}_{k})-e_{i}(t))||\\&\quad =||E^{-1}(\varepsilon )Q^{-T}(\varepsilon )\frac{1}{\Gamma (\alpha )}\int _{t^{i}_{k}}^{t}(t-s)^{\alpha -1} D^{\alpha }Q^{T}(\varepsilon )E(\varepsilon )\bar{e}_{i}(s)ds||\\&\quad =||E^{-1}(\varepsilon )Q^{-T}(\varepsilon )\frac{1}{\Gamma (\alpha )}\int _{t^{i}_{k}}^{t}(t-s)^{\alpha -1} Q^{T}(\varepsilon )E(\varepsilon )D^{\alpha }e_{i}(s)ds||\\&\quad \le ||E^{-1}(\varepsilon )Q^{-T}(\varepsilon )||\frac{1}{\Gamma (\alpha )}\int _{t^{i}_{k}}^{t}(t-s)^{\alpha -1} ||Q^{T}(\varepsilon )E(\varepsilon )D^{\alpha }e_{i}(s)||ds \\&\quad \le \frac{\hbar ||E^{-1}(\varepsilon )Q^{-T}(\varepsilon )||(t-t^{i}_{k})^{\alpha }}{\Gamma (\alpha +1)}. \end{aligned} \end{aligned}$$
(23)

For \( t\in [t_{k}^{i},t_{k+1}^{i})\), under the event-triggering mechanism (6), it yields

$$\begin{aligned} \begin{aligned} ||\bar{e}_{i}(t) ||\ge (\gamma _{i}\eta _{i}( t)+\vartheta _{i}e^{T}_{i}(t_{k}^{i})e_{i}(t_{k}^{i}))^{\frac{1}{2}}. \end{aligned} \end{aligned}$$
(24)

Let \(\Delta = t^i_{k+1} -t_{k}^{i}\). Together (23) with (24), it follows that,

$$\begin{aligned} \begin{aligned} (\gamma _{i}\eta _{i}(t^i_{k+1})+\vartheta _{i}e^{T}_{i}(t_{k}^{i})e_{i}(t_{k}^{i}))^{\frac{1}{2}}\le \frac{\hbar || E^{-1}(\varepsilon )Q(\varepsilon )|| \Delta ^{\alpha }}{\Gamma (\alpha +1)}. \end{aligned} \end{aligned}$$

By means of Lemma 6, we can obtain that

$$\begin{aligned} \begin{aligned} \Delta \ge \bigg (\frac{\Gamma (\alpha +1)(\gamma _{i}\eta _{i}(t_{k+1}^{i}) +\vartheta _{i}|| e_{i}(t_{k}^{i})||^{2})^{\frac{1}{2}}}{\hbar ||E^{-1} (\varepsilon )Q(\varepsilon )||}\bigg )^{\frac{1}{\alpha }}>0. \end{aligned} \end{aligned}$$

This means that, under the designed event-triggering mechanism(6), the Zeno phenomenon will not occur. \(\square \)

5 A Simulation Example

Here, a simulation example is provided to demonstrate the correctness of the theoretical results of this paper.

Consider a 3-dimensional FSPCNs with 3 nodes, where the dynamic of nodes is described by Lorenz system:

$$\begin{aligned} \begin{aligned} E(\varepsilon )D^{\alpha }x_{i}(t)=f(x_{i}(t))+d\sum _{j=1}^{3}b_{ij}\Gamma x_{j}(t),\ \ \ \\ \end{aligned} \end{aligned}$$
(25)

where \(i,j=1,2,3\); \(d=0.02\); \(\varepsilon =0.27\), \(\varepsilon ^{\prime }=0.3\); \(B=(b_{ij})_{3\times 3}\) and \(\Gamma \) are chosen as

$$\begin{aligned}{} & {} B=\left[ \begin{array}{ccc} 2.4&{}-5.2&{}2.8\\ 0&{}1.0&{}-1.0\\ -2.3&{}0.4&{}1.9\\ \end{array} \right] ,\ \ \Gamma =\left[ \begin{array}{ccc} 1.52&{}0&{}0\\ 0&{}0.351&{}0\\ 0&{}0&{}3.105\\ \end{array} \right] . \\{} & {} f(x_{i}(t))=\left[ \begin{array}{ccc} -10&{}10&{}0\\ 28&{}-1&{}0\\ 0&{}0&{}-8/3\\ \end{array} \right] \left[ \begin{array}{c} x_{i1}(t)\\ x_{i2}(t)\\ x_{i3}(t)\\ \end{array} \right] +\left[ \begin{array}{c} 0\\ -x_{i1}(t)x_{i3}(t)\\ x_{i1}(t)x_{i2}(t)\\ \end{array} \right] . \end{aligned}$$

Network (25) is considered as the drive network, the corresponding response network is

$$\begin{aligned} \begin{aligned} E(\varepsilon )D^{\alpha }y_{i}(t)=f(y_{i}(t))+d\sum _{j=1}^{3}{\hat{b}}_{ij}(t)\Gamma y_{j}(t)+u_{i}(t), \end{aligned} \end{aligned}$$
(26)

where the parameters \(\alpha \), d, \(\Gamma \) are same as that in the drive network (25). In addition, the update law of the observer \({\hat{b}}_{ij}(t)\) is described by

$$\begin{aligned} \begin{aligned} D^{\alpha }{\hat{b}}_{ij}(t)=&-\rho \text {sgn}(\widetilde{b}_{ij}(t))(|{\hat{b}}_{ij}(t)| +5.2) \\&-d\xi _{ij}e^{T}_{i}(t)Q^{T}(\varepsilon )\Gamma y_{j}(t) \end{aligned} \end{aligned}$$

where \(\rho =0.1\); \(\xi _{ij}=1\), \(i, j=1, 2, 3\); \(Q_{1}=\left[ {\begin{array}{cc} 1.3105 &{} 0.061 \\ 0.061 &{}1.7092\\ \end{array} } \right] \), \(Q_{2}=[0.68\ \ 0.566]\), \(Q_{3}=[5.433]\), d and \(\Gamma \) are same as that in (25).

1). When \(u_i(t)=0\), the trajectories of the error \(e_{ij}(t)\) and topology observer \({\hat{b}}_{ij}(t)\) are depicted in Fig. 2 (a) and (b), respectively. It is easy to see that, network (25) and (26) can not achieve the synchronization, and the topology of network (25) can not also be identified in finite time.

Fig. 2
figure 2

The curves of \(e_{ij}(t)\) and \({\hat{b}}_{ij}(t)\) without controller

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2). For DETM (6) with (7), set \(c=1.004\), other parameters are shown in Table 2. For system (7), take initial values \(\eta _{1}(0)=1.8\), \(\eta _{2}(0)=2.7\), \(\eta _{3}(0)=2.0\) of the internal dynamic variable \(\eta (t)\).

Table 2 Parameter values in (6) with (7)
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Let \(t_0=0\). Take the initial values \(x_1(0)=(-1.320,-2.140, 1.035)^T\), \(x_2(0)= (-3.070,-1.852,-2.041)^T\), \(x_3(0)=(-1.331,2.508,-0.901)^{T}\) for network (25), and \(y_1(0)=(-5.330,-5.503, 2.761)^T\), \(y_2(0)=( -3.791,-1.030,-3.547)^T\), \(y_3(0)=(-0.842,0.571,-1.415)^{T}\) for network (26). Under above parameters, the system states at three different orders are plotted in Figs. 3, 4, 5, 6, 7 and 8.

\(\textcircled {1}.\) Fractional order \(\alpha =0.97\). The Chaotic attractor of Lorenz system is shown in Fig. 3. By solving the inequality (ii) in Theorem 1, the control gain K in the controller (5) can be calculated as

\(K=\left[ {\begin{array}{ccc} 2.71 &{} 2.10 &{} 4.25\\ -0.14 &{}-14.12&{}-7.06\\ 1.40 &{}-4.61&{}-7.51\\ \end{array} } \right] \).

And, on the basis of Theorem 1, the setting time \(T(e_i(t_0))\) is calculated as \(T(e_i(t_0))=4.7041\). The corresponding simulation results are depicted in Fig. 4.

\(\textcircled {2}\). Fractional order \(\alpha =0.8\). The Chaotic attractor of Lorenz system is displayed in Fig. 5. The control gain K can be computed as

\(K=\left[ {\begin{array}{ccc} -0.43 &{} 3.27 &{} 1.20\\ -1.65 &{}-5.72&{}-8.24\\ 2.25 &{}-3.28&{}2.12\\ \end{array} } \right] \).

The setting time \(T(e_i(t_0))\) is calculated as \(T(e_i(t_0))=3.3156\) based Theorem 1. The corresponding simulation results are shown in Fig. 6.

Fig. 3
figure 3

Chaotic attractor of Lorenz system with \(\alpha =0.97\)

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Fig. 4
figure 4

In 2) \(\textcircled {1} \alpha =0.97\): a The event intervals based on DETM (6); b The trajectories of \(x_{ij}(t)\) and \(y_{ij}(t)\); c The trajectories of \(e_{ij}(t)\); d The trajectories of \(\hat{b}_{ij}(t)\); \(i,j=1,2,3\)

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Fig. 5
figure 5

Chaotic attractor of Lorenz system with \(\alpha =0.8\)

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Fig. 6
figure 6

In 2) \(\textcircled {2} \alpha =0.8\): a The event intervals based on DETM (6); b The trajectories of \(x_{ij}(t)\) and \(y_{ij}(t)\); c The trajectories of \(e_{ij}(t)\); d The trajectories of \(\hat{b}_{ij}(t)\); \(i,j=1,2,3\)

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\(\textcircled {3}\). Fractional order \(\alpha =0.7\). Figure 7 shows the corresponding Chaotic attractor of Lorenz system. By solving the inequality (ii) in Theorem 1, the control gain K in the controller (5) can be derived as

\(K=\left[ {\begin{array}{ccc} 4.13 &{} 3.55 &{} 1.95\\ -0.07 &{}-8.14&{}-4.68\\ 3.07 &{}-1.77&{}-2.58\\ \end{array} } \right] \).

The setting time \(T(e_i(t_0))\) is calculated as \(T(e_i(t_0))=5.258\). The corresponding simulation results are displayed in Fig. 8.

From Figs. 3, 4, 5, 6, 7 and 8, we can see that, in the cases of three different values of fractional order \(\alpha \), the synchronization error \(e_{ij}(t)\) converges to zero in finite, which indicates that the systems (25) and (26) can achieve the finite-time synchronization, and the topology in network (25) is identified successfully.

Fig. 7
figure 7

Chaotic attractor of Lorenz system with \(\alpha =0.7\)

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Fig. 8
figure 8

In 2) \(\textcircled {3} \alpha =0.7\): a The event intervals based on DETM (6); b The trajectories of \(x_{ij}(t)\) and \(y_{ij}(t)\); c The trajectories of \(e_{ij}(t)\); d The trajectories of \(\hat{b}_{ij}(t)\); \(i,j=1,2,3\)

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3). In [25], the event-triggering mechanism is designed as

$$\begin{aligned} t_{k+1}^{i}=\inf _{t>t^{i}_{k}}\big \{t: \bar{e}^{T}_{i}(t)\bar{e}_{i}(t) \ge \vartheta _{i} e^{T}_{i}(t_{k}^{i})e_{i}(t_{k}^{i})+e^{-\varsigma t}\big \}. \end{aligned}$$

Take fractional order \(\alpha =0.97\), and simulate system (25) and (26) under controller in [25] and the above parameters. The results are depicted in Fig. 9. In addition, the corresponding event-triggered intervals and times by using the control mechanism in [25] and the control mechanism in this example, which are recorded in Table 2. Compared with [25], in this example, the event-triggering times are reduced obviously, and the event-triggered interval is magnified. This indicates that the designed DETM (6) can save communication resources more effectively than the event-triggering mechanism in [25] and has better control effect (Table 3).

Table 3 Event-triggering times and intervals in [25] and the example
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Fig. 9
figure 9

\(\alpha =0.97\): a The event intervals based on DETM [25]; b The trajectories of \(x_{ij}(t)\) and \(y_{ij}(t)\); c The trajectories of \(e_{ij}(t)\); d The trajectories of \(\hat{b}_{ij}(t)\); \(i,j=1,2,3\)

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Fig. 10
figure 10

Corollary 1\(\alpha =0.7\): a The event intervals based on DETM (6); b The trajectories of \(x_{ij}(t)\) and \(y_{ij}(t)\); c The trajectories of \(e_{ij}(t)\); d The trajectories of \(\hat{b}_{ij}(t)\); \(i,j=1,2,3\)

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Fig. 11
figure 11

Corollary 2\(\alpha =0.7\): a The event intervals based on DETM (6); b The trajectories of \(x_{ij}(t)\) and \(y_{ij}(t)\); c The trajectories of \(e_{ij}(t)\); d The trajectories of \(\hat{b}_{ij}(t)\); \(i,j=1,2,3\)

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4). Taking \(\alpha =0.97\), based on the above parameters, we verify the results of Corollary 1 and 2. The corresponding results are shown in Figs. 10 and 11. Figure 10 shows the system trajectory with \(\varepsilon =0\), and Fig. 11 shows the system trajectory with \(\varepsilon =1\).

From Figs. 10 and 11, it can be seen that, the synchronization error \(e_{ij}(t)\) converges to zero in finite time. This implies that \(\varepsilon =0\) and \(\varepsilon =1\), the systems (25) and (26) can achieve the finite-time synchronization. On the basis of Corollary 1, the setting time \(T(e_i(t_0))\) is calculated as \(T(e_i(t_0))=4.6913\), and from Corollary 2, it can be get \(T(e_i(t_0))=6.4287\). In addition, the identification trajectories of \({\hat{b}}_{ij}(t)\) is depicted in Figs. 10d and 11d. From Figs. 10d and 11d, we conclude that the topology in network (25) is identified successfully.

6 Conclusion

In this paper, we have discussed the topology identification and synchronization of FSPCNs in finite time via DETC. A topology observer and a DETM-based controller have been designed to derive the topology identification and synchronization in finite time. Moreover, it has been proved that Zeno behavior does not exhibit for the designed DETM.

In the future, it is a meaning endeavor to investigate the finite-time topology identification and synchronization of variable-order Fractional complex networks with discontinuous internal dynamics.