1 Introduction

Complex dynamic networks (CDNs) are networks consisted of a great deal of nodes in a specific topology, which are widely used in a large amount of real systems, such as metabolic networks, mobile communication networks, urban transportation networks, neural networks etc [1,2,3,4,5,6]. As a result, CDNs have become an attractive research topic in practical engineering applications.

CDNs have many interesting dynamic behaviors, such as space-time chaos, spiral waves, and synchronization. In particular, synchronization, as a collective behavior of CDNs, has attracted more and more scholars to explore and discuss in the field of physics, biology, and other research [7,8,9]. Therefore, in recent years, plenty of research has been done on various analyses of network synchronization, including dissipative synchronization, exponential synchronization, asymptotic synchronization, finite-time synchronization, etc. In literature [10], the sliding mode control method was used to study the asymptotic synchronization of time-delayed CDN. [11] researched how CDN achieve exponential synchronization under feedback control and periodic intermittent noise. The literature [12] demonstrated that the CDN achieves finite-time synchronization problem under the influence of sampled-data control. Among them, several studies have been conducted on the exponential synchronization problem of CDNs [13, 14].

Through the study of CDN, scholars have found that CDN is not a fixed structure and is vulnerable to various disturbances from the external environment. Therefore, in order to overcome this problem, various control strategies are proposed to synchronize it, such as pulse control, sliding mode control, sampled-data control, event-triggered control, etc. Among them, literature [15] studied the global asymptotic synchronization of nonlinear complex neural networks under the control of event-triggered time delay pulses. Hu and others in the literature [16] studied CDN quasi-synchronization of event-triggered control in a time-varying topology framework. In [17], authors studied the global \({H_\infty }\) synchronization of a class of aperiodic CDNs by sampled-data control. In literature [18], global asymptotic synchronization of non-equivalent CDNs with time delay and unknown perturbations was studied by sliding mode control. However, in the normal operation of CDN, a large number of data packets are sent to the communication channels, which consumes a lot of communication resources, so an event-triggered control (ETC) strategy is proposed.

In recent decades, the ETC strategy is a control strategy for data transmission according to the triggering conditions, that can only be transmitted when the conditions of event-triggered are met, so this control strategy has been extensively explored and researched by scholars. This method was first put forward in the literature [19] for the study of system stability. The research method is to transmit the data to the actuator to complete the control task when the error signal violates the given triggering condition. It is found that the ETC strategy can efficiently reduce the number of samples, save communication resources and reduce communication burden. Therefore, researchers have increasingly focused on the synchronization of CDN via the ETC strategy, and have obtained a number of interesting findings [20,21,22,23,24,25,26,27,28]. However, in much literature, the study of ETC is static, that is, triggering conditions are set in advance [29, 30]. In addition, in the existing literature, the authors combined ETC with adaptive control to consider system consistency, such as [31]. In fact, in recent decades, adaptive control strategy has received more and more attention as a control strategy that adapts to the evolution results of the controlled network [32, 33]. As is known to all, in the actual network, the internal system of the CDN is inevitably interfered by the external environment and therefore has a certain uncertainty [34, 35]. Besides, time delay is inevitable [36, 37]. In addition, because the ETC has reduced the number of sampling, so it makes the controller more flexible under adaptive control.

In view of the above discussion, this paper proposes a dynamic adaptive ETC scheme, and studies the exponential synchronization of CDN with uncertain time delay by designing a suitable adaptive ETC control scheme. At the same time, according to the matrix inequality technique, the conditions for exponential synchronization are obtained. The main features are expressed as follows:

  1. 1.

    Due to the sensitivity of the internal system to the external environment, CDN with delay uncertainty is considered, which is more common. According to the designed adaptive law, the event-triggered conditions can change dynamically.

  2. 2.

    Putting forward a new dynamic ETC, and the results of existing literature [38,39,40] are compared with the method proposed in this paper, which avoids the tedious transmission signal, reduces the number of sampling, and effectively saves communication resources.

  3. 3.

    According to the stability of Lyapunov, the conditions that there is no Zeno behavior are obtained, the Zeno behavior is avoided, and the exponential synchronization with time delay CDN is realized.

The rest of the article is arranged below. The second part designs an ETC control strategy and gives some assumptions, definition, and lemma. The third part combines the previous knowledge to draw the main conclusions. The fourth part verifies the advantages of the designed ETC through two simulation examples. Finally, the conclusion is given.

2 Problem Formulation and Preliminaries

In the paper, the continuous-time CDN with time delays containing M coupled nodes is taken into account below:

$$\begin{aligned} {{\dot{x}}_i}(t) = A{x_i}(t) + l({x_i}(t - \tau (t))) + \sum \limits _{j = 1}^M {{s_{ij}}\Gamma {x_j}(t)} + {u_i}(t), \end{aligned}$$
(1)

where \(i = 1,\;2, \ldots ,\;M;\;\) \({x_i}(t) \in {R^n}\) denotes the state variable of the i th dynamic node; \({u_i}\left( t \right) \in {R^n}\) is the control input of the i th node; as well as \(\Gamma \in {R^{n \times n}}\) is a known matrix denoting the inner-coupling. \(\tau (t)\) is time delay, and \(\tau (t) \in (0\,,\tau )\), \(l({x_i}(t - \tau (t)))\) is a nonlinear function. \(A \in {R^{n \times n}}\) is a known matrix.

Assumption 1

In the convention, the weighted matrix \(S = {({s_{ij}})_{M \times M}}\), satisfies that all elements are non-negative that \({s_{ij}} \ge 0\), \(i \ne j\), otherwise \({s_{ij}} = 0\). Moreover, \({s_{ii}} = - \sum \limits _{j = 1,j \ne i}^M {{s_{ij}}} \).

The objective of this article is to synchronize the CDN. Let the synchronization error vector be \({v_i}(t) = {x_i}(t) - {\bar{v}}(t)\), here, the corresponding response system \({\bar{v}}(t)\) is proposed, which stands for the state trajectory of the isolated node, and the condition \(\dot{\bar{ v}}(t) = Av(t) + l({\bar{v}}(t - \tau (t)))\) is met. Therefore, the error system can be represented below:

$$\begin{aligned} {{\dot{v}}_i}(t) = A{v_i}(t) + {\tilde{l}}({v_i}(t - \tau (t))) + \sum \limits _{j = 1}^M {{s_{ij}}\Gamma {v_j}(t) + {u_i}(t)}, \end{aligned}$$
(2)

where \({\tilde{l}}({v_i}(t - \tau (t))) = l({x_i}(t - \tau (t))) - l({{\bar{v}}_i}(t - \tau (t)))\).

It is worth noting that in practical applications, the communication among nodes will be affected. To this end, an event-triggered strategy is proposed to determine whether the current signal should be sent to reduce the waste of communication resources.

Remark 1

Although the existing literatures have proposed to adopt the ETC strategy to synchronize CDNs, most of them are static and need to exchange information with adjacent nodes, which may waste resources and cause information blockage. Therefore, this paper proposes dynamic ETC strategy, which greatly reduces the sampling times and saves resources.

In addition, the triggering time of the basic condition is defined as the following formula:

$$\begin{aligned} t_{k + 1}^i = \inf \{ t > t_k^i|{\varsigma _i}(t) + {\sigma _i}|{v_i}(t){|^2} - |{v_i}(t_k^i) - {v_i}(t){|^2} \le 0\}, \nonumber \\ (i = 1,\;2,\; \ldots ,\;M), (k = 0,\;1,\; \ldots ,\;\infty ), \end{aligned}$$
(3)

where \({\sigma _i} \in (0,1)\), and \({\varsigma _i}(t)\;(i = 1,\;2,\; \ldots ,\;M)\) mean a dynamic parameter and meet the following relation:

$$\begin{aligned} {{\dot{\varsigma }} _i}(t) = - \alpha {\varsigma _i}(t) + {\sigma _i}|{v_i}(t){|^2} - |{v_i}(t_k^i) - {v_i}(t){|^2}, \end{aligned}$$

and \(\alpha > 0\).

Besides, parameters \(\alpha \) and \(\sigma _i\) indicate the tightness of the triggering process and the attenuation rate of the filter, respectively.

Remark 2

Furthermore, the event-triggered mechanism in this paper has a dynamic entry, that is \({\varsigma _i}(t)\), whereas [31] has only a static entry. In addition, the event-triggered control policy only uses its own state and is completed if and only if the triggering condition is true. Therefore, dynamic adaptive ETC can reduce the number of triggers more effectively.

Remark 3

Zeno behavior refers to the control that is triggered an infinite number of times in a finite period of time under the event-triggered control. And in the subsequent proof of Theorem 2, it can be seen that Zeno behavior is avoided.

According to the above discussion, the controller designed is expressed as follows:

$$\begin{aligned} {u_i}(t) = - {K_i}{v_i}(t_k^i) - {\rho _i}sign({v_i}(t)){\mu _i}(t). \end{aligned}$$
(4)

Remark 4

The significance of considering sign function in control design is to ensure the stability and reliability of the system. In this paper, the sign function is used to judge whether a variable reaches a certain threshold value, so as to trigger the corresponding event or control strategy.

Here, \({K_i}\) represents the gain matrix, and \({\mu _i}(t)\) stands for adaptive strength, and satisfies the expression:

$$\begin{aligned} {{\dot{\mu }} _i}(t) = \gamma v_i^{\textrm{T}}(t)sign({v_i}(t)). \end{aligned}$$

Remark 5

In order to study exponential synchronization of CDN, adaptive ETC is proposed in this paper. This strategy is dynamic and introduces adaptive intensity. Furthermore, in contrast with the existing results in [41, 42], the designed controller is more general.

Substituting the above Eq. (4) into Eq. (2), we get the following equation:

$$\begin{aligned} {{\dot{v}}_i}(t) = A{v_i}(t) + {\tilde{l}}({v_i}(t - \tau (t))) + \sum \limits _{j = 1}^M {{s_{ij}}\Gamma {v_j}(t)} - {K_i}{v_i}(t_k^i) - {\rho _i}sign({v_i}(t)){\mu _i}(t). \end{aligned}$$
(5)

Remark 6

The Kronecker product is an operation between two matrices of any size, and it is a special form of tensor basis. Given two matrices \(A \in {R^{m \times n}}\) and \(B \in {R^{p \times q}}\), then the Kronecker product of these two matrices is a partitioned matrix of space \({R^{mp \times nq}}\)

$$\begin{aligned} A \otimes B = \left[ {\begin{array}{*{20}{c}} {{a_{11}}B}&{} \cdots &{}{{a_{1n}}B}\\ \vdots &{} \ddots &{} \vdots \\ {{a_{m1}}B}&{} \cdots &{}{{a_{mn}}B} \end{array}} \right] . \end{aligned}$$

Using the Kronecker product, we can simplify the above equation to:

$$\begin{aligned} \dot{v}(t) = Av(t) + {\tilde{l}}(v(t - \tau (t))) + (S \otimes \Gamma )v(t) - Kv({t_k}) - \rho sign(v(t))\mu (t). \end{aligned}$$
(6)

Definition 1

[43] If the error system (5) is exponentially stable and satisfies the relation:

$$\begin{aligned} \left\| {{v_i}(t)} \right\| \le \beta {e^{ - \alpha t}}, \end{aligned}$$

where \(\beta > 0\), \(\alpha > 0\), then the CDN is exponential synchronization.

Assumption 2

In the nonlinear function \(l( \cdot )\), for any \({f_1}\), \({f_2}\), there exists a constant \(l > 0\), so that the relation holds: \(||l({f_1}) - l({f_2})|| \le l||{f_1} - {f_2}||\).

Lemma 1

[44] Given two vectors \(a,b \in {R^n}\), any constant \(\eta > 0\) and a positive definite matrix \(W \in {R^{n \times n}}\), the following inequality holds:

$$\begin{aligned} 2{a^{\textrm{T}}}b \le \eta {a^{\textrm{T}}}Wa + \frac{1}{\eta }{b^{\textrm{T}}}{W^{ - 1}}b. \end{aligned}$$

3 Main Results

This part is on account of dynamic event-triggered mechanism, the stability of error system is analyzed. First of all, the sufficient condition of exponential synchronization is explored based on Lyapunov function theory. Then it is clear that the CDN can avoid Zeno behavior via the designed dynamic event-triggered controller.

Set

$$\begin{aligned}{} & {} {\theta _i}(t) = {v_i}(t_k^i) - {v_i}(t),\\{} & {} \delta (t) = {\left[ {\begin{array}{*{20}{c}} {{v^{\textrm{T}}}(t)}&{{v^{\textrm{T}}}(t - \tau (t))}&{{\theta ^{\textrm{T}}}(t)}&{{\xi ^{\textrm{T}}}(t)} \end{array}} \right] ^{\textrm{T}}},\\{} & {} {\varsigma (t) = {\left[ {\begin{array}{*{20}{c}} {\varsigma _1^{\frac{1}{2}}(t)}&{\varsigma _2^{\frac{1}{2}}(t)}&\cdots&{\varsigma _M^{\frac{1}{2}}(t)} \end{array}} \right] ^{\textrm{T}}},}\\{} & {} \theta (t) = {\left[ {\begin{array}{*{20}{c}} {\theta _1^{\textrm{T}}(t)}&{\theta _2^{\textrm{T}}(t)}&{\ldots }&{\theta _M^{\textrm{T}}(t)} \end{array}} \right] ^{\textrm{T}}}.\\ \end{aligned}$$

Theorem 1

For the given parameters \(\beta ,\,\sigma ,\,l > 0\) and matrices \(A,\;S,\;\Gamma \in {R^{n \times n}}\), if there is matrix \(G \in {R^{n \times n}}>0\) and positive number \(\lambda \), satisfying inequality below:

$$\begin{aligned} \Psi = \left[ {\begin{array}{*{20}{c}} {2A + \beta I + 2\sigma I + G - \lambda I}&{}0&{}{ - (S \otimes \Gamma )}&{}0\\ * &{}{\frac{{{l^2}}}{\beta }I}&{}0&{}0\\ * &{} * &{}{K - 2I}&{}0\\ * &{} * &{} * &{}{{W_{11}}} \end{array}} \right] < 0 \end{aligned}$$
(7)

where \({W_{11}} = diag( {-\lambda _1}{-\alpha _1}+ 1,\,{-\lambda _2}{-\alpha _2}+ 1,\, \ldots ,\,{-\lambda _M}{-\alpha _M}+ 1)\),

\(K = diag\{ {K_1},\;{K_2},\; \ldots ,\;{K_M}\} \),

then the uncertain CDN with time delay can achieve exponential synchronization under ETC.

Proof

The Lyapunov function is constructed below:

$$\begin{aligned} V(t) = \sum \limits _{i = 1}^3 {{V_i}(t)}, \end{aligned}$$
(8)

where

$$\begin{aligned}{} & {} {V_1}(t) = \sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){v_i}(t)},\\{} & {} {V_2}(t) = \sum \limits _{i = 1}^M {{{\varsigma _i}(t)}},\\{} & {} {V_3}(t) = \int _0^t {{e^{\lambda (p - t)}}{v^{\textrm{T}}}(p)Gv(p)dp}.\\ \end{aligned}$$

Deriving the above equations, it can be obtained that:

$$\begin{aligned} \begin{array}{r@{~}l} {{\dot{V}}_1}(t) = &{} \sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){{\dot{v}}_i}(t)} + \sum \limits _{j = 1}^M {\dot{v}_i^{\textrm{T}}(t){v_i}(t)} \\ = &{} 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)[A{v_i}(t) + {\tilde{l}}({v_i}(t - \tau (t))) + \sum \limits _{j = 1}^M {{s_{ij}}\Gamma {v_j}(t) + {u_i}(t)} ]}, \end{array} \end{aligned}$$
(9)
$$\begin{aligned} {\dot{V}_2}(t) = \sum \limits _{i = 1}^M {{{{{\dot{\varsigma }} }_i}(t)}} = \sum \limits _{i = 1}^M {[ - {\alpha _i}{{\varsigma _i}}(t) + {\sigma _i}v_i^{\textrm{T}}(t){v_i}(t) - \theta _i^{\textrm{T}}(t){\theta _i}(t)]}, \end{aligned}$$
(10)
$$\begin{aligned} {\dot{V}_3}(t) = - \lambda {V_3}(t) + {v^{\textrm{T}}}(t)Gv(t). \end{aligned}$$
(11)

Using Assumption 2 and Lemma 1, we can scale the inequality to:

$$\begin{aligned} \begin{array}{r@{~}l} &{}2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){\tilde{l}}({v_i}(t - \tau (t)))} \\ \le &{} \beta \sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){v_i}(t)} + \frac{1}{\beta }{{{\tilde{l}}}^{\textrm{T}}}({v_i}(t - \tau (t))){\tilde{l}}({v_i}(t - \tau (t)))\\ \le &{} \beta \sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){v_i}(t)} + \frac{{{l^2}}}{\beta }v_i^{\textrm{T}}(t - \tau (t)){v_i}(t - \tau (t)). \end{array} \end{aligned}$$
(12)

By using the equation \({v_i}(t) = {v_i}(t_k^i) - {\theta _i}(t)\), the formula can be obtained:

$$\begin{aligned} \begin{array}{r@{~}l} &{}2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)\sum \limits _{j = 1}^M {{s_{ij}}\Gamma {v_j}(t)} } \\ = &{} 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)\sum \limits _{j = 1}^M {{s_{ij}}\Gamma ({v_j}(t_k^i) - {\theta _j}(t))} } \\ = &{} 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)\sum \limits _{j = 1}^M {{s_{ij}}\Gamma {v_j}(t_k^i)} } - 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)\sum \limits _{j = 1}^M {{s_{ij}}\Gamma {\theta _j}(t)} } \\ = &{} - 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)\sum \limits _{j = 1}^M {{s_{ii}}\Gamma {v_j}(t_k^i)} } - 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)\sum \limits _{j = 1}^M {{s_{ij}}\Gamma {\theta _j}(t)} } \\ \le &{} - 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)\sum \limits _{j = 1}^M {{s_{ij}}\Gamma {\theta _j}(t)} }, \end{array} \end{aligned}$$
(13)

and

$$\begin{aligned} \begin{array}{r@{~}l} &{}2\sum \limits _{j = 1}^M {v_i^{\textrm{T}}(t){u_i}(t)} \\ = &{} 2\sum \limits _{j = 1}^M {v_i^{\textrm{T}}(t)( - {K_i}{v_i}(t_k^i) - {\rho _i}sign({v_i}(t)){\mu _i}(t))} \\ = &{} - 2\sum \limits _{j = 1}^M {v_i^{\textrm{T}}(t){K_i}{v_i}(t_k^i) - 2\sum \limits _{j = 1}^M {v_i^{\textrm{T}}(t){\rho _i}sign({v_i}(t)){\mu _i}(t)} } \\ = &{} 2\sum \limits _{j = 1}^M {(v_i^{\textrm{T}}(t_k^i) - v_i^{\textrm{T}}(t)){K_i}{v_i}(t_k^i)} - 2\sum \limits _{j = 1}^M {v_i^{\textrm{T}}(t_k^i){K_i}{v_i}(t_k^i)} - 2\sum \limits _{j = 1}^M {{\rho _i}|{v_i}(t)|{\mu _i}(t)} \\ \le &{} {K_i}||{v_i}(t_k^i) - {v_i}(t)|{|^2} + {K_i}||{v_i}(t_k^i)|{|^2} - 2{K_i}||{v_i}(t_k^i)|{|^2} - 2\sum \limits _{j = 1}^M {{\rho _i}|{v_i}(t)|{\mu _i}(t)}. \end{array} \end{aligned}$$
(14)

By combining Eqs. (12–14) to (9), it is available that:

$$\begin{aligned} \begin{array}{r@{~}l} {{\dot{V}}_1}(t) \le &{} 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)A{v_i}(t)} + \beta \sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){v_i}(t)} + \frac{{{l^2}}}{\beta }\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t - \tau (t)){v_i}(t - \tau (t))} \\ &{} - 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)\sum \limits _{j = 1}^M {{s_{ij}}\Gamma {\theta _j}(t)} } + {K_i}||{\theta _i}(t)|{|^2} - {K_i}||{v_i}(t_k^i)|{|^2}\\ \le &{} 2{v^{\textrm{T}}}(t)Av(t) + \frac{{{l^2}}}{\beta }{v^{\textrm{T}}}(t - \tau (t))v(t - \tau (t)) - 2{v^{\textrm{T}}}(t)(S \otimes \Gamma )v(t)\\ &{}+ {\theta ^{\textrm{T}}}(t)K\theta (t) + \beta {v^{\textrm{T}}}(t)v(t). \end{array} \end{aligned}$$
(15)

In addition, we can also obtain:

$$\begin{aligned} \begin{array}{r@{~}l} &{}{{\dot{V}}_2}(t) + {{\dot{V}}_3}(t) \\ = &{} \sum \limits _{i = 1}^M {({\sigma _i}v_i^{\textrm{T}}(t){v_i}(t)} - \theta _i^{\textrm{T}}(t){\theta _i}(t) - {\alpha _i}{{\varsigma _i}(t)}) - \lambda {V_3}(t) + {v^{\textrm{T}}}(t)Gv(t)\\ = &{} \sigma {v^{\textrm{T}}}(t)v(t) - {\theta ^{\textrm{T}}}(t)\theta (t) - \lambda {V_3}(t) + {v^{\textrm{T}}}(t)Gv(t) - \alpha {\varsigma (t)}\\ = &{} {v^{\textrm{T}}}(t)(\sigma I + G)v(t) - {\theta ^{\textrm{T}}}(t)\theta (t) - \lambda {V_3}(t) - \alpha {\varsigma (t)}. \end{array} \end{aligned}$$
(16)

Moreover, according to the triggering condition (3), the formula can easily obtained:

$$\begin{aligned} {{\varsigma _i}(t) + {\sigma _i}v_i^{\textrm{T}}(t){v_i}(t) - \theta _i^{\textrm{T}}(t){\theta _i}(t) > 0}. \end{aligned}$$
(17)

By summarizing the inequalities (8–17), we can get the following inequality:

$$\begin{aligned} \begin{array}{r@{~}l} &{}\dot{V}(t) - \lambda V(t)\\ \le &{} 2{v^{\textrm{T}}}(t)Av(t) + \beta {v^{\textrm{T}}}(t)v(t) + \frac{{{l^2}}}{\beta }{v^{\textrm{T}}}(t - \tau (t))v(t - \tau (t)) - 2{v^{\textrm{T}}}(t)(S \otimes \Gamma )\theta (t)\\ &{} + {\theta ^{\textrm{T}}}(t)K\theta (t) + 2{v^{\textrm{T}}}(t)(\sigma I + G)v(t) - {\theta ^{\textrm{T}}}(t)\theta (t) - \lambda {V_3}(t) - \lambda \sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){v_i}(t)} \\ &{} - \lambda \sum \limits _{i = 1}^M {{{\varsigma _i}(t)}} - \lambda {V_3}(t) + \sum \limits _{i = 1}^M {{{\varsigma _i}(t)}} + \sum \limits _{i = 1}^M {{\sigma _i}v_i^{\textrm{T}}(t){v_i}(t)} - \sum \limits _{i = 1}^M {\theta _i^{\textrm{T}}(t){\theta _i}(t)}- \alpha {\varsigma (t) }\\ \le &{} {v^{\textrm{T}}}(t)(2A + \beta I + 2\sigma I + G - \lambda I)v(t) + \frac{{{l^2}}}{\beta }{v^{\textrm{T}}}(t - \tau (t))v(t - \tau (t))\\ &{} - 2{v^{\textrm{T}}}(t)(S \otimes \Gamma )\theta (t) + {\theta ^{\textrm{T}}}(t)(K - 2I)\theta (t) + ({\lambda _i} + 1){\varsigma (t)}\\ = &{} {\delta ^{\textrm{T}}}(t)\Psi \delta (t). \end{array} \end{aligned}$$
(18)

According to the conditions known by the Theorem 1, the formula can be get: \(\dot{V}(t) - \lambda V(t) < 0\).

And, it is not hard to figure out that: \(V(t) - V(0){e^{ - \lambda t}} \le 0\).

Thus, there is:

$$\begin{aligned} V(t) \le V(0){e^{ - \lambda t}} \le (||v(0)|{|^2} + \sum \limits _{i = 1}^M {{{\varsigma _i}(0)}} ){e^{ - \lambda t}}. \end{aligned}$$
(19)

Also known by \(||v(t)|{|^2} = {V_1}(t) < V(t) \le (||v(0)|{|^2} + \sum \limits _{i = 1}^M {{{\varsigma _i}(0)}} ){e^{ - \lambda t}}\).

Namely

$$\begin{aligned} ||v(t)|| \le \sqrt{||v(0)|{|^2} + \sum \limits _{i = 1}^M {{{\varsigma _i}(0)}} } {e^{ - \frac{{\lambda t}}{2}}}. \end{aligned}$$
(20)

Based on the concept of exponential synchronization given above, we can draw a conclusion that the system (2) realizes exponential synchronization. \(\square \)

Remark 7

From the above conclusion, it can be concluded that the error system v(t) is exponentially convergent, namely \(||v(t)|| \le \sqrt{||v(0)|{|^2} + \sum \limits _{i = 1}^M {{{\varsigma _i}}(0)} } {e^{ - \frac{{\lambda t}}{2}}}\). And its norm is affected by M and \(\lambda \).

Theorem 2

After the whole of conditions in Theorem 1 are met, the system under ETC will avoid the Zeno behavior.

Proof

First, according to equation \({\theta _i}(t) = {v_i}(t_k^i) - {v_i}(t)\), it can receive that:

$$\begin{aligned} \begin{array}{r@{~}l} {{{\dot{\theta }} }_i}(t) = &{} - [A{v_i}(t) + {\tilde{l}}({v_i}(t - \tau (t))) + \sum \limits _{j = 1}^M {{s_{ij}}\Gamma {v_j}(t) + {u_i}(t)} ]\\ = &{} - [A{v_i}(t_k^i) - A{\theta _i}(t) + {\tilde{l}}({v_i}(t - \tau (t))) + \sum \limits _{j = 1}^M {{s_{ij}}\Gamma {v_j}(t) + {u_i}(t)} ]. \end{array} \end{aligned}$$
(21)

Use the proof by contradiction.

Suppose that the system has Zeno behavior, namely there is a time \(\Delta T_i^*\) which makes \(\mathop {\lim }\limits _{k \rightarrow \infty } (t_{k + 1}^i - t_k^i) = \Delta T_i^*\) as well as \(\mathop {\lim }\limits _{k \rightarrow \infty } t_k^i = \Delta T_i^*\) true.

In addition, the condition \({[l(x) - l(y) - M(x - y)]^{\textrm{T}}}[l(x) - l(y) - N(x - y)] \le 0\) satisfied by the nonlinear function \({\tilde{l}}( \cdot )\) gives the inequality below:

$$\begin{aligned} ||{\tilde{l}}({v_i}(t - \tau (t)))|| \le \gamma ||{v_i}(t)|| \le \gamma {\bar{w}}, \end{aligned}$$
(22)

where \(\gamma = \frac{1}{2}[||M + N|| + \sqrt{{\lambda _{\max }}{{(M - N)}^{\textrm{T}}}(M - N)} ]\), \({\bar{w}} = \sqrt{||v(0)|{|^2} + \sum \limits _{i = 1}^M {{{\varsigma _i(0)}} }} {e^{ - \frac{{\lambda t}}{2}}}\).

Moreover, there exists a number \({\varphi _i>0}\), makes \(||{u_i}(t)|| \le {\varphi _i}\), \(||{v_j}(t)|| \le {\varphi _i}\).

Therefore, combining the above inequalities, it can be obtained:

$$\begin{aligned} \begin{array}{r@{~}l} &{} ||{{{\dot{\theta }} }_i}(t)||\\ \le &{} ||A||||{v_i}(t_k^i)|| + ||A||||{\theta _i}(t)|| + ||\sum \limits _{j = 1}^M {{s_{ij}}\Gamma {v_j}(t)} || + ||{u_i}(t)||\\ &{} + ||{\tilde{l}}({v_i}(t - \tau (t)))||\\ \le &{} ||A||||{v_i}(t_k^i)|| + ||A||||{\theta _i}(t)|| + {{{\bar{\varphi }} }_i}, \end{array} \end{aligned}$$
(23)

where \({{\bar{\varphi }} _i} = \gamma {\bar{w}} + {\bar{s}}||\Gamma ||\sum \limits _{j = 1}^M {{v_j}(t)} + {\varphi _i}\).

Integral the above equation to get:

$$\begin{aligned} ||{\theta _i}(t)|| \le \left( ||{v_i}(t_k^i)|| + \frac{{{{{\bar{\varphi }} }_i}}}{{||A||}}\right) ({e^{||A||(t - t_k^i)}} - 1). \end{aligned}$$
(24)

The event-triggered mechanism shows that, when the sufficient condition of the triggering condition is not met, it can be obtained:

$$\begin{aligned} ||{\theta _i}(t)|| \le {\varphi _i}{e^{ - ({\lambda _i} + 1)\frac{t}{2}}}, \end{aligned}$$
(25)

here, \({\varphi _i} = \sqrt{{\varsigma _i}(0) + {\sigma _i}||{v_i}(0)|{|^2}} \).

Therefore, we can obtain the inequality as follows:

$$\begin{aligned} \left( ||{v_i}(t_k^i)|| + \frac{{{{{\bar{\varphi }} }_i}}}{{||A||}}\right) ({e^{||A||(t - t_k^i)}} - 1) \le&{\varphi _i}{e^{ - ({\lambda _i} + 1)\frac{t}{2}}}. \end{aligned}$$
(26)

From this we can easily derive inequalities:

$$\begin{aligned} t - t_k^i \le&\frac{1}{{||A||}}\ln \left( \frac{{{\varphi _i}}}{{||{v_i}(t_k^i)|| + \frac{{{{{\bar{\varphi }} }_i}}}{{||A||}}}}{e^{ - ({\lambda _i} + 1)\frac{t}{2}}} + 1 \right) . \end{aligned}$$
(27)

Make \({\kappa _0} = \frac{1}{{||A||}}\ln \left( \frac{{{\varphi _i}}}{{||{v_i}(t_k^i)|| + \frac{{{{{\bar{\varphi }} }_i}}}{{||A||}}}}{e^{ - ({\lambda _i} + 1)\frac{{{T_0}}}{2}}} + 1 \right) > 0\), that is, there is a large enough positive integer \({N_0}\), making \(t_k^i \in [{T_0} - {\kappa _0},\,{T_0}],\quad \forall k \ge {N_0}\) is established.

Assume that \(t_k^i\) is on behalf of the k th triggering time of node \({v_i}\). Then, set \(t_{k + 1}^i\) and \({\tilde{t}}_{k + 1}^i\) denote the next triggering moment decided by the triggering condition (3) and Eq. (27) above, respectively.

As a result, it can be easily obtained that:

$$\begin{aligned} \begin{array}{r@{~}l} &{}t_{k + 1}^i - t_k^i \ge {\tilde{t}}_{k + 1}^i - t_k^i \\ = &{} \frac{1}{{||A||}}\ln \left( \frac{{{\varphi _i}}}{{||{v_i}(t_k^i)|| + \frac{{{{{\bar{\varphi }} }_i}}}{{||A||}}}}{e^{ - ({\lambda _i} + 1)\frac{{{\tilde{t}}_{k + 1}^i}}{2}}} + 1 \right) \\ \ge &{} \frac{1}{{||A||}}\ln \left( \frac{{{\varphi _i}}}{{||{v_i}(t_k^i)|| + \frac{{{{{\bar{\varphi }} }_i}}}{{||A||}}}}{e^{ - ({\lambda _i} + 1)\frac{{t_{k + 1}^i}}{2}}} + 1 \right) \\ \ge &{} \frac{1}{{||A||}}\ln \left( \frac{{{\varphi _i}}}{{||{v_i}(t_k^i)|| + \frac{{{{{\bar{\varphi }} }_i}}}{{||A||}}}}{e^{ - ({\lambda _i} + 1)\frac{{{T_0}}}{2}}} + 1 \right) \\ > &{} {\kappa _0}. \end{array} \end{aligned}$$
(28)

This result is inconsistent with the result of equation (27), and the hypothesis is not valid. Therefore, Zeno behavior can be avoided.

In addition, without considering nonlinear CDNs, system (2) can be converted to:

$$\begin{aligned} {\dot{v}_i}(t) = A{v_i}(t) + \sum \limits _{j = 1}^M {{s_{ij}}\Gamma {v_j}} + {u_i}(t). \end{aligned}$$
(29)

\(\square \)

Corollary 1

For the given parameters \(\beta ,\,\sigma > 0\) and matrices \(A,\;S,\;\Gamma \in {R^{n \times n}}\), if there is a positive number \(\lambda \), the following matrix is true:

$$\begin{aligned} \Omega = \left[ {\begin{array}{*{20}{c}} {2A - 2K + {\bar{\gamma }} K{K^{\textrm{T}}} + \sigma I + \lambda I}&{}{ - (S \otimes \Gamma )}&{}0\\ *&{}{(\frac{1}{{{\bar{\gamma }} }} - 1)I}&{}0\\ *&{}*&{}{{{{\bar{W}}}_{11}}} \end{array}} \right] < 0 \end{aligned}$$
(30)

where \({{\bar{W}}_{11}} = diag( - {\alpha _1} + {\lambda _1},\, - {\alpha _2} + {\lambda _2},\, \ldots \, - {\alpha _M} + {\lambda _M}) \),

\(K = diag\{ {K_1},\;{K_2},\; \ldots ,\;{K_M}\} \),

then the CDN can achieve exponential synchronization under ETC. As well as the Zeno behavior can be avoided.

Proof

First, construct the Lyapunov function below:

$$\begin{aligned} V(t) = {V_1}(t) + {V_2}(t). \end{aligned}$$
(31)

Here, \({V_1}(t) = \sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)} {v_i}(t)\), \({V_2}(t) = \sum \limits _{i = 1}^M {{{{{\bar{\varsigma }} }_i}(t)}} \).

The derivative of V(t) can be obtained from the above equation:

$$\begin{aligned} {\dot{V}_1}(t) = 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)} (A{v_i}(t) + \sum \limits _{j = 1}^M {{s_{ij}}\Gamma {v_j}(t)} + {u_i}(t)), \end{aligned}$$
(32)
$$\begin{aligned} {\dot{V}_2}(t) = \sum \limits _{i = 1}^M {{( - {\alpha _i}{{{\bar{\varsigma }} }_i}(t)} + {\sigma _i}v_i^{\textrm{T}}(t){v_i}(t) - \theta _i^{\textrm{T}}(t){\theta _i}(t))}. \end{aligned}$$
(33)

From Lemma 1 and Eq. (eq4), it can be easily obtained:

$$\begin{aligned} \begin{array}{r@{~}l} &{}2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){u_i}(t)} \\ = &{} 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)( - {K_i}{v_i}(t_k^i) - {\rho _i}sign({v_i}(t)){\mu _i}(t))} \\ = &{} - 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){K_i}{v_i}(t_k^i) - 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){\rho _i}sign({v_i}(t)){\mu _i}(t)} } \\ = &{} - 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){K_i}({\theta _i}(t) + {v_i}(t))} - 2\sum \limits _{i = 1}^M {{\rho _i}|{v_i}(t)|{\mu _i}(t)} \\ \le &{} - 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){K_i}{v_i}(t)} + 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){K_i}{\theta _i}(t)} \\ \le &{} - 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){K_i}{v_i}(t)} + \sum \limits _{i = 1}^M {{\bar{\gamma }} v_i^{\textrm{T}}(t){K_i}K_i^{\textrm{T}}{v_i}(t)} + \sum \limits _{i = 1}^M {\frac{1}{{{\bar{\gamma }} }}\theta _i^{\textrm{T}}(t){\theta _i}(t)}. \end{array} \end{aligned}$$
(34)

Thus, it can be get:

$$\begin{aligned} \begin{array}{r@{~}l} &{}{{\dot{V}}_1}(t) \\ \le &{} 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)A{v_i}(t)} - 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t)} \sum \limits _{j = 1}^M {{s_{ij}}\Gamma {\theta _j}(t)} - 2\sum \limits _{i = 1}^M {v_i^{\textrm{T}}(t){K_i}{v_i}(t)} \\ &{} + \sum \limits _{i = 1}^M {{\bar{\gamma }} v_i^{\textrm{T}}(t){K_i}K_i^{\textrm{T}}{v_i}(t)} + \sum \limits _{i = 1}^M {\frac{1}{{{\bar{\gamma }} }}\theta _i^{\textrm{T}}(t){\theta _i}(t)} \\ \le &{} 2{v^{\textrm{T}}}(t)Av(t) - 2{v^{\textrm{T}}}(t)\Gamma \theta (t) - 2{v^{\textrm{T}}}(t)Kv(t) + {\bar{\gamma }} {v^{\textrm{T}}}(t){K_i}K_i^{\textrm{T}}v(t) \\ &{}+ \frac{1}{{{\bar{\gamma }} }}{\theta ^{\textrm{T}}}(t)\theta (t). \end{array} \end{aligned}$$
(35)

Combining the (31), (33), (35), the inequality can be obtained below:

$$\begin{aligned} \begin{array}{r@{~}l} &{}\dot{V}(t) - \lambda V(t)\\ \le &{} 2{v^{\textrm{T}}}(t)Av(t) - 2{v^{\textrm{T}}}(t)\Gamma \theta (t) - 2{v^{\textrm{T}}}(t)Kv(t) + {\bar{\gamma }} {v^{\textrm{T}}}(t){K_i}K_i^{\textrm{T}}v(t)\\ &{} + \frac{1}{{{\bar{\gamma }} }}{\theta ^{\textrm{T}}}(t)\theta (t)- {\alpha {\bar{\varsigma }} (t) + \sigma {v^{\textrm{T}}}(t)v(t) - {\theta ^{\textrm{T}}}(t)\theta (t) - \lambda {v^{\textrm{T}}}(t)v(t) - \lambda {\bar{\varsigma }} (t)}\\ = &{} {{{\bar{\delta }} }^{\textrm{T}}}(t)\Omega {\bar{\delta }} (t), \end{array} \end{aligned}$$
(36)

where \({\bar{\delta }} (t) = {\left[ {\begin{array}{*{20}{c}} {{v^{\textrm{T}}}(t)}&{{\theta ^{\textrm{T}}}(t)}&{{{{\bar{\xi }} }^{\textrm{T}}}(t)} \end{array}} \right] ^{\textrm{T}}}\), \({\bar{\varsigma }} (t) = {\left[ {\begin{array}{*{20}{c}} {{\bar{\varsigma }} _1^{\frac{1}{2}}(t)}&{{\bar{\varsigma }} _2^{\frac{1}{2}}(t)}&\cdots&{{\bar{\varsigma }} _M^{\frac{1}{2}}(t)} \end{array}} \right] ^{\textrm{T}}}\).

According to Schur’s lemma, it is easy to get \(\dot{V}(t) - \lambda V(t) < 0\).

Integrating the above equation can get:

$$\begin{aligned} V(t) \le V(0){e^{ - \lambda t}} \le (||v(0)|{|^2} + {\sum \limits _{i = 1}^M {{{{\bar{\varsigma }} }_i}(0)}} ){e^{ - \lambda t}}. \end{aligned}$$
(37)

Because of \(||v(t)|{|^2} \le V(t)\), it can be reached that:

$$\begin{aligned} ||v(t)|| \le \sqrt{||v(0)|{|^2} + \sum \limits _{i = 1}^M {{{{{\bar{\varsigma }} }_i}(0)}} } {e^{ - \frac{{\lambda t}}{2}}}. \end{aligned}$$
(38)

It can be seen that CDN achieves exponential synchronization. Furthermore, by referring to the proof of Theorem 2, it can be obtained that Zeno behavior in CDN (29) can be avoided. \(\square \)

Remark 8

Corollary 1 studies exponential synchronization in linear and delay-free CDNs, making the results more general.

4 Numerical Examples

We demonstrate the superiority of the proposed control method through two examples.

Example 1

Here an uncertain CDN with time delay with three nodes and two topologies is considered, the system is expressed as follows:

$$\begin{aligned} {\dot{x}_i}(t) = A{x_i}(t) + l({x_i}(t - \tau (t))) + \sum \limits _{j = 1}^M {{s_{ij}}\Gamma {x_j}(t)} + {u_i}(t). \end{aligned}$$

In the above equation, let \(A = \left[ {\begin{array}{*{20}{c}} { - 3}&{}0&{}0&{}0&{}0&{}0\\ 0&{}{ - 3}&{}0&{}0&{}0&{}0\\ 0&{}0&{}{ - 3}&{}0&{}0&{}0\\ 0&{}0&{}0&{}{ - 3}&{}0&{}0\\ 0&{}0&{}0&{}0&{}{ - 3}&{}0\\ 0&{}0&{}0&{}0&{}0&{}{ - 3} \end{array}} \right] \), \(\Gamma = \left[ {\begin{array}{*{20}{c}} 0.5&{}0\\ 0&{}0.5 \end{array}} \right] \), \(S = \left[ {\begin{array}{*{20}{c}} { - 1}&{}0&{}1\\ 2&{}{ - 2}&{}0\\ 0&{}2&{}{ - 2} \end{array}} \right] \).

In addition, the nonlinear function is chosen as:

$$\begin{aligned} l({x_i}(t)) = \left[ {\begin{array}{*{20}{c}} { - 0.5{x_{i1}}(t) + \tan (0.2{x_{i1}}(t)) + 0.2{x_{i2}}(t)}\\ {0.95{x_{i2}}(t) - \tan (0.75{x_{i2}}(t))} \end{array}} \right] . \end{aligned}$$

And based on the Assumptions and Lemma 1 given above, the value of the parameters can be chosen as: \(\alpha \mathrm{{ = }}3\), \(\sigma \mathrm{{ = }}0.8\), \(\lambda \mathrm{{ = }}1.5\), \(l = 2\).

Moreover, we select the initial values of the system state below: \({v_1}(0) = {\left[ {\begin{array}{*{20}{c}} 2&{ - 2.5} \end{array}} \right] ^{\textrm{T}}}\), \({v_2}(0) = {\left[ {\begin{array}{*{20}{c}} {1.5}&{ - 1} \end{array}} \right] ^{\textrm{T}}}\), \({v_3}(0) = {\left[ {\begin{array}{*{20}{c}} {3.5}&{ - 2} \end{array}} \right] ^{\textrm{T}}}\). And by using LMI to calculate the matrix in Theorem 1, the control gain matrices can be expressed as follows:

$$\begin{aligned} K = \left[ {\begin{array}{*{20}{c}} { 0.6728}&{}{0.7783}&{}0&{}0&{}0&{}0\\ {0.7783}&{}{ 0.6766}&{}0&{}0&{}0&{}0\\ 0&{}0&{}{ - 1.4678 }&{}{1.0975}&{}0&{}0\\ 0&{}0&{}{1.0975}&{}{ 0.5974}&{}0&{}0\\ 0&{}0&{}0&{}0&{}{ 0.6728}&{}{0.7783}\\ 0&{}0&{}0&{}0&{}{0.7783}&{}{ 0.6766} \end{array}} \right] . \end{aligned}$$

Besides, parameters in the controller and system can be set to \(\gamma = 1\), \(\alpha = 3\).

Figures 1, 2, 3 and 4 show the simulation results for the sake of demonstrating the superiority of the results obtained. Figure 1 indicates the state dynamic trajectory of each node in the CDN system. Figure 2 displays the state trajectories of the synchronization errors, which indicates convergence to a range. Figures 3 and 4 show the dynamic trajectories of adjective law and adaptive strength respectively. As a consequence, from above four figs, it is observed that each node of the CDN under the control of the deaigned ETC can realize synchronization. Figure 5 demonstrates that without a controller, the system is not stable. Not only that, Fig. 6 shows the event interval of each node, which greatly reducing the number of information transfers. Besides, Fig. 7 illustrates the error bound.

Fig. 1
figure 1

Trajectories of inputs

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Fig. 2
figure 2

Trajectories of synchronization errors

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Fig. 3
figure 3

Trajectories of adaptive laws

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Fig. 4
figure 4

Trajectories of adaptive strength

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Fig. 5
figure 5

Trajectories of synchronization errors without controller

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Fig. 6
figure 6

Trajectories of event times and intervals

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Fig. 7
figure 7

Trajectories of error bound

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Remark 9

In the same time, compared with the literature [45], under the event-triggered mechanism, Table 1 shows that the event-triggered control proposed in this paper can better reduce the number of triggers.

Remark 10

Reference [6] studied the quasi-synchronization problem of a periodic time-varying complex dynamic network under the event-triggered control strategy. In contrast, the adaptive event-triggered control strategy proposed in this paper can reduce the triggering times more effectively and avoid unnecessary resource waste.

Example 2

As a typical nonlinear system, Chua’s circuit system is used to verify the reliability of the above conclusions. Let us consider a CDN with 5 nodes and 3 topologies. Here, the system parameters are set to:

\(A = \left[ {\begin{array}{*{20}{c}} { - 5}&{}1&{}0&{}{ - 1}&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}0&{}{ - 1}&{}{ - 1}&{}{ - 1}\\ 1&{}{ - 5}&{}0&{}0&{}{ - 1}&{}0&{}0&{}1&{}1&{}0&{}0&{}0&{}{ - 1}&{}0&{}0\\ 0&{}0&{}{ - 5}&{}0&{}0&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}{ - 1}&{}0&{}{ - 1}\\ { - 1}&{}0&{}0&{}{ - 5}&{}0&{}0&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0\\ 0&{}{ - 1}&{}0&{}0&{}{ - 5}&{}0&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}{ - 5}&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 1&{}0&{}0&{}0&{}0&{}0&{}{ - 5}&{}0&{}0&{}1&{}0&{}0&{}{ - 1}&{}0&{}0\\ 0&{}1&{}0&{}0&{}0&{}0&{}0&{}{ - 5}&{}0&{}0&{}0&{}0&{}0&{}0&{}1\\ 0&{}1&{}0&{}0&{}0&{}0&{}0&{}0&{}{ - 5}&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}1&{}0&{}0&{}0&{}1&{}0&{}0&{}{ - 5}&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}1&{}1&{}0&{}0&{}0&{}0&{}0&{}{ - 5}&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}{ - 5}&{}0&{}0&{}0\\ { - 1}&{}{ - 1}&{}{ - 1}&{}0&{}0&{}0&{}{ - 1}&{}0&{}0&{}0&{}0&{}0&{}{ - 5}&{}1&{}1\\ { - 1}&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}1&{}{ - 5}&{}0\\ { - 1}&{}0&{}{ - 1}&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}1&{}0&{}{ - 5} \end{array}} \right] \), \(\Gamma = \left[ {\begin{array}{*{20}{c}} 1&{}0&{}0\\ 0&{}1&{}0\\ 0&{}0&{}1 \end{array}} \right] \), \( S = \left[ {\begin{array}{*{20}{c}} { - 1}&{}1&{}0&{}0&{}0\\ 1&{}{ - 2}&{}1&{}0&{}0\\ 1&{}0&{}{ - 2}&{}1&{}0\\ 1&{}0&{}0&{}{ - 2}&{}1\\ 1&{}1&{}0&{}0&{}{ - 2} \end{array}} \right] \).

Furthermore, we choose the nonlinear function as:

$$\begin{aligned} l({x_i}(t)) = \left[ {\begin{array}{*{20}{c}} { - 10({x_{i1}}(t) - {x_{i2}}(t) + \chi ({x_{i1}}(t)))}\\ {{x_{i1}}(t) - {x_{i2}}(t) + {x_{i3}}(t)}\\ { - 14.87{x_{i2}}(t)} \end{array}} \right] , \end{aligned}$$

where \(\chi ({x_{i1}}(t)) = - 0.68 - 0.295(|{x_{i1}}(t) + 1| - |{x_{i1}}(t) - 1|)\).

It is similar to the Example 1, given the conditions, we chose the other parameters of the system as: \(\alpha = 1\), \(\sigma = 0.5\), \(\lambda = 1\), \(l = 2\). Not only that, we set \({x_1}(0) = {\left[ {\begin{array}{*{20}{c}} 2&{ - 3.4}&{0} \end{array}} \right] ^{\textrm{T}}}\), \({x_2}(0) = {\left[ {\begin{array}{*{20}{c}} 0&4&{ - 2} \end{array}} \right] ^{\textrm{T}}}\), \({x_3}(0) = {\left[ {\begin{array}{*{20}{c}} { - 4.5}&{ - 2.5}&5 \end{array}} \right] ^{\textrm{T}}}\), \({x_4}(0) = {\left[ {\begin{array}{*{20}{c}} { - 1}&{3.5}&{ - 4} \end{array}} \right] ^{\textrm{T}}}\) and \({x_5}(0) = {\left[ {\begin{array}{*{20}{c}} {1}&{ - 0.5}&{1.5} \end{array}} \right] ^{\textrm{T}}}\) as the initial conditions.

Table 1 The comparison of the event times and intervals (Example 1)
Full size table

The control gain matrices obtained by calculating the LMI in the MATLAB toolbox and some parameters therein are shown below: \(\gamma = 0.5\), and

$$\begin{aligned} {K_{11}} = \left[ {\begin{array}{*{20}{c}} { 0.0939}&{}{0.0835}&{}{0.1370}\\ {0.0835}&{}{ 0.0789}&{}{0.1257}\\ {0.1370}&{}{0.1257}&{}{ 0.1719} \end{array}} \right] , {K_{22}} = \left[ {\begin{array}{*{20}{c}} { 0.1639}&{}{0.1704}&{}{0.3040}\\ {0.1704}&{}{ 0.1372}&{}{0.2692}\\ {0.3040}&{}{0.2692}&{}{ 0.3087} \end{array}} \right] , \\ {K_{33}} = \left[ {\begin{array}{*{20}{c}} { 0.1180}&{}{0.2002}&{}{0.2088}\\ {0.2002}&{}{ 0.2170}&{}{0.2822}\\ {0.2822}&{}{0.2822}&{}{ 0.2471} \end{array}} \right] , {K_{44}} = \left[ {\begin{array}{*{20}{c}} {0.3275}&{}{0.3253}&{}{0.4040}\\ {0.3253}&{}{0.1960}&{}{0.3022}\\ {0.4040}&{}{0.3022}&{}{0.2938} \end{array}} \right] , \\ {K_{55}} = \left[ {\begin{array}{*{20}{c}} { 0.0840}&{}{0.1727}&{}{0.1274}\\ {0.1727}&{}{ 0.2388}&{}{0.2378}\\ {0.1274}&{}{0.2378}&{}{ 0.1660} \end{array}} \right] . \end{aligned}$$

In terms of the above parameters, the simulation is carried out. As illustrated in Figs. 8, 9, 10, 11, and 12, the trajectories of states as well as the trajectories of the error system are stable under ETC. When Fig. 13 depicts a system that is not stable without a controller. Besides, the adaptive law and adaptive strength are shown in Figs. 14 and 15 respectively. As well as Fig. 16 expresses the event intervals and Fig. 17 shows the error bound.

Fig. 8
figure 8

a Trajectories of inputs for the first node. b Trajectories of errors for the first node

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Fig. 9
figure 9

a Trajectories of inputs for the second node. b Trajectories of errors for the second node

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Fig. 10
figure 10

a Trajectories of inputs for the third node. b Trajectories of errors for the third node

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Fig. 11
figure 11

a Trajectories of inputs for the fourth node. b Trajectories of errors for the fourth node

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Fig. 12
figure 12

a Trajectories of inputs for the fifth node. b Trajectories of errors for the fifth node

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Fig. 13
figure 13

Trajectories of synchronization errors without controller

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Fig. 14
figure 14

Trajectories of adjective law

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Fig. 15
figure 15

Trajectories of adjective strength

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Fig. 16
figure 16

Trajectories of event intervals

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Fig. 17
figure 17

Trajectories of error bound

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5 Conclusions

In this paper, the exponential synchronization problem with time delay uncertain CDNs has been discussed. However, because CDN is susceptible to interference from external environment, its system is affected. Therefore, we have proposed an adaptive dynamic event-triggered control strategy to better adapt to different environmental changes and system states and reduce possible instability problems. In addition, it can save the sampling times and reduce the information transmission frequency effectively. Then, Lyapunov function and inequality technique have been used to realize the synchronization of CDN under the control of ETC, and sufficient conditions to make certain that the exponential synchronization of CDN have been obtained. In addition, it has also been proved that the system can avoid Zeno behavior. In the end, two simulation examples have been used to prove the reliability of the theory. In the future research, we will compare with reference [46] to study the exponential congruence problem of complex dynamic networks with multi-coupling delays.

Remark 11

The limitation of adaptive event-triggered control strategy is that in some cases, the system cannot accurately judge when the event is triggered. This may be because the system does not have enough information to judge, or because the triggering conditions of the event are very complex and variable.

Remark 12

Although the control strategy proposed in this paper can effectively reduce the number of triggers, the trigger of the event is not monitored in real time, and the real-time occurrence of the event cannot be guaranteed. Therefore, the effect of adaptive control will be affected. In future work, in order to improve such limitations, more sensor data will be considered. By increasing the type and quantity of sensor data, more information can be provided to support the decision of event-triggered control.