Synchronization of Singular Perturbation Complex Networks with an Event-Triggered Delayed Impulsive Control

Abstract

This paper investigates the synchronization problem of singularly perturbed complex networks with time delays, in which a novel event-triggered delayed impulsive control strategy is developed. To conserve limited communication bandwidth, a dynamic event-triggered mechanism is proposed based on a Lyapunov function construction, while incorporating both delay and singular perturbation parameter $\mathit{\epsilon }$ information to avoid ill conditioning. Unlike conventional triggering approaches, the proposed mechanism only requires the Lyapunov function to decrease at impulsive instants, thereby relaxing the constraint on the energy function. Moreover, an impulse-assisted variable $\mathit{\theta }$ is introduced to adjust the event-triggered threshold according to the intensity of impulsive control, which reduces the triggering frequency while ensuring synchronization. By employing stability theory and the singular perturbation method, a singular perturbation parameter $\mathit{\epsilon }$-dependent Lyapunov function is constructed to derive sufficient synchronization conditions and provide the design of the impulsive gain matrix. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed approach.

1. Introduction

The advancement of information technology has established complex dynamic networks (CDNs) as a cornerstone research topic in control science and engineering, with applications spanning social networks, transportation systems, power grids [1], and neural networks [2,3,4]. Synchronization [5], a fundamental collective behavior ensuring cooperative operation, is vital for tasks like distributed optimization, multi-agent coordination, and maintaining smart grid stability. However, practical networks often encounter challenges such as random noise [6], time delays [7], nonlinear couplings, and singular perturbations, which significantly complicate the achievement of synchronization.

The presence of small parameters in real-world systems often gives rise to multi-time-scale dynamics, mathematically described by singular perturbation theory [8]. These systems exhibit an interaction between slow and fast dynamical modes, a feature common in power grids, aerospace control, and chemical reactors [9]. For instance, in electromechanical systems, electrical variables evolve much faster than mechanical ones. Similarly, aerospace systems show a separation between slow positional dynamics and rapid attitude changes. This inherent time-scale disparity complicates control design, as standard methods often fail to adequately manage the interplay between the slow and fast subsystems. Consequently, investigating synchronization in complex networks that incorporate singular perturbations—referred to as singularly perturbed complex networks (SPCNs)—holds significant theoretical and practical importance [10,11,12].

Predominant approaches to synchronization control have largely focused on strategies employing continuous control signals [13]. However, such continuous strategies typically demand substantial communication bandwidth and result in high energy consumption. In numerous practical applications, such as bank deposit regulation [14] or satellite orbit control [15], reliance on continuous inputs is neither efficient nor reliable. In these contexts, impulsive control presents an effective alternative, where control actions are applied only at discrete instants, thereby avoiding continuous intervention and offering considerable savings in communication resources.

The past decade has seen widespread adoption of impulsive control for achieving synchronization in CDNs. It is noteworthy that in networked environments, frequent sampling can lead to data collisions and packet loss. Therefore, the impact of an impulsive jump at a given time may is a function not only of the immediate circumstances but also of historical status information. This insight has motivated the development of delayed impulsive control frameworks, an area that has attracted increasing research interest [16,17,18,19]. For example, the work in [20] established a delay-dependent criterion within an impulsive control framework to ensure the effect of time delays on system stability. Utilizing an extended Halanay inequality, Ref. [21] explored the synchronization of the effects of delayed impulsive control on memristive recurrent neural networks. In [22], consensus in leader–follower nonlinear multi-agent systems was studied under decentralized delayed impulsive control. These collective efforts highlight the critical need to incorporate time delays into the analysis and design of impulsive controllers.

Conventional impulsive control strategies are predominantly based on time-triggered protocols, requiring control updates at fixed intervals. This periodic execution can lead to inefficient resource usage, as impulses might be applied unnecessarily even when the system’s behavior does not require correction. Event-triggered control (ETC) mechanisms have emerged as a promising solution to mitigate this issue. A specific event, which is related to the system state, serves as the necessary condition for initiating any control action. ETC strategies significantly reduce the number of transmissions and conserve communication resources, garnering substantial research attention [23,24,25,26]. By fusing impulsive control with event-triggered mechanisms, researchers have developed event-triggered impulsive control for application in multi-agent systems. In this framework, the controller maintains the most recently received system state information until the next triggering event. Although continuous monitoring is still necessary, no control signals are transmitted between consecutive triggering instants, which markedly reduces the communication burden in multi-agent systems. For instance, Ref. [27] designed an event-triggered impulsive controller to investigate synchronization in memristive neural networks, where the triggering mechanism was subject to an upper bound on impulsive intervals. To remove this constraint, Ref. [28] proposed a Lyapunov-based event-triggered impulsive mechanism addressing stability issues in dynamical systems.

In practical networked environments, time delays are inevitably introduced due to factors such as limited bandwidth and data congestion. Addressing this, the study in [29] analyzed the risk of instability delayed configurations under event-triggered impulsive control, where time delays were handled using the Lyapunov–Razumikhin method. Building on this, Ref. [30] further scrutinized the system’s behavior under bounded external inputs and investigated the input-to-state stability of delayed systems under event-triggered impulsive control. Despite these advancements, most existing results on delayed systems have concentrated on a single time scale. However, the synchronization of singularly perturbed complex networks with dual time scales, using event-triggered impulsive control, remains an area that is largely unexplored, providing the motivation for this work.

To address the gaps identified above, this paper proposes a novel event-triggered delayed impulsive control (ETDIC) protocol specifically designed for the synchronization of SPCNs exhibiting two-time-scale [31] characteristics. The contributions of this work are structured as follows: Firstly, the impulsive controller is co-designed with explicit consideration of the singular perturbation parameter $\mathit{\epsilon }$ and delay information ${\mathit{\tau }}_k$, aiming to minimize communication costs. This $\mathit{\epsilon }$-dependent design is pivotal in avoiding numerical ill-conditioning, ensuring the derived stability conditions are effective. Secondly, compared with [32], a dynamic event-triggered mechanism is designed with respect to a relaxed Lyapunov function. This relaxation is that the Lyapunov function is only required to decrease at the impulsive instants, not continuously. Thirdly, different from [33], an impulse-assisted variable is incorporated into the triggering condition. This variable dynamically modulates the triggering threshold based on the instantaneous strength of the impulsive control. Specifically, a stronger impulse permits a higher triggering threshold, resulting in a more flexible impulse sequence. This co-design strategy effectively balances the trade-off between achieving desired synchronization performance and minimizing the communication burden.

2. Preliminaries

2.1. Problem Formulation

For a nonlinear stochastic coupled network (SPCN) with N nodes, the dynamical behavior of node i is governed by

$$E_{\mathit{\epsilon }}{\dot{x}}_i\left(t\right)=A{x}_i\left(t\right)+Bf\left(x_i\left(t\right)\right)+c\sum _{j=1}^N{d}_{ij}\Gamma x_j(t-{\mathit{\tau }}_0)+u_i\left(t\right)$$

(1)

where

$$E_{\mathit{\epsilon }}=\left[\begin{array}{cc}{I}_{n_1}& 0\\ 0& \mathit{\epsilon }{I}_{n_2}\end{array}\right],A=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right],B=\left[\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\right],$$

$i=1,2,\cdots ,N$, $x_i\left(t\right)={[x_i^1\left(t\right),x_i^2\left(t\right)]}^T$ represents the state of the ith node, which is divided into slow and fast sub-states, denoted by $x_i^1\left(t\right)={[x_{i1}^1\left(t\right),x_{i2}^1\left(t\right),\cdots ,x_{i{n}_1}^1\left(t\right)]}^T\in R^{n_1}$ and $x_i^2\left(t\right)={[x_{i1}^2\left(t\right),x_{i2}^2\left(t\right),\cdots ,x_{i{n}_2}^2\left(t\right)]}^T\in R^{n_2}$, respectively ($n_1+n_2=n$). And $0<\mathit{\epsilon }\ll 1$ means the singular perturbation parameter. A and $B\in {\mathbb{R}}^n$ are known system matrices. The nonlinear functions are defined as $f\left(x_i\left(t\right)\right)={[f^1\left(x_i^1\left(t\right)\right),f^2\left(x_i^2\left(t\right)\right)]}^T$, where $f^1\left(x_i^1\left(t\right)\right)={\left[\begin{array}{cccc}{f}_1^1\left(x_{i1}^1\left(t\right)\right)& f_2^1\left(x_{i2}^1\left(t\right)\right)& \cdots & f_{n_1}^1\left(x_{i{n}_1}^1\left(t\right)\right)\end{array}\right]}^T$ and $f^2\left(x_i^2\left(t\right)\right)={[f_1^2\left(x_{i1}^2\left(t\right)\right)f_2^2\left(x_{i2}^2\left(t\right)\right)\cdots f_{n_2}^2\left(x_{i{n}_2}^2\left(t\right)\right)]}^T$. ${\mathit{\tau }}_0$ is a coupling constant delay. The internal coupling matrix $\Gamma =$ diag$\{F_1,F_2\}>0$, the coupling strength $c>0$, and the connectivity matrix $D={\left(d_{ij}\right)}_{N\times N}$. If nodes i and j ($i\ne j$) are connected, then $d_{ij}=d_{ji}>0$; otherwise, $d_{ij}=0$. To ensure a zero-row sum, the diagonal is constructed as

$$d_{ii}=-\sum _{j=1,j\ne i}^N{d}_{ij}.$$

Assumption 1. 

The network topology is considered to be time-invariant, undirected, and connected.

The network’s synchronization is studied against a dynamically postulated isolated node, which acts as a synchronization reference. Its dynamics are governed by

$$E_{\mathit{\epsilon }}\dot{y}\left(t\right)=Ay\left(t\right)+Bf\left(y\left(t\right)\right).$$

(2)

where $y\left(t\right)={\left[\begin{array}{cc}{y}^1\left(t\right),& y^2\left(t\right)\end{array}\right]}^T$ denotes the state of the target, of which the slow and fast states are $y^1\left(t\right)={[y_1^1\left(t\right),y_2^1\left(t\right),\cdots ,y_{n_1}^1\left(t\right)]}^T,y^2\left(t\right)={[y_1^2\left(t\right),y_2^2\left(t\right),\cdots ,y_{n_2}^2\left(t\right)]}^T$. The nonlinear function is $f\left(y\left(t\right)\right)={[f^1\left(y^1\left(t\right)\right),f^2\left(y^2\left(t\right)\right)]}^T\in {\mathbb{R}}^n$, where $f^1\left(y^1\left(t\right)\right)=[f_1^1\left(y_1^1\left(t\right)\right),f_2^1\left(y_2^1\left(t\right)\right),\cdots ,f_{n_1}^1\left(y_{n_1}^1\left(t\right)\right)],f^2\left(y^2\left(t\right)\right)=[f_1^2\left(y_1^2\left(t\right)\right),f_2^2\left(y_2^2\left(t\right)\right),\cdots ,f_{n_2}^2\left(y_{n_2}^2\left(t\right)\right)]$.

2.2. $\mathit{\epsilon }$-Dependent Impulsive Controller

Given the limited communication bandwidth, a discrete and efficient event-triggered delay impulsive control (ETDIC) scheme is developed. Furthermore, by incorporating the inherent two-time-scale characteristics of the system, the control gain is designed in an $\mathit{\epsilon }$-dependent form to utilize the information of the singular perturbation parameter.

Define the error vector as ${\mathit{\eta }}_i\left(t\right)=x_i\left(t\right)-y\left(t\right)$; then, the proposed ETDIC strategy is designed as

$$u_i\left(t\right)=F_{k\mathit{\epsilon }}{\mathit{\eta }}_i(t_k^{-}-{\mathit{\tau }}_k)-{\mathit{\eta }}_i\left(t_k^{-}\right).$$

(3)

where $F_{k\mathit{\epsilon }}$ denotes the impulsive gain matrix subject to a later design, and ${\mathit{\tau }}_k$ represents the time delay induced by the event-triggering mechanism. Suppose $t_k-{\mathit{\tau }}_k\in [t_{k-1},t_k)$, and $\left\{t_k\right\}$ is the timing of the impulse generated by the event-triggering strategy satisfying $0<t_1<t_2<\cdots <t_k\left(k\in Z_{+}\right)$, ${lim}_{k\to \infty }{t}_k=\infty$. ${\mathit{\eta }}_i\left(t_k^{-}\right)={lim}_{\Delta \to 0^{-}}{\mathit{\eta }}_i(t_k+\Delta )$ and ${\mathit{\eta }}_i\left(t_k\right)={\mathit{\eta }}_i\left(t_k^{+}\right)={lim}_{\Delta \to 0^{+}}{\mathit{\eta }}_i(t_k+\Delta )$. That is, $t=t_k$ and ${\mathit{\eta }}_i\left(t_k\right)$ are right-hand continuous. The impulse sequence is shown in Figure 1.

The error system can be written as

$$\left\{\begin{array}{c}{E}_{\mathit{\epsilon }}{\dot{\mathit{\eta }}}_i\left(t\right)=A{\mathit{\eta }}_i\left(t\right)+Bf\left({\mathit{\eta }}_i\left(t\right)\right)+c\sum _{j=1}^N{d}_{ij}\Gamma {\mathit{\eta }}_j(t-{\mathit{\tau }}_0),\hfill \\ \hfill t\ne t_k,t\ge t_0\hfill \\ {\mathit{\eta }}_i\left(t_k\right)=F_{k\mathit{\epsilon }}{\mathit{\eta }}_1(t_k^{-}-{\mathit{\tau }}_k), t=t_k,k\in Z_{+}\hfill \end{array}\right.$$

(4)

where $f\left({\mathit{\eta }}_i\left(t\right)\right)=f\left(x_i\left(t\right)\right)-f\left(y\left(t\right)\right)$.

For convenience, let $\mathit{\eta }\left(t\right)={[{\mathit{\eta }}_1^T\left(t\right),{\mathit{\eta }}_2^T\left(t\right),\cdots ,{\mathit{\eta }}_N^T\left(t\right)]}^T$, $f(\mathit{\eta }\left(t\right))={[f^T\left({\mathit{\eta }}_1\left(t\right)\right),f^T\left({\mathit{\eta }}_2\left(t\right)\right),\cdots ,f^T\left({\mathit{\eta }}_N\left(t\right)\right)]}^T$. Then, the error dynamic (4) can be rewritten as

$$\left\{\begin{array}{c}\begin{array}{c}(I_N\otimes E_{\mathit{\epsilon }})\dot{\mathit{\eta }}\left(t\right)=(I_N\otimes A)\mathit{\eta }\left(t\right)+(I_N\otimes B)f\left(\mathit{\eta }\left(t\right)\right)\hfill \\ +c(D\otimes \Gamma )\mathit{\eta }(t-{\mathit{\tau }}_0),t\ne t_{\mathrm{k}}\hfill \\ \mathit{\eta }\left(t_{\mathrm{k}}\right)=(I_N\otimes F_{k\mathit{\epsilon }})\mathit{\eta }(t_{\mathrm{k}}^{-}-{\mathit{\tau }}_k),t=t_{\mathrm{k}},k\in Z_{+}\hfill \end{array}\hfill \end{array}\right.$$

(5)

Definition 1. 

If there exist constants $0<\overline{\mathit{\epsilon }}\ll 1$, $\mathit{\delta }>0$, $T>0$, $Q>1$, the SPCN (1) is globally exponentially synchronized with the virtual synchronization target (2), so that any $\mathit{\epsilon }\in (0,\overline{\mathit{\epsilon }}]$ and the initial condition $\mathit{\eta }\left(t_0\right)$ satisfy

$$\left|\mathit{\eta }\left(t\right)\right|⩽Q∥\mathit{\eta }\left(t_0\right)∥e^{-\mathit{\delta }\left(t-t_0\right)},\forall t>T.$$

Assumption 2. 

Then, the Lipschitz condition of nonlinear function $f_i(·)$ is satisfied, namely, for any given positive scalar $h_i$, $i=1,2,...,n$, ${\mathit{\theta }}_1,{\mathit{\theta }}_2\in R$, it follows that

$$|f_i\left({\mathit{\theta }}_1\right)-f_i\left({\mathit{\theta }}_2\right)|\le h_i|{\mathit{\theta }}_1-{\mathit{\theta }}_2|.$$

Lemma 1. 

For any vectors $\mathbf{x}$ and $\mathbf{y}$, any positive definite matrix $W>0$, and any scalar $s>0$, the following inequality is valid:

$$2{\mathbf{x}}^T\mathbf{y}\le s{\mathbf{x}}^{T}W\mathbf{x}+{\displaystyle \frac{1}{s}}{\mathbf{y}}^{T}W\mathbf{y}$$

Lemma 2. 

The following inequality, involving a scalar ε and symmetric matrices $S_1,S_2,S_3$ of suitable dimensions, must be satisfied:

$$\begin{array}{cc}\hfill S_1& \ge 0,\hfill \\ \hfill S_1+\overline{\mathit{\epsilon }}{S}_2& >0,\hfill \\ \hfill S_1+\overline{\mathit{\epsilon }}{S}_2+{\left(\overline{\mathit{\epsilon }}\right)}^2{S}_3& >0\hfill \end{array}$$

Then, for all $\mathit{\epsilon }\in (0,\overline{\mathit{\epsilon }}]$, the inequality $S_1+\mathit{\epsilon }{S}_2+{\mathit{\epsilon }}^2{S}_3>0$ holds.

Lemma 3. 

The existence of a function $V:R^n\to R_{+}$ and positive constants ensure $c_1,c_2,m,\mathit{\delta },\mathit{\lambda },\mathit{\beta },G_1,$ and $G_2$ are assumed, and sequences ${\mathit{\sigma }}_k,{\mathit{\theta }}_k,t_k,{\mathit{\tau }}_k\in (0,{\mathit{\sigma }}_k/(\mathit{\lambda }+\mathit{\delta })]$ for $k\in Z_{+}$, $\overline{\mathit{\theta }}>0$, and ${\mathit{\theta }}_0=0$ satisfy the following conditions:

$$\begin{array}{c}\begin{array}{cc}\hfill \left(C_1\right)& c_1{\parallel \mathit{\eta }\parallel }^m\le V(t,\mathit{\eta })\le c_2{\parallel \mathit{\eta }\parallel }^m,(t,\mathit{\eta })\in [t_0-\mathit{\tau },+\infty )\times R^{Mn}\hfill \end{array}\hfill \\ \begin{array}{cc}\hfill \left(C_2\right)& D^{+}V(t,\mathit{\eta }\left(t\right))\le \mathit{\delta }V(t,\mathit{\eta }\left(t\right)),t\ne t_k\hfill \end{array}\hfill \\ \begin{array}{cc}\hfill \left(C_3\right)& V(t_k,\mathit{\eta }\left(t_k\right))\le e^{-({\mathit{\phi }}_k+\mathit{\lambda }{\mathit{\tau }}_k)}V(t_k^{-}-{\mathit{\tau }}_k,\mathit{\eta }(t_k^{-}-{\mathit{\tau }}_k)),k\in Z_{+}.\hfill \end{array}\hfill \end{array}$$

Meanwhile, the impulse sequence $\left\{t_k\right\}$ can be generated by the following event-triggered mechanism:

$$\begin{array}{cc}\hfill t_k=inf\{t>& t_{k-1}:V(t,\mathit{\eta }\left(t\right))\ge e^{{\mathit{\sigma }}_k+g({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }})-\mathit{\lambda }t}\hfill \\ \hfill & \times \left(e^{\mathit{\lambda }{t}_{k-1}}V(t_{k-1},\mathit{\eta }\left(t_{k-1}\right))\vee e^{\mathit{\lambda }{t}_0}{V}_0\right)\}\hfill \end{array}$$

(6)

where

$$g({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }})=\left\{\begin{array}{cc}\mathit{\beta }(tanh({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }})),\hfill & {\mathit{\theta }}_{k-1}>\overline{\mathit{\theta }}\hfill \\ 0,\hfill & {\mathit{\theta }}_{k-1}\le \overline{\mathit{\theta }}\hfill \end{array}\right.$$

(7)

In addition, parameters ${\mathit{\sigma }}_k,{\mathit{\theta }}_k$ satisfy

$$\underset{k\to +\infty }{lim}\sum _{i=1}^k{\mathit{\sigma }}_i\to \infty ,$$

(8)

$${\mathit{\sigma }}_1+g({\mathit{\theta }}_0-\overline{\mathit{\theta }})\le G_1,$$

(9)

$${\mathit{\sigma }}_k+g({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }})+\sum _{j=1}^{k-1}({\mathit{\sigma }}_j+g({\mathit{\theta }}_{j-1}-\overline{\mathit{\theta }})-{\mathit{\phi }}_j)\le G_2,k\ge 2.$$

(10)

Then, it can be concluded that system (5) is globally exponentially stable.

Proof. 

For simplify, let $V\left(t\right)=V(t,\mathit{\eta }(t\left)\right).$ First, select the following auxiliary functions:

$$v\left(t\right)=\left\{\begin{array}{cc}{e}^{\mathit{\lambda }(t-t_0)}V\left(t\right),\hfill & t\in [t_0,+\infty )\hfill \\ V_{t_0},\hfill & t\in [t_0-\mathit{\tau },t_0).\hfill \end{array}\right.$$

(11)

Substituting the auxiliary function from (11) into the event-triggered mechanism (6) yields the reformulated expression:

$$\begin{array}{c}\hfill t_k=inf\left\{t>t_{k-1}:\nu \left(t\right)\ge e^{{\mathit{\sigma }}_k+g({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }})}\overline{\nu }\left(t_{k-1}\right)\right\},\end{array}$$

where $\overline{v}\left(t_{k-1}\right)=v\left(t_{k-1}\right)\vee v_{t_0}$ and initial value $v_{t_0}=V_{t_0}$. Noted that at instants $t_k,k\in Z$, there are two cases: the event can occur, $\overline{v}\left(t_{k-1}\right)=v\left(t_{k-1}\right)$, and the event cannot occur if $\overline{v}\left(t_{k-1}\right)=v_{t_0}$, respectively.

When $t=t_0$, we consider

$$v\left(t_0\right)\le v_{t_0}\le e^{{\mathit{\sigma }}_1+g({\mathit{\theta }}_0-\overline{\mathit{\theta }})}{v}_{t_0}\le e^{G_1}{v}_{t_0}.$$

Since the number of triggering events is not known in advance, the analysis must consider the following three potential cases.

Case 1: The number of triggers is zero. In this case, no event is triggered. Defining $G=max\{G_1,G_2\}$, based on function (11), we have

$$v\left(t\right)<e^{{\mathit{\alpha }}_1+g({\mathit{\sigma }}_0-\overline{\mathit{\theta }})}\overline{v}\left(t_0\right)=e^{{\mathit{\sigma }}_1+g({\mathit{\theta }}_0-\overline{\mathit{\theta }})}{V}_{t_0}\le e^G{V}_{t_0}$$

Case 2: The number of triggers is finite. In this case, the impulse sequence $\left\{t_k\right\}$ satisfies $0<t_1<t_2<\cdots <t_N$. According to function (11), when $k=1$, we obtain

$$v\left(t_1^{-}\right)=e^{{\mathit{\sigma }}_1+g({\mathit{\theta }}_0-\overline{\mathit{\theta }})}\overline{v}\left(t_0\right)\le G_1$$

Thus, for $t\in [t_0,t_1)$, we have

$$v\left(t\right)\le e^{{\mathit{\sigma }}_1+g({\mathit{\theta }}_0-\overline{\mathit{\theta }})}\overline{v}\left(t_0\right)\le e^{{\mathit{\sigma }}_1+g({\mathit{\theta }}_0-\overline{\mathit{\theta }})}{V}_{t_0}$$

When $t=t_1$, according to condition (C3),

$$v\left(t_1\right)\le e^{-{\mathit{\theta }}_1}v(t_1^{-}-{\mathit{\tau }}_1)\le e^{-{\mathit{\theta }}_1}v\left(t_1^{-}\right)\le e^{{\mathit{\sigma }}_1+g({\mathit{\theta }}_0-\overline{\mathit{\theta }})-{\mathit{\phi }}_1}{V}_{t_0}$$

For $t\in [t_1,t_2)$, we obtain

$$\begin{array}{cc}\hfill v\left(t\right)& \le e^{{\mathit{\sigma }}_2+g({\mathit{\theta }}_1-\overline{\mathit{\theta }})}\overline{v}\left(t_1\right)\hfill \\ \hfill & \le (e^{{\mathit{\sigma }}_2+g({\mathit{\theta }}_1-\overline{\mathit{\theta }})}\vee e^{{\sum }_{j=1}^2({\mathit{\sigma }}_j+g({\mathit{\theta }}_{j-1}-\overline{\mathit{\theta }}))-{\mathit{\phi }}_1})V_{t_0}\hfill \end{array}$$

According to condition $\left(C_3\right)$, when $t=t_2$, we obtain

$$\begin{array}{cc}\hfill v\left(t_2\right)& \le e^{-{\mathit{\phi }}_2}v(t_2^{-}-{\mathit{\tau }}_2)\le e^{-{\mathit{\phi }}_2}v\left(t_2^{-}\right)\hfill \\ \hfill & \le (e^{{\mathit{\sigma }}_2+g({\mathit{\theta }}_1-\overline{\mathit{\theta }})-{\mathit{\phi }}_2}\vee e^{{\sum }_{j=1}^2({\mathit{\sigma }}_j+g({\mathit{\theta }}_{j-1}-\overline{\mathit{\theta }})-{\mathit{\phi }}_j)})V_{t_0}\hfill \end{array}$$

When $k=3$, we obtain

$$v\left(t_3^{-}\right)=e^{{\mathit{\sigma }}_3+g({\mathit{\theta }}_2-\overline{\mathit{\theta }})}\overline{v}\left(t_2\right)$$

For $t\in [t_2,t_3)$,

$$\begin{array}{cc}\hfill v\left(t\right)\le & e^{{\mathit{\sigma }}_3+g({\mathit{\theta }}_2-\overline{\mathit{\theta }})}\overline{v}\left(t_2\right)\hfill \\ \hfill \le & (e^{{\mathit{\sigma }}_3+g({\mathit{\theta }}_2-\overline{\mathit{\theta }})}\vee e^{{\sum }_{j=2}^3({\mathit{\sigma }}_j+g({\mathit{\theta }}_{j-1}-\overline{\mathit{\theta }}))-{\mathit{\phi }}_2}\hfill \\ & \vee e^{{\mathit{\sigma }}_3+g({\mathit{\theta }}_2-\overline{\mathit{\theta }})+{\sum }_{j=1}^2({\mathit{\sigma }}_j+g({\mathit{\theta }}_{j-1}-\overline{\mathit{\theta }})-{\mathit{\phi }}_j)})V_{t_0}\hfill \end{array}$$

Define ${\rm Y}={max}_{k\in Z_{+}}\{{\mathit{\sigma }}_k+g({\mathit{\theta }}_k-\overline{\mathit{\theta }})+{\sum }_{j=1}^{k-1}({\mathit{\sigma }}_{k-j}+g({\mathit{\theta }}_{k-j+1}-\overline{\mathit{\theta }})-{\mathit{\phi }}_{k-j})\}$. By mathematical induction, for $t\in [t_{k-1},t_k),k=1,2,\dots ,N$,

$$\begin{array}{cc}\hfill v\left(t\right)& \le e^{{\mathit{\sigma }}_k+g({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }})}\overline{v}\left(t_{k-1}\right)\hfill \\ & \le (e^{{\mathit{\sigma }}_k+g({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }})}\vee e^{{\rm Y}})V_{t_0}\hfill \end{array}$$

Moreover, since in Case 2 the number of triggering events is finite, no events are triggered after $t_N$. Therefore, for $t\in [t_N,+\infty )$,

$$\begin{array}{c}\hfill v\left(t\right)\le e^{{\mathit{\sigma }}_{N+1}+g({\mathit{\theta }}_N-\overline{\mathit{\theta }})}\overline{v}\left(t_{k-1}\right)\le (e^{{\mathit{\sigma }}_{N+1}+g({\mathit{\theta }}_N-\overline{\mathit{\theta }})}\vee e^{{\rm Y}})V_{t_0}\end{array}$$

Based on the conclusion of Case 1, we obtain

$$v\left(t\right)\le e^G{V}_{t_0},t\ge t_0.$$

Case 3: The number of triggers is infinite. Under this circumstance, the impulsive sequence $t_k$ fulfills $0<t_1<t_2<\cdots <t_w<\cdots$. A primary requirement is to demonstrate the absence of Zeno behavior.

When $k=1$, it yields

$$v\left(t_1^{-}\right)=e^{{\mathit{\sigma }}_1+g({\mathit{\theta }}_0-\overline{\mathit{\theta }})}{v}_{t_0}>v_{t_0}\ge v\left(t_0\right)$$

Thus, we define $t_0^{*}=sup\{t\in [t_0,t_1):v\left(t\right)\le v_{t_0}\}$. Then, $v\left(t_0^{*}\right)=v_{t_0}$, and $v_{t_0}\le v\left(t\right)\le v\left(t_1^{-}\right)$$\forall t\in [t_0^{*},t_1)$, which implies for $\mathit{\theta }\in [-{\mathit{\tau }}_0,0]$,

$$v(t+\mathit{\theta })\le \left\{\begin{array}{cc}v\left(t_1^{-}\right),\hfill & t+\mathit{\theta }\in [t_0,t_1)\hfill \\ v_{t_0},\hfill & t+\mathit{\theta }\in [t_0-{\mathit{\tau }}_0,t_0)\hfill \end{array}\right.$$

$$\le \left\{\begin{array}{cc}{e}^{{\mathit{\sigma }}_1+g({\mathit{\theta }}_0-\overline{\mathit{\theta }})}v\left(t\right),\hfill & t+\mathit{\theta }\in [t_0,t_1)\hfill \\ v\left(t\right),\hfill & t+\mathit{\theta }\in [t_0-{\mathit{\tau }}_0,t_0)\hfill \end{array}\right.$$

$$\le e^{G}v\left(t\right),t\in [t_0^{*},t_1)$$

When $k=2$,

$$v\left(t_2^{-}\right)=e^{{\mathit{\sigma }}_2+g({\mathit{\theta }}_1-\overline{\mathit{\theta }})}v\left(t_1\right)\ge v\left(t_1\right)$$

Define $t_1^{*}=sup\{t\in [t_1,t_2):v\left(t\right)\le v_{t_1}\}$. Then, $v\left(t_1^{*}\right)=v_{t_1}$, and $v_{t_0}\le v\left(t_1^{-}\right)\le v\left(t\right)\le v\left(t_2^{-}\right)$$\forall t\in [t_1^{*},t_2)$, proving that for $\mathit{\theta }\in [-{\mathit{\tau }}_0,0]$,

$$v(t+\mathit{\theta })\le \left\{\begin{array}{cc}v\left(t_2^{-}\right),\hfill & t+\mathit{\theta }\in [t_1,t_2)\hfill \\ v\left(t_1^{-}\right),\hfill & t+\mathit{\theta }\in [t_0,t_1)\hfill \\ v_{t_0},\hfill & t+\mathit{\theta }\in [t_0-{\mathit{\tau }}_0,t_0)\hfill \end{array}\right.$$

$$\le \left\{\begin{array}{cc}{e}^{{\mathit{\sigma }}_2+g({\mathit{\theta }}_1-\overline{\mathit{\theta }})}v\left(t\right),\hfill & t+\mathit{\theta }\in [t_1,t_2)\hfill \\ e^{{\mathit{\sigma }}_1+g({\mathit{\theta }}_0-\overline{\mathit{\theta }})}v\left(t\right),\hfill & t+\mathit{\theta }\in [t_0,t_1)\hfill \\ v\left(t\right),\hfill & t+\mathit{\theta }\in [t_0-{\mathit{\tau }}_0,t_0)\hfill \end{array}\right.$$

$$\le e^{G}v\left(t\right),t\in [t_1^{*},t_2)$$

By recursive induction, for every $k\in {\mathbb{Z}}_{+}$, there corresponds $t_{k-1}^{*}=sup\{t\in [t_{k-1},t_k):v\left(t\right)\le v_{t_{k-1}}\}$ with $v\left(t_{k-1}^{*}\right)=v_{t_{k-1}}$ and $v_{t_0}\le v\left(t_{k-1}^{-}\right)\le v\left(t\right)\le v\left(t_k^{-}\right)$ for $t\in [t_{k-1}^{*},t_k)$, which implies that for $\mathit{\theta }\in [-{\mathit{\tau }}_0,0]$,

$$v(t+\mathit{\theta })\le \left\{\begin{array}{cc}v\left(t_k^{-}\right),\hfill & t+\mathit{\theta }\in [t_{k-1},t_k)\hfill \\ v\left(t_{k-1}^{-}\right),\hfill & t+\mathit{\theta }\in [t_{k-2},t_{k-1})\hfill \\ ⋮\hfill \\ v\left(t_1^{-}\right),\hfill & t+\mathit{\theta }\in [t_0,t_1)\hfill \\ v_{t_0},\hfill & t+\mathit{\theta }\in [t_0-{\mathit{\tau }}_0,t_0)\hfill \end{array}\right.$$

$$\le \left\{\begin{array}{c}{e}^{{\mathit{\sigma }}_k+g\left({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }}\right)}\nu \left(t\right),t+\mathit{\theta }\in [t_{k-1},t_k)\hfill \\ e^{{\mathit{\sigma }}_{k-1}+g\left({\mathit{\theta }}_{k-2}-\overline{\mathit{\theta }}\right)}\nu \left(t\right),t+\mathit{\theta }\in [t_{k-2},t_{k-1})\hfill \\ ...\hfill \\ e^{{\mathit{\sigma }}_1+g\left({\mathit{\theta }}_0-\overline{\mathit{\theta }}\right)}\nu \left(t\right),t+\mathit{\theta }\in [t_0,t_1)\hfill \\ v\left(t\right),t+\mathit{\theta }\in [t_0-{\mathit{\tau }}_0,t_0)\hfill \\ \le e^{G}v\left(t\right),t\in [t_{k-1}^{*},t_k)\hfill \end{array}\right.$$

From the definition of $v\left(t\right)$, we obtain

$$e^{\mathit{\lambda }(t+\mathit{\theta }-t_0)}V(t+\mathit{\theta })\le e^{G+\mathit{\lambda }(t-t_0)}V\left(t\right).$$

For any $t\in [t_{k-1}^{*},t_k)$, $k\in Z_{+}$:

$$e^{\mathit{\lambda }\mathit{\theta }}V(t+\mathit{\theta })\le e^{G}V\left(t\right).$$

Using condition $\left(C_2\right)$ to compute the right-hand upper Dini derivative $v\left(t\right)$ on $[t_{k-1},t_k)$ reveals

$$\begin{array}{cc}\hfill D^{+}v\left(t\right)& \le \mathit{\lambda }{e}^{\mathit{\lambda }(t-t_0)}V\left(t\right)+e^{\mathit{\lambda }(t-t_0)}{D}^{+}V\left(t\right)\hfill \\ \hfill & =(\mathit{\lambda }+\mathit{\delta })e^{\mathit{\lambda }(t-t_0)}V\left(t\right)\hfill \\ \hfill & =(\mathit{\lambda }+\mathit{\delta })v\left(t\right),t\in [t_{k-1},t_k).\hfill \end{array}$$

(12)

Integrating both sides of (12) over $[t_{k-1},t_k)$ yields

$$\begin{array}{c}\hfill \left\{\begin{array}{c}v\left(t_k^{-}\right)\le e^{(\mathit{\lambda }+\mathit{\delta })(t_k-t_{k-1})}v\left(t_{k-1}\right)\hfill \\ e^{{\mathit{\sigma }}_k+g({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }})}v\left(t_{k-1}\right)\le e^{(\mathit{\lambda }+\mathit{\delta })(t_k-t_{k-1})}v\left(t_{k-1}\right)\hfill \end{array}\right.\end{array}$$

Thus, we obtain

$$t_k-t_{k-1}\ge {\displaystyle \frac{{\mathit{\sigma }}_k+g({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }})}{\mathit{\lambda }+\mathit{\delta }}}$$

From conditions (7) and (8), we know ${\mathit{\sigma }}_k+g({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }})>0$, which implies $t_k-t_{k-1}^{*}>0$, thus excluding Zeno behavior.

With this configuration, where events are triggered infinitely often, subsequent proofs similar to Case 2 yield

$$v\left(t\right)\le e^{G}V\left(t_0\right),t\ge t_0.$$

Furthermore, we obtain

$$V\left(t\right)\le e^{G}V\left(t_0\right)e^{-\mathit{\lambda }(t-t_0)},t\ge t_0.$$

Combining with condition $\left(C_1\right)$, one obtains

$$\begin{array}{c}\hfill |\mathit{\eta }(t,t_0,\mathit{\eta }\left(t_0\right))|\le \sqrt[m]{{\displaystyle \frac{c_2{e}^G}{c_1}}}\parallel \mathit{\eta }\left(t_0\right)\parallel e^{-{\displaystyle \frac{\mathit{\lambda }}{m}}(t-t_0)}.\end{array}$$

In conclusion, the global exponential stability of system (5) is obtained. The proof is complete.    □

Remark 1. 

Conventional event-triggered mechanisms are typically designed for continuous-time dynamics. In contrast, for networks with impulsive controllers, the so-called “beating phenomenon” makes such approaches infeasible. To address this, the triggering rule constructed using a Lyapunov function (6) is developed in this paper. Specifically, unlike conventional approaches that require the Lyapunov function to decrease monotonically, the proposed mechanism only enforces a decrease at discrete triggering instants, without imposing restrictions on the continuous impulsive intervals. The corresponding constraints are specified in conditions ($C_2$) and ($C_3$) of Lemma 3. In particular, the parameter δ characterizes a maximum growth limit established for the Lyapunov function throughout the impulsive intervals, while condition ($C_3$) reveals the effect of delayed impulses on the stability of the system.

Remark 2. 

The parameter θ, which appears in Equation (17) of Theorem 1, can be regarded as an impulse-assisted variable that characterizes the influence of the impulsive controller on the system stabilization process. A larger value of θ indicates a stronger effect of the impulsive controller, resulting in a faster convergence rate of the system, whereas a smaller value leads to slower convergence. Based on this observation, the impulse-assisted variable θ is systematically incorporated into the event-triggered mechanism (6). In particular, when the parameter θ exceeds its upper bound $\overline{\mathit{\theta }}$ at $t=t_k$, the function $g({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }})$ increases, which enlarges the triggering threshold. Consequently, the number of triggering instants is reduced, thereby achieving a more efficient utilization of communication resources.

2.3. $\mathit{\epsilon }$-Dependent Impulsive Event-Triggered Mechanism

An $\mathit{\epsilon }$-dependent impulsive event-triggered mechanism to determine the impulse sequence $\left\{t_k\right\}$ in (3) is designed as

$$t_k=inf\{t>t_{k-1}:\Lambda (t,\mathit{\eta }\left(t\right))\le 0\}.$$

(13)

where

$$\begin{array}{cc}\hfill \Lambda (t,\mathit{\eta }(t\left)\right)=& e^{{\mathit{\sigma }}_k+g({\mathit{\theta }}_{k-1}-\overline{\mathit{\theta }})-\mathit{\lambda }t}[e^{\mathit{\lambda }{t}_{k-1}}{\mathit{\eta }}^T\left(t_{k-1}\right)(I_N\hfill \\ \hfill & \otimes P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }})\mathit{\eta }\left(t_{k-1}\right)-{\mathit{\eta }}^T\left(t\right)(I_N\otimes P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }})\mathit{\eta }\left(t\right)]\hfill \end{array}$$

with $P_{\mathit{\epsilon }}$ as the candidate matrix to be determined.

Remark 3. 

Since the dynamic network (1) under consideration exhibits a two-time-scale property, the presence of a small singular perturbation parameter ε can easily induce ill-conditioned numerical characteristics. To address this issue, the event-triggered condition (13) in this paper is designed in a singular perturbation parameter ε-dependent form, which allows for full utilization of the system’s information. Consequently, the subsequently designed control gain $F_{k\mathit{\epsilon }}$ can achieve a broader range of applicability.

3. Main Results

This section formulates criteria ensuring the global stabilization of the error dynamics described by (5) for $\mathit{\epsilon }\in (0,\overline{\mathit{\epsilon }}]$ and determines the impulsive gains.

Theorem 1. 

Under Assumption 1, if there exist positive parameters $\mathit{\epsilon },\mathit{\delta },\mathit{\lambda },\mathit{\beta },G_1,G_2,{\mathit{\theta }}_k,\mathit{\zeta }$, ${\mathit{\tau }}_k\in (0,({\mathit{\sigma }}_k/[\mathit{\lambda }+\mathit{\delta }])]$, $k\in Z_{+},$ some matrices $P_{\mathit{\epsilon }},X_k$ and a diagonal matrix $W>0$ satisfy conditions (8)–(10) and the following linear matrix inequalities:

$$P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}>0$$

(14)

$$D^{T}D\otimes {\Gamma }^T{P}_{\mathit{\epsilon }}\Gamma \le \mathit{\zeta }{I}_N\otimes P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}$$

(15)

$$\left[\begin{array}{cc}\Theta & P_{\mathit{\epsilon }}B\\ *& -W\end{array}\right]<0$$

(16)

$$\left[\begin{array}{cc}-e^{-({\mathit{\theta }}_k+\mathit{\lambda }{\mathit{\tau }}_k)}{P}_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}& X_k^T\\ *& -P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}\end{array}\right]<0$$

(17)

where

$$\begin{array}{c}\Theta =P_{\mathit{\epsilon }}A+A^T{P}_{\mathit{\epsilon }}^T+H^{T}WH+(c+c\mathit{\zeta }{e}^{G+\mathit{\lambda }{\mathit{\tau }}_0}-\mathit{\delta })P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }},\\ P_{\mathit{\epsilon }}=\left[\begin{array}{cc}{P}_1+\mathit{\epsilon }{P}_2& P_3\\ \mathit{\epsilon }{P_3}^T& P_4+\mathit{\epsilon }{P}_5\end{array}\right],\\ H=\mathrm{diag}\{h_1,h_2,\dots ,h_n\},\end{array}$$

Then, the exponential synchronization between the SPCN (1) and the virtual target (2) is ensured by the ETDIC scheme (3) for $\mathit{\epsilon }\in (0,\overline{\mathit{\epsilon }}]$, where the impulsive control gain matrix $F_{k\mathit{\epsilon }}$ is designed as

$$F_{k\mathit{\epsilon }}={\left(P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}\right)}^{-1}{X}_k,k\in Z_{+}.$$

(18)

Proof. 

Select a singular perturbation parameter $\mathit{\epsilon }$-dependent Lyapunov function candidate to stabilize the error dynamics (5):

$$V\left(t\right)=\sum _{i=1}^N{\mathit{\eta }}_i^T\left(t\right)P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}{\mathit{\eta }}_i\left(t\right).$$

(19)

For $t\in [t_{k-1},t_k)$, computing the upper right Dini derivative of $V\left(t\right)$ along the trajectory of (5) yields

$$\begin{array}{cc}\hfill D^{+}V\left(t\right)& =2\sum _{i=1}^N{\mathit{\eta }}_i^T\left(t\right)P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}{\dot{\mathit{\eta }}}_i\left(t\right)\hfill \\ \hfill & =\sum _{i=1}^N{\mathit{\eta }}_i^T\left(t\right)(P_{\mathit{\epsilon }}A+A^T{P}_{\mathit{\epsilon }}){\mathit{\eta }}_i\left(t\right)\hfill \\ \hfill & +2\sum _{i=1}^N{\mathit{\eta }}_i^T\left(t\right)P_{\mathit{\epsilon }}Bf\left({\mathit{\eta }}_i\left(t\right)\right)\hfill \\ \hfill & +2c\sum _{i=1}^N\sum _{j=1}^N{d}_{ij}{\mathit{\eta }}_i^T\left(t\right)P_{\mathit{\epsilon }}\Gamma {\mathit{\eta }}_j(t-{\mathit{\tau }}_0).\hfill \end{array}$$

(20)

Based on Lemma 1 and Assumption 1, one has

$$\begin{array}{cc}\hfill 2{\mathit{\eta }}_i^T\left(t\right)P_{\mathit{\epsilon }}Bf\left({\mathit{\eta }}_i\left(t\right)\right)& \le {\mathit{\eta }}_i^T\left(t\right)P_{\mathit{\epsilon }}B{W}^{-1}{B}^T{P}_{\mathit{\epsilon }}{\mathit{\eta }}_i\left(t\right)\hfill \\ \hfill & +f^T\left({\mathit{\eta }}_i\left(t\right)\right)Wf\left({\mathit{\eta }}_i\left(t\right)\right)\hfill \\ \hfill & \le {\mathit{\eta }}_i^T\left(t\right)(P_{\mathit{\epsilon }}B{W}^{-1}{B}^T{P}_{\mathit{\epsilon }}\hfill \\ \hfill & +H^{T}WH){\mathit{\eta }}_i\left(t\right)\hfill \end{array}$$

Similarly, we can obtain

$$2c\sum _{i=1}^N\sum _{j=1}^N{\mathit{\eta }}_i^T\left(t\right)d_{ij}{P}_{\mathit{\epsilon }}\Gamma {\mathit{\eta }}_j(t-{\mathit{\tau }}_0)=2c{\mathit{\eta }}^T\left(t\right)(D\otimes P_{\mathit{\epsilon }}\Gamma )\mathit{\eta }(t-{\mathit{\tau }}_0)$$

$$\le c{\mathit{\eta }}^T\left(t\right)(I_N\otimes P_{\mathit{\epsilon }})\mathit{\eta }\left(t\right)+c{\mathit{\eta }}^T(t-{\mathit{\tau }}_0)(D^{T}D\otimes \Gamma P_{\mathit{\epsilon }}\Gamma )\mathit{\eta }(t-{\mathit{\tau }}_0)$$

According to condition $\left(C_2\right)$ in Lemma 3, $D^{+}V(t,\mathit{\eta }\left(t\right))\le \mathit{\delta }V(t,\mathit{\eta }\left(t\right))$ for $t\ne t_k$, we have that for $\mathit{\theta }\in [-{\mathit{\tau }}_0,0)$ and $t\in [t_{k-1},t_k)$, when $V\left(t\right)$ satisfies

$$e^{\mathit{\lambda }\mathit{\theta }}V(t+\mathit{\theta },\mathit{\eta }(t+\mathit{\theta }))\le e^{G}V(t,\mathit{\eta }\left(t\right)),$$

it follows that the following inequality holds:

$$\sum _{i=1}^N{\mathit{\eta }}_i^T(t-{\mathit{\tau }}_0)P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}{\mathit{\eta }}_i(t-{\mathit{\tau }}_0)\le e^{G+\mathit{\lambda }{\mathit{\tau }}_0}V\left(t\right)$$

From condition (15), we obtain: $c{\mathit{\eta }}^T(t-{\mathit{\tau }}_0)(D^{T}D\otimes \Gamma P_{\mathit{\epsilon }}\Gamma )\mathit{\eta }(t-{\mathit{\tau }}_0)\le c\mathit{\zeta }V(t-{\mathit{\tau }}_0)\le c\mathit{\zeta }{e}^{G+\mathit{\lambda }{\mathit{\tau }}_0}V\left(t\right).$

Substituting these inequalities into (20) and considering condition (16) yields

$$\begin{array}{cc}\hfill D^{+}V\left(t\right)& \le \sum _{i=1}^N{\mathit{\eta }}_i^T\left(t\right)[P_{\mathit{\epsilon }}A+A^T{P}_{\mathit{\epsilon }}\hfill \\ \hfill & +P_{\mathit{\epsilon }}B{W}^{-1}{B}^T{P}_e+H^{T}WH\hfill \\ \hfill & +(c+c\mathit{\zeta }{e}^{G+\mathit{\lambda }{\mathit{\tau }}_0})P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}]{\mathit{\eta }}_i\left(t\right)\hfill \\ \hfill & \le \mathit{\delta }V\left(t\right).\hfill \end{array}$$

(21)

Furthermore, condition (17) implies

$$\left[\begin{array}{cc}-e^{-({\mathit{\theta }}_k+\mathit{\lambda }{\mathit{\tau }}_k)}{P}_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}& X_k^T\\ *& -P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}\end{array}\right]<0\iff$$

$$\left[\begin{array}{cc}{I}_n& F_{k{\mathit{\epsilon }}^T}\\ 0& I_n\end{array}\right]\left[\begin{array}{cc}-e^{-({\mathit{\theta }}_k+\mathit{\lambda }{\mathit{\tau }}_k)}{P}_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}& X_k^T\\ *& -P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}\end{array}\right]\left[\begin{array}{cc}{I}_n& 0\\ F_{k\mathit{\epsilon }}& I_n\end{array}\right]<0\iff$$

$$\left[\begin{array}{cc}-e^{-({\mathit{\theta }}_k+\mathit{\lambda }{\mathit{\tau }}_k)}{P}_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}+F_{k{\mathit{\epsilon }}^T}{P}_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}{F}_{k\mathit{\epsilon }}& 0\\ *& -P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}\end{array}\right]<0$$

$$-e^{-({\mathit{\theta }}_k+\mathit{\lambda }{\mathit{\tau }}_k)}{P}_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}+{F_{k\mathit{\epsilon }}}^T{P}_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}{F}_{k\mathit{\epsilon }}<0$$

Combining system (5), we formulate

$$\begin{array}{cc}\hfill V\left(t_k\right)& =\sum _{i=1}^N{\mathit{\eta }}_i^T\left(t_k\right)P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}{\mathit{\eta }}_i\left(t_k\right)\hfill \\ \hfill & =\sum _{i=1}^N{\mathit{\eta }}_i^T(t_k-{\mathit{\tau }}_k)F_{k{\mathit{\epsilon }}^T}{P}_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}{F}_{k\mathit{\epsilon }}{\mathit{\eta }}_i(t_k^{-}-{\mathit{\tau }}_k)\hfill \\ \hfill & \le e^{-({\mathit{\theta }}_k+\mathit{\lambda }{\mathit{\tau }}_k)}V(t_k^{-}-{\mathit{\tau }}_k).\hfill \end{array}$$

(22)

By comparison, we find that Equations (13), (19), (21), and (22) satisfy the complete set of conditions for Lemma 3. Therefore, error system (5) attains global exponential stability, which ensures global exponential synchronization between the SPCN (1) and the virtual target (2) and rules out Zeno behavior. This concludes the proof.    □

Remark 4. 

The event-triggered impulsive control (ETDIC) strategy proposed in this study aims to achieve asymptotic stability of the system. Owing to the impulsive controller only acting at discrete impulse instants, this approach exhibits inherent robustness against short-term service interruptions. It is noteworthy that, compared with finite-time stability strategies, the asymptotic stability method demonstrates certain limitations in convergence speed. However, inspired by [34], which provides an effective positive method for handling DoS attacks, the current approach in our paper still has room for improvement in countering sustained service interruptions, which is planned for our future research.

Based on the above analysis, the following algorithm is presented. Algorithm 1 provides a well-structured framework for applying the proposed control strategy to achieve the synchronization of SPCN, aiming to achieve exponential synchronization of complex networks while simultaneously optimizing communication resources under the proposed event-triggered mechanism.

Algorithm 1 Event-Triggered Delayed Impulsive Control for Synchronization of SPCN

Initialization:    1. Initialize parameters of SPCN: system matrices A, B and $\Gamma$, parameter $\mathit{\epsilon }$, coupling delay ${\mathit{\tau }}_0$, network topology matrix D, and coupling strength c, convergence rate $\mathit{\delta }$.    2. Set parameters of event-triggering: $\mathit{\beta }$, upper bound of impulse-assisted variable $\overline{\mathit{\theta }}$, ${\mathit{\theta }}_k$, and $\mathit{\zeta }$.    3. Construct the form of Lyapunov function matrices $P_{\mathit{\epsilon }}$, $X_{\mathit{\epsilon }}$ satisfying stability conditions. Main Loop:    4. for $t\ge t_0$ do    5.       Solve the network dynamics (1) and (2) to obtain the error system $\mathit{\eta }\left(t\right)$ (5).    6.       Calculate the value of the event-triggered condition $\Lambda (t,\mathit{\eta }(t\left)\right)$ defined in Equation (13).    7.        if $\Lambda (t,\mathit{\eta }(t\left)\right)>0$ then    8.             $t_k\leftarrow t$ (Record the current time as the triggering instant.)    9.             Solve the LMIs (14)–(17) to obtain matrices $P_{\mathit{\epsilon }}$, $X_{\mathit{\epsilon }}$, $F_{k\mathit{\epsilon }}$.    10.             Dynamically adjust the next triggering threshold ${\mathit{\sigma }}_{k+1}$ based on ${\mathit{\theta }}_k$ and $\overline{\mathit{\theta }}$ to regulate triggering frequency.    11.              Apply control: Applying $u_i\left(t\right)=F_{k\mathit{\epsilon }}{\mathit{\eta }}_i(t_k^{-}-{\mathit{\tau }}_k)-{\mathit{\eta }}_i\left(t_k^{-}\right)$ into error system (5).    12.        end if    13. end for

4. Simulation Example

Example 1. 

This section demonstrates the main results via a numerical example involving an SPCN comprising one leader and five followers. The system matrices are specified as follows:

$$A_{11}=\left[\begin{array}{cc}2& 0\\ -1& 1\end{array}\right],A_{12}=\left[\begin{array}{c}1\\ -1\end{array}\right],$$

$$A_{21}=\left[\begin{array}{cc}1& 0\end{array}\right],A_{22}=0.2,$$

$$B_{11}=\left[\begin{array}{cc}0.5& -1\\ 0& 0.5\end{array}\right],B_{12}=\left[\begin{array}{c}0\\ -0.4\end{array}\right],$$

$$B_{21}=\left[\begin{array}{cc}0.6& 0\end{array}\right],B_{22}=0.4,$$

$$\Gamma =\mathit{diag}(0.2,0.4,0.8).$$

Additional parameters were chosen as follows: $\mathit{\epsilon }=0.3$, $c=0.6$, $\mathit{\zeta }=8$, $\mathit{\delta }=14$, and ${\mathit{\tau }}_0=0.4$. The coupling configuration matrix was selected as

$$D=\left[\begin{array}{ccccc}-2& 1& 0& 1& 0\\ 1& -2& 1& 0& 0\\ 0& 1& -2& 0& 1\\ 1& 0& 0& -1& 0\\ 0& 0& 1& 0& 1\end{array}\right],$$

and

$${\mathit{\theta }}_k=\left\{\begin{array}{cc}0.5,\hfill & \mathit{mod}(k,3)=0\hfill \\ 3,\hfill & \mathit{mod}(k,3)=1\hfill \\ 6,\hfill & \mathit{mod}(k,3)=2.\hfill \end{array}\right.$$

The state trajectories and error trajectories of the system without dynamic ETDIC are shown in Figure 2 and Figure 3.

According to the synchronous criteria of Theorem 1, the required controller gains $F_{k\mathit{\epsilon }}$ and $P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}$ can be obtained as follows:

$$\begin{array}{cc}\hfill F_{k\mathit{\epsilon }}& =\left[\begin{array}{ccc}0.3076& -0.0013& 0.0002\\ -0.0013& 0.2950& 0.0002\\ 0.0002& 0.0002& 0.3072\end{array}\right],\mathrm{mod}(k,3)=0\hfill \\ \hfill F_{k\mathit{\epsilon }}& =\left[\begin{array}{ccc}0.3076& -0.0013& 0.0002\\ -0.0013& 0.2950& 0.0002\\ 0.0002& 0.0002& 0.3072\end{array}\right],\mathrm{mod}(k,3)=1\hfill \\ \hfill F_{k\mathit{\epsilon }}& =\left[\begin{array}{ccc}0.3076& -0.0013& 0.0002\\ -0.0013& 0.2950& 0.0002\\ 0.0002& 0.0002& 0.3072\end{array}\right],\mathrm{mod}(k,3)=2\hfill \\ \hfill P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}& =\left[\begin{array}{ccc}4.3658& 2.0295& 0.2419\\ 2.0295& 24.1247& -0.4547\\ 0.2419& -0.4547& 3.5342\end{array}\right]\hfill \end{array}$$

Then, the parameters are selected as ${\mathit{\sigma }}_k=0.01$, $\mathit{\beta }=1$, and $\mathit{\alpha }=0.06$, with the initial conditions

$$\begin{array}{cc}\hfill y\left(0\right)& ={\left[\begin{array}{ccc}-1.30& -0.80& -0.70\end{array}\right]}^T,\hfill \\ \hfill x_1\left(0\right)& ={\left[\begin{array}{ccc}1.95& 1.20& 1.05\end{array}\right]}^T,\hfill \\ \hfill x_2\left(0\right)& ={\left[\begin{array}{ccc}1.22& 0.70& -0.62\end{array}\right]}^T,\hfill \\ \hfill x_3\left(0\right)& ={\left[\begin{array}{ccc}2.20& -2.16& 0.00\end{array}\right]}^T,\hfill \\ \hfill x_4\left(0\right)& ={\left[\begin{array}{ccc}1.30& 0.80& 0.70\end{array}\right]}^T,\hfill \\ \hfill x_5\left(0\right)& ={\left[\begin{array}{ccc}1.83& 1.05& -0.93\end{array}\right]}^T.\hfill \end{array}$$

Based on the above parameters, the evolution of the error dynamics is described by (5), governed by the ETDIC strategy (3). The state trajectory and error evolution of the system is shown in Figure 4 and Figure 5. Additionally, the number of triggering events under the proposed event-triggered mechanism (13) is shown in Figure 6. To illustrate the effectiveness of the proposed method, the data transmission rate is calculated as follows:

$$\mathit{Data}\mathit{transmission}\mathit{rate}={\displaystyle \frac{\mathit{The}\mathit{number}\mathit{of}\mathit{transmitted}\mathit{data}}{\mathit{The}\mathit{number}\mathit{of}\mathit{all}\mathit{sampled}\mathit{data}}}.$$

According to the formula, the proposed ETDIC produces a total of 47 triggering instants over 0.15 s (the sampling period is 0.001 s), resulting in a data transmission rate of 32%. For comparison, we present the following static event-triggered mechanism:

$$\begin{array}{cc}\hfill \Lambda (t,\mathit{\eta }(t\left)\right)=& e^{\mathit{\sigma }-\mathit{\lambda }t}\left[e^{\mathit{\lambda }{t}_{k-1}}{\mathit{\eta }}^T\left(t_{k-1}\right)\left(I_N{P}_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}\right)\mathit{\eta }\left(t_{k-1}\right)-{\mathit{\eta }}^T\left(t\right)(I_N\otimes P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }})\mathit{\eta }\left(t\right)\right],\hfill \end{array}$$

where the triggering threshold $e^{\mathit{\sigma }}$ is a constant. Then, Figure 7 presents the number of event triggers with the static event-triggered mechanism, which produces 70 triggering instants over the same duration, corresponding to a higher data transmission rate of 47.3%. This clearly shows that the proposed method (13) can save network communication resources. The data transmission rates are summarized in

Table 1. The data transmission rate with event-triggering conditions.

	Proposed ETDIC	Static Event-Triggered Mechanism
Data transmission rate	32%	47.3%

.

Example 2. 

In this example, a network consisting of one leader’s electric circuit and three followers’ electric circuits is considered to verify the effectiveness of the proposed method under ETDIC. The diagram of each circuit is shown in Figure 8, and its dynamic is shown below:

$$\left\{\begin{array}{c}L{\dot{I}}_{Li}=U_i-{\nu }_{ci}-I_{Li}\left(t\right)R_1,\hfill \\ C_i{\dot{\nu }}_{ci}\left(t\right)=\mathit{\eta }{U}_i+I_{Li}\left(t\right)-{\nu }_{ci}/R_2,\hfill \end{array}\right.$$

The circuit parameters are as follows: resistors $R_1=2\Omega$ and $R_2=5\Omega$, inductance $L=0.1\mathrm{H}$, and parameter $\mathit{\eta }=0.5$. In applying singular perturbation theory, the singular perturbation parameter is defined as $\mathit{\epsilon }=C_i=0.06$, and the nonlinearities are removed. The slow-state variable is defined as $x_{1i}\left(t\right)=L{I}_{Li}$, the fast-state variable as $x_{2i}\left(t\right)=v_{Ci}$, and the control input as $u_i\left(t\right)=U_i$. Furthermore, the networked electrical system has a time constant ${\mathit{\tau }}_0=0.4$. Based on the above analysis, the closed-loop system can be described by

$$E_{\mathit{\epsilon }}{\dot{x}}_i\left(t\right)=A{x}_i\left(t\right)+c\sum _{j=1}^N{d}_{ij}\Gamma x_j(t-{\mathit{\tau }}_0)+u_i\left(t\right)$$

where

$$E_{\mathit{\epsilon }}=\left[\begin{array}{cc}1& 0\\ 0& 0.06\end{array}\right],A=\left[\begin{array}{cc}-20& -1\\ 10& -0.4\end{array}\right],\Gamma =\mathit{diag}(0.2,0.4,0.8).$$

The remaining parameters were selected with the following values: $c=0.2$, $\mathit{\zeta }=8$, $\mathit{\delta }=10$, and ${\mathit{\tau }}_0=0.4$. Furthermore, the coupling configuration matrix is defined as

$$D=\left[\begin{array}{ccc}-2& 1& 1\\ 1& -1& 0\\ 1& 0& -1\end{array}\right],$$

and

$${\mathit{\theta }}_k=\left\{\begin{array}{cc}5,\hfill & \mathit{mod}(k,3)=0\hfill \\ 3,\hfill & \mathit{mod}(k,3)=1\hfill \\ 6,\hfill & \mathit{mod}(k,3)=2\hfill \end{array}\right.$$

According to the synchronization criteria in Theorem 1, the required controller gains $F_{k\mathit{\epsilon }}$ and $P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}$ can be obtained as follows:

$$\begin{array}{cc}\hfill F_{k\mathit{\epsilon }}& =\left[\begin{array}{cc}0.0472& -0.0003\\ -0.0022& 0.0496\end{array}\right],\mathrm{mod}(k,3)=0\hfill \\ \hfill F_{k\mathit{\epsilon }}& =\left[\begin{array}{cc}0.1249& -0.0005\\ -0.0014& 0.1284\end{array}\right],\mathrm{mod}(k,3)=1\hfill \\ \hfill F_{k\mathit{\epsilon }}& =\left[\begin{array}{cc}0.0287& -0.0013\\ -0.0012& 0.0388\end{array}\right],\mathrm{mod}(k,3)=2\hfill \\ \hfill P_{\mathit{\epsilon }}{E}_{\mathit{\epsilon }}& =\left[\begin{array}{cc}11.1434& 1.3934\\ 1.3434& 0.2950\end{array}\right]\hfill \end{array}$$

Furthermore, the parameters were chosen as ${\mathit{\sigma }}_k=0.01$, $\mathit{\beta }=1$, and $\mathit{\alpha }=0.06$. The initial conditions are

$$\begin{array}{cc}\hfill y\left(0\right)& ={\left[\begin{array}{cc}-1.30& -0.80\end{array}\right]}^T,\hfill \\ \hfill x_1\left(0\right)& ={\left[\begin{array}{cc}1.95& 1.20\end{array}\right]}^T,\hfill \\ \hfill x_2\left(0\right)& ={\left[\begin{array}{cc}1.22& 0.70\end{array}\right]}^T,\hfill \\ \hfill x_3\left(0\right)& ={\left[\begin{array}{cc}2.20& -2.16\end{array}\right]}^T.\hfill \end{array}$$

Based on the above analysis, Figure 9 presents the state trajectory of the electric circuit system, while Figure 10 shows its synchronization error. Furthermore, the triggering instants of the proposed method are depicted in Figure 11. A total of 166 triggering instants occurred within 1 s. According to the formula for calculating the transmission percentage, this corresponds to a rate of $16.6\%$.

5. Conclusions

In this work, a novel event-triggered delayed impulsive control (ETDIC) scheme is designed to solve the synchronization problem of stochastic coupled networks (SPCNs) with coupling delays. A dynamic event-triggering mechanism grounded on a Lyapunov function is introduced, significantly lowering communication frequency and saving network resources. Notably, an impulse-assisted parameter is incorporated into the mechanism to dynamically adjust the event-triggered threshold, enabling the impulsive sequence to depend on the control strength and thereby balance synchronization performance and impulsive frequency. Furthermore, by employing Lyapunov stability theory and singular perturbation techniques, sufficient conditions for exponential synchronization of SPCN are established, while avoiding ill conditioning. It is worth mentioning that this work only considers network communication delay, whereas the privacy preservation of dynamic networks is also crucial during communication, which constitutes an important direction for future research.