Consensus Control for Stochastic Multi-Agent Systems with Markovian Switching via Periodic Dynamic Event-Triggered Strategy

Abstract

The consensus problem in stochastic multi-agent systems (MASs) with Markovian switching is addressed by proposing a novel distributed dynamic event-triggered (DDET) technique based on periodic sampling to reduce information transmission. Unlike traditional event-triggered control, the proposed periodic sampling-based DDET method is characterized by the following three advantages: (1) The need for continuous monitoring of the event trigger is eliminated. (2) Zeno behavior in stochastic MASs is effectively prevented. (3) Communication costs are significantly reduced. Based on this, sufficient conditions for achieving consensus in the mean-square sense are derived using Lyapunov–Krasovskii functions, providing a solid theoretical foundation for the proposed strategy. The effectiveness of the proposed DDET control is validated through two numerical examples.

1. Introduction

In recent years, advancements in computing and sensor technology have greatly expanded the applications of MASs, which consist of a large number of agents. Consequently, research on distributed coordinated control of MASs has gained significant attention due to its potential in various fields such as the cooperative control of unmanned airborne vehicles (UAVs) [1], and automated highway systems [2]. Cooperative control endeavors to devise and deploy appropriate control mechanisms for all participating agents, thereby fostering their collaboration to achieve various objectives such as consensus, formation tracking, cooperative defense, and flocking. Among them, consensus has garnered considerable attention in recent research. The goal of consensus is to develop techniques that enable all agents to converge to a common state [3,4,5,6,7]. In this paper, we investigate the problem of consensus for a unique class of MASs.

In reality, MASs are subject to various random disturbances that have adverse effects on the communication between individual agents and the dynamics of the MASs [8,9,10,11]. Therefore, it is crucial to take into account the impact of stochastic noise on the consensus of MASs. For a class of uncertain stochastic nonlinear MASs, almost sure consensus is explored through time-varying feedback in [8]. In [9], the event-triggered bipartite consensus of stochastic nonlinear MASs with uncertain dead-zone input is investigated. In [10], the impulsive consensus of stochastic MASs under semi-Markovian switching topology is explored. Additionally, proportional–integral control protocols are designed in [11] for the leader-following consensus problems in stochastic dynamical MASs. Furthermore, in practical engineering works, unexpected switching often occurs in the dynamics of MASs [12,13] and the connection structures [14,15,16,17], which is attributed to the constraint on communication channels and the rapid change in the environment. The problem of distributed consensus tracking facing a set of modified nonlinear MASs is resolved in [13] using a pure-feedback form and directed communication networks. The stochastic scaled consensus issue in the almost surely and mean-square sense of stochastic MASs in a Markovian switching topology is settled in [16]. In this paper, the stochastic MASs with concurrently Markovian switching system matrices and communication topologies are investigated. Our work takes into account additional factors beyond those considered in the mentioned works, making it more closely aligned with real-world scenarios.

It is widely recognized that the agents in MASs rely on a network to communicate with their neighbors for the exchange of information about their surroundings, which enables them to send measurement outputs and update control inputs. The conventional techniques of consensus control are premised on continual agent-to-agent communication [18,19,20,21]. In practice, however, the network of communication is generally undesirable for purposes due to constraints on bandwidth and high hardware costs. Differently, sampled-data control is integrated into the consensus control strategy for MASs. Ref. [22] utilized impulsive consensus method to solve the limited bandwidth problem of MASs. In [23], the authors focused on the estimation of time cost for stochastic consensus of second-order nonlinear MASs through the pinning protocols. Ref. [24] dealt with prespecified-time bipartite consensus for leaderless and leader-following MASs via intermittent control. Unlike the traditional time-driven mechanism of sampling control [22,23,24], whose sampling behavior must be executed at a fixed interval, compared with the traditional time-driven mechanism of sampling control [22,23,24], the event-triggered strategy, with predetermined trigger conditions based on the system state, significantly reduces information transfer and lowers costs and resource consumption [25,26,27,28,29,30]. For example, ref. [25] investigated the consensus problem of discrete time-varying stochastic MASs via a stastic event-based mechanism. In [26], the event-triggered consensus of stochastic MASs was explored. In [27], the issue of bipartite synchronization for coupled delayed neural networks with cooperative–competitive interaction was first assessed via an event-triggered control mechanism. In [28], the secure consensus of linear MASs under event-triggered control subject to a scaling deception attack was analyzed. In [29], the consensus control of MASs using the DET mechanism was examined. The consensus of multiple neural networks was investigated by applying the DET control framework in [30]. Nonetheless, a drawback of the findings as mentioned above is that the event generator must run constantly, and it might suffer from Zeno phenomena. Then, an event-triggering mechanism based on periodic sampling was established and implemented in MASs [31,32,33]. This method addresses the concern about Zeno behavior.

In [31,32], the static event-triggered (SET) control techniques were applied to address the consensus issue of MASs. In contrast to SET control, the DDET control may significantly suppress the transmission of data without any severe disruption caused to the controller operation. Focusing on a certain type of stochastic switching system susceptible to semi-Markovian processes, an adaptive event-triggered mechanism based on periodic sampling was explored in [33]. In this paper, the DDET consensus control methods based on periodic sampling focus on dealing with the mean-square consistency of Markovian switching stochastic MASs.

According to the existing analysis, a unique DDET mechanism immune from Zeno behavior is introduced to achieve the consistency of stochastic MASs with Markovian switching. Also, a generalized technique of stochastic analysis is developed. The main contributions of this study are summarized as follows:

• It should be noted that most of the existing consensus control strategies focus on MASs that communicate over a fixed linked graph [4,6,7,9,34]. Therefore, it is essential to address the consensus problem for the MASs with switching topologies. In this study, a continuous-time Markovian process is conducted to represent the system matrices of the stochastic MASs and their communication topologies. Unlike in [13,14,15,16], the DDET strategy as proposed in this study allows the simultaneous switch of system matrices and topologies according to the exact environmental conditions, thus enhancing the adaptivity and flexibility of the system.
• Several DET sampled-data consensus control approaches have been developed for stochastic MASs [33]. In this study, a novel DDET consensus control strategy based on periodic sampling is presented to achieve consensus in stochastic MASs. There are two key challenges to overcome in introducing this DET mechanism. One is to design an appropriate DET scheme to avoid the need for continuously monitoring the triggered condition, and the other is to exclude Zeno behavior. Unlike [19,20,21,27,31,32,35], this approach significantly reduces the communication and computation burden while providing a more flexible and adaptive solution for the consensus control of stochastic MASs.
• The sufficient conditions for achieving consensus in the stochastic MASs with a DET strategy based on periodic sampled-data in a mean-square sense are determined through Lyapunov–Krasovskii functionals and Linear matrix inequalities. The conditions required to ensure the consistent performance of the system under the context of switching system matrices and topologies are proposed, which provides a theoretical basis for the chosen strategy.

The article continues with the following structure. Section 2 provides a brief introduction to graph theory, outlines the system model, and presents the DET protocol. Section 3 establishes consensus in a mean-square sense. Section 4 provides two illustrative simulations. Finally, in Section 5, the work is concluded.

Notations: The collection ${\mathbb{R}}^r$ is made up of r-dimensional real vectors, whereas the collection $\mathbb{Z}$ is made up of non-negative integers. The Kronecker product is described by ⊗. The identity matrix with N dimensions and the diagonal matrix are each represented by $I_N$ and $diag\left\{\cdots \right\}$ individually. Positive definite (positive semi-definite) matrix A is shown by the symmetric matrix $A\succ 0$ $\left(A⪰0\right)$, and ${\lambda }_{max}(A)$ is the maximum eigenvalue of matrix A. $x=col\left(x_1,x_2,\cdots ,x_N\right),x_i\in {\mathbb{R}}^r$, is a column vector of dimension $Nr$, and $\Vert x\Vert ={\left(x^{\top }x\right)}^{1/2}$ is the two-norm of the vector x. ${\mathbf{1}}_N={\left[1,1,\cdots ,1\right]}^{\top }\in {\mathbb{R}}^N$. Probability and expectation, denoted by the symbols $\mathbf{Pr}$ and $\mathbb{E}$, are both described in a complete probability space.

2. Model Description and Preliminaries

This section is divided into three parts. In the first part, a mechanism for describing the switching of system topology is proposed. Next, the second part provides a detailed introduction to the switching stochastic MASs model. Then, the third part elaborates on the design process of the event-triggered consensus controller.

Table 1. Symbols and descriptions.

Symbol	Definition
$\sigma (\tau )$	The Markovian switching signal
$\mathcal{S}$	The state space of the Markovian process $\left\{\sigma (\tau ),\tau \ge 0\right\}$
ℏ	The sampling period
${\tau }_k^i\hslash$	The k-th event-triggered instant for agent i
${\tau }_k^i\hslash +lh$	The nearest sampling time to the current time $\tau$, $l\in \mathbb{Z}$
${\tau }_{k^{\prime }}^j\hslash$	${\tau }_{k^{\prime }}^j\hslash =max\left\{\tau |\tau \in \left\{{\tau }_k^j\hslash ,k=0,1,\dots \right\},\tau \le {\tau }_k^i\hslash +l\hslash \right\}$
${\delta }_i\left({\tau }_k^i\hslash +l\hslash \right)$	${\delta }_i\left({\tau }_k^i\hslash +l\hslash \right)=x_i\left({\tau }_k^i\hslash \right)-x_i\left({\tau }_k^i\hslash +l\hslash \right)$
$e_i\left({\tau }_k^i\hslash +l\hslash \right)$	$e_i\left({\tau }_k^i\hslash +l\hslash \right)={\sum }_{j\in {\mathcal{N}}_i^{\sigma (\tau )}}{a}_{ij}^{\sigma (\tau )}\left(x_i\left({\tau }_k^i\hslash \right)-x_j\left({\tau }_{k^{\prime }}^j\hslash \right)\right)$
$\breve{x}(\tau )$	$\breve{x}(\tau )={\displaystyle \frac{1}{N}}{\sum }_{j=1}^N{x}_j(\tau )$
$z_i(\tau )$	The consensus error of i-th agent $z_i(\tau )=x_i(\tau )-\breve{x}(\tau )$
$z(\tau )$	The consensus error of system (2)
$\iota (\tau )$	The piecewise function $\iota (\tau )=\tau -q\hslash$

summarizes the important symbols and their definitions that appear later in this paper.

A. 

Markovian process and graph theory

The weighted graph $\mathcal{G}=\left\{\mathcal{V},\mathcal{E},\mathcal{A}\right\}$ defines the relationship topology of N agents, where $\mathcal{V}=\left\{1,\dots ,N\right\}$ implies the agents collection and $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ refers to the edges set. If data may be passed from agent j to agent i, $\left(i,j\right)\in \mathcal{E}$ is termed an edge, i, j$\in \mathcal{V}$. If j is a neighbour of i, then ${\mathcal{N}}_i=\left\{j|\left(j,i\right)\in \mathcal{E}\right\}$ represents the set of all neighbours of i. It can be seen from an undirected graph that if $\left(i,j\right)\in \mathcal{E}$, then $\left(j,i\right)\in \mathcal{E}$. Define $\mathcal{A}=\left[a_{ij}\right]\in {\mathbb{R}}^{N\times N}$ as the adjacency matrix of $\mathcal{G}$, which is given as $a_{ij}>0$ if $j\in {\mathcal{N}}_i$, $a_{ij}=0$ otherwise, and $a_{ii}=0$, $i,j\in \mathcal{V}$, and $\mathcal{L}=\left[l_{ij}\right]\in {\mathbb{R}}^{N\times N}$ as the Laplacian matrix of $\mathcal{G}$ with $l_{ij}=-a_{ij}$ for $i\ne j$, and $l_{ii}={\sum }_{j\ne i}{a}_{ij}$, $i,j\in \mathcal{V}$.

In this work, we assume that the connection topology is continuously varying across s distinct graphs ${\mathcal{G}}_1,\dots ,{\mathcal{G}}_s$. At time instant $\tau$, the topology selected is denoted by ${\mathcal{G}}_{\sigma (\tau )}=\left(\mathcal{V},{\mathcal{E}}_{\sigma (\tau )},{\mathcal{A}}_{\sigma (\tau )}\right)$, in which the switching signal $\sigma (\tau )$ is governed by a continuous-time Markovian process. Every component of the finite state space $\mathcal{S}=\left\{1,\dots ,s\right\}$ of the Markovian process $\left\{\sigma (\tau ),\tau \ge 0\right\}$ is associated with one of the potential relationship topologies, i.e., $\sigma (\tau )=m$ if and only if the m-th graph ${\mathcal{G}}_m$ is selected at the instant of time $\tau$. The transition probability is represented as

$$\mathbf{Pr}\left\{\sigma (\tau +\overline{\mathsf{\Delta }})=n|\sigma (\tau )=m\right\}=\left\{\begin{array}{cc}{\pi }_{mn}\overline{\mathsf{\Delta }}+o(\overline{\mathsf{\Delta }}),\hfill & m\ne n,\hfill \\ 1+{\pi }_{mn}\overline{\mathsf{\Delta }}+o(\overline{\mathsf{\Delta }}),\hfill & m=n,\hfill \end{array}\right.$$

(1)

where $\overline{\mathsf{\Delta }}>0$, $o(\overline{\mathsf{\Delta }})$ means that ${lim}_{\overline{\mathsf{\Delta }}\to 0}{\displaystyle \frac{o(\overline{\mathsf{\Delta }})}{\overline{\mathsf{\Delta }}}}=0$, ${\pi }_{mn}\le 0$ is the transition rate from mode m at time $\tau$ to mode n at time $\tau +\overline{\mathsf{\Delta }}$, ${\pi }_{mm}=-{\sum }_{n\ne m}{\pi }_{mn}\ge 0$, $m\ne n$, and $\forall m,n\in \mathcal{S}$.

B. 

Stochastic MASs model

Consider a category of stochastic MASs that consists of N agents with switching topologies ${\mathcal{G}}_{\sigma (\tau )}$. The following is a description of the dynamics of agent i:

$$\begin{array}{c}\mathrm{d}{x}_i(\tau )=\left(A_{\sigma (\tau )}{x}_i(\tau )+B_{\sigma (\tau )}{u}_i(\tau )\right)\mathrm{d}\tau +g(x_i(\tau ))\mathrm{d}\omega (\tau ),i\in \mathcal{V},\hfill \end{array}$$

(2)

in which $x_i(\tau )\in {\mathbb{R}}^r$ stands for the state vector; $A_{\sigma (\tau )}\in {\mathbb{R}}^{r\times r},B_{\sigma (\tau )}\in {\mathbb{R}}^{r\times p}$ individually are the system state matrix and the system control input matrix; $\omega (\tau )$ is the one-dimension Brownian motion, established on a complete probability space $(\mathsf{\Omega },\mathcal{F};\mathbf{Pr})$ with a filtration ${\left\{{\mathcal{F}}_{\tau }\right\}}_{\tau \ge {\tau }_0}$ meets the norm needs; and $g(x_i(\tau )):{\mathbb{R}}^r\to {\mathbb{R}}^r$ is the partial diffusion term related to stochastic perturbations.

The goal of this research is to present a reliable control technique such that the stochastic MASs (2) can reach consensus in a mean-square sense while also reducing the amount of unnecessary transmission. The analysis of stochastic convergence relies heavily on the lemmas below.

Lemma 1 

([36]). Consider a Markovian switching stochastic system

$$\mathrm{d}x=f(\tau ,x,\sigma (\tau ))\mathrm{d}\tau +g(\tau ,x,\sigma (\tau ))\mathrm{d}\omega (\tau ),$$

(3)

where $\omega (\tau )$ is a standard Brownian process. The infinitesimal generator of a positive definite $V\left(\tau ,x(\tau ),\sigma \right)\in {\mathcal{C}}^{1,2}\left({\mathbb{R}}^{+}\times {\mathbb{R}}^r\times \mathcal{S}\right)$ is

$$\mathcal{L}V(x,\tau ,i)={\displaystyle \frac{\mathsf{\partial }V}{\mathsf{\partial }\tau }}+{\displaystyle \frac{\mathsf{\partial }V}{\mathsf{\partial }x}}f(\tau ,x,i)+{\displaystyle \frac{1}{2}}tr\left\{g^{\top }(\tau ,x,i){\displaystyle \frac{{\mathsf{\partial }}^{2}V}{\mathsf{\partial }{x}^2}}g(\tau ,x,i)\right\}+\sum _{j=1}^{\mathcal{S}}{\pi }_{ij}V(\tau ,x,j),$$

(4)

then

$$\mathbb{E}V(\tau ,x,\sigma )=\mathbb{E}V\left({\tau }_0,x\left({\tau }_0\right),\sigma \right)+\mathbb{E}{\int }_{{\tau }_0}^{\tau }\mathcal{L}V(s,x(s),\sigma )\mathrm{d}s,$$

(5)

all $0\le {\tau }_0<\tau <\infty$, provided that $\mathbb{E}{\int }_{{\tau }_0}^{\tau }\mathcal{L}V(s,x(s),\sigma )\mathrm{d}s$ exist.

After that, the following consensus definition will be presented, which is intended to identify the control purpose of this research.

Definition 1 

([37]). Under Markovian switching topologies ${\mathcal{G}}_{\sigma (\tau )}$, the consensus of MASs (2) under control protocol $u_i(\tau )$ can be achieved if

$$\underset{\tau \to \infty }{lim}\mathbb{E}\Vert x_i(\tau )-x_j(\tau ){\Vert }^2=0,i,j\in \mathcal{V},$$

(6)

is valid for any initial distribution $\sigma ({\tau }_0)\in \mathcal{S}$ and any initial conditions.

Designing a DDET control mechanism for system (2) while achieving consensus in the mean-square sense (6) is the desired outcome. Before designing the DET consensus protocol, some lemmas and assumptions should be introduced firstly.

Assumption 1 

([38]). Every graph ${\mathcal{G}}_m$ is undirected and connected, $m\in \mathcal{S}$.

Assumption 2 

([39]). The vector functions $g(x_i)$ satisfies

$$\Vert g(x_i)-g\left(x_j\right)\Vert ⩽\rho \Vert x_i-x_j\Vert ,i,j\in \mathcal{V},$$

(7)

where $\rho >0$ is a known constant.

Remark 1. 

In (2), the vector function $\mathrm{d}\omega$ details the random impacts in the control environment, and the term $g(x_i)$ indicates the impact strength of the stochastic effects, which are intended to be connected to the state $x_i$. The identical ω is understood to mean the fact that the full MASs is used in the same circumstance. The value of the random variable varies with $g(x_i)$ over time for distinct agents i, which may be used to represent the numerous noise-filled Chua’s circuit network. Future considerations might include alternative random processes ${\omega }_i$ for various agents i.

Lemma 2 

(Jensen inequality [40]). For any constant symmetric matrix $U\in {\mathbb{R}}^{n\times n}$, a scalar $\hslash >0$, and a vector function $y(·):[-\hslash ,0]\to$ ${\mathbb{R}}^n$, if the integrals in the following are well defined, then

$$-\hslash {\int }_{\tau -\hslash }^{\tau }{y}^{\top }(s)Uy(s)\mathrm{d}s⩽-{\int }_{\tau -\hslash }^{\tau }{y}^{\top }(s)\mathrm{d}sU{\int }_{\tau -\hslash }^{\tau }y(s)\mathrm{d}s.$$

(8)

Lemma 3 

(The It$\widehat{\mathrm{o}}$ isometry [41]). If $\varphi (\tau ,\omega )$ is bounded and elementary, then

$$\mathbb{E}\left[{\left({\int }_{{\tau }_0}^{\tau }\varphi (\tau ,\omega )\mathrm{d}\omega (\tau )\right)}^2\right]=\mathbb{E}\left[{\int }_{{\tau }_0}^{\tau }\varphi {(\tau ,\omega )}^2\mathrm{d}\tau \right].$$

(9)

Lemma 4 

([32]). The subsequent inequality can be applied for symmetric matrices $H\succ 0$, Y, and any constant β:

$$-Y{H}^{-1}Y⪯{\beta }^{2}H-2\beta Y.$$

(10)

C. 

Dynamic event-triggered consensus protocol

Establishing a DDET control method for the system (2) is the focus of this section. In addition to this, a criterion about the existence and uniqueness of solutions for the event-triggered control system is presented.

In order to avoid the continuous monitoring and Zeno behavior in the stochastic MASs (2), for agent i, we define a sequence of triggering time instants $\left\{{\tau }_k^i\hslash \right\}\subseteq \{0,\hslash ,2\hslash ,\cdots \}$, where $i\in \mathcal{V}$, ${\tau }_k^i\in \mathbb{Z}$, $k=0,1,2,\cdots$, and $\hslash \in (0,1)$ is the sampling period, using some prescribed trigger function. We assume that agent i samples the information of its neighbors periodically and updates the controller at its own triggering time instants. Then, it broadcasts its information to its neighbors. As a result, the event-based controller may be implemented as

$$u_i(\tau )=K_{\sigma (\tau )}\sum _{j\in {\mathcal{N}}_i^{\sigma (\tau )}}{a}_{ij}^{\sigma (\tau )}\left(x_j\left({\tau }_{k^{\prime }}^j\hslash \right)-x_i\left({\tau }_k^i\hslash \right)\right),\tau \in \left[{\tau }_k^i\hslash +l\hslash ,{\tau }_k^i\hslash +l\hslash +\hslash \right),$$

(11)

where $K_{\sigma (\tau )}$ is a control gain matrix to be determined later, ${\tau }_k^i\hslash$ is the k-th event-triggered instant for agent i, ${\tau }_{k^{\prime }}^j\hslash =max\left\{\tau |\tau \in \left\{{\tau }_k^j\hslash ,k=0,1,\dots \right\},\tau \le {\tau }_k^i\hslash +l\hslash \right\}$, and ${\tau }_k^i\hslash +lh$ denotes the nearest sampling time to the current time $\tau$, $l\in \mathbb{Z}$. Figure 1 shows an example of information transfer between agent i and its neighbor agent j. Specifically, at each sampling instant, the i-th agent can receive the latest state information from the triggering time of its neighboring j-th agent. Particularly, the execution times of agent are determined by the following DET mechanism with ${\tau }_0\hslash =0$:

$$\begin{array}{cc}\hfill {\tau }_{k+1}^i\hslash ={\tau }_k^i\hslash +\underset{l\ge 1}{min}\{l\hslash \mid {\delta }_i^{\top }\left({\tau }_k^i\hslash +l\hslash \right)& {\daleth }_{\sigma (\tau )}{\delta }_i\left({\tau }_k^i\hslash +l\hslash \right)\hfill \\ \hfill & >{\alpha }_i{e}_i^{\top }\left({\tau }_k^i\hslash +l\hslash \right){\daleth }_{\sigma (\tau )}{e}_i\left({\tau }_k^i\hslash +l\hslash \right)+{\zeta }_i^{-1}{\psi }_i(\tau )\},\hfill \end{array}$$

(12)

with

$$\begin{array}{cc}\hfill {\dot{\psi }}_i(\tau )=& -{\gamma }_i{\psi }_i(\tau )+{\xi }_i\left[{\alpha }_i{e}_i^{\top }\left({\tau }_k^i\hslash +l\hslash \right){\daleth }_{\sigma (\tau )}{e}_i\left({\tau }_k^i\hslash +l\hslash \right)-{\delta }_i^{\top }\left({\tau }_k^i\hslash +l\hslash \right){\daleth }_{\sigma (\tau )}{\delta }_i\left({\tau }_k^i\hslash +l\hslash \right)\right],\hfill \end{array}$$

(13)

where ${\daleth }_{\sigma (\tau )}$ is the positive definite matrices to be designed later; ${\alpha }_i$, ${\zeta }_i$, ${\gamma }_i$, and ${\xi }_i$ are the given positive parameters with ${\gamma }_i>{\zeta }_i^{-1}$; ${\tau }_k^i\hslash +l\hslash$ denotes the currently sampled instant; ${\delta }_i\left({\tau }_k^i\hslash +l\hslash \right)=x_i\left({\tau }_k^i\hslash \right)-x_i\left({\tau }_k^i\hslash +l\hslash \right)$; and $e_i\left({\tau }_k^i\hslash +l\hslash \right)={\sum }_{j\in {\mathcal{N}}_i^{\sigma (\tau )}}{a}_{ij}^{\sigma (\tau )}\left(x_i\left({\tau }_k^i\hslash \right)-x_j\left({\tau }_{k^{\prime }}^j\hslash \right)\right)$, $i\in \mathcal{V}$. When the consensus protocol is associated with the DET mechanism (12), the diagram of the overall control system (2) is shown in Figure 2.

For the sake of analysis, several assumptions are made as follows: (i) the sampling period for all agents is synchronized by a clock; (ii) the state of each agent is periodically sampled, and the sampling time sequence is $\left\{0,\hslash ,2\hslash ,\dots \right\}$; (iii) the sampled data of agent i are transmitted to its neighbor only when the event-triggered condition is violated, and the event-triggered time sequence of agent i is $\left\{{\tau }_0^i\hslash ,{\tau }_1^i\hslash ,{\tau }_2^i\hslash ,\dots \right\}$, where ${\tau }_0^i\hslash =0$ is the initial time.

Remark 2. 

It follows from above assumptions that the event-triggered time sequence is a subsequence of the sampling time sequence, namely, $\left\{{\tau }_0^i\hslash ,{\tau }_1^i\hslash ,{\tau }_2^i\hslash ,\dots \right\}\subseteq \{0,\hslash ,2\hslash ,\dots \}$, which means that the minimum inter-event time ${min}_k\left\{{\tau }_{k+1}^i\hslash -{\tau }_k^i\hslash \right\},\forall i$, is lower bounded by the sampling period ℏ. Hence, Zeno behavior does not occur.

Remark 3. 

Time-triggered mechanisms, as a consequence, have the potential to result in the needless waste of resources due to the absence of specific restrictions for selecting a suitable triggered interval. In contrast, an event-triggered mechanism that uses the zero-order hold (ZOH) approach minimizes the transmission burden by maintaining a constant control input, which is built according to the state information at the most recent triggered instant, until the next triggered instant. Furthermore, as shown in Equation (13), since ${\psi }_i(\tau )>0$, which has been proven in [28] to be a positive scalar function, it can be interpreted as a dynamic threshold of the DET control scheme (12). The dynamic threshold helps decrease the number of triggers compared to the static case [25,26], suggesting that triggers occur less frequently overall.

Remark 4. 

In contrast to the widely used and developed traditional SET mechanism [25], the DET scheme is adopted by setting dynamically flexible threshold parameters, which allows for a good trade-off between the communication resources and the required control performance. The suggested DET technique (12) also uses a fixed-time sampling method. The sampled signal will be delivered if and only if the DET condition (12) is met.

Setting up an error system whose stochastic stability is realized by the control input (11). Define $\breve{x}={\displaystyle \frac{1}{N}}{\sum }_{j=1}^N{x}_j$, $z_i=x_i-\breve{x}$ and $z=col\left(z_1,\cdots ,z_N\right)$. The compact form of z is $z=\left(\mathcal{M}\otimes I_r\right)x$, where $\mathcal{M}=I_N-{\displaystyle \frac{1}{N}}{\mathbf{1}}_N{\mathbf{1}}_N^{\top }⪰0$, $x=col\left(x_1,x_2,\cdots ,x_N\right)$. It is easy to know that $z=\mathbf{0}$ if and only if $x_1=\cdots =x_N$. Thus, z is called the consensus error vector. Noticeably, $\forall m\in \mathcal{S}$, ${\mathcal{L}}_m\mathcal{M}={\mathcal{L}}_m=\mathcal{M}{\mathcal{L}}_m$ and ${\mathcal{M}}^2=\mathcal{M}$ are guaranteed by Assumption 1.

Notice that $\left[{\tau }_k^i\hslash ,{\tau }_{k+1}^i\hslash \right)$ can be divided into $\left[{\tau }_k^i\hslash ,{\tau }_{k+1}^i\hslash \right)={\bigcup }_{q={\tau }_k^i}^{{\tau }_{k+1}^i-1}\left[q\hslash ,(q+1)\hslash \right)$, $q\in \mathbb{Z}$. For $\tau \in \left[{\tau }_k^i\hslash ,{\tau }_{k+1}^i\hslash \right)\cap \left[q\hslash ,(q+1)\hslash \right)$, define the piecewise function $\iota (\tau )=\tau -q\hslash$, where $0\le \iota (\tau )<\hslash$. It is easily known that $\iota (\tau )$ is piecewise linear with the derivative $\dot{\iota }(\tau )=1$ at $\tau \ne q\hslash$ and ${lim}_{\tau \to q{\hslash }^{+}}\iota (\tau )=0$, ${lim}_{\tau \to q{\hslash }^{-}}\iota (\tau )=\hslash$. Suppose $\sigma (\tau )=m,m\in \mathcal{S}$, then the error system together with (2) and (11) can be written as

$$\mathrm{d}z(\tau )=\left[\left(I_N\otimes A_m\right)z(\tau )-\left({\mathcal{L}}_m\otimes B_m{K}_m\right)\left(\delta \left(\tau -\iota (\tau )\right)+z\left(\tau -\iota (\tau )\right)\right)\right]\mathrm{d}\tau +\mathsf{{\rm Y}}\left(z(\tau )\right)\mathrm{d}\omega (\tau ),$$

(14)

where $\delta \left(\tau -\iota (\tau )\right)=col\left({\delta }_1\left(t-\iota (\tau )\right),{\delta }_2\left(\tau -\iota (\tau )\right),\dots ,{\delta }_N\left(\tau -\iota (\tau )\right)\right)$ and $\mathsf{{\rm Y}}\left(z(\tau )\right)=\left(\mathcal{M}\otimes I_r\right)\mathsf{\Phi }\left(x(\tau )\right)$ with $\mathsf{\Phi }\left(x(\tau )\right)=col\left(g\left(x_1(\tau )\right),g\left(x_2(\tau )\right),\dots ,g\left(x_N(\tau )\right)\right)$. Denoted $g_1(\tau )=\left(I_N\otimes A\right)z(\tau )-\left({\mathcal{L}}_m\otimes B{K}_m\right)\left(z\left(\tau -\iota (\tau )\right)-\delta \left(\tau -\iota (\tau )\right)\right)$.

3. Consensus Stability Analysis

Based on the above analysis, in this section, we will provide sufficient conditions for the realization of consistency for system (2) based on the DETC method (11).

Theorem 1. 

Under Assumptions 1 and 2 and consensus protocol (11), for given scalar $0<\hslash <1$, the consensus of MASs (2) can be achieved in a mean-square sense, and the corresponding periodic sampling-based DET consensus algorithm is shown in Algorithm 1, if there exist positive parameters ${\alpha }_i$, ${\zeta }_i$, ${\gamma }_i$, ${\xi }_i$, and matrix $K_m$ and positive-definite matrices ${\daleth }_m$, $Q_m$, $U_1$, $U_2$, and $U_3$ such that ${\gamma }_i>{\zeta }_i^{-1}$, $i\in \mathcal{V}$ and, for all $m\in \mathcal{S}$, the following matrix inequality holds:

$${\mathcal{Q}}_m=\left[\begin{array}{ccccc}{\mathcal{Q}}_m^{11}& I_{Nr}& {\mathcal{Q}}_m^{13}& \mathit{O}& {\mathcal{Q}}_m^{15}\\ \ast & {\mathcal{Q}}_m^{22}& \mathit{O}& \mathit{O}& \mathit{O}\\ \ast & \ast & {\mathcal{Q}}_m^{33}& {\mathcal{Q}}_m^{34}& {\mathcal{Q}}_m^{35}\\ \ast & \ast & \ast & {\mathcal{Q}}_m^{44}& \mathit{O}\\ \ast & \ast & \ast & \ast & {\mathcal{Q}}_m^{55}\end{array}\right]\prec 0,$$

(15)

where

$$\begin{array}{cc}& {\mathcal{Q}}_m^{11}=I_N\otimes \left(A_m^{\top }{Q}_m+Q_m{A}_m+\sum _{n=1}^s{\pi }_{mn}{Q}_n-{\displaystyle \frac{\hslash }{2}}{U}_3\right),\hfill \\ & {\mathcal{Q}}_m^{13}=-{\mathcal{L}}_m\otimes Q_m{B}_m{K}_m+{\displaystyle \frac{\hslash }{2}}{I}_N\otimes U_3-{\hslash }^2{\mathcal{L}}_m\otimes A_m^{\top }{U}_3{B}_m{K}_m,\hfill \\ & {\mathcal{Q}}_m^{15}=-{\mathcal{L}}_m\otimes Q_m{B}_m{K}_m-{\hslash }^2{\mathcal{L}}_m\otimes A_m^{\top }{U}_3{B}_m{K}_m,\hfill \\ & {\mathcal{Q}}_m^{22}=-{\left[\delta I_{Nr}+I_N\otimes \left(U_1+U_2+{\hslash }^2{A}_m^{\top }{U}_3{A}_m\right)\right]}^{-1},\delta ={\rho }^2\left(\underset{m\in \mathcal{S}}{max}{\lambda }_{max}(Q_m)+{\hslash }^2{\lambda }_{max}(U_3)\right),\hfill \\ & {\mathcal{Q}}_m^{33}=(1+\xi ){\mathcal{L}}_m^{\top }\mathsf{\Xi }\aleph {\mathcal{L}}_m\otimes {\daleth }_m+{\hslash }^2{\mathcal{L}}_m^{\top }{\mathcal{L}}_m\otimes K_m^{\top }{B}_m^{\top }{U}_3{B}_m{K}_m-\hslash I_N\otimes U_3,{\mathcal{Q}}_m^{34}={\displaystyle \frac{\hslash }{2}}{I}_N\otimes U_3,\hfill \\ & {\mathcal{Q}}_m^{35}=(1+\xi ){\mathcal{L}}_m^{\top }\mathsf{\Xi }\aleph {\mathcal{L}}_m\otimes {\daleth }_m+{\hslash }^2{\mathcal{L}}_m^{\top }{\mathcal{L}}_m\otimes K_m^{\top }{B}_m^{\top }{U}_3{B}_m{K}_m,{\mathcal{Q}}_m^{44}=-I_N\otimes \left(U_1+{\displaystyle \frac{\hslash }{2}}{U}_3\right),\hfill \\ & {\mathcal{Q}}_m^{55}=-(1+\xi )\mathsf{\Xi }\otimes {\daleth }_m+(1+\xi ){\mathcal{L}}_m^{\top }\mathsf{\Xi }\aleph {\mathcal{L}}_m\otimes {\daleth }_m+{\hslash }^2{\mathcal{L}}_m^{\top }{\mathcal{L}}_m\otimes K_m^{\top }{B}_m^{\top }{U}_3{B}_m{K}_m,\hfill \\ & \aleph =diag\left\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N\right\},\mathsf{\Xi }=diag\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_N\right\}.\hfill \end{array}$$

Proof of Theorem 1. 

Construct the following Lyapunov–Krasovskii functional candidate as

$$V=V_1+V_2+V_3+V_4,$$

(16)

where

$$\begin{array}{cc}\hfill & V_1(\tau )=z^{\top }(\tau )\left(I_N\otimes Q_{\sigma (\tau )}\right)z(\tau ),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & V_2(\tau )={\int }_{\tau -\hslash }^{\tau }{z}^{\top }(s)\left(I_N\otimes U_1\right)z(s)\mathrm{d}s+{\int }_{\tau -\iota (\tau )}^{\tau }{z}^{\top }(s)\left(I_N\otimes U_2\right)z(s)\mathrm{d}s,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & V_3(\tau )=\hslash {\int }_{-\hslash }^0{\int }_{\tau +\beta }^{\tau }\left[g_1^{\top }(s)\left(I_N\otimes U_3\right)g_1(s)+{\mathsf{{\rm Y}}}^{\top }(z(s))\left(I_N\otimes U_3\right)\mathsf{{\rm Y}}(z(s))\right]\mathrm{d}s\mathrm{d}\beta ,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & V_4(\tau )=\sum _{i=1}^N{\psi }_i(\tau ).\hfill \end{array}$$

(20)

At time $\tau$, there exists $q,k\in \mathbb{Z}$ such that $\tau \in \left[{\tau }_k^i\hslash ,{\tau }_{k+1}^i\hslash \right)\cap \left[q\hslash ,(q+1)\hslash \right)$, $q\in \left\{{\tau }_k^i,{\tau }_k^i+1,\cdots ,{\tau }_{k+1}^i-1\right\}$. Without loss of generality, assume that $\sigma (\tau )=m\in \mathcal{S}$. By Lemma 1, the weak infinitesimal operator $\mathcal{L}$ of $V_1$ (17) along (14) can be written as

$$\begin{array}{cc}\hfill \mathcal{L}{V}_1(\tau )\le & 2z^{\top }(\tau )\left(I_N\otimes Q_m{A}_m\right)z(\tau )-z^{\top }(\tau )\left({\mathcal{L}}_m\otimes Q_m{B}_m{K}_m\right)\left[\delta (\tau -\iota (\tau ))+z(\tau -\iota (\tau ))\right]\hfill \\ \hfill & +\sum _{n=1}^s{\pi }_{mn}{z}^{\top }(\tau )\left(I_N\otimes Q_n\right)z(\tau )+{\mathsf{{\rm Y}}}^{\top }(z(\tau ))\left(I_N\otimes Q_m\right)\mathsf{{\rm Y}}(z(\tau )).\hfill \end{array}$$

(21)

Based on Assumptions 1 and $\mathsf{{\rm Y}}\left(z(\tau )\right)=\left(\mathcal{M}\otimes I_r\right)\mathsf{\Phi }\left(x(\tau )\right)$, one poses

$$\begin{array}{cc}\hfill {\mathsf{{\rm Y}}}^{\top }(z)\left(I_N\otimes Q_m\right)\mathsf{{\rm Y}}(z)=& {\mathsf{\Phi }}^{\top }(x)\left(\mathcal{M}\otimes Q_m\right)\mathsf{\Phi }(x)\hfill \\ \hfill =& {\left(\mathsf{\Phi }(x)-{\mathbf{1}}_N\otimes g(\breve{x})\right)}^{\top }\left(\mathcal{M}\otimes Q_m\right)\left(\mathsf{\Phi }(x)-{\mathbf{1}}_N\otimes g(\breve{x})\right)\hfill \\ \hfill \le & \underset{m\in \mathcal{S}}{max}{\lambda }_{max}(Q_m){\rho }^2{z}^{\top }z.\hfill \end{array}$$

(22)

Then, substituting (22) into (21), it is derived that

$$\begin{array}{cc}\hfill \mathcal{L}{V}_1(\tau )\le & z^{\top }(\tau )\left[I_N\otimes \left(Q_m{A}_m+A_m^{\top }{Q}_m\right)+\sum _{n=1}^s{\pi }_{mn}\left(I_N\otimes Q_n\right)+\underset{m\in \mathcal{S}}{max}{\lambda }_{max}(Q_m){\rho }^2{I}_{Nr}\right]z(\tau )\hfill \\ \hfill & -z^{\top }(\tau )\left({\mathcal{L}}_m\otimes Q_m{B}_m{K}_m\right)\left[\delta (\tau -\iota (\tau ))+z(\tau -\iota (\tau ))\right].\hfill \end{array}$$

(23)

Next, for $V_2\left(t\right)$ (18) and $V_3\left(t\right)$ (19), we have

$$\mathcal{L}{V}_2(\tau )=z^{\top }(\tau )\left[I_N\otimes (U_1+U_2)\right]z(\tau )-z^{\top }(\tau -\hslash )\left(I_N\otimes U_1\right)z(\tau -\hslash ),$$

(24)

and

$$\begin{array}{cc}\hfill \mathcal{L}{V}_3\left(\tau \right)=& {\hslash }^2\left[g_1^{\top }(\tau )\left(I_N\otimes U_3\right)g_1(\tau )+{\mathsf{{\rm Y}}}^{\top }(z(\tau ))\left(I_N\otimes U_3\right)\mathsf{{\rm Y}}(z(\tau ))\right]\hfill \\ \hfill & -\hslash {\int }_{\tau -\hslash }^{\tau }\left[g_1^{\top }(s)\left(I_N\otimes U_3\right)g_1(s)+{\mathsf{{\rm Y}}}^{\top }(z(s))\left(I_N\otimes U_3\right)\mathsf{{\rm Y}}(z(s))\right]\mathrm{d}s.\hfill \end{array}$$

(25)

Similar to (22), we have

$$\begin{array}{c}\hfill {\mathsf{{\rm Y}}}^{\top }(z)\left(I_N\otimes U_3\right)\mathsf{{\rm Y}}(z)\le {\lambda }_{max}(U_3){\rho }^2{z}^{\top }z.\end{array}$$

(26)

By using Lemma 2 and $0<\hslash <1$, the term $-{\int }_{\tau -\hslash }^{\tau }{g}_1^{\top }(s)\left(I_N\otimes U_3\right)g_1(s)\mathrm{d}s$ in the equation $\mathcal{L}{V}_3\left(\tau \right)$ (25) can be scaled as

$$\begin{array}{cc}\hfill -& {\int }_{\tau -\hslash }^{\tau }{g}_1^{\top }(s)\left(I_N\otimes U_3\right)g_1(s)\mathrm{d}s\hfill \\ \hfill & \le -{\displaystyle \frac{1}{\iota (\tau )}}{\int }_{\tau -\iota (\tau )}^{\tau }{g}_1^{\top }(s)\mathrm{d}s\left(I_N\otimes U_3\right){\int }_{\tau -\iota (\tau )}^{\tau }{g}_1(s)\mathrm{d}s-{\displaystyle \frac{1}{\hslash -\iota (\tau )}}{\int }_{\tau -\hslash }^{\tau -\iota (\tau )}{g}_1^{\top }(s)\mathrm{d}s\left(I_N\otimes U_3\right){\int }_{\tau -\hslash }^{\tau -\iota (\tau )}{g}_1(s)\mathrm{d}s\hfill \\ \hfill & \le -{\int }_{\tau -\iota (\tau )}^{\tau }{g}_1^{\top }(s)\mathrm{d}s\left(I_N\otimes U_3\right){\int }_{\tau -\iota (\tau )}^{\tau }{g}_1(s)\mathrm{d}s-{\int }_{\tau -\hslash }^{\tau -\iota (\tau )}{g}_1^{\top }(s)\mathrm{d}s\left(I_N\otimes U_3\right){\int }_{\tau -\hslash }^{\tau -\iota (\tau )}{g}_1(s)\mathrm{d}s.\hfill \end{array}$$

(27)

Moreover, by Lemma 3, we can obtain

$$\begin{array}{c}\hfill \mathbb{E}\left[{\int }_{\tau -\iota (\tau )}^{\tau }{\mathsf{{\rm Y}}}^{\top }(z(s))\left(I_N\otimes U_3\right)\mathsf{{\rm Y}}(z(s))\mathrm{d}s\right]=\mathbb{E}\left[{\int }_{\tau -\iota (\tau )}^{\tau }{\mathsf{{\rm Y}}}^{\top }(z(s))\mathrm{d}\omega \left(I_N\otimes U_3\right){\int }_{\tau -\iota (\tau )}^{\tau }\mathsf{{\rm Y}}(z(s))\mathrm{d}\omega \right],\end{array}$$

(28)

and

$$\begin{array}{c}\hfill \mathbb{E}\left[{\int }_{\tau -\hslash }^{\tau -\iota (\tau )}{\mathsf{{\rm Y}}}^{\top }(z(s))\left(I_N\otimes U_3\right)\mathsf{{\rm Y}}(z(s))\mathrm{d}s\right]=\mathbb{E}\left[{\int }_{\tau -\hslash }^{\tau -\iota (\tau )}{\mathsf{{\rm Y}}}^{\top }(z(s))\mathrm{d}\omega \left(I_N\otimes U_3\right){\int }_{\tau -\hslash }^{\tau -\iota (\tau )}\mathsf{{\rm Y}}(z(s))\mathrm{d}\omega \right].\end{array}$$

(29)

From (25)–(29), it has

$$\begin{array}{cc}\hfill \mathbb{E}\left[\mathcal{L}{V}_3\left(\tau \right)\right]\le & \mathbb{E}[{\hslash }^2{g}_1^{\top }(\tau )\left(I_N\otimes U_3\right)g_1(\tau )+{\hslash }^2{\lambda }_{max}(U_3)z^{\top }(\tau )z(\tau )-\hslash {\int }_{\tau -\iota (\tau )}^{\tau }{g}_1^{\top }(s)\mathrm{d}s\left(I_N\otimes U_3\right){\int }_{\tau -\iota (\tau )}^{\tau }{g}_1(s)\mathrm{d}s\hfill \\ \hfill & -\hslash {\int }_{\tau -\iota (\tau )}^{\tau }{\mathsf{{\rm Y}}}^{\top }(z(s))\mathrm{d}\omega \left(I_N\otimes U_3\right){\int }_{\tau -\iota (\tau )}^{\tau }\mathsf{{\rm Y}}(z(s))\mathrm{d}\omega -\hslash {\int }_{\tau -\hslash }^{\tau -\iota (\tau )}{g}_1^{\top }(s)\mathrm{d}s\left(I_N\otimes U_3\right){\int }_{\tau -\hslash }^{\tau -\iota (\tau )}{g}_1(s)\mathrm{d}s\hfill \\ \hfill & -\hslash {\int }_{\tau -\hslash }^{\tau -\iota (\tau )}{\mathsf{{\rm Y}}}^{\top }(z(s))\mathrm{d}\omega \left(I_N\otimes U_3\right){\int }_{\tau -\hslash }^{\tau -\iota (\tau )}\mathsf{{\rm Y}}(z(s))\mathrm{d}\omega ].\hfill \end{array}$$

(30)

In addition, by using AM-GM inequality $a^2+b^2\ge {\displaystyle \frac{1}{2}}{(a+b)}^2$, $a,b\in \mathbb{R}$, it follows that

$$\begin{array}{cc}\hfill -& {\int }_{\tau -\iota (\tau )}^{\tau }{g}_1^{\top }(s)\mathrm{d}s\left(I_N\otimes U_3\right){\int }_{\tau -\iota (\tau )}^{\tau }{g}_1(s)\mathrm{d}s-{\int }_{\tau -\iota (\tau )}^{\tau }{\mathsf{{\rm Y}}}^{\top }(z(s))\mathrm{d}\omega \left(I_N\otimes U_3\right){\int }_{\tau -\iota (\tau )}^{\tau }\mathsf{{\rm Y}}(z(s))\mathrm{d}\omega \hfill \\ \hfill & \le -{\displaystyle \frac{1}{2}}{\int }_{\tau -\iota (\tau )}^{\tau }\mathrm{d}{z}^{\top }\left(I_N\otimes U_3\right){\int }_{\tau -\iota (\tau )}^{\tau }\mathrm{d}z,\hfill \end{array}$$

(31)

and

$$\begin{array}{cc}\hfill -& {\int }_{\tau -\hslash }^{\tau -\iota (\tau )}{g}_1^{\top }(s)\mathrm{d}s\left(I_N\otimes U_3\right){\int }_{\tau -\hslash }^{\tau -\iota (\tau )}{g}_1(s)\mathrm{d}s-{\int }_{\tau -\hslash }^{\tau -\iota (\tau )}{\mathsf{{\rm Y}}}^{\top }(z(s))\mathrm{d}\omega \left(I_N\otimes U_3\right){\int }_{\tau -\hslash }^{\tau -\iota (\tau )}\mathsf{{\rm Y}}(z(s))\mathrm{d}\omega \hfill \\ \hfill & \le -{\displaystyle \frac{1}{2}}{\int }_{\tau -\hslash }^{\tau -\iota (\tau )}\mathrm{d}{z}^{\top }\left(I_N\otimes U_3\right){\int }_{\tau -\hslash }^{\tau -\iota (\tau )}\mathrm{d}z.\hfill \end{array}$$

(32)

Then, substituting (31) and (32) into (30), it can be obtained that

$$\begin{array}{cc}\hfill \mathbb{E}\left[\mathcal{L}{V}_3\left(\tau \right)\right]\le & {\hslash }^2\mathbb{E}\left[g_1^{\top }(\tau )\left(I_N\otimes U_3\right)g_1(\tau )+{\lambda }_{max}(U_3)z^{\top }(\tau )z(\tau )\right]\hfill \\ \hfill & -{\displaystyle \frac{\hslash }{2}}\mathbb{E}\left[{\int }_{\tau -\iota (\tau )}^{\tau }\mathrm{d}{z}^{\top }\left(I_N\otimes U_3\right){\int }_{\tau -\iota (\tau )}^{\tau }\mathrm{d}z+{\int }_{\tau -\hslash }^{\tau -\iota (\tau )}\mathrm{d}{z}^{\top }\left(I_N\otimes U_3\right){\int }_{\tau -\hslash }^{\tau -\iota (\tau )}\mathrm{d}z\right]\hfill \\ \hfill =& {\hslash }^2\mathbb{E}\left[g_1^{\top }(\tau )\left(I_N\otimes U_3\right)g_1(\tau )+{\lambda }_{max}(U_3)z^{\top }(\tau )z(\tau )\right]\hfill \\ \hfill & -{\displaystyle \frac{\hslash }{2}}\mathbb{E}[{\left(z(\tau )-z\left(\tau -\iota (\tau )\right)\right)}^{\top }\left(I_N\otimes U_3\right)\left(z(\tau )-z\left(\tau -\iota (\tau )\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{\hslash }{2}}\mathbb{E}\left[{\left(z\left(\tau -\iota (\tau )\right)-z(\tau -\hslash )\right)}^{\top }\left(I_N\otimes U_3\right)\left(z\left(\tau -\iota (\tau )\right)-z(\tau -\hslash )\right)\right].\hfill \end{array}$$

(33)

One can immediately obtain by (13) along with $V_4$ (20) that

$$\begin{array}{cc}\hfill \mathcal{L}{V}_4\left(\tau \right)=& -\sum _{i=1}^N{\gamma }_i{\psi }_i(\tau )+\sum _{i=1}^N\left[{\xi }_i{\alpha }_i{e}_i^{\top }\left(\tau -\iota (\tau )\right){\daleth }_m{e}_i\left(\tau -\iota (\tau )\right)-{\delta }_i^{\top }\left(\tau -\iota (\tau )\right){\daleth }_m{\delta }_i\left(\tau -\iota (\tau )\right)\right]\hfill \\ \hfill =& -\sum _{i=1}^N{\gamma }_i{\psi }_i(\tau )+e^{\top }\left(\tau -\iota (\tau )\right)(\mathsf{\Xi }\aleph \otimes {\daleth }_m)e\left(\tau -\iota (\tau )\right)-{\delta }^{\top }\left(\tau -\iota (\tau )\right)\left(\mathsf{\Xi }\otimes {\daleth }_m\right)\delta \left(\tau -\iota (\tau )\right),\hfill \end{array}$$

(34)

where $\mathsf{\Xi }$ and ℵ have previously been defined, and $e(\tau -\iota (\tau ))=col\left(e_1\left(\tau -\iota (\tau )\right),e_2\left(\tau -\iota (\tau )\right),\dots ,e_N\left(\tau -\iota (\tau )\right)\right)$. Moreover, because of $e(\tau -\iota (\tau ))=\left({\mathcal{L}}_m\otimes I_n\right)\left[z(\tau -\iota (\tau ))+\delta (\tau -\iota (\tau ))\right]$, it immediately holds that

$$\begin{array}{cc}\hfill \mathcal{L}{V}_4\left(\tau \right)=& -\gamma \sum _{i=1}^N{\psi }_i(\tau )+{\left[z(\tau -\iota (\tau ))+\delta (\tau -\iota (\tau ))\right]}^{\top }\left({\mathcal{L}}_m^{\top }\mathsf{\Xi }\aleph {\mathcal{L}}_m\otimes {\daleth }_m\right)\left[z(\tau -\iota (\tau ))+\delta (\tau -\iota (\tau ))\right]\hfill \\ \hfill & -\xi {\delta }^{\top }\left(\tau -\iota (\tau )\right)\left(\mathsf{\Xi }\otimes {\daleth }_m\right)\delta (\tau -\iota (\tau )).\hfill \end{array}$$

(35)

From DET mechanism (12), there holds

$$\begin{array}{cc}\hfill & {\delta }^{\top }\left(\tau -\iota (\tau )\right)\left(\mathsf{\Xi }\otimes {\daleth }_m\right)\delta \left(\tau -\iota (\tau )\right)\hfill \\ \hfill & \le {\left[z\left(\tau -\iota (\tau )\right)+\delta \left(\tau -\iota (\tau )\right)\right]}^{\top }\left({\mathcal{L}}_m^{\top }\mathsf{\Xi }\aleph {\mathcal{L}}_m\otimes {\daleth }_m\right)\left[z\left(\tau -\iota (\tau )\right)+\delta \left(\tau -\iota (\tau )\right)\right]+\sum _{i=1}^N{\zeta }_i^{-1}{\psi }_i(\tau ).\hfill \end{array}$$

(36)

Thus, combining (16), (23), (24), (33), (35) and (36), we obtain

$$\begin{array}{c}\hfill \mathbb{E}\left[\mathcal{L}V\left(\tau \right)\right]\le {\chi }^{\top }(\tau ){\mathcal{Q}}_m\chi (\tau )+\sum _{i=1}^N\left({\zeta }_i^{-1}-{\gamma }_i\right){\psi }_i(\tau ),\end{array}$$

(37)

where $\chi (\tau )=col\left(z(\tau ),-{({\mathcal{Q}}_m^{22})}^{-1}z(\tau ),z\left(\tau -\iota (\tau )\right),z(\tau -\hslash ),\delta \left(\tau -\iota (\tau )\right)\right)$ and ${\mathcal{Q}}_m$ is defined in Theorem 1. From ${\zeta }_i^{-1}<{\gamma }_i$, we know

$$\begin{array}{c}\hfill \mathbb{E}\left[\mathcal{L}V\left(\tau \right)\right]\le {\chi }^{\top }(\tau ){\mathcal{Q}}_m\chi (\tau ),\end{array}$$

(38)

then, according to the matrix inequality (15), it can be ensured that

$$\begin{array}{c}\hfill \mathbb{E}\left[\mathcal{L}V\left(\tau \right)\right]<0,\tau \in \left[q\hslash ,(q+1)\hslash \right),\end{array}$$

(39)

which yields

$$\begin{array}{c}\hfill \mathbb{E}\left[\mathcal{L}V\left(\tau \right)\right]<-\epsilon \mathbb{E}\left[{\Vert z(\tau )\Vert }^2\right],\tau \in \left[q\hslash ,(q+1)\hslash \right),\end{array}$$

(40)

for a sufficiently small $\epsilon >0$. To make the notation more concise, let ${\tau }_q=q\hslash$, ${\tau }_{q+1}=(q+1)\hslash$. Applying Lemma 1 and Dynkins formula leads to

$$\begin{array}{c}\hfill \mathbb{E}\left[V\left({\tau }_{q+1}^{-}\right)\right]-\mathbb{E}\left[V\left({\tau }_q\right)\right]\le -\epsilon \mathbb{E}\left[{\int }_{{\tau }_q}^{{\tau }_{q+1}^{-}}{\Vert z(s)\Vert }^2\mathrm{d}s\right].\end{array}$$

(41)

Similarly, we arrive at

$$\begin{array}{cc}\hfill \mathbb{E}\left[V\left({\tau }_q^{-}\right)\right]-\mathbb{E}\left[V\left({\tau }_{q-1}\right)\right]& \le -\epsilon \mathbb{E}\left[{\int }_{{\tau }_{q-1}}^{{\tau }_q^{-}}{\Vert z(s)\Vert }^2\mathrm{d}s\right],\hfill \end{array}$$

(42)

$$\cdots \cdots$$

$$\begin{array}{cc}\hfill \mathbb{E}\left[V\left({\tau }_1^{-}\right)\right]-\mathbb{E}\left[V\left(0\right)\right]& \le -\epsilon \mathbb{E}\left[{\int }_0^{{\tau }_1^{-}}{\Vert z(s)\Vert }^2\mathrm{d}s\right].\hfill \end{array}$$

(43)

Since $V_2(\tau )\ge 0$ and ${\int }_{\tau -\iota (\tau )}^{\tau }{z}^{\top }(s)\left(I_N\otimes U_2\right)z(s)\mathrm{d}s=0$ at the sampling instant $\tau =q\hslash$, it follows from (16) that ${lim}_{\tau \to q{\hslash }^{-}}{V}_2(\tau )\ge V_2{\left(\tau \right)}_{\mid \tau =q\hslash }$, which implies that

$$\begin{array}{c}\hfill \mathbb{E}\left[V\left({\tau }_q^{-}\right)\right]\ge \mathbb{E}\left[V\left({\tau }_q\right)\right].\end{array}$$

(44)

Thus, for the above inequalities (41)–(44), it yields

$$\begin{array}{c}\hfill \sum _{q=0}^{\infty }\mathbb{E}\left[{\int }_{{\tau }_q}^{{\tau }_{q+1}^{-}}{\Vert z(s)\Vert }^2\mathrm{d}s\right]\le {\epsilon }^{-1}\mathbb{E}\left[V\left(0\right)\right].\end{array}$$

(45)

That is,

$$\begin{array}{c}\hfill \underset{T\to \infty }{lim}\mathbb{E}\left[{\int }_0^T{\Vert z(s)\Vert }^2\mathrm{d}s\right]<\infty ,\end{array}$$

(46)

which implies that ${lim}_{\tau \to \infty }\mathbb{E}\left[{\Vert z(\tau )\Vert }^2\right]=0$ by using the Fubini’s theorem. Therefore, under consensus control protocol (11) and Definition 1, the MASs (2) achieves consensus. The proof of Theorem 1 is completed.    □

Algorithm 1: Periodic sampling-based DET consensus control operation

Remark 5. 

During the execution of Algorithm 1, the appropriate scalars ${\alpha }_i$, ${\zeta }_i$, ${\gamma }_i$, ${\xi }_i$ and $K_m$ should be chosen such that the inequality constraint ${\gamma }_i>{\zeta }_i^{-1}$ and (15) hold.

The DET control is thought about in the argument above. The system (2) also can realize the consensus by the following SET control mechanism:

$$\begin{array}{c}\hfill {\tau }_{k+1}^i\hslash ={\tau }_k^i\hslash +\underset{l\ge 1}{min}\left\{l\hslash |{\delta }_i^{\top }\left({\tau }_k^i\hslash +l\hslash \right){\widehat{\daleth }}_{\sigma (\tau )}{\delta }_i\left({\tau }_k^i\hslash +l\hslash \right)>{\alpha }_i{e}_i^{\top }\left({\tau }_k^i\hslash +l\hslash \right){\widehat{\daleth }}_{\sigma (\tau )}{e}_i\left({\tau }_k^i\hslash +l\hslash \right)+b_i{\mathrm{e}}^{-c_i\tau }\right\},\end{array}$$

(47)

where ${\widehat{\daleth }}_{\sigma (\tau )}\succ 0$ are to be designed and $b_i$ and $c_i$, $i\in \mathcal{V}$, are the designed parameters. Next, we obtain the outcome shown below.

Similar to Theorem 1, we establish the consensus results of system (2) by SET mechanism (47) in the mean-square sense in the following corollary.

Corollary 1. 

Under Assumptions 1 and 2 and consensus protocol (47), for given scalar $1>\hslash >0$, the consensus of MASs (2) can be achieved in a mean-square sense if there exist positive parameters $b_i$ and $c_i$, $i\in \mathcal{V}$, and matrix $K_m$, and positive-definite matrices ${\widehat{Q}}_m$, ${\widehat{U}}_1$, ${\widehat{U}}_2$, and ${\widehat{U}}_3$ ${\widehat{\daleth }}_m$ such that for all $m\in \mathcal{S}$, the following matrix inequality holds:

$${\mathcal{Q}}_m=\left[\begin{array}{ccccc}{\widehat{\mathcal{Q}}}_m^{11}& I_{Nr}& {\widehat{\mathcal{Q}}}_m^{13}& \mathit{O}& {\widehat{\mathcal{Q}}}_m^{15}\\ \ast & {\widehat{\mathcal{Q}}}_m^{22}& \mathit{O}& \mathit{O}& \mathit{O}\\ \ast & \ast & {\widehat{\mathcal{Q}}}_m^{33}& {\widehat{\mathcal{Q}}}_m^{34}& {\widehat{\mathcal{Q}}}_m^{35}\\ \ast & \ast & \ast & {\widehat{\mathcal{Q}}}_m^{44}& \mathit{O}\\ \ast & \ast & \ast & \ast & {\widehat{\mathcal{Q}}}_m^{55}\end{array}\right]\prec 0,$$

(48)

where

$$\begin{array}{cc}& {\widehat{\mathcal{Q}}}_m^{11}=I_N\otimes \left(A_m^{\top }{\widehat{Q}}_m+{\widehat{Q}}_m{A}_m+\sum _{n=1}^s{\pi }_{mn}{\widehat{Q}}_n-{\displaystyle \frac{\hslash }{2}}{\widehat{U}}_3\right),\hfill \\ & {\widehat{\mathcal{Q}}}_m^{13}=-{\mathcal{L}}_m\otimes {\widehat{Q}}_m{B}_m{K}_m+{\displaystyle \frac{\hslash }{2}}{I}_N\otimes {\widehat{U}}_3-{\hslash }^2{\mathcal{L}}_m\otimes A_m^{\top }{\widehat{U}}_3{B}_m{K}_m,\hfill \\ & {\widehat{\mathcal{Q}}}_m^{15}=-{\mathcal{L}}_m\otimes {\widehat{Q}}_m{B}_m{K}_m-{\hslash }^2{\mathcal{L}}_m\otimes A_m^{\top }{\widehat{U}}_3{B}_m{K}_m,\hfill \\ & {\widehat{\mathcal{Q}}}_m^{22}=-{\left[\delta I_{Nr}+I_N\otimes \left({\widehat{U}}_1+{\widehat{U}}_2+{\hslash }^2{A}_m^{\top }{\widehat{U}}_3{A}_m\right)\right]}^{-1},\hfill \\ & {\widehat{\mathcal{Q}}}_m^{33}={\mathcal{L}}_m^{\top }\aleph {\mathcal{L}}_m\otimes {\widehat{\daleth }}_m+{\hslash }^2{\mathcal{L}}_m^{\top }{\mathcal{L}}_m\otimes K_m^{\top }{B}_m^{\top }{\widehat{U}}_3{B}_m{K}_m-\hslash I_N\otimes {\widehat{U}}_3,{\widehat{\mathcal{Q}}}_m^{34}={\displaystyle \frac{\hslash }{2}}{I}_N\otimes {\widehat{U}}_3,\hfill \\ & {\widehat{\mathcal{Q}}}_m^{35}={\mathcal{L}}_m^{\top }\aleph {\mathcal{L}}_m\otimes {\widehat{\daleth }}_m+{\hslash }^2{\mathcal{L}}_m^{\top }{\mathcal{L}}_m\otimes K_m^{\top }{B}_m^{\top }{\widehat{U}}_3{B}_m{K}_m,{\widehat{\mathcal{Q}}}_m^{44}=-I_N\otimes \left({\widehat{U}}_1+{\displaystyle \frac{\hslash }{2}}{\widehat{U}}_3\right),\hfill \\ & {\widehat{\mathcal{Q}}}_m^{55}=-I_N\otimes {\widehat{\daleth }}_m+{\mathcal{L}}_m^{\top }\aleph {\mathcal{L}}_m\otimes {\widehat{\daleth }}_m+{\hslash }^2{\mathcal{L}}_m^{\top }{\mathcal{L}}_m\otimes K_m^{\top }{B}_m^{\top }{\widehat{U}}_3{B}_m{K}_m.\hfill \end{array}$$

An efficient method for designing the consensus controller gain matrix is shown below, and it is founded on the findings of Theorem 1.

Theorem 2. 

For given scalars $1>\hslash >0$ and $\mu >0$, consider that Assumptions 1 and 2 hold and there exist positive parameters ${\alpha }_i$, ${\zeta }_i$, ${\gamma }_i$, ${\xi }_i$, and matrix ${\tilde{K}}_m$ and positive-definite matrices ${\tilde{\daleth }}_m$, ${\tilde{Q}}_m$, ${\tilde{U}}_m^1$, ${\tilde{U}}_m^2$, and ${\tilde{U}}_m^3$ such that ${\gamma }_i>{\zeta }_i^{-1}$, $i\in \mathcal{V}$, and, for all $m\in \mathcal{S}$, the following matrix inequality holds:

$$\left[\begin{array}{cccc}{\tilde{\mathsf{\Xi }}}_m^1& {\tilde{\mathsf{\Xi }}}_m^2& {\tilde{\mathsf{\Xi }}}_m^3& {\tilde{\mathsf{\Xi }}}_m^4\\ \ast & {\mu }^2{\tilde{U}}_m^3-2\mu {\tilde{Q}}_m& \mathit{O}& \mathit{O}\\ \ast & \ast & -I_{Nr}& \mathit{O}\\ \ast & \ast & \ast & {\tilde{\mathsf{\Xi }}}_m^5\end{array}\right]\prec 0,$$

(49)

where

$${\tilde{\mathsf{\Xi }}}_m^1=\left[\begin{array}{cccc}{\tilde{\mathsf{\Xi }}}_m^{11}& {\tilde{\mathsf{\Xi }}}_m^{12}& \mathit{O}& {\tilde{\mathsf{\Xi }}}_m^{14}\\ \ast & {\tilde{\mathsf{\Xi }}}_m^{22}& {\tilde{\mathsf{\Xi }}}_m^{23}& {\tilde{\mathsf{\Xi }}}_m^{24}\\ \ast & \ast & {\tilde{\mathsf{\Xi }}}_m^{33}& \mathit{O}\\ \ast & \ast & \ast & {\tilde{\mathsf{\Xi }}}_m^{44}\end{array}\right],$$

(50)

$$\begin{array}{cc}& {\tilde{\mathsf{\Xi }}}_m^{11}=I_N\otimes \left({\tilde{Q}}_m{A}_m^{\top }+A_m{\tilde{Q}}_m+{\pi }_{mm}{\tilde{Q}}_m+{\tilde{U}}_m^1+{\tilde{U}}_m^2-{\displaystyle \frac{\hslash }{2}}{\tilde{U}}_m^3\right),\hfill \\ & {\tilde{\mathsf{\Xi }}}_m^{12}=-{\mathcal{L}}_m\otimes B_m{\tilde{K}}_m+{\displaystyle \frac{\hslash }{2}}{I}_N\otimes {\tilde{U}}_m^3-{\hslash }^2{\mathcal{L}}_m\otimes {\tilde{Q}}_m{A}_m^{\top }{\tilde{Q}}_m^{-1}{\tilde{U}}_m^3{\tilde{Q}}_m^{-1}{B}_m{\tilde{K}}_m,\hfill \\ & {\tilde{\mathsf{\Xi }}}_m^{14}=-{\mathcal{L}}_m\otimes B_m{\tilde{K}}_m-h^2{\mathcal{L}}_m\otimes {\tilde{Q}}_m{A}_m^{\top }{\tilde{Q}}_m^{-1}{\tilde{U}}_m^3{\tilde{Q}}_m^{-1}{B}_m{\tilde{K}}_m,\hfill \\ & {\tilde{\mathsf{\Xi }}}_m^{22}=(1+\xi ){\mathcal{L}}_m^{\top }\mathsf{\Xi }\aleph {\mathcal{L}}_m\otimes {\tilde{\daleth }}_m-\hslash I_N\otimes {\tilde{U}}_m^3,\hfill \\ & {\tilde{\mathsf{\Xi }}}_m^{23}={\displaystyle \frac{\hslash }{2}}{I}_N\otimes {\tilde{U}}_m^3,{\tilde{\mathsf{\Xi }}}_m^{24}=(1+\xi ){\mathcal{L}}_m^{\top }\mathsf{\Xi }\aleph {\mathcal{L}}_m\otimes {\tilde{\daleth }}_m+{\hslash }^2{\mathcal{L}}_m^{\top }{\mathcal{L}}_m\otimes {\tilde{K}}_m^{\top }{B}_m^{\top }{\tilde{Q}}_m^{-1}{\tilde{U}}_m^3{\tilde{Q}}_m^{-1}{B}_m{\tilde{K}}_m,\hfill \\ & {\tilde{\mathsf{\Xi }}}_m^{33}=-I_N\otimes \left({\tilde{U}}_m^1+{\displaystyle \frac{\hslash }{2}}{\tilde{U}}_m^3\right),{\tilde{\mathsf{\Xi }}}_m^{44}=-(1+\xi )\mathsf{\Xi }\otimes {\tilde{\daleth }}_m+(1+\xi ){\mathcal{L}}_m^{\top }\mathsf{\Xi }\aleph {\mathcal{L}}_m\otimes {\tilde{\daleth }}_m,\hfill \end{array}$$

$$\begin{array}{cc}& {\tilde{\mathsf{\Xi }}}_m^2=\left[\hslash {\tilde{Q}}_m{A}_m^{\top },\hslash {\mathcal{L}}_m^{\top }\otimes {\tilde{K}}_m^{\top }{B}_m^{\top },\mathit{O},\hslash {\mathcal{L}}_m^{\top }\otimes {\tilde{K}}_m^{\top }{B}_m^{\top }\right],{\tilde{\mathsf{\Xi }}}_m^3=\sqrt{\delta }\left(I_N\otimes {\tilde{Q}}_m\right),\hfill \\ & {\tilde{\mathsf{\Xi }}}_m^4=\left[\sqrt{{\pi }_{m1}}\left(I_N\otimes {\tilde{Q}}_m\right),\dots ,\sqrt{{\pi }_{mm-1}}\left(I_N\otimes {\tilde{Q}}_m\right),\sqrt{{\pi }_{mm+1}}\left(I_N\otimes {\tilde{Q}}_m\right),\dots ,\sqrt{{\pi }_{ms}}\left(I_N\otimes {\tilde{Q}}_m\right)\right],\hfill \\ & {\tilde{\mathsf{\Xi }}}_m^5=diag\left\{I_N\otimes {\tilde{Q}}_1,I_N\otimes {\tilde{Q}}_2,\dots ,I_N\otimes {\tilde{Q}}_{m-1},I_N\otimes {\tilde{Q}}_{m+1},\dots ,I_N\otimes {\tilde{Q}}_s\right\}.\hfill \end{array}$$

Then, under consensus protocol (11), the MASs (2) achieve consensus in the mean-square sense. The controller gain matrices $K_m$ and the event-triggered parameter matrices ${\daleth }_m$ are presented in the following format: $K_m={\tilde{K}}_m{\tilde{P}}_m^{-1},{\daleth }_m={\tilde{P}}_m^{-1}{\tilde{\daleth }}_m{\tilde{P}}_m^{-1},m\in \mathcal{S}$.

Remark 6. 

The DET requirement, together with the Markovian switching systems and topologies, are both taken into consideration by Theorem 2, which offers a sufficient condition to assure consensus for stochastic MASs (2). In addition to this, both the consensus gain matrices and the event-triggered parameter matrices are gained.

4. Simulation Examples

In this section, two illustrative examples are provided to show the validity of the developed theoretical results and the created DDET control. Consider six nondriven and undamped harmonic oscillators with random noise [42,43,44]:

$$\left\{\begin{array}{c}\mathrm{d}{x}_{i1}(\tau )=x_{i2}(\tau )\mathrm{d}\tau +x_{i1}(\tau )\mathrm{d}\omega (\tau ),\hfill \\ \mathrm{d}{x}_{i2}(\tau )=\left(-{\beta }_{\sigma (\tau )}{x}_{i1}(\tau )+u_i(\tau )\right)\mathrm{d}\tau +x_{i2}\mathrm{d}\omega (\tau ),i=1,2,\dots ,6\hfill \end{array}\right.$$

(51)

where $x_{i1}(\tau )\in \mathbb{R}$ is the position of oscillator i, $x_{i2}(\tau )\in$ $\mathbb{R}$ is its velocity, and ${\beta }_{\sigma (\tau )}>0$ is the frequency of the oscillaors. The Markovian process $\sigma$ jumps among modes $\left\{1,2,3\right\}$ and ${\beta }_1=0.35$, ${\beta }_2=0.65$, ${\beta }_3=0.75$. Three undirected graphs are displayed in Figure 3, and the highlighting network architecture ${\mathcal{G}}_{\sigma (\tau )}$ alternates between them at random.

Remark 7. 

The coordinated control of coupled harmonic oscillators has aroused considerable interest because of its successful application in many multi-agent networks, such as for cooperative patrolling, mapping, sampling or surveillance. Subsequently, fruitful results regarding the understanding of the coordinated control of coupled harmonic oscillators have emerged in the last decade. For example, [42] investigated the problem of the bipartite consensus control of coupled harmonic oscillators in directed network topology, in which positive and negative interaction weights coexist, and a distributed bipartite consensus control protocol is proposed. Ref. [43] was concerned with resilient control design for the synchronization of multiple harmonic oscillators coupled via a vulnerable network suffering from denial-of-service (DoS) attacks. Ref. [44] solved the average cluster synchronization control problem of networked harmonic oscillators under DoS attacks. Given the broad applications of harmonic oscillators, we verify that the proposed DET consensus control based on periodic sampling is highly effective and practical.

Example 1. 

We set $\hslash =0.01$ and employ the LMI toolbox to address the inequality (15). The simulation is performed with a sampling interval of 0.001 s. The outcomes are presented in Figure 4, Figure 5 and Figure 6. As shown in Figure 4a, the state trajectories and the error trajectories depicted in Figure 4b indicate that the system (51) achieves mean-square consensus using the distributed DET control (12) within roughly 5 s, providing strong evidence of the effectiveness of our control strategy. Additionally, Figure 5a,b and Figure 6a illustrate the control inputs $u_i$, the mode of Markovian switching, and the triggering instances for each agent under the DET control law (12). These figures demonstrate that the time intervals between successive triggering instances increase while the frequency of the triggerings decreases, underscoring the efficacy of our DET approach.

Example 2. 

The system (51) successfully reaches consensus when employing the distributed SET control methodology (47) within approximately 8 s. Figure 7a,b, along with Figure 6b, depict the state trajectories, error trajectories, and triggering instances for each agent under the SET control framework (47). The parameters chosen for this analysis include $b_1=0.001$; $b_2=0.003$; $b_3=0.002$; $b_4=0.0009$; $b_5=-0.006$; $b_5=-0.007$; $c_1=300$; $c_2=350$; $c_3=332$; $c_4=320$; $c_5=300$; and $c_5=300$.

From Figure 4 and Figure 7, it can be observed that under the DET control (12), the stochastic MASs achieves consensus faster compared to the SET control (47). Additionally, In

Table 2. Triggering numbers for Examples 1 and 2.

Agent	1	2	3	4	5	6
the DET control law (12)	36	24	18	40	26	44
the SET control law (47)	160	202	179	167	215	162

, the triggering number of each agent is separately listed under the DET control law (12) and the SET control law (47). It can be seen that the dynamic excitation times are far less than the static excitation times. Thus, it can be concluded that the DET control approach employed in this study requires less time and fewer control updates to ensure system performance compared to the SET control approach in [31,32], resulting in a significant reduction in energy costs.

5. Conclusions

In this paper, a novel DET consensus control strategy is developed for stochastic MASs with Markovian switching system matrices and topologies. The proposed strategy combines the advantages of DET control and stochastic switched systems and topologies to achieve consensus. Additionally, a continuous-time Markovian process is conducted to model the switching of the MASs and the communication topology among the agents, enhancing the adaptivity and flexibility of the system. Simulation results demonstrate that the proposed strategy outperforms traditional time-triggered control strategies in terms of the communication and computation requirements, providing a flexible and adaptive solution for consensus control in MASs. Overall, the proposed strategy is validated as a promising approach for achieving consensus control in MASs.

In future studies, it is important to focus on addressing more complex and larger-scale MASs by extending the proposed strategy. This could involve investigating advanced control techniques and optimization algorithms to further enhance the performance and scalability of the proposed approach. Additionally, the application of the proposed strategy to real-world scenarios and practical implementations is another avenue for future research. By validating the strategy in real-world settings, its effectiveness and applicability can be further assessed and refined. Overall, the development and refinement of consensus control strategies for MASs are essential for advancing the capabilities and applications of MASs in various domains.