Consensus of heterogeneous second-order nonlinear multi-agent systems via pinning sampling control

INTRODUCTION

Over the past few decades, with the widespread application of dynamical systems, increasing attention has been devoted to investigating their dynamic behaviors. Several theoretical results have been reported, including consensus tracking control for multi-agent systems (MASs) with linear nodal dynamics ^[1], robust non-fragile $$ H_\infty $$ filtering for fuzzy fractional order systems ^[2], Hopf bifurcation in multiple-delayed predator-prey systems ^[3], and clustering of genetic regulatory networks ^[4]. Among these studies, consensus control has attracted significant interest ^[5,6] and is the central focus of this work.

MASs represent a prevalent class of dynamical systems with broad applications in robot cooperation ^[7], sensor networks ^[8], military applications ^[9], and other fields. Every agent in a MAS can be modeled as a node in a network, and consensus can be achieved with efficiently and flexibly through the exchange of local state information. In early studies on consensus in MASs, most researchers focused on position information as an influencing factor ^[10,11,12]. In subsequent research, to more accurately characterize agent dynamics, velocity information was incorporated alongside position information. This led to the development of second-order MASs ^[13,14,15]. Lu et al.^[13] studied predefined-time consensus of second-order MASs subject to external disturbances using a predefined-time observer. Zhao et al.^[14] proposed an event-triggered control strategy to investigate consensus in second-order discrete-time MASs influenced by the eigenvalues of the Laplacian matrix. Meng et al.^[15] examined consensus in a double-integrator MAS under an undirected connected communication topology. The inclusion of velocity information has promoted the study of MAS models. In most existing studies, MAS dynamics are assumed to be homogeneous. However, such an assumption is often idealized, as agents in practical systems rarely exhibit identical dynamics. Compared with homogeneous systems, heterogeneous systems are more general and have higher theoretical significance. For heterogeneous MASs, simple linear feedback controllers cannot achieve complete consensus due to the inherent differences among agent dynamics. Achieving complete consensus would therefore require complex and costly control strategies. Accordingly, this paper focuses on quasi-consensus for second-order heterogeneous MASs, where the consensus errors are guaranteed to converge to a bounded range.

External controllers are essential for accelerating and facilitating consensus in heterogeneous systems. Common approaches include adaptive control ^[16], intermittent control ^[17], sampling control ^[18,19], impulsive control ^[20], and pinning control ^[21,22], which are used to address consensus problems in heterogeneous MASs. Specifically, adaptive control provides a strategy for handling time-varying parameters by continuously adjusting them through an adaptive law, thereby effectively regulating the system. In ^[16], optimal adaptive consensus tracking control for nonlinear MASs with prescribed performance is discussed, where the parameters in the adaptive protocol are adjusted based on the received output information. In practical digital computer systems, continuous signals cannot be directly utilized; instead, signal data must undergo processes such as discretization, sampling, compression, and decompression before they can be processed. Sampling control methods are widely used due to their ease of design and effective implementation in computer-controlled digital systems. For MAS, selecting an appropriate sampling control scheme is not only practical for data communication and controller execution, but also effective in significantly reducing system resource consumption. In ^[18], consensus of MASs is investigated using dynamic output feedback based on sampled data. Unlike impulsive control, which affects nodes only at discrete instants, sampling control can exert influence over continuous intervals based on sampled information. In ^[20], a consensus control strategy is proposed for a novel heuristic nonlinear MAS. Compared with previous studies, this research explores a control protocol that integrates both saturation effects and an impulsive control mechanism, demonstrating high efficiency, low cost, and broad applicability. In practice, to reduce control costs, it is common to control only a subset of agents rather than the entire system; this approach is referred to as pinning control. Hao et al.^[21] investigated group consensus of MASs subject to external disturbances by constructing two pinning control algorithms. In ^[22], to address consensus in second-order nonlinear MASs under attacks, actuator failures, and integral quadratic constraints, an event-triggered pinning scheme based on local information was designed. To further reduce costs, sampling control is often combined with pinning control, thereby reducing both control effort and the number of controlled nodes.

In addition to the above control methods, there is another type of controller, namely event-triggered control, which has received significant attention in recent years. Such a controller determines the triggering instants based on predefined event conditions and can achieve consensus while reducing resource consumption and cost by effectively decreasing the number of sampling instances. For example, Yang et al.^[23] adopted event-triggered control in their research on nonlinear MASs under limited communication resources. Li et al.^[24] also presented an event-triggered control strategy for time-varying MASs to address parameter uncertainties within a given finite horizon and deception attacks. More recently, to better reflect practical scenarios, dynamic event-triggered control strategies have been proposed ^[25,26]. Furthermore, studies ^[22] and ^[27] have combined pinning control and sampling control with event-triggered control to reach consensus in MASs. In ^[22], the authors mainly focus on cluster consensus in the presence of acyclic denial-of-service (DoS) attacks, actuator failures, and integral quadratic constraints. In ^[27], the synchronization problem of first-order MASs with time-varying control gains is investigated under a synchronous periodic sampling framework. However, it does not consider factors such as network attacks or actuator faults; instead, it mainly addresses the synchronization control of nonlinear agents based on sampled data in directed topologies. In addition, in ^[28], the authors propose both an observer-based event-triggered controller and an observer-based full asynchronous event-triggered controller to study the bipartite consensus problem of MASs under unknown false data injection attack. Event-triggered control has been widely applied in the field of control systems. It can be used not only in consensus studies of MASs but also in investigating annular finite-time $$ H_\infty $$ performance in complex networks^[29].

In this work, by leveraging the advantages of event-triggered control, sampling control, and pinning control, two novel hybrid controllers are designed to achieve consensus in heterogeneous second-order nonlinear MASs. The effectiveness of the proposed methods is demonstrated through numerical examples. Furthermore, the main contributions of this paper are summarized below.

(ⅰ) At present, there is limited literature on consensus in heterogeneous second-order MASs. In this paper, by studying such systems, sufficient criteria for achieving consensus are established.

(ⅱ) Although hybrid controllers have made great progress in the past decade, there remains room for improvement in their application to heterogeneous second-order MASs. By integrating event-triggered control, sampling control, and pinning control, the control duration, the number of sampling instances, and the number of controlled nodes are reduced, thereby saving resources.

(ⅲ) The sampling control adopted in this paper effectively avoids Zeno behavior. It is important to note that since triggering instants occur only at sampling instants, the lower bound of the inter-event times is equal to the sampling period.

PRELIMINARIES

In this section, the notation and models for a heterogeneous second-order MAS comprising one leader and N followers are presented.

MAIN RESULTS

This section designs pinning sampling controllers to achieve consensus in heterogeneous second-order nonlinear MASs (1) and (2). The investigation is divided into two subsections: consensus under sampling control and consensus under hybrid event-triggered sampling control. Furthermore, a detailed derivation of the consensus process for the MASs is given below.

Leader-follower consensus under sampling control

Define the consensus errors as $$ \hat{x}_i(t)=x_i(t)-x_{N+1}(t) $$ and $$ \hat{v}_i(t)=v_i(t)-v_{N+1}(t) $$, for $$ i\in \mathcal{N} $$. Then, the error dynamics are given by

(5) $$ \begin{equation} \begin{aligned} \left\{ \begin{array}{l} \dot{\hat{x}}_i(t)=\hat{v}_i(t), \\ \dot{\hat{v}}_i(t)=h_i(t, x_i(t), v_i(t))-h_{N+1}(t, x_{N+1}(t), v_{N+1}(t))-u_i(t), \\ \end{array} \right. \end{aligned} \end{equation} $$

for $$ i\in \mathcal{N} $$. The sampling period is h, and the agent states are available at sampling instants $$ \mathcal{T}=\{T_l\}_{l \in \mathbb{Z}_+} $$, where $$ T_l=T_0+lh $$ with $$ T_0=0 $$. Then, the sampling control law $$ u_i^1(t) $$ is designed as

(6) $$ \begin{equation} \begin{aligned} u_i^1(t)=&\alpha(t)[\Sigma_{j=1}^Na_{ij}(x_j(lh)-x_i(lh))-\varrho_i(x_i(lh)-x_{N+1}(lh))]\\ &+\beta(t)[\Sigma_{j=1}^Na_{ij}(v_j(lh)-v_i(lh))-\varrho_i(v_i(lh)-v_{N+1}(lh))], \end{aligned} \end{equation} $$

where $$ i\in \mathcal{N} $$. The functions $$ \alpha(t)\geq0 $$ and $$ \beta(t)\geq0 $$ denote bounded control gains, and $$ \varrho_i>0 $$ represents the pinning control gain. The flowchart for sampling control $$ u_i^1(t) $$ is shown in Figure 1.

Figure 1. Flowchart of sampling control $$ u_i^1(t) $$.

Given that $$ x(t)=(x_1^T(t), x_2^T(t), \cdots, x_N^T(t))^T $$, $$ v(t)=(v_1^T(t), v_2^T(t), \cdots, v_N^T(t))^T $$, $$ \hat{x}(t)=(\hat{x}_1^T(t), \hat{x}_2^T(t), \cdots, \hat{x}_N^T(t))^T $$, $$ \hat{v}(t)=(\hat{v}_1^T(t), \hat{v}_2^T(t), \cdots, \hat{v}_N^T(t))^T $$, $$ H(t, x(t), v(t))=(h_1^T(t, x_1, v_1), h_2^T(t, x_2, v_2), \cdots, h_N^T(t, x_N, v_N))^T $$, $$ \hat{H}(t, x_{N+1}(t), v_{N+1}(t))=((h_1-h_{N+1})^T(t, x_{N+1}, v_{N+1}), (h_2-h_{N+1})^T(t, x_{N+1}, v_{N+1}), \cdots, (h_N-h_{N+1})^T(t, x_{N+1}, v_{N+1}))^T $$, $$ {\Upsilon=diag\{\varrho_1, \varrho_2, \cdots, \varrho_N\}} $$, system (5) can be rewritten as

(7) $$ \begin{equation} \begin{aligned} \left\{ \begin{array}{l} \dot{\hat{x}}(t)=\hat{v}(t), \\ \dot{\hat{v}}(t)=H(t, x(t), v(t))-H(t, X_{N+1}(t), V_{N+1}(t))+\hat{H}(t, X_{N+1}(t), V_{N+1}(t))\\ \; \; \; \; \; \; \; \; -\alpha(t)((L+\Upsilon)\otimes I_n)\cdot\hat{x}(lh)-\beta(t)((L+\Upsilon)\otimes I_n)\cdot\hat{v}(lh), \\ \end{array} \right. \end{aligned} \end{equation} $$

where $$ X_{N+1}=1_N\otimes x_{N+1} $$, $$ V_{N+1}=1_N\otimes v_{N+1} $$.

Assumption 1. The function H in (7) satisfies the global Lipschitz condition, that is,

$$ \begin{equation} \nonumber \|H(t, x, v)-H(t, y, z)\|\leq\rho_1\|x-y\|+\rho_2\|v-z\|, \forall x, y, v, z\in \mathbb{R}^n, \end{equation} $$

where $$ \rho_1\geq0 $$ and $$ \rho_2\geq0 $$.

Assumption 2. The leader state is bounded, which implies that the function $$ \hat{H} $$ in (7) is also bounded under any initial condition. Specifically, there is a constant $$ \chi>0 $$ such that $$ \sup_{-h\leq s\leq 0}\|\hat{H}(t+s, x_{N+1}(t+s), v_{N+1}(t+s))\|=\chi $$.

Remark 1. The global Lipschitz condition in Assumption 1 is a standard assumption for deriving explicit stability criteria. In practice, for locally Lipschitz nonlinear systems, this condition can be interpreted as requiring a sufficiently large Lipschitz constant valid over the bounded operational region ensured by the closed-loop system. For systems exhibiting unbounded nonlinear growth, techniques such as input saturation may be introduced to guarantee boundedness.

Remark 2. The leader's state may correspond to a stable equilibrium, periodic orbit, or chaotic attractor. Therefore, it is reasonable to assume that the leader state is bounded. Since $$ \hat{H} $$ is a continuous function of the leader state, it is also bounded.

Remark 3. As for pinning control, the selection of pinned nodes is crucial. If the communication topology is disconnected, at least one node in each connected component should be pinned. Otherwise, any node can be selected as a pinned node. The pinning strength $$ \varrho_i $$ can then be assigned accordingly.

Theorem 1. Under Assumptions 1-2 and controller (6), the MASs (1) and (2) can achieve quasi-consensus exponentially if there exist a constant $$ \theta>0 $$ and a positive definite matrix $$ J>0 $$ satisfying

(8) $$\zeta_1(t)+\zeta_2(t)\leq-\theta<0,$$

and

(9) $$C-J>0,$$

where

$$ {\zeta_1(t)}=\frac{1}{\|\Xi\|}\max\{1+\|J\|\cdot(3\rho_1+\rho_2+1)+(\alpha(t)+\beta(t)\rho_1^2)h\cdot\|J(L+\Upsilon)\|+2\beta(t) h\cdot(1+\rho_1)\cdot\|J(L+\Upsilon)\|+(\alpha(t)+\beta(t))\beta(t) h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|-\alpha(t)\cdot\lambda_{\min}[J(L+\Upsilon)+(L+\Upsilon)J], \|J\|\cdot(3\rho_2+\rho_1+1)+(\alpha(t)+\beta(t)\rho_2)h\cdot\|J(L+\Upsilon)\|+2\beta(t) h\cdot(1+\rho_2)\cdot\|J(L+\Upsilon)\|+(\alpha(t)+\beta(t))\beta(t) h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|+2\lambda_{\max}(J)-\beta(t)\cdot\lambda_{\min}[J(L+\Upsilon)+(L+\Upsilon)J]\}\leq0 $$, $$ {\zeta_2(t)}=\frac{1}{\|\Xi\|}\max\{2\alpha(t)\beta(t) h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|, (\alpha(t)+2\beta(t)^2\cdot\|L+\Upsilon\|)\cdot\|J(L+\Upsilon)\|\cdot h\}\geq0 $$, $$ C=\min{\{\underline{\beta}, \underline{\alpha}\}}\cdot [J(L+\Upsilon)+(L+\Upsilon)J] $$, $$ \underline{\alpha}=\min\{\alpha(t)\}>0 $$, and $$ \underline{\beta}=\min\{\beta(t)\}>0 $$.

Proof. For $$ t\in[lh, (l+1)h) $$, define $$ V_1(t)=\hat{z}^T(t)(\Xi\otimes I_n)\hat{z}(t) $$ with $$ \hat{z}(t)=(\hat{x}^T(t), \hat{v}^T(t))^T $$ and $$ \Xi=\begin{pmatrix} C & J\\J & J\end{pmatrix} $$. We calculate $$ \dot{V_1}(t) $$ and substitute the trajectories of (7) into $$ \dot{V_1}(t) $$,

(10) $$ \begin{equation} \begin{aligned} \dot{V}_1(t)=&2\hat{x}^T(t)(C\otimes I_n)\dot{\hat{x}}(t)+2\hat{x}^T(t)(J\otimes I_n)\dot{\hat{v}}(t)+2\hat{v}^T(t)(J\otimes I_n)\dot{\hat{x}}(t)+2\hat{v}^T(t)(J\otimes I_n)\dot{\hat{v}}(t)\\ =&2\hat{x}^T(t)(C\otimes I_n)\hat{v}(t)+2\hat{v}^T(t)(J\otimes I_n)\hat{v}(t)+2\hat{x}^T(t)(J\otimes I_n)[H(t, x(t), v(t))\\ &+\hat{H}(t, x_{N+1}(t), v_{N+1}(t))-\alpha(t)((L+\Upsilon)\otimes I_n)\hat{x}(lh)-\beta(t)((L+\Upsilon)\otimes I_n)\hat{v}(lh)]\\ &+2\hat{v}^T(t)(J\otimes I_n)-H(t, X_{N+1}(t), V_{N+1}(t))(H(t, x(t), v(t))-H(t, X_{N+1}(t), V_{N+1}(t))\\ &+\hat{H}(t, x_{N+1}(t), v_{N+1}(t))-\alpha(t)((L+\Upsilon)\otimes I_n)\hat{x}(lh)-\beta(t)((L+\Upsilon)\otimes I_n)\hat{v}(lh)). \end{aligned} \end{equation} $$

Then, we obtain

(11) $$ \begin{equation} \begin{aligned} &2\hat{x}^T(t)(J\otimes I_n)(H(t, x(t), v(t))-H(t, X_{N+1}(t), V_{N+1}(t)))\\ \leq&2\|\hat{x}(t)\|\cdot\|J\|\cdot\|H(t, x(t), v(t))-H(t, X_{N+1}(t), V_{N+1}(t))\|\\ \leq&\|J\|\cdot[(2\rho_1+\rho_2)\cdot\|\hat{x}(t)\|^2+\rho_2\cdot\|\hat{v}(t)\|^2], \end{aligned} \end{equation} $$

and

(12) $$2\hat{x}^T(t)(J\otimes I_n)\hat{H}(t, x_{N+1}(t), v_{N+1}(t))\leq\|J\|\cdot(\|\hat{x}(t)\|^2+\chi^2).$$

Furthermore, we have

(13) $$ \begin{equation} \begin{aligned} &2\hat{x}^T(t)(J\otimes I_n)[-\alpha(t)((L+\Upsilon)\otimes I_n)\hat{x}(lh)]\\ =&2\alpha(t)\hat{x}^T(t)(J(L+\Upsilon)\otimes I_n)\int_{t-\tau(t)}^t\dot{\hat{x}}(s)ds-2\alpha(t)\hat{x}^T(t)(J(L+\Upsilon)\otimes I_n)\hat{x}(t)\\ \leq&\alpha(t)h\cdot\|J(L+\Upsilon)\|\cdot(\|\hat{x}(t)\|^2+(\sup\limits_{-h\leq s\leq 0}\|\hat{v}(t+s)\|)^2)-\alpha(t)\cdot \lambda_{\min}[J(L+\Upsilon)\\ &+(L+\Upsilon)J]\cdot\|\hat{x}(t)\|^2, \end{aligned} \end{equation} $$

and

(14) $$ \begin{equation} \begin{aligned} &2\hat{x}^T(t)(J\otimes I_n)[-\beta(t)((L+\Upsilon)\otimes I_n)\hat{v}(lh)]\\ =&2\beta(t)\hat{x}^T(t)(J(L+\Upsilon)\otimes I_n)\int_{t-\tau(t)}^t\dot{\hat{v}}(s)ds-2\beta(t)\hat{x}^T(t)(J(L+\Upsilon)\otimes I_n)\hat{v}(t)\\ \leq&2\beta(t)h\rho_1\cdot\|J(L+\Upsilon)\|\cdot\|\hat{x}(t)\|^2+\beta(t)h\cdot\|J(L+\Upsilon)\|\cdot(2\|\hat{x}(t)\|^2+\rho_2^2\cdot\|\hat{v}(t)\|^2+\chi^2)\\ &+\beta(t)\alpha(t)h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|\cdot(\|\hat{x}(t)\|^2+(\sup\limits_{-h\leq s\leq0}\|\hat{x}(t+s)\|))^2+\beta^2(t)h\\ &\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|(\|\hat{x}(t)\|^2+(\sup\limits_{-h\leq s\leq0}\|\hat{v}(t+s)\|)^2)-2\beta(t)\hat{x}^T(t)(J(L+\Upsilon)\otimes I_n)\hat{v}(t). \end{aligned} \end{equation} $$

Similarly, we obtain the following deduction,

(15) $$ \begin{equation} \begin{aligned} &2\hat{v}^T(t)(J\otimes I_n)(H(t, x(t), v(t))-H(t, X_{N+1}(t), V_{N+1}(t)))\\ \leq&\|J\|\cdot[(2\rho_2+\rho_1)\cdot\|\hat{v}(t)\|^2+\rho_1\cdot\|\hat{x}(t)\|^2], \end{aligned} \end{equation} $$

(16) $$2 \hat{v}^T(t)\left(J \otimes I_n\right) \hat{H}\left(t, x_{N+1}(t), v_{N+1}(t)\right) \leq \quad\|J\| \cdot\left(\|\hat{v}(t)\|^2+\chi^2\right),$$

(17) $$ \begin{equation} \begin{aligned} &2\hat{v}^T(t)(J\otimes I_n)[-\alpha(t)((L+\Upsilon)\otimes I_n)\hat{x}(lh)]\\ \leq&\alpha(t)h\cdot\|J(L+\Upsilon)\|\cdot(\|\hat{v}(t)\|^2+(\sup\limits_{-h\leq s\leq 0}\|\hat{v}(t+s)\|)^2)-2\alpha(t)\hat{v}^T(t)(J(L+\Upsilon)\otimes I_n)\hat{x}(t), \end{aligned} \end{equation} $$

and

(18) $$ \begin{equation} \begin{aligned} &2\hat{v}^T(t)(J\otimes I_n)[-\beta(t)((L+\Upsilon)\otimes I_n)\hat{v}(lh)]\\ \leq&2\beta(t)h\rho_2\cdot\|J(L+\Upsilon)\|\cdot\|\hat{v}(t)\|^2+\beta(t)h\cdot\|J(L+\Upsilon)\|\cdot(2\|\hat{v}(t)\|^2+\rho_1^2\cdot\|\hat{v}(t)\|^2+\chi^2)\\ &+\beta(t)\alpha(t)h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|\cdot(\|\hat{v}(t)\|^2+(\sup\limits_{-h\leq s\leq0}\|\hat{x}(t+s)\|))^2+\beta^2(t)h\\ &\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|(\|\hat{v}(t)\|^2+(\sup\limits_{-h\leq s\leq0}\|\hat{v}(t+s)\|))^2-\beta(t)\lambda_{\min}[J(L+\Upsilon)\\ &+(L+\Upsilon)J]\|\hat{v}(t)\|^2. \end{aligned} \end{equation} $$

Applying (11)-(18) yields

(19) $$\dot{V}_1(t)\leq\zeta_1(t)V_1(t)+\zeta_2(t)\sup\limits_{-h\leq s\leq0}V_1(t+s)+\zeta_3,$$

where $$ \zeta_3=\frac{(2\|J\|+10h\cdot\|J(L+\Upsilon)\|)\chi^2}{\|\Xi\|} $$.

By Theorem 2.3 in ^[30], we obtain $$ V_1(t)\leq\frac{\zeta_3}{\theta}+V_1(0) $$. Consequently,

(20) $$\|\hat{z}(t)\|^2 \leq \frac{V_1(t)}{\lambda_{\min }(\Xi)} \leq \frac{\frac{\zeta_3}{\theta}+V_1(0)}{\lambda_{\min }(\Xi)},$$

Therefore, the heterogeneous second-order MASs (1) and (2) can realize quasi-consensus.

Remark 4. Notably, the matrix blocks C and J in matrix $$ \Upsilon $$ are $$ N \times N $$ square matrices. By the Schur complement lemma ^[31], it follows that $$ \Xi>0 $$ is equivalent to $$ C>J>0 $$. Therefore, to ensure $$ \Xi>0 $$, the condition $$ \Delta=\min{\{\underline{\beta}, \underline{\alpha}\}}\cdot [J(L+\Upsilon)+(L+\Upsilon)J]-J>0 $$ should hold.

Remark 5. The coupling strengths $$ \alpha(t) $$ and $$ \beta(t) $$ used in this paper are time-varying, meaning that they can change over time. Compared to constant coupling strengths, time-varying coupling strengths are more suitable for describing dynamic interactions among agents. In practice, the coupling strength between agents is often influenced by factors such as relative distance and energy, both of which vary over time.

Lead-following consensus under event-triggered control

In this subsection, a hybrid control strategy that combines the controller in (6) with an event-triggered mechanism is represented as follows

(21) $$ \begin{equation} \begin{aligned} u_i^2(t)=&\alpha(t)[\Sigma_{j=1}^Na_{ij}(\tilde{x}_j(t)-\tilde{x}_i(t))-\varrho_i(\tilde{x}_i(t)-x_{N+1}(lh))]\\ &+\beta(t)[\Sigma_{j=1}^Na_{ij} (\tilde{v}_j(t)-\tilde{v}_i(t))-\varrho_i(\tilde{v}_i(t)-v_{N+1}(lh))], \end{aligned} \end{equation} $$

where $$ i\in \mathcal{N} $$ and $$ t\in[lh, (l+1)h) $$. The sampled states are defined as $$ \tilde{x}_i(t)=x_i(t_{k_i}^ih) $$ and $$ \tilde{v}_i(t)=v_i(t_{k_i}^ih) $$ for $$ t\in[t_{k_i}^ih, t_{k_{i}+1}^ih) $$. The functions $$ \alpha(t)\geq0 $$ and $$ \beta(t)\geq0 $$ represent bounded time-varying control gains, and $$ \varrho_i $$ is a positive constant representing the pinning control gain. The sequence $$ \{t_{k_{i}}h\}_{k_{i} \in \mathbb{Z}_+}\subseteq \mathcal{T} $$ represents the event-triggered instants, which occur only at selected sampling moments. The flowchart of the event-triggered control $$ u_i^2(t) $$ is shown in Figure 2.

Figure 2. Flowchart of the event-triggered control $$ u_i^2(t) $$.

Define $$ \delta_i^x(t)=x_i(t)-x_i(t_{{k_i}}^ih) $$ and $$ \delta_i^v(t)=v_i(t)-v_i(t_{{k_i}}^ih) $$ as the measurement errors for $$ t\in [t_{{k_i}}^ih, t_{{k_{i}+1}}^ih) $$. Let $$ \delta^x(t)=({\delta_1^x}^T(t), {\delta_2^x}^T(t), \cdots, {\delta_N^x}^T(t))^T $$ and $$ \delta^v(t)=({\delta_1^v}^T(t), {\delta_2^v}^T(t), \cdots, {\delta_N^v}^T(t))^T $$. Then, we get

(22) $$ \begin{equation} \begin{aligned} \left\{ \begin{array}{l} \dot{\hat{x}}(t)=\hat{v}(t), \\ \dot{\hat{v}}(t)=H(t, x(t), v(t))-H(t, X_{N+1}(t), V_{N+1}(t))+\hat{H}(t, X_{N+1}(t), V_{N+1}(t))\\ \; \; \; \; \; \; \; \; -\alpha(t)((L+\Upsilon)\otimes I_n)\cdot\hat{x}(lh)-\beta(t)((L+\Upsilon)\otimes I_n)\cdot\hat{v}(lh)\\ \; \; \; \; \; \; \; \; +\alpha(t)((L+\Upsilon)\otimes I_n)\cdot\delta^x(lh)+\beta(t)((L+\Upsilon)\otimes I_n)\cdot\delta^v(lh), \\ \end{array} \right. \end{aligned} \end{equation} $$

where $$ t\in [lh, (l+1)h) $$, $$ X_{N+1}=1_N\otimes x_{N+1} $$, and $$ V_{N+1}=1_N\otimes v_{N+1} $$.

Furthermore, the triggering instant $$ t_{k_i}^ih $$ is determined as follows

(23) $$ \begin{equation} \begin{aligned} t_{k_{i+1}}^ih=&t_{k_i}^ih+\inf\Big{\{}mh\big{|}\bar{\alpha}^2\|\delta_i^x(t_{k_i}^ih+mh)\|^2+\bar{\beta}^2\|\delta_i^v(t_{k_i}^ih+mh)\|^2\geq\sigma_i(\|\hat{x}_i(t_{k_i}^ih+mh)\|^2\\ &+\|\hat{v}_i(t_{k_i}^ih+mh)\|^2)\Big{\}}, \end{aligned} \end{equation} $$

where $$ \bar{\alpha}=\max\{\alpha(t)\}>0 $$, $$ \bar{\beta}=\max\{\beta(t)\}>0 $$, and $$ \bar{\sigma}=\max\{\sigma_i\} $$. Notably, the triggering instants occur only at discrete sampling instants. Therefore, the minimum interval between two consecutive triggering instants is the sampling period h, which naturally excludes Zeno behavior.

Theorem 2. Base on Assumptions 1-2 and controller (21), the MASs (1) and (2) can realize quasi-consensus exponentially if there is a constant $$ \theta>0 $$ and a matrix $$ J>0 $$ satisfying

(24) $$\tilde{\zeta}_1(t)+\zeta_2(t)\leq-\theta<0,$$

and

(25) $$C-J>0,$$

where

$$ \tilde{\zeta}_1(t)=\frac{1}{\|\Xi\|}\max\{1+\|J\|\cdot(3\rho_1+\rho_2+1)+(\alpha(t)+\beta(t)\rho_1^2)h\cdot\|J(L+\Upsilon)\|+2\beta(t)h\cdot(1+\rho_1)\cdot \|J(L+\Upsilon)\|+4\|JL\|+(\alpha(t)+\beta(t))\beta(t)h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|+2\bar{\sigma}\beta(t)h\cdot\|J(L+\Upsilon)\|\cdot\|L\| -\alpha(t)\cdot\lambda_{\min}[J(L+\Upsilon)+(L+\Upsilon)J, \|J\|\cdot(3\rho_2+\rho_1+1)+(\alpha(t)+\beta(t)\rho_2)h\cdot\|J(L+\Upsilon)\|+2\beta(t)h\cdot(1+\rho_2)\cdot\|J(L+\Upsilon)\|+4\|JL\|+(\alpha(t) +\beta(t))\beta(t)h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|+2\lambda_{\max}(J)+2\bar{\sigma}\beta(t)h\cdot\|J(L+\Upsilon)\|\cdot\|L\| -\beta(t)\cdot\lambda_{\min}[J(L+\Upsilon)+(L+\Upsilon)J]\}\leq0 $$ and $$ \zeta_2(t)=\frac{1}{\|\Xi\|}\max\{2\alpha(t)\beta(t)h\cdot \|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|, (\alpha(t)+2\beta^2(t)\cdot\|L+\Upsilon\|)\cdot\|J(L+\Upsilon)\|\cdot h\}\geq0 $$, $$ C=\min{\{\underline{\beta}, \underline{\alpha}\}}\cdot [J(L+\Upsilon)+(L+\Upsilon)J] $$, $$ \underline{\alpha}=\min\{\alpha(t)\}>0 $$, and $$ \underline{\beta}=\min\{\beta(t)\}>0 $$.

Proof. For $$ t\in[lh, (l+1)h) $$, define $$ V_2(t)=\hat{z}^T(t)(\Xi\otimes I_n)\hat{z}(t) $$ with $$ \Xi=\begin{pmatrix} C & J\\J & J\end{pmatrix} $$ and $$ \hat{z}(t)=(\hat{x}^T(t), \hat{v}^T(t))^T $$. Then

(26) $$ \begin{equation} \begin{aligned} V_2(t)=&\begin{pmatrix} \hat{x}^T(t) & \hat{v}^T(t)\end{pmatrix}\Bigg{[}\begin{pmatrix} C & J\\J & J\end{pmatrix}\otimes I_n\Bigg{]}\begin{pmatrix} \hat{x}(t)\\ \hat{v}(t)\end{pmatrix}\\ =&\hat{x}^T(t)(C\otimes I_n)\hat{x}(t)+2\hat{x}^T(t)(J\otimes I_n)\hat{v}(t)+\hat{v}^T(t)(J\otimes I_n)\hat{v}(t). \end{aligned} \end{equation} $$

Taking the derivative of $$ \dot{V_2}(t) $$ and substituting the trajectories from (22) into $$ \dot{V_2}(t) $$, we obtain

(27) $$ \begin{equation} \begin{aligned} \dot{V}_2(t)=&2\hat{x}^T(t)(C\otimes I_n)\dot{\hat{x}}(t)+2\hat{x}^T(t)(J\otimes I_n)\dot{\hat{v}}(t)+2\hat{v}^T(t)(J\otimes I_n)\dot{\hat{x}}(t)+2\hat{v}^T(t)(J\otimes I_n)\dot{\hat{v}}(t)\\ =&2\hat{x}^T(t)(C\otimes I_n)\hat{v}(t)+2\hat{v}^T(t)(J\otimes I_n)\hat{v}(t)+2\hat{x}^T(t)(J\otimes I_n)[H(t, x(t), v(t))\\ &-H(t, X_{N+1}(t), V_{N+1}(t))+\hat{H}(t, x_{N+1}(t), v_{N+1}(t))-\alpha(t)((L+\Upsilon)\otimes I_n)\hat{x}(lh)\\ &-\beta(t)((L+\Upsilon)\otimes I_n)\hat{v}(lh)+\alpha(t)((L+\Upsilon)\otimes I_n)\delta^x(lh)+\beta(t)((L+\Upsilon)\otimes I_n)\cdot\delta^v(lh)]\\ &+2\hat{v}^T(t)(J\otimes I_n)[H(t, x(t), v(t))-H(t, X_{N+1}(t), V_{N+1}(t))+\hat{H}(t, x_{N+1}(t), v_{N+1}(t))\\ &-\alpha(t)((L+\Upsilon)\otimes I_n)\hat{x}(lh)-\beta(t)((L+\Upsilon)\otimes I_n)\hat{v}(lh)+\alpha(t)((L+\Upsilon)\otimes I_n)\\ &\delta^x(lh)+\beta(t)((L+\Upsilon)\otimes I_n)\cdot\delta^v(lh)]. \end{aligned} \end{equation} $$

Then, we have

(28) $$ \begin{equation} \begin{aligned} &2\hat{x}^T(t)(J\otimes I_n)[\alpha(t)((L+\Upsilon)\otimes I_n)\delta^x(lh)+\beta(t)(L\otimes I_n)\delta^v(lh)]\\ &+2\hat{v}^T(t)(J\otimes I_n)[\alpha(t)((L+\Upsilon)\otimes I_n)\delta^x(lh)+\beta(t)(L\otimes I_n)\delta^v(lh)]\\ \leq&2\cdot\|J(L+\Upsilon)\|\cdot\|\hat{x}(t)+\hat{v}(t)\|\cdot\|\alpha(t)\delta^x(lh)+\beta(t)\delta^v(lh)\|\\ \leq&2\|J(L+\Upsilon)\|(\|\hat{x}(t)\|^2+\|\hat{v}(t)\|^2+\bar{\alpha}^2\cdot\|\delta^x(lh)\|^2+\bar{\beta}^2\cdot\|\delta^v(lh)\|^2). \end{aligned} \end{equation} $$

In addition, we have

(29) $$ \begin{equation} \begin{aligned} &2\hat{x}^T(t)(J\otimes I_n)[-\beta(t)((L+\Upsilon)\otimes I_n)\hat{v}(lh)]\\ =&2\beta(t)\hat{x}^T(t)(J(L+\Upsilon)\otimes I_n)\int_{t-\tau(t)}^t\dot{\hat{v}}(s)ds-2\beta\hat{x}^T(t)(J(L+\Upsilon)\otimes I_n)\hat{v}(t)\\ \leq&2\beta(t)h\rho_1\cdot\|J(L+\Upsilon)\|\cdot\|\hat{x}(t)\|^2+\beta(t)h\cdot\|J(L+\Upsilon)\|\cdot[2(1+\|L\|)\cdot\|\hat{x}(t)\|^2\\ &+\rho_2^2\cdot\|\hat{v}(t)\|^2+\chi^2]+\beta(t)\alpha(t)h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|\cdot(\|\hat{x}(t)\|^2+(\sup\limits_{-h\leq s\leq0}\|\hat{x}(t+s)\|)^2)\\ &+\beta^2(t)h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|(\|\hat{x}(t)\|^2+(\sup\limits_{-h\leq s\leq0}\|\hat{v}(t+s)\|)^2)-2\beta(t)\hat{x}^T(t)\\ &(J(L+\Upsilon)\otimes I_n)\hat{v}(t)+\beta(t)h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|(\bar{\alpha}^2\cdot\|\delta^x(lh)\|^2+\bar{\beta}^2\cdot\|\delta^v(lh)\|^2), \end{aligned} \end{equation} $$

and

(30) $$ \begin{equation} \begin{aligned} &2\hat{v}^T(t)(J\otimes I_n)[-\beta(t)((L+\Upsilon)\otimes I_n)\hat{v}(lh)]\\ =&2\beta(t)\hat{v}^T(t)(J(L+\Upsilon)\otimes I_n)\int_{t-\tau(t)}^t\dot{\hat{v}}(s)ds-2\beta\hat{v}^T(t)(J(L+\Upsilon)\otimes I_n)\hat{v}(t)\\ \leq&2\beta(t)h\rho_2\cdot\|J(L+\Upsilon)\|\cdot\|\hat{v}(t)\|^2+\beta(t)h\cdot\|J(L+\Upsilon)\|\cdot[2(1+\|L\|)\cdot\|\hat{v}(t)\|^2\\ &+\rho_1^2\cdot\|\hat{v}(t)\|^2+\chi^2]+\beta(t)\alpha(t)h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|\cdot(\|\hat{v}(t)\|^2\\ &+(\sup\limits_{-h\leq s\leq0}\|\hat{x}(t+s)\|))^2+\beta^2(t)h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|(\|\hat{v}(t)\|^2+(\sup\limits_{-h\leq s\leq0}\|\hat{v}(t+s)\|))^2\\ &+\beta(t)h\cdot\|J(L+\Upsilon)\|\cdot\|L+\Upsilon\|\cdot(\bar{\alpha}^2\cdot\|\delta^x(lh)\|^2+\bar{\beta}^2\cdot\|\delta^v(lh)\|^2)\\ &-\beta(t)\cdot\lambda_{\min}[J(L+\Upsilon)+(L+\Upsilon)J]\cdot\|\hat{v}(t)\|^2. \end{aligned} \end{equation} $$

Applying (11)-(13), (15)-(17), and (28)-(30), and combining these with the event-triggered strategy (23), we obtain

(31 $$\dot{V}_2(t) \leq \tilde{\zeta}_1(t) V_2(t)+\zeta_2(t) \sup\limits_{-h \leq s \leq 0} V_2(t+s)+\zeta_3,$$

where $$ {\zeta_3(t)}=\frac{(2\|J\|+10h\cdot\|J(L+\Upsilon)\|)\chi^2}{\|\Xi\|} $$.

The remainder of the proof is similar to that of Theorem 1. Therefore, the heterogeneous second-order MASs (1) and (2) can realize quasi-consensus.

Remark 6. Since heterogeneous systems are considered in this paper, quasi-consensus is analyzed in both Theorem 1 and Theorem 2. According to (20), the upper bound of the error $$ \|\hat{z}(t)\| $$ is related to the constant $$ \chi $$ and the initial conditions. When the system is homogeneous (i.e., $$ \chi = 0 $$), the upper bound of the error becomes smaller and the system achieves improved consensus performance.

Remark 7. The threshold parameters $$ \sigma_i>0 $$ in (23) play a critical role in balancing consensus performance against communication and computational resource usage. Their selection is guided by the following trade-off consideration. A larger $$ \sigma_i $$ makes the triggering condition more difficult to satisfy, leading to fewer control updates but potentially a larger ultimate bound for the quasi-consensus error. Conversely, a smaller $$ \sigma_i $$ yields more frequent updates and closer tracking performance, approaching that of purely time-driven sampling control, at the cost of increased resource consumption. In this study, the parameters $$ \sigma_i $$ are chosen as fixed values. A principled selection method can be derived from the stability analysis. Specifically, examining the term $$ 2\bar{\sigma}\beta(t)h\|J(L+\Gamma)\|\|L\| $$ in the coefficient $$ \tilde{\zeta}_1(t) $$ in Theorem 2 reveals that $$ \sigma_i $$ directly influences the negativity required for $$ \tilde{\zeta}_1(t)+\zeta_2(t)\leq-\theta<0 $$. Therefore, to ensure stability, $$ \sigma_i $$ cannot be arbitrarily large and must satisfy constraints imposed by other system parameters. The values used in the simulations ($$ \sigma_i=\{10, 20, 10, 20, 10\} $$) are chosen heuristically to satisfy these conditions while demonstrating the effectiveness of the event-triggered mechanism. In practice, these parameters can be tuned offline to achieve a desired trade-off or adapted online within the stability bounds.

SIMULATIONS

This section provides two examples to explain the above results. Specifically, Theorem 1 and Theorem 2 are illustrated in Example 1 and Example 2, respectively. In addition, both connected and disconnected graphs are discussed in the two examples. Consider systems (1) and (2) with $$ N=6 $$ and $$ n=2 $$, where the sixth node is the leader and the remaining five nodes are followers. The initial conditions are given as $$ x_1(0)=(-5, 2)^T $$, $$ x_2(0)=(4, 3)^T $$, $$ x_3(0)=(2, 1)^T $$, $$ x_4(0)=(0, -7)^T $$, $$ x_5(0)=(-3, 3)^T $$, $$ x_6(0)=(5, 0)^T $$, and $$ v_1(0)=(-5, 2)^T $$, $$ v_2(0)=(4, 3)^T $$, $$ v_3(0)=(4, 3)^T $$, $$ v_4(0)=(0, -7)^T $$, $$ v_5(0)=(2, 1)^T $$, $$ v_6(0)=(-2, 3)^T $$. The nonlinear functions are defined as $$ h_1(t)=\frac{1}{10}\sin(x_1(t)) $$, $$ h_2(t)=\frac{1}{20}[\cos(x_2(t))+\sin(v_2(t))] $$, $$ h_3(t)=\frac{1}{10}\cos(v_3(t)) $$, $$ h_4(t)=\frac{1}{10}\sin(v_4(t)) $$, $$ h_5(t)=\frac{1}{10}\cos(x_5(t)) $$, and $$ h_6(t)=\frac{1}{10}\tanh(x_6(t)) $$. The sampling period is $$ h=10^{-5} $$, $$ \alpha(t)=\left\{\begin{aligned} 75, t\in[2kh, h+2kh), \\ 70, t\in[h+2kh, 4kh), \end{aligned} \right. $$ and $$ \beta(t)=\left\{\begin{aligned} 70, t\in[2kh, h+2kh), \\ 75, t\in[h+2kh, 4kh), \end{aligned} \right. $$ where $$ k\in \mathbb{Z} $$. Furthermore, the parameters are set as $$ \sigma_1=10 $$, $$ \sigma_2=20 $$, $$ \sigma_3=10 $$, $$ \sigma_4=20 $$, and $$ \sigma_5=10 $$ in (23). The topologies of followers are considered under both connected and disconnected cases. The corresponding Laplacian matrices are given by

$$ L_1 $$ and $$ L_2 $$ correspond to Figure 3 and Figure 4, respectively.

Figure 3. Second-order heterogeneous MASs over a connected graph.

Figure 4. Second-order heterogeneous MASs over a disconnected graph.

Moreover, the pinning strength matrices $$ \Upsilon_1 $$ and $$ \Upsilon_2 $$ are selected defined as follows, corresponding to Figure 3 and Figure 4, respectively. According to Remark 3, node 2 is selected as the pinning node with pinning gain $$ \varrho_2=1 $$ for Figure 3. Nodes 1 and 5 are selected as pinning nodes with gains $$ \varrho_1=\varrho_5=2 $$ for Figure 4. Therefore, the $$ \Upsilon_1 $$ and $$ \Upsilon_1 $$ matrices are expressed as $$ \Upsilon_1 = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 & 0\\ 0 & 1 &0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 \end{array}} \right] $$ and $$ \Upsilon_2 = \left[ {\begin{array}{*{20}{c}} 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 \end{array}} \right] $$. It is calculated that $$ \max{\{\zeta_1(t)\}}=-0.2448 $$, $$ \max{\{\zeta_2(t)\}}=0.1908 $$, and $$ \max{\{\tilde{\zeta}_1(t)\}}=-0.2001 $$ when $$ J=\frac{1}{10}\mathsf{I}_5 $$. Finally, the condition $$ C-J>0 $$ must be verified. For the connected and disconnected graphs corresponding to Figures 3 and 4, C is given as $$ \begin{bmatrix} 28 & -14&0 &-14 &0 \\ -14 & 42 &0 &0 &-14 \\ 0 & 0 & 14 & 0 &-14 \\ -14 & 0 & 0 &14 &0 \\ 0 & -14 & -14 &0 &28 \end{bmatrix} $$ and $$ \begin{bmatrix} 42 & 0 & 0 & -14 & 0\\ 0 & 14 & -14 & 0 & 0\\ 0 & -14 & 28 & 0 & -14\\ -14 & 0 & 0 & 14 & 0\\ 0 & 0 & -14 & 0 & 42 \end{bmatrix} $$, respectively. By simple calculation, it can be confirmed that $$ C-J $$ is positive definite in both cases.

Example 1 Lead-following consensus under time-driven sampling control

In this example, consensus of systems (1)-(2) is simulated using controller (6), which does not include event-triggered control. Figures 5 and 6 show the consensus error trajectories of the position and velocity states of six nodes in a connected network. Correspondingly, Figures 7 and 8 sketch the consensus error trajectories for a disconnected network. As shown in Figures 5-8, the heterogeneous second-order MASs eventually achieve consensus.

Figure 5. Error trajectories of (A) the first component and (B) the second component of the nodes' position over the connected graph.

Figure 6. Error trajectories of (A) the first component and (B) the second component of the nodes' velocity over the connected graph.

Figure 7. Error trajectories of (A) the first component and (B) the second component of the nodes' position over the disconnected graph.

Figure 8. Error trajectories of (A) the first component and (B) the second component of the nodes' velocity over the disconnected graph.

In the following, the error trajectories of the position and velocity states of the nodes under the disconnected topology are presented. In a disconnected graph, the leader node needs to pin at least one node in each connected component to ensure that all nodes can achieve consensus. Under the same initial conditions in Example 1, a comparison between connected and disconnected graphs is conducted. The results show that consensus is achieved by all nodes at approximately $$ t=4.5 $$.

Remark 8. The sampling period h is a fundamental design parameter that significantly affects both the theoretical guarantees and the practical performance of the control system. Its role is twofold and closely integrated within the closed-loop dynamics. First, from a stability perspective, h is not merely an implementation constant but appears intrinsically within the sufficient conditions for consensus. In the subsequent Lyapunov analysis, the discretization errors introduced by the sampled-data control, such as $$ \hat{x}(t)-\hat{x}(lh) $$, lead to terms in the derivative of the Lyapunov functional that scale linearly with h, such as terms proportional to $$ \alpha(t)h\|J(L+\Gamma)\| $$. Consequently, the coefficients $$ \zeta_1(t) $$ and $$ \zeta_2(t) $$ in the core stability condition (8) of Theorem 1 become explicit functions of h. This establishes a direct theoretical link: increasing h tightens the stability condition, effectively imposing an upper bound on the allowable sampling period for provable exponential convergence. Second, from a resource utilization perspective, h defines the fundamental temporal resolution of the system. It dictates the fixed update interval of the pure sampling controller and, more importantly, sets the minimum possible inter-event time in the event-triggered strategy, thereby determining the maximum achievable communication frequency. Thus, the choice of h reflects an inherent trade-off between stability margin robustness (favored by a smaller h) and the inherent demand on communication and computational resources (favored by a larger h). The following analysis formally characterizes this dual dependence.

Example 2 Lead-following consensus under event-triggered control

In this example, the consensus of the systems (1) and (2) is simulated via the event-triggered control strategy (21). Figure 9 shows the triggering instants over the connected graph when the nodes in the MASs reach consensus. From Figure 9, it can be observed that the number of triggering instants evolves from sparse to dense, which is related to the designed triggering function (23). Moreover, Figures 10 and 11 show that the error trajectories of the MASs converge to consensus under the event-triggered mechanism. Correspondingly, Figures 12 and 13 present the consensus error trajectories under the event-triggered mechanism over the disconnected network. Figure 14 shows the triggering instants over the disconnected graph when the nodes in the MASs reach consensus.

Figure 9. The triggering instants of control (21) over the connected graph.

Figure 10. Error trajectories of (a) the first component and (b) the second component of the nodes' positions over the connected graph under control 21.

Figure 11. Error trajectories of (A) the first component and (B) the second component of the nodes' velocities over the connected graph under control 21.

Figure 12. Error trajectories of (A) the first component and (B) the second component of the nodes' positions over the disconnected graph under control 21.

Figure 13. Error trajectories of (A) the first component and (B) the second component of the nodes' velocities over the disconnected graph under control 21.

Figure 14. The triggering instants of control 21 over the disconnected graph.

Remark 9. It is worth noting from Figures 9 and 14 in this example that Zeno behavior does not occur, although the triggering becomes more frequent over time. Since sampling control is applied in both control schemes designed in this paper, and triggering instants are restricted to occur only at sampling instants, Zeno behavior can be naturally excluded.

The difference between Example 1 and Example 2 is the error trajectories of each node under different controllers. The controller used in Example 1 is a sampling controller, whereas Example 2 employs a sampling-based event-triggered control mechanism. The results shown in the above figures indicate that the nodes can achieve consensus under both controllers. In addition, the system considered in this paper is a heterogeneous MAS, meaning that the dynamics of each agent are different. As a result, achieving complete consensus is more challenging. In the numerical simulations, the consensus errors of the position and velocity states remain within a small upper bound, indicating that the heterogeneous MAS can still achieve satisfactory consensus.

Remark 10. The controller in Theorem 2 extends the sampling-based controller in Theorem 1 by introducing an event-triggered mechanism, such that control updates are applied only at triggering instants, thereby reducing communication costs to some extent. The controller in this subsection is shown in (21). Since the consensus error eventually approaches zero, i.e., $$ \sigma_i(\|\hat{x}_i(t_{k_i}^ih+mh)\|^2+\|\hat{v}_i(t_{k_i}^ih+mh)\|^2)\rightarrow0 $$ when $$ t\rightarrow \infty $$, the number of triggering instants evolves from sparse to dense. However, because triggering instants occur only at discrete sampling instants, Zeno behavior is naturally avoided.

Remark 11. Research on consensus in MASs remains an active and significant area, with various methodologies continuously being developed. For instance, literature ^[20] studied a consensus strategy for a novel nonlinear MAS under DoS attacks, employing a saturated impulsive control mechanism and analyzing stability via Lyapunov theory and matrix measure theory. Literature ^[2] proposed a robust non-fragile $$ H_\infty $$ filtering method for fuzzy fractional order systems, and addressed the $$ H_\infty $$ control problem based on linear matrix inequalities. Literature ^[32] investigated the $$ H_\infty $$ control problem for fractional-order neural networks under spoofing attacks using an improved memory-based event-triggered strategy. In contrast, the present work focuses on the consensus problem of heterogeneous second-order nonlinear MASs and proposes two distinct control strategies. Specifically, a time-driven sampling controller and an event-triggered controller are designed, both incorporating time-varying control gains. In addition, matrix theory is used to derive consensus criteria for heterogeneous second-order MASs under both sampling control and event-triggered control. Zeno behavior is naturally excluded because periodic sampling is included in the control strategy.

CONCLUSION

This work investigates the consensus problem for heterogeneous second-order MASs. To ensure consensus, the consensus problem in MASs was transformed into a stability problem of the corresponding error system. Subsequently, sufficient criteria for achieving quasi-consensus were derived using stability theory. It is worth noting that two control protocols were designed in this study. The difference between the two control strategies lies in whether an event-triggered mechanism is incorporated. The second controller introduces an event-triggered mechanism based on the design of the first controller, which could further reduce control cost. In addition, Zeno behavior does not occur, as a sampling mechanism is applied in the control strategies. The minimum lower bound of the event-triggered intervals is equal to the sampling period. Finally, numerical examples were presented to validate the derived theoretical results. The current analysis assumes the absence of external disturbances. However, the quasi-consensus framework is inherently robust to bounded disturbances, as such disturbances can be regarded as additional bounded inputs in the error dynamics, leading to an increase in the ultimate error bound. Future work will explicitly incorporate and analyze the effects of persistent external disturbances and design corresponding disturbance rejection mechanisms. Furthermore, the study of fully distributed algorithms and security control^[29] for second-order networked systems remains both challenging and important, and these topics will be investigated in future work.