Quasi-Consensus of Fractional-Order Multi-Agent Systems with Mixed Delays and External Disturbance via Impulsive Pinning Control

Abstract

In this brief, the quasi-consensus issue is examined for fractional-order multi-agent systems (FROMASs) subject to both mixed delays and external disturbances, employing an impulsive pinning control (IPC) strategy. Unlike mainstream pinning strategies with fixed nodes or static rules, a dynamic pinning mechanism based on consensus error distances is proposed, which adaptively adjusts the set of pinned nodes at each impulsive instant. By integrating the Razumikhin method, Lyapunov stability theory, and the comparison system approach, sufficient conditions for achieving quasi-consensus of FROMASs are established. Moreover, the convergence bound of consensus errors is quantitatively estimated and shown to be explicitly determined by the intensity of external disturbances. This paper organically integrates the dynamic node pinning mechanism, FRO impulsive control, and Razumikhin stability analysis. It effectively handles mixed delays and external disturbances while significantly reducing control costs. Finally, numerical simulations show that the synchronization errors are strictly bounded within the threshold $M=13.6372$, which effectively validates both the proposed control scheme and the theoretical analysis.

1. Introduction

As a fundamental branch of distributed control theory, the cooperative control of multi-agent systems (MASs) has attracted extensive research interest and witnessed substantial developments over the past few decades [1,2,3,4]. Consensus control, serving as a cornerstone of cooperative control, aims to design distributed protocols that drive the states or outputs (e.g., position, velocity) of all agents in a network to converge asymptotically to a common value. Theoretical advancements in this domain have been successfully applied to a range of critical fields, including congestion control in communication networks, formation control of autonomous aerial vehicles, and cooperative maneuvering of underwater vehicles, thereby demonstrating both significant theoretical value and practical engineering importance [5,6,7].

Beyond engineering systems, the dynamics of information and behavior diffusion are crucial for understanding a wide array of natural and social complex systems. Recent interdisciplinary studies have revealed that many real-world networks underlying these processes, including social networks, neural networks, and urban dynamics, exhibit notable hierarchical and self-similar structures. Such structures can be effectively described using fractal geometry [8,9,10]. This fractal architecture profoundly influences system dynamics in terms of robustness, efficiency, and propagation patterns. Consequently, investigating consensus or synchronization phenomena in networks with fractal properties is not only a challenge in control theory but also a subject of broad scientific significance, with potential applications ranging from epidemiology to urban planning.

In recent years, driven by the increasing demand for modeling accuracy in complex systems, fractional-order (FRO) calculus has attracted considerable interest within the control research field, owing to its distinct ability to model materials and processes that exhibit memory and hereditary characteristics. Moreover, the existing literature [11] offers substantial methodological support for research on FRO systems. In contrast to integer-order models, FRO operators provide a more precise description of dynamic behaviors in practical systems, such as the locomotion of autonomous robots over unmodeled terrains (e.g., muddy roads) [12] and the flight of unmanned aerial vehicles through complex airflow [13]. Consequently, research on FROMASs has emerged as a natural and critical extension of its integer-order counterpart, propelling the study of consensus problems into a new frontier. Preliminary yet substantial results have already been established in the existing literature, addressing critical aspects such as stability analysis and controller design for both linear and nonlinear FROMASs [14,15,16,17,18]. For instance, leveraging spectral graph theory and FRO system theory, the authors in [14] performed a thorough investigation into the consensus challenge associated with FROMASs characterized by general linear dynamics. In [16], addressing the challenge of severe actuator faults and time delays, authors developed a low-complexity consensus control scheme for FROMASs by combining neural network-based approximation with an adaptive control framework. Based on an event-triggered mechanism, the authors designed a pinning controller to achieve asymptotic consensus for discrete stochastic FROMASs [17]. In [18], by employing both aligned and distributed boundary measurement formulations, the authors devised two boundary observer-based control strategies that ensured consensus of the FROMASs.

However, the consensus control strategies for FROMASs discussed above predominantly fall within the framework of continuous-time control. This paradigm necessitates persistent communication and state updates among agents, which is often impractical in resource-constrained networks. In contrast, impulsive control emerges as a highly efficient alternative, wherein the control action is exerted solely at discrete instants. This mechanism substantially alleviates the communication burden and energy consumption. Furthermore, impulsive control has garnered significant attention from researchers in the field of FROMASs due to its inherent advantages, such as rapid response, superior convergence performance, and enhanced system robustness [19,20,21,22]. For instance, in [20], an impulsive controller was devised to achieve consensus for FROMASs under both fixed and switching topology. In [21], the authors further considered scenarios where the order of differentiation in FRO derivative is subject to change with each update. To address this, they introduced the variable-order FROMASs and achieved consensus via the impulsive control scheme.

It is noteworthy that although the aforementioned MASs enhance control efficiency via impulsive control, they necessitate controlling all agents within the network. Given that practical complex systems often comprise a vast number of nodes, exerting full control over all of them is typically challenging, if not infeasible. To address this, researchers have introduced the concept of pinning control into the impulsive control framework, leading to the IPC. This approach achieves the desired collective behavior by controlling only a subset of agents. While research on the consensus of integer-order MASs using IPC has witnessed substantial development [23,24,25], studies exploring its application to FROMASs remain scarce, presenting a significant research gap.

In practical applications, systems are inevitably subject to multiple complex disturbances. On the ond hand, time delays commonly arise due to shared and limited communication bandwidth, as well as constraints in the computational capacity of the system hardware. Such delays can not only degrade system performance but may also compromise stability. As a result, stability analysis of delayed systems has attracted considerable interest as an important research topic [26,27,28]. Regarding a specific category of stochastic systems subject to combined delays, Liu et al. [27] developed an asynchronous impulsive control scheme to achieve bipartite synchronization. Besides, in the context of FRO systems, Fan et al. [28] addressed the synchronization for FRO memristive neural networks with multiple delays. They introduced the concept of time-varying function matrix projective synchronization and devised a hybrid controller, whose distinctive feature is that the impulsive intervals are independent of the delay bounds. On the other hand, practical systems are susceptible to unknown external disturbances during operation, arising from factors such as electromagnetic interference, measurement noise, and model mismatches [29,30,31]. These disturbances can significantly degrade system performance. To achieve adaptive synchronization for FRO chaotic systems subject to external disturbances, Derakhshannia et al. [31] introduced a sliding mode control methodology that incorporates a disturbance observer. Wang et al. [32] utilized Mittag–Leffler functions to propose two consensus conditions related to system structure and control parameters, thereby resolving the consensus issue of FROMASs under external disturbances on directed networks.

Although existing studies have separately investigated impulsive control of FRO systems or consensus with time delays, the consensus problem for FROMASs subject to both mixed delays and external disturbances under an impulsive framework remains, to the best of our knowledge, an open challenge. Prevailing approaches often overlook the combined effects of distributed delays and external disturbances, or fail to fully leverage the distinctive properties of FRO impulsive dynamics. The existing studies on FRO impulsive control mostly assume that impulsive control must be applied to all nodes in the network [19,20,21]. This assumption is neither economical nor feasible in large-scale systems. Furthermore, existing IPC typically relies on a fixed set of pinned nodes, lacking a mechanism for dynamic adjustment. It is these identified gaps that motivate the present work. While our prior work in [33] investigated impulsive control for MASs with delays, the study introduces, for the first time, a dynamic pinning mechanism into the framework of FROMASs. Furthermore, although we explored IPC for FROMASs in [34], the concurrent consideration of mixed delays and external disturbances, analyzed via the Razumikhin method, represents a novel aspect not addressed in previous research.

Building upon the above analysis, this brief is dedicated to investigating the quasi-consensus for a category of FROMASs subject to both mixed delays and external disturbances under IPC. The main contributions presented in this study can be outlined in the following manner:

• Existing studies on delayed FROMASs primarily focus on systems with a single type of time delay [26,35] and often disregard external disturbances [36,37]. In contrast, this paper proposes a FROMASs that simultaneously incorporates mixed delays and external disturbances. This unified modeling framework overcomes the limitations of existing research, which tends to address a single delay in isolation or neglect disturbances.
• Different from the static pinning schemes commonly adopted in existing literature for FROMASs [38,39,40], a novel IPC protocol is designed to achieve quasi-consensus. The proposed scheme incorporates a dynamic pinning rule, where the set of pinned nodes is adaptively adjusted based on the distance of real-time synchronization errors.
• Based on the Razumikhin condition, this paper establishes a comparison system framework for the analysis of FRO impulsive systems. This framework effectively unifies the handling of analytical challenges arising from the interplay of impulsive actions, FRO dynamics, and external disturbances. Consequently, it leads to sufficient conditions for achieving quasi-consensus and provides a quantitative estimate for the convergence bound of errors.

The structure of this paper is as follows. The theoretical knowledge of FRO differentials, graph theory and lemmas is introduced in Section 2. Some sufficient conditions with the designed pinning impulsive are proposed in Section 3. The numerical example in Section 4 is to verify the validity of obtained the results. Section 5 concludes this paper.

Table 1. Notation.

Symbol	Meaning
${\mathbb{N}}^{+}$	Set of positive integers
$\mathcal{N}$	$\{1,2,\dots ,N\}$
$\mathbb{R}$ and ${\mathbb{R}}^{+}$	Set of real numbers and set of positive real numbers
${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times n}$	The n-dimensional real space and the set of $n\times n$ real matrices
$|·|$	Euclidean norm
$D^{\alpha +}$	The upper right-hand Dini derivative
${\lambda }_{max}\left(A\right)$	Largest eigenvalue of matrix A
$\mathrm{diag}\{·\}$	Diagonal matrix
⊗	Kronecker product

provides a summary of the primary notations employed in this brief.

2. Preliminaries

2.1. Graph Theory

Graph theory provides a suitable framework for modeling the communication within MASs, where agents correspond to nodes and their communication links to edges. Consider an undirected graph represented by $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$, which contains N nodes. Here, $\mathcal{V}=\mathcal{N}$ denotes the set of nodes, and $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ is the set of edges. The weighted adjacency matrix is defined as $\mathcal{A}={\left[a_{ij}\right]}_{N\times N}$, with $a_{ij}>0$ if an edge $(i,j)\in \mathcal{E}$ exists (indicating a path between nodes i and j), and $a_{ij}=0$ otherwise. The degree matrix is given by $\mathcal{D}=\mathrm{diag}\{d_1,d_2,\dots ,d_N\}$, where each entry $d_i={\sum }_{j=1}^N{a}_{ij}$. Subsequently, the Laplacian matrix is obtained as $L={\left[l_{ij}\right]}_{N\times N}=\mathcal{D}-\mathcal{A}$. Finally, the set of neighbors for a node ${\mathcal{V}}_i$ is defined as ${\mathbb{N}}_i=\{{\mathcal{V}}_j\in \mathcal{V}:(i,j)\in \mathcal{E}\}$.

When the graph $\mathcal{G}$ contains a leader, then the leader is labeled by zero, which can send information to followers. The communication topology including both the leader and followers is denoted by graph $\mathcal{G}$. Let $H={\left[h_{ij}\right]}_{N\times N}=L+\mathcal{C}$, where $\mathcal{C}=\mathrm{diag}\{c_1,c_2,\dots ,c_N\}$. When there exists an edge between the follower i and the leader, then $c_i$ = 1, otherwise $c_i$ = 0. As a result,

$$h_{ij}=\left\{\begin{array}{c}{l}_{ij},i\ne j,\hfill \\ l_{ij}+c_i,i=j.\hfill \end{array}\right.$$

2.2. Caputo Fractional Derivative and Mittag–Leffler Function

One may view the Caputo derivative as serving a role analogous to the fractal derivative. This connection places FRO models within a broader domain of nonstandard calculus, such as the q-deformed derivative associated with Tsallis statistics [41]. Notably, such approaches have led to fundamental dynamical equations, for instance, the nonlinear Fokker–Planck equation proposed by A.R. Plastino and A. Plastino [42,43], which provides analytical expressions for information spreading in fractal systems and can therefore serve as a key reference benchmark for the this work. Besides, within the realm of FROMASs, the Caputo FRO operator is fundamental for both system modeling and analytical examination. A principal advantage of the Caputo derivative is that it requires initial conditions of the same form for FRO differential equations as those used for integer-order differential equations, which generally offer more transparent physical interpretations [44]. Hence, this work employs the Caputo derivative as the primary analytical tool.

Given any function $y:\mathbb{R}\to \mathbb{R}$, the Caputo fractional derivative [44] of y is given by:

$${}_{t_0}{}^{C}D_t^{\alpha }y\left(t\right)={\displaystyle \frac{1}{\Gamma (m-\alpha )}}{\int }_{t_0}^t{(t-\mathfrak{z})}^{m-\alpha -1}{y}^{\left(m\right)}\left(\mathfrak{z}\right)d\mathfrak{z},$$

where $m-1<\alpha <m$, $m\in {\mathbb{N}}^{+},\Gamma \left(\mathfrak{z}\right)={\int }_0^{\infty }{t}^{\mathfrak{z}-1}{e}^{-t}dt$ is the Gamma function, and $t_0$ is the initial time. For convenience, denote $D^{\alpha }y\left(t\right){=}_{t_0}^C{D}_t^{\alpha }y\left(t\right)$.

For any $\alpha ,\beta >0$, the Mittag–Leffler function [44] is given by

$$E_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{+\infty }{\displaystyle \frac{z^k}{\Gamma (\alpha k+\beta )}}.$$

If $\beta =1$, then the one parameter Mittag–Leffler function can be obtained:

$$E_{\alpha }\left(z\right)=\sum _{k=0}^{+\infty }{\displaystyle \frac{z^k}{\Gamma (\alpha k+1)}}=E_{\alpha ,1}\left(z\right).$$

2.3. Model Description

Consider the following nonlinear FROMAS with internal and coupling delays:

$$\begin{array}{cc}\hfill & D^{\alpha }{x}_i\left(t\right)=f(x_i\left(t\right),x_i(t-{\tau }_1\left(t\right)))+\sum _{j\in \mathcal{N}}{a}_{ij}G(x_i(t-{\tau }_2\left(t\right))-x_j(t-{\tau }_2\left(t\right)))\hfill \\ \hfill & +u_i\left(t\right)+w_i\left(t\right),t\ge t_0,\hfill \end{array}$$

(1)

where $0<\alpha <1$ is the order of system (1). $x_i\left(t\right)\in {\mathbb{R}}^n$ and $u_i\left(t\right)\in {\mathbb{R}}^n$ denote the state and control input of follower $i,i\in \mathcal{N}$, respectively. $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a continuous nonlinear function. ${\tau }_1\left(t\right)\in (0,{\overline{\tau }}_1]$ denotes the internal delay and ${\tau }_2\left(t\right)\in (0,{\overline{\tau }}_2]$ represents the information transmission delay among any agent i and j, where ${\overline{\tau }}_1$ and ${\overline{\tau }}_2$ are known constants. Besides, let $\tau =max\{{\overline{\tau }}_1,{\overline{\tau }}_2\}$. $G=\mathrm{diag}(g_1,g_2,\dots ,g_n)$ represents the internal coupling of agents. The external disturbance of agent i is denoted by $w_i\left(t\right)\in {\mathbb{R}}^n$. Assume that $x_i\left(r\right)={\phi }_i\left(r\right),r\in [-\tau ,0]$ is the initial conditions of system (1).

On the other hand, the dynamics governing the leader agent are formulated as follows:

$$\begin{array}{c}\hfill D^{\alpha }{x}_0\left(t\right)=f(x_0\left(t\right),x_0(t-{\tau }_1\left(t\right))),t\ge t_0,\end{array}$$

(2)

where $x_0\left(t\right)\in {\mathbb{R}}^n$ is the state of leader, and the function f as defined in (1). Similarly, we assume that $x_0\left(r\right)={\phi }_0\left(r\right)$ as the initial condition of the leader.

Remark 1.

Different from [26,35] which considers only a single type of delay, this work incorporates both the self-delay of agents and the coupling delays among them, establishing a mixed delay model characterized by two independent time-varying delay parameters ${\tau }_1\left(t\right)$ and ${\tau }_2\left(t\right)$. Furthermore, in contrast to the [45,46], external disturbance $w_i\left(t\right)$ is also integrated into the FROMAS framework. Consequently, the proposed model exhibits enhanced generality and practical relevance for engineering applications.

To achieve leader-following impulsive consensus in the FROMAS, we propose the following controller based on an IPC strategy:

$$u_i\left(t\right)=\sum _{k=1}^{+\infty }{b}_k\varpi (k,i)(\sum _{j\in \mathcal{N}}{a}_{ij}(x_i\left(t\right)-x_j\left(t\right))+c_i(x_i\left(t\right)-x_0\left(t\right)))\delta (t-t_k)$$

(3)

where $\delta (·)$ denotes the Dirac function. $b_k\in \mathbb{R}$ is impulsive gain. In addition, let ${\left\{t_k\right\}}_{k=1}^{+\infty }$ be the impulsive sequence with $t_0<t_1<t_2<\dots$ and ${lim}_{k\to +\infty }{t}_k=+\infty$. Furthermore, $\varpi (k,i)$ is the traction switch where

$$\begin{array}{c}\hfill \varpi (k,i)=\left\{\begin{array}{c}1,i\in \aleph \left(t_k\right),\hfill \\ 0,i\notin \aleph \left(t_k\right).\hfill \end{array}\right.\end{array}$$

(4)

Here the set $\aleph \left(t_k\right)$ identifies the indices corresponding to agents that are pinned during the impulsive moments $t_k$.

Remark 2.

Note that the pinning impulsive protocol (3) is used at a different impulsive control time $t_k$, $k\in {\mathbb{N}}^{+}$. Different agent sets can be used at each impulsive instant, which is more universal than the common pinning impulsive control with a fixed pinning set. In FROMASs, the consensus error trajectories ${\vartheta }_i\left(t\right)$ of individual agents differ at each impulsive instant. Therefore, to accelerate consensus convergence, an intuitive and effective strategy is to prioritize the application of control to those agents with larger error norms $|{\vartheta }_i\left(t\right)|$ at the impulsive instants. Specifically, at the instant $t=t_k^{-}$, the norm of the consensus error $|{\vartheta }_i\left(t_k^{-}\right)|$, which quantifies the deviation between follower $x_i\left(t\right)$ and leader $x_0\left(t\right)$, can be sorted in descending order across all agents as $|{\vartheta }_{q_1}\left(t_k^{-}\right)|\ge |{\vartheta }_{q_2}\left(t_k^{-}\right)|\ge \dots \ge |{\vartheta }_{q_N}\left(t_k^{-}\right)|$, where the sequence ${\left\{q_i\right\}}_{i=1}^{l_k}$ denotes the relabeled index after sorting. Accordingly, the agents selected for control based on a specified condition regarding the sorted sequence above are termed the pinning agents. Assume that $l_k$$(l_k<N)$ agents are selected as the pinning agents, and define the corresponding index set as $\aleph \left(t_k\right)=\{q_1,q_2,\dots ,q_{l_k}\}$. Let $\nu \left(t_k^{-}\right)=min\left\{\right|{\vartheta }_i\left(t_k^{-}\right){|}^2:i\in \aleph \left(t_k\right)\}$ and $\mu \left(t_k^{-}\right)=max\left\{\right|{\vartheta }_i\left(t_k^{-}\right){|}^2:i\notin \aleph \left(t_k\right)\}$. It then follows directly that $\mu \left(t_k^{-}\right)<\nu \left(t_k^{-}\right)$.

Remark 3.

To intuitively illustrate the IPC rule based on error distance, Figure 1 depicts its dynamic adjustment process. The figure comprises four blue circles representing follower nodes and one green circle denoting the leader node. Solid lines indicate communication links among followers, while dashed lines represent the error distances between followers and the leader. Assuming two nodes are pinned at each impulsive instant (i.e., $l_k=2$), it can be observed that at time $t=t_{k-1}$, nodes 1 and 3 exhibit the farthest distances from the leader and are thus pinned. Correspondingly, at time $t=t_k$, the set of pinned nodes is dynamically adjusted to nodes 1 and 2. To enhance the operability and reproducibility of the proposed dynamic pinning rule, the Algorithm 1 is provided.

Algorithm 1 Dynamic Pinning Rule

Require:  1: States of all followers $x_i\left(t_k^{-}\right)$ and the leader $x_0\left(t_k^{-}\right)$.  2: Number of pinning agents $l_k<N$. Ensure: The index set of pinned agents $\aleph \left(t_k\right)$.  3: Calculate the consensus error norm: $|{\vartheta }_i\left(t_k^{-}\right)|=|x_i\left(t_k^{-}\right)-x_0\left(t_k^{-}\right)|$.  4: Sort the followers in descending order of their error norms $|{\vartheta }_i\left(t_k^{-}\right)|$. Obtain an ordered list of indices $\{q_1,q_2,\dots ,q_N\}$ such that: $|{\vartheta }_{q_1}\left(t_k^{-}\right)| \ge |{\vartheta }_{q_2}\left(t_k^{-}\right)| \ge \dots \ge |{\vartheta }_{q_N}\left(t_k^{-}\right)|$.  5: Select the first $l_k$ agents from the sorted list. Formally, the pinned set is: $\aleph \left(t_k\right)=\{q_1,q_2,\dots ,q_{l_k}\}$.

As a result, the leader-following FROMAS with the designed controller (3) can be described as:

$$\left\{\begin{array}{c}{D}^{\alpha }{x}_0\left(t\right)=f(x_0\left(t\right),x_0(t-{\tau }_1\left(t\right))),t\ge t_0,\hfill \\ D^{\alpha }{x}_i\left(t\right)=f(x_i\left(t\right),x_i(t-{\tau }_1\left(t\right)))+\sum _{j\in \mathcal{N}}{a}_{ij}G(x_i(t-{\tau }_2\left(t\right))-x_j(t-{\tau }_2\left(t\right)))+w_i\left(t\right),t\ne t_k,\hfill \\ \Delta x_i\left(t_k\right)={\displaystyle \frac{b_k}{\Gamma (\alpha +1)}}\varpi (k,i)(\sum _{j\in \mathcal{N}}{a}_{ij}(x_i\left(t_k^{-}\right)-x_j\left(t_k^{-}\right))+c_i(x_i\left(t_k^{-}\right)-x_0\left(t_k^{-}\right))),t=t_k,\hfill \end{array}\right.$$

(5)

where $\Delta x_i\left(t_k\right)=x_i\left(t_k^{+}\right)-x_i\left(t_k^{-}\right)$, $x\left(t_k^{+}\right)={lim}_{t\to t_k^{+}}x\left(t\right)$ and $x\left(t_k^{-}\right)={lim}_{t\to t_k^{-}}x\left(t\right)$. Besides, assume that $x_i\left(t\right)$ is right-hand continuous, i.e., $x\left(t_k^{+}\right)=x\left(t_k\right)$.

Remark 4.

Unlike the integral-order MASs with impulsive control, it can observe from (5) that the gains of impulsive control, i.e., ${\displaystyle \frac{b_k}{\Gamma (\alpha +1)}}$, depend not only on the designed impulsive gains $b_k$ but also on the order of systems α.

Let ${\vartheta }_i\left(t\right)=x_i\left(t\right)-x_0\left(t\right)$, ${\vartheta }_i(t-{\tau }_1\left(t\right))=x_i(t-{\tau }_1\left(t\right))-x_0(t-{\tau }_1\left(t\right))$, $i\in \mathcal{N}$ denotes the error state of system (5). Then, the error dynamics can be described by

$$\left\{\begin{array}{c}{D}^{\alpha }{\vartheta }_i\left(t\right)=\tilde{f}({\vartheta }_i\left(t\right),{\vartheta }_i(t-{\tau }_1\left(t\right)))+\sum _{j=1}^N{l}_{ij}G{\vartheta }_j(t-{\tau }_2\left(t\right))+w_i\left(t\right),t\ne t_k\hfill \\ {\vartheta }_i\left(t_k\right)=\left(1+{\psi }_k\varpi (k,i)\right)(\sum _{j\in \mathcal{N}}{a}_{ij}({\vartheta }_i\left(t_k^{-}\right)-{\vartheta }_j\left(t_k^{-}\right))+c_i{\vartheta }_i\left(t_k^{-}\right)),t=t_k\hfill \\ {\vartheta }_i\left(r\right)={\phi }_i\left(r\right)-{\phi }_0\left(r\right)={\Phi }_i\left(r\right),-\tau \le r\le 0,\hfill \end{array}\right.$$

(6)

where $\tilde{f}({\vartheta }_i\left(t\right), {\vartheta }_i(t-{\tau }_1\left(t\right)))=f(x_i\left(t\right), x_i(t-{\tau }_1\left(t\right)))-f(x_0\left(t\right), x_0(t-{\tau }_1\left(t\right)))$, ${\displaystyle \frac{b_k}{\Gamma \left(\alpha +1\right)}}={\psi }_k$.

Let $\vartheta \left(t\right)={({\vartheta }_1^T\left(t\right),{\vartheta }_2^T\left(t\right),\dots ,{\vartheta }_N^T\left(t\right))}^T$, and $w\left(t\right)={(w_1^T\left(t\right),w_2^T\left(t\right),\dots ,w_N^T\left(t\right))}^T$, then a compact formulation of system (6) is given as follows

$$\left\{\begin{array}{c}{D}^{\alpha }\vartheta \left(t\right)=\tilde{f}(\vartheta \left(t\right), \vartheta (t-{\tau }_1\left(t\right)))+(L\otimes G)\vartheta (t-{\tau }_2\left(t\right))+w\left(t\right),t\ne t_k\hfill \\ \vartheta \left(t_k\right)=\left(I_{Nn}+{\psi }_k(\left({\Omega }_{k}H\right)\otimes I_n)\right)\vartheta \left(t_k^{-}\right),t=t_k\hfill \\ \vartheta \left(r\right)=\Phi \left(r\right),-\tau \le r\le 0,\hfill \end{array}\right.$$

(7)

where ${\Omega }_k=\mathrm{diag}\{\varpi (k,1),\varpi (k,2),\dots ,\varpi (k,N)\}$ is the indicator matrix consist of $\varpi (k,i)$ at each impulsive instants $t_k$, which also indicates that whether agent i is a pinning agent or not.

To derive the consensus conditions for the FROMAS, it is essential to establish the following definitions, lemmas, and assumptions.

Definition 1.

If there exists a constant $M\in [0,+\infty )$ such that

$$\underset{t\to +\infty }{lim}|x_i\left(t\right)-x_0\left(t\right)|\le M,i\in \mathcal{N},$$

then the FROMAS is said to achieve quasi-consensus with the error bound M.

Remark 5.

Asymptotic consensus requires that the system state errors converge asymptotically to zero, i.e., $\underset{t\to +\infty }{lim}|x_i\left(t\right)-x_0\left(t\right)|=0$. However, in practical systems, persistent external disturbances $w_i\left(t\right)$ prevent the state errors from converging asymptotically to zero. Therefore, this paper adopts the more practically relevant concept of quasi-consensus, whereby the errors are ultimately confined within a computable bound M, which is determined by the intensity of the disturbances.

Definition 2.

For any impulsive sequence ${\left\{t_k\right\}}_{k=1}^{+\infty }$, if there exists positive number Δ and π such that

$${\displaystyle \frac{t-t_0}{\pi }}-\Delta \le \sigma (t_0,t)\le {\displaystyle \frac{t-t_0}{\pi }}+\Delta ,$$

where $\sigma (t_0,t)$ is the total number of impulses in $(t_0,t]$. Then, π and Δ, respectively, are called the average impulsive interval and the elasticity number.

Assumption 1.

For function $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ in systems (1) and (2), there exist two constants $k_1,k_2>0$ for any $x_1\left(t\right), x_2\left(t\right)\in {\mathbb{R}}^n$ such that

$$\begin{array}{cc}\hfill {\left(x_1\left(t\right)-x_2\left(t\right)\right)}^T& \left(f(x_1\left(t\right),(x_1\left(t\right)-{\tau }_1\left(t\right)))-f(x_2\left(t\right),(x_2\left(t\right)-{\tau }_1\left(t\right)))\right)\hfill \\ \hfill & \le k_1|x_1\left(t\right)-x_2{\left(t\right)|}^2+k_2{|x_1(t-{\tau }_1\left(t\right))-x_2(t-{\tau }_1\left(t\right))|}^2.\hfill \end{array}$$

Assumption 2.

The external disturbances $w_i\left(t\right)$ satisfy $|w_i\left(t\right)|\le ϵ<+\infty ,i\in \mathcal{N}$.

Remark 6.

The Assumption 1 is a Lipschitz condition commonly used in the analysis to handle the nonlinear function. Furthermore, the Assumption 2 requires the external disturbances $w_i\left(t\right)$ to be uniformly bounded. Both assumptions are widely adopted widely in the literature [19,30]

Lemma 1

([44]). For any $0<\alpha <1$ and $z>0$, $E_{\alpha }\left(z\right)$ is a monotone increasing function.

Lemma 2

([47]). For any $x\left(t\right)\in {\mathbb{R}}^n$, the following inequality holds

$$D^{\alpha +}\left(x^T\left(t\right)Px\left(t\right)\right)\le 2x^T\left(t\right)P{D}^{\alpha +}x\left(t\right),$$

where $0<\alpha <1$, $P\in {\mathbb{R}}^{n\times n}$ is a positive definite symmetric matrix.

Lemma 3

([44]). Consider the following FRO differential systems:

$$\begin{array}{c}\hfill D^{\alpha }\Theta \left(t\right)=p\Theta \left(t\right)+q\left(t\right),\end{array}$$

(8)

where $0<\alpha <1$, $p\in \mathbb{R}$, $\Theta \left(t\right)\in {\mathbb{R}}^n$ and $q\left(t\right)$ is a continuous function. Then, the solution of system (8) can be derived as:

$$\begin{array}{c}\hfill \Theta \left(t\right)=\Theta \left(t_0\right)E_{\alpha }\left(p{(t-t_0)}^{\alpha }\right)+{\int }_{t_0}^t{(t-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t-\mathfrak{z})}^{\alpha }\right)q\left(\mathfrak{z}\right)d\mathfrak{z}.\end{array}$$

Lemma 4

([48]). Assume that $\alpha >0,$$\beta >0$ and $p\in \mathbb{R}.$ Then,

$${\int }_{t_0}^t{E}_{\alpha ,\beta }\left[p{\nu }^{\alpha }\right]{\nu }^{\beta -1}d\nu ={(t-t_0)}^{\beta }{E}_{\alpha ,\beta +1}\left[p{(t-t_0)}^{\alpha }\right].$$

Specially, let $\alpha =\beta$, the following equality holds

$${\int }_{t_0}^t{E}_{\alpha ,\alpha }\left[p{\nu }^{\alpha }\right]{\nu }^{\alpha -1}d\nu ={(t-t_0)}^{\alpha }{E}_{\alpha ,\alpha +1}\left[p{(t-t_0)}^{\alpha }\right].$$

Lemma 5

([44]). For any $0<\alpha <1$ and $t\in R,t>0$, we have

$$\underset{t\to +\infty }{lim}{E}_{\alpha }\left(t\right)\le \underset{t\to +\infty }{lim}{\displaystyle \frac{1}{\alpha }}{e}^{t^{\frac{1}{\alpha }}}.$$

Lemma 6

([49]). For $x,$$y\in {\mathbb{R}}^n$, $\xi >0$, the following inequality holds:

$$2x^{T}y\le \xi x^{T}x+{\xi }^{-1}{y}^{T}y.$$

3. Quasi-Consensus of FROMASs

In this section, the quasi-consensus of FROMASs with mixed delays and external disturbance is investigated. The derivation of sufficient quasi-consensus conditions is accomplished through the application of the Razumikhin approach and the comparison system method.

Theorem 1.

Provided that Assumptions 1 and 2 hold true, if there exist constants ${\xi }_1,{\xi }_2>0$, $\eta \ge 1$ and $0<\varrho <1$ such that the following condition hold:

$$\chi =\varrho E_{\alpha }\left(p{\pi }^{\alpha }\right)\in (0,1),$$

(9)

where

$$\begin{array}{cc}\hfill & \varrho =\underset{k\in {\mathbb{N}}^{+}}{sup}\left\{{\varrho }_k\right\},p=\rho +2k_2\eta +{\xi }_2^{-1}\eta ,\hfill \\ \hfill & \rho =2k_1+{\xi }_1+{\xi }_2{\lambda }_{max}\left(L^{T}L\right){\lambda }_{max}\left(G^{T}G\right),\hfill \end{array}$$

here ${\varrho }_k=1+\frac{l_k}{N}\left({\theta }^2-1\right)\in \left(0,1\right)$, $\theta ={sup}_{k\in {\mathbb{N}}^{+},i\in \mathcal{N}}\left\{{\theta }_{k,i}\right\}$, and ${\theta }_{k,i}=\left(1+{\psi }_k\varpi (k,i){\sum }_{j\in \mathcal{N}}{h}_{ij}\right)$, $\forall i\in \mathcal{N}$, $l_k$ is defined as the maximum allowable number of pinning agents, then system (5) can reach the quasi-consensus, and the error bound can be estimated as

$$M=\sqrt{{ϵ}^2{\pi }^{\alpha }{E}_{\alpha ,\alpha +1}\left(p{\pi }^{\alpha }\right){\xi }_1^{-1}}.$$

Proof. 

Employ the Lyapunov function candidate given by:

$$V(t,\vartheta \left(t\right))={\vartheta }^T\left(t\right)\vartheta \left(t\right).$$

For $t\in [t_{k-1},t_k)$, calculating the $\alpha$-order derivatives of $V(t,\vartheta (t\left)\right)$ along the solutions of system (7) and from Lemma 2 and Assumption 1, we have

$$\begin{array}{cc}\hfill & D^{\alpha +}V(t,\vartheta \left(t\right))=D^{\alpha +}{\vartheta }^T\left(t\right)\vartheta \left(t\right)\hfill \\ \hfill & \le 2{\vartheta }^T\left(t\right)[\tilde{f}(\vartheta \left(t\right),\vartheta (t-{\tau }_1\left(t\right)))+(L\otimes G)\vartheta (t-{\tau }_2\left(t\right))+w\left(t\right)]\hfill \\ \hfill & \le 2{\vartheta }^T\left(t\right)\tilde{f}(\vartheta \left(t\right),\vartheta (t-{\tau }_1\left(t\right)))+2{\vartheta }^T\left(t\right)(L\otimes G)\vartheta (t-{\tau }_2\left(t\right))\hfill \\ \hfill & +2{\vartheta }^T\left(t\right)w\left(t\right)\hfill \\ \hfill & \le 2k_1{\left|\vartheta \left(t\right)\right|}^2+2k_2{\left|\vartheta (t-{\tau }_1\left(t\right))\right|}^2+2{\vartheta }^T\left(t\right)(L\otimes G)\vartheta (t-{\tau }_2\left(t\right))\hfill \\ \hfill & +2{\vartheta }^T\left(t\right)w\left(t\right).\hfill \end{array}$$

(10)

From Lemma 6, we can get

$$\begin{array}{c}\hfill 2e^T\left(t\right)w\left(t\right)\le {\xi }_1{\vartheta }^T\left(t\right)\vartheta \left(t\right)+{\xi }_1^{-1}{w}^T\left(t\right)w\left(t\right).\end{array}$$

(11)

and

$$\begin{array}{cc}\hfill 2{\vartheta }^T\left(t\right)(L\otimes G)\vartheta (t-{\tau }_2\left(t\right))& \le {\xi }_2{\vartheta }^T\left(t\right){(L\otimes G)}^T(L\otimes G)\vartheta \left(t\right)\hfill \\ \hfill & +{\xi }_2^{-1}{\vartheta }^T(t-{\tau }_2\left(t\right))\vartheta (t-{\tau }_2\left(t\right)).\hfill \end{array}$$

(12)

where ${\xi }_1>0$ and ${\xi }_2>0$ denote the disturbance trade-off parameter and the coupling delayed trade-off parameter, respectively.

Combining with (10)–(12), it can be derived that,

$$\begin{array}{cc}\hfill & D^{\alpha +}V(t,\vartheta \left(t\right))\hfill \\ \hfill & \le 2k_1{\left|\vartheta \left(t\right)\right|}^2+2k_2{\left|\vartheta (t-{\tau }_1\left(t\right))\right|}^2+{\xi }_2{\vartheta }^T\left(t\right){(L\otimes G)}^T(L\otimes G)\vartheta \left(t\right)\hfill \\ \hfill & +{\xi }_2^{-1}{\vartheta }^T(t-{\tau }_2\left(t\right))\vartheta (t-{\tau }_2\left(t\right))+{\xi }_1{\vartheta }^T\left(t\right)\vartheta \left(t\right)+{\xi }_1^{-1}{w}^T\left(t\right)w\left(t\right)\hfill \\ \hfill & \le 2k_1{\vartheta }^T\left(t\right)\vartheta \left(t\right)+2k_2{\vartheta }^T(t-{\tau }_1\left(t\right))\vartheta (t-{\tau }_1\left(t\right))\hfill \\ \hfill & +{\xi }_1{\vartheta }^T\left(t\right)\vartheta \left(t\right)+{\xi }_2{\lambda }_{max}\left(L^{T}L\right){\lambda }_{max}\left(G^{T}G\right){\vartheta }^T\left(t\right)\vartheta \left(t\right)\hfill \\ \hfill & +{\xi }_1^{-1}{w}^T\left(t\right)w\left(t\right)+{\xi }_2^{-1}{\vartheta }^T(t-{\tau }_2\left(t\right))\vartheta (t-{\tau }_2\left(t\right))\hfill \\ \hfill & \le (2k_1+{\xi }_1+{\xi }_2{\lambda }_{max}\left(L^{T}L\right){\lambda }_{max}\left(G^{T}G\right))V(t,\vartheta \left(t\right))\hfill \\ \hfill & +2k_{2}V(t,\vartheta (t-{\tau }_1\left(t\right)))+{\xi }_2^{-1}V(t,\vartheta (t-{\tau }_2\left(t\right)))+{\xi }_1^{-1}{w}^T\left(t\right)w\left(t\right).\hfill \end{array}$$

(13)

Let $\rho =2k_1+{\xi }_1+{\xi }_2{\lambda }_{max}\left(L^{T}L\right){\lambda }_{max}\left(G^{T}G\right)>0$, it follows from (13) that

$$\begin{array}{cc}\hfill D^{\alpha +}V(t,\vartheta \left(t\right))& \le \rho V(t,\vartheta \left(t\right))+2k_{2}V(t,\vartheta (t-{\tau }_1\left(t\right)))\hfill \\ \hfill & +{\xi }_2^{-1}V(t,\vartheta (t-{\tau }_2\left(t\right)))+{\xi }_1^{-1}{w}^T\left(t\right)w\left(t\right)\hfill \\ \hfill & \le \rho V(t,\vartheta \left(t\right))+(2k_2+{\xi }_2^{-1})\underset{t-\tau \le r\le t}{sup}V(r,\vartheta \left(r\right))+{\xi }_1^{-1}{w}^T\left(t\right)w\left(t\right).\hfill \end{array}$$

(14)

Drawing on the Razumikhin method for delayed systems from reference [50], there exists a Razumikhin dependence factor $\eta \ge 1$ satisfying the corresponding Razumikhin condition, the following can be further derived:

$$\underset{t-\tau \le r\le t}{sup}V(r,\vartheta \left(r\right))\le \eta V(t,\vartheta \left(t\right)).$$

Then, we have

$$\begin{array}{cc}\hfill & D^{\alpha +}V(t,\vartheta \left(t\right))\le (\rho +2k_2\eta +{\xi }_2^{-1}\eta )V(t,\vartheta \left(t\right))+{\xi }_1^{-1}{w}^T\left(t\right)w\left(t\right)\hfill \\ \hfill & \le pV(t,\vartheta \left(t\right))+{\xi }_1^{-1}{w}^T\left(t\right)w\left(t\right)\hfill \end{array}$$

(15)

where $p=\rho +2k_2\eta +{\xi }_2^{-1}\eta$.

When $t=t_k$, from the Remark 2 and (6), we can further obtain

$$\begin{array}{cc}\hfill V(t_k,\vartheta \left(t_k\right))& =\sum _{i\in \aleph \left(t_k\right)}{\vartheta }_i^T\left(t_k\right){\vartheta }_i\left(t_k\right)+\sum _{i\notin \aleph \left(t_k\right)}{\vartheta }_i^T\left(t_k\right){\vartheta }_i\left(t_k\right)\hfill \\ \hfill & \le \sum _{i\in \aleph \left(t_k\right)}{\vartheta }_i^T\left(t_k\right){\vartheta }_i\left(t_k\right)+{\varrho }_k\sum _{i\notin \aleph \left(t_k\right)}{\vartheta }_i^T\left(t_k^{-}\right){\vartheta }_i\left(t_k^{-}\right)\hfill \\ \hfill & +(1-{\varrho }_k)(N-l_k)\mu \left(t_k^{-}\right)\hfill \\ \hfill & \le \sum _{i\in \aleph \left(t_k\right)}{\vartheta }_i^T\left(t_k^{-}\right){(1+{\psi }_k\varpi (k,i)\sum _{j\in \mathcal{N}}{h}_{ij})}^2{\vartheta }_i\left(t_k^{-}\right)+{\varrho }_k\sum _{i\notin \aleph \left(t_k\right)}{\vartheta }_i^T\left(t_k^{-}\right){\vartheta }_i\left(t_k^{-}\right)\hfill \\ \hfill & +(1-{\varrho }_k)({\displaystyle \frac{N}{l_k}}-1)l_k\nu \left(t_k^{-}\right)\hfill \\ \hfill & \le {\theta }^2\sum _{i\in \aleph \left(t_k\right)}{\vartheta }_i^T\left(t_k^{-}\right){\vartheta }_i\left(t_k^{-}\right)+{\varrho }_k\sum _{i\notin \aleph \left(t_k\right)}{\vartheta }_i^T\left(t_k^{-}\right){\vartheta }_i\left(t_k^{-}\right)\hfill \\ \hfill & +({\varrho }_k-{\theta }^2)\sum _{i\in \aleph \left(t_k\right)}{\vartheta }_i^T\left(t_k^{-}\right){\vartheta }_i\left(t_k^{-}\right)\hfill \\ \hfill & \le {\varrho }_{k}V(t_k^{-},\vartheta \left(t_k^{-}\right)).\hfill \end{array}$$

(16)

To facilitate the analysis, a corresponding FRO impulsive comparison system is established for any given positive parameter $\varsigma$:

$$\left\{\begin{array}{c}{D}^{\alpha +}\mathcal{W}(t,\vartheta \left(t\right))=p\mathcal{W}(t,\vartheta \left(t\right))+{\zeta }_1^{-1}{w}^T\left(t\right)w\left(t\right)+\varsigma ,t\ne t_k,\hfill \\ \mathcal{W}(t,\vartheta \left(t\right))={\varrho }_k\mathcal{W}(t_k^{-},\vartheta \left(t_k^{-}\right))+\varsigma ,t=t_k,\hfill \\ \mathcal{W}(r,\vartheta \left(r\right))={\Phi }^2\left(r\right),-\tau \le r\le 0.\hfill \end{array}\right.$$

(17)

Given that $\varsigma >0$, it follows that $0<V(t,\vartheta (t\left)\right)\le \mathcal{W}(t,\vartheta (t\left)\right)$. Hence, if it can be proven that $\mathcal{W}(t,\vartheta (t\left)\right)$ asymptotically converges to a constant $M^2$ as time approaches infinity, then $V(t,\vartheta (t\left)\right)$ consequently converges to the same constant $M^2$.

So, from Lemma 3 and (17), for $t\in [t_{k-1},t_k)$ we get

$$\begin{array}{cc}\hfill \mathcal{W}(t,\vartheta (t\left)\right)& =\mathcal{W}(t_{k-1},\vartheta \left(t_{k-1}\right))E_{\alpha }\left(p{(t-t_{k-1})}^{\alpha }\right)\hfill \\ \hfill & +{\int }_{t_{k-1}}^t{(t-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t-\mathfrak{z})}^{\alpha }\right)({\xi }_1^{-1}{w}^T\left(\mathfrak{z}\right)w\left(\mathfrak{z}\right)+\varsigma )d\mathfrak{z}\hfill \\ \hfill & =\mathcal{W}(t_{k-1},\vartheta \left(t_{k-1}\right))E_{\alpha }\left(p{(t-t_{k-1})}^{\alpha }\right)+{\int }_{t_{k-1}}^t{(t-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t-\mathfrak{z})}^{\alpha }\right)q\left(\mathfrak{z}\right)d\mathfrak{z},\hfill \end{array}$$

(18)

where $q\left(\mathfrak{z}\right)={\xi }_1^{-1}{w}^T\left(\mathfrak{z}\right)w\left(\mathfrak{z}\right)+\varsigma$.

Together (17) with (18), we can obtain that for $t\in [t_0,t_1)$

$$\begin{array}{cc}\hfill & \mathcal{W}(t,\vartheta \left(t\right))=\mathcal{W}(t_0,\vartheta \left(t_0\right))E_{\alpha }\left(p{(t-t_0)}^{\alpha }\right)+{\int }_{t_0}^t{(t-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t-\mathfrak{z})}^{\alpha }\right)q\left(\mathfrak{z}\right)d\mathfrak{z},\hfill \end{array}$$

and for $t=t_1$, we have

$$\begin{array}{cc}\hfill \mathcal{W}(t_1,\vartheta \left(t_1\right))& \le {\varrho }_{1}V(t_1^{-},\vartheta \left(t_1^{-}\right))\hfill \\ \hfill & \le {\varrho }_{1}V(t_0,\vartheta \left(t_0\right))E_{\alpha }\left(p{(t_1-t_0)}^{\alpha }\right)\hfill \\ \hfill & +{\varrho }_1{\int }_{t_0}^{t_1}{(t_1-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t_1-\mathfrak{z})}^{\alpha }\right)q\left(\mathfrak{z}\right)d\mathfrak{z}.\hfill \end{array}$$

Then, for $t\in [t_1,t_2)$, one has

$$\begin{array}{cc}\hfill \mathcal{W}(t,\vartheta (t\left)\right)& \le \mathcal{W}(t_1,\vartheta \left(t_1\right))E_{\alpha }\left(p{(t-t_1)}^{\alpha }\right)+{\int }_{t_1}^t{(t-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t-\mathfrak{z})}^{\alpha }\right)q\left(\mathfrak{z}\right)d\mathfrak{z}\hfill \\ \hfill & \le {\varrho }_1\mathcal{W}(t_1^{-},\vartheta \left(t_1^{-}\right))E_{\alpha }\left[p{(t-t_1)}^{\alpha }\right]+{\int }_{t_1}^t{(t-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t-\mathfrak{z})}^{\alpha }\right)q\left(\mathfrak{z}\right)d\mathfrak{z}\hfill \\ \hfill & \le {\varrho }_1\mathcal{W}(t_0,\vartheta \left(t_0\right))E_{\alpha }\left[p{(t_1-t_0)}^{\alpha }\right]E_{\alpha }\left[p{(t-t_1)}^{\alpha }\right]\hfill \\ \hfill & +{\varrho }_1{\int }_{t_0}^{t_1}{(t_1-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t_1-\mathfrak{z})}^{\alpha }\right)q\left(\mathfrak{z}\right)d\mathfrak{z}{E}_{\alpha }\left[p{(t-t_1)}^{\alpha }\right]\hfill \\ \hfill & +{\int }_{t_1}^t{(t-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t-\mathfrak{z})}^{\alpha }\right)q\left(\mathfrak{z}\right)d\mathfrak{z}.\hfill \end{array}$$

In general, let $\varrho ={sup}_{k\in {\mathbb{N}}^{+}}\left\{{\varrho }_k\right\}$ and $t\in [t_{k-1},t_k),$ it is easy to get

$$\begin{array}{cc}\hfill \mathcal{W}(t,\vartheta (t\left)\right)& \le \mathcal{W}(t_{k-1},\vartheta \left(t_{k-1}\right))E_{\alpha }\left(p{(t-t_{k-1})}^{\alpha }\right)\hfill \\ \hfill & +{\int }_{t_{k-1}}^t{(t-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t-\mathfrak{z})}^{\alpha }\right)q\left(\mathfrak{z}\right)dv\hfill \\ \hfill & \le {\varrho }_k\mathcal{W}(t_{k-1}^{-},\vartheta \left(t_{k-1}^{-}\right))E_{\alpha }\left(p{(t-t_{k-1})}^{\alpha }\right)\hfill \\ \hfill & +{\int }_{t_{k-1}}^t{(t-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t-\mathfrak{z})}^{\alpha }\right)q\left(\mathfrak{z}\right)d\mathfrak{z}\hfill \\ \hfill & \le \mathcal{W}(t_0,\vartheta \left(t_0\right)){\varrho }^{k-1}\prod _{i=1}^{k-1}{E}_{\alpha }\left[p{(t_i-t_{i-1})}^{\alpha }\right]E_{\alpha }\left[p{(t-t_{k-1})}^{\alpha }\right]\hfill \\ \hfill & +\sum _{i=1}^{k-1}{\varrho }^{(k-1-i)}{\int }_{t_{i-1}}^{t_i}{(t_i-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t_i-\mathfrak{z})}^{\alpha }\right)q\left(\mathfrak{z}\right)d\mathfrak{z}\prod _{i=1}^{k-1}{E}_{\alpha }\left(p{(t_i-t_{i-1})}^{\alpha }\right)\hfill \\ \hfill & \times E_{\alpha }\left[p{(t-t_{k-1})}^{\alpha }\right]+{\int }_{t_{k-1}}^t{(t-\mathfrak{z})}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t-\mathfrak{z})}^{\alpha }\right)q\left(\mathfrak{z}\right)d\mathfrak{z}.\hfill \end{array}$$

(19)

By using Lemma 4 and Assumption 2, and let $\varsigma \to 0$, then we get that

$$\begin{array}{cc}\hfill \mathcal{W}(t,\vartheta (t\left)\right)& =V(t,\vartheta (t\left)\right)\hfill \\ \hfill & \le V(t_0,\vartheta \left(t_0\right)){\varrho }^{k-1}\prod _{i=1}^{k-1}{E}_{\alpha }\left[p{(t_i-t_{i-1})}^{\alpha }\right]E_{\alpha }\left[p{(t-t_{k-1})}^{\alpha }\right]\hfill \\ \hfill & +\sum _{i=1}^{k-1}{\varrho }^{(k-1-i)}{\int }_{t_{i-1}}^{t_i}{(t_i-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t_i-\ell )}^{\alpha }\right)q(\ell )d\ell \prod _{i=1}^{k-1}{E}_{\alpha }\left(p{(t_{i+1}-t_i)}^{\alpha }\right)\hfill \\ \hfill & \times E_{\alpha }\left[p{(t-t_{k-1})}^{\alpha }\right]+{\int }_{t_{k-1}}^t{(t-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }\left(p{(t-\ell )}^{\alpha }\right)q(\ell )d\ell \hfill \\ \hfill & \le V(t_0,\vartheta \left(t_0\right)){\varrho }^{k-1}\prod _{i=1}^{k-1}{E}_{\alpha }\left[p{(t_i-t_{i-1})}^{\alpha }\right]E_{\alpha }\left[p{(t-t_{k-1})}^{\alpha }\right]\hfill \\ \hfill & +{\xi }_1^{-1}{ϵ}^2\sum _{i=1}^{k-1}{\varrho }^{(k-1-i)}{(t_i-t_{i-1})}^{\alpha }{E}_{\alpha ,\alpha +1}\left[p{(t_i-t_{i-1})}^{\alpha }\right]\prod _{i=1}^{k-1}{E}_{\alpha }\left(p{(t_{i+1}-t_i)}^{\alpha }\right)\hfill \\ \hfill & \times E_{\alpha }\left[p{(t-t_{k-1})}^{\alpha }\right]+{\xi }_1^{-1}{ϵ}^2{\left(t-t_{k-1}\right)}^{\alpha }{E}_{\alpha ,\alpha +1}\left[p{(t-t_{k-1})}^{\alpha }\right].\hfill \end{array}$$

(20)

By using Lemma 5 and $V(t_0,\vartheta \left(t_0\right))={\left|V\right|}^2$, one can conclude that

$$\begin{array}{cc}\hfill V(t,\vartheta (t\left)\right)& \le V(t_0,\vartheta \left(t_0\right)){\left(\varrho E_{\alpha }\left(p{\pi }^{\alpha }\right)\right)}^{\sigma (t,t_0)}{\displaystyle \frac{1}{\alpha }}{e}^{{\left(pt\right)}^{\frac{1}{\alpha }}}\hfill \\ \hfill & +{\xi }_1^{-1}{ϵ}^{2}k{\pi }^{\alpha }{\left(\varrho E_{\alpha }\left(p{\pi }^{\alpha }\right)\right)}^{\sigma (t,t_0)}{\displaystyle \frac{1}{\alpha }}{e}^{{\left(pt\right)}^{\frac{1}{\alpha }}}+{\xi }_1^{-1}{ϵ}^2{\pi }^{\alpha }{E}_{\alpha ,\alpha +1}\left(p{\pi }^{\alpha }\right)\hfill \\ \hfill & \le {\left|V\right|}^2{\displaystyle \frac{1}{\alpha }}{e}^{{\left(pt\right)}^{\frac{1}{\alpha }}}+{\xi }_1^{-1}{ϵ}^{2}k{\pi }^{\alpha }{\displaystyle \frac{1}{\alpha }}{e}^{\left(pt\right)\frac{1}{\alpha }}){\left(\varrho E_{\alpha }\left(p{\pi }^{\alpha }\right)\right)}^{{\displaystyle \frac{t-t_0}{\pi }}-\Delta }\hfill \\ \hfill & +{\xi }_1^{-1}{ϵ}^2{\pi }^{\alpha }{E}_{\alpha ,\alpha +1}\left[p{\pi }^{\alpha }\right]\hfill \\ \hfill & \le {\left(\right|V|}^2{\displaystyle \frac{1}{\alpha }}{e}^{{\left(pt\right)}^{\frac{1}{\alpha }}}+{\xi }_1^{-1}{ϵ}^{2}k{\pi }^{\alpha }{\displaystyle \frac{1}{\alpha }}{e}^{\left(pt\right)\frac{1}{\alpha }}{)\left(\chi \right)}^{\frac{t-t_0}{\pi }-\Delta }+{\xi }_1^{-1}{ϵ}^2{\pi }^{\alpha }{E}_{\alpha ,\alpha +1}\left[p{\pi }^{\alpha }\right]\hfill \\ \hfill & \le {\left(\right|V|}^2{\displaystyle \frac{1}{\alpha }}{e}^{{\left(pt\right)}^{\frac{1}{\alpha }}}+{\xi }_1^{-1}{ϵ}^{2}k{\pi }^{\alpha }{\displaystyle \frac{1}{\alpha }}{e}^{\left(pt\right)\frac{1}{\alpha }}){\chi }^{-\Delta }{e}^{\frac{ln\chi }{\pi }(t-t_0)}\hfill \\ \hfill & +{\xi }_1^{-1}{ϵ}^2{\pi }^{\alpha }{E}_{\alpha ,\alpha +1}\left[p{\pi }^{\alpha }\right],\hfill \end{array}$$

(21)

where ${\displaystyle \frac{ln\chi }{\pi }}<0$ since $\chi =\varrho E_{\alpha }\left(p{\pi }^{\alpha }\right)\in (0,1)$. Consequently, the relationship expressed in (21) leads to

$$\underset{t\to +\infty }{lim}V(t,\vartheta \left(t\right))=M^2={ϵ}^2{\pi }^{\alpha }{E}_{\alpha ,\alpha +1}\left(p{\pi }^{\alpha }\right){\xi }_1^{-1},$$

and it is clear that M is a bounded constant. Therefore, quasi-consensus is attainable for the FROMASs under the proposed control scheme (3), which completes the demonstration. □

Remark 7.

In comparison with the Krasovskii technique utilized in earlier works [16,51] for FROMASs with delay, where a relatively complicated Krasovskii functional must be constructed, this study applies the Razumikhin method, enabling the stability analysis to proceed with merely a common positive definite function.

Remark 8.

When the number of pinned nodes $l_k=N$, the control action reduces to applying impulsive control to all nodes, i.e., the entire network is pinned, which aligns with the scenarios studied in [19,20,21]. In this case, it is clear that ${\varrho }_k={\theta }^2$. Although the compression effect of impulsive control becomes stronger, applying control to all nodes is generally neither practical nor economical under resource-constrained conditions. On the other hand, if $l_k=0$ (i.e., ${\varrho }_k=1$), no impulsive control is exerted on the system because pinning is only activated at impulsive instants. Consequently, the dynamic pinning strategy proposed in this work, which flexibly adjusts $l_k$, not only maintains control performance but also effectively adapts to resource-limited environments, demonstrating notable practical advantages.

Remark 9.

Although existing studies [38,39,40] have applied pinning control to FRO systems, they generally employ a fixed set of pinned nodes. This static strategy fails to adequately account for the memory effects inherent in FROMASs and the complex transient behavior induced by impulsive actions, thereby limiting its performance and adaptability. In contrast, this paper proposes an intelligent strategy that dynamically adjusts the pinned nodes based on error distances, thereby enhancing control performance. Moreover, the control action is exerted only at discrete impulsive instants, constituting an IPC scheme. Compared with the continuous pinning control methods adopted in [40,52,53], the proposed approach achieves comparable control objectives with higher efficiency and lower energy consumption.

Remark 10.

Compared with the models in references [36,37], this work incorporates external disturbances, making the FROMAS description more realistic. Furthermore, unlike the simplified treatment in [54,55,56] where asymptotic stability is obtained merely by restricting the disturbances to be bounded by a constant, we construct a FRO comparison system to rigorously derive sufficient conditions for quasi-consensus and quantitatively estimate the convergence bound of the consensus errors. If no external disturbance acts on the FROMASs, i.e., $w_i\left(t\right)=0$ (and consequently $ϵ=0$), it follows from Theorem 1 that the convergence error bound becomes $M=0$, and the FROMASs achieve asymptotic consensus. This clearly illustrates that quasi-consensus can be regarded as a natural extension of asymptotic consensus in the presence of bounded external disturbances. Such a relationship underscores the practical relevance of the model established in this paper for characterizing real-world system dynamics.

If the pinned nodes of the FROMASs are randomly selected at each impulsive instant, the following Corollary 1 can be derived.

Corollary 1.

Under the same assumptions of Theorem 1, if the pinning is chosen randomly and the condition (9) holds where ${\varrho }_k=1+{\displaystyle \frac{l_k}{N}}\left({\theta }^2-1\right)\in \left(0,1\right)$ can be replaced by ${\rho }_k={\lambda }_{max}^2(I_{Nn}+{\psi }_k\left({\Omega }_{k}H\right)\otimes I_n)\in \left(0,1\right)$, then system (1) can achieve the quasi-consensus with the error bound $M=\sqrt{{ϵ}^2{\pi }^{\alpha }{E}_{\alpha ,\alpha +1}\left(p{\pi }^{\alpha }\right){\xi }_1^{-1}}$.

Proof. 

This proof mainly focuses on the difference of estimation about $V(t,\vartheta (t\left)\right)$ at impulsive instant $t_k$. We utilize the identical Lyapunov candidate function $V(t,\vartheta \left(t\right))={\vartheta }^T\left(t\right)\vartheta \left(t\right)$ introduced in Theorem 1 for the subsequent analysis. The pinning rules here is selecting agents randomly at any impulsive instant $t_k,k\in {\mathbb{N}}^{+}$, then we have

$$\begin{array}{cc}\hfill V(t_k,\vartheta \left(t_k\right))& ={\vartheta }^T\left(t_k^{-}\right){(I_{Nn}+{\psi }_k\left({\Omega }_{k}H\right)\otimes I_n)}^T(I_{Nn}+{\psi }_k\left({\Omega }_{k}H\right)\otimes I_n)\vartheta \left(t_k^{-}\right)\hfill \\ \hfill & \le {\lambda }_{max}^2\left(I_{Nn}+{\psi }_k\left({\Omega }_{k}H\right)\otimes I_n\right)V(t_k^{-},\vartheta \left(t_k^{-}\right))\hfill \\ \hfill & ={\rho }_{k}V(t_k^{-},\vartheta \left(t_k^{-}\right)),\hfill \end{array}$$

(22)

The proof is therefore concluded, as the remaining arguments follow analogously to those presented for Theorem 1 and are thus omitted for brevity. □

4. Numerical Simulation

To illustrate and validate the theoretical findings presented earlier, a simulation case is conducted in this section. The considered FROMAS operates within a three-dimensional space, with its topology comprising a single leader (agent 0) and four followers (agents $1,2,3$, and 4). The communication topology among the agents is depicted in Figure 2, and their dynamics can be described as follows:

$$\left\{\begin{array}{cc}\hfill D^{\alpha }{x}_0\left(t\right)& =f(x_0\left(t\right),x_0(t-{\tau }_1\left(t\right))),\hfill \\ \hfill D^{\alpha }{x}_i\left(t\right)& =f(x_i\left(t\right),x_i(t-{\tau }_1\left(t\right)))+\sum _{j\in \mathcal{N}}{a}_{ij}G(x_i(t-{\tau }_2\left(t\right))-x_j(t-{\tau }_2\left(t\right)))\hfill \\ \hfill & +u_i\left(t\right)+w_i\left(t\right),\hfill \end{array}\right.$$

(23)

where for $i\in \mathcal{N}\cup \left\{0\right\}, x_i\left(t\right)={[x_{i1}\left(t\right),x_{i2}\left(t\right),x_{i3}\left(t\right)]}^T$ and $f(x_i\left(t\right), x_i(t-{\tau }_1\left(t\right)))={[\tanh (x_{i1}\left(t\right)+x_{i1}(t-{\tau }_1\left(t\right))), \tanh (x_{i2}\left(t\right)+x_{i2}(t-{\tau }_1\left(t\right))), \tanh (x_{i3}\left(t\right)+x_{i3}(t-{\tau }_1\left(t\right)))]}^T$. Let ${\tau }_1\left(t\right)=0.01\left|sin\left(t\right)\right|,{\tau }_2\left(t\right)=0.02\left|sin\left(t\right)\right|$, so $\tau =\max \{{\overline{\tau }}_1,{\overline{\tau }}_2\}=\{0.01,0.02\}=0.02$. In addition, we select $w_i\left(t\right)={[3\sin \left(t\right),\cos \left(t\right),4\cos \left(t\right)]}^T$, so $ϵ=4$.

From Figure 2, we know that

$$\mathcal{A}=\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right],L=\left[\begin{array}{cccc}1& -1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 1& -1\\ 0& 0& -1& 0\end{array}\right],$$

$$\mathcal{C}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right],H=\left[\begin{array}{cccc}2& -1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 2& -1\\ 0& 0& -1& 0\end{array}\right].$$

The initial states of followers are chosen as

$$\begin{array}{cc}\hfill x\left(0\right)& =\left[x_1\left(0\right),x_2\left(0\right),x_3\left(0\right),x_4\left(0\right)\right]\hfill \\ \hfill & =\left[\begin{array}{cccc}-12& -12& -11& 12\\ 10& -12& 13& 14\\ 14& 13& 14& -12\end{array}\right],\hfill \end{array}$$

and the initial states of leader is chosen as $x_0\left(0\right)={(4, 0 ,-2)}^T$. Furthermore, the impulsive control gain is set to $0.6$, with the corresponding impulsive sequence shown in Figure 3.

Choose ${\xi }_1=0.1, {\xi }_2=0.1$ and $T=0.026$. Based on Theorem 1, after simple calculation, we get that $\varrho =0.5326$, $p=12.4873$, $\chi =\varrho E_{\alpha }\left(p{\pi }^{\alpha }\right)=0.7582$. It can be verified that all conditions specified in Theorem 1 are satisfied. Moreover, the bound of the system error is evaluated as $M=13.6372$. In the simulation, the step size is set to $0.1$ and the total duration to 40 time units. The impulsive control gain is chosen as $b_k=0.6$, and the number of pinned nodes is set to $l_k=2$.

Under the specified parameters, Figure 4 and Figure 5, respectively, present the evolution of consensus errors and the agent pinning status at impulsive instants in the FROMASs. The simulation results presented in Figure 4 demonstrate that the consensus errors between the four followers and the leader converge to a steady state after approximately $t=7.5$, with the steady-state error bounded by $M=13.6372$. This confirms that the FROMASs successfully achieve quasi-consensus. Furthermore, Figure 5 illustrates that at each impulsive instant, two nodes are selected for pinning control, and the set of pinned agents is adaptively adjusted based on real-time error information.

Remark 11.

When only impulsive control is applied to the FROMASs without pinning, i.e., all nodes are controlled, Figure 6 depicts the resulting error trajectories under the classical impulsive control strategy. Figure 7 illustrates the scenario in which every node is controlled at each impulsive instant. From Figure 6 and Figure 7, it can be observed that although impulsive control alone can achieve quasi-consensus more rapidly, controlling all nodes is impractical when resources are constrained.

Remark 12.

It is worth noting that if no external disturbance is present (i.e., $w_i\left(t\right)=0$), the FROMASs reduce to the case without external disturbance. Under this condition, the system achieves asymptotic consensus with the proposed controller (3), and the corresponding error trajectories are shown in Figure 8. Evidently, the errors eventually converge to zero, demonstrating the attainment of asymptotic consensus.

5. Conclusions

This paper has solved the quasi-consensus problem of leader-following FROMASs with mixed delays and external disturbance. By employing the Lyapunov–Razumikhin method, this study has established conditions for quasi-consensus in FROMASs under IPC, with the selection of pinned agents following a dynamic priority rule based on the distance of consensus errors. It is important to note that the impact of impulsive control on the performance of the controlled systems is determined by both the designed impulsive gains and the order of the FROMASs. Results have shown that the errors of systems are eventually converging to a bounded constant related to the bounded external disturbances, and to obtain the estimated error bound, a comparison FRO impulsive system has been constructed. To validate the established theoretical findings, a simulation case is presented as a final demonstration. This result has provided important theoretical support for addressing key challenges in a class of engineered cooperative systems. Furthermore, by situating our FRO impulsive control problem within the broader context of dynamics on fractal networks, this work opens dialogues with fields like statistical physics, epidemiology, and computational social science. The comparative study between the performance of designed control protocols in engineered fractal networks and the emergent efficiency of self-organized natural networks (e.g., social or neural networks) remains a fertile ground for future research, promising deeper insights into the principles of robust coordination in complex systems. Representative applications include ensuring stable operation for large-scale multi-robot systems swarm on unstructured soft terrains under constrained control resources, and achieving smooth cooperative flight for multi-UAV systems in complex dynamic airflow. Given the prevalent stochastic noise in communication channels and measurement processes, as well as the increasingly stringent industrial demands on control precision and response speed, our future research will focus on the prescribed-time control problem of FROMASs subject to stochastic disturbances.