Leader-Following Formation Control for Discrete-Time Fractional Stochastic Multi-Agent Systems by Event-Triggered Strategy

Abstract

Fractional differential equations, which are non-local and can better describe memory and genetic properties, are widely used to describe various physical, chemical, and biological phenomena. Therefore, the multi-agent systems based on discrete-time fractional stochastic models are established. First, some followers are selected for pinning control. In order to save resources and energy, an event-triggered based control mechanism is proposed. Second, under this control mechanism, sufficient conditions on the interaction graph and the fractional derivative order such that formation control can be achieved are given. Additionally, influenced by noise, the multi-agent system completes formation control in the mean square. In addition to that, these results are equally applicable to the discrete-time fractional formation problem without noise. Finally, the example of numerical simulation is given to prove the correctness of the results.

1. Introduction

In recent years, multi-agent systems have been instrumental in the field of robotics, involving multiple agents with autonomous decision-making and collaborative abilities. These systems enable the collective accomplishment of complex tasks through communication and cooperation. They are applied in diverse areas like unmanned aerial vehicle formations, robot collaborations, autonomous vehicles, and distributed sensor systems [1,2].

The key issue of multi-agent systems is the formation and adaptation in their development. Formation refers to a group of agents maintaining specific relative positions and exhibiting coordinated behavior while performing tasks. These formations can be either static, with agents maintaining fixed relative positions, or dynamic, allowing agents to flexibly adjust their positions during task execution. Solving formation problems is crucial for enhancing system efficiency, robustness, and collaborative capabilities. Solving the formation problem in multi-agent systems presents several challenges. First, determining the optimal formation configuration that satisfies task constraints and objectives is a complex optimization problem [3,4]. It involves considering factors such as agent capabilities [5], communication constraints [6], obstacle avoidance [7], and task-specific requirements [8]. Second, ensuring robustness and fault tolerance in formation is crucial, as agents may face uncertainties, failures, or communication disruptions during task execution [9,10]. Developing algorithms and strategies that can handle these uncertainties and maintain formation integrity is a significant research challenge. Finally, scalability is another important consideration, as the number of agents and the complexity of tasks increase. Efficient algorithms and communication protocols are required to handle large-scale multi-agent systems. Therefore, it is highly useful to study the problem of formation control in multi-agent systems.

Recently, the exploration of formation control in fractional systems has garnered considerable interest due to their capacity to capture intricate dynamics with memory and long-range dependencies [11,12]. Fractional systems, characterized by non-integer derivatives, have demonstrated promising applications across diverse domains, including control systems [13], signal processing [14], and optimization. By extending fractional dynamics to multi-agent systems, the formation control challenge in fractional multi-agent systems has emerged as a captivating and sought-after research area. Unlike conventional integer-order systems, fractional-order dynamics introduce memory effects and long-range interactions among agents, allowing for more sophisticated and nuanced formation control strategies [15,16,17]. Various control techniques have been employed and refined in the realm of fractional formation control, such as sliding-mode control [18,19], double-integrator control [20], and both-and-observer control [21]. Thus, building on these prior studies and the benefits of fractional-order control, the domain of fractional multi-agent system formation holds the potential to revolutionize numerous applications.

In practical scenarios, the prevalence of attacks and uncertainties necessitates considering the impact of the resulting perturbations on systems. As a result, many studies have utilized stochastic differential equations with noise terms to model systems. While the research on continuous-time stochastic systems has been significant, the control strategies proposed for continuous systems are often too idealized for direct application in real-world settings. Consequently, discrete-time stochastic systems have garnered considerable attention due to their enhanced realism and practicality. The researchers in Ref. [22] delved into finite-time $H_{\infty }$ state estimation for discrete time-delayed stochastic systems using communication protocols. Additionally, Ref. [23] explored recursive fusion estimation for stochastic discrete-time-varying complex networks under stochastic communication protocols. To address energy consumption concerns in scenarios with limited resources, energy-efficient controllers are in high demand. Moreover, the fixed allocation of bandwidth can constrain communication between nodes in networks. To conserve resources and bandwidth, event-triggered control mechanisms are widely implemented in discrete-time stochastic systems. In Ref. [24], an event-triggered law was developed for discrete stochastic systems to ensure system stability. Ref. [25] investigated event-triggered control problems in the presence of packet loss. Furthermore, novel approaches for addressing event-triggered control problems in discrete-time stochastic systems were presented in the context of the pinning synchronization problem in Ref. [26] and the quasi-synchronization in probability problem in Ref. [27]. Compared to systems with continuous-time dynamics, discrete-time systems are more computationally implementable. Additionally, many real-world systems are not accurately captured by continuous dynamics. References [28,29,30] have explored intriguing questions concerning the consensus stability of discrete-time multi-agent systems. Notably, significant progress has been made in event-triggered control for integer discrete-time stochastic systems. This paper extends event-triggered control from integer discrete-time stochastic systems to fractional systems with time delay, with results applicable to both types of systems.

Inspired by the above, this paper investigates leader-following formation control for discrete-time fractional stochastic multi-agent systems by the event-triggered strategy. In summary, the contributions of this paper can be outlined as:

(1)

In contrast with the existing discrete-time literature [31,32], a novel fractional stochastic discrete-time multi-agent system model is introduced.

(2)

Unlike the traditional discrete-time fractional stochastic systems [33,34], this paper uses a control law with an event-triggering mechanism, which conserves resources and saves bandwidth.

(3)

Pinning controllers and the corresponding sufficient conditions are provided to achieve mean square pinning consensus in the fractional stochastic networks.

(4)

Differently from the traditional synchronous control problems [27,35,36], the formation control proposed is structurally more flexible and expandable. When the ratio formation structure is taken as a zero vector, it is the ordinary asymptotic synchronization problem.

The paper is structured as follows: Section 2 introduces some preliminaries. Section 3 presents the system model and main results. Section 4 illustrates a numerical example. Finally, the conclusion is presented in Section 5.

2. Preliminaries

Definition 1

([37]). The discrete-time difference operator of order α of function $f(·)$ is defined by

$${\nabla }_T^{\alpha }f\left(k\right)=\frac{1}{T^{\alpha }}\sum _{r=0}^{[k/T]}{(-1)}^r\left(\begin{array}{c}\alpha \hfill \\ r\hfill \end{array}\right)f(k-rT),\alpha >0,$$

where 0 is the initial time. Take $T=1$ for convenience, then ${\nabla }^{\alpha }f\left(k\right)={\sum }_{r=0}^k{(-1)}^r\left(\begin{array}{c}\alpha \hfill \\ r\hfill \end{array}\right)f(k-$$r)$.

Lemma 1

([37]). For $\alpha \in (0,1)$, the function $w_r^{\alpha }={(-1)}^{r+1}\left(\begin{array}{c}\alpha \hfill \\ r\hfill \end{array}\right)$ satisfies the properties $w_0^{\alpha }=-1,w_{r-1}^{\alpha }>w_r^{\alpha }$ for $r\in {\mathbb{N}}^{+}\setminus \left\{1\right\},{lim}_{r\to +\infty }{w}_r^{\alpha }=0$ and ${\sum }_{r=1}^{+\infty }{w}_r^{\alpha }=1$.

Lemma 2

([38]). Consider a sequence of nonnegative random variables ${\left\{V\left(k\right)\right\}}_{k\ge 0}$ with $\mathbb{E}\left\{V\right(0\left)\right\}<\infty$. Let

$$\begin{array}{c}\hfill \mathbb{E}\left\{V(k+1)\right\}\le \left(1-c_1\left(k\right)\right)\mathbb{E}V\left(k\right)+c_2\left(k\right)\end{array}$$

where

$$\begin{array}{cc}& c_1\left(k\right)\ge 0,c_2\left(k\right)\ge 0,\forall k\hfill \\ & \sum _{k=0}^{\infty }{c}_2\left(k\right)<\infty ,\sum _{k=0}^{\infty }{c}_1\left(k\right)=\infty \hfill \\ & \underset{k\to \infty }{lim}\frac{c_2\left(k\right)}{c_1\left(k\right)}=0.\hfill \end{array}$$

Then, $V\left(k\right)$ converges to zero in the mean square, i.e.,

$$\underset{k\to \infty }{lim}\mathbb{E}V\left(k\right)=0.$$

Lemma 3

([39]). Given a convergent series $\left\{l_m\mid l_m\in \mathbb{R},m=1,2,\dots \right\}$, then ${\left({\sum }_{m=1}^{+\infty }{l}_m{z}_m\right)}^{T}Q\left({\sum }_{m=1}^{+\infty }{l}_m{z}_m\right)\le \left({\sum }_{m=1}^{+\infty }{l}_m\right){\sum }_{m=1}^{+\infty }{l}_m{z}_m^{T}Q{z}_m$ where $z_m\in {\mathbb{R}}^n,Q\in {\mathbb{R}}^{n\times n}$ is a positive semidefinite matrix.

3. Problem Formulation

Consider the following discrete-time fractional stochastic multi-agent systems composed of the leader

$${\nabla }^{\alpha }{x}_0(k+1)=0,$$

(1)

and N followers

$${\nabla }^{\alpha }{x}_i(k+1)=\sigma \left(k\right)\zeta \left(k\right)+u_i\left(t\right),i=1,2,\dots ,N,$$

(2)

where $k\in \mathbb{N},\alpha \in (0,1),x_i\left(k\right)={\left[x_{i1}\left(k\right),x_{i2}\left(k\right),\dots ,x_{in}\left(k\right)\right]}^T\in {\mathbb{R}}^n$ is the state of the i th agent at time k; $\zeta \left(k\right)={\left[{\zeta }_1\left(k\right),{\zeta }_2\left(k\right),\dots ,{\zeta }_m\left(k\right)\right]}^T\in {\mathbb{R}}^m$ are system noises on $\left(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathrm{P}\right)$ satisfying the following statistical properties:

$$\begin{array}{cc}\hfill & E\left\{{\zeta }_i\left(k\right)\right\}=0,\hfill \\ \hfill & E\left\{{∥{\zeta }_i\left(k\right)∥}^2\right\}={\mu }^2,\hfill \\ \hfill & E\left\{{\zeta }_i\left(k\right){\zeta }_j\left(k\right)\right\}=0,i\ne j,i,j=1,2,\dots ,m\hfill \end{array}$$

(3)

and $\sigma (·):\mathbb{R}\to {\mathbb{R}}^{n\times m}$ is continuous nonlinear noise intensity function.

Assumption 1.

Assume noise intensity function $\sigma \left(k\right)$ satisfied

$$\sum _{k=1}^{\infty }{∥\sigma \left(k\right)∥}^2<+\infty ,\underset{k\to +\infty }{lim}∥\sigma \left(k\right)∥=0.$$

Defining a formation using a vector $h={\left[h_1^T,\dots ,h_N^T\right]}^T\in {\mathbb{R}}^{Nn}$, where $h_i\in {\mathbb{R}}^n$ represents the desired relative position of agent i with respect to the leader, the leader-following system is deemed to have achieved formation h in a mean square sense if

$$\underset{k\to \infty }{lim}\mathbb{E}∥x_i\left(k\right)-x_0\left(k\right)-h_i∥=0,i=1,2,...,N.$$

(4)

For this purpose, the control law is designed as follows:

$$u_i\left(k\right)=c\sum _{j=1}^N{g}_{ij}\Gamma (x_j\left(k\right)-h_j)-d_i\Gamma \left(x_i\left(k\right)-x_0\left(k\right)-h_i\right)-\sum _{r=1}^{k+1}{\omega }_r^{\alpha }{h}_i+h_i,$$

(5)

where $c>0$ represents the coupling strength. The matrix $\Gamma =diag\left\{{\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n\right\}$ is the internal linking matrix with ${\gamma }_i>0$. The matrix $G=\left[g_{ij}\right]\in {\mathbb{R}}^{N\times N}$ denotes the coupling configuration matrix, where $g_{ii}=-{\sum }_{j=1,j\ne i}^N{g}_{ij}$. If agent i can transmit information to agent j, then $g_{ij}>0$, ensuring ${\sum }_{j=1}^N{g}_{ij}=0$ for all $i=1,2,\dots ,N$. A directed path from agent $i_1$ to $i_p$ is represented by a sequence of edges $\left(i_1,i_2\right),\left(i_2,i_3\right),\dots ,\left(i_{p-1},i_p\right)$. Drawing inspiration from pinning control concepts, only a small subset of agents are selected for control. Without loss of generality, control actions are applied to the first l agents. This implies $d_i>0$ for $i=1,\dots ,l$, and $d_i=0$ for $i=l+1,\dots ,N$.

According to Definition 1 and Equation (5), then Equations (1) and (2) can be rewritten as

$$\begin{array}{cc}\hfill x_0(k+1)=& \sum _{r=1}^{k+1}{\omega }_r^{\alpha }{x}_0(k+1-r)\hfill \\ \hfill x_i(k+1)=& \sum _{r=1}^{k+1}{\omega }_r^{\alpha }{x}_i(k+1-r)+\sigma \left(k\right)\zeta \left(k\right)+c\sum _{j=1}^N{g}_{ij}\Gamma (x_j\left(k\right)-h_j)\hfill \\ \hfill & -d_i\Gamma \left(x_i\left(k\right)-x_0\left(k\right)-h_i\right)-\sum _{r=1}^{k+1}{\omega }_r^{\alpha }{h}_i+h_i,\hfill \end{array}$$

(6)

where the function $w_r^{\alpha }={(-1)}^{r+1}\left(\begin{array}{c}\alpha \hfill \\ r\hfill \end{array}\right)$ represents a heavy-tailed discretization following a power-law distribution. In the context of a multi-agent system modeled by a discrete-time difference equation with specific “time-delays”, where $x_i(k+1)$ relies on all historical states $\left\{x_i\left(0\right),x_i\left(1\right),\dots ,x_i\left(k\right)\right\}$, it is essential to consider that the impact of historical states on $x_i(k+1)$ may diminish as the time-delay increases. In this scenario, the function ${\omega }_r^{\alpha }$ is selected as the weight function.

Remark 1.

Unlike in the literature [40,41,42] in the original text on system modeling, which investigates integer-order time delay for discrete systems, this paper considers fractional-order distributed time delay. The propagation delays are distributed over a period of time and may influence the control system to oscillate more. Hence, it is mandatory to investigate the formation control systems with fractional-order distributed delay. Additionally, differing from the literature [32,34] in the original text on system modeling, the systems mentioned in this paper add noise term $\zeta \left(k\right)$, which considers the effect of disturbance in real life, having real significance.

To minimize the frequency of controller updates and communication overhead, a distributed event-triggered strategy is employed for the system described in Equation (6). Assuming that the first l nodes in the network are under control ($1\le l<N$), a pinning event-triggered control system is formulated as:

$$\begin{array}{cc}\hfill x_i(k+1)=& \sum _{r=1}^{k+1}{\omega }_r^{\alpha }{x}_i(k+1-r)+c\sum _{j=1}^N{g}_{ij}\Gamma (x_j\left(k_s^i\right)-h_j)\hfill \\ \hfill & +\sigma \left(k\right)\zeta \left(k\right)+h_i-\sum _{r=1}^{k+1}{\omega }_r^{\alpha }{h}_i-d_i\Gamma \left(x_i\left(k_s^i\right)-x_0\left(k_s^i\right)-h_i\right),\hfill \\ \hfill & k\in \left[k_s^i,k_{s+1}^i\right),i=1,2,\dots ,l,\hfill \\ \hfill x_i(k+1)=& \sum _{r=1}^{k+1}{\omega }_r^{\alpha }{x}_i(k+1-r)+h_i-\sum _{r=1}^{k+1}{\omega }_r^{\alpha }{h}_i+c\sum _{j=1}^N{g}_{ij}\Gamma (x_j\left(k_s^i\right)-h_i)+\sigma \left(k\right)\zeta \left(k\right),\hfill \\ \hfill & k\in \left[k_s^i,k_{s+1}^i\right),i=l+1,l+2,\dots ,N,\hfill \\ \hfill \end{array}$$

(7)

where the time sequence $k_0^i,k_1^i,\dots ,k_s^i,k_{s+1}^i,\dots$ represents the event triggering time sequence for agent i, which is generated by the following event-triggered mechanism.

Remark 2.

The pinning control adopted in this paper only requires controlling a subset of nodes to achieve control over the entire system. Compared to non-pinning control [43], the control strategy proposed in this paper is simpler and more efficient. In addition, unlike traditional continuous control [44], event-triggered control only transmits and updates information when triggered, saving resources and bandwidth.

Let the combined measurement of the systems Equation (5) be

$$q_i\left(k\right)=c\sum _{j=1}^N{g}_{ij}\Gamma (x_j\left(k\right)-h_i)-d_i\Gamma \left(x_i\left(k\right)-x_0\left(k\right)-h_i\right),$$

(8)

for $i=1,2,...,N$, and the measurement error be

$$e_i\left(k\right)=q_i\left(k_s^i\right)-q_i\left(k\right),i=1,2,\dots ,N$$

(9)

The sequence $k_0^i,k_1^i,\dots ,k_s^i,k_{s+1}^i,\dots$ for agent i is determined by the triggering condition specified in Theorem 1:

$${\varphi }_i\left(e_i\left(k\right),q_i\left(k\right)\right)=0,i=1,2,\dots ,N.$$

(10)

4. Main Result

Let $y_i\left(k\right)=x_i\left(k\right)-x_0\left(k\right)-h_i,i=1,2,\dots ,N$. From Equation (7),

$$\begin{array}{cc}\hfill y_i(k+1)=& \sum _{r=1}^{k+1}{\omega }_r^{\alpha }{y}_i(k+1-r)+\sigma \left(k\right)\zeta \left(k\right)+c\sum _{j=1}^N{g}_{ij}\Gamma (x_j\left(k_s^i\right)-h_j)-d_i\Gamma y_i\left(k_s^i\right),\hfill \\ \hfill & k\in \left[k_s^i,k_{s+1}^i\right),i=1,2,\dots ,l,\hfill \\ \hfill y_i(k+1)=& \sum _{r=1}^{k+1}{\omega }_r^{\alpha }{y}_i(k+1-r)+\sigma \left(k\right)\zeta \left(k\right)+c\sum _{j=1}^N{g}_{ij}\Gamma (x_j\left(k_s^i\right)-h_i)\hfill \\ \hfill & k\in \left[k_s^i,k_{s+1}^i\right),i=l+1,l+2,\dots ,N.\hfill \end{array}$$

(11)

Let $f_r^{\alpha }\triangleq {\omega }_{r+1}^{\alpha }$, then

$$\begin{array}{c}{\displaystyle \sum _{r=1}^{k+1}}{\omega }_r^{\alpha }{x}_i(k+1-r)={\omega }_1^{\alpha }{x}_i\left(k\right)+{\displaystyle \sum _{r=2}^{k+1}}{\omega }_r^{\alpha }{x}_i(k+1-r)\\ =\alpha x_i\left(k\right)+{\displaystyle \sum _{r=1}^k}{\omega }_{r+1}^{\alpha }{x}_i(k-r)=\alpha x_i\left(k\right)+{\displaystyle \sum _{r=1}^k}{f}_r^{\alpha }{x}_i(k-r).\end{array}$$

(12)

$\mathrm{By}\mathrm{the}\mathrm{gefinition}\mathrm{of}{q}_i\left(k\right)\mathrm{and}{e}_i\left(k\right),\mathrm{we}\mathrm{obtain}$

$$\begin{array}{cc}\hfill y_i(k+1)=& q_i\left(k\right)+e_i\left(k\right)+\sigma \left(k\right)\zeta \left(k\right)+\alpha y_i\left(k\right)+\sum _{r=1}^k{f}_r^{\alpha }{y}_i(k-r),\hfill \\ \hfill & k\in \left[k_s^i,k_{s+1}^i\right),i=1,2,\dots ,N,\hfill \end{array}$$

(13)

where $d_i=0$, $i=l+1,l+2,...,N$.

Denote

$$\begin{array}{c}y\left(k\right)={\left(y_1^{\mathrm{T}}\left(k\right),y_2^{\mathrm{T}}\left(k\right),\dots ,y_N^{\mathrm{T}}\left(k\right)\right)}^{\mathrm{T}},\\ e\left(k\right)={\left(e_1^{\mathrm{T}}\left(k\right),e_2^{\mathrm{T}}\left(k\right),\dots ,e_N^{\mathrm{T}}\left(k\right)\right)}^{\mathrm{T}},\end{array}$$

(14)

and $q\left(k\right)={\left(q_1^{\mathrm{T}}\left(k\right),q_2^{\mathrm{T}}\left(k\right),\dots ,q_N^{\mathrm{T}}\left(k\right)\right)}^{\mathrm{T}}$, then

$$q\left(k\right)=\left(\right(cG-D)\otimes \Gamma )y\left(k\right)$$

(15)

where $D=diag\left\{d_1,d_2,\dots ,d_l,0,\dots ,0\right\}$.

Thus,

$$\begin{array}{cc}\hfill y(k+1)=& ((cG-D)\otimes \Gamma )y\left(k\right)+e\left(k\right)+\alpha y\left(k\right)+\sum _{r=1}^k{f}_r^{\alpha }y(k-r)+\left(1_N\otimes \sigma \left(k\right)\right)\zeta \left(k\right)\hfill \\ \hfill =& [\alpha I_{n\times N}+(cG-D)\otimes \Gamma ]y\left(k\right)+e\left(k\right)+\sum _{r=1}^k{f}_r^{\alpha }y(k-r)+\left(1_N\otimes \sigma \left(k\right)\right)\zeta \left(k\right).\hfill \end{array}$$

(16)

Theorem 1.

Under Assumption 1, the system Equation (6) will realize formation control if there exists ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3>0$ such that

$$\begin{array}{cc}\hfill H\triangleq & (1-{\eta }_2)I_{n\times N}-2\alpha {\eta }_1\left(c\frac{G+G^T}{2}-D\right)\otimes \Gamma -{\eta }_1{(cG-D)}^T(cG-D)\otimes \Gamma \Gamma >0,\hfill \end{array}$$

(17)

where ${\eta }_1=1+{\epsilon }_1+{\epsilon }_2$, ${\eta }_2={\eta }_1{\alpha }^2+4(1-\alpha )(1+\frac{1}{{\epsilon }_1}+{\epsilon }_3)$ and the triggering condition as follows

$${\varphi }_i\left(e_i\left(k\right),q_i\left(k\right)\right)={∥e_i\left(k\right)∥}^2-\frac{\left(1-{\nu }_i\right){\lambda }_{min}\left(H\right)}{{\eta }_3{\lambda }_{max}\left(A\right)}{∥q_i\left(k\right)∥}^2\le 0,i=1,2,\dots ,N$$

(18)

where ${\eta }_3=1+\frac{1}{{\epsilon }_2}+\frac{1}{{\epsilon }_3}$, $A={((cG-D)\otimes \Gamma )}^{\mathrm{T}}((cG-D)\otimes \Gamma )$ and $0<{\nu }_i<1$.

Proof. 

Construct the Lyapunov function as follows:

$$V\left(k\right)=V_1\left(k\right)+V_2\left(k\right)+V_3\left(k\right),$$

(19)

with

$$\begin{array}{cc}\hfill & V_1\left(k\right)=y^T\left(k\right)y\left(k\right),\hfill \\ \hfill & V_2\left(k\right)=\sum _{r=1}^{+\infty }{f}_r^{\alpha }\sum _{j=k-r}^{k-1}{y}^T\left(j\right)y\left(j\right),\hfill \\ \hfill & V_3\left(k\right)=\sum _{r=k+1}^{+\infty }{f}_r^{\alpha }\sum _{j=k-r}^{k-1}{y}^T\left(j\right)y\left(j\right),\hfill \end{array}$$

(20)

where $V_2\left(k\right),V_3\left(k\right)<+\infty$ according to Lemma 1. Then

$$\begin{array}{cc}\hfill & \mathbb{E}\left[\Delta V_1\left(k\right)\right]\hfill \\ \hfill =& \mathbb{E}\left\{y^T\left(k\right)(\alpha I_{n\times N}+{(cG-D)}^T\otimes \Gamma )\right.\left[\left(\alpha I_{n\times N}+(cG-D)\otimes \Gamma \right)y\left(k\right)\right.\hfill \\ \hfill & +2\sum _{r=1}^k{f}_r^{\alpha }y(k-r)+2\left(1_N\otimes \sigma \left(k\right)\right)\zeta \left(k\right)+2e\left(k\right)]\hfill \\ \hfill & +{\left(\sum _{r=1}^k{f}_r^{\alpha }y(k-r)\right)}^T\left(\sum _{r=1}^k{f}_r^{\alpha }y(k-r)+2\left(1_N\otimes \sigma \left(k\right)\right)\zeta \left(k\right)+2e\left(k\right)\right)\hfill \\ \hfill & +e^T\left(k\right)\left[e\left(k\right)+2\left(1_N\otimes \sigma \left(k\right)\right)\zeta \left(k\right)\right]\left.+{\zeta }^T\left(k\right){\left(1_N\otimes \sigma \left(k\right)\right)}^T\left(1_N\otimes \sigma \left(k\right)\right)\zeta \left(t\right)\right\}.\hfill \end{array}$$

(21)

$\mathrm{From}\mathrm{the}\mathrm{properties}\mathrm{of}\mathrm{the}\mathrm{stochastic}\mathrm{process}\left\{\zeta \right(k),k\ge 0\}$ in Equation (3), then

$$\begin{array}{cc}\hfill & \mathbb{E}\left[\Delta V_1\left(k\right)\right]\hfill \\ \hfill =& \mathbb{E}\left\{y^T\left(k\right)(\alpha I_{n\times N}+{(cG-D)}^T\otimes \Gamma )\right.\left[\left(\alpha I_{n\times N}+(cG-D)\otimes \Gamma \right)y\left(k\right)\right.\hfill \\ \hfill & +2\sum _{r=1}^k{f}_r^{\alpha }y(k-r)+2e\left(k\right)]+{\left(\sum _{r=1}^k{f}_r^{\alpha }y(k-r)\right)}^T\left(\sum _{r=1}^k{f}_r^{\alpha }y(k-r)\right.\hfill \\ \hfill & \left.+2e\left(k\right)\right)+e^T\left(k\right)e\left(k\right)\left.+{\zeta }^T\left(k\right){\left(1_N\otimes \sigma \left(t\right)\right)}^T\left(1_N\otimes \sigma \left(t\right)\right)\zeta \left(k\right)\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{y^T\left(k\right)\left[\left(1+{\epsilon }_2+{\epsilon }_3\right)\right.\right.\left(\alpha I_{n\times N}+{(cG-D)}^T\otimes \Gamma \right)\hfill \\ \hfill & \left.\times \left(\alpha I_{n\times N}+(cG-D)\otimes \Gamma \right)-I_{n\times N}\right]y\left(k\right)\hfill \\ \hfill & +\left(1+\frac{1}{{\epsilon }_1}+{\epsilon }_3\right){\left(\sum _{r=1}^{+\infty }{f}_r^{\alpha }y(k-r)\right)}^T\left(\sum _{r=1}^{+\infty }{f}_r^{\alpha }y(k-r)\right)\hfill \\ \hfill & +\left(1+\frac{1}{{\epsilon }_1}+{\epsilon }_3\right){\left(\sum _{r=k+1}^{+\infty }{f}_r^{\alpha }y(k-r)\right)}^T\left(\sum _{r=k+1}^{+\infty }{f}_r^{\alpha }y(k-r)\right)\hfill \\ \hfill & \left.+e^T\left(k\right)e\left(k\right)+{\zeta }^T\left(k\right){\left(1_N\otimes \sigma \left(t\right)\right)}^T\left(1_N\otimes \sigma \left(t\right)\right)\zeta \left(k\right)\right\}.\hfill \end{array}$$

(22)

Moreover,

$$\begin{array}{cc}\hfill & \mathbb{E}\left[\Delta V_2\left(k\right)\right]\hfill \\ \hfill =& \mathbb{E}\left[\sum _{r=1}^{+\infty }{f}_r^{\alpha }\sum _{j=k+1-r}^k{y}^T\left(j\right)y\left(j\right)-\sum _{r=1}^{+\infty }{f}_r^{\alpha }\sum _{j=k-r}^k{y}^T\left(j\right)y\left(j\right)\right]\hfill \\ \hfill =& \mathbb{E}\left[\sum _{r=1}^{+\infty }{f}_r^{\alpha }{y}^T\left(k\right)y\left(k\right)-\sum _{r=1}^{+\infty }{f}_r^{\alpha }{y}^T(k-r)y(k-r)\right].\hfill \end{array}$$

(23)

According to Lemma 1, then ${\sum }_{r=1}^{+\infty }{f}_r^{\alpha }=1-\alpha \in (0,1)$. From Lemma 3, then

$$\sum _{r=1}^{+\infty }{f}_r^{\alpha }{y}^T(k-r)y(k-r)\ge \frac{1}{1-\alpha }{\left(\sum _{r=1}^{+\infty }{f}_r^{\alpha }y(k-r)\right)}^T\left(\sum _{r=1}^{+\infty }{f}_r^{\alpha }y(k-r)\right)$$

(24)

Thus,

$$\mathbb{E}\left[\Delta V_2\left(k\right)\right]\le \mathbb{E}\left[y^T\left(k\right)y\left(k\right)-\frac{1}{1-\alpha }{\left(\sum _{r=1}^{+\infty }{f}_r^{\alpha }y(k-r)\right)}^T\left(\sum _{r=1}^{+\infty }{f}_r^{\alpha }y(k-r)\right)\right].$$

(25)

Furthermore,

$$\begin{array}{cc}\hfill & \mathbb{E}\left[\Delta V_3\left(k\right)\right]\hfill \\ \hfill =& \mathbb{E}\left[\sum _{r=k+2}^{+\infty }{f}_r^{\alpha }\sum _{j=k+1-r}^k{y}^T\left(j\right)y\left(j\right)-\sum _{r=k+1}^{+\infty }{f}_r^{\alpha }\sum _{j=k-r}^{k-1}{y}^T\left(j\right)y\left(j\right)\right]\hfill \\ \hfill \le & \mathbb{E}\left[\sum _{r=k+1}^{+\infty }{f}_r^{\alpha }{y}^T\left(k\right)y\left(k\right)-\sum _{r=k+1}^{+\infty }{f}_r^{\alpha }{y}^T(k-r)y(k-r)\right]\hfill \\ \hfill \le & \mathbb{E}\left[y^T\left(k\right)y\left(k\right)-\frac{1}{1-\alpha }{\left(\sum _{r=k+1}^{+\infty }{f}_r^{\alpha }y(k-r)\right)}^T\left(\sum _{r=k+1}^{+\infty }{f}_r^{\alpha }y(k-r)\right)\right].\hfill \end{array}$$

(26)

Combining Equations (19)–(26) gives

$$\begin{array}{cc}\hfill & \mathbb{E}\{\Delta V(k\left)\right\}\hfill \\ \hfill =& \mathbb{E}\left\{\Delta V_1\left(k\right)+\Delta V_2\left(k\right)+\Delta V_3\left(k\right)\right\}\hfill \\ \hfill \le & \mathbb{E}\left\{-y^{\mathrm{T}}\left(k\right)Hy\left(k\right)+e^{\mathrm{T}}\left(k\right)e\left(k\right)\right\}\hfill \\ \hfill \le & -{\lambda }_{min}\left(H\right)\mathbb{E}\left\{{\parallel y\left(k\right)\parallel }^2\right\}+\mathbb{E}\left\{{\parallel e\left(k\right)\parallel }^2\right\}+Nm{\mu }^2{∥\sigma \left(t\right)∥}^2.\hfill \end{array}$$

(27)

Then, from Equation (18),

$$\begin{array}{cc}\hfill {\parallel e\left(k\right)\parallel }^2& \le \sum _{i=1}^N{∥e_i\left(k\right)∥}^2\hfill \\ \hfill & \le \frac{\left(1-{\nu }_i\right){\lambda }_{min}\left(H\right)}{{\eta }_3{\lambda }_{max}\left(A\right)}\sum _{i=1}^N{∥q_i\left(k\right)∥}^2\hfill \\ \hfill & =\frac{\left(1-{\nu }_i\right){\lambda }_{min}\left(H\right)}{{\eta }_3{\lambda }_{max}\left(A\right)}{\parallel q\left(k\right)\parallel }^2\hfill \\ \hfill & =\frac{\left(1-{\nu }_i\right){\lambda }_{min}\left(H\right)}{{\eta }_3{\lambda }_{max}\left(A\right)}{\parallel ((cW-\gamma \left(k\right)D)\otimes \Gamma )y\left(k\right)\parallel }^2\hfill \\ \hfill & \le \frac{(1-\nu ){\lambda }_{min}\left(H\right)}{{\eta }_3}{\parallel y\left(k\right)\parallel }^2,\hfill \end{array}$$

(28)

where $\nu =max\left\{{\nu }_i\right\}$. Therefore,

$$\mathbb{E}\{\Delta V\left(k\right)\}\le -\epsilon \mathbb{E}\left\{{∥y\left(k\right)∥}^2\right\}+Nm{\mu }^2{∥\sigma \left(t\right)∥}^2.$$

(29)

where $\epsilon =\nu {\lambda }_{min}\left(H\right)>0$.

Then, from Assumption 1, we can obtain $Nm{\mu }^2{∥\sigma \left(k\right)∥}^2>0$, $Nm{\sum }_{k=1}^{\infty }{∥\sigma \left(k\right)∥}^2<+\infty$, ${\sum }_{k=1}^{\infty }\epsilon =+\infty$ and ${lim}_{k\to +\infty }\frac{\sigma \left(k\right)}{\epsilon }=0$, which satisfies the condition of Lemma 2. Thus, from Lemma 2, ${lim}_{k\to \infty }\mathbb{E}∥x_i\left(k\right)-x_0\left(k\right)-h_i∥=0,\forall i=1,2,\dots ,N$. □

5. Simulation

The directed communication graph given by Figure 1 is composed of four agents. The communication topology shown in Figure 1 has a spanning tree, with agent 1 as the root node. Let $N=4,n=3,\alpha =0.88,\Gamma =diag\{1,1,1\},d_1=d_2=d_3=d_4=0.2,c=0.02,\sigma \left(k\right)=\frac{3}{{(k+1)}^2}$, and the initial position is as follows:

$$\begin{array}{cc}& x_0\left(0\right)={\left(\begin{array}{ccc}1.1\hfill & -0.9\hfill & 0.3\hfill \end{array}\right)}^T\hfill \\ & x_1\left(0\right)={\left(\begin{array}{ccc}1.8\hfill & 0.9\hfill & -1.1\hfill \end{array}\right)}^T,x_2\left(0\right)={\left(\begin{array}{ccc}0.7\hfill & -0.6\hfill & -1.1\hfill \end{array}\right)}^T,\hfill \\ & x_3\left(0\right)={\left(\begin{array}{ccc}-0.9\hfill & 1.1\hfill & -1.4\hfill \end{array}\right)}^T,x_4\left(0\right)={\left(\begin{array}{ccc}1.4\hfill & -0.8\hfill & -0.6\hfill \end{array}\right)}^T.\hfill \end{array}$$

Choose the formation vector

$$h=\left(\begin{array}{cccc}3& -3& 5& -5\\ -3& 3& 4& -4\\ -3& 3& 3& -3\end{array}\right)$$

for the leader-following system. For the triggering condition (17),

$${∥e_i\left(k\right)∥}^2\le \frac{\left(1-{\nu }_i\right){\lambda }_{min}\left(H\right)}{{\eta }_3{\lambda }_{max}\left(A\right)}{∥q_i\left(k\right)∥}^2,i=1,2,\dots ,N$$

It can be calculated that $H=\left[\begin{array}{cccc}-3.7600& 0.0048& 0.0056& 0\\ 0.0048& -3.7475& 0.0056& 0\\ 0.0056& 0.0056& -3.7475& 0.0048\\ 0& 0& 0.0048& -3.7360\end{array}\right]\otimes I_3<0$, which satisfies the condition of Theorem 1. Note that the definitions of H and A are independent of ${\mu }_i$, so regardless of the sizes of H and A, it is always possible to find three sets suitable ${\mu }_i\in (0,1)$ and ${\eta }_3$ such that $\frac{\left(1-{\mu }_i\right){\lambda }_{min}\left(H\right)}{{\eta }_3{\lambda }_{max}\left(A\right)}=0.75$, $\frac{\left(1-{\mu }_i\right){\lambda }_{min}\left(H\right)}{{\eta }_3{\lambda }_{max}\left(A\right)}=0.9$, and $\frac{\left(1-{\mu }_i\right){\lambda }_{min}\left(H\right)}{{\eta }_3{\lambda }_{max}\left(A\right)}=1.05$, respectively. Therefore, we can assume that the event-triggering conditions used in this simulation are ${∥e_i\left(k\right)∥}^2\le 0.75∥q_i\left(k\right)∥$, ${∥e_i\left(k\right)∥}^2\le 0.9∥q_i\left(k\right)∥$, and ${∥e_i\left(k\right)∥}^2\le 1.05∥q_i\left(k\right)∥$, separately. Obviously, from Figure 2, it can be observed that the state of the error system tends to zero over time, indicating that systems (1) and (2) achieve formation control under the requirements of Theorem 1. Figure 3, Figure 4 and Figure 5 show the moments of updating the control law; it can be seen that as the trigger parameter increases, the number of controller updates becomes fewer, revealing that the larger the threshold value, the fewer the number of updates to the controller. Thus, event-triggered control reduces the update frequency compared to continuous control, which requires real-time monitoring of agent information, saving the resources needed for monitoring and interacting information. Figure 6 demonstrates that the followers eventually stabilize at a fixed position and maintain a constant relative position with the leader, thus achieving formation control. Figure 7, Figure 8, Figure 9 and Figure 10 depict their specific positions in space, showing that the followers stabilize over time, indicating successful formation control.

6. Conclusions

We studied leader-following formation control for discrete-time fractional stochastic multi-agent systems by an event-triggered strategy. First, a new discrete-time fractional stochastic system is modeled, which with stochastic terms can better characterize systems operating under the influence of disturbances and cases of restricted communication. Second, an event-triggered control law is proposed for controlling partial followers, with the aim of making all the agents eventually reach their expected positions and achieve a certain shape of motion. Finally, the theory of Lyapunov’s energy equation is employed to prove that under the impact of disturbance, the formation error eventually tends to zero in the mean square, and formation control can be achieved. However, the formation formed in this way can only achieve translational motion in the end, so the affine formation control with higher maneuverability and flexibility will be studied in the future.