Event-Triggered Time-Varying Formation Tracking Control for Multi-Agent Systems with a Switching-Directed Topology

Abstract

This study investigates the problem of time-varying formation tracking (TVFT) control involving event-triggered and switching topological mechanisms. Specifically, TVFT is evaluated with a consensus analysis and deduced via the use of linear matrix inequality techniques combined with Lyapunov stability theory. This strategy obtains sufficient conditions for system stability and the feedback and coupling gains. In addition, the TVFT compensational signals are presented in two cases to enhance the algorithm’s applicability. Given that ideal multi-agent systems (MASs) should be highly flexible and resilient, we propose a co-design algorithm that strikes a balance between the need for a lower communication frequency and a reduction in the state disagreements of agents. Finally, the effectiveness of the theoretical analysis is demonstrated through 3D figures and comparison tables, from which it can be concluded that the communication frequency of the MAS was clearly reduced on the basis of ensuring consensus performance via applying the algorithm proposed in this paper.

1. Introduction

The study of formation tracking control for multi-agent systems (MASs) is significant for practical applications, such as marine exploration [1], reconnaissance [2], localization [3], satellite service [4], and drag reduction [5]. Typical mobile robot platforms, such as wheeled robots [6] and unmanned aerial vehicles (UAVs) [7], achieve the desired formation tracking via the use of the leader–follower control strategy. In this strategy, the desired formation preserves a predefined geometric configuration and tracks a certain trajectory generated via an actual or virtual leader. An ideal MAS should have high flexibility, resilience, robustness, and cooperation while conserving energy. Specifically, such systems should have an appropriate communication strategy for realizing interactions between agents. They should also possess a flexible switching communication topology that adjusts to the actual environment and a suitable working frequency to ensure the system’s continuous operation.

Behavioral-, virtual structure/virtual leader-, and leader–follower-based approaches are typical formation control strategies [8,9,10] that can be categorized as special cases of consensus-based approaches [11]. Through local neighboring interactions, the consensus control problem of MASs is fundamental in formation control. For instance, Zheng et al. [12] considered the consensus problem of hybrid MASs, and Xiao et al. [13] studied the formation control problem for low-order MASs in which consensus-based strategies were utilized. Considering more complex tasks in practical applications, such as formation flying and cooperative guidance cases, the expected formation should be time-varying [14,15,16]. Therefore, Deng et al. [17] introduced an adaptive practical optimal time-varying formation tracking (TVFT) protocol. However, a fixed-topology structure cannot adapt to unreliable agents’ communication due to random events, such as possible faults in individual sensor nodes, channel blocking, and spiteful packet dropping. Thus, the ideal topology of MASs should be diversified and switchable; the literature contains various works on switching topologies [18,19,20,21,22]. The authors of [23] proposed a switching topology protocol scheme that effectively reaches the global practical leader–follower consensus under any initial conditions. Moreover, the communication between agents in undirected interaction topologies is bidirected, which means that twice the energy resources are consumed compared to directed interaction topologies. Switching-directed topologies were applied to time-varying formation control theory for MASs in [24]. However, although the interaction topologies in the above-mentioned literature were switched and directed, they were mainly determined via a switching signal from external events rather than the states of the MASs. Assuming that the topology can be determined via a predefined function, the system performance may be improved through switching the directed topology, which leads to the primary motivation of this study.

In addition, communication can be conventionally maintained via the use of a time-triggered scheme with equidistant sampling, which consumes too much power due to persistent periodic radio signals. Furthermore, time-triggered schemes are computationally expensive, as continual updates of sampled signals are usually transformed into radio signals. Obviously, when there is no significant change in the equidistant sampling signals to be transmitted, the later-transmitted signals contain hardly any additional valuable information. If these relatively redundant sampling signals are allowed to be discarded or no data are sampled at settled intervals, the power consumption and computational complexity will be reduced; thus, event-triggered strategies in which agents communicate and update the sampled data only when well-designed event-triggered conditions are satisfied have been conceived [25,26,27,28]. Event-triggered mechanisms usually execute trigger rules via event generators, but they generate extra computations, leading to the consideration of their complexity and physical realizability. E. Aranda-Escolástico et al. [29] summarized the merits and drawbacks of event-based control from a technical perspective. In accordance with threshold functions, Zhang, X. M. et al. [30] classified typical event-triggered mechanisms into four types: static, dynamic, and stochastic event-triggered control and self-triggered sampling mechanisms. Dynamic event-triggered control generally generates fewer data samples than the static version; however, static event-triggered control is more practical to execute. Ali Kazemy et al. [31] designed two static event-triggered mechanisms in both a sensor–controller channel and a controller–actuator channel. Important properties of MASs, such as their energy-saving properties and their static and dynamic performance, are correlated with event-triggered criteria. Karl Johan Åström et al. [32] investigated several event-based sampling methods and compared their performance by means of steady-state variance and average sampling period analysis; the results showed the effects of different criteria and different parameters with the same criterion. For event-triggered control in MASs, it should be noted that each agent has individual event-triggered instants for communication; the design of event-triggered conditions and control protocols is an active research area. For example, an event-triggered output feedback control method for MASs with linear dynamics was presented in [33], where each agent used information from its individual states at the current instant, the last event-triggered instants for the next event-triggered instant, and the states of the neighbors at the last time instant for the design of the control protocol. The authors of [34] developed a dynamic event-triggered communication mechanism for MASs in which the threshold parameter in the event-triggered condition could be dynamically adjusted according to a designed rule. Ali Azarbahram et al. [35] proposed an event-triggered data transmission strategy for networked systems via the use of the dead-zone constraint. The authors of [36] presented a consensus on MASs with arbitrarily switching communication topologies. Therefore, inspired by the references above, we developed a scheme with a switching-directed topology and an event-triggered mechanism that applied TVFT control for MASs.

In this study, a TVFT control strategy that used an event-triggered mechanism and switching-directed topology, which allowed the MAS to achieve the desired astringency and energy-saving performance, was investigated. The main motivations of this study were the provision of a detailed theoretical basis for formation control applications, such as those mentioned above, or other applications in which the control object can be modeled as time-varying functions, as well as to make a breakthrough in terms of energy consumption and resilience performance.

The main contributions of our work are summarized as follows:

(1) An event-triggered control protocol based on an MAS in which the agents communicate according to the local state difference between the real instants and the most recently triggered instants is presented. The state errors of the communication are limited depending on the designed parameter during the communication period.

(2) A new automatic switching-directed topology design is proposed, and it can be applied when the system does not need specific switching signals. This topology further optimizes the astringency performance based on information exchange.

(3) A novel switching-directed topology strategy based on event-triggered mechanisms is developed to ensure that an MAS can carry out TVFT. The main TVFT and consensus analysis are formulated and expressed via LMIs.

The remainder of this study is organized as follows. Section 2 presents the problem statement in relation to TVFT control. Section 3 introduces and verifies the event-triggered control protocol, switching-directed topology design, and compensation function. Section 4 presents a co-design algorithm, and Section 5 demonstrates the performance of the proposed scheme through two numerical examples. Finally, Section 6 concludes this work.

2. Preliminaries and Problem Statement

2.1. Basic Concepts of Graph Theory

Let $G=\left\{O,E,W\right\}$ be a weighted directed graph with N nodes, where $O=\left\{o_1,o_2,\dots ,o_N\right\}$ denotes a set of nodes and $E\subseteq \left\{(o_i,o_j):o_i,o_j\in O,i\ne j\right\}$ is a set of directed edges $e_{ij}=(o_i,o_j)\left(i\ne j\right)$ so that node $o_j$ can obtain information from node $o_i$. $W=\left[w_{ij}\right]\in {\mathbb{R}}^{N\times N}$ represents a weighted adjacency matrix for which $w_{ij}=1$, if $(o_i,o_j)\in E$. Otherwise, $w_{ij}=0$. Let $N_i=\left\{o_j\in O:e_{ji}\in E\right\}$ be the neighbor set of node $o_i$. The in-degree of node $o_i$ is defined as ${\mathrm{deg}}_{in}\left(o_i\right)={\sum }_{j=1}^N{w}_{ij}$, and the out-degree of node $o_i$ is defined as ${\mathrm{deg}}_{out}\left(o_i\right)={\sum }_{j=1}^N{w}_{ji}$.

We assume that $\sigma \left(t\right):[0,+\infty )\to \left\{1,2,\dots ,p\right\}$ is a switching signal and that $p\in \mathbb{N}$ is the total number of possible graphs. The index of the topology at time t is represented by the value of $\sigma \left(t\right)$. Let $w_{ij}^{\sigma \left(t\right)}\left(i,j\in \left\{1,2,\dots ,N\right\}\right)$ be the weight at $\sigma \left(t\right)$. $N_i^{\sigma \left(t\right)},G_{\sigma \left(t\right)},andlet{W}_{\sigma \left(t\right)}$ represent the neighbor set of agent i, the directed topology, and the weighted adjacency matrix at $\sigma \left(t\right)$, respectively.

2.2. Problem Formulation

We consider a second-order MAS and suppose that the dynamics of agent i are described by

$${\dot{x}}_i\left(t\right)=A{x}_i\left(t\right)+B{u}_i\left(t\right)$$

(1)

where $i=1,2,\dots ,N$, and $x_i\left(t\right)\in {\mathbb{R}}^n$ and $u_i\left(t\right)\in {\mathbb{R}}^m$ denote the state and the control input of the ith agent, respectively. $A\in {\mathbb{R}}^{n\times n},B\in {\mathbb{R}}^{n\times m}$ are known matrices.

The leader node is labeled as 0, with the following dynamics:

$${\dot{x}}_0\left(t\right)=A{x}_0\left(t\right)+B{u}_0\left(t\right).$$

(2)

For $x\left(t\right)={\left[x_1\left(t\right),x_2\left(t\right),\dots ,x_N\left(t\right)\right]}^T$ and $u\left(t\right)={\left[u_1\left(t\right),u_2\left(t\right),\dots ,u_N\left(t\right)\right]}^T$, MAS (1) can be written in a compact form as follows:

$$\dot{x}\left(t\right)=\left(I_N\otimes A\right)x\left(t\right)+\left(I_N\otimes B\right)u\left(t\right).$$

(3)

Definition 1.

MAS (1) is said to achieve TVFT $h\left(t\right)={\left[h_1\left(t\right),h_2\left(t\right),\dots ,h_N\left(t\right)\right]}^T$ if, for any given bounded initial states, the following is satisfied:

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

(4)

Remark 1.

One of the reasons to use leader–follower-based approaches is the assumption of the position of the leader to be the coordinate origin of the local coordinate systems. In this case, the followers can accept the relative positions of the leader from the desired formation $h\left(t\right)$ and interact with each other to maintain their formation as a rigid body. Then, the agents’ positions in a global coordinate system can be deduced via the leader’s position in this coordinate system and the desired formation $h\left(t\right)$.

This study proposes a triggered scheme that was inspired by the event-triggered designs in [37]. Figure 1 depicts the framework of an MAS with an event-triggered communication scheme, where the solid line represents full transmission, for which the signal must be transmitted at each instant. The dotted line represents partial transmission, i.e., the signal is only transmitted when the event generator is triggered. In contrast to [38], it is observed that the individual followers upload interactive information to the communication network when the event generators in neighbor followers are triggered; rather than using a mode that accepts information at the local triggered instants, the system can sense variation more sensitively, and it can avoid interference effects by means of the special switching-directed topology strategy that will be introduced in Section 4 of this study.

We consider the following condition:

$$t_{k+1}^i=inf\left\{t\mid {e_i}^T\left(t\right){\Omega }_i{e}_i\left(t\right)>{\epsilon {\theta }_i}^T\left(t\right){\Omega }_i{\theta }_i\left(t\right)\right\}$$

(5)

where ${\theta }_i\left(t\right)=x_i\left(t\right)-h_i\left(t\right)-x_0\left(t\right)$, $e_i\left(t\right)=x_i\left(t_k^i\right)-h_i\left(t_k^i\right)-x_i\left(t\right)+h_i\left(t\right)={\theta }_i\left(t_k^i\right)-{\theta }_i\left(t\right)$, $\epsilon \in (0,1)$ is a designed parameter, $t_k^i$ is the latest event-triggered instant of follower i, and $\Omega$ is a following-designed positive definite weighting matrix.

Remark 2.

It is not difficult to see that the event-triggered instant that updates the system’s state is determined via the measurement error between the triggered instant and the actual instant in the communication process. During the time in which the error is within the allowable range, no communication is conducted. One can observe in Figure 1 that the event generators only need to accept information from agents locally; the hardware structure is simple, and it has low computational difficulty. The event-triggered instants are decided via the norms of measurement errors $e_i\left(t\right)$ and tracking errors ${\theta }_i\left(t\right)$ in the form of $∥\mathit{a}∥={\mathit{a}}^{\mathrm{T}}\Omega \mathit{a},\Omega >0$, where $\mathit{a}$ represents a vector, and the weight matrix is obtained using the restrictions and optimization conditions; thus, one can select the desired weight matrix for one’s specific needs.

Next, we consider the event-triggered TVFT protocol for MAS (1) with switching topologies.

$$\begin{array}{cc}\hfill u_i\left(t\right)=& K_i^1\left(x_i\left(t\right)-h_i\left(t\right)-x_0\left(t\right)\right)+v_i\left(t\right)\hfill \\ \hfill & +K_{ij}^2\sum _{j\in N_i}\left(x_j\left(t_k^j\right)-x_i\left(t_k^i\right)-\left(h_j\left(t_k^j\right)-h_i\left(t_k^i\right)\right)\right)\hfill \end{array}$$

(6)

where $K_i^1\in {\mathbb{R}}^{m\times n}$ and $K_{ij}^2\in {\mathbb{R}}^{m\times n}\left(i\ne j\right)$ denote the feedback gain matrices and coupling gain matrices with appropriate dimensions to be designed later, respectively. $h\left(t\right)$ is a constant vector that describes the desired formation.

Substituting ${\theta }_i$ and ${\theta }_j$ into protocol (6), one can obtain

$$\begin{array}{cc}\hfill u_i\left(t\right)& =K_i^1{\theta }_i\left(t\right)+K_{ij}^2\sum _{j=1}^N{w}_{ij}^{\sigma \left(t\right)}\left({\theta }_j\left(t_k^j\right)-{\theta }_i\left(t_k^i\right)\right)+v_i\left(t\right)\hfill \\ \hfill & =K_i^1{\theta }_i\left(t\right)+\sum _{j=1}^N{w}_{ij}^{\sigma \left(t\right)}{K}_{ij}^2\left(\left({\theta }_j\left(t\right)+e_j\left(t\right)\right)-\left({\theta }_i\left(t\right)+e_i\left(t\right)\right)\right)+v_i\left(t\right)\hfill \end{array}$$

(7)

and

$$\begin{array}{cc}\hfill u\left(t\right)=& K^1\theta \left(t\right)+\left(W_{\sigma \left(t\right)}\otimes K^2\right)\left(\theta \left(t\right)+e\left(t\right)\right)\hfill \\ \hfill & -{\mathrm{diag}}_N\left\{\sum _{j=1}^N\left(w_{ij}^{\sigma \left(t\right)}{I}_m\right)K_{ij}^2\right\}\left(\theta \left(t\right)+e\left(t\right)\right)+v\left(t\right)\hfill \end{array}$$

(8)

where $\theta \left(t\right)={\left[{\theta }_1\left(t\right),{\theta }_2\left(t\right),\dots ,{\theta }_N\left(t\right)\right]}^T$, $h\left(t\right)={\left[h_1\left(t\right),h_2\left(t\right),\dots ,h_N\left(t\right)\right]}^T$, $e\left(t\right)=\left[e_1\left(t\right),\right.{\left.e_2\left(t\right),\dots ,e_N\left(t\right)\right]}^T$, $K^1={\mathrm{diag}}_N\left\{K_i^1\right\}\in {\mathbb{R}}^{mN\times nN}$, $K^2={\left[K_{ij}^2\right]}_{N\times N}\in {\mathbb{R}}^{mN\times nN},$$K_{ij}^2\left(i=j\right)=0$.

In order to simplify the calculations, let $K_{ij}=K_{ji}\left(i\ne j\right)$, and we define the following:

$\begin{array}{l}{\chi }_i=\mathrm{diag}\left\{\underset{i-1}{\underbrace{0,\dots }},\underset{i}{1},\underset{N-i}{\underbrace{\dots ,0}}\right\}\in {\mathbb{R}}^{N\times N}\\ {\chi }_i^1=\mathrm{diag}\left\{\underset{i-1}{\underbrace{0,\dots }},\underset{i}{1},\underset{N-i}{\underbrace{\dots ,0}}\right\}\otimes I_m\in {\mathbb{R}}^{mN\times mN}\\ {\chi }_i^2=\mathrm{diag}\left\{\underset{i-1}{\underbrace{0,\dots }},\underset{i}{1},\underset{N-i}{\underbrace{\dots ,0}}\right\}\otimes I_n\in {\mathbb{R}}^{nN\times nN}.\end{array}$

Then, protocol (8) can be described as:

$$\begin{array}{cc}\hfill u\left(t\right)=& K^1\theta \left(t\right)\hfill \\ \hfill & +\left(\sum _{j=1}^N{\stackrel{`}{W}}_j^{\sigma \left(t\right)}{K}^2{\chi }_j^2-\sum _{j=1}^N{\chi }_j^1\left(W_{\sigma \left(t\right)}\otimes I_n\right)K^2{\chi }_j^2\right)\left(\theta \left(t\right)+e\left(t\right)\right)+v\left(t\right)\hfill \end{array}$$

(9)

where ${\stackrel{`}{W}}_j^{\sigma \left(t\right)}={\mathrm{diag}}_N\left\{w_{ij}^{\sigma \left(t\right)}{I}_m\right\}=\mathrm{diag}\left\{w_{1j}^{\sigma \left(t\right)}{I}_m,\dots ,w_{Nj}^{\sigma \left(t\right)}{I}_m\right\}\in {\mathbb{R}}^{mN\times mN},j=1,2,\dots ,N$.

Remark 3.

The Kronecker product that contains the variable $K_{ij}$ cannot be directly dealt with via the LMI toolbox; as a result, the corresponding Kronecker product with $K_{ij}$ in protocol (8) is transformed into the matrix product in (9).

Substituting (9) into (3), the MAS with an event-triggered communication scheme can be rewritten as:

$$\begin{array}{cc}\hfill \dot{x}\left(t\right)=& \left(I_N\otimes A\right)x\left(t\right)+\left(I_N\otimes B\right)u\left(t\right)\hfill \\ \hfill =& \left(I_N\otimes A\right)x\left(t\right)\hfill \\ \hfill & +\left(I_N\otimes B\right)\left(K^1+\sum _{j=1}^N{\stackrel{`}{W}}_j^{\sigma \left(t\right)}{K}^2{\chi }_j^2-\sum _{j=1}^N{\chi }_j^1\left(W_{\sigma \left(t\right)}\otimes I_n\right)K^2{\chi }_j^2\right)\left(\theta \left(t\right)+e\left(t\right)\right)\hfill \\ \hfill & +\left(I_N\otimes B\right)v\left(t\right).\hfill \end{array}$$

(10)

Because $\theta \left(t\right)=x\left(t\right)-h\left(t\right)-1_N\otimes x_0\left(t\right)$, the MAS (10) can be transformed into

$$\begin{array}{cc}\hfill \dot{\theta }\left(t\right)=& \dot{x}\left(t\right)-\dot{h}\left(t\right)-1_N\otimes {\dot{x}}_0\left(t\right)\hfill \\ \hfill =& \left(I_N\otimes A\right)x\left(t\right)\hfill \\ \hfill & +\left(I_N\otimes B\right)\left(K^1+\sum _{j=1}^N{\stackrel{`}{W}}_j^{\sigma \left(t\right)}{K}^2{\chi }_j^2-\sum _{j=1}^N{\chi }_j^1\left(W_{\sigma \left(t\right)}\otimes I_n\right)K^2{\chi }_j^2\right)\left(\theta \left(t\right)+e_i\left(t\right)\right)\hfill \\ \hfill & +\left(I_N\otimes B\right)v\left(t\right)-\dot{h}\left(t\right)-1_N\otimes {\dot{x}}_0\left(t\right)\hfill \\ \hfill =& \left(I_N\otimes A\right)\theta \left(t\right)\hfill \\ \hfill & +\left(I_N\otimes B\right)\left(K^1+\sum _{j=1}^N{\stackrel{`}{W}}_j^{\sigma \left(t\right)}{K}^2{\chi }_j^2-\sum _{j=1}^N{\chi }_j^1\left(W_{\sigma \left(t\right)}\otimes I_n\right)K^2{\chi }_j^2\right)\left(\theta \left(t\right)+e\left(t\right)\right)\hfill \\ \hfill & +\left(I_N\otimes A\right)h\left(t\right)+\left(I_N\otimes A\right)\left(1_N\otimes x_0\left(t\right)\right)\hfill \\ \hfill & +\left(I_N\otimes B\right)v\left(t\right)-\dot{h}\left(t\right)-1_N\otimes {\dot{x}}_0\left(t\right).\hfill \end{array}$$

(11)

Then, we consider the following two cases:

Case 1. For $rank\left(B\right)=min\left\{m,n\right\}$, there exists a compensational function $v\left(t\right)$ such that

$$\begin{array}{c}\hfill \dot{h}\left(t\right)=\left(I_N\otimes A\right)h\left(t\right)+\left(I_N\otimes B\right)v\left(t\right)-\left(I_N\otimes B\right)\left(1_N\otimes u_0\left(t\right)\right).\end{array}$$

(12)

Substituting Equation (12) into system (11), one has

$$\begin{array}{c}\hfill \dot{\theta }\left(t\right)=\left(\mathcal{A}+\mathcal{B}{K}^1+\sum _{j=1}^N\mathcal{B}{\Phi }_1-\sum _{j=1}^N\mathcal{B}{\Phi }_2\right)\theta \left(t\right)+\mathcal{B}\left(K^1+\sum _{j=1}^N{\Phi }_1-\sum _{j=1}^N{\Phi }_2\right)e\left(t\right)\end{array}$$

(13)

where $\mathcal{A}=I_N\otimes A,\mathcal{B}=I_N\otimes B,{\Phi }_1={\stackrel{`}{W}}_j^{\sigma \left(t\right)}{K}^2{\chi }_j^2,{\Phi }_2={\chi }_j^1\left(W_{\sigma \left(t\right)}\otimes I_n\right)K^2{\chi }_j^2$.

Case 2. For $rank\left(B\right)=n$, there exists a matrix $\tilde{B}\in {\mathbb{R}}^{m\times n}$ such that $\tilde{B}B=I_n$. In addition, $v_i\left(t\right)$ is the TVFT compensational signal given by

$$\begin{array}{c}\hfill v_i\left(t\right)=-\tilde{B}\left(A{h}_i\left(t\right)-{\dot{h}}_i\left(t\right)-B{u}_0\left(t\right)\right).\end{array}$$

(14)

Then, one has

$$\begin{array}{c}\hfill v\left(t\right)=-\left(I_N\otimes \tilde{B}\right)\left(\left(I_N\otimes A\right)h\left(t\right)-\dot{h}\left(t\right)-\left(I_N\otimes B\right)\left(1_N\otimes u_0\left(t\right)\right)\right)\end{array}$$

(15)

where the time-varying state formation $h\left(t\right)$ can be described as any function.

Substituting Equation (15) of the compensational signal $v\left(t\right)$ into system (11), formula (13) is obtained.

Remark 4.

It can be deduced that Case 2 is a special case of Case 1; when the conditions of Case 2 are satisfied, the compensational signal $v\left(t\right)$ can be obtained more conveniently. We consider the sufficient condition for the system to achieve the expected formation rather than the necessary condition. The method mentioned in this study can not be feasible for all systems, yet studying the system that meets the conditions above is still significant.

Definition 2.

Let the disagreement function for system (1) be defined as follows:

$$\Delta \left(x\left(t\right),h\left(t\right),\sigma \left(t\right)\right)=\frac{1}{N}\sum _{i=1}^N\sum _{j\in N_i^{\sigma \left(t\right)}}{∥\left(x_j\left(t\right)-h_j\left(t\right)\right)-\left(x_i\left(t\right)-h_i\left(t\right)\right)∥}^2$$

(16)

where Δ reflects the formation errors with certain directed topologies in the MAS and affects the parameters that will be designed later.

Remark 5.

The partial neighbor comprehensive error can not accurately reflect the consensus of multiple agents. We believe that the more partial information we calculate, the greater the accuracy of the consistent performance evaluation we can obtain.

Lemma 1

([39]). Let ${\mu }_i^{\sigma \left(t\right)}={\mathrm{deg}}_{in}\left(o_i\right)={\sum }_{j=1}^N{w}_{ij}^{\sigma \left(t\right)}$, ${\phi }_i^{\sigma \left(t\right)}={\mathrm{deg}}_{out}\left(o_i\right)={\sum }_{j=1}^N{w}_{ji}^{\sigma \left(t\right)}$ be the in-degree and out-degree of agent i under the directed topology $G_{\sigma \left(t\right)}$, respectively, and let $M_{\sigma \left(t\right)}={\mathrm{diag}}_N\left\{{\mu }_i^{\sigma \left(t\right)}+{\phi }_i^{\sigma \left(t\right)}\right\}$. Then, we obtain

$$\begin{array}{c}\hfill \Delta \left(x\left(t\right),h\left(t\right),\sigma \left(t\right)\right)=\frac{1}{N}{\theta }^T\left(t\right)\left(\left(M_{\sigma \left(t\right)}-W_{\sigma \left(t\right)}-W_{\sigma \left(t\right)}^T\right)\otimes I_n\right)\theta \left(t\right).\end{array}$$

(17)

Proof. 

Using (16), we can obtain

$$\begin{array}{cc}\hfill \Delta =& \frac{1}{N}\sum _{i=1}^N\sum _{j\in N_i}{∥\left(x_j\left(t\right)-h_j\left(t\right)\right)-\left(x_i\left(t\right)-h_i\left(t\right)\right)∥}^2\hfill \\ \hfill =& \frac{1}{N}\sum _{i=1}^N\sum _{j\in N_i}{∥{\theta }_j\left(t\right)-{\theta }_i\left(t\right)∥}^2\hfill \\ \hfill =& \frac{1}{N}\sum _{i=1}^N\sum _{j=1}^N{w}_{ij}^{\sigma \left(t\right)}\left({∥{\theta }_j\left(t\right)∥}^2+{∥{\theta }_i\left(t\right)∥}^2-2∥{\theta }_j\left(t\right)∥∥{\theta }_i\left(t\right)∥\right)\hfill \\ \hfill =& \frac{1}{N}\left\{\sum _{i=1}^N\sum _{j=1}^N{w}_{ij}^{\sigma \left(t\right)}\left({∥{\theta }_j\left(t\right)∥}^2\right)+\sum _{i=1}^N\sum _{j=1}^N{w}_{ij}^{\sigma \left(t\right)}\left({∥{\theta }_i\left(t\right)∥}^2\right)\right.\hfill \\ \hfill & \left.-\sum _{i=1}^N\sum _{j=1}^N{w}_{ij}^{\sigma \left(t\right)}\left(∥{\theta }_j\left(t\right)∥∥{\theta }_i\left(t\right)∥\right)-\sum _{i=1}^N\sum _{j=1}^N{w}_{ij}^{\sigma \left(t\right)}\left(∥{\theta }_i\left(t\right)∥∥{\theta }_j\left(t\right)∥\right)\right\}\hfill \\ \hfill =& \frac{1}{N}{\theta }^T\left(t\right)\left(\left(M_{\sigma \left(t\right)}-W_{\sigma \left(t\right)}-W_{\sigma \left(t\right)}^T\right)\otimes I_n\right)\theta \left(t\right).\hfill \end{array}$$

(18)

This completes the proof of the lemma.    □

3. Main Results

This section proposes a procedure for determining the gain matrices in protocol (7) for MAS (1) to achieve TVFT.

Theorem 1.

Suppose that the feedback gains $K_i^1$ and the coupling gains $K_{ij}^2$ are known constant matrices and that ε is a given constant scalar. If there exists a set of positive definite symmetric matrices $P={\mathrm{diag}}_N\left\{P_i\right\}=I_N\otimes P_1\in {\mathbb{R}}^{nN\times nN},\Omega ={\mathrm{diag}}_N\left\{{\Omega }_i\right\}\in {\mathbb{R}}^{nN\times nN},P_1\in {\mathbb{R}}^{n\times n},{\Omega }_i\in {\mathbb{R}}^{n\times n},i=1,2,\dots ,N$ satisfying the following inequalities:

$$\begin{array}{c}\hfill {\Pi }_1\left(\sigma \right)=\left[\begin{array}{ccc}{\Xi }_1+\epsilon P\Omega P& \ast & \ast \\ {{\Xi }_2}^T& -P\Omega P& \ast \\ \varphi P& 0& -\gamma I\end{array}\right]<0\end{array}$$

(19)

where ${\Xi }_1=He\left(\mathcal{A}P+\mathcal{B}{K}^{1}P+{\sum }_{j=1}^N\mathcal{B}{\Phi }_3-{\sum }_{j=1}^N\mathcal{B}{\Phi }_4\right),{\Xi }_2=\mathcal{B}{K}^{1}P+{\sum }_{j=1}^N\mathcal{B}{\Phi }_3-{\sum }_{j=1}^N\mathcal{B}{\Phi }_4,$ ${\Phi }_3={\Phi }_{1}P={\stackrel{`}{W}}_j^{\sigma \left(t\right)}{K}^{2}P{\chi }_j^2,{\Phi }_4={\Phi }_{2}P={\chi }_j^1\left(W_{\sigma \left(t\right)}\otimes I_n\right)K^{2}P{\chi }_j^2,\varphi =I_n\otimes \left(M_{\sigma \left(t\right)}-W_{\sigma \left(t\right)}\right.{\left.-W_{\sigma \left(t\right)}^T\right)}^{\frac{1}{2}}$; then, the MAS (1) is able to achieve TVFT $h\left(t\right)$ using an event-triggered protocol (7) with condition (5).

Proof. 

We select the following Lyapunov functional for system (15):

$$\begin{array}{c}\hfill V\left(t\right)={\theta }^T\left(t\right)Q\theta \left(t\right)\end{array}$$

(20)

where $Q=P^{-1}$. Taking the derivative of $V\left(t\right)$ concerning t along system (15) yields

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)=& {\dot{\theta }}^T\left(t\right)Q\theta \left(t\right)+{\theta }^T\left(t\right)Q\dot{\theta }\left(t\right)\hfill \\ \hfill =& {\theta }^T\left(t\right)QP{\left(\mathcal{A}+\mathcal{B}{K}^1+\sum _{j=1}^N\mathcal{B}{\Phi }_1-\sum _{j=1}^N\mathcal{B}{\Phi }_2\right)}^{T}Q\theta \left(t\right)\hfill \\ \hfill & +{\theta }^T\left(t\right)Q\left(\mathcal{A}+\mathcal{B}{K}^1+\sum _{j=1}^N\mathcal{B}{\Phi }_1-\sum _{j=1}^N\mathcal{B}{\Phi }_2\right)PQ\theta \left(t\right)\hfill \\ \hfill & +e^T\left(t\right)QP{\left(K^1+\sum _{j=1}^N{\Phi }_1-\sum _{j=1}^N{\Phi }_2\right)}^T{\mathcal{B}}^{T}Q\theta \left(t\right)\hfill \\ \hfill & +{\theta }^T\left(t\right)Q\mathcal{B}\left(K^1+\sum _{j=1}^N{\Phi }_1-\sum _{j=1}^N{\Phi }_2\right)PQe\left(t\right)\hfill \\ \hfill =& {\xi }^T\left(t\right)Q\left[\begin{array}{cc}{\Xi }_1& {\Xi }_2\\ {{\Xi }_2}^T& 0\end{array}\right]Q\xi \left(t\right)\hfill \end{array}$$

(21)

where $\xi \left(t\right)={\left({\theta }^T\left(t\right),e^T\left(t\right)\right)}^T$. From the event-triggered condition (5), we can obtain

$$\begin{array}{c}\hfill -\epsilon {\theta }^T\left(t\right)\Omega \theta \left(t\right)⩽e^T\left(t\right)\Omega e\left(t\right)-\epsilon {\theta }^T\left(t\right)\Omega \theta \left(t\right)⩽0.\end{array}$$

(22)

Adding the terms $e^T\left(t\right)\Omega e\left(t\right)-\epsilon {\theta }^T\left(t\right)\Omega \theta \left(t\right)$ and $ϵ\Delta \left(x\left(t\right),h\left(t\right)\right)$ to both sides of (20) and applying Lemma 1, the following equality holds:

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)& -{e_i}^T\left(t\right)\Omega e_i\left(t\right)+\epsilon {{\theta }_i}^T\left(t\right)\Omega {\theta }_i\left(t\right)+ϵ\Delta \left(x\left(t\right),h\left(t\right)\right)\hfill \\ \hfill =& {\xi }^T\left(t\right)Q\left[\begin{array}{cc}{\Xi }_1& {\Xi }_2\\ {{\Xi }_2}^T& 0\end{array}\right]Q\xi \left(t\right)-{e_i}^T\left(t\right)\Omega e_i\left(t\right)+\epsilon {{\theta }_i}^T\left(t\right)\Omega {\theta }_i\left(t\right)\hfill \\ \hfill & +{\gamma }^{-1}\frac{1}{N}{\theta }^T\left(t\right)\left(\left(M_{\sigma \left(t\right)}-W_{\sigma \left(t\right)}-W_{\sigma \left(t\right)}^T\right)\otimes I_{2n}\right)\theta \left(t\right)\hfill \\ \hfill =& {\xi }^T\left(t\right)Q\left[\begin{array}{cc}{\Xi }_1& {\Xi }_2\\ {{\Xi }_2}^T& 0\end{array}\right]Q\xi \left(t\right)-{\xi }^T\left(t\right)Q\left[\begin{array}{cc}-\epsilon P\Omega P& 0\\ 0& P\Omega P\end{array}\right]Q\xi \left(t\right)\hfill \\ \hfill & +{\xi }^T\left(t\right)Q\left[\begin{array}{cc}{\gamma }^{-1}\frac{1}{N}P\left(\left(M_{\sigma \left(t\right)}-W_{\sigma \left(t\right)}-W_{\sigma \left(t\right)}^T\right)\otimes I_{2n}\right)P& \ast \\ 0& 0\end{array}\right]Q\xi \left(t\right)\hfill \end{array}$$

(23)

where $ϵ$ yields $ϵ={\gamma }^{-1}$, and

$$\begin{array}{c}\hfill \dot{V}\left(t\right)-{e_i}^T\left(t\right)\Omega e_i\left(t\right)+\epsilon {{\theta }_i}^T\left(t\right)\Omega {\theta }_i\left(t\right)+ϵ\Delta \left(x\left(t\right),h\left(t\right)\right)\triangleq {\xi }^T\left(t\right)Q{\Pi }_{2}Q\xi \left(t\right)\end{array}$$

(24)

By using the Schur complement to inequality (19), we have

$$\begin{array}{c}\hfill {\Pi }_2<0.\end{array}$$

(25)

Then,

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)& -{e_i}^T\left(t\right)\Omega e_i\left(t\right)+\epsilon {{\theta }_i}^T\left(t\right)\Omega {\theta }_i\left(t\right)\hfill \\ \hfill & +ϵ\frac{1}{N}{\theta }^T\left(t\right)\left(\left(M_{\sigma \left(t\right)}-W_{\sigma \left(t\right)}-W_{\sigma \left(t\right)}^T\right)\otimes I_{2n}\right)\theta \left(t\right)\hfill \\ \hfill =& {\xi }^T\left(t\right)Q{\Pi }_{2}Q\xi \left(t\right)\hfill \\ \hfill <& {\xi }^T\left(t\right)Q\left(t_{min}I\right)Q\xi \left(t\right)\hfill \\ \hfill =& t_{min}{\lambda }_{min}^{-1}\left(P\right){\xi }^T\left(t\right)Q\left({\lambda }_{min}\left(P\right)I\right)Q\xi \left(t\right)\hfill \\ \hfill <& t_{min}{\lambda }_{min}^{-1}\left(P\right){\xi }^T\left(t\right)Q\left[\begin{array}{cc}P& 0\\ 0& 0\end{array}\right]Q\xi \left(t\right)\hfill \\ \hfill =& t_{min}{\lambda }_{min}^{-1}\left(P\right)V\left(t\right)<0\hfill \end{array}$$

(26)

where $t_{min}$ yields ${\Pi }_2⩽t_{min}I$, and varies by algorithm, which can also be obtained directly by MATLAB 2022a.

Then, one has

$$\begin{array}{c}\hfill \dot{V}\left(t\right)⩽t_{min}{\lambda }_{min}^{-1}\left(P\right)V\left(t\right)+{e_i}^T\left(t\right)\Omega e_i\left(t\right)-\epsilon {{\theta }_i}^T\left(t\right)\Omega {\theta }_i\left(t\right)-ϵ\Delta \end{array}$$

(27)

Applying (22) to (27), the following inequality holds:

$$\begin{array}{c}\hfill \dot{V}\left(t\right)⩽t_{min}{\lambda }_{min}^{-1}\left(P\right)V\left(t\right)<0.\end{array}$$

(28)

Then, we can obtain

$$\begin{array}{c}\hfill ∥V\left(t\right)∥⩽e^{-\kappa t}∥V\left(0\right)\left(t\right)∥,\end{array}$$

(29)

where $\kappa =t_{min}{\lambda }_{min}^{-1}\left(P\right)$. From (22), it can be found that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\theta \left(t\right)=\underset{t\to \infty }{lim}\left(x\left(t\right)-h\left(t\right)-x_0\left(t\right)\otimes 1_N\right)=0,\end{array}$$

(30)

which proves that Definition 1 is satisfied. Namely, system (1) can achieve TVFT if the conditions of Theorem 1 are satisfied.    □

Theorem 2.

For given constants $\gamma >0$ and $\epsilon >0$, if there exist positive definite symmetric matrices $R={\mathrm{diag}}_N\left\{R_i\right\}\in {\mathbb{R}}^{mN\times nN},S=\left[S_{ij}\right],S_{ij}\in {\mathbb{R}}^{m\times n},T\in {\mathbb{R}}^{nN\times nN}$ satisfying the following inequality:

$$\begin{array}{c}\hfill {\overline{\Pi }}_1\left(\sigma \right)=\left[\begin{array}{ccc}{\overline{\Xi }}_1+\epsilon T& \ast & \ast \\ {{\overline{\Xi }}_2}^T& -T& \ast \\ \varphi P& 0& -\gamma I\end{array}\right]<0\end{array}$$

(31)

where ${\overline{\Xi }}_1=He\left(\mathcal{A}P+\mathcal{B}R+{\sum }_{j=1}^N\mathcal{B}{\stackrel{`}{W}}_j^{\sigma \left(t\right)}S{\chi }_j^2-{\sum }_{j=1}^N\mathcal{B}{\chi }_j^1\left(W_{\sigma \left(t\right)}\otimes I_n\right)S{\chi }_j^2\right),$ ${\overline{\Xi }}_2=\mathcal{B}R+{\sum }_{j=1}^N\mathcal{B}{\stackrel{`}{W}}_j^{\sigma \left(t\right)}S{\chi }_j^2-{\sum }_{j=1}^N\mathcal{B}{\chi }_j^1\left(W_{\sigma \left(t\right)}\otimes I_n\right)S{\chi }_j^2$; then, MAS (1) with

$$\begin{array}{c}\hfill K^1=R{P}^{-1},K^2=S{P}^{-1},\Omega =P^{-1}T{P}^{-1}\end{array}$$

(32)

can achieve TVFT.

Proof. 

By defining $R=K^{1}P,S=K^{2}P$ and $T=P\Omega P$, it is obvious that

$$\begin{array}{cc}\hfill {\Xi }_1=& He\left(\mathcal{A}P+\mathcal{B}{K}^{1}P+\sum _{j=1}^N\mathcal{B}{\stackrel{`}{W}}_j^{\sigma \left(t\right)}{K}^{2}P{\chi }_j^2-\sum _{j=1}^N\mathcal{B}{\chi }_j^1\left(W_{\sigma \left(t\right)}\otimes I_n\right)K^{2}P{\chi }_j^2\right)\hfill \\ \hfill =& He\left(\mathcal{A}P+\mathcal{B}R+\sum _{j=1}^N\mathcal{B}{\stackrel{`}{W}}_j^{\sigma \left(t\right)}S{\chi }_j^2-\sum _{j=1}^N\mathcal{B}{\chi }_j^1\left(W_{\sigma \left(t\right)}\otimes I_n\right)S{\chi }_j^2\right)={\overline{\Xi }}_1\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\overline{\Xi }}_2=& \mathcal{B}{K}^{1}P+\sum _{j=1}^N\mathcal{B}{\stackrel{`}{W}}_j^{\sigma \left(t\right)}{K}^{2}P{\chi }_j^2-\sum _{j=1}^N\mathcal{B}{\chi }_j^1\left(W_{\sigma \left(t\right)}\otimes I_n\right)K^{2}P{\chi }_j^2\hfill \\ \hfill =& \mathcal{B}R+\sum _{j=1}^N\mathcal{B}{\stackrel{`}{W}}_j^{\sigma \left(t\right)}S{\chi }_j^2-\sum _{j=1}^N\mathcal{B}{\chi }_j^1\left(W_{\sigma \left(t\right)}\otimes I_n\right)S{\chi }_j^2={\overline{\Xi }}_2\hfill \end{array}$$

(34)

It is easy to see that LMI (31) equals LMI (19). Thus, Theorem 1 is proven.

This completes the proof.    □

Remark 6.

The matrix inequalities in Theorem 1 are not yet linear, and the matrix inequalities in Theorem 2 are non-unique linear matrix inequalities that can be solved via the LMI technologies in the MATLAB toolbox.

4. Optimization and Co-Design Problem

This section presents the optimization and co-design problem in order to provide the parameters of the controllers and event generators.

From inequalities (22) and (29), we infer that the components $\dot{V}\left(t\right),V\left(t\right),\Delta$, and ${e_i}^T\left(t\right)\Omega e_i\left(t\right)-\epsilon {{\tilde{\theta }}_i}^T\left(t\right)\Omega {\tilde{\theta }}_i\left(t\right)$ in (27) are all bounded. To some extent, if the value of the component $ϵ\Delta$ is relatively large, namely, $\gamma$ is relatively small, we easily obtain a relatively small value of $\Delta$, such that the system has a good consensus. As a result, we provide the optimization problem next.

For system (1) and its corresponding event-triggered TVFT protocol (7) with an event-triggered condition (5), if the optimal problem

$$\begin{array}{c}\hfill \underset{P,R,S,T}{min}\gamma \end{array}$$

(35)

under the constraints

$$\begin{array}{c}\hfill \gamma >\underline{\gamma },R>0,S>0,T>0\end{array}$$

(36)

meets the LMI conditions (31), then system (1) can achieve TVFT, and $\underline{\gamma }$ denotes a certain threshold.

In Theorem 2, one can notice that if $\epsilon$ is an uncertain variable in (31), then (31) is a nonlinear matrix inequality. The following objective function is defined to co-design the parameters $\epsilon ,\Omega ,K^1,K^2$ so as to formulate an optimal issue that can be directly addressed via the MATLAB toolbox, to reduce the communication load, and to ensure the desired performance of the consensus in an MAS:

$$\begin{array}{c}\hfill {\Gamma }^{\left(d\right)}=-{\varrho }_1{\gamma }^{\left(d-1\right)}{t}_{min}^{\left(d-1\right)}+{\varrho }_2{\left({\epsilon }^{\left(d-1\right)}\right)}^{-1}\end{array}$$

(37)

where $\epsilon ={\epsilon }_0+\Delta \epsilon ·d$, d denotes the iterations, ${\varrho }_1>0$ and ${\varrho }_2>0$ are the prescribed weighting coefficients, and $t_{min}^{\left(d\right)}$ and ${\gamma }^{\left(d\right)}$ stand for $t_{min}$ and $\gamma$ in the dth iteration, respectively.

Remark 7.

Given that too much communication will increase the energy consumption, a low communication trigger frequency is desired. From the event-triggered condition (5), one can conclude that the communication trigger frequency is mainly influenced by the value of ε, which can be acquired from the iterative algorithm; the larger the parameter ε, the lower the trigger frequency. Moreover, a ridiculously large value of ε will cause ridiculously low communication frequency, which will reduce the consensus performance of MASs; thus one can set the proportion of the weight coefficients $\varrho 1$ and $\varrho 2$ to balance the communication frequency and state consensus of agents. In relative terms, the larger $\varrho 1$ is, the better the consensus performance and the smaller the amount of communication will be.

Based on the objective function (37), an iterative algorithm for designing an event-triggered controller will be proposed.

To further make the system adjust itself in the case of existing interference, a switching signal scheme for the graphs is set as follows:

$$\begin{array}{c}\hfill \sigma \left(t^{+}\right)=\underset{\mathsf{\ell }}{\arg max}\left\{\Delta \left(\overline{x}\left(t\right),\overline{h}\left(t\right),\mathsf{\ell }\right)\right\},\end{array}$$

(38)

where $\overline{x}\left(t\right)={\left(x_1\left(t_k^1\left(t\right)\right),\dots ,x_N\left(t_k^N\left(t\right)\right)\right)}^T,\overline{h}\left(t\right)={\left(h_1\left(t_k^1\left(t\right)\right),\dots ,h_N\left(t_k^N\left(t\right)\right)\right)}^T$ can be obtained from the communication network, $\mathsf{\ell }=1,\dots ,o$, and o represents the number of switching topologies.

Remark 8.

When the system takes the consistency error as the basis for a decision, excessive information and calculation may increase the system’s energy consumption. Thus, reducing the system’s complexity in this aspect is a problem worth considering in the future.

5. Numerical Examples

This section verifies the effectiveness of the proposed switching-directed topology design and event-triggered control protocol using two numerical examples. In Example 1, a third-order MAS with one leader and five followers shows that TVFT is achieved with the triggered condition (3) and the formation protocol (4). In Example 2, a system with abrupt errors that occur at certain moments is addressed to show further event-triggered performance and the feasibility of TVFT while the topology is switching. The interaction topologies of the MASs in the following two examples are illustrated in Figure 2, where ${\mathcal{N}}_i^{1,\dots ,4},i=1,\dots ,5$ denote the four neighbors of the ith agent, and $G_i\left(i=1,\dots ,7\right)$ are the marks of the interaction topologies.

Example 1.

Let the initial state of system (1) with the leader (2) be $x_1={\left(5.2,2.9,3.7\right)}^{\mathrm{T}},x_2={\left(7.3,5.6,4.2\right)}^{\mathrm{T}},x_3={\left(7.4,5.2,4.94\right)}^{\mathrm{T}},x_4={\left(5.5,1.7,4.5\right)}^{\mathrm{T}},x_5={\left(2.2,1.1,1.7\right)}^{\mathrm{T}}$, and $x_0={\left(2.8,8.4,18.3\right)}^{\mathrm{T}}$, which are randomly produced. The dynamics of the MAS with one leader and five followers are given by (1) and (2), with $A=\left[\begin{array}{ccc}0& 1& 1\\ 1& 2& 1\\ -2& -6& -3\end{array}\right],B=\left[\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]$, and $u_0\left(t\right)=0$.

These five followers have to hold an assigned pentagon structure; meanwhile, the structure is time-varying, and the followers rotate around the leader, which is treated as the center of the structure at a constant angular velocity. The corresponding formation is described by $h_i=\left[\begin{array}{c}-15cos\left(\omega t+\frac{2\left(i-1\right)\pi }{5}\right)\\ 15sin\left(\omega t+\frac{2\left(i-1\right)\pi }{5}\right)\\ 30cos\left(\omega t+\frac{2\left(i-1\right)\pi }{5}\right)\end{array}\right]$.

If the predefined time-varying formation $h\left(t\right)={\left[h_1\left(t\right),h_2\left(t\right),\dots ,h_N\left(t\right)\right]}^T$ is achieved, the five followers will keep the formation; they will be regularly distributed at the vertices of the pentagon and will rotate with a constant angular velocity of $1rad/s$.

Given that $rank\left(B\right)=3$, we choose $\tilde{B}=\left[\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$ such that $\tilde{B}B=I_n$. According to Case 2, the compensational signal $v_i\left(t\right)$ can be derived from (15). Then, to obtain the event-triggered parameters $\epsilon$ and $\Omega$, the feedback gains $K_i^1$, and the coupling gains $K_{ij}^2$, the co-design strategy will be considered.

Let the initial value of $\epsilon$ be ${\epsilon }_0=0.3$, and we set the step increment of $\epsilon$ to $\Delta \epsilon =0.01$. We set the weighting coefficients to ${\varrho }_1=1,{\varrho }_2=200$. By applying Algorithm 1, it can be found that the TVFT problem is feasible, and the obtained event-triggered parameters $\epsilon$ and $\Omega$, the feedback gains $K_i^1$, and the coupling gains $K_{ij}^2$ are given by  

$\epsilon =0.36$,   $\Omega =I_N\otimes \left[\begin{array}{ccc}0.1268& 0.0921& 0.0680\\ 0.0921& 0.8318& 0.1218\\ 0.0680& 0.1218& 0.0964\end{array}\right]\times {10}^{-5}$,

$K^1=I_N\otimes \left[\begin{array}{ccc}2.8792& 21.3816& 2.0100\\ 0.8487& 21.6464& 8.2230\\ -9.2215& -17.6966& -12.5415\end{array}\right]$,

$K_{ij}^2=\left[\begin{array}{ccc}0.1232& 0.1826& 0.0827\\ 0.6548& 1.8953& 1.0887\\ -0.8235& -1.9527& -1.0437\end{array}\right],i,j=1,2,\dots ,5,i\ne j$.

Algorithm 1 Co-design algorithm.

Input: ${\epsilon }_0$, $\Delta \epsilon$;

Output: $\epsilon ,K^1,K^2,\Omega$;

1: Choose a sufficiently small ${\epsilon }_0$ as the initial value of $\epsilon$ and a suitable $\Delta \epsilon$ as the step increment of $\epsilon$. 2: Solve the LMIs (31) with $\epsilon ={\epsilon }_0$. If it is feasible, determine $t_{min},\gamma ,R,S,T$ and move to Step 3. Otherwise, update $\epsilon =\epsilon +\Delta \epsilon$ and repeat Step 2. 3: Calculate (37). If ${\Gamma }^{\left(d\right)}<{\Gamma }^{\left(d-1\right)}$, set $d=d+1$ and update $t_{min}=t_{min}^{\left(d\right)},\gamma ={\gamma }^{\left(d\right)},R=R^{\left(d\right)},S=S^{\left(d\right)},T=T^{\left(d\right)}$. Otherwise, keep $t_{min},\gamma ,R,S,T$ and exit. 4: Compute $K^1,K^2,\Omega$ according to (32). 5: Output $\epsilon ,K^1,K^2,\Omega$ and exit.

Let ${\varsigma }_i\left(t\right)=∥x_i\left(t\right)-h_i\left(t\right)-x_0\left(t\right)∥$ be the formation tracking errors. The corresponding simulation results are depicted in Figure 3, Figure 4 and Figure 5. Figure 3 illustrates the state trajectory of the MAS with one leader and five followers under the protocol (4) and triggered condition (3) within 10 s. The state snapshots of the followers at different time instants $(t=2,6,10$ s) are marked and located at the five vertices of the pentagon in 3D coordinates. Figure 3 highlights that the followers rotate in a small circle and track the leader as the center, carrying out a large circular motion. Figure 4 presents the formation tracking errors of the MAS within 10 s, revealing that the formation tracking errors are exponentially stable. The evolution of the switching topology and the triggered instants for the followers are presented in Figure 5 and Figure 6, respectively. It is observed that (1) the system with the switching-directed topology and event-triggered mechanism is feasible and (2) the control updates are reduced with the changing communication requirements.

Example 2.

Let the initial state of system (1) with the leader (2) be $x_1={\left(0.52,0.23,0.48\right)}^{\mathrm{T}},x_2={\left(0.49,0.54,0.31\right)}^{\mathrm{T}},x_3={\left(0.22,0.59,0.02\right)}^{\mathrm{T}},x_4={\left(0.35,0.36,0.31\right)}^{\mathrm{T}},x_5={\left(0.02,0.05,0.07\right)}^{\mathrm{T}}$, and $x_0={\left(0.02,0.05,0.05\right)}^{\mathrm{T}}$, which are randomly produced. The dynamics of the five followers and one leader are described using (1) and (2), where $A=\left[\begin{array}{ccc}0& 2& 0\\ 0& 0& 2\\ -1& -2& -2\end{array}\right],B=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$, and $u_0\left(t\right)=0$.

The formation is described by $h_i=\left[\begin{array}{c}sin\left(\omega t+\frac{2\left(i-1\right)\pi }{5}\right)\\ cos\left(\omega t+\frac{2\left(i-1\right)\pi }{5}\right)\\ -sin\left(\omega t+\frac{2\left(i-1\right)\pi }{5}\right)\end{array}\right]$, and let $\omega =2$ such that the five followers maintain a rotation with a constant angular velocity of 2 rad/s.

Given that $rank\left(B\right)=1$ according to Case 1, there exists a compensational function $v\left(t\right)=\left(I_N\otimes \left[\begin{array}{ccc}0& 0& 1\end{array}\right]\right)\left[\left(I_N\otimes A\right)h\left(t\right)-\dot{h}\left(t\right)\right]$ such that equality (12) holds, and the compensational signal is obtained.

Then, to obtain the event-triggered parameters $\epsilon$ and $\Omega$, the feedback gains $K_i^1$, and the coupling gains $K_{ij}^2$, we first consider the co-design strategy. Setting the appropriate values is critical in practical applications. For example, an excessively large feedback gain can cause vibration, so the range of gains was limited in this experiment. Thus, let the initial value of $\epsilon$ be ${\epsilon }_0=0.3$ and let the step increment of $\epsilon$ be $\Delta \epsilon =0.01$. We set the weighting coefficients to ${\varrho }_1=1,{\varrho }_2=700$ via applying Algorithm 1, and we find that the time-varying formation problem is feasible. The obtained event-triggered parameters $\epsilon$ and $\Omega$, the feedback gains $K_i^1$, and the coupling gains $K_{ij}^2$ are given by

$\epsilon =0.49$, $\Omega =I_N\otimes \left[\begin{array}{ccc}0.0903& 0.1667& 0.0992\\ 0.1667& 0.6409& 0.4853\\ 0.0992& 0.4853& 0.4417\end{array}\right]$,

$K^1=I_N\otimes \left[\begin{array}{ccc}-0.1937& -2.0416& -1.8354\end{array}\right]$,

$K_{ij}^2=\left[\begin{array}{ccc}0.0703& -0.1211& 0.1693\end{array}\right],i,j=1,2,\dots ,5,i\ne j$.

Next, we introduce a disturbance $\zeta \left(t\right)={\sum }_{i=1}^I{\chi }_{{\iota }_i}^1{\int }_{t_i}^{\infty }{\varpi }_i\delta \left(t\right)$ into the system, where $\iota$ denotes the $\iota$th agent that is disturbed at the ith interference period, and I is the number of interference behaviors. Let $I=2,{\iota }_1=5,t_1=6,{\varpi }_1={\left(0.08,0.26,0.80\right)}^T,{\iota }_2=3,$$t_2=8,{\varpi }_2={\left(0.66,0.72,0.52\right)}^T$.

The corresponding simulation results are presented in Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. Figure 7 depicts the state trajectory of the MAS with one leader and five followers under the protocol (4) and the triggered condition (3) within 20 s, with disturbances at $t=6$ s and $t=8$ s. In this case, the followers keep the circular formation and finally adjust to a normal condition. Figure 8 depicts the formation tracking errors of the MAS within 20 s. The evolution of switching signals and the triggered instants for the followers are presented in Figure 9 and Figure 10, respectively, revealing that the topology’s structure and the event generator adjust accordingly at the disturbed instant. More specifically, the system chose and triggered topology $G_7$ approximately 6 s later, and it chose and triggered topology $G_5$ approximately 8 s later, so it could reduce the information sent from the disturbed followers 3 and 5, which meant that the switching-directed topology strategy and the event-triggered mechanism that we designed are valid. Figure 11 depicts state trajectory snapshots of the MAS. Figure 11a shows that the system almost achieved the desired time-varying formation at 3 s. Figure 11b,c show that the system was disturbed at 6 s and 8 s. Figure 11d shows that the system returned to the desired state, which illustrates that the desired time-varying formation with the predefined disturbance was achieved. The formation tracking errors of the disturbed followers 3 and 5 under different communication mechanisms are reported in

Table 1. The formation tracking errors of the disturbed follower 3 with different communication mechanisms.

Time (s)	No Communication	Fixed Topology	Switching Topology
2	0.5862	0.5748	0.5748
4	0.0662	0.0605	0.0605
6	0.0433	0.0398	0.0398
8	0.8647	0.8643	0.8647
10	0.3746	0.3564	0.3308
12	0.0660	0.0578	0.0351
14	0.0256	0.0266	0.0338

and

Table 2. The formation tracking errors of the disturbed follower 5 with different communication mechanisms.

Time (s)	No Communication	Fixed Topology	Switching Topology
2	0.0631	0.0541	0.0541
4	0.0259	0.0271	0.0271
6	1.0902	1.0903	1.0903
8	0.6843	0.6627	0.6130
10	0.0676	0.0535	0.0563
12	0.0364	0.0339	0.0351
14	0.0353	0.0337	0.0338

, respectively. The no-communication case is listed in the tables to illustrate that the coupling feedback was valid, and a comparison of fixed-topology and switching-topology schemes should be made on the basis thereof. The results suggest that the MAS performed better with interactive communication than without communication. In addition, the MAS under the switching topology mechanism had better resilience than that of the fixed topology. More specifically, the formation tracking errors of follower 3 decreased by $7.18\%$ at 10 s, and the formation tracking errors of follower 5 decreased by $7.50\%$ at 8 s.

Let the average communication frequency of an MAS be $\overline{f}=Num\left(T\right)·T^{-1},$$T=t_2-t_1$, where $t_1$ is the selected initial instant, $t_2$ is the selected last instant of the time duration, T is the time duration, and $Num$ denotes the sum of the communication trigger times of all followers within T seconds. Let the mean firing interval $\tau$ be $\tau ={\overline{f}}^{-1}$. We define the consensus achievement time Tim as

$$Tim=inf\left\{t\mid \underset{\mathsf{\ell }}{max}\left\{\Delta \left(x\left(t\right),h\left(t\right),\mathsf{\ell }\right)\right\}<\varkappa ·\underset{\mathsf{\ell }}{max}\left\{\Delta \left(x\left(t_0\right),h\left(t_0\right),\mathsf{\ell }\right)\right\},t_1<t<t_2\right\}-t_1,$$

where $\varkappa$ is an index parameter related to the degree of consensus, and $t_0$ is the initial instant.

Table 3. A comparison of previous work without the proposed algorithm and the current work with the proposed algorithm.

	$\left({\mathit{t}}_1,{\mathit{t}}_2\right)$	$\overline{\mathit{f}}$	$\mathit{\tau }$	Time $\left(\mathit{\varkappa }=0.001\right)$
Previous work	(0.5)	20.24	0.49	3.89
	(6.10)	18.89	0.53	4.89
Current work	(0.5)	14.83	0.67	4.01
	(6.10)	15.25	0.66	5.37

shows a comparison with previous work in which the proposed algorithm was not used and the current work with the proposed algorithm in terms of the average communication frequency $\overline{f}$, the mean triggered interval $\tau$, and the consensus achievement time $Tim$. As the table shows, the mean triggered intervals clearly increased, but there was no significant change in consensus. The last column of the table shows the increase in consensus achievement time, which reflects that the optimization of the trigger frequency may reduce the consensus effect; nevertheless, its influence can be neglected in certain cases.

6. Conclusions

A TVFT control problem for an MAS based on event triggering and the switching of topological mechanisms was investigated in this study. The proposed communication topology can be automatically switched according to the latest system state that is updated at the triggered instants. The sufficient conditions for the system’s stability were obtained, and the feedback gain and coupling gain were obtained via solving an optimal problem under LMI constraints. Two examples verified the validity of the theoretical analysis.

The delay and packet dropout problems are unavoidable in practice [40]; to enhance the applicability of this study, event-triggered TVFT control with switching topologies, packet losses, and time delays will be developed in the future.