Distributed event-triggered consensus control for a class of nonlinear multi-agent systems under switching topologies

Abstract

A distributed event-triggered consensus control protocol is proposed for a class of nonlinear multi-agent systems with switching topologies. The nonlinear dynamics and switching topologies are considered simultaneously to adapt to some actual scenarios. To economize the communication resources, a distributed event-triggered mechanism is established to decrease the consumption of communication resources. The new event-triggered mechanism is constructed by the information of distributed errors, which can fully utilize communication relationships among agents to control the timing of events. A class of negative exponential functions is added to the event-triggered condition as the event-triggered mechanism is applied for the presented multi-agent systems. The design of event-triggered condition can reduce the occurring frequency of events when the adjusting time of control scheme begins at an initial small amount period of time. Finally, the simulation results are provided, which verify the feasibility and effectiveness of the designed event-triggered consensus control strategy for the presented multi-agent systems with switching topologies.

Keywords

Consensus control multi-agent systems switching topologies event-triggered mechanism nonlinear dynamics

Introduction

Multi-agent systems (MASs) have attracted a great deal of attention in the past few decades, due to their promising performance in various pragmatic applications (Bhowmick and Panja, 2019; Li, 2015; Liu et al., 2022; Olfati-Saber et al., 2007; Wei et al., 2021; Wen et al., 2017; Zheng et al., 2021), which have received dramatic advances by the endeavors of researchers. One important problem growing out of MASs is to coordinate the states of all agents to accomplish the control assignment by employing distributed protocols, which is known as the consensus control issue. The consensus control can be divided into leader-follower consensus and leaderless consensus. The achievement of consensus control strategies primarily relies on the mutual communication among agents, which is commonly called graph theory. On the other hand, according to the direction of the information interaction among agents, the corresponding topological graph can be divided into directed graph and undirected graph. For instance, the consensus control issue was studied for a class of MASs with stochastic noises and undirected topology when the MASs suffered from the influence of binary-valued communication in Zhao et al. (2019). In Ni et al. (2021), a fixed-time consensus control approach was proposed for second-order MASs with a directed topology graph, and a nonsingular terminal sliding mode consensus control scheme was developed to realize the goal of tracking control.

In reality, the communication relationships among agents maybe changeless, which is called the fixed topology structure. However, to satisfy the requirement of certain natural and pragmatic coupling systems, the communication relationships need to be alterable among agents. Therefore, to cope with the transformation of various modes, the concept of switching has been studied in many fields (Li et al., 2020; Su and Huang, 2012; Sun et al., 2022). Accordingly, the switching topologies theory has been widely studied in the area of MASs (Lu et al., 2017; Saboori and Khorasani, 2014; Zhai et al., 2021). For instance, the issue of uncertain switching topology was studied for a class of MASs with time delays (Savino et al., 2016), and the variations of topology structure were modeled by utilizing Markov jumping with uncertain transition rates. Moreover, the issue of distributed consensus control was investigated for a class of linear MASs with switching topologies in Wen and Zheng (2019), and the approach of multiple Lyapunov functions was proposed satisfying the characteristics of Laplacian matrices. In Razaq et al. (2020), the property of average dwell time was used to govern the variation of topology structures which relaxed the bound of classical dwell time for switching instances. Based on the local design and approach of multiple Lyapunov functions, the consensus tracking goal was achieved for a class of nonlinear MASs. Unfortunately, most of results about the consensus control assume that it is continuous between the communication and the sensing of information, which may result in some unnecessary waste of communication resources to be temporarily used.

To overcome intermittent communication difficulties and reduce communication burdens, the event-triggered control (ETC), as an effective control technique, can enable the manipulation of continuous-time systems under intermittent communication, as well as the occupation of communication resources can be reduced simultaneously. Therefore, the ETC technique has been broadly concerned in recent decades (Li et al., 2021a, 2021b; Nowzari et al., 2019; Wang et al., 2022; Yang et al., 2021; Yu et al., 2022; Zegers et al., 2022; Zhang et al., 2020; Zhou et al., 2021a, 2021b). In particular, the ETC mechanism is normally composed by threshold and evaluation mechanism. For instance, in Hao et al. (2021), the consensus control issue of heterogeneous linear MASs under connected digraphs was studied, and the information transmissions of event-triggered and topology switchings were considered. By using the distributed information, the distributed ETC scheme was developed in Qian et al. (2020), and the issue of distributed event-triggered adaptive consensus control was investigated for a class of MASs with disturbances. In addition, it is disadvantageous for system if the event is triggered frequently at the initial stage of tuning. Hence, to avoid the occurring of this issue, an initial value is added to the design of event-triggered condition, and the initial value is unprevailing for the adjusting process of system after a period of time. Based on the design approach of event-triggered condition, the controller has full adjustment time at the beginning, such that the value of controller avoids frequent changes. Therefore, this control scheme can eliminate the frequent jitter of the actuator to some extent.

Summarized by the above discussions, the event-triggered consensus control issue is studied for a class of nonlinear MASs with switching topologies in this paper. Based on the information of the distributed errors, the distributed ETC scheme is designed to reduce the communication burdens. The frequent jitter of actuator can be avoided at the initial transient time based on the proposed event-triggered mechanism. The property of average dwell time is considered to broaden the communication network requirements. The main contributions of this paper are summarized in the fllowing: (1) First, a distributed ETC scheme is proposed in virtue of the distributed error, and the communication burdens of MASs are reduced. Then, an attenuation function is added to the threshold value design of event-triggered condition in this paper, such that the controller can avoid frequent changes at the initial operating period. (2) Second, to adapt the control design scheme of some actual MASs, the influences of nonlinear dynamics and switching topologies are considered for the presented MASs in the implementation of event-triggered consensus control scheme. (3) Finally, the property of average dwell time, as adjusting criteria of switching signals, is utilized to govern the switching sequence for MASs with topology graph.

The rest of this paper is introduced as follows. The relevant preliminaries are described in the second section. The third section carries out the design of the distributed event-triggered consensus control scheme. The simulation example is shown in the fourth section. The fifth section gives the conclusions of this paper.

Notation: $λ_{min} (\cdot)$ and $λ_{max} (\cdot)$ represent the minimum and maximum eigenvalues of matrix $[\cdot]$ , respectively. The Kronecker product is shown as the symbol ⊗. $N$ expresses the positive integer. $inf (x)$ and $sup (x)$ stand for the infimum and supremum of $x$ , respectively. $I_{N} \in R^{n \times n}$ is an identity matrix, and $I_{m} \in R^{m}$ is an identity column vector. The acronym col indicates the column vector. The symbol $| | \cdot | |$ represents 2-norm of vector.

Preliminaries

Graph theory

In the consensus control scheme of the presented MASs, the communication among agents is indispensable. The communication relationships are described by the graph theory, and also the theory of switching topologies is required in this paper. Specifically, the MASs are constituted of $(n + 1)$ agents, and the $(n + 1)$ -th agent is defined as the leader agent. Let $\overset{⌣}{G} ≜ {G_{1}, G_{2}, \dots, G_{k}}$ , where $k > 1$ , and $k \in N$ . To expediently adjust the communication topology graph, a signal function $ϑ (t)$ is introduced. The mapping relationship of $ϑ (t)$ is defined as $ϑ (t) : [0, + \infty) \to 1, 2, \dots, k$ . Then, $G_{ϑ (t)}$ represents the topology graph at the time $t$ , and $G_{ϑ (t)} \in \overset{⌣}{G}$ for all $t \geq 0$ . The symbol $a_{ij}^{ϑ (t)}$ expresses the connection between the two agents, and as $a_{ij}^{ϑ (t)} = 0$ , it expresses that agent $i$ cannot receive the efficient information from agent $j$ , otherwise $a_{ij}^{ϑ (t)} = 1$ . The augmented adjacency matrix is expressed as $A_{ϑ (t)} \overset{Δ}{=} [a_{ij}^{ϑ (t)}] \in R^{(n + 1) \times (n + 1)}$ . Define that $d_{i}^{ϑ (t)} \overset{Δ}{=} \sum_{j = 1}^{j = n + 1} a_{ij}^{ϑ (t)}$ , and the degree matrix is expressed as $D_{ϑ (t)} \overset{Δ}{=} diag {d_{1}^{ϑ (t)}, d_{2}^{ϑ (t)}, \dots, d_{(n + 1)}^{ϑ (t)}}$ . Then, the Laplacian matrix is expressed as $L_{ϑ (t)} \overset{Δ}{=} D_{ϑ (t)} - A_{ϑ (t)}$ . Suppose that the relational expression $[t_{k}, t_{k + 1}]$ is an infinite time-interval sequence, where $σ_{0}$ denotes a positive constant. The time-interval sequences need to satisfy: $t_{1} > 0$ , $in f_{k \in N} (t_{k + 1} - t_{k}) > σ_{0}$ , and $li m_{k \to + \infty} t_{k} = + \infty$ , over which the communication graph is time-invariant, where $t_{1}, t_{2}, \dots$ represent the switching time point of topologies graph, respectively.

In this paper, the issue of switching topologies is studied in the presented MASs, and the concept of average dwell time is utilized, which allows the fast switching to relax the bound of dwell time constraint. At first, the concept of average dwell time is provided in the following.

Definition 1 : (Liberzon, 2003) For an arbitrary switching signal $ϑ (t)$ , there exists a constant $σ_{a} > 0$ , if the inequality

N_{ϑ} (t, σ) \leq N_{0} + (t - σ) / σ_{a}

(1)

holds for all $t > σ > 0$ and the scalar $N_{0} > 0$ , where $N_{ϑ} (t, σ)$ denotes the number of mode switches for $ϑ (t)$ at the interval $(σ, t)$ , and $N_{0}$ is known as the bound of $ϑ (t)$ , then $σ_{a}$ can be called the average dwell time.

Based on the graph theory and the definition of Laplacian matrix, $L_{ϑ (t)}$ can be expressed as

ℒ_{ϑ (t)} ≜ [\begin{matrix} {\overset{⌣}{ℒ}}_{ϑ (t)} & {\overset{⌣}{b}}_{ϑ (t)} \\ 0_{n}^{T} & 0 \end{matrix}]

(2)

where ${\overset{⌣}{ℒ}}_{ϑ (t)}$ denotes the Laplacian matrix for MASs without the $(n + 1)$ -th agent, and ${\overset{⌣}{b}}_{ϑ (t)} ≜ {[a_{1 (n + 1)}^{ϑ (t)}, a_{2 (n + 1)}^{ϑ (t)}, \dots, a_{n (n + 1)}^{ϑ (t)}]}^{T}$ . 0_n stands for the $n$ -order column vector, where the elements are all equal to zero.

Remark 1: In this paper, the $(n + 1)$ -th agent is defined as leader’s agent, it generally assumes that leaders do not receive information from neighbor’s agents. Therefore, the element of $(n + 1)$ -order of Laplacian matrix of $G_{ϑ (t)}$ equals to zero. Therefore, the eigenvalues of Laplacian matrix $L_{ϑ (t)}$ are indeterminate. Based on the above analyses, we need to establish the matrix ${\hat{ℒ}}_{ϑ (t)} ≜ {\overset{⌣}{ℒ}}_{ϑ (t)} + diag {a_{1 (n + 1)}^{ϑ (t)}, a_{2 (n + 1)}^{ϑ (t)}, \dots, a_{n (n + 1)}^{ϑ (t)}}$ .

Problem statement

In this section, a class of MASs with one leader’s agent and $n$ followers’ agents is presented, and the influence of switching topologies is considered to adapt to the special circumstances. At first, the system model dynamics of $i$ -th follower’s agent is represented as follows

{\overset{\cdot}{x}}_{i} = A x_{i} + B u_{i} + B ξ (x_{i})

(3)

where $x_{i} \in R^{m}$ represents the state vector, $u_{i}$ expresses the control input, and the nonlinear function is defined as $ξ (x_{i})$ . Moreover, $A \in R^{m \times m}$ and $B \in R^{m \times p}$ are constant parameter matrices, $m$ and $p$ are positive integers. Meanwhile, the relation $p \leq m$ is valid.

Next, the leader agent’s model is constructed in the following

{\overset{\cdot}{x}}_{n + 1} = A x_{n + 1} + B ξ (x_{n + 1}) + Bf (t)

(4)

where $x_{n + 1} \in R^{m}$ denotes the state vector of leader’s agent, $ξ (x_{n + 1})$ expresses the nonlinear function, $f (t)$ expresses a known bounded function describing the external control inputs. Moreover, to realize the control scheme, an assumption is given as follows

Assumption 1: For the topology graph $G_{ϑ (t)}$ , it contains a spanning tree rooted at node $n + 1$ .

Under Assumption 1, it can be obtained that for each $t \geq 0$ , the inequalities

Q^{ϑ (t)} {\hat{L}}_{ϑ (t)} + {\hat{L}}_{ϑ (t)} Q^{ϑ (t)} > 0

(5)

are feasible for some positive-definite matrices $Q^{ϑ (t)}$ . Besides, it denotes that $k_{0} \overset{Δ}{=} λ_{max}$ $(Q^{ϑ (t)}) / λ_{min} (Q^{ϑ (t)})$ and ${\overset{⌣}{λ}}_{\min} ≜ \min {λ_{\min} (Q^{ϑ (t)} {\hat{ℒ}}_{ϑ (t)} + {\hat{ℒ}}_{ϑ (t)} Q^{ϑ (t)})$ } for later use.

For the context of MASs, the communication among agents is essential to complete the control task. Based on the communication relationships among $(n + 1)$ agents for the switching topologies, the distributed error of $i$ -th agent is defined as follows

s_{i} (t) = \sum_{j = 1}^{N + 1} a_{ij}^{ϑ (t)} (x_{i} - x_{j})

(6)

Before completing the design of the next control scheme, the following lemma is introduced, which describes the connection between two real matrixes with quadratic forms.

Lemma 1: (Dennis, 2009) There exist semi-definite positive matrices $W \in R^{r \times r}$ and $Y \in R^{nr}$ , a positive-definite matrix $D \in R^{n \times n}$ and a symmetric matrix $S \in R^{n \times n}$ , such that the following inequality can be satisfied

Y^{T} (S \otimes W) Y \geq λ_{min} (D^{- 1} S) Y^{T} (D \otimes W) Y

(7)

Lemma 2: (Heinonen, 2005) For nonlinear function $ξ (x)$ , there exist constants $l_{ξ}$ satisfying

∥ ξ (x) - ξ (y) ∥ \leq l_{ξ} ∥ x - y ∥

(8)

where $l_{ξ}$ is the Lipschitz constant.

Distributed event-triggered control scheme

In this section, the process of designing the distributed event-triggered consensus control is provided. The distributed event-triggered errors can be represented as below

Z_{i} = s_{i} (t_{i}) - s_{i} (t_{i}^{k}), t_{i} \in [t_{i}^{k}, t_{i}^{k + 1})

(9)

Then, the next triggered time $t_{i}^{k + 1}$ is identified by

t_{i}^{k + 1} = \inf {t > t_{i}^{k} | ∥ Z_{i} (t) ∥ \geq ℵ_{i} ∥ {\tilde{x}}_{i} (t_{i}^{k + 1}) ∥ + e^{- t_{i}^{k + 1}}}

(10)

where $ℵ_{i} \overset{Δ}{=} 1 / \sqrt{∥ Λ^{- 1} B B^{T} Λ^{- 1} ∥}$ , and $Λ$ is a positive-definite matrix.

Remark 2: To avoid the frequent updates of control input signals at an initial period of time of MASs operation, the event-triggered design can guarantee that control signals have a period of buffered time to adjust the system states. The negative exponential functions have larger initial values to increase the size of the event-triggered threshold during the initial phase of system regulation. Based on the larger event-triggered threshold, the event-triggered mechanism avoided triggered frequently. Hence, the control input signals will be remained for a longer period of time. Furthermore, the control input signal can avoid frequent updates, so the actuators avoid frequent jitter to a certain extent. If the precision requirement of the system is very high, indeed the usage of negative exponential function brings some disadvantages that the parameter precision of the system is lowered by the inevitable errors. Fortunately, this error is very small after a long time.

Event-triggered consensus control design

To accomplish the tracing consensus control objective and reduce usage of the communication resource, the ETC scheme is used to develop a class of distributed controller. Then, based on the distributed information of MASs with switching topologies, the distributed event-triggered controller is designed as

u_{i} = - μ K \sum_{j = 1}^{n + 1} a_{i j}^{ϑ (t)} (\overset{⌣}{x_{i}} - \overset{⌣}{x_{j}}) + f (t)

(11)

where $K \in R^{p \times m}$ stands for the gain matrix to be determined, $μ$ is a positive constant, ${\overset{⌣}{x}}_{i} ≜ x_{i} (t_{i}^{k})$ , and ${\overset{⌣}{x}}_{j} ≜ x_{j} (t_{i}^{k})$ .

Furthermore, according to the Kronecker product operation, the model of MASs can be expressed as

\overset{\cdot}{x} = (I_{N} \otimes A) x + (I_{N} \otimes B) u + (I_{N} \otimes B) ξ (x)

(12)

Similar to the operation of equation (12), to guarantee the proper calculation, the dynamics model of leader agent can also be transformed as the following equation

{\overset{\cdot}{\bar{x}}}_{n + 1} = (I_{N} \otimes A) x_{n + 1} + (I_{N} \otimes B) ξ (x_{n + 1}) + (I_{N} \otimes B) f (t)

(13)

where ${\bar{x}}_{n + 1} \overset{Δ}{=} \bar{I} \otimes x_{n + 1}$ , and $\bar{I} \in R^{N}$ denotes the identity column vector.

Remark 3 : The tracing consensus control scheme is proposed in this paper, such that the follower agents need to trace the leader agent. Hence, the state tracking error needs to be considered. However, the state dimensions of leader agent only satisfies the state dimensions of one follower agent. Based on the graph theory and Laplacian matrix, (3) can be transformed as the model of all follower’s agent (12). Therefore, the dynamics model of leader agent accordingly needs to be transformed to ensure the rationality of calculation.

The state tracking error can be defined as $\tilde{x} \overset{Δ}{=} x - x_{n + 1}$ , and the derivative of $\tilde{x}$ can be represented in the following

\begin{matrix} \overset{\cdot}{\tilde{x}} = (I_{N} \otimes A) \tilde{x} + (I_{N} \otimes B) u + (I_{N} \otimes B) ξ (x) \\ - (I_{N} \otimes B) ξ (x_{n + 1}) - (I_{N} \otimes B) f (t) \end{matrix}

(14)

Based on the above discussions, the following theorem can be obtained.

Theorem 1: A class of nonlinear MASs (3) with the leader dynamic model (4) is considered. Considering the distributed controller (11), if there exist positive scalars $α > 0$ and $β > 0$ , positive-definite matrix $Λ$ and $K \overset{Δ}{=} B^{T} Λ^{- 1}$ , such that the following matrix inequality is satisfied

A Λ + Λ A^{T} + 2 l_{ξ} B^{T} Λ - α B B^{T} + β Λ \leq 0

(15)

and the tracing consensus errors are bounded. Moreover, the average dwell time constraint needs to satisfy the following condition

σ_{a} > σ_{a}^{*} = \ln k_{0} / β

(16)

Proof. At first, the Lyapunov candidate function is constructed as

V = {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) \tilde{x}

(17)

Next, the derivative of equation (17) is obtained as follows

\overset{\cdot}{V} = {\overset{\cdot}{\tilde{x}}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) \tilde{x} + {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) \overset{\cdot}{\tilde{x}}

(18)

Based on equation (14), we can get

\begin{matrix} \overset{\cdot}{V} = ((I_{N} \otimes A) \tilde{x} + (I_{N} \otimes B) u + (I_{N} \otimes B) ξ (x) \\ - (I_{N} \otimes B) ξ (x_{n + 1}) - (I_{N} \otimes B) f (t))^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) \tilde{x} \\ + {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) ((I_{N} \otimes A) \tilde{x} + (I_{N} \otimes B) u \\ + (I_{N} \otimes B) ξ (x) - (I_{N} \otimes B) ξ (x_{n + 1}) - (I_{N} \otimes B) f (t)) \\ = {\tilde{x}}^{T} (I_{N} \otimes A)^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) \tilde{x} \\ + u^{T} (I_{N} \otimes B)^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) \tilde{x} \\ + {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) (I_{N} \otimes A) \tilde{x} \\ + {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) (I_{N} \otimes B) u \\ - 2 {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) (I_{N} \otimes B) f (t) \\ + 2 {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) (I_{N} \otimes B) ξ (x) \\ - 2 {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) (I_{N} \otimes B) ξ (x_{n + 1}) \end{matrix}

(19)

Then, based on the Lipschitz condition of Assumption 2, one has

\begin{matrix} \overset{\cdot}{V} \leq {\tilde{x}}^{T} (I_{N} \otimes A)^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) \tilde{x} \\ + u^{T} (I_{N} \otimes B)^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) \tilde{x} \\ + {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) (I_{N} \otimes A) \tilde{x} \\ + {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) (I_{N} \otimes B) u \\ - 2 {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) (I_{N} \otimes B) f (t) \\ + 2 {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) (I_{N} \otimes B) l_{ξ} \tilde{x} \end{matrix}

(20)

Next, according to equation (11), one has

\begin{array}{l} \dot{V} \leq {\tilde{x}}^{T} ({(I_{N} \otimes A)}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) \\ + (Q^{ϑ (t)} \otimes Λ^{- 1}) (I_{N} \otimes A)) \tilde{x} \\ + 2 {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) (I_{N} \otimes B) l_{ξ} \tilde{x} . \\ - μ {\overset{⌣}{\tilde{x}}}^{T} K {({\hat{ℒ}}_{ϑ (t)} \otimes I_{m})}^{T} {(I_{N} \otimes B)}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) \tilde{x} \\ - μ {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) (I_{N} \otimes B) K ({\hat{ℒ}}_{ϑ (t)} \otimes I_{m}) \overset{⌣}{\tilde{x}} \\ = {\tilde{x}}^{T} [Q^{ϑ (t)} \otimes (A^{T} Λ^{- 1} + Λ^{- 1} A + 2 l_{ξ} Λ^{- 1} B)] \tilde{x} \\ - μ {\tilde{x}}^{T} [(Q^{ϑ (t)} {\hat{ℒ}}_{ϑ (t)} + {\hat{ℒ}}_{ϑ (t)} Q^{ϑ (t)}) \otimes Λ^{- 1} B K) \overset{⌣}{\tilde{x}} \end{array}

(21)

The gain matrix $K \overset{Δ}{=} B^{T} Λ^{- 1}$ is substituted into equation (21), it gets

\begin{array}{l} \dot{V} \leq {\tilde{x}}^{T} [Q^{ϑ (t)} \otimes (A^{T} Λ^{- 1} + Λ^{- 1} A + 2 l_{ξ} Λ^{- 1} B)] \tilde{x} \\ - μ {\tilde{x}}^{T} [(Q^{ϑ (t)} {\hat{ℒ}}_{ϑ (t)} + {\hat{ℒ}}_{ϑ (t)} Q^{ϑ (t)}) \otimes Λ^{- 1} B B^{T} Λ^{- 1}] \overset{⌣}{\tilde{x}} \end{array}

(22)

Based on equation (9), it yields that

\begin{matrix} \overset{\cdot}{V} \leq {\tilde{x}}^{T} [Q^{ϑ (t)} \otimes (A^{T} Λ^{- 1} + Λ^{- 1} A + 2 l_{ξ} Λ^{- 1} B)] \tilde{x} \\ - μ {\tilde{x}}^{T} [(Q^{ϑ (t)} {\hat{L}}_{ϑ (t)} + {\hat{L}}_{ϑ (t)} Q^{ϑ (t)}) \otimes Λ^{- 1} B B^{T} Λ^{- 1}] \tilde{x} \\ - μ {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1} B B^{T} Λ^{- 1}) Z \end{matrix}

(23)

Then, according to equation (7), it holds

\begin{array}{l} \dot{V} \leq {\tilde{x}}^{T} [Q^{ϑ (t)} \otimes (A^{T} Λ^{- 1} + Λ^{- 1} A + 2 l_{ξ} Λ^{- 1} B - μ {\overset{⌣}{λ}}_{\min} Λ^{- 1} B B^{T} Λ^{- 1})] \tilde{x} \\ - μ {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1} B B^{T} Λ^{- 1}) Z \end{array}

(24)

Based on the event-triggered condition equations (10) and (15), equation (24) can be transformed as

\overset{\cdot}{V} \leq - β {\tilde{x}}^{T} (Q^{ϑ (t)} \otimes Λ^{- 1}) \tilde{x}

(25)

Hence, it gets from equation (25) that

\overset{\cdot}{V} \leq - β V

(26)

Defining $\bar{t}$ is an arbitrary positive constant, $N_{ϑ} (\bar{t}, 0)$ expresses the frequency of discontinuities of switching signal $σ (t)$ over the time-interval $(0, \bar{t})$ . Then, on account of equation (26), it gets

V (\bar{t}) \leq V (0) e^{- β t}

(27)

as $N_{ϑ} (\bar{t}, 0) = 0$ , and $t_{i}, i = 1, \dots, N_{ϑ} (\bar{t}, 0)$ , denotes the switching time points of $σ (t)$ over $(0, \bar{t})$ with $t_{1} < \dots < t_{N_{ϑ} (\bar{t}, 0)}$ . For $N_{ϑ} (\bar{t}, 0) = 1$ , from equation (26) and the fact

V (t_{i}) \leq k_{0} V (t_{i}^{-})

(28)

it can be obtained that

V (\bar{t}) \leq k_{0} V (0) e^{- β \bar{t}}

(29)

For $N_{ϑ} (\bar{t}, 0) \geq 2$ , the following relation can be got

\begin{matrix} V (\bar{t}) \leq V (t_{N_{ϑ} (\bar{t}, 0)}) e^{- β (\bar{t} - t_{N_{ϑ} (t, 0)})} \\ \leq k_{0} V (t_{N_{ϑ} (\bar{t}, 0)}) e^{- β (\bar{t} - t_{N_{ϑ} (t, 0) - 1})} \end{matrix}

(30)

Based on inductive reasoning, it has

V (\bar{t}) \leq k_{0}^{N_{ϑ} (\bar{t}, 0)} V (0) e^{- β \bar{t}}

(31)

By Definition 1, it yields that

V (\bar{t}) \leq M_{1} e^{- (β - \frac{\ln k_{0}}{σ_{a}}) \bar{t}}

(32)

where $M_{1} \overset{Δ}{=} k_{0}^{N_{0}} V (0)$ . Therefore, the condition $σ_{a} > \ln k_{0} / β$ implies $β > \ln k_{0} / σ_{a}$ . Thus, it follows from equation (32) that the tracking consensus control is achieved for the considered MASs.

Furthermore, the event-triggered control mechanism is applied for the tracking consensus control in this paper. To guarantee the feasibility and effectiveness of the proposed event-triggering control mechanism, the Zeno behavior needs to be excluded. In other words, the phenomenon of an infinite number of events needs to be avoided in a finite time interval. The following theorem is given by the relevant results.

Theorem 2: Considering the proposed control scheme and the event-triggered mechanism in this paper. Let Assumptions 1–2 be satisfied. If the tracing errors are bounded by the designed parameters in Theorem 1, it ensures that no agent exhibits Zeno behavior under the event-triggered condition (10).

Proof: To prove that the Zeno behavior does not exist, the time evolution of $Z_{i} (t)$ is considered. Thus, the derivative of event-triggered error equation (6) in $[t_{i}^{k}, t_{i}^{k + 1})$ is given as follows

{\overset{\cdot}{Z}}_{i} (t) = ∥ \sum_{j = 1}^{N + 1} a_{ij}^{ϑ (t)} (A x_{i} - A x_{j} + B u_{i} - B u_{j}) ∥

(33)

Next, the upper right-hand derivative of $Z_{i} (t)$ is

\begin{matrix} D^{+} ∥ Z_{i} (t) ∥ \leq \frac{Z_{i}^{T} (t) Z_{i} (t)}{∥ Z_{i} (t) ∥} \leq ∥ {\overset{\cdot}{Z}}_{i} (t) ∥ \\ \leq ∥ A s_{i} (t) + \sum_{j = 1}^{N + 1} a_{ij}^{ϑ (t)} BK (c_{i} s_{i} (t_{k}) - c_{j} s_{j} (t_{k})) ∥ \\ \leq ∥ A ∥ ∥ Z_{i} (t) ∥ + ℵ_{s} \end{matrix}

(34)

where $ℵ_{s} \overset{Δ}{=} ∥ A s_{i} (t_{k}) + \sum_{j = 1}^{N + 1} a_{ij}^{ϑ (t)} BK (c_{i} s_{i} (t_{k}) - c_{j} s_{j} (t_{k})) ∥$ . Then, the derivative of event-triggered error equation (6) at $[t_{i}^{k}, t_{i}^{k + 1})$ satisfies the above relation. Furthermore, integrating both sides of equation (34), one has

\begin{matrix} ∥ Z_{i} (t) ∥ \leq \int_{t_{i}^{k}}^{t} \exp ((∥ A ∥ (t - s)) ℵ_{s} ds, \\ \leq \frac{ℵ_{s}}{∥ A ∥} (\exp (∥ A ∥ (t - t_{i}^{k})) - 1) \end{matrix}

(35)

Moreover, based on the event-triggered condition equation (10), it yields that

ℵ_{i} \leq \frac{ℵ_{s}}{∥ A ∥} (\exp (∥ A ∥ (t_{i}^{k + 1} - t_{i}^{k})) - 1)

(36)

thus, it has

T_{i}^{k} \geq \frac{1}{∥ A ∥} \ln (\frac{ℵ_{i} ∥ A ∥}{ℵ_{s}} + 1) > 0

(37)

where $T_{i}^{k} \overset{Δ}{=} t_{i}^{k + 1} - t_{i}^{k}$ . Therefore, the event-triggered time interval $T_{i}^{k}$ is bounded by a positive scalar. In other words, there is no Zeno behavior.

Remark 4 : Note that it requires no Zeno behavior exists at $[t_{i}^{k}, t_{i}^{k + 1})$ , and hence the expressions (33) and (34) are derived in the proof of Theorem 2. To be specific, the Zeno behaviors do not exist at the continuous and discrete points by using equations (33) and (34). Moreover, it is worth noting that the distributed ETC mechanism has some distinctive features. The distributed ETC scheme is established by the distributed error of MASs. Therefore, the proposed ETC mechanism is fully distributed by the global information of MASs, and triggering time of each agent is independent.

Numerical example

In this section, a numerical example is provided to demonstrate the effectiveness of the proposed event-triggered consensus control approach. It assumes that the presented MASs have five agents, and the fifth agent is defined as the leader’s agent, and the communication topologies among agents are shown in Figure 1. The influence of switching topologies is considered in this paper, so the switching signal $ϑ (t)$ is needed to adjust the switching sequence of the topology, as shown in Figure 2.

Figure 1.

Communication switching topologies.

Figure 2.

Switching signal function $ϑ (t)$ .

The parameters of MASs model are provided as follows

\begin{matrix} x_{i} (t) = c ol (x_{i 1}, x_{i 2}), \\ ℏ_{i} (x_{i}) = c ol (0.86 \cos (x_{i 1}), 0.95 \cos (x_{i 2})), \\ f (t) = c ol (0, 0.5 \cos (0.5 t) + 0.25 \sin (0.25 t)) \end{matrix}

The state space matrices can be chosen as

A = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}], B = [\begin{matrix} 1.25 & 0 \\ 0 & 1.35 \end{matrix}]

To satisfy Assumption 1, the following values are set: $λ_{max} (Φ^{ϑ (t)}) = 6.525$ , $λ_{min} (Φ^{ϑ (t)}) = 0.25$ , and $k_{0} = 26.1$ . The parameter of matrix inequation (15) and control gain parameter are defined as $β = 3.1$ and $μ = 100$ , respectively. The average dwell time of the switching signal satisfies $σ_{a} > σ_{a}^{*} = \ln k_{0} / β = 1.0522$ .

Besides, the initial states of MASs are given as $x_{1} (0) = [0.26, 1.27]^{T}$ , $x_{2} (0) = [0.86, - 0.15]^{T}$ , $x_{3} (0) = [- 1.34, 0.27]^{T}$ , $x_{4} (0) = [1.43, - 0.7]^{T}$ , $x_{5} (0) = [[2, 0.25]^{T}$ . By solving the conditions of Theorems 1–2, the corresponding results, determined by the distributed event-triggered consensus control scheme, are plotted in Figures 3 –10. Specifically, Figure 3 shows the trajectories of the states $x_{11}$ , $x_{21}$ , $x_{31}$ , $x_{41}$ , and $x_{51}$ . The tracking responses of the states $x_{12}$ , $x_{22}$ , $x_{32}$ , $x_{42}$ , and $x_{52}$ are displayed in Figure 4. Therefore, it is testified that the tracing consensus control objective is realized from Figures 3 and 4. Next, the controller responses of $u_{i 1}$ and $u_{i 2}$ are provided in Figures 5–6, respectively, and the triggered time intervals are shown in Figures 7 –10. In contrast to traditional time transmission, the control signal information is transmitted all the time. From Figures 5–6, the controller signals are updated intermittently, and it observably economizes on the communication resources from Figures 7 to 10. The Zeno behavior not appears based on the proposed control scheme. Finally, it confirms that the developed event-triggered consensus control scheme is effective and feasible via the simulation results.

Figure 3.

The tracking trajectory of states $x_{i 1}$ , $i$ = $1, 2, 3, 4, 5$ .

Figure 4.

The tracking trajectory of states $x_{i 2}$ , $i$ = $1, 2, 3, 4, 5$ .

Figure 5.

The controller responses of $u_{i 1}$ , $i$ = $1, 2, 3, 4$ .

Figure 6.

The controller responses of $u_{i 2}$ , $i$ = $1, 2, 3, 4$ .

Figure 7.

The trigger-time interval of $1$ -st agent.

Figure 8.

The trigger-time interval of $2$ -nd agent.

Figure 9.

The trigger-time interval of $3$ -rd agent.

Figure 10.

The trigger-time interval of $4$ -th agent.

Conclusion

This paper has studied the distributed event-triggered consensus control problem for a class of MASs with switching topologies. To guarantee that the proposed strategy could be applied to some complex systems, both the nonlinear functions and switching topologies have been simultaneously considered in the presented MASs. A distributed event-triggered mechanism has been proposed, which can reduce the usage of communication resources. Moreover, the Zeno behavior has been avoided for the proposed distributed ETC scheme. With the aid of the Lyapunov inequality theory, the solvable inequality conditions have been derived for the MASs with switching topologies to deal with the consensus tracking issue. The effectiveness and feasibility of the proposed distributed event-triggered consensus control approach have been testified via a numerical example. In future work, how to further improve the ETC scheme to adapt to more complex cases of the MASs is an interesting topic.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the National Natural Science Foundation of China under Grants 62222310, 61973131, in part by the Research Fund for the Taishan Scholar Project of Shandong Province of China, and in part by the Fujian Outstanding Youth Science Fund under Grant 2020J06022.

ORCID iD

Yanzheng Zhu

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