Edge-Event-Triggered Synchronization for Multi-Agent Systems with Nonlinear Controller Outputs

Abstract

This paper addresses the synchronization problem of multi-agent systems with nonlinear controller outputs via event-triggered control, in which the combined edge state information is utilized, and all controller outputs are nonlinear to describe their inherent nonlinear characteristics and the effects of data transmission in digital communication networks. First, an edge-event-triggered policy is proposed to implement intermittent controller updates without Zeno behavior. Then, an edge-self-triggered solution is further investigated to achieve discontinuous monitoring of sensors. Compared with the previous event-triggered mechanisms, our policy design considers the controller output nonlinearities. Furthermore, the system’s inherent nonlinear characteristics and networked data transmission effects are combined in a unified framework. Numerical simulations demonstrate the effectiveness of our theoretical results.

1. Introduction

In recent years, distributed coordinated control of multi-agent systems has attracted a lot of attention from researchers [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. The main attention is focused on two aspects: one is how to more accurately describe the models of multi-agent systems, and the other is how to design the information interaction mechanisms between agents to achieve control objectives. For these two aspects, researchers have not only studied the coordinated control of multi-agent systems with various dynamics, but also combined the design of data interaction mechanisms and the effects of communication networks, sensors and controllers, for instance, data packet loss, sensor noises, external disturbances, etc.

Recently, event- and self-triggered sampling [21,22,23,24,25,26,27,28,29,30,31,32,33] have been introduced into multi-agent coordinated control [32,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51], and its sampling times are determined by event-triggered conditions rather than by synchronous or asynchronous digital clock signals. In comparison with traditional periodic sampled-data control, distributed event-triggered control performs data sampling according to the needs of agents, avoiding redundant sampling at unnecessary times, thereby saving system resources. According to the information acquisition methods of the sensors equipped by agents, distributed event-triggered sampling can usually be divided into node-based [34,35,36,38,44,52,53] and edge-based [39,40,41,42,43,45,46,47,54,55]. These two methods are suitable for different scenarios. For the node-based event-triggered control, the sensors assembled in each agent are required to sample each individual’s absolute state information and communicate with neighbor individuals. For instance, employing GPS and inertial navigation equipments to obtain its absolute velocity and position information in the given inertial reference frame. For edge-based event-triggered control, it is applicable in the scenarios where the absolute state information is difficult to measure. For instance, in the deep sea or deep space, each AUV or spacecraft employs sonar or LiDAR to measure the relative velocity and position between itself and its neighbors to achieve multi-agent coordination goals. In [39,45,46,47,54,55], all relevant edge state information of each agent is acquired and processed together, which is different from the way of asynchronously acquiring and processing edge state information [40,41,42,43,45]. However, most of the previous results have not considered the controller output nonlinearities. On one hand, the controller device may have inherent nonlinear characteristics that cause the output signals to be distorted. For instance, the nonlinear characteristics of the control circuits may lead to signal output distortion and saturation. On the other hand, the employment of quantizers may also lead to the distortion of controller output information.

In this paper, we provide edge-event- and edge-self-triggered control solutions to solve the synchronization problem of Lipschitz nonlinear multi-agent systems with controller output nonlinearities, in which the edge states associated with each individual are processed together. Besides, we take the effects of controller output nonlinearities into account. It does not only include the nonlinear characteristics of controllers, but also the effects of nonlinear coding required for data transmission in digital communication networks. To avoid the requirement for sensor continuous scanning due to continuous monitoring of measurement errors in the proposed event-triggered condition, an edge-self-triggered policy is proposed to save sensor resource consumption. In the designed event-triggered policy, neighboring individuals do not need to employ communication networks to transmit data. However, in the proposed self-triggered policy, a small amount of data communication is necessary, which brings savings in sensor resources. There exists a trade-off between communication resource consumption and sensor resource consumption between the two policies, which are suitable for different application scenarios. Nonlinear dynamic models can be used to describe a large number of actual systems in engineering scenarios, for example, various types of nonlinear circuits, spring-mass-vehicle devices, inverted pendulums, etc. The research of nonlinear multi-agent system coordination control problems can provide solutions for such scenarios. The proposed triggering mechanisms can save resources consumed by interaction, and the considered output nonlinear fluctuation can evaluate the impact of communication transmissions and actuator nonlinearities. Although there have been some results on the coordination of nonlinear multi-agent systems based on event-triggered control [45,54,55], the influences of data quantization and actuator nonlinearities on systems have not been considered. Our work evaluates the applicability of event-triggered mechanisms in such scenarios. The novelties of this study are summarized in the following.

• The proposed event- and self-triggered policies take the effects of controller output nonlinearities into account, not only including controllers’ inherent nonlinear properties, but also the signal distortion caused by network data transmission. In comparison with edge-based distributed event-triggered control in [39,40,41,42,43,45,46,47,54,55], our policies consider non-ideal digital signal transmission in actual application scenarios, paving the way for digital applications of distributed sampled-data control.
• The proposed event- and self-triggered schemes have trade-offs in the scheduling of system resources. The event-triggered policy does not require digital signal transmission between neighboring individuals but requires all sensors to continuously obtain edge state information to evaluate triggering conditions and implement intermittent controller updates. The proposed self-triggered policy avoids continuous acquisition of edge state information and requires a small amount of information transmission between neighboring individuals. The trade-off between these two policies is also suitable for scenarios with different levels of resource schedulability. Furthermore, in comparison with the previous distributed self-triggered approaches [34,39,41,42,46,47,52], the interactive control inputs in our algorithm are quantized information, which is a more efficient solution for saving communication bandwidth.

The rest of this article is arranged as follows. Section 2 gives preliminaries on graph theory and problem formulation. Section 3 provides convergence analysis of the proposed edge-event-triggered policy. Section 4 further designs the edge self-triggered control algorithm. Section 5 shows the results on numerical simulations. Finally, Section 6 concludes this paper.

2. Preliminaries and Problem Formulation

2.1. Preliminaries on Graph Theory

For an undirected graph $\mathcal{G}$ consisting of m edges and N nodes, it has the form $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with node set $\mathcal{V}=\{v_1,v_2,\cdots ,v_n\}$ and edge set $\mathcal{E}=\{(v_i,v_j):v_i,v_j\in \mathcal{V}\}$. The set of neighbors of node i is denoted by $N_i=\{v_j\in \mathcal{V}:(v_j,v_i)\in \mathcal{E}\}$, where $|N_i|$ represents the cardinality of $N_i$, i.e., $2m=|N_1|+|N_2|+\cdots +|N_N|$. Then, we introduce several important matrices in the graph theory. The matrix which relates the nodes to the edges is called incidence matrix, denoted by $H=\left\{h_{ki}\right\}\in {\mathbb{R}}^{m\times N}$ [47]. By choosing arbitrary orientation for all edges in the undirected graph $\mathcal{G}$, the entries of its incidence matrix are defined as $h_{ki}=1$ if the kth edge sinks at node i, $h_{ki}=-1$ if the kth edge leaves node i, or $h_{ki}=0$ otherwise. Therefore, the Laplacian matrix is represented as $L=H^{T}WH$ and the edge Laplacian matrix is $L_E=H{H}^T$, where $W=\mathrm{diag}(w_1,w_2,\cdots ,w_m)$ and each $w_p>0$, $p=1,2,\cdots ,m$, is the weight of pth edge. Moreover, the matrices $L_E$ and $H^{T}H$ share the common nonzero eigenvalues.

For the edge $(v_i,v_j)\in \mathcal{E}$ with the orientation from agents $v_j$ to $v_i$, let $y_p\left(t\right)=y_{(i,j)}\left(t\right)=x_i\left(t\right)-x_j\left(t\right)$ be the corresponding edge state.

2.2. Problem Formulation

We consider a multi-agent system with a group of agents with nonlinear dynamics, which are described as

$$\begin{array}{c}\hfill x_i^{\prime }\left(t\right)=f\left(x_i\right)+u_i\left(t\right),i\in \mathcal{V}\end{array}$$

(1)

where $x_i\left(t\right)\in {\mathbb{R}}^n$ is the state vector of the ith agent, $u_i\left(t\right)\in {\mathbb{R}}^n$ is the control input, and $f\left(x_i\right)\in {\mathbb{R}}^n$ is the inherent nonlinear dynamics.

Definition 1.

The nonlinear function $f\left(x\right)$ in system (1) satisfies the Lipschitz condition in $x\in {\mathbb{R}}^n$, i.e.,

$$\begin{array}{c}\hfill \parallel f\left(x_i\right)-f\left(x_j\right)\parallel \le \mathcal{L}\parallel x_i-x_j\parallel ,x_i,x_j\in {\mathbb{R}}^n,\end{array}$$

(2)

where $\mathcal{L}$ represents the Lipschitz constant.

In this article, the main goal is to design edge-based event- and self-triggered policies to solve the synchronization problem of nonlinear multi-agent systems. Assume that for each agent the given event conditions are constantly monitored to determine whether to update, and the event time sequence of each agent $v_i$ is set as $t_0^i,t_1^i,t_2^i,\cdots$. For agent $v_i$, denote the latest sampling edge states of ${\epsilon }_p$ by ${\widehat{y}}_p^i\left(t\right)={\widehat{y}}_p^i\left(t_k^i\right)=y_p\left(t_k^i\right)$, $t\in [t_k^i,t_{k+1}^i)$. For each agent $v_i$, $i=1,2,\cdots ,N$, we design the control protocol with nonlinear controller outputs as follows,

$$\begin{array}{c}\hfill u_i\left(t\right)=-g\left(c\sum _{p\in {\mathcal{E}}_i}{h}_{ip}{w}_p{\widehat{y}}_p^i\left(t\right)\right),t\in [t_k^i,t_{k+1}^i),\end{array}$$

(3)

where the control gain $c>0$. Here, the nonlinear vector function $g\left(\mathcal{Y}\right):{\mathbb{R}}^n\to {\mathbb{R}}^n$ has the form $g\left(\mathcal{Y}\right)$=${\left[g_1{\left({\mathcal{Y}}_1\right)}^T,g_2{\left({\mathcal{Y}}_2\right)}^T,\cdots ,g_N{\left({\mathcal{Y}}_N\right)}^T\right]}^T$. Here, for any $q=1,2,\cdots ,N$, $g_q\left({\mathcal{Y}}_q\right)$ satisfies the following properties.

P1.

$g_q\left({\mathcal{Y}}_q\right)=0$ if and only if ${\mathcal{Y}}_q=0$.

P2.

$-g_q\left({\mathcal{Y}}_q\right)=g_q(-{\mathcal{Y}}_q)$.

P3.

$g_1{\mathcal{Y}}_q\le g_q\left({\mathcal{Y}}_q\right)\le k_2{\mathcal{Y}}_q$ with the positive scalars ${\kappa }_1<{\kappa }_2$.

Besides, the following inequalities can be derived based on the properties P1–P3.

• $k_1^2{\mathcal{Y}}_q^2\le g_q{\left({\mathcal{Y}}_q\right)}^2\le k_2^2{\mathcal{Y}}_q^2$.
• $k_1{\mathcal{Y}}_q^2\le {\mathcal{Y}}_q{g}_q\left({\mathcal{Y}}_q\right)\le k_2{\mathcal{Y}}_q^2$.
• $\frac{k_1}{k_2}{g}_q{\left({\mathcal{Y}}_q\right)}^2\le {\mathcal{Y}}_q{g}_q\left({\mathcal{Y}}_q\right)\le \frac{k_2}{k_1}{g}_q{\left({\mathcal{Y}}_q\right)}^2$.

For system (1), its graph $\mathcal{G}$ satisfies the following assumption and lemma.

Assumption 1.

The interaction topology $\mathcal{G}$ of system (1) is connected.

Lemma 1.

For a connected graph $\mathcal{G}$, its Laplacian matrix L and edge Laplacian matrix $L_E$ satisfy ${\lambda }_2{\xi }^T\xi \le {\xi }^{T}L\xi \le {\lambda }_n{\xi }^T\xi$ and ${\overline{\lambda }}_2{\zeta }^T\zeta \le {\zeta }^T{L}_E\zeta \le {\overline{\lambda }}_n{\zeta }^T\zeta$, for any real vectors $\xi ,\zeta \in {\mathbb{R}}^n$, where ${\lambda }_2$, ${\overline{\lambda }}_2$ and ${\lambda }_n$, ${\overline{\lambda }}_n$ are the minimum and maximum non-zero eigenvalues of L, $L_E$, respectively, with the properties ${\xi }^T{1}_n=0$ and ζ.

Remark 1.

The proposed controller output nonlinearities mainly include two aspects: one is the inherent nonlinear characteristics of controllers, such as the nonlinear oscillation characteristics of the control circuit. The second is the nonlinear encoding required for network transmission of information, such as quantization transmission. These influencing factors can be described by nonlinear functions that satisfy the properties P1–P3.

3. Edge-Event-Triggered Policy

We now introduce the following event-triggered condition. For $i=1,2,\cdots ,N$, denote $r_i\left(t\right)={\sum }_{p\in {\mathcal{E}}_i}{h}_{ip}{w}_p{y}_p\left(t\right)$ and ${\widehat{r}}_i\left(t_k^i\right)={\widehat{r}}_i\left(t\right)={\sum }_{p\in {\mathcal{E}}_i}{h}_{ip}{w}_p{\widehat{y}}_p$. Define the state measurement error by

$$\begin{array}{c}\hfill e_i\left(t\right)={\widehat{r}}_i\left(t\right)-r_i\left(t\right),t\in [t_k^i,t_k^{i+1}).\end{array}$$

(4)

Each triggering instant $t_k^i$ is determined by

$$\begin{array}{c}\hfill t_{k+1}^i=inf\left\{t>t_k^i:{\overline{f}}_i\left(t\right)\ge 0\right\},i\in \mathbb{N},\end{array}$$

(5)

where

$$\begin{array}{c}\hfill {\overline{f}}_i\left(t\right)=\parallel e_i\left(t\right){\parallel }^2-{\sigma }_i{\parallel g\left(c{\widehat{r}}_i\left(t_k^i\right)\right)\parallel }^2-\frac{{\beta }_1}{{\beta }_2+{\beta }_3{t}^2}.\end{array}$$

(6)

Here, ${\sigma }_i$ and ${\beta }_i$, $i=1,2,\cdots ,N$ are positive constants to be designed.

Denote $\tilde{f}\left(y_p\right)=f\left(x_i\right)-f\left(x_j\right)$, the edge state vector $y\left(t\right)$ satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{y}\left(t\right)=& F\left(y\right)-c(H\otimes I_n)G\left(c\widehat{r}\right)\hfill \\ \hfill =& F\left(y\right)-c(H\otimes I_n)G\left(c(r+e)\right),\hfill \end{array}\end{array}$$

(7)

where $x\left(t\right)={[x_1^T\left(t\right)x_2^T\left(t\right)\cdots x_N^T\left(t\right)]}^T$, $y\left(t\right)={[y_1^T\left(t\right)y_2^T\left(t\right)\cdots y_m^T\left(t\right)]}^T$, $e\left(t\right)={[e_1^T\left(t\right)e_2^T\left(t\right)\cdots e_N^T\left(t\right)]}^T$, $F\left(y\right)={[\tilde{f}{\left(y_1\right)}^T\tilde{f}{\left(y_2\right)}^T\cdots \tilde{f}{\left(y_m\right)}^T]}^T$, and $G\left(c\widehat{r}\right)={[g{\left(c{\widehat{r}}_1\right)}^{T}g{\left(c{\widehat{r}}_2\right)}^T\cdots g{\left(c{\widehat{r}}_m\right)}^T]}^T$. Here, $r\left(t\right)=(H^{T}W\otimes I_n)y\left(t\right)$, where $r\left(t\right)={[r_1{\left(t\right)}^T{r}_2{\left(t\right)}^T\cdots r_N{\left(t\right)}^T]}^T$.

The following theorem presents some sufficient conditions for achieving synchronism tracking of system (1) with event-triggered information.

Theorem 1.

Suppose that the underlying topology $\mathcal{G}$ of the Lipschitz nonlinear multi-agent system (1) with control input (3) and triggering condition (6) is connected. If the nonlinear function $g(·)$ satisfies properties P1–P3 and there exist positive scalars a, $b_1$, $b_2$, c, ${\beta }_1$, ${\beta }_2$, ${\beta }_3$, $k_1$, $k_2$, and ${\sigma }_i$, $i=1,2,\cdots ,N$, such that the defined parameter $S_3>0$ in (16). Then, the system (1) achieves state synchronization asymptotically for any initial condition. Moreover, all triggering time intervals are lower bounded by a positive scalar such that all agents achieve Zeno-free triggering.

Proof. 

Choose the following Lyapunov function.

$$\begin{array}{c}\hfill V\left(t\right)=y{\left(t\right)}^T(W\otimes I_n)y\left(t\right),\end{array}$$

(8)

Taking the time derivative of $V\left(t\right)$ along the trajectory of (7), which yields

$$\begin{array}{cc}\hfill \frac{dV\left(t\right)}{dt}=& 2y{\left(t\right)}^T(W\otimes I_n)F\left(y\right)-2y{\left(t\right)}^T(WH\otimes I_n)G\left(c\widehat{r}\right)\hfill \\ \hfill \le & a{w}_{max}y{\left(t\right)}^T(W\otimes I_n)y\left(t\right)+\frac{1}{a}F{\left(y\right)}^{T}F\left(y\right)\hfill \\ \hfill & -2{\left[(H^{T}W\otimes I_n)y\left(t\right)\right]}^{T}G\left(c\widehat{r}\right)\hfill \\ \hfill \le & \left(a{w}_{max}+\frac{{\mathcal{L}}^2}{a{w}_{min}}\right)y{\left(t\right)}^T(W\otimes I_n)y\left(t\right)\hfill \\ \hfill & -2\widehat{r}{\left(t\right)}^{T}G\left(c\widehat{r}\right)+2e{\left(t\right)}^{T}G\left(c\widehat{r}\right),\hfill \end{array}$$

(9)

where we use the fact that $2y{\left(t\right)}^T(W\otimes I_n)F\left(y\right)\le ay{\left(t\right)}^T(W^2\otimes I_n)y\left(t\right)+\frac{1}{a}F{\left(y\right)}^{T}F\left(y\right)$ for any positive real constant a to derive the first inequality, and the second one takes advantage of the Lipschitz condition of the function $f\left(y\right)$. However, a has to be fixed due to the $S_1$.

Furthermore, the triangle inequality ${\xi }^T\zeta \le \frac{b}{2}{\xi }^T\xi +\frac{1}{2b}{\zeta }^T\zeta$ holds for any real vectors $\xi$, $\zeta$ and positive scalar b, which implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill r{\left(t\right)}^{T}r\left(t\right)=& {[\widehat{r}\left(t\right)-e\left(t\right)]}^T[\widehat{r}\left(t\right)-e\left(t\right)]\hfill \\ \hfill \le & (1+b_1)\widehat{r}{\left(t\right)}^T\widehat{r}\left(t\right)+(1+\frac{1}{b_1})e{\left(t\right)}^{T}e\left(t\right),\hfill \end{array}\end{array}$$

(10)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 2e{\left(t\right)}^{T}G\left(c\widehat{r}\right)\le & b_{2}e{\left(t\right)}^{T}e\left(t\right)+\frac{1}{b_2}G{\left(c\widehat{r}\right)}^{T}G\left(c\widehat{r}\right)\hfill \\ \hfill \le & \left(b_2{\sigma }_{max}+\frac{1}{b_2}\right)G{\left(c\widehat{r}\right)}^{T}G\left(c\widehat{r}\right)\hfill \\ \hfill & +\frac{b_2{\beta }_{1}N}{{\beta }_2+{\beta }_3{t}^2}\hfill \\ \hfill \le & \left(b_2{\sigma }_{max}{k}_2^2{c}^2+\frac{k_2^2{c}^2}{b_2}\right)\widehat{r}{\left(t\right)}^T\widehat{r}\left(t\right)\hfill \\ \hfill & +\frac{b_2{\beta }_{1}N}{{\beta }_2+{\beta }_3{t}^2},\hfill \end{array}\end{array}$$

(11)

where the final inequality uses that the nonlinear vector function $G\left(c\widehat{r}\right)$ satisfies $G{\left(c\widehat{r}\right)}^2\le k_2^2{\left(c\widehat{r}\left(t\right)\right)}^2$. Noting that ${\sigma }_{max}=ma{x}_{i\in \{1,2,\cdots ,N\}}{\sigma }_i$. Obtained from the inequality (10), one gets

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill -\widehat{r}{\left(t\right)}^T\widehat{r}\left(t\right)\le & -\frac{1}{1+b_1}r{\left(t\right)}^{T}r\left(t\right)+\frac{1}{b_1}e{\left(t\right)}^{T}e\left(t\right)\hfill \\ \hfill \le & -\frac{1}{1+b_1}r{\left(t\right)}^{T}r\left(t\right)+\frac{{\sigma }_{max}}{b_1}G{\left(c\widehat{r}\right)}^{T}G\left(c\widehat{r}\right)\hfill \\ \hfill & +\frac{{\beta }_{1}N}{b_1({\beta }_2+{\beta }_3{t}^2)}\hfill \\ \hfill \le & -\frac{1}{1+b_1}r{\left(t\right)}^{T}r\left(t\right)+\frac{{\sigma }_{max}{k}_2^2{c}^2}{b_1}\widehat{r}{\left(t\right)}^T\widehat{r}\left(t\right)\hfill \\ \hfill & +\frac{{\beta }_{1}N}{b_1({\beta }_2+{\beta }_3{t}^2)},\hfill \end{array}\end{array}$$

(12)

Then, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill -\widehat{r}{\left(t\right)}^T\widehat{r}\left(t\right)\le & -\frac{b_1}{(1+b_1)(b_1+{\sigma }_{max}{k}_2^2{c}^2)}r{\left(t\right)}^{T}r\left(t\right)\hfill \\ \hfill & +\frac{{\beta }_{1}N}{(b_1+{\sigma }_{max}{k}_2^2{c}^2)({\beta }_2+{\beta }_3{t}^2)}.\hfill \end{array}\end{array}$$

(13)

The inequality (9) yields

$$\begin{array}{cc}\hfill V^{\prime }(t)\le & (a{w}_{max}+\frac{{\mathcal{L}}^2}{a{w}_{min}})y{(t)}^T(W\otimes I_n)y(t)\hfill \\ \hfill & +(b_2{\sigma }_{max}{k}_2^2{c}^2+\frac{k_2^2{c}^2}{b_2})\widehat{r}{(t)}^T\widehat{r}(t)\hfill \\ \hfill & -2\widehat{r}{(t)}^{T}G(c\widehat{r})+\frac{b_2{\beta }_{1}N}{{\beta }_2+{\beta }_3{t}^2}\hfill \\ \hfill \le & (a{w}_{max}+\frac{{\mathcal{L}}^2}{a{w}_{min}})y{(t)}^T(W\otimes I_n)y(t)\hfill \\ \hfill & -[2k_{1}c-(b_2{\sigma }_{max}{k}_2^2{c}^2+\frac{k_2^2{c}^2}{b_2})]\widehat{r}{(t)}^T\widehat{r}(t)\hfill \\ \hfill & +\frac{b_2{\beta }_{1}N}{{\beta }_2+{\beta }_3{t}^2}\hfill \\ \hfill \le & \mathcal{K}y{(t)}^T(W\otimes I_n)y(t)-S_{1}r{(t)}^{T}r(t)\hfill \\ \hfill & +\frac{{\beta }_{1}N}{{\beta }_2+{\beta }_3{t}^2}{S}_2,\hfill \end{array}$$

(14)

where

$$\left\{\begin{array}{c}\mathcal{K}=a{w}_{max}+\frac{{\mathcal{L}}^2}{a{w}_{min}},\hfill \\ S_1=\frac{2b_1{b}_2{k}_{1}c-b_1{b}_2^2{\sigma }_{max}{k}_2^2{c}^2-b_1{k}_2^2{c}^2}{b_2(1+b_1)(b_1+{\sigma }_{max}{k}_2^2{c}^2)},\hfill \\ S_2=\frac{1+b_1}{b_1}{S}_1\hfill \end{array}\right.$$

(15)

with the nonlinear vector function $\widehat{r}{\left(t\right)}^{T}G\left(c\widehat{r}\right)\ge k_{1}c\widehat{r}{\left(t\right)}^T\widehat{r}\left(t\right)$. Let

$$\begin{array}{c}\hfill S_3={\overline{\lambda }}_2{w}_{min}{S}_1-\mathcal{K}.\end{array}$$

(16)

The inequality (14) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V^{\prime }\left(t\right)\le & (\mathcal{K}-{\overline{\lambda }}_2{w}_{min}{S}_1)y{\left(t\right)}^T(W\otimes I_n)y\left(t\right)\hfill \\ \hfill & +\frac{{\beta }_{1}N}{{\beta }_2+{\beta }_3{t}^2}{S}_2\hfill \\ \hfill \le & -S_{3}V\left(t\right)+\frac{{\beta }_{1}N}{{\beta }_2+{\beta }_3{t}^2}{S}_2,\hfill \end{array}\end{array}$$

(17)

Therefore, according to the comparison principle in [56], it follows that $V\left(t\right)\le \overline{V}\left(t\right)$, ${\overline{V}}^{\prime }\left(t\right)=-S_3\overline{V}\left(t\right)+\frac{{\beta }_{1}N}{{\beta }_2+{\beta }_3{t}^2}{S}_2$. Here, $\frac{{\beta }_{1}N}{{\beta }_2+{\beta }_3{t}^2}{S}_2\in {\mathcal{L}}^{\infty }$ and ${lim}_{t\to \infty }\frac{{\beta }_{1}N}{{\beta }_2+{\beta }_3{t}^2}{S}_2\to 0$, then ${lim}_{t\to \infty }{\overline{V}}^{\prime }\left(t\right)=0$ [47]. Therefore, the synchronization is achieved.

Next, we show that the considered system does not exhibit Zeno behavior. If there is a lower bound on the inter-event intervals of any agent under the event-triggered condition (6), Zeno-free triggering can be achieved. For each pair of adjacent agents $v_i$ and $v_j$, define an auxiliary vector function ${\mathcal{U}}_p\left(t\right)={\sum }_{p\in {\mathcal{E}}_i}{h}_{ip}{w}_p[g\left(c{\widehat{r}}_i\right)-g\left(c{\widehat{r}}_j\right)]$ and ${\mathsf{\Pi }}_p\left(t\right)=\mathcal{L}\parallel {\widehat{r}}_i\left(t\right)\parallel +\parallel {\mathcal{U}}_p\left(t\right)\parallel$. Here, ${\mathcal{U}}_p\left(t\right)$ and ${\mathsf{\Pi }}_p\left(t\right)$ lie between any two consecutive event instants of agent $v_i$ or $v_j$. Let ${\mathcal{U}}_k^p={max}_{t\in [t_k^i,t_{k+1}^i)}\parallel {\mathcal{U}}_p\left(t\right)\parallel$ and ${\mathsf{\Pi }}_k^p={max}_{t\in [t_k^i,t_{k+1}^i)}\parallel {\mathsf{\Pi }}_p\left(t\right)\parallel$. The measurement error vector $e_i\left(t\right)$ satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel {\dot{e}}_i\left(t\right)\parallel =& \parallel {\dot{r}}_i\left(t\right)\parallel \hfill \\ \hfill \le & \parallel \sum _{p\in {\mathcal{E}}_i}{h}_{ip}{w}_p[f\left(x_i\right)-f\left(x_j\right)]\parallel +{\mathcal{U}}_k^p\hfill \\ \hfill \le & \mathcal{L}\parallel \sum _{p\in {\mathcal{E}}_i}{h}_{ip}{w}_p{y}_p\left(t\right)]\parallel +{\mathcal{U}}_k^p\hfill \\ \hfill \le & \mathcal{L}\parallel e_i\left(t\right)\parallel +{\mathsf{\Pi }}_k^p.\hfill \end{array}\end{array}$$

(18)

As $\parallel e_i\left(t_k^i\right)\parallel =0$ holds, $\frac{d}{dt}{\parallel e_i\left(t\right)\parallel }^2$ satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d}{dt}{\parallel e_i\left(t\right)\parallel }^2\le & 2\parallel {\dot{e}}_i\left(t\right)\parallel \parallel e_i\left(t\right)\parallel \hfill \\ \hfill \le & (2\mathcal{L}+\beta )\parallel e_i{\left(t\right)\parallel }^2+\frac{{\left({\mathsf{\Pi }}_k^p\right)}^2}{\beta },\hfill \end{array}\end{array}$$

(19)

with the scalar $\beta >0$, which implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel e_i\left(t\right){\parallel }^2\le & {\int }_{t_k^i}^t\frac{{\mathsf{\Pi }}_k^p{e}^{(2\mathcal{L}+\beta )(t-s)}}{\beta }ds\hfill \\ \hfill =& \frac{{\mathsf{\Pi }}_k^p\left(e^{(2\mathcal{L}+\beta )(t-t_k^i)}-1\right)}{\beta (2\mathcal{L}+\beta )}.\hfill \end{array}\end{array}$$

(20)

Applying the condition (6), the next event does not occur until ${\parallel e\left(t\right)\parallel }^2={\sigma }_i{\parallel g\left(c{\widehat{r}}_i\left(t_k^i\right)\right)\parallel }^2+\frac{{\beta }_1}{{\beta }_2+{\beta }_3{t}^2}$, which yields that the inter-event time interval is lower bounded by ${\tau }_k^i=t-t_k^i$ with

$$\begin{array}{c}\hfill \frac{{\mathsf{\Pi }}_k^p\left(e^{(2\mathcal{L}+\beta )t_k^i}-1\right)}{\beta (2\mathcal{L}+\beta )}={\sigma }_i{\parallel g\left(c{\widehat{r}}_i\right)\parallel }^2+\frac{{\beta }_1}{{\beta }_2+{\beta }_3{t}^2}.\end{array}$$

(21)

Therefore, it follows that

$$\begin{array}{c}\hfill {\tau }_k^i>\frac{1}{2\mathcal{L}+\beta }ln\left(1+\frac{{\sigma }_i\beta (2\mathcal{L}+\beta ){\parallel g\left(c{\widehat{r}}_i\right)\parallel }^2}{{\mathsf{\Pi }}_k^p}\right).\end{array}$$

(22)

Equation (22) holds because the function $\frac{{\beta }_1}{{\beta }_2+{\beta }_3{t}^2}>0$ until $t\to \infty$. Therefore, no Zeno behavior occurs. The proof is completed. □

4. Edge-Self-Triggered Policy

In the event-triggered formulation, it becomes apparent that continuous monitoring of the measurement error norm $\parallel e\left(t\right)\parallel$ is required to check condition (6). In the following self-trigged control algorithm, this requirement is relaxed. Specifically, the next time $t_{k+1}^i$ at which control law is updated is predetermined at the previous event time $t_k^i$ and no state or error measurement is required between the control update instants. Define an increasing positive sequence $k_0^i,k_1^i,\cdots ,k_{\overline{q}}^i$, taking into account the effects of neighboring agent updates, with $t_k^i=t_{k_0^i}$ and $t_{k_{\overline{q}}^i}\le t_{k+1}^i$. By the inequality (20), the measurement error $e_i\left(t\right)$ satisfies

$$\begin{array}{c}\hfill \parallel e_i\left(t\right){\parallel }^2\le \frac{{\mathsf{\Pi }}_k^p\left(e^{(2\mathcal{L}+\beta )(t-t_k^i)}-1\right)}{\beta (2\mathcal{L}+\beta )},\end{array}$$

(23)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{U}}_p\left(t\right)=& \sum _{p\in {\mathcal{E}}_i}{h}_{ip}{w}_p\left(g\left(c{\widehat{r}}_i\right)-g\left(c{\widehat{r}}_j\right)\right),\hfill \\ \hfill {\mathcal{U}}_k^p=& \underset{t\in [t_k^i,t_{k+1}^i)}{max}\parallel {\mathcal{U}}_p\left(t\right)\parallel ,\hfill \end{array}\end{array}$$

(24)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathsf{\Pi }}_p\left(t\right)=& \mathcal{L}\parallel {\widehat{r}}_i\left(t\right)\parallel +\parallel {\mathcal{U}}_p\left(t\right)\parallel ,\hfill \\ \hfill {\mathsf{\Pi }}_k^p=& \underset{t\in [t_k^i,t_{k+1}^i)}{max}\parallel {\mathsf{\Pi }}_p\left(t\right)\parallel .\hfill \end{array}\end{array}$$

(25)

By (6) and (23), the self-triggered function is as follows,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{f}\left({\tau }_k^i\right)=& \frac{{\mathsf{\Pi }}_k^p\left(e^{(2\mathcal{L}+\beta ){\tau }_k^i}-1\right)}{\beta (2\mathcal{L}+\beta )}-{\sigma }_i{\parallel g\left(c{\widehat{r}}_i\right)\parallel }^2\hfill \\ \hfill & -\frac{{\beta }_1}{{\beta }_2+{\beta }_3{(t_k^i+{\tau }_k^i)}^2},\hfill \end{array}\end{array}$$

(26)

For each $i=1,2,\cdots ,N$, the self-triggered rule defines the next update time as follows. If there is a ${\tau }_k^i>0$ such that $\tilde{f}\left({\tau }_k^i\right)=0$, the next update time $t_{k+1}^i$ take place at most ${\tau }_k^i$ time units after $t_k^i$. Moreover, agent $v_i$ also checks the condition (26) when its neighbors updated. Otherwise, if the inequality $\tilde{f}\left({\tau }_k^i\right)<0$ holds for all ${\tau }_k^i>0$, then agent $v_i$ waits until the next update of the control law of one of its neighbors to recheck its condition.

According to the above analysis, we give Algorithm 1 and Theorem 2 as follows.

Algorithm 1: Edge-Self-Triggered Control Algorithm

Step 1: Set the algorithm execution time $\mathcal{T}$. At initial time $t=t_0$, let $k=0$, $t_k^i=t_0$, $q=0$ and $k_q^i=0$. Let ${\widehat{r}}_i\left(t\right)=r_i\left(t_0\right)$; update $u_i\left(t_0\right)=-g\left(c{\widehat{r}}_i\right)$; compute ${\mathcal{U}}_k^p$ and ${\mathsf{\Pi }}_k^p$ by (24) and (25); compute maximum allowable value ${\tau }_k^i$ by (26) such that $\tilde{f}\left({\tau }_k^i\right)=0$. Step 2: When $0<t\le \mathcal{T}$, check the events related to agents $v_i$, $v_j$; perform the following steps: – If any agent updated before $t_k^i+{\tau }_k^i$, let $q=q+1$; recompute ${\mathcal{U}}_k^p$ and ${\mathsf{\Pi }}_k^p$ by (24) and (25), such that $\tilde{f}\left({\tau }_k^i\right)=0$; – If there is no agent updated before $t_k^i+{\tau }_k^i$, the event related to agent $v_j$, occurs at $t_k^i+{\tau }_k^i$; let $k=k+1$; update the event instant $t_k^i$; let ${\widehat{r}}_i\left(t\right)=r_i\left(t_k^i\right)$; update $u_i\left(t_k^i\right)=-g\left(c{\widehat{r}}_i\right)$; let $q=0$; compute ${\mathcal{U}}_k^p$ and ${\mathsf{\Pi }}_k^p$ by (24) and (25); compute maximum allowable value ${\tau }_k^i$ by (26) such that $\tilde{f}\left({\tau }_k^i\right)=0$. Step 3: When $t>\mathcal{T}$, jump out of Algorithm 1.

Theorem 2.

Under the assumptions and conditions in Theorem 1, the system (1) with the control law (3) and Algorithm 1 can achieve synchronization for any initial condition. Furthermore, Zeno-free triggering can be achieved for all agents.

Remark 2.

The proposed edge-event- and edge-self-triggered policies are suitable for different scenarios with different system resources. The proposed event-triggered policy employs sensors to continuously measure edge state information to generate measurement errors and does not require communication resources, for instance, each individual employs LiDAR, ultrasonic rangefinder, and sonar to measure the relative position and relative speed of all neighbors. However, the proposed self-triggered policy can avoid continuous monitoring of these sensors, but it requires communication networks for information exchange. Therefore, these two policies include a trade-off between the scheduling of sensor resources and communication resources, which can be suitable for the scenarios with different levels of resource utilization.

Remark 3.

Compared with the previous results on self-triggered control [34,39,41,42,46,47,52], the interactive control input information of our algorithm contains the considered nonlinear factors, which include the influence of the logarithmic quantizer on the interactive information. This setting can save the communication bandwidth. This is also the theoretical expansion of the distributed self-triggered mechanism in these scenarios.

5. Simulations

This section provides two numerical simulations to verify the effectiveness of the proposed policies. For a connected nonlinear multi-agent system with 4 agents and 4 edges, its connected topology is shown in Figure 1.

Figure 1. Connected topology.

Let the nonlinear dynamics of each agent be expressed as follows,

$$\begin{array}{c}\hfill {\dot{x}}_i\left(t\right)=\left(x+2cos\left(x\right)\right)sin\left(t\right)+u_i\left(t\right)\end{array}$$

where $x_i\left(t\right)$=$[x_{i,1}$$x_{i,2}{]}^T$. For the controller, taking into account its inherent nonlinearity and the nonlinear effects caused by data communication quantization, the inherent nonlinear function and quantization function are provided respectively. Let the controllers’ inherent nonlinearity function $\mathcal{F}(·)$ be

$$\left\{\begin{array}{cc}\hfill {\mathcal{F}}_1\left(c{\widehat{r}}_{i,1}\right)=& c{\widehat{r}}_{i,1}(1-0.3cos(0.5c{\widehat{r}}_{i,1}+0.5sin\left(c{\widehat{r}}_{i,1}\right)\hfill \\ & +0.2c{\widehat{r}}_{i,1}cos\left(0.5c{\widehat{r}}_{i,1}\right)+0.5sin\left(3c{\widehat{r}}_{i,1}\right)\left)\right),\hfill \\ \hfill {\mathcal{F}}_2\left(c{\widehat{r}}_{i,2}\right)=& c{\widehat{r}}_{i,2}\left(1-0.3cos\left(0.5c{\widehat{r}}_{i,2}+sin\left(c{\widehat{r}}_{i,2}\right)\right)\right).\hfill \end{array}\right.$$

Let the logarithmic quantizer function $g_q\left({\mathcal{F}}_q(·)\right)$, $q=1,2,$ satisfy

$$g\left({\mathcal{F}}_q(·)\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}{\mathcal{F}}_q(·)=0,\hfill \\ \mathrm{sign}\left({\mathcal{F}}_q(·)\right)e^{∆⌊\frac{ln|{\mathcal{F}}_q(·)|}{∆}+\frac{1}{2}⌋},\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

with the quantization parameter $∆=0.3$. Obviously, the nonlinear functions $g_q\left({\mathcal{F}}_q(·)\right)$, $q=1,2,$ satisfy the properties P1–P3 with $k_1=0.7$ and $k_2=1.3$. The aforementioned vector functions $\mathcal{F}(·)$ and $g(·)$ correspond to Figure 2. Let the weight matrix be $W=\mathrm{diag}(0.85,0.97,1.12,0.78)$ with $w_{min}=0.78$ and $w_{max}=1.12$. Let the triggering thresholds be ${\sigma }_i=0.1+0.01i$, $i=1,2,3,4$. Let ${\beta }_1=0.02$ and ${\beta }_2={\beta }_3=1$. The control gain is given by $c=6.78$. It can be verified that all parameter settings meet the conditions of Theorem 1. The initial values of all agents are given by $[x_1{\left(0\right)}^T$$x_2{\left(0\right)}^T$$x_3{\left(0\right)}^T$$x_4{\left(0\right)}^T{]}^T$=$[-2$ 6 4 $-4$$-6$ 8 8 ${-2]}^T$. The results of edge-event- and edge-self-triggered policies are illustrated in Figure 3 and Figure 4 and Table 1 and Table 2, respectively. Here, we show the evolutionary state trajectories, the update times of all controllers, and the update values of all controllers. In subgraphs (a) of Figure 3 and Figure 4, Xset-1 and Xset-2 represent the values of $[x_{1,1}$$x_{2,1}$$x_{3,1}$$x_{4,1}{]}^T$, and $[x_{1,2}$$x_{2,2}$$x_{3,2}$$x_{4,2}{]}^T$, respectively. It can be observed that the proposed control policies can prompt all agents to achieve state synchronization when the controllers are updated intermittently and the considered system includes controller output nonlinearities. Furthermore, Table 2 also shows the effectiveness of the trade-off between different system resources in the scenarios with nonlinear controller outputs, that is, the event-triggered policy has fewer sampling times, but requires continuous employment of sensors; the self-triggered policy has more sampling times and requires communication networks, but can save sensor resources.

Figure 2. The functions of controller output nonlinearities. (a) The controllers’ inherent nonlinearity function $\mathcal{F}(·)$. (b) The logarithmic quantizer function $g(·)$.

Figure 3. Simulation results of edge-event-triggered control. (a) State trajectories of all individuals. (b) Update times of all controllers. (c) Update values of all controllers.

Figure 4. Simulation results of edge-self-triggered control. (a) State trajectories of all individuals. (b) Update times of all controllers. (c) Update values of all controllers.

Table 1. Comparisons of the triggering numbers of each agent in the time interval $t\in [0,10]$.

Agents	${\mathit{v}}_1$	${\mathit{v}}_2$	${\mathit{v}}_3$	${\mathit{v}}_4$	Total Numbers	Average Time Intervals
The edge-event-triggered policy	35	35	23	20	113	0.354
The edge-self-triggered policy	71	97	145	48	361	0.111

Table 2. Comparisons of the trade-off between different system resources in $t\in [0,10]$.

System Resources	Sampling Times	Communication Times	Controller Update Times
The edge-event-triggered policy	Continuous	0	113
The edge-self-triggered policy	361	361	361

6. Conclusions

This paper investigates the synchronization problem of nonlinear multi-agent systems with controller output nonlinearities based on edge-event- and edge-self-triggered policies. The combined edge state is utilized by all controllers, and it is affected by the controllers’ inherent nonlinear characteristics and the quantization transmission of communication networks. The event-triggered strategy ensures the intermittent update of all controllers instead of continuous execution, while the self-triggered strategy further prevents sensors from continuously capturing information, thereby saving sensor resources. Not only are the proposed policies effective, but also the trade-off of different policies for different system resources is effective. Our future work on multi-agent systems will involve the performance design, network security problem, and switching topologies.