Distributed Optimization for Second-Order Multi-Agent Systems over Directed Networks

Abstract

This paper studies a generalized distributed optimization problem for second-order multi-agent systems (MASs) over directed networks. Firstly, an improved distributed continuous-time algorithm is proposed. By using the linear transformation method and Lyapunov stability theory, some conditions are obtained to guarantee all agents’ states asymptotically reach the optimal solution. Secondly, to reduce unnecessary communication transmission and control cost, an event-triggered algorithm is designed. Moreover, the convergence of the algorithm is proved, and the Zeno behavior can be avoided based on strict theoretic analysis. Finally, one example is given to verify the good performance of the proposed algorithms.

1. Introduction

In recent years, the MASs [1] with swarm intelligence have been applied in many fields, such as attitude alignment and consensus [2], automated highway systems [3], flocking [4], formation control [5,6], and so on. In practical applications, MASs should optimize the global objective and reach consensus under appropriate control protocols. Different from traditional optimization methods, distributed optimization can effectively avoid the dependence of centralized control to improve privacy protection [7], reliability [8] and scalability [9].

The major assignment of distributed optimization is to design appropriate distributed control protocols, which not only enable agents to achieve efficient swarm behaviors through local collaborations but also optimize the global objective. Therefore, the distributed optimization of MASs is widely used in the maximization of social welfare [10], optimization of allocate resources [11], energy management of microgrids [12,13,14,15], and sensor networks [16]. According to different information updating and transmission rules, existing distributed optimization models can be generally divided into two categories: discrete-time models and continuous-time models. Early researches mainly focused on discrete-time models [17,18,19,20,21,22,23]. Subsequently, the distributed optimization problem for MASs with continuous-time models was considered because it can effectively avoid the analysis of time complexity in discrete-time algorithms with the assistance of modern control theory. At present, some meaningful results have been obtained. For instance, an objective optimization problem for MASs over strongly connected weighted balanced networks was studied in [24]. Considering the constraints of control input, an optimization algorithm with input saturation was designed in [25]. To shorten the convergence time, the finite-time and fixed-time optimization algorithms were proposed in [26,27], respectively.

Most distributed optimization problems were considered based on first-order or second-order MASs. In [28], a first-order distributed optimization problem with an information restriction policy was studied. In [29,30], the first-order distributed optimization problems over directed network structure were discussed. In [31,32], some gradient-based algorithms based on second-order MASs with undirected networks were proposed. In addition, several distributed optimization problems with exogenous disturbances were considered in [33]. However, most distributed optimization algorithms for second-order MASs were proposed based on undirected network structures. How to design distributed optimization algorithms on directed networks is challenging. This is one of the motives of this paper.

In practical application, agents are usually limited by communication resources. Continuous communication inevitably leads to high energy consumption. To reduce unnecessary communication, the event-triggered control technique was employed in the design of the distributed optimal control protocol. For example, some event-triggered subgradient and consensus algorithms were proposed to address optimization problems for first-order MASs over undirected networks in [34,35], respectively. In [36], an event-triggered zero-gradient-sum algorithm over directed networks was proposed. Additionally, a distributed optimization problem for second-order MASs with undirected networks was considered based on the event-triggered communication algorithm in [37]. To our knowledge, the distributed optimization problems for second-order MASs over directed networks have not been considered by using an event-triggered communication algorithm.

Based on the above discussion, this paper considers the distributed optimization problem for second-order MASs over directed networks. Both continuous-time and event-triggered communication protocols are proposed. The main contributions of this paper can be broadly summarized as follows:

(1)

Motivated by [38,39], this paper considers a more general distributed optimization problem, which can be regarded as a generalization of the existing one in [33,37]. The global optimization objective is the weighted sum of local objective functions, which can adjust the global optimization objective more flexibly. Thus, it has a wider application prospect.

(2)

A directed network is constructed based on the considered optimization problem, and a second-order continuous protocol is proposed according to the constructed directed network. Some sufficient conditions are obtained to ensure the solvability of the considered optimization problem.

(3)

In order to reduce the limited resource cost, an event-triggered communication protocol is designed. Combined with the linear transformation method and Lyapunov stability theory, we can prove that the optimization problem can be solved under the proposed protocols.

The remaining parts of this paper are organized as follows. Some preliminaries are presented in Section 2. In Section 3, the continuous-time communication protocol is proposed. In Section 4, an improved event-triggered protocol is given. A simulation example is presented in Section 5. Finally, a conclusion is given in Section 6.

Notations. Let ${\mathbb{R}}^N$ and ${\mathbb{R}}^{N\times n}$ denote the N-dimensional Euclidean space and $N\times n$ dimensional matrices. $I_n$ is the identity matrix with $n\times n$ dimension. $1_N$ and $0_N$ denote the column vectors with N ones and N zeros, respectively. ⊗ and $\parallel ·\parallel$ denote the Kronecker product and the Euclidean norm, respectively. $\nabla f(·)$ represents the gradient of function $f(·)$. ${\mathcal{Q}}^T$$\left(x^T\right)$ is the transposition of matrix $\mathcal{Q}$ (vector x), and ${\mathcal{Q}}^{-1}$ is the inverse of matrix $\mathcal{Q}$. ${\mathbb{N}}_{+}=\{1,2,\cdots \}$ denotes the positive integer set.

2. Preliminaries

2.1. Graph Theory

Consider the MASs with N agents, and the interactions among agents are described by a directed graph $\mathcal{G}=\{\mathcal{V},\mathcal{E},\mathcal{A}\}$, in which $\mathcal{V}=\{v_1,v_2,\cdots ,v_N\}$ is a set of finite vertices and $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ is an edge set. For a directed edge $(v_i,v_j)\in \mathcal{E}$, $v_i$ and $v_j$ represent the tail vertex and head vertex, respectively. The neighbor set of vertex $v_i$ is denoted as ${\mathcal{N}}_i=\{v_j\in \mathcal{V}:(v_j,v_i)\in \mathcal{E}\}$. $\mathcal{A}=\left[a_{ij}\right]\in R^{N\times N}$ is the weighted adjacency matrix of graph $\mathcal{G}$, where $a_{ij}>0$ if $(v_j,v_i)\in \mathcal{E}$, and $a_{ii}=0$. The graph $\mathcal{G}$ is called strongly connected if there is a directed path between any two different nodes. The in-degree matrix of the graph $\mathcal{G}$ is defined as $\mathcal{D}=\mathrm{diag}\{d_1,d_2,\cdots ,d_N\}$, where $d_i={\sum }_{j\in {\mathcal{N}}_i}{a}_{ij}$. The corresponding Laplacian matrix $L=\left[l_{ij}\right]$∈${\mathbb{R}}^{N\times N}$ is defined by $L=\mathcal{D}-\mathcal{A}$. A digraph $\mathcal{G}$ is called detail-balanced [40] if there exists a positive vector $\vartheta =({\vartheta }_1,{\vartheta }_2$, ⋯,${\vartheta }_N{)}^T$, such that ${\vartheta }_i{a}_{ij}={\vartheta }_j{a}_{ji}$ for $i,j=1,2,\cdots ,N$.

2.2. Basic Definition

Definition 1.

The function $f(·):{\mathbb{R}}^n\to \mathbb{R}$ is called $s—$strongly convex if for any $x_1,x_2\in {\mathbb{R}}^n,$$x_1\ne x_2$ there exists $s>0$, such that ${(x_1-x_2)}^T(\nabla f\left(x_1\right)-\nabla f\left(x_2\right))\ge s{\parallel x_1-x_2\parallel }_2^2$.

Definition 2.

The function $f(·):{\mathbb{R}}^n\to \mathbb{R}$ is called the gradient $\nabla f_i(·)$ m-Lipschitz if for any $x_1,x_2\in {\mathbb{R}}^n,x_1\ne x_2$ there exists $m>0$, such that $\parallel \nabla f\left(x_1\right)-\nabla f\left(x_2\right)\parallel \le$m$\parallel x_1-x_2\parallel$.

2.3. Problem Statement

In this paper, we mainly study the following optimization problem in a distributed method

$$\underset{\mathbf{x}\in {\mathbb{R}}^n}{min}\tilde{f}\left(\mathbf{x}\right)=\sum _{i=1}^N{\xi }_i{f}_i\left(\mathbf{x}\right),$$

(1)

where $\mathbf{x}\in {\mathbb{R}}^n$ is the decision vector and $f_i(·):{\mathbb{R}}^n\to \mathbb{R}$ is the local cost function of agent $v_i$. ${\xi }_i\in (0,1)$ is a weight parameter satisfying ${\displaystyle \sum _{i=1}^N}{\xi }_i=1$, and $\tilde{f}\left(\mathbf{x}\right)$ is the global optimization function. Our aim is to solve this optimization problem (1) by proposing some distributed algorithms.

Remark 1.

In existing works [26,27,33,36,37,41,42], the global optimization function is given by$\underset{\mathbf{x}\in {\mathbb{R}}^n}{min}\tilde{f}\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^N}{f}_i\left(\mathbf{x}\right)$. However, the optimization problem (1) is considered in this paper. When we choose${\xi }_i=\frac{1}{N}$, $i=1,2,\cdots ,N$, the optimization problem (1) becomes$\underset{\mathbf{x}\in {\mathbb{R}}^n}{min}\tilde{f}\left(\mathbf{x}\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^N}{f}_i\left(\mathbf{x}\right)$, which has the same optimal solution as$\underset{\mathbf{x}\in {\mathbb{R}}^n}{min}\tilde{f}\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^N}{f}_i\left(\mathbf{x}\right)$. Therefore, the optimization problem (1) can be regarded as a generalization of the existing one. Furthermore, the optimization problem (1) is equivalent to the optimization problem$\underset{\mathbf{x}\in {\mathbb{R}}^n}{min}$$\tilde{F}\left(\mathbf{x}\right)={\sum }_{i=1}^N{\theta }_i{f}_i\left(\mathbf{x}\right)$with${\theta }_i>0$. If we let$\Theta ={\sum }_{i=1}^N{\theta }_i$and${\xi }_i={\theta }_i/\Theta$, it has$\tilde{F}\left(\mathbf{x}\right)=\Theta {\displaystyle \sum _{i=1}^N}{\xi }_i{f}_i\left(\mathbf{x}\right)$with${\sum }_{i=1}^N{\xi }_i=1$. As Θ is a constant, it has the same optimal solution as problem (1). Hence, the optimization problem (1) is more general and has more extensive applications.

For the optimization problem (1), we use $x_i\in {\mathbb{R}}^n$ to represent the estimation of the decision vector $\mathbf{x}$ of the ith agent. Denote $x^T={[x_1^T,x_2^T,\cdots ,x_N^T]}^T\in {\mathbb{R}}^{Nn}$. Assume that the network topology $\mathcal{G}$ among agents is directed and strongly connected, and the corresponding Laplacian matrix is denoted by $\mathcal{L}$. Then, the problem (1) can be written as the following problem:

$$\begin{array}{c}\hfill \underset{x\in {\mathbb{R}}^{Nn}}{min}F\left(x\right)={\displaystyle \sum _{i=1}^N}{\xi }_i{f}_i\left(x_i\right),\mathrm{s}.\mathrm{t}.(\mathcal{L}\otimes I_n)x=0.\end{array}$$

(2)

The proof is similar to the one in [43].

To solve the above problem, the network topology $\mathcal{G}$ needs to designed carefully. Based on the weighted parameters ${\xi }_i$ for $i=1,\cdots ,N$, we need to construct a directed graph to satisfy the distributed optimization problem (2). In the literature [41], a directed detail-balanced network construction method was proposed, which is also feasible in this paper. Hence, the specific construction algorithm is omitted here. In the constructed network, the adjacency matrix $\mathcal{A}$ needs to satisfy the following two conditions:

(1).

The directed graph $\mathcal{G}$ is detail-balanced with the weight $\xi ={[{\xi }_1,\cdots ,{\xi }_N]}^T$. That means ${\xi }_i{a}_{ij}={\xi }_j{a}_{ji}$, for $\forall i,j=1,2,\cdots ,N$.

(2).

The directed graph $\mathcal{G}$ is strongly connected.

Let $\Xi =\mathrm{diag}\{{\xi }_1,{\xi }_2,\cdots ,{\xi }_N\}$ and $\mathbf{G}$ be a new undirected graph where the adjacency matrix is $\mathbf{A}={\left[{\xi }_i{a}_{ij}\right]}_{N\times N}$, so that the corresponding Laplacian matrix is $\mathcal{W}=\Xi \mathcal{L}=\left[w_{ij}\right]\in {\mathbb{R}}^{N\times N}$. Then, the matrix $\mathcal{W}$ is a symmetric matrix, and its eigenvalues are represented by ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_N$, satisfying $0={\lambda }_1<{\lambda }_2\le \cdots \le {\lambda }_N$. Define $\mathcal{Q}=(q_1,{\mathcal{Q}}_2)\in {\mathbb{R}}^{N\times N}$, in which

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{q}_1=\frac{1_N}{\sqrt{N}},{\mathcal{Q}}_2\in {\mathbb{R}}^{N\times (N-1)},\hfill \\ q_1^T{\mathcal{Q}}_2=0_{N-1}^T,{\mathcal{Q}}_2^T{\mathcal{Q}}_2=I_{N-1},\hfill \end{array}\right.\end{array}$$

and $\mathcal{Q}$ satisfying ${\mathcal{Q}}^T\mathcal{Q}=I_N$. The matrix $\mathcal{Q}$ is an orthogonal matrix which satisfies ${\mathcal{Q}}^T\mathcal{W}\mathcal{Q}=\mathrm{diag}\{{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_N\}$.

In order to analyze the above problem, the following Assumption bout the objective function is widely used, and details can be found in works [24,33,37].

Assumption 1.

The objective function $f_i(·):{\mathbb{R}}^n\to \mathbb{R}$ of agent i is differentiable and $s_i$-strongly convex on ${\mathbb{R}}^n$.

Assumption 2.

The gradient $\nabla f_i(·)$ is $m_i$-Lipschitz.

Assumption 3.

The network topology $\mathcal{G}$ is directed strongly connected and ξ-detailed balanced.

3. The Continuous-Time Communication Protocol

In this section, we assume that the communication among agents is continuous. Based on directed graph $\mathcal{G}$, the following algorithm is presented

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_i\left(t\right)={\xi }_i{y}_i\left(t\right),\hfill \\ {\dot{y}}_i\left(t\right)=-k_1{y}_i\left(t\right)-k_2\nabla f_i\left(x_i\left(t\right)\right)-k_3{h}_i\left(t\right),\hfill \\ {\dot{h}}_i\left(t\right)=-k_4{\displaystyle \sum _{j\in {\mathcal{N}}_i}}{a}_{ij}(x_j\left(t\right)-x_i\left(t\right)),i=1,2,\cdots ,N,\hfill \end{array}\right.\end{array}$$

(3)

where $x_i\left(t\right),y_i\left(t\right)\in {\mathbb{R}}^n$ denote the position and the velocity states of agent i, respectively. $k_1,k_2,k_3$, and $k_4$ are positive parameters. $h_i\left(t\right)\in {\mathbb{R}}^n$ is an auxiliary variable, and the initial value satisfies ${\displaystyle \sum _{i=1}^N}{\xi }_i{h}_i\left(0\right)=0$.

Remark 2.

The distributed optimal algorithm (3) not only contains the consensus term $-k_4{\displaystyle \sum _{j\in {\mathcal{N}}_i}}$$a_{ij}(x_j\left(t\right)-x_i\left(t\right))$ and optimization term $-k_2\nabla f_i\left(x_i\left(t\right)\right)$ but also has auxiliary variable $k_3{h}_i\left(t\right)$. Motivated by [37,42], the auxiliary variable $h_i\left(t\right)$, as the integral of $-k_4$${\displaystyle \sum _{j\in {\mathcal{N}}_i}}$$a_{ij}$$(x_j\left(t\right)-x_i\left(t\right))$, is designed for ensuring that the equilibrium point of (3) is the optimal value through the transmission of local information.

Remark 3.

Compared with [37], each item of (3) has a weighted coefficient to efficiently adjust the influence strength on the distributed optimization algorithm. It is well known that the sensitivity of the second-order MASs is higher than that of the first-order MASs, so the stability analysis becomes more difficult. The algorithm (3) is suitable for second-order MASs because of their improved flexibility and higher fineness. In addition, different from the algorithm in [42], the dynamics of $y_i\left(t\right)$ do not use the position information of agents in this paper. Thus, the calculation consumption is much simpler than the existing one.

Because of the strict convexity of function $f_i(·)$, the objective function $F(·)$ is also a differentiable and strongly convex function. Based on Assumption 1, the optimization problem (1) has a unique optimal solution

$$\begin{array}{c}\hfill x^{*}\triangleq arg{\displaystyle \underset{x\in {\mathbb{R}}^n}{min}}f\left(x\right),-\infty <x^{*}<+\infty .\end{array}$$

According to the definition of Laplacian $\mathcal{L}$, the distributed optimization algorithm (3) can be rewritten as the following compact form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}\left(t\right)=(\Xi \otimes I_n)y\left(t\right),\hfill \\ \dot{y}\left(t\right)=-k_{1}y\left(t\right)-k_2\nabla \overline{f}\left(x\left(t\right)\right)-k_{3}h\left(t\right),\hfill \\ \dot{h}\left(t\right)=k_4(\mathcal{L}\otimes I_n)x\left(t\right),\hfill \end{array}\right.\end{array}$$

(4)

where $x\left(t\right)={[x_1^T\left(t\right),x_2^T\left(t\right),\cdots ,x_N^T\left(t\right)]}^T,y\left(t\right)=[y_1^T\left(t\right),y_2^T\left(t\right)$, ⋯, $y_N^T{\left(t\right)]}^T$, $\nabla \overline{f}\left(x\left(t\right)\right)=[{(\nabla f_1\left(x_1\left(t\right)\right))}^T,(\nabla f_2$$\left(x_2\left(t\right)\right){)}^T$, ⋯, ${(\nabla f_N\left(x_N\left(t\right)\right))}^T{]}^T$, $h\left(t\right)=[h_1^T\left(t\right),h_2^T\left(t\right),\cdots ,$$h_N^T{\left(t\right)]}^T$.

Assumption 4.

For algorithm (3), ${\displaystyle \sum _{i=1}^N}{\xi }_i{h}_i\left(0\right)=0$ is satisfied, and there exist parameters $k_1,k_2>0,0<k_4<\frac{k_1^3{ϵ}_0^2(1-{ϵ}_0)}{2k_3{\lambda }_N}$, such that ${\pi }_1=k_1{k}_2{ϵ}_0\check{\xi }\check{s}-k_3{k}_4{\lambda }_N>0$ and ${\pi }_2=k_1(1-{ϵ}_0)>0$, where $\check{\xi }=min\{{\xi }_1,{\xi }_2,\cdots ,{\xi }_N\}$, $\check{s}=min\{s_1,s_2,\cdots ,s_N\}$, and ${ϵ}_0>0$ is a constant.

Lemma 1.

For algorithm (3), if Assumptions 1 and 2 are satisfied, then there is a unique solution for any initial value on $[0,+\infty )$.

Proof. 

Based on the strict convexity of function $f_i(·)$, one can easily know that there exists one solution for algorithm (3) at least. Therefore, we only need to prove the uniqueness of the solution. Let $P\left(t\right)={[x^T\left(t\right),y^T\left(t\right),h^T\left(t\right)]}^T$, then one can obtain

$$\dot{P}\left(t\right)=\left[\begin{array}{c}(\Xi \otimes I_n)y\left(t\right)\\ -k_{1}y\left(t\right)-k_2\nabla \overline{f}\left(x\left(t\right)\right)-k_{3}h\left(t\right)\\ k_4(\mathcal{L}\otimes I_n)x\left(t\right)\end{array}\right].$$

It is noticed that $\nabla \overline{f}\left(x\left(t\right)\right)$ is Lipschitz continuous, then the solution of $P\left(t\right)$ exists. Assume there are two different solutions of Equation (4), which are $P_a\left(t\right)$ and $P_b\left(t\right)$ satisfying initial condition $P_a\left(0\right)=P_b\left(0\right)$. Based on the Assumption 1, we can obtain that

$$\begin{array}{cc}\hfill \parallel {\dot{P}}_a\left(t\right)-{\dot{P}}_b\left(t\right)\parallel \le & \parallel \Xi \otimes I_n\parallel \parallel y_a\left(t\right)-y_b\left(t\right)\parallel +k_1\parallel y_a\left(t\right)-y_b\left(t\right)\parallel \parallel +k_2\parallel \nabla {\overline{f}}_a\left(x\left(t\right)\right)\hfill \\ \hfill & -\nabla {\overline{f}}_b\left(x\left(t\right)\right)\parallel +k_3\parallel h_a\left(t\right)-h_b\left(t\right)\parallel \parallel +k_4\parallel \mathcal{L}\otimes I_n\parallel \parallel x_a\left(t\right)-x_b\left(t\right)\parallel \hfill \\ \hfill \le & \widehat{\xi }\parallel y_a\left(t\right)-y_b\left(t\right)\parallel +k_1\parallel y_a\left(t\right)-y_b\left(t\right)\parallel \parallel +k_2\parallel \nabla {\overline{f}}_a\left(x\left(t\right)\right)-\nabla {\overline{f}}_b\left(x\left(t\right)\right)\parallel \hfill \\ \hfill & +k_3\parallel h_a\left(t\right)h_b\left(t\right)\parallel +k_4\parallel \mathcal{L}\parallel \parallel x_a\left(t\right)-x_b\left(t\right)\parallel \hfill \\ \hfill \le & \iota \parallel P_a\left(t\right)-P_b\left(t\right)\parallel ,\hfill \end{array}$$

where $\widehat{\xi }=max\{{\xi }_1,{\xi }_2,\cdots ,{\xi }_N\}$ and $\iota =\widehat{\xi }+k_1+k_2+k_3$$+k_4\parallel \mathcal{L}\parallel$. Additionally, then, it has

$$\begin{array}{cc}\hfill \frac{1}{2}{\parallel P_a\left(t\right)-P_b\left(t\right)\parallel }^2& \le {\int }_0^t\frac{d}{ds}(\frac{1}{2}\parallel P_a\left(s\right)-P_b\left(s\right){\parallel }^2)ds\hfill \\ \hfill & \le {\int }_0^t\iota {\parallel P_a\left(s\right)-P_b\left(s\right)\parallel }^{2}ds.\hfill \end{array}$$

By using the Gronwall inequality, one has

$$\begin{array}{c}\hfill \frac{1}{2}{\parallel P_a\left(t\right)-P_b\left(t\right)\parallel }^2\le 0,\end{array}$$

which means that $P_a\left(t\right)=P_b\left(t\right)$ for $\forall t\in [0,+\infty )$. Therefore, Lemma 1 is proved. □

If Assumption 1 and equality ${\displaystyle \sum _{i=1}^N}{\xi }_i{h}_i\left(0\right)=0$ holds, then there exists an equilibrium point $(x^{*},y^{*},h^{*})$ of algorithm (4). Therefore, the following equation is satisfied for the $(x^{*},y^{*},h^{*})$:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(\Xi \otimes I_n)y^{*}=0,\hfill \\ -k_1{y}^{*}-k_2\nabla \overline{f}\left(x^{*}\right)-k_3{h}^{*}=0,\hfill \\ k_4(\mathcal{L}\otimes I_n)x^{*}=0.\hfill \end{array}\right.\end{array}$$

(5)

Since $\mathcal{G}$ is strongly connected and detail-balanced, then it has ${(\xi \otimes I_n)}^T(\mathcal{L}\otimes I_n)=0$. Left multiplying ${(\xi \otimes I_n)}^T$ for the third equation of (5), it has ${(\xi \otimes I_n)}^T\dot{h}\left(t\right)=k_4{(\xi \otimes I_n)}^T(\mathcal{L}\otimes I_n)x\left(t\right)=0$; equivalently, ${\displaystyle \sum _{i=1}^N}{\xi }_i{\dot{h}}_i\left(t\right)=0$. In combination with ${\displaystyle \sum _{i=1}^N}{\xi }_i{h}_i\left(0\right)=0$, it has

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^N}{\xi }_i{h}_i\left(t\right)={\displaystyle \sum _{i=1}^N}{\xi }_i{h}_i\left(0\right)=0,\forall t\ge 0.\end{array}$$

Because $\Xi$ is an inevitable diagonal matrix, then the first equation of (5) implies $y^{*}=0$. Substituting $y^{*}=0$ into the second equation of (5), and left multiplying ${(\xi \otimes I_n)}^T$, it has $k_2{(\xi \otimes I_n)}^T\nabla f\left(x^{*}\right)=0$, that is ${\displaystyle \sum _{i=1}^N}{\xi }_i\nabla f_i\left(x_i^{*}\right)=0$. Hence, $x^{*}$ is an optimal value of (2).

We define $\widehat{x}\left(t\right)=x\left(t\right)-x^{*},\widehat{y}\left(t\right)=y\left(t\right)-y^{*},\widehat{h}\left(t\right)=h\left(t\right)-h^{*}$ and $\varphi \left(t\right)=\nabla \overline{f}(\widehat{x}\left(t\right)+x^{*})-\nabla \overline{f}\left(x^{*}\right)$, then the system (4) is described as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\widehat{x}}\left(t\right)=(\Xi \otimes I_n)\widehat{y}\left(t\right),\hfill \\ \dot{\widehat{y}}\left(t\right)=-k_1\widehat{y}\left(t\right)-k_2\varphi \left(t\right)-k_3\widehat{h}\left(t\right),\hfill \\ \dot{\widehat{h}}\left(t\right)=k_4(\mathcal{L}\otimes I_n)\widehat{x}\left(t\right).\hfill \end{array}\right.\end{array}$$

(6)

Denote $\check{x}\left(t\right)=({\mathcal{Q}}^T\otimes I_n)\widehat{x}\left(t\right)$,$\check{y}\left(t\right)=({\mathcal{Q}}^T\Xi \otimes I_n)\widehat{y}\left(t\right)$ and $\check{h}\left(t\right)=({\mathcal{Q}}^T\Xi \otimes I_n)\widehat{h}\left(t\right)$, it has

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\check{x}}\left(t\right)=\check{y}\left(t\right),\hfill \\ \dot{\check{y}}\left(t\right)=-k_1\check{y}\left(t\right)-k_2({\mathcal{Q}}^T\Xi \otimes I_n)\varphi \left(t\right)-k_3\check{h}\left(t\right),\hfill \\ \dot{\check{h}}\left(t\right)=k_4({\mathcal{Q}}^T\mathcal{W}\mathcal{Q}\otimes I_n)\check{x}\left(t\right).\hfill \end{array}\right.\end{array}$$

(7)

Then, the variables of equality (7) is divided as follows $\check{x}\left(t\right)={({\check{x}}_1^T\left(t\right),{\check{x}}_{2:N}^T\left(t\right))}^T$, $\check{y}\left(t\right)=({\check{y}}_1^T\left(t\right),$${\check{y}}_{2:N}^T{\left(t\right))}^T$, and $\check{h}\left(t\right)=({\check{h}}_1^T\left(t\right)$, ${\check{h}}_{2:N}^T{\left(t\right))}^T$, in which ${\check{x}}_1\left(t\right),{\check{y}}_1\left(t\right),{\check{h}}_1\left(t\right)\in {\mathbb{R}}^n$ and ${\check{x}}_{2:N}\left(t\right)$, ${\check{y}}_{2:N}\left(t\right)$, ${\check{h}}_{2:N}\left(t\right)\in {\mathbb{R}}^{(N-1)\times n}$. Therefore, equality (7) can be decomposed as follows

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\check{x}}}_1\left(t\right)={\check{y}}_1\left(t\right),\hfill \\ {\dot{\check{y}}}_1\left(t\right)=-k_1{\check{y}}_1\left(t\right)-k_2(q_1^T\Xi \otimes I_n)\varphi \left(t\right),\hfill \\ {\dot{\check{h}}}_1\left(t\right)=0,\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\check{x}}}_{2:N}\left(t\right)={\check{y}}_{2:N}\left(t\right),\hfill \\ {\dot{\check{y}}}_{2:N}\left(t\right)=-k_1{\check{y}}_{2:N}\left(t\right)-k_2({\mathcal{Q}}_2^T\Xi \otimes I_n)\varphi \left(t\right)-k_3{\check{h}}_{2:N}\left(t\right),\hfill \\ {\dot{\check{h}}}_{2:N}\left(t\right)=k_4({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{x}}_{2:N}\left(t\right).\hfill \end{array}\right.\end{array}$$

Remark 4.

Different from the dynamics in [37], the parameter $k_3$ is added to adjust the changing rate of $h_i\left(t\right)$ in the proposed dynamics. Hence, the application scope of the second-order system is generalized. In the meantime, the matrix transformation from system (6) to system (7) solves the difficulties caused by the asymmetry of directed network topology.

Theorem 1.

Suppose that Assumptions 1–4 hold, then the optimization problem (1) can be solved via distributed algorithm (3).

Proof. 

Construct the following Lyapunov function

$$\begin{array}{c}\hfill V_1\left(t\right)=W_1\left(t\right)+W_2\left(t\right)+k_2{W}_3\left(t\right),\end{array}$$

(8)

where

$$\begin{array}{cc}\hfill W_1\left(t\right)=& \frac{k_1^2{ϵ}_0}{2}{\check{x}}_1^T\left(t\right){\check{x}}_1\left(t\right)+k_1{ϵ}_0{\check{x}}_1^T\left(t\right){\check{y}}_1\left(t\right)+\frac{1}{2}{\check{y}}_1^T\left(t\right){\check{y}}_1\left(t\right),\hfill \\ \hfill W_2\left(t\right)=& \frac{k_1^2{ϵ}_0}{2}{\check{x}}_{2:N}^T\left(t\right){\check{x}}_{2:N}\left(t\right)+k_1{ϵ}_0{\check{x}}_{2:N}^T\left(t\right){\check{y}}_{2:N}\left(t\right)+\frac{1}{2}{\check{y}}_{2:N}^T\left(t\right){\check{y}}_{2:N}\left(t\right)\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & +k_3{\check{x}}_{2:N}^T\left(t\right){\check{h}}_{2:N}\left(t\right)+\frac{k_1{k}_3{ϵ}_0}{2k_4}{\check{h}}_{2:N}^T\left(t\right)({\left({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\right)}^{-1}\otimes I_n){\check{h}}_{2:N}\left(t\right),\hfill \end{array}$$

(10)

and

$$\begin{array}{c}\hfill W_3\left(t\right)=F\left(x\left(t\right)\right)-{(\nabla F\left(x^{*}\right))}^{T}x\left(t\right)-F\left(x^{*}\right).\end{array}$$

(11)

Note that $W_1\left(t\right),W_2\left(t\right)>0$ if ${ϵ}_0<1$. $W_3\left(t\right)\ge 0$ and $W_3\left(t\right)=0$ if and only if $\check{x}=0$. Then, $V_1\left(t\right)\ge 0$ and $V_1\left(t\right)=0$ if and only if $\check{x}=0,\check{y}=0,\check{h}=0$. Take the time derivation of $W_1\left(t\right)$, $W_2\left(t\right)$ and $W_3\left(t\right)$, it has

$$\begin{array}{cc}\hfill {\dot{W}}_1\left(t\right)=& k_1^2{ϵ}_0{\check{x}}_1^T\left(t\right){\dot{\check{x}}}_1\left(t\right)+k_1{ϵ}_0{\dot{\check{x}}}_1^T\left(t\right){\check{y}}_1\left(t\right)+k_1{ϵ}_0{\check{x}}_1^T\left(t\right){\dot{\check{y}}}_1\left(t\right)+{\check{y}}_1^T\left(t\right){\dot{\check{y}}}_1\left(t\right)\hfill \\ \hfill =& -k_1(1-{ϵ}_0){\check{y}}_1^T\left(t\right){\check{y}}_1\left(t\right)-k_2{\check{y}}_1^T\left(t\right)(q_1^T\Xi \otimes I_n)\varphi \left(t\right)-k_1{k}_2{ϵ}_0{\check{x}}_1^T\left(t\right)(q_1^T\Xi \otimes I_n)\varphi \left(t\right),\hfill \\ \hfill {\dot{W}}_2\left(t\right)=& k_1^2{ϵ}_0{\check{x}}_{2:N}^T\left(t\right){\dot{\check{x}}}_{2:N}\left(t\right)+k_1{ϵ}_0{\dot{\check{x}}}_{2:N}^T\left(t\right){\check{y}}_{2:N}\left(t\right)+k_1{ϵ}_0{\check{x}}_{2:N}^T\left(t\right){\dot{\check{y}}}_{2:N}\left(t\right)+{\check{y}}_{2:N}^T\left(t\right){\dot{\check{y}}}_{2:N}\left(t\right)\hfill \\ \hfill & +k_3{\dot{\check{x}}}_{2:N}^T\left(t\right){\check{h}}_{2:N}\left(t\right)+k_3{\check{x}}_{2:N}^T\left(t\right){\dot{\check{h}}}_{2:N}\left(t\right)+\frac{k_1{k}_3{ϵ}_0}{k_4}{\check{h}}_{2:N}^T\left(t\right)({\left({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\right)}^{-1}\otimes I_n){\dot{\check{h}}}_{2:N}\left(t\right)\hfill \\ \hfill =& -k_1(1-{ϵ}_0){\check{y}}_{2:N}^T\left(t\right){\check{y}}_{2:N}\left(t\right)-k_2{\check{y}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\Xi \otimes I_n)\varphi \left(t\right)\hfill \\ \hfill & -k_1{k}_2{ϵ}_0{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\Xi \otimes I_n)\varphi \left(t\right)+k_3{k}_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{x}}_{2:N}\left(t\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\dot{W}}_3\left(t\right)=& (\nabla F{\left(x\left(t\right)\right)}^T\dot{x}\left(t\right)-{(\nabla F\left(x^{*}\right))}^T\dot{x}\left(t\right)\hfill \\ \hfill =& {\displaystyle \sum _{i=1}^N}{\xi }_i{(\nabla f_i({\widehat{x}}_i\left(t\right)+x^{*}))}^T{\xi }_i{y}_i\left(t\right)-{\displaystyle \sum _{i=1}^N}{\xi }_i{(\nabla f_i\left(x^{*}\right))}^T{\xi }_i{y}_i\left(t\right)\hfill \\ \hfill =& {\displaystyle \sum _{i=1}^N}{\xi }_i{(\nabla f_i({\widehat{x}}_i\left(t\right)+x^{*})-f_i\left(x^{*}\right))}^T{\xi }_i{\widehat{y}}_i\left(t\right)\hfill \\ \hfill =& {\widehat{y}}^T\left(t\right)({\Xi }^2\otimes I_n)\varphi \left(t\right)\hfill \\ \hfill =& {\check{y}}^T\left(t\right)(Q^T\Xi \otimes I_n)\varphi \left(t\right).\hfill \end{array}$$

Therefore, the time derivation of $V_1\left(t\right)$ is obtained as

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)=& {\dot{W}}_1\left(t\right)+{\dot{W}}_2\left(t\right)+k_2{\dot{W}}_3\left(t\right)\hfill \\ \hfill =& -k_1(1-{ϵ}_0){\check{y}}^T\left(t\right)\check{y}\left(t\right)-k_1{k}_2{ϵ}_0{\check{x}}^T\left(t\right)({\mathcal{Q}}^T\Xi \otimes I_n)\varphi \left(t\right)\hfill \\ \hfill & +k_3{k}_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{x}}_{2:N}\left(t\right).\hfill \end{array}$$

Since

$$\begin{array}{cc}\hfill & k_3{k}_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{x}}_{2:N}\left(t\right)\le k_3{k}_4{\lambda }_N{\parallel {\check{x}}_{2:N}\left(t\right)\parallel }^2,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & -k_1{k}_2{ϵ}_0{\check{x}}^T\left(t\right)({\mathcal{Q}}^T\Xi \otimes I_n)\varphi \left(t\right)\hfill \\ \hfill =& -k_1{k}_2{ϵ}_0{\displaystyle \sum _{i=1}^N}{\xi }_i{\widehat{x}}_i^T\left(t\right)(\nabla f_i({\widehat{x}}_i\left(t\right)+x^{*})-\nabla f_i\left(x^{*}\right))\hfill \\ \hfill \le & -k_1{k}_2{ϵ}_0{\displaystyle \sum _{i=1}^N}{s}_i{\xi }_i{\widehat{x}}_i^T\left(t\right){\widehat{x}}_i\left(t\right)\hfill \\ \hfill \le & -k_1{k}_2{ϵ}_0\check{\xi }\check{s}{\parallel \widehat{x}\left(t\right)\parallel }^2.\hfill \end{array}$$

As $\mathcal{Q}$ is an identity orthogonal matrix, and $\parallel \check{x}\left(t\right)\parallel =\parallel \widehat{x}\left(t\right)\parallel$, then one has $\parallel \check{x}{\left(t\right)\parallel }^2=\parallel {\check{x}}_1{\left(t\right)\parallel }^2+{\parallel {\check{x}}_{2:N}\left(t\right)\parallel }^2$.

Therefore, it has that

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)\le & -k_1(1-{ϵ}_0){\check{y}}^T\left(t\right)\check{y}\left(t\right)-k_1{k}_2{ϵ}_0\check{\xi }\check{s}\parallel \widehat{x}\left(t\right){\parallel }^2+k_3{k}_4{\lambda }_N{\parallel {\check{x}}_{2:N}\left(t\right)\parallel }^2\hfill \\ \hfill =& -{\pi }_1\parallel {\check{x}}_{2:N}{\left(t\right)\parallel }^2-{\pi }_2\parallel \check{y}{\left(t\right)\parallel }^2-{\pi }_3{\parallel {\check{x}}_1\left(t\right)\parallel }^2,\hfill \end{array}$$

where ${\pi }_1=k_1{k}_2{ϵ}_0\check{\xi }\check{s}-k_3{k}_4{\lambda }_N$, ${\pi }_2=k_1(1-{ϵ}_0)$, and ${\pi }_3=k_1{k}_2{ϵ}_0\check{\xi }\check{s}$. By comparing the above inequalities, we can easily know that ${\pi }_3>0$. Thus, we only need to prove ${\pi }_i>0$, $i=1,2$. From the condition of Theorem 1, ${\dot{V}}_1\left(t\right)\le 0$, and ${\dot{V}}_1\left(t\right)=0$ if and only if $\check{y}=0,\check{x}=0,\check{h}=0$. Therefore, consensus can be achieved. □

Remark 5.

The distributed optimization problem for MASs with directed networks is more difficult than those on undirected networks, which are mainly reflected in two aspects. The first aspect is the symmetry of the Laplacian matrix of the network topology. The symmetry of the Laplacian matrix of undirected topology can be fully utilized as in [33,37], but the Laplacian matrix is usually asymmetric for directed topology. Therefore, to solve the optimization problem on directed networks, other more complicated methods need to be adopted. The second aspect is that the optimization problem is usually only with the sum of local objective functions and without the convex hull. However, in order to solve more general distributed optimization problems, the directed network needs to be designed carefully in this paper.

Remark 6.

It is not difficult to see that the above algorithm relies on continuous-time communication, which contains distinct shortcomings. In practical application, a continuous-time algorithm inevitably needs a lot of communication costs. As we all know, the bandwidth, transmission energy, and communication frequency are limited by the current industrial level and cannot be used indefinitely. Therefore, in the next subsection, we propose a new algorithm in order to improve this drawback.

4. The Event-Triggered Communication Protocol

Based on directed communication topology $\mathcal{G}$, the following algorithm is presented

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_i\left(t\right)={\xi }_i{y}_i\left(t\right),\hfill \\ {\dot{y}}_i\left(t\right)=-k_1{y}_i\left(t\right)-k_2\nabla f_i\left(x_i\left(t\right)\right)-k_3{h}_i\left(t\right),\hfill \\ {\dot{h}}_i\left(t\right)=-k_4{\displaystyle \sum _{j\in {\mathcal{N}}_i}}{a}_{ij}(x_j\left(t_k\right)-x_i\left(t_k\right)),t\in [t_k,t_{k+1}),\hfill \end{array}\right.\end{array}$$

(12)

where $x_i\left(t_k\right)$ denotes the kth event instant of agent i for $k\in {\mathbb{N}}_{+}$. Other parameters are consistent with the ones in algorithm (3).

Before analyzing the distributed optimal algorithm, the measurement error $e_i\left(t\right)$ of agent i is defined as $e_i\left(t\right)=x_i\left(t_k\right)-x_i\left(t\right),t\in [t_k,t_{k+1})$. Hence, with the definition of $e_i\left(t\right)$, the distributed optimal algorithm (12) for $t\in [t_k,t_{k+1})$ can be rewritten as follows

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_i\left(t\right)={\xi }_i{y}_i\left(t\right),\hfill \\ {\dot{y}}_i\left(t\right)=-k_1{y}_i\left(t\right)-k_2\nabla f_i\left(x_i\left(t\right)\right)-k_3{h}_i\left(t\right),\hfill \\ {\dot{h}}_i\left(t\right)=-k_4{\displaystyle \sum _{j\in {\mathcal{N}}_i}}{a}_{ij}[(x_j\left(t\right)-x_i\left(t\right))+(e_j\left(t\right)-e_i\left(t\right))].\hfill \end{array}\right.\end{array}$$

(13)

According to the definition of Laplacian $\mathcal{L}$, the distributed optimal algorithm can be rewritten as the following compact matrix–vector form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}\left(t\right)=(\Xi \otimes I_n)y\left(t\right),\hfill \\ \dot{y}\left(t\right)=-k_{1}y\left(t\right)-k_2\nabla \overline{f}\left(x\left(t\right)\right)-k_{3}h\left(t\right),\hfill \\ \dot{h}\left(t\right)=k_4(\mathcal{L}\otimes I_n)(x\left(t\right)+e\left(t\right)),\hfill \end{array}\right.\end{array}$$

(14)

where $x\left(t\right)={[x_1^T\left(t\right),x_2^T\left(t\right),\cdots ,x_N^T\left(t\right)]}^T$, $y\left(t\right)=[y_1^T\left(t\right)$,$y_2^T\left(t\right)$, ⋯,$y_N^T{\left(t\right)]}^T$, $\nabla \overline{f}\left(x\left(t\right)\right)=[{(\nabla f_1\left(x_1\left(t\right)\right))}^T,$$(\nabla f_2$$\left(x_2\left(t\right)\right){)}^T,\cdots$, ${(\nabla f_N\left(x_N\left(t\right)\right))}^T{]}^T$, $h\left(t\right)=[h_1^T\left(t\right),h_2^T\left(t\right),\cdots ,$$h_N^T{\left(t\right)]}^T$, $e\left(t\right)=$$[e_1^T\left(t\right)$, $e_2^T\left(t\right),\cdots$, $e_N^T{\left(t\right)]}^T$.

We perform a similar coordination transformation as before. Define $\widehat{x}\left(t\right)=x\left(t\right)-x^{*},\widehat{y}\left(t\right)=y\left(t\right)-y^{*},\widehat{h}\left(t\right)=h\left(t\right)-h^{*}$ and $\varphi \left(t\right)=\nabla \overline{f}\left(x\left(t\right)\right)-\nabla \overline{f}\left(x^{*}\right)$, then the new transformed system is described as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\widehat{x}}\left(t\right)=(\Xi \otimes I_n)\widehat{y}\left(t\right),\hfill \\ \dot{\widehat{y}}\left(t\right)=-k_1\widehat{y}\left(t\right)-k_2\varphi \left(t\right)-k_3\widehat{h}\left(t\right),\hfill \\ \dot{\widehat{h}}\left(t\right)=k_4(\mathcal{L}\otimes I_n)(\widehat{x}\left(t\right)+e\left(t\right)).\hfill \end{array}\right.\end{array}$$

(15)

Denote $\check{x}\left(t\right)=({\mathcal{Q}}^T\otimes I_n)\widehat{x}\left(t\right),\check{y}\left(t\right)=({\mathcal{Q}}^T\Xi \otimes I_n)\widehat{y}\left(t\right),\check{h}\left(t\right)=({\mathcal{Q}}^T\Xi \otimes I_n)\widehat{h}\left(t\right)$ and $\check{e}\left(t\right)=({\mathcal{Q}}^T\otimes I_n)e\left(t\right)$. Then, we divide the variables as follows: $\check{x}\left(t\right)={[{\check{x}}_1^T\left(t\right),{\check{x}}_{2:N}^T\left(t\right)]}^T,$$\check{y}\left(t\right)=[{\check{y}}_1^T\left(t\right),$${\check{y}}_{2:N}^T{\left(t\right)]}^T,\check{h}\left(t\right)={[{\check{h}}_1^T\left(t\right),{\check{h}}_{2:N}^T\left(t\right)]}^T$ and $\check{e}\left(t\right)=[{\check{e}}_1^T\left(t\right)$, ${\check{e}}_{2:N}^T{\left(t\right)]}^T$, where ${\check{x}}_1\left(t\right),$${\check{y}}_1\left(t\right),$${\check{h}}_1\left(t\right),{\check{e}}_1\left(t\right)\in {\mathbb{R}}^n$ and ${\check{x}}_{2:N}\left(t\right)$, ${\check{y}}_{2:N}\left(t\right)$, ${\check{h}}_{2:N}\left(t\right)$, ${\check{e}}_{2:N}\left(t\right)\in {\mathbb{R}}^{(N-1)n}$. Hence, equality (15) can be rewritten as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\check{x}}}_1\left(t\right)={\check{y}}_1\left(t\right),\hfill \\ {\dot{\check{y}}}_1\left(t\right)=-k_1{\check{y}}_1\left(t\right)-k_2(q_1^T\Xi \otimes I_n)\varphi \left(t\right),\hfill \\ {\dot{\check{h}}}_1\left(t\right)=0,\hfill \end{array}\right.\end{array}$$

(16)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\check{x}}}_{2:N}\left(t\right)={\check{y}}_{2:N}\left(t\right),\hfill \\ {\dot{\check{y}}}_{2:N}\left(t\right)=-k_1{\check{y}}_{2:N}\left(t\right)-k_2({\mathcal{Q}}_2^T\Xi \otimes I_n)\varphi \left(t\right)-k_3{\check{h}}_{2:N}\left(t\right),\hfill \\ {\dot{\check{h}}}_{2:N}\left(t\right)=k_4({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n)({\check{x}}_{2:N}\left(t\right)+{\check{e}}_{2:N}\left(t\right)).\hfill \end{array}\right.\end{array}$$

(17)

Assumption 5.

For MAS (12), ${\displaystyle \sum _{i=1}^N}{\xi }_i{h}_i\left(0\right)=0$ is satisfied, and there exist parameters $k_1,$$k_2>0,$$0<k_4<\frac{k_1^3{ϵ}_0^2(1-{ϵ}_0)}{2k_3{\lambda }_N}$, such that ${\vartheta }_1=k_1(1-{ϵ}_0)+\frac{ϵk_1}{k_2}-\frac{(ϵ+k_1{k}_2)k_3}{4k_2{\zeta }_2}-ϵk_4{\lambda }_N{\zeta }_5-\frac{{ϵ}^2{k}_4^2{\lambda }_N^2}{4{\zeta }_6}>0$, ${\vartheta }_2=k_3ϵ-\frac{k_3^2{ϵ}_0^2}{4{\zeta }_1}-\frac{(ϵ+k_1{k}_2)k_3{\zeta }_2}{k_2}-ϵk_2\widehat{\xi }\widehat{m}{\zeta }_3-\frac{ϵk_2{k}_4{\lambda }_N}{4{\zeta }_4}-\frac{{ϵ}^2{k}_4^2}{4{\zeta }_7}>0$, and ${\sigma }_1=k_1{k}_2{ϵ}_0\check{\xi }\check{s}-k_1^2{\zeta }_1-\frac{ϵk_2\widehat{\xi }\widehat{m}}{4{\zeta }_3}-ϵk_2{k}_4{\lambda }_N{\zeta }_4-\frac{ϵk_4{\lambda }_N}{4{\zeta }_5}-k_3{k}_4{\lambda }_N-k_3{k}_4(\frac{1}{2}+{\mu }_0)>0$, where ${\zeta }_i,i=1,2,\cdots ,7$ are positive constants, $\widehat{\xi }=max\{{\xi }_1$, ${\xi }_2,\cdots ,{\xi }_N\}$, $\check{s}=min\{s_1,s_2,$$\cdots ,$$s_N\}$, $\widehat{m}=max\{m_1,m_2,\cdots ,m_N\}$, and ϵ and ${ϵ}_0$ are two positive constants, and ${\mu }_0>0$ is a positive number for adjustment. Moreover, the event-trigger instantly satisfies

$$\begin{array}{c}\hfill t_{k+1}={\displaystyle \underset{t}{inf}}\{t>t_k{|\parallel e\left(t\right)\parallel }^2\ge \eta {\displaystyle \sum _{i=1}^N}({\displaystyle \sum _{j=1}^N}{w}_{ij}{\parallel x_j\left(t\right)-x_i\left(t\right)\parallel }^2\left)\right\},\end{array}$$

(18)

where $\eta =\frac{2k_3{k}_4{\mu }_0}{k_3{k}_4{\lambda }_N+2(k_1^2{\zeta }_1+{\zeta }_6+k_2^2{\zeta }_7)}$.

Theorem 2.

Suppose that Assumptions 1–3 and 5 hold, then the optimization problem (1) can be solved via distributed algorithm (12).

Proof. 

Choosing the same Lyapunov function as (8). Based on equalities (9)–(11), (16), and (17), we take the time derivation of $W_1\left(t\right),W_2\left(t\right)$, and $W_3\left(t\right)$ for $t\in [t_k,t_{k+1})$, as follows:

$$\begin{array}{cc}\hfill {\dot{W}}_1\left(t\right)=& -k_1(1-{ϵ}_0){\check{y}}_1^T\left(t\right){\check{y}}_1\left(t\right)-k_2{\check{y}}_1^T\left(t\right)(q_1^T\Xi \otimes I_n)\varphi \left(t\right)\hfill \\ \hfill & -k_1{k}_2{ϵ}_0{\check{x}}_1^T\left(t\right)(q_1^T\Xi \otimes I_n)\varphi \left(t\right),\hfill \\ \hfill {\dot{W}}_2\left(t\right)=& -k_1(1-{ϵ}_0){\check{y}}_{2:N}^T\left(t\right){\check{y}}_{2:N}\left(t\right)-k_2{\check{y}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\Xi \otimes I_n)\varphi \left(t\right)\hfill \\ \hfill & -k_1{k}_2{ϵ}_0{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\Xi \otimes I_n)\varphi \left(t\right)+k_1{k}_3{ϵ}_0{\check{h}}_{2:N}^T\left(t\right){\check{e}}_{2:N}^T\left(t\right)\hfill \\ \hfill & +k_3{k}_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{x}}_{2:N}\left(t\right)+k_3{k}_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{e}}_{2:N}\left(t\right),\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\dot{W}}_3\left(t\right)={\check{y}}^T\left(t\right)(Q^T\Xi \otimes I_n)\varphi \left(t\right).\end{array}$$

Therefore, the time derivation of $V_1\left(t\right)$ for $t\in [t_k,t_{k+1})$ is obtained as

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)=& {\dot{W}}_1\left(t\right)+{\dot{W}}_2\left(t\right)+k_2{\dot{W}}_3\left(t\right)\hfill \\ \hfill =& -k_1(1-{ϵ}_0){\check{y}}^T\left(t\right)\check{y}\left(t\right)-k_1{k}_2{ϵ}_0{\check{x}}^T\left(t\right)({\mathcal{Q}}_2^T\Xi \otimes I_n)\varphi \left(t\right)\hfill \\ \hfill & +k_3{k}_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{x}}_{2:N}\left(t\right)+k_1{k}_3{ϵ}_0{\check{h}}_{2:N}^T\left(t\right){\check{e}}_{2:N}^T\left(t\right)\hfill \\ \hfill & +k_3{k}_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{e}}_{2:N}\left(t\right).\hfill \end{array}$$

By using the Young’s inequality with $ϵ>0$, $ab\le \frac{1}{4ϵ}{a}^2+ϵb^2$ for $\forall a,b>0$, one has

$$\begin{array}{c}\hfill k_1{k}_3{ϵ}_0{\check{h}}_{2:N}^T\left(t\right){\check{e}}_{2:N}^T\left(t\right)\le \frac{k_3^2{ϵ}_0^2}{4{\zeta }_1}\parallel {\check{h}}_{2:N}{\left(t\right)\parallel }^2+k_1^2{\zeta }_1{\parallel {\check{e}}_{2:N}\left(t\right)\parallel }^2.\end{array}$$

Then, we consider the following Lyapunov function

$$\begin{array}{c}\hfill V_2\left(t\right)=W_4\left(t\right)+\frac{ϵ}{2k_2}{\check{y}}_1^T\left(t\right){\check{y}}_1\left(t\right)+ϵW_1\left(t\right),\end{array}$$

where

$$\begin{array}{c}\hfill W_4\left(t\right)=\frac{ϵ}{2k_2}{\check{y}}_{2:N}^T\left(t\right){\check{y}}_{2:N}\left(t\right)+ϵ{\check{y}}_{2:N}^T\left(t\right){\check{h}}_{2:N}\left(t\right)+\frac{ϵk_2}{2}{\check{h}}_{2:N}^T\left(t\right){\check{h}}_{2:N}\left(t\right).\end{array}$$

(19)

Note that $W_4\left(t\right)\ge 0$ and $W_4\left(t\right)=0$ if and only if ${\check{y}}_{2:N}\left(t\right)=0$ and ${\check{h}}_{2:N}\left(t\right)=0$. Take the time derivation of $W_4\left(t\right)$ and $V_2\left(t\right)$ for $t\in [t_k,t_{k+1})$, respectively, one has

$$\begin{array}{cc}\hfill {\dot{W}}_4\left(t\right)=& \frac{ϵ}{k_2}{\check{y}}_{2:N}^T\left(t\right){\dot{\check{y}}}_{2:N}\left(t\right)+ϵ{\dot{\check{y}}}_{2:N}^T\left(t\right){\check{h}}_{2:N}\left(t\right)+ϵ{\check{y}}_{2:N}^T\left(t\right){\dot{\check{h}}}_{2:N}\left(t\right)+ϵk_2{\check{h}}_{2:N}^T\left(t\right){\dot{\check{h}}}_{2:N}\left(t\right)\hfill \\ \hfill =& -\frac{ϵk_1}{k_2}{\check{y}}_{2:N}^T\left(t\right){\check{y}}_{2:N}\left(t\right)-\frac{(ϵ+k_1{k}_2)k_3}{k_2}{\check{y}}_{2:N}^T\left(t\right){\check{h}}_{2:N}\left(t\right)-ϵ{\check{y}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\Xi \otimes I_n)\varphi \left(t\right)\hfill \\ \hfill & -ϵk_2{\check{h}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\Xi \otimes I_n)\varphi \left(t\right)+ϵk_2{k}_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{h}}_{2:N}\left(t\right)\hfill \\ \hfill & +ϵk_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{y}}_{2:N}\left(t\right)+ϵk_4{\check{y}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{e}}_{2:N}\left(t\right)\hfill \\ \hfill & -k_3ϵ{\check{h}}_{2:N}^T\left(t\right){\check{h}}_{2:N}\left(t\right)+ϵk_2{k}_4{\check{h}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{e}}_{2:N}\left(t\right).\hfill \end{array}$$

Therefore, the time derivation of $V_2\left(t\right)$ for $t\in [t_k,t_{k+1})$ is obtained as

$$\begin{array}{cc}\hfill {\dot{V}}_2\left(t\right)=& {\dot{W}}_4\left(t\right)+\frac{ϵ}{k_2}{\check{y}}_1^T\left(t\right){\dot{\check{y}}}_1\left(t\right)+ϵ{\dot{W}}_1\left(t\right)\hfill \\ \hfill =& -\frac{ϵk_1}{k_2}{\check{y}}^T\left(t\right)\check{y}\left(t\right)-\frac{(ϵ+k_1{k}_2)k_3}{k_2}{\check{y}}_{2:N}^T\left(t\right){\check{h}}_{2:N}\left(t\right)-k_3ϵ{\check{h}}_{2:N}^T\left(t\right){\check{h}}_{2:N}\left(t\right)\hfill \\ \hfill & -ϵk_2{\check{h}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\Xi \otimes I_n)\varphi \left(t\right)+ϵk_2{k}_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{h}}_{2:N}\left(t\right)\hfill \\ \hfill & +ϵk_4{\check{y}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{e}}_{2:N}\left(t\right)+ϵk_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{y}}_{2:N}\left(t\right)\hfill \\ \hfill & +ϵk_2{k}_4{\check{h}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{e}}_{2:N}\left(t\right).\hfill \end{array}$$

By using Young’s inequality, it has

$$\begin{array}{cc}\hfill & -\frac{(ϵ+k_1{k}_2)k_3}{k_2}{\check{y}}_{2:N}^T\left(t\right){\check{h}}_{2:N}\left(t\right)\le \frac{(ϵ+k_1{k}_2)k_3}{4k_2{\zeta }_2}\parallel {\check{y}}_{2:N}{\left(t\right)\parallel }^2+\frac{(ϵ+k_1{k}_2)k_3{\zeta }_2}{k_2}{\parallel {\check{h}}_{2:N}\left(t\right)\parallel }^2,\hfill \\ \hfill & -ϵk_2{\check{h}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\Xi \otimes I_n)\varphi \left(t\right)\le \frac{ϵk_2\widehat{\xi }\widehat{m}}{4{\zeta }_3}\parallel \check{x}\left(t\right){\parallel }^2+ϵk_2\widehat{\xi }\widehat{m}{\zeta }_3{\parallel {\check{h}}_{2:N}\left(t\right)\parallel }^2,\hfill \\ \hfill & ϵk_2{k}_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{h}}_{2:N}\left(t\right)\le \frac{ϵk_2{k}_4{\lambda }_N}{4{\zeta }_4}\parallel {\check{h}}_{2:N}\left(t\right){\parallel }^2+ϵk_2{k}_4{\lambda }_N{\zeta }_4{\parallel {\check{x}}_{2:N}\left(t\right)\parallel }^2,\hfill \\ \hfill & ϵk_4{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{y}}_{2:N}\left(t\right)\le \frac{ϵk_4{\lambda }_N}{4{\zeta }_5}\parallel {\check{x}}_{2:N}\left(t\right){\parallel }^2+ϵk_4{\lambda }_N{\zeta }_5{\parallel {\check{y}}_{2:N}\left(t\right)\parallel }^2,\hfill \\ \hfill & ϵk_4{\check{y}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{e}}_{2:N}\left(t\right)\le \frac{{ϵ}^2{k}_4^2{\lambda }_N^2}{4{\zeta }_6}\parallel {\check{y}}_{2:N}\left(t\right){\parallel }^2+{\zeta }_6{\parallel {\check{e}}_{2:N}\left(t\right)\parallel }^2,\hfill \end{array}$$

and

$$\begin{array}{c}\hfill ϵk_2{k}_4{\check{h}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{e}}_{2:N}\left(t\right)\le \frac{{ϵ}^2{k}_4^2}{4{\zeta }_7}\parallel {\check{h}}_{2:N}\left(t\right){\parallel }^2+k_2^2{\zeta }_7{\parallel {\check{e}}_{2:N}\left(t\right)\parallel }^2.\end{array}$$

Let $V_3\left(t\right)=V_1\left(t\right)+V_2\left(t\right)$, then it is clear that

$$\begin{array}{cc}\hfill {\dot{V}}_3\left(t\right)=& {\dot{V}}_1\left(t\right)+{\dot{V}}_2\left(t\right)\hfill \\ \hfill \le & -{\vartheta }_1\parallel {\check{y}}_{2:N}{\left(t\right)\parallel }^2-{\vartheta }_2\parallel {\check{h}}_{2:N}{\left(t\right)\parallel }^2-{\vartheta }_3\parallel {\check{x}}_{2:N}{\left(t\right)\parallel }^2-{\vartheta }_4\parallel {\check{y}}_1{\left(t\right)\parallel }^2-{\vartheta }_5{\parallel {\check{x}}_1\left(t\right)\parallel }^2\hfill \\ \hfill & +k_3{k}_4\left[{\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{e}}_{2:N}\left(t\right)+\frac{k_1^2{\zeta }_1+{\zeta }_6+k_2^2{\zeta }_7}{k_3{k}_4}{\parallel {\check{e}}_{2:N}\left(t\right)\parallel }^2\right],\hfill \end{array}$$

where ${\vartheta }_1=k_1(1-{ϵ}_0)+\frac{ϵk_1}{k_2}-\frac{(ϵ+k_1{k}_2)k_3}{4k_2{\zeta }_2}-ϵk_4{\lambda }_N{\zeta }_5-\frac{{ϵ}^2{k}_4^2{\lambda }_N^2}{4{\zeta }_6}$, ${\vartheta }_2=k_3ϵ-\frac{k_3^2{ϵ}_0^2}{4{\zeta }_1}-\frac{(ϵ+k_1{k}_2)k_3{\zeta }_2}{k_2}$$-ϵk_2\widehat{\xi }\widehat{m}{\zeta }_3-\frac{ϵk_2{k}_4{\lambda }_N}{4{\zeta }_4}-\frac{{ϵ}^2{k}_4^2}{4{\zeta }_7}$, ${\vartheta }_3=k_1{k}_2{ϵ}_0\check{\xi }\check{s}-k_1^2{\zeta }_1-\frac{ϵk_2\widehat{\xi }\widehat{m}}{4{\zeta }_3}-ϵk_2{k}_4{\lambda }_N{\zeta }_4-\frac{ϵk_4{\lambda }_N}{4{\zeta }_5}-k_3{k}_4{\lambda }_N$, ${\vartheta }_4=k_1(1-{ϵ}_0)+\frac{ϵk_1}{k_2}$, and ${\vartheta }_5=k_1{k}_2{ϵ}_0\check{\xi }\check{s}-k_1^2{\zeta }_1-\frac{ϵk_2\widehat{\xi }\widehat{m}}{4{\zeta }_3}$.

Next, we let $\overline{r}\left(t\right)=r\left(t\right)+\frac{k_1^2{\zeta }_1+{\zeta }_6+k_2^2{\zeta }_7}{k_3{k}_4}{\parallel {\check{e}}_{2:N}\left(t\right)\parallel }^2$, where $r={\check{x}}_{2:N}^T\left(t\right)({\mathcal{Q}}_2^T\mathcal{W}{\mathcal{Q}}_2\otimes I_n){\check{e}}_{2:N}\left(t\right)$. As $\check{x}\left(t\right)=({\mathcal{Q}}^T\otimes I_n)\widehat{x}\left(t\right)$ and $\check{e}\left(t\right)=({\mathcal{Q}}^T\otimes I_n)e\left(t\right)$, it has

$$\begin{array}{c}\hfill r\left(t\right)={\widehat{x}}^T\left(t\right)(\mathcal{W}\otimes I_n)e\left(t\right).\end{array}$$

Moreover, according to the above definition $\widehat{x}\left(t\right)=x\left(t\right)-x^{*}$, one has

$$\begin{array}{c}\hfill r\left(t\right)={(x\left(t\right)-(1_N\otimes x^{*}))}^T(\mathcal{W}\otimes I_n)e\left(t\right).\end{array}$$

Based on $1_N^T\mathcal{W}=0$, it has

$$\begin{array}{cc}\hfill r\left(t\right)=& x^T\left(t\right)(\mathcal{W}\otimes I_n)e\left(t\right)\hfill \\ \hfill \le & \frac{1}{2}{x}^T\left(t\right)(\mathcal{W}\otimes I_n)x\left(t\right)+\frac{1}{2}{e}^T\left(t\right)(\mathcal{W}\otimes I_n)e\left(t\right).\hfill \end{array}$$

Based on $e^T\left(t\right)(\mathcal{W}\otimes I_n)e\left(t\right)\le {\lambda }_N{\parallel \check{e}\left(t\right)\parallel }^2$, let

$$\begin{array}{c}\hfill \parallel \check{e}{\left(t\right)\parallel }^2\le \frac{2k_3{k}_4{\mu }_0{x}^T\left(t\right)(\mathcal{W}\otimes I_n)x\left(t\right)}{k_3{k}_4{\lambda }_N+2(k_1^2{\zeta }_1+{\zeta }_6+k_2^2{\zeta }_7)},\end{array}$$

where ${\mu }_0>0$ is an arbitrary positive parameter.

Therefore, it has

$$\begin{array}{cc}\hfill \overline{r}\left(t\right)\le & (\frac{1}{2}{\lambda }_N+\frac{k_1^2{\zeta }_1+{\zeta }_6+k_2^2{\zeta }_7}{k_3{k}_4}){\parallel \check{e}\left(t\right)\parallel }^2+\frac{1}{2}{x}^T\left(t\right)(\mathcal{W}\otimes I_n)x\left(t\right)\hfill \\ \hfill \le & (\frac{1}{2}+{\mu }_0)x^T\left(t\right)(\mathcal{W}\otimes I_n)x\left(t\right).\hfill \end{array}$$

Consequently, it is clear that

$$\begin{array}{cc}\hfill {\dot{V}}_3\left(t\right)\le & -{\vartheta }_1\parallel {\check{y}}_{2:N}{\left(t\right)\parallel }^2-{\vartheta }_2\parallel {\check{h}}_{2:N}{\left(t\right)\parallel }^2-{\sigma }_1\parallel {\check{x}}_{2:N}{\left(t\right)\parallel }^2-{\vartheta }_4\parallel {\check{y}}_1{\left(t\right)\parallel }^2-{\sigma }_2{\parallel {\check{x}}_1\left(t\right)\parallel }^2.\hfill \end{array}$$

Therefore, the agent i can use event-triggered sampling control (18) to determine whether to store and update sampling data.

From the condition of Theorem 2, ${\dot{V}}_3\left(t\right)\le 0$, and ${\dot{V}}_3\left(t\right)=0$ if and only if $\check{y}\left(t\right)=0,$$\check{x}\left(t\right)=0,{\check{h}}_{2:N}\left(t\right)=0$. Therefore, the state $x\left(t\right)$ of the system can be asymptotically uniform and converge to the optimal solution of the optimization problem. □

Remark 7.

Compared with existing work in [37], the selection of the Lyapunov function is different. In our paper, the positive function $W_3\left(t\right)$, inspired by the one in [44], is added to broaden the limitation of related coefficients in the dynamics.

Remark 8.

In this event-triggered algorithm, the global information $e\left(t\right)$ and $x\left(t\right)$ are used by each agent in the trigger function. To avoid using global information, the proposed algorithm will be improved in our future work.

Theorem 3.

For the MAS (12), if all conditions of Theorem 2 are satisfied, then there is no Zeno behavior.

Proof. 

For $t\in [t_k,t_{k+1})$, $k\in {\mathbb{N}}_{+}$, the the upper right-hand derivative of $\parallel e_i\left(t\right)\parallel$ is given by

$$\begin{array}{c}\hfill D^{+}\parallel e_i\left(t\right)\parallel \le \parallel {\dot{e}}_i\left(t\right)\parallel ={\xi }_i\parallel y_i\left(t\right)\parallel .\end{array}$$

(20)

According to equality (12), it has

$$\begin{array}{cc}\hfill & y_i\left(t\right)-y_i\left(t_k\right)\hfill \\ \hfill =& {\int }_{t_k}^t(-k_1{\xi }_i{y}_i\left(s\right)-k_2\nabla f_i\left(x_i\left(s\right)\right)-k_3{h}_i\left(s\right))ds\hfill \\ \hfill =& e^{-k_1{\xi }_i(t-t_k)}{y}_i\left(t_k\right)-e^{-k_1{\xi }_{i}t}{k}_2{\int }_{t_k}^t{e}^{k_1{\xi }_{i}s}\nabla f_i\left(x_i\left(s\right)\right)ds-e^{-k_1{\xi }_{i}t}{k}_3{\int }_{t_k}^t{e}^{k_1{\xi }_{i}s}{h}_i\left(s\right)ds.\hfill \end{array}$$

As the functions $e^{k_1{\xi }_{i}t}\nabla f_i\left(x_i\left(t\right)\right)$ and $e^{k_1{\xi }_{i}t}{h}_i\left(t\right)$ are continuous and differentiable on the closed interval $[t_k,t]$, then there exist at least two points $\alpha$ and $\beta$ in $[t_k,t]$, which makes the following two formulas hold, respectively,

$$\begin{array}{c}\hfill {\int }_{t_k}^t{e}^{k_1{\xi }_{i}s}\nabla f_i\left(x_i\left(s\right)\right)ds=e^{k_1{\xi }_i\alpha }\nabla f_i\left(x_i\left(\alpha \right)\right)(t-t_k),\end{array}$$

(21)

and

$$\begin{array}{c}\hfill {\int }_{t_k}^t{e}^{k_1{\xi }_{i}s}{h}_i\left(s\right)ds=e^{k_1{\xi }_i\beta }{h}_i\left(\beta \right)(t-t_k).\end{array}$$

(22)

Hence, it has

$$\begin{array}{c}\hfill \parallel y_i\left(t\right)\parallel \le 2\parallel y_i\left(t_k\right)\parallel +\parallel k_2\nabla f_i\left(x_i\left(\alpha \right)\right)+k_3{h}_i\left(\beta \right)\parallel (t-t_k).\end{array}$$

Denote ${\Delta }_i\left(t\right)={\displaystyle \underset{t_k\le s\le t}{max}}\{{\xi }_i\parallel k_2\nabla f_i\left(x_i\left(\alpha \right)\right)+h_i\left(\beta \right)\parallel (s-t_k)\}$.

Since $e\left(t_k\right)=0$, considering equality (20), we can obtain

$$\begin{array}{c}\hfill \parallel e_i\left(t\right)\parallel \le (2{\xi }_i\parallel y_i\left(t_k\right)\parallel +{\Delta }_i\left(\varpi \right))(t-t_k),\end{array}$$

where $\varpi \in [t_k,t]$ is a constant. The event instants of agent i are determined by equality (18), and the next event will not be triggered before $\parallel e_i{\left(t\right)\parallel }^2=\eta {\sum }_{j=1}^N{w}_{ij}{\parallel x_j\left(t\right)-x_i\left(t\right)\parallel }^2$. Therefore, it has

$$\begin{array}{cc}\hfill \parallel e_i\left(t_{k+1}\right)\parallel & =\sqrt{\eta {\displaystyle \sum _{j=1}^N}{w}_{ij}(\parallel (x_j\left(t_{k+1}\right)-x_i\left(t_{k+1}\right)){\parallel }^2)}\hfill \\ \hfill & \le (2{\xi }_i\parallel y_i\left(t_k\right)\parallel +{\Delta }_i\left(\varpi \right))(t_{k+1}-t_k).\hfill \end{array}$$

(23)

From equality (23), we can easily obtain that $t_{k+1}-t_k\ge \frac{\sqrt{\eta {\displaystyle \sum _{j=1}^N}{w}_{ij}(\parallel (x_j\left(t_{k+1}\right)-x_i\left(t_{k+1}\right)){\parallel }^2)}}{2{\xi }_i\parallel y_i\left(t_k\right)\parallel +{\Delta }_i\left(\varpi \right)}$. As the network topology $\mathcal{G}$ is strongly connected, directed, and detail-balanced,

$$\begin{array}{c}\hfill {\displaystyle \sum _{j=1}^N}{w}_{ij}(\parallel (x_j\left(t_{k+1}\right)-x_i\left(t_{k+1}\right)){\parallel }^2)=0\end{array}$$

if and only if

$$\begin{array}{c}\hfill \parallel (x_j\left(t_{k+1}\right)-x_i\left(t_{k+1}\right))\parallel =0,i,j=1,2,\dots ,N.\end{array}$$

That means all agents states achieve consensus, and the communication between agents is no longer needed. Then, $t_{k+1}-t_k>0$ before all agents achieve consensus. Therefore, Zeno behavior can be excluded. □

Remark 9.

The optimization problem with event-triggered communication and data rate constraint was investigated in [18]. To preprocess the information, a vector-valued quantizer with finite quantization levels was introduced. This gives us a good research direction. In [18], the considered system is a first-order linear difference equation, and the optimization problem is modeled as the sum of all agents’ local convex cost functions. However, in our paper, we consider a second-order multi-agent system, and the optimization problem is the weighted sum of agents’ local convex cost functions.

5. Numerical Example

In this section, an economic dispatch example is used to verify the performance of the proposed algorithms (3) and (12).

We consider a power system which contains 8 generators and 22 loads. The active power generated by the ith generator is denoted by $x_i={[x_{i1},x_{i2}]}^T$, and the allocation weight is represented by ${\xi }_i,i=1,2,\cdots ,8$. The communication topology is constructed as shown in Figure 1.

Figure 1. The communication topology.

The cost functions are given by the quadratic function [26] $f_i\left(x_i\left(t\right)\right)={\alpha }_{i1}{x}_{i1}{\left(t\right)}^2+{\alpha }_{i2}{x}_{i2}{\left(t\right)}^2+{\beta }_{i1}{x}_{i1}\left(t\right)+{\beta }_{i2}{x}_{i2}\left(t\right)+{\eta }_i$, in which ${\alpha }_{i1},{\alpha }_{i2},{\beta }_{i1},{\beta }_{i2}$, and ${\eta }_i$ represent the cost coefficients. Therefore, the economic dispatch problem is given by

$$\begin{array}{c}\hfill {\displaystyle \underset{\mathbf{x}\in {\mathbb{R}}^2}{min}}\tilde{f}\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^8}{\xi }_i{f}_i\left(\mathbf{x}\right)\end{array}$$

(24)

In the simulation, we let $k_1=100$, $k_2=8$, $k_3=0.2$, and $k_4=0.022$ and choose the initial states as $x_1\left(0\right)={[1,-0.2]}^T$, $x_2\left(0\right)={[0.3,0.5]}^T$, $x_3\left(0\right)={[0.4,-0.2]}^T$, $x_4\left(0\right)={[-0.3,-0.4]}^T$, $x_5\left(0\right)={[1,-0.9]}^T$, $x_6\left(0\right)={[0.8,-0.5]}^T$, $x_7\left(0\right)={[-0.4,0.3]}^T$, and $x_8\left(0\right)={[0,-1]}^T$. The cost coefficients and the allocation weights are given in Table 1.

Table 1. The coefficients.

No.	${\mathit{\alpha }}_{\mathit{i}1}$	${\mathit{\alpha }}_{\mathit{i}2}$	${\mathit{\beta }}_{\mathit{i}1}$	${\mathit{\beta }}_{\mathit{i}2}$	${\mathit{\eta }}_{\mathit{i}}$	${\mathit{\xi }}_{\mathit{i}}$
1	0.1	0.1	0.2	0	−7.9	0.0625
2	0.2	0.2	0	0	0	0.125
3	0.1	0.1	0.2	0	20	0.1875
4	0.2	0.1	0	0	−7	0.125
5	0.1	0.2	0	0	5	0.125
6	0.1	0.1	0	−0.2	9	0.1875
7	0.1	0.2	0.2	0.1	0	0.0625
8	0.1	0.1	−0.2	0	0	0.125

Case 1.

For the optimization problem (24), we use the proposed algorithm (3) with the above parameters. Through calculation, all conditions of Theorem 1 are satisfied. By using MATLAB software for simulation, the states of $x_{i1}\left(t\right)$ and $x_{i2}\left(t\right)$ are given in Figure 2 and Figure 3, respectively. It is noted that all agents’ states reach consensus and gradually converge to the optimal solution $x^{*}={[-0.164,0.105]}^T$. Figure 4 and Figure 5 show the trajectories of $y_{i1}\left(t\right)$ and $y_{i2}\left(t\right)$, respectively. Obviously, $y_{i1}\left(t\right)$ and $y_{i2}\left(t\right)$ gradually converge to 0. Figure 6 and Figure 7 show the evolution trajectories of $h_{i1}\left(t\right)$ and $h_{i2}\left(t\right)$, respectively. From Figure 6 and Figure 7, it can be seen that the condition of ${\displaystyle \sum _{i=1}^8}{\xi }_i{h}_i\left(t\right)=0$ is always satisfied. Moreover, $h_{ij}\left(t\right)$ gradually converge to some constants. Based on the third equation of (3), that means all agents’ states asymptotically reach the same value. The evolution trajectory of the objective function $F\left(t\right)$ is shown in Figure 8.

Figure 2. The states of $x_{i1}\left(t\right)$.

Figure 3. The states of $x_{i2}\left(t\right)$.

Figure 4. The trajectories of $y_{i1}\left(t\right)$.

Figure 5. The trajectories of $y_{i2}\left(t\right)$.

Figure 6. The trajectories of $h_{i1}\left(t\right)$.

Figure 7. The trajectories of $h_{i2}\left(t\right)$.

Figure 8. The cost function $F\left(t\right)$.

Case 2.

For the optimization problem (24), we use the proposed algorithm (12). All parameters are the same as the ones in Case 1. Through calculation, all conditions of Theorem 2 are also satisfied. The states of $x_{i1}\left(t\right)$ and $x_{i2}\left(t\right)$ are given in Figure 9 and Figure 10, respectively, where the optimal solution $x^{*}$ is asymptotically reachable. The trajectories of $y_{i1}\left(t\right)$ and $y_{i2}\left(t\right)$ are presented in Figure 11 and Figure 12, respectively. Apparently, they all asymptotically converge to 0. Figure 13 and Figure 14 show the trajectories of $h_{i1}\left(t\right)$ and $h_{i2}\left(t\right)$, respectively. The condition of ${\displaystyle \sum _{i=1}^8}{\xi }_i{h}_i\left(t\right)=0$ is always satisfied. Figure 15 and Figure 16 describe the trajectories of $x_{i1}\left(t_k\right)$ and $x_{i2}\left(t_k\right)$, respectively. It can be shown that $x_{i1}\left(t_k\right)$ and $x_{i2}\left(t_k\right)$ oscillate to the same value. Figure 17 shows the evolution trajectory of the objective function $F\left(t\right)$. Figure 18 shows the event-triggered instants, in which 1 and 0 represent trigger and non-trigger, respectively.

Figure 9. The states of $x_{i1}\left(t\right)$.

Figure 10. The states of $x_{i2}\left(t\right)$.

Figure 11. The trajectories of $y_{i1}\left(t\right)$.

Figure 12. The trajectories of $y_{i2}\left(t\right)$.

Figure 13. The trajectories of $h_{i1}\left(t\right)$.

Figure 14. The trajectories of $h_{i2}\left(t\right)$.

Figure 15. The trajectories of $x_{i1}\left(t_k\right)$.

Figure 16. The trajectories of $x_{i2}\left(t_k\right)$.

Figure 17. The cost function $F\left(t\right)$.

Figure 18. Event-triggered signals.

According to the comparison between Cases 1 and 2, we find that the continuous-time communication algorithm (3) and the event-triggered communication algorithm (12) can make the states of MAS achieve consensus and asymptotically reach the optimal solution of the optimization problem (24). That means these two algorithms are feasible. However, under the same conditions, the event-triggered communication algorithm can effectively reduce the limited resource cost of the agents while slowing down the convergence rate of the system. Therefore, in practical application, selecting the optimal method still needs the analysis of specific problems.

Through simulation, we can verify that the distributed optimization problem (1) can be solved asymptotically under the proposed continuous-time communication and event-triggered control protocols. This means that the optimal solution can be obtained as $t\to +\infty$. In a finite time, only an approximate solution of the optimization problem can be obtained. However, in some practical applications, such as the economic dispatch of smart grids, the optimal allocation is required in a finite time. Therefore, the distributed optimization algorithms with finite-time convergence are worth studying further.

6. Conclusions

In this paper, compared with the existing work, we considered a more general distributed optimization problem based on the second-order MASs over directed networks. A special directed network was carefully designed, and then an improved distributed optimization algorithm with continuous-time communication was proposed. By using the Lyapunov stability theory, some conditions were obtained to ensure the asymptotical convergence of the proposed algorithm. Furthermore, we designed an event-triggered control protocol to reduce the burden of communication and analyze the convergence of the event-triggered control algorithm. In addition, the Zeno behavior can be avoided. Finally, some numerical simulations were presented to verify the validity of the results. In our future work, the finite-time distributed optimization problem over directed networks will be considered.