Average Consensus Tracking of Weight-Balanced Multi-Agent Systems via Sampled Data

Abstract

This paper studies the average consensus tracking problem of first-order multi-agent systems under directed topologies. A control protocol is constructed based on the neighbors’ agent sampled information. With the help of the matrix theory and frequency domain analysis, the necessary and sufficient condition can be induced for the average consensus tracking of multi-agent systems, although the weight-balanced condition is not satisfied. In addition, a numerical simulation example is given to illustrate the effectiveness of the proposed results.

1. Introduction

There are some interesting group behavior phenomena in the biological world, such as birds flying in formation, fish swimming and dancing, cattle gathering and migrating, fireflies glowing synchronously, etc. In these group coordination behaviors, individuals in the group cooperate through certain cooperation rules [1], causing the group to present wonderful and spectacular coordinated behaviors. Inspired by biological phenomena in nature, researchers proposed the concept of artificial multi-agent systems (MASs). Intelligent agent generally refers to a physical or abstract entity with autonomous capabilities [2]. It can perceive its own environment and use information in the knowledge system to respond appropriately to the environment. Whether it is hardware or software, such as drones, satellites, robots, birds in the air, fish in the water, etc., all can be called intelligent agents. It is generally believed that an intelligent agent has the following basic characteristics: (1) autonomy: an intelligent agent can operate independently without direct intervention from other agents or humans in the network. It can sense changes in the external environment through sensors and can adjust its own status and behavior; (2) communication capability: the agent can exchange information with other entities through communication language; (3) computing capability: the agent can use existing information to perform relevant reasoning and calculations; and (4) adaptability: agents can adapt to new environments by learning knowledge, accumulating experience, and modifying their behavior.

For the coordinated control of multiagent systems (MASs), the consensus problem has always been a hot research topic of the control community in the past several years, due to its extensive applications in many areas, such as flocking control, sensor networks, and formation control [3,4,5].

Multi-agent system refers to a type of complex system that is composed of a group of such agents that are associated through communication and other methods. Its purpose is to achieve a common goal through cooperation between multiple agents with simple functions [6]. In engineering applications, it takes a lot of energy to continuously update and receive information [7]. There are clear advantages to using multi-agent system technology in large systems [8]. First, due to the ability of parallel perception and execution, the multi-agent system has a faster operation speed and higher execution efficiency. Secondly, when one or more agents fail, the system degrades normally, thus improving the efficiency of the system [9,10,11]. Thirdly, the reliability and robustness of the multi-agent system; the multi-agent system can add agents when necessary, and is scalable and flexible. Finally, the cost of a system composed of multiple simple agents is far lower than a single centralized complex system [12]. Multi-agent systems have been widely adopted in many application fields such as military system control, intelligent traffic control, and power system dispatching because of the beneficial advantages they provide.

In recent years, with the rapid development of computer technology and communication technology, collaborative control of multi-agent systems has received more and more widespread attention and its theoretical results have become more and more abundant. The main issues involved include consensus, coordinated tracking, formation, control, distributed optimization, distributed average tracking, etc. Among them, distributed average tracking (DAT) is also called “dynamic average consensus”, which was first proposed in the literature by Richard Murray [13,14]. The core of the DAT problem is to design a distributed algorithm for the agents in the system so that all agents can perform tasks independently and track the average of a set of time-varying reference signals using only local information. DAT technology can be used to estimate the parameters of complex systems, which has significant application effects in multi-core microprocessor task migration and dynamic area formation tracking control.

This study focuses on the average consensus tracking problem, which requires each agent to track the average of all agents under a given reference input. Obviously, the average consensus tracking problem is different from the conventional average consensus [15,16,17,18], which requires all agents to converge to the average of their initial values. Freeman et al. conducted a comparative study exploring proportional (P) average consensus tracking and proportional integral (PI) average consensus tracking for solving static and dynamic average consensus trajectory problems. Research results show that the PI algorithm has a better steady-state response when facing constant or slowly changing inputs [19]. Based on [19], researchers such as Bai conducted a detailed analysis of the structural characteristics of the PI algorithm. This algorithm successfully achieves smooth, average, consistent tracking of a constant reference input and extends the PI algorithm into a general formula. By applying internal model principles, this algorithm enables the agent to effectively track certain types of time-varying inputs [10]. Shan and Liu studied a first-order multi-agent system whose reference input is a constant [20,21]. They explored the problem of average consensus tracking when there are additional interconnections between agents without reference inputs and proposed the design of a PI average consensus tracking algorithm. Under the condition of symmetrically connected topology, by using the generalized Nyquist criterion [22], Shao Tian proposed an average consensus algorithm for first-order multi-agent systems, covering the cases both with and without communication delays [23]. This algorithm works well in situations with unparalleled reference inputs, but the requirement is that each agent must accurately obtain the numbers of all agents and all reference inputs.

DAT is sometimes seen as a generalization of the consensus problem and the coordinated tracking problem. However, since the average value of tracking is a function of multiple time-varying signals and is unknown to each agent, this problem has its own unique challenges and meaning. Next, we illustrate the challenge of studying the DAT problem by comparing the key differences between DAT and average consensus.

The difference between DAT and average consensus is as follows: Consider a multi-agent system composed of $n$ agents, let $x_i(t)$ be the state of agent $i$ at time $t$. Average consensus means that the states of all agents converge to the average of the initial states. That is $\underset{t\to \infty }{\lim }‖x_i(t)-\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{x}_i(0)}‖=0$. In DAT, the control goal is to make the states of all agents track the average value of the previous set of reference signals, that is $\underset{t\to \infty }{\lim }‖x_i(t)-\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{r}_i(t)}‖=0$. Therefore, by setting $r_i\equiv x_i(0)$ the average consensus is a special case of the DAT problem. Note that, in the average consensus problem, the convergence state is static, while in DAT the convergence state $\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{r}_i}(t)$ is time varying. It is well known that, even for a single system, controlling the system to a time-varying trajectory is more challenging than controlling it to a static point. Control algorithms designed for the latter are generally not suitable for the former. For example, consider a first-order integrator system $\dot{x}(t)=u(t)$. If we want to stabilize it to the static equilibrium point $\overline{x}$, then a simple feedback control can be implemented. For example, $u_i(t)=-(x-\overline{x})$ can drive the system asymptotically to $\overline{x}$. However, if $\overline{x}(t)$ is a time-varying trajectory, then the controller $u_i(t)=-(x-\overline{x})$ will no longer work. The above example shows that it is not easy to change the static value $x_i(0)$ in average consensus to the dynamic value $\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{r}_i}(t)$ in DAT. In addition, the requirement for the distributed nature of the DAT control law makes the problem more challenging.

In [24,25], the distributed average tracking was studied, in which a distributed control protocol is designed to achieve the average tracking objective. However, we should point out that the developed results in [24,25] need a restrictive condition that the communication topology is balanced. If the balanced condition is not satisfied, then results in [24,25] do not hold.

Inspired by the above discussion, we aim to investigate the distributed average problems of multi-agent systems under any digraph based on sampled data. The contributions of this paper include three aspects.

(1)

We design a distributed control protocol based on the sampled information. With this protocol being employed, the average consensus tracking problems can be solved for multi-agent systems.

(2)

We can develop the necessary and sufficient conditions for the average consensus tracking. It is shown that an upper bound of the sampling period can be computed and the average consensus tracking objective can be reached if, and only if, the sampling period is less than the upper bound of the sampling period.

(3)

Compared with [23,24], our developed results do not need the weight-balanced condition. It greatly improves the results of [23,24].

Notations: Denote $1_n={\left[1,1,\dots ,1\right]}^T\in {ℝ}^m$, $0_n={\left[0,0,\dots ,0\right]}^T\in {ℝ}^m$, and $diag\left\{{\delta }_1\right.,{\delta }_2$$\left.\dots ,{\delta }_m\right\}$ as a diagonal matrix whose diagonal elements are ${\delta }_1,{\delta }_2,\dots ,{\delta }_m$. For a real number $a\in ℝ$, $\left|a\right|$ denotes the absolute value of $a$. For a complex number $a\in ℂ$, it can be expressed as $a=\mathrm{Re}\left(a\right)+\mathrm{Im}\left(a\right)\iota$, where $\mathrm{Re}\left(a\right)$ and $\mathrm{Im}\left(a\right)$ represent the real part and the imaginary part of $a$, respectively, and $\iota$ is the imaginary unit. Additionally, $\left|a\right|=\sqrt{\mathrm{Re}{\left(a\right)}^2+\mathrm{Im}{\left(a\right)}^2}$ denotes the module value of $a.$

2. Preliminaries and Problem Description

2.1. Graph Theory

First, we will introduce some basic knowledge about graph theory in order to solve consensus problems in multi-agent systems.

Let $G=(V,E,A)$ be a weighted graph, where $V=\{1,2,\dots ,n\}$ is the node set, $E\subseteq V\times V$ is the edge set, and $A=[a_{ij}]\in R^{n\times n}$ is the weighted adjacency matrix. If the node $i$ communicates with the node $j$, i.e., $(j,i)\in E$, the corresponding coupling weights are $a_{ij}> 0$ and $a_{ij}=0$, otherwise the neighbors of agent $i$ are denoted by $N_i=\{j\in V:(j,i)\in E\}$. If ${\displaystyle \sum { }_{j=1}^n}{a}_{ij}={\displaystyle \sum { }_{j=1}^n}{a}_{ji},i\in V$, then $G$ is called balanced. The Laplacian matrix $L$ is denoted by $L=[l_{ij}]\in R^{n\times n}$, where $l_{ii}={\displaystyle \sum a_{ij}}$ and $l_{ij}=-a_{ij}$.

In a directed graph, if there is a path from node $i$ to node $j$, we say that node $i$ is reachable to node $j$. Furthermore, we say a node is globally reachable if it can be reached from any other node in the directed graph. If every node in a directed graph is globally reachable, then the directed graph is said to be strongly connected [26].

We will provide some notations of digraphs. An $n$th order signed digraph $G$ is construed by a node set $V=\left\{\left.v_1,v_2,\dots ,v_n\right\}\right.$, an edge set $\epsilon \subseteq \left\{(v_j,v_i):v_j,v_i\in V\right\}$, and an adjacency weight matrix $A=\left[a_{ij}\right]\in {ℝ}^{n\times n}$, whose element $a_{ij}$ meets $a_{ij}\ne 0\iff (v_j,v_i)$$\in \epsilon$. The edge $(v_j,v_i)$ denotes that node $v_i$ can receive information from node $v_j$ [19]. In addition, node $v_j$ is called the neighbor of node $v_i$. All neighbors of $v_i$ are denoted as $N(i)=\left\{v_j:(v_j,v_i)\in \epsilon \right\}$. A directed path $P$ with $m$ edges is represented as

$$P=\left\{(v_{k0},v_{k1}),(v_{k1},v_{k2}),\dots ,(v_{km-1},v_{km})\right\}.$$

2.2. Agent’s Model

Consider a multi-agent system with $n$ agents whose dynamics can be described by

$${\dot{x}}_i(t)=u_i(t),i=1,2,\dots ,n,$$

(1)

where the state and control input of the agents are denoted as $x_i(t)\in R$ and $u_i(t)\in R$, respectively.

The goal of this paper is to design a control protocol that achieves average consensus tracking for any reference input ${\varphi }_i,i=1,2,\dots ,n$.

$$\underset{t\to \infty }{\lim }{x}_i(t)=\frac{1}{n}{\displaystyle \sum _{j=1}^n{\varphi }_j}.$$

(2)

To achieve the average consensus tracking, a general assumption on topology is necessary.

Assumption 1.

The digraph $G$ is strongly connected.

In the following, we will introduce two useful lemmas that will be used below.

Lemma 1

[25]. For a square matrix $H,N,P,Q$ with n dimensions, if $HP=PH$, then.

$$\left|\begin{array}{cc}H& N\\ P& Q\end{array}\right|=\left|HQ-PN\right|.$$

(3)

Lemma 2

[25]. Given a complex coefficient polynomial of order two as follows:

$$g(s)=s^2+({\xi }_1+i{\sigma }_1)s+{\xi }_0+i{\sigma }_0$$

(4)

where ${\xi }_1,{\sigma }_1,{\xi }_0$, and ${\sigma }_0$ are real constants. Then, $g(s)$ is stable if, and only if, ${\xi }_1>0$ and ${\xi }_1{\sigma }_1{\sigma }_0+{\xi }_1^2{\xi }_0-{\sigma }_0^2>0$.

3. Average Consensus Tracking Results

In this section, we aim to study the average consensus tracking problem for first-order multi-agent systems with directed topologies. Let $T=t_{k+1}-t_k$ represent the sampling interval, and the control input is based on zero-order hold. We propose the following distributed protocol as:

$$\begin{array}{l}{u}_i(t)=\gamma [{\varphi }_i-x_i(t)]+k_p{\displaystyle \sum _{j\in N_i}\det (L_{ii})}{a}_{ij}[x_j(t_k)-x_i(t_k)]+k_I{\eta }_i(t)\\ {\dot{\eta }}_i(t)={\displaystyle \sum _{j\in N_i}\det (L_{ii})a_{ij}}[x_j(t_k)-x_i(t_k)],t\in (t_k,t_{k+1})\end{array}$$

(5)

where ${\varphi }_i\in R$ is the constant reference input of agent $i$, ${\eta }_i\in R$ is the internal state of agent $i$, and the control gains are $\gamma >0$, $k_p>0$, and $k_i>0$. In the average consensus tracking protocol (5), in order to eliminate the agent’s tracking error, we introduce ${\eta }_i(t)$. It can be seen that $x_j(t_k)$ and $x_i(t_k)$ are the information measured at the sampling moment.

Let $L$ represent the Laplacian matrix of $G$. We can validate that $L$ satisfies $L=\Delta -A$. Define $L_{ii}\in {ℝ}^{\left(n-1\right)\times \left(n-1\right)}\forall i\in 1,2,\dots ,n$ as the matrix that is induced from $L$ by removing its $i$th row and $i$th column, with which we correspondingly denote $M=diag\{\det (L_{11}),\det (L_{22}),\dots ,\det (L_{nn})\}$.

Define ${\theta }_i={\displaystyle {\sum }_{j\in N_i}{a}_{ij}[x_j(t_k)-x_i(t_k)]},\omega =\gamma {\phi }_i+M{k}_p{\theta }_i(t_k)+k_I{\eta }_i(t_k)$ and it is obvious that $\omega$ is a constant at $t_k$. Then, we obtain

$$\begin{array}{l}{\dot{x}}_i(t)=-\gamma x_i(t)+f(t)\\ {\eta }_i(t)={\eta }_i(t_k)+(t-t_k)M{\theta }_i(t_k)\end{array}$$

(6)

where

$$f(t)=M{k}_I{\theta }_i(t_k)(t-t_k)+\omega$$

Based on (6), we can obtain

$$\begin{array}{c}{e}^{\gamma t}{x}_i(t)-e^{\gamma t_k}{x}_i(t_k)={\displaystyle {\int }_{t_k}^t{e}^{\gamma \tau }f(\tau )}d\tau \\ x_i(t)=e^{-\gamma (t-t_k)}{x}_i(t_k)+{\displaystyle {\int }_{t_k}^t{e}^{-\gamma (t-\tau )}}[M{k}_I{\theta }_i(t_k)(\tau -t_k)+\gamma {\varphi }_i+k_{p}M{\theta }_i(t_k)+k_I{\eta }_i(t_k)]d\tau \end{array}$$

(7)

It follows from (7) that

$$\begin{array}{l}{x}_i(t)=e^{-\gamma (t-t_k)}{x}_i(t_k)+{\displaystyle {\int }_{t_k}^t{e}^{-\gamma (t-\tau )}}{k}_{I}M{\theta }_i(t_k)(\tau -t_k)d\tau +[1-e^{-\gamma (t-t_k)}]{\varphi }_i+\frac{k_I[1-e^{-\gamma (t-t_k)}]}{\gamma }{\eta }_i(t_k)+\frac{k_p[1-e^{-\gamma (t-t_k)}]}{\gamma }M{\theta }_i(t_k)\\ =e^{-\gamma (t-t_k)}{x}_i(t_k)+[1-e^{-\gamma (t-t_k)}]{\varphi }_i+\frac{k_I[1-e^{-\gamma (t-t_k)}]}{\gamma }{\eta }_i(t_k)+\frac{k_p[1-e^{-\gamma (t-t_k)}]}{\gamma }M{\theta }_i(t_k)\\ +\frac{1}{\gamma }{\displaystyle {\int }_{t_k}^{t}M{k}_I{\theta }_i}(t_k)(\tau -t_k)d{e}^{-\gamma (t-\tau )}\\ =e^{-\gamma (t-t_k)}{x}_i(t_k)+[1-e^{-\gamma (t-t_k)}]{\varphi }_i+\frac{k_I[1-e^{-\gamma (t-t_k)}]}{\gamma }{\eta }_i(t_k)+\frac{k_p[1-e^{-\gamma (t-t_k)}]}{\gamma }M{\theta }_i(t_k)+\frac{1}{\gamma }{k}_I(t-t_k)M{\theta }_i(t_k)-\frac{k_{I}M{\theta }_i(t_k)}{{\gamma }^2}[1-e^{-\gamma (t-t_k)}]\end{array}$$

Therefore, we have

$$\begin{array}{l}{x}_i(t)=e^{-\gamma (t-t_k)}{x}_i(t_k)+[(k_p-\frac{k_I}{\gamma })(1-e^{-\gamma (t-t_k)})+k_I(t-t_k)]\frac{M{\theta }_i(t_k)}{\gamma }+\frac{k_I(1-e^{-\gamma (t-t_k)})}{\gamma }{\eta }_i(t_k)+(1-e^{-\gamma (t-t_k)}){\varphi }_i\\ {\eta }_i(t)={\eta }_i(t_k)+(t-t_k){\theta }_i(t_k)\end{array}$$

(8)

Defining $\alpha =e^{-\gamma T},\beta =k_p-\frac{k_I}{\gamma }$. From Equation (8), we can obtain

$$\left\{\begin{array}{c}{x}_i(k+1)=\alpha x_i(k)+\frac{\beta (1-\alpha )+k_{I}T}{\gamma }M{\theta }_i(k)+\frac{k_I(1-\alpha )}{\gamma }{\eta }_i(k)+(1-\alpha ){\varphi }_i\\ {\eta }_i(k+1)={\eta }_i(k)+TM{\theta }_i(k)\end{array}\right.$$

(9)

Defining $x(k)={[x_1^T(k),x_2^T(k),\dots ,x_n^T(k)]}^T$, ${\overline{\eta }}_i(k)={\eta }_i(k)+\frac{\gamma }{k_I}{\varphi }_i$, and $\overline{\eta }(k)=[$${\left.{\eta }_1^T(k),{\eta }_2^T(k),\dots ,{\eta }_n^T(k)\right]}^T$, (9) can be re-expressed in matrix form

$$\left[\begin{array}{c}x(k+1)\\ \overline{\eta }(k+1)\end{array}\right]=\left[\begin{array}{cc}\alpha I_n-\frac{\beta (1-\alpha )+k_{I}T}{\gamma }ML& \frac{k_I(1-\alpha )}{\gamma }{I}_n\\ -TML& I_n\end{array}\right]\left[\begin{array}{c}x(k)\\ \overline{\eta }(k)\end{array}\right].$$

(10)

Now, we deduce the average consensus tracking conditions under strongly connected topology.

Theorem 1.

Consider a first-order system (1) whose topology satisfies Assumption 1. With${\displaystyle \sum { }_{i=1}^n}{\eta }_i\left(0\right)=0$, the system (1) can achieve the average consensus tracking objective if, and only if:

$${\stackrel{⌢}{\xi }}_{i,1}>0 \mathrm{and} {\stackrel{⌢}{\xi }}_{i,1}{\stackrel{⌢}{\sigma }}_{i,1}{\stackrel{⌢}{\sigma }}_{i,0}+{\stackrel{⌢}{\xi }}_{i,1}^2{\stackrel{⌢}{\xi }}_{i,0}-{\stackrel{⌢}{\sigma }}_{i,0}^2>0,$$

(11)

where${\stackrel{⌢}{\xi }}_{i,1}=\frac{2\gamma \mathrm{Re}(u_i)}{k_{I}T{\left|u_i\right|}^2}+\frac{2\beta }{k_{I}T}+\frac{2\alpha }{1-\alpha },{\stackrel{⌢}{\sigma }}_{i,1}=-\frac{2\gamma \mathrm{Im}(u_i)}{k_{I}T{\left|u_i\right|}^2},$

$${\stackrel{⌢}{\xi }}_{i,0}=\frac{(2+2\alpha )\gamma \mathrm{Re}(u_i)}{k_I(1-\alpha )T{\left|u_i\right|}^2}-\frac{2\beta }{k_{I}T}+\frac{\alpha +1}{1-\alpha },{\stackrel{⌢}{\sigma }}_{i,0}=\frac{(2+2\alpha )\gamma \mathrm{Im}(u)}{k_I(\alpha -1)T{\left|u_i\right|}^2}$$

and$u_i,i=2,\dots ,n$are the non-zero eigenvalues of$L$.

Proof.

Firstly, we analyze the consensus convergence of first-order multi-agent systems (10). □

Let ${\psi }_i(k)=x_i(k)-x_1(k),{\phi }_i(k)={\overline{\eta }}_i(k)-{\overline{\eta }}_1(k),i=2,\dots ,n$. Then, the consensus of system (12) is equivalent to $\left|{\psi }_i(k)\right|\to 0,\left|{\phi }_i(k)\right|\to 0,$, i.e., the consensus problem of the system can be transformed into the stability problem of the system.

With $\psi (k)={\left[{\psi }_2(k),\dots ,{\psi }_n(k)\right]}^T,\phi (k)={\left[{\phi }_2(k),\dots ,{\phi }_n(k)\right]}^T$, and $z(k)=\left[{\psi }^T(k)\right.,$${\left.{\phi }^T(k)\right]}^T$, we obtain

$$z(k+1)=\Delta z(k)$$

(12)

where

$$\Delta =\left[\begin{array}{cc}\alpha I_{n-1}-\frac{\beta (1-\alpha )+k_{I}T}{\gamma }\tilde{L}& \frac{k_I(1-\alpha )}{\gamma }{I}_{n-1}\\ -T\tilde{L}& I_{n-1}\end{array}\right]$$

$$\tilde{L}=\left[\begin{array}{ccc}\det \left(L_{22}\right)l_{22}-l_{12}& \dots & \det \left(L_{22}\right)l_{2n}-l_{1n}\\ \dots & \dots & \dots \\ \det \left(L_{nn}\right)l_{n2}-l_{12}& \dots & \det \left(L_{nn}\right)l_{nn}-l_{1n}\end{array}\right]$$

Perform basic column and row transformations on the Laplacian matrix as follows:

$$\left[\begin{array}{cccc}1& 0& \dots & 0\\ -1& 1& \dots & 0\\ ⋮& ⋮& ⋮& ⋮\\ -1& 0& \dots & 1\end{array}\right]ML\left[\begin{array}{cccc}1& 0& \dots & 0\\ 1& 1& \dots & 0\\ ⋮& ⋮& ⋮& ⋮\\ 1& 0& \dots & 1\end{array}\right]=\left[\begin{array}{cc}0& p^T\\ 0_n& \tilde{L}\end{array}\right]$$

where $0_n={\left[0,\dots ,0\right]}^T,p={\left[\det \left(L_{11}\right)l_{12},\det \left(L_{11}\right)l_{13},\dots ,\det \left(L_{11}\right)l_{1n}\right]}^T$. Eigenvalues of $\overline{L}$ are all the eigenvalues of $L$ except for 0.

Now, we analyze the roots of the following equation:

$$\det \left(\lambda I_{2n-2}-\Delta \right)=0$$

(13)

and it follows from Lemma 1 that (13) is equivalent to

$$\det \left\{\left(\lambda -\alpha \right)\left(\lambda -1\right)I_{n-1}+\left[\beta \left(\lambda -1\right)\left(1-\alpha \right)+k_{I}T\left(\lambda -1\right)+k_I\left(1-\alpha \right)T\right]\frac{\tilde{L}}{\gamma }\right\}=0$$

(14)

Equation (14) equals

$$\prod _{i=2}^n\left\{\left(\lambda -\alpha \right)\left(\lambda -1\right)I_{n-1}+\left[\beta \left(\lambda -1\right)\left(1-\alpha \right)+k_{I}T\left(\lambda -1\right)+k_I\left(1-\alpha \right)T\right]\frac{u_i}{\gamma }\right\}=0$$

(15)

where $u_i,i=2,\dots ,n$ are the non-zero eigenvalues of $L$, i.e., all the eigenvalues of $\overline{L}$.

We then study the roots $u_i,i=2,\dots ,n$ of the following equation:

$$\left(\lambda -\alpha \right)\left(\lambda -1\right)I_{n-1}+\left[\beta \left(\lambda -1\right)\left(1-\alpha \right)+k_{I}T\left(\lambda -1\right)+k_I\left(1-\alpha \right)T\right]\frac{u_i}{\gamma }=0$$

(16)

Taking $s=\frac{\lambda +1}{\lambda -1}$, then (16) can be transformed into

$$f\left(s\right)=s^2+\left({\stackrel{⌢}{\xi }}_{i,1}+i{\stackrel{⌢}{\sigma }}_{i,1}\right)s+{\stackrel{⌢}{\xi }}_{i,0}+i{\sigma }_{i,0}.$$

(17)

According to Lemma 2, (11) holds when the roots of Equation (17) lie on the left half of the complex plane. When agents converge asymptotically to consensus, then the topology satisfies Assumption 1. Second, we show that the average of constant reference inputs is equivalent to the final consensus value. Since the stationary consensus is reached asymptotically, when $k\to \infty ,x_1(k)=\dots =x_n(k)$, for $i=1,2,\dots ,n$

$$x_i(k)={\varphi }_i+\frac{k_I}{\gamma }{\eta }_i(k),$$

i.e.,

$$\left\{\begin{array}{c}{x}_i(k)={\varphi }_1+\frac{k_I}{\gamma }{\eta }_1(k)\\ x_i(k)={\varphi }_2+\frac{k_I}{\gamma }{\eta }_2(k)\\ \dots \\ x_i(k)={\varphi }_n+\frac{k_I}{\gamma }{\eta }_n(k)\end{array}\right.$$

Then, we have

$$x_i(k)=\frac{1}{n}({\varphi }_1+{\varphi }_2+\dots {\varphi }_n)+\frac{k_I}{n\gamma }({\eta }_1(k)+{\eta }_2(k)+\dots {\eta }_n(k)).$$

(18)

From the equation ${\eta }_i\left(k+1\right)={\eta }_i\left(k\right)+{\displaystyle {\sum }_{j\in N_i}{a}_{ij}}\left[x_j(t_k)-x_i(t_k)\right]$, we obtain ${\eta }_1\left(k\right)+$${\eta }_2\left(k\right)+\dots {\eta }_n\left(k\right)={\eta }_1\left(0\right)+{\eta }_2\left(0\right)+\dots {\eta }_n\left(0\right)$.

Noting that ${\displaystyle {\sum }_{i=1}^n{n}_i(0)}=0$, it follows from (18) that, when the agent asymptotically tracks the average consensus, i.e., $x_i(k)=\frac{1}{n}\left({\varphi }_1+{\varphi }_2+\dots {\varphi }_n\right)$, $k\to \infty$. Theorem 1 is proved.

Remark 1.

Based on Theorem 1, we can solve the average consensus tracking problems of multi-agent systems under digraphs no matter whether the digraphs are weight-balanced or not. It significantly improves the results of [25,26], in which the strict condition on weight balance is not needed any more.

4. Simulation Examples

In this section, a simulation example is provided to demonstrate the correctness of the theoretical results we developed. We consider a directed graph of a multi-agent system with eight agents, whose topology is a strongly connected network, to describe $G$ in Figure 1. The Laplacian matrix L is given as

$$L=\left[\begin{array}{cccccccc}2& -2& 0& 0& 0& 0& 0& 0\\ 0& 2& -2& 0& 0& 0& 0& 0\\ 0& 0& 2& -2& 0& 0& 0& 0\\ 0& 0& 0& 2& -2& 0& 0& 0\\ 0& 0& 0& -2& 4& -2& 0& 0\\ 0& 0& 0& 0& 0& 2& -2& 0\\ 0& 0& 0& 0& 0& 0& 2& -2\\ -2& 0& 0& 0& 0& 0& 0& 2\end{array}\right]$$

According to $L$, we compute the matrix $M$ as follows:

$$M=\left[\begin{array}{cccccccc}128& & & & & & & \\ & 128& & & & & & \\ & & 128& & & & & \\ & & & 256& & & & \\ & & & & 128& & & \\ & & & & & 128& & \\ & & & & & & 128& \\ & & & & & & & 128\end{array}\right]$$

Figure 1. Strongly connected directed graph $G$.

Figure 1. Strongly connected directed graph $G$.

With L and M, we can develop that the eigenvalues of $ML$ are $0, 0.3275 +$$0.7021j, 0.3275 - 0.7021j,$$1.8564 + 0.4826j, 1.8564 - 0.4826j, 1.1073 + 0.9456j$$, 1.1073 - 0.9456j, 3.4177$. The control parameters are chosen as $\gamma =1,k_p=1$. Without a loss of generality, the initial states of eight agents are selected as $x_i=$ ${\left[-10,20,30,40,-10,20,30,40\right]}^T$ and the initial conditions of ${\eta }_i(t)$ are given by ${\eta }_0=$ ${\left[\begin{array}{cccccccc}3& -4& 7& -6& 3& -4& 7& -6\end{array}\right]}^T$, which meet the criteria that ${\displaystyle {\sum }_{i=1}^n{\eta }_i}(0)=$$0$. The constant reference inputs are $\varphi ={\left[\begin{array}{cccccccc}10& 20& 40& 50& 10& 20& 40& 50\end{array}\right]}^T$. Based on condition (11), we can obtain that the average consensus tracking objective can be achieved by using the control protocol (5) if, and only if, the sampled period $T$ satisfies $T<0.28$(s).

The state evolutions of all nodes is plotted in Figure 2 and Figure 3. From Figure 2 and Figure 3, we can see that all agents can achieve the average consensus tracking if the sampled period $T$ = 0.25 holds, and the states of all agents are divergent if the sampled period $T$ = 0.3 holds, which satisfy the results of Theorem 1.

5. Conclusions

In this paper, we have investigated the average consensus tracking problems of multi-agent systems under arbitrary digraphs. We have proposed a distributed control protocol based on the sampled data. With this protocol being used, we have given the necessary and sufficient conditions for average consensus tracking. It is shown that the multi-agent system can achieve the average consensus tracking if, and only if, the sampled period satisfies the specified condition. We have introduced a simulation example to illustrate the effectiveness of our derived results. It is shown that the average consensus tracking objective can be achieved by using our proposed control protocol if the appropriate sampled period is selected.

In our future work, we aim to study the distributed average tracking issues of multi-agent systems under arbitrary digraphs based on the event-triggered method.