Sampled-data based average consensus of second-order integral multi-agent systems: Switching topologies and communication noises

doi:10.1016/j.automatica.2013.02.004

Automatica

Volume 49, Issue 5, May 2013, Pages 1458-1464

https://doi.org/10.1016/j.automatica.2013.02.004 Get rights and content

Abstract

A distributed sampled-data based protocol is proposed for the average consensus of second-order integral multi-agent systems under switching topologies and communication noises. Under the proposed protocol, it is proved that sufficient conditions for ensuring mean square average consensus are: the consensus gain satisfies the stochastic approximation type condition and the communication topology graph at each sampling instant is a balanced graph with a spanning tree. Moreover, if the consensus gain takes some particular forms, the proposed protocol can solve the almost sure average consensus problem as well. Compared with the previous work, the distinguished features of this paper lie in that: (1) a sampled-data based stochastic approximation type protocol is proposed for the consensus of second-order integral multi-agent systems; (2) both communication noises and switching topologies are simultaneously considered; and (3) average consensus can be reached not only in the mean square sense but also in the almost sure sense.

Introduction

Recently, distributed consensus of multi-agent systems has attracted considerable attention from the control community. Generally speaking, this problem can be studied from two aspects: the agent’s dynamics and the communication network among agents. From the viewpoint of agent’s dynamics, recent work includes the first-order integral dynamics (Olfati-Saber & Murray, 2004), second-order integral dynamics (Ren, 2008), linear time-invariant dynamics (Cheng, Hou, Lin, Tan, & Zhang, 2011), and nonlinear and uncertain dynamics (Cheng et al., 2010, Hou et al., 2009), to name a few. Studies on communication network can be further divided into the communication topology and communication constraint. Communication topology describes how the agent connects to each other, which includes the undirected/directed topology in terms of the direction of information flow and the fixed/switching topology in terms of time course. Communication constraint represents various occasions occurring in realistic communication environments, such as time delays, packet loss and communication noises.

This paper mainly focuses on the communication noise because it inevitably occurs in reality. In the literature, there are some published papers concerning this problem. Among them, the stochastic approximation type consensus protocol has shown a great potential to counteract the effect of noises. For example, Huang and Manton (2009) studied the consensus of discrete-time first-order integral multi-agent systems with communication noises under a fixed undirected topology. They introduced the concepts of mean square and almost sure consensus, and showed that a stochastic approximation type gain was able to reduce the detrimental effect of noises. Extensions to the switching topology were made in their follow-up work (Huang, Dey, Nair, & Manton, 2010). Li and Zhang (2010) proved that the stochastic approximation type gain was not only sufficient but also necessary for ensuring mean square consensus of the discrete-time first-order integral multi-agent systems. Liu, Xie, and Zhang (2011) proposed a mean square/almost sure consensus protocol based on the stochastic approximation type gain for the discrete-time first-order integral multi-agent system with the co-existence of communication noises and time delays. It is noted that most recent work concerning communication noises studies the first-order integral multi-agent system. However, many mechanical systems are modeled by the second-order integral dynamics after using feedback linearization. In the authors’ previous work (Cheng, Hou, Tan, & Wang, 2011), necessary and sufficient conditions were given for the mean square consensus of continuous-time second-order integral multi-agent systems under fixed topology and communication noises. However, some interesting topics such as the switching topology and almost sure consensus are still left open, which is the motivation of the study presented in this paper.

Because of the wide use of computers in modern control engineering, a sampled-data based average consensus protocol is proposed for second-order integral multi-agent systems under switching topologies and communication noises. This protocol is composed of two parts: the agent’s own state feedback and the relative states between the agent and its neighbor agents. Due to the existence of communication noises, relative states cannot be obtained accurately, which requires a stochastic approximation type consensus gain to counteract the noise effect. To the best of the authors’ knowledge, there is no publication considering the stochastic approximation type protocol for consensus of second-order integral multi-agent systems based on the sampled information. It is proved that sufficient conditions for ensuring the mean square average consensus are the consensus gain satisfies the stochastic approximation condition and the topology graph at each sampling instant is a balanced graph with a spanning tree. Moreover, if the consensus gain has a specific form, the almost sure average consensus can also be reached. This paper is not a trivial extension to the previous work (Cheng, Hou, Tan et al., 2011, Huang et al., 2010, Huang and Manton, 2009) because: (1) the main results in Cheng, Hou, Tan et al. (2011) heavily rely on the explicit state transition matrix of the closed-loop dynamics of multi-agent systems. In this paper, due to the consideration of switching topology, the closed-loop dynamics is a time-varying difference equation which results in the difficulty in obtaining the state transition matrix explicitly; and (2) the agent studied in Huang et al. (2010) and Huang and Manton (2009) is described by the first-order integral dynamics. Then the state matrix in the closed-loop dynamics is just the Laplacian matrix of the topology graph and the corresponding convergence analysis is mainly based on the property of the graph’s Laplacian matrix. However, this case does not happen in second-order integral multi-agent systems.

The following notations will be used throughout this paper: $1_{n} = (1, 1, \dots, 1) \in R^{n}; 0_{n} = (0, 0, \dots, 0) \in R^{n}; I_{n}$ denotes the $n \times n$ dimensional identity matrix; $Θ$ denotes the zero matrix with appropriate sizes ( $Θ_{n}$ denotes the $n \times n$ dimensional zero matrix); $\otimes$ denotes the Kronecker operator; $Z^{+}$ denotes the set of positive integers; and $N$ denotes the set of natural numbers. For a given matrix $X, X^{T}$ denotes its transpose; ${‖ X ‖}_{2}$ denotes its 2-norm and ${‖ X ‖}_{F}$ denotes its Frobenius norm. diag $(\cdot)$ denotes a block diagonal matrix formed by its inputs. For the random variable $x, E (x)$ denotes its mathematical expectation; $D (x)$ denotes its variance. For two random variables $x_{1}$ and $x_{2}, C o v (x_{1}, x_{2})$ denotes the covariances of $x_{1}$ and $x_{2}$ .

Section snippets

Problem formulation and preliminaries

Firstly, the communication topology of multi-agent systems is usually modeled by the weighted digraph $G = (V_{G}, E_{G}, A_{G})$ where $V_{G} = {v_{1}, \dots, v_{N}}$ denotes the node set, $E_{G} \subseteq V_{G} \times V_{G}$ denotes the edge set, and $A_{G} = [α_{i j}] \in R^{N \times N}$ denotes the weighted adjacency matrix. Node $v_{i}$ represents agent $i$ . A directed edge $e_{i j} = (v_{i}, v_{j})$ means that there is a communication channel from agent $j$ to agent $i$ . Let $N_{i} = {v_{j} \in V_{G} | e_{i j} \in E_{G}}$ denote the set of neighbors of node $v_{i}$ . It is assumed that there is no self-loop in $G$ , i.e., $e_{i i} \notin E_{G} i = 1, \dots, N$ . $α$

Sampled-data based consensus protocol

Due to the wide use of computer-based control, only the sampled data at each sampling instant is available for designing the agent’s controller. That is the $i$ th agent’s controller $u_{i} (t)$ can only use $y_{x i j} (k T)$ and $y_{v i j} (k T)$ where $T$ is the sampling interval and $k T \leq t < (k + 1) T (k \in N, i = 1, \dots, N, j \in N_{i} (k T))$ . For any signal $f (\cdot)$ , its value at the $k$ th sampling instant $f (k T)$ is denoted by $f [k]$ for simplicity. By adopting the zero-order holder strategy $u_{i} (t) = u_{i} [k] (k T \leq t < (k + 1) T)$ , the agent’s behavior at sampling

Main results

Substituting (5) into (4) obtains $\bar{Z} [k + 1] = (I_{N} \otimes (\bar{A} + \bar{B} K_{1}) + a [k] L_{G} [k] \otimes \bar{B} K_{2}) \bar{Z} [k] - a [k] (Σ [k] \otimes (\bar{B} K_{2} T_{z}^{- 1})) N [k],$ where $\bar{Z} [k] = {({\bar{z}}_{1}^{T} [k], \dots, {\bar{z}}_{N}^{T} [k])}^{T} \in R^{2 N}, L_{G} [k]$ is the Laplacian matrix of graph $G [k], N [k] = {(n_{x 11} [k], n_{v 11} [k], \dots, n_{x 1 N} [k], n_{v 1 N} [k], \dots \dots, n_{x N N} [k], n_{v N N} [k])}^{T} \in R^{2 N^{2}}$ and $Σ [k] = diag ({\tilde{α}}_{1} [k], {\tilde{α}}_{2} [k], \dots, {\tilde{α}}_{N} [k]) \in R^{N \times N^{2}}, {\tilde{α}}_{i} [k] = (α_{i 1} [k], α_{i 2} [k], \dots, α_{i N} [k]) \in R^{1 \times N}$ .

To study the dynamical behavior of (6), the following conditions on communication topology, communication noise and consensus gain are given.

(C1)
Topology condition: $G [k]$

Conclusion and future work

This paper studies the stochastic average consensus of second-order integral multi-agent systems from two aspects. From the communication network aspect, the communication noise is considered and communication topology is assumed to be time-varying. From the aspect of agent’s dynamics, only the sampled-data based information is used to design a consensus protocol. Some sufficient conditions have been obtained to ensure the mean square/almost sure average consensus.

In the future, more effort is

Long Cheng received the B.S. degree (with honors) in control engineering from Nankai University, Tianjin, China, in July 2004 and the Ph.D. degree (with honors) in control theory and control engineering from the Institute of Automation, Chinese Academy of Sciences, Beijing, China, in July 2009. Currently, he is an Associate Professor at the Laboratory of Complex Systems and Intelligent Science, Institute of Automation, Chinese Academy of Sciences. His current research interests include

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Yunpeng Wang received the B.S. degree in information and computing science from Shandong University, China, in July 2010. He is currently working towards the Ph.D. degree in control theory and control engineering at the State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences. He is also with the University of Chinese Academy of Sciences, Beijing. His current research interests include multi-agent systems, human–robot interaction, and brain–computer interface.

Zeng-Guang Hou received the B.E. and M.E. degrees in electrical engineering from Yanshan University (formerly North-East Heavy Machinery Institute), Qinhuangdao, China, in 1991 and 1993, respectively, and the Ph.D. degree in electrical engineering from the Beijing Institute of Technology, Beijing, China, in 1997. He is now a Professor in the Laboratory of Complex Systems and Intelligent Science, Institute of Automation, Chinese Academy of Sciences. His current research interests include neural networks, optimization algorithms, robotics, and intelligent control systems.

Min Tan received the B.S. degree in control engineering from Tsinghua University, Beijing, China, in 1986 and the Ph.D. degree in control theory and control engineering from the Institute of Automation, Chinese Academy of Sciences, Beijing, in 1990. He is a Professor in the Laboratory of Complex Systems and Intelligent Science, Institute of Automation, Chinese Academy of Sciences. His research interests include advanced robot control, multirobot systems, biomimetic robots, and manufacturing systems.

Zhiqiang Cao received the B.S. and M.S. degrees from Shandong University of Technology, China, in 1996 and 1999, respectively. In 2002, he received the Ph.D. degree in control theory and control engineering from the Institute of Automation, Chinese Academy of Sciences, Beijing, China. He is currently an associate professor in the State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences. His research interests include multirobot coordination, intelligent control and biomimetic robots.

^☆: This work has been partially supported by the National Natural Science Foundation of China (Grants 61004099, 61273326, 61225017, 61273352, and 61105105), the Beijing Natural Science Foundation (Grant 4123105), the Beijing Nova Program (Grant Z121101002512066), and the start-up fund for the recipient of Presidential Award of Chinese Academy of Sciences, and the Early Career Development Award of SKLMCCS. The material in this paper was partially presented at the 10th World Congress on Intelligent Control and Automation (WCICA 2012), July 6–8 2012, Beijing, China. This paper was recommended for publication in revised form by Associate Editor Hideaki Ishii under the direction of Editor Ian R. Petersen.

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Brief paperSampled-data based average consensus of second-order integral multi-agent systems: Switching topologies and communication noises☆

Abstract

Introduction

Section snippets

Problem formulation and preliminaries

Sampled-data based consensus protocol

Main results

Conclusion and future work

Automatica

Automatica

Automatica

Automatica

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International Journal of Control

Neural-network-based adaptive leader-following control for multi-agent systems with uncertainties

IEEE Transactions on Neural Networks

Necessary and sufficient conditions for consensus of double-integrator multi-agent systems with measurement noises

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Brief paper
Sampled-data based average consensus of second-order integral multi-agent systems: Switching topologies and communication noises☆