Consensus of second-order multi-agent systems via impulsive control using sampled hetero-information

doi:10.1016/j.automatica.2013.06.014

Automatica

Volume 49, Issue 9, September 2013, Pages 2881-2886

https://doi.org/10.1016/j.automatica.2013.06.014 Get rights and content

Abstract

In this paper, the consensus problem for second-order multi-agent systems using impulsive control with sampled hetero-information is investigated. Necessary and sufficient conditions for sampling interval to achieve consensus for the second-order multi-agent systems are obtained. Analysis for the upper bound of sampling interval, convergence performance, and communication cost of the proposed control protocols are also discussed. Some numerical examples are given to demonstrate the effectiveness of the proposed control protocols.

Introduction

During the past few years, there have been increasing research activities in the field of consensus analysis for networked multi-agent systems (MAS) due to its wide applications. The distributed consensus problem originated from particles swarm model (Vicsek, Czirok, B-Jacob, Cohen, & Schochet, 1995). Afterwards, many valuable results have been obtained with different particular features, such as transmission delay (Sun, Wang, & Xie, 2006), limited communication date rate (Li, Fu, Xie, & Zhang, 2011), noisy communication channel (Huang & Manton, 2009) and switching topology (Sun, 2011).

In the literature dealing with the consensus problem, agents are commonly considered to be governed by first-order dynamics (Sun, 2011, Wang and Xiao, 2010). Recently, the study for consensus of second-order agent systems has attracted more attention since the second-order dynamics are more realistic to describe real MAS. A variety of second-order consensus algorithms have been investigated and a number of results have been obtained (Gao and Wang, 2011, Gao et al., 2009, Guan et al., 2012, Hong et al., 2006, Ren, 2008, Yu et al., 2010).

The sampling operation is one of the indispensable steps to accomplish digital communication. Therefore, the effect of sampling period and sampling delay on MAS has been well studied in recent years (Gao et al., 2009, Hayakawa et al., 2006, Liu et al., 2012).

The hybrid impulsive control approach is an effective control strategy in synchronization of chaotic systems and complex networks (Guan et al., 2000, Guan et al., 2005, Liu and Wang, 2008, Yang, 2001). In our previous work in Guan et al. (2012), both the position and velocity information are sampled and transmitted at each sampling instant simultaneously. However, it would lead to high communication cost, such as bandwidth or transmission power. Actually, it might not be necessary for achieving consensus. Motivated by the above consideration, two impulsive control protocols are designed for consensus of second-order MAS using sampled hetero-information in this paper. The contributions of this paper can be summarized as follows. First, the necessary and sufficient conditions for sampling interval to achieve consensus are obtained for both protocols, and it is shown that both protocols result in a higher upper bound of allowable sampling interval under some conditions. Second, the asymptotical decay rate is provided to evaluate the convergence performance. Finally, the communication cost of the proposed protocol is given based on asymptotical decay rate.

Section snippets

Preliminaries and problems formulation

The following notations are used throughout this article. Denote $Z$ as the set of all positive integers and $R^{n}$ as the set of real vectors with dimension $n$ . Given a complex number $λ \in C, R e (λ), I m (λ), | λ |$ are the real part, the imaginary part and the modulus of $λ$ respectively, $i$ denotes the imaginary unit. In addition, $I_{N}$ is the identity matrix with order $N, 0_{N}$ denotes the square matrix with order $N$ with each of its element to be zero, $ρ (\cdot)$ represents the spectral radium of a matrix and $\otimes$ is the

Consensus analysis of MAS (3)

Define $f (γ_{i}, k) = {1 + {(1 - p_{2} γ_{i})}^{k}} / {1 - {(1 - p_{2} γ_{i})}^{k}}, i = 2, 3, \dots, N$ . Then we can get the following theorem.

Theorem 3.1

MAS (3) achieves consensus asymptotically if and only if the communication graph $G$ contains a directed spanning tree and the sampling interval $h < min_{γ_{i}} \frac{2 p_{2} α_{i}}{p_{1} (1 + β_{i}^{2})}, i = 2, 3, \dots, N,$ where $α_{i} = R e (f (γ_{i}, k)) > 0, β_{i} = I m (f (γ_{i}, k))$ .

Proof

The system (5) is an impulsive switching linear system with the switching period $k h$ . Denote $y (l) = {(\tilde{x} (l), \tilde{v} (l))}^{T}$ , then over the first period $k h, y (k) = B_{2} y (k - 1) = B_{2} B_{1} y (k - 2) = \dots = B_{2} B_{1}^{k - 1} y (0)$ . Define $S =$

Illustration examples

Example 1

Consider MAS (3) with directed communication graph shown in Fig. 1. The eigenvalues of the Laplacian matrix are $γ_{1} = 0, γ_{2} = γ_{3} = 1, γ_{4} = 3, γ_{5} = γ_{6} = 1.2929 \pm 0.7071 i, γ_{7} = γ_{8} = 2.7071 \pm 0.7071 i$ . Let $p_{1} = 1, p_{2} = 0.25$ , which will be used throughout the simulation. Given $k = 3$ , then from Theorem 3.1, it is found that consensus can be achieved asymptotically if and only if $h < 0.4481$ . Fig. 2 shows that the MAS can achieve consensus when $h = 0.38$ but cannot when $h = 0.48$ .

Example 2

In this example, the comparison between MAS (3) and MAS with

Conclusions

In this paper, the consensus problem of second-order MAS has been considered for directed and undirected topologies using sampled hetero-information. Based on the two proposed impulsive control protocols, some sufficient and necessary conditions for consensus have been obtained. Analysis for the upper bound of sampling interval, the convergence performance, and the communication cost are also given. The theoretical results are verified by some numerical examples.

Li Ding received the B.A.Sc. and Ph.D. degrees from Wuhan University of Science and Technology, Wuhan, China and Huazhong University of Science and Technology, Wuhan, China, in 2003 and 2010, respectively. He is currently an associate professor in the Department of Automation, School of Power and Mechanical Engineering, Wuhan University, Wuhan, China. His research interests include networked control systems, complex networks and multi-agent systems.

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Pian Yu was born in Wuhan, China, in 1990. She is currently an undergraduate from Department of Automation, School of Power and Mechanical Engineering, Wuhan University. Her research interests are focused on distributed and cooperative control in multi-agent systems.

Zhi-Wei Liu received the B.S. degree from Southwest Jiaotong University, Chengdu, China, 2004, and the Ph.D. degree from Huazhong University of Science and Technology, Wuhan, China in 2011. Currently, he is a postdoctoral scholar at Wuhan University, Wuhan, China. His research interests include complex networks and multi-agent systems.

Zhi-Hong Guan received Ph.D. degree in Automatic Control Theory and Its Application from the South China University of Technology, Guangzhou, China, in 1994. From 1994 to 1996, he was a Postdoctoral Fellow with the Faculty of Electrical Engineering and Communications, the South China University of Technology.

From 1979 to 1994, he was with the Jianghan Petroleum Institute, Jingzhou, China, where he was Lecturer, Associate Professor, and then Full Professor of Mathematics and Automatic Control. Since December 1997, he has been Full Professor of the Department of Control Science and Engineering, Executive Associate Director of the Centre for Nonlinear and Complex Systems and Director of the Control and Information Technology in the Huazhong University of Science and Technology (HUST), Wuhan, China. Since January 2011, he has been a Huazhong Leading Professor at HUST.

Since 1999, he has held visiting positions at the Harvard University, USA, the Central Queensland University, Australia, the Loughborough University, UK, the National University of Singapore, the University of Hong Kong, and the City University of Hong Kong. Currently, he is an Associate Editor of the Journal of Control Theory and Applications, the International Journal of Nonlinear Systems and Application, the Journal of Applied Mathematics, and severs as a Member of the Committee of Control Theory of the Chinese Association of Automation, Executive Committee Member of the Hubei Province Association of Automation. His research interests include complex systems and complex networks, impulsive and hybrid control systems, networked control systems, multi-agent systems and robot networks.

Gang Feng received the B.Eng and M.Eng. degrees in Automatic Control from Nanjing Aeronautical Institute, China in 1982 and in 1984 respectively, and the Ph.D. degree in Electrical Engineering from the University of Melbourne, Australia in 1992.

He has been with City University of Hong Kong since 2000 where he is at present a Chair Professor. He is a ChangJiang Chair professor at Nanjing University of Science and Technology, awarded by Ministry of Education, China. He was lecturer/senior lecturer at School of Electrical Engineering, University of New South Wales, Australia, 1992–1999. He was awarded an Alexander von Humboldt Fellowship in 1997–1998, and the IEEE Transactions on Fuzzy Systems Outstanding Paper Award in 2007. His current research interests include networked control systems, robot networks, and intelligent systems & control.

Prof. Feng is an IEEE Fellow and has been an associate editor of IEEE Trans. Automatic Control, IEEE Trans. on Fuzzy Systems, Mechatronics, IEEE Trans. on Systems, Man & Cybernetics, Part C, Journal of Control Theory and Applications, and the Conference Editorial Board of IEEE Control System Society.

^☆: This work was partially supported by the National Natural Science Foundation of China under Grants 61170024, 61272114, 61073026, 61272069, and the Hong Kong Research Grant Council under GRF grant with grant number CityU 113209. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Wei Ren under the direction of Editor Frank Allgöwer.

¹: Tel.: +86 27 68773985; fax: +86 27 68773985.

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Brief paperConsensus of second-order multi-agent systems via impulsive control using sampled hetero-information☆

Abstract

Introduction

Section snippets

Preliminaries and problems formulation

Consensus analysis of MAS (3)

Illustration examples

Conclusions

Automatica

Automatica

Automatica

Automatica

Automatica

Sampled-data based consensus of continuous-time multi-agent systems with time-varying topology

IEEE Transactions on Automatic Control

Consensus of multi-agent systems based on sampled-data control

International Journal of Control

On impulsive control of a periodically forced chaotic pendulum system

IEEE Transactions on Automatic Control

On hybrid impulsive and switching systems and application to nonlinear control

IEEE Transactions on Automatic Control

Coordination and consensus of networked agents with noisy measurements: stochastic algorithms and asymptotic behavior

SIAM Journal on Control and Optimization

Brief paper
Consensus of second-order multi-agent systems via impulsive control using sampled hetero-information☆