Elsevier

Neurocomputing

Volume 138, 22 August 2014, Pages 86-91
Neurocomputing

Distributed output feedback consensus of discrete-time multi-agent systems

https://doi.org/10.1016/j.neucom.2014.02.043Get rights and content

Abstract

This paper investigates the problem of distributed output feedback consensus for multi-agent systems. Both fixed topology and stochastic switching topology are considered. It is assumed that each agent updates its state according to the output information of itself and its neighbors. We obtain some necessary and sufficient conditions of consensusability based on algebraic graph theory and Markov jump linear system theory. Algorithms are given to derive the allowable control gains. Simulation results are presented to show the effectiveness of the theoretical results.

Introduction

Recently, a lot of research effort has been put into the study of multi-agent systems, especially the distributed consensus problem. Applications of the distributed consensus include unmanned air vehicles, formation control of mobile robots, distributed computation and estimation, wireless sensor network [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. The main idea of consensus problem is to make a group of agents converge to a common value by designing some protocols or algorithms. In [1], two consensus protocols for multi-agent systems with or without time-delays were introduced and some necessary and sufficient or sufficient conditions of consensus were obtained. In [5], the authors studied some distributed consensus problems and their applications systematically, where the results are mainly based on algebraic graph theory and matrix analysis. The containment control problem for linear multi-agent systems was addressed in [8], where all the internal agents converge to the convex hull spanned by some boundary agents. While in [9], the authors studied the output consensus problem for heterogeneous uncertain linear multi-agent systems based on the output regulation theory. More recent progress in distributed multi-agent coordination can be found in [10].

In the aforementioned work on distributed consensus of multi-agent systems, the interaction topologies are fixed or switching in a deterministic framework. However, the stochastic phenomena are frequently appear in the practice due to the disturbances. Some stochastic models, such as Markov chain, have been used to describe the practical systems [11], [12]. For multi-agent systems, the stochastic model, especially the Markov chain, is often used to describe the interaction topologies among the agents. It is assumed that there are finite topologies among which the interaction topology at each moment switches stochastically with some probabilities [13], [14], [15]. In [13], the authors studied the static stabilization problem for a decentralized discrete-time single-integrator network with Markovian switching topologies. In [14], the mean square consensusability problem for a network of double-integrator agents with Markovian switching topologies was studied. While in [15], the authors studied the distributed discrete-time coordinated tracking problem for multi-agent systems with Markovian switching topologies where the transition probabilities are equal.

In the early literature, the researchers analyzed the consensus problem are mainly based on algebraic graph theory and matrix theory. When the structure of topology becomes complicated or the interaction topologies are stochastic switching, it is difficult to analyze the conditions of consensus from the perspective of the topology structure directly. The linear transformation is employed to overcome this difficult, which transfers the consensusability problem of multi-agent system to the stability problem of linear systems [16], [17], [18]. In [16], the authors studied the consensusability of multi-agent systems with input delays by the model transformation, where the methods to compute the maximum input delay were given. In [17], a sufficient condition of consensus for multi-agent system with random delays governed by a Markov chain was obtained, where the results given are in terms of a set of linear matrix inequalities. While in [18], some necessary and sufficient conditions for consensusability of linear multi-agent systems were obtained, where the consensus protocol is based on the outputs of the agents. However, the interaction topology in [18] is fixed. It is necessary to extend the results in [18] to the case of stochastic switching topology.

In this paper, we deal with the distributed consensus problem for discrete-time multi-agent systems. Both fixed topology and stochastic switching topologies are considered. The outputs are used by the agents to update their state information. Some necessary and sufficient conditions of consensusability are obtained based on linear system theory, algebraic graph theory and matrix theory. Algorithms are given to design the allowable control gains.

Notation: Let R and N represent, respectively, the real number set and the nonnegative integer set. Denote the spectral radius of the matrix M by ρ(M). Suppose that A, BRp×p. Let AB (respectively, AB) denote that AB is symmetric positive semi-definite (respectively, symmetric positive definite). Denote the determinant of the matrix A by |A|. For a scalar α, |α| represents the modulus of α (α is a complex number) or the absolute value of α (α is a real number). “” represents the Kronecker product of matrices. In denotes the n×n identity matrix. R(·) and I(·) represent the real part and imaginary part of a complex number, respectively. Let 1n and 0m×n denote, respectively, the n×1 column vector with all components equal to 1 and m×n zero matrix.

Section snippets

Problem formulation and preliminaries

We introduce the graph theory notions similar to [19], firstly. Let G=(V,E,A) be a directed graph of order n, where V and E denote, respectively, the node set and the edge set. An edge (i,j)E if agent j can obtain the information from agent i. We say agent i is a neighbor of agent j. A=[aij]Rn×n is the adjacency matrix associated with G, where aij>0 if (i,j)E, otherwise, aij=0. The (nonsymmetrical) Laplacian matrix L associated with A and hence G is defined as L=[lij]Rn×n, where lii=j=1,ji

Consensusability analysis

The output feedback consensusability problem for multi-agent systems was studied in [18] where the interaction topology is fixed. Both continuous-time case and discrete-time case were considered. However, the authors did not give the consensusability conditions for discrete-time case. Here, we first give a consensusability condition for one-dimension multi-agent system with fixed topology, that is, the system (4) only has one mode.

Simulation results

In this section, we give two examples to show the effectiveness of the presented results. In the following examples, for simplicity, we let aijθ[k]=1 (or aij=1) if (i,j)Eθ[k] (or E), otherwise, aijθ[k]=0 (or aij=0).

Example 1

For fixed topology case, the interaction topology is as shown in Fig. 1, which includes four nodes. It can be seen that the graph G1 has a directed spanning tree. Set A=1.1(>1), B=0.7, C=1.3. By solving the minimization problem in Algorithm 1 and using the Matlab LMI toolbox, we

Conclusion

We have studied the output feedback consensus problem for multi-agent systems with both fixed and stochastic switching topologies. Based on algebraic graph theory and Markov jump system theory, some necessary and sufficient conditions of consensusability have been obtained. Simulation examples have been given to show the usefulness of the presented results. Furthermore, the method of how to design the control gain has been given. The transition probabilities in this paper are assumed to be

Acknowledgment

The work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2010-0009373), in part by the National Natural Science Foundation of China under Grant nos. 61203056, 61104007, 61374061, 61273123.

Huanyu Zhao received his B.S. degree in mathematics and applied mathematics from Hubei University in 2005, and Ph.D. degree in control science and engineering from Nanjing University of Science and Technology in 2011. From May 2010 to November 2010, he was an exchange Ph.D. student supported by Nanjing University of Science and Technology with Department of Electrical and Computer Engineering, Utah State University. Since 2011, he has been with Faculty of Electronic and Electrical Engineering,

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    Huanyu Zhao received his B.S. degree in mathematics and applied mathematics from Hubei University in 2005, and Ph.D. degree in control science and engineering from Nanjing University of Science and Technology in 2011. From May 2010 to November 2010, he was an exchange Ph.D. student supported by Nanjing University of Science and Technology with Department of Electrical and Computer Engineering, Utah State University. Since 2011, he has been with Faculty of Electronic and Electrical Engineering, Huaiyin Institute of Technology. He is currently doing postdoctoral research with Department of Electrical Engineering, Yeungnam University, Republic of Korea. His research interests include Markov switching systems, cooperative control of multi-agent systems, and game theory.

    Ju H. Park received his Ph.D. degree in electronics and electrical engineering from POSTECH, Pohang, Republic of Korea, in 1997. From May 1997 to February 2000, he was a Research Associate in ERC-ARC, POSTECH. In March 2000, he joined Yeungnam University, Kyongsan, Republic of Korea, where he is currently the Chuma Chair Professor. From December 2006 to December 2007, he was a Visiting Professor in the Department of Mechanical Engineering, Georgia Institute of Technology. Prof. Park׳s research interests include robust control and filtering, neural networks, complex networks, multi-agent systems, and chaotic systems. He has published a number of papers in these areas. Prof. Park severs as an Editor of International Journal of Control, Automation and Systems. He is also an Associate Editor/Editorial Board member for several international journals, including IET Control Theory and Applications, Applied Mathematics and Computation, Journal of the Franklin Institute, Journal of Applied Mathematics and Computing, etc.

    Yulin Zhang received his B.S. degree in automatic control from Nanjing University of Science and Technology in 1992 and M.S. degree in power electronics and power transmission from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2002, respectively, and the Ph.D. degree in light industry information technology and engineering from Jiangnan University in 2009. He has been with Huaiyin Institute of Technology, Huai׳an, China, where he is a Professor with Faculty of Electronic and Electrical Engineering. He has published more than 30 international journal and conference papers. His current research interests include data fusion, intelligent control.

    Hao Shen received his B.S. degree in communication engineering from Anhui University of Technology in 2006, and the Ph.D. degree in control theory and control engineering from Nanjing University of Science and Technology in 2011. He is currently with the School of Electrical Engineering and Information, Anhui University of Technology, China. He has been doing postdoctoral research with Department of Electrical Engineering, Yeungnam University, Republic of Korea, since March 2013. His current research interests include stochastic hybrid systems, network control systems, robust control and filtering, nonlinear systems.

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