Average consensus in networks of dynamic agents with uncertain topologies and time-varying delays
Introduction
The study of information flow and interaction among multiple dynamic agents plays an important role in understanding the coordinated movements of these agents. A critical problem for coordinated control is to design appropriate protocols and algorithms such that the group of agents can reach consensus on the shared information in the presence of limited and unreliable information exchange, dynamically changing interaction topologies as well as communication delays.
Consensus problems have a long history in the field of computer science, particularly in automata theory and distributed computation [1]. In many applications involving multi-agent/multi-robot systems, such as unmanned vehicles [2], robot manipulators [3], [4], sensor networks [5], [6], and satellite clusters so on [7], groups of dynamic agents need to agree upon certain quantities of interest. Such quantities might or might not be related to the motion of the individual agents. As a result, it is important to address agreement problems in their general form for networks of dynamic agents with directed information flow under link failure and creation as well as communication delays.
In recent years, many researchers have studied consensus problems from various perspectives (see [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18] and references therein). Particularly, consensus problems are addressed under a variety of assumptions on the network topology (fixed or switching), presence or lack of communication delays, and directed or undirected network information flow in [14]. Consensus problems under dynamically changing interaction topologies was studied in [15].
In general, unmodelled delay effects in a feedback mechanism may destabilize an otherwise stable system, which has been well documented in the literature. In multi-agent systems, time-varying delays may arise naturally due to the congestion of the communication channels, the asymmetry of interactions, and the finite transmission speed. What's more, the system uncertainties may exist in many cases. To the best of our knowledge, little has been known about consensus problems when communication is affected by both time-varying delays and uncertainties.
The objective of this paper is to study average consensus problems in directed networks with uncertain topologies as well as time-varying delays. Due to the existence of uncertainties and time-varying delays, the methods in [14], [16] cannot be applied. In this paper, we shall employ a linear matrix inequality (LMI) method to deal with this problem. We will prove that the group of dynamic agents can reach average consensus asymptotically for suitable time-varying delays and uncertainties if the network topology is strongly connected and balanced. The main results are given in terms of simple and feasible linear matrix inequalities (LMIs), from which the allowable upper bounds of time-varying delays and uncertainties can be easily obtained by using Matlab's LMI Tool Box. Finally, numerical examples are worked out to illustrate the theoretical results.
This paper is organized as follows. Section 2 contains the problem formulation, Section 3 presents the main results. Three numerical examples are presented in Section 4. The conclusion is given in Section 5.
Throughout this paper, is a finite index set. The notation ⁎ represents the elements below the main diagonal of a symmetric matrix. AT means the transpose of the matrix A. In is an n×n identity matrix. We say if is positive definite, where X and Y are symmetric matrices of same dimensions. refers to the Euclidean norm for vectors.
Section snippets
Problem statement
Let be a weighted diagraph (directed graph) of order n () , where is the set of nodes, is the set of edges, and is a weighted adjacency matrix. A directed edge of G is denoted by . . Moreover, we assume for all . The set of neighbors of node vi is denoted by . The number of elements in the set Ni is called in-degree of node vi. Similarly, the number of elements in the set is called
Analysis of average consensus
In this section, we will consider the average consensus problem for systems (2a), (2b), (2c), respectively. It is proved that under appropriate conditions the system asymptotically achieves average consensus for allowable upper bound h of the time-varying delay and appropriate uncertainties. Some feasible LMIs are also established to determine the allowable upper bounds of delays and uncertainties that guarantees the average consensus of the system.
Numerical examples
Example 1 Consider a digraph G1 with 0–1 weights shown in Fig. 1. For the given , using Matlab's LMI Toolbox to solve Eq. (6) yields the allowable values of h for different values of d in the following table. Empty Cell d=0 d=0.1 d=0.5 d=0.9 d unknown h 0.4178 0.4090 0.3926 0.3474 0.3684
Conclusion
This paper has considered the average consensus problems in networks of dynamic agents with uncertain topologies as well as time-varying communication delays. By introducing an LMI method, we proved that all the nodes in the network can reach average consensus asymptotically for an allowable upper bound of communication delays and appropriate uncertainties if the network topology is strongly connected and balanced. Particularly, some feasible linear matrix inequalities have been established to
Acknowledgment
The authors thank the reviewers for their valuable suggestions and comments on this paper. This paper was supported by The National Natural Science Foundations of China (60704039 and 61174217) and The Natural Science Foundations of Shandong Province (ZR2010AL002 and JQ201119).
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