Formation tracking of the second-order multi-agent systems using position-only information via impulsive control with input delays
Introduction
In recent years, the multi-agent system (MAS) has drawn significant attention owing to its various applications in many areas, such as sensor networks [1], multiple mobile robots [2], unmanned air vehicles [3]. The research topics of MAS include consensus [4], [5], [6], rendezvous [7], [8], flocking [9], [10], formation [11], [12], [13], [14] and so on, among which formation control has been a hot topic [15], [16]. It has appealed to more and more researchers due to its broad range of applications in autonomous unmanned vehicles, navigation of spacecrafts and the cooperative control of mobile robot systems. Formation control problem in the presence of a leader which is called formation tracking problem [17] can be seen as an extension of the general formation control problem.
Over the past decades, the formation tracking problems have been studied by many researchers. In the existing literatures, many different control methods have been utilized including the sliding mode control [18], [19], adaptive control [20], [21], [22], model predictive control [23], [24], feedback control [25] and so on. The decentralized sliding mode estimators were introduced to achieve decentralized formation tracking of multiple autonomous vehicles and multiple robot system in [18], [19], respectively. The adaptive control laws were proposed to solve the formation tracking problem of electrically driven multiple mobile robots and multiple fully actuated surface vessels in [20], [21], respectively. A distributed adaptive control approach based on backstepping technique was proposed to achieve the consensus tracking and the formation control of nonholonomic mobile robots in [22]. The model predictive control was adopted to study the formation control problem for mobile robots in active target tracking problem in [23]. The model predictive control was proposed to solve the two sub-problems: trajectory control and formation control problems in [24]. The output-feedback cooperative control was applied to the formation tracking problem of unicycle-type mobile robots in [25].
The control methods mentioned above have the common feature of requiring continuous control. However, utilizing continuous control may result in much energy consumption, large amounts of communication bandwidth and short communication lifetime. Unlike the continuous control method, the impulsive control is only exerted at impulsive instants and has advantages over the above continuous control in some aspects, such as smaller control efforts, less energy consumption, less occupation of communication bandwidth and simpler implementation.
To the best of the authors’ knowledge, there are no results concerning with the formation tracking of the second-order MASs by utilizing the impulsive control with or without input delays. Therefore, the formation tracking of second-order MAS with and without input delay via impulsive control are studied in this paper. Both the directed graph and undirected graph are considered. Considering that the velocity states of agents are usually difficult to be measured in practice and the position states of agents can be obtained more easily, instead, a novel impulsive algorithm with position-only information is proposed to achieve the formation tracking for the second-order MAS. The necessary and sufficient conditions are derived for the formation tracking of the MAS with and without input delay, respectively. These results give the guidelines to design the impulsive formation tracking algorithm for the second-order MAS. The numerical examples are given to illustrate our theoretical results.
The rest of the paper is organized as follows. Some preliminaries of algebraic graph theory and matrix theory, as well as the model description and the definition of formation tracking are given in Section 2. The convergence analysis of the proposed impulsive algorithms with and without input delay under both the directed and undirected graph are formulated in Section 3. The illustrative numerical examples are given in Section 4. Some conclusions are finally drawn in Section 5.
Notations: The notations used in this paper are fairly standard. and denote the sets of real numbers, complex numbers and positive integers, respectively. and denote the n-dimensional Euclidean space and the space of -dimensional matrices with real entries, respectively. For and are the real part and the imaginary of , respectively. The superscript ‘T’ stands for the transpose of a matrix. The notation refers to the vector norm. The notation denotes the Kronecker product of matrices. is a column vector with all its elements equal to one, is a column vector with all its elements equal to zero, is a N-dimensional identity matrix, refers to a diagonal matrix, or a block diagonal matrix. denotes the spectral radius of square matrix P.
Section snippets
Preliminaries and problem formulation
In this section, some preliminaries of algebraic graph theory and matrix theory, as well as the model descriptions and the definition of formation tracking of the system are given.
To begin with, some basic concepts and lemmas on the algebraic graph theory and the matrix theory are briefly introduced.
A directed graph is described with a set of nodes , a set of edges and the weighted adjacency matrix . Node i represents the agent, an ordered pair
Impulsive algorithm without input delay
In this subsection, the formation tracking problem under the proposed impulsive algorithm with both directed and undirected graph are considered.
Since the output of the system are the position states, while the velocity states are unavailable, only the position information is used to design the control input , to be more specifically, only the relative positions to its neighbors are utilized to obtain the control input for each agent.
The impulsive formation tracking algorithm is designed as
Numerical simulations
In this section, some numerical examples are given to show the effectiveness of our theoretical results. Consider a second-order MAS consisting of one leader and five followers.
Assume that G is a directed graph. The Laplacian matrix L of the digraph G is given by . Let , i.e., the follower 1 can obtain the information of the leader. Thus, , and the matrix . It is easy to see that the graph
Conclusions
In this paper, we have investigated the formation tracking problem of the second-order MAS, where each agent can be modeled as an identical oscillator. The impulsive formation tracking algorithms with the position-only information have been proposed to achieve the formation tracking for second-order MAS for the first time. In addition, only the velocity is regulated by the control force. Moreover, the input delay has been considered in this paper. Based on the stability theory of impulsive
Acknowledgements
This work is supported by the National Natural Science Foundation of China under Grants 61374171 and 61074124, and the Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT1245.
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