Elsevier

Neurocomputing

Volume 397, 15 July 2020, Pages 244-252
Neurocomputing

Distributed time-varying group formation control for generic linear systems with observer-based protocols

https://doi.org/10.1016/j.neucom.2020.01.065Get rights and content

Abstract

In this paper, the time-varying group formation control for linear multi-agent systems under directed communication topology is investigated from an observer viewpoint. Different from the existing works on the time-varying group formation, the groups herein could have a cyclic partition, which is more common in real applications than the topology with the acyclic groups. The leaderless time-varying group formation problem is studied first. An observer-based distributed protocol is presented for each agent, where the observer is used to estimate the unmeasurable state utilizing the output information. Then, to broaden the scope of applications, we further study the leader-following case, in which there exists a leader with nonzero and bounded input for each subgroup. To tackle this problem, we take the input of the leader as a disturbance, and develop the new forms of control protocols with nonlinear functions. Furthermore, for both cases, it is shown that under the formation feasible conditions, the desired time-varying group formation can be achieved if the strong enough intra-group coupling is selected and the corresponding digraph of each subgraph contains a directed spanning tree. Finally, two simulation examples are given.

Introduction

Recently, there has been a surge tremendous of interest in studying the problems related to cooperative control of multi-agent systems (MASs), due to the fact that compared with a single agent, multiple agents could accomplish more complex tasks. Various subjects on cooperative control of MASs, such as formation and consensus/synchronization, emerge [1], [2], [3], [4]. Among them, formation control is a fundamental and critical issue, which tries to drive the states of all agents to keep certain expected shape. As is well-known, formation control of MASs has wide potential applications, e.g., target enclosing [5], surveillance [6], and navigation [7]. In theoretical research of the formation control, one of the challenges arises from controlling the systems utilizing only the partial and relative information.

Over the last decades, three well-known formation control approaches have been proposed, namely leader-following based approach, virtual structure based approach, and behavior based approach [8]. These methods only apply to the problem considering fixed formations. However, in the real-world applications, in order to adapt to the dynamical changing environment, the formation of MASs is required to be changing. Therefore, there is a strong motivation to investigate the time-varying formation control problem, where the desired formation can be changed in the premise of guarantee the system stability. As compared to the fixed formation control of MASs, time-varying formation could cover broader areas and has a better ability of obstacle avoidance. The research results on time-varying formation control can be found in [9], [10], [11]. In [9], a dencentralized controller-observer approach for time-varying centroid and formation control of multi-robot systems is proposed. In [10], the authors study time-varying formation control for collaborative heterogeneous MASs, including unmanned grounded vehicles and unmanned aerial vehicles.

It is noteworthy to highlight that in the aforementioned works, there is only a single group of agents with the purpose of achieving a specified formation. In real applications, there are many circumstances where the agents in the MASs are required to be divided into several subgroups to achieve a few different distributed tasks, e.g., cooperative searching and obstacle avoidance for multiple areas and multi-target enclosing [12], [13]. The group formation problems arise in such scenarios, where the agents in the same subgroup try to reach one formation, and several different formations may exist. To the best of our knowledge, few existing works are concerned with the time-varying group formation with the exception in [13], [14]. The sufficient conditions and feasible constraints required to accomplish multiple time-varying sub-formations are provided in [13], [14] for second-order MASs and general linear MASs, respectively. In these two works, the groups must have an acyclic partition, which means that there exist no paths from any subgroup to itself. Obviously, such a condition is a little strict and restricts the applications in the real world. In addition to the strict topology condition, the control protocols proposed in [13], [14] are dependent on the full information of the state variables, which are difficult to access directly in practice. A widely used approach to overcome this difficulty is to estimate the unknown state information by designing the distributed observers by using the measured system outputs [15], [16], [17].

Inspired by the above discussions, this paper focuses on designing the observer-based distributed protocols to achieve the desired time-varying group formation control for the generic linear systems under a directed communication topology. Compared with the works on a single group of agents [9], [10], [11], the problem under consideration is more complicated, due to the existence of the couplings among different subgroups. Furthermore, since the desired formation is time-varying, the derivative of the formation information should be brought into both the design and analysis of the control protocols. The contributions of this paper are concluded as follows.

  • 1.

    Both leaderless and leader-following group formation control problem are studied. In both cases, the local state observer is designed for each agent by using the measurable input and output information of its own. Moreover, considering the time-varying property of the desired formation, an extra term with respect to the derivative of the formation information is introduced into the designed distributed control protocol. In the leader-following case, in order to deal with the non-zero and bounded input of the leader, we introduce a nonlinear function in the control protocol. Rigorous analyses are provided for both cases to show that the desired time-varying group formation can be realized based on the sufficient conditions with respect to the topology connectivity, the intra-group coupling strengths, and the formation feasible condition.

  • 2.

    Different from the works [13], [14] on time-varying group formation under a directed communication graph with groups having an acyclic partition, our results are presented based on a weaker connectivity condition where the groups could have a cyclic partition. Under this circumstance, the corresponding Laplacian matrix cannot be written as the lower-triangular Frobenius normal structure in the light of the partition of groups such that the approaches proposed in [13], [14] are not available. We overcome this difficulty by partially borrowing ideas from [18], [19], which develop the framework for time-varying group/cluster consensus problems.

The rest of the paper is structured below. Section 2 provides a brief summary of some basic concepts and formulates the problem to consider. In Section 3, two main results for the leaderless and leader-following time-varying group formation problems are presented. Simulation examples are provided in Section 4. Section 5 gives a brief conclusion.

Notations: R+ refers to the set of all positive real numbers. Notation ⊗ denotes the Kronecker product. ‖x‖ denotes the Euclidean norm of a vector x and ‖X‖ is the induced 2-norm of a square matrix X. Denote by 1(0) the all-one (all-zero) column vector of compatible dimensions. In represents the identity matrix with n dimensions. For a symmetric matrix M, we use M > 0 (M ≥ 0) to represent a positive definite (semi-positive definite) matrix. The maximum and minimum eigenvalues of M, are denoted, respectively, by λmax(M) and λmin(M). Let diag{E1,,Ep} represent the block diagonal matrix, where the ith diagonal block is the square matrix Ei. For any vector αRm, diag{α} refers to a diagonal matrix, where the ith diagonal element is the ith component of α.

Section snippets

Graph notation

We use a weighted directed graph G to conveniently describe the communication topology among agents. Let G={V,E,A} be a weighted digraph of order N with a finite nonempty set of nodes V={1,2,,N}, a set of edges EV×V, and a weighted adjacency matrix A=[aij]RN×N. The edge (j,i)E means that there exists a directed edge from agent j to agent i but not vice versa. aij denotes the weight of edge (j, i). Note that aij ≠ 0 if (j,i)E, and aij=0 otherwise. We assume that there exist no self-loops

Main results

This section aims to develop the distributed formation protocols and provide the sufficient conditions to deal with the problem given in Definition 1. Two different cases, leaderless time-varying group formation control and leader-following one with leaders of bounded nonzero inputs, are shown in order.

Simulation results

To verify the effectiveness of the proposed results, we aim to present two numerical examples in this section.

Example 1

Consider two groups of agents V1={1,2,3} and V2={4,5,6,7} with the communication topology showing in Fig. 1(a). Choose the system parameters asA=[0220],B=[0110],C=[10]T.Set the initial states for seven agents as x1(0)=[2,1.5]T, x2(0)=[0.5,3.5]T, x3(0)=[1,2.5]T, x4(0)=[1,0.5]T, x5(0)=[0,3]T, x6(0)=[0,1.5]T, x7(0)=[2.5,1]T. The desired formation is given byhi(t)={[sin(t2(i1)π/3),μ1cos(t

Conclusions

The time-varying group formation for leaderless multi-agent systems with linear dynamics has been investigated first. Then, a special case where the leader with non-zero and bounded input is pinned to each group has been studied. For both cases, the observer-based distributed control protocols have been designed by using the measurable output information rather than the state information. It has been shown that the desired formation can be achieved under certain sufficient conditions with

CRediT authorship contribution statement

Man Li: Methodology, Writing - original draft, Writing - review & editing. Qichao Ma: Writing - review & editing. Chongjian Zhou: Conceptualization, Writing - original draft. Jiahu Qin: Funding acquisition, Writing - review & editing. Yu Kang: Software.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was supported in part by the National Natural Science Foundation of China under Grant 61922076, Grant 61725304, Grant 61873252, and Grant 61673361 and in part by the Fok Ying-Tong Education Foundation for Young Teachers in Higher Education Institutions of China under Grant 161059.

Man Li received the B.S. degree in electrical engineering and automation from the Jiangnan University, Wuxi, China, in 2016. She is currently pursuing the Ph.D. degree in control science and engineering at the University of Science and Technology of China, Hefei, China. Her current research interests include consensus problems and optimal control of multiagent systems.

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    Man Li received the B.S. degree in electrical engineering and automation from the Jiangnan University, Wuxi, China, in 2016. She is currently pursuing the Ph.D. degree in control science and engineering at the University of Science and Technology of China, Hefei, China. Her current research interests include consensus problems and optimal control of multiagent systems.

    Qichao Ma received the B.S. degree in electronic information science and technology from the China University of Mining and Technology, Xuzhou, China, in 2013, and the Ph.D. degree in control science and engineering from the University of Science and Technology of China (USTC), Hefei, China, in 2019. He is currently a postdoctoral researcher in control science and engineering at USTC. His current research interests include multiagent systems and complex networks.

    Chongjian Zhou received both the Bachelor’s Degree and the Master’s Degree in control science and engineering from the University of Science and Technology of China, Hefei, China. His research interests include consensus in multiagent systems and synchronization of complex dynamical networks.

    Jiahu Qin received the first Ph.D. degree in control science and engineering from the Harbin Institute of Technology, Harbin, China, in 2012, and the second Ph.D. degree in systems and control from Australian National University, Canberra, Australia, in 2014. He is currently a Professor with the Department of Automation, University of Science and Technology of China, Hefei, China. His current research interests include consensus and coordination in multi-agent systems, as well as complex network theory and its application.

    Yu Kang received the Dr. Eng. degree in control theory and control engineering from the University of Science and Technology of China, Hefei, China, in 2005. From 2005 to 2007, he was a Post-Doctoral Fellow with the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China. He is currently a Professor with the Department of Automation, University of Science and Technology of China. His current research interests include adaptive/robust control, variable structure control, mobile manipulator, and Markovian jump systems.

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