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Time-varying output formation control for high-order linear

time-invariant swarm systems

Xiwang Dong

a,b

, Zongying Shi

, Geng Lu

, Yisheng Zhong

⇑

School of Automation Science and Electronic Engineering, Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing 100191, PR China

Department of Automation, TNlist, Tsinghua University, Beijing 100084, PR China

article info

Article history:

Received 4 December 2013

Received in revised form 21 November 2014

Accepted 30 November 2014

Available online 6 December 2014

Keywords:

Time-varying formation

Output formation

Formation feasibility

High-order

Swarm system

abstract

Formation control of swarm systems has gained considerable attention from scientiﬁc

communities due to its potential applications in various areas. In practical applications,

the dynamics of each agent may be of high order and only the outputs of all agents are

required to achieve time-varying formations. Therefore, this paper focuses on time-varying

output formation control problems for high-order linear time-invariant swarm systems

with directed interaction topologies. A general output formation protocol is proposed

based on the relative outputs of neighboring agents. Necessary and sufﬁcient conditions

for swarm systems to achieve time-varying output formations are presented using a con-

sensus based approach. An explicit expression of the output formation reference function is

given. For a swarm system, whether or not a desired output formation is feasible is a crucial

problem. Based on partial stability theory, necessary and sufﬁcient conditions for output

formation feasibility are derived. Approaches to expand the feasible time-varying output

formation set and an algorithm to design the protocol for swarm systems to achieve

time-varying output formation are presented respectively. Finally, theoretical results are

demonstrated by numerical simulations.

1. Introduction

The past decades have witnessed the increasing research interest in cooperative control of swarm systems. It has been

revealed that cooperative control of swarm systems has broad applications in various areas, such as ﬂocking [25,30], consen-

sus [12,24] and formation control [8,10]. Although individual agent can be used to accomplish various tasks, great beneﬁts

including high efﬁciency, low cost and easy maintenance, just to name a few, can be achieved by having a group of agents

work in formation. To achieve formation control, many approaches have been proposed in robotics community, such as lea-

der–follower [4], virtual structure [17] and behavior [1] based approaches. However, Beard et al. [2] pointed out that leader–

follower, behavior and virtual structure based formation control approaches have their own weaknesses. For example, lea-

der–follower approaches lack robustness due to the existence of the leader, and behavior approaches are difﬁcult to analyze

mathematically, etc.

Recently, consensus problems for linear time-invariant (LTI) swarm systems have been investigated extensively, and lots of

results have been obtained for ﬁrst-order, second-order and high-order LTI swarm systems [5,13,14,18–20,22,26,28,32,33].

http://dx.doi.org/10.1016/j.ins.2014.11.047

This work was supported by the National Natural Science Foundation of China and the Fundamental Research Funds for the Central Universities under

Grants 61374034, 61210012, 61174067, 61333011 and YWF-14-RSC-101.

⇑

Corresponding author. Tel.: +86 10 62772573.

E-mail address: zys-dau@mail.tsinghua.edu.cn (Y. Zhong).

Information Sciences 298 (2015) 36–52

Contents lists available at ScienceDirect

Information Sciences

journal homepage: www.elsevier.com/locate/ins

Consensus requires that the states/outputs of agents reach an agreement. With the development of consensus theory, more and

more researchers realized that consensus approaches can be used to deal with formation control problems. Ren [29] applied a

consensus protocol to formation control of second-order swarm systems and showed that leader–follower, behavior and vir-

tual structure based approaches in the literature can be uniﬁed in the general framework of consensus based approach, and the

weaknesses of the these approaches can be overcome. Therefore it is signiﬁcant to study the formation control problem for

swarm systems using the consensus based approach.

Based on the consensus strategy, Chen and Tian [3] proposed a formation controller for second-order swarm systems

using the backstepping approach. Finite-time consensus protocols were applied to study the formation control problems

of ﬁrst-order and second-order swarm systems in [35,7]. Xie and Wang [36] transformed formation control problems for sec-

ond-order swarm systems with undirected interaction topologies into consensus problems, and revealed that if the interac-

tion topologies are connected then the desired formations can be achieved. Using consensus based approaches, Lafferriere

et al. [16] studied formation control problems for swarm systems with a special high-order LTI model which can be regarded

as a series of second-order models. Lin et al. [21] addressed formation feasibility problems for ﬁrst-order swarm systems

with undirected interaction topologies. Formation problems for general high-order LTI swarm systems were considered

by Ma and Zhang [23]. Dong et al. [6] proposed necessary and sufﬁcient conditions for high-order LTI swarm systems with

time delays to achieve time-varying state formations.

The authors in [3,6,7,16,21,23,29,35,36] only discussed state formation problems. In many practical applications, each

agent can only obtain output information of its neighbors rather than state information, and only the outputs of all agents

are required to achieve predeﬁned formations. Output formation stability problems for general high-order LTI swarm sys-

tems were investigated by Fax and Murray [8]. Based on the work of [8], Porﬁri et al. [27] studied output formation and

tracking stability problems for general high-order LTI swarm systems. However, both [8,27] only addressed output formation

stability problems and did not consider the output formation feasibility problems. For a swarm system, whether or not a

given output formation is feasible is a crucial problem. In addition, the formations in [8,16,23,27] are assumed to be

time-invariant, which means that those results cannot be used to deal with time-varying formation problems. To the best

of our knowledge, time-varying output formation and formation feasibility problems for general high-order LTI swarm sys-

tems are still open.

In the current paper, time-varying output formation analysis and design problems, and output formation feasibility prob-

lems for general high-order LTI swarm systems with directed interaction topologies are studied respectively. A consensus

based output formation protocol is presented ﬁrstly. Then output formation problems are transformed into output consensus

problems. Necessary and sufﬁcient conditions for high-order LTI swarm systems to achieve time-varying output formation

are proposed and an explicit expression of the time-varying output formation reference function is given. Moreover, neces-

sary and sufﬁcient conditions for output formation feasibility are derived based on partial stability theory, where the feasible

time-varying output formation set can be expanded. Finally, an algorithm to design the protocol for swarm systems to

achieve time-varying output formation is proposed.

Compared with the existing work on formation control, the novel features of the current paper are threefold. Firstly, both

output formation analysis and feasibility problems for general high-order LTI swarm systems with directed interaction

topologies are investigated. In [3,7,21,29,35,36], the dynamics of each agent is of low order. Refs. [8,16,23,27] addressed for-

mation problems for high-order swarm systems, but they did not consider output formation feasibility problems. Secondly,

the output formation can be time-varying while the formations in [8,16,23,27] are time-invariant, and a description of the

feasible time-varying output formation set is also shown in the current paper. Thirdly, an explicit expression of the time-

varying output formation reference function is given. An algorithm to determine the gain matrices in the protocol for swarm

systems to achieve time-varying output formation is also presented. However, protocol design problems were not dealt with

in [8,27].

The rest of this paper is organized as follows. In Section 2, basic concepts and useful results on graph theory and partial

stability theory are introduced respectively, and the problem to be investigated is formulated. In Section 3, criteria for swarm

systems to achieve time-varying output formation are derived and an explicit expression of the time-varying output forma-

tion reference function is given. In Section 4, conditions for output formation feasibility and an algorithm to design the pro-

tocol are presented respectively. Numerical simulations are shown in Section 5. Finally, Section 6 concludes the whole work.

Throughout this paper, for simplicity of notation, let 0 be a zero matrix of appropriate size with zero vector and zero num-

ber as special cases. 1 denotes a column vector of appropriate size with 1 as its elements. I stands for an identity matrix with

appropriate dimension, and  represents Kronecker product. Given any real matrix R and any k 2 C, let

¼ diagfR; Rg and

ReðkÞI ImðkÞI

ImðkÞI ReðkÞI



2. Preliminaries and problem description

In this section, ﬁrstly, basic concepts and results on graph theory and partial stability theory are introduced respectively.

Then the problem description is given.

X. Dong et al. / Information Sciences 298 (2015) 36–52

2.1. Basic concepts and results on graph theory

A directed graph of order N can be denoted by G ¼fS; E; Wg, where S ¼fs

; s

; ...; s

g is the set of nodes,

E # fðs

; s

Þ : i – j; s

; s

2 Sg is the set of edges and W ¼½w

2R

NN

is the adjacency matrix with nonnegative weights w

An edge of G is represented by e

¼ðs

; s

Þ. The set of neighbors of node s

is denoted by N

¼ s

2 S : ðs

; s

Þ2E



. The in-degree

of node s

is deﬁned as deg

ðs

Þ¼

j¼1

. The degree matrix of G is denoted by D ¼ diag deg

ðs

Þ; i ¼ 1; 2; ...; N

. w

> 0if

and only if e

2 E and w

¼ 0 for all i 2 1; 2; ...; N

. The Laplacian matrix of G can be written as L ¼ D  W.Adirected path

from node s

to s

is a sequence of ordered edges such that for k ¼ 1; 2; ...; l  1; ðs

; s

kþ1

Þ2E. A directed graph is said to have

a spanning tree if there exists at least one node having a directed path to all the other nodes. More details on graph theory can

be found in [9]. The following lemma is useful in analyzing time-varying output formation problems of swarm systems.

Lemma 1 [28]. Let L 2 R

NN

be the Laplacian matrix of a directed graph G, then

(i) L has at least one zero eigenvalue, and 1 is the associated eigenvector; that is, L1 ¼ 0;

(ii) If G has a spanning tree, then 0 is a simple eigenvalue of L, and all the other N  1 eigenvalues have positive real parts.

2.2. Basic concepts and results on partial stability theory

Consider the following system

xðtÞ¼

xðtÞ;

yðtÞ¼

CxðtÞ;

(

ð1Þ

where

xðtÞ

ðtÞ;



ðtÞ



and

A ¼

;

C ¼ I 0

½:

Deﬁnition 1. System (1) is said to be asymptotically stable with respect to

yðtÞ if for any bounded initial state

xð0Þ; lim

t!1

yðtÞ¼0.

Based on the observability of system (1), the following result about partial stability can be obtained, which will be used to

prove the main results concerning the time-varying output formation.

Lemma 2 [33]. If ð

;

Þ is completely observable, then system (1) is asymptotically stable with respect to

yðtÞ if and only if

Ais

Hurwitz.

2.3. Problem description

Consider a swarm system consisting of N agents with an interaction topology represented by the directed graph G. For

i; j 2f1; 2; ...; Ng, agent i can be regarded as the node s

in G. The interaction from agent i to agent j and the corresponding

interaction strength can be denoted by the edge e

and weight w

respectively. It is assumed that G has a spanning tree. The

dynamics of agent i ði 2f1; 2; ...; NgÞ is described by

ðtÞ¼Ax

ðtÞþBu

ðtÞ;

ðtÞ¼Cx

ðtÞ;



ð2Þ

where x

ðtÞ2R

; u

ðtÞ2R

; y

ðtÞ2R

denote the state, control input and output of agent i respectively. It is assumed that

rankðCÞ¼q. It should be pointed out that the assumption that C is of full row rank means that the elements in the output

ðtÞ are independent of each other. If C is not of full row rank, then some of the elements in y

ðtÞ can be expressed as linear

combination of the others, hence involves no new information. In this case, those elements in y

ðtÞ which are linearly depen-

dent on others can be ignored.

A time-varying output formation is speciﬁed by a vector hðtÞ¼ h

ðtÞ; h

ðtÞ; ...; h

ðtÞ

2 R

Deﬁnition 2. Swarm system (2) is said to achieve time-varying output formation hðtÞ if for any given bounded initial states,

there exists a vector-valued function rðtÞ2R

such that

lim

t!1

ðy

ðtÞh

ðtÞrðtÞÞ ¼ 0 ði ¼ 1; 2; ...; NÞ;

where rðtÞ is called an output formation reference function.

38 X. Dong et al. / Information Sciences 298 (2015) 36–52

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