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www.ietdl.org

Published in IET Control Theory and Applications

Received on 11th November 2013

Revised on 16th March 2014

Accepted on 18th March 2014

doi: 10.1049/iet-cta.2013.1007

Special Issue on Recent Developments in

Networked Control and Estimation

ISSN 1751-8644

Time-varying formation control for high-order linear

swarm systems with switching interaction topologies

Xiwang Dong, Zongying Shi, Geng Lu,Yisheng Zhong

Department of Automation,TNlist,Tsinghua University, Beijing 100084, People’s Republic of China

E-mail: szy@mail.tsinghua.edu.cn

Abstract: Time-varying formation control problems for high-order linear time-invariant swarm systems with switching inter-

action topologies are investigated. A general formation control protocol is proposed ﬁrstly. Then using a consensus based

approach, necessary and sufﬁcient conditions for swarm systems with switching interaction topologies to achieve a given

time-varying formation are presented. An explicit expression of the time-varying formation reference function is given. It is

revealed that the switching interaction topologies have no effect on the formation reference function and the motion modes

of the formation reference can be speciﬁed. Furthermore, necessary and sufﬁcient conditions for formation feasibility are

presented. An approach to expand the feasible formation set is given and an algorithm to design the protocol for swarm

systems with switching interaction topologies to achieve time-varying formations is provided. Finally, numerical simulations

are presented to demonstrate theoretical results.

1 Introduction

Recently, formation control of swarm systems has attracted

considerable attention because of its broad potential appli-

cations in civilian and military areas such as load trans-

portation [1], radiation detection and contour mapping

[2], target search and localisation [3], reconnaissance [4],

surveillance [5], telecommunication relay [6] and so on.

In the past decades, several formation control approaches

have been proposed in the robotics community, such as

behaviour [7], leader-follower [8] and virtual structure [9]

based approaches and so on. However, Beard et al.[10]

pointed out that behaviour, leader-follower and virtual struc-

ture based formation control approaches have their own

weaknesses. For example, behaviour based approaches are

difﬁcult to be analysed mathematically, and leader-follower

approaches lack of robustness because of the existence of

the leader, just to name a few.

In recent years, consensus for linear time-invariant (LTI)

swarm systems have been studied extensively (see, e.g.

[11–18] and references therein). With the development of

consensus theory, more and more researchers ﬁnd that con-

sensus approaches can be used to deal with formation control

problems. Using consensus based approaches, Ren [19] dis-

cussed formation control problems for second-order swarm

systems, and revealed that behaviour, leader-follower and

virtual structure based approaches can be treated as special

cases of consensus based approaches and the weaknesses of

the these approaches can be overcome. In [20], a consensus

based formation control strategy was applied to a multi-robot

swarm system. Xiao and Wang [21] investigated ﬁnite-time

formation control problems for ﬁrst-order swarm systems

based on consensus approaches. Sufﬁcient conditions for

second-order swarm systems with undirected interaction

topologies to achieve formations were presented in [22].

Consensus based formation control problems for second-

order swarm systems with time delays were addressed in

[23]. Chen et al.[24] discussed formation control prob-

lems for ﬁrst-order and second swarm systems with bounded

input, disturbance and time delays.

In practical applications, many swarm systems are of high

order, so formation control problems for high-order swarm

systems make more sense. Based on consensus approaches,

Lafferriere et al.[25] proposed a necessary and sufﬁcient

condition for swarm systems with a special high-order LTI

model, which can be regarded as a series of second-order

models, to achieve formations. Fax and Murray [26] dis-

cussed formation stability problems for general high-order

LTI swarm systems. Formation stability problems for gen-

eral high-order LTI swarm systems with ﬁxed and periodic

switching undirected interaction topologies were studied in

Porﬁri et al.[27]. However, both Fax and Murray [26] and

Porﬁri et al.[27] only considered formation stability prob-

lems and did not consider the formation feasibility problems.

For a swarm system, whether or not a given formation is

feasible is a crucial problem. For general high-order LTI

swarm systems, Ma and Zhang [28] proposed a necessary

and sufﬁcient condition for formation feasibility. However,

the formation considered in [28] is time-invariant and the

feasible formation set is very limited. Moreover, the inter-

action topologies in [28] are assumed to be ﬁxed. In practical

applications, the interaction topologies may be switching

because of the existing of interaction channel failures and

creations among agents. When interaction topologies are

switching, both the analysis and design for cooperative

control of swarm systems become much complicated and

2162 IET Control Theory Appl., 2014, Vol. 8, Iss. 18, pp. 2162–2170

www.ietdl.org

challenging than the ﬁxed case [29]. To the best of our

knowledge, time-varying formation analysis and feasibility

problems for general high-order LTI swarm systems with

switching interaction topologies have not been investigated.

In this paper, time-varying formation control problems

for general high-order LTI swarm systems with switching

interaction topologies are dealt with. Firstly, a consensus

based formation protocol is presented. Then the formation

problems are transformed into consensus problems. Nec-

essary and sufﬁcient conditions for high-order LTI swarm

systems with switching interaction topologies to achieve

time-varying formation are proposed. An explicit expression

of the time-varying formation reference function is given,

and it is shown that switching interaction topologies have

no effect on the formation reference function and the motion

modes of the formation reference can be assigned. More-

over, necessary and sufﬁcient conditions for time-varying

formation feasibility are presented, where the feasible for-

mation set can be expanded. Finally, an algorithm to design

the protocol for swarm systems with switching interaction

topologies to achieve time-varying formation is proposed.

Compared with the existing works on formation control,

the novel features of the current paper are threefold. Firstly,

both formation analysis and feasibility problems for general

high-order LTI swarm systems with switching interaction

topologies are discussed. In [19–27], formation feasibility

problems were not considered. The interaction topologies in

[28] were ﬁxed. In [19–24], the dynamics of each agent is

restricted to be low-order. Secondly, the formation can be

time-varying whereas the formations in [25–28] are time-

invariant, and a description of the feasible formation set

is also shown. Thirdly, an explicit expression of the time-

varying formation reference function is given, and it is

shown that the motion modes of the formation reference

can be speciﬁed. Moreover, an algorithm to determine the

gain matrices in the protocol is presented. However, protocol

design problems were not dealt with in [26, 27].

The rest of this paper is organised as follows. In Section 2,

basic concepts and useful results on graph theory are intro-

duced and the problem to be investigated is formulated. In

Section 3, necessary and sufﬁcient conditions to achieve

time-varying formation are presented. In Section 4, neces-

sary and sufﬁcient conditions for formation feasibility are

proposed and an algorithm to design the protocol is given.

Numerical simulations are shown in Section 5. Finally,

Section 6 concludes the whole work.

Throughout this paper, for simplicity of notation, let 0

denote zero matrices of appropriate size with zero vectors

and zero number as special cases, and 1

be a column vector

of size N with 1 as its elements. Let I

represent an identity

matrix with dimension N , and ⊗ denote Kronecker product.

2 Preliminaries and problem description

In this section, basic concepts and results on graph theory

are introduced and the problem description is presented.

2.1 Basic concepts and results on graph theory

An undirected graph G consists of a node set Q =

{

, q

, ..., q

}

, an edge set E ⊆



, q

) : q

, q

∈ Q, i = j



and a symmetric adjacency matrix W =[w

]∈R

N ×N

with

non-negative elements w

.Anedge of G is denoted by

= (q

, q

). The adjacency elements associated with the

edges of G are positive, i.e. w

> 0 if and only if q

∈ E.

Moreover, w

= 0 for all i ∈

{

1, 2, ..., N

}

. The set of neigh-

bours of node q

is denoted by N



∈ Q : (q

, q

) ∈ E



The in-degree of node q

is deﬁned as deg

) =



j=1

. The degree matrix of G is denoted by D =

diag



deg

), i = 1, 2, ..., N



. The Laplacian matrix of G

is deﬁned as L = D − W . An undirected graph is said to

be connected if there is a path from each node to every

other nodes. More details on graph theory can be found in

[30]. The following lemma is useful in analysing formation

problems of swarm systems.

Lemma 1 [30]: Let L ∈ R

N ×N

be the Laplacian matrix of an

undirected graph G, then

(i) L has at least one zero eigenvalue, and 1

is the

associated eigenvector; that is, L1

= 0; and

(ii) If G is connected, then 0 is a simple eigenvalue of L,

and all the other N − 1 eigenvalues are real and positive.

2.2 Problem description

Consider a swarm system with N agents. Suppose that each

agent has the LTI dynamics described by

˙x

(t) = Ax

(t) + Bu

(t) (1)

where i = 1, 2, ..., N , x

(t) ∈ R

is the state, u

(t) ∈ R

the control input. The interaction topology of the swarm

system can be described by an undirected graph G, and each

agent can be treated as a node in G. For i, j ∈

{

1, 2, ..., N

}

the interaction channel from agent i to agent j is denoted

by the edge q

, and the corresponding interaction strength

is denoted by w

Assumption 1: B is of full column rank.

A time-varying formation is speciﬁed by a vector h(t) =

(t), h

(t), ..., h

(t)]

∈ R

with h

(t)(i = 1, 2, ..., N )

piecewise continuously differentiable.

Deﬁnition 1: Swarm system (1) is said to achieve

time-varying formation h(t) if there exists a vector-valued

function r(t) ∈ R

such that

lim

t→∞

(t) − h

(t) − r(t)) = 0 (i = 1, 2, ..., N )

where r(t) is called a formation reference function.

Remark 1: The formation deﬁnition speciﬁed by vectors has

been used a lot in previous works on formation control such

as [19, 20, 22, 24, 25, 27] and so on. Deﬁnition 1 presents a

general framework and it can be veriﬁed that the deﬁnitions

in [19, 20, 22, 24, 25, 27] can be treated as special cases of

Deﬁnition 1.

Deﬁnition 2: If there exist control inputs u

(t)(i = 1,

2, ..., N ) such that swarm system (1) achieves time-varying

formation h(t), then the formation h(t) is feasible for swarm

system (1).

Deﬁnition 3: Swarm system (1) is said to achieve consensus

if there exists a vector-valued function c(t) ∈ R

such that

lim

t→∞

(t) − c(t)) = 0 (i = 1, 2, ..., N )

where c(t) is called a consensus function.

IET Control Theory Appl., 2014, Vol. 8, Iss. 18, pp. 2162–2170 2163

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