Research Article Formation Control for Unmanned Aerial...

Research ArticleFormation Control for Unmanned Aerial Vehicles withDirected and Switching Topologies

Yahui Qi, Shaolei Zhou, Yuhang Kang, and Shi Yan

Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai 264000, China

Correspondence should be addressed to Yahui Qi; [email protected]

Received 15 April 2016; Revised 2 August 2016; Accepted 15 August 2016

Academic Editor: Mahmut Reyhanoglu

Copyright © 2016 Yahui Qi et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Formation control problems for unmanned aerial vehicle (UAV) swarm systems with directed and switching topologies areinvestigated. A general formation control protocol is proposed firstly. Then, by variable transformation, the formation problemis transformed into a consensus problem, which can be solved by a novel matrix decomposition method. Sufficient conditionsto achieve formation with directed and switching topologies are provided and an explicit expression of the formation referencefunction is given. Furthermore, an algorithm to design the gainmatrices of the protocol is presented. Finally, numerical simulationsare provided to illustrate the effectiveness of the theoretical results.

1. Introduction

In the past decades, unmanned aerial vehicles (UAVs) havebeen widely used in civilian and military areas, such assurveillance and reconnaissance [1, 2] and target search andlocalization [3]. Since the performance of a team of UAVsworking cooperatively exceeds the performance of individualUAVs, formation control of UAVs is of importance and hasreceived a lot of attention.

The formation control of UAVs has been studied withmany different methods, such as leader–follower [4], behav-ior [5], and virtual structure-based [6] approaches. Recently,with the development of consensus theory [7–15], somerelated methods are also used to deal with the formationcontrol problems of UAVs. Consensus means that all agentsreach a common state.The results in [16] show that consensusapproaches can be used to deal with formation control prob-lems, and leader–follower, behavior, and virtual structure-based formation control approaches are special cases ofconsensus-based approaches.

Based on consensus method, Abdessameud and Tayebi[17] proposed controllers for UAV swarm systems to achieveformation in the presence of communication delays. A con-sensus protocol together with an output feedback lineariza-tionmethod is presented in [18] such that theUAVswarm sys-tems can achieve partially time-varying formation. Besides,

indoor and outdoor flight experiments for quadrotor swarmsystems to achieve formation by consensus approaches arecarried out in [19] and [20], respectively. Based on consensustheory, we know that the achievement of formation dependson not only the individual UAV dynamics but also the struc-ture of the networks between UAVs which can be modeledby directed and undirected graphs. However, the interactiontopologies between UAVs in [19, 20] are assumed to be fixed.In practical applications, the interaction topologies of UAVswarm systemsmay be switching due to the fact that the com-munication channel may fail and new channels may be cre-ated during flight. Time-varying formation control for UAVswarm systems and high-order LTI systems with switchinginteraction topologies are studied by Dong [20, 21], but thetopologies are assumed to be undirected. To the best of ourknowledge, there is still work to do on formation control ofUAV swarm systems with directed and switching topologies.

In this paper, we aim to solve the formation problem ofUAV swarm systems with directed and switching topologies.Compared with the existing results, the assumptions of thecommunication topology are quite general.The remainder ofthis paper is organized as follows. In Section 2, some neces-sary concepts and useful results on graph theory are summa-rized and the problem formulation is given. Main theoreticalresults are proposed in Section 3. In Section 4, a numericalsimulation is presented. Section 5 is the conclusion.

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2016, Article ID 7657452, 8 pageshttp://dx.doi.org/10.1155/2016/7657452

2 International Journal of Aerospace Engineering

2. Preliminaries and Problem Description

2.1. Notations and Graph Theory. In this paper, the followingnotations will be used. 𝑅𝑛×𝑛 and 𝐶

𝑛×𝑛 denote the set of 𝑛 × 𝑛

real and complex matrices, respectively. For 𝜇 ∈ 𝐶, the realpart is Re(𝜇). ⊗ denotes the Kronecker product. 𝐼

𝑛is the iden-

titymatrix of order 𝑛. For a squarematrix𝐴, 𝜆(𝐴) denotes theeigenvalues of matrix 𝐴. 𝐴 > 0 (𝐴 ≥ 0) means that 𝐴 is pos-itive definite (positive semidefinite). max{𝜆(𝐴)} (min{𝜆(𝐴)})denotes the largest (smallest) eigenvalue of the matrix 𝐴.

A directed graph 𝐺 = (V,E,A) contains the vertex setV = {1, 2, . . . , 𝑁}, the directed edges setE ⊆ V×V, and theadjacency matrix A = [𝑎

𝑖𝑗]𝑁×𝑁

with nonnegative elements𝑎𝑖𝑗. 𝑎𝑖𝑗

= 1 if there is a directed edge from vertex 𝑗 to 𝑖; 𝑎𝑖𝑗

= 0,otherwise. The Laplacian matrix of the graph 𝐺 is defined as𝐿 = [𝐿

𝑖𝑗]𝑁×𝑁

, where 𝐿𝑖𝑖

= ∑𝑗 =𝑖

𝑎𝑖𝑗and 𝐿

𝑖𝑗= −𝑎𝑖𝑗(𝑖 = 𝑗).

Zero is an eigenvalue of 𝐿with the eigenvector 1𝑁. A directed

graph is said to have a spanning tree if there is a vertex suchthat there is a directed path from this vertex to every othervertex.

Lemma 1 (see [8]). Zero is a simple eigenvalue of 𝐿 and all theother nonzero eigenvalues have positive real parts if and only ifthe graph has a directed spanning tree.

2.2. ProblemDescription. ConsiderUAV swarm systemswith𝑁UAVs.The interaction topology of theUAV swarm systemscan be described by a directed graph𝐺, in whichUAV 𝑖 can bedenoted by a vertex and the interaction channel fromUAV 𝑖 toUAV 𝑗 can be denoted by an edge. Comparedwith the attitudedynamics, the trajectory dynamics of each UAV have muchlarger time constants, which means the attitude controllerand trajectory controller can be designed separately. On theformation level, only trajectory control needs to be consid-ered. Therefore, in this brief, the dynamics of each UAV canbe described by the following double integrator [18, 21, 22]:

��𝑖(𝑡) = V

𝑖(𝑡) ,

V𝑖(𝑡) = 𝑢

𝑖(𝑡) ,

(1)

where 𝑖 = 1, 2, . . . , 𝑁, 𝑥𝑖(𝑡) ∈ 𝑅

𝑛 and V𝑖(𝑡) ∈ 𝑅

𝑛 denotethe position and velocity vectors of UAV 𝑖, respectively,and 𝑢

𝑖(𝑡) ∈ 𝑅

𝑛 are the control inputs. In the following, forsimplicity of description, it is assumed that 𝑛 = 1, if nototherwise specified.

Therefore, UAV swarm systems (1) can be rewritten as

��𝑖(𝑡) = 𝐴𝜉

𝑖(𝑡) + 𝐵𝑢

𝑖(𝑡) , (2)

where 𝜉𝑖(𝑡) = [𝑥

𝑖(𝑡), V𝑖(𝑡)]𝑇, 𝐴 = [

0 1

0 0], 𝐵 = [

0

1].

A formation is specified by a vector ℎ(𝑡) = [ℎ𝑇

1(𝑡), ℎ𝑇

2(𝑡),

. . . , ℎ𝑇

𝑁(𝑡)]𝑇

∈ 𝑅2𝑁 with ℎ

𝑖(𝑡) = [ℎ

𝑖𝑥(𝑡), ℎ𝑖V(𝑡)]𝑇 (𝑖 = 1,

2, . . . , 𝑁) continuously differentiable and ℎ𝑖V(𝑡) being the

derivative of ℎ𝑖𝑥(𝑡). Let ℎ

𝑥(𝑡) = [ℎ

𝑇

𝑥1(𝑡), ℎ𝑇

𝑥2(𝑡), . . . , ℎ

𝑇

𝑥𝑁(𝑡)]𝑇

and let ℎV(𝑡) = [ℎ𝑇

V1(𝑡), ℎ𝑇

V2(𝑡), . . . , ℎ𝑇

V𝑁(𝑡)]𝑇; then one has that

if ℎV(𝑡) are not equal to zeros, the formation is time-varying.

Definition 2 (see [21]). UAV swarm systems (2) are saidto achieve formation ℎ(𝑡) if there exists a function 𝑟(𝑡) =

[𝑟𝑥(𝑡), 𝑟V(𝑡)] ∈ 𝑅

2 with 𝑟V(𝑡) being the derivative of 𝑟𝑥(𝑡) suchthat

lim𝑡→∞

(𝜉𝑖(𝑡) − ℎ

𝑖(𝑡) − 𝑟 (𝑡)) = 0, 𝑖 = 1, 2, . . . , 𝑁, (3)

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

In this paper, the communication topology ismolded by adirected graph andwe assume that the communication topol-ogy is time-varying. Let �� = {𝐺

1

, 𝐺2

, . . . , 𝐺𝑝

}, 𝑝 ≥ 1, be theset of all possible directed topologies.We define the switchingsignal 𝜎(𝑡), where 𝜎(𝑡) : [0, +∞) → P = {1, 2, . . . , 𝑝}.0 = 𝑡0< 𝑡1< 𝑡2< ⋅ ⋅ ⋅ denote the switching instants of 𝜎(𝑡).

Let 𝐺𝜎(𝑡) ∈ �� be the communication topology at time 𝑡. 𝐿𝜎(𝑡)

stands for the corresponding Laplacian matrix of 𝐺𝜎(𝑡).

Assumption 3. Each possible graph 𝐺𝜎(𝑡)

∈ �� is fixed andcontains a directed spanning tree.

Let 𝜆𝜎(𝑡)𝑖

(𝑖 = 1, 2, . . . , 𝑁) be the eigenvalues of the Lapla-cian matrix 𝐿

𝜎(𝑡). Without loss of generality, it is assumedthat Re(𝜆𝜎(𝑡)

1) ≤ Re(𝜆𝜎(𝑡)

2) ≤ ⋅ ⋅ ⋅ ≤ Re(𝜆𝜎(𝑡)

𝑁). Furthermore,

from Lemma 1, one can obtain that 𝜆𝜎(𝑡)

1= 0 and 0 ≤

Re(𝜆𝜎(𝑡)2

) ≤ ⋅ ⋅ ⋅ ≤ Re(𝜆𝜎(𝑡)𝑁

). Let 𝜆min = min{𝜆𝑚𝑖

(∀𝑚 ∈ P; i =

2, 3, . . . , 𝑁)}.Consider the following formation protocol:

𝑢𝑖(𝑡) = 𝐾

1(𝜉𝑖(𝑡) − ℎ

𝑖(𝑡))

+ 𝐾2

𝑁

∑

𝑗=1

𝑎𝑖𝑗((𝜉𝑗(𝑡) − ℎ

𝑗(𝑡)) − (𝜉

𝑖(𝑡) − ℎ

𝑖(𝑡)))

+ ℎ𝑖V (𝑡) , 𝑖 = 1, 2, . . . , 𝑁,

(4)

where 𝑖 = 1, 2, . . . , 𝑁, 𝐾1∈ 𝑅1×2 and 𝐾

2∈ 𝑅1×2 are constant

gain matrices, and 𝑎𝑖𝑗is defined as in Section 2.1. Let 𝜉(𝑡) =

[𝜉𝑇

1(𝑡), 𝜉𝑇

2(𝑡), . . . , 𝜉

𝑇

𝑁(𝑡)]𝑇, let ℎ

𝑥(𝑡) = [ℎ

𝑇

1𝑥(𝑡), ℎ𝑇

2𝑥(𝑡), . . . ,

ℎ𝑇

𝑁𝑥(𝑡)]𝑇, and let ℎV(𝑡) = [ℎ

𝑇

1V(𝑡), ℎ𝑇

2V(𝑡), . . . , ℎ𝑇

𝑁V(𝑡)]𝑇. Under

protocol (4), the UAV swarm systems (2) can be written in acompact closed-loop form as follows:

�� (𝑡) = (𝐼𝑁

⊗ (𝐴 + 𝐵𝐾1) − 𝐿𝜎(𝑡)

⊗ (𝐵𝐾2)) 𝜉 (𝑡)

− (𝐼𝑁

⊗ 𝐵𝐾1− 𝐿𝜎(𝑡)

⊗ (𝐵𝐾2)) ℎ (𝑡)

+ (𝐼𝑁

⊗ 𝐵) ℎV (𝑡) .

(5)

This brief mainly investigates how to design the gainmatrices in protocol (4) for the UAV swarm systems (5) toachieve the formation ℎ(𝑡).

International Journal of Aerospace Engineering 3

3. Main Results

Let 𝑧𝑖(𝑡) = 𝜉

𝑖(𝑡) − ℎ

𝑖(𝑡), 𝑧(𝑡) = [𝑧

𝑇

1(𝑡), 𝑧𝑇

2(𝑡), . . . , 𝑧

𝑇

𝑁(𝑡)]𝑇. Then

the UAV swarm systems can be rewritten as follows:

�� (𝑡) = (𝐼𝑁

⊗ (𝐴 + 𝐵𝐾1) − 𝐿𝜎(𝑡)

⊗ 𝐵𝐾2) 𝑧 (𝑡)

+ (𝐼𝑁

⊗ 𝐴) ℎ (𝑡) − (𝐼𝑁

⊗ 𝐼𝑛) ℎ (𝑡)

+ (𝐼𝑁

⊗ 𝐵) ℎV (𝑡) .

(6)

As for ℎ𝑖𝑥(𝑡) = ℎ

𝑖V(𝑡), one can obtain that

(𝐼𝑁

⊗ 𝐴) ℎ (𝑡) − (𝐼𝑁

⊗ 𝐼𝑛) ℎ (𝑡) + (𝐼

𝑁⊗ 𝐵) ℎV (𝑡) = 0. (7)

Thus (6) can be further rewritten as

�� (𝑡) = (𝐼𝑁

⊗ (𝐴 + 𝐵𝐾1) − 𝐿𝜎(𝑡)

⊗ 𝐵𝐾2) 𝑧 (𝑡) . (8)

It holds directly that UAV swarm systems (2) withdirected and switching topologies achieve formation ℎ(𝑡) ifand only if system (8) achieves consensus.

Before the consensus analysis of system (8), the followinglemmas and definition are introduced.

Lemma 4 (see [23]). For a Laplacian matrix 𝐿 of graph𝐺 anda full row rank matrix 𝐸 defined as

𝐸 =

[[[[[[

[

1 −1 0 ⋅ ⋅ ⋅ 0

0 1 −1 ⋅ ⋅ ⋅ 0

.

.

.... d

.

.

....

0 0 0 1 −1

]]]]]]

]

, (9)

there exists a matrix𝑀 such that 𝐿 = 𝑀𝐸. Further, if the graphhas a directed spanning tree, 𝑀 is of full column rank and theeigenvalues of 𝐸𝑀 are equal to the nonzero eigenvalues of 𝐿.

Lemma 5 (see [24]). Suppose that the eigenvalues of 𝐴 ∈

𝑅𝑁×𝑁 have positive real parts; then there exists a positive

definite matrix 𝑄 > 0 such that

𝐴𝑇

𝑄 + 𝑄𝐴 > 0. (10)

Definition 6. For a switching signal 𝜎(𝑡) over time interval[0, 𝑡), the average dwell time of the switching signal is definedas 𝜏𝑎= 𝑡/(𝑁

𝜎(𝑡) + 1), where𝑁

𝜎(𝑡) denotes the number of the

switches.

Remark 7. In [11, 25], the definition of the average dwell timeof a switching signal 𝜎(𝑡) over time interval [0, 𝑡) can bedescribed as follows. If there exist two positive numbers 𝑁

0

and 𝜏𝑎such that 𝑁

𝜎(𝑡) ≤ 𝑁

0+ 𝑡/𝜏𝑎, where 𝑁

𝜎(𝑡) denotes the

number of the switches, 𝜏𝑎is called the average dwell time.

It is inaccurate to give the definition by an inequality, but,according to Definition 6, it can be seen that 𝑁

𝜎(𝑡) ≤ 𝑡/𝜏

𝑎.

From Lemma 4, one can obtain that, for each 𝐿(𝑖), 𝑖 ∈

P, there exists a matrix 𝑀(𝑖) such that 𝐿

(𝑖)

= 𝑀(𝑖)

𝐸.Given Assumption 3 and Lemma 1, it can be known that the

eigenvalues of each 𝐸𝑀(𝑖) (𝑖 ∈ P) have positive real parts.

Based on Lemma 5, one can obtain that there exist positivedefinite matrices 𝑄(𝑖) such that

(𝐸𝑀(𝑖)

− 𝛼𝐼)𝑇

𝑄(𝑖)

+ 𝑄(𝑖)

(𝐸𝑀(𝑖)

− 𝛼𝐼) > 0, (11)

where 𝛼 < 𝜆min. Further, one can obtain that

(𝐸𝑀(𝑖)

)𝑇

𝑄(𝑖)

+ 𝑄(𝑖)

𝐸𝑀(𝑖)

> 2𝛼𝑄(𝑖)

, 𝑖 ∈ P. (12)

Let 𝜃𝑖(𝑡) = 𝑧

𝑖+1(𝑡) − 𝑧

𝑖(𝑡), 𝑖 = 1, 2, . . . , 𝑁, and let 𝜃(𝑡) =

[𝜃𝑇

1(𝑡), 𝜃𝑇

2(𝑡), . . . , 𝜃

𝑇

𝑁(𝑡)]𝑇. One can obtain that 𝜃(𝑡) = (𝐸 ⊗

𝐼𝑛)𝑧(𝑡), where 𝐸 is defined as in Lemma 4.Premultiplying both sides of (8) by (𝐸 ⊗ 𝐼

𝑛) leads to

�� (𝑡) = (𝐼𝑁−1

⊗ (𝐴 + 𝐵𝐾1) − 𝐸𝑀

𝜎(𝑡)

⊗ 𝐵𝐾2) 𝜃 (𝑡) , (13)

where 𝑀𝜎(𝑡)

= 𝐿𝜎(𝑡)

𝐸𝑇

(𝐸𝐸𝑇

)−1.

According to the definition of 𝜃(𝑡), it is obvious that𝑧1(𝑡) = 𝑧

2(𝑡) = ⋅ ⋅ ⋅ = 𝑧

𝑁(𝑡) if and only if 𝜃(𝑡) = 0. So if

system (13) converges to zero, system (8) achieves consensusand UAV swarm systems (2) with directed and switchingtopologies achieve formation ℎ(𝑡).

Theorem 8. Suppose that Assumption 3 holds. The formationproblem of UAV swarm systems (2) with directed and switchingtopologies can be solved by controller (4) if there exists a positivedefinite matrix 𝑃 such that

(𝐴 + 𝐵𝐾1)𝑇

𝑃 + 𝑃 (𝐴 + 𝐵𝐾1) − 2𝛼𝑃𝐵𝐵

𝑇

𝑃 + 𝛽𝑃 ≤ 0, (14)where 𝛼 < 𝜆min, 𝛽 > ln ℎ/𝜏

𝑎, ℎ = 𝜑

1/𝜑2, 𝜑1

=

max𝑖∈P{𝜆(𝑄

(𝑖)

)}, 𝜑2= min

𝑖∈P{𝜆(𝑄(𝑖)

)}, and 𝑄(𝑖) satisfies (12).

The feedback matrix is designed as 𝐾2= 𝐵𝑇

𝑃.

Proof. Consider the following piecewise Lyapunov candidateof system (13):

𝑉 (𝑡) = 𝜃 (𝑡)𝑇

(𝑄𝜎(𝑡)

⊗ 𝑃) 𝜃 (𝑡) , (15)

where 𝑃 is a solution of inequality (14) and 𝑄𝜎(𝑡) are feasible

solutions of (12).

Note that the communication topology is fixed for 𝑡 ∈

[𝑡𝑖, 𝑡𝑖+1

), 𝑖 = 0, 1, . . .. Then, the derivation of this Lyapunovcandidate along the trajectory of system (13) within theinterval is

�� (𝑡) = 𝜃 (𝑡)𝑇

(𝐼𝑁−1

⊗ (𝐴 + 𝐵𝐾1) − 𝐸𝑀

𝜎(𝑡)

⊗ 𝐵𝐾2)𝑇

⋅ (𝑄𝜎(𝑡)

⊗ 𝑃) 𝜃 (𝑡) + 𝜃 (𝑡)𝑇

(𝑄𝜎(𝑡)

⊗ 𝑃)

⋅ (𝐼𝑁−1

⊗ (𝐴 + 𝐵𝐾1) − 𝐸𝑀

𝜎(𝑡)

⊗ 𝐵𝐾2)

⋅ 𝜃 (𝑡) .

(16)

Substituting 𝐾2= 𝐵𝑇

𝑃 into (16) yields

�� (𝑡) = 𝜃 (𝑡)𝑇

(𝑄𝜎(𝑡)

⊗ ((𝐴 + 𝐵𝐾1)𝑇

𝑃 + 𝑃 (𝐴 + 𝐵𝐾1))

− ((𝐸𝑀𝜎(𝑡)

)𝑇

𝑄𝜎(𝑡)

+ 𝑄𝜎(𝑡)

𝐸𝑀𝜎(𝑡)

) ⊗ 𝑃𝐵𝐵𝑇

𝑃)

⋅ 𝜃 (𝑡) .

(17)


It then follows from (12) that

�� (𝑡) ≤ 𝜃 (𝑡)𝑇

(𝑄𝜎(𝑡)

⊗ ((𝐴 + 𝐵𝐾1)𝑇

𝑃 + 𝑃 (𝐴 + 𝐵𝐾1) − 2𝛼𝑃𝐵𝐵

𝑇

𝑃))

⋅ 𝜃 (𝑡) .

(18)

Based on (14), one has

�� (𝑡) < −𝛽𝜃 (𝑡)𝑇

(𝑄𝜎(𝑡)

⊗ 𝑃) 𝜃 (𝑡) . (19)

Thus, from (15), one can obtain that

𝑉 (𝑡) < 𝑒−𝛽(𝑡−𝑡𝑖)𝑉 (𝑡

𝑖) . (20)

Note that the communication topology switches at 𝑡 = 𝑡𝑖;

then one can get

𝑉 (𝑡𝑖) ≤ ℎ𝑉 (𝑡

−

𝑖) , (21)

where ℎ = 𝜑1/𝜑2, 𝜑1

= max𝑖∈P{𝜆(𝑄

(𝑖)

)}, and 𝜑2

=

min𝑖∈P{𝜆(𝑄

(𝑖)

)}.Thus, when 𝑡 ∈ [𝑡

𝑖, 𝑡𝑖+1

), from (20) and (21), one has

𝑉 (𝑡) < 𝑒−𝛽(𝑡−𝑡𝑖)ℎ𝑉 (𝑡

−

𝑖) < 𝑒−𝛽(𝑡−𝑡𝑖)ℎ𝑒

−𝛽(𝑡𝑖−𝑡𝑖−1)𝑉 (𝑡𝑖−1

)

< 𝑒−𝛽𝑡

ℎ𝑖

𝑉 (0) .

(22)

Since 𝑖 ≤ 𝑁𝜎(𝑡) ≤ 𝑡/𝜏

𝑎,

𝑉 (𝑡) < 𝑒−(𝛽−ln ℎ/𝜏𝑎)𝑡𝑉 (0) . (23)

From (15), one can obtain that

𝑉 (0) ≤ 𝜙1‖𝜃 (0)‖

2

,

𝜙2‖𝜃 (𝑡)‖

2

≤ 𝑉 (𝑡) ,

(24)

where 𝜙1= 𝜑1max{𝜆(𝑃)} and 𝜙

2= 𝜑2min{𝜆(𝑃)}.

According to (23) and (24), one has

‖𝜃 (𝑡)‖2

≤𝜙1

𝜙2

𝑒−(𝛽−ln ℎ/𝜏𝑎)𝑡

‖𝜃 (0)‖2

. (25)

Note that 𝛽 > ln ℎ/𝜏𝑎; one has 𝜃(𝑡) → 0 as 𝑡 → ∞. This

means that the consensus problem of system (8) is solved.Furthermore, formation for UAV swarm systems (2) withdirected and switching topologies is achieved.

Corollary 9. If UAV swarm systems (2) achieve formationℎ(𝑡), the formation center function 𝑟(𝑡) can be determined asfollows:

𝑟 (𝑡) = 𝑒(𝐴+𝐵𝐾1)(𝑡−𝑡𝑖) (𝑝

𝜎(𝑡)𝑇

⊗ 𝐼) (𝑥 (𝑡𝑖) − ℎ (𝑡

𝑖)) ,

𝑡 ∈ [𝑡𝑖, 𝑡𝑖+1

) ,

(26)

where 𝑝𝜎(𝑡) is the left eigenvector of 𝐿

𝜎(𝑡) associated witheigenvalue 0 and (𝑝

𝜎(𝑡)

)𝑇1𝑁

= 1.

Proof. From [7], there exists a left eigenvector 𝑝𝜎(𝑡) of 𝐿

𝜎(𝑡)

associated with eigenvalue 0 and (𝑝𝜎(𝑡)

)𝑇1𝑁

= 1. For 𝑡 ∈

[𝑡𝑖, 𝑡𝑖+1

), 𝐿𝜎(𝑡) is fixed and so is 𝑝𝜎(𝑡).

Premultiplying both sides of (8) by ((𝑝𝜎(𝑡)

)𝑇

⊗𝐼) results in

((𝑝𝜎(𝑡)

)𝑇

⊗ 𝐼) �� (𝑡)

= (𝐴 + 𝐵𝐾1) ((𝑝𝜎(𝑡)

)𝑇

⊗ 𝐼) 𝑧 (𝑡) .

(27)

Based on Definition 2, one can obtain that

lim𝑡→∞

(𝑧 (𝑡) − (1𝑁

⊗ 𝐼) 𝑟 (𝑡)) = 0. (28)

Premultiplying both sides by ((𝑝𝜎(𝑡)

)𝑇

⊗ 𝐼), one has

lim𝑡→∞

(((𝑝𝜎(𝑡)

)𝑇

⊗ 𝐼) 𝑧 (𝑡) − 𝑟 (𝑡)) = 0. (29)

Therefore, a formation center function can be

𝑟 (𝑡) = ((𝑝𝜎(𝑡)

)𝑇

⊗ 𝐼) 𝑧 (𝑡) . (30)

It follows from (27) and (30) that

𝑟 (𝑡) = (𝐴 + 𝐵𝐾1) 𝑟 (𝑡) . (31)

Thus, (26) can be obtained.

Remark 10. As can be seen, the formation center is discon-tinuous due to the switching of the communication topology.In addition, 𝐾

1can be used to design the motion modes

of the formation center function. If 𝐾1

= 0, protocol (4)becomes a totally distributed controller. 𝐾

2has no effect on

the formation center function.

Remark 11. Compared with [21, 22], the interaction topolo-gies are more common. Formation for UAV swarm systemswith directed and switching topologies is solved. Further-more, the gainmatrix was designed by solving an LMI, whichis simpler than solving an algebraic Riccati equation in [22].In fact, undirected topologies are just special cases of directedtopologies. So the algorithms presented in this paper areapplicable to those cases in [21, 22].

Based on the above results, a design procedure of protocol(4) can be summarized as follows. First, choose 𝐾

1to design

the motion modes of the formation center by assigning theeigenvalues of (𝐴+𝐵𝐾

1).Then design𝐾

2using the conclusion

of Theorem 8.

4. Examples

In this section, we provide an example to illustrate theeffectiveness of the above theoretical results. UAV swarm


G1 G

2

G3

G4

11

1 1

2

22

2

3 3

3 3

4

4

4

4

Figure 1: Communication topologies.

1 2 3 4 5 6 7 8 9 100Time (s)

1

1.5

2

2.5

3

3.5

4

𝜎(t)

Figure 2: Switching signal 𝜎(𝑡).

systems consisting of four agents are considered. The systemmatrices are defined as

𝑥𝑖=

[[[[[

[

𝑥𝑖1

𝑥𝑖2

𝑥𝑖3

𝑥𝑖4

]]]]]

]

,

𝐴 =

[[[[[

[

0 1 0 0

0 0 0 0

0 0 0 1

0 0 0 0

]]]]]

]

,

𝐵 =

[[[[[

[

0 0

1 0

0 0

0 1

]]]]]

]

,

(32)

where 𝑥𝑖1, 𝑥𝑖2, 𝑥𝑖3, and 𝑥

𝑖4stand for east position, east

velocity, north position, and north velocity. The directedcommunication topologies are given in Figure 1. Clearly, eachtopology contains a directed spanning tree. The switchingsignal is shown in Figure 2.

Thus, we can obtain 𝜆min = 1 and then choose 𝛼 = 0.9.Further, we can get that 𝜑

1= 3.4175, 𝜑

2= 0.2009, and ℎ =

17.0075. FromFigure 2, we can get that the average dwell timeis 1.25 s and then choose 𝛽 = 5.

Assign the eigenvalues of (𝐴 + 𝐵𝐾1) at (±𝑖, ±𝑖); we get

𝐾1= [

−1 0 0 0

0 0 −1 0] . (33)

Solve LMI (14) with 𝛼 = 0.9 and 𝛽 = 5; a feasible solutioncan be obtained. Accordingly, we can get

𝐾2= [

0.6565 1.3431 0 0

0 0 0.6565 1.3431] . (34)


UAV 1UAV 2UAV 3

UAV 4Center

1 2 3 4 5 6 7 8 9 100Time (s)

−6

−4

−2

0

2

4

6

8Po

sitio

n (m

)

(a) East position

UAV 1UAV 2UAV 3

UAV 4Center

−8

−6

−4

−2

0

2

4

6

Velo

city

(m/s

)

1 2 3 4 5 6 7 8 9 100Time (s)

(b) East velocity

UAV 1UAV 2UAV 3

UAV 4Center

1 2 3 4 5 6 7 8 9 100Time (s)

−6

−4

−2

0

2

4

6

8

Posit

ion

(m)

(c) North position

UAV 1UAV 2UAV 3

UAV 4Center

−6

−4

−2

0

2

4

6Ve

loci

ty (m

/s)

1 2 3 4 5 6 7 8 9 100Time (s)

(d) North velocity

Figure 3: Difference of UAV state and time-varying formation.

Choose the following time-varying formation:

ℎ𝑖(𝑡) =

[[[[[[[[[[[[[[[

[

3 sin(0.2𝑡 +(𝑖 − 1) 𝜋

2)

0.6 cos(0.2𝑡 + (𝑖 − 1) 𝜋

2)

8 cos(0.5𝑡 + (𝑖 − 1) 𝜋

2)

−4 sin(0.5𝑡 +(𝑖 − 1) 𝜋

2)

]]]]]]]]]]]]]]]

]

(𝑖 = 1, 2, 3, 4) .

(35)

If ℎ(𝑡) is achieved, both the positions and velocities ofthe four UAVs locate at the vertexes of a rotating parallel-ogram, respectively. Choose initial states of four UAVs as𝑥1(0) = [3 0 4 0]

𝑇, 𝑥2(0) = [−2 0 −5 0]

𝑇, 𝑥3(0) =

[0 0 −1 0]𝑇, and 𝑥

4(0) = [5 0 −5 0]

𝑇.Figure 3 shows the trajectories of the difference of UAV

states and time-varying formation, which are denoted bysolid line, dotted line, dash-dotted line, and dashed line. Andthe bold dotted line denotes the formation center trajectory.It is obvious that the differences achieve consensus afterabout 𝑡 = 7 s and converge to the formation center. FromDefinition 2, one can obtain that the time-varying formationproblem is solved. Figure 4 shows the snapshots of four UAV


UAV 1UAV 2

UAV 3UAV 4

−5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1−6East position (m)

−4

−2

0

2

4

6

8

10

12N

orth

pos

ition

(m)

(a) 𝑡 = 4 s

UAV 1UAV 2

UAV 3UAV 4

−12

−10

−8

−6

−4

−2

0

2

4

6

Nor

th p

ositi

on (m

)

3 4 5 6 7 8 92East position (m)

(b) 𝑡 = 6 s

−8

−6

−4

−2

0

2

4

6

Nor

th p

ositi

on (m

)

−3 −2 −1 0 1 2 3−4East position (m)

UAV 1UAV 2

UAV 3UAV 4

(c) 𝑡 = 8 s

−7 −6 −5 −4 −3 −2 −1−8East position (m)

−4

−2

0

2

4

6

8

10

12

14N

orth

pos

ition

(m)

UAV 1UAV 2

UAV 3UAV 4

(d) 𝑡 = 10 s

Figure 4: Snapshots of UAV positions.

positions at different time. It can be seen that, after 𝑡 = 6 s, theUAV swarm systems achieve a time-varying parallelogramformation.Therefore, the time-varying formation is achievedunder the directed and switching topologies.

5. Conclusions

Formation problems for UAV swarm systems with directedand switching topologies are studied. The average dwell timeof the switching topologies is introduced, based on whichan LMI-based method to design the protocol is proposed.Though the UAV swarm systems can achieve the specifiedformationwith the presentedmethod, there are still problemsin real application. As mentioned in Assumption 3, each ofthe switching topologies is supposed to have a spanning tree,

whichmay not be applicable, so there is still work to do in ourfuture work.

Competing Interests

The authors declare that they have no competing interests.

References

[1] T. Kopfstedt, M. Mukai, M. Fujita, and C. Ament, “Controlof formations of UAVs for surveillance and reconnaissancemissions,” in Proceedings of the 17th IFAC World Congress, pp.6–11, Seoul, South Korea, July 2008.

[2] N. Nigam, S. Bieniawski, I. Kroo, and J. Vian, “Control ofmultiple UAVs for persistent surveillance: algorithm and flight


test results,” IEEE Transactions on Control Systems Technology,vol. 20, no. 5, pp. 1236–1251, 2012.

[3] D. Pack, P. DeLima, G. Toussaint, and G. York, “Cooperativecontrol of UAVs for localization of intermittently emittingmobile targets,” IEEE Transactions on Systems, Man, and Cyber-netics B, vol. 39, no. 4, pp. 959–970, 2009.

[4] Z. Lin, W. Ding, G. Yan, C. Yu, and A. Giua, “Leader-followerformation via complex Laplacian,”Automatica, vol. 49, no. 6, pp.1900–1906, 2013.

[5] T. Balch and R. C. Arkin, “Behavior-based formation controlfor multirobot teams,” IEEE Transactions on Robotics andAutomation, vol. 14, no. 6, pp. 926–939, 1998.

[6] M. A. Lewis and K.-H. Tan, “High precision formation controlof mobile robots using virtual structures,” Autonomous Robots,vol. 4, no. 4, pp. 387–403, 1997.

[7] W. Ren and E. Atkins, “Distributed multi-vehicle coordinatedcontrol via local information exchange,” International Journal ofRobust and Nonlinear Control, vol. 17, no. 10-11, pp. 1002–1033,2007.

[8] W. Ren, “On consensus algorithms for double-integratordynamics,” IEEE Transactions on Automatic Control, vol. 53, no.6, pp. 1503–1509, 2008.

[9] Z. Li, Z. Duan, G. Chen, and L. Huang, “Consensus ofmultiagent systems and synchronization of complex networks:a unified viewpoint,” IEEE Transactions on Circuits and Systems.I. Regular Papers, vol. 57, no. 1, pp. 213–224, 2010.

[10] S. Su and Z. Lin, “Distributed consensus control of multi-agentsystems with higher order agent dynamics and dynamicallychanging directed interaction topologies,” IEEE Transactions onAutomatic Control, vol. 61, no. 2, pp. 515–519, 2016.

[11] I. Saboori and K. Khorasani, “𝐻∞

consensus achievementof multi-agent systems with directed and switching topologynetworks,” IEEE Transactions on Automatic Control, vol. 59, no.11, pp. 3104–3109, 2014.

[12] Y. Zhu, C. Chen, X. Guan, and X. Luo, “Finite-time consensusof leader-follower multi-agent systems with fixed and switch-ing topologies,” in Proceedings of the 11th World Congress onIntelligent Control and Automation (WCICA ’14), pp. 226–231,Shenyang, China, July 2014.

[13] Y. Su and J. Huang, “Stability of a class of linear switchingsystems with applications to two consensus problems,” IEEETransactions on Automatic Control, vol. 57, no. 6, pp. 1420–1430,2012.

[14] G. Wen, W. Yu, J. Cao, G. Hu, and G. Chen, “Consensuscontrol of switching directed networks with general linear nodedynamics,” in Proceedings of the 9th Asian Control Conference(ASCC ’13), pp. 1–6, IEEE, Istanbul, Turkey, June 2013.

[15] G. Wen and V. Ugrinovskii, “Distributed consensus of linearmulti-agent systems with switching directed topologies,” inProceedings of the Australian Control Conference, pp. 146–151,IEEE, Canberra, Australia, November 2014.

[16] W. Ren, “Consensus strategies for cooperative control of vehicleformations,” IET ControlTheory & Applications, vol. 1, no. 2, pp.505–512, 2007.

[17] A. Abdessameud and A. Tayebi, “Formation control of VTOLunmanned aerial vehicles with communication delays,” Auto-matica, vol. 47, no. 11, pp. 2383–2394, 2011.

[18] J. Seo, Y. Kim, S. Kim, and A. Tsourdos, “Consensus-basedreconfigurable controller design for unmanned aerial vehicleformation flight,” Proceedings of the Institution of MechanicalEngineers, Part G: Journal of Aerospace Engineering, vol. 226, no.7, pp. 817–829, 2012.

[19] M. Turpin,N.Michael, andV.Kumar, “Decentralized formationcontrol with variable shapes for aerial robots,” in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA ’12), pp. 23–30, IEEE, St. Paul, Minn, USA, May 2012.

[20] B. Yu, X. Dong, Z. Shi, and Y. Zhong, “Formation controlfor quadrotor swarm systems: algorithms and experiments,” inProceedings of the 32nd Chinese Control Conference (CCC ’13),pp. 7099–7104, Xi’an, China, July 2013.

[21] Y. Zhou, X. Dong, G. Lu, and Y. Zhong, “Time-varyingformation control for unmanned aerial vehicles with switchinginteraction topologies,” in Proceedings of the International Con-ference on Unmanned Aircraft Systems (ICUAS ’14), pp. 1203–1209, Orlando, Fla, USA, May 2014.

[22] X. Dong, B. Yu, Z. Shi, and Y. Zhong, “Time-varying formationcontrol for unmanned aerial vehicles: theories and applica-tions,” IEEE Transactions on Control Systems Technology, vol. 23,no. 1, pp. 340–348, 2015.

[23] S. Zhou, W. Liu, Q. Wu, and G. Yin, “Leaderless consensus oflinear multi-agent systems: matrix decomposition approach,”in Proceedings of the 7th International Conference on IntelligentHuman-Machine Systems and Cybernetics (IHMSC ’15), pp. 327–331, Hangzhou, China, August 2015.

[24] C. T. Cben, Linear SystemTheory andDesign, Holt, Rinehart andWinston, Inc, New York, NY, USA, 1970.

[25] D. Liberzon, Switching in Systems and Control, Birkhauser,Boston, Mass, USA, 2003.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Research Article Formation Control for Unmanned Aerial...

Documents

Transcript of Research Article Formation Control for Unmanned Aerial...