Formation tracking control for time-delayed multi-agent systems with second-order dynamics

Han Liang,Dong Xiwang,Li Qingdong,Ren Zhang

Science and Technology on Aircraft Control Laboratory,School of Automation Science and Electrical Engineering,Beihang University,Beijing 100083,China

Formation tracking control for time-delayed multi-agent systems with second-order dynamics

Han Liang,Dong Xiwang,Li Qingdong*,Ren Zhang

Science and Technology on Aircraft Control Laboratory,School of Automation Science and Electrical Engineering,Beihang University,Beijing 100083,China

Formation tracking control;Multiple unmanned aerial vehicles;Second-order dynamics;Time-delayed multi-agent systems;Time-varying formation

In this paper,formation tracking control problems for second-order multi-agent systems(MASs)with time-varying delays are studied,specifically those where the position and velocity of followers are designed to form a time-varying formation while tracking those of the leader.A neighboring relative state information based formation tracking protocol with an unknown gain matrix and time-varying delays is presented.The formation tracking problems are then transformed into asymptotically stable problems.Based on the Lyapunov-Krasovskii functional approach,conditions sufficient for second-order MASs with time-varying delays to realize formation tracking are examined.An approach to obtain the unknown gain matrix is given and,since neighboring relative velocity information is difficult to measure in practical applications,a formation tracking protocol with time-varying delays using only neighboring relative position information is introduced.The proposed results can be used on target enclosing problems for MASs with second-order dynamics and time-varying delays.An application for target enclosing by multiple unmanned aerial vehicles(UAVs)is given to demonstrate the feasibility of theoretical results.

1.Introduction

In recent years,formation control techniques of multi-agent systems(MASs)have received lots of attention in a variety of research areas such as unmanned aerial vehicle(UAV)formation flying,1spacecraft formations,2and cooperative localization.3Currently,formation control techniques can be classified into different categories,the most representative of which are formation tracking control,4formation enclosing control,5and formation-containment control.6

Consensus-based formation control approaches have been widely used in solving the MASs problems.Ref.7introduced leader-follower,virtual structure,and behavior-based formation control methods as typical consensus-based approaches,and a series of distributed consensus-based formation control methods were presented for MASs with second-order dynamics.A finite-time formation control strategy for MASs with first-order dynamics was developed in Ref.8.Ref.9proposedconditions sufficient for MASs with second-order dynamics to achieve time-varying formations.Conditions necessary for first-order MASs with fixed and undirected topologies to realize a rigid formation are presented in Ref.10.A formation control approach that considered heterogeneous communication delays was presented for MASs with second-order dynamics in Ref.11.Formation control problems for MASs with second-order dynamics and time-varying delays were investigated in Ref.12.In Ref.13,a leader-follower based formation control approach for second-order MASs with time-varying delays and nonlinear dynamics was developed in which both the fixed and switching topologies were considered.An adaptive iterative learning based formation control protocol was proposed for MASs with second-order dynamics in Ref.14.A time-varying formation control approach for MASs with second-order dynamics and jointly connected topologies is presented in Ref.15.The aforementioned methods used neighboring relative position and velocity information simultaneously.However,in contrast to the measurement of relative position,which can be obtained by ultrasonic or infrared distance sensors,there are few means to measure relative velocity,and the measuring sensor is quite expensive.Thus,neighboring relative position information based methods are highly applicable to MAS formation control problems.This is explored in Ref.16with a distributed position estimation based formation control strategy for MASs with first-order dynamics.Ref.17demonstrates a similar formation control method using local relative distance measurement for MASs with a group of mobile autonomous agents.

The above approaches provide theoretical support for practical use of MASs.With the development of UAV technology,more and more researchers develop formation control methods for multiple UAV systems.Formation stabilization schemes using undirected interaction topologies to realize the formation in the presence of constant or time-varying delays for multiple UAV systems are proposed in Ref.18.A virtual structure-based formation control method was presented for multiple UAVs with communication delays in Ref.19.In Ref.20,an output feedback linearization approach with a consensus-based protocol was developed for multiple UAV systems to achieve time-varying formation.Using only the measurement of relative position and orientation, a leader-follower structure-based formation control method is presented for multiple UAV systems in Ref.21.A hybrid formation control method that considered formation maintenance and collision avoidance is addressed in Ref.22.

The aforementioned works focused on the formation control problems of MASs,which mainly consider formation maintenance or stabilization.However,formation tracking problems where the followers are required to form a timevarying formation while tracking the trajectory of the leader are more common in practical applications.Ref.23studied the formation tracking problems for MASs with first-order dynamics and introduced a virtual leader that provided the position trajectory for followers.The target enclosing problem can be considered as a special case formation tracking problem with one leader.In Ref.5,a distributed target enclosing strategy for MASs with first-order dynamics and switching topologies was proposed.A differential game approach was used to solve the formation tracking problem with collision avoidance for first-order MASs in Ref.24.In Ref.25,complex Laplacian based formation tracking strategies were applied to MASs with first-order and second-order dynamics.It is assumed that nonlinear dynamics,26model uncertainty,27incomplete information,28unknown disturbance,29and time-varying delays30are inevitable in any practical control system.

The goal of this paper was to study formation tracking control problems for second-order MASs with time-varying delays.A time-delayed formation tracking protocol using neighboring relative state information and, using a Lyapunov-Krasovskii functional approach,conditions for time-delayed MASs with second-order dynamics suf ficient to realize formation tracking are proposed.In addition,an approach to determine the unknown gain matrix in the neighboring relative state information based protocol is presented.Considering the dif ficulty of measuring relative velocity information,a neighboring relative position information based time-delayed formation tracking protocol is designed.Using the Lyapunov-Razumikhin functional method,conditions sufficient for time-delayed MASs with second-order dynamics to achieve formation tracking are given.Numerical examples for multiple UAVs to enclose a moving target using the developed methods are presented to demonstrate application.

Compared with previous research,the contributions of this paper are threefold.First,it considers the effects of time-varying delays on formation tracking control of MASs with second-order dynamics from a practical perspective.Second,instead of requiring all the agents to achieve a predefined formation,the leader provides the trajectory reference to partial followers,and the states of followers are designed to form a predefined time-varying formation while tracking those of the leader.Third,this paper studies the formation tracking control methods of second-order MASs,which are more common than first-order MASs in practical applications.

The paper is organized as follows:Mathematical preliminaries(a basic introduction about graph theory)and the problem statement are introduced in Section 2.In Section 3,a neighboring relative state information based formation tracking protocol is presented along with an approach to obtain the unknown gain matrix in the protocol.A neighboring relative position information based formation tracking protocol is proposed in Section 4.Numerical simulation results are given in Section 5,which are followed by conclusions in Section 6.

Throughout this article,⊗is used to represent the Kronecker product.

2.Preliminaries and problem description

2.1.Graph theory

2.2.Problem description

Let us consider a second-order MAS withNagents on a directed graphG.The individual agentiis represented as the nodesiinG.Fori，j∈ ｛1，2，...，N｝,the interaction channel from agentito agentjis represented by the edgesijand the nonnegative adjacency interaction strength is represented bywji.Suppose that there areN-1 followers and one leader.LetF= ｛1，2，...，N-1｝ denote the follower subscript set.The dynamics of the followeri(i∈F)can be described by

where pi(t)∈Rn,vi(t)∈Rnand ui(t)∈Rnare the position,velocity and control input vectors of the followeri,respectively;αp∈ R and αv∈ R are damping constants.The dynamics of the leaderNcan be described by

where pN(t)∈Rnand vN(t)∈Rnare the position and velocity of the leaderN.For simplicity of presentation,assumen=1 unless otherwise specified.However,all the research results hereafter are still effective forn＞1 by using the Kronecker product.

Definition 1.For any given bounded initial states,MAS Eq.(1)is said to realize time-varying formation tracking if

Definition 2.For any given bounded initial states,MAS Eq.(1)is said to realize consensus tracking if

Remark 2.From Definition 2,one can obtain that if hi(t)≡0(i∈F),MAS Eq.(1)realizes point formation tracking,which is also known as consensus tracking.Therefore,the consensus tracking problem is just a special case of the formation tracking problem.

According to the neighboring property of the leader and followers,the Laplacian matrix L has the form of

where L1∈ R(N-1)×(N-1)and L2∈ R(N-1)×1.The interaction topologies among followers are denoted byGF.

Assumption 1.There exists at least one directed path from the leader to each follower.

3.Formation tracking control using neighboring relative state information

This section examines a formation tracking protocol with unknown gain matrix and time-varying delays based on neighboring relative state information.Also included are a method to design the unknown gain matrix and the conditions suff icient for second-order MASs with time-varying delays to realize formation tracking based on the Lyapunov-Krasovskii functional approach.

wherei∈F; α =[αp，αv];K=[k11，k12]are constant gain matrix;and τ(t)with 0≤τ(t)≤τ0and|˙τ(t)|≤q＜1 is the time-varying delay.

Lemma 2.For any given bounded initial states,MAS Eq.(1)with protocol Eq.(5)achieves time-varying formation tracking if

Proof.Note that L1is nonsingular.It has

From Eqs.(6)and (9),itcan be obtained that limt→∞ξF(t)=0 is equal to

that is

where Icis a column vector of appropriate dimension with 1 as its element.

From Eq.(11),it can be obtained MAS Eq.(1)with timevarying delays achieves time-varying formation tracking under protocol Eq.(5)if and only if limt→∞ξF(t)=0. □

Taking the time derivative of Eq.(9),one can obtain that

Sufficient conditions for MAS Eq.(1)with time-varying delays to realize formation tracking follow.

Lemma 333.Letη(t)∈ Rnbe a vector-valued function with first-order continuous-derivative entries.Then for any vectorsC1∈Rn,C2∈RnandX=XT≥0,the following integral inequality holds

whereψ(t)=[ηT(t)，ηT(t- τ(t))]T.

Theorem 1.Under time-delayed protocols Eq.(5),MAS Eq.(1)with time-varying delays realizes formation tracking if the following conditions are satisfied simultaneously:

(i)

(ii)There exist2×2positive symmetric matricesPF=PTF＞0,QF=QTF＞0,SF=STF＞0,T1andT2suchthatfor i∈ ｛1，2，...，N-1｝the following LMI is feasible:

Proof.If condition(i)holds,one has

Consider the stability of the following systems:

Choose the following Lyapunov-Krasovskii functional candidate:

Taking the time derivative ofVFialong the trajectory of Eq.(20),one has

From Eqs.(21)–(25),it can be obtained that

Remark 3.Ifthe gain matrix K isunknown,then Πi＜ 0(i=1，2，...，N-1)become nonlinear matrix inequalities.Note that the unknown gain matrix K cannot be obtained in Theorem 1.

Proof.In order to obtain controller gain K,choose

Based on Schur complement, Πi＜ 0(i=1，2，...，N-1)are equivalent to

Choose T1=-PFand T2=QF.Then,W-1Fhas the following form:

It holds that

Based on the above results,a method to obtain the unknown gain matrix in the protocol Eq.(5)is given in the following theorem.

4.Formation tracking control using neighboring relative position information

This section discusses MAS Eq.(1)with single damping constant where αp=0.A neighboring relative position information based formation tracking protocol is followed by the sufficient conditions for second-order MASs with timevarying delays to achieve formation tracking according to the Lyapunov-Krasovskii functional approach.

Consider the following formation tracking protocol with time-varying delays:

The following lemma holds directly based on a similar analysis of Lemma 2.

Lemma 4.MAS Eq.(1)with time-varying delays achieves time-varying formation tracking if

Sufficient conditions for MAS Eq.(1)with the time-varying delays to realize formation tracking follow:

Lemma 5.For any real vectorsW1andW2with same dimension,it can be obtained that

whereΔpis any positive definite matrix with suitable dimension.

Theorem 3.MAS Eq.(1)realizes time-varying formation tracking under protocol Eq.(32)with time-varying delays if the following conditions are satisfied simultaneously:

where w is a constant with w＞1.

Proof.According to Eqs.(17)and(19),it can be obtained that condition(i)guarantees that

Choose the following Lyapunov functional candidate:

According to Lemma 5,one can obtain

one can obtain

Remark 4.Communication is limited in practical environments.In contrast to measurement of relative velocity information,relative position information is more easily obtained by sensors such as radar,ultrasonic,and infrared distance.Therefore,neighboring relative position formation based formation tracking protocol is more effective for MASs.

5.Applications in target enclosing of multiple UAVs

The following two examples about target enclosing of multiple UAVs are presented to illustrate the effectiveness of the obtained theoretical results.Fig.1 shows the target enclosing process of multiple UAVs,where the leader UAV is marked by ‘L”.

The dynamics of UAVs can be divided into attitude dynamics and trajectory dynamics.The time constants for the attitude dynamics are much smaller than those for trajectory dynamics;35therefore,the formation control system of UAVs can be decoupled into an outer-loop system and an innerloop system.The inner-loop is used to track attitude while the outer-loop is used to guide a UAV toward the specified position with a specified velocity.Fig.2 shows the two-loop control structure for a multiple UAV system.The design of the outer-loop is our concern in this paper.The obtained theoretical results can be used to deal with formation tracking control problems of a multiple UAV system with timevarying delays.

The dynamics of each UAV are described by

The time-varying formation for the followers is defined as

The interaction topologies of a multiple UAV system in the examples are given in Fig.3.In this example,we assume that the interaction topologies have 0-1 weights.The control objective is to require that followers maintain a time-varying circular parallelogram formation while enclosing the moving leader.Let the initial position and velocity of each UAV be given bypkj(0)=15+ ∈(k=1，2，3，4，5;j=1，2) andvkj(0)=10+ ∈(k=1，2，3，4，5;j=1，2),respectively,where ∈represents a random distribution value on the interval(0，1).Choose the time-varying delay as τ(t)=0.2+0.1sint.In the following figures,the initial states of five UAVs are represented by circle markers,state trajectories of the leader and followers are solid lines,and final states of the leader and followers are represented by triangle markers and asterisk markers,respectively.Coordinatexand Coordinateyrepresent the position components in thexandydirections,respectively.Velocityxand Velocityyrepresent the velocity components of thexandydirections,respectively.

5.1.Target enclosing using neighboring relative state information

Choose the constant gains as αp=0.01 and αv=-1.15.Assume that gain K is unknown,applying the Feasp solver in MAYLAB LMI Toolbox to solve the LMI inequality in Theorem 2,one can gettheunknown gain matrix K=I⊗[-1.5256，-0.5872].The formation tracking feasibility conditions in Theorem 1 are satisfied.

Fig.4 shows the state trajectories of the five UAVs within 60 s.Fig.5 shows the state snapshots of the five UAVs att=60 s.From Figs.4 and 5,it can be obtained that the states of the four followers form a time-varying circular parallelogram formation,and the state of the leader stays in the center of the parallelogram.Therefore,the target enclosing of multiple UAVs is realized.

5.2.Formation tracking using neighboring relative position information

According to the interaction topologies in Fig.2,it can be obtained that~λ=1.309.Choosetheconstantgain as αv=-2.5 andw=1.00001.The formation tracking feasibility conditions in Theorem 3 are satisfied.

Fig.6 shows the state trajectories of the five UAVs within 60 s.Fig.7 shows the state snapshots of the five UAVs att=60 s.From Figs.6 and 7,it can be obtained that the states of the four followers form a time-varying circular parallelogram formation and the state of the leader stays in the center of the parallelogram.Therefore,the target enclosing of multiple UAVs is achieved.

6.Conclusions

(1)Formation tracking control problems for second-order MASs with time-varying delays were studied in this paper.

(2)A formation tracking protocol with time-varying delays was constructed using neighboring relative state information.Based on the approach of replacing variables,a method to obtain the unknown gain matrix in the neighboring relative state information based protocol was proposed.Due to the difficulty of measuring neighboring relative velocity information,a neighboring relative position information based protocol was designed.

(3)Sufficient conditions for second-order MASs with timevarying delays to realize formation tracking were presented.

(4)Simulations showed that the obtained results were effective for target enclosing problems of multiple UAVs.

(5)Further designs of formation tracking control methods for MASs with system uncertainty and external disturbance should be developed in the future.

Acknowledgments

This study was co-supported by the National Natural Science Foundation ofChina (Nos.61333011,91216304 and 61121003).

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24 March 2016;revised 19 July 2016;accepted 26 August 2016

Available online 7 December 2016

Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.

E-mail addresses:lianghan@buaa.edu.cn(L.Han),xwdong@buaa.edu.cn(X.Dong),liqingdong@buaa.edu.cn(Q.Li),renzhang@buaa.edu.cn(Z.Ren).

Peer review under responsibility of Editorial Committee of CJA.