Containment control for heterogeneous multi-agent systems with time delay via an output regulation approach

WEI Wenjun， LÜ Jiahui， Huang Julong， GE Junde

(1. School of Automation and Electrical Engineering， Lanzhou Jiaotong University， Lanzhou 730070, China；2. Key Lab of Opti-Electronic Technology and Intelligent Control of Ministry of Education，Lanzhou Jiaotong University， Lanzhou 730070, China；3. China Energy Engineering Group Gansu Electric Power Design Institute Co., Ltd.， Lanzhou 730050, China)

Abstract： This paper presents the containment analysis and design of heterogeneous linear multi-agent systems (MAS) with time-delay under the output regulation. The leaders are treated as exosystems and an modified output regulation error is designed, which can deal with more than one leader in containment control, then the containment problem will be turned into an output regulation problem. A novel analysis framework of the output regulation is proposed to design a dynamic state feedback control law for containment error and distributed observer when the agents cannot receive external system signal, which guarantees the convergence of all follower agents to the dynamic convex hull spanned by the leaders. The system stability for time-delay containment is proved by the output regulation method instead of the Lyapunov method. Finally, a numerical example is given to illustrate the validity of the theoretical results.

Key words： heterogeneous multi-agent systems； output regulation； linear time-delay systems； containment control

0 Introduction

Multi-agent systems (MAS) are developed on the basis of the study of collective movement of animals and birds. In recent years, it has attracted wide attention of scholars. The control of MAS has been widely applied in the areas of robot cooperative work, unmanned aerial vehicle(UAV) formation, wireless sensor network, smart grid, etc[1-4]. The control objectives of the system mainly include multi-agent formation, aggregation, clustering and so on, in which consensus is the most important basic research problem. As a special situation of consensus, the output regulation has developed rapidly in recent years and becomes a hot topic in the field of multi-agent. The output regulation problem (ORP) aims to design a distributed feedback control law for a multi-agent system to drive the tracking error of each follower to the origin asymptotically while rejecting some classes of external disturbance[5]. There have been two methods for the study of the ORP, namely distributed internal model method and distributed observer method. The robust ORP of linear MAS was presented in Ref.[6], which adopted the internal model method to realize the tracking and interference suppression that was generated by external systems. The problem of cooperative semi-global robust ORP for a class of minimum phase nonlinear uncertain MAS was discussed in Ref.[7]. By using the method of internal model, it changed the cooperative semi-global robust ORP into a cooperative semi-global robust stabilization problem called the augmented system. Through internal model approach in Ref.[8], the studies were done on the cooperative global robust ORP for a class of nonlinear MAS with switching network. By using the second method, the distributed observer could approximate the leader’s system parameters and provide the leader’s signal for followers, thus the cooperative ORP of discrete-time linear MAS was solved[9].

According to the number of leaders in MAS, the system can be divided into single-leader and multi-leader situations. The containment control is a typical case of multi-leader whose essence is that a group of followers, under the guidance of multiple leaders, achieve and maintain the minimum geometric space (convex hull) produced by leaders[10]. The problem of containment control first appeared in Ref.[11], and then many research results appeared. The multi-agent containment of first-order agent and second-order agent under directed graph proposed a containment control protocol based on static leader and dynamic leader, which were presented in Refs.[12-13]. The containment control for Lagrange MAS with many dynamic leaders was studied in Ref.[14], which had nonlinear uncertainly and external disturbances in the digraph. Subsequently, a lot of research has been done for multi-agent input saturation. Su et al. realized input saturated containment control of MAS under semi-global state and output feedback protocol by adopting a low-gain feedback method[15]. The containment control of MAS under topology switching by combining state and output feedback was fulfilled by containment protocol with low gain feedback in Ref.[16].

In practice, time-delay is unavoidable in MAS, and extremely large time-delay may break down the stability of a normal system. The research about containment control for MAS with time delay has just started. For the multiple static or dynamic leaders, containment control for second-order MAS with time-varying delays has given a linear matrix inequality (LMI) method to realize the conditions for containment time-delay control[17]. The method of containment control for heterogeneous multi-agent systems with time-delay and fixed time-delay is given in Refs.[18-19]. The Lyapunov-Krasovskii method was introduced to prove the stability of containment control problem for MAS with time-delay in Refs.[20-22], but this method was complicated and required advanced mathematical skills.

Inspired by Refs.[24-25] which presented the time-delay issues for linear systems by ORP method, we will research the containment control of linear heterogeneous MAS in Ref.[23] plus condition time delay by ORP method.

To our knowledge, there has been no study on leader-following containment control issue of linear heterogeneous multi-agent with state time-delay by an ORP approach.We have solved the problem, which is the first contribution in our work. Based on ORP theory, a time-delay containment control law is constructed to solve the problem of multi-leader time-delay containment control. A dynamic observer is also designed to estimate leaders’ states quickly when some agents cannot obtain leaders’ information. The second contribution is that a novel concise proof method is applied to prove the stability of time-delay containment control, which is different from the general Lyapunov method. In addition, the dynamics of followers is heterogeneous while the MAS of multiple leaders have directed fixed topology.

1 Preliminaries

1.1 Graph theory

Graphs are used to express the communication relationship among the MAS. For a group ofnagents, the corresponding directed graphGcan be represented by a weighted digraphG={V,ε,A}, where a node set isV={1,2,…,n}, and edge set isε⊆V×V. A directed edge of graphGisε(i,j), which means agentican transmit information to agentjand also, agentiis a neighbour of agentj.Ni={j∈V∶(i,j)∈ε,i≠j} represent the set of neighbours of agenti. The adjacency matrixA=[aij]∈Rn×nassociated withGis defined asaij>0 if (i,j)∈ε, otherwiseaij=0. It is called the weighted adjacency matrix ofG. The Laplace matrixL=[lij]∈Rn×nof graphGis defined as

(1)

1.2 Problem statement

We will study the containment control problem of time-delay heterogeneous MAS based on output regulation. Firstly, several definitions are given.

Definition1[26]The MAS considered a group ofn+magents, if an agent has no neighbor, it is called a leader, otherwise it is called a follower.

Definition2[13]A setC⊆Rnis a convex if (1-z)x+zy∈C, for anyx,y∈Candz(0

Co(X)=

(2)

The MAS consists ofnheterogeneous agents as the followers with linear dynamics andmhomogeneous agents as the leaders. The linear time-delay dynamics of theith follower are given by

(3)

wherexi(t-τi)⊆Rnandui(t)∈Rpare the state input and control input of theith follower, respectively;τ0=0 andτi>0(i=1,…,n) are the delays in the system state,τmin=min{τ1,…,τn};AiandBiare constant matrices of appropriate dimensions. The dynamics of the leaders are given by

(4)

wherevk(t)∈Rnis the state of thekth leader.

Based on the output regulation theory, the containment control problem of time-delay heterogeneous systems (3)-(4) is defined as follows.

Problem1Given time-delay heterogeneous systems (1)-(2) realize containment control if the control protocol designed for each follower can guarantee that all followers will converge to the convex hull spanned by the leaders ast→∞, that is, ∀i∈F,

(5)

we will introduce the containment control error vector of theith follower as[23]

(6)

According to graphGand the Laplace matrixL, the overall containment control error vector can be rewritten as

(7)

(8)

(9)

(10)

ast→∞, Eq.(10) denotes the convex combination of all leaders.

Now we describe the linear cooperative output regulation problem as follows.

Problem2Given the time-delay heterogeneous systems (3)-(4), and the containment control error (6), we find a distributed dynamic state time-delay feedback control protocol such that the following properties hold.

In order to deal with the problem, we give some assumptions as follows：

Assumption1Shas eigenvalues with positive real parts.

Assumption3The linear matrix equation

S=Ai+BiUi,i∈F,

(11)

has solutionsUi.

Assumption4The directed graphGis strongly connected.

2 Some technical lemmas

In practical applications, some agents in an MAS cannot obtain the signals of the external system, that is, the information of the leader. Thus, the state of the external system needs to be estimated by the observer. Here, we consider the distributed state time-delay feedback control protocol for theith follower agent as

ui(t)=Ki,1xi(t-τi)+Ki,2ηi(t),

(12)

whereKi,1，Ki,2∈Rp×Nare gain matrices;γis a positive constant;ηi(t)∈RNis the state observer, which denotes the status of the external system.

The closed-loop system of the time-delay heterogeneous system (3)-(4), the containment control error (6) and the distributed state time-delay feedback control protocol can be rewritten as

(13)

First, the M-matrice is used to prove that the followers of MAS should assemble to the convex hull. Some properties of the matrixΘkdefined in Eq.(8) are listed as follows:

Lemma1[23]Given assumption 4 holds,Θkis positive definite also,Θkis a nonsingularM-matrix. Then we have the following results.

i) The real parts of all eigenvalues ofΘkare positive;

(14)

ast→∞. After some manipulation, we obtain

(15)

Then Eq.(15) can be written as

x(t-τ)=

(16)

From Eq.(16), we can get

(17)

Note that we have

(18)

Inspired by Refs.[24-25], the following lemma is proposed for the matrix equations for the output regulation time-delay system.

Lemma3Considering the closed-loop system (13), assumingAcisHurwitz, if a constant matrixXcexists, it should obey the following linear matrix equations fork∈R, that is,

(19)

(20)

(21)

Lemma4[27]Given that the communication network of the system is fixed and directed, and satisfies that in a multi-agent system, there is at least one leader with a directed path to each follower, all followers (starting from any initial state) will converge asymptotically into the static convex hull of the leader state if and only if either of the following equivalent conditions is satisfied.

i)τ∈(0,τmin) withτmin=π/(2λk)=λmax(L).

ii) The Nyquist plot ofΓ(s)=e-τs/shas a zero encirclement around -1/λk， ∀k>1.

Moreover, forτ=τmin, the system has a globally asymptotically stable oscillatory solution with frequencyω=λk. By adding control parameters to the control protocol to change the eigenvalue of matrixL, the containment control of the system can be guaranteed according to the actual requirements.

Currently, most studies on the ORP focus on the consensus problem with only one leader agent, which corresponds to one solution of linear matrix equation. Due to the multi-leader containment control problem studied in this work, many pairs of equations are needed to satisfy lemma 3.

Ki,2=Ui-Ki,1,i∈F.

(22)

From Eq.(11) we have

(23)

whereK1=diag{K1,1,K2,1,…,Km,1}，K2=diag{K1,2,K2,2,…,Km,2},A=diag{A1,A2,…,Am} andB=diag{B1,B2, …,Bm}.

3 Main results

Theorem1Under assumptions 1-3, through the state time-delay feedback control protocol Eq.(12), the problem of output regulation time-delay containment control can be solved, whereKi,1∈Rp×N,i∈F, andKi,2is given by Eq.(22).

Proof： By using the control protocol (12), the state of theith follower agent can be written as

(24)

Also the state observer of the follower agent can be written as

(25)

By denoting

the system (13) can be formed as

(26)

(27)

Now we verify property 2 of problem 2. Inspired by Ref.[7], we use the Kronecker product property, (A⊗B)(C⊗D)=(AC)⊗(BD).

Under assumption 3, let

(28)

By matrix Eq.(19) and e(IN⊗A)t=IN⊗eAt, we have

In(ΘkIn)⊗(Ine-Sτ)=

(29)

where e-Sτis time-delay factor,Ine-Sτ=In, also

(30)

For the first part of Eq.(19), we have

(31)

Also,

(32)

Remark1By solving problem 2, from lemma 2 it follows that the follower agents (3) will converge to the convex hull formed by the leaders (4) ast→∞.

(33)

4 Numerical example

On the basis of theoretical analysis, this section uses simulation examples to verify the control of time-delay heterogeneous MAS based on output regulation method. The heterogeneous MAS is composed of seven agents as shown in Fig.1, where the nodes 1-4 denote the heterogeneous followers and 5-7 denote the homogeneous leader agents.

Fig.1 Interaction network topology graph G

The matrices corresponding to the dynamic equation of follower systems (3) are given by

By solving Eq.(11), we obtain

The Laplace matrixLof Fig.1 is

according to lemma 4,τmin=0.08.

Under the distributed state time-delay feedback control protocol Eq.(12), we get the results, as shown in Figs.2 and 3, respectively. The time-delay factorτ=0.01.

(a) xi,1(t)

(b) xi,2(t)

Fig.2 shows that the containment control is realized by using the distributed state time-delay feedback control protocol Eq.(12) with time-delay coefficientτ=0.01. The curves trend of control errors Eq.(21) is shown in Fig.3 under these circumstances, from which the containment errors turn to zero gradually.

(a) ei,1(t)

(b) ei,2(t)

Then we give the simulation results when the delay coefficients areτ=0.08, 0.105, respectively, as shown in Figs.4-7.

As seen from Figs.4 and 5, when the delay coefficientτ=0.08, MAS can achieve containment control, but the time required for containment control is slightly longer and the convergence rate of tracking containment control error is slower. From Figs.6 and 7, it can be found that when the delay coefficientτ=0.105, MAS can barely achieve the containment control. At this time, it takes a long time to realize the containment control, and the tracking containment control error is difficult to converge to zero asymptotically.

(a) xi,1(t)

(b) xi,2(t)

(a) ei,1(t)

(b) ei,2(t)

(a) xi,1(t)

(b) xi,2(t)

(a) ei,1(t)

(b) ei,2(t)

5 Conclusions

By using a distributed dynamic state time-delay feedback control protocol, the containment control of heterogeneous MAS with time-delay based on ORP method has been studied. A containment control error function of output regulation is designed, which makes the dynamic tracking error of the system be zero at last, and solves the containment control of multi-leader agents. Finally, under any initial conditions, the trajectories of all follower agents converge to be in the convex hull of the leader’s trajectory. The time-delay is assumed to be constant in this work. Nonlinear dynamics and time-varying delay will be considered in the same problem in the future work.