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Research article

Consensus control of multi-agent systems with delays

  • Received: 13 June 2024 Revised: 16 July 2024 Accepted: 05 August 2024 Published: 14 August 2024
  • This paper concerns the consensus problem of linear time-invariant multi-agent systems (MASs) with multiple state delays and communicate delays. Consensus control is widely applied in spacecraft formation, sensor networks, robotic manipulators, autonomous vehicles, and others. By introducing a linear transformation, the consensus problem of the delayed MAS under an undirected network was converted into a robust asymptotic stability problem associated with the eigenvalues of the normalized Laplacian matrix of the network. By means of the argument principle and optimization technologies, a numerical controller design method was presented for the delayed MAS to reach consensus. The effectiveness of the proposed approach was illustrated by some numerical examples. The proposed approach may be applied to multi-agent systems with distributed delays.

    Citation: Yi Gong. Consensus control of multi-agent systems with delays[J]. Electronic Research Archive, 2024, 32(8): 4887-4904. doi: 10.3934/era.2024224

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  • This paper concerns the consensus problem of linear time-invariant multi-agent systems (MASs) with multiple state delays and communicate delays. Consensus control is widely applied in spacecraft formation, sensor networks, robotic manipulators, autonomous vehicles, and others. By introducing a linear transformation, the consensus problem of the delayed MAS under an undirected network was converted into a robust asymptotic stability problem associated with the eigenvalues of the normalized Laplacian matrix of the network. By means of the argument principle and optimization technologies, a numerical controller design method was presented for the delayed MAS to reach consensus. The effectiveness of the proposed approach was illustrated by some numerical examples. The proposed approach may be applied to multi-agent systems with distributed delays.



    Consensus control of MASs is one of the basic problems in the control community, having attracted considerable attention. The basic idea of the consensus control problem is to design a distributed protocol for each agent by using the information of the agent and its neighbors such that all the agents in the network asymptotically reach a common value [1]. In recent years, consensus control has been widely applied in spacecraft formation, sensor networks, robotic manipulators, autonomous vehicles, and others (see for instance, [2] and the references therein).

    In practice, MASs are often accompanied with time delays that arise in sensor response or information transmission, which might cause negative effects on the consensus performance or, even worse, make the system fail to achieve consensus [3]. Therefore, the consensus problem of MASs with time delays is a challenging topic that is worthy of investigating.

    In recent years, many research works involving consensus problem of linear time-invariant MASs in the presence of time delays have been developed. These works have been conducted under different perspectives. For example, based on the frequency domain analysis for MASs with communication delays, a large number of works have dedicated to seeking the consensus margin for system parameters such that the consensus can be achieved within this range [4,5,6,7,8,9,10,11,12,13,14]. Among these, the delay margin problem was investigated in [4,5,6,7,8,9], while the gain margin problem was studied in [10,11,12]. Using Lyapunov functional and linear matrix inequality technologies, sufficient conditions for the consensus of delayed MASs were derived in [15,16,17,18,19]. By means of the algebraic Riccati equation, a delay-dependent consensus protocol was proposed by [20]. In [21], approaches including Lyapunov theorems and the Nyquist stability criterion were used to study consensus algorithms for MASs with both communication and input delays. Using z-transformation and Routh criterion, a necessary and sufficient condition for the consensus of discrete-time MASs with communication delay was provided by [22]. Based on the impulsive observer and piecewise Lyapunov functional, sufficient conditions in terms of linear matrix inequalities for consensus of MASs with single delay were derived in [23]. Besides, leader-follower consensus problems of MASs with time delays were investigated in [24] and [25]. In [26], the consensus problems of MASs with time-varying delays were considered. In [27], the consensus problems of MASs with the random input delay were discussed.

    Overall, the existing works have been focused on communication delays in the information transmission or a single delay in system interior.

    In practical MASs, multiple time delays might arise in system states caused by mechanical transmission, sensor sensing, hydraulic process, and others. These state delays affect the system performance and complicate the problem analysis. For general linear MASs with multiple state delays, it is still challenging work to design a consensus protocol. Motivated by the above statements, this paper investigates the consensus problem of general continuous-time linear MASs with multiple state delays and communication delays. We consider a multi-agent networks with N agents intercommunicated by a fixed undirected networks graph; the dynamics of agent i is given by

    ˙xi(t)=md=0Adxi(tτd)+Bui(t),yi(t)=Cxi(t), (1.1)

    where xiRp and uiRq are system states and the control input of agent i, respectively, AdRp×p for d=0,1,...,m are system matrices, and BRp×q is input matrix, 0=τ0<τ1<...<τm are time delays.

    Let us give the output feedback control protocol

    ui(t)=md=0(KddijNiaij[yi(tτdτc)yj(tτdτc)]), (1.2)

    where τc>0 represents the communication delay, aij is the entries of the network adjacency matrix, KdRq×p for d=0,1,...,m are feedback gain matrices to be determined.

    Considerations on the structure of protocol (1.2) are illustrated as two aspects: On the one hand, the agent i receives a set of available historical information with respect to its neighbors; on the other hand, for general linear systems with multiple delays, the more information of delayed states being used, the more possible it is to stabilize it.

    The consensus problem in this paper is: for the delayed MAS (1.1) with the protocol structure (1.2), determining the gain matrices K0,...,Km such that limtxi(t)xj(t)=0 for i,j=1,2,...,N.

    Along the line of [28], we investigate consensus problem of the delayed MAS (1.1) with the protocol structure (1.2). By introducing a linear transformation, we will show that the consensus of N agents is equivalent to the simultaneous asymptotic stability of N1 delayed subsystems associated with the eigenvalues of normalized Laplacian matrix of the network. We prove that a sufficient condition for the consensus of the delayed MASs is that a linear delay system with an uncertain parameter is robust stable, where the uncertain parameter covers the interval bounded by the maximum and minimum eigenvalues of the normalized Laplacian matrix. Then, using the argument principle and the properties of state transition matrix, we derive a sufficient condition for the consensus of the delayed MAS. Based on the obtained condition, we construct an optimization scheme to numerically solve the feedback gain matrices.

    The main contributions of this paper are summarized as follows:

    1) Compared with the existing works, this paper considers multiple state delays in linear MASs and designs a distributed consensus protocol with neighbor historical output information.

    2) By the argument principle and optimization technologies, a sufficient condition is derived for the consensus of the MAS with multiple state delays. Based on this, an optimization-based method for calculating the consensus gain matrices is presented.

    The rest of this paper is organized as follows. Section 2 introduces the basic material on graph theory and the stability criteria based on the argument principle needed in this paper. The main results are given in Section 3. In Section 4, simulation examples are presented to illustrate the accuracy of the theoretical results. Section 5 is a brief conclusion.

    Notations: Let Rp be the p-dimensional Euclidean space, Rp×q represents the p×q real matrix, x denotes Euclidean norm of matrix xRp×q, Ip denotes p×p identity matrix, 1p denotes p-dimensional column vector with entries all being 1, 0p×q denotes p×q matrix with all the elements being zero. The notation diag{} denotes a block-diagonal matrix. The notation denotes the Kronecker product. The notations ker(W) and im(W) denote the kernel and image of matrix W, respectively. The notation a denotes the smallest integer that is greater than or equal to aR.

    Let a simple graph (no self-loops or multiple edges) G={V,E,A} denote the undirected communication topology between MASs with the set of vertices V={1,2,...,N} and the set of edges EV×V. The ith vertex represents agent i, and the edge (i,j) denotes the communication channel between agent i and agent j. The set E{(i,j):i,jV} is the edge set. The set of neighbors of the ith agent is denoted by Ni={jV|(i,j)E}. A=[aij]RN×N is called the weighted adjacency matrix of G with nonnegative elements, where aij=aji>0 if (i,j)E and ij; aij=0 otherwise. The degree of the ith vertex is denoted by Nj=1aij and the degree matrix D=diag{d1,d2,...,dN}. The Laplacian matrix L of G is defined by L=DA. Note that L1N=0N. Let L:=D1L be the normalized Laplacian matrix associated with G by replacing the original weight aij with a new weight aijdi. For an undirected graph, L is a symmetric, positive semi-definite matrix, and all its eigenvalues are nonnegative. The eigenvalues of L can be arranged as 0=λ1(L)<λ2(L)...λN(L). Throughout this paper, we consider connected and undirected network graph.

    For preparation, we review some results in [28] that provide stability criteria for linear time-invariant delay systems described as

    ˙x(t)=md=0Adxi(tτd). (2.1)

    Lemma 2.1. Consider the linear delay system (2.1) with characteristic equation

    p(s)=det(sIpmd=0Adeτds)=0,

    let s be an unstable characteristic root of p(s), then |s|r=md=0Ad. Furthermore, system (2.1) is asymptotically stable if and only if

    p(s)0forsl,andΔlargp(s)=0,

    where l=l1l2 is a closed half-circumference with radius r, l1={s:s=iw,rwr,i2=1} and l2={s:|s|=r,π/2argsπ/2}, the notation argp(s) stands for the argument of complex function p(s), and Δlargp(s) stands for the change of the argument as the complex variable s traverses the contour l once in the positive direction.

    The explanation for Lemma 2.1 is as follows. First, we obtain that all the unstable characteristic roots of p(s) are contained in a region Ω, which is enclosed by the half-circumference l (see Figure 1). Then, by means of the argument principle, the number of the unstable roots located in Ω is equal to Δlargp(s)/2π. Hence, by calculating Δlargp(s), we can check the stability of system (2.1). The following Algorithm 1 is to check the stability of a linear delay system described by (2.1).

    Figure 1.  Explanation for Lemma 2.1.

    In this section, we first prove that delayed MAS (1.1) can achieve consensus if and only if N1 delayed systems associated with the eigenvalues of the Laplacian matrix of the network are simultaneously asymptotically stable. With the protocol (1.2), the overall closed-loop dynamics of the delayed MAS (1.1) can be written as

    ˙x(t)=md=0(INAd)x(tτd)+md=0(LBKdC)x(tτdτc), (3.1)

    Algorithm 1 An algorithm to check the stability of linear delay system
      1: Calculate the upper bound r of the unstable roots and get the half-circumference l.
      2: Take a sufficiently large integer nl to discretize l as uniformly as possible in the clockwise direction, and record these nodes as {sk}nlk=1.
      3: For k=1,...,nl, calculate complex value p(sk), check whether p(sk)=0 by evaluating its magnitude satisfies |p(sk)|δ1 with the preassigned tolerance δ1. If it holds, i.e., p(sk)=0, then the system is not asymptotically stable and stop the algorithm. Otherwise, we continue.
      4: Compute Δlargp(s) along the ordered node {sk}nlk=1 by checking |Δlargp(s)|δ2 with the preassigned tolerance δ2. If Δlargp(s)=0, the system is asymptotically stable, otherwise is not stable.

    where x(t)=[x1(t)T,x2(t)T,...,xN(t)T]TRNp represents the aggregate state of the delayed MAS.

    The matrix L is a real symmetric matrix and there exists an orthogonal matrix URN×N such that UTLU=Λ:=diag{0,λ2(L),λ3(L),...,λN(L)}, where 0<λ2(L)λ3(L)...λN(L). By using the linear transformation ˜x(t)=(UTIp)x(t), the overall closed-loop dynamics (3.1) becomes

    ˙˜x(t)=md=0(INAd)˜x(tτd)+md=0(ΛBKdC)˜x(tτdτc). (3.2)

    Theorem 3.1. Under an undirected and connected network graph, the delayed MAS (1.1) with the protocol (1.2) achieves consensus if and only if the following N1 delayed subsystems are simultaneously asymptotically stable:

    ˙˜xi(t)=md=0Ad˜xi(tτd)+λi(L)md=0BKdC˜xi(tτdτc),i=2,3,...,N. (3.3)

    Proof. Since ker(L)=im(1N), xi(t)xj(t)0 if and only if x(t)im(1NIp)=ker(LIp). This holds if and only if (LIp)x(t)0, i.e., (UΛIp)˜x(t)0. Since U is nonsingular, this holds if and only if (ΛIp)˜x(t)0, which is equivalent to ˜xi(t)0 for i=2,3,...,N.

    In order to seek the gain matrices K0,...,Km such that each delayed subsystem of (3.3) is asymptotically stable, we first consider the stability of the following linear delay system with uncertain parameters:

    ˙η(t)=md=0Adη(tτd)+(α+Δα)md=0BKdCη(tτdτc), (3.4)

    where

    α:=0.5(λ2(L)+λN(L))and|Δα|¯Δα:=0.5(λN(L)λ2(L)). (3.5)

    Remark 3.2. In system (3.4), the range of continuous parameter α+Δα covers the closed interval [λ2(L),λN(L)]. It is obvious that if system (3.4) is robust asymptotically stable, then delayed subsystems (3.3) are simultaneously asymptotically stable. Therefore, if there are gain matrices K0,...,Km such that system (3.4) is robust asymptotically stable, then delayed MAS (1.1) can achieve consensus by using these gain matrices.

    In the rest of this section, we will mainly discuss the robust stability problem of system (3.4); its nominal model is

    ˙η(t)=md=0Adη(tτd)+αmd=0BKdCη(tτdτc). (3.6)

    Before giving the main results results in this paper, the following lemmas and definitions are useful.

    Lemma 3.3. ([28]) The nominal system (3.6) is asymptotically stable if and only if

    0||F(t)||2dtβ, (3.7)

    where β is a finite constant, and F(t) represents the state transition matrix of system (3.6), which is the solution of the matrix differential equation

    ˙F(t)=md=0AdF(tτd)+αmd=0BKdCF(tτdτc),F(0)=Ip,F(t)=0p×pfort<0. (3.8)

    Lemma 3.4. Denote the characteristic polynomial of system (3.4) by

    ˜P(s)=det(sIpmd=0Adeτds(α+Δα)md=0BKdCeτdsτcs), (3.9)

    let s be an unstable characteristic root, i.e., ˜P(s)=0, then

    |s|r=md=0(||Ad||+λN(L)||B||||Kd||||C||). (3.10)

    Proof. Since s is an unstable root, Res0, |eτds|1, |eτdsτcs|1, τd,τc0. Define

    W(s):=md=0Adeτds+(α+Δα)md=0BKdCeτdsτcs,

    then ˜P(s)=det(sIpW(s))=0, this implies that s is an eigenvalue of matrix W(s), and it can be deduced that

    |s|=|λk(W(s))|||W(s)||=||md=0Adeτds+(α+Δα)md=0BKdCeτdsτcs||md=0||Ad|||eτds|+(α+Δα)md=0||BKdC|||eτdsτcs|md=0(||Ad||+λN(L)||BKdC||)r.

    Definition 3.5. A semicircular region Ω is defined as Ω={s:Res0,|s|r}, where r is given by (3.10). The boundary of region Ω is denoted as l.

    Next, we present the main results for the robust stability of system (3.4).

    Lemma 3.6. System (3.4) is asymptotically stable if and only if

    ˜P(s)0,forsl (3.11)

    and

    larg˜P(s)=0. (3.12)

    where ˜P(s) is defined by (3.9) and l is given by Definition (3.5).

    Proof. The results can be derived by applying Lemma 2.1 to system (3.4).

    Theorem 3.7. System (3.4) is robust asymptotically stable, if

    1) The nominal system (3.6) is asymptotically stable;

    2) The condition

    supsl||R1(s)||md=0||BKdC||<1¯Δα (3.13)

    holds, where

    R(s):=sIpmd=0Adeτdsαmd=0BKdCeτdsτcs, (3.14)

    and l is given by Definition 3.5.

    Proof. Since the nominal system (3.6) is asymptotically stable, R(s)1 exists for sl. Let P(s)=det(R(s)) be the characteristic polynomial of the nominal system (3.6), let D(s)=Δαmd=0BKdCeτdsτcs. For sl, we have

    ˜P(s)=det(sIpmd=0Adeτds(α+Δα)md=0BKdCeτdsτcs)=det(R(s)D(s))=det(R(s))det(IpR1(s)D(s))=P(s)det(IpR1(s)D(s)).

    Thus,

    ˜P(s)=P(s)det(IpR1(s)D(s)). (3.15)

    According to Lemma 3.4 and Definition 3.5, all the unstable characteristic roots of ˜P(s) lie in region Ω. It is sufficient to check whether ˜P(s) has roots in Ω for the stability of system (3.4). Using (3.15), for sl,

    ˜P(s)=P(s)pk=1[1λk(R1(s)D(s))]. (3.16)

    It can be known that the eigenvalue λk(R1(s)D(s)) is continuous for sl. Since the nominal system (3.6) is asymptotically stable, P(s)0 for sl. With the condition (3.13), for sl and k=1,...,p, we have

    |λk(R1(s)D(s))|||R1(s)D(s)||||R1(s)||||D(s)||=||R1(s)||||Δαmd=0BKdCeτdsτcs||||R1(s)|||Δα|md=0||BKdC|||eτdsτcs|supsl||R1(s)||¯Δαmd=0||BKdC||<1,

    i.e.,

    |λk(R1(s)D(s))|<1. (3.17)

    By means of (3.16) and (3.17), we can deduce that for sl,

    ˜P(s)0. (3.18)

    Furthermore, using (3.16) and (3.17), we have that for sl,

    Δlarg˜P(s)=ΔlargP(s)+pk=1Δlarg[1λk(R1(s)D(s))]=0+pk=10=0, (3.19)

    where ΔlargP(s)=0 since system (3.6) is asymptotically stable.

    Using (3.18), (3.19), and Lemma 3.6, it can be deduced that system (3.4) is robust asymptotically stable, and the proof is completed.

    As is illustrated in the foregoing, if system (3.4) is robust asymptotically stable, it is sufficient to obtain that the delayed MAS (1.1) achieves consensus. Combining Lemma 3.3 and Theorem 3.7, we have the following corollary for the consensus of the delayed MAS (1.1).

    Corollary 3.8. Under an undirected and connected network graph represented by a Laplacian matrix L, the delayed MAS (1.1) with the protocol (1.2) achieves consensus if there are gain matrices K0,K1,...,Km such that

    1) 0||F(t)||2dtβ, and

    2) supsl||R1(s)||md=0||BKdC||<1¯Δα.

    where F(t) is subject to (3.8), R(s) is defined in (3.14), and l is given by Definition 3.5.

    For the sake of convenience, we introduce a decision variable K=[K0,K1,...,Km] to stand for all the gain matrices to be determined. Since F(t) and R(s) are functions of K, let us re-denote them as F(K,t) and R(K,s), respectively. Our goal is to calculate K such that the two conditions in Corollary 3.8 are satisfied. Based on this, we construct an optimization problem that minimizes the objective function

    minKJ(K)=T0||F(K,t)||F2dt (3.20)

    subject to equality constraints

    {˙F(K,t)=md=0AdF(K,tτd)+αmd=0BKdF(K,tτdτc),F(K,t)=0p×pfort<0andF(K,0)=Ip, (3.21)

    and inequality constraints

    supsl||R1(K,s)||Fmd=0||BKdC||F<1¯Δα, (3.22)
    md=0||Kd||Fb, (3.23)

    where R(K,s) is given by (3.14), α and ¯Δα are given by (3.5), T is a sufficient large positive constant, b is a positive constant, and the symbol ||||F represents the Frobenius norm.

    Remark 3.9. For solving the optimization problem, T is taken as a finite. Assume that there exists a feasible solution Kf such that the nominal system (3.6) is asymptotically stable, then we can find a constant T0 and a sufficient small value ϵ such that

    |T0||F(Kf,t)||F20||F(Kf,t)||F2|ϵ,

    for TT0. In addition, if ||F(Kf,t)||F decreases faster (the transient process is very short), T0 may not be vary large. Hence, for a sufficiently large constant T, an optimal solution K that minimizes the integral T0||F(K,t)||F2dt can compress the integral 0||F(K,t)||F2dt to the finite.

    Remark 3.10. The constraint (3.23) is necessary, since in engineering practice the cost of the control law should be constrained. On the other hand, this constraint is also used to determine the radius of the semicircle l. According to Lemma 3.4, we have

    r=md=0||Ad||F+bλN(L)||B||F||C||F. (3.24)

    In order to obtain a numerical solution on K, it is necessary to discrete the above optimization problem. Assume that the numerical solution gives a sequence of approximated values {F1(K),F2(K), ...,FM(K)} of {F(K,t1),F(K,t2),...,F(K,tM)} of Eq (3.8) on certain equidistant step-values {tn=nh} with the step-size h=T/M, where M is a positive integer. Let vd=τdh1, δd=vdτdh1, 0δd<1, nd=(τd+τc)h1, εd=nd(τd+τc)h1, 0εd<1, for d=0,...,m. Besides, take a sufficiently small step-size hl to discretize l as uniformly as possible in the clockwise direction, record these nodes as {sk}nlk=1, and denote Nl={1,2,...,nl}. In this paper, these discrete points on l are taken as {sk=iw:w=r+khl,k=0,1,...,2rhl}{sk=rejθ:θ=π2+khl,k=0,1,...,πhl},

    Thus, combining the Euler scheme with the linear interpolation, a numerical solvable version for the optimization problem with the objective function (3.20) subject to the constraints (3.21) and (3.22) is obtained as follows.

    minKJn(K)=Mn=0||Fn(K)||F2h (3.25)

    subject to

    Fn+1(K)=Fn(K)+h[md=0AdFnvd+δd(K)+αmd=0BKdFnnd+εd(K)],Fnvd+δd(K)=(1δd)Fnvd(K)+δdFnvd+1(K)fornvd+δd>0,Fnnd+εd(K)=(1εd)Fnnd(K)+εdFnnd+1(K)fornnd+εd>0,Fnvd+δd(K)=0p×pfornvd+δd<0,Fnvd+δd(K)=Ipfornvd+δd=0,Fnnd+εd(K)=0p×pfornnd+εd<0,Fnnd+εd(K)=Ipfornnd+εd=0. (3.26)
    supkNl||R1(K,sk)||Fmd=0||BKdC||F<1¯Δα, (3.27)

    and

    md=0||Kd||Fb. (3.28)

    Thus, the optimization problem with the objective function (3.20) subject to the constraints (3.21) and (3.22) is reduced to one with the objective function (3.25) subject to the constraints (3.26) and (3.27). This is a nonlinear minimization problem with the quadratic equality constraints and a non-convex inequality constraint.

    Remark 3.11. The numerical method that is used for the discretization of the optimization problem is not limited to the Euler method; other numerical methods such as Runge-Kutta or Multi-step methods can also be applied.

    Remark 3.12. For the inequality constraint (3.27), it should be noticed that R1(K,s) is well defined only when the nominal system (3.6) is asymptotically stable. Hence, an initial value of K that stabilizes system (3.6) is needed for the iteration of the optimization algorithm. Such an initial value can be obtained by minimizing the objective function (3.25) subject to the constraints (3.26).

    Remark 3.13. We note that the constraints may describe a non-convex feasibility region, and thus multiple local solutions can arise [29]. Conventional numerical techniques are efficient at locating an optimum but they do not necessarily locate the global optimum unless some a priori knowledge is available. It is necessary for checking whether the optimal solution K stabilizes the nominal system (3.6).

    Algorithm 2 Computing consensus gain matrices for delayed MAS
      1: Given a starting point K0, a sufficient large constant T, a weight constant ρ.
      2: Solve the optimization problem with objective function (3.25) subject to constraints (3.26) and obtain KI={KI0,KI1,...,KIm}.
      3: For the KI obtained in Step 2, use Algorithm 1 to check whether system (3.6) is asymptotically stable. If KI stabilizes the system, then go to Step 4. Otherwise, increase T or choose another K0 and go to Step 2.
      4: Let KI be the starting point, solve the optimization problem with objective function (3.25) subject to constraints (3.26) and (3.27) and obtain K={K0,K1,...,Km}.
      5: For the K obtained in step 4, use Algorithm 1 to check whether system (3.6) is asymptotically stable. If K stabilizes the system, then stop the algorithm. Otherwise, increase T or choose another K0 and go to Step 2.

    In view of the above statements, we propose an algorithm to numerically compute the consensus gain matrices K as follows.

    Remark 3.14. Our results have a close relationship with the second smallest eigenvalue λ2(L) and the largest eigenvalue λN(L) of the normalized Laplacian matrix L. Denote the lower bound of λ2(L) by bl and the upper bound of λN(L) by bu. From Remark 3.2, we know that as long as Theorem 3.7 holds for

    α=0.5(bl+bu)andΔα=0.5(bubl),

    then delayed MAS (1.1) is consensusable. It is easy to obtain the upper bound of λN(L) that

    λN(L)||L||=2.

    For the lower bound of λ2(L), we have the estimation as follows.

    Lemma 3.15. ([30]) Let G be an undirected and connected simple graph with N vertices, let e(G) denote the edge connectivity of G, i.e., the minimal number of edges whose removal would result in losing connectivity of the graph G, and let λ2(L) denote the second smallest eigenvalue associated with the Laplacian matrix of G. Then

    λ2(L)2e(G)(1cosπN)2(1cosπN).

    Remark 3.16. The proposed algorithm based on Theorem 3.1 is less conservative than the existing methods based on LMI. Theorem 3.1 is a sufficient and necessary condition for asymptotic stability of system (3.3). However, the results based on LMI are only sufficient conditions for asymptotic stability of system (3.3).

    In this section, we provide two examples to illustrate the main results. For both examples, we consider a network graph with the Laplacian matrix and the normalized Laplacian matrix as follows.

    L=[5122110020202002],L=[10.20.40.4110010101001],

    where the eigenvalues of L are λ1(L)=0,λ2(L)=λ3(L)=1,λ4(L)=2.

    For solving the optimization problems in Algorithm 2, we use the interior point method, which is included by the OPTI Toolbox of the MATLAB.

    Example 4.1. Consider the dynamics of agent i given by

    ˙xi(t)=A0xi(t)+A1xi(tτ1)+A2xi(tτ2)+Bui(t),yi(t)=Cxi(t),

    where

    A0=[0100],A1=[00.100],A2=[000.10],B=[21],C=[0.20.1],
    τ1=1,τ2=2,τc=0.5.

    Let the controller structure be

    ui(t)=2d=0(KddijNiaij[yi(tτdτc)yj(tτdτc)])

    with the constraint

    ||K0||F+||K1||F+||K2||F10.

    By calculating, we obtain the radius r=11.20. Let the step-size be h=hl=0.01 and the initial value be K0=01×3.

    For T=10, using Algorithm 2 we obtain the gains

    K0=1.1654,K1=0.0321,K2=0.0192.

    For T=20, we obtain the gains

    K0=1.2054,K1=0.0024,K2=0.0089.

    Using Lemma 2.1, we check that the subsystems (3.3) with the above two groups of gains are both asymptotically stable, i.e., the delayed MAS (1.1) achieves consensus. By taking the initial values as x1(t)=[50,20]T, x2(t)=[50,20]T, x3(t)=[100cos(t),60et]T, x4(t)=[100+sin(t),60et]T for t[2.5,0], we draw the trajectories of the delayed MAS (1.1) with the gain matrices K0=1.2054,K1=0.0024,K2=0.0089 in Figures 2 and 3. From the figures, we can see that the consensus is achieved within 10 seconds.

    Figure 2.  Example 1: trajectories of xi1(t), i=1,2,3,4.
    Figure 3.  Example 1: trajectories of xi2(t), i=1,2,3,4.

    Let the controller structure be

    ui(t)=K0dijNiaij[yi(tτc)yj(tτc)],

    where ||K0||F10.

    For T=20, we obtain the gain K0=1.2166. Using Lemma 2.1, we check that the delayed MAS (1.1) can also achieve consensus with this gain.

    Example 4.2. Consider the dynamics of agent i given by

    ˙xi(t)=A0xi(t)+A1xi(tτ1)+A2xi(tτ2)+Bui(t),yi(t)=Cxi(t),

    where

    A0=[0.2000000110110110],A1=[0.8010000000000000],A2=[0.10.100.200.20.1000.100.300.30.10],B=[0.10.10.10.1],
    C=[0.20.1000.200.10.1],τ1=0.2,τ2=0.3,τc=0.1.

    Let the controller structure be

    ui(t)=2d=0(KddijNiaij[yi(tτdτc)yj(tτdτc)])

    with the constraint

    ||K0||F+||K1||F+||K2||F10.

    By calculating, we obtain the radius r=5.62. Let the step-size h=kl=0.01 and the initial value be K0=01×6.

    For T=10, using Algorithm 2, we obtain the gain matrices

    K0=[0.22461.8883],K1=[0.08511.3535],K2=[0.19691.3399].

    For T=20, we obtain the gain matrices

    K0=[0.22281.8868],K1=[0.08701.3536],K2=[0.19921.3411].

    Using Lemma 2.1 we check that the delayed MAS (1.1) achieves consensus. By taking the initial values as x1(t) = [5,5cos(t),12et,13(t+1)]T, x2(t) = [5,5+sin(t),12et,15(t+1)]T, x3(t) = [10,10(t+1),20cos(t),20et]T, x4(t) = [10,10(t+1)2,20,20+t]T for t[2.5,0], we draw the differences of corresponding components between xi(t) and x1(t) with the second group of gain matrices above in Figure 4. From the figure, we can see that the consensus is achieved within 15 seconds.

    Figure 4.  Example 2: differences of corresponding components between xi(t) and x1(t), i=2,3,4.

    Let the controller structure be

    ui(t)=K0dijNiaij[yi(tτc)yj(tτc)],

    where ||K0||F10.

    For T=20, we obtain the gain matrix K0=[0.52553.1183]. Using Lemma 2.1, we check that the delayed MAS (1.1) achieve consensus.

    Remark 4.3. In principle, the greater T is, the better effect of K is for the consensus problem. However, a great value of T needs more computational effort. In the implementation of solving the optimization problem with objective function (3.25) subject to constraints (3.26) and (3.27), when T is large enough, the gain matrices change a little. The examples show the argument.

    Remark 4.4. The examples show that, under the controller structure with less feedback term (e.g., K=K0), the delayed MAS can also achieve consensus as long as there exists a feasible solution for K such that the Corollary 3.8 holds. Hence, we can flexibly try different controller structures and solve the gain matrices by Algorithm 2.

    This paper investigated the consensus problem of linear MASs with multiple state delays and communication delay under an undirected network graph. The consensus problem of N agents was converted into the simultaneously asymptotically stable problem of N1 delayed subsystems associated with the eigenvalues of normalized Laplacian matrix by employing a linear transformation. We presented a sufficient condition for the consensus of delayed MASs in the form of the integral performance and inequalities, which were derived by integral stability criteria for linear delay systems and argument principle. Based on the obtained condition, we formulated an optimization-based framework to numerically solve the consensus gain matrices. The numerical examples demonstrated that the proposed Algorithm is efficient for the consensus of the delayed MASs.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The author declares there is no conflicts of interest.



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