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Sufficient and Necessary Conditions for Leader-Following Consensus of Second-Order Multi-Agent Systems via Intermittent Sampled Control

Ziyang Wang, Yuanzhen Feng*, Zhengxin Wang, Cong Zheng

School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

* Corresponding Author: Yuanzhen Feng. Email: email

Computers, Materials & Continua 2025, 83(3), 4835-4853. https://doi.org/10.32604/cmc.2025.063179

Abstract

Continuous control protocols are extensively utilized in traditional MASs, in which information needs to be transmitted among agents consecutively, therefore resulting in excessive consumption of limited resources. To decrease the control cost, based on ISC, several LFC problems are investigated for second-order MASs without and with time delay, respectively. Firstly, an intermittent sampled controller is designed, and a sufficient and necessary condition is derived, under which state errors between the leader and all the followers approach zero asymptotically. Considering that time delay is inevitable, a new protocol is proposed to deal with the time-delay situation. The error system’s stability is analyzed using the Schur stability theorem, and sufficient and necessary conditions for LFC are obtained, which are closely associated with the coupling gain, the system parameters, and the network structure. Furthermore, for the case where the current position and velocity information are not available, a distributed protocol is designed that depends only on the sampled position information. The sufficient and necessary conditions for LFC are also given. The results show that second-order MASs can achieve the LFC if and only if the system parameters satisfy the inequalities proposed in the paper. Finally, the correctness of the obtained results is verified by numerical simulations.

Keywords

Intermittent sampled control; leader-following consensus; time delay; second-order multi-agent system

1  Introduction

In recent years, MASs have always been at the forefront of control and network science research. The rapid development of MASs owes mainly to their widespread applications in formation control of UAVs [1], spacecraft attitude coordination control [2], distributed control in microgrids [3], and so on.

Serving as a basis for the control of MASs, studies on consensus are of great practical and theoretical significance. Up to now, many meaningful research results have been obtained. For example, studies [4] and [5] discussed the consensus in first-order MASs under switching and dynamically changing topologies, respectively. Since second-order dynamics is more usual than first-order in applications, consensus on second-order MASs has drawn the interest of an increasing number of researchers. Authors in [6] proposed several second-order algorithms to solve the consensus problems. The consensus of heterogeneous second-order nonlinear MASs was studied in [79], whereas studies [10] and [11] discussed the consensus problems for discrete-time MASs under directed information exchange.

Under the continuous control framework, communication between agents must always be maintained. The intermittent control strategy was proposed to overcome the excessive consumption of limited resources in continuous control. Using the intermittent control technique, studies [12] and [13] studied the consensus problems of linear and nonlinear MASs, respectively. Intermittent control was also used in [1416] to study the second-order consensus of time-delay MASs. The distributed consensus problem for leader-following MASs was investigated in [17] via directed intermittent communication.

Although intermittent control can shorten the controller’s running time, the frequency of information transmission cannot be decreased. Sampled-based intermittent controllers were introduced to improve control efficiency and solve second-order consensus problems [1820]. LFC in second-order MASs was analyzed in [18] by designing an intermittent controller based on a filter with relative state information. In particular, the topology graph in [18] is undirected; however, it is virtually impossible to guarantee bidirectional communication between agents. Therefore, studying MASs with directed topologies of one-way communication is more practical. Reference [19] discussed the consensus of directed second-order MASs by employing current and sampled information. For the containment control problem, the proof of consensus for MASs has been given in [20] under a directed communication topology. Studies [19] and [20] both used the current and sampled state information to design the intermittent communication control protocols.

Sampled data can be used to design control protocols when real-time information is unavailable. For second-order leaderless MASs, periodic intermittent control protocols using sampled information were designed in [21] and [22]. Study [21] considered a consensus protocol with sampled position and velocity information. A protocol with sampled and past sampled position information was proposed in [22] to overcome the dependence on velocity information. By analyzing the relationship of the system parameters, sufficient and necessary conditions have been obtained to ensure the consensus of MASs.

It should be pointed out that none of the above references explored the LFC problems for directed MASs by taking an intermittent sampled control strategy. Based on these analyses, it is valuable to delve deeply into the intermittent sampled LFC control problems for second-order MASs under weighted directed graphs.

The key contributions made by this paper are outlined below. Firstly, a distributed intermittent controller is proposed, which guarantees the second-order LFC and effectively reduces the energy loss of the MAS. Secondly, time delay is considered when designing the controller. Based on the relations of the communication bandwidth, the sampling period, and the time delay, sufficient and necessary conditions for LFC are derived by analyzing the error dynamics of the MAS. Furthermore, following the idea in [23], an intermittent controller based only on sampled position information is designed, under which LFC can still be reached if and only if control parameters, time delay, communication bandwidth, and sampling period satisfy certain conditions.

Before proceeding, we need to introduce some symbols: R, Rn, Rn×n, and N indicate the set of real numbers, the n-dimensional real vector space, the n×n real matrix set, and the natural number set, respectively. Let 0 stand for an appropriate dimension zero vector or zero matrix, and In refers to the n×n identity matrix. For a complex number u, its real part, imaginary part, and modulus are denoted by R(μ), J(μ), and |μ|, respectively. z means the Euclidean norm of a vector z. |X| denotes the determinant of a square matrix X. i and diag() are the imaginary unit and the diagonal matrix, respectively.

2  Preliminaries

This part will briefly introduce some necessary basics.

2.1 Graph Theory

Basic graph theory is introduced by considering a MAS, which includes one leader and N followers.

G~ represents the network topology of the considered MAS. G = {W, E, A} is a subgraph of G~, which describes the data exchange among all the followers, where W={ω1,ω2,,ωN}, E, and A are the set of follower nodes, the set of directed edges, and the adjacency matrix, respectively. A directed path from node ωj to ωi in G is composed of a series of directed edges (ωi,ωi1),(ωi1,ωi2),,(ωil,ωj), where ωik(k=1,2,,l) are distinct from each other.

If there is a node (defined as the root node) such that directed paths exist from this node to every other node in the graph, we say that the directed graph has a directed spanning tree. If ejiE, then aij>0; or else aij=0; aii=0 for all i=1,2,,N. L=[lij]RN×N denotes the Laplacian matrix associated with graph G, where lii=j=1,jiNaij and lij=aij(ij). D = diag(d1,d2,,dN) is utilized to state whether the followers can receive the information from the leader. di>0 means that the ith follower can obtain the information from the leader directly; otherwise, di=0.

2.2 Useful Lemmas

Lemma 1 ([24]): L + D is nonsingular if G~ has a spanning tree where the leader node is the root node.

Lemma 2 ([25]): Consider a block matrix S=[S11S12S21S22], det(S)= det(S11S22S21S12) if S11 and S21 are commutable, where S11,S12,S21,S22Rn×n.

Lemma 3 ([26]): Consider a polynomial of order two with complex coefficients: l(s)=s2+(p1+iq1)s+p2+iq2, where p1,q1,p2, and q2 are real constants. Then, l(s) is stable if and only if p1>0 and p1q1q2+p12p2q2>0.

Lemma 4 ([27]): For a third-order complex coefficient polynomial: m(s)=s3+(p1+iq1)s2+(p2+iq2)s+p3+iq3, where pi and qi(i=1,2,3) are real constants. Then, m(s) is stable if and only if p1>0, p1q1q2+p12p2q22p1p3>0, and p1|q1p2p30p1q2p301q1p2q30p1q2p3||q2p300p1q2p301q1p2q30p1q2p3|>0.

2.3 Problem Formulation

Consider a second-order MAS, which includes N followers and one leader. The ith follower’s dynamical model can be described as

{x˙i(t)=vi(t)v˙i(t)=ui(t),i=1,2,,N(1)

where xi(t)R, vi(t)R, and ui(t)R are the ith agent’s position, velocity, and control input, respectively. Moreover, the leader’s dynamics can be expressed by

{x˙0(t)=v0(t)v˙0(t)=0,i=1,2,,N(2)

where x0(t)R and v0(t)R stand for the leader’s position and velocity, respectively.

Definition 1: For any xi(0) and vi(0) (i=0,1,2,,N), if there exists a control input ui(t) which makes limt|xi(t)x0(t)|=0 and limt|vi(t)v0(t)|=0 hold, then the second-order MAS (1)(2) is said to attain the LFC.

The following ISC protocol is introduced to reduce energy consumption:

ui(t)={γ1(j=1,jiNaij(xj(tk)xi(tk))+di(x0(tk)xi(tk)))+γ2(j=1,jiNaij(vj(tk)vi(tk))+di(v0(tk)vi(tk))), t(tk,tk+δ],0, t(tk+δ,tk+1],i=1,2,,N(3)

where γ1>0 and γ2>0 are the coupling gains, δ, tk, and T are the communication bandwidth, the sampling instant, and the sampling period, respectively. And the sampling instants satisfy tk+1tk=T, limktk=+, 0=t0<t1<<tk<, and 0<δ<T.

Remark 1: Continuous control protocols, which are widely applied in the study of consensus-related problems of second-order MASs, will cause excessive consumption of limited resources. Controllers using current and sampled state information were designed in [1820] to decrease the control cost. Reference [18] considered the LFC problems of undirected MASs; however, the obtained results do not apply to MASs under directed graphs. References [19] and [20] address the leaderless consensus of MASs in directed topologies. The controller (3) proposed in this paper only uses the sampled data to deal with the LFC problems of directed MASs.

3  Leader-Following Consensus without Time Delay

In this part, ISC protocol (3) is applied to study the LFC problem of MAS (1)(2).

Let ζi(t)=xi(t)x0(t) and ηi(t)=vi(t)v0(t). With (3), we obtain that

{ζ˙i(t)=ηi(t)η˙i(t)={γ1(j=1Nlijζj(tk)+diζi(tk))γ2(j=1Nlijηj(tk)+diηi(tk)), t(tk,tk+δ]0, t(tk+δ,tk+1], i=1,2,,N.(4)

Let ζ(t)=[ζ1(t),ζ2(t),,ζN(t)]T,η(t)=[η1(t),η2(t),ηN(t)]T and H = L + D, (4) can be converted to

{ζ˙(t)=η(t)η˙(t)={γ1Hζ(tk)γ2Hη(tk), t(tk,tk+δ]0, t(tk+δ,tk+1], i=1,2,,N.(5)

Eq. (5) can be further reworded to

ξ˙(t)={P1ξ(t)+P2ξ(tk), t(tk,tk+δ]P1ξ(t), t(tk+δ,tk+1](6)

where ξ(t)=[ζT(t),ηT(t)]T, P1=[0IN00], P2= [00γ1Hγ2H].

For matrix H, there is a nonsingular matrix K such that H=KJK1, and J denotes the Jordan form corresponding to H. Let ξ^(t)=[K100K1]ξ(t), we can derive that

ξ^˙(t)={P1ξ^(t)+P~2ξ^(tk), t(tk,tk+δ]P1ξ^(t), t(tk+δ,tk+1](7)

where P~2=[00γ1Jγ2J]. From Lemma 1, if G is directed, some of the eigenvalues of H may be complex and J = diag(J1,J2,,Jw), where Jv(v=1,2,,w) have the following form: Jv=[μv00010000001μv]Nv×Nv, in which μv are the nonzero eigenvalues of H with algebraic multiplicity Nv(v=1,2,,w) and N1+N2++Nw=N. Besides, if G is undirected, J becomes a diagonal matrix with all diagonal elements being positive.

Before proceeding, the following assumption and proposition need to be given.

Assumption 1: G~ is a digraph and it has a directed spanning tree with the leader as the root node.

Proposition 1: When Assumption 1 holds, system (1)(2) with protocol (3) can reach LFC if and only if limtξ^i(t)=0(i=1,2,,N), and ξ^i(t) is the ith component of ξ^(t).

Proof of Proposition 1 (Necessity): If limtxi(t)=x0(t) and limtvi(t)=v0(t) for i=1,2,,N, then ζi(t)0 and ηi(t)0 as t for i=1,2,,N. Thus, the system (7) exhibits asymptotic stability.

(Sufficiency): If limtξ^i(t)=0 for i=1,2,,N, then limtξ(t)=0 since K is nonsingular. It means that system (1)(2) realizes LFC. □

Corollary 1: Under Assumption 1, MAS (1)(2) can reach LFC if and only if the following N systems are asymptotically stable:

h˙i(t)={A1hi(t)+μiA2hi(tk), t(tk,tk+δ]A1hi(t), t(tk+δ,tk+1](8)

where hi(t)=[ξ^i(t),ξ^i+N(t)]T, ξ^i(t) and ξ^i+N(t) represent the ith and the (i+N)th components of ξ^(t), respectively; A1=[0100] and A2=[00γ1γ2].

Proof of Corollary 1 (Necessity): If MAS (1)(2) reaches LFC, it follows from Proposition 1 that limtξ^i(t)=0 for i=1,2,,N. For t(tk,tk+1], because the variables in the system (8) are the first terms of the Jordan blocks in (7), it is straightforward to get that limthi(t)=0 for i=1,2,,N, which means that system (8) is asymptotically stable.

(Sufficiency): When the N systems in the (8) are asymptotically stable, then limthi(t)=0 for i=1,2,,N. Based on the characteristics of the Jordan form, the LFC of (8) is determined by the diagonal entries. Thus, limtξ^i(t)=0 holds for i=1,2,,N. □

Although Proposition 1 and Corollary 1 provide some conditions for ensuring LFC, they do not show how consensus behavior is affected by relevant parameters of the system. Therefore, the following theorem is given to show the relationship among them.

Theorem 1: With (3), MAS (1)(2) can reach the LFC if and only if

{δ<2γ2γ1T<Ξi(9)

where Ξi=[δ2(γ1δ2γ2)3|μi|4+4δ(2γ2γ1δ)2(μi)|μi|2]/[γ1δ2(2γ2γ1δ)2|μi|4+16γ12(μi)], μi is the eigenvalue of H.

Proof of Theorem 1: From (8), we can obtain

hi(t)={eA1(ttk)hi(tk)+μitkteA1(tn)A2hi(tk)dn, t(tk,tk+δ]eA1(ttkδ)hi(tk+δ), t(tk+δ,tk+1].(10)

When t=tk+δ, it is obvious that hi(tk+δ)=eA1δhi(tk)+μitktk+δeA1(tk+δn)A2hi(tk)dn.

Solving the first equation of (10) and substituting its solution into the second equation of (10), it can be obtained that

hi(t)={Ei(ttk)hi(tk), t(tk,tk+δ]Fi(ttkδ)hi(tk), t(tk+δ,tk+1](11)

where Ei(s)=[γ1μis2/21γ2μis2/2sγ1μisγ2μis1], Fi(s)=[Q1i(s)Q2i(s)γ1μiδ1γ2μiδ] with Q1i(s)=1γ1μiδ(δ/2+s) and Q2i(s)=s+δγ2μiδ(s+δ/2).

Hence, we have

hi(t)={Ei(ttk)Fik(Tδ)hi(t0), t(tk,tk+δ]Fi(ttkδ)Fik(Tδ)hi(t0), t(tk+δ,tk+1].(12)

When tk<ttk+1, we can easily know that Ei(ttk) and Fi(ttkδ) are bounded. hi(t)0 if and only if all the eigenvalues of Fi(Tδ) have modulus less than 1. Then, let |λI2Fi(Tδ)|=0, by Lemma 2, we get

λ2+a1λ+a2=0(13)

where a1=μiδ(αT+βαδ/2)2 and a2=αμiδ2/2βμiδ+1. In order to analyze the Schur stability of Eq. (13), let λ=(m+1)/(m1), then

m2+b1m+b2=0(14)

where b1=(2γ2γ1δ)/(γ1T) and b2=δ/T+4/(γ1μiδT)2γ2/(γ1T)1. It follows from Lemma 3 that (14) is asymptotically stable if and only if (9) holds. Based on the aforementioned analysis, (9) guarantees that hi(t)0 for i=1,2,,N. So, MAS (1)(2) reaches LFC under protocol (3). The proof is complete.   □

Corollary 2: When T=δ, system (1)(2) achieves the LFC if and only if

{δ<2γ2/γ1Ξi>1(15)

where Ξi=[T(γ1T2γ2)3|μi|4+4(2γ2γ1T)2(μi)|μi|2]/[γ1T2(2γ2γ1T)2|μi|4+16γ12(μi)].

Proof of Corollary 2: When T=δ, system (1)(2) becomes a general sampled system. It is straightforward to get condition (15) from (9).  □

Undirected graphs can be considered unique directed graphs. For undirected graphs, the subsequent corollary can be derived.

Corollary 3: If the communication topology of all the followers is undirected, system (1)(2) can reach the LFC if and only if

{δ<2γ2/γ1T<Ξi(16)

where Ξi=[δ2(γ1δ2γ2)3|μi|+4δ(2γ2γ1δ)2]/[γ1δ2(2γ2γ1δ)2|μi|].

Proof of Corollary 3: When G is undirected, the eigenvalues μi(i=1,2,,N) of H are real. Then, (μi)=μi and (μi)=0 for i=1,2,,N. Thus, (9) is equivalent to (16) for undirected graphs, then LFC is reached if and only if (16) holds.   □

4  Leader-Following Consensus with Time Delay

It is widely known that time delay has always been present in actual systems and can not be ignored. For time delay situations, a kind of ISC algorithm is proposed.

ui(t)={γ1(j=1,jiNaij(xj(tkι)xi(tkι))+di(x0(tkι)xi(tkι)))+γ2(j=1,jiNaij(vj(tkι)vi(tkι))+di(v0(tkι)vi(tkι))), t(tk,tk+δ]0, t(tk+δ,tk+1], i=1,2,,N(17)

where ι represents the time delay and 0<ι<T.

Remark 2. Since time delay always exists in engineering applications, based on [18], we further consider the time-delay system and propose a time-delay controller that only uses the sampled information. Literature [13] and [14] considered the leaderless consensus of linear and nonlinear MASs, respectively, where intermittent control protocols were designed using sampled information. As a comparison, this paper discusses the ISC problem of second-order leader-following MASs. An ISC protocol (17) is designed for MAS (1)(2) with time delay to achieve LFC.

With (17), by using procedures similar to those in Section 3, we obtain that

ξ˙(t)={P1ξ(t)+P2ξ(tkι), t(tk,tk+δ]P1ξ(t), t(tk+δ,tk+1].(18)

Let ξ^(t)=[K100K1]ξ(t), system (18) can be cast to

ξ^˙(t)={P1ξ^(t)+P~2ξ^(tkι), t(tk,tk+δ]P1ξ^(t), t(tk+δ,tk+1].(19)

Let hi(t)=[ξ^i(t),ξ^i+N(t)]T. (19) can be further reformulated as

h˙i(t)={A1hi(t)+μiA2hi(tkι), t(tk,tk+δ]A1hi(t), t(tk+δ,tk+1].(20)

Theorem 2: When Assumption 1 holds, MAS (1)(2) with the protocol (17) achieves the LFC if and only if one of the following conditions hold:

1) ι< min{δ,Tδ}or δ<ι<tδ(δ<t/2),

{δ<2γ2/γ12ιT<Φ1i/Φ2i(21)

where Φ1i=δ2(γ1δ2γ2+2γ1ι)3|μi|4+4δ(2γ2γ1δ2γ1ι)2(μi)|μi|2 and Φ2i=γ1δ2(2γ2γ1δ2γ1ι)2|μi|4+16γ12(μi).

2) Tδ<ι<T(δ>T/2) or ι> max{δ,Tδ},

{T2ι/3+2γ2/(3γ1)>0p12p2q22p1p3>0Λi>0(22)

where Λi=p1|q1p2p30p1q2p301q1p2q30p1q2p3||q2p300p1q2p301q1p2q30p1q2p3| with p1=2γ1T(γ2γ1ι)+3,q1=0,p2=4(μi)/(γ1δT|μi|2)+2[(Tι)22γ2]/(γ1T)+4γ2(Tι)/(γ1δT)+2(δ+2ι)/T5,q2=q3=4(μi)/(γ1δT|μi|2) and p3=4(μi)/(γ1δT|μi|2)+2[(Tι)2γ2]/(γ1T)+4γ2/(γ1δ)4γ2ι/(γ1δT)+(δ+ι)/T3.

Proof of Theorem 2: Based on Eq. (20), we deduce that

hi(t)={eA1(ttk)hi(tk)+tkteA1(tn)A2ihi(tkι)dn, t(tk,tk+δ]eA1(ttkδ)hi(tk+δ), t(tk+δ,tk+1].(23)

Therefore, hi(tk+δ)=eA1δhi(tk)+tktk+δeA1(tk+δn)A2ihi(tkι)dn. For t(tk+δ,tk+1], it is obvious that hi(t)=Φ3(ttkδ)hi(tk)+μiΦ4(ttkδ)hi(tkι) where Φ3(s)=[γ1δ(δ/2+s)γ2δ(δ/2+s)γ1δγ2δ] and Φ4(s)=[1s+δ01]. Then

hi(t)={Φ1(ttk)hi(tk)+μiΦ2(ttk)hi(tkι), t(tk,tk+δ]Φ3(ttkδ)hi(tk)+μiΦ4(ttkδ)hi(tkι), t(tk+δ,tk+1](24)

where Φ1(s)=[1s01] and Φ2(s)=[γ1s2/2γ2s2/2γ1sγ2s].

Let h^i(t)=[hi(t),hi(tι)]T. According to the relationship of ι, δ, and T, the following four cases are discussed to obtain the detailed expressions of hi(tkι).

Case 1: ι< min{δ,Tδ}.

For t[tk,tk+ι), hi(t)=Φ1(ttk)hi(tk)+μiΦ2(ttk)hi(tkι), hi(tι)=Φ3(tιtk1δ)hi(tk1)+μiΦ4(tιtk1δ)hi(tk1ι). For t[tk+ι,tk+δ), hi(t)=Φ1(ttk)hi(tk)+μiΦ2(ttk)hi(tkι), hi(tι)=Φ1(tιtk)hi(tk)+μiΦ2(tιtk)hi(tkι). For t[tk+δ,tk+δ+ι), hi(t)=Φ3(ttkδ)hi(tk)+μiΦ4(ttkδ)hi(tkι), hi(tι)=Φ1(tιtk)hi(tk)+μiΦ2(tιtk)hi(tkι). For t[tk+δ+ι,tk+1), hi(t)=Φ3(ttkδ)hi(tk)+μiΦ4(ttkδ)hi(tkι), hi(tι)=Φ3(tιtkδ)hi(tk)+μiΦ4(tιtkδ)hi(tkι).

By letting V1i(s)=[E1i(s)F1i(s)Φ3(sι+Tδ)μiΦ4(sι+Tδ)], V2i(s)=[Φ1(s)μiΦ2(s)Φ1(sι)μiΦ2(sι)], V3i(s)=[Φ3(s)μiΦ4(s)Φ1(sι)μiΦ2(sι)], V4i(s)=[Φ3(sδ)μiΦ4(sδ)Φ3(sδι)μiΦ4(sδι)], E1i(s)=Φ1(s)Φ3(Tδ)+μiΦ2(s)Φ4(Tδι) and F1i(s)=μi(Φ1(s)Φ4(Tδ)+Φ2(s)Φ4(Tιδ)), we have hi(tk+1)=Φ3(Tδ)hi(tk)+μiΦ2(ttk)hi(tkι) and hi(tk+1ι)=Φ3(Tδι)hi(tk)+μiΦ4(ttk)hi(tkι).

Therefore, for t[tk,tk+ι), h^i(t)=V1i(ttk)h^i(tk1)=V1i(ttk)V4ik1(T)h^i(t0). For t[tk+ι,tk+δ), h^i(t)=V2i(ttk)h^i(tk1)=V2i(ttk)V4ik(T)h^i(t0). For t[tk+δ,tk+δ+ι), h^i(t)=V3i(ttk)h^i(tk)=V3i(ttk)V4ik(T)h^i(t0). For t[tk+δ+ι,tk+1), h^i(t)=V4i(ttk)h^i(tk)=V4i(ttk)V4ik(T)h^i(t0).

Case 2: When Tδ<ι<δ(δ>T/2), the analysis is similar to that in Case 1.

Let V6i(s)=[E2i(s)F2i(s)Φ3(sι+Tδ)μiΦ4(s+Tιδ)], V7i(s)=[Φ3(sδ)μiΦ4(sδ)Φ1(sι)μiΦ2(sι)], V5i(s)=[E2i(s)F2i(s)Φ1(s+Tι)μiΦ2(sι+T)], E2i(s)=Φ1(s)Φ3(Tδ)+μiΦ2(s)Φ4(Tι) and F2i(s)=μi(Φ2(s)Φ2(Tι)+Φ1(s)Φ4(Tδ)).

For t[tk,tk1+ι+δ), h^i(t)=V5i(ttk)h^i(tk1)=V5i(ttk)V7ik1(T)h^i(t0). For t[tk1+ι+δ,tk+ι), h^i(t)=V6i(ttk)h^i(tk1)=V6i(ttk)V7ik1(T)h^i(t0). For t[tk+ι,tk+δ), h^i(t)=V2i(ttk)h^i(tk)=V2i(ttk)V7ik(T)h^i(t0). For t[tk+δ,tk+1), h^i(t)=V7i(ttk)h^i(tk)=V7i(ttk)V7ik1(T)h^i(t0).

Case 3: δ<ι<Tδ(δ<T/2).

Let V8i(s)=[E3i(s)F3i(s)Φ3(sι+Tδ)μiΦ4(sι+Tδ)], E3i(s)=Φ3(sδ)Φ3(Tδ)+μiΦ4(sδ)Φ3(Tιδ) and F3i(s)=μi(Φ3(sδ)Φ4(Tδ)+Φ4(sδ)Φ4(Tιδ)).

For t[tk,tk+δ), h^i(t)=V1i(ttk)h^i(tk1)=V1i(ttk)V4ik1(T)h^i(t0). For t[tk+δ,tk+ι), h^i(t)=V8i(ttk)h^i(tk1)=V8i(ttk)V4ik1(T)h^i(t0). For t[tk+ι,tk+δ+ι), h^i(t)=V7i(ttk)h^i(tk)=V7i(ttk)V4ik(T)h^i(t0). For t[tk+δ+ι,tk+1), h^i(t)=V4i(ttk)h^i(tk)=V4i(ttk)V4ik(T)h^i(t0).

Case 4: ι> max{δ,Tδ}.

Let V9i(s)=[E2i(s)F2i(s)Φ1(s+Tι)μiΦ2(sι)], V10i(s)=[E2i(s)F2i(s)Φ3(s+Tιδ)μiΦ4(s+Tιδ)], V11i(s)=[E4i(s)F4i(s)Φ3(sι+Tδ)μiΦ4(sι+Tδ)], E4i(s)=Φ3(sδ)Φ3(Tδ)+μiΦ1(Tι)Φ4(sδ) and F4i(s)=μi(Φ3(sδ)Φ4(Tδ)+Φ1(Tι)Φ4(sδ)).

For t[tk,tk1+δ+ι), h^i(t)=V9i(ttk)h^i(tk1)=V9i(ttk)V7ik1(T)h^i(t0). For t[tk1+δ+ι,tk+δ), h^i(t)=V10i(ttk)h^i(tk1)=V10i(ttk)V7ik1(T)h^i(t0). For t[tk+δ,tk+δ+ι), h^i(t)=V11i(ttk)h^i(tk1)=V11i(ttk)V7ik1(T)h^i(t0). For t[tk+δ+ι,tk+1), h^i(t)=V7i(ttk)h^i(tk)=V7i(ttk)V7ik(T)h^i(t0).

If t(tk,tk+1], we have that Vji(ttk)(j=1,2,, 11,j4,7) are all bounded. h(t)0 if and only if h^(t)0, in other words, h(t)0 is equivalent to the modulus of λ(V4i(T)) or λ(V7i(T)) being less than 1.

For Cases 1 and 3, let |λI4V4i(T)|=0, from Lemma 2, we have

λ4+c1λ3+c2λ2=0(25)

where c1=μiδ(γ1Tγ1ι+γ2γ1δ/2)2 and c2=μiδ(γ1ιγ2+γ1δ/2)+1. Obviously, V4i has two eigenvalues λ1=λ2=0. Let λ=(m+1)/(m1), we get

m2+d1m+d2=0(26)

where d1=2(γ2/γ1δ/2ι)/T and d2=4/(γ1δTμi2)+2(δ/2+ιγ2)/T1. It is apparent that the modulus of λ is less than 1 if and only if (m)<0. So, a sufficient and necessary condition for hi(t)0 is that all the roots of (26) have negative real parts. According to Lemma 3, the system (26) is stable if and only if (21) holds. The LFC of MAS (1)(2) with protocol (17) can be reached if and only if (21) holds.

For Cases 2 and 4, let |λI4V7i(T)|=0, based on Lemma 2, we get

λ4+e1λ3+e2λ2+e3λ=0(27)

where e1=γ1μi(ιT)(2+ιT)/2, e2=μi(ιT)(2γ2γ1(1+ιT))+(γ1δ/2+γ1(Tδ)+γ2)δμi and e3=μi(Tι)(γ2γ1δ+γ1(Tι))+(γ1δ/2γ2)δμi. Clearly, λ1=0 is an eigenvalue of V7i. Let λ=(m+1)/(m1), we have

m3+f1m2+f2m+f3=0(28)

where f1=μi[γ2(Tι)+γ1(Tι)2/2]2, f2=μi(Tι)[γ12γ2γ1(Tι)]+μiδ(γ1Tγ1δ/2)+1 and f3=μi(Tι)[γ2γ1δ+γ1(Tι)/2]μi(γ2γ1δ/2). hi(t)0 holds if and only if all the roots of (28) have negative real parts. By Lemma 4, the polynomial in (28) is stable if and only if (22) holds. The proof is complete. □

Remark 3: In Theorem 2, sufficient and necessary conditions are provided under which system (1)(2) achieves LFC with protocol (17). From the relationship of ι, δ, and Tδ, four cases are considered, i.e., ι< min{δ,Tδ}, δ<ι<tδ(δ<t/2), Tδ<ι<δ(δ>T/2), and ι> max{δ,Tδ}. Notably, for 0=ι< min{δ,Tδ}, the condition (21) reduces to (9). Thus, Theorem 1 is a special case that fits within the framework of Theorem 2.

Remark 4: The second-order LFC of MAS (1)(2) with time delay can be reached if and only if (21) holds. In Case 1, we can first set the values of γ1, γ2, and ι, then select the value of δ satisfying δ>ι. By combining the network topology, we can select the range of the sampling period T so that (21) is satisfied. Finally, further refine the extent of the sampling period such that ι< min{δ,Tδ}. In the other three cases, the parameters can be chosen similarly.

Corollary 4: For the undirected topology, the second-order MAS (1)(2) reaches the LFC if and only if one of the following is satisfied:

1) ι< min{δ,Tδ}or δ<ι<tδ(δ<t/2),

{δ<2γ2/γ12ιT<Φ1i/Φ2i(29)

where Φ1i=δ2(γ1δ2γ2+2γ1ι)3μi+4δ(2γ2γ1δ2γ1ι)2 and Φ2i=γ1δ2(2γ2γ1δ2γ1ι)2μi.

2) Tδ<ι<δ(δ>T/2) or ι> max{δ,Tδ},

{T2ι/3+2γ2/(3γ1)>0p12p2p1p3>0Λi>0(30)

where Λi=p1|q1p2p30p10p301q1p200p10p3||0p300p10p301q1p200p10p3| with p1=2γ1γ2T2γ12ιT+3,p2=4/(γ1δTμi)+2(Tι)2/(γ1T)+4γ2(Tι)/(γ1δT)4γ2/(γ1T)+2δ/T+4ι/T5 and p3=4/(γ1δTμi)+2(Tι)2/(γ1T)+4γ2/(γ1δ)4γ2ι/(γ1δT)2γ2/(γ1T)+(δ+ι)/T3.

Proof of Corollary 4: For undirected graph G, we can easily get (μi)=μi and (μi)=0 (i=1,2,,N). So, (21) and (22) are equivalent to (29) and (30), respectively. LFC can be achieved if and only if inequalities (29) and (30) hold.   □

5  Leader-Following Consensus Based Only on Position Information

Obtaining velocity information is difficult in engineering applications, so we propose the following ISC protocol based only on position information.

ui(t)={γ1(j=1,jiNaij(xj(tk)xi(tk))+di(x0(tk)xi(tk)))γ2(j=1,jiNaij(xj(tkι)xi(tkι))+di(x0(tkι)xi(tkι))), t(tk,tk+δ]0, t(tk+δ,tk+1], i=1,2,,N.(31)

Remark 5: Velocity information was used in designing the controllers in [18,19] and [21]. However, it is usually hard to get the velocity information. When velocity is unavailable, systems need to rely on position data to reach consensus. This motivates us to explore whether LFC can still be reached when protocols are designed based only on position data. In fact, most studies regard the time delay as a detrimental factor due to its significant impact on system stability (examples include references [4,1416], and [21]). When velocity information is absent, the LFC cannot be achieved by relying solely on position information. In such cases, incorporating time delay information can be beneficial for reaching consensus. Reference [23] proved that under some circumstances, time delay is beneficial to achieving the consensus of MASs, in which a continuous time-delay control algorithm was designed to reach second-order consensus. ISC protocol (31) employing the time-delay sampled position information is proposed in this paper for second-order MASs to achieve LFC, which can effectively reduce the energy loss of MASs and solve the problem that the velocity information is unavailable.

After similar calculations as in Theorem 2, we obtain that

h˙i(t)={A1hi(t)+μiB1hi(tk)+μiB2hi(tkι), t(tk,tk+δ]A1hi(t), t(tk+δ,tk+1](32)

where B1=[00γ10] and B2=[000γ2].

Theorem 3: Suppose that Assumption 1 holds. MAS (1)(2) with control protocol (31) reaches LFC if and only if one of the following holds:

1) ι< min{δ,Tδ}or δ<ι<tδ(δ<t/2),

{δ<2γ2ι/(γ1γ2)T<Δ1i(33)

where Δ1i=[4(μi)/((γ1γ2)|μi|2δ)]16(μi)(γ1γ2)|μi|2δ/[2γ2ι/(γ1γ2)δ]2.

2) Tδ<ι<δ(δ>T/2) or ι> max{δ,Tδ},

{p1>0p1q1q2+p12p2q22p1p3>0Δ2i>0(34)

where Δ2i=p1|q1p2p30p1q2p301q1p2q30p1q2p3||q2p300p1q2p301q1p2q30p1q2p3| with p1=[γ2δ(δ/2T)γ2/2(ιT)2γ1/2δ2](μi)/(κ|μi|2/2)3, q1=[γ2δ(δ/2T)γ2/2(ιT)2δ2γ1/2](μi)/(κ|μi|2/2), p2=4[[(γ1γ2)δT+γ2δ2](μi)+[2(μi)2(μi)]|μi|2]/(κ|μi|4)+3, q2=4[[(γ1γ2)δT+γ2δ2]|μi|2(μi)+2(μi)(μi)]/(κ|μi|4), p3=2[[(γ1+γ2)δ(δ/2T)+γ2/2(Tι)γ1δγ1T](μi)|μi|2+2[2(μi)2(μi)]]/(κ|μi|4), q3=2[(γ1+γ2)δ(δ/2T)+γ2/2(Tι)γ1δγ1T](μi)|μi|2+4(μi)(μi)]/(κ|μi|4) and κ=γ1δ/4(Tδ2T2δι+2Tι)(2γ2δγ1T2+γ1ιT).

Proof of Theorem 3: The steps of the proof for this part are akin to that of Theorem 2, so it will not be repeated here.   □

Remark 6: By following the same line as in Remark 4, we can select proper parameters for (33) and (34) to be satisfied under which LFC in MAS (1)(2) can still be reached.

Corollary 5: For undirected graphs, MAS (1)(2) with the protocol (31) can achieve the LFC if and only if one of the following conditions is valid:

1) ι< min{δ,Tδ} or δ<ι<tδ(δ<t/2),

{δ<2γ2ι/(γ1γ2)T<Δ1i(35)

where Δ1i=4/((γ1γ2)μiδ).

2) Tδ<ι<δ(δ>T/2) or ι> max{δ,Tδ},

{p1>0p12p2p1p3>0Δ2i>0(36)

where Δ2i=p1|0p2p30p10p3010p200p10p3||0p300p10p3010p200p10p3| with p1=[γ2δ(Tδ/2)+γ2/2(ι22Tι+T2)γ1/2δ2]μi/(κ|μi|2/2)3, p2=4[(γ1γ2)δT+γ2δ2+μi2]/(κ|μi|3)+3 and p3=2[[(γ1+γ2)δ(δ/2T)+γ2/2(Tι)γ1δγ1T]|μi|+2]/(κ|μi|2).

Proof of Corollary 5: A similar proof follows from Corollary 4.   □

Remark 7: Both [19] and [20] used the current and sampled state information to design the control protocols, while this paper only uses the sampled information to improve the control efficiency further. In addition, considering that velocity information is difficult to obtain, this paper proposes an ISC protocol (31) that only utilizes the sampled position information, and the sufficient and necessary conditions for LFC are obtained. This approach can avoid the control cost resulting from the velocity measurement.

6  Simulation

Consider a second-order MAS, which includes 5 followers (nodes 15) and 1 leader (node 0). Different from the undirected graph in [18], this paper employs a weighted directed graph, whose communication topology is illustrated in Fig. 1. x~i and v~i(i=1,2,,5) stand for position errors and velocity errors between the followers and the leader, respectively. To show the effectiveness of the obtained theories, we select several scenarios from the proof of Theorems 1–3 as representatives for numerical simulations.

images

Figure 1: Communication graph of MAS (1)(2)

Example 1: Consider system (1), (2) with protocol (3). According to Theorem 1, we choose γ1=0.7, γ2=1.5, and δ=0.5 in the simulation. Obviously, the first inequality in (9) holds. We get T<1.62 from the second inequality of (9). Thus, MAS (1)(2) can reach LFC if and only if 0.5<T<1.62. The position and velocity errors between the followers and the leader are presented in Figs. 2 and 3, respectively. LFC in MAS (1)(2) can be achieved when T=1.1. However, it cannot be reached when T=1.63.

images

Figure 2: State errors between the followers and the leader when T=1.1

images

Figure 3: State errors between the followers and the leader when T=1.63

Example 2: In this example, simulations in two different cases (Cases 1 and 2 in Theorem 2) are given to check the validity of the theoretical results.

When ι< min{δ,Tδ}, we select γ1=0.5, γ2=1.9 and ι=0.1. Since ι< min{δ,Tδ}, δ=0.3 is selected. According to (21), 0.4<T<1.12 should be satisfied. From Fig. 4, LFC can be achieved when T=0.8.

images

Figure 4: The error trajectories of system (1)(2) with protocol (17) in Case 1

When Tδ<ι<δ(δ>T/2), γ1=0.3, γ2=1.5 and ι=0.5 are chosen to verify the correctness of Case 2 in Theorem 2. From (22), we have 0.6<T<1.1 when δ=0.6. Fig. 5 shows the position and velocity error trajectories when T=0.9.

images

Figure 5: The error trajectories of system (1)(2) with protocol (17) in Case 2

Under protocol (17), the position and velocity errors between all the followers and the leader tend to 0 when time delay exists, which means that MAS (1)(2)achieves LFC asymptotically.

Example 3: The other two different cases (Cases 3 and 4 in Theorem 3) are considered to verify the effectiveness of protocol (31) in this example.

When δ<ι<Tδ(δ<T/2), we choose γ1=2.1, γ2=1.4, ι=0.5, and δ=0.4. From (33) in Theorem 3, we get 0.9<T<2.11. Fig. 6 illustrates that LFC is reached when T=1.

images

Figure 6: The error trajectories of system (1)(2) with protocol (31) in Case 3

When ι> max{δ,Tδ}, γ1=2.2, γ2=1.8, ι=0.65, and δ=0.54 are chosen. By (34), MAS (1)(2) can reach consensus if and only if 0.54<T<1.19. It can be observed from Fig. 7 that LFC is achieved when T=0.94.

images

Figure 7: The error trajectories of system (1)(2) with protocol (31) in Case 4

In these two cases, the state errors between all the followers and the leader converge to 0, even if the time delay exists. That is to say, MAS (1)(2) with protocol (31) achieves LFC.

7  Conclusion

Second-order LFC problems in MASs are investigated in this work. Firstly, a new ISC protocol is proposed, and consensus conditions are analyzed to ensure the LFC. A sufficient and necessary condition dependent on coupling gain, sampling period, communication bandwidth, and network structure is obtained. Besides, input delay is taken into account, and a time-delay protocol is proposed. According to the relationship of system parameters, four cases are discussed separately, and it is concluded that MAS can reach the LFC if and only if system parameters satisfy certain conditions. Furthermore, considering it is hard to get the velocity information, an intermittent sampled protocol only with position information is proposed. A sufficient and necessary condition is also acquired to guarantee the LFC. In the future, we will further investigate the second-order LFC of MASs with quantized communication or stochastic switching topologies, etc.

Acknowledgement: The authors are grateful to all the editors and anonymous reviewers for their comments and suggestions.

Funding Statement: This research was supported by the National Natural Science Foundation of China under Grants 62476138 and 42375016.

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: Ziyang Wang; draft manuscript preparation: Ziyang Wang and Yuanzhen Feng; funding acquisition and supervision: Zhengxin Wang and Yuanzhen Feng; review: Cong Zheng. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: Not applicable.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

Abbreviations

ISC Intermittent sampled control
LFC Leader-following consensus
MASs Multi-agent systems

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Cite This Article

APA Style
Wang, Z., Feng, Y., Wang, Z., Zheng, C. (2025). Sufficient and Necessary Conditions for Leader-Following Consensus of Second-Order Multi-Agent Systems via Intermittent Sampled Control. Computers, Materials & Continua, 83(3), 4835–4853. https://doi.org/10.32604/cmc.2025.063179
Vancouver Style
Wang Z, Feng Y, Wang Z, Zheng C. Sufficient and Necessary Conditions for Leader-Following Consensus of Second-Order Multi-Agent Systems via Intermittent Sampled Control. Comput Mater Contin. 2025;83(3):4835–4853. https://doi.org/10.32604/cmc.2025.063179
IEEE Style
Z. Wang, Y. Feng, Z. Wang, and C. Zheng, “Sufficient and Necessary Conditions for Leader-Following Consensus of Second-Order Multi-Agent Systems via Intermittent Sampled Control,” Comput. Mater. Contin., vol. 83, no. 3, pp. 4835–4853, 2025. https://doi.org/10.32604/cmc.2025.063179


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