# Delay-Distribution-Dependent Consensus for Second-Order Leader-Follower Nonlinear Multiagent Systems via Pinning Control

Authors: Hongjie Li; Ming Chen; Shigen Shen; Lin Li
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper investigates the consensus problem for second-order leader-follower nonlinear multiagent systems with general network topologies. A pinning control algorithm is proposed, where it includes time-varying delay information. By using the information of delay-partition and delay-distribution and constructing an appropriate Lyapunov-Krasovskii functional, the consensus criteria are derived to achieve leader-follower consensus for multiagent systems, which are in the form of linear inequalities that can be solved by employing the semidefinite programme method. Moreover, this paper addresses what kind of agents and how many agents should be pinned. Two numerical examples are presented to further demonstrate the effectiveness of the proposed approach.


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## Body

## 1. Introduction

The coordinated and cooperative control of multiple autonomous vehicles is a problem which has been much researched in recent years due to its applications in the formation control of unmanned air vehicles, the cooperative control of mobile robots, the design of distributed sensor networks, and so on. Particularly, consensus in multiagent systems, which means all the agents can reach an agreement on certain concern or interest, has been extensively studied in the past few years [1–8].A particularly interesting topic is the consensus of a group of agents with a leader, where the leader is a special agent whose motion is independent of all the other agents and thus is followed by all the other ones, and such a problem is commonly called leader-following consensus problem [9–13]. Leader-following consensus problem is investigated in [9], and the corresponding conclusion is that if all the agents are jointly connected with their leader, their states would converge to that of the leader. Reference [11] provides a rigorous proof for the consensus using an extension of LaSalle’s invariance principle. A neighbor-based observer design approach is proposed in [12] and discusses a group of mobile agents with an active leader moving with an unknown velocity. In [13], the leader-following consensus problem is considered by designing distributed controllers using local information to ensure that all the agents follow the leader. It is well known that time delay often causes undesirable dynamic behaviors such as oscillation, performance degradation, and instability of the network [14–16]; therefore, various approaches to consensus analysis for multiagent systems with time delay have been investigated in the literature [17, 18]. In [17], two different cases of coupling topologies are investigated for coordination problem of multiagent system with time-varying coupling delays. Reference [18] studies the leader-following consensus for a multiagent system with a varying-velocity leader and time-varying delays. However, less consideration has been paid to the delay-dependent consensus conditions for multiagent systems with time-varying delays. It is noted that all physical systems are nonlinear in nature [19] and that even different agents may have different dynamics. It is very challenging task to discuss consensus problem of nonlinear multiagent systems [20, 21]. Reference [21] proposes leader-following consensus algorithms based on pinning control for second-order nonlinear multiagent systems and provides a pinned-agent selection scheme to determine what kind of followers and how many followers should be pinned. Considering that it is practically impossible to apply control actions to all agents in a large-scale multiagent system, and some authors develop consensus algorithms based on pinning control [22–25]. It is well known that pinning control is an effective method to solve the problem of nonlinear system, it would be beneficial to apply these control techniques to study the consensus problem of multiagent system. In many practical systems, such as networked control systems, the probability distribution of the time delay in the interval is an important characteristic for the network conditions, and its probability distribution can be measured by the statistical method [26–28]. Therefore, the information of probability distribution of the delay should be employed in the model. To the best of the authors’ knowledge, no result has been reported for the consensus problem for second-order leader-follower nonlinear multiagent systems via pinning control, when both the information of variation range of the time delay and the information of variation probability of the time delay in an interval can be observed, which motivates the present study.Motivated by the above discussions, in this paper, we investigate the consensus problem for second-order leader-follower nonlinear multiagent systems. Based on the probability distribution of delay, a new type of multiagent system model with stochastic parameters in the coefficient matrices is proposed. A pinning control algorithm is proposed, and the delay-partition-dependent and delay-distribution-dependent consensus criteria are derived by combining delay-partitioning method and Lyapunov-Krasovskii functional method. Moreover, this paper addresses what kind of agents and how many agents should be pinned.The main contributions of this paper can be summarized as follows. (1) Based on the probability distribution of delay, a novel multiagent system model with stochastic parameters in the coefficient matrices is introduced. (2) The new pinning control algorithm is proposed, and this paper gives what kind of agents and how many agents should be pinned.The rest of this paper is organized as follows. In Section2, problem formulation and preliminaries are briefly outlined. In Section 3, the consensus criteria are derived to achieve leader-follower consensus for multiagent systems. In Section 4, two simulation examples are provided to show the advantages of the obtained results, and some conclusions are drawn in Section 5.Notation.
The notation used in the paper is fairly standard.ℝn denotes the n-dimensional Euclidean space, and ℝn×m is a set of real n×m matrices. The notation X>0 (resp., X<0), for X∈ℝn×n, means that the matrix X is real symmetric positive definite (resp., negative definite). diag{⋯} stands for a block-diagonal matrix. ∥·∥ denotes the Euclidean norm in ℝn. The superscript “T” stands for matrix transposition. E{·} stands for mathematical expectation. The Kronecker product of matrices Q∈ℝm×n and R∈ℝp×q is a matrix in ℝmp×nq and denoted as  Q⊗R. In this paper, if not explicitly stated, matrices are assumed to have compatible dimensions.


## 2. Preliminaries

Letg={v,ε,A} be a weighted diagraph of order N, with the set of nodes v={v1,v2,…,vN}, an edge set ε⊆v×v, and a weighted adjacency matrix A=(aij)N×N with nonnegative elements, directed edge eij in network g, is denoted by the ordered pair of nodes (vi,vj), which means that node vj can receive information from node vi. The elements of the adjacency matrix A are defined as aij>0 if and only if there is a directed edge (vj,vi) in g; otherwise aij=0. We assume that aii=0 for all i∈v. The neighbor set of node i is defined by Ni={j∈v∣(vj,vi)∈ε}, and the in-degree and out-degree of node i are defined as
(1)degin(i)=∑j=1,j≠iN‍aij,degout(i)=∑j=1,j≠iN‍aji.
A diagraph is called balanced if degin(i)=degout(i) for all i∈v.The Laplacian matrixL=(lij)N×N associated with the adjacency matrix A is defined as
(2)lij=-aij(i≠j)lii=-∑j=1,j≠iN‍lij(i,j=1,2,…,N)
which ensures the diffusion property that ∑j=1Nlij=0.A directed path from nodevi to vj in g is a sequence of edges ((vi,vi1),(vi1,vi2),…,(vil,vj)) in the directed graph which is strongly connected if for any two distinct nodes vj and vi, there exists a directed path form node vj to vi. A directed graph has a directed spanning tree if there exists at least one node called root which has a directed path to all other nodes.The dynamics of the virtual leader is described by the following second-order nonlinear system:(3)x˙r(t)=vr(t),v˙r(t)=f(t,xr(t),vr(t)),
where xr(t)=(xr1(t),xr2(t),…,xrn(t))T∈ℝn and vr(t)=(vr1(t),vr2(t),…,vrn(t))T∈ℝn are the position and velocity states of the virtual leader, respectively. f(t,xr(t),vr(t)) = (f1(t,xr(t),vr(t)),f2(t,xr(t),vr(t)),…,fn(t,xr(t),vr(t)))T is a nonlinear vector-valued continuous bounded function.The second-order dynamics of multiagent systems consisting ofN-coupled autonomous agents is given as follows:
(4)x˙i(t)=vi(t)+u1i(t),v˙i(t)=f(t,xi(t),vi(t))+u2i(t)(i=1,2,…,N),
where xi(t)=(xi1(t),xi2(t),…,xin(t))T∈ℝn and vi(t)=(vi1(t),vi2(t),…,vin(t))T∈ℝn are the position and velocity states of agent i, respectively. u1i(t) and u2i(t) are the control inputs for agent i to be designed.Definition 1.
The multiagent systems (4) are said to achieve second-order leader-following consensus if, for any initial conditions, the state of each following agent i(i=1,2,…,N) satisfies
(5)limt→∞∥xi(t)-xr(t)∥=0limt→∞∥vi(t)-vr(t)∥=0.Motivated by the coupling rule used in [17], the consensus algorithm is designed based on pinning control such that all agents in nonlinear multiagent systems (4) can asymptotically follow the virtual leader (3). Without loss of generality, rearrange the order of all agents, and the sort method will be given in Section 3. Let the first l(1≤l<N) agents in multiagent systems (4) be controlled, and the controllers have the following form:
(6)u1i(t)=-∑j∈Ni‍aij{(xi(t)-xj(t))ggggggg+(xi(t-τ(t))-xj(t-τ(t)))}-δi{(xi(t-τ(t))-xr(t-τ(t)))(xi(t)-xr(t))ggggg+(xi(t-τ(t))-xr(t-τ(t)))}u2i(t)=-∑j∈Ni‍aij{(vi(t)-vj(t))ggggggg+(vi(t-τ(t))-vj(t-τ(t)))}-δi{(vi(t-τ(t))-vr(t-τ(t)))(vi(t)-vr(t))fff+(vi(t-τ(t))-vr(t-τ(t)))},
where the local feedback gains satisfy δi>0(i=1,2,…,l) and δi=0(i=l+1,l+2,…,N). Also τ(t)∈[0,τ] is the bounded time-varying delay, and τ is a positive scalar.For notational convenience, we define(7)x(t)=(x1T(t),x2T(t),…,xNT(t))T∈ℝNn,v(t)=(v1T(t),v2T(t),…,vNT(t))T∈ℝNn,1N=(1,1,…,1)T∈ℝN,F(t,x(t),v(t))=(fT(t,x1(t),v1(t)),fT(t,x2(t),v2(t)),…,dddfdfT(t,xN(t),vN(t)))T∈ℝNnδ=diag{δ1,δ2,…,δl,0,…,0}∈ℝN×N.Letξ(t)=x(t)-1N⊗xr(t), η(t)=v(t)-1N⊗vr(t). Combining (3) and (4), one has
(8)ξ˙(t)=x˙(t)-1N⊗x˙r(t)=v(t)-[(L+δ)⊗In](ξ(t)+ξ(t-τ(t)))-1N⊗vr(t)=η(t)-[(L+δ)⊗In](ξ(t)+ξ(t-τ(t))),η˙(t)=v˙(t)-1N⊗v˙r(t)=F(t,x(t),v(t))-1N⊗f(t,xr(t),vr(t))-[(L+δ)⊗In](η(t)+η(t-τ(t))).Based on (8), the augment system can be obtained as follows:
(9)[ξ˙(t)η˙(t)]T=[-(L+δ)⊗InINn0-(L+δ)⊗In][ξ(t)η(t)]+[-(L+δ)⊗In00-(L+δ)⊗In][ξ(t-τ(t))η(t-τ(t))]+[0F(t,x(t),v(t))-1N⊗f(t,xr(t),vr(t))].Setting(10)e(t)=[ξ(t)η(t)]A=[-(L+δ)IN0-(L+δ)]F~=[0F-]Aτ=[-(L+δ)00-(L+δ)]F-=F(t,x(t),v(t))-1N⊗f(t,xr(t),vr(t)),
then the system (9) can be rewritten in the following compact form:
(11)e˙(t)=(A⊗In)e(t)+(Aτ⊗In)e(t-τ(t))+F~.Remark 2.
In many practical systems, such as networked control systems, the probability distribution of the delay in the interval is an important characteristic. Inspired by [26–28], in this paper, it is assumed that the probability of the delay appearing in some intervals can be observed. Then new multiagent systems with stochastic time-varying delays can be proposed.For a given positive integerM and a partition 0=τ0<τ1<τ2⋯<τM=τ, we assume that the probabilities of τ(t)∈[τi-1,τi)(i=1,2,…,N) are known. Define the set
(12)Ωi={t:τ(t)∈[τi-1,τi)}(i=1,2,…,M).It can easily be seen that⋃‍Ωi=R+, Ωi⋂‍Ωj=ϕ(i≠j). Define
(13)τi(t)={τ(t)t∈Ωi0t∈-Ωiβi(t)={1t∈Ωi0t∈-Ωi.Assumption 3.
Stochastic variableβ(t):=[β1(t),β2(t),…,βN(t)]T is a distributed sequence with
(14)Prob{βi(t)=1}=E{βi(t)}=βi∑i=1M‍βi=1.Remark 4.
By some computations, we derived from Assumption3 that
(15)E(βi(t)-βi)2=βi(1-βi)E{βi(t)-βi}=0(i=1,2,…,M)E{(βi(t)-βi)(βj(t)-βj)}=-βiβj(i,j=1,2,…,M;i≠j).By using the new functionsβi(t)(i=1,2,…,M), system (11) can be rewritten as
(16)e˙(t)=[𝒜+∑i=1M-1‍(βi(t)-βi)𝒜τi]ξ(t)+F~,
where
(17)𝒜=[A,β1Aτ,0,β2Aτ,0,…,0,βM-1Aτ,0,βMAτ]⊗In𝒜τi=[0N,(2i-1)N,Aτ,0N,(2M-2i-1)N,-Aτ],ξT(t)=[eT(t),eT(t-τ1(t)),eT(t-τ1),fffeT(t-τ2(t)),…,eT(t-τM-1),eT(t-τM(t))].LetC([-τ,0];ℝn) denote the family of continuous functions φ from [-τ,0] to ℝn. It is not difficult to show that, given any φ∈C([-τ,0];ℝn), the system (16) has a unique continuous solution e(t) on t≥-τ. Clearly, e(t) is a continuous ℝn-valued stochastic process on t∈[-τ,0]. Let et={e(t+θ):-τ≤θ≤0} for t≥0. Then {et} is a C([-τ,0];ℝn)-valued Markov process. Its infinitesimal operator L, acting on functional V:C([-τ,0];ℝn)×[0,+∞)→ℝ, is defined by
(18)LV(et,t)=limΔ→0+1Δ[E(V(et+Δ,t+Δ)∣et)-V(et,t)].The following lemmas and assumption are needed to derive our main results.Lemma 5.
For matricesA, B, C, and D with appropriate dimensions and a positive scalar α, by the definition of Kronecker product, the following properties can be proved:
(19)(αA)⊗B=A⊗(αB),(A+B)⊗C=A⊗C+B⊗C,(A⊗B)(C⊗D)=(AC)⊗(BD),(A⊗B)T=AT⊗BT.Lemma 6.
LetR∈ℝn×n satisfy R>0. If a ℝn-valued function x(t) is integral on the interval [a,b], then
(20)-∫ab‍xT(s)Rx(s)ds≤-1b-a∫ab‍xT(s)dsR∫ab‍x(s)ds.Assumption 7.
For the nonlinear functionf(t,·,·) and for all x,v,y,z∈ℝn, there exist nonnegative constants ρ1 and ρ2, such that the following inequality holds:
(21)∥f(t,x,v)-f(t,y,z)∥2≤ρ1∥x-y∥2+ρ2∥v-z∥2.


## 3. Main Results

Theorem 8.
Suppose that Assumption7 holds. Second-order leader-following consensus is achieved under the pinning controllers (6), if for some given positive scalars τi and βi∈[0,1](i=1,2,…,M) there exist some positive matrices P=diag{P,P}, Qi=diag{Qi,Qi}(i=1,2,…,M-1), Rj=diag{Rj,Rj}(j=1,2,…,M+1) and a positive α>0 such that the following linear matrix inequality holds:(22)Φ=[Φ11***⋯**Φ21Φ22**⋯**00Φ33*⋯**Φ410Φ43Φ44⋯**⋮⋮⋮⋮⋱⋮⋮Φ2M,1000⋯Φ2M,2M*0Φ2M+1,20Φ2M+1,4⋯Φ2M+1,2MΦ2M+1,2M+1]<0,where
(23)Φ11=[Ω1+αρ1INP-(L+δ)TΔP-Δ(L+δ)Ω1+αρ1IN],Φ21=[Ω2-β1(L+δ)TΔ0Ω2],Φ22=[β1(L+δ)TΔ(L+δ)-R1τ100β1(L+δ)TΔ(L+δ)-R1τ1],Φ33=[-Q1-R2τ2-τ100-Q1-R2τ2-τ1],Φ41=[Ω3-β2(L+δ)TΔ0Ω3],Φ43=[R2τ2-τ100R2τ2-τ1],Φ44=[R2τ2-τ1+β2(L+δ)TΔ(L+δ)00R2τ2-τ1+β2(L+δ)TΔ(L+δ)],Φ2M,1=[Ω4-βM(L+δ)TΔ0Ω4],Φ2M,2M=[Ω500Ω5],Φ2M+1,2=[P-Δ(L+δ)00P-Δ(L+δ)],Φ2M+1,4=[-β1Δ(L+δ)00-β1Δ(L+δ)],Φ2M+1,2M=[-βMΔ(L+δ)00-βMΔ(L+δ)],Φ2M+1,2M+1=[-αIN+Δ00-αIN+Δ],Δ=∑i=1M‍(τi-τi-1)Ri+τMRM+1,Ω1=-P(L+δ)-(L+δ)TP+∑i=1M-1‍Qi-R1τ1-RM+1τM+(L+δ)TΔ(L+δ),Ω2=-β1(L+δ)TP+R1τ1+β1(L+δ)TΔ(L+δ),Ω3=-β2(L+δ)TP+β2(L+δ)TΔ(L+δ),Ω4=-βM(L+δ)TP+RM+1τM+βM(L+δ)TΔ(L+δ),Ω5=-RM+1τM-RMτM-τM-1+(βM-2∑i,j=1;i≠jM-1‍βiβj)(L+δ)TΔ(L+δ).Proof.
Consider the following Lyapunov function candidate:(24)V(t,et)=V1(t,et)+V2(t,et)+V3(t,et),
where
(25)V1(t,et)=eT(t)(P⊗In)e(t),V2(t,et)=∑i=1M-1‍∫t-τit‍eT(s)(Qi⊗In)e(s)ds,V3(t,et)=∑i=1M‍∫t-τit-τi-1‍∫st‍e˙T(v)(Ri⊗In)e˙(v)dvds+∫t-τMt‍e˙T(v)(RM+1⊗In)e˙(v)dvds.
Taking time derivative ofVi(t,et)(i=1,2,3) along the trajectory (16) yields
(26)LV1(t,et)=2eT(t)(P⊗In)𝒜ξ(t)+2eT(t)(P⊗In)F~=2eT(t)(P⊗In)𝒜ξ(t)+2eT(t)×(P[0IN]⊗In)F-,(27)LV2(t,et)=∑i=1M-1‍[eT(t)(Qi⊗In)e(t)fssss-eT(t-τi)(Qi⊗In)e(t-τi)],(28)LV3(t,et)=e˙T(t)(Δ⊗In)e˙(t)-∑i=1M‍∫t-τit-τi-1‍e˙T(s)(Ri⊗In)e˙(s)ds-∫t-τMt‍e˙T(s)(RM+1⊗In)e˙(s)ds,
where Δ=∑i=1M(τi-τi-1)Ri+τMRM+1
By Lemma6, the following inequalities hold:
(29)-∫t-τit-τi-1‍e˙T(s)(Ri⊗In)e˙(s)ds≤-1τi-τi-1∫t-τit-τi-1‍e˙T(s)ds(Ri⊗In)∫t-τit-τi-1‍e˙(s)ds≤-1τi-τi-1∫t-τi(t)t-τi-1‍e˙T(s)ds(Ri⊗In)∫t-τi(t)t-τi-1‍e˙(s)ds=-1τi-τi-1ξT(t)[04(i-1)Nn,2NnI2Nn-I2Nn04(M-i)Nn,2Nn]×(Ri⊗In)[04(i-1)Nn,2NnI2Nn-I2Nn04(M-i)Nn,2Nn]Tξ(t),-∫t-τMt‍e˙T(s)(RM+1⊗In)e˙(s)ds≤-1τM∫t-τMt‍e˙T(s)ds(RM+1⊗In)∫t-τMt‍e˙(s)ds≤-1τM∫t-τM(t)t‍e˙T(s)ds(RM+1⊗In)∫t-τM(t)t‍e˙(s)ds=-1τMξT(t)[I2Nn04(M-1)Nn,2Nn-I2Nn]×(RM+1⊗In)[I2Nn04(M-1)Nn,2Nn-I2Nn]Tξ(t).
From (16) and (28), we can obtain
(30)e˙T(t)(Δ⊗In)e˙(t)=ξT(t)(𝒜T+∑i=1M-1‍(βi(t)-βi)𝒜τiT)×(Δ⊗In)(𝒜+∑i=1M-1‍(βi(t)-βi)𝒜τi)ξ(t)+2ξT(t)(𝒜T+∑i=1M-1‍(βi(t)-βi)𝒜τiT)(Δ⊗In)F~+F~T(Δ⊗In)F~=ξT(t)𝒜TΔ𝒜ξ(t)+2∑i=1M-1‍(βi(t)-βi)ξT(t)×𝒜T(Δ⊗In)𝒜τiξ(t)+2∑i≠j‍(βi(t)-βi)(βj(t)-βj)ξT(t)𝒜τiT(Δ⊗In)𝒜τiξ(t)+2∑i=1M-1‍(βi(t)-βi)2ξT(t)𝒜τiT(Δ⊗In)𝒜τiξ(t)+2ξT(t)(𝒜T+∑i=1M-1‍(βi(t)-βi)𝒜τiT)(Δ⊗In)F~+F~TF~.
Taking the mathematical expectation on both sides of (30) yields
(31)E{e˙T(t)(Δ⊗In)e˙(t)}=ξT(t)(∑i,j=1;i≠jM-1‍𝒜T(Δ⊗In)𝒜-2∑i,j=1;i≠jM-1‍βiβj𝒜τiT(Δ⊗In)𝒜τifffggggggfffff+∑i=1M-1‍βi(1-βi)𝒜τiT(Δ⊗In)𝒜τi∑i=1M-1‍)ξ(t)+2ξT(t)𝒜T(Δ⊗In)[0IN]F-+F-TF-.
By Assumption3, we can obtain the following inequality:
(32)F-TF-=∥F(t,x(t),v(t))-1N⊗f(t,xr(t),vr(t))∥2=∑i=1N‍∥f(t,xi(t),vi(t))-f(t,xr(t),vr(t))∥2≤ρ1∑i=1N‍∥xi(t)-xr(t)∥2+ρ2∑i=1N‍∥vi(t)-vr(t)∥2=ρ1∥x(t)-1N⊗xr(t)∥2+ρ2∥v(t)-1N⊗vr(t)∥2=ρ1ξT(t)ξ(t)+ρ2ηT(t)η(t)=eT(t)[ρ1INn00ρ2INn]e(t).
For a givenα>0, the following inequality holds based on (32):
(33)αeT(t)[ρ1INn00ρ2INn]e(t)-αF-TF-≥0.
It follows from (26)–(33) that
(34)LV(t,et)≤2eT(t)(P⊗In)𝒜ξ(t)+2eT(t)(P[0IN]⊗In)F-+∑i=1M-1‍{eT(t)(Qi⊗In)e(t)-eT(t-τi)(Qi⊗In)e(t-τi)}+e˙T(t)(Δ⊗In)e˙(t)-∑i=1M‍1τi-τi-1ξT(t)[04(i-1)Nn,2NnI2Nn-I2Nn04(M-i)Nn,2Nn]×(Ri⊗In)[04(i-1)Nn,2NnI2Nn-I2Nn04(M-i)Nn,2Nn]Tξ(t)-1τMξT(t)[I2Nn04(M-1)Nn,2Nn-I2Nn]×(RM+1⊗In)[I2Nn04(M-1)Nn,2Nn-I2Nn]Tξ(t)+αeT(t)[ρ1INn00ρ2INn]e(t)-αF-TF-=ξT(t)(Ψ⊗In)ξ(t)+2eT(t)(P[0IN]⊗In)F-+2ξT(t)𝒜T(Δ⊗In)[0IN]F-+αeT(t)[ρ1INn00ρ2INn]e(t)+(1-α)F-TF-,
where
(35)Ψ=[Φ-11***⋯*Φ21Φ22**⋯*00Φ33*⋯*Φ410Φ43Φ44⋯*⋮⋮⋮⋮⋱⋮Φ2M,1000⋯Φ2M,2M]Φ-11=Φ11-α[ρ1INn00ρ2INn].
Letζ(t)=[ξT(t),F-T]T, and the inequality (34) can further be written as
(36)LV(t,et)≤ζT(t){[Ψ*ΓΦ2M+1,2M+1]⊗In}ζ(t)=ζT(t)(Φ⊗In)ζ(t)<0,
where
(37)Γ=[0,Φ2M+1,2,Φ2M+1,4,…,0,Φ2M+1,2M].
From (22), we have LV(t,et)<0. That is, ∥ξ(t)∥→0, ∥η(t)∥→0 as t→∞. Therefore, the states of pinning-controlled multiagent systems (4) globally asymptotically approach state of virtual leader (3). That is, ∥xi(t)-xr(t)∥→0, ∥vi(t)-vr(t)∥→0 as t→∞. So the second-order leader-following consensus in pinning-controlled multiagent systems (4) is achieved. The proof is completed.Remark 9.
It can be seen from Theorem8 that the feasibility of (22) dependents on not only τi and δ but also the probability distribution of the delay βi, and more information of time delays is involved (22).Let us consider one special case. Whenf(t,·,·)≡0, then the virtual leader (3) reduces to
(38)x˙r(t)=vr(t)v˙r(t)=0,
which indicates that the reference velocity is constant.Each agent in multiagent systems is described by(39)x˙i(t)=vi(t)+u1i(t),v˙i(t)=u2i(t)(i=1,2,…,N).The pinning controllers are given as follows:(40)u1i(t)=-∑j∈Ni‍aij{(xi(t)-xj(t))+(xi(t-τ(t))-xj(t-τ(t)))}-δi{(xi(t)-xr(t))-(xi(t-τ(t))-xr(t-τ(t)))}u2i(t)=-∑j∈Ni‍aij{(vi(t)-vj(t))-(vi(t-τ(t))-vj(t-τ(t)))}-δi{(vi(t)-vr(t))-(vi(t-τ(t))-vr(t-τ(t)))},
where the parameters δi are defined in Section 2.Letξ(t)=x(t)-1N⊗xr(t), η(t)=v(t)-1N⊗vr(t). Then we can obtain from (38)–(40) that
(41)e˙(t)=(A⊗In)e(t)+(Aτ⊗In)e(t-τ(t)).From Theorem8 and its proof, we have the following corollary.Corollary 10.
For some given positive scalarsβi∈[0,1](i=1,2,…,M), there exist some positive matrices P=diag{P,P}, Qi=diag{Qi,Qi}(i=1,2,…,M-1), Rj=diag{Rj,Rj}(j=1,2,…,M+1) such that the following linear matrix inequality holds:
(42)[Φ11***⋯*Φ21Φ22**⋯*00Φ33*⋯*Φ410Φ43Φ44⋯*⋮⋮⋮⋮⋱⋮Φ2M,1000⋯Φ2M,2M]<0.
Then the second-order leader-following consensus is achieved under the adaptive pinning controllers (40). That is, xi(t)→xr(0)+vr(0)t and vi(t)→vr(0), as t→∞.Up to this point, a question arises naturally: how to choose a set of pinned agents such that pinning conditions (22) and (42) are satisfied?Proposition 11 (see [21]).
For a diagraphg, let v and D denote the node set of g and the pinned-node set, respectively. All nodes in v/D should have access to the pinned-node set D. That is, for any node i∈v/D, one can always find node j∈D such that there exists a directed path from node j to node i.According to Proposition11, we give the following procedure to select a set of pinning-agents and to design the parameters δi  (i=1,2,…,l).(1)
Define a degree-difference vectordegdif(i)=degout-degin(i)  (i=1,2,…,N).(2)
Pick all agents with zero in-degrees as pinned agents, and rearrange the remaining agents in descending order according to their degree-differences.(3)
Find the minimum number of agentsl0 which satisfies (22) or (42), and let l=l0.(4)
From Theorem8 and Corollary 10, by using LMI Toolbox in MATLAB, we can obtain δi(i=1,2,…,l0).(5)
Ifδi(i=1,2,…,l0) is not good enough for practical use, add more agents to the pinned-agent set, and repeat STEP (4) until we find proper δi.


## 4. Numerical Results

In this section, two numerical examples are given to verify the effectiveness of the proposed pinning control techniques.Example 12.
We consider multiagent systems consisting of six agents described by(43)x˙i(t)=vi(t)+u1i(t),v˙i(t)=f(t,xi(t),vi(t))+u2i(t)(i=1,2,…,6),
where
(44)f(t,xi(t),vi(t))=2cos(2t)-55sin(xi(t))-105vi(t)u1i(t)=-∑j∈Ni‍aij[(xi(t)-xj(t))-(xi(t-τ(t))-xj(t-τ(t)))]-δi[(xi(t)-xr(t))-(xi(t-τ(t))-xr(t-τ(t)))]u2i(t)=-∑j∈Ni‍aij[(vi(t)-vj(t))-(vi(t-τ(t))-vj(t-τ(t)))]-δi[(vi(t)-vr(t))-(vi(t-τ(t))-vr(t-τ(t)))],
where xi(t)∈ℝ and vi(t)∈ℝ are the position and velocity states of agent i, respectively.
The interaction diagraph of multiagent systems (43) is shown in Figure 1. We examine what agents should be pinned, and note that agent 6 has zero in-degree and that the out-degree of agents 2 is bigger than their in-degree. According to Proposition 11, agent 6 should be pinned first. Because its state is not affected by others, agent 2 can be chosen as pinned candidates, furthermore. Based on the pinned-agent selection scheme, we rearrange six agents and the new order is 6, 2, 4, 5, 3, 1. By some calculation, we can obtain ρ1=0.1, ρ2=0.2, for given M,  τi(i=1,2,…,M-1) and βj(j=1,2,…,M) (when M=1, let τ0=0; when M=2, let τ0=0, τ1=0.1, β1=0.1, β2=0.9; when M=3, let τ0=0, τ1=0.1, τ2=0.2, β1=0.1, β2=0.1, β3=0.8 ). By employing the LMI Toolbox in Theorem 8, we can obtain Table 1, it can be seen that (1) the minimum number of pinned agents is three, and we chose agents 6, 2, and 4 as pinned agents; (2) the allowable upper bound τ increases with increasing M. If we can obtain more information of time delay, it will lead to a larger allowable upper bound of time delay. When we chose agents 6, 2, and 4 as pinned agents, the evolutions of positions and velocities of six agents are shown in Figures 2, 3, 4, and 5.Table 1
The allowable upper bound of time delay for differentM.
System
Pinning nodes
M
=
1
M
=
2
M
=
3
Nonlinear system
6, 2, 4
0.3207
0.3363
0.3626
Linear system
6, 2, 4
0.3287
0.3463
0.3787Figure 1
The topology structure of the six agents.Figure 2
The position curvesxi(t)(i=1,2,…,6) for nonlinear multiagent systems (43) with τ0=0, τ1=0.3.Figure 3
The velocity curvesvi(t)(i=1,2,…,6) for nonlinear multiagent systems (43) with τ0=0, τ1=0.3.Figure 4
The position curvesxi(t)(i=1,2,…,6) for nonlinear multiagent systems (43) with τ0=0, τ1=0.2, τ2=0.3,  β1=0.1, and β2=0.9.Figure 5
The velocity curvesvi(t)(i=1,2,…,6) for nonlinear multiagent systems (43) with τ0=0, τ1=0.2, τ2=0.3, β1=0.1, and β2=0.9.Example 13.
We consider the following multiagent systems consisting of six agents described by(45)x˙i(t)=vi(t)+u1i(t)v˙i(t)=u2i(t)(i=1,2,…,6),
where
(46)u1i(t)=-∑j∈Ni‍aij[(xi(t)-xj(t))-(xi(t-τ(t))-xj(t-τ(t)))]-δi[(xi(t)-xr(t))-(xi(t-τ(t))-xr(t-τ(t)))]u2i(t)=-∑j∈Ni‍aij[(vi(t)-vj(t))-(vi(t-τ(t))-vj(t-τ(t)))]-δi[(vi(t)-vr(t))-(vi(t-τ(t))-vr(t-τ(t)))],
where xi(t)∈ℝ and vi(t)∈ℝ are the position and velocity states of agent i, respectively.
Similar to Example12, by using the LMI Toolbox in Corollary 10, we can obtain Table 1. When we chose agents 6, 2, and 4 as pinned agents, the evolutions of positions and velocities of six agents are shown in Figures 6, 7, 8, and 9.Figure 6
The velocity curvesvi(t)(i=1,2,…,6) for nonlinear multiagent systems (45) with τ0=0, τ1=0.3.Figure 7
The velocity curvesvi(t)(i=1,2,…,6) for nonlinear multiagent systems (45) with τ0=0, τ1=0.3.Figure 8
The position curvesxi(t)(i=1,2,…,6) for nonlinear multiagent systems (45) with τ0=0, τ1=0.2, τ2=0.3, β1=0.1, and β2=0.9.Figure 9
The velocity curvesvi(t)(i=1,2,…,6) for nonlinear multiagent systems (45) with τ0=0, τ1=0.2, τ2=0.3, β1=0.1, and β2=0.9.


## 5. Conclusions

By employing the information of the probability distribution of the time delay, this paper investigates the consensus problem for second-order leader-follower nonlinear multiagent systems with general network topologies. Different from the common assumptions on the time delay in the existing literatures, it is assumed in this paper that the delay is random and its probability distribution is known a prior. Based on graph theory, a pinning control algorithm is proposed, and the consensus criteria are derived to achieve leader-follower consensus for multiagent systems with nonlinear second-order dynamics. Moreover, this paper addresses what kind of agents and how many agents should be pinned.



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