# Distributed Node-to-Node Consensus of Multiagent Systems with Delayed Nonlinear Dynamics and Intermittent Communications

Authors: Kexin Jia; Hongjie Li; Wenqiang Zheng; Qinyou Mou; Jiajun Shao
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The paper is concerned with the problem of distributed node-to-node consensus of multiagent systems with delayed nonlinear dynamics and communication constraints. A new kind of consensus protocol based only on the intermittent measurements of neighboring agents is proposed to make each follower track the corresponding leader asymptotically. Based on the Lyapunov stability theory andM-matrix theory, some novel and simple criteria are derived for node-to-node consensus of multiagent systems. It is shown that consensus can be reached if the communication time duration is larger than the corresponding threshold value. Finally, a numerical example is provided to demonstrate the effectiveness of the obtained theoretical results.


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## Body

## 1. Introduction

Distributed control of multiagent systems has recently attracted various scientific communities ranging from mathematics to electronic engineering [1, 2], which is partly due to its broad applications in sensor networks, formation of unmanned vehicles, clustering of satellites, and so on. An important issue in distributed control of multiagent systems is consensus control, which can be described as only using the local relative information to make the whole system reach a state agreement [3].Recently, there are many fruitful results on distributed consensus of multiagent systems, according to the presence or absence of a leader in the considered multiagent systems; within the context of multiagent systems, the commonly studied consensus problems can be classified into three categories: leaderless consensus, leader-following, and containment control. Leaderless consensus means that the states of considered multiple agents achieve a prior unknown agreement [4–6]. Leader-following consensus is also called distributed consensus tracking, which will converge to a designed state trajectory which is generated by a virtual or real leader [7–10]. Distributed containment control of multiagent systems with multiagent systems with multiple leaders has been addressed in the literature [11–15], where the control goal is to make the states of each follower converge to a convex combination of the states of all leaders. It should be noted that there is a common assumption in existing works on containment control that there is no interaction among the leaders. However, in some real applications, there do exist interactions between the neighboring leaders and the state of a given follower needs to track that of a particular leader not the convex combination of the states of all leaders. Motivated by this observation, the notion of distributed node-to-node consensus is introduced in [16, 17]. Specifically, it is assumed that there are two levels in the multiagent systems: the leader’s and the follower’s levels; the dynamics of each leader may only be affected by those of its neighbors in the leader’s levels, while the dynamics of each follower may be affected by those of its neighbors in both leader’s and follower’s levels. It should be noticed that most of the aforementioned works in multiagent systems are derived on a key assumption that information is transmitted continuously among the agents. However, this assumption may not be ensured in some practical cases due to communication disturbances and sensor range limitations [18, 19]. Thus it is reasonable to assume that each agent can sense its neighbors only intermittently. It is worth noting that both impulsive control and intermittent control are discontinuous controls. The difference of them is that impulsive control is activated only at some isolated points [20–22], while intermittent control is activated during some intervals and does not work during the other intervals. On the other hand, impulsive control and intermittent control have attracted more interest recently due to their easy implementation in engineering control and being more efficient and useful in real-life applications. On the other hand, in order to guarantee stability in complex networks, adaptive strategies to appropriately tune the strengths of the interconnections among network nodes have been proposed [23–25]; it can be proved that the fixed coupling strength is larger than those needed in practice. Therefore, it may be conservative; adaptive strategies are proposed and can effectively overcome these shortcomings, which can appropriately tune the strength with the dynamic evolution of the network. Finally, the characteristic of time-delayed coupling is very common in biological and physical systems; some of the time delays are trivial and so can be ignored, while some others cannot be ignored, such as in long distance communication and traffic congestions. Therefore, time delays should be modeled in order to simulate more realistic networks [26, 27].Motivated by these observations, this paper is devoted to investigate distributed node-to-node consensus problem for a class of multiagent systems with delayed nonlinear dynamics by adaptive intermittent control. The main contributions of this paper can be summarized as follows. (1) The multiagent systems in the present framework consist of two levels, that is, the leader’s level and the follower’s level. (2) The agent dynamics are extended to delayed nonlinear form. (3) A novel adaptive intermittent control protocol is designed, where only a small fraction of followers can sense their corresponding leaders on some disconnected intervals.Notation. The notation used here is fairly standard except where otherwise stated. Rn denotes the n-dimensional Euclidean space and Rn×m is a set of real n×m matrices. In and 1N represent the n×n identity matrices and the n-dimensional column vector with all entries equal to one. · refers to the Euclidean norm. Let diag⁡{⋯} stand for a block-diagonal matrix. Symbol · denotes the Euclidean norm. MT represents the transpose of the matrix M. For a real symmetric matrix A, λmax⁡(A) and λmin⁡(A) represent its largest and smallest eigenvalues, respectively. Notation ⊗ denotes the Kronecker product. In this paper, if not explicitly stated, matrices are assumed to have compatible dimensions.


## 2. Problem Formulation and Preliminaries

Letg=v,ε,G be a directed graph with the set of nodes v=v1,v2,…,vN, the set of directed links ε⊆v×v, and a weighted adjacency matrix G=(Gij)N×N with nonnegative adjacency elements Gij. A directed graph has or contains a directed spanning tree if there exists a node called root such that there exists a directed path from this node to every other node. The Laplacian matrix L=(lij)N×N of the weighted directed network is defined as Lij=-Gij(i≠j); Lii=∑j=1,j≠iNGij.Definition 1.
A nonsingular real square matrixA is called an M-matrix if all of its off-diagonal elements are nonpositive, and all elements of A-1 are nonnegative.Lemma 2.
Suppose that matrixA∈Rn×n has nonpositive off-diagonal elements; then A is anM-matrix if and only if there exists a positive diagonal matrix Ξ=diag⁡{1/θ1,1/θ2,…,1/θn}, such that ATΞ+ΞA>0, where Aθ=1N and θ=[θ1,θ2,…,θn]T.Lemma 3.
Suppose that nonnegative functiony(t), t∈[-τ,+∞), satisfies (1)y˙t≤-c1yt+c2yt-τt≥0;then (2)yt≤y0τe-rtt≥0,where c1>c2>0, y0τ=max-τ≤s≤0⁡{ys}, and r is the unique solution of -r=-c1+c2erτ.Lemma 4.
Suppose that nonnegative functiony(t), t∈[-τ,+∞), satisfies (3)y˙t≤c1yt+c2yt-τt≥0;then (4)yt≤y0τec1+c2tt≥0,where c1 and c2 are two positive constants and y(0)τ=max-τ≤s≤0⁡{y(s)}.Consider that multiagent systems consist of the leader’s and the follower’s level; it is assumed that each level containsN coupled agents. The dynamics of the ith agent in the leader’s level are described as follows:(5)x˙it=fxit,xit-τ,t-c∑j=1NLijxjti=1,2,…,N,where xi(t)∈Rn is the position state of the ith leader. f(·,·,t):Rn×Rn×R+→Rn is a continuously differentiable vector-valued function representing the inherent delayed nonlinear dynamics of agent i. τ>0 is the time delay. c>0 represents the coupling strength. L=[lij]N×N is the Laplacian matrix of communication topology. Furthermore, the evolution of the ith follower is given as(6)x^˙it=fx^it,x^it-τ,t-c∑j=1Nlijx^jt+uiti=1,2,…,N,where x^i(t)∈Rn is the position state of the ith follower. ui(t) is the control input acting agent i.It is commonly assumed in most existing works that all information is transmitted continuously among the agents. However, in some real situations agents may only communicate with their neighbors over some disconnected time intervals due to the unreliability of communication channels, failure of physical devices; thus the adaptive intermittent consensus protocol is designed as follows:(7)uit=cditxit-x^itt∈kT,kT+δ0t∈kT+δ,k+1T,where the pinning link di(t)>0 if and only if the ith follower can directly sense the ith leader; otherwise di(t)=0; its updating law is given later. The communication time duration δ satisfies τ<δ≤T.Remark 5.
What kind of agents and how many agents should be pinned? The selection criteria are as follows. Firstly, divide the digraph of the network intom components, where each component contains a directed tree; select the root nodes in these components as pinned nodes. There are two reasons: (1) the related message of the leader can be transmitted to each follower; (2) the consensus can be achieved if and only if the root node of each component is pinned, which means that the minimum number of pinning controllers is m. If we select the root nodes in these components as pinned nodes, the following consensus is not achieved; we need to rearrange the remaining nodes in descending order according to the differences of out-degrees and in-degrees and gradually increase the amount of control node, until the consensus is achieved. More detailed discussion can be found in the literatures [28, 29].Definition 6.
The distributed node-to-node consensus in multiagent systems (5) and (6) is said to be achieved if, for any initial conditions, the following holds:(8)limt→∞⁡xit-x^it=0i=1,2,…,N.


## 3. Main Result

In this section, we deal with node-to-node consensus problem for multiagent systems with delayed nonlinear dynamics by adaptive intermittent control. By defining a Lyapunov function and utilizing a combination of stability theory and Kronecker products, some simple conditions are established to ensure node-to-node consensus of multiagent systems (5) and (6). The following assumption is needed to derive the main results.Assumption 7.
For the nonlinear functionf(·,·,t):Rn×Rn×R+→Rn in (5) and (6), there exist two positive constants L1,L2>0 such that(9)fx1,x2,t-fy1,y2,t2≤L1x1-y12+L2x2-y22,where x1,x2,y1,y2∈Rn.Substituting (7) into (6), one has(10)x^˙it=fx^it,x^it-τ,t-c∑j=1Nlijx^jt+cditxit-x^itt∈kT,kT+δfx^it,x^it-τ,t-c∑j=1Nlijx^jtt∈kT+δ,k+1T.For convenience, systems (5) and (10) can be rewritten as(11)x˙t=fxt,xt-τ,t-cL⊗Inxt,x^˙t=fx^t,x^t-τ,t-cL⊗Inx^t-cDt⊗Inx^t-xtt∈kT,kT+δfx^t,x^t-τ,t-cL⊗Inx^tt∈kT+δ,k+1T,where (12)xt=x1Tt,x2Tt,…,xNTtT,x^t=x^1Tt,x^2Tt,…,x^NTtT,fxt,xt-τ,t=fTx1t,x1t-τ,t,fTx2t,x2t-τ,t,…,fTxNt,xNt-τ,tT,fx^t,x^t-τ,t=fTx^1t,x^1t-τ,t,fTx^2t,x^2t-τ,t,…,fTx^Nt,x^Nt-τ,tT,Dt=diag⁡d1t,d2t,…,dlt,0,…,0.Letet=x^t-xt; then the following error system can be obtained from (11) such that(13)e˙t=fet,et-τ,t-cL⊗Inet-cDt⊗Inett∈kT,kT+δfet,et-τ,t-cL⊗Inett∈kT+δ,k+1T,where(14)fet,et-τ,t=fx^t,x^t-τ,t-fxt,xt-τ,t.It is not hard to verify that node-to-node consensus for multiagent systems (5) and (6) will be achieved if and only if zero equilibrium point of error system (13) is globally attractive.Theorem 8.
Suppose that Assumption7 holds; then distributed node-to-node consensus problem for multiagent systems (5) and (6) is achieved under the following distributed adaptive intermittent control protocol:(15)ut=-cDt⊗Inett∈kT,kT+δ0t∈kT+δ,k+1Tand the updating law di(t)(i=1,2,…,l) is defined as(16)d˙it=ca0iea1iteTtΞ⊗Inett∈kT,kT+δ0t∈kT+δ,k+1Tif the following conditions hold:(17)1+L1-2cλmin⁡L¯θmin⁡+a1<0,(18)1+L1-cλmin⁡S-b1<0,(19)δ>rτ+b1+L2Tr+b1+L2,where (20)Ξ=diag⁡1θ1,1θ2,…,1θN,θ=θ1,θ2,…,θnT,Lθ=1N,θmax⁡=max1≤i≤N⁡θi,θmin⁡=min1≤i≤N⁡θi,L¯=ΞL^+L^TΞ2,L^=L+D∗,D∗=diag⁡d1∗,d2∗,…,dl∗,0,…,0,S=Ξ-1ΞL+LTΞ,-r=-a1+L2erτ,a1=min1≤i≤N⁡a1i.Proof.
Choose the following Lyapunov function candidate:(21)Vt=12eTtΞ⊗Inet+12∑i=1le-a1itdit-d∗2a0i,where Ξ=diag{1/θ1,1/θ2,…,1/θN}, θ=[θ1,θ2,…,θn]T satisfies Lθ=1N. di∗(i=1,2,…,l) are some undetermined sufficiently large positive constants.
(1) Fort∈kT,kT+δand an arbitrarily given k∈N, taking the derivative of Vt along the trajectories of system (13) can be given as follows:(22)V˙t=eTtΞ⊗Ine˙t-12∑i=1la1ie-a1itdit-di∗2a0i+∑i=1le-a1itdit-di∗a0id˙it=eTtΞ⊗Infet,et-τ,t-ceTtΞL+LTΞ2⊗Inet-ceTtΞDt+DtTΞ2⊗Inet-12∑i=1la1ie-a1itdit-di∗2a0i+∑i=1ldit-di∗eTtΞ⊗Ine˙t=eTtΞ⊗Infet,et-τ,t-ceTtL¯⊗Inet-12∑i=1la1ie-a1itdit-di∗2a0i≤12eTtΞ⊗Inet+12fTet,et-τ,tΞ⊗Infet,et-τ,t-ceTtL¯⊗Inet-12∑i=1la1ie-a1itdit-di∗2a0i≤12eTtΞ⊗Inet+12L1eTtΞ⊗Inet+12L2eTt-τΞ⊗Inet-τ-cλmin⁡L¯θmin⁡eTtΞ⊗Inet-12∑i=1la1ie-a1itdit-di∗2a0i=121+L1-2cλmin⁡L¯θmin⁡eTtΞ⊗Inet-12∑i=1la1ie-a1itdit-di∗2a0i+12L2eTt-τΞ⊗Inet-τ≤121+L1-2cλmin⁡L¯θmin⁡+a1eTtΞ⊗Inet-12a1eTtΞ⊗Inet-12a1∑i=1le-a1itdit-di∗2a0i+12L2eTt-τΞ⊗Inet-τ+12L2∑i=1le-a1it-τdit-τ-di∗2a0i≤-a1Vt+L2Vt-τ,where(23)a1=min1≤i≤N⁡a1i,L¯=ΞL^+L^TΞ2,L^=L+D∗,D∗=diag⁡d1∗,d2∗,…,dl∗,0,…,0,θmin⁡=min1≤i≤N⁡θi.
(2) Fort∈kT+δ,k+1T with arbitrarily given k∈N, taking the time derivative Vt along the trajectories of (13), we have(24)V˙t=eTtΞ⊗Ine˙t-12∑i=1la1ie-a1itdit-di∗2a0i=eTtΞ⊗Infet,et-τ,t-ceTtL~⊗Inet-12∑i=1la1ie-a1itdit-di∗2a0i≤12eTtΞ⊗Inet+12fTet,et-τ,tΞ⊗Infet,et-τ,t-ceTtL~⊗Inet-12∑i=1la1ie-a1itdit-di∗2a0i≤121+L1eTtΞ⊗Inet+12L2eTt-τΞ⊗Inet-τ-ceTtL~⊗Inet-12∑i=1la1ie-a1itdit-di∗2a0i,where L~=ΞL+LTΞ/2.
Note that, for an arbitrarily givent∈[kT+δ,(k+1)T), the matrix ΞL+LTΞ,Ξ is normal; for any given z∈RN, it is easily derived from the generalized eigenvalue theory that(25)λmin⁡SzTΞz≤zTΞL+LTΞz,where S=Ξ-1ΞL+LTΞ; then one has(26)12λmin⁡SeTtΞ⊗Inet≤eTtL~⊗Inet.
By considering (18), (24), and (26), we can obtain(27)V˙t≤121+L1eTtΞ⊗Inet+12L2eTt-τΞ⊗Inet-τ-12cλmin⁡SeTtΞ⊗Inet-12∑i=1la1ie-a1itdit-di∗2a0i≤121+L1-cλmin⁡S-b1eTtΞ⊗Inet+12b1eTtΞ⊗Inet+12b1∑i=1le-a1itdit-di∗2a0i+12L2eTt-τΞ⊗Inet-τ+12L2∑i=1le-a1it-τdit-τ-di∗2a0i≤b1Vt+L2Vt-τ,where b1>0.
Based on the above analysis and according to Lemma3, when t∈[0,δ), we can obtain(28)Vt≤max-τ≤s≤0⁡Vse-rt=V0τe-rt,where r is the positive solution of -r=-a1+L2e-rτ.
Whent∈[δ,T), by using Lemma 4, one has(29)Vt≤maxδ-τ≤s≤δ⁡Vseb1+L2t-δ≤max-τ≤s≤0⁡Vsmaxδ-τ≤t≤δ⁡e-rteb1+L2t-δ=V0τe-rδ-τ+b1+L2t-δ.
ConsiderT-τ>δ; then(30)Vtτ=maxT-τ≤t≤T⁡Vt≤maxδ-τ≤s≤δ⁡VsmaxT-τ≤t≤T⁡eb1+L2t-δ≤maxδ-τ≤s≤δ⁡Vseb1+L2T-δ≤max-τ≤s≤0⁡Vse-rδ-τ+b1+L2T-δ=V0τe-Δ,where Δ=r(δ-τ)-(b1+L2)(T-δ).
Next, for allk∈N, we will prove that VkTτ≤V0τe-kΔ. One has(31)VkTτ=maxkT-τ≤t≤kT⁡Vt≤maxk-1T+δ-τ≤s≤k-1T+δ⁡VsmaxkT-τ≤t≤kT⁡eb1+L2t-k-1T-δ=maxk-1T+δ-τ≤s≤k-1T+δ⁡Vseb1+L2T-δ≤maxk-1T-τ≤s≤k-1T⁡Vsmaxk-1T+δ-τ≤t≤k-1T+δ⁡e-rt-k-1Teb1+L2T-δ=maxk-1T-τ≤s≤k-1T⁡Vse-rδ-τ+b1+L2T-δ=maxk-1T-τ≤s≤k-1T⁡Vse-Δ=⋯≤max-τ≤s≤0⁡Vse-kΔ=V0τe-kΔ.
For anyt>0, there exists a positive integer k0, such that k0T≤t≤(k0+1)T; then one has the following.
Whent∈[k0T,k0T+δ), we can obtain(32)Vt≤maxk0T-τ≤s≤k0T⁡Vse-rt-k0T≤max-τ≤s≤0⁡Vse-k0Δ≤max-τ≤s≤0⁡VseΔ1-t/T=V0τeΔe-Δ/Tt.
Whent∈[k0T+δ,(k0+1)T), we have(33)Vt≤maxk0T+δ-τ≤s≤k0T+δ⁡Vseb1+L2t-k0T-δ≤maxk0T-τ≤s≤k0T⁡Vsmaxk0T+δ-τ≤t≤k0T+δ⁡e-rt-k0Teb1+L2t-k0T-δ=maxk0T-τ≤s≤k0T⁡Vse-rδ-τet-k0+1T+T-δ≤V0τe-k0Δe-rδ-τ+b1+L2T-δ=V0τe-k0+1Δ≤V0τe-Δ/Tt.
From (32) and (33), we can obtain(34)Vt≤K0e-Δ/Tt,where K0=max{1,eΔ}, which indicates that the states of agents exponentially converge to node-to-node consensus. This completes the proof.Remark 9.
It can be seen from (34) that the convergence speed is related to the value of Δ; by some simple computation, we have (35)Δ=rδ-rτ-b1+L2T+b1+L2δ=r+b1+L2δ-rτ-b1+L2T;from the above equation, we can see that the convergence speed will accelerate with the increase of δ. Therefore, a trade-off has to be made between the convergence rate and the control cost.


## 4. A Numerical Example

In this section, a numerical example is provided to verify the analysis results.Example 1.
Suppose that there are five leaders in the considered multiagent systems; the communication topologies among the leaders and the followers are given in Figure1; the solid lines indicate the neighboring relationship among the agents in the same level, while the dashed lines are the pinning links; the corresponding Laplacian matrix L and the pinning matrix D∗ are given as follows:(36)L=4-3-100-35-200-1-230000-23-100000,D∗=00000000000010000001000001.
The nonlinear dynamics are described asf(xi(t),xi(t-τ),t)=cos⁡0.5xi1t+cos⁡0.5xi1t-τ,0.25sin2xi2t+0.25sin2xi2t-τT; it can be verified that the intrinsic nonlinear dynamics of the agents obey Assumption 7 with L1=L2=0.5.
By some simple computation, we have(37)θ=1.02730.93640.30.13080.1,Ξ=diag0.97351.0683.33337.647110;then we can obtain θmin⁡=0.1, λmin⁡(L¯)=0.8816, and λmin⁡(S)=-0.8499. Taking c=11.4, T=1, and τ=0.1, by using condition (17) in Theorem 8, we can obtain a1<0.51; let a1=0.5099. By using condition (18), we have b1>11.1319, here taking b1=11.14. Then we can obtain the communication δ>0.9646, taking δ=0.9650. It is assumed that the agents communicate with their neighbors only when t∈⋃k∈N[k,k+0.9650), which indicates that the average communication rate is 0.9650. The position state trajectories of the leaders and the followers are shown in Figures 2–6, which indicate that node-to-node consensus in multiagent systems (5) and (6) is indeed guaranteed. The position state error trajectories of agents are shown in Figure 7, which further demonstrate the effectiveness of the obtained criteria. The curve of adaptive intermittent control is shown in Figure 8; it is noted that the control u(t)=0 in time interval ⋃k∈N[k+0.9650,k+1). The trajectory of pinning feedback gain di(t)(i=3,4,5) under the updating laws (16) is shown in Figure 9. The simulation has confirmed the effectiveness of our theoretical results.Figure 1
Communication topology of multiagent systems (5) and (6).Figure 2
The curves of position statesx1(t) and x^1t.Figure 3
The curves of position statesx2t and x^2t.Figure 4
The curves of position statesx3(t) and x^3(t).Figure 5
The curves of position statesx4(t) and x^4(t).Figure 6
The curves of position statesx5(t) and x^5(t).Figure 7
The position state error trajectories ofe(t).Figure 8
The trajectory of adaptive intermit controlu(t).Figure 9
The trajectory of pinning feedback gainsdi(t)(i=3,4,5).


## 5. Conclusions

In this paper, we have investigated node-to-node consensus of multiagent systems with delayed nonlinear dynamics. The coordination goal is to make each follower track the corresponding leader asymptotically, where only a small fraction of followers can sense the states of their corresponding leaders on some disconnected time intervals. Finally, a simulation example has been employed to demonstrate the effectiveness of the results. Further research topics include the extension of our results to second-order or high-order multiagent systems with delayed nonlinear dynamics and communication constraints.



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