# Consensus Learning Control for Leader-Following Nonlinear Multiagent Systems with Control Delay

Authors: Xiongfeng Deng; Xiuxia Sun; Shuguang Liu
Journal: Wireless Communications and Mobile Computing
 (2019)

---

## Abstract

In this paper, the consensus tracking problem of leader-following nonlinear control time-delay multiagent systems with directed communication topology is addressed. An improved high-order iterative learning control scheme with time-delay is proposed, where the local information between agents is considered. The uniformly global Lipschitz condition is applied to deal with the nonlinear dynamics. Then, a sufficient condition is driven, which guarantees that all the following agents track the trajectory of leader. Also, the convergence of proposed control protocol is analyzed by the norm theory. Finally, two cases are provided to illustrate the validity of theoretical results.


---

## Body

## 1. Introduction

In recent years, cooperative control problems of multiagent systems have been paid considerable attention owing to their applications in multiple unmanned aerial vehicle systems, mobile robot systems, and sensor networks [1–3]. The key problem of cooperative control is consensus, which means that the states of a group of agents arrive at the agreement under a designed control law.In the existing literature, the consensus problems of linear multiagent systems, such as the leader-following consensus tracking problem [4, 5], the random packet dropout problem [6], and the finite-time formation problem [7], have been widely considered. However, compared with the linear multiagent systems, the consensus problems of nonlinear multiagent systems are more popular for researchers. For example, the robust tracking problem of heterogeneous nonlinear multiagent systems with bounded disturbances was developed in [8]. In [9], the consensus problem of nonlinear multiagent systems with unknown control directions was addressed, and the quantized consensus protocol for second-order nonlinear multiagent systems was studied in [10]. In addition, the formation control problems of nonlinear multiagent systems [11, 12] and the tracking problem of high-order nonlinear multiagent systems [13] were investigated as well. It should be pointed out that the consensus problems considered in the above papers do not take into consideration the case of time delay.In practice, the time delay occurred due to the impact of physical factors of sensors. It is of great significance to study the control problem of time-delay systems. In [14], Lyapunov matrices for a type of time-delay systems were designed. In [15], the stabilization problem of time-delay switched control systems was analyzed, and a compensation approach for a time-delay system was proposed in [16]. By now, the consensus problems of multiagent systems with time delay have been investigated in some papers. In [17], an event-triggered consensus protocol for the leader-following first-order linear multiagent systems with time-varying delays was proposed. In [18], the consensus problem of second-order linear multiagent systems with communication delay was addressed. In [19, 20], the consensus problems of general linear multiagent systems with time-delay were studied, while the consensus problem of delayed linear multiagent systems was fully considered and studied in [21]. Moreover, the leader-following consensus problem of nonlinear multiagent systems with mixed delays and the formation control problem of multiagent systems with time-varying delays were addressed [22, 23]. Different from [17–20], the case of control delay was considered in [24]. Based on the above papers, it is not difficult to see that there are few efforts focusing on the consensus problems of multiagent systems with control delay. Hence, the first motivation of this work is to discuss the control delay problem of multiagent systems.As we know, the iterative learning control is based upon the idea that the performance of a system that performs the same task repeatedly can be improved by learning from previous iterations [25]. Currently, the method has been used to achieve the consensus problem of multiagent systems. In [26], the optimal iterative learning control protocol for the consensus tracking of multiagent systems was presented, while the event-triggered iterative learning control protocol for the same problem was designed in [27]. In [28], the formation control of multiagent systems with iterative learning control was studied, and the finite-time consensus problem of multiagent systems with iterative learning control scheme was discussed in [29]. Furthermore, the distributed adaptive iterative learning control scheme [30], the sliding mode iterative learning control approach [31], and the high-order iterative learning control protocol [32] were also designed to deal with the consensus of multiagent systems. The results obtained in the above literature indicate that the iterative learning control in solving the consensus problem of multiagent systems is effective. Accordingly, the second motivation of this work is to consider the iterative learning control for the consensus problem.Inspired by the above analysis, the consensus tracking problem of leader-following nonlinear multiagent systems with control delay is investigated in this paper. The iterative learning control approach is introduced to design the control protocol. The main contributions of this work are summarized as follows: (i) compared with [17–20], the consensus problem of nonlinear multiagent systems with control delay under directed topology is investigated. It is assumed that the delay of control input is fixed; (ii) different from [32], an improved high-order iterative learning control protocol with time delay is designed, where the local information between agents is considered in control protocol design. In addition, the uniformly global Lipschitz condition is applied to process the nonlinear terms; (iii) based on the norm theory, the convergence of the proposed control protocol is verified, and a sufficient condition is driven. The condition can guarantee all the following agents to track the trajectory of the leader.The rest of this paper is organized as follows. In Section2, the graph theory, some useful definitions, and lemmas are introduced. In Section 3, the problem formulation on the leader-following nonlinear multiagent systems with control delay is described. The control protocol design and convergence analysis are shown in Section 4, and simulation examples are provided in Section 5. Finally, some conclusions are briefly drawn in Section 6.


## 2. Preliminaries

### 2.1. Graph Theory

A multiagent system which consists ofn agents can be defined as a graph G. Let V=v1,⋯,vn be a node set and let E=(i,j),i,j∈V,andi≠j be a directed edge set. The adjacency matrix is defined as A=aij∈Rn×n, where aij>0 if and only if (i,j)∈E and aij=0 otherwise. It is assumed that aii=0. Let Ni=vj:(vj,vi)∈E represent the set of neighbors of agent i. The Laplacian matrix L is denoted by L=D-A, where D=diagd1,⋯,dn with di=∑j=1Niaij.In this paper, an augmented graphG- consists of n following agents and one leader agent is considered. The communication between following agents and the leader is defined as B=diagb1,⋯,bn. If the agent i can obtain the information of the leader, then bi>0 and bi=0 otherwise.


### 2.2. Definitions and Lemmas

Definition 1 (see [33]).
For a functionh(t):0,T→Rn, the λ norm is defined as(1)htλ=supt∈0,Te-λtht,λ>0The following property forλ norm is held.Property 2.
For the functionsf(t),h(t):0,T→Rn, if h(t)=∫0tea(t-s)f(s)ds, there exists(2)htλ≤1-ea-λTλ-aftλand if h(t)=∫0t∫0σea(σ-s)f(s)dsdσ, then we have(3)htλ≤1-e-λTλ1-ea-λTλ-aftλ,0<a<λLemma 3 (Bellman-Gronwall lemma).
Suppose that the functionsx(t) and y(t) are real continuous functions on the interval 0,T; if there exists(4)xt≤c+∫0taxs+bysdsthen we have(5)xt≤ceat+∫0tbyseat-sdswhere a≥0 and b,c∈R.Lemma 4 (see [34]).
For a real positive seriesann∈1,⋯,∞, if it satisfies(6)an≤β1an-1+β2an-2+⋯+βNaN-2+φn=N+1,N+2,⋯where βm≥0m=1,⋯,N, φ≥0 and there exists β=∑i=1Nβm<1, then we have(7)limn→∞an≤φ1-β


## 2.1. Graph Theory

A multiagent system which consists ofn agents can be defined as a graph G. Let V=v1,⋯,vn be a node set and let E=(i,j),i,j∈V,andi≠j be a directed edge set. The adjacency matrix is defined as A=aij∈Rn×n, where aij>0 if and only if (i,j)∈E and aij=0 otherwise. It is assumed that aii=0. Let Ni=vj:(vj,vi)∈E represent the set of neighbors of agent i. The Laplacian matrix L is denoted by L=D-A, where D=diagd1,⋯,dn with di=∑j=1Niaij.In this paper, an augmented graphG- consists of n following agents and one leader agent is considered. The communication between following agents and the leader is defined as B=diagb1,⋯,bn. If the agent i can obtain the information of the leader, then bi>0 and bi=0 otherwise.


## 2.2. Definitions and Lemmas

Definition 1 (see [33]).
For a functionh(t):0,T→Rn, the λ norm is defined as(1)htλ=supt∈0,Te-λtht,λ>0The following property forλ norm is held.Property 2.
For the functionsf(t),h(t):0,T→Rn, if h(t)=∫0tea(t-s)f(s)ds, there exists(2)htλ≤1-ea-λTλ-aftλand if h(t)=∫0t∫0σea(σ-s)f(s)dsdσ, then we have(3)htλ≤1-e-λTλ1-ea-λTλ-aftλ,0<a<λLemma 3 (Bellman-Gronwall lemma).
Suppose that the functionsx(t) and y(t) are real continuous functions on the interval 0,T; if there exists(4)xt≤c+∫0taxs+bysdsthen we have(5)xt≤ceat+∫0tbyseat-sdswhere a≥0 and b,c∈R.Lemma 4 (see [34]).
For a real positive seriesann∈1,⋯,∞, if it satisfies(6)an≤β1an-1+β2an-2+⋯+βNaN-2+φn=N+1,N+2,⋯where βm≥0m=1,⋯,N, φ≥0 and there exists β=∑i=1Nβm<1, then we have(7)limn→∞an≤φ1-β


## 3. Problem Formulation

In this section, a type of leader-following nonlinear multiagent systems with control delay is considered. The dynamics of theith following agent at the kth iteration are described as(8)x˙ikt=viktv˙ikt=fzikt,t+Bzikt,tuikt-τyikt=xikt,viktT=gzikt,twhere xik(t)∈R, vik(t)∈R, and uik(t)∈R are the position, velocity, and control input of the ith following agent, respectively; yik(t)∈R2 is the specified output; zik(t)=xik(t),vik(t)T and t∈0,T; τ<T is known time delay. The nonlinear functions f(·,·):R2×[0,T]→R and B(·,·):R2×[0,T]→R are piecewise continuous on the interval 0,T; g(·,·):R2×[0,T]→R2 is differentiable in z and t, with gz(·,·) and gt(·,·).The vector form of (8) is written as(9)z˙kt=fzkt,t+Bzkt,tukt-τykt=gzkt,twhere zk(t)=z1k(t)T,⋯,znk(t)TT, yk(t)=y1k(t)T,⋯,ynk(t)TT, uk(t)=u1k(t-τ),⋯,unk(t-τ)T, fzk(t),t=f1Tz1k(t),t,⋯,fnTznk(t),tT with fizik(t),t=zik(t),fzik(t),tT, Bzk(t),t=diagB1z1k(t),t,⋯,Bnznk(t),t with Bizik(t),t=0,Bzik(t),tT, and gzk(t),t=gTz1k(t),t,⋯,gTznk(t),tT.The dynamics of leader agent are given as(10)x˙ot=votv˙ot=fzot,t+Bzot,tuot-τyot=xot,votT=gzot,twhere zo(t)=xo(t),vo(t)T; xo(t)∈R, xo(t)∈R, uo(t)∈R, and yo(t)∈R2 are the position, velocity, and control input and output of the leader agent, respectively. It is assumed that the input uo(t) is bounded; that is, there exists uo(t)≤buo with buo being a positive constant.The following assumptions are provided for the system (9).Assumption 5.
The functionsf, B, g, gzk, and gtk for zk are uniformly globally Lipschitz on the interval 0,T; that is, there exists constant hα such that(11)αz1kt,t-αz2kt,t≤hαz1kt-z2ktwhere α∈f,B,g,gzk,gtk.Assumption 6.
The functionsgz and B are bounded with bounds bgz and bB that are denoted by(12)bgz=maxsupk→∞,t∈-τ,T-τgzkzkt+τ,t+τbB=maxsupk→∞,t∈-τ,T-τBzkt+τ,t+τAssumption 7.
The resetting condition is considered for each iteration; that is,(13)zk0=1n⊗zo0,k=0,1,⋯where zo(0) is the initial state of leader, ⊗ represents the Kronecker product, and 1n=1,⋯,1T.Remark 8.
In Assumption5, the uniformly globally Lipschitz condition is satisfied for the nonlinear dynamics. However, the radial basis function neural network is applied to approximate the nonlinear dynamics in [30, 35]. Due to the fact that the number of hidden layers impacts the approximation accuracy of neural network, the method is not adopted in this paper.In this paper, the objective is to find a control schemeuk(t) such that the output of all the following agents can track the trajectory of leader as k tends to infinity; that is, limk→∞⁡xik(t)=xo(t) and limk→∞⁡vik(t)=vo(t) as t∈0,T for i=1,⋯,n.


## 4. Main Results

In this section, the consensus problem of leader-following nonlinear multiagent systems with control delay is discussed, and the control protocol design and convergence analysis will be described.Consider the multiagent systems (8) and (10); the consensus tracking error of the following agent i at the kth iteration is defined as(14)εikt=∑j∈Niaijyjkt-yikt+biyot-yiktand we have(15)εkt=H-ektwhere H-=H⊗I2, H=L+B, ek(t)=yo(t)-yk(t), and yo(t)=1n⊗yo(t); I2 is the unit matrix with 2 dimensions and t∈0,T.For the sake of the discussion simplification, some notations are provided as follows.(16)αkt=αzkt,t,αot=αzot,t,δαkt=αot-αktαzkt=∂αxt,t∂xtxt=zkt,αtkt=∂αxt,t∂txt=zktαzot=∂αxot,t∂xotxot=zot,αtot=∂αxot,t∂txot=zotwhere α represents a function concerned and zo(t)=1n⊗zo(t).According to the multiagent systems (8) and (10), an improved high-order iterative learning control protocol with time delay is presented as(17)uk+1t=ult+∑m=1NQmtεlt+τ+∑m=1NRmtε˙lt+τ+∑m=1NSmt∫-τtεls+τdswhere εl(t+τ)=H-el(t+τ), t∈-τ,T-τ; l=k-m+1, N≥1 is the order of the control protocol; Qm(t), Rm(t), and Sm(t) are learning matrices with appropriate dimensions. In addition, it is supposed that Qm(t)H-, Rm(t)H-, and Sm(t)H- are bounded and their bounds are defined as bQ, bR, and bS, respectively; that is,(18)bQ=maxt∈-τ,T-τQmtH-,bR=maxt∈-τ,T-τRmtH-,bS=maxt∈-τ,T-τSmtH-Remark 9.
The introduction of learning matricesQm(t), Rm(t), and Sm(t) is to adjust the performance of control protocol (17). Through adjusting these learning matrices, the desired control effect can be obtained. However, as can be seen from the below analysis, the learning matrices Qm(t) and Sm(t) do not affect the convergence result of control protocol (17). A sufficient condition with Rm(t) will be shown in Theorem 10.Based on the above analysis, we have Theorem10.Theorem 10.
Consider the multiagent systems (9) and (10) under directed topology and suppose that Assumptions 5–7 are satisfied, and the control protocol (17) is applied. If there exists(19)∑m=1NPmt=Inand positive numbers βm satisfy(20)Pmt-RmtH-gzzkt,tBzkt,t≤βm∑m=1Nβm<1then the output of all following agents can track the trajectory of leader agent; that is, limk→∞⁡xik(t)=xo(t) and limk→∞⁡vik(t)=vo(t) as t∈0,T for i=1,⋯,n.Proof.
Considering (9) and (10), the tracking error is defined as(21)elt+τ=yot+τ-ylt+τ=δglt+τwhere yo(t+τ)=1n⊗gzo(t+τ),t+τ, yl(t+τ)=gTz1l(t+τ),t+τ,⋯,gTznl(t+τ),t+τT, and t∈-τ,T-τ.
Then, the derivative of (21) is (22)e˙lt+τ=y˙ot+τ-y˙lt+τ=gzot+τz˙ot+τ+gtot+τ-gzlt+τz˙lt+τ+gtlt+τ=gzot+τfzot+τ,t+τ+Bzot+τ,t+τuot+gtot+τ-gzlt+τfzlt+τ,t+τ+Bzlt+τ,t+τult-gtlt+τ=gzlt+τBlt+τδult+gzot+τfot+τ+Bot+τuot-gzlt+τflt+τ+Blt+τuot+δgtlt+τwhere zl(t+τ)=z1l(t+τ)T,⋯,znl(t+τ)TT, ul(t)=u1l(t),⋯,unl(t)T, uo(t)=1n⊗uo(t), fl(t+τ)=f1Tz1l(t+τ),t+τ,⋯,fnTznl(t+τ),t+τT, fo(t+τ)=1n⊗fozo(t+τ),t+τ, Bl(t+τ)=diagB1z1l(t+τ),t+τ,⋯,Bnznl(t+τ),t+τ, Bo(t+τ)=In⊗Bozo(t+τ),t+τ, and fozo(t+τ),t+τ=zo(t+τ),fzo(t+τ),t+τT, Bozo(t+τ),t+τ=0,Bzo(t+τ),t+τT.
Substituting (21) and (22) into (17), then we have(23)δuk+1t=uot-uk+1t=∑m=1NPmtδult-∑m=1NQmtH-δglt+τ-∑m=1NSmtH-∫-τtδgls+τds-∑m=1NRmtH-gzlt+τBlt+τδult+gzot+τfot+τ+Bot+τuot-gzlt+τflt+τ+Blt+τuot+δgtlt+τ=∑m=1NPmt-RmtH-gzlt+τBlt+τδult-∑m=1NQmtH-δglt+τ-∑m=1NRmtH-δgzlt+τfot+τ+Bot+τuot+δgtlt+τ-∑m=1NRmtH-gzlt+τfot+τ-flt+τ-∑m=1NSmtH-∫-τtδgls+τds-∑m=1NRmtH-gzlt+τBot+τ-Blt+τuotwhere uo(t)=1n⊗uo(t) and the condition (19) is considered.
Taking norms on both sides of (23) yields(24)δuk+1t≤∑m=1NPmt-RmtH-gzlt+τBlt+τδult+∑m=1NQmtH-δglt+τ+∑m=1NRmtH-δgzlt+τfot+τ+Bot+τuot+δgtlt+τ+∑m=1NRmtH-gzlt+τfot+τ-flt+τ+∑m=1NSmtH-∫-τtδgls+τds+∑m=1NRmtH-gzlt+τBot+τ-Blt+τuot≤∑m=1Nβmδult+∑m=1NbShg∫-τtδzls+τds+∑m=1NbQhg+bRbDbuohgz+hgt+bgzhf+bgzbuohBδzlt+τ=∑m=1Nβmδult+∑m=1Nγ1δzlt+τ+∑m=1Nγ2∫-τtδzls+τdswhere the condition (20) is applied; hg, hgz, hgt, hf, and hB are Lipschitz constants of corresponding functions; δzl(t+τ)=zo(t+τ)-zl(t+τ), bD=maxsupk→∞,t∈[-τ,T-τ]⁡fo(t+τ)+Bo(t+τ), γ1=bQhg+bRbDbuohgz+hgt+bgzhf+bgzbuohB, and γ2=bShg.
According to Assumption7, one gets(25)δzlt+τ=zo0-zl0+∫-τtz˙os+τ-z˙ls+τds≤∫-τtfos+τ-fls+τ+Bos+τ-Bls+τuos+Bls+τδulsds≤∫-τthf+buohBδzls+τ+bBδulsds=∫-τtγ3δzls+τ+bBδulsdswhere γ3=hf+buohB.
Applying Lemma3, (25) is written as(26)δzlt+τ≤bB∫-τteγ3t-sδulsds
Substituting (26) into (24), then we obtain(27)δuk+1t≤∑m=1Nβmδult+∑m=1NbBγ1∫-τteγ3t-sδulsds+∑m=1NbBγ2∫-τt∫-τσeγ3σ-sδulsdsdσ
Multiply both sides of (27) by e-λt, one has(28)supt∈-τ,T-τe-λtδuk+1t≤∑m=1Nβm+bBγ11-eγ3-λt+τλ-γ3+bBγ21-e-λt+τλ1-eγ3-λTλ-γ3supt∈-τ,T-τe-λtδultwhere Definition 1 is applied.
Hence, we get(29)δuk+1t-τλ≤∑m=1Nβ-mδult-τλwhere β-m=βm+bBγ11-e(γ3-λ)T/λ-γ3+bBγ21-e-λT/λ1-e(γ3-λ)T/λ-γ3 and t∈0,T.
Selectλ sufficiently large such that ∑m=1Nβ-m<1. In view of Lemma 4, we have(30)limk→∞δukt-τ=0
Furthermore, we get from (26)(31)zot-zktλ≤bB1-eγ3-λTλ-γ3δukt-τλ
Hence, it is obviously obtained from (30) and (31) that limk→∞⁡zo(t)-zk(t)λ=0. Considering the definitions of zo(t) and zk(t), we have limk→∞⁡xik(t)=xo(t) and limk→∞⁡vik(t)=vo(t) as t∈0,T for i=1,⋯,n, which means that all the following agents can track the trajectory of the leader on the interval 0,T. The proof is completed.


## 5. Simulation Analysis

In this section, we will provide two cases to illustrate the effectiveness of theoretical results in this paper.Consider a class of leader-following nonlinear multiagent systems with control delay. The directed communication topology consists of four following agents and one leader agent (labeled as 0) is shown in Figure1.Figure 1
Communication topology.The weighted adjacency matrices are(32)A=0000100001001010,B=diag1,1,0,0The dynamics of four following agents are described as(33)z˙ikt=viktcos⁡xiktvikt+0sin⁡xikt+1uikt-τyikt=xikt,viktTwhere τ is fixed delay time and i=1,2,3,4.The dynamics of leader are given as(34)z˙ot=votcos⁡xotvot+0sin⁡xot+1uot-τyot=xot,votTThe parameters are set as follows: the initial position and velocity of four agents are given asx(0)=-0.6,0.3,1.3,-1.0T and v(0)=0.7,-1.2,-0.4,1.5T, respectively; the simulation time t∈0,8 and the delay time τ=0.2; the initial states of leader agent are xo(0)=vo(0)=0; the input of leader agent uo(t)=sin(πt) and the iteration number kmax=50.In addition, we have from (20)(35)gzzt,t=∂gzt,t∂zt=I8The following two cases are provided to check the validity of our results.Case 1 (consensus analysis withN=2).
In this case, letP1=0.75I4, P2=0.25I4, R1=0.6I4⊗11×2, and R1=0.3I4⊗11×2.
Due toP1(t)+P2(t)=0.75I4+0.25I4=I4, β1=P1(t)-R1(t)H-gzBmax=0.5732, β2=P2(t)-R2(t)H-gzBmax=0.2748, β1+β2=0.8480<1, which implies that the convergence conditions (19) and (20) are satisfied. The simulation results for Case 1 at the 50th iteration are shown in Figures 2, 3, 4, and 5.
The tracking trajectories of position and velocity are shown in Figures2 and 3, which indicate that the consensus problem of following-leader nonlinear multiagent systems with control delay (8) and (10) can be achieved by using the proposed high-order iterative learning control scheme (17). Although the control signal of multiagent systems is impacted by the time delay, the problem is solved via the control protocol proposed in this paper. The tracking errors of four following agents at the 50th iteration are given in Figure 4, and the control input curves at the 50th iteration are shown in Figure 5.Figure 2
Position tracking trajectories for Case1.Figure 3
Velocity tracking trajectories for Case1.Figure 4
Tracking errors of four agents for Case1.Figure 5
Control input for Case1.Case 2 (comparison analysis withN=1 and N=2).
In this case, letP1=I4 and R1=0.6I4⊗11×2 for N=1. Due to P1=I4 and β1=P1(t)-R1(t)H-gzBmax=0.5732<1, this means that the convergence conditions (19) and (20) are guaranteed. The simulation results for Case 2 at the 50th iteration are shown in Figure 6.
The comparison results of position and velocity are shown in Figure6. Due to the consideration of control protocols with different orders, the control effects have changed to some extent. Compared with the first-order control protocol, the control effects of second-order control protocol are better. Moreover, from Figure 6, it can be seen that the overshoot of position and velocity results using the second-order control protocol is smaller, and the convergence speed is faster.Figure 6
State comparison results of four agents for Case2.
(a) (b) (c) (d)


## 6. Conclusions

In this paper, we addressed the consensus tracking problem of leader-following nonlinear control time-delay multiagent systems with iterative learning control. An improved high-order iterative learning control protocol was proposed. Then, a sufficient condition was given to guarantee all the following agents to track the trajectory of the leader. Through simulation analysis, the consensus problem of nonlinear multiagent systems with control delay can be achieved by using the presented high-order learning control scheme. Meanwhile, the simulation results also implied that the higher the order, the better the control effect.



---

*Source: 2035683-2019-03-04.xml*