# Distributed Coordination for a Class of High-Order Multiagent Systems Subject to Actuator Saturations by Iterative Learning Control

Authors: Nana Yang; Suoping Li
Journal: Complexity
 (2022)

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## Abstract

This paper investigates a distributed coordination control for a class of high-order uncertain multiagent systems. Under the framework of iterative learning control, a novel fully distributed learning protocol is devised for the coordination problem of MASs including time-varying parameter uncertainties as well as actuator saturations. Meanwhile, the learning updating laws of various parameters are proposed. Utilizing Lyapunov theory and combining with Graph theory, the proposed algorithm can make each follower track a leader completely over a limited time interval even though each follower is without knowing the dynamics of the leader. Moreover, the extension to formation control is made. The validity and feasibility of the algorithm are verified conclusively by two examples.


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## Body

## 1. Introduction

Coordination of multiagent systems (MASs) draws a lot of attention because of its wide applications in UAV, biological systems, sensor networks, and so forth. An important problem in coordination is to develop the distributed protocols, which specifies information exchange among agents, such that the group as a whole can agree on a common quantity. This kind of problem is called consensus [1–5]. For this consensus realization, a leader-follower consensus [6, 7] has been extensively studied by many scholars. Another problem of coordination is formation control; the fault-tolerant leader-following formation control and cluster formation control were discussed in [8, 9], respectively. However, in the real world, the dynamics of systems are usually uncertain. Therefore, the consensus or/and formation control for uncertain MASs become a hot topic in the field of control.Generally speaking, an effective scheme to dispose parameter uncertainties is adaptive control [10–19]. An adaptive neural network approximation scheme was utilized to design the control protocols of the first-, second-, and high-order uncertain MASs in [10–12]. References [13, 14] used the adaptive idea to design the fully distributed control protocols by adjusting the protocols gains, so that the global information dependent on topology matrix can be avoided, and the promising technique is more and more popular among researchers for different purposes [15–19], where adaptive iterative learning control is utilized to handle the consensus for MASs over a finite time interval [20–24].It is noteworthy that the abovementioned studies did not consider actuator saturations in the MASs. Actually, due to the limited actuator capability, actuator saturations may exist in most physical systems and cause the system instability. Consequently, it is useful to take into account actuator saturations in system analysis. Recently, a large number of work about MASs with actuator saturations emerge [25–32]. References [25–29] studied consensus algorithms under one-dimensional system framework when time goes to infinity and [30–32] tackled the consensus problem by iterative learning control (ILC). In [30], authors addressed the neural network consensus for the second-order MASs subject to saturation input; authors in [31] proposed the fully distributed learning scheme and made use of the properties of saturation function to solve actuator saturations; the leader-follower consensus for the first-order MASs with input saturation was researched in [32]. Thereinto, the dynamic of the leader was known to each follower in [31, 32].Based on the aforementioned observations, we have successfully combined the fully distributed algorithm applying ILC over the finite time interval to address the coordination for a class of high-order MASs. The major contributions are highlighted as follows:(i)
Differing from the adaptive consensus of MASs with actuator saturations, the learning consensus researched in this paper can make each follower track the leader absolutely over, where the parameter uncertainties are time-varying(ii)
In contrast with the adaptive ILC consensus algorithm available, the algorithm in the paper is more complicated as well as each follower is without knowing the dynamic of the leader(iii)
The consensus problem is extended to the formation control for a class of high-order MASsThe rest is arranged as follows. Some useful preliminary results and problem formulation are presented in Section2. Section 3 is the protocols design and consensus analysis. The extension to formation control is given in Section 4. Two examples for illustration are taken in Sections 5 and conclusion is drawn in Sections 6, respectively.


## 2. Problem Formulation

Here, the MASs withN followers and one leader are considered under a repetitive environment. The models are(1)followers:x˙ikt=Axik+θitξixikt+satuikt,u∗i=1,2,…,N,leader:x˙0t=Ax0t+fx0t,t,where xikt∈Rn is the state vector of the i th follower and uikt∈Rn is the input vector of the i th follower; θit is an unknown continuous time-varying parameter, which shows the uncertainty in system models for each follower agent; ξixik∈Rn is a known smooth nonlinear vector valued function; A∈Rn×n; satuijk,uj∗ is the saturation function [31], j=1,2,…,n, and satuik,u∗=satui1k,u1∗,satui2k,u2∗,…,satuink,un∗T is a saturation vector function; x0t∈Rn is the state vector of the leader, and fx0t,t=f01x0t,t,f02x0t,t,…,f0nx0t,tT∈Rn is an unknown but bounded vector valued nonlinear function.Assumption 1.
It is assumed thatf0jx0t⩽ηj with ηj being an unknown positive constant. Denote η¯=η1,η2,…,ηnT.Remark 1.
From the above, it is known that each follower is without knowing the dynamic of the leader.
For thei th follower, the consensus error is(2)δik=xik−x0.
In this paper, we aim to find suitable protocolsuik,i=1,2,…,N,0≤t≤T and the updating laws of parametric uncertainties so that all the followers can uniformly track the leader over 0,T as k approaches to infinity, that is to say,(3)limk⟶∞δik=limk⟶∞xik−x0t=0,i=1,2,…,N.Assumption 2.
The state vector of each follower and the leader satisfyxik0=xik−1t and x00=x0t.Remark 2.
From assumption2, we have δik0=δik−1t.
To design the distributed protocols, the distribute error is(4)eikt=∑j=1Naijxjkt−xikt+bix0t−xikt.
The compact forms are(5)δk=xk−1N⊗x0,ek=−L+Bxk−1N⊗x0=−H⊗Inδk,where ek=e1kT,e2kT,…,eNkTT∈RNn and δk=δ1kT,δ2kT,…,δNkTT∈RNn; H=L+B is a symmetric positive definite matrix, and the communication topology of the paper is the same as that in [31].


## 3. Learning Control Protocols Design and Consensus Analysis

The error dynamic can be calculated as(6)δ˙ik=Aδik+θitξik+satuik,u∗−fx0,t.Then, the learning protocol is devised as(7)uikt=c^ikteik−θ^iktξixik+ϖμikt+υikt,where c^ikt∈R is a time-varying gain; θ^ikt∈R is to estimate θit; ϖ>0; μikt is to deal with the saturation term in (1); and υikt∈Rn is to offset fx0t,t.υikt, the learning adaptive laws of c^ikt and μikt, is devised as(8)υikt=η^ikttanheiktTPη^iktΔk+1,(9)c^˙ikt=ρieiktTPeikt,c^ik0=c^ik−1T,c^i00=0,and(10)μ˙ikt=ϖeik−δukTδuk2μkTIN⊗Pμkμikt,μik0=μik−1T,where η^ikt is the estimation of η¯ and P is symmetric positive definite; ρi>0; δuk=uk−satuk, uk=colu1k,u1k,…,uNk∈RNn, satuk=colsatu1k,satu2k,…,satuNk∈RNn, and μk=colμ1k,μ2k,…,μNk∈RNn.Furthermore, the learning-based updating law ofθit is devised as(11)θ^ikt=satθ^i,∗kt,θ^i,∗kt=satθ^i,∗k−1t−mieikTPξik,θ^i−1t=satθ^i,∗−1t=0,t∈0,T,with mi>0.Meanwhile,(12)η^˙ikt=ΓiPeik,η^ik0=η^ik−1T,where η^ikt=η^i1kt,η^i2kt,…,η^inktT;Γi=diagγi1,γi2,…,γin,γij>0, i=1,2,…,N and j=1,…,n; η^i00=η^i100,η^i200,…,η^in00T, η^ij00>0.Remark 4.
It is obvious thatη^ikt is nonnegative, which is guaranteed from the updating law (12).
Thei th error dynamic becomes(13)δ˙ik=Aδik+θ˜iktξik+c^ikeik+ϖμikt−δuik+υik−fx0,t,where θ˜ikt=θit−θ^ikt and δuik=uik−satuik.
Thus,(14)δ˙k=IN⊗Aδk+Θ˜kt⊗Inξk−C^ktH⊗Inδk+ϖμk−δuk+υk−1N⊗fx0,t,where(15)Θ˜kt=diagθ˜1kt,θ˜2kt,…,θ˜Nkt,ξk=ξ1kT,ξ2kT,…,ξNkTT∈RNn,υkt=colυ1kt,υ2kt,…,υNkt∈RNnandC^kt=diagc^1kt,c^2kt,…,c^Nkt.Theorem 1.
For the MASs (1), under assumptions1 and 2, the protocols (7)-(8) as well as adaptive learning laws (9)-(12) guarantee that each follower can completely track the leader over 0,T along the iteration axis, i.e., limk⟶∞δik=0, i=1,…,N. At the same time, the boundednesses of signals involved are obtained.Proof.
The proof falls into three parts. At thek th iteration, a Lyapunov candidate is established as(16)Vk=δkTH⊗Pδk+∑i=1N1mi∫0tθ˜ikτ2dτ+∑i=1N1ρic˜ikt2+∑i=1NμiktTPμikt+∑i=1Nη˜iktTΓi−1η˜ikt,where c˜ik=ci−c^ikt and η˜ikt=η¯−η^ikt,i=1,2,…,N.
At first, the difference betweenVk and Vk−1 is(17)ΔVkt=Vkt−Vk−1t=δkTH⊗Pδk−δk−1tTH⊗Pδk−1t+∑i=1N1mi∫0tθ˜ik2−θ˜ik−12dτ+∑i=1N1ρic˜ik2−c˜ik−12+∑i=1NμiktTΓi−1μikt−∑i=1Nμik−1tTΓi−1μik−1t+∑i=1Nη˜iTΓi−1η˜ik−∑i=1Nη˜ik−1TΓi−1η˜ik−1.
From the error dynamic (14), we have(18)δkTH⊗Pδk=2∫0tδkTH⊗Pδ˙kdτ+δk0TH⊗Pδk0=∫0tδkTH⊗PA+ATPδkdτ+2ϖ∫0tδkTH⊗Pμkdτ−2∫0tδkTH⊗Pδukdτ−2∫0tδkTHCkτH⊗Inδk+2∫0tδkTHΘ˜kτ⊗Pξkdτ+2∫0tδkTH⊗Pυkdτ−2∫0tδkTH⊗P1N⊗fx0dτ+δk0TH⊗Pδk0,where(19)−2∫0tδkTH⊗Pδukdτ≤∫0tδkTH2⊗P2δkdτ+∫0tδukTδukdτ,and(20)∑i=1N1mi∫0tθ˜ik2−θ˜ik−12dτ=∑i=1N1mi∫0tθ^ik−1−θ^ik2θi−θ^ik+θ^ik−θ^ik−1dτ⩽∑i=1N2mi∫0tθ˜ikθ^ik−1−θ^ikdτ⩽2∑i=1N∫0tθ˜ikeikTPξikdτ=−2∫0tδkTHΘ˜k⊗Pξkdτ.
Simultaneously,(21)∑i=1N1ρic˜ik2−c˜ik−12=∑i=1N2ρi∫0tc˜ikc˜˙ikdτ+∑i=1N1ρic˜ik02−c˜ik−1t2=−2∫0tδkTHCH⊗Pδkdτ+2∫0tδkTHC^kτH⊗Pδkdτ+∑i=1N1ρic˜ik02−c˜ik−1t2,where C=diagc1,c1,…,cN.
In addition,(22)∑i=1NμiktTPμikt−∑i=1Nμik−1tTPμik−1t=2∑i=1N∫0tμikTPμ˙ikdτ−∑i=1Nμik−1tTPμik−1t+∑i=1Nμik0TPμik0=−2ϖ∫0tδkTH⊗Pμkdτ−∫0tδukTδukdτ−∑i=1Nμik−1tTPμik−1t+∑i=1Nμik0TPμik0,and(23)∑i=1Nη˜ikTΓi−1η˜ik−∑i=1Nη˜ik−1TΓi−1η˜ik−1=2∑i=1N∫0tη˜ikTΓi−1η˜˙ikdτ−∑i=1Nη˜ik−1tTΓi−1η˜ik−1t+∑i=1Nη˜ik0TΓi−1η˜ik0=−2∑i=1N∫0tη˜ikTPeikdτ−∑i=1Nη˜ik−1tTΓi−1η˜ik−1t+∑i=1Nη˜ik0TΓi−1η˜ik0.
Substituting (18)–(23) into (17) yields(24)ΔVkt⩽∫0tδkTH⊗PA+ATP−2HCH⊗P−H2⊗P2δkdτ+δk0TH⊗Pδk0−δk−1tTH⊗Pδk−1t+∑i=1N1ρic˜ik02−c˜ik−1t2+∑i=1Nμik0TPμik0−∑i=1Nμik−1tTPμik−1t+∑i=1Nη˜ik0TΓi−1η˜ik0−∑i=1Nη˜ik−1tTΓi−1η˜ik−1t+2∫0tδkTH⊗Pυkdτ−2∫0tδkTH⊗P1N⊗fx0dτ−2∑i=1N∫0tη˜ikTPeikdτ.
Due to (8) and Property 1 in [20],(25)2∫0tδkTH⊗Pυkdτ−2∫0tδkTH⊗P1N⊗fx0dτ−2∑i=1N∫0tη˜ikTPeikdτ=−2∑i=1N∫0teikTPυikdτ+2∑iN∫0teikTPfx0dτ−2∑i=1N∫0tη˜ikTPeikdτ≤−2∑i=1N∫0teikTPυikdτ+2∑iN∫0teikTPη¯dτ−2∑i=1N∫0teikTPη˜ikdτ=2∑i=1N∫0teikTPη^ik−eikTPη^iktanheikTPη^ikΔk+1dτ⩽2NTεΔk+1.
Therefore,(26)ΔVkt⩽∫0tδkTH⊗PA+ATP−2HCH⊗P−H2⊗P2δkdτ+δk0TH⊗Pδk0−δk−1tTH⊗Pδk−1t+∑i=1N1ρic˜ik02−c˜ik−1t2+∑i=1Nμik0TPμik0−∑i=1Nμik−1tTPμik−1t+∑i=1Nη˜ik0TΓi−1η˜ik0−∑i=1Nη˜ik−1tTΓi−1η˜ik−1t+2NεΔk+1.
On account of the positiveness ofH, C, and P, set δ˜k=FT⊗Inδk and F is an orthogonal matrix:(27)ΔVk≤∑i=1NλiH∫0tδ˜ikTPA+ATP−2cminλminHλminP−λmax2Hλmax2PIδ˜ikdτ+2NTεΔk+1+δk0TH⊗Pδk0−δk−1tTH⊗Pδk−1t+∑i=1N1ρic˜ik02−c˜ik−1t2+∑i=1Nμik0TPμik0−∑i=1Nμik−1tTPμik−1t+∑i=1Nη˜ik0TΓi−1η˜ik0−∑i=1Nη˜ik−1tTΓi−1η˜ik−1t,where λiH>0 shows the eigenvalue of H and cmin=min1≤i≤Nci. For the sufficient large constant cmin>0, the inequality PA+ATP−2cminλminHλminP−λmax2Hλmax2P≤−σI with σ>0 always holds.
Whent=T, it follows that(28)ΔVkT⩽−σλminH∑i=1N∫0TδikTδikdτ+2NTεΔk+1,which results in(29)VkT⩽Vk−1T+2NTεΔk+1.
In the second place, let us prove boundednesses of signals involved. On the basis of the definition ofVk results,(30)Vk=ΔVk+Vk−1⩽−σλminH∑i=1N∫0tδikTδikdτ+δk0TH⊗Pδk0−δk−1TH⊗Pδk−1+δk−1TH⊗Pδk−1+∑i=1N1mi∫0tθ˜ik−12dτ+∑i=1Nμik−1TPμik−1+∑i=1Nμik0TPμik0−∑i=1Nμik−1TPμik−1+∑i=1Nη˜ik−1TΓi−1η˜ik−1+∑i=1Nη˜ik0TΓi−1η˜ik0−∑i=1Nη˜ik−1TΓi−1η˜ik−1+∑i=1N1ρic˜ik−12+∑i=1N1ρic˜ik02−c˜ik−12+2NTεΔk+1.
That is,(31)Vk⩽Vk−1T+2NTεΔk+1.
It can be obtained from (29) and (31) that(32)Vk⩽Vk−1T+2NTεΔk+1⩽Vk−2T+2NTεΔk+2NTεΔk+1⋮⩽V0T+2NTε∑l=2k+1Δl.
Aslimk⟶∞2NTε∑l=2k+1Δl⩽4NTεa [20], 2NTε∑l=2k+1Δl is uniformly bounded, ∀k∈Z+. Denote 2NTε∑l=2k+1Δl⩽S with S>0. Hence,(33)Vk⩽V0T+S.
If the finiteness ofV0T is attained, the uniform boundedness of Vkt is followed. And then, we will show the finiteness of V0. It obtained that(34)V0t=δ0TH⊗Pδ0+∑i=1Nμi0tTPμi0+∑i=1N1ρic˜i02+∑i=1N1mi∫0tθ˜i02dτ+∑i=1Nη˜i0TΓi−1η˜i0,and(35)V˙0⩽δ0TH⊗PA+ATP−2HCH⊗P−H2⊗P2δ0+∑i=1N1miθi2t+2δ0TH⊗Pυ0−2δ0TH⊗P1N⊗fx0−2∑i=1Nη˜i0TPei0,where(36)2δ0TH⊗Pυ0−2δ0TH⊗P1N⊗fx0−2∑i=1Nη˜i0TPei0dτ⩽2∑i=1Nei0TPη^i0−ei0TPη^i0tanhei0TPη^i0Δ1⩽2Nεa.
Thus,(37)V˙0⩽−σλminH∑i=1Nδi0Tδi0+∑i=1N1miθi2t+2Nεa⩽∑i=1N1miθi2t+2Nεa.
Sinceθit is continuous, the boundedness of it obtained 0,T. Then,(38)V˙0t⩽M0′,where M0′=1/mminmaxt∈0,T1≤i≤Nθi2t+2Nεa and qmin=min1≤i≤Nqi,mmin=min1≤i≤Nmi. Meanwhile,(39)V0=V00+∫0tV˙0dτ⩽δ00TH⊗Pδ00+∑i=1N1ρici2+∑i=1Nμi00TPμi00+∑i=1Nη¯TΓi−1η¯+TM0′<∞.
V0t is finite; it is followed that V0T is bounded. Therefore, the uniformly boundedness of Vkt is obtained over 0,T, ∀k∈Z+. Furthermore, it is inspired by the definition of Vk; it obtained the uniform boundednesses of δkt, μikt, c^ikt, and ηikt. The updating law (11) indicates that θ^ikt is bounded. From (7), we can calculate the uniform boundedness of uikt. So, the boundednesses of signals involved are gained.
At last, we prove the property of learning consensus. As we know,(40)VkT=V0T+∑l=1kΔVlT.
From (28), it can yield(41)VkT⩽V0T−σλminH∑l=1k∑i=1N∫0TδilTδildτ+2NTε∑l=2k+1Δl.
SinceVkT is positive, V0T is bounded, and the series 2NTε∑l=2k+1Δl is convergent, and the series ∑l=1k∑i=1N∫0tδilTδildτ is convergent. Consequently, limk⟶∞∑i=1N∫0tδikTδikdτ=0. It is easy to achieve limk⟶∞∫0TδikTδikdτ=0, i=1,2,…,N. From (13), δ˙ikt is bounded on 0,T. At last, from Barbalat-like Lemma, limk⟶∞δikt=0 holds uniformly over 0,T, i.e., limk⟶∞xikt−x0t=0. In other words, each follower can completely track the leader over 0,T.Remark 5.
The conditionPA+ATP−2cminλminHλminP−λmax2Hλmax2PI≤−σI with σ>0 is only for the analysis purpose; as a matter of fact, it is not utilized in the design of protocols. Accordingly, the distributed learning control protocols are fully distributed and the consensus for the MASs is solved faultlessly even if each follower is without knowing the dynamic of the leader.


## 4. Formation Control of the MASs

The formation control of the MASs (1) is concerned here. If the followers and leader form a formation at a certain distance over0,T, we can say that the formation control is achieved.Let us define(42)x¯i1k=xi1k−Δi,where Δi is the expected formation vector for the i th follower relative to the leader.The formation error is(43)δi1kt=x¯i1kt−x01t.And,δilkt is the same as δilkt defined in (2), l=2,3,…,n.Like that, the problem of formation can be reformulated as the consensus problem, i.e.,limk⟶∞δik=0.Simultaneously, the neighborhood formation errors are(44)ei1k=∑j=1Naijx¯j1k−x¯i1k+bix01−x¯i1k,eilk=∑j=1Naijxjlk−xilk+bix0l−xilk,l=2,3,…,n.Assumption 3.
For each follower,x¯i1k0=x¯i1k−1T,xilk0=xilk−1T,i=1,2,…,N and l=2,3,…,n; for the leader, x00=x0T. Then, it follows that δik0=δik−1T.Theorem 2.
For the MASs with graphG¯, under assumptions 1 and 3, N followers represented by (1) under the protocols (7)-(8) with learning-based updating laws (9)-(12) with the local neighborhood formation errors (44) can make the followers form the desired formation in the iteration domain on t∈0,T. The variables involved are bounded.


## 5. Simulation

In this part, two examples are provided to validate the validity and practicability for the fully distributed learning protocol of this paper. As mentioned in [33], the LC oscillator system can be considered as a MASs. Therefore, here, let us consider the MASs consisting of six followers as well as one leader. Figure 1 shows the communication graph.Figure 1
Topology graph (0 indicates the leader).Then, we have(45)L=3−1−10−10−12000−1−102−10000−1100−1000100−10001,B=1000000000000000000000000.Example 1.
Consider the MASs (1) withA=0,1;−1,0, θit=cos2πit, ξikt=0.01sinxi1k3,0.01sinxi2k3T, i=1,2,3,4,5,6. x0t=sint,costT. In simulations, we choose a=1×104, T=2π; −25,25 is the lower and upper bounds of the saturation function.Case 1.
Consensus of the MASs
The designed parameters are chosen asϖ=2:(46)q1=0.2,q2=0.3,q3=0.1,q4=0.3,q5=0.5,q6=0.1;m1=0.1,m2=0.3,m3=0.1,m4=0.1,m5=0.1,m6=0.2,ρ1=ρ2=ρ3=ρ4=ρ5=ρ6=0.1,Γ1=diag0.2,0.3,Γ2=diag0.2,0.4,Γ3=diag0.3,0.2,Γ4=diag0.3,0.2,Γ5=diag0.4,0.2,Γ6=diag0.5,0.3,P=diag1,1,c=3,x10=1,2T,x20=1,1.5T,x30=3,1T,x40=1.5,2T,x50=1,1T,x60=2,3T,η^100=1,2T,η^200=2,1T,η^300=1,1T,η^400=1.5,2T,η^500=2,1T,η^600=1,1.5T,μ100=1,2T,μ200=2,1T,μ300=1,1T,μ400=1.5,2T,μ500=2,1T,μ600=1,1.5T.
After 30 cycles, Figures2–4 show the simulation results. Even if the dynamic of the leader is unknown to each follower and there exit actuator saturations in the dynamic of system, it can be seen from Figures 2 and 3 that six followers can perfectly track the leader, and in Figure 4, it is evident the signals involved are bounded. The results fit into Theorem 1.Figure 2
Consensus trajectories of six followers for Case1 at 30th iteration.Figure 3
Consensus errors of six followers for Case1.Figure 4
Responses of six followers for Case1.Case 2.
Formation control of the MASs.
SelectΔ1=−0.1,Δ2=−0.2,Δ3=−0.3,Δ4=−0.4,Δ5=−0.5,Δ6=−0.6. Other values involved are the same as in Case 1.
Figures5–7 depict the results of formation control for the MASs (1). From Figure 5, we know that the agents form the desired formation. Figure 6 shows that the errors converge to zero over 0,2π, and Figure 7 illustrates that the signals involved in the closed-loop system are bounded. The results obtained align with Theorem 2.Figure 5
Formation trajectories of agents for Case2 at 50th iteration.Figure 6
Formation errors of six followers for Case2.Figure 7
Responses of six followers for Case2.Example 2.
In the LC oscillator system [33] with six follower oscillators and one leader oscillator, each LC oscillator is governed by(47)dvitdt=1Ccit,dcitdt=−1Lvit,where L, C, cit, and vit denote the inductance, capacitor, current, and voltage; i=1,2,…,6. Under the repetitive environment, we study the fully distributed adaptive ILC for the LC oscillator system (47); hence, the input is applied to each oscillator (47) and affected by actuator saturations. Furthermore, assume that that the disturbances are time-varying linearly parameterized uncertainties θitξixik. Under these circumstances, system (47) is rewritten as(48)x˙ik=Axik+θitξixik,t+satuikt,where A=01/C−1/L0, xikt=xi1ktxi2kt=viktcikt, ξikxik,t=5xi2kcosxi1k3xi1kcosxi2k, θit=0.1sin2πit, and k is the iteration index. In addition, x0t=sinπt,cosπtT.
Selecta=8×106, T=2, C=1/2, and L=1/4; the bounds of saturation functions are −35,35. The designed parameters are chosen as ϖ=1.3: (49)q1=q2=q3=q4=q5=q6=1,m1=m2=m3=m4=m5=m6=2,ρ1=ρ2=ρ3=ρ4=ρ5=ρ6=1.5,Γ1=Γ2=Γ3=Γ4=Γ5=Γ6=diag1,0.2,P=diag4,5,c=3,x10=x20=x30=x40=x50=x60=−1,−1.5T,η^100=η^200=η^300=η^400=η^500=η^600=2,3T,μ100=1,2T,μ200=2,1T,μ300=1,1T,μ400=1.5,2T,μ500=2,1T,μ600=1,1.5T.
Figures8–10 demonstrate the results of consensus control for 60 iterations, respectively. We can see that six follower oscillators can perfectly track the leader oscillator from Figures 8 and 9, even if each follower oscillator is without knowing the dynamic of the leader oscillator, and the signals involved are bounded. The results fit into Theorem 1.Figure 8
Consensus trajectories of six follower oscillators at 60th iteration.Figure 9
Consensus errors of six follower oscillators.Figure 10
Responses of six follower oscillators.


## 6. Conclusions

We have solved the fully distributed learning coordination problem of a class of high-order nonlinear MASs by adaptive ILC in this study. With the help of algebraic graph theory, Barlat-like lemma, and Lyapunov theory, the perfect consensus tracking as well as the formation control problem has been resolved over0,T. At last, two examples testify the effectiveness and efficiency of the algorithm devised in the paper.



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