# Iterative Learning Consensus Control for Nonlinear Partial Difference Multiagent Systems with Time Delay

Authors: Cun Wang; Xisheng Dai; Kene Li; Zupeng Zhou
Journal: Complexity
 (2021)

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## Abstract

This paper considers the consensus control problem of nonlinear spatial-temporal hyperbolic partial difference multiagent systems and parabolic partial difference multiagent systems with time delay. Based on the system’s own fixed topology and the method of generating the desired trajectory by introducing virtual leader, using the consensus tracking error between the agent and the virtual leader agent and neighbor agents in the last iteration, an iterative learning algorithm is proposed. The sufficient condition for the system consensus error to converge along the iterative axis is given. When the iterative learning numberk approaches infinity, the consensus error in the sense of the L2 norm between all agents in the system will converge to zero. Furthermore, simulation results illustrate the effectiveness of the algorithm.


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## Body

## 1. Introduction

As sensor technology advances, communication and network technology, when solving complex practical problems, the ability of communication and coordination between various agents in a multiagent system (MAS) is used to solve the problem that a single agent cannot complete complex task [1, 2]. In recent years, MASs’s cooperative control has been widely used in the industry, such as formation control for drones [3], satellite attitude control [4], and congestion control of communication networks [5]. At present, the main problems are collaborative control of MASs cluster control [6], formation control [7], and consensus control [8]. Among them, the consensus problem of MASs is its most fundamental problem. The research on the MASs consensus problem is mainly through designing appropriate controllers for agents, so that all agents must be consensus in some given state or output [9].In recent years, the consensus of MASs has been extensively studied from different perspectives [10]. In [11], Vicsek et al. first designed a classic consensus model and verified its effectiveness through simulation examples. In [12, 13], Ren studied the consensus problem of discrete and continuous models of MASs. In [14], Olfati-Saber et al. handled the progressive consensus of the leaderless MASs. Subsequently, based on the characteristics of the MASs, scholars designed more control algorithms to achieve consensus control of the MASs. In practical applications, there are many tasks that need to be executed repeatedly, and coordination should be carried out among several independent individuals [15]. For this reason, we need to study MASs. For example, in the coordinated control of a multimanipulator system with repeated operations on a production line, it is impossible to fully realize the control of the tracking desired trajectory within a given finite-time control [16]. For example, the periodic operation of multisatellite systems cannot ignore the influence of changes in the internal parameters of the system and external disturbances, and the application of the sliding mode control strategy cannot guarantee the stability of such systems [17]. Due to the limitation of information obtained by each agent, the expected information (or trajectory) of the leader specified by the topology diagram is only available to some agents. For each agent, this requires that the designed distributed algorithm only allows the use of local information obtained by certain agents to track the desired trajectory as perfectly as possible. Iterative learning control (ILC) has great advantages in dealing with MASs that continuously repeat tasks within the finite time interval [18].Other control methods [19–21] (such as feedback control, adaptive control, and robust control) are compared with ILC. Many formers consider a continuous process system; that is, the operation of the system will continue to run over time. ILC is to complete a given task in a period of time and repeat it continuously. Therefore, compared with other control methods, in designing ILC algorithms, not only the time axis but also the iteration axis must be considered. The main control goal is to achieve convergence along the iteration axis, which is different from other control methods that only consider the time axis. In other words, ILC is a typical 2D process, and this characteristic also causes the theoretical analysis method of ILC algorithm to be different from other control methods. The controller of the ILC algorithm has a simple structure, and the ILC algorithm is effective for complex dynamic systems (such as uncertain, nonlinear, difficult to model, and missing information). It was proposed by Arimoto et al. in 1984 and after more than 30 years of development. There are many results in the research on ILC [22–25]. Because ILC has the above advantages, it provides an effective and feasible method for solving the consensus control problem of networked MASs in the finite-time interval. For example, in [26], the ILC method is used to solve the consensus problem that the leader with packet loss follows the nonlinear MASs. In [27], in order to solve the consensus tracking problem of the heterogeneous MASs, a corresponding ILC protocol was designed. In [28], P-type and PI-type ILC update algorithms are designed to solve the consensus tracking problem of a class of nonlinear fractional MASs. In [29], ILC technology is used to solve the consensus tracking problem in singular MASs. In [30], the problem of iterative learning consensus control in a continuous-time MAS is studied, and, during the learning process, the test length of each iteration is allowed to change randomly.With the rapid development of digital sensors and digital controllers, in practical applications, although the system itself is a continuous process, since the channel bandwidth is limited in the communication system, the controller can only apply the sampled data obtained at discrete times [31]. Therefore, related research on discrete systems is necessary. Many scholars have begun to study the issues related to discrete-time MASs [32–36]. In [34], a discrete form of ILC algorithm was designed for the discrete MASs consensus tracking problem with random noise. In [35], the consensus tracking problem of a class of nonlinear discrete MASs is studied when the iteration length changes randomly. In [36], with the help of ILC algorithm, the consensus tracking problem of discrete-time MASs with cooperative competition network is solved.Analysis of the above literatures reveals that the current results of using ILC algorithms to solve the consensus problems of MASs are mostly studies on ordinary differential MASs, which study the convergence of all agents in the system in the time domain. However, in practice, MASs have spatial dynamic behaviors, such as biological systems, chemical reactions, food networks, and other processes that depend on time and space [37]. The distributed parameter MASs consider the convergence of both the time domain and space domain, which has been of certain practical significance for the research of MASs related problems. Recently, there have been reports on related research on distributed parameter MASs. For example, in [38], the consensus of the continuous distributed parameter model MASs was achieved based on the Lyapunov function method. In [39], an ILC algorithm was proposed for continuous distributed parameter model MASs. In [40], the consensus problem of continuous distributed parameter MASs with time delay is studied. In [41], a second-order ILC algorithm is proposed for a class of consensus problems with continuous distributed parameter MASs with time delay.Considering these factors, in this paper, a class of nonlinear first-order hyperbolic partial difference MASs and parabolic partial difference MASs with time delay are studied in the finite-time interval. Based on the fixed topology for the MASs, an ILC algorithm is proposed with a method for the virtual leader agent to generate the desired trajectory. This algorithm uses the tracking error between any agent with the virtual leader agent and neighbor agents in the last iteration and combines the characteristics of the MASs communication topology to continuously modify the previous control law to obtain the ideal control law. When the iterative learning indexk approaches infinity, in the sense of the L2 norm, the consensus error between all agents in the MASs will converge to zero.The main contributions of this paper are summarized as follows:(1)
It can be noted that the works in [37–41] present a type of consensus problem in the study of continuous distributed parameter MASs. However, due to the limited channel bandwidth in the communication system, the controller can only apply the sampling data obtained at discrete moments. Therefore, it can be considered that the consensus control of discrete distributed parameter MASs is more practical.(2)
In the works in [38–41], the consensus control problem of parabolic distributed parameter MASs is studied. Since hyperbolic system is an important system type, a kind of consensus control of hyperbolic distributed parameter MASs has been studied. When the leader agent number is unknown, complete consensus control of all agents in the system is achieved within a finite time.(3)
From the analysis of the works in [38–42], in reality, the MASs have relatively complex nonlinearities and time delay. Therefore, an ILC algorithm is designed, which contains consensus errors between any two agents. After the system has been iteratively learned enough times, the consensus problem of nonlinear hyperbolic partial difference MASs and parabolic partial difference MASs with time delay is better solved.The remainder of the article is organized as follows. In Section2, we consider the iterative learning consensus control for nonlinear first-order hyperbolic partial difference MASs with time delay. In Section 3, we discuss iterative learning consensus control of the nonlinear parabolic partial difference MASs with time delay. In Section 4, two examples illustrate the results. Section 5 gives the conclusion.Notations. This article uses the following symbols: IN is the N×N dimension unit matrix, 1N is a column vector of N order, and each element is 1. ⊗ denotes the Kronecker product. For a function fζ, ⋅ is expressed as f=∑ζ=0Zf2ζ1/2, where 0≤ζ≤Z, and ζ=0,1,2,…,Z. Let λ0<λ<1 be a positive constant, and, for the binary function ψζ,h:0,Z×0,H⟶ℝ,ζ=0,1,2,…,Z,h=0,1,2,…,H, its λ norm is expressed as ψλ2=sup0≤h≤Hψ⋅,h2λh.Using the knowledge of graph theory to describe the interrelationship between any two agents, the corresponding graph theory knowledge is as follows: letG′=V′,E′,A′ represent a weighted directed graph, where V′=1,2,…,N is a collection of nodes, and E′⊆V′×V′ is the set of edges. A′=ai,j represents the adjacency matrix of graph G; if node j to node i has an edge, then aj,i>0, and this indicates that agent j can accept the information of agent i; otherwise, aj,i=0. Define the Laplacian matrix of the graph G as LG′=D′−A′, degree matrix D′=diagd1in,d2in,…,dNin, djin=∑i=1Nai,j be node j.


## 2. Iterative Learning Consensus Control for Nonlinear Hyperbolic Partial Difference Mass with Time Delay

Consider a class of nonlinear hyperbolic partial difference MASs composed ofN homogeneous discrete multiple agents, which run repeatedly in the finite-time interval. The dynamic model of the i-th agent is(1)△2wi,kζ,h=ah△1wi,kζ+1,h+fwi,kζ,h−τ,ui,kζ,h,h,yi,kζ,h=bwi,kζ,h,h+chui,kζ,h,where ζ,h are discrete variables of space and time, respectively. Z and H are given integers, and 0≤ζ≤Z, 0⩽h⩽H. k is the number of iterations, and i∈1,2,…,N indicates the i-th agent. ah,ch are uncertain bound real sequences, and ah>0. wi,kζ,h,ui,kζ,h,yi,kζ,h∈ℝ, respectively, denote the state of the i-th agent in system (1), control input, and control output. τ is the time delay. Let the initial state function be φ⋅; when h∈−τ,0, wi,kζ,h=φi,kζ,h. f:ℝ×ℝ×0,H⟶ℝ,b:ℝ×0,H⟶ℝ are both nonlinear functions.The differential form of the first expression in system (1) is defined as follows:(2)△2wi,kζ,h=wi,kζ,h+1−wi,kζ,h,△1wi,kζ+1,h=wi,kζ+1,h−wi,kζ,h.Assumption 1.
GraphG contains the spanning tree, and the virtual leader agent is the root of the spanning tree.Assumption 2.
For allk, the initial boundary condition of the i-th agent in system (1) is given as(3)wi,k0,h=0=wi,kZ,h,i=1,2,…,N,−τ≤h≤H,(4)wi,kζ,h=φiζ,h,i=1,2,…,N,−τ≤h≤0,0≤ζ≤Z.Assumption 3.
The dynamic characteristics of system (1) are reversible, ensuring that there is a single desired control input such that the state and output of system (1) are expected.Assumption 4.
The nonlinear functionsf,b, respectively, satisfy the following uniform global Lipschitz condition:(5)fw1ζ,h−τ,u1ζ,h,h−fw2ζ,h−τ,u2ζ,h,h,≤lfw1⋅,h−τ−w2⋅,h−τ+u1⋅,h−u2⋅,h,bw1ζ,h,h−bw2ζ,h,h⩽lbw1⋅,h−w2⋅,h,where lf and lb are constants.Remark 1.
Agents can only get information from virtual leader and neighbor agents, while virtual leaders cannot get information from other agents.Lemma 1 (see [43]).
LetAv, Bv, and Cv be real sequences defined for v≥0 satisfying(6)Av+1≤BvAv+Cv,Bv≥0,v≥0.
Then,(7)Ah≤∏v=0h−1BvA0+∑v=1h−1Cv∏s=v+1h−1Bs,h≥0.In this paper, the trajectory of the virtual leader agentyrζ,h is the desired trajectory of the discrete distributed parameter MAS (see (1)) ydζ,h. Because MAS has a distributed structure, only a few agents in the system are allowed to directly obtain the required trajectory information. The agents in system (1) which will directly obtain the desired trajectory information depend on the communication topology between the multiple agents.The ultimate goal of this paper is that each agent in the MAS is able to completely track the motion trajectory of the virtual leader agent in the finite-time interval and finally achieve the consensus control of the MAS. However, due to the uncertainty of system (1), it is not easy to directly find the appropriate control input udζ,h. Therefore, we use the ILC method to gradually find the appropriate control input udζ,h to satisfy(8)limk⟶∞yi,k⋅,h−yj,k⋅,h2=0,i,j∈1,2,…,N,0≤h≤H,where yi,kζ,h is the output of the i-th agent in the k-th iteration of system (1). This equation means that after the MAS learns with an ILC algorithm, in system (1), the consensus error between any two agents is able to converge to zero.Based on the characteristics of information transfer between agents, the following ILC algorithm is designed:(9)ui,k+1ζ,h=ui.kζ,h+γh∑j=1Nai,jei,kζ,h−ej,kζ,h+siei,kζ,h,i,j∈1,2,…,N,0≤h≤H,where γh is the learning gain, k is the number of iterations. ei,kζ,h=yrζ,h−yi,kζ,h is the tracking error of the i-th agent, and ej,kζ,h=yrζ,h−yj,kζ,h is the tracking error of the j-th agent. si reflects the communication relationship between the i-th agent and virtual leader agents. si>0 indicates that the i-th agent can communicate with virtual leader agent, and si=0 is the opposite.For the simplicity of the subsequent convergence analysis, the compact form of system (1) is as follows:(10)△2wkζ,h=ah△1wkζ+1,h+fwkζ,h−τ,ukζ,h,h,ykζ,h=bwkζ,h,h+chukζ,h,where(11)wkζ,h=wT1,kζ,h,wT2,kζ,h,…,wTN,kζ,hT∈RN,ukζ,h=uT1,kζ,h,uT2,kζ,h,…,uTN,kζ,hT∈RN,ykζ,h=yT1,kζ,h,yT2,kζ,h,…,yTN,kζ,hT∈RN.Define the consensus error of system (10) as(12)ekζ,h=1Nyrζ,h−ykζ,h,0≤h≤H,where yrζ,h is the trajectory of virtual leader agent and ekζ,h=eT1,kζ,h,eT2,kζ,h,…,eTN,kζ,hT∈RN.Further, the compact form corresponding to ILC law (see (9)) is can be written as(13)uk+1ζ,h=ukζ,h+γhL+S1Nekζ,h,where L∈ℝN×N is the Laplacian matrix of graph G and S≜diags1,s2,…,sN.After system (10) iterates k times, the corresponding difference form is expressed as(14)△2wkζ,h=wkζ,h+1−wkζ,h,(15)△1wkζ+1,h=wkζ+1,h−wkζ,h.Theorem 1.
Ifγh in the ILC law (see (13)) satisfies 1−γhL+S1N2≤0.5,0≤h≤H and satisfies Assumptions 1–4 at the same time, and when the number of iterations k⟶∞, the consensus error of system (10) can converge to zero in the following sense:(16)limk⟶∞yi,k⋅,h−yj,k⋅,h2=0,i,j∈1,2,…,N,0≤h≤H.Proof.
From equations (12) and (13), after the k-th iteration, the consensus error of system (10) can be written as(17)ek+1ζ,h=1Nyrζ,h−yk+1ζ,h,=ekζ,h−bwk+1ζ,h,h−bwkζ,h,h−ctuk+1ζ,h−ukζ,h,=ekζ,h−bwk+1ζ,h,h−bwkζ,h,h−ctγhL+S1Nekζ,h,=1−chlhekζ,h−bk+1−bk,where(18)lh=γtL+S1N,bk=bwkζ,h,h,bk+1=bwk+1ζ,h,h.
Immediately after the two sides of equation (17) are squared, one obtains(19)ek+12ζ,h=1−chlhekζ,h−bk+1−bk2,≤21−chlh2ek2ζ,h+2bk+1−bk2.
Summing fromζ=0 to Z on both sides of inequality (19), one obtains(20)∑ζ=0Zek+12ζ,h≤∑ζ=0Z21−chlh2ek2ζ,h+∑ζ=0Zbk+1−bk2,=21−chlh2ek⋅,h2+2lbwk+1⋅,h−wk⋅,h2⩽2λclek⋅,h2+2lbw˜k⋅,h2,where(21)λcl=sup0≤h≤H1−chlh2,w˜k⋅,h=wk+1ζ,h−wkζ,h.
It can be seen from inequality (20) that, to prove the convergence of consensus error ek⋅,h2 of system (10), it is necessary to estimate w˜k⋅,h. Thus, by the first equation in system (10) and equation (14), one can derive(22)wkζ,h+1=ah△1wkζ+1,h+wkζ,h+fwkζ,h−τ,ukζ,h,h.
Further, during thek+1-th iteration, the agents state can also be written as(23)wk+1ζ,h+1=ah△1wk+1ζ+1,h+wk+1ζ,h+fwk+1ζ,h−τ,uk+1ζ,h,h.
Next, after subtracting equation (22) from equation (23) and combining with equation (15), we have the following results:(24)w˜kζ,h+1=ahw˜kζ+1,h−w˜kζ,h+w˜kζ,h+fwk+1ζ,h−τ,uk+1ζ,h,h−fwkζ,h−τ,ukζ,h,h.
The two sides of equation (24) are squared at the same time. Since a+b+c≤3a2+3b2+3c2, equation (24) can be rewritten as(25)w˜kζ,h+12=ahw˜kζ+1,h−w˜kζ,h+w˜kζ,h+fwk+1ζ,h−τ,uk+1ζ,h,h−fwkζ,h−τ,ukζ,h,h2≤3a2hw˜kζ+1,h−w˜kζ,h2+3w˜k2ζ,h+3fwk+1ζ,h−τ,uk+1ζ,h,h−fwkζ,h−τ,ukζ,h,h2.
Summing fromζ=0 to Z on both sides of inequality (25) and using Lipschitz condition (5), one obtains(26)∑ζ=0Zw˜kζ,h+12≤3a2h∑ζ=0Zw˜kζ+1,h−w˜kζ,h2+3∑ζ=0Zw˜k2ζ,h+3∑ζ=0Zfwk+1ζ,h−τ,uk+1x,h,h−fwkζ,h−τ,ukζ,h,h2,≤3a2h∑ζ=0Zw˜kζ+1,h−w˜kζ,h2+3w˜k⋅,h2+3lfw˜k⋅,h−τ2+u˜k⋅,h2≜Σ1+Σ2,where Σ1=3a2h∑ζ=0Zw˜kζ+1,h−w˜kζ,h2+3w˜k⋅,h2,Σ2=3lfw˜k⋅,h−τ2+u˜k⋅,h2. Combining the boundary condition (3), Σ1 in inequality (26) can be written as(27)Σ1=3a2h∑ζ=0Zw˜kζ+1,h−wkζ,h2+3w˜k⋅,h2,≤3a2h∑ζ=0Z2w˜k2ζ+1,h+2w˜k2ζ,h+3w˜k⋅,h2≤12a2h∑ζ=0Zw˜k2ζ,h−6a2hw˜k21,h+3w˜k⋅,h2≤12a2h+3w˜k⋅,h2.
Substituting (27) into (26), one has(28)w˜k⋅,h+12≤12a2h+3w˜k⋅,h2+3lfw˜k⋅,h−τ2+u˜k⋅,h2m1w˜k⋅,h2+m2w˜k⋅,h−τ2+u˜k⋅,h2,where m1=sup0≤h≤H12a2h+3,m2=3lf. Combining Lemma 1 and the initial value condition (4), we have the following results:(29)∑ζ=0Zw˜k2ζ,h≤∑s=0h−1m2∑ζ=0Zw˜k2ζ,s−τ+∑ζ=0Zu˜k2ζ,sm1h−s−2.
So, we can see the relationship betweenw˜kζ,h and u˜kζ,h. Next, choosing ILC algorithm (13), one obtains(30)u˜kζ,h2=uk+1ζ,h−ukζ,h2≤l2ek2ζ,h,where l=sup0≤h≤HγhL+S1N.
In addition, by inequality (30), inequality (29) can be further written as(31)∑ζ=0Zw˜k2ζ,h≤∑s=0h−1m2∑ζ=0Zw˜k2ζ,s−τ+l2∑ζ=0Zek2ζ,sm1ζ−s−2.
Multiplying both sides of inequality (31) by λh0<λ<1, (31) can be rewritten as follows:(32)∑ζ=0Zw˜k2ζ,hλh≤∑s=0h−1m2∑ζ=0Zw˜k2ζ,s−τλh+l2∑ζ=0Zek2x,sλhm1h−s−2.
Estimating statew˜kζ,s−τ with time delay in inequality (32) and letting Σ3=m2∑s=0h−1∑ζ=0Zw˜k2ζ,s−τλhm1h−s−2, from the known conditions, when h∈−τ,0, there is wkζ,h=φkζ,h, and one has(33)Σ3=m2λτ+2∑m=0h−τ−1λm1h−m−τ−2φ˜kλ2+m2λτ+2∑m=0h−τ−1λm1h−m−τ−2w˜kλ2.
By the initial value conditionφ˜kλ2=0, inequality (33) can be written as(34)Σ3=m2λτ+2∑m=0h−τ−1λm1h−m−τ−2w˜kλ2≤m2λτ+21−m1λw˜kλ2.
Substituting inequality (32) into inequality (34), one has(35)w˜kλ2≤m2λτ+21−m1λw˜kλ2+m2l2λ2∑s=0h−1m1λh−s−2ekλ2=h2λτ+21−m1λw˜kλ2+m2l2λ21−m1λekλ2.
Sinceλ0<λ<1 is sufficiently small, 0<1−m1λ<1 can be obtained. Inequality (35) can be written as(36)w˜kλ2⩽m2l2λ21−m1λ−m2λτ+2ekλ2≤ekλ2.
Multiplying both sides of inequality (20) by λh0<λ<1, one obtains(37)ek+1λ2≤2λclekλ2+2lbw˜kλ2.
Substituting inequality (37) into inequality (36), one obtains(38)ek+1λ2≤2λcl+2lbekλ2.
Lettingρ=2λcl+2lb,λcl=sup0≤h≤H1−chlh22λcl<1 can be obtained from the condition 1−γhL+S1N2≤0.5,0≤h≤H, of Theorem 1. Therefore, ρ<1. After recursion by inequality (38), one gets(39)ek+1λ2≤ρekλ2≤⋯≤ρke1λ2.
Then, by the contraction mapping principle, one obtainslimk⟶∞ek+1λ2=0.
Further, we have the following results:(40)0≤ek⋅,h2=λ−hλhek⋅,h2,≤sup0⩽h⩽Hek⋅,h2λhλ−H=λ−Hekλ2.
Thus, one has(41)limk⟶∞ek⋅,h2=0,0≤h≤H.
Next, combining equation (22), the i-th agent consensus tracking error can be rewritten as follows:(42)limk⟶∞ei,k⋅,h2=0,i=1,2,…,N,0≤h≤H.
Further, due toei,k⋅,h2=yr⋅,h−yi,k⋅,h2, one gets(43)yi,k⋅,h−yj,k⋅,h2=yr⋅,h−yj,k⋅,h−yr⋅,h−yi,k⋅,h2,=ej,k⋅,h−ei,k⋅,h2≤ej,k⋅,h2+ei,k⋅,h2.
Finally, based on equation (42) and inequality (43), one obtains(44)limk⟶∞yi,k⋅,h−yj,k⋅,h2=0,i,j∈1,2,…,N,0≤h≤H.
The proof of Theorem1 is completed.


## 3. Iterative Learning Consensus Control for Nonlinear Parabolic Partial Difference Mass with Time Delay

In this section, consider a nonlinear parabolic partial difference MAS (45) composed of N homogeneous discrete multiple agents, which run repeatedly within the finite-time interval, where the i-th agent’s dynamic equation is(45)△2wi,kζ,h=ah△12wi,kζ−1,h+fwi,kζ,h−τ,ui,kζ,h,h,yi,kζ,h=bwi,kζ,h,h+chui,kζ,h,where ζ,h are discrete variables of space and time, respectively. Z and H are given integers, and 0≤ζ≤Z, 0≤h≤H. k is the number of iterations, and i∈1,2,…,N indicates the i-th agent. ah,ch are uncertain bound real sequences, and ah>0. wi,kζ,h,ui,kζ,h,yi,kζ,h∈ℝ, respectively, denote the state of the i-th agent in system (41), control input, and control output. τ is the time delay; let the initial state function be φ⋅; when h∈−τ,0, wi,kζ,h=φi,kζ,h. f:ℝ×ℝ×0,H⟶ℝ,b:ℝ×0,H⟶ℝ are both nonlinear functions.The differential form of the first expression in system (45) is defined as(46)△2wi,kζ,h=wi,kζ,h+1−wi,kζ,h,△12wi,kζ−1,h=wi,kζ+1,h−2wi,kζ,h+wi,kζ−1,h.Here, it is still necessary to achieve consensus error between any two agents in system (45) to satisfy limk⟶∞yi,k⋅,h−yj,k⋅,h2=0. Under the conditions of Assumptions 1–4, we use ILC algorithm (13) to achieve this goal.For the simplicity of the subsequent convergence analysis, the compact form of system (45) is as follows:(47)△2wkζ,h=ah△12wkζ−1,h+fwkζ,h−τ,ukζ,h,h,ykζ,h=bwkζ,h,h+chukζ,h,where(48)wkζ,h=wT1,kζ,h,wT2,kζ,h,…,wTN,kζ,hT∈RN,ukζ,h=uT1,kζ,h,uT2,kζ,h,…,uTN,kζ,hT∈RN,ykζ,h=yT1,kζ,h,yT2,kζ,h,…,yTN,kζ,hT∈RN.After system (45) iterates k times, the corresponding difference form is expressed as(49)△2wkζ,h=wkζ,h+1−wkζ,h,(50)△12wkζ−1,h=wkζ+1,h−2wkζ,h+wkζ−1,h.Theorem 2.
Ifγh is in use, ILC algorithm (13) satisfies 1−γhL+S1N2≤0.5,0≤h≤H and satisfies Assumptions 1–4 at the same time; then when the iteration index k⟶∞, in the following sense, consensus error for the nonlinear parabolic partial difference system (47) can converge to zero; that is,(51)limk⟶∞yi,k⋅,h−yj,k⋅,h2=0,i,j∈1,2,…,N,0≤h≤H.Proof.
To prove the convergence for system (47) consensus error ek⋅,h2, we need to estimate w˜k⋅,h. By equation (49) and the first equation in system (47), one has(52)wkζ,h+1=at△12wkζ−1,h+wkζ,h+fwkζ,h−τ,ukζ,h,h.
Similar to equation (52), during the k+1-th iteration, the agent’s state can also be written as(53)wk+1ζ,h+1=ah△12wk+1ζ−1,h+wk+1ζ,h+fwk+1ζ,h−τ,uk+1ζ,h,h.
Next, subtracting equation (52) from equation (53) and combining with equation (50), one has(54)w˜kζ,h+1=ah△12wk+1ζ−1,h−△12wkζ−1,h+w˜kζ,h+fwk+1ζ,h−τ,uk+1ζ,h,h−fwkζ,h−τ,ukζ,h,h,=ahw˜kζ+1,h−2w˜kζ,h+w˜kζ1,h+w˜kζ,h+fwk+1ζ,h−τ,uk+1ζ,h,h−fwkζ,h−τ,ukζ,h,h.
Here, squaring both sides of equation (54) and by a+b+c≤3a2+3b2+3c2, one obtains(55)w˜kζ,h+12=ahw˜kζ+1,h−2w˜kζ,h+w˜kζ−1,h+w˜kζ,h+fwk+1ζ,h−τ,uk+1ζ,h,h−fwkζ,h−τ,ukζ,h,h2,≤3a2hw˜kζ+1,h−2w˜kζ,h+w˜kζ−1,h2+3w˜k2ζ,h+3fwk+1ζ,h−τ,uk+1ζ,h,h−fwkζ,h−τ,ukζ,h,h2.
Summing fromζ=0 to Z on both sides of inequality (55) and using Lipschitz condition (5), we have the following results:(56)∑ζ=0Zw˜kζ,h+12≤∑ζ=0Z3a2hw˜kζ+1,h−2w˜kζ,h+w˜kζ−0,h2+∑ζ=0Z3fwk+1ζ,h−τ,uk+1ζ,h,h−fwkζ,h−τ,ukζ,h,h2+∑ζ=0Z3w˜k2ζ,h,≤∑ζ=0Z3a2hw˜kζ+1,h−2w˜kζ,h+w˜kζ−0,h2+3w˜k⋅,h2+3lfw˜k⋅,h−τ2+u˜k⋅,h2≜Σ1′+Σ2′.where(57)Σ1′=3a2h∑ζ=0Zw˜kζ+1,h−2w˜kζ,h+w˜kζ−1,h2+3w˜k⋅,h2,Σ2′=3lfw˜k⋅,h−τ‖2+u˜k⋅,h2.
By the boundary condition (4), the following inequality holds:(58)Σ1′=3a2h∑ζ=0Zw˜kζ+1,h−2w˜kζ,h+w˜kζ−1,h2+3w˜k⋅,h2≤3a2h∑ζ=0Z3w˜k2ζ+1,h+12w˜k2x,h+3w˜k2ζ−1,h+3w˜k⋅,h2≤54a2h∑ζ=0Zw˜k2ζ,h−9a2tw˜k21,h−9a2hw˜k2ζ,h+3w˜k⋅,h2≤54a2h+3w˜k⋅,h2.
Substituting (58) into (56), one has (59)w˜k⋅,h+12≤54a2h+3w˜k⋅,h2+3lfw˜k⋅,h−τ2+u˜k⋅,h2,≤M1w˜k⋅,h2+M2w˜k⋅,h−τ2+u˜k⋅,h2,where M1=sup0≤h≤H54a2h+3,M2=3lf. Combining Lemma 1 and the initial value condition (4), one has(60)∑ζ=0Zw˜k2ζ,h≤∑s=0h−1M2∑ζ=0Zw˜k2ζ,s−τ+∑ζ=0Zu˜k2ζ,sM1h−s−2.
Here, we found the relationship betweenw˜kζ,h2 and u˜kζ,h2. Following on, by the ILC law (see (13)), one gets(61)u˜kζ,h2=uk+1ζ,h−ukζ,h2≤l2ek2ζ,h,where l=sup0≤h≤HγhL+S1N. It is observed that the output equation of system (45) is the same as that of system (1), which gives the following results:(62)∑ζ=0Zek+12ζ,h≤∑ζ=0Z21−chlh2ek2ζ,h+∑ζ=0Zbk+1−bk2=21−chlh2ek⋅,h2+2lbwk+1⋅,h−wk⋅,h2≤2λclek⋅,h2+2lbw˜k⋅,h2,where(63)λcl=sup0≤h≤H1−chlh2,w˜k⋅,h=wk+1ζ,h−wkζ,h.
In addition, by inequality (61), inequality (60) can be further written as(64)∑ζ=0Zw˜k2ζ,h≤∑s=0h−1M2∑ζ=0Zw˜k2ζ,s−τ+l2∑ζ=0Zek2ζ,sM1h−s−2.
Multiplying both sides of inequality (64) by λh0<λ<1, one obtains(65)∑ζ=0Zw˜k2ζ,hλh≤∑s=0h−1M2∑ζ=0Zw˜k2ζ,s−τλh+l2∑ζ=0Zek2ζ,sλhM1h−s−2.
Estimating statew˜kζ,s−τ with time delay in inequality (65) and letting Σ3′=M2∑s=0h−1∑ζ=0w˜k2ζ,s−τλhM1h−s−2, from the known conditions, when h∈−τ,0, there is wkζ,h=φkζ,h, and one has(66)Σ3′=M2λτ+2∑m=0h−τ−1λM1h−m−τ−2φ˜kλ2+M2λτ+2∑m=0h−τ−1λM1h−m−τ−2w˜kλ2.
By the initial value conditionφ˜kλ2=0, inequality (66) can be written as(67)Σ3′=M2λτ+2∑m=0h−τ−1λM1h−m−τ−2w˜kλ2≤M2λτ+21−M1λw˜kλ2.
Substituting inequality (65) into inequality (67), one has(68)w˜kλ2≤M2λτ+21−M1λw˜kλ2+M2l2λ2∑s=0h−1M1λh−s−2ekλ2=M2λτ+21−M1λw˜kλ2+M2l2λ21−M1λekλ2.
Sinceλ0<λ<1 is sufficiently small, 0<1−M1λ<1 can be obtained. Inequality (68) can be written as(69)w˜kλ2≤M2l2λ21−M1λ−M2λτ+2ekλ2≤ekλ2.
Because the output equation of system (45) is the same as that of system (1), one gets(70)ek+1λ2≤2λclekλ2+2lbw˜kλ2.
Substituting inequality (70) into inequality (69), we have the following results:(71)ek+1λ2≤2λcl+2lbekλ2.
Letρ′=2λcl+2lb, and 2λcl<1 can be obtained from the condition of Theorem 2. Therefore, ρ′<1. After recursion by inequality (71), one has(72)ek+1λ2≤ρ′ekλ2≤⋯≤ρ′ke1λ2.
Finally, using proof method similar to that in formulas (40)–(43) in Section 2, one obtains(73)limk⟶∞yi,k⋅,h−yj,k⋅,h2=0,i,j∈1,2,…,N,0≤h≤H.
This completes the proof of Theorem2.


## 4. Simulations

In order to verify the effectiveness of the ILC scheme proposed in this paper for the nonlinear partial difference MASs with time delay, two specific numerical examples are given as follows.Take the corresponding Laplacian matrix:(74)L=2−10−12−10−12.The nonlinear functionsf,b in systems (77) and (79) are(75)fwi,kζ,h−τ,ui,kζ,h,h=−0.1coswi,kζ,h−τ+ui,kζ,h,bwi,kζ,h,h=vhsinwi,kx,h,where vh=0.1+e−6h,τ=0.1.After selecting the appropriate learning gain matrix,1−γhL+S1N2≤0.5 is easy to calculate, which satisfies the condition for Theorems 1 and 2. Applying the designed ILC algorithm,(76)uk+1ζ,h=ukζ,h+γhL+S1Nekζ,h.Example 1.
The dynamic model of thei-th agent in the nonlinear hyperbolic partial difference MAS with time delay is(77)△2wi,kζ,h=at△1wi,kζ+1,h+fwi,kζ,h−τ,ui,kζ,h,h,yiζ,h=bwi,kζ,h,h+chui,kζ,h,where i=1,2,3, ζ,h∈1,15×0,250,Z=15,H=250, while Z and H are given integers, and ah=0.21−1/15h+1, ch=1. Let the desired output of system (77) be (the virtual leader’s motion trajectory)(78)yr1ζ,h=4sinh−12ζ−11−e−2.5Z−ζ.
The simulation diagram and results of Example1 can be obtained, as shown in Figures 1–3.Figure 1
Actual output of three agents and the virtual leader agent in nonlinear hyperbolic partial MASs. (a) Virtualyτ(ζ, h). (b) Agent1 y1(ζ, h)(k = 30). (c) Agent2 y2(ζ, h)(k = 30). (d) Agent3 y3(ζ, h)(k = 30).
(a)(b)(c)(d)Figure 2
Error surface between the three agents and the virtual leader agent in nonlinear hyperbolic partial MASs. (a) Agent1 error surface (k = 30). (b) Agent2 error surface (k = 30). (c) Agent3 error surface (k = 30).
(a)(b)(c)Figure 3
Changes in the consensus error between the three agents and the virtual leader agent in nonlinear hyperbolic partial MASs.Example 2.
The dynamic model of thei-th agent in the nonlinear parabolic partial difference MASs is(79)△2wi,kζ,h=ah△12wi,kζ−1,h+fwi,kζ,h−τ,ui,kζ,h,h,yi,kζ,h=bwi,kζ,h,h+ctui,kζ,h,where i=1,2,3, ζ,h∈1,15×0,250,Z=15,H=250, while Z and H are given integers, and ah=0.2−1/20h+1, ch=1. Let the desired output of this system be (the virtual leader’s motion trajectory)(80)yr2ζ,h=2sinζ51−e−2Z−ζsinh.
The simulation diagram and results of Example2 can be obtained, as shown in Figures 4–6.
Finally, the simulation results are shown in Figures1–6. Next, we analyze the simulation results.
Figure1 depicts the actual output changes of the three agents and virtual leaders yr1ζ,h in hyperbolic partial MASs when iterating 30 times k=30. Figure 4 depicts the actual output changes of the three agents and virtual leaders yr2ζ,h in nonlinear parabolic partial MASs when iterating 30 times k=30. It can be seen that system (77) and the three other agents in system (79) track the output changes of the virtual leader. It can be seen from Figures 1–3 that the control goal has been basically achieved.
Figures2 and 5 show error surfaces between the three agents and virtual leader agent in nonlinear hyperbolic partial MASs and nonlinear parabolic partial MASs when iterating 30 times k=30, respectively. In Figures 2 and 4, it can be seen that the absolute value for largest consensus error surface is less than 3×10−10. Therefore, from Figures 2 and 4, it can be seen that the three agents in system (77) and system (79), respectively, track the motion trajectories of the respective virtual leader agent.
Figures3 and 6 show the variation of the consensus error of three agents in nonlinear hyperbolic partial MASs and nonlinear parabolic partial MASs in the sense of L2 norm. From Figures 3 and 6, it can be seen that the consensus error of the three agents in system (77) and system (79) along the iterative axis gradually converges to zero in the sense of L2 norm.
From Figures1–6, within the finite-time interval 0,T, when the designed ILC algorithm is used in nonlinear hyperbolic partial MASs and parabolic partial MASs, respectively, in the sense of L2 norm, the consensus error along the iteration axis between any two agents in system (77) and system (79) can gradually converge to zero. Therefore, in this paper, simulation results illustrate the effectiveness the ILC algorithm designed in nonlinear hyperbolic partial MASs and parabolic partial MASs with time delay, respectively.Figure 4
Actual output of three agents and the virtual leader agent in nonlinear parabolic partial MASs. (a) Virtualyτ(ζ, h). (b) Agent1 y1(ζ, h)(k = 30). (c) Agent2 y2(ζ, h)(k = 30). (d) Agent3 y3(ζ, h)(k = 30).
(a)(b)(c)(d)Figure 5
Error surface between each of the three agents in nonlinear parabolic partial MASs. (a) Agent1 error surface (k = 30). (b) Agent2 error surface (k = 30). (c) Agent2 error surface (k = 30).
(a)(b)(c)Figure 6
Changes in the consensus error between the three agents and the virtual leader agent in nonlinear parabolic partial MASs.


## 5. Conclusions

The consensus control problem for nonlinear hyperbolic partial MASs and nonlinear parabolic partial MASs with time delay is studied in this paper, taking into consideration the space-time discrete problem of agents. Based on the fixed topology of the MASs, an ILC algorithm is proposed by applying a method for the virtual leader agent to generate the desired trajectory. This algorithm uses the tracking error between any agent and the virtual leader and neighbor agents in the last iteration and combines the characteristics of the system communication topology to continuously modify the previous control law to obtain the ideal control law. When the iterative learning indexk approaches infinity, in the sense of the L2 norm, the consensus error between all agents in the system can converge to zero. Furthermore, simulation results illustrate the effectiveness of the algorithm. In the future, the iterative learning consensus control problem of discrete linear MASs with random noise and measurement saturation can be further carried out, and, for the topological structure between multiple agents, the topology of the iterative axis can be studied.



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