# Iterative Learning Tracking Control of Nonlinear Multiagent Systems with Input Saturation

Authors: Bingyou Liu; Zhengzheng Zhang; Lichao Wang; Xing Li; Xiongfeng Deng
Journal: Complexity
 (2021)

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## Abstract

A tracking control algorithm of nonlinear multiple agents with undirected communication is studied for each multiagent system affected by external interference and input saturation. A control design scheme combining iterative learning and adaptive control is proposed to perform parameter adaptive time-varying adjustment and prove the effectiveness of the control protocol by designing Lyapunov functions. Simulation results show that the high-precision tracking control problem of the nonlinear multiagent system based on adaptive iterative learning control can be well realized even when the input is saturated. Finally, the validity of the proposed algorithm is verified by numerical analysis.


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## Body

## 1. Introduction

A multiagent control system is formed by multiple agents with independent action capabilities to complete a complex task through local cooperation and interaction [1–4]. In recent years, multiagent control has achieved great breakthroughs due to the rapid development of coordinated control of dynamic systems. Multiagent systems have a wide range of applications in many engineering fields, such as formation control of various robots, positioning of distributed sensor networks, and multisatellite collaborative control [5–10].The problem of consistency tracking is the basis of research on coordinated control, and it is the most important and basic problem of multiagent control. This problem has very important engineering value and theoretical significance. A number of scholars have carried out in-depth research on consistency tracking and achieved great results in consistency tracking [11, 12], distributed tracking positioning [13], swarm [14], and other group behavior control problems.According to the consistency problem, multiagent control system can be divided into leader and nonleader control systems. However, in practical engineering applications, most of the systems are nonlinear regardless of with or without leader control system. Therefore, research on the consistency of nonlinear multiagent control system has more important practical significance. In [15], a fuzzy logic system is used as a feedforward compensation device to study the consistency problem of multiple agents with unknown mixed nonlinearity regardless of direction. In [16], a distributed group tracking algorithm for a class of nonlinear multiple agents is proposed to solve the problem of distributed group tracking. On the basis of [16], [17] investigated the problem of distributed robust asymptotic finite-time tracking control. This protocol does not need the navigator to input accurate boundary and does not need to know the exact boundary of the dynamic following agent in contrast to other control protocols. In [18, 19], an adaptive distributed control strategy is proposed to deal with the influence of uncertainty and nonlinear dynamics in a nonlinear multi-intelligent system. Another work [20] evaluated the consistency problem of the Lupin clustering of nonlinear agents through adaptive sliding mode variable control based on [21–23] which considered the control problem of switched nonlinear multiagent systems and achieved the consensus tracking under the presented control strategies. However, our results are still different compared with the above two articles. Compared with the event-triggered control problem of multiagent systems and the finite-time control problem of multiagent systems, our results are only studied from the consensus control problem. The main reason is that the consensus control problem is the basic problem of the multiagent systems. Besides, we introduce the iterative learning control method to solve the consensus problem, which can achieve control effects from two perspectives: time domain and iteration domain.In an actual control system, the controller controls the object through the actuator. However, the system itself determines that the output of the actuator and its rate of change cannot be arbitrarily changed, so the saturation of the input amplitude is inevitable [24–26]. The performance of multiagent control may be degraded, and it may even become unstable due to strict constraints on speed, overload, and so forth. Many scholars have endeavored to solve the input saturation problem. In [27], a new PI protocol is designed, in which a more rigorous adaptive coupling saturation function is constructed to deal with the multiagent system with relative state saturation constraints. The study in [28] developed a distributed low-gain feedback control, which solves the problem of semiglobal and global multiagent control of input saturation. The study in [29] established a sliding mode based on literature [28] to distribute controller for handling input-saturated second-order multiagent distributed robust global control problems. The study in [30] continued to improve the semiglobal multiagent control of input saturation by using low-gain and inverse saturation methods based on literature [28]. The study in [31] used Lyapunov theory to design an adaptive iterative learning algorithm with time-varying coupled-gain full-saturation parameter update to solve the problem of multiagent coordinated learning control with input saturation. However, most scholars have rarely investigated the relationship between input saturation and nonlinearity.Iterative learning is similar to the process and characteristics of human learning. Iterative learning constantly corrects the current signal by the deviation between the output signal and the given signal of the controlled system, thereby improving the tracking performance of the controlled system; as such, it is widely used in uncertain, nonlinear, and complex intelligent systems. In [32–34], distributed iterative learning for nonlinear follower is studied. In [35, 36], an iterative control protocol is designed for each follower for the unknown input gain function of the system, so each follower can follow the leader with high accuracy. The authors of [37–40] investigated the first-order, second-order, and even higher-order models of multiple agents based on iterative learning. Based on previous research, we use adaptive iterative control to solve the high-precision tracking problem of nonlinear multiple agents in undirected communication.Inspired by the above analysis, we discuss the iterative learning control approach for tracking control of leader-following nonlinear multiagent systems with input saturation. Each following agent is modeled as first-order nonlinear dynamics, and external disturbance is considered. The main contributions of this paper are summarized as follows:(i)
A class of multiagent systems with nonlinear dynamics and external disturbance are addressed in this work, which are more general than those in previous studies [17–21]. The design of control protocol will be further complicated due to the introduction of nonlinear dynamics.(ii)
In comparison with [31, 32], the present work discusses an adaptive control protocol with iterative learning control for nonlinear multiagent systems. Moreover, the adaptive updating laws for time-varying parameters are designed.(iii)
A new Lyapunov function is structured to check the validity of the proposed control protocol. The tracking problem of leader-following nonlinear multiagent systems with input saturation can be comprehensively solved using the presented iterative learning control protocol.The rest of the paper is designed as follows: Section2 introduces basic knowledge and related theorems about graph theory. Section 3 reports the consensus of the proposed methods and analysis-related issues. Section 4 presents a nonlinear multiagent adaptive iterative control protocol with input saturation and analyzes system convergence. Section 5 discusses the simulation results. Section 6 provides the conclusion and summarizes the full text.


## 2. Preliminaries

### 2.1. Graph Theory

On the basis of graph theory, information topology among agents can be described as an undirected graphG=V,E,A, with the set of nodes V=v1,…,vn and the set of edges E=i,j,i,j∈V,andi≠j. The weighted adjacency matrix is defined as A=aij∈Rn×n, in which aij=aji>0 if i,j∈E; otherwise, aij=aji=0. aii=0 is assumed. The set of neighbors of node i is defined as Ni=vj:vj,vi∈E. The Laplacian matrix of G is denoted by ℒ=D−A, where D=diagd1,…,dn with di=∑j=1naij. Graph G is connected if a path exists between any two nodes.Augmented graphG¯ consists of n following agents whose information topology graph is G and one leader agent is concerned. Let bi denote the connection matrix between agent i and the leader. If agent i obtains the information of the leader, then bi>0; otherwise, bi=0. Hence, the connection matrix between the leader and following agents can be defined as ℬ=diagb1,…,bn. ℋ=ℒ+ℬ is a matrix associated with G¯.Lemma 1.
(see [31]). If graph G¯ is connected, then the symmetric matrix ℋ associated with G¯ is positive definite.


### 2.2. Useful Definitions and Lemmas

Some useful definitions and lemmas are provided as follows to facilitate subsequent analysis.Definition 1.
(see [32]). Convergent series sequence Δk is denoted by Δk=c/km, where k∈Z+, c>0, and m∈Z+≥2 are the parameters to be designed.Lemma 2.
(see [32]). For a given sequence c/km, where k∈Z+, c>0, and m∈Z+≥2, limk⟶∞∑j=1kc/jm≤2c.Lemma 3.
(see [33]). For any b∈R>0 and ζ>0, the hyperbolic tangent function satisfies 0≤b−btanhb/ζ≤qζ, where q=0.2785.


## 2.1. Graph Theory

On the basis of graph theory, information topology among agents can be described as an undirected graphG=V,E,A, with the set of nodes V=v1,…,vn and the set of edges E=i,j,i,j∈V,andi≠j. The weighted adjacency matrix is defined as A=aij∈Rn×n, in which aij=aji>0 if i,j∈E; otherwise, aij=aji=0. aii=0 is assumed. The set of neighbors of node i is defined as Ni=vj:vj,vi∈E. The Laplacian matrix of G is denoted by ℒ=D−A, where D=diagd1,…,dn with di=∑j=1naij. Graph G is connected if a path exists between any two nodes.Augmented graphG¯ consists of n following agents whose information topology graph is G and one leader agent is concerned. Let bi denote the connection matrix between agent i and the leader. If agent i obtains the information of the leader, then bi>0; otherwise, bi=0. Hence, the connection matrix between the leader and following agents can be defined as ℬ=diagb1,…,bn. ℋ=ℒ+ℬ is a matrix associated with G¯.Lemma 1.
(see [31]). If graph G¯ is connected, then the symmetric matrix ℋ associated with G¯ is positive definite.


## 2.2. Useful Definitions and Lemmas

Some useful definitions and lemmas are provided as follows to facilitate subsequent analysis.Definition 1.
(see [32]). Convergent series sequence Δk is denoted by Δk=c/km, where k∈Z+, c>0, and m∈Z+≥2 are the parameters to be designed.Lemma 2.
(see [32]). For a given sequence c/km, where k∈Z+, c>0, and m∈Z+≥2, limk⟶∞∑j=1kc/jm≤2c.Lemma 3.
(see [33]). For any b∈R>0 and ζ>0, the hyperbolic tangent function satisfies 0≤b−btanhb/ζ≤qζ, where q=0.2785.


## 3. Problem Formulation

Considering a class of leader-following first-order nonlinear multiagent systems with input saturation, the dynamics of theith following agent as the kth iteration are described as(1)x˙ikt=fxikt,t+satuikt+dikt,where xikt∈R and uikt∈R are the position and control input of the ith agent, respectively; fxikt,t represents the continuous nonlinear function; dikt is the unknown but bounded external disturbance; that is, dikt≤di∗ with di∗ as an unknown positive constant; k denotes the iteration number; and t∈0,T. satuikt is the saturation function defined as(2)satuikt=signuiktu¯,uikt>u¯,uikt,uikt≤u¯,where u¯>0 is the upper bound of the saturation function and it is prespecified. The dynamics of the leader are given as(3)x˙0t=fx0t,t+u0t,where x0t∈R and u0t∈R are the state and control input of the leader agent, respectively. In addition, we assume that the information of the leader is only accepted by a portion of following agents.On the basis of multiagent systems (1) and (3), the tracking error of the ith following agent is defined as(4)eikt=x0t−xikt.Definition 2.
The consensus tracking problem of leader-following nonlinear multiagent systems with input saturation (1) and (3) is achieved if limk⟶∞xikt=x0t for i=1,…,n over the interval 0,T.Assumption 1.
The alignment initial condition, that is,xik0=xik−1T for i=1,…,n, is considered, and the trajectory of the leader is spatially closed; namely, x00=x0T. eik0=eik−1T for i=1,…,n can be easily obtained.Assumption 2.
Functionfxt,t is uniform global Lipschitz in x over the interval 0,T; that is, constant κf exists, such that(5)fx1t,t−fx2t,t≤κfx1t−x2t.Remark 1.
For the initial condition setting, three cases are mostly applied in existing literature. The first one is the alignment initial condition, similar to Assumption1 in this study. The second one is the resetting initial condition, that is, xk0=xo for each iteration, where xo is a given constant. The last one is that the initial values are different for each iteration, that is, xk0=xok, where xok is a constant of change. The alignment initial condition is considered for the convenience of analysis. For the two other cases, the analysis approach is similar to that for the alignment initial condition.
The control objective of this work is to design an appropriate control schemeuikt for i=1,…,n and adaptive updating laws such that all following agents can track the trajectory of the leader over the interval 0,T as the iteration number k tends to infinity.


## 4. Control Protocol Design and Convergence Analysis

In this section, the tracking problem of leader-following nonlinear multiagent systems with input saturation is introduced. The adaptive control protocol is designed on the basis of the iterative learning control approach. The adaptive updating laws are also presented. A new Lyapunov function is developed to analyze the convergence of the proposed control protocol.


### 4.1. Distributed Iterative Learning Control Protocol Design

In view of multiagent systems (1) and (3), the consensus tracking error is described as(6)εikt=∑j∈Niaijxjkt−xikt+bix0t−xikt.Considering the definition ofεikt, the vector form of Equation (6) can be expressed as(7)εkt=ℋ1nx0t−xkt=ℋekt,where εkt=ε1kt,…,εnktT, ekt=1nx0t−xkt, xkt=x1kt,…,xnktT, ℋ=ℒ+ℬ, and 1n=1,…,1T.Remark 2.
We only investigate the states of each agent asxikt∈R and x0t∈R. For the case of xikt∈Rp and x0t∈Rp, we can easily obtain the results by utilizing the Kronecker product. Thus, Equation (7) can be written as(8)εkt=ℋ⊗Ipekt,where ⊗ represents the Kronecker product. All related results can be changed by applying the Kronecker product operation.
On the basis of Equation (7), the derivative of εkt is as follows:(9)ε˙kt=ℋe˙kt=ℋ1nx˙0t−x˙kt.
Substitutingx˙0t and x˙kt into Equation (9) yields(10)ε˙kt=ℋ1nfx0t,t−fxkt,t+1nu0t−satukt−dkt,where fxkt,t=fx1kt,t,…,fxnkt,tT, satukt=satu1kt,…,satunktT, and dkt=d1kt,…,dnktT.
Hence, the adaptive iterative learning control protocol for each following agent is designed as(11)uikt=u0t+αεikt+υikt+ρikttanhρiktεiktΔk,and the adaptive updating laws for υikt and ρikt are designed as(12)υ˙ikt=αεikt−δuktTδukt2υkt2υikt,υik0=υik−1T,(13)ρ˙ikt=λiεiktρik0=ρik−1T,ρi0t>0,where α>0 and λi>0 are the constants to be designed, υikt and ρikt are the time-varying parameters, and δukt=ukt−satukt with ukt=u1kt,…,unktT, satukt=satu1kt,…,satunktT, and υkt=υ1kt,…,υnktT.
The vector form of the control protocol (11) is written as(14)ukt=1nu0t+αεkt+υkt+σkt,where σkt=σ1kt,…,σnktT with σikt=ρikttanhρiktεikt/Δk.Remark 3.
In the control protocol (11), time-varying parameters υikt and ρikt are considered. The purpose of designing υikt is to compensate saturation error δuikt, and the purpose of designing ρikt is to eliminate unknown disturbance dikt. By introducing υikt and ρikt, the desired tracking control problem of leader-following nonlinear multiagent systems with input saturation ((1) and (3)) can be comprehensively solved over the interval 0,T.


### 4.2. Convergence Analysis

Theorem1 presents the main result of this study.Theorem 1.
Leader-following first-order nonlinear multiagent systems with input saturation ((1) and (3)) under Assumptions 1 and 2 are considered. Let graph G¯ be connected and the adaptive iterative learning control protocol (11) and adaptive updating laws (12) and (13) be adopted; then, all following agents can track the trajectory of the leader agent over the interval 0,T as the iteration number k tends to infinity; namely, limk⟶∞xikt=x0t for i=1,…,n.Proof.
A Lyapunov function candidate is defined as(15)Vkt=12εktTℋ−1εkt+12υktTυkt+∑i=1n12λiρikt−ρ02,where ρ0>0 is a constant to be confirmed later.
Hence, the difference betweenVkt and Vk−1t is given as(16)ΔVkt=Vkt−Vk−1t,=12εktTℋ−1εkt−12εk−1tTℋ−1εk−1t+12υktTυkt−12υk−1tTυk−1t+∑i=1n12λiρikt−ρ02−∑i=1n12λiρik−1t−ρ02.
Notably,(17)12εktTℋ−1εkt=12εk0Tℋ−1εk0+∫0tεkτTℋ−1ε˙kτdτ.
Substituting Equation (10) into Equation (17) yields(18)12εktTℋ−1εkt=12εk0Tℋ−1εk0+∫0tεkτT1nfx0τ,τ−fxkτ,τdτ+∫0tεkτT1nu0τ−satukτdτ−∫0tεkτTdkτdτ.
Considering Assumption2, we have(19)∫0tεkτT1nfx0τ,τ−fxkτ,τdτ≤κfℋ∑i=1n∫0teikτeikτdτ,=κ∫0tεkτTℋℋT−1εkτdτ,where κ=κfℋ.
Given thatsatukt=ukt−δukt and considering Equation (14), we have(20)∫0tεkτT1nu0τ−satukτdτ=∫0tεkτT1nu0τ−ukτ+δukτdτ=−α∫0tεkτTεkτdτ−α∫0tεkτTυkτdτ−∫0tεkτTσkτdτ+∫0tεkτTδukτdτ≤−α−12∫0t∫0tεkτTεkτdτ−α∫0tεkτTυkτdτ+12∫0tδukτTδukτdτ−∑i=1n∫0tρikτεikτtanhρikτεikτΔkdτ,where εktTδut≤εktTεkt/2+δuktTδukt/2 is applied.
Meanwhile,(21)−∫0tεkτTdkτdτ=−∑i=1n∫0tεikτdikτdτ≤∑i=1n∫0tdi∗εikτdτ.
Substituting Equations (19)–(21) into Equation (18) yields(22)12εktTℋ−1εkt≤12εk0Tℋ−1εk0−∫0tεkτTBεkτdτ−α∫0tεkτTυkτdτ+12∫0tδukτTδukτdτ−∑i=1n∫0tρikτεikτtanhρikτεikτΔkdτ+∑i=1n∫0tdi∗εikτdτ,where B=α−1/2In−κℋℋT−1.
Analogically, we obtain(23)12υktTυkt=12υk0Tυk0+∫0tυkτTυ˙kτdτ=12υk0Tυk0+α∫0tεkτTυkτdτ−12∫0tδukτTδukτdτ,(24)∑i=1n12λiρikt−ρ02=∑i=1n12λiρik0−ρ02+∑i=1n∫0t1λiρikτ−ρ0ρ˙ikτdτ,=∑i=1n12λiρik0−ρ02+∑i=1n∫0tρikτεikτdτ−ρ0∑i=1n∫0tεikτdτ,where adaptive updating laws υ˙ikt and ρ˙ikt are adopted.
Therefore, Equations (22)–(24) yield the following:(25)ΔVkt≤12εk0Tℋ−1εk0−12εk−1tTℋ−1εk−1t+12υk0Tυk0−12υk−1tTυk−1t+∑i=1n12λiρik0−ρ02−∑i=1n12λiρik−1t−ρ02−∫0tεkτTBεkτdτ+∑i=1n∫0tρikτεikτdτ−∑i=1n∫0tρikτεikτtanhρikτεikτΔkdτ−∑i=1n∫0tρ0−di∗εikτdτ,and ρikt≥0 from the adaptive updating law (13); then, Lemma 3 shows that(26)∑i=1n∫0tρikτεikτdτ−∑i=1n∫0tρikτεikτtanhρikτεikτΔkdτ,≤nTqΔk.
Furthermore,ρ0 is sufficiently large such that ρ0>maxdi∗ for i=1,…,n. Let θminB represent the minimum singular value of B. Based on (25) and (26), we obtain(27)ΔVkt≤12εk0Tℋ−1εk0−12εk−1tTℋ−1εk−1t+12υk0Tυk0−12υk−1tTυk−1t,+∑i=1n12λiρik0−ρ02−∑i=1n12λiρik−1t−ρ02−θminB∫0tεkτTεkτdτ+nTqΔk.
On the basis of Equation (27),(28)Vkt=Vk−1t+ΔVkt,≤12εk0Tℋ−1εk0+12υk0Tυk0+∑i=1n12λiρik0−ρ02−θminB∫0tεkτTεkτdτ+nTqΔk.
Considering Assumption1, εk0=εk−1T can be easily obtained. We also achieve υik0=υik−1T and ρik0=ρik−1T from Equations (12) and (13), respectively. We obtain(29)Vkt≤12εk−1TTℋ−1εk−1T+12υk−1TTυk−1T+∑i=1n12λiρik−1T−ρ02−θminB∫0tεkτTεkτdτ+nTqΔk,=Vk−1T−θminB∫0tεkτTεkτdτ+nTqΔk.
Lett=T. On the basis of Equation (29), we acquire(30)VkT≤Vk−1T−θminB∫0TεkτTεkτdτ+nTqΔk.
Equation (30) and Lemma 2 yield(31)VkT≤V1T−θminB∑j=2k∫0TεjτTεjτdτ+nTq∑j=2kΔj,≤V1T−θminB∑j=2k∫0Tεjτεjτdτ+2nTqc≤V1T+2nTqc.
In view of Equation (31), the boundedness of VkT is guaranteed for any iteration provided that V1T is bounded. Appendix A shows the boundedness of V1t.
The boundedness implies thatV1T is bounded. Hence, VkT is bounded from Equation (31) for k∈Z+. Equation (29) indicates that Vkt is uniformly bounded over the interval 0,T.
On the basis of Equation (31),(32)VkT≤V1T−θminB∑j=2k∫0TεjτTεjτdτ+2nTqc.
Series∑j=2k∫0Tεjτεjτdτ is convergent due to the positiveness of VkT and boundedness of V1T. Hence, limk⟶∞∫0TεkτTεkτ=0 can be easily obtained. Accordingly, by adopting the Barbalat-like lemma [34], we have limk⟶∞εkt=0 uniformly over the interval 0,T. Therefore, limk⟶∞ekt=0 based on Equation (7) suggests that limk⟶∞eikt=0 is uniform over the interval 0,T for i=1,…,n. Thus, we have limk⟶∞xikt=x0t for i=1,…,n, indicating that all following agents can uniformly track the trajectory of the leader over the interval 0,T. The proof is completed.


## 4.1. Distributed Iterative Learning Control Protocol Design

In view of multiagent systems (1) and (3), the consensus tracking error is described as(6)εikt=∑j∈Niaijxjkt−xikt+bix0t−xikt.Considering the definition ofεikt, the vector form of Equation (6) can be expressed as(7)εkt=ℋ1nx0t−xkt=ℋekt,where εkt=ε1kt,…,εnktT, ekt=1nx0t−xkt, xkt=x1kt,…,xnktT, ℋ=ℒ+ℬ, and 1n=1,…,1T.Remark 2.
We only investigate the states of each agent asxikt∈R and x0t∈R. For the case of xikt∈Rp and x0t∈Rp, we can easily obtain the results by utilizing the Kronecker product. Thus, Equation (7) can be written as(8)εkt=ℋ⊗Ipekt,where ⊗ represents the Kronecker product. All related results can be changed by applying the Kronecker product operation.
On the basis of Equation (7), the derivative of εkt is as follows:(9)ε˙kt=ℋe˙kt=ℋ1nx˙0t−x˙kt.
Substitutingx˙0t and x˙kt into Equation (9) yields(10)ε˙kt=ℋ1nfx0t,t−fxkt,t+1nu0t−satukt−dkt,where fxkt,t=fx1kt,t,…,fxnkt,tT, satukt=satu1kt,…,satunktT, and dkt=d1kt,…,dnktT.
Hence, the adaptive iterative learning control protocol for each following agent is designed as(11)uikt=u0t+αεikt+υikt+ρikttanhρiktεiktΔk,and the adaptive updating laws for υikt and ρikt are designed as(12)υ˙ikt=αεikt−δuktTδukt2υkt2υikt,υik0=υik−1T,(13)ρ˙ikt=λiεiktρik0=ρik−1T,ρi0t>0,where α>0 and λi>0 are the constants to be designed, υikt and ρikt are the time-varying parameters, and δukt=ukt−satukt with ukt=u1kt,…,unktT, satukt=satu1kt,…,satunktT, and υkt=υ1kt,…,υnktT.
The vector form of the control protocol (11) is written as(14)ukt=1nu0t+αεkt+υkt+σkt,where σkt=σ1kt,…,σnktT with σikt=ρikttanhρiktεikt/Δk.Remark 3.
In the control protocol (11), time-varying parameters υikt and ρikt are considered. The purpose of designing υikt is to compensate saturation error δuikt, and the purpose of designing ρikt is to eliminate unknown disturbance dikt. By introducing υikt and ρikt, the desired tracking control problem of leader-following nonlinear multiagent systems with input saturation ((1) and (3)) can be comprehensively solved over the interval 0,T.


## 4.2. Convergence Analysis

Theorem1 presents the main result of this study.Theorem 1.
Leader-following first-order nonlinear multiagent systems with input saturation ((1) and (3)) under Assumptions 1 and 2 are considered. Let graph G¯ be connected and the adaptive iterative learning control protocol (11) and adaptive updating laws (12) and (13) be adopted; then, all following agents can track the trajectory of the leader agent over the interval 0,T as the iteration number k tends to infinity; namely, limk⟶∞xikt=x0t for i=1,…,n.Proof.
A Lyapunov function candidate is defined as(15)Vkt=12εktTℋ−1εkt+12υktTυkt+∑i=1n12λiρikt−ρ02,where ρ0>0 is a constant to be confirmed later.
Hence, the difference betweenVkt and Vk−1t is given as(16)ΔVkt=Vkt−Vk−1t,=12εktTℋ−1εkt−12εk−1tTℋ−1εk−1t+12υktTυkt−12υk−1tTυk−1t+∑i=1n12λiρikt−ρ02−∑i=1n12λiρik−1t−ρ02.
Notably,(17)12εktTℋ−1εkt=12εk0Tℋ−1εk0+∫0tεkτTℋ−1ε˙kτdτ.
Substituting Equation (10) into Equation (17) yields(18)12εktTℋ−1εkt=12εk0Tℋ−1εk0+∫0tεkτT1nfx0τ,τ−fxkτ,τdτ+∫0tεkτT1nu0τ−satukτdτ−∫0tεkτTdkτdτ.
Considering Assumption2, we have(19)∫0tεkτT1nfx0τ,τ−fxkτ,τdτ≤κfℋ∑i=1n∫0teikτeikτdτ,=κ∫0tεkτTℋℋT−1εkτdτ,where κ=κfℋ.
Given thatsatukt=ukt−δukt and considering Equation (14), we have(20)∫0tεkτT1nu0τ−satukτdτ=∫0tεkτT1nu0τ−ukτ+δukτdτ=−α∫0tεkτTεkτdτ−α∫0tεkτTυkτdτ−∫0tεkτTσkτdτ+∫0tεkτTδukτdτ≤−α−12∫0t∫0tεkτTεkτdτ−α∫0tεkτTυkτdτ+12∫0tδukτTδukτdτ−∑i=1n∫0tρikτεikτtanhρikτεikτΔkdτ,where εktTδut≤εktTεkt/2+δuktTδukt/2 is applied.
Meanwhile,(21)−∫0tεkτTdkτdτ=−∑i=1n∫0tεikτdikτdτ≤∑i=1n∫0tdi∗εikτdτ.
Substituting Equations (19)–(21) into Equation (18) yields(22)12εktTℋ−1εkt≤12εk0Tℋ−1εk0−∫0tεkτTBεkτdτ−α∫0tεkτTυkτdτ+12∫0tδukτTδukτdτ−∑i=1n∫0tρikτεikτtanhρikτεikτΔkdτ+∑i=1n∫0tdi∗εikτdτ,where B=α−1/2In−κℋℋT−1.
Analogically, we obtain(23)12υktTυkt=12υk0Tυk0+∫0tυkτTυ˙kτdτ=12υk0Tυk0+α∫0tεkτTυkτdτ−12∫0tδukτTδukτdτ,(24)∑i=1n12λiρikt−ρ02=∑i=1n12λiρik0−ρ02+∑i=1n∫0t1λiρikτ−ρ0ρ˙ikτdτ,=∑i=1n12λiρik0−ρ02+∑i=1n∫0tρikτεikτdτ−ρ0∑i=1n∫0tεikτdτ,where adaptive updating laws υ˙ikt and ρ˙ikt are adopted.
Therefore, Equations (22)–(24) yield the following:(25)ΔVkt≤12εk0Tℋ−1εk0−12εk−1tTℋ−1εk−1t+12υk0Tυk0−12υk−1tTυk−1t+∑i=1n12λiρik0−ρ02−∑i=1n12λiρik−1t−ρ02−∫0tεkτTBεkτdτ+∑i=1n∫0tρikτεikτdτ−∑i=1n∫0tρikτεikτtanhρikτεikτΔkdτ−∑i=1n∫0tρ0−di∗εikτdτ,and ρikt≥0 from the adaptive updating law (13); then, Lemma 3 shows that(26)∑i=1n∫0tρikτεikτdτ−∑i=1n∫0tρikτεikτtanhρikτεikτΔkdτ,≤nTqΔk.
Furthermore,ρ0 is sufficiently large such that ρ0>maxdi∗ for i=1,…,n. Let θminB represent the minimum singular value of B. Based on (25) and (26), we obtain(27)ΔVkt≤12εk0Tℋ−1εk0−12εk−1tTℋ−1εk−1t+12υk0Tυk0−12υk−1tTυk−1t,+∑i=1n12λiρik0−ρ02−∑i=1n12λiρik−1t−ρ02−θminB∫0tεkτTεkτdτ+nTqΔk.
On the basis of Equation (27),(28)Vkt=Vk−1t+ΔVkt,≤12εk0Tℋ−1εk0+12υk0Tυk0+∑i=1n12λiρik0−ρ02−θminB∫0tεkτTεkτdτ+nTqΔk.
Considering Assumption1, εk0=εk−1T can be easily obtained. We also achieve υik0=υik−1T and ρik0=ρik−1T from Equations (12) and (13), respectively. We obtain(29)Vkt≤12εk−1TTℋ−1εk−1T+12υk−1TTυk−1T+∑i=1n12λiρik−1T−ρ02−θminB∫0tεkτTεkτdτ+nTqΔk,=Vk−1T−θminB∫0tεkτTεkτdτ+nTqΔk.
Lett=T. On the basis of Equation (29), we acquire(30)VkT≤Vk−1T−θminB∫0TεkτTεkτdτ+nTqΔk.
Equation (30) and Lemma 2 yield(31)VkT≤V1T−θminB∑j=2k∫0TεjτTεjτdτ+nTq∑j=2kΔj,≤V1T−θminB∑j=2k∫0Tεjτεjτdτ+2nTqc≤V1T+2nTqc.
In view of Equation (31), the boundedness of VkT is guaranteed for any iteration provided that V1T is bounded. Appendix A shows the boundedness of V1t.
The boundedness implies thatV1T is bounded. Hence, VkT is bounded from Equation (31) for k∈Z+. Equation (29) indicates that Vkt is uniformly bounded over the interval 0,T.
On the basis of Equation (31),(32)VkT≤V1T−θminB∑j=2k∫0TεjτTεjτdτ+2nTqc.
Series∑j=2k∫0Tεjτεjτdτ is convergent due to the positiveness of VkT and boundedness of V1T. Hence, limk⟶∞∫0TεkτTεkτ=0 can be easily obtained. Accordingly, by adopting the Barbalat-like lemma [34], we have limk⟶∞εkt=0 uniformly over the interval 0,T. Therefore, limk⟶∞ekt=0 based on Equation (7) suggests that limk⟶∞eikt=0 is uniform over the interval 0,T for i=1,…,n. Thus, we have limk⟶∞xikt=x0t for i=1,…,n, indicating that all following agents can uniformly track the trajectory of the leader over the interval 0,T. The proof is completed.


## 5. Simulation Analysis

A numerical example is provided in this section to illustrate the effectiveness of the proposed control protocol (11). Figure 1 presents the undirected communication topology consisting of five following agents and one leader (labeled as 0).Figure 1
Communication topology.The weighted adjacency matrices from Figure1 are as follows: A=0100010100010110010000100, and ℬ=diag0,1,0,0,0. The dynamics of the five following agents are described as(33)x˙1kt=x1ktsinx1kt+satu1kt+d1kt,x˙2kt=sinx2kt+satu2kt+d2kt,x˙3kt=x3ktcosx3kt+satu3kt+d3kt,x˙4kt=sinx4ktcosx4kt+satu4kt+d4kt,x˙5kt=cosx5kt1−sinx5kt+satu5kt+d5kt,where the disturbance of the ith following agent is dikt=a1sina2ti=1,2,3,4,5, where a1 and a2 are arbitrary positive real numbers and a1,a2∈0,2.The dynamics of the leader agent are given asx˙0t=x0tsinx0t−2cos2πt.Let the initial states of the leader agent and five following agents bex00=0.1, x10=−0.5, x20=1.0, x30=−0.7, x40=0.6, and x50=1.2, respectively. The simulation time t∈0,2 and iteration number kmax=50. Other parameters are selected as follows: c=1, m=2, u¯=5, α=1.5, and ρ0=3; λ1=0.5, λ2=1.5, λ3=1.0, λ4=2.0, and λ5=2.5; and υ0=0.1,0.1,0.1,0.1,0.1T and ρ0=0.2,0.2,0.2,0.2,0.2T.Figures2–4 display the simulation results for 50 iterations considering the adaptive iterative learning control protocol (11) and adaptive updating laws (12) and (13).Figure 2
Tracking trajectories of five agents.Figure 3
Maximum absolute errors of five agents.Figure 4
Saturation control inputs of the last iteration.From Figure3, the control inputs are constrained, but the tracking problem is solved. From another perspective, the designed control protocol is implied to be effective. Figure 4 depicts the response of saturation inputs at the 50th iteration.A comparative simulation result is presented to further clarify the effectiveness of the design method. Considering the designed control protocol of this paper (Scheme 1), the control law of [7] (Scheme 2) and the control law without input saturation (Scheme 3) are presented. The simulation results are shown in Figure 5.Figure 5
Comparison results with other approaches. (a), (b), (c), (d), and (e).
(a)(b)(c)(d)(e)Figure5 indicates that the tracking control problem can be solved using Schemes 2 and 3. However, compared with the control protocol of this paper (Scheme 1), the control protocol designed in this article can obtain better control effects. Despite the fact that the control input of the multiagent system is saturated, better control effects can be achieved by applying the proposed control protocol.Remark 4.
In the process of simulation analysis, the undirected graph is considered in our paper. In fact, similar results can also be obtained applying the undirected graph; only the analysis process is different. Through the simulation analysis, it is not difficult to find that the iterative learning control method has better advantages in solving complex nonlinear problems. Therefore, some complex control issues with iterative learning control method will be considered in our follow-up work.Remark 5.
In the simulation analysis part, the alignment initial condition, that is,xik0=x00,i=1,2,…,n for xik0=x00,i=1,2,…,n, is considered. In fact, the same initial state and different initial states can also be considered, that is, xik0=x00,i=1,2,…,n or xik0≠x00,i=1,2,…,n, which are also the direction and measure for improvement. Although the paper did not analyze the other two situations, similar results can be obtained by applying the redesigned control strategy.


## 6. Conclusion

In this work, an adaptive iterative learning control is designed for high-accuracy tracking of input saturated nonlinear multi-intelligence systems. A new Lyapunov function is constructed and simulated, which strictly proves that the tracker can accurately monitor the leader’s trajectory in time and iteration domains. Even under the conditions of input saturation and system nonlinearity, the proposed control protocol can ensure that each follower agent can track accurately the leader’s trajectory. In future work, we will study the high-precision tracking problem of input saturated high-order nonlinear multiagent adaptive iteration system.



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