# RBFNN-Based Nonuniform Trajectory Tracking Adaptive Iterative Learning Control for Uncertain Nonlinear System with Continuous Nonlinearly Input

Authors: Chunli Zhang; Xu Tian; Lei Yan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper proposes an adaptive iterative learning control (AILC) method for uncertain nonlinear system with continuous nonlinearly input to solve different target tracking problem. The method uses the radial basis function neural network (RBFNN) to approximate every uncertain term in systems. A time-varying boundary layer, a typical convergent series are introduced to deal with initial state error and unknown bounds of errors, respectively. The conclusion is that the tracking error can converge to a very small area with the number of iterations increasing. All closed-loop signals are bounded on finite-time interval0,T. Finally, the simulation result of mass-spring mechanical system shows the correctness of the theory and validity of the method.


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## Body

## 1. Introduction

The research of the nonuniform trajectory is an interesting problem. Two new AILC methods for first-order hybrid parametric systems and high-order nonlinear hybrid parameter systems were proposed in the literature studies [1, 2], respectively. Recently, AILC was presented, the literature [3] proposed a nonuniform target tracking AILC method, and the literature [4] proposed a fault-tolerant ILC technique for mobile robot nonrepetitive target tracking with output constraints. An ILC for a flapping wing micro aerial vehicle under distributed disturbances was proposed in the literature [5]. It can be seen from the above literature studies that solving the nonuniform target tracking problem for uncertain nonlinear systems is an important problem.Adaptive control is used to handle system control problem about uncertainties. Adaptive control schemes learn uncertainties by adaptive laws. NN and FLS are used in the method as function approximators, for example, the paper [6, 7]. The literature [8] could complete the varying control tasks by designing an adaptive fuzzy ILC for the uncertain nonlinear system. Based on RBF neural network approximation, the paper [9] proposed AILC for nonlinear pure-feedback systems to solve the nonuniform target tracking problem. The uniform AILC frame for uncertain nonlinear system was proposed in the literature [10], by Lyapunov theory, and it could prove the convergence. It should be noted that Lyapunov function-based AILC played an important role in dealing with the time-varying parameter [8, 10, 11]. However, initial state error problem is a challenging one as they need to converge to zero for keeping stability. The literature studies [8, 9, 12] considered this problem recently. It is an important problem for AILC.Because the actuator’s physical is limited, the control input widely exists continuous nonlinearly such that system performance can be deteriorated. In the literature studies [13–22], control performance could change by using different technique recently. The literature [15] solved the question of adaptive stabilization for the time-delay system. The literature [16] developed an adaptive backstepping method of uncertain nonlinear systems about nonsymmetric dead-zone. As yet, there is no report from the literature for the AILC of nonlinear systems with continuous nonlinearly input and initial state error. This is a problem that needs to be solved urgently.In this paper, the nonuniform trajectory tracking issue is discussed for the uncertain nonlinear systems with continuous nonlinearly input and initial state error. The contributions of the proposed control method are presented as follows:(i)
The nonuniform trajectory tracking issue is studied for uncertain nonlinear systems under continuous nonlinearly input and initial state error issues.(ii)
The AILC method is used to uncertain nonlinear systems. The RNFNN is introduced to learn unknown dynamic. A convergence order is introduced to solve the unknown bound and nonuniform target tracking problem.Finally, simulation results of the mass-spring mechanical system are given to verify the validity of the designed controller.This paper is organized as follows: in Section2, the system description and related concepts are given in detail. The main results are presented in Section 3. A simulation is shown in Section 4. Section 5 is the conclusion.


## 2. System Description and Related Concepts

### 2.1. System Model

The following nonlinear systems are considered:(1)x˙1,k=x2,k+f1x¯1,k,x˙j,k=xj+1,k+fjx¯j,k,x˙n,k=Nuk+fnx¯n,k,yk=x1,k,where x¯j,k=x1,k,…,xj,kT∈Rj, and x=x¯n is the state that is measured. Nuk∈R represents the actuator characteristics, and yk∈R is the system output. fjx¯j,k, j=1,2,…,n, are smooth unknown nonlinear functions.Nuk represents continuous nonlinearly input with N0=0 which belongs to g1,g2, i.e.,(2)g1uk≤Nuk≤g2uk,ifuk≥0,g1uk≥Nuk≥g2uk,ifuk≤0.Assumption 1.
g1 and g2 are unknown nonzero positive parameters.Remark 1.
This is a reasonable assumption, because many system constraints can satisfy this condition.
Designing an AILC lawukt on 0,T to make the output ykt following the target trajectory yr,kt is the control objective of this paper; that is to say, limk⟶∞ykt−yr,kt≤ϱ, where ϱ is a very small positive number. All closed-loop signals are guaranteed to be bounded. yr,kt is the smooth desired target. k is the iteration index.


### 2.2. Convergent Series Sequence

The following definition and lemma are used in the controller design process.Definition 1 (see [19]).
Δk is a series convergent sequence which is shown as(3)Δk=akl,where k=1,2,…, a and l satisfies a>0∈R, l≥2∈N.Lemma 1 (see [19]).
For1/kl, where k=1,2,…,l≥2, the following inequality holds:(4)limk⟶∞∑j=1k1jl≤2.


### 2.3. Description of RBFNN Approximation

(see [23])(5)Fjx¯j,k=WTζjx¯j,k+δjx¯j,k,where ζjx¯j,k=s1x¯j,k,…,slx¯j,kT:Ω⟶Rm are known, and l>1 is the number of NN node. sjx¯j,k1≤j≤l are Gaussian basis functions, sjx¯j,k=exp−x¯j,k−μj2/η2, μj∈Ω, and η>0 are the center and the width of sjx¯j,k, respectively. The truth weight W=w1,…,wlT is given as follows:(6)W=argminW^∈Rlsupx¯j,k∈ΩFjx¯j,k−W^Tζx¯j,k,and δx¯j,k are the NN approximation errors. The following assumption holds.Assumption 2.
On a compact setΩ, δjx¯j,k are bounded and δjx¯j,k≤θj, where θj1≤j≤n are unknown and θj≥0.Remark 2.
This is a general assumption for the approximation error of the neural network. The approximation error can be arbitrarily decreased by increased NN node number. So, this assumption is reasonable. The following assumptions4−j,j=1,2,…,n are also reasonable.


### 2.4. Time-Varying Boundary Layer

The following functionszjϕ,k are introduced to deal with initial state errors:(7)zjϕ,k=zj,k−ϕj,ktsatzj,kϕj,kt,ϕj,kt=εj,ke−ηt,where zj,k and zjϕ,k are variables of t, εj,k are sequences of convergent series, and the saturation function sat is given as follows:(8)satzj,kϕj,kt=1,ifzj,k>ϕj,kt,zj,kϕj,kt,ifzj,k≤ϕj,kt,−1,ifzj,k<−ϕj,kt,where ϕj,kt are the time-varying boundary layers. The control objective can be achieved by this. See the literature [8] for specific analysis.Assumption 3.
The initial state errors must satisfyzj,k0=εj,k for some known positive parameters εj,k, j=1,…,n, where zj,k0 are initial state errors at each iteration of state errors which are given later in this paper.Remark 3.
Under normal circumstances, the initial state errors at each iteration are zero, small, or fixed. Compared with the assumptions of the initial state in other articles, the assumptions in this paper are more relaxed and easy to be satisfied.


## 2.1. System Model

The following nonlinear systems are considered:(1)x˙1,k=x2,k+f1x¯1,k,x˙j,k=xj+1,k+fjx¯j,k,x˙n,k=Nuk+fnx¯n,k,yk=x1,k,where x¯j,k=x1,k,…,xj,kT∈Rj, and x=x¯n is the state that is measured. Nuk∈R represents the actuator characteristics, and yk∈R is the system output. fjx¯j,k, j=1,2,…,n, are smooth unknown nonlinear functions.Nuk represents continuous nonlinearly input with N0=0 which belongs to g1,g2, i.e.,(2)g1uk≤Nuk≤g2uk,ifuk≥0,g1uk≥Nuk≥g2uk,ifuk≤0.Assumption 1.
g1 and g2 are unknown nonzero positive parameters.Remark 1.
This is a reasonable assumption, because many system constraints can satisfy this condition.
Designing an AILC lawukt on 0,T to make the output ykt following the target trajectory yr,kt is the control objective of this paper; that is to say, limk⟶∞ykt−yr,kt≤ϱ, where ϱ is a very small positive number. All closed-loop signals are guaranteed to be bounded. yr,kt is the smooth desired target. k is the iteration index.


## 2.2. Convergent Series Sequence

The following definition and lemma are used in the controller design process.Definition 1 (see [19]).
Δk is a series convergent sequence which is shown as(3)Δk=akl,where k=1,2,…, a and l satisfies a>0∈R, l≥2∈N.Lemma 1 (see [19]).
For1/kl, where k=1,2,…,l≥2, the following inequality holds:(4)limk⟶∞∑j=1k1jl≤2.


## 2.3. Description of RBFNN Approximation

(see [23])(5)Fjx¯j,k=WTζjx¯j,k+δjx¯j,k,where ζjx¯j,k=s1x¯j,k,…,slx¯j,kT:Ω⟶Rm are known, and l>1 is the number of NN node. sjx¯j,k1≤j≤l are Gaussian basis functions, sjx¯j,k=exp−x¯j,k−μj2/η2, μj∈Ω, and η>0 are the center and the width of sjx¯j,k, respectively. The truth weight W=w1,…,wlT is given as follows:(6)W=argminW^∈Rlsupx¯j,k∈ΩFjx¯j,k−W^Tζx¯j,k,and δx¯j,k are the NN approximation errors. The following assumption holds.Assumption 2.
On a compact setΩ, δjx¯j,k are bounded and δjx¯j,k≤θj, where θj1≤j≤n are unknown and θj≥0.Remark 2.
This is a general assumption for the approximation error of the neural network. The approximation error can be arbitrarily decreased by increased NN node number. So, this assumption is reasonable. The following assumptions4−j,j=1,2,…,n are also reasonable.


## 2.4. Time-Varying Boundary Layer

The following functionszjϕ,k are introduced to deal with initial state errors:(7)zjϕ,k=zj,k−ϕj,ktsatzj,kϕj,kt,ϕj,kt=εj,ke−ηt,where zj,k and zjϕ,k are variables of t, εj,k are sequences of convergent series, and the saturation function sat is given as follows:(8)satzj,kϕj,kt=1,ifzj,k>ϕj,kt,zj,kϕj,kt,ifzj,k≤ϕj,kt,−1,ifzj,k<−ϕj,kt,where ϕj,kt are the time-varying boundary layers. The control objective can be achieved by this. See the literature [8] for specific analysis.Assumption 3.
The initial state errors must satisfyzj,k0=εj,k for some known positive parameters εj,k, j=1,…,n, where zj,k0 are initial state errors at each iteration of state errors which are given later in this paper.Remark 3.
Under normal circumstances, the initial state errors at each iteration are zero, small, or fixed. Compared with the assumptions of the initial state in other articles, the assumptions in this paper are more relaxed and easy to be satisfied.


## 3. Adaptive Iterative Learning Controller Design and Convergence Analysis

In this section, we will discuss the detailed controller design process, main conclusion, and specific proof process.


### 3.1. Adaptive Iterative Learning Controller Design

The following Lemma2 is used in the controller design process.Lemma 2.
For the controlleruk and the error znϕ,k, the following inequality holds:(9)znϕ,kNuk≤gznϕ,kuk,g∈g1,g2.Proof.
If both sides of equation (2) multiply uk, we obtain(10)g1uk2≤ukNuk≤g2uk2.
If both sides of equation (10) multiply znϕ,k2, we obtain(11)g1uk2znϕ,k2≤ukNukznϕ,k2≤g2uk2znϕ,k2,i.e.,(12)g1znϕ,kuk2≤znϕ,kznϕ,kukNuk≤g2znϕ,kuk2.
Ifuk satisfies znϕ,kuk≤0, then(13)znϕ,kNuk≤g1znϕ,kuk.
Ifuk satisfies znϕ,kuk≥0, then(14)znϕ,kNuk≤g2znϕ,kuk.
So, the result is obtained.
The specific process about designing controller is given as follows.Step 1.
Letz1,k=x1,k−yr,k, z2,k=x2,k−α1,k, F1x¯1,k=f1x¯1,k, α1,k is the virtual controller. Introduce error function z1ϕ,k and z2ϕ,k by Section 2.4 to deal with initial state error as(15)z1ϕ,k=z1,k−ϕ1,ktsatz1,kϕ1,kt,ϕ1,kt=ϵ1,ke−η1t;z2ϕ,k=z2,k−ϕ2,ktsatz2,kϕ2,kt,ϕ2,kt=ϵ2,ke−η2t.
Recall that(16)x˙1,k=x2,k+F1x¯1,k.
The derivative ofz1ϕ,k is as follows:(17)z˙1ϕ,k=z2,k+α1,k+F1x¯1,k−y˙r,k−sgnz1ϕ,kϕ˙1,k.
According to Section2.3, by RBFNN, F1x¯1,k is approximated and approximate error δ1x¯1,k is as follows:(18)F1x¯1,k=W1Tζ1x¯1,k+δ1x¯1,k,where W1 is optimal weight vector.
DenoteN1=ωM12, which is needed later, Δk=a/kl, a>0, and l≥2. The virtual controller is taken as(19)α1,k=−W^1,kTζ1x¯1,k−N^1,k1Δkz1ϕ,k+y˙r,k−η1z1,k.
When (18) and (19) are substituted into (17), we have(20)z˙1ϕ,k=z2,k−N^1,k1Δkz1ϕ,k+W1Tζ1x¯1,k+δ1x¯1,k−W^1,kTζ1x¯1,k−η1z1,k−sgnz1ϕ,kϕ˙1,kt=z2ϕ,k−N^1,k1Δkz1ϕ,k−W˜1,kTζ1x¯1,k+δ1x¯1,k+ϕ2,ktsatz2,kϕ2,kt−η1z1,k−sgnz1ϕ,kϕ˙1,kt,where estimated W1 and N1 are W^1,k and N^1,k, respectively. W˜1,k=W^1,k−W1 and N˜1,k=N^1,k−N1 are the errors of estimated parameters. The last two terms of (20) can be changed to(21)−η1z1,k−sgnz1ϕ,kϕ˙1,kt=−η1z1ϕ,k−η1ϕ1,ktsatz1,kϕ1,kt−sgnz1ϕ,kϕ˙1,kt=−η1z1ϕ,k.
By (20), (21) can be rewritten as(22)z˙1ϕ,k=z2ϕ,k−N^1,k1Δkz1ϕ,k−η1z1ϕ,k+W˜1Tζ1x¯1,k+δ1x¯1,k+ϕ2,ktsatz2,kϕ2,kt.
Letω1=δ1x¯1,k+ϕ2,ktsatz2,kt/ϕ2,kt, then (22) becomes(23)z˙1ϕ,k=z2ϕ,k−N^1,k1Δkz1ϕ,k−η1z1ϕ,k+W˜1Tζ1x¯1,k+ω1.Assumption 4.
1: the bounded termω1 satisfies ω1≤ωM1, where ωM1 is a positive parameter.
Take the following nonnegative function:(24)V1,k=12z1ϕ,k2+12W˜1,kTΓ11−1W˜1,k+12Γ21−1N˜1,k2,where Γ11 and Γ21 are the symmetric positive definite matrixes. The derivative of V1,k along (23) is as follows:(25)V˙1,k=z1ϕ,kz2ϕ,k−η1z1ϕ,k2−W˜1,kTΓ11−1Γ11ζ1x¯1,kz1ϕ,k−W^˙1,k−N^1,k1Δkz1ϕ,k2+ω1,kz1ϕ,k+Γ21−1N˜1,kN^˙1,k≤z1ϕ,kz2ϕ,k−η1z1ϕ,k2−W˜1,kTΓ11−1Γ11ζ1x¯1,kz1ϕ,k−W^˙1,k−N^1,k1Δkz1ϕ,k2+1ΔkωM12z1ϕ,k2+14Δk+Γ21−1N˜1,kN^˙1,k=z1ϕ,kz2ϕ,k−η1z1ϕ,k2−W˜1,kTΓ11−1Γ11ζ1x¯1,kz1ϕ,k−W^˙1,k−N˜1,kΓ21−1Γ211Δkz1ϕ,k2−N^˙1,k+14Δk.
In previous equation,mn≤1/rm2+1/4n2r is used, where r=Δk.Step 2.
j: 2≤j≤n−1. Denote Fjx¯j,k=fjx¯j,k−∑l=1j−1∂αj−1,k/∂xl,kx˙l,kx¯l,k, Nj=ωMj2, which is given later. zj+1,k=xj+1,k−αj,k, similar to Step 1, and zjϕ,k and zj+1ϕ,k by Section 2.4 are introduced as follows:(26)zjϕ,k=zj,k−ϕj,ktsatzj,kϕj,kt,ϕj,kt=εj,ke−ηjt,zj+1ϕ,k=zj+1,k−ϕj+1,ktsatzj+1,kϕj+1,kt,ϕj+1,kt=εj+1,ke−ηj+1t.
The derivative ofzjϕ,k is as follows:(27)z˙jϕ,k=zj+1,k+αj,k+fjx¯j,k−α˙j−1,k−sgnzjϕ,ktϕ˙j,k,
whereα˙j−1,k=∑l=1j−1∂αj−1,k/∂xl,kxl+1,k+flx¯l,k+∂αj−1,k/∂W^j−1,kW^˙j−1,k+∂αj−1,k/∂N^j−1,kN^˙j−1,k+αj−1,k/∂t. Denote Fjx¯j,k=fjx¯j,k−∑l=1j∂αj−1,k/∂xj,kxj+1,k+fjx¯j,k, Pj−1,k=∂αj−1,k/∂W^j−1,kW^˙j−1,k+∂αj−1,k/∂N^j−1,kN^˙j−1,k+αj−1,k/∂t, then (27) can be rewritten as(28)z˙jϕ,k=zj+1,k+αj,k+Fjx¯j,k−Pj−1,k−sgnzjϕ,ktϕ˙j,k.
According to Section2.3, Fjx¯j,k can become by the RBFNN(29)Fjx¯j,k=WjTζjx¯j,k+δjx¯j,k,where δjx¯j,k are the approximation errors and Wj are optimal weight vectors.
The virtual controller is chosen as follows:(30)αj,k=−zj−1ϕ,k−W^j,kTζjx¯j,k−N^j,k1Δkzjϕ,k+Pj−1,k−ηjzj,k.
When (29) and (30) are substituted into (28), we have(31)z˙jϕ,k=zj+1,k−zj−1ϕ,k−N^j,k1Δkzjϕ,k−ηjzj,k−sgnzjϕ,ktϕ˙j,k+WjTζjx¯j,k+δjx¯j,k−W^j,kTζjx¯j,k=zj+1,k−zj−1ϕ,k−N^j,k1Δkzjϕ,k−W˜j,kTζjx¯j,k+δjx¯j,k−ηjzj,k−sgnzjϕ,kϕ˙j,k=zj+1ϕ,k−zj−1ϕ,k−N^j,k1Δkzjϕ,k−W˜j,kTζjx¯j,k+δjx¯j,k+ϕj+1ϕ,ktsatzj+1ϕ,kϕj+1,kt−ηjzj,k−sgnzjϕ,ktϕ˙j,k,where W^j,k and N^j,k are the estimated parameters of Wj and Nj, respectively. W˜j,k=W^j,k−Wj and N˜j,k=N^j,k−Nj are estimated parameter errors. The last two terms of (31) can be changed as(32)−ηjzj,k−sgnzjϕ,kϕ˙j,kt=−ηjzjϕ,k−ηjϕj,ktsatzj,kϕj,kt−sgnzjϕ,kϕ˙j,kt=−ηjzjϕ,k.
Letωj=δjx¯j,k+ϕj+1ϕ,ktsatzj+1ϕ,k/ϕj+1,kt, then (31) becomes(33)z˙jϕ,k=zj+1ϕ,k−zj−1ϕ,k−ηjzjϕ,k−N^j,k1Δkzjϕ,k+W˜j,kTζjx¯j,k+ωj.Assumption 5.
j: ωj satify ωj≤ωMj, here ωMj is unknown.
The following positive definite function is chosen:(34)Vj,k=Vj−1,k+12zjϕ,k2+12W˜j,kTΓ1j−1W˜j,k+12Γ2j−1N˜j,k2.
The derivative ofVj,k is as follows by substituting (33):(35)V˙j,k≤zjϕ,kzj+1ϕ,k−∑l=1jηlzl,k2−∑l=1jW˜l,kTΓ1l−1Γ1lζjx¯l,kzlϕ,k−W^˙l,k−∑l=1jN˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+j14Δk.Step 3.
n, define zn,k=xn,k−αn−1,k, znϕ,k by Section 2.4 is introduced as follows:(36)znϕ,k=zn,k−ϕn,ktsatzn,kϕn,kt,ϕn,kt=εn,ke−ηnt.
The derivative ofznϕ,k is as follows:(37)z˙nϕ,k=Nuk+fx¯n,k−α˙n−1,k−sgnznϕ,kϕ˙n,kt,where α˙n−1,k=∑l=1n−1∂αn−1,k/∂xl,kxl+1,k+flx¯l,k+∂αn−1,k/∂W^n−1,kW^˙n−1,k+∂αn−1,k/∂N^n−1,kN^˙n−1,k+αn−1,k/∂t. Denote Fnx¯n,k=fnx¯n,k−∑l=1n∂αn−1,k/∂xl,kxl+1,k+flx¯l,k, Pn−1,k=∂αn−1,k/∂W^n−1,kW^˙n−1,k+∂αn−1,k/∂N^n−1,kN^˙n−1,k+αn−1,k/∂t, and then (37) can be rewritten as follows:(38)z˙nϕ,k=Nuk+Fx¯n,k−Pn−1,k−sgnznϕ,kϕ˙n,kt.
Fnx¯n,k can be rewritten by RBFNN approximation as follows:(39)Fnx¯n,k=WnTζnx¯n,k+δnx¯n,k,where δnx¯n,k is the approximation error and Wn is an optimal weight vector.
Then, (38) can be rewritten by substituting (39):(40)z˙nϕ,k=Nuk+WnTζnx¯n,k+δnx¯n,k−Pn−1,k−sgnznϕ,kϕ˙n,kt.
Lyapunov function is chosen as follows:(41)Vn,k=Vn−1,k+12znϕ,k2+12W˜n,kTΓ1n−1W˜n,k+12Γ2n−1N˜n,k2+12Γ4−1gϱ˜k2,where estimated Wn, Nn and ϱ=1/g are W^n,k, N^n,k and ϱ^kt, respectively. W˜n,k=W^n,k−Wn, N˜n,k=N^n,k−Nn, and ϱ˜kt=ϱ^kt−ϱ are parameter estimation errors. And Γ4 is learning gain to be designed.
Using the derivative ofVn,k along (40), then we have(42)V˙n,k≤zn−1ϕ,kznϕ,k−∑l=1n−1ηlzlϕ,k2−∑l=1n−1W˜l,kTΓ1l−1Γ1lζlx¯l,kzlϕ,k−W^˙l,k−∑l=1n−1N˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n−114Δk+znϕ,kz˙nϕ,k+W˜n,kTΓ1n−1W^˙n,k+Γ2n−1N˜n,kN^˙n,k+Γ4−1gϱ˜kϱ^˙k≤zn−1ϕ,kznϕ,k−∑l=1n−1ηjzlϕ,k2−∑l=1n−1W˜l,kTΓ1l−1Γ1lζlx¯l,kzlϕ,k−W^˙l,k−∑l=1n−1N˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n−114Δk+uk+g−1uk+WnTζnx¯n,k+δnx¯n,k−Pn−1,k−sgnznϕ,kϕ˙n,ktznϕ,k+W˜n,kTΓ1n−1W^˙n,k+Γ2n−1N˜n,kN^˙n,k+Γ4−1gϱ˜kϱ^˙k.
Chooseu1,k=−zn−1ϕ,k−W^n,kTζnx¯n,k−N^n,k1/Δkzn,k+Pn−1,k−ηnzn,k.
Take the actual controller as follows:(43)uk=u1,k+u2,k,where u2,k is needed to design for dealing with g.
Then, (42) becomes(44)V˙n,k≤zn−1ϕ,kznϕ,k−∑l=1n−1ηlzlϕ,k2−∑l=1n−1W˜l,kTΓ1l−1Γ1lζlx¯l,kzlϕ,k−W^˙l,k−∑l=1n−1N˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n−114Δk+−zn−1ϕ,k−W˜n,kTζnx¯n,k−N^n,k1Δkzn,k−ηnzn,k+g−1u1,k+u2,k+u2,k+δnx¯n,k−sgnznϕ,kϕ˙n,ktznϕ,k+W˜n,kTΓ1n−1W^˙n,k+Γ2n−1N˜n,kN^˙n,k+Γ4−1gϱ˜kϱ^˙k.
Whenωn=δnx¯n,k, the following equation holds:(45)−ηnzn,k−sgnznϕ,kϕ˙n,kt=−ηnznϕ,k−ηnϕn,ktsatzn,kϕn,kt−sgnznϕ,kϕ˙n,kt=−ηnznϕ,k,and then we have(46)V˙n,k≤−∑l=1nηlzlϕ,k2−∑l=1n−1W˜l,kTΓ1l−1Γ1lζlx¯l,kzlϕ,k−W^˙l,k−∑l=1n−1N˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n−114Δk+−W˜n,kTζnx¯n,k−N^n,k1Δkznϕ,k+gu2,k+g−1u1,k+ωnznϕ,k+W˜n,kTΓ1n−1W^˙n,k+Γ2n−1N˜n,kN^˙n,k+Γ4−1gϱ˜kϱ^˙k.
Takeu2,k=−satzn,k/ϕn,ktϱ^kt+1u1,k, and then (46) becomes(47)V˙n,k≤−∑l=1nηlzlϕ,k2−∑l=1n−1W˜l,kTΓ1l−1Γ1lζlx¯l,kzlϕ,k−W^˙l,k−∑l=1n−1N˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n−114Δk+−W˜n,kTζnx¯n,k−N^n,k1Δkznϕ,k+ωn−gsatzn,kϕn,ktϱ^kt+1u1,k+g−1u1,k+W˜n,kTΓ1n−1W^˙n,k+Γ2n−1N˜n,kN^˙n,k+Γ4−1gϱ˜kϱ^˙k.Assumption 6.
n: bounded ωn satisfies ωn≤ωMn, here ωMn is an unknown parameter.
Choose the AILC laws as follows:(48)W^˙j,k=Γ1jζjx¯j,kziϕ,k,j=1,…,m,(49)N^˙j,k=Γ2j1Δkzjϕ,k2,j=1,…,m,(50)ϱ^˙kt=Γ4u1,kznϕ,k,and then we obtain(51)V˙n,k≤−∑l=1nηlzlϕ,k2−∑l=1nW˜l,kTΓ1l−1Γ1lζlx¯l,kzlϕ,k−W^˙l,k−∑l=1nN˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n14Δk−Γ4−1gϱ˜kΓ4u1,kznϕ,k−ϱ^˙k=−∑l=1nηlzl,k2+n14Δk.


### 3.2. Convergence Analysis

According to the above design process, we can get the following conclusion expressed by Theorem1. The following assumption is needed.Assumption 7.
For anyk, as t=0, W^j,k0=W^j,k−1T; N^j,k0=N^j,k−1T,j=1,…,m, ϱ^k0=ϱ^k−1T.Remark 4.
This is the general setting of initial state in ILC.Theorem 1.
Under Assumptions1–7, by controller (43) and adaptive parameter estimation laws (48)–(50) for system (1), we get all signals are bounded, and(52)limk⟶∞zlϕ,kt=0,l=1,2,…,n,i.e., limk⟶∞z1ϕ,kt=0, z1,kt<ϕ1,∞t+ς as k⟶∞, where ς is a small positive parameter.Proof.
By Section2.4, zϕ,k02=0≤zϕ,kT2, where zϕ,k=z1ϕ,k,z2ϕ,k,…,znϕ,kT. By (41), we get(53)Vn,kzϕ,k0,W^kT,N^kT,ϱ^kT≤Vn,k0+∫0TV˙n,kdt,where W^k=W^1,k,W^2,k,…,N^n,kT and N^k=N^1,k,N^2,k,…,N^n,kT.
(51) is substituted into (53), and we get(54)Vn,kzϕ,k0,W^kT,N^kT,ϱ^kT≤Vn,10−∑j=1k∑l=1m∫0Tηlzlϕ,j2dt+n14T∑j=1kΔj.
DenoteV0,k=Vn,10+n1/4T∑j=1kΔj, and then (54) can be rewritten as(55)∑l=1k∑l=1n∫0Tηlzlϕ,j2dt≤V0,k−Vn,kzϕ,k0,W^kT,N^kT,ϱ^kT.
By (4), limk⟶∞V0,k≤Vn,10+2an1/4T, then V0,k is bounded, and by Assumption 7 and (54), Vn,k0 is bounded. Vn,kzk0,W^kT,N^kT,ϱ^kT≥0, so(56)limk⟶∞∑l=1n∫0Tηlzlϕ,k2dt=0.
By (41), for any k, Vn,kt=Vn,k0+∫0tV˙n,kτdτ, by (51), then(57)Vn,kt≤Vn,k0−∑l=1n∫0tηlzlϕ,kτ2dτ+tn14Δk.
By (56), ∑l=1n∫0tηlzlϕ,kτ2dτ is bounded. By Definition 1, Δk is bounded, and t∈0,T, tn1/4Δk is also bounded. From above all, for any k, Vn,kt is bounded, then we have xj,k, and W^kt and N^kt are bounded. By (43), uk is bounded. By (33), z˙jϕ,k is bounded, zjϕ,k is continuous uniformly, according to Barbalat lemma, zjϕ,k⟶0, when k⟶∞, i.e., k⟶∞, z1ϕ,k⟶0, then z1ϕ,k<ς. z1,k=z1ϕ,k+sgnz1ϕ,kϕ1,kt≤z1ϕ,k+ϕ1,kt<ϕ1,kt+ς, so k⟶∞, z1,k<ϕ1,∞t+ς, and the proof is finished.


## 3.1. Adaptive Iterative Learning Controller Design

The following Lemma2 is used in the controller design process.Lemma 2.
For the controlleruk and the error znϕ,k, the following inequality holds:(9)znϕ,kNuk≤gznϕ,kuk,g∈g1,g2.Proof.
If both sides of equation (2) multiply uk, we obtain(10)g1uk2≤ukNuk≤g2uk2.
If both sides of equation (10) multiply znϕ,k2, we obtain(11)g1uk2znϕ,k2≤ukNukznϕ,k2≤g2uk2znϕ,k2,i.e.,(12)g1znϕ,kuk2≤znϕ,kznϕ,kukNuk≤g2znϕ,kuk2.
Ifuk satisfies znϕ,kuk≤0, then(13)znϕ,kNuk≤g1znϕ,kuk.
Ifuk satisfies znϕ,kuk≥0, then(14)znϕ,kNuk≤g2znϕ,kuk.
So, the result is obtained.
The specific process about designing controller is given as follows.Step 1.
Letz1,k=x1,k−yr,k, z2,k=x2,k−α1,k, F1x¯1,k=f1x¯1,k, α1,k is the virtual controller. Introduce error function z1ϕ,k and z2ϕ,k by Section 2.4 to deal with initial state error as(15)z1ϕ,k=z1,k−ϕ1,ktsatz1,kϕ1,kt,ϕ1,kt=ϵ1,ke−η1t;z2ϕ,k=z2,k−ϕ2,ktsatz2,kϕ2,kt,ϕ2,kt=ϵ2,ke−η2t.
Recall that(16)x˙1,k=x2,k+F1x¯1,k.
The derivative ofz1ϕ,k is as follows:(17)z˙1ϕ,k=z2,k+α1,k+F1x¯1,k−y˙r,k−sgnz1ϕ,kϕ˙1,k.
According to Section2.3, by RBFNN, F1x¯1,k is approximated and approximate error δ1x¯1,k is as follows:(18)F1x¯1,k=W1Tζ1x¯1,k+δ1x¯1,k,where W1 is optimal weight vector.
DenoteN1=ωM12, which is needed later, Δk=a/kl, a>0, and l≥2. The virtual controller is taken as(19)α1,k=−W^1,kTζ1x¯1,k−N^1,k1Δkz1ϕ,k+y˙r,k−η1z1,k.
When (18) and (19) are substituted into (17), we have(20)z˙1ϕ,k=z2,k−N^1,k1Δkz1ϕ,k+W1Tζ1x¯1,k+δ1x¯1,k−W^1,kTζ1x¯1,k−η1z1,k−sgnz1ϕ,kϕ˙1,kt=z2ϕ,k−N^1,k1Δkz1ϕ,k−W˜1,kTζ1x¯1,k+δ1x¯1,k+ϕ2,ktsatz2,kϕ2,kt−η1z1,k−sgnz1ϕ,kϕ˙1,kt,where estimated W1 and N1 are W^1,k and N^1,k, respectively. W˜1,k=W^1,k−W1 and N˜1,k=N^1,k−N1 are the errors of estimated parameters. The last two terms of (20) can be changed to(21)−η1z1,k−sgnz1ϕ,kϕ˙1,kt=−η1z1ϕ,k−η1ϕ1,ktsatz1,kϕ1,kt−sgnz1ϕ,kϕ˙1,kt=−η1z1ϕ,k.
By (20), (21) can be rewritten as(22)z˙1ϕ,k=z2ϕ,k−N^1,k1Δkz1ϕ,k−η1z1ϕ,k+W˜1Tζ1x¯1,k+δ1x¯1,k+ϕ2,ktsatz2,kϕ2,kt.
Letω1=δ1x¯1,k+ϕ2,ktsatz2,kt/ϕ2,kt, then (22) becomes(23)z˙1ϕ,k=z2ϕ,k−N^1,k1Δkz1ϕ,k−η1z1ϕ,k+W˜1Tζ1x¯1,k+ω1.Assumption 4.
1: the bounded termω1 satisfies ω1≤ωM1, where ωM1 is a positive parameter.
Take the following nonnegative function:(24)V1,k=12z1ϕ,k2+12W˜1,kTΓ11−1W˜1,k+12Γ21−1N˜1,k2,where Γ11 and Γ21 are the symmetric positive definite matrixes. The derivative of V1,k along (23) is as follows:(25)V˙1,k=z1ϕ,kz2ϕ,k−η1z1ϕ,k2−W˜1,kTΓ11−1Γ11ζ1x¯1,kz1ϕ,k−W^˙1,k−N^1,k1Δkz1ϕ,k2+ω1,kz1ϕ,k+Γ21−1N˜1,kN^˙1,k≤z1ϕ,kz2ϕ,k−η1z1ϕ,k2−W˜1,kTΓ11−1Γ11ζ1x¯1,kz1ϕ,k−W^˙1,k−N^1,k1Δkz1ϕ,k2+1ΔkωM12z1ϕ,k2+14Δk+Γ21−1N˜1,kN^˙1,k=z1ϕ,kz2ϕ,k−η1z1ϕ,k2−W˜1,kTΓ11−1Γ11ζ1x¯1,kz1ϕ,k−W^˙1,k−N˜1,kΓ21−1Γ211Δkz1ϕ,k2−N^˙1,k+14Δk.
In previous equation,mn≤1/rm2+1/4n2r is used, where r=Δk.Step 2.
j: 2≤j≤n−1. Denote Fjx¯j,k=fjx¯j,k−∑l=1j−1∂αj−1,k/∂xl,kx˙l,kx¯l,k, Nj=ωMj2, which is given later. zj+1,k=xj+1,k−αj,k, similar to Step 1, and zjϕ,k and zj+1ϕ,k by Section 2.4 are introduced as follows:(26)zjϕ,k=zj,k−ϕj,ktsatzj,kϕj,kt,ϕj,kt=εj,ke−ηjt,zj+1ϕ,k=zj+1,k−ϕj+1,ktsatzj+1,kϕj+1,kt,ϕj+1,kt=εj+1,ke−ηj+1t.
The derivative ofzjϕ,k is as follows:(27)z˙jϕ,k=zj+1,k+αj,k+fjx¯j,k−α˙j−1,k−sgnzjϕ,ktϕ˙j,k,
whereα˙j−1,k=∑l=1j−1∂αj−1,k/∂xl,kxl+1,k+flx¯l,k+∂αj−1,k/∂W^j−1,kW^˙j−1,k+∂αj−1,k/∂N^j−1,kN^˙j−1,k+αj−1,k/∂t. Denote Fjx¯j,k=fjx¯j,k−∑l=1j∂αj−1,k/∂xj,kxj+1,k+fjx¯j,k, Pj−1,k=∂αj−1,k/∂W^j−1,kW^˙j−1,k+∂αj−1,k/∂N^j−1,kN^˙j−1,k+αj−1,k/∂t, then (27) can be rewritten as(28)z˙jϕ,k=zj+1,k+αj,k+Fjx¯j,k−Pj−1,k−sgnzjϕ,ktϕ˙j,k.
According to Section2.3, Fjx¯j,k can become by the RBFNN(29)Fjx¯j,k=WjTζjx¯j,k+δjx¯j,k,where δjx¯j,k are the approximation errors and Wj are optimal weight vectors.
The virtual controller is chosen as follows:(30)αj,k=−zj−1ϕ,k−W^j,kTζjx¯j,k−N^j,k1Δkzjϕ,k+Pj−1,k−ηjzj,k.
When (29) and (30) are substituted into (28), we have(31)z˙jϕ,k=zj+1,k−zj−1ϕ,k−N^j,k1Δkzjϕ,k−ηjzj,k−sgnzjϕ,ktϕ˙j,k+WjTζjx¯j,k+δjx¯j,k−W^j,kTζjx¯j,k=zj+1,k−zj−1ϕ,k−N^j,k1Δkzjϕ,k−W˜j,kTζjx¯j,k+δjx¯j,k−ηjzj,k−sgnzjϕ,kϕ˙j,k=zj+1ϕ,k−zj−1ϕ,k−N^j,k1Δkzjϕ,k−W˜j,kTζjx¯j,k+δjx¯j,k+ϕj+1ϕ,ktsatzj+1ϕ,kϕj+1,kt−ηjzj,k−sgnzjϕ,ktϕ˙j,k,where W^j,k and N^j,k are the estimated parameters of Wj and Nj, respectively. W˜j,k=W^j,k−Wj and N˜j,k=N^j,k−Nj are estimated parameter errors. The last two terms of (31) can be changed as(32)−ηjzj,k−sgnzjϕ,kϕ˙j,kt=−ηjzjϕ,k−ηjϕj,ktsatzj,kϕj,kt−sgnzjϕ,kϕ˙j,kt=−ηjzjϕ,k.
Letωj=δjx¯j,k+ϕj+1ϕ,ktsatzj+1ϕ,k/ϕj+1,kt, then (31) becomes(33)z˙jϕ,k=zj+1ϕ,k−zj−1ϕ,k−ηjzjϕ,k−N^j,k1Δkzjϕ,k+W˜j,kTζjx¯j,k+ωj.Assumption 5.
j: ωj satify ωj≤ωMj, here ωMj is unknown.
The following positive definite function is chosen:(34)Vj,k=Vj−1,k+12zjϕ,k2+12W˜j,kTΓ1j−1W˜j,k+12Γ2j−1N˜j,k2.
The derivative ofVj,k is as follows by substituting (33):(35)V˙j,k≤zjϕ,kzj+1ϕ,k−∑l=1jηlzl,k2−∑l=1jW˜l,kTΓ1l−1Γ1lζjx¯l,kzlϕ,k−W^˙l,k−∑l=1jN˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+j14Δk.Step 3.
n, define zn,k=xn,k−αn−1,k, znϕ,k by Section 2.4 is introduced as follows:(36)znϕ,k=zn,k−ϕn,ktsatzn,kϕn,kt,ϕn,kt=εn,ke−ηnt.
The derivative ofznϕ,k is as follows:(37)z˙nϕ,k=Nuk+fx¯n,k−α˙n−1,k−sgnznϕ,kϕ˙n,kt,where α˙n−1,k=∑l=1n−1∂αn−1,k/∂xl,kxl+1,k+flx¯l,k+∂αn−1,k/∂W^n−1,kW^˙n−1,k+∂αn−1,k/∂N^n−1,kN^˙n−1,k+αn−1,k/∂t. Denote Fnx¯n,k=fnx¯n,k−∑l=1n∂αn−1,k/∂xl,kxl+1,k+flx¯l,k, Pn−1,k=∂αn−1,k/∂W^n−1,kW^˙n−1,k+∂αn−1,k/∂N^n−1,kN^˙n−1,k+αn−1,k/∂t, and then (37) can be rewritten as follows:(38)z˙nϕ,k=Nuk+Fx¯n,k−Pn−1,k−sgnznϕ,kϕ˙n,kt.
Fnx¯n,k can be rewritten by RBFNN approximation as follows:(39)Fnx¯n,k=WnTζnx¯n,k+δnx¯n,k,where δnx¯n,k is the approximation error and Wn is an optimal weight vector.
Then, (38) can be rewritten by substituting (39):(40)z˙nϕ,k=Nuk+WnTζnx¯n,k+δnx¯n,k−Pn−1,k−sgnznϕ,kϕ˙n,kt.
Lyapunov function is chosen as follows:(41)Vn,k=Vn−1,k+12znϕ,k2+12W˜n,kTΓ1n−1W˜n,k+12Γ2n−1N˜n,k2+12Γ4−1gϱ˜k2,where estimated Wn, Nn and ϱ=1/g are W^n,k, N^n,k and ϱ^kt, respectively. W˜n,k=W^n,k−Wn, N˜n,k=N^n,k−Nn, and ϱ˜kt=ϱ^kt−ϱ are parameter estimation errors. And Γ4 is learning gain to be designed.
Using the derivative ofVn,k along (40), then we have(42)V˙n,k≤zn−1ϕ,kznϕ,k−∑l=1n−1ηlzlϕ,k2−∑l=1n−1W˜l,kTΓ1l−1Γ1lζlx¯l,kzlϕ,k−W^˙l,k−∑l=1n−1N˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n−114Δk+znϕ,kz˙nϕ,k+W˜n,kTΓ1n−1W^˙n,k+Γ2n−1N˜n,kN^˙n,k+Γ4−1gϱ˜kϱ^˙k≤zn−1ϕ,kznϕ,k−∑l=1n−1ηjzlϕ,k2−∑l=1n−1W˜l,kTΓ1l−1Γ1lζlx¯l,kzlϕ,k−W^˙l,k−∑l=1n−1N˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n−114Δk+uk+g−1uk+WnTζnx¯n,k+δnx¯n,k−Pn−1,k−sgnznϕ,kϕ˙n,ktznϕ,k+W˜n,kTΓ1n−1W^˙n,k+Γ2n−1N˜n,kN^˙n,k+Γ4−1gϱ˜kϱ^˙k.
Chooseu1,k=−zn−1ϕ,k−W^n,kTζnx¯n,k−N^n,k1/Δkzn,k+Pn−1,k−ηnzn,k.
Take the actual controller as follows:(43)uk=u1,k+u2,k,where u2,k is needed to design for dealing with g.
Then, (42) becomes(44)V˙n,k≤zn−1ϕ,kznϕ,k−∑l=1n−1ηlzlϕ,k2−∑l=1n−1W˜l,kTΓ1l−1Γ1lζlx¯l,kzlϕ,k−W^˙l,k−∑l=1n−1N˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n−114Δk+−zn−1ϕ,k−W˜n,kTζnx¯n,k−N^n,k1Δkzn,k−ηnzn,k+g−1u1,k+u2,k+u2,k+δnx¯n,k−sgnznϕ,kϕ˙n,ktznϕ,k+W˜n,kTΓ1n−1W^˙n,k+Γ2n−1N˜n,kN^˙n,k+Γ4−1gϱ˜kϱ^˙k.
Whenωn=δnx¯n,k, the following equation holds:(45)−ηnzn,k−sgnznϕ,kϕ˙n,kt=−ηnznϕ,k−ηnϕn,ktsatzn,kϕn,kt−sgnznϕ,kϕ˙n,kt=−ηnznϕ,k,and then we have(46)V˙n,k≤−∑l=1nηlzlϕ,k2−∑l=1n−1W˜l,kTΓ1l−1Γ1lζlx¯l,kzlϕ,k−W^˙l,k−∑l=1n−1N˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n−114Δk+−W˜n,kTζnx¯n,k−N^n,k1Δkznϕ,k+gu2,k+g−1u1,k+ωnznϕ,k+W˜n,kTΓ1n−1W^˙n,k+Γ2n−1N˜n,kN^˙n,k+Γ4−1gϱ˜kϱ^˙k.
Takeu2,k=−satzn,k/ϕn,ktϱ^kt+1u1,k, and then (46) becomes(47)V˙n,k≤−∑l=1nηlzlϕ,k2−∑l=1n−1W˜l,kTΓ1l−1Γ1lζlx¯l,kzlϕ,k−W^˙l,k−∑l=1n−1N˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n−114Δk+−W˜n,kTζnx¯n,k−N^n,k1Δkznϕ,k+ωn−gsatzn,kϕn,ktϱ^kt+1u1,k+g−1u1,k+W˜n,kTΓ1n−1W^˙n,k+Γ2n−1N˜n,kN^˙n,k+Γ4−1gϱ˜kϱ^˙k.Assumption 6.
n: bounded ωn satisfies ωn≤ωMn, here ωMn is an unknown parameter.
Choose the AILC laws as follows:(48)W^˙j,k=Γ1jζjx¯j,kziϕ,k,j=1,…,m,(49)N^˙j,k=Γ2j1Δkzjϕ,k2,j=1,…,m,(50)ϱ^˙kt=Γ4u1,kznϕ,k,and then we obtain(51)V˙n,k≤−∑l=1nηlzlϕ,k2−∑l=1nW˜l,kTΓ1l−1Γ1lζlx¯l,kzlϕ,k−W^˙l,k−∑l=1nN˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n14Δk−Γ4−1gϱ˜kΓ4u1,kznϕ,k−ϱ^˙k=−∑l=1nηlzl,k2+n14Δk.


## 3.2. Convergence Analysis

According to the above design process, we can get the following conclusion expressed by Theorem1. The following assumption is needed.Assumption 7.
For anyk, as t=0, W^j,k0=W^j,k−1T; N^j,k0=N^j,k−1T,j=1,…,m, ϱ^k0=ϱ^k−1T.Remark 4.
This is the general setting of initial state in ILC.Theorem 1.
Under Assumptions1–7, by controller (43) and adaptive parameter estimation laws (48)–(50) for system (1), we get all signals are bounded, and(52)limk⟶∞zlϕ,kt=0,l=1,2,…,n,i.e., limk⟶∞z1ϕ,kt=0, z1,kt<ϕ1,∞t+ς as k⟶∞, where ς is a small positive parameter.Proof.
By Section2.4, zϕ,k02=0≤zϕ,kT2, where zϕ,k=z1ϕ,k,z2ϕ,k,…,znϕ,kT. By (41), we get(53)Vn,kzϕ,k0,W^kT,N^kT,ϱ^kT≤Vn,k0+∫0TV˙n,kdt,where W^k=W^1,k,W^2,k,…,N^n,kT and N^k=N^1,k,N^2,k,…,N^n,kT.
(51) is substituted into (53), and we get(54)Vn,kzϕ,k0,W^kT,N^kT,ϱ^kT≤Vn,10−∑j=1k∑l=1m∫0Tηlzlϕ,j2dt+n14T∑j=1kΔj.
DenoteV0,k=Vn,10+n1/4T∑j=1kΔj, and then (54) can be rewritten as(55)∑l=1k∑l=1n∫0Tηlzlϕ,j2dt≤V0,k−Vn,kzϕ,k0,W^kT,N^kT,ϱ^kT.
By (4), limk⟶∞V0,k≤Vn,10+2an1/4T, then V0,k is bounded, and by Assumption 7 and (54), Vn,k0 is bounded. Vn,kzk0,W^kT,N^kT,ϱ^kT≥0, so(56)limk⟶∞∑l=1n∫0Tηlzlϕ,k2dt=0.
By (41), for any k, Vn,kt=Vn,k0+∫0tV˙n,kτdτ, by (51), then(57)Vn,kt≤Vn,k0−∑l=1n∫0tηlzlϕ,kτ2dτ+tn14Δk.
By (56), ∑l=1n∫0tηlzlϕ,kτ2dτ is bounded. By Definition 1, Δk is bounded, and t∈0,T, tn1/4Δk is also bounded. From above all, for any k, Vn,kt is bounded, then we have xj,k, and W^kt and N^kt are bounded. By (43), uk is bounded. By (33), z˙jϕ,k is bounded, zjϕ,k is continuous uniformly, according to Barbalat lemma, zjϕ,k⟶0, when k⟶∞, i.e., k⟶∞, z1ϕ,k⟶0, then z1ϕ,k<ς. z1,k=z1ϕ,k+sgnz1ϕ,kϕ1,kt≤z1ϕ,k+ϕ1,kt<ϕ1,kt+ς, so k⟶∞, z1,k<ϕ1,∞t+ς, and the proof is finished.


## 4. Simulation

In this section, a mass-spring mechanical system is considered to show the effectiveness of the proposed controller.m¯ is a mass, and assume that resistive force caused by friction is zero. The external force uk drives the mass, which is a control variable. yk is the displacement from a reference position, and the motion equation of the system with continuous nonlinearly input is as follows:(58)m¯y¨k+Fmsyk=Nuk,where t∈0,π and Fms⋅ is the restoring force of the spring. k denotes the iteration index.We definex1,k=yk, x2,k=y˙k, and m¯=1 which transform (58) into the state-space form(59)x˙1,k=x2,k,x˙2,k=Nuk−Fmsx1,k.The restoring force of the spring can be modeled as(60)Fmsx1,k=kx1,k∑i=0qaix1,ki.In the system, we havek=1, a0=0, a1=a2=a3=a4=1, and q=4.Continuous nonlinearly inputNu is shown as Nu=0.5+0.1sinuu. System objective is that the output of (59) can follow the reference trajectory yr,k on 0,π when k⟶∞. In the different target case, yr,k=gksin2t as k is even, and yr,k=gkcost as k is odd, where gk=rand0,1.By Theorem2, the AILC is chosen as(61)α1,k=−W^1,kTζ1x¯1,k−N^1,k1Δkz1ϕ,k+y˙r−η1z1,k,uk=u1,k+u2,k,u1,k=−z1ϕ,k−W^2,kTζ2x¯2,k−N^2,k1Δkz2,k+P1,k−η2z2,k,u2,k=−satzn,kϕn,ktϱ^kt+1u1,k.where P1,k=∂α1,k/∂W^1,kW^˙1,k+∂α1,k/∂N^1,kN^˙1,k+α1,k/∂yry˙r+α1,k/∂y˙ry¨r and the parameter estimation laws are represent by (48)–(50), where η1=10, η2=5, Δk=a/k2, a=100/3, Γ11=diag1,0.1,…,0.1︸30, Γ21=1, Γ12=diag0.1,1,0.1,…,0.1︸29, Γ22=1, Γ3=1. The variables x1,k, x2,k are used as the inputs of the RBFNNS; the neural network consists of 31 neurons, the center of basis function uniform coverage of −1,1, and the width of basis function is set to 1.Here, choose the parameters and initial value as follows:z1,k0=1/k2, z2,k0=1/k2, W^1,10=0.01,0,…,0︸30T, W^2,10=0.01,0,…,0︸30T, N^1,10=−0.1, N^2,10=−0.1, ϱ^10=0.1. Taking k=40, Figures 1–5 are the simulation figures.Figure 1
Curve graph ofz1,k along k.Figure 2
Curve graph ofuk along k.Figure 3
Curve graph ofW^1,k, W^2,k along k.
(a)(b)Figure 4
Curve graph ofN^1,k, N^2,k along k.
(a)(b)Figure 5
Curve graph ofϱ^k along k.Figure1 shows the convergence of error. Moreover, on the interval 0,π, Figures 2–5 give that uk, W^1,k, W^2,k, N^1,k, N^2,k, ϱk are bounded. Figures 1–5 confirm the validity of control scheme which is developed in this article from the simulation results.


## 5. Conclusions

The different target tracking problem for unknown nonlinear systems with continuous nonlinearly input is solved in this paper. We introduce the RBFNN to deal with the uncertain dynamics. The problems of approximation error and initial state error can be efficiently solved by suitable means. This paper can keep all signals being bounded on0,T, and errors can converge with the number of iterations increasing. Finally, the effectiveness of the proposed control method is verified via a simulation example.



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*Source: 6858023-2021-08-09.xml*