# FLS-Based Nonuniform Trajectory Tracking AILC for Uncertain Nonlinear Systems with Nonsymmetric Dead-Zone Input and Initial State Error

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

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## Abstract

This paper proposes an AILC method for uncertain nonlinear system to solve different target tracking problems. The method uses fuzzy logic systems (FLS) to approximate every uncertain term in systems. All closed-loop signals are bounded on0,T according to the Lyapunov theory. A time-varying boundary layer and a typical convergent series are introduced to handle initial state error, unknown bounds of errors, and nonuniform target tracking, respectively. The result is that the tracking error’s norm can converge to a small neighborhood along iteration increasing asymptotically. Finally, the simulation results of mass-spring mechanical system show 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. The paper [1] proposed a new ILC law for first-order hybrid parametric system and the paper [2] proposed a novel AILC method for nonlinear hybrid parameter systems. Recently, AILC is presented; a nonuniform target tracking AILC method was proposed in the paper [3]. The paper [4] proposed a fault-tolerant ILC technique for mobile robots’ nonrepetitive target tracking with output constraints. From the above literature analysis, solving the nonuniform target tracking problem for uncertain mechanical nonlinear system is an important problem.Adaptive control is used to handle system’s control problem about uncertainties which is a challenging problem. Adaptive control schemes learn uncertainties by adaptive laws. NN and FLS are used in the method as function approximators, such as the papers [5–8]. The literature [9] could complete the varying control tasks by designing an adaptive fuzzy ILC for uncertain nonlinear system. Based on RBF neural network approximation, the literature [10] 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 paper [11]; by Lyapunov theory, it can prove the convergence. It should be noted that Lyapunov function-based AILC plays an important role in dealing with the time-varying parameter in the literatures [9, 11, 12]. But initial state error is a challenging one as they need to converge to zero for keeping stability. Only the papers [9, 10, 13] considered this problem recently it is an important problem for AILC.Due to the physical limitations of actuators, in real systems, control inputs are often constrained, such as dead-zone inputs. However, these constraints may damage the performance of the system. In the papers [14–23], control performance could be changed by using different techniques recently. The question of adaptive stabilization for time-delay system had been solved in the literature [16]. The paper [17] 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 nonsymmetric dead-zone 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 nonsymmetric dead-zone 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 nonsymmetric dead-zone input and initial state error issues.(ii)
The AILC method is used to uncertain nonlinear systems. FLS 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 mass-spring mechanical system are given to verify the validity of the designed controller.The remainder of this paper is organized as follows: in Section2, the system description and related concepts are given in detail. The controller design process and main results are presented in Section 3. A simulation is shown in Section 4. Section 5 is the conclusion.


## 2. System Description

### 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=Γuk+fnx¯n,k,yk=x1,k,where x¯j,k=x1,k,…,xj,kT∈Rj, x=x¯n is the state that is measured; Γuk∈R represents the actuator characteristics and yk∈R is the system output; fjx¯j,k, j=1,2,…,n, are smooth unknown nonlinear functions.Here, the functionΓuk represents actuator output with nonsymmetric dead-zone and can be expressed as(2)Γuk=mruk−br,ifuk≥br,0,if−bl<uk<br,mluk+bl,ifuk≤−bl,where mr and ml present the right and left slopes, respectively; br and bl present the breakpoint of dead-zone. The nonsymmetric dead-zone can be rewritten as a combination of a line and a disturbance-like term [18].(3)Γuk=mtuk+dt,where(4)mt=m,ifuk≤0,mr,ifuk>0,dt=−mrbr,ifuk≥br,−mtuk,if−bl<uk<br,mlbl,ifuk≤−bl.Assumption 1.
Parametersml, mr, bl, and br are uncertain constants. There exists an unknown ν satisfing 0<ν≤minml,mr, which is small enough. The unknown constant d¯ is the upper bound of dt.
Designing an AILC lawukt on 0,T to make the output ykt following the target trajectory yr,kt is our control objective, that is to say, limk⟶∞ykt−yr,kt≤ϱ, where ϱ is the small positive error. Guarantee that the closed-loop signals are bounded. yr,kt is the smooth desired target. k is the iteration index.


### 2.2. Convergent Series Sequence

The following definition and lemma can be used in the design process.Definition 1.
(see [20]). Δk is a series convergent sequence which is shown as(5)Δk=akl,where k=1,2,…; a and l are the needed parameters, satisfing a>0∈R, l≥2∈N.Lemma 1 (see [20]).
For1/kl, where k=1,2,…, l≥2, the inequality as follows holds:(6)limk⟶∞∑j=1k1jl≤2.


### 2.3. Description of Fuzzy Logic System (FLS)

The FLS has a good approximation property [24]. For a smooth fx∈R can be approximated by(7)fx=W∗TSx+δx,where δx is the fuzzy approximated error, W∗T is the optimal weight vector satisfing(8)W∗T=argminW∗∈ΩW∗supx∈Dxf^x|W−fx,where Dx represents the set of x and f^x|W that is estimated f, shown by(9)f^x|W=WTSx,with an adjustable W, and ΩW∗ is given as follows:(10)ΩW∗=W∗|W∗≤WM∗,with WM∗ being a positive constant.Assumption 2.
In this paper, the following inherent approximation errorδjx¯j,k is assumed to be bounded with δjx¯j,k≤θj, where the unknown parameters θj1≤j≤n denote the smallest upper bounds of δjx¯j,k with θj≥0.


### 2.4. Time-Varying Boundary Layer

Introduce the functionzjϕ,k as follows to deal with initial state errors:(11)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 is a series convergence sequence; the saturation function sat is given as(12)satzj,kϕj,kt=1,ifzj,k>ϕj,kt,zj,kϕj,kt,if−ϕj,kt≤zj,k≤ϕj,kt,−1,ifzj,k<−ϕj,kt,where ϕj,kt is the time-varying boundary layer. The control objective can be achieved by this. See the literature [9] for specific analysis.Assumption 3.
The initial state errors must satisfyzj,k0=εj,k for some positive parameters εj,k which are known; j=1,…,m, zj,k0 are given later in this paper.


## 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=Γuk+fnx¯n,k,yk=x1,k,where x¯j,k=x1,k,…,xj,kT∈Rj, x=x¯n is the state that is measured; Γuk∈R represents the actuator characteristics and yk∈R is the system output; fjx¯j,k, j=1,2,…,n, are smooth unknown nonlinear functions.Here, the functionΓuk represents actuator output with nonsymmetric dead-zone and can be expressed as(2)Γuk=mruk−br,ifuk≥br,0,if−bl<uk<br,mluk+bl,ifuk≤−bl,where mr and ml present the right and left slopes, respectively; br and bl present the breakpoint of dead-zone. The nonsymmetric dead-zone can be rewritten as a combination of a line and a disturbance-like term [18].(3)Γuk=mtuk+dt,where(4)mt=m,ifuk≤0,mr,ifuk>0,dt=−mrbr,ifuk≥br,−mtuk,if−bl<uk<br,mlbl,ifuk≤−bl.Assumption 1.
Parametersml, mr, bl, and br are uncertain constants. There exists an unknown ν satisfing 0<ν≤minml,mr, which is small enough. The unknown constant d¯ is the upper bound of dt.
Designing an AILC lawukt on 0,T to make the output ykt following the target trajectory yr,kt is our control objective, that is to say, limk⟶∞ykt−yr,kt≤ϱ, where ϱ is the small positive error. Guarantee that the closed-loop signals are bounded. yr,kt is the smooth desired target. k is the iteration index.


## 2.2. Convergent Series Sequence

The following definition and lemma can be used in the design process.Definition 1.
(see [20]). Δk is a series convergent sequence which is shown as(5)Δk=akl,where k=1,2,…; a and l are the needed parameters, satisfing a>0∈R, l≥2∈N.Lemma 1 (see [20]).
For1/kl, where k=1,2,…, l≥2, the inequality as follows holds:(6)limk⟶∞∑j=1k1jl≤2.


## 2.3. Description of Fuzzy Logic System (FLS)

The FLS has a good approximation property [24]. For a smooth fx∈R can be approximated by(7)fx=W∗TSx+δx,where δx is the fuzzy approximated error, W∗T is the optimal weight vector satisfing(8)W∗T=argminW∗∈ΩW∗supx∈Dxf^x|W−fx,where Dx represents the set of x and f^x|W that is estimated f, shown by(9)f^x|W=WTSx,with an adjustable W, and ΩW∗ is given as follows:(10)ΩW∗=W∗|W∗≤WM∗,with WM∗ being a positive constant.Assumption 2.
In this paper, the following inherent approximation errorδjx¯j,k is assumed to be bounded with δjx¯j,k≤θj, where the unknown parameters θj1≤j≤n denote the smallest upper bounds of δjx¯j,k with θj≥0.


## 2.4. Time-Varying Boundary Layer

Introduce the functionzjϕ,k as follows to deal with initial state errors:(11)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 is a series convergence sequence; the saturation function sat is given as(12)satzj,kϕj,kt=1,ifzj,k>ϕj,kt,zj,kϕj,kt,if−ϕj,kt≤zj,k≤ϕj,kt,−1,ifzj,k<−ϕj,kt,where ϕj,kt is the time-varying boundary layer. The control objective can be achieved by this. See the literature [9] for specific analysis.Assumption 3.
The initial state errors must satisfyzj,k0=εj,k for some positive parameters εj,k which are known; j=1,…,m, zj,k0 are given later in this paper.


## 3. AILC Design and Convergence Analysis

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, and α1,k is a virtual controller. Introduce the error function z1ϕ,k and z2ϕ,k from Section 2.4 to deal with initial state errors as(13)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(14)x˙1,k=x2,k+F1x¯1,k.
Derivatez1ϕ,k as follows:(15)z˙1ϕ,k=z2,k+α1,k+F1x¯1,k−y˙r,k−sgnz1ϕ,kϕ˙1,k.
According toSection 2.3, by FLS, F1x¯1,k is approximated as follows:(16)F1x¯1,k=W1TS1x¯1,k+δ1x¯1,k,where δ1x¯1,k is the approximation error and W1 is the optimal weight vector.
DefineN1=ωM12, which is needed later; Δk=a/kl, a>0, and l≥2. The virtual controller is taken as(17)α1,k=−W^1,kTS1x¯1,k−N^1,k1Δkz1ϕ,k+y˙r,k−η1z1,k.
Substitute equations (16) and (17) into equation (15), then(18)z˙1ϕ,k=z2,k−N^1,k1Δkz1ϕ,k+W1TS1x¯1,k+δ1x¯1,k−W^1,kTS1x¯1,k−η1z1,k−sgnz1ϕ,kϕ˙1,kt=z2ϕ,k−N^1,k1Δkz1ϕ,k−W˜1,kTS1x¯1,k+δ1x¯1,k+ϕ2,ktsatz2,kϕ2,kt−η1z1,k−sgnz1ϕ,kϕ˙1,kt,where the 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 equation (18) can be changed as(19)−η1z1,k−sgnz1ϕ,kϕ˙1,kt=−η1z1ϕ,k−η1ϕ1,ktsatz1,kϕ1,kt−sgnz1ϕ,kϕ˙1,kt=−η1z1ϕ,k.
Equations (18) and (19) can be rewritten as(20)z˙1ϕ,k=z2ϕ,k−N^1,k1Δkz1ϕ,k−η1z1ϕ,k+W˜1TS1x¯1,k+δ1x¯1,k+ϕ2,ktsatz2,kϕ2,kt.
Letω1=δ1x¯1,k+ϕ2,ktsatz2,kt/ϕ2,kt, then equation (20) becomes(21)z˙1ϕ,k=z2ϕ,k−N^1,k1Δkz1ϕ,k−η1z1ϕ,k+W˜1TS1x¯1,k+ω1.Assumption 4.
The bounded termω1 satisfies ω1≤ωM1, where ωM1 is a positive parameter.
Take the following nonnegative function:(22)V1,k=12z1ϕ,k2+12W˜1,kTΓ11−1W˜1,k+12Γ21−1N˜1,k2,where Γ11 and Γ21 are symmetric matrices and are positive. Derivate V1,k according to equation (21), then(23)V˙1,k=z1ϕ,kz2ϕ,k−η1z1ϕ,k2−W˜1,kTΓ11−1Γ11S1x¯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,kTΓ11−1Γ11S1x¯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,kTΓ11−1Γ11S1x¯1,kz1ϕ,k−W^˙1,k−N˜1,kΓ21−1Γ211Δkz1ϕ,k2−N^˙1,k+14Δk.
In the previous equation,mn≤1/rm2+1/4n2r is used, where r=Δk.Step 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, the same to Step 1; zjϕ,k and zj+1ϕ,k from Section 2.4 are introduced as(24)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.
Derivatezjϕ,k as follows:(25)z˙jϕ,k=zj+1,k+αj,k+fjx¯j,k−α˙j−1,k−sgnzjϕ,ktϕ˙j,k,where α˙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 equation (25) can be rewritten as(26)z˙jϕ,k=zj+1,k+αj,k+Fjx¯j,k−Pj−1,k−sgnzjϕ,ktϕ˙j,k.
According toSection 2.3, Fjx¯j,k by FLS can become(27)Fjx¯j,k=WjTSjx¯j,k+δjx¯j,k,where δjx¯j,k is the approximation error and Wj is the optimal weight vector.
The virtual controller is taken as(28)αj,k=−zj−1ϕ,k−W^j,kTSjx¯j,k−N^j,k1Δkzjϕ,k+Pj−1,k−ηjzj,k.
Equations (27) and (28) are substituted into equation (26), then we have(29)z˙jϕ,k=zj+1,k−zj−1ϕ,k−N^j,k1Δkzjϕ,k−ηjzj,k−sgnzjϕ,ktϕ˙j,k+WjTSjx¯j,k+δjx¯j,k−W^j,kTSjx¯j,k=zj+1,k−zj−1ϕ,k−N^j,k1Δkzjϕ,k−W˜j,kTSjx¯j,k+δjx¯j,k−ηjzj,k−sgnzjϕ,kϕ˙j,k=zj+1ϕ,k−zj−1ϕ,k−N^j,k1Δkzjϕ,k−W˜j,kTSjx¯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 equation (30) can be changed as(30)−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 equation (29) becomes(31)z˙jϕ,k=zj+1ϕ,k−zj−1ϕ,k−ηjzjϕ,k−N^j,k1Δkzjϕ,k+W˜j,kTSjx¯j,k+ωj.Assumption j.
ωj satisfies ωj≤ωMj; here, ωMj is unknown.
Positive definite function is chosen:(32)Vj,k=Vj−1,k+12zjϕ,k2+12W˜j,kTΓ1j−1W˜j,k+12Γ2j−1N˜j,k2.
DerivateVj,k according to equation (31),(33)V˙j,k≤zjϕ,kzj+1ϕ,k−∑l=1jηlzl,k2−∑l=1jW˜l,kTΓ1l−1Γ1lSjx¯l,kzlϕ,k−W^˙l,k−∑l=1jN˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+j14Δk.Step n.
Definezn,k=xn,k−αn−1,k; znϕ,k from Section 2.4 is introduced as(34)znϕ,k=zn,k−ϕn,ktsatzn,kϕn,kt,ϕn,kt=εn,ke−ηnt.
Derivateznϕ,k as follows:(35)z˙nϕ,k=Γuk+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 by equation (3), then equation (35) can be rewritten as(36)z˙nϕ,k=mtuk+Fx¯n,k−Pn−1,k−sgnznϕ,kϕ˙n,kt+dt,from which Fnx¯n,k can be rewritten as follows:(37)Fnx¯n,k=WnTSnx¯n,k+δnx¯n,k,where δnx¯n,k is the approximation error and Wn is an optimal weight vector.
Letu1,k=−zn−1ϕ,k−W^n,kTSnx¯n,k−N^n,k1/Δkzn,k+Pn−1,k−ηnzn,k. Take the actual controller as(38)uk=u1,k+u2,k,where u2,k is designed to compensate for unknown input gain mt.
Then, according to equations (37) and (38), equation (36) can be rewritten as(39)z˙nϕ,k=uk+mt−1uk+WnTSnx¯n,k+δnx¯n,k−Pn−1,k−sgnznϕ,kϕ˙n,kt+dt=−zn−1ϕ,k−W˜n,kTSnx¯n,k−N^n,k1Δkzn,k+mtu2,k+mt−1u1,k+δnx¯n,k+dt−ηnzn,k−sgnznϕ,kϕ˙n,kt,where W^n,k and N^n,k are the estimated parameters of Wn and Nn, respectively. W˜n,k=W^n,k−Wn and N˜n,k=N^n,k−Nn are the estimated parameter errors.
The last two terms of equation (39) can be changed as(40)−ηnzn,k−sgnznϕ,kϕ˙n,kt=−ηnznϕ,k−ηnϕn,ktsatzn,kϕn,kt−sgnznϕ,kϕ˙n,kt=−ηnznϕ,k.
Then,(41)z˙nϕ,k=−zn−1ϕ,k−ηnznϕ,k−W˜n,kTSnx¯n,k−N^n,k1Δkzn,k+mtu2,k+mt−1u1,k+δnx¯n,k+dt.
Takeu2,k=−satzn,k/ϕn,ktρ^kt+1u1,k, where ρ^kt is the estimation of uncertain parameter ρ=1/mt and let ωn=δnx¯n,k+dt, then equation (41) becomes(42)z˙nϕ,k=−zn−1ϕ,k−ηnznϕ,k−W˜n,kTSnx¯n,k−N^n,k1Δkznϕ,k−mtsatzn,kϕn,ktρ^kt+1u1,k+mt−1u1,k+ωn.Assumption n.
Because of assumptions1 and 2, ωn is bounded such that ωn≤ωMn with ωMn being unknown.
Adaptive learning laws are taken as follows:(43)W^˙j,k=Γ1jSjx¯j,kzjϕ,k,j=1,…,m,(44)N^˙j,k=Γ2j1Δkzjϕ,k2,j=1,…,m,(45)ρ^˙kt=Γ3u1,kznϕ,k,where Γ3 is the learning gain to be designed.
Positive definite function is chosen:(46)Vn,k=Vn−1,k+12znϕ,k2+12W˜n,kTΓ1n−1W˜n,k+12Γ2n−1N˜n,k2+12Γ3−1mtρ˜k2.
DerivateVn,k according to equation (42) and substitute equations (43)–(45) into it, then(47)V˙n,k≤zn−1ϕ,kznϕ,k−∑l=1n−1ηlzlϕ,k2−∑l=1n−1W˜l,kTΓ1l−1Γ1lSlx¯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+Γ3−1mtρ˜kρ^˙k≤−∑l=1nηlzlϕ,k2−∑l=1nW˜l,kTΓ1l−1Γ1lSlx¯l,kzlϕ,k−W^˙l,k−∑l=1n−1N˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n−114Δk−N^n,k1Δkznϕ,k2−mtρ^ktu1,kznϕ,k+u1,kznϕ,k+1ΔkωNn2znϕ,k2+14Δk+Γ2n−1N˜n,kN^˙n,k+Γ3−1mtρ˜kρ^˙k=−∑l=1nηlzlϕ,k2−∑l=1nW˜l,kTΓ1l−1Γ1lSlx¯l,kzlϕ,k−W^˙l,k−∑l=1nN˜l,kΓ2l−1Γ2l1Δkzlϕ,k2−N^˙l,k+n14Δk−Γ3−1mtρ˜kΓ3u1,kznϕ,k−ρ^˙k=−∑l=1nηlzl,k2+n14Δk.
The assumption below is needed.Assumption 5.
For anyk, as t=0,N^j,k0=N^j,k−1T, W^j,k0=W^j,k−1T;j=1,…,m, ρ^k0=ρ^k−1T.Theorem 1.
Under assumptions1, 2, 3, 4–n, and 5, design controller equation (38), and adaptive parameter estimation laws (43)–(45) for system (1), we get all signals bounded, and(48)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.
FromSection 2.4, zϕ,k02=0≤zϕ,kT2, where zϕ,k=z1ϕ,k,z2ϕ,k,…,znϕ,kT. By (46), we get(49)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, N^k=N^1,k,N^2,k,…,N^n,kT.
Substituting equation (47) into equation (49), we get(50)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, then equation (50) can be rewritten as(51)∑l=1k∑l=1n∫0Tηlzlϕ,j2dt≤V0,k−Vn,kzϕ,k0,W^kT,N^kT,ρ^kT.
By equation (6), limk⟶∞V0,k≤Vn,10+2an1/4T, then V0,k is bounded; by assumption 5 and equation (50), Vn,k0 can be bounded. Vn,kzk0,W^kT,N^kT,ρ^kT≥0, so(52)limk⟶∞∑l=1n∫0Tηlzlϕ,k2dt=0.
By equation (46), for any k, Vn,kt=Vn,k0+∫0tV˙n,kτdτ, by (47), then(53)Vn,kt≤Vn,k0−∑l=1n∫0tηlzlϕ,kτ2dτ+tn14Δk.
By equation (52), ∑l=1n∫0tηlzlϕ,kτ2dτ is bounded. By Definition 1, Δk is bounded, and t∈0,T, so tn1/4Δk is also bounded. From above all, for any k, Vn,kt is bounded, then we have xj,k; W^kt and N^kt are bounded. By equation (38), uk is bounded. By equation (31), 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, 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; 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; the motion equation of the system with nonsymmetric dead-zone input is as follows:(54)m¯y¨k+Fmsyk=Γuk,where t∈0,π, Fms⋅ is the spring’s restoring force; k denotes the iteration index.We definex1,k=yk, x2,k=y˙k, and m¯=1, so equation (54) can be transformed into the state-space form(55)x˙1,k=x2,k,x˙2,k=Γuk−Fmsx1,k.The spring’s restoring force can be modeled as(56)Fmsx1,k=kx1,k∑i=0qaix1,ki.In the system, we havek=1, a0=0, a1=a2=a3=a4=1, and q=4.The nonsymmetric dead-zone is shown as(57)Γuk=uk−1,ifuk≥1,0,if−3<uk<1,uk+3,ifuk≤−3.System objective is that the output of system (54) 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 Theorem1, the AILC is chosen as(58)α1,k=−W^1,kTS1x¯1,k−N^1,k1Δkz1ϕ,k+y˙r,k−η1z1,k,uk=u1,k+u2,k,u1,k=−z1ϕ,k−W^2,kTS2x¯2,k−N^2,k1Δkz2,k+P1,k−η2z2,k,u2,k=−satz2,kϕ2,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 estimation laws are represented by equations (43)–(45), where η1=10, η2=1, Δk=a/k2, a=100/3, Γ11=diag0.1,0.1,0.1,0.1,0.1, Γ21=1, Γ12=diag0.1,0.1,0.1,0.1,0.1, Γ22=1, and Γ3=1. The inputs of the FLS are x1,k, x2,k; the membership function of every variable is given as follows:(59)μAi1=exp−yi+aibi2,μAi2=exp−yi+0.5×aibi2,μAi3=exp−yibi2,μAi4=exp−yi−0.5×aibi2,μAi5=exp−yi−aibi2,where i=1,…,4, y1=x1,k, y2=x2,k, a1=a2=2, and b1=b2=2, and the fuzzy rules are given as(60)Rtj:ifytisAtj,fsx¯sisBsj,where s=1,2: as s=1, t=1,2,3, as s=2, t=1,2,4. The period of sampling is 0.01s.Here, choose the parameters and initial values as follows:z1,k0=1/k2, z2,k0=1/k2, W^1,10=0.01,0,0,0,0T, W^2,10=0.01,0,0,0,0T, N^1,10=−0.1, N^2,10=−0.1, and 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.Figure 4
Curve graph ofN^1,k, N^2,k along k.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, and ρ^k 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 nonsymmetric dead-zone input is solved. Introduce FLS to deal with the uncertain dynamics, and the problem of approximation error and initial state error can be efficiently solved by suitable means. This paper can keep all signals being bounded on0,T; errors can converge to a small set along iteration increasing. Simulation study proves the correctness of AILC method of this paper.



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*Source: 3351074-2021-08-19.xml*