# Adaptive Asymptotic Tracking Control for a Class of Uncertain Input-Delayed Systems with Periodic Time-Varying Disturbances

Authors: Xiaoman Yan; Chunsheng Zhang; Dewen Cao; Jian Wu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, the problem of adaptive asymptotic tracking control for a class of uncertain systems with periodic time-varying disturbances and input delay is studied. By combining Fourier series expansion (FSE) with radial basis function neural network (RBFNN), a hybrid function approximator is used to learn the functions with periodic time-varying disturbances. At the same time, the dynamic surface control technique with a nonlinear filter is used to avoid the “complexity explosion” problem in the process of traditional backstepping technology. Ultimately, all closed-loop signals are guaranteed to be semiglobally uniformly bounded, and the given reference signal can be asymptotically tracked by the output signals of system. A simulation example is given to verify the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

In recent years, with the deepening of the research on uncertain nonlinear control systems, more and more scholars have carried out indepth research on this problem and achieved remarkable results. The uncertainties appear in nonlinear systems in strict-feedback form [1–6], pure-feedback form [7, 8], multiple-input-multiple-output (MIMO) form [9–11], time-delay form [12, 13], switched form [14–19], and discrete-time form [20, 21]. Aiming at different forms of system, different methods are proposed to solve the problem of system control. For example, the adaptive gain scheduling backstepping sliding mode control method [1] is proposed for a class of strict-feedback systems. The problem of predetermined performance control for a class of pure-feedback uncertain nonlinear systems with input saturation [7] is studied. For a class of uncertain MIMO nonlinear systems with given tracking performance, the error-driven nonlinear feedback design method [10] is proposed to improve the dynamic performance of fuzzy adaptive moving surface control. The dynamic output feedback fault-tolerant controller [12] is designed to solve the problems of fault estimation and fault-tolerant controller design for a class of discrete-time fuzzy systems. The adaptive fuzzy tracking control for a class of switched uncertain nonlinear systems [14] is studied. For a class of uncertain strict-feedback nonlinear systems, the output feedback adaptive neural tracking control [15] is studied.In particular, the study of uncertain nonlinear systems with periodic perturbations is also a challenging problem [22–26]. Since RBFNN does not use periodic perturbation functions as inputs, it is difficult to deal with such systems in traditional ways. Therefore, some new methods are proposed to solve this problem. In particular, the study of uncertain nonlinear systems with periodic perturbations is also a challenging problem [27] is studied. This article [28] investigates the sampled data stabilization problem of a class of switched nonlinear systems. To guarantee that all states of the closed-loop system (CLS) are bounded, a new allowable sampling period is deduced.It is well known that the traditional backstepping technology has the problem of “complexity explosion.” To avoid the problem, many scholars have introduced dynamic surface control technology and achieved some results [29–34]. For a class of fractional-order nonlinear systems with uncertain parameters, the design of adaptive dynamic surface control [30] is studied. For a class of single input single output strict-feedback fractional-order uncertain nonlinear systems, the dynamic surface control method based on fractional-order filter [32] is proposed to avoid the inherent “complexity explosion” problem in the backstepping design process. The problem of adaptive fuzzy dynamic surface control [33] is studied for a class of nonstrict feedback nonlinear systems with unknown virtual control coefficients and full state constraints. The dynamic surface control technology [34] is used to solve the “complexity explosion” problem and combines the Nussbaum function to solve the problem of unknown control direction.It is worth noting that, for uncertain systems with periodic perturbations, the asymptotic tracking control problem is not well solved. It is because the asymptotic tracking control problem of uncertain systems with periodic perturbations has not been well solved. Aiming at this difficulty, the control method proposed in this paper solves the asymptotic tracking control problem of this kind of system well. The main work is as follows: (1) in this paper, the problem of adaptive asymptotic tracking control for a class of nonlinear uncertain systems with periodic time-varying disturbances and input delay is studied. To solve this problem, we propose a new adaptive control scheme. (2) To ensure the feasibility of the control scheme, the FSE-RBFNN approximator is used to approximate the unknown disturbance function. (3) An adaptive dynamic surface control technique with a nonlinear filter is designed to avoid the “complexity explosion” problem in the traditional backstepping design process. (4) Based on the designed adaptive controller, all closed-loop signals are ultimately guaranteed to be semiglobally uniformly bounded. It is proved that the output signal can asymptotically track the given reference signal. Finally, a simulation example is given to verify the effectiveness of the proposed control scheme.


## 2. Problem Description and Preliminaries

A class of periodic perturbed nonlinear systems with strict-feedback is considered:(1)x˙i=xi+1+fix¯i,ωit+dit,i=1,…,n−1,x˙n=ut−τt+fnx¯n,ωnt+dnt,y=x1,for i=1,…,n, x¯i=x1,…,xiT is the system state variable, y∈R is the output of the system, u is the control input of the system, τt represents the time-varying delay, fi·:ℝi+qi⟶ℝ is the unknown continuous function, ωit is the unknown continuous time-varying disturbance with known period Ti, i.e.,ωit+Ti=ωit,0,+∞⟶ℝqi, and dit is the disturbance function.Remark 1.
For the actual control systems, such as industrial control systems, communication systems, and aerospace control systems, the disturbance and time-delay are common problems. These problems are well solved, and the performance of the controlled systems will have a great improvement.
Control objective: an adaptive dynamic surface control algorithm based on a neural network is designed to ensure that the output signal of system (1) can track the reference signal ydt asymptotically, and all the closed-loop signals are semiglobally uniformly bounded.
To achieve the above control objectives, we make the following assumptions on system (1):Assumption 1 (see [32]): reference trajectory ydt is bounded, and its derivatives y˙dt and y¨dt are continuous and boundedAssumption 2 (see [35]): for i=1,2,…,n, there exists an unknown constant Ni, which makes the following inequality holds dit≤Ni
In this paper, the mixed function approximator based on FSE-RBFNN in reference [28] is used to model giEi,ωit as follows:(2)giEi,ωit=WTSEi,γTφt+δEi,t.
A hybrid neural network approximator is designed(3)HEi,t=WTSEi,γTφt,δEi,t=WTSEi,γTφt+δωt−WTSEi,γTφt+δbEi,ωt,δωt≤εω,δbEi,ωt≤εb,where Ei⊂ΩE⊂ℝl is a measured signal, ωit is an unknown continuous disturbance vector with known period T, ωit=v1t,…,vqt∈Ωω⊂ℝq, ΩE and Ωω are compact sets, optimal weight vector W=argminw^∈ℝhsupEi,ωt∈ΩgEi,ωt−W^TSEi,ωt, W=ω1,…,ωhT, δbEi,ωt is the inherent NN approximation error with the minimum upper bound εb>0, which can be decreased by increasing the NN node number p, SEi,ωt=s1Ei,ωt,…,shEi,ωt is a known smooth vector value function, its component siEi,ωt=exp−ζ−vi2/ζ2i=1,…,h,ζ=EiT,ωTtT, vi∈Ω is the center of SEi,ωt, ζ>0 is the width of SEi,ωt, γ=γ1,…,γq∈ℝs×q is a constant matrix, γi is a vector composed of s coefficients expanded by fourier series (s is an odd number), φt=φ1t,…,φstT, φ1t=1, φ2jt=2sin2πjt/T, and φ2j+1t=2cos2πjt/T, j=1,…,s−1/2, its n-th derivative is smooth and bounded, the upper bound εω is the minimum truncation error of δωt.Lemma 1 (see [36]).
ForEi,ωt∈Ω, the approximation error δbEi,t in (2) satisfies the following condition:(4)δEi,t≤ε,where ε is the minimum upper bound of δEi,t and it can be reduced arbitrarily by increasing the values of s and h.
Generally speaking, the parametersW and γ are unknown. It needs to be estimated in the controller design. Suppose W^ and γ^ are, respectively, the estimates of W and γ, and the estimated error is W˜=W−W^,γ˜=γ−γ^.Lemma 2 (see [36]).
For approximator (2), the estimated error can be expressed as follows:(5)WTSEi,γTφt−W^TSEi,γ^Tφt=W˜TS^−S^′γ^Tφt+W^TS^′γ˜Tφt+b,where S^=SEi,γ^Tφt,S^′=s^1′,…,s^h′∈ℝq×h,ands^i′=∂siEi,ω/∂ω|ω=γ^Tφt,i=1,…,h. The remainder b is bounded(6)b≤γFφtW^TS^′F+Wφtγ^TS^′F+W1.Lemma 3 (see [37]).
Forς>0 and λ∈ℝ, the following inequality holds:(7)0≤λ−λ2λ2+ς2≤ς.Lemma 4 (see [38]).
Assume thata is an unknown nonzero variable. Vt0,t and Rt are smooth functions defined in the time interval t0,t, and Vt0,tf is nonnegative, Rt0 is bounded. In addition, NRt=eR2tRt is a dynamic gain function, ξ is a normal number, and ξ¯ is a bounded variable. If the following inequality holds(8)Vt0,t≤e−ξt∫t0taNRcR˙ceξcdc+e−ξt∫t0tR˙ceξcdc+ξ¯,it can be concluded that Rt,e−ξt∫t0tR˙ceξcdc,Vt0,t and e−ξt∫t0taNRcR˙ceξcdc must be bounded on the interval t0,tf. Furthermore, when tf=∞, the obtained closed-loop system is also bounded.Lemma 5 (see [38]).
For the following equation(9)m^˙t=−am^t+dk¯t,where a>0 and d>0 are the design constants, k¯t is a positive function. If the initial value m^0 is nonnegative, then for any t>0, m^t≥0 must be established.


## 3. Control Design and Stability Analysis

The main conclusions of this paper can be summarized as the following theorem.Theorem 1.
Consider a closed-loop system (1), including the nonlinear filter (10), the actual controller (20), and the adaptive law (29). Based on Assumption 1 and Assumption 2, for any initial condition satisfying V0≤c, where c is a positive design parameter, there exists design parameters ki,ηi,βi,ξji=1,…,n,j=1,…,n−1 and invertible matrices ΓWiandΓγi, such that(i)
All the closed-loop signals remain semiglobally uniformly ultimate bounded (SGUUB)(ii)
The given reference signal can be asymptotically tracked by the output signal of the system
The proof of Theorem1 is divided into two parts: adaptive controller design and stability analysis of the closed-loop system.
In the following, we propose an adaptive NN dynamic surface control scheme based on the backstepping design process. To order to avoid the “complexity explosion” problem, the following nonlinear filters are constructed:(10)ιis˙i=−ιiei−ιiM^i2eiM^i2ei2+μ2,si0=αi0,i=1,…,n−1,where ei=si−αi is the i-th boundary layer error and ιi and μ are the positive constants. To compensate for the influence of input delay, the following system [35] is introduced:(11)λ˙i=λi+1−piλi,i=2,…,n−1,λ˙n=−pnλn+ut−τt−ut,where pi>1i=2,…,n is optional known parameter, and the initial condition of the system is λ0=0.
In the next part, an adaptive NN tracking controller is designed based on the backstepping method. Firstly, the following coordinate transformations are introduced:(12)z1=x1−yd,zi=xi−si−1−λi,i=2,…,n−1.Step 1: according to (12), we can get the following result:(13)z˙1=x˙1−y˙d=x2+f1x¯1,ω1t+d1t−y˙d=z2+s1+λ2+f1x¯1,ω1t+d1t−y˙d.The Lyapunov functionV1 is constructed as follows:(14)V1=12z12+12η1θ˜12+12W˜1TΓw1−1W˜1+12trγ˜1TΓγ1−1γ˜1+12β1N˜12,whereη1>0 and β1>0 are the design parameters, the estimation errors θ˜1=θ1−θ^1,N˜1=N1−N^1,θ^1 and N^1 represent the estimation of θ1 and N1, respectively. Γw1>0 and Γγ1>0 are the invertible matrices of the design. Then, we can get(15)V˙1=z1z2+e1+α1+λ2+f1x¯1,ω1t+d1t−y˙d−1η1θ˜1θ^˙1−W˜1TΓw1−1W^˙1−trγ˜1TΓγ1−1γ^˙1−1β1N˜1N^˙1.According to Lemma3, we have(16)z1f1x¯1,ω1t−y˙d+λ2≤z12f1x¯1,ω1t−y˙d+λ22z12f1x¯1,ω1t−y˙d+λ22+μ2+μ.Letg1E1,ω1t=z1f1x¯1,ω1t−y˙d+λ22/z12f1x¯1,ω1t−y˙d+λ22+μ2, according to (2) and Lemma 1, it can be concluded that(17)g1E1,ω1t=W1TS1E1,γ1Tφ1t+δ1E1,ω1t,δ1E1,ω1t≤ε1,whereE1=x1,y˙d, ε1 denotes the upper bound of δ1E1,ω1t. Substituting (16) and (17) into (15), we can get(18)V˙1≤z1z2+e1+α1+λ2+W1TS1E1,γ1Tφ1tz1+ε1z1+d1t−1η1θ˜1θ^˙1−W˜1TΓw1−1W^˙1−trγ˜1TΓγ1−1γ^˙1−1β1N˜1N^˙1+μ.Using Young’s inequality, one has(19)z1e1≤12z12+12e12,z1z2≤12z12+12z22.The design of virtual controllerα1 is as follows:(20)α1=Lstϖ1α¯1,whereLst· is a dynamic gain function in Lemma 4, the adaptive law of ϖ1 is given in (29), and α¯1 is the equivalent unit of virtual controller α1(21)α¯1=−k1+32z1−W^1TS1E1,γ^1Tφ1tz1−θ^1z13v12z14v12+μ2−z1N^12z12N^12+μ2,where the definition ofv1 will give in (24), k1 is a positive parameter. Then, substituting (19), (20), and (21) into (18), we can get the following inequality:(22)V˙1≤z1−k1z1+W1TS1E1,γ1Tφ1tz1−W^1TS1E1,γ^1Tφ1tz1+ε1z1−θ^1z13v12z14v12+μ2−z1N^12z12N^12+μ2+d1t−1η1θ˜1θ^˙1−W˜1TΓw1−1W^˙1−trγ˜1TΓγ1−1γ^˙1−1β1N˜1N^˙1+μ+12e12+12z22+z1Lstϖ1α¯1−z1α¯1.According to Lemma2, we have(23)V˙1≤z1−k1z1+W˜1TS^1−S^1′γ^1Tφ1tz1+W^1TS^1′γ˜1Tφ1tz1+b1z1+ε1z1−θ^1z13v12z14v12+μ2−z1N^12z12N^12+μ2+d1t−1η1θ˜1θ^˙1−W˜1TΓw1−1W^˙1−trγ˜1TΓγ1−1γ^˙1−1β1N˜1N^˙1+μ+12e12+12z22+z1Lstϖ1α¯1−z1α¯1.By using Lemmas2 and 3, we can get(24)b1+ε1z12≤γ1Fφ1tW^1TS^1′F+W1φ1tγ^1TS^1′+W11+ε1z12≤θ1z12v1≤θ1z14v12z14v12+μ2+θ1μ,(25)z1d1t≤z1d1t≤z1N1=z1N^1+z1N˜1≤z12N^12z12N^12+μ2+μ+z1N˜1,where(26)θ1=γ1F2+W12+W11+ε12,v1=φ1tW^1TS^1′F2+φ1tγ^1TS^1′2+1.Combining (23)–(25), the following inequality holds:(27)V˙1≤z1−k1z1+W˜1TS^1−S^1′γ^1Tφ1tz1+W^1TS^1′γ˜1Tφ1tz1−θ˜1z13v12z14v12+μ2+θ1+2μ+z1N˜1−1η1θ˜1θ^˙1−W˜1TΓw1−1W^˙1−trγ˜1TΓγ1−1γ^˙1−1β1N˜1N^˙1+12e12+12z22+z1Lstϖ1α¯1−z1α¯1,whereW^1TS^1′γ˜1Tφ1t=trW^1TS^1′γ˜1Tφ1t. Then, one has(28)V˙1≤−k1z12+12e12+12z22+θ1+2μ+z1Lstϖ1α¯1−z1α¯1+W˜1Tz12S^1−S^1′γ^1Tφ1t−Γw1−1W^˙1+θ˜1z14v12z14v12+μ2−1η1θ^.1+N˜1z1−1β1N^˙1+trγ˜1TW^1TS^1′φ1tz12−Γγ1−1γ^˙1.Based on (28), the adaptive laws are constructed as follows:(29)θ^˙1=η1z14v12z14v12+μ2−ρ1θ^1,W^˙1=ΓW1z12S^1−S^1′γ^1Tφ1t−σ1W^1,γ^˙1=Γγ1z12W^1TS^1′φ1t−κ1γ^1,N^˙1=β1z1−o1N^1,ϖ˙1=−D1z1α¯1,whereρ1,σ1,κ1,o1,andD1 are positive design parameters. Then, substituting (29) into (28) produces(30)V˙1≤−k1z12+12e12+12z22+θ1+2μ+ρ1η1θ˜1θ^1+σ1W˜1TW^1+κ1trγ˜1Tγ^1+o1β1N˜1N^1+1D1ϖ˙1t−1D1Lstϖ1tϖ˙1t.Using Young’s inequality, the following inequalities hold:(31)ρ1η1θ˜1θ^1≤−ρ12η1θ˜12+ρ12η1θ12,σ1W˜1TW^1≤−σ12W˜12+σ12W12,κ1trγ˜1Tγ^1≤−κ12γ˜1F2+κ12γ1F2,o1β1N˜1N^1≤−o12β1N˜12+o12β1N12.Therefore, substituting (31) into (30), we have(32)V˙1≤−k1z12+12e12+12z22−ρ12η1θ˜12−σ12W˜12−κ12γ˜1F2−o12β1N˜12+A1+1D1ϖ˙1t−1D1Lstϖ1tϖ˙1t,where(33)A1=θ1+2μ+ρ12η1θ12+σ12W12+κ12γ1F2+o12β1N12.Step 2: similar to (13), then the derivative of zi is(34)z˙i=x˙i−s˙i−1−λ˙i=xi+1+fix¯i,ωit+dit−s˙i−1−λ˙i+1+piλi=zi+1+si+fix¯i,ωit+dit−s˙i−1+piλi=zi+1+ei+αi+fix¯i,ωit+dit−s˙i−1+piλi.The Lyapunov functionVi is constructed(35)Vi=Vi−1+12zi2+12ηiθ˜i2+12W˜iTΓwi−1W˜i+12trγ˜iTΓγi−1γ˜i+12βiN˜i2,whereηi>0 and βi>0 are the design parameters, the estimation errors θ˜i=θi−θ^i,N˜i=Ni−N^i,θ^i and N^i represent the estimation of θi and Ni, respectively. Γwi>0 and Γγi>0 are the invertible matrices of the design. Similar to (15), one has(36)V˙i=V˙i−1+zizi+1+ei+αi+fix¯i,ωit+dit−s˙i−1+piλi−1ηiθ˜iθ^˙i−W˜iTΓwi−1W^˙i−trγ˜iTΓγi−1γ^˙i−1βiN˜iN^˙i.Similar to (16), the following inequality can be obtained:(37)zifix¯i,ωit−s˙i−1+piλi≤zi2fix¯i,ωit−s˙i−1+piλi2zi2fix¯i,ωit−s˙i−1+piλi2+μ2+μ.Let(38)giEi,ωit=zifix¯i,ωit−s˙i−1+piλi2zi2fix¯i,ωit−s˙i−1+piλi2+μ2,then according to (2) and Lemma 1, we can get the following conclusions:(39)giEi,ωit=WiTSiEi,γiTφit+δiEi,ωit,δiEi,ωit≤εi,whereEi=x1,…,xi,s˙1,…,s˙i−1,λ2,…,λi and εi denotes the upper bound of δiEi,ωit. Then, substituting (37) and (39) into (36), one has(40)V˙i≤V˙i−1+zizi+1+ei+αi+WiTSiEi,γiTφitzi+εizi+dit−1ηiθ˜iθ^˙i−W˜iTΓwi−1W^˙i−trγ˜iTΓγi−1γ^˙i−1βiN˜iN^˙i+μ.Using Young’s inequality, we have(41)ziei≤12zi2+12ei2,zizi+1≤12zi2+12zi+12.The design of virtual controllerαi is designed as follows:(42)αi=Lstϖitα¯i,whereLst· is a dynamic gain function in Lemma 4, the adaptive law of ϖi is given in (51), and α¯i is the equivalent unit of virtual controller αi(43)α¯i=−ki+32zi−W^iTSiEi,γ^iTφitzi−θ^izi3vi2zi4vi2+μ2−ziN^i2zi2N^i2+μ2,where the definition ofvi will give in formula (46), ki is the positive parameter of the design. Substituting (41)–(43) into (40), the following inequality holds:(44)V˙i≤V˙i−1+zi−ki+12zi+WiTSiEi,γiTφitzi−W^iTSiEi,γ^iTφitzi+εizi−θ^izi3vi2zi4vi2+μ2−ziN^i2zi2N^i2+μ2+dit−1ηiθ˜iθ^˙i−W˜iTΓwi−1W^˙i−trγ˜iTΓγi−1γ^˙i−1βiN˜iN^˙i+μ+12ei2+12zi+12+ziLstϖiα¯i−ziα¯i.According to Lemma2, we have(45)V˙i≤V˙i−1+zi−ki+12zi+W˜iTS^i−S^i′γ^iTφitzi−W^iTS^i′γ˜iTφitzi+bizi+εizi−θ^izi3vi2zi4vi2+μ2−ziN^i2zi2N^i2+μ2+dit−1ηiθ˜iθ^˙i−W˜iTΓwi−1W^˙i−trγ˜iTΓγi−1γ^˙i−1βiN˜iN^˙i+μ+12ei2+12zi+12+ziLstϖiα¯i−ziα¯i.By using Lemmas2 and 3, we can get(46)bi+εizi2≤γiFφitW^iTS^i′F+Wiφitγ^iTS^i′+Wi1+εizi2≤θizi2vi≤θizi4vi2zi4vi2+μ2+θiμ,(47)zidit≤zidit≤ziNi=ziN^i+ziN˜i≤zi2N^i2zi2N^i2+μ2+μ+ziN˜i,where(48)θi=γiF2+Wi2+Wi1+εi2,vi=φitW^iTS^i′F2+φitγ^iTS^i′2+1.Substituting (46) and (47) into (45), the following inequality holds:(49)V˙i≤V˙i−1+zi−ki+12zi+W˜iTS^i−S^i′γ^iTφitzi+W^iTS^i′γ˜iTφitzi−θ˜izi3vi2zi4vi2+μ2+θi+2μ+ziN˜i−1ηiθ˜iθ^˙i−W˜iTΓwi−1W^˙i−trγ˜iTΓγi−1γ^˙i−1βiN˜iN^˙i+12ei2+12zi+12+ziLstϖiα¯i−ziα¯i,whereW^iTS^i′γ˜iTφit=trW^iTS^i′γ˜iTφit. Then, one has(50)V˙i≤V˙i−1−ki+12zi2+12ei2+12zi+12+θi+2μ+ziLstϖiα¯i−ziα¯i+W˜iTzi2S^i−S^i′γ^iTφit−Γwi−1W^˙i+θ˜izi4vi2zi4vi2+μ2−1ηiθ^˙i+N˜izi−1βiN^˙i.Based on (50), the adaptive laws are constructed as follows:(51)θ^˙i=ηizi4vi2zi4vi2+μ2−ρiθ^i,W^˙i=ΓWizi2S^i−S^i′γ^iTφit−σiW^i,γ^˙i=Γγizi2W^iTS^i′φit−κiγ^i,N^˙i=βizi−oiN^i,ϖ˙i=−Diziα¯i,whereρi,σi,κi,oi,andDi are positive design parameters. Then, substituting (51) into (50), one has(52)V˙i≤V˙i−1−ki+12zi2+12ei2+12zi+12+θi+2μ+ρiηiθ˜iθ^i+σiW˜iTW^i+κitrγ˜iTγ^i+oiβiN˜iN^i+1Diϖ˙it−1DiLstϖitϖ˙it.Using Young’s inequality, the following inequalities hold:(53)ρiηiθ˜iθ^i≤−ρi2ηiθ˜i2+ρi2ηiθi2,σiW˜iW^iT≤−σi2W˜i2+σi2Wi2,κitrγ˜iTγ^i≤−κi2γ˜iF2+κi2γiF2,oiβiN˜iN^i≤−oi2βiN˜i2+oi2βiNi2,Substituting (53) into (52), we can get the following inequality:(54)V˙i≤−∑j=1ikjzj2+∑j=1i12ej2+12zi+12−∑j=1iρj2ηjθ˜j2−∑j=1iσj2W˜j2−∑j=1iκj2γ˜jF2−∑j=1ioj2βjN˜j2+Ai+Σj=1i1Djϖ˙jt−∑j=1i1DjLstϖjtϖ˙jt,where(55)Ai=∑j=1iθj+2μ+∑j=1iρj2ηjθj2+∑j=1iσj2Wj2+∑j=1iκj2γjF2+∑j=1ioj2βjNj2.Stepn: similar to (13) and (34), we have(56)z˙n=x˙n−s˙n−1−λ˙n=ut−τt+fnx¯n,ωnt+dnt−s˙n−1+pnλn−ut−τt+ut=fnx¯n,ωnt+dnt−s˙n−1+pnλn+ut.
The Lyapunov functionVn is constructed as follows:(57)Vn=Vn−1+12zn2+12ηnθ˜n2+12W˜nTΓwn−1W˜n+12trγ˜nTΓγn−1γ˜n+12βnN˜n2,where ηn>0 and βn>0 are the design parameters, the estimation errors θ˜n=θn−θ^n,N˜n=Nn−N^n,θ^n and N^n represent the estimation of θn and Nn, respectively. Γwn>0 and Γγn>0 are the invertible matrices of the design. According to (57), it can be concluded that(58)V˙n=V˙n−1+znfnx¯n,ωnt+dnt−s˙n−1+pnλn+ut−1ηnθ˜nθ^˙n−W˜nTΓwn−1W^˙n−trγ˜nTΓγn−1γ^˙n−1βnN˜nN^˙n.
According to Lemma3, one has(59)znfnx¯n,ωnt−s˙n−1+pnλn≤zn2fnx¯n,ωnt−s˙n−1+pnλn2zn2fnx¯n,ωnt−s˙n−1+pnλn2+μ2+μ.
Let(60)gnEn,ωnt=znfnx¯n,ωnt−s˙n−1+pnλn2zn2fnx¯n,ωnt−s˙n−1+pnλn2+μ2.
Then, according to (2) and Lemma 1, we have(61)gnEn,ωnt=WnTSnEn,γnTφnt+δnEn,ωnt,δnEn,ωnt≤εn,where En=x1,…,xn,s˙1,…,s˙n−1,λ2,…,λn, εn denotes the upper bound of δnEn,ωnt. Combining (58)–(61), the following inequality holds(62)V˙n≤V˙n−1+znu+WnTSnEn,γnTφntzn+εnzn+dnt−1ηnθ˜nθ^˙n−W˜nTΓwn−1W^˙n−trγ˜nTΓγn−1γ^˙n−1βnN˜nN^˙n+μ.
The controlleru is designed as follows:(63)u=Lstϖntu¯,where Lst· is a dynamic gain function in Lemma 4, the adaptive law of ϖn will give in (72), and u¯ is the equivalent unit of virtual controller u:(64)u¯=−kn+12zn−W^nTSnEn,γ^nTφntzn−θ^nzn3vn2zn4vn2+μ2−znN^n2zn2N^n2+μ2,where the definition of vn will give in (67), kn is the positive parameter of the design.
Similar to (44), the following inequality holds(65)V˙n≤V˙n−1+zn−kn+12zn+WnTSnEn,γnTφntzn−W^nTSnEn,γ^nTφntzn+εnzn−θ^nzn3vn2zn4vn2+μ2−znN^n2zn2N^n2+μ2+dnt−1ηnθ˜nθ^˙n−W˜nTΓwn−1W^˙n−trγ˜nTΓγn−1γ^˙n−1βnN˜nN^˙n+μ+znLstϖnα¯n−znα¯n.
According to Lemma2, we have(66)V˙n≤V˙n−1+zn−kn+12zn+W˜nTS^n−S^n′γ^nTφntzn+W^nTS^n′γ˜nTφntzn+bnzn+εnzn−θ^nzn3vn2zn4vn2+μ2−znN^n2zn2N^n2+μ2+dnt−1ηnθ˜nθ^˙n−W˜nTΓwn−1W^˙n−trγ˜nTΓγn−1γ^˙n−1βnN˜nN^˙n+μ+znLstϖnu¯−znu¯.
By using Lemmas2 and 3, we can get(67)bn+εnzn2≤γnFφntW^nTS^n′F+Wnφntγ^nTS^n′+Wn1+εnzn2≤θnzn2vn≤θnzn4vn2zn4vn2+μ2+θnμ,(68)zndnt≤zndnt≤znNn=znN^n+znN˜n≤zn2N^n2zn2N^n2+μ2+μ+znN˜n,where(69)θn=γnF2+Wn2+Wn1+εn2,vn=φntW^nTS^n′F2+φntγ^nTS^n′2+1.
Substituting (67) and (68) into (66) produces(70)V˙n≤V˙n−1+zn−kn+12zn+W˜nTS^n−S^n′γ^nTφntzn+W^nTS^n′γ˜nTφntzn−θ˜nzn3vn2zn4vn2+μ2+θn+2μ+znN˜n−1ηnθ˜nθ^˙n−W˜nTΓwn−1W^˙n−trγ˜nTΓγn−1γ^˙n−1βnN˜nN^˙n+znLstϖnu¯−znu¯,where W^nTS^n′γ˜nTφnt=trW^nTS^n′γ˜nTφnt. Then, we can get the following inequality:(71)V˙n≤V˙n−1−kn+12zn+θn+2μ+znLstϖnu¯n−znu¯+W˜nTzn2S^n−S^n′γ^nTφnt−Γwn−1W^˙n+θ˜nzn4vn2zn4vn2+μ2−1ηnθ^˙n+N˜nzn−1βnN^˙n+trγ˜nTW^nTS^n′φntzn2−Γγn−1γ^˙n.
Based on (71), the adaptive laws are constructed as follows:(72)θ^˙n=ηnzn4vn2zn4vn2+μ2−ρnθ^n,W^˙n=ΓWnzn2S^n−S^n′γ^nTφnt−σnW^n,γ^˙n=Γγnzn2W^nTS^n′φnt−κnγ^n,N^˙n=βnzn−onN^n,ϖ˙n=−Dnznα¯n,where ρn,σn,κn,on,andDn are positive design parameters. Substituting (72) into (71), the following inequality can be obtained:(73)V˙n≤V˙n−1−kn+12zn2+θn+2μ+ρnηnθ˜nθ^n+σnW˜nTW^n+κntrγ˜nTγ^n+onβnN˜nN^n+1Dnϖ˙nt−1DnLstϖntϖ˙nt.
Using Young’s inequality, the following inequalities hold:(74)ρnηnθ˜nθ^n≤−ρn2ηnθ˜n2+ρn2ηnθn2,σnW˜nTW^n≤−σn2W˜n2+σn2Wn2,κntrγ˜nTγ^n≤−κn2γ˜nF2+κn2γnF2,onβnN˜nN^n≤−on2βnN˜n2+on2βnNn2.
Substituting (74) into (73), one has(75)V˙n≤−∑j=1nkjzj2−∑j=1nρj2ηjθ˜j2−∑j=1nσj2W˜j2−∑j=1nκj2γ˜jF2−∑j=1noj2βjN˜j2+An+∑j=1n1Djϖ˙jt−∑j=1n1DjLstϖjtϖ˙jt,where(76)An=∑j=1nθj+2μ+∑j=1nρj2ηjθj2+∑j=1nσj2Wj2+∑j=1nκj2γjF2+∑j=1noj2βjNj2.
For boundary layer errorei=si−αi,i=1,…,n−1, using the method of differentiation, we have(77)e˙i=−ei−M^i2eiM^i2ei+μ2+Gizi,…,zi+1,e1,…,ei,θ^1,…,θ^i+1,W^1,…,W^i+1,γ^1,…,γ^i+1,M^1,…,M^i,yr,y˙r,y¨r,
G1· and Gi· are continuous functions and defined as(78)G1·=−α1=−∂α1∂x1x˙1−∂α1∂θ^1θ^˙1−∂α1∂W^1W^˙1−∂α1∂γ^1γ^˙1−∂α1∂ydy˙d−∂α1∂y^dy¨d,Gi·=−αi=−∑j=1i∂αi∂xjx˙j−∂αi∂θ^iθ^˙i−∂αi∂ei−1e˙i−1−∂αi∂M^i−1M^˙i−1−∂αi∂W^iW^˙i−∂αi∂γ^iγ^˙i−∂αi∂yry˙r−∂αi∂y^ry¨r,i=2,…,n−1.
The compact setsΩ1 and Ω2 are defined as follows:(79)Ω1=yd,y˙d,y¨dT:yd2+y˙d2+y¨d2≤G0,Ω2=Vt≤c,where G0 is a positive constant. It is noting that the set Ω1×Ω2 is compact in R4n+1. There is an unknown positive number Mi on Ω1×Ω2, such that Gi≤Mi. Therefore, according to (71) and (72), we use its estimated value M^i,i=1,…,n−1.
The Lyapunov functionV is constructed as follows:(80)V=Vn+∑i=1n−112ei2+∑i=1n−112ξiM˜i2,where ξi is the design parameter. Then, the derivative of V is(81)V˙=V˙n+∑i=1n−1eie˙i−∑i=1n−11ξiM˜iM^˙i≤V˙n−∑i=1n−1ei2−∑i=1n−1M^i2ei2M^i2ei2+μ2+∑i=1n−1Miei−∑i=1n−11ξiM˜iM^˙i.
According to Lemma3, we have(82)Miei=M^iei+M˜iei≤M^i2ei2M^i2ei2+μ2+μ+M˜iei.
Substituting (82) into (81), the following inequality holds:(83)V˙≤V˙n−∑i=1n−1ei2+n−1μ−∑i=1n−11ξiM˜iM^˙i−ξiei.
The adaptive law is constructed as follows:(84)M^˙i=ξiei−ξ¯iM^i,i=1,…,n−1,where ξi and ξ¯i are the positive parameters. Substituting (84) into (83), the following inequality holds:(85)V˙≤V˙n−∑i=1n−1ei2+∑i=1n−1ξ¯iξiM˜iM^i+n−1μ.
Using Young’s inequality, the following inequality holds:(86)M˜nM^n≤−12M˜n2+12Mn2,i=1,…,n−1.
Substituting (86) into (85) produces(87)V˙≤−∑j=1nkjzj2+∑j=1nρj2ηjθ˜j2+∑j=1nσj2W˜j2+∑j=1nκj2γ˜jF2+∑j=1noj2βjN˜j2+∑j=1n−1ξ¯j2ξjM˜n2+∑j=1n−1ξ¯j2ξjMn2+n−1μ+An+∑j=1n1Djϖ˙jt−∑j=1n1DjLstϖjtϖ˙jt≤−εV+∑j=1n1Djϖ˙jt−∑j=1n1DjLstϖjtϖ˙jt+χ,where(88)χ=An+∑j=1n−1ξ¯j2ξjMn2+n−1μ,ε=min2kj,ρj,σj,κj,oj,ξ¯j.
By integrating (87), we can get the following inequality:(89)Vt≤−e−εt∫t0t∑j=1nDjLstϖjιϖ˙jιeειdι−e−εt∫t0t∑j=1n1Djϖ˙jιeειdι+χε,where χ/ε is a bounded variable and V0e−εt is bounded. Then, applying Lemma 4 to (89), we can know that all signals in the closed-loop system are bounded. Then, based on (63), we can get ut−τt−u≤σΔ, where σΔ is a positive constant. Let Lyapunov function(90)Vλ=12∑j=1nλj2.
According to (18), the derivative of Vλ can be obtained as follows:(91)V˙λ≤∑j=2nλjλj+1−pjλj+λn−pnλn+ut−τt−ut≤∑j=2np¯jλj2+σΔ2≤∑j=2np¯jλj2+σλ,from Young’s inequality λjλj+1≤1/2λj2+1/2λj+12,λnσΔ≤1/2λn2+1/2σΔ2. It can be concluded from (91) that p¯2=p2−1/2,p¯j=pj−1,j=2,…,n,p¯n=pn−1,σλ=σΔ2/2.
Therefore, it can be shown thatλi is bounded by (91). From Lemma 5 and (51), W^iT is nonnegative. Therefore, we can get the following result:(92)ϖ˙it=Diki+32zi2+DiW^iTSiEi,γ^iTφitzi2+θ^iDizi4vi2zi4vi2+μ2+Dizi2N^i2zi2N^i2+μ2≤Diki+32zi2.
On the interval0,t, the integral of (92) can be obtained as follows:(93)∫0tDiki+32zi2ιdι≤ϖit−ϖi0.
According to Lemma4, ϖit is bounded, and using Barbalat’s Lemma, we can get(94)limt⟶∞zi=0,i=1,…,n.
Furthermore, it can be obtained that(95)limt⟶∞z1=y−yr=0.
Thus, the asymptotic tracking of the closed-loop system is achieved.
To show the design idea of the proposed control scheme, the design procedure for the second-order system is presented in Figure1.Figure 1
Schematic representation of the proposed adaptive neural control scheme.Remark 2.
Compared with the control scheme proposed in this paper, similar design works have also been carried out in [39–41]. The main differences can be summarized as follows. Firstly, the uncertain input-delayed systems with periodic time-varying disturbances are considered in this paper. Secondly, the problem of adaptive asymptotic tracking control for a class of nonlinear uncertain systems with periodic time-varying disturbances and input delay is solved by developing a new adaptive control scheme based on the FSE-RBFNN approximator.


## 4. Simulation Example

In this section, consider the following second-order system:(96)x˙1=x2+x12ω12+x1ω1x12ω12+10+sint,x˙2=ut−τt+sinx1x2ω2e−x12x22ω22+cost,y=x1,where τ=0.04+0.01sint,ω1=sin0.5t, and ω2=cos0.5t are unknown time-varying disturbances.In simulation, the reference signalyd=cos0.5t and the initial condition of the system is set to be x10=1.1,x20=0.078,θ^10=0.8,θ^20=0.8M^10=0.99,ϖ10=0.9,andϖ20=0.5. We choose the numbers of FSE components as p1=p2=3. The centres of radial basis functions (RBFs) evenly cover the compact sets −4,4×−4,4 and −2,2×−2,2×−1,1×−1,1, and the width of RBFs is 0.9 and the number of nodes of the neural network is 225. The design parameters are specified as k1=14.8,k2=39,η1=η2=5,ρ1=ρ2=5,σ1=σ2=5,κ1=5,κ2=10,β1=5,β2=2,o1=1,o2=2,D1=0.1,D2=7,μ=5 and ξ1=3,ξ¯i=0.6. The adaptive gains are chosen as ΓW1=ΓW1=10I,Γγ1=Γγ1=10I,N^1=1,andN^2=0.1. The simulation time is 50 seconds, and the CPU time for the proposed algorithm is about 2 minutes for system (96). The simulation results are shown in Figures 2–8. Figure 2 shows that the system output can asymptotically track the given reference signal, and the boundedness of the closed-loop signals can be seen from Figures 3–8.Figure 2
Trajectories of system output signaly and reference signal yd.Figure 3
Trajectory of control signalu.Figure 4
Trajectories of adaptive lawsϖ^1,ϖ^2 and M^1.Figure 5
Trajectories of adaptive lawsθ^1 and θ^2.Figure 6
Trajectories of adaptive lawsN^1 and N^2.Figure 7
Trajectories of adaptive lawsγ^1 and γ^2.Figure 8
Trajectories of adaptive lawsW^1 and W^2.Remark 3.
To show the superior performance of our control scheme, for system (96), the simulation results by using the method proposed in [32] have been shown in Figures 9 and 10. From Figures 2 and 9, it can be seen that the control scheme presented in this paper has a better control performance.Figure 9
Trajectories of system output signaly and reference signal yd using method [32].Figure 10
Trajectory of control signalu using method [32].


## 5. Conclusions

In this paper, the adaptive asymptotic tracking control for a class of periodic time-varying disturbances uncertain systems with input delay is studied. To solve this problem, a dynamic surface technique based on a hybrid approximator is proposed. It is proved that the tracking error asymptotically converges to zero, and all the closed-loop signals are semiglobally uniformly bounded. Finally, the simulation example validates that the proposed design scheme is feasible. The control scheme proposed in this paper is a theoretical result, and from the simulation example, it can be seen that from the implementation and computational point of view, the feasibility of the proposed approach is guaranteed. However, how to test their technique against a realistic system is a more challenging problem, which is the main disadvantage of the proposed method and will be discussed in further research.



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