# Adaptive Control Design with Assigned Tracking Accuracy for a Class of Nonlinearly Parameterized Input-Delayed Systems

Authors: Dajie Yao; Jing Li; Jian Wu
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

This paper addresses the adaptive control problem of a class of nonlinear systems with unknown parameters and input delay, and the tracking accuracy of the controlled system is assigned a priori. The Pade approximation method is introduced to deal with the problem from the input delay. By creating a group of nonnegative functions, an appropriate controller is designed with the backstepping technology. It is shown that under the obtained controller, the boundedness of all the closed-loop signals is guaranteed, and the tracking error especially can converge to the accuracy assigned a priori. Finally, a simulation example is given to verify the effectiveness of the proposed scheme.


---

## Body

## 1. Introduction

In the past few decades, the nonlinear parameterized systems played an important role in the engineering field, and many scholars paid more attention to these systems (see, e.g., [1–4]). The backstepping technique especially which was originally proposed in [5, 6] provides a systematic design method for these systems. Adaptive control method is often employed to study these control systems (see, e.g., [7–11]). The authors in [7] study the adaptive control for the whole nonlinear parameterized system using uncontrollable linearization. The authors address the adaptive control of compensation for uncertain nonlinear parameters in robot manipulator in [8]. The adaptive control problem of nonlinear multi-input and multioutput (MIMO) systems is presented in [11].It is well known that the input delay phenomenon exists in many fields of engineering applications. Input delay has a bad effect on the output of the system and even damages the stability of the system. Therefore, it is very important to study the control problem of this system, and lots of interesting results have been obtained in recent years (see, e.g., [12–20]). Based on the constructed integral Lyapunov function, the authors in [13] study a class of unknown nonlinear time-delay systems by using the online approximation model of wavelet neural network. For a class of MIMO nonlinear delay systems, the authors address the adaptive neural control problem in [15]. A suboptimal learning control scheme has been proposed for nonlinear parametric time-delay systems in [19], and the high convergence speed of the trajectory tracking performance is achieved.Although some related works have been reported for uncertain nonlinear systems with the given tracking accuracy (see, e.g., [21, 22]). It is noted that no control schemes have been proposed to solve the given accuracy control problem for nonlinear parametric system with input-delay phenomenon. The work of this article is to try to solve this issue. The framework of this paper is summarized as follows. (1) According to the given system, the Pade approximation method is introduced to deal with the problem from the input delay. (2) The appropriate adaptive controller is developed based on the nonnegative functions by using the backstepping design method. (3) A simulation example is given to verify the correctness of the proposed control scheme.


## 2. Preliminaries

Some symbols and lemmas are introduced in this section.R denotes one-dimensional real value space. Ri denotes i-dimensional Euclidean space. Ci is a continuous function of ith order. ∗^ is a estimate of ∗ and ∗~ is the estimate error.To design the desired control scheme, the following two functions are introduced:(1)Nn,σm=m-σnn!,m>σ0,m≤σand(2)Sn,σm=sgn⁡m,m>σ1-2cosn⁡π2sinn⁡π4σm+σ,m≤σ,where σ is the given precision and n is an integer.Lemma 1 (see [21, 22]).
FunctionsNn,σ(m) and Sn,σ(m) have the following properties.
(1)Nn,σ(m)∈Cn and Sn,σ(m)∈Cn, where Cn is a set of continuous functions that are differentiable up to the order n.
(2)Nn,σ(m) is a function that is not less than zero, and Nn,σ(m)=0 if and only if m≤σ. The j-th order derivative of Nn+1,σ(m) is (3)djNn+1,σmdmj=Nn-j+1,σmSn-j+1,σmj,j=1,…,n.
(3)Nj,σ(m)[Sj,σ(m)]2= Nj,σ(m), where j=1,2,…,n.The following lemmas are used to design the desired controller.Lemma 2 (see[7]).
IfM(s,t) is any continuous real valued function, there are four scalar smoothing functions, for example, x(s)≥0, y(t)≥0, a(s)≥1, and b(t)≥1 satisfying the following relationships: (4)Ms,t≤xs+yt,Ms,t≤asbt,where s,t∈Rn.Lemma 3 (see[7]).
If the following inequality is true,(5)ρix1,…,xi,ξ≤x1+⋯+xiχix1,x2,…,xi,ξ,and at the same time the two functions ϕi(x1,…,xi)≥1 and ϑi(ξ)≥1 are true, of course, the following inequality is also true: (6)χix1,x2,…,xi,ξ≤ϕix1,…,xiϑiξ,where i=1,2,…,n, ξ is a number that does not change over time, ρi(x1,…,xi,ξ):Ri×R→R and ρi(0,…,0,ξ)=0, χi(·) is a function which is continuous and nonnegative, and ϑi(ξ) does not change.
Think of∑i=1nϑi(ξ) as a whole, and replace it with h. The following inequality can be obtained: (7)ρix1,…,xi,ξ≤x1+⋯+xiϕix1,…,xih,where h≥1 and it is also a constant.Lemma 4 ((Barbalat’s lemma) [21]).
If the first derivative ofg(x) is consistent, and it is also differentiable, and in the meantime x→∞, g(x) is limited, then x→∞,g(x)=0.Lemma 5 ((Young’s inequality) [21]).
For∀(x,y)∈R2, the following inequality is true: (8)xy≤υssxs+1tυtyt,where υ>0,s>1,t>1, and (s-1)(t-1)=1.


## 3. Problem Statement, Controller Design, and Analysis Process

### 3.1. Problem Statement

The nonlinearly parameterized system with input delay considered in this paper is described as follows:(9)x˙i=bixi+1+Qix¯i,θi,i=1,…,n-1x˙n=bnut-τ+Qnx¯n,θny=x1,where x¯i=[x1,x2,…,xi]T∈Ri, i=1,2,…,n. y denotes the output of the system. u(t) denotes the control input of the system, and τ denotes delay time. Qi(x¯i,θi) is a known smooth nonlinear function containing unknown parameter θi, and bi is an unknown constant, i=1,2,…,n.Assumption 6.
The reference trajectoryyr(t) and its derivative yr(i)(t) are continuous and bounded i=1,2,…,n.Assumption 7.
Althoughbi is an unknown constant, its sign is determined. In order to lose generality, it is assumed that bi>0 holds all the time. Meanwhile it is supposed that there are bmin>0 and bmax>0 which satisfy bmin≤bi≤bmax all the time for i=1,2,…,n.The purpose of this paper is to design an adaptive controller which can make the output of the system around the reference signalyr(t) within a given precision σ>0.Remark 8.
Compared with our control problem statement and [23], the main difference is that in this paper, we design the desired adaptive controller making the tracking error within a given accuracy according to the actual demand. To achieve this control issue, some nonnegative switching functions are introduced in each backstepping design, and the control performance analysis is achieved by using Barbalat’s lemma.In order to solve the problem from the input delay, Pade approximation approach proposed in [24] is introduced: (10)Lut-τ=exp⁡-τvLut=exp⁡-τv/2exp⁡τv/2Lut≈1-τv/21+τv/2Lut,where L[u(t)] is the Laplace transform of u(t), and v is Laplace variable. For further analysis, a new variable xn+1 is proposed which conforms to the following relation: (11)Lut1-τv/21+τv/2=Lxn+1t-Lut,and then the following equation is obtained: (12)u-τ2u˙=xn+1+τ2x˙n+1-u-τ2u˙.That is, the following equation can be obtained:(13)x˙n+1=-ωxn+1+2ωu,where ω=2/τ.According to the above information, we can rewrite system (9) as(14)x˙i=bixi+1+Qix¯i,θi,i=1,…,n-1x˙n=bnxn+1-bnu+Qnx¯n,θnx˙n+1=-ωxn+1+2ωuy=x1.It is noted that xn+1 is not the real variable of the system. It leads to an uncertain ω. The specific analysis process appears in step n.Remark 9.
Recently the more general result has been reported in [25] where the global uniform input-to-state stabilization of nonlinear switched systems with time-varying and periodic dynamics is present. The differences of our approach in comparison with [25] can be presented as follows. (i) In [25], an interconnected switched system without time delay has been considered. However, in interconnection (14), the input delay case is considered, and then the Pade approximation method is introduced to design the adaptive controller. (ii) Different from [25], for interconnection system (14), we consider the practical tracking control problem; that is, the desired adaptive controller makes the tracking error within a given accuracy which can be assigned according to the actual demand.


### 3.2. Controller Design and Analysis Process

In this subsection, we present a new adaptive controller design method to ensure the control precision and stability of the whole controlled system by using the backstepping technique.The error variables are defined as follows:(15)z1=x1-yrzi=xi-αi-1,i=2,3,…,n-1zn=xn-αn-1+bnωxn+1,where αi-1 are virtual controllers which will be designed in each backstepping process.To design the desired adaptive controller, the backsteping algorithm is presented as follows. Let us introduce the detailed design process from Step1 to Step n.Step 1.
According to the error description in (15), and combining with system (14), we can have(16)z˙1=b1x2+Q1x¯1,θ1-y˙r.Choose the nonnegative function as follows:(17)V1=n!Nn+1,σz1b1,and the following equation is obtained:(18)dV1dt=dNn+1,σz1dz1z˙1n!b1=Nn,σz1Sn,σz1n!b1b1x2+Q1x¯1,θ1-y˙r.The following inequalities can be obtained by using Lemma3 and the Young’s inequality:(19)Nn,σz1Sn,σz1n!b1Q1x1¯,θ1≤Nn,σz1Sn,σz12n!η1b1φ1x¯1≤Nn,σz1Sn,σz12n!12η1b12+12φ12x¯1=Nn,σz1Sn,σz12n!h1+ϕ1x¯1and(20)-Nn,σz1Sn,σz1n!b1y˙r≤Nn,σz1Sn,σz12n!b1y˙r2+1≤Nn,σz1Sn,σz12n!1bminy˙r2+1,where h1=1/2(η1/b1)2 and ϕ1(x¯1)=1/2φ12(x¯1). So (18) can be changed into(21)dV1dt≤Nn,σz1Sn,σz1n!z2+α1+h1Sn,σz1+ϕ1x¯1Sn,σz1-1bminy˙r2+1Sn,σz1.The above information is used to design the first virtual controller as(22)α1=-k1+14Nn,σz1Sn,σz1n!-h^1Sn,σz1-ϕ1x¯1Sn,σz1-σ+1Sn,σz1+1bminy˙r2+1Sn,σz1,where k1>0, and it is an invariant.Substituting (22) into (21) yields(23)dV1dt≤Nn,σz1Sn,σz1n!z2+h1-h^1Sn,σz1-k1+14Nn,σz1Sn,σz1n!-σ+1Sn,σz1≤-k1+14Nn,σz1n!2+h~1Sn,σz12Nn,σz1n!+Nn,σz1n!z2-σ+1,where h~1=h1-h^1, and the third property of Lemma 1 is used in (23).Furthermore, chooseV¯1=V1+1/2λ1h~12 and(24)h^˙1=λ1Nn,σz1Sn,σz12n!.Obviously, the following inequalities can be gained:(25)dV¯1dt=dV1dt-1λ1h~1h^˙1≤-k1+14Nn,σz1n!2+h~1Sn,σz12Nn,σz1n!+Nn,σz1n!z2-σ+1-1λ1h~1h^˙1≤-k1+14Nn,σz1n!2+Nn,σz1n!z2-σ+1-1λ1h~1h^˙1-λ1Nn,σz1Sn,σz12n!≤-k1+14Nn,σz1n!2+Nn,σz1n!z2-σ+1,where λ1>0 is an adaptive coefficient.Step i   ( i = 2,3 , … , n - 1 ) . We can see the following equation from (14) and (15):(26)z˙i=x˙i-α˙i-1=bixi+1+Qix¯i,θi-α˙i-1,where α˙i-1=∑j=1i-1∂αi-1/∂h^jh^˙j+∑j=1i-1∂αi-1/∂xjQjx¯j,θj+bjxj+1+∑j=0i-1∂αi-1/∂yr(j)yr(j+1).Choose the nonnegative functions as follows:(27)Vi=V¯i-1+Nn-i+2,σzin-i+1!bi,V¯i=Vi+12λih~i2,and we can get(28)dVidt=dV¯i-1dt+dNn-i+2,σzidziz˙in-i+1!bi=dV¯i-1dt+Nn-i+1,σziSn-i+1,σzin-i+1!bi×bixi+1+Qix¯i,θi-∑j=0i-1∂αi-1∂yrjyrj+1-∑j=1i-1∂αi-1∂h^jh^˙j-∑j=1i-1∂αi-1∂xjQjx¯j,θj+bjxj+1.By using the Young’s inequality, we have(29)Nn-i+1,σziSn-i+1,σzin-i+1!biQix¯i,θi≤Nn-i+1,σziSn-i+1,σzi2n-i+1!ηibiφix¯i≤Nn-i+1,σziSn-i+1,σzi2n-i+1!×12ηibi2+12φi2x¯i=Nn-i+1,σziSn-i+1,σzi2n-i+1!×12ηibi2+ϕix¯i,with ϕi(x¯i)=1/2φi2(x¯i),(30)-Nn-i+1,σziSn-i+1,σzin-i+1!bi∑j=0i-1∂αi-1∂yjryrj+1+∑j=1i-1∂αi-1∂h^jh^j˙≤Nn-i+1,σziSn-i+1,σzi2n-i+1!1bmin∑j=0i-1∂αi-1∂yjryrj+12+1+1bmin∑j=1i-1∂αi-1∂h^jh^j˙2+1,and(31)-Nn-i+1,σziSn-i+1,σzin-i+1!bi∑j=1i-1∂αi-1∂xjQjx¯j,θj≤Nn-i+1,σziSn-i+1,σzi2∑j=1i-1∂αi-1∂xjηjbiφjx¯j≤Nn-i+1,σziSn-i+1,σzi2×∑j=1i-112ηjbi2+12∂αi-1∂xjφjx¯j2=Nn-i+1,σziSn-i+1,σzi2∑j=1i-112ηjbi2+Nn-i+1,σziSn-i+1,σzi2∑j=1i-1∂αi-1∂xj2ϕjx¯j,where ϕj(x¯j)=1/2φj2(x¯j).Substituting (29)-(31) into (28) yields(32)dVidt≤dV¯i-1dt+Nn-i+1,σziSn-i+1,σzin-i+1!×xi+1+Sn-i+1,σzi12ηibi2+ϕix¯i+βi+Sn-i+1,σzi∑j=1i-112ηjbi2+∂αi-1∂xj2ϕjx¯j=dV¯i-1dt+Nn-i+1,σziSn-i+1,σzin-i+1!×zi+1+αi+hiSn-i+1,σzi+βi+Sn-i+1,σzi∑j=1i∂αi-1∂xj2ϕjx¯j,where hi=∑j=1i1/2(ηj/bi)2, and βi=Sn-i+1,σ(zi){1/bmin∑j=0i-1∂αi-1/∂y(j)r[(yr(j+1))2+1] + 1/bmin∑j=1i-1∂αi-1/∂h^j[(h^j˙)2+1]+bmax/bmin∑j=1i-1(xj+12+1)}.Based on (32), the i-th virtual controller is designed as(33)αi=-ki+54Nn-i+1,σziSn-i+1,σzin-i+1!-h^iSn-i+1,σzi-βi-σ+1Sn-i+1,σzi-Sn-i+1,σzi∑j=1i∂αi-1∂xj2ϕjx¯j,where ki>0 is a design parameter.Then, we can have(34)dVidt≤dV¯i-1dt+Nn-i+1,σziSn-i+1,σzin-i+1!×zi+1+hiSn-i+1,σzi-h^iSn-i+1,σzi-ki+54Nn-i+1,σziSn-i+1,σzin-i+1!-σ+1Sn-i+1,σzi≤-∑j=1ikjNn-j+1,σzjn-j+1!2+Nn-i+1,σziSn-i+1,σzi2n-i+1!h~i-14Nn-i+2,σzi-1n-i+2!2+Nn-i+2,σzi-1n-i+2!zi-σ+1-54Nn-i+1,σzin-i+1!2+Nn-i+1,σzin-i+1!zi+1-σ+1.From the above information, chooseV¯i=Vi+1/λih~i2 and then we can gain(35)dV¯idt=dVidt-1λih~ih^˙i≤-∑j=1ikjNn-j+1,σzjn-j+1!2-14Nn-i+2,σzi-1n-i+2!2+Nn-i+2,σzi-1n-i+2!zi-σ+1-54Nn-i+1,σzin-i+1!2+Nn-i+1,σzin-i+1!zi+1-σ+1+Nn-i+1,σziSn-i+1,σ2n-i+1!h~i-1λih~ih^˙i≤-∑j=1ikjNn-j+1,σzjn-j+1!2+Gi-14Nn-i+1,σzin-i+1!2+Nn-i+1,σzin-i+1!zi+1-σ+1,where Gi=-1/4[Nn-i+2,σ(zi-1)(n-i+2)!]2 + Nn-i+2,σ(zi-1)(n-i+2)![zi-(σ+1)]-[Nn-i+1,σ(zi)(n-i+1)!]2. The i-th adaptation law is designed as(36)h^˙i=λiNn-i+1,σziSn-i+1,σzi2n-i+1!,where λi>0.Next, we show thatGi≤0 is true all the time.Whenzi≤(σ+1), Gi≤0 is clearly right. When zi>(σ+1), we can use Young’s inequality to prove it; that is,(37)Gi≤zi-σ+12-Nn-i+1,σzin-i+1!2=zi-σ+12-zi-σn-i+1n-i+1!n-i+1!2=zi-σ+12-zi-σ2n-i+1≤zi-σ2-zi-σ2n-i+1=zi-σ21-zi-σ2n-i.So, Gi≤0 is true all the time.From the above information, the inequality is clearly true as follows:(38)dV¯idt≤-∑j=1ikjNn-j+1,σzjn-j+1!2-14Nn-i+1,σzin-i+1!2+Nn-i+1,σzin-i+1!zi+1-σ+1.Step n . Choose V¯n=Vn+1/2λnh~n2, Vn=V¯n-1+N2,σ(zn)/bn, where h~n=hn-h^n, hn=1/2∑j=1n1/2(ηj/bn)2 is an unknown constant, and λn>0 is a design parameter.The actual controller is designed as(39)u=-kn+1N1,σznS1,σzn-h^nS1,σzn-βn-S1,σzn∑j=1n∂αn-1∂xj2ϕjx¯j,where βn=S1,σ(zn){1/bmin∑j=0n-1∂αn-1/∂y(j)r[(yr(j+1))2+1]+1/bmin∑j=1n-1∂αn-1/∂h^j[(h^j˙)2 + 1]+bmax/bmin∑j=1n-1(xj+12+1)}, and the n-th adaptive law is designed as(40)h^˙n=λnN1,σznS1,σzn2.Then, we gain(41)dV¯ndt=dV¯n-1dt+N1,σznS1,σznbnx˙n-α˙n-1+bnωx˙n+1-1λnh~nh^˙n≤dV¯n-1dt+N1,σznS1,σz1bnbnxn+1-bnu+Qnx¯n,θn+bnω-ωxn+1+2ωuN1,σznS1,σznβn-1λnh~nh^˙n≤-∑j=1nkjNn-j+1,σzjn-j+1!2-14N2,σzn-12!2-N1,σzn2+N2,σzn-12!zn-σ+1≤-∑j=1nkjNn-j+1,σzjn-j+1!2.(41) indicates that the nonnegative function V is not increasing, and at the same time we can see that all the closed-loop signals are bounded. Furthermore, according to Lemma 4, the conclusion of (41) also shows that z1t≤σ when t→∞.Theorem 10.
Under Assumptions6 and 7 and considering the virtual controllers (22) and (33), the actual controller (39), and the parameters adaption laws (24), (36), and (40) for system (9), we can have(i)
all the closed-loop signals are bounded,(ii)
the tracking errorz1t≤σ holds as t→∞.


## 3.1. Problem Statement

The nonlinearly parameterized system with input delay considered in this paper is described as follows:(9)x˙i=bixi+1+Qix¯i,θi,i=1,…,n-1x˙n=bnut-τ+Qnx¯n,θny=x1,where x¯i=[x1,x2,…,xi]T∈Ri, i=1,2,…,n. y denotes the output of the system. u(t) denotes the control input of the system, and τ denotes delay time. Qi(x¯i,θi) is a known smooth nonlinear function containing unknown parameter θi, and bi is an unknown constant, i=1,2,…,n.Assumption 6.
The reference trajectoryyr(t) and its derivative yr(i)(t) are continuous and bounded i=1,2,…,n.Assumption 7.
Althoughbi is an unknown constant, its sign is determined. In order to lose generality, it is assumed that bi>0 holds all the time. Meanwhile it is supposed that there are bmin>0 and bmax>0 which satisfy bmin≤bi≤bmax all the time for i=1,2,…,n.The purpose of this paper is to design an adaptive controller which can make the output of the system around the reference signalyr(t) within a given precision σ>0.Remark 8.
Compared with our control problem statement and [23], the main difference is that in this paper, we design the desired adaptive controller making the tracking error within a given accuracy according to the actual demand. To achieve this control issue, some nonnegative switching functions are introduced in each backstepping design, and the control performance analysis is achieved by using Barbalat’s lemma.In order to solve the problem from the input delay, Pade approximation approach proposed in [24] is introduced: (10)Lut-τ=exp⁡-τvLut=exp⁡-τv/2exp⁡τv/2Lut≈1-τv/21+τv/2Lut,where L[u(t)] is the Laplace transform of u(t), and v is Laplace variable. For further analysis, a new variable xn+1 is proposed which conforms to the following relation: (11)Lut1-τv/21+τv/2=Lxn+1t-Lut,and then the following equation is obtained: (12)u-τ2u˙=xn+1+τ2x˙n+1-u-τ2u˙.That is, the following equation can be obtained:(13)x˙n+1=-ωxn+1+2ωu,where ω=2/τ.According to the above information, we can rewrite system (9) as(14)x˙i=bixi+1+Qix¯i,θi,i=1,…,n-1x˙n=bnxn+1-bnu+Qnx¯n,θnx˙n+1=-ωxn+1+2ωuy=x1.It is noted that xn+1 is not the real variable of the system. It leads to an uncertain ω. The specific analysis process appears in step n.Remark 9.
Recently the more general result has been reported in [25] where the global uniform input-to-state stabilization of nonlinear switched systems with time-varying and periodic dynamics is present. The differences of our approach in comparison with [25] can be presented as follows. (i) In [25], an interconnected switched system without time delay has been considered. However, in interconnection (14), the input delay case is considered, and then the Pade approximation method is introduced to design the adaptive controller. (ii) Different from [25], for interconnection system (14), we consider the practical tracking control problem; that is, the desired adaptive controller makes the tracking error within a given accuracy which can be assigned according to the actual demand.


## 3.2. Controller Design and Analysis Process

In this subsection, we present a new adaptive controller design method to ensure the control precision and stability of the whole controlled system by using the backstepping technique.The error variables are defined as follows:(15)z1=x1-yrzi=xi-αi-1,i=2,3,…,n-1zn=xn-αn-1+bnωxn+1,where αi-1 are virtual controllers which will be designed in each backstepping process.To design the desired adaptive controller, the backsteping algorithm is presented as follows. Let us introduce the detailed design process from Step1 to Step n.Step 1.
According to the error description in (15), and combining with system (14), we can have(16)z˙1=b1x2+Q1x¯1,θ1-y˙r.Choose the nonnegative function as follows:(17)V1=n!Nn+1,σz1b1,and the following equation is obtained:(18)dV1dt=dNn+1,σz1dz1z˙1n!b1=Nn,σz1Sn,σz1n!b1b1x2+Q1x¯1,θ1-y˙r.The following inequalities can be obtained by using Lemma3 and the Young’s inequality:(19)Nn,σz1Sn,σz1n!b1Q1x1¯,θ1≤Nn,σz1Sn,σz12n!η1b1φ1x¯1≤Nn,σz1Sn,σz12n!12η1b12+12φ12x¯1=Nn,σz1Sn,σz12n!h1+ϕ1x¯1and(20)-Nn,σz1Sn,σz1n!b1y˙r≤Nn,σz1Sn,σz12n!b1y˙r2+1≤Nn,σz1Sn,σz12n!1bminy˙r2+1,where h1=1/2(η1/b1)2 and ϕ1(x¯1)=1/2φ12(x¯1). So (18) can be changed into(21)dV1dt≤Nn,σz1Sn,σz1n!z2+α1+h1Sn,σz1+ϕ1x¯1Sn,σz1-1bminy˙r2+1Sn,σz1.The above information is used to design the first virtual controller as(22)α1=-k1+14Nn,σz1Sn,σz1n!-h^1Sn,σz1-ϕ1x¯1Sn,σz1-σ+1Sn,σz1+1bminy˙r2+1Sn,σz1,where k1>0, and it is an invariant.Substituting (22) into (21) yields(23)dV1dt≤Nn,σz1Sn,σz1n!z2+h1-h^1Sn,σz1-k1+14Nn,σz1Sn,σz1n!-σ+1Sn,σz1≤-k1+14Nn,σz1n!2+h~1Sn,σz12Nn,σz1n!+Nn,σz1n!z2-σ+1,where h~1=h1-h^1, and the third property of Lemma 1 is used in (23).Furthermore, chooseV¯1=V1+1/2λ1h~12 and(24)h^˙1=λ1Nn,σz1Sn,σz12n!.Obviously, the following inequalities can be gained:(25)dV¯1dt=dV1dt-1λ1h~1h^˙1≤-k1+14Nn,σz1n!2+h~1Sn,σz12Nn,σz1n!+Nn,σz1n!z2-σ+1-1λ1h~1h^˙1≤-k1+14Nn,σz1n!2+Nn,σz1n!z2-σ+1-1λ1h~1h^˙1-λ1Nn,σz1Sn,σz12n!≤-k1+14Nn,σz1n!2+Nn,σz1n!z2-σ+1,where λ1>0 is an adaptive coefficient.Step i   ( i = 2,3 , … , n - 1 ) . We can see the following equation from (14) and (15):(26)z˙i=x˙i-α˙i-1=bixi+1+Qix¯i,θi-α˙i-1,where α˙i-1=∑j=1i-1∂αi-1/∂h^jh^˙j+∑j=1i-1∂αi-1/∂xjQjx¯j,θj+bjxj+1+∑j=0i-1∂αi-1/∂yr(j)yr(j+1).Choose the nonnegative functions as follows:(27)Vi=V¯i-1+Nn-i+2,σzin-i+1!bi,V¯i=Vi+12λih~i2,and we can get(28)dVidt=dV¯i-1dt+dNn-i+2,σzidziz˙in-i+1!bi=dV¯i-1dt+Nn-i+1,σziSn-i+1,σzin-i+1!bi×bixi+1+Qix¯i,θi-∑j=0i-1∂αi-1∂yrjyrj+1-∑j=1i-1∂αi-1∂h^jh^˙j-∑j=1i-1∂αi-1∂xjQjx¯j,θj+bjxj+1.By using the Young’s inequality, we have(29)Nn-i+1,σziSn-i+1,σzin-i+1!biQix¯i,θi≤Nn-i+1,σziSn-i+1,σzi2n-i+1!ηibiφix¯i≤Nn-i+1,σziSn-i+1,σzi2n-i+1!×12ηibi2+12φi2x¯i=Nn-i+1,σziSn-i+1,σzi2n-i+1!×12ηibi2+ϕix¯i,with ϕi(x¯i)=1/2φi2(x¯i),(30)-Nn-i+1,σziSn-i+1,σzin-i+1!bi∑j=0i-1∂αi-1∂yjryrj+1+∑j=1i-1∂αi-1∂h^jh^j˙≤Nn-i+1,σziSn-i+1,σzi2n-i+1!1bmin∑j=0i-1∂αi-1∂yjryrj+12+1+1bmin∑j=1i-1∂αi-1∂h^jh^j˙2+1,and(31)-Nn-i+1,σziSn-i+1,σzin-i+1!bi∑j=1i-1∂αi-1∂xjQjx¯j,θj≤Nn-i+1,σziSn-i+1,σzi2∑j=1i-1∂αi-1∂xjηjbiφjx¯j≤Nn-i+1,σziSn-i+1,σzi2×∑j=1i-112ηjbi2+12∂αi-1∂xjφjx¯j2=Nn-i+1,σziSn-i+1,σzi2∑j=1i-112ηjbi2+Nn-i+1,σziSn-i+1,σzi2∑j=1i-1∂αi-1∂xj2ϕjx¯j,where ϕj(x¯j)=1/2φj2(x¯j).Substituting (29)-(31) into (28) yields(32)dVidt≤dV¯i-1dt+Nn-i+1,σziSn-i+1,σzin-i+1!×xi+1+Sn-i+1,σzi12ηibi2+ϕix¯i+βi+Sn-i+1,σzi∑j=1i-112ηjbi2+∂αi-1∂xj2ϕjx¯j=dV¯i-1dt+Nn-i+1,σziSn-i+1,σzin-i+1!×zi+1+αi+hiSn-i+1,σzi+βi+Sn-i+1,σzi∑j=1i∂αi-1∂xj2ϕjx¯j,where hi=∑j=1i1/2(ηj/bi)2, and βi=Sn-i+1,σ(zi){1/bmin∑j=0i-1∂αi-1/∂y(j)r[(yr(j+1))2+1] + 1/bmin∑j=1i-1∂αi-1/∂h^j[(h^j˙)2+1]+bmax/bmin∑j=1i-1(xj+12+1)}.Based on (32), the i-th virtual controller is designed as(33)αi=-ki+54Nn-i+1,σziSn-i+1,σzin-i+1!-h^iSn-i+1,σzi-βi-σ+1Sn-i+1,σzi-Sn-i+1,σzi∑j=1i∂αi-1∂xj2ϕjx¯j,where ki>0 is a design parameter.Then, we can have(34)dVidt≤dV¯i-1dt+Nn-i+1,σziSn-i+1,σzin-i+1!×zi+1+hiSn-i+1,σzi-h^iSn-i+1,σzi-ki+54Nn-i+1,σziSn-i+1,σzin-i+1!-σ+1Sn-i+1,σzi≤-∑j=1ikjNn-j+1,σzjn-j+1!2+Nn-i+1,σziSn-i+1,σzi2n-i+1!h~i-14Nn-i+2,σzi-1n-i+2!2+Nn-i+2,σzi-1n-i+2!zi-σ+1-54Nn-i+1,σzin-i+1!2+Nn-i+1,σzin-i+1!zi+1-σ+1.From the above information, chooseV¯i=Vi+1/λih~i2 and then we can gain(35)dV¯idt=dVidt-1λih~ih^˙i≤-∑j=1ikjNn-j+1,σzjn-j+1!2-14Nn-i+2,σzi-1n-i+2!2+Nn-i+2,σzi-1n-i+2!zi-σ+1-54Nn-i+1,σzin-i+1!2+Nn-i+1,σzin-i+1!zi+1-σ+1+Nn-i+1,σziSn-i+1,σ2n-i+1!h~i-1λih~ih^˙i≤-∑j=1ikjNn-j+1,σzjn-j+1!2+Gi-14Nn-i+1,σzin-i+1!2+Nn-i+1,σzin-i+1!zi+1-σ+1,where Gi=-1/4[Nn-i+2,σ(zi-1)(n-i+2)!]2 + Nn-i+2,σ(zi-1)(n-i+2)![zi-(σ+1)]-[Nn-i+1,σ(zi)(n-i+1)!]2. The i-th adaptation law is designed as(36)h^˙i=λiNn-i+1,σziSn-i+1,σzi2n-i+1!,where λi>0.Next, we show thatGi≤0 is true all the time.Whenzi≤(σ+1), Gi≤0 is clearly right. When zi>(σ+1), we can use Young’s inequality to prove it; that is,(37)Gi≤zi-σ+12-Nn-i+1,σzin-i+1!2=zi-σ+12-zi-σn-i+1n-i+1!n-i+1!2=zi-σ+12-zi-σ2n-i+1≤zi-σ2-zi-σ2n-i+1=zi-σ21-zi-σ2n-i.So, Gi≤0 is true all the time.From the above information, the inequality is clearly true as follows:(38)dV¯idt≤-∑j=1ikjNn-j+1,σzjn-j+1!2-14Nn-i+1,σzin-i+1!2+Nn-i+1,σzin-i+1!zi+1-σ+1.Step n . Choose V¯n=Vn+1/2λnh~n2, Vn=V¯n-1+N2,σ(zn)/bn, where h~n=hn-h^n, hn=1/2∑j=1n1/2(ηj/bn)2 is an unknown constant, and λn>0 is a design parameter.The actual controller is designed as(39)u=-kn+1N1,σznS1,σzn-h^nS1,σzn-βn-S1,σzn∑j=1n∂αn-1∂xj2ϕjx¯j,where βn=S1,σ(zn){1/bmin∑j=0n-1∂αn-1/∂y(j)r[(yr(j+1))2+1]+1/bmin∑j=1n-1∂αn-1/∂h^j[(h^j˙)2 + 1]+bmax/bmin∑j=1n-1(xj+12+1)}, and the n-th adaptive law is designed as(40)h^˙n=λnN1,σznS1,σzn2.Then, we gain(41)dV¯ndt=dV¯n-1dt+N1,σznS1,σznbnx˙n-α˙n-1+bnωx˙n+1-1λnh~nh^˙n≤dV¯n-1dt+N1,σznS1,σz1bnbnxn+1-bnu+Qnx¯n,θn+bnω-ωxn+1+2ωuN1,σznS1,σznβn-1λnh~nh^˙n≤-∑j=1nkjNn-j+1,σzjn-j+1!2-14N2,σzn-12!2-N1,σzn2+N2,σzn-12!zn-σ+1≤-∑j=1nkjNn-j+1,σzjn-j+1!2.(41) indicates that the nonnegative function V is not increasing, and at the same time we can see that all the closed-loop signals are bounded. Furthermore, according to Lemma 4, the conclusion of (41) also shows that z1t≤σ when t→∞.Theorem 10.
Under Assumptions6 and 7 and considering the virtual controllers (22) and (33), the actual controller (39), and the parameters adaption laws (24), (36), and (40) for system (9), we can have(i)
all the closed-loop signals are bounded,(ii)
the tracking errorz1t≤σ holds as t→∞.


## 4. Simulation Example

Consider the following nonlinear system:(42)x˙1=b1x2+Q1x¯1,θ1x˙2=b2ut-τ+Q2x¯2,θ2y=x1,where x¯2=[x1,x2]T∈R2 and y denotes the system state vector and the system output. u is the system control input and τ is the input delay. In this example, we suppose that the reference trajectory yr=cos⁡(2t) is given a priori.It is shown that Assumptions6 and 7 hold. Here a set of known functions are Q1(x¯1,θ1)=sin(x1θ1) and Q2(x¯2,θ2)=sin⁡(x1x2θ2)+2. By using Lemmas 2 and 3, the following functions are acquired φ1(x¯1)=x1+2 and φ2(x¯1)=x1x2+2. By adopting the proposed control algorithm, the actual and virtual controllers are, respectively, designed as (39) with n=2 and (22), and the adaptive laws are selected as (40) with n=2 and (24).The initial conditions are chosen asx1(0)=1.5,x2(0)=0.5 and h^1(0)=h^2(0)=0. The rest of the design parameters are selected as k1=5,k2=10 and λ1=2,λ2=5. The input delay is chosen as τ=0.0027. The tracking accuracy is prescribed as σ=0.02; that is, yt-yrt≤0.02 as t→∞.The simulated results are presented in Figures1–5. From Figure 1, the output state y tracks the reference trajectory yr with the prescribed accuracy 0.02 quickly. From Figure 2, the tracking error z1 falls into a priori known domain finally. The actual control signal u(t) is displayed in Figure 5. The adaptation laws are shown in Figures 3 and 4.Figure 1
The reference trajectoryyr(t) and the output signal y(t).Figure 2
The tracking error signalz1(t).Figure 3
The adaptation lawh^1.Figure 4
The adaptation lawh^2.Figure 5
The control inputu(t).


## 5. Conclusions

This paper deals with the tracking control problem of nonlinear parameterized system with input time delay. A new method about the adaptive control is proposed by using the backstepping technique. Some special nonnegative functions are introduced to design desired controller such that the tracking error of the delayed system satisfies the assigned tracking accuracy, and all the closed-loop signals are bounded. The simulation is given to verify the effectiveness of the proposed method.



---

*Source: 8602719-2019-02-03.xml*