# Adaptive Dynamic Surface Control for a Class of Nonlinear Pure-Feedback Systems with Parameter Drift

Authors: Cheng He; Jian Wu; Jin Ying; Jiyang Dai; Zhe Zhang; Liangxing Jiang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In order to solve the problem of unknown parameter drift in the nonlinear pure-feedback system, a novel nonlinear pure-feedback system is proposed in which an unconventional coordinate transformation is introduced and a novel unconventional dynamic surface algorithm is designed to eliminate the problem of “calculation expansion” caused by the use of backstepping in the pure-feedback system. Meanwhile, a sufficiently smooth projection algorithm is introduced to suppress the parameter drift in the nonlinear pure-feedback system. Simulation experiments demonstrate that the designed controller ensures the global and ultimate boundedness of all signals in the closed-loop system and the appropriate designed parameters can make the tracking error arbitrarily small.


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## Body

## 1. Introduction

In recent years, nonlinear systems have been widely studied by researchers at home and abroad. Isidori [1] proposed a precise linearization technique, Krstic et al. [2] proposed backstepping, and Astolfi et al. [3] put forward adaptive control methods. After that, backstepping adaptive control has been widely developed and applied [4, 5]. In the majority of studies on nonlinear system tracking control problems [6, 7], the global or semiglobal asymptotic stability and the gradual stability of tracking error are considered as the ultimate control target of nonlinear systems. However, at present, there is no uniform method to analyze nonlinear systems in the control theory of nonlinear systems.Since nonlinear systems are different from linear systems, it is difficult to be analyzed in the way that linear systems are discussed. However, nonlinear pure-feedback systems [8, 9] are more general systems compared with lower triangle nonlinear systems, which can reflect the actual physical situation while the nonaffine pure-feedback system is one of the complicated systems. According to the literature [10, 11], there are two nonaffine functions in the pure-feedback system: system control input and system state variables. Liu [12] proposed a novel coordinate transformation in the research on the tracking problem of pure-feedback systems and achieved global asymptotic stability of tracking error using the backstepping method. Zeng et al. [13] proposed an active sliding mode disturbance rejection control of a single-input single-output nonaffine nonlinear system. However, the research of Zeng et al. is only applicable to a second-order system. For systems larger than second-order, this method cannot work. In this regard, Liu [14] proposed nonconventional coordinate transformation based on [12] and introduced a first-order auxiliary controller to solve the nonaffine control input problem of n-order system. In engineering applications, system parameters change due to various conditions, so system parameters are variables rather than constants. However, in most of the tracking control theory studies of nonlinear systems [15, 16], we often set system parameters as constants, which reduces the practicality of the tracking control theory of nonlinear systems, so it is necessary to study the influence of unknown parameter drift on system stability. For the uncertainty of parameterization, when there is external disturbance or unmodeled dynamics, parameter drift will occur, and the adaptive controller will deteriorate the system and even lead to system instability [17]. In this case, the boundedness of the estimated value of parameters cannot be guaranteed, causing divergence of other signals in the closed-loop system. A common method for dealing with this situation is to add a robust term (leakage term) to the parameter adaptive law or use a projection operator [18].In this paper, a sufficiently smooth projection algorithm [19] is introduced to deal with the parameter drift in the designed nonlinear pure-feedback system. Since nonaffine functions exist in the system state variables xi and the system control input u in the nonlinear pure-feedback system, the conventional coordinate transformation [2] will increase the difficulty of designing the controllers, so the nonconventional coordinate transformation proposed by [14] is introduced. Due to the nonaffine structure of the pure-feedback system, it is difficult to design the virtual controllers αi, and the conventional dynamic surface algorithm [20] is difficult to be directly used in the pure-feedback system. Thus, in this paper, a novel nonconventional dynamic surface algorithm is designed to reduce the complexity of designing the controllers for the nonlinear pure-feedback system. A nonlinear pure-feedback system with unknown parameters θi is designed, in which unknown parameters are added compared with [21] in order to solve the control problem of parameter uncertainty in the pure-feedback system, which is a further supplement to nonlinear pure-feedback systems. In [21], the unknown parameters are set to θ∗, which limits the selection of system parameters, thus constraining the whole system. However, the unknown parameters are θi in this paper, so the nonlinear pure-feedback system designed is more common. In this paper, backstepping method, nonconventional dynamic surface algorithm, Nussbaum function [22], and sufficiently smooth projection algorithm are combined so that the designed controller can achieve the uniform and ultimate boundedness of all signals in the closed-loop system.Main contributions of this paper are summarized as follows:(1)
The parametric drift concept is first proposed in pure-feedback systems. In order to solve the parameter drift problem of the nonlinear pure-feedback system, a sufficiently smooth projection algorithm is introduced.(2)
A novel nonconventional dynamic surface algorithm is designed according to the nonconventional coordinate transformation to eliminate the problem of “calculation expansion” in the pure-feedback system.(3)
The virtual controllersαi, the actual controller w, and the adaptive laws θ^i are designed so that the tracking error of the nonlinear pure-feedback system with parameter drift converges to a region of the origin, and all signals in the closed-loop system are uniform and ultimately bounded.Other parts of this paper are arranged as follows: In Section 2, problem statement and preliminaries are presented. In the third section, aiming at the tracking control problem of the nonlinear pure-feedback system with parameter drift, the virtual controllersαi, the actual controller w, and the parameter adaptive laws θ^i are designed and semiglobal uniform ultimate boundedness of all signals in the closed-loop system are proved. In Section 4, the correctness of the proposed control scheme is proved by simulation study. The conclusion is presented in Section 5.


## 2. Problem Statement and Preliminaries

### 2.1. Systems Description

Consider the following nonlinear pure-feedback system:(1)x˙it=fixi+1t,x¯it+ϕiTx¯it,tθi+dix¯it,t,x˙nt=fnbut,x¯nt+ϕnTx¯nt,tθn+dnx¯nt,t,yt=x1t,where 1≤i≤n−1, x¯i=x1,x2,…,xiT∈Ri with i=1,2,…,n, u∈R, and y∈R are system state variables, system control input, and system output, respectively. The nonlinear functions fixi+1,x¯i are smooth nonaffine known functions; the nonlinear function fnbu,x¯n is the smooth nonaffine known function with control input coefficient b, among which b is a unknown nonzero constant; ϕix¯i,t are known smooth functions, where ϕi0,…,0=0; θi=θ1,…,θhT∈Rh are unknown parameters; and dix1,…,xi are unmodeled dynamics and external disturbances related to state variables.In system (1), the control coefficient b is added compared with [11, 12]. The unknown parameters θi∈Rh are added in system (1) compared with [21], where i represents the ith order subsystem and h represents the number of unknown parameters in the ith subsystem. In [21], the number of parameters in each subsystem is equal, while in actual engineering systems, the number of parameters in each subsystem cannot be equal. The unknown parameters θi∈Rh designed in this paper just solve this defect, that is, different h is set according to different i.For easy analysis and calculation,fixi+1,x¯i are simplified as fi and fnbu,x¯n is simplified as fn.In order to avoid the control problem of system (1) in the design of the controller, the following assumptions are made.Remark 1.
Since the control coefficientb is an unknown nonzero constant, the positive and negative signs are unknown, i.e., the control direction is unknown.Assumption 1.
The following equation is made workable at any time:(2)∂fi∂xi+1≠0,i=1,…,n,∂fn∂u≠0.Remark 2.
Assumption1 is weaker than the assumed condition ∃κ∈R+, ∂fixi+1,x¯i/∂xi+1>κ in [11].
Because the inputu is characterized by being nonaffine. Therefore, the first-order auxiliary system of the controller is designed as follows:(3)∂fn∂xn+1u˙=w,where w is the undetermined function and xn+1=u.Assumption 2.
Unknown parameters are bounded and there is a bounded setΩ(4)Ω=θ:θ≤θ0,where θ0 is a known positive constant.Assumption 3.
d
i
x
¯
i
,
t are bounded and their known upper bounds satisfy(5)dix¯i,t≤d¯ix¯i,i=1,2,…,n,where d¯ix¯i are positive constants correlated with x¯i.Assumption 4.
Smooth nonaffine functionsfixi+1,x¯i meet(6)0<∂fixi+1,x¯i∂xi+1≤Ai,i=1,2,…,n−1,where Ai are positive constants.Lemma 1.
If smooth bounded functionsfixi+1,x¯i meet assumption 4, xi+1 are bounded.Proof.
According to mean value theorem, the following equation is obtained:(7)fixi+1,x¯i−fi0,x¯i=∂fiς,x¯i∂ςxi+1−0,where ς is the arbitrary point of 0,xi+1 interval. It can be known from assumption 4 that ∂fixi+1,x¯i/∂xi+1 are bounded and fixi+1,x¯i are also bounded; thus, xi+1 are bounded with i=1,2,…,n.Assumption 5.
The smooth reference trajectoryr is bounded and the n+1th order is derivable.


### 2.2. Preliminaries

In this paper, a sufficiently smooth projection algorithm [19] is introduced to the nonlinear pure-feedback system, as shown in (8)–(11).(8)Projdμ,θ^=μ−η1η24σ2+2σθ0m+1θ02∇pdθ^,pdθ^=θ^Tθ^−θ02,η1=pdm+1θ^,pdθ^>0,0,others,η2=12∇pdθ^Tμ+12∇pdθ^Tμ2+δ2,where σ>0 and δ>0 are arbitrary positive constants.Ifθ^0∈Ω, the algorithm has the following properties:(1)
θ^t≤θ0+σ,∀t≥0(2)
θ˜TProjdμ,θ^≥θ˜Tμ(3)
Projdμ,θ^ is m-order differentiableDefinition 1.
If the continuous functionN⋅ satisfies (9), it is called the Nussbaum function [22].(9)lims⟶+∞sup1s∫0sNξdξ=+∞,lims⟶+∞inf1s∫0sNξdξ=−∞,where ξ2cosξ is adopted as the Nussbaum function in this paper.Lemma 2 (see [23]).
IfV⋅ and ξ⋅ are smooth functions defined in 0,tf and Vt≥0 with ∀t∈0,tf, N⋅ is even smooth Nussbaum function, if the following inequality holds:(10)Vt≤C0+e−C1t∫0tNξτ+1ξ˙τeC1τdτ,∀t∈0,tf,where C1>0 and C0 are certain appropriate constants, then Vt, ξt and ∫0tNξτ+1ξ˙τdτ are bounded in 0,tf.


## 2.1. Systems Description

Consider the following nonlinear pure-feedback system:(1)x˙it=fixi+1t,x¯it+ϕiTx¯it,tθi+dix¯it,t,x˙nt=fnbut,x¯nt+ϕnTx¯nt,tθn+dnx¯nt,t,yt=x1t,where 1≤i≤n−1, x¯i=x1,x2,…,xiT∈Ri with i=1,2,…,n, u∈R, and y∈R are system state variables, system control input, and system output, respectively. The nonlinear functions fixi+1,x¯i are smooth nonaffine known functions; the nonlinear function fnbu,x¯n is the smooth nonaffine known function with control input coefficient b, among which b is a unknown nonzero constant; ϕix¯i,t are known smooth functions, where ϕi0,…,0=0; θi=θ1,…,θhT∈Rh are unknown parameters; and dix1,…,xi are unmodeled dynamics and external disturbances related to state variables.In system (1), the control coefficient b is added compared with [11, 12]. The unknown parameters θi∈Rh are added in system (1) compared with [21], where i represents the ith order subsystem and h represents the number of unknown parameters in the ith subsystem. In [21], the number of parameters in each subsystem is equal, while in actual engineering systems, the number of parameters in each subsystem cannot be equal. The unknown parameters θi∈Rh designed in this paper just solve this defect, that is, different h is set according to different i.For easy analysis and calculation,fixi+1,x¯i are simplified as fi and fnbu,x¯n is simplified as fn.In order to avoid the control problem of system (1) in the design of the controller, the following assumptions are made.Remark 1.
Since the control coefficientb is an unknown nonzero constant, the positive and negative signs are unknown, i.e., the control direction is unknown.Assumption 1.
The following equation is made workable at any time:(2)∂fi∂xi+1≠0,i=1,…,n,∂fn∂u≠0.Remark 2.
Assumption1 is weaker than the assumed condition ∃κ∈R+, ∂fixi+1,x¯i/∂xi+1>κ in [11].
Because the inputu is characterized by being nonaffine. Therefore, the first-order auxiliary system of the controller is designed as follows:(3)∂fn∂xn+1u˙=w,where w is the undetermined function and xn+1=u.Assumption 2.
Unknown parameters are bounded and there is a bounded setΩ(4)Ω=θ:θ≤θ0,where θ0 is a known positive constant.Assumption 3.
d
i
x
¯
i
,
t are bounded and their known upper bounds satisfy(5)dix¯i,t≤d¯ix¯i,i=1,2,…,n,where d¯ix¯i are positive constants correlated with x¯i.Assumption 4.
Smooth nonaffine functionsfixi+1,x¯i meet(6)0<∂fixi+1,x¯i∂xi+1≤Ai,i=1,2,…,n−1,where Ai are positive constants.Lemma 1.
If smooth bounded functionsfixi+1,x¯i meet assumption 4, xi+1 are bounded.Proof.
According to mean value theorem, the following equation is obtained:(7)fixi+1,x¯i−fi0,x¯i=∂fiς,x¯i∂ςxi+1−0,where ς is the arbitrary point of 0,xi+1 interval. It can be known from assumption 4 that ∂fixi+1,x¯i/∂xi+1 are bounded and fixi+1,x¯i are also bounded; thus, xi+1 are bounded with i=1,2,…,n.Assumption 5.
The smooth reference trajectoryr is bounded and the n+1th order is derivable.


## 2.2. Preliminaries

In this paper, a sufficiently smooth projection algorithm [19] is introduced to the nonlinear pure-feedback system, as shown in (8)–(11).(8)Projdμ,θ^=μ−η1η24σ2+2σθ0m+1θ02∇pdθ^,pdθ^=θ^Tθ^−θ02,η1=pdm+1θ^,pdθ^>0,0,others,η2=12∇pdθ^Tμ+12∇pdθ^Tμ2+δ2,where σ>0 and δ>0 are arbitrary positive constants.Ifθ^0∈Ω, the algorithm has the following properties:(1)
θ^t≤θ0+σ,∀t≥0(2)
θ˜TProjdμ,θ^≥θ˜Tμ(3)
Projdμ,θ^ is m-order differentiableDefinition 1.
If the continuous functionN⋅ satisfies (9), it is called the Nussbaum function [22].(9)lims⟶+∞sup1s∫0sNξdξ=+∞,lims⟶+∞inf1s∫0sNξdξ=−∞,where ξ2cosξ is adopted as the Nussbaum function in this paper.Lemma 2 (see [23]).
IfV⋅ and ξ⋅ are smooth functions defined in 0,tf and Vt≥0 with ∀t∈0,tf, N⋅ is even smooth Nussbaum function, if the following inequality holds:(10)Vt≤C0+e−C1t∫0tNξτ+1ξ˙τeC1τdτ,∀t∈0,tf,where C1>0 and C0 are certain appropriate constants, then Vt, ξt and ∫0tNξτ+1ξ˙τdτ are bounded in 0,tf.


## 3. Main Results

The backstepping design is based on the following nonconventional coordinate transformation [14]:(11)z1=x1−r,zi=x˙i−1−qi−1,where i=2,3,…,n+1, z1∈R, r, and qi−1 are the tracking error, smooth reference trajectory signal, and error state variables, respectively.Remark 3.
In the conventional coordinate transformation [2], xi+1 are made as the virtual controller of the ith order subsystem, as shown in (15).(12)z1=x1−r,zi=xi−ai−1.
Unlike the standard dynamic surface algorithm [20], the nonconventional dynamic surface algorithm designed in this paper is as follows.
Define the first-order low-pass filter formula.(13)τ1q˙1+q1=α1,∂fi−1∂xiτiq˙i+q˙=αi,q10=α10,qi0=αi0,where i=2,…,n, τi are the filter time constants.
The filter error equation is defined as follows:(14)y1=q1−α1,∂fi−1∂xiyi=∂fi−1∂xiqi−αi,where i=2,…,n, αi are the ith virtual controllers. The nonconventional coordinate transformation cooperates with the nonconventional dynamic surface algorithm to solve the problem of “calculation expansion” caused by the application of the nonconventional coordinate transformation in the nonlinear pure-feedback systems.Remark 4.
The first-order low-pass filter of standard dynamic surface algorithm [20](15)τi+1q˙i+1+qi+1=αi.
The filter error of standard dynamic surface algorithm(16)yi=qi−ai−1.Step 1.
According to (1) and (11), we obtain(17)z˙1=x˙1−r˙=z2+y1+α1−r˙.
Design the virtual controllerα1 as(18)α1=−c1z1+r˙,where c1 is a positive design constant.
According to (14), the following equation is obtained:(19)y1=q1−α1.
By derivingy1 with respect to time, we can obtain(20)y˙1=q˙1−α˙1=−y1τ1+c1z2+y1+α1−r˙−r¨=−y1τ1+B1z1,z2,y1,r,r˙,r¨,where B1z1,z2,y1,r,r˙,r¨ is a continuous function, the maximum of B1⋅ is set to be D1.
The Lyapunov function is defined as(21)V1=12z12+12y12.
According to (17), (20) and Young’s inequality, we have(22)V˙1≤−c1−12z12−1τ1−1+δ2y12+z1z2+D122δ,where y1⋅B1≤δy12/2+D12/2δ, δ is a positive constant.Step 2.
According to (1) and (11), we have(23)z˙2=∂f1∂x2x˙2+∂f1∂x1x˙1+∂ϕ1∂x1x˙1θ1+∂d1∂x1x˙1−q˙1=∂f1∂x2z3+y2+α2∂f1/∂x2+∂f1∂x1x˙1+∂ϕ1∂x1x˙1θ1+∂d1∂x1x˙1−q˙1.
Design the virtual controllerα2 as(24)α2=−c2z2−z1−∂f1∂x1x˙1−∂ϕ1∂x1x˙1θ^1−d¯x1122εz2+q˙1,where c2 is a positive design constant, θ^1 is the estimation of θ1, and d¯x112z2/2ε is a nonlinear damping term which is used to process the disturbance term in (23).
Design the adaptive law as(25)θ^˙1=γProjdμ1,θ^1,(26)μ1=∂ϕ1∂x1x˙1z2.
Substituting (24) into (23) yields(27)z˙2=−c2z2−z1+∂f1∂x2z3+∂f1∂x2y2+∂ϕ1∂x1x˙1θ˜1+∂d1∂x1x˙1−d¯x1122εz2,where θ˜1=θ1−θ^1.
According to (14), we can obtain(28)∂f1∂x2y2=∂f1∂x2q2−α2.
By derivingy2 with respect to time, we can obtain(29)y˙2=−y2τ2+B2,where B2z1,z2,z3,y1,y2,r,r˙,r¨ is a continuous function and the maximum of B2⋅ is set to be D2.
The Lyapunov function is defined as(30)V2=V1+12z22+12γθ˜1Tθ˜1+12y22.
According to (22), (25), (27), (28), and Young’s inequality, we have(31)V˙2≤V˙1−c2z22−z1z2+∂f1∂x2z2z3+∂f1∂x2y2z2+ε2+θ˜1∂ϕ1∂x1x˙1z2−Projdμ1,θ^1−y22τ2+δy222+D222δ.
According to the properties (2) of the sufficiently smooth projection algorithm [20], the following equation is obtained:(32)V˙2≤−c1−12z12−c2−A122z22−∑i=121τi−1+δ2yi2+∑i=12Di22δ+∂f1∂x2z2z3+ε2,where y2⋅B2≤δy22/2+D22/2δ and ε is a positive constant.Stepi 3≤i≤n−1.
According to (1) and (11), we can obtain(33)z˙i=∂fi−1∂xix˙i+∑j=1i−1∂fi−1∂xjx˙j+∑j=1i−1∂ϕi−1∂xjx˙jθi−1+∑j=1i−1∂di−1∂xjx˙j−q˙i−1=∂fi−1∂xizi+1+yi+αi∂fi−1/∂xi+∑j=1i−1∂fi−1∂xjx˙j+∑j=1i−1∂ϕi−1∂xjx˙jθi−1+∑j=1i−1∂di−1∂xjx˙j−q˙i−1.
Design the virtual controllerαi as(34)αi=−cizi−∂fi−2∂xi−1zi−1−∑j=1i−1∂fi−1∂xjx˙j−∑j=1i−1∂ϕi−1∂xjx˙jθ^i−1−i−1d¯xi−1i−122εzi+q˙i−1,where ci is a positive design constant, θ^i−1 is the estimation of θi−1, and i−1zid¯xi−1i−12/2ε is the nonlinear damping term which is used to process the disturbance term in (33).
Design the adaptive law as(35)θ^˙i−1=γProjdμi−1,θ^i−1,(36)μi−1=∑j=1i−1∂ϕi−1∂xjx˙jzi.
Substituting (34) into (33) yields(37)z˙i=−cizi−∂fi−2∂xi−1zi−1+∂fi−1∂xizi+1+∂fi−1∂xiyi+∑j=1i−1∂ϕi−1∂xjx˙jθ˜i−1+∑j=1i−1∂di−1∂xjx˙j−i−1d¯xi−1i−122εzi,where θ˜i−1=θi−1−θ^i−1.
Based on the filter error equation (14), we can obtain(38)∂fi−1∂xiyi=∂fi−1∂xiqi−αi.
By derivingyi with respect to time, the following equation is obtained:(39)y˙i=−yiτi+Bi,where Biz1,…,zi+1,y1,…,yi,r,r˙,r¨ is a continuous function and the maximum of Bi⋅ is set to be Di.
The Lyapunov function is defined asVi=Vi−1+zi2/2+θ˜i−1Tθ˜i−1/2γ+yi2/2.
According to (35), (37), (39) and Young’s inequality, we have(40)V˙i≤V˙i−1−cizi2−∂fi−2∂xi−1zi−1zi+∂fi−1∂xizizi+1+∂fi−1∂xiyizi+θ˜i−1∑j=1i−1∂ϕi−1∂xjx˙jzi−Projdμi−1,θ^i−1+∑j=1i−1∂di−1∂xjx˙jzi−i−1d¯x¯i−1i−122εzi2−yi2τi+δyi22+Di22δ.
By the properties (2) of the sufficiently smooth projection algorithm [20], the following equation is obtained:(41)V˙i≤−c1−12z12−∑j=2icj−Aj22zj2−∑j=1i1τj−1+δ2yi2+∑j=1iDj22δ+∂fi−1∂xizizi+1+ii−14ε,where yi⋅Bi≤δyi2/2+Di2/2δ.Stepn.
According to (1) and (11), we can obtain(42)z˙n=∂fn−1∂xnx˙n+∑j=1n−1∂fn−1∂xjx˙j+∑j=1n−1∂ϕn−1∂xjx˙jθn−1+∑j=1n−1∂dn−1∂xjx˙j−q˙n−1=∂fn−1∂xnzn+1+yn+αn∂fn−1/∂xn+∑j=1n−1∂fn−1∂xjx˙j+∑j=1n−1∂ϕn−1∂xjx˙jθn−1+∑j=1n−1∂dn−1∂xjx˙j−q˙n−1.
Design the virtual controllerαn as(43)αn=−cnzn−∂fn−2∂xn−1zn−1−∑j=1n−1∂fn−1∂xjx˙j−∑j=1n−1∂ϕn−1∂xjx˙jθ^n−1−n−1d¯xn−1n−122εzn+q˙n−1,where cn is a positive design constant, θ^n−1 is the estimation of θn−1, n−1znd¯xn−1n−12/2ε is the nonlinear damping term which is used to process the disturbance term in (42).
Design the adaptive law as(44)θ^˙n−1=γProjdμn−1,θ^n−1,(45)μn−1=∑j=1n−1∂ϕn−1∂xjx˙jzn.
Substituting (43) into (42) yields(46)z˙n=−cnzn−∂fn−2∂xn−1zn−1+∂fn−1∂xnzn+1+∂fn−1∂xnyn+∑j=1n−1∂ϕn−1∂xjx˙jθ˜n−1+∑j=1n−1∂dn−1∂xjx˙j−n−1d¯xn−1n−122εzn,where θ˜n−1=θn−1−θ^n−1.
According to (14), we can obtain(47)∂fn−1∂xnyn=∂fn−1∂xnqn−αn,
By derivingyn with respect to time, the following equation is obtained:(48)y˙n=−ynτn+Bn,where Bnz1,…,zn+1,y1,…,yn,r,r˙,r¨ is a continuous function and the maximum of Bn⋅ is set to be Dn.
The Lyapunov function is defined asVn=Vn−1+zn2/2+θ˜n−1Tθ˜n−1/2γ+yn2/2.
According to (44), (46), (48), and Young’s inequality, we have(49)V˙n≤V˙n−1−cnzn2−∂fn−2∂xn−1zn−1zn+∂fn−1∂xnznzn+1+∂fn−1∂xnynzn+δyn22+Dn22δ+θ˜n−1∑j=1n−1∂ϕn−1∂xjx˙jzn−Projdμn−1,θ^n−1+∑j=1n−1∂dn−1∂xjx˙jzn−n−1d¯x¯n−1n−122εzn2−yn2τn.
By the properties (2) of the sufficiently smooth projection algorithm [20], the following equation is obtained:(50)V˙n≤−c1−12z12−∑j=2ncj−Aj22zj2−∑j=1n1τj−1+δ2yj2+∑j=1nDj22δ+∂fn−1∂xnznzn+1+nn−14ε,where yn⋅Bn≤δyn2/2+Dn2/2δ.
Stepn+1. According to (1) and (11), we can obtain(51)z˙n+1=∂fn∂xn+1x˙n+1+∑j=1n∂fn∂xjx˙j+∑j=1n∂ϕn∂xjx˙jθn+∑j=1n∂dn∂xjx˙j−q˙n=∂fn∂xn+1u˙+∑j=1n∂fn∂xjx˙j+∑j=1n∂ϕn∂xjx˙jθn+∑j=1n∂dn∂xjx˙j−q˙n.Remark 5.
The control coefficientb is included in ∂fn/∂xn+1, so it is unnecessary to analyze the unknown parameter b independently, which is also the advantage of introducing nonconventional coordinate transformation in this paper.
Design the actual controller as(52)w=Nξαn+1,(53)αn+1=cn+1zn+1+∂fn−1∂xnzn+∑j=1n∂fn∂xjx˙j+∑j=1n∂ϕn∂xjx˙jθ^n+nd¯xnn22εzn+1−q˙n,where cn+1 is a positive design constant, θ^n is the estimation of θn, and nzn+1d¯xnn2/2ε is the nonlinear damping term which is used to process the disturbance term in (51).
The adaptive law of the parameterξ is designed as(54)ξ˙=zn+1αn+1.
The adaptive law of parameterθ^n is designed to be(55)θ^˙n=γProjdμn,θ^n,(56)μn=∑j=1n∂ϕn∂xjx˙jzn+1.
Substituting (52) into (51) yields(57)z˙n+1=Nξαn+1+∑j=1n∂fn∂xjx˙j+∑j=1n∂ϕn∂xjx˙jθn+∑j=1n∂dn∂xjx˙j−q˙n.where θ˜n=θn−θ^n.Theorem 1.
For system (1), under the condition that Assumptions 1–5 are satisfied, the first-order low-pass filter (13), the control law (52), and the parameter adaptive laws (26), (35), (44), (54), and (55) are adopted to make the tracking error of the system converge to a region of the origin, and all signals in the closed-loop system are uniformly and ultimately bounded.Proof.
The Lyapunov function is defined asV=Vn+zn+12/2+θ˜nTθ˜n/2γ.
According to (50), (55), (57) and Young’s inequality, we can get(58)V˙=V˙n+ξ˙−ξ˙+zn+1Nξαn+1+∑j=1n∂fn∂xjx˙jzn+1+∑j=1n∂ϕn∂xjx˙jθnzn+1+∑j=1n∂dn∂xjx˙jzn+1−q˙nzn+1−1γθ˜nTθ^˙n≤−c1−12z12−∑j=2ncj−Aj22zj2−∑j=1n1τj−1+δ2yj2−cn+1zn+12+∑j=1nDj22δ+nn−14ε+Nξξ˙+ξ˙+θ˜n∑j=1n∂ϕn∂xjx˙jzn+1−Projdμn,θ^n+nε2.
Based on the property (2) of the sufficiently smooth projection algorithm, we can obtain(59)V˙≤−c1−12z12−∑j=2ncj−Aj22zj2−∑j=1n1τj−1+δ2yj2+∑j=1nDj22δ+nn+14ε+Nξξ˙+ξ˙−cn+1zn+12,where the parameter is designed to satisfy(60)δ>0,ε>0,c1>12,ci>Ai22,i=2,…,n,1τi>1+δ2,τi>0,i=1,…,n.
Make(61)a0=minc1−12,cj−Aj22,cn+11τ1−1+δ2,1τj−1+δ2,j=2,…,n,b0=∑j=1nDj22δ,then (59) can be rewritten to(62)V˙≤−a0∑i=1n+1zi2+∑i=1nyi2+b0+nn+14ε+Nξξ˙+ξ˙≤−k0V+∑i=1nθiTθi2γ+nn+14ε+Nξξ˙+ξ˙+b0,where k0=min2a0,1.
When both ends of (62) are multiplied by the exponent ek0t, we can get(63)ddtek0tVt≤∑i=1nθiTθi2γ+nn+14ε+Nξξ˙+ξ˙+b0ek0t.
After the integration of both ends of (63) in 0,t, we can get(64)Vt≤e−k0tV0+e−k0t∫0t∑i=1nθiTθi2γ+nn+14ε+Nξ+1ξ˙+b0ek0τdτ=e−k0tV0+e−k0t∫0tNξ+1ξ˙ek0τdτ+∫0t∑i=1nθiTθi2γ+nn+14ε+b0e−k0t−τdτ.
Since the adaptive lawθ^˙i utilizes a sufficiently smooth projection algorithm, it can be known from the properties of a sufficiently smooth projection algorithm (1) that θ^i is bounded and thus θ˜i is also bounded, which means the ∫0t∑i=1nθiTθi/2γ+nn+1/4ε+b0e−k0t−τdτ is bounded.
It is noted that (64) can be written as(65)Vt≤V0+e−k0t∫0tNξ+1ξ˙ek0τdτ+C0.
According to Lemma2, it can be known that in the interval 0,tf, Vt, ξt, and e−k0t∫0tNξ+1ξ˙ek0τdτ are bounded, and it can be known from Lemma 1 that f1,…,fn are bounded and satisfies assumption 4; thus, x1,…,xn+1 are bounded. When tf tends to ∞, all signals in the closed-loop system are uniformly and ultimately bounded. The upper bound of e−k0t∫0tNξ+1ξ˙ek0τdτ is set to be CN, then (64) can be written as(66)Vt≤e−k0tV0+CN+C0.
For arbitrarily givenv≥2V0e−k0t+CN+C0, there is T; make it when t≥T, z≤v, z=z1,…,zn+1T. Appropriate parameters are selected so that the tracking error z1 can be adjusted to small region of the origin:(67)z1≤2V0e−k0t+CN+C0.


## 4. Illustrative Example

Consider the following second-order nonlinear pure-feedback system:(68)x˙1=x1+x2+x235+θ11x12+θ12x13+d1x¯1,t,x˙2=x1x2+bu+u37+θ21x1x2+dx¯2,t,z1=x1−r.Corresponding to system (1), we have(69)ϕ1Tx¯1,t=x12,x13,θ1=θ11;θ12,ϕ2Tx¯2,t=x1x22,0,θ2=θ21;θ22,d1x¯1,t=0.5x12cos1.5t,d2x¯2,t=0.7x12+x22sin3t,r=0.5sinπt+0.5sint.Set the design parameters in the above control scheme as(70)c1=2.2,c2=5,c3=5,τ1=0.01,τ2=0.01,δ=0.1,ε=0.1,A1=2,D1=D2=2,x10=0.5,x20=0.3,θ0=2,θ1=0.2sint;0.2cos1.5t,θ2=0.1sint;0.1,θ^10=0.1;0.1,θ^20=0.1;0.1,ζ0=0,u0=0,q10=0,q20=0.The control coefficient is set to beb=1,t≤5−1,others.The simulation results are shown in the following figures. Figure1 shows the system output y and the reference signal r. It can be seen that the system output y can track the reference signal r well even when the parameters θ1, θ2, and control coefficient b change. Figure 2 shows the control input u, indicating that the system input curve is asymptotically bounded. Figure 3 shows the estimated curve of the parameter θ1. Figure 4 shows the estimated curve of the parameter θ2. Figure 5 shows that functions ξ and Nξ are bounded. Figure 6 shows the system state variables x1 and x2. Figure 7 shows the error between the system output y and the reference signal r.Figure 1
The curve of tracking performance.Figure 2
Actual control lawut.Figure 3
The estimated curve of parameterθ1.Figure 4
The estimated curve of parameterθ2.Figure 5
The curve ofξ and Nξ.Figure 6
The curve of system state variablesx1 and x2.Figure 7
The curve of system error.It can be seen from Figures1–7 that system state variables x1 and x2 are bounded, actual controller u is bounded, θ^11 and θ^12 are bounded, θ^21 and θ^22 are bounded, ξ and Nξ are bounded, and error is bounded, so all signals in the closed-loop system are bounded.


## 5. Conclusion

In this paper, the adaptive tracking control for a class of nonlinear pure-feedback systems is studied to solve the parameter drift problem. Different from the conventional backstepping method, the nonconventional coordinate transformation is introduced and the nonconventional dynamic surface algorithm is designed to solve the problem “calculation expansion” in the pure-feedback system, and the sufficiently smooth projection algorithm is introduced to solve the unknown parameter drift problem in the pure-feedback system so that the designed controller can make all signals of the closed-loop system globally bounded. Finally, the correctness of the algorithm in this paper is verified by simulation. The nonlinear pure-feedback system considered in this paper has the defect, that is,fixi+1,x¯i are known functions, but fixi+1,x¯i are often not accurately modeled in engineering applications, so fixi+1,x¯i are smooth unknown functions. In future, the radial basis function (RBF) will be used to approximate the unknown functions fixi+1,x¯i, so that the adaptive control for a class of nonlinear pure-feedback systems with parameter drift is more extensive and practical.



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*Source: 3628658-2020-01-03.xml*