# Approximation-Based Fixed-Time Adaptive Tracking Control for a Class of Uncertain Nonlinear Pure-Feedback Systems

Authors: Cheng He; Jian Wu; Jiyang Dai; Zhe Zhang; Libin Xu; Pinwei Li
Journal: Complexity
 (2020)

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## Abstract

This paper examines approximation-based fixed-time adaptive tracking control for a class of uncertain nonlinear pure-feedback systems. Novel virtual and actual controllers are designed that resolve the meaninglessness of virtual and actual controllers at the origin and in the negative domain, and the sufficient condition for the system to have semiglobal fixed-time stability is also provided. Radial basis function neural networks are introduced to approximate unknown functions for solving the fixed-time control problem of unknown nonlinear pure-feedback systems, and the mean value theorem is used to solve the problem of nonaffine structure in nonlinear pure-feedback systems. The controllers designed in this paper ensure that all signals in the closed-loop system are semiglobally uniform and ultimately bounded in a fixed time. Two simulation results show that appropriate design parameters can limit the tracking error within a region of the origin in a fixed time.


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## Body

## 1. Introduction

Nonlinear pure-feedback systems [1, 2] are more common than general strict feedback nonlinear systems or nonlinear systems with input affine structure. Such systems have been studied widely in recent years because they can reflect more accurately the working conditions of actual engineering systems owing to the nonaffine structure of their control inputs and state variables.In 1988, David S. Broomhead proposed the radial basis function (RBF) neural network [3], which is used widely in pattern recognition, signal processing, and control system theory and application because of its simple structure and generalizability. Many scholars have used it to address the uncertainties of nonlinear systems based on its approximation ability, which has been demonstrated by many researchers [4–7]. Kanellakopoulos et al. [8, 9] proposed the backstepping method and the backstepping adaptive control scheme for a class of strict feedback nonlinear systems in 1991. The adaptive control of RBF neural networks has attracted extensive attention [10–18], resulting in the stability analysis method of RBF neural networks based on the Lyapunov method [19–21]. Finite-time stability technology can be traced back to the 1960s [22], and it developed rapidly in the 1990s, owing primarily to the improvement of finite-time Lyapunov theory [23] and homogeneous systems [24]. Much has been achieved in recent years on finite-time stability [25–30]. A finite-time control system ensures that a nonlinear system converges in a finite time, but the convergence time is related to the initial state of the system. For this reason, fixed-time control systems have been proposed [31–34] that ensure that the upper limit of the fixed convergence time is no longer related to the initial state of the system but only depends on the design parameters. Since Polyakov et al. first proposed fixed-time stability control [33], nonlinear fixed-time control has developed rapidly and has been studied by many scholars. For example, in [35], a fixed-time control with generalized directional topology was proposed for nonlinear multiagent systems; in [36], a prescribed performance fixed-time recurrent neural network control was proposed for a class of uncertain nonlinear systems; in [37], a fast fixed-time nonsingular terminal sliding mode control was proposed to solve the problem of chaos suppression of power systems; and in [38], a fixed-time observer was proposed to detect distributed faults of nonlinear multiagent systems.In [35–40], strict feedback nonlinear systems are considered primarily, and the fixed-time control problem of more common nonlinear systems is not solved. In this paper, the fixed-time control problem is solved based on nonlinear pure-feedback systems, and the sufficient condition and design steps for semiglobal fixed-time stability are provided.The principal contributions of this paper are as follows:(1)
The fixed-time control algorithm proposed in [35–40] does not solve the problem of the nonaffine structure of the control input ut. This paper applies fixed-time control theory in nonlinear pure-feedback systems to solve this problem.(2)
The controller designed in [40–43] has a power function similar to z2q−1, 0<q<1. Not selecting q properly results in singularity. A novel fixed-time controller is designed in this paper to solve this problem.(3)
An RBF neural network control algorithm is introduced to approximate the unknown functionsfi⋅ to overcome the difficulty of modeling accurately and solving the problem of interference in nonlinear pure-feedback systems.The remainder of this paper is organized as follows. Section2 presents the problem description and preliminaries. Section 3 proposes fixed-time adaptive neural tracking control using backstepping, adaptive neural networks, and Lyapunov functions for a class of unknown nonlinear pure-feedback systems to solve the problem of fixed-time tracking control for nonlinear systems with nonaffine structure. In Section 4, all signals in the closed-loop system are proved to be semiglobally uniform and ultimately bounded. In Section 5, the proposed control scheme is proved to be effective through simulation experiments. Section 6 draws conclusions.


## 2. Problem Description and Preliminaries

### 2.1. Problem Description

Consider the following nonlinear pure-feedback system:(1)x˙it=fix¯it,xi+1t+dix¯i,t,x˙nt=fnx¯nt,utdnx¯n,t,yt=x1t,where 1≤i≤n−1, x¯i=x1t,…,xitT∈Ri with i=1,…,n, ut∈R, and yt∈R are state variables, system input, and system output, respectively. fi⋅ are unknown but smooth nonaffine functions, and di⋅ are unknown but bounded disturbances.According to the mean value theorem [44], fix¯i,xi+1=fix¯i,xi0+hiμixi+1−xi0 and fnx¯n,u=fnx¯n,xn0+hnμnu−xn0, where hiμi:=hix¯i,xμi=∂fix¯i,xμi/∂xμi with i=1,...,n, xμi=μixi+1+1−μixi0 with 0<μi<1, and xi0 is a known quantity at the given time t0. System (1) can be written as(2)x˙i=fix¯i,xi0+hiμi⋅xi+1−xi0+dix¯i,t,x˙n=fnx¯n,xn0+hnμn⋅u−xn0+dnx¯n,t,y=x1.Remark 1.
It can be seen from system (2) that the mean value theorem separates the nonaffine structure from fi⋅ in system (1).
This paper aims to design a fixed-time controller that can meet the fixed-time control requirements in nonlinear pure-feedback systems, enabling the system outputy to track the reference signal yd in a fixed time. All of the signals of the closed-loop system are semiglobally uniform and ultimately bounded.Assumption 1 (see [45]).
Unknown smooth nonlinear functionshi⋅ are bounded, and there are known positive constants b and c with 0<b≤hi⋅<c<∞, ∀x¯i,xi+1∈Ri×R. Without loss of generality, we assume that 0<b≤hi⋅, i=1,...,n.
For ease of calculation, vector functions are defined asy¯d,i=yd,yd1,...,ydiT, i=1,...,n, where ydi is the ith derivative of yd.Assumption 2.
Reference signal vector functionsy¯di are known smooth continuous bounded functions. y¯di∈Ωdi⊂Ri+1 with i=1,...,n, where Ωdi are known compact sets and reference signal yd is an n-order differentiable and bounded function.


### 2.2. Fixed Time

Definition 1.
Consider the following nonlinear system:(3)x˙t=ft,x,x0=x0,where x∈Rn and f:R+×Rn⟶Rn, and assume that the origin is an equilibrium point.Lemma 1 (see [46]).
If there exist design constantsϕ1>0, ϕ2>0, α∈1,+∞, and β∈0,1 such that(4)V˙x≤−ϕ1Vαx−ϕ2Vβx,where Vx is a continuous differentiable positive definite function, then system (3) is global fixed-time stable, and the fixed convergence time satisfies(5)T≤Tmax:=1ϕ1α−1+1ϕ21−β.Lemma 2 (see [40]).
If there exist design constantsϕ1>0, ϕ2>0, α∈1,∞, β∈0,1, τ∈0,∞, and ϖ∈0,1 such that(6)V˙x≤−ϕ1Vαx−ϕ2Vβx+τ,then the origin of system (3) is practical fixed-time stable and the fixed-time T can be estimated by(7)T≤Tmax:=1ϕ1ϖα−1+1ϕ2ϖ1−β.The residual set of the solution of system (3) is given by(8)x∈Vx≤minτ1−ϖϕ11/α,τ1−ϖϕ21/β.Lemma 3 (see [47]).
Letx1,x2,...,xn≥0. Then,(9)∑i=1nxiρ≥∑i=1nxiρ,if0<ρ≤1,∑i=1nxiρ≥n1−ρ∑i=1nxiρ,if1<ρ≤∞.Lemma 4 (see [48]).
For any variablex∈R and any positive constant κ, the following relationship holds:(10)0≤x<κ+x2x2+κ2.Lemma 5.
Fory≥x>0, x, y∈R and any positive constant χ, then the following is satisfied(11)yy+χ≥xx+χ.Proof.
(12)
y
y
+
χ
−
x
x
+
χ
=
y
+
χ
y
+
χ
−
x
+
χ
x
+
χ
−
χ
y
+
χ
+
χ
x
+
χ
=
y
+
χ
−
x
+
χ
+
χ
x
+
χ
−
χ
y
+
x
≥
0.


### 2.3. Neural Network

An RBF neural network [11, 49] is applied in this paper to approximate arbitrary continuous functions. The mathematical expression of an RBF neural network is as follows:(13)φ^=WTSZ,where W=w1,w2,…,wlT∈Rl is the weight vector, l>1 is the number of nodes of the neural network, Z∈ΩZ⊂Rq is the input of the RBF neural network, q is the input dimension of the RBF neural network, SZ=s1Z,s2Z,…,slZT∈Rl is the basis vector function, and siZ is the output of the ith node. A Gaussian function is always chosen as siZ, i.e., siZ=exp−Z−ξiTZ−ξi/ri2, i=1,...,l, where ri is the width of the base function and ξi=ξi1,ξi2,…,ξiqT is the center of the basis function. With a sufficient number l of nodes selected, an RBF neural network can approximate an arbitrary continuous function φZ in a compact set ΩZ∈Rq with arbitrary accuracy ε.(14)φZ=W∗TSZ+δZ,∀Z∈ΩZ∈Rq,where δZ is the approximation error with δZ≤ε and W∗ is the given ideal constant weight vector, which is defined as(15)W∗=argminW∈RlsupZ∈ΩZφZ−φ^Z.In this paper, letθi=maxWi∗2/b,i=1,2,...,n with θ˜i=θi−θ^i, where θ^i are the estimates of the unknown constants θi, Wi∗ are the ideal weight vectors of the RBF neural network, b is a positive design parameter, and ⋅ is the norm.Remark 2.
b is related to Assumption 1.Assumption 3 (see [40]).
There are unknown constantsQi that make θ˜i≤Qi<∞, i=1,2,...,n.Lemma 6 (see [50]).
Consider the Gaussian function (13). SZ has an upper bound such that(16)SZ≤∑k=0∞3qk+2q−1e−2p2k2/r2:=s,where p=1/2mini≠jξi−ξj.Lemma6 has been proved in [50, 51]. Since ∑k=0∞3qk+2q−1e−2p2k2/r2 is convergent, s is a limited value. In addition, s is independent of the neural network node numbers l and the neural network inputs Z.


## 2.1. Problem Description

Consider the following nonlinear pure-feedback system:(1)x˙it=fix¯it,xi+1t+dix¯i,t,x˙nt=fnx¯nt,utdnx¯n,t,yt=x1t,where 1≤i≤n−1, x¯i=x1t,…,xitT∈Ri with i=1,…,n, ut∈R, and yt∈R are state variables, system input, and system output, respectively. fi⋅ are unknown but smooth nonaffine functions, and di⋅ are unknown but bounded disturbances.According to the mean value theorem [44], fix¯i,xi+1=fix¯i,xi0+hiμixi+1−xi0 and fnx¯n,u=fnx¯n,xn0+hnμnu−xn0, where hiμi:=hix¯i,xμi=∂fix¯i,xμi/∂xμi with i=1,...,n, xμi=μixi+1+1−μixi0 with 0<μi<1, and xi0 is a known quantity at the given time t0. System (1) can be written as(2)x˙i=fix¯i,xi0+hiμi⋅xi+1−xi0+dix¯i,t,x˙n=fnx¯n,xn0+hnμn⋅u−xn0+dnx¯n,t,y=x1.Remark 1.
It can be seen from system (2) that the mean value theorem separates the nonaffine structure from fi⋅ in system (1).
This paper aims to design a fixed-time controller that can meet the fixed-time control requirements in nonlinear pure-feedback systems, enabling the system outputy to track the reference signal yd in a fixed time. All of the signals of the closed-loop system are semiglobally uniform and ultimately bounded.Assumption 1 (see [45]).
Unknown smooth nonlinear functionshi⋅ are bounded, and there are known positive constants b and c with 0<b≤hi⋅<c<∞, ∀x¯i,xi+1∈Ri×R. Without loss of generality, we assume that 0<b≤hi⋅, i=1,...,n.
For ease of calculation, vector functions are defined asy¯d,i=yd,yd1,...,ydiT, i=1,...,n, where ydi is the ith derivative of yd.Assumption 2.
Reference signal vector functionsy¯di are known smooth continuous bounded functions. y¯di∈Ωdi⊂Ri+1 with i=1,...,n, where Ωdi are known compact sets and reference signal yd is an n-order differentiable and bounded function.


## 2.2. Fixed Time

Definition 1.
Consider the following nonlinear system:(3)x˙t=ft,x,x0=x0,where x∈Rn and f:R+×Rn⟶Rn, and assume that the origin is an equilibrium point.Lemma 1 (see [46]).
If there exist design constantsϕ1>0, ϕ2>0, α∈1,+∞, and β∈0,1 such that(4)V˙x≤−ϕ1Vαx−ϕ2Vβx,where Vx is a continuous differentiable positive definite function, then system (3) is global fixed-time stable, and the fixed convergence time satisfies(5)T≤Tmax:=1ϕ1α−1+1ϕ21−β.Lemma 2 (see [40]).
If there exist design constantsϕ1>0, ϕ2>0, α∈1,∞, β∈0,1, τ∈0,∞, and ϖ∈0,1 such that(6)V˙x≤−ϕ1Vαx−ϕ2Vβx+τ,then the origin of system (3) is practical fixed-time stable and the fixed-time T can be estimated by(7)T≤Tmax:=1ϕ1ϖα−1+1ϕ2ϖ1−β.The residual set of the solution of system (3) is given by(8)x∈Vx≤minτ1−ϖϕ11/α,τ1−ϖϕ21/β.Lemma 3 (see [47]).
Letx1,x2,...,xn≥0. Then,(9)∑i=1nxiρ≥∑i=1nxiρ,if0<ρ≤1,∑i=1nxiρ≥n1−ρ∑i=1nxiρ,if1<ρ≤∞.Lemma 4 (see [48]).
For any variablex∈R and any positive constant κ, the following relationship holds:(10)0≤x<κ+x2x2+κ2.Lemma 5.
Fory≥x>0, x, y∈R and any positive constant χ, then the following is satisfied(11)yy+χ≥xx+χ.Proof.
(12)
y
y
+
χ
−
x
x
+
χ
=
y
+
χ
y
+
χ
−
x
+
χ
x
+
χ
−
χ
y
+
χ
+
χ
x
+
χ
=
y
+
χ
−
x
+
χ
+
χ
x
+
χ
−
χ
y
+
x
≥
0.


## 2.3. Neural Network

An RBF neural network [11, 49] is applied in this paper to approximate arbitrary continuous functions. The mathematical expression of an RBF neural network is as follows:(13)φ^=WTSZ,where W=w1,w2,…,wlT∈Rl is the weight vector, l>1 is the number of nodes of the neural network, Z∈ΩZ⊂Rq is the input of the RBF neural network, q is the input dimension of the RBF neural network, SZ=s1Z,s2Z,…,slZT∈Rl is the basis vector function, and siZ is the output of the ith node. A Gaussian function is always chosen as siZ, i.e., siZ=exp−Z−ξiTZ−ξi/ri2, i=1,...,l, where ri is the width of the base function and ξi=ξi1,ξi2,…,ξiqT is the center of the basis function. With a sufficient number l of nodes selected, an RBF neural network can approximate an arbitrary continuous function φZ in a compact set ΩZ∈Rq with arbitrary accuracy ε.(14)φZ=W∗TSZ+δZ,∀Z∈ΩZ∈Rq,where δZ is the approximation error with δZ≤ε and W∗ is the given ideal constant weight vector, which is defined as(15)W∗=argminW∈RlsupZ∈ΩZφZ−φ^Z.In this paper, letθi=maxWi∗2/b,i=1,2,...,n with θ˜i=θi−θ^i, where θ^i are the estimates of the unknown constants θi, Wi∗ are the ideal weight vectors of the RBF neural network, b is a positive design parameter, and ⋅ is the norm.Remark 2.
b is related to Assumption 1.Assumption 3 (see [40]).
There are unknown constantsQi that make θ˜i≤Qi<∞, i=1,2,...,n.Lemma 6 (see [50]).
Consider the Gaussian function (13). SZ has an upper bound such that(16)SZ≤∑k=0∞3qk+2q−1e−2p2k2/r2:=s,where p=1/2mini≠jξi−ξj.Lemma6 has been proved in [50, 51]. Since ∑k=0∞3qk+2q−1e−2p2k2/r2 is convergent, s is a limited value. In addition, s is independent of the neural network node numbers l and the neural network inputs Z.


## 3. Design of a Fixed-Time Controller

The backstepping design coordinate transformation is as follows:(17)z1=x1−yd,zi=xi−αi−1,i=2,...,n,where αi is the virtual controller of the ith subsystem.An RBF neural network is used in this paper to approximate the unknown functionsf^iZi:(18)f^iZi=Wi∗TSiZi+δiZi,and the inequalities involved in the following text are as follows:(19)ziδiZi≤bki3zi2+εi24bki3,(20)ziWi∗TSiZi≤bθi2ηi2SiTZiSiZizi2+ηi22,(21)−bki221+β/212zi21+β/2+bθ˜i22γ1+β/2≤−2bki212zi2+bθ˜i22γ1+β/2,(22)−bki121+α/212zi21+α/2+bθ˜i22γ1+α/2≤−bki121+α/212zi2+bθ˜i22γ1+α/2,where Z1=x1,θ^1,yd,y˙d∈ΩZ1⊂R3+1 and Zi=x1,x2,...,xi,θ^1,θ^2,...,θ^i,y¯di∈ΩZi⊂R3i+1 with 2≤i≤n are input vectors and ηi, ki1, ki2, ki3, and γ are positive design parameters.Step 1.
According toz1=x1−yd and (2), we have(23)z˙1=f1x¯1,x10+h1μ1x2−x10+d1x¯1,t−y˙d.
Construct a Lyapunov function as(24)V1=12z12+bθ˜122γ.
Time differentiation ofV1 yields(25)V˙1=z1f1x¯1,x10+z1h1μ1x2−x10+z1d1−z1y˙d−bθ˜1θ^˙1γ.
Substitutingz2=x2−α1 into (25) yields(26)V˙1=z1f1x¯1,x10+z1h1μ1z2+z1h1μ1α1−z1h1μ1x10+z1d1−z1y˙d−bθ˜1θ^˙1γ.
The virtual controllerα1 is defined as(27)α1=−k11Sz1,1−k12Sz1,2−θ^12η12S1TZ1S1Z1z1−k13z1+x10,where k11, k12, k13, and η1 are positive design parameters.
In (27), Sz1,1 and Sz1,2 are defined as(28)Sz1,1=z12α+1/2z1,z1≥ε10,∑j=1najz12jε102−j+α+1/2,z1<ε10,Sz1,2=z12β+1/2z1,z1≥ε10,∑j=1najz12jε102−j+β+1/2,z1<ε10,where ε10 is a positive design parameter and coefficients aj, j=1,2,...,n, are calculated using the following equation:(29)11⋯1112⋯n−1n02×1⋯n−1n−2nn−1⋮⋮⋱⋮⋮00⋯∏j=0n−2n−1−j∏j=0n−2n−j×a1a2a3⋮an=b1b2b3⋮bn,where b1=1, b2=3/4, b3=3/43/4−1, …, and bn=∏j=0n−23/4−j.Remark 3.
One of the main contributions of this paper is to design suitable virtual controllersαi so that nonlinear pure-feedback systems meet the requirement of fixed-time control. In [40–43], virtual controllers designed exhibit similar power functions z2α−1, where 0<α<1. If α is not appropriately selected, it will make z unsolvable at the origin and in the negative domain. As 0<α<1, power exponents 2α−1∈−1,1, which leads to the possibility of negative power exponents. For example, if 2α−1=−1/3, z−1/3 is unsolvable at z=0, that is, 0−1/3 does not exist. Suppose 2α−1=q1/q2∈−1,1, where q2 is an even number; then, zq1/q2 is unsolvable at the negative domain. For example, if q1/q2=1/2, z1/2 is unsolvable at the negative domain, that is, −11/2 does not exist. The controller designed in this paper overcomes the aforementioned defect and promotes the application of fixed-time control in more common nonlinear systems.
Substitutingα1 into (26) yields(30)V˙1≤z1f^1Z1+z1h1μ1z2−bθ˜1θ^˙1γ−bk11z1Sz1,1−bk12z1Sz1,2−bθ^12η12S1TZ1S1Z1z12−bk13z12,where f^1Z1=f1x¯1,x10+d1−y˙d.
The RBF neural network (18) is utilized to approximate f^1Z1 and introduce inequalities (19) and (20), following which (30) can be rewritten as(31)V˙1≤bθ˜1γγ2η12S1TZ1S1Z1z12−θ^˙1+σ1+z1h1μ1z2−bk11z1Sz1,1−bk12z1Sz1,2,where σ1=η12/2+ε12/4bk13.
The adaptive lawθ^1 is then defined as(32)θ^˙1=γ2η12S1TZ1S1Z1z12−λθ^1,where λ is a positive design parameter.
Combining with Assumption3 and substituting (32) into (31), one can obtain(33)V˙1≤−bk11z1Sz1,1−bk12z1Sz1,2−bk1121+α/2bθ˜122γ1+α/2−bk1221+β/2bθ˜122γ1+β/2+z1h1μ1z2+C1,where(34)β1=bk11bQ12γ1+α/2+bk12bQ12γ1+β/2,λbθ˜1θ^1γ≤λbγθ122−θ˜122≤λb2γθ12,
According to (21), (22), and (28), if z1≥ε10, then (33) is rewritten as(35)V˙1≤−bk1121+α/212z12+bθ˜122γ1+α/2−2bk1212z12+bθ˜122γ1+β/2+z1h1μ1z2+C1.
According to (21), (22), and (28), if z1<ε10, then (33) is rewritten as(36)V˙1≤bk11z121+α/2−bk11z1∑j=1najz12jε102−j+1+α/2+bk12z121+β/2−bk12z1∑j=1najz12jε102−j+1+β/2−bk1121+α/212z12+bθ˜122γ1+α/2−2bk1212z12+bθ˜122γ1+β/2+z1h1μ1z2+C1.Remark 4 (see [52]).
Based on (28), when z1<ε10, there is an additional term in (36): bk11z121+α/2−bk11z1∑j=1najz12jε102−j+1+α/2+bk12z121+β/2−bk12z1∑j=1najz12jε102−j+1+β/2. Note that if z1<ε10, then this additional term is obviously limited by some smaller constant ε11, so the structure of (35) is retained, while the constant term C1 only slightly increases. Owing to page limitations and to avoid repetitive discussions, we will omit this part in the rest of the analysis.Step 2.
According toz2=x2−α1, we have(37)z˙2=f2x¯2,x20+h2μ2x3−x20−α˙1+d2x¯2,t,where(38)α˙1=∂α1∂x1x˙1+∂α1∂ydy˙d+∂α1∂θ^1θ^˙1.
Construct a Lyapunov function as(39)V2=V1+12z22+bθ˜222γ.
The derivative ofV2 is written as(40)V˙2≤−bk1121+α/212z12+bθ˜122γ1+α/2−2bk1212z12+bθ˜122γ1+β/2+z2f^2Z2+z2h2μ2z3+z2h2μ2α2−z2h2μ2x20−z2∂α1∂θ^1θ^˙1−bθ˜2θ^˙2γ+z2M1Z2+C1,where(41)f^2Z2=f2x¯2,x20−∂α1∂x1x˙1−∂α1∂ydy˙d+z1h1μ1+d2−M1Z2.Remark 5.
M
1
Z
2 is a smooth function that is used to overcome the design difficulty of θ^˙1∂α1/∂θ^1.
The virtual controllerα2 is defined as(42)α2=−k21Sz2,1−k22Sz2,2−θ^22η22S2TZ2S2Z2z2−k23z2+x20,where k21, k22, k23, and η2 are positive design parameters.
In (42), Sz2,1 and Sz2,2 are defined as(43)Sz2,1=z22α+1/2z2,z2≥ε20,∑j=1najz22jε202−j+α+1/2,z2<ε20,Sz2,2=z22β+1/2z2,z2≥ε20,∑j=1najz22jε202−j+β+1/2,z2<ε20,where ε20 is a positive design parameter.
Choose the adaptive lawθ^2 as(44)θ^˙2=γ2η22S2TZ2S2Z2z22−λθ^2.
Combining (18)–(20) and Assumption 3, substituting (42) and (44) into (40), and adopting the same design method as in Step 1 yields (45) as a rewriting of (40).(45)V˙2≤−bk21z2Sz2,1−bk2121+α/2bθ˜222γ1+α/2−bk22z2Sz2,2−bk2221+β/2bθ˜222γ1+β/2−bk1121+α/212z12+bθ˜122γ1+α/2−2bk1212z12+bθ˜122γ1+β/2+∑j=12Cj+z2h2μ2z3+z2M1Z2−∂α1∂θ^1θ^˙1,where(46)λbθ˜2θ^2γ≤λbγθ222−θ˜222≤λb2γθ22,β2=bk21bQ22γ1+α/2+bk22bQ22γ1+β/2,σ2=η222+ε224bk23,C2=σ2+β2+λbθ222γ.
According to (21) and (22) and Remark 4, we consider only z2≥ε20. Then, (45) can be written as(47)V˙2≤−b21+α/2∑j=12kj112zj2+bθ˜j22γ1+α/2−2b∑j=12kj212zj2+bθ˜j22γ1+β/2+∑j=12Cj+z2h2μ2z3+z2M1Z2−∂α1∂θ^1θ^˙1.
It can be seen from (47) that defining the design smooth function M1Z2 to overcome the design difficulty of ∂α1/∂θ^1θ^˙1 is one of the difficulties of designing the controllers in this paper.
From Lemmas4–6 and (32), it follows that(48)−z2∂α1∂θ^1θ^˙1≤−z2∂α1∂θ^1γ2η12S1TZ1S1Z1z12−λθ^1≤z22∂α1/∂θ^12γ/2η12S1TZ1S1Z1z122z22∂α1/∂θ^12γ/2η12S1TZ1S1Z1z122+ς2,12+z2∂α1∂θ^1λθ^1≤z22∂α1/∂θ^12γ/2η12s2z122z22∂α1/∂θ^12γ/2η12s2z122+ς2,12+z2∂α1∂θ^1λθ^1.
Therefore,M1Z2 can be defined as(49)M1Z2=−λ∂α1∂θ^1θ^1−z2∂α1/∂θ^12γ/2η12s2z122z22∂α1/∂θ^12γ/2η12s2z122+ς2,12,with the result that(50)z2M1Z2−∂α1∂θ^1θ^˙1≤0.
Substituting (50) into (47) yields(51)V˙2≤−b21+α/2∑j=12kj112zj2+bθ˜j22γ1+α/2−2b∑j=12kj212zj2+bθ˜j22γ1+β/2+∑j=12Cj+z2h2μ2z3.Step 3.
3
≤
k
≤
n
−
1
. According to zk=xk−αk−1, we have(52)z˙k=fkx¯k,xk0+hkμkxk+1−xk0+dkx¯k,t−α˙k−1,where(53)α˙k−1=∑j=1k−1∂αk−1∂xjx˙j+∂αk−1∂y¯d,k−1y¯˙d.k−1+∑j=1k−1∂αk−1∂θ^jθ^˙j.
Constructing a Lyapunov function asVk=Vk−1+zk2/2+bθ˜k2/2γ yields(54)V˙k≤−b21+α/2∑j=1k−1kj112zj2+bθ˜j22γ1+α/2−2b∑j=1k−1kj212zj2+bθ˜j22γ1+β/2+∑j=1k−1Cj+zkf^kZk+zkhkμkzk+1+zkhkμkαk−zkhkμkxk0−zk∑j=1k−1∂αk−1∂θ^jθ^˙j−bθ˜kθ^˙kγ+zkMk−1Zk,where(55)f^kZk=fkx¯k,xk0−∑j=1k−1∂αk−1∂xjx˙j−∂αk−1∂yd,k−1y¯˙d,k−1+zk−1hk−1μk−1+dkx¯k,t−Mk−1Zk.
Define virtual controllerαk as(56)αk=−kk1Szk,1−kk2Szk,2−θ^k2ηk2SkTZkSkZkzk−kk3zk+xk0,where kk1, kk2, kk3, and ηk are positive design parameters.
In (56), Szk,1 and Szk,2 are defined as(57)Szk,1=zk2α+1/2zk,zk≥εk0,∑j=1najzk2jεk02−j+α+1/2,zk<εk0,Szk,2=zk2β+1/2zk,zk≥εk0,∑j=1najzk2jεk02−j+β+1/2,zk<εk0,zkMk−1Zk−∑j=1k−1∂αk−1∂θ^jθ^˙j≤0where εk0 is a positive design parameter.
Define the adaptive lawθ^k as(58)θ^˙k=γ2ηk2SkTZkSkZkzk2−λθ^k.
Combining (18)–(20) and Assumption 3 and substituting (56) and (58) into (54) enable (54) to be rewritten as(59)V˙k≤−b21+α/2∑j=1k−1kj112zj2+bθ˜j22γ1+α/2−bkk1zkSzk,1−bkk121+α/2bθ˜k22γ1+α/2−2b∑j=1k−1kj112zj2+bθ˜j22γ1+β/2−bkk2zkSzk,2−bkk221+β/2bθ˜k22γ1+β/2+∑j=1kCj+zkhkμkzk+1+zkMk−1Zk−∑j=1k−1∂αk−1∂θ^jθ^˙j,where(60)λbθ˜kθ^kγ≤λbγθk22−θ˜k22≤λb2γθk2,βk=bkk1bQk2γ1+α/2+bkk2bQk2γ1+β/2,σk=ηk22+εk24bkk3,Ck=σk+βk+λbθk22γ.
In accordance with (21) and (22) and Remark 4, we consider only zk≥εk0. Then, (59) can be written as(61)V˙k≤−b21+α/2∑j=1kkj112zj2+bθ˜j22γ1+α/2−2b∑j=1kkj212zj2+bθ˜j22γ+∑j=1kCj+zkhkμkzk+1+zkMk−1Zk−∑j=1k−1∂αk−1∂θ^jθ^˙j.
Lemmas4–6 and (58) yield(62)−zk∑j=1k−1∂αk−1∂θ^jθ^˙j≤−zk∑j=1k−1∂αk−1∂θ^jγ2ηj2SjTZjSjZjzj2−λθ^j≤∑j=1k−1zk2∂αk−1/∂θ^j2γ/2ηj2s2zj2zk2∂αk−1/∂θ^j2γ/2ηj2s2zj22+ςk,j2+λzk∑j=1k−1∂αk−1∂θ^jθ^j.
Therefore, the smooth functionMk−1Zk can be defined as(63)Mk−1Zk=−∑j=1k−1zk∂αk−1/∂θ^j2γ/2ηj2s2zj22zk2∂αk−1/∂θ^j2γ/2ηj2s2zj22+ςk,j2−λ∑j=1k−1∂αk−1∂θ^jθ^j,with the result that(64)zkMk−1Zk−∑j=1k−1∂αk−1∂θ^jθ^˙j≤0.
Substituting (64) into (61) yields(65)V˙k≤−b21+α/2∑j=1kkj112zj2+bθ˜j22γ1+α/2−2b∑j=1kkj212zj2+bθ˜j22γ1+β/2+∑j=1kCj+zkhkμkzk+1.Step 4.
According tozn=xn−αn−1, we have(66)z˙n=fnx¯n,xn0+hnμnu−xn0+dnx¯n,t−α˙n−1,where(67)α˙n−1=∑j=1n−1∂αn−1∂xjx˙j+∂αn−1∂y¯d,n−1y¯˙d.n−1+∑j=1n−1∂αn−1∂θ^jθ^˙j.
Constructing a Lyapunov function asVn=Vn−1+zn2/2+bθ˜n2/2γ yields(68)V˙n≤−b21+α/2∑j=1n−1kj112zj2+bθ˜j22γ1+α/2−2b∑j=1n−1kj212zj2+bθ˜j22γ1+β/2+∑j=1n−1Cj+znf^nZn+znhnμnu−znhnμnxn0−zn∑j=1n−1∂αn−1∂θ^jθ^˙j−bθ˜nθ^˙nγ+znMn−1Zn,where(69)f^nZn=fnx¯n,xn0−∑j=1n−1∂αn−1∂xjx˙j−∂αn−1∂y¯d,n−1y¯˙d,n−1+zn−1hn−1μn−1+dnx¯n,t−Mn−1Zn.
Define an actual controlleru as(70)u=−kn1Szn,1−kn2Szn,2−θ^n2ηn2SnTZnSnZnzn−kn3zn+xn0,where kn1, kn2, kn3, and ηn are positive design parameters.
In (70), Szn,1 and Szn,2 are defined as(71)Szn,1=zn2α+1/2zn,zn≥εn0,∑j=1najzn2jεn02−j+α+1/2,zn<εn0,Szn,2=zn2β+1/2zn,zn≥εn0,∑j=1najzn2jεn02−j+β+1/2,zn<εn0,where εn0 is a positive design parameter.
The adaptive lawθ^n is defined as(72)θ^˙n=γ2ηn2SnTZnSnZnzn2−λθ^n.
Combining (18)–(20) and Assumption 3 and substituting (70) and (72) into (68) enable (68) to be rewritten as(73)V˙n≤−b21+α/2∑j=1n−1kj112zj2+bθ˜j22γ1+α/2−bkn1znSzn,1−bkn121+α/2bθ˜n22γ1+α/2−2b∑j=1n−1kj212zj2+bθ˜j22γ1+β/2−bkn2znSzn,2−bkn221+β/2bθ˜n22γ1+β/2+∑j=1nCj+znMn−1Zn−∑j=1n−1∂αk−1∂θ^jθ^˙j,where(74)λbθ˜nθ^nγ≤λbγθn22−θ˜n22≤λb2γθn2,βn=bkn1bQn2γ1+α/2+bkn2bQn2γ1+β/2,σn=ηn22+εn24bkn3,Cn=σn+βn+λbθn22γ.
According to (21) and (22) and Remark 4, we consider only zn≥εn0. Then, (73) can be written as(75)V˙n≤−b21+α/2∑j=1nkj112zj2+bθ˜j22γ1+α/2−2b∑j=1nkj212zj2+bθ˜j22γ1+β/2+∑j=1nCj+znMn−1Zn−∑j=1n−1∂αk−1∂θ^jθ^˙j.
The treatment ofMn−1Zn is similar to that of (62) and (63), with Mn−1Zn being defined as(76)Mn−1Zn=−∑j=1n−1zn∂αn−1/∂θ^j2γ/2ηj2s2zj22zn2∂αn−1/∂θ^j2γ/2ηj2s2zj22+ςn,j2−λ∑j=1n−1∂αn−1∂θ^jθ^j,with the result that(77)znMn−1Zn−∑j=1n−1∂αn−1∂θ^jθ^˙j≤0.
Substituting (77) into (75) yields(78)V˙n≤−b21+α/2∑j=1nkj112zj2+bθ˜j22γ1+α/2−2b∑j=1nkj212zj2+bθ˜j22γ1+β/2+∑j=1nCj,where ∑j=1nCj=τ.
Makingϕ1=min21+α/2bk11,bk21,...,bkn1 and ϕ2=min2bk12,bk22,...,bkn2, in accordance with Lemma 3, yields(79)−b21+α/2∑j=1nkj112zj2+bθ˜j22γ1+α/2≤−ϕ1∑j=1n12zj2+bθ˜j22λ1+α/2,(80)−2b∑j=1nkj212zj2+bθ˜j22γ1+β/2≤−n1−β/2ϕ2∑j=1n12zj2+bθ˜j22γ1+β/2,and then substituting (79) and (81) into (78) yields(81)V˙n≤−ϕ1Vn1+α/2−n1−β/2ϕ2Vn1+β/2+τ.
Up to this point, the design of the controller is finished.


## 4. Stability Analysis

Theorem 1.
If system (1) satisfies Assumptions 1–3 and uses the virtual controllers (27), (42), and (56), the actual controller (70), and the adaptive laws (32), (44), (58), and (72), all signals in the closed-loop system are semiglobally uniform and ultimately bounded, and the upper limit of the fixed convergence time is irrelevant to the initial state.In accordance with Lemma2, design proper parameters kj1>0,kj2>0, kj3>0, and Cj>0, j=1,...,n, to make (81) satisfy the following situation.Case 1.
IfVn>τ/1−ϖϕ12/1+α, ϖ∈0,1, then (81) can be written as(82)V˙n≤−ϖϕ1Vn1+α/2−n1−β/2ϕ2Vn1+β/2,and in that manner, the solution of system (1) converges on the compact set(83)x∈Vnx≤τ1−ϖϕ12/1+α.
The fixed convergence time is(84)T≤Tmax:=2ϖϕ11−α+2nβ−1/2ϕ2β−1.Case 2.
IfVn≥τnβ−1/2/1−ϖϕ22/1+β, then (81) can be written as(85)V˙n≤−ϕ1Vn1+α/2−ϖn1−β/2ϕ2Vn1+β/2,and in that manner, the solution of system (1) converges on the compact set(86)x∈Vnx≤τnβ−1/21−ϖϕ22/1+β.
The fixed convergence time is(87)T≤Tmax:=2ϕ11−α+2nβ−1/2ϖϕ2β−1.
When cases 1 and 2 are combined, the solution of system (1) converges on(88)x∈Vnx≤minτ1−ϖϕ12/1+α,τnβ−1/21−ϖϕ22/1+β.
The fixed convergence time is(89)Ts≤Tmax:=2ϖϕ11−α+2nβ−1/2ϖϕ2β−1.
It can be seen from (82) and (85) that Vn is bounded, with the result that zj and θ˜j are bounded. As θ^j=θj−θ˜j, θ^j is also bounded, where j=1,...,n. Because z1=x1−yd and z1 and yd are bounded, x1 is bounded. Because α1 is a function of z1, yd, y˙d, and θ^1, α1 is bounded. Because z2=x2−α1, x2 is bounded. Similarly, αj−1 and xj, j=1,...,n, are bounded. Therefore, all signals in system (1) are bounded.Remark 6.
The fixed-time control algorithm in this paper is different from previous control algorithms. The principle differences are as follows:(1)
The fixed-time control algorithm proposed in [35–40] does not solve the problem of the nonaffine structure of the control input ut. The fixed-time control algorithm proposed here solves that problem.(2)
Some systems’fi⋅ structure is complex, which interferes with direct usage of fi⋅ for designing controllers. An RBF neural network is used here to approximate unknown functions fi⋅, thereby obviating the need to know the structure of fix¯i,xi+1 and avoiding the difficult design problem of controllers derived from complex system structures.


## 5. Simulation Results

In this section, two samples are studied to verify the effectiveness of the controller designed in the paragraphs above.


### 5.1. Mathematical Example

Consider the following nonlinear pure-feedback system:(90)x˙1=1−e−x11+e−x2+x23+x2e−1−x12+d1x¯1,t,x˙2=x12x2+2u+u3x12+x22+d2x¯2,t,y=x1,where x1 and x2 are the system state variables, u is the system control input, y is the system output, d1x¯1,t=0.7x12cos1.5t and d2x¯2,t=0.5x12+x22sin3t 0.5 are the external disturbance terms, and yd=0.5sin1.5t+cos0.5t is the reference signal. It can be seen from system (90) that the state variables and the control input 2u+u3x12+x22 have the nonaffine structures. The simulation study aims to design a fixed-time controller based on system (90) that ensures that the output signal y can track the reference signal yd.For system (90), the fixed-time controller is defined as follows:(91)α1=−k11Sz1,1−k12Sz1,2−θ^12η12S1TZ1S1Z1z1−k13z1+x10,u=−k21Sz2,1−k22Sz2,2−θ^22η22S2TZ2S2Z2z2−k23z2+x20,θ^˙1=γ2η12S1TZ1S1Z1z12−λθ^1,θ^˙2=γ2η22S2TZ2S2Z1z22−λθ^2,where a1=5/4, a2=−1/4, z1=x1−yd, z2=x2−α1, Z1=x1,θ^1,yd,y˙d, and Z2=x1,x2,θ^1,y˙d,y¨d.Choose initial conditions asx10,x20T=0.5,0.5T and θ^10,θ^20T=0,0T. The design parameters are chosen as follows: α=1/3, β=2, k11=0.2, k12=0.2, k13=1, k21=0.2, k22=0.2, k23=1, γ=5, η1=0.25, η2=0.25, ε10=0.001, ε20=0.001, λ=0.1, x10=0.2, and x20=0.5. We set the width of the RBF neural network as four. W1S1Z1 and W2S2Z2 contain seven and five nodes, respectively. The center of the Gaussian function is set as(92)ξ1=−2−1.5−1011.52−2−1.5−1011.52−2−1.5−1011.52−2−1.5−1011.52,ξ2=−2−1012−2−1012−2−1012−3−2023−3−2023−3−2023−3−2023.Figure1 shows the system output y and the reference signal yd. It can be seen that y can track reference signals yd effectively. Figures 2–4 show the state variables x1 and x2, the actual controller u, and the adaptive parameters θ^1 and θ^2.Figure 1
System outputy and reference signal yd.Figure 2
State variablesx1 and x2.Figure 3
Actual controlu.Figure 4
Adaptive parametersθ^1 and θ^2.For a further simulation study, we selected two sets of different data to verify the tracking performance of system (90). It can be seen from Figure 5 that the error between the system output y and the reference signal yd increases with an increase in the α value. It can be seen from Figure 6 that the error between the system output y and the reference signal yd does not change significantly with an increase in the β value.Figure 5
Tracking errory−yd.Figure 6
Tracking errory−yd.Remark 7.
In [40–43], because the controller has the power function z2α−1, α can select only specific values; otherwise, the power function z2α−1 is unsolvable at the origin and in the negative range. In this paper, α can be any number between zero and one.
It can be seen from Figures1–6 that the state variables x1 and x2, the actual controller u, and the adaptive parameters θ^1 and θ^2 are bounded, with the consequence that all signals in the closed-loop system (90) are bounded.


### 5.2. Physical Example

Consider the following electromechanical system [41]:(93)x˙1=x2,x˙2=e21x2x32+e22sinx1+e23x2−e23x22x33,x˙3=e31u+e32x2+e33x32−e33x22sinx3,y=x1,where e21=1/M, e22=−N/M, e23=−B/M, e31=1/L, e32=−KB/L, and e33=−R/L are system parameters. The descriptions of M, N, B, L, KB, and R can be found in [53]. yd=sin0.5t+0.5sint is the reference signal. The system parameters are chosen as follows: M=0.0642, N=1.1408, B=0.0181, L=0.025, KB=0.9, and R=5.0. The simulation study aims to design a fixed-time controller based on system (93) to enable the output signal y to track the reference signal yd.For system (93), the fixed-time controller is defined as follows:(94)α1=−k11Sz1,1−k12Sz1,2+y˙d,α2=−k21Sz2,1−k22Sz2,2−θ^22η22S2TZ2S2Z2z2−k23z2+x20,u=−k31Sz3,1−k32Sz3,2−θ^32η32S3TZ3S3Z3z3−k33z3+x30,θ^˙2=γ2η22S2TZ2S2Z1z22−λθ^2,θ^˙3=γ2η32S3TZ3S3Z3z32−λθ^3,where a1=45/32, a2=−18/32, a3=5/32, and zi=xi−αi−1 with i=1,2,3.To verify the superiority of our designed controller, the tracking performance of this controller is compared with those of a fixed-time controller [40] and a finite-time controller [41]. The same design parameters as [40] are used, as follows: k11=10, k12=10, k21=10, k22=10, k31=10, k32=10, k23=10, k33=10, α=97/101, β=107/100, x10=0.1, x20=0.1, x30=0.1, ε10=10−5, ε20=10−5, and ε30=10−5. The initial conditions are selected as x10,x20,x30T=0.15,0.15,0.15T and θ^20,θ^30=0.3,0.3T. The design parameters of the neural network are the same as [41].Figure7 shows the system output y and the reference signal yd. As can be seen from Figure 7, the system output y can track reference signals yd effectively. Figures 8–10 show the state variables, the actual controller, and the adaptive parameters, respectively. Figure 11 shows the tracking error comparison between the fixed-time controller in this paper and the fixed-time controller in [40]. Figure 12 shows the tracking error comparison between the fixed-time controller in this paper and the finite-time controller in [41]. It can be seen from the simulation diagram that the performance of the fixed-time controller proposed in this paper is superior.Figure 7
System outputy and reference signal yd.Figure 8
State variablesx1, x2, and x3.Figure 9
Actual controlu.Figure 10
Adaptive parametersθ^2 and θ^3.Figure 11
Tracking errory−yd.Figure 12
Tracking errory−yd.It can be seen from Figures7–12 that the state variables x1 and x2, the actual controller u, and the adaptive parameters θ^1 and θ^2 are bounded, with the result that all signals in the closed-loop system (93) are bounded.


## 5.1. Mathematical Example

Consider the following nonlinear pure-feedback system:(90)x˙1=1−e−x11+e−x2+x23+x2e−1−x12+d1x¯1,t,x˙2=x12x2+2u+u3x12+x22+d2x¯2,t,y=x1,where x1 and x2 are the system state variables, u is the system control input, y is the system output, d1x¯1,t=0.7x12cos1.5t and d2x¯2,t=0.5x12+x22sin3t 0.5 are the external disturbance terms, and yd=0.5sin1.5t+cos0.5t is the reference signal. It can be seen from system (90) that the state variables and the control input 2u+u3x12+x22 have the nonaffine structures. The simulation study aims to design a fixed-time controller based on system (90) that ensures that the output signal y can track the reference signal yd.For system (90), the fixed-time controller is defined as follows:(91)α1=−k11Sz1,1−k12Sz1,2−θ^12η12S1TZ1S1Z1z1−k13z1+x10,u=−k21Sz2,1−k22Sz2,2−θ^22η22S2TZ2S2Z2z2−k23z2+x20,θ^˙1=γ2η12S1TZ1S1Z1z12−λθ^1,θ^˙2=γ2η22S2TZ2S2Z1z22−λθ^2,where a1=5/4, a2=−1/4, z1=x1−yd, z2=x2−α1, Z1=x1,θ^1,yd,y˙d, and Z2=x1,x2,θ^1,y˙d,y¨d.Choose initial conditions asx10,x20T=0.5,0.5T and θ^10,θ^20T=0,0T. The design parameters are chosen as follows: α=1/3, β=2, k11=0.2, k12=0.2, k13=1, k21=0.2, k22=0.2, k23=1, γ=5, η1=0.25, η2=0.25, ε10=0.001, ε20=0.001, λ=0.1, x10=0.2, and x20=0.5. We set the width of the RBF neural network as four. W1S1Z1 and W2S2Z2 contain seven and five nodes, respectively. The center of the Gaussian function is set as(92)ξ1=−2−1.5−1011.52−2−1.5−1011.52−2−1.5−1011.52−2−1.5−1011.52,ξ2=−2−1012−2−1012−2−1012−3−2023−3−2023−3−2023−3−2023.Figure1 shows the system output y and the reference signal yd. It can be seen that y can track reference signals yd effectively. Figures 2–4 show the state variables x1 and x2, the actual controller u, and the adaptive parameters θ^1 and θ^2.Figure 1
System outputy and reference signal yd.Figure 2
State variablesx1 and x2.Figure 3
Actual controlu.Figure 4
Adaptive parametersθ^1 and θ^2.For a further simulation study, we selected two sets of different data to verify the tracking performance of system (90). It can be seen from Figure 5 that the error between the system output y and the reference signal yd increases with an increase in the α value. It can be seen from Figure 6 that the error between the system output y and the reference signal yd does not change significantly with an increase in the β value.Figure 5
Tracking errory−yd.Figure 6
Tracking errory−yd.Remark 7.
In [40–43], because the controller has the power function z2α−1, α can select only specific values; otherwise, the power function z2α−1 is unsolvable at the origin and in the negative range. In this paper, α can be any number between zero and one.
It can be seen from Figures1–6 that the state variables x1 and x2, the actual controller u, and the adaptive parameters θ^1 and θ^2 are bounded, with the consequence that all signals in the closed-loop system (90) are bounded.


## 5.2. Physical Example

Consider the following electromechanical system [41]:(93)x˙1=x2,x˙2=e21x2x32+e22sinx1+e23x2−e23x22x33,x˙3=e31u+e32x2+e33x32−e33x22sinx3,y=x1,where e21=1/M, e22=−N/M, e23=−B/M, e31=1/L, e32=−KB/L, and e33=−R/L are system parameters. The descriptions of M, N, B, L, KB, and R can be found in [53]. yd=sin0.5t+0.5sint is the reference signal. The system parameters are chosen as follows: M=0.0642, N=1.1408, B=0.0181, L=0.025, KB=0.9, and R=5.0. The simulation study aims to design a fixed-time controller based on system (93) to enable the output signal y to track the reference signal yd.For system (93), the fixed-time controller is defined as follows:(94)α1=−k11Sz1,1−k12Sz1,2+y˙d,α2=−k21Sz2,1−k22Sz2,2−θ^22η22S2TZ2S2Z2z2−k23z2+x20,u=−k31Sz3,1−k32Sz3,2−θ^32η32S3TZ3S3Z3z3−k33z3+x30,θ^˙2=γ2η22S2TZ2S2Z1z22−λθ^2,θ^˙3=γ2η32S3TZ3S3Z3z32−λθ^3,where a1=45/32, a2=−18/32, a3=5/32, and zi=xi−αi−1 with i=1,2,3.To verify the superiority of our designed controller, the tracking performance of this controller is compared with those of a fixed-time controller [40] and a finite-time controller [41]. The same design parameters as [40] are used, as follows: k11=10, k12=10, k21=10, k22=10, k31=10, k32=10, k23=10, k33=10, α=97/101, β=107/100, x10=0.1, x20=0.1, x30=0.1, ε10=10−5, ε20=10−5, and ε30=10−5. The initial conditions are selected as x10,x20,x30T=0.15,0.15,0.15T and θ^20,θ^30=0.3,0.3T. The design parameters of the neural network are the same as [41].Figure7 shows the system output y and the reference signal yd. As can be seen from Figure 7, the system output y can track reference signals yd effectively. Figures 8–10 show the state variables, the actual controller, and the adaptive parameters, respectively. Figure 11 shows the tracking error comparison between the fixed-time controller in this paper and the fixed-time controller in [40]. Figure 12 shows the tracking error comparison between the fixed-time controller in this paper and the finite-time controller in [41]. It can be seen from the simulation diagram that the performance of the fixed-time controller proposed in this paper is superior.Figure 7
System outputy and reference signal yd.Figure 8
State variablesx1, x2, and x3.Figure 9
Actual controlu.Figure 10
Adaptive parametersθ^2 and θ^3.Figure 11
Tracking errory−yd.Figure 12
Tracking errory−yd.It can be seen from Figures7–12 that the state variables x1 and x2, the actual controller u, and the adaptive parameters θ^1 and θ^2 are bounded, with the result that all signals in the closed-loop system (93) are bounded.


## 6. Conclusion

A novel virtual controller and a novel actual controller designed in this paper solve the singularity problem of the virtual controller and the actual controller at the origin and in the negative domain. The fixed-time controller designed is applied to nonlinear pure-feedback systems and solves the problem of nonaffine structure. The controller enables the system output to track the reference signal in a fixed time and also to make the tracking error converge to a region of the origin in a fixed time. The simulation results prove the efficiency of the controller designed in this paper. We intend to apply the approximation-based fixed-time adaptive tracking control for a class of uncertain nonlinear pure-feedback systems in time-delay systems in the future.



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*Source: 4205914-2020-04-27.xml*