# Composite Learning Prescribed Performance Control of Nonlinear Systems

Authors: Fang Zhu; Wei Xiang; Chunzhi Yang
Journal: Complexity
 (2021)

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## Abstract

This paper investigates a composite learning prescribed performance control (PPC) scheme for uncertain strict-feedback system. Firstly, a prescribed performance boundary (PPB) condition is developed for the tracking error, and the original system is transformed into an equivalent one by using a transformation function. In order to ensure that the tracking error satisfies the PPB, a sufficient condition is given. Then, a control scheme of PPC combined with neural network (NN) and backstepping technique is proposed. However, the unknown functions cannot be guaranteed to estimate accurately by this method. To solve this problem, predictive errors are defined by applying online recorded date and instantaneous date. Furthermore, novel composite learning laws are proposed to update NN weights based on a partial persistent excitation (PE) condition. Subsequently, the stability of the closed-loop system is guaranteed and all signals are kept bounded by using composite learning PPC method. Finally, simulation results verify the effectiveness of the proposed methods.


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## Body

## 1. Introduction

In recent decades, many scholars are studying the stability tracking control of strict-feedback nonlinear systems [1–5], such as the inverted model and the one-link manipulator. Noting that the structures of these nonlinear systems are cascaded, an effective control method, namely, backstepping technology is developed; however, an inherent shortcoming of this method consists in repeated differentiation of virtual control inputs, which leads to “explosion-complexity.” A feasible approach to solve this problem is to estimate the virtual control inputs together with their derivatives. Consequently, filter command control approach was introduced in [1]. And, an adaptive dynamic surface control (ADSC) was proposed in [6]. The design feature of ADSC is that it does not need to calculate the derivative of virtual controller, and it can relax the smoothness requirements of system models and referenced signals. In [7], Yu et al. used a first-order Levant differentiator to discuss the finite-time tracking control problem of a class of nth order strict-feedback systems. However, the uncertain parameter or unknown function does not consider whether it can be accurately estimated. Recently, some composite learning approaches were proposed in [8, 9]. Under an interval excitation (IE) condition, composite learning needs to use online recorded data to construct a prediction error and then use prediction error and tracking error to construct composite learning update law. So, the unknown parameter is estimated accurately. In addition to the accurate estimation of unknown parameters, how to estimate the unknown functions accurately needs further research.However, most of the methods in the above literatures focus on how to ensure that the tracking error converges to a small neighborhood of zero, which is a steady-state research. However, some practical systems, such as near space vehicles (NSV) [10], need to consider both steady-state performance and transient performance. In order to solve this problem, prescribed performance control (PPC) [11–18] as one of the solutions has received extensive attention. The basic idea of the PPC method is to design the performance function (PPF) and the error transfer function so that the tracking error can be always limited within the performance function boundary (PPB) by the PPF. In [15], an adaptive neural PPC is proposed for NSVs with input nonlinearity. In order to achieve the convergence of tracking error with small overshoot, Bu et al. [16] constructed a new PPF and error transformation function (ETF). Compared with traditional PPC, the control effect of the proposed method has been improved in [16]. Xiang and Liu [17] employed the same PPF and ETF in [16] to design a fuzzy adaptive tracking control for uncertain nonlinear systems with unknown control gain signs. Wang et al. [18] used finite-time performance function to discuss the performance problem of tracking error under actuator faults and external disturbances. Inspired by [18], can we construct a new simple finite-time performance function to achieve the predetermined performance of tracking error? Moreover, fuzzy logic system (FLS) or neural network (NN) is used to estimate the unknown function, but literatures [11–16] did not further verify whether the unknown function is effectively estimated. Therefore, it is a challenge to effectively estimate the unknown function under the condition of ensuring the steady-state and transient performance of the tracking error. Usually, in order to achieve effective estimation of unknown functions, it is necessary to obtain online recorded data together with instantaneous data to generate prediction errors [19–23]. In this paper, in order to realize the output state to track the recurrent reference signal and the tracking error satisfies PPB, prediction errors are generated by using the partial persistent excitation (PE) condition in [23]. The main contributions of this paper are summarized as follows. (1) A new finite-time performance function is developed. (2) The proposed composite learning PPC method can guaranteed the small convergence overshoot of tracking error. (3) Learning-based parameter adaptive laws constructed with prediction error can be used to accurately estimate unknown nonlinear functions.The remainder of this paper is organized as follows. Some preliminaries are listed in Section2. In Section 3, two types of adaptive neural network PPC method are discussed. The simulation results are explained in Section 4. Finally, brief conclusion is presented in Section 5.


## 2. Preliminaries

### 2.1. System Description

Consider the followingnth-order strict-feedback nonlinear system:(1)x˙i=fix¯i+xi+1,i=1,2,…,n−1,x˙n=fnx¯n+u,y=x1,where x¯i=x1,x2,…,xiT∈Ri (x1=x¯1 and x=x¯n) are the measurable state vectors, u∈R is the control input, y∈R is the output, and fix¯i, i=1,2,…,n, are smooth functions.Assumption 1.
The reference signalxd is recurrent and bounded. xd and its derivative x˙d are continuous and available.Assumption 2.
fix¯i, i=1,2,…,n, are unknown and bounded.
Usually, the backstepping technique is used to design the controlleru for system (1). The tedious derivative calculation of the virtual control law is required, which will cause the problem of “explosion-complexity.” To avoid such problems, the following first-order filter is employed in this paper.Lemma 1.
For the first-order filter,(2)τζ˙+ζ=α,where τ is time design constant and ζ0=α0. Define an error as ε=ζ−α. According to [5], if the appropriate parameter τ is chosen, then there exists positive constant κ such that ε≤κ.In this paper, we employed NN to approximatefix¯i. According to the approximation property of RBFNNs [20], the approximation form of fix¯i can be expressed as(3)f^ix¯i=φfiTx¯iθ^fi,where θ^fi=θ^fi1,θ^fi2,…,θ^fiιiT∈Ωθi⊂Rιi is weight vector, ιi is node number, and φfix¯i=φfi1x¯i,φfi2x¯i,…,φfiιix¯iT∈Rιi is basis function vector. Here, φfijx¯i:Ωx¯i↦R,j=1,2,…,ιi, are(4)φfijx¯i=exp−x¯i−cjTx¯i−cj2bj2,where cj=cj1,cj2,…,cjiT∈Ri and bj are the center vector and width of the basis function, respectively.By [20, 21], choosing appropriate cj,bj, and some sufficiently large integer ι, there exists ideal weight vector θfi∗∈Rιi such that(5)fix¯i=φfiTx¯iθfi∗+εfix¯i,where εfix¯i is the approximate error, which satisfies εfix¯i≤εfi∗, where εfi∗ is a positive constant. So, we can rewrite fix¯i in (5) as(6)fix¯i=φfiTx¯iθ˜fi+φfiTx¯iθ^fi+εfix¯i,where θ˜fi=θfi∗−θ^fi.Define the tracking errore1=x1−xd; substituting (6) into (1), we obtain(7)e˙1=φf1Tx¯1θ^f1+φf1Tx¯1θ˜f1+x2+εf1x¯1−x˙d,x˙i=φfiTx¯iθ^fi+φfiTx¯iθ˜fi+xi+1+εfix¯i,i=2,…,n−1,x˙n=φfnTx¯nθ^fn+φfnTx¯nθ˜fn+u+εfnx¯n,y=x1.Remark 1.
Basis functionsφfix¯i, i=1,2,…,n, in (7) satisfy maxφfix¯i,φ.fix¯i≤χi, where χi are positive constants independent of x¯i and ιi. For detailed proof, see [6].Definition 1 (see [23]).
A bounded signalφt∈Rm is said to satisfy the persistent excitation PE condition if there exist constants τd and σ such that ∫t−τdtφsφTsds≥σI, ∀t∈R+, where I is an appropriate identity matrix.Lemma 2 (partialPE condition) (see [23]).
Consider the RBF networkφfiTx¯iθ^fi (3) with centers placed on a regular lattice to cover Ωx¯i. If the trajectory of x¯i is recurrent, then there exists a regression subvector φf¯ix¯i that almost always satisfies PE condition.Remark 2.
Since the reference signalxd is recurrent, it is necessary to prove that the trajectories of x1,x2,…,xn are recurrent in a finite time T, which mainly avoids the case that learning occurs only when time approach to infinity. Moreover, notice that system states x1,x2,…,xn are measurable, so the following operation is feasible for t≥T when recurrent trajectory of xi is proved, which is (i) to reselect center vector c¯j′=cj′1,cj′2,…,cj′i∈Ri such that c¯j′∈Ux¯i,ε, where the value of ε is the minimum distance between any two center vectors in RBF (3) and (ii) construct a regression subvector φf¯ix¯i=φf¯i1x¯i,φf¯i2x¯i,…,φf¯iιi′x¯iT∈Rιi′ based on the new center vector c¯j′, where ιi′<ιi.Remark 3.
It should be emphasized that the whole regression vectorsφfix¯i do not satisfy the PE condition (since the NN input x¯i=x1,x2,…,xiT cannot visit every center of the whole RBF network persistently). So, when the control scheme guarantee that all system states are bounded and NN inputs x¯i are recurrent for t≥T, we need to reset the center vector so that the regression subvector φf¯ix¯i meet partial PE condition according to Lemma 2. Therefore, after time T, the effective learning knowledge of the unknown function fix¯i is obtained in φf¯iTx¯iθ^f¯i, not φfiTxiθ^fi.Lemma 3 (recurrent trajectory, (see [23])).
ωt is called a recurrent trajectory if there exists a positive constant γ such that ωt−ωdt<γ, where ωd is recurrent signal.


### 2.2. Prescribed Performance

Firstly, define a finite-time performance function (FTPF)pt,T,λ0,λ∞ as(8)pt,T,λ0,λ∞=a4t4+a3t3+a2t2+a1t+λ0,t<T,λ∞,t≥T,where T is predefined settling time, λ0 is the initial value of the FTPF, and λ∞ is the boundary value of the FTPF after T. To ensure that the FTPF pt,T,λ0,λ∞ is continuous and derivative at T, parameters a1,a2,a3, and a4 are designed as follows:(9)a1=4λ∞−λ0T,a2=6λ0−λ∞T2,a3=4λ∞−λ0T3,a4=λ0−λ∞T4.Define the tracking error variablee1=x1−xd, and set e1 satisfies the following prescribed performance boundary (PPB):(10)pt,T,a0,a∞<e1<pt,T,b0,b∞,where pt,T,a0,a∞ and pt,T,b0,b∞ are FTPFs and a0,b0,a∞,b∞ satisfy a∞<a0<b0 and a∞<b∞<b0.The transformation variablez1, which corresponding to PPB (10), is defined as follows:(11)z1=lnρt1−ρt,where ρt=e1−pt,T,a0,a∞/pt,T,b0,b∞−pt,T,a0,a∞.Lemma 4 (see [17]).
Ifz1 is bounded, e1 can be limited within PPB (10).The time derivative ofz1 becomes(12)z˙1=r1φf1Tx¯1θ^f1+φf1Tx¯1θ˜f1+x2+εf1x¯1−x˙d+Π,where(13)r1=1ρt1−ρtpt,T,b0,b∞−pt,T,a0,a∞>0,Π=pt,T,a0,a∞p˙t,T,b0,b∞−p˙t,T,a0,a∞pt,T,b0,b∞−ep˙t,T,b0,b∞−p˙t,T,a0,a∞pt,T,b0,b∞−pt,T,a0,a∞.Remark 4.
Compared with the traditional predetermined performance function [24], the proposed prescribed performance function (8) in this paper can guarantee that e can be limited within PPB (10) after t≥T. Meanwhile, we can obtain the following results: (i) r1=1/ρt1−ρtpt,T,b0,b∞−pt,T,a0,a∞>0 and (ii) r˙1/r12 tends to a small neighborhood of zero. Therefore, z12/2r1 will be used in Lyapunov function to explore the boundedness of z1.


### 2.3. Control Aim

The aim of this paper is (i) to design the composite prescribed performance control method so that the tracking errore1 satisfies PPB (10), (ii) to construct prediction errors by using historical data for neural weights update, and (iii) to compare with an adaptive neural network PPC method to show the effectiveness of the proposed control method in this paper.


## 2.1. System Description

Consider the followingnth-order strict-feedback nonlinear system:(1)x˙i=fix¯i+xi+1,i=1,2,…,n−1,x˙n=fnx¯n+u,y=x1,where x¯i=x1,x2,…,xiT∈Ri (x1=x¯1 and x=x¯n) are the measurable state vectors, u∈R is the control input, y∈R is the output, and fix¯i, i=1,2,…,n, are smooth functions.Assumption 1.
The reference signalxd is recurrent and bounded. xd and its derivative x˙d are continuous and available.Assumption 2.
fix¯i, i=1,2,…,n, are unknown and bounded.
Usually, the backstepping technique is used to design the controlleru for system (1). The tedious derivative calculation of the virtual control law is required, which will cause the problem of “explosion-complexity.” To avoid such problems, the following first-order filter is employed in this paper.Lemma 1.
For the first-order filter,(2)τζ˙+ζ=α,where τ is time design constant and ζ0=α0. Define an error as ε=ζ−α. According to [5], if the appropriate parameter τ is chosen, then there exists positive constant κ such that ε≤κ.In this paper, we employed NN to approximatefix¯i. According to the approximation property of RBFNNs [20], the approximation form of fix¯i can be expressed as(3)f^ix¯i=φfiTx¯iθ^fi,where θ^fi=θ^fi1,θ^fi2,…,θ^fiιiT∈Ωθi⊂Rιi is weight vector, ιi is node number, and φfix¯i=φfi1x¯i,φfi2x¯i,…,φfiιix¯iT∈Rιi is basis function vector. Here, φfijx¯i:Ωx¯i↦R,j=1,2,…,ιi, are(4)φfijx¯i=exp−x¯i−cjTx¯i−cj2bj2,where cj=cj1,cj2,…,cjiT∈Ri and bj are the center vector and width of the basis function, respectively.By [20, 21], choosing appropriate cj,bj, and some sufficiently large integer ι, there exists ideal weight vector θfi∗∈Rιi such that(5)fix¯i=φfiTx¯iθfi∗+εfix¯i,where εfix¯i is the approximate error, which satisfies εfix¯i≤εfi∗, where εfi∗ is a positive constant. So, we can rewrite fix¯i in (5) as(6)fix¯i=φfiTx¯iθ˜fi+φfiTx¯iθ^fi+εfix¯i,where θ˜fi=θfi∗−θ^fi.Define the tracking errore1=x1−xd; substituting (6) into (1), we obtain(7)e˙1=φf1Tx¯1θ^f1+φf1Tx¯1θ˜f1+x2+εf1x¯1−x˙d,x˙i=φfiTx¯iθ^fi+φfiTx¯iθ˜fi+xi+1+εfix¯i,i=2,…,n−1,x˙n=φfnTx¯nθ^fn+φfnTx¯nθ˜fn+u+εfnx¯n,y=x1.Remark 1.
Basis functionsφfix¯i, i=1,2,…,n, in (7) satisfy maxφfix¯i,φ.fix¯i≤χi, where χi are positive constants independent of x¯i and ιi. For detailed proof, see [6].Definition 1 (see [23]).
A bounded signalφt∈Rm is said to satisfy the persistent excitation PE condition if there exist constants τd and σ such that ∫t−τdtφsφTsds≥σI, ∀t∈R+, where I is an appropriate identity matrix.Lemma 2 (partialPE condition) (see [23]).
Consider the RBF networkφfiTx¯iθ^fi (3) with centers placed on a regular lattice to cover Ωx¯i. If the trajectory of x¯i is recurrent, then there exists a regression subvector φf¯ix¯i that almost always satisfies PE condition.Remark 2.
Since the reference signalxd is recurrent, it is necessary to prove that the trajectories of x1,x2,…,xn are recurrent in a finite time T, which mainly avoids the case that learning occurs only when time approach to infinity. Moreover, notice that system states x1,x2,…,xn are measurable, so the following operation is feasible for t≥T when recurrent trajectory of xi is proved, which is (i) to reselect center vector c¯j′=cj′1,cj′2,…,cj′i∈Ri such that c¯j′∈Ux¯i,ε, where the value of ε is the minimum distance between any two center vectors in RBF (3) and (ii) construct a regression subvector φf¯ix¯i=φf¯i1x¯i,φf¯i2x¯i,…,φf¯iιi′x¯iT∈Rιi′ based on the new center vector c¯j′, where ιi′<ιi.Remark 3.
It should be emphasized that the whole regression vectorsφfix¯i do not satisfy the PE condition (since the NN input x¯i=x1,x2,…,xiT cannot visit every center of the whole RBF network persistently). So, when the control scheme guarantee that all system states are bounded and NN inputs x¯i are recurrent for t≥T, we need to reset the center vector so that the regression subvector φf¯ix¯i meet partial PE condition according to Lemma 2. Therefore, after time T, the effective learning knowledge of the unknown function fix¯i is obtained in φf¯iTx¯iθ^f¯i, not φfiTxiθ^fi.Lemma 3 (recurrent trajectory, (see [23])).
ωt is called a recurrent trajectory if there exists a positive constant γ such that ωt−ωdt<γ, where ωd is recurrent signal.


## 2.2. Prescribed Performance

Firstly, define a finite-time performance function (FTPF)pt,T,λ0,λ∞ as(8)pt,T,λ0,λ∞=a4t4+a3t3+a2t2+a1t+λ0,t<T,λ∞,t≥T,where T is predefined settling time, λ0 is the initial value of the FTPF, and λ∞ is the boundary value of the FTPF after T. To ensure that the FTPF pt,T,λ0,λ∞ is continuous and derivative at T, parameters a1,a2,a3, and a4 are designed as follows:(9)a1=4λ∞−λ0T,a2=6λ0−λ∞T2,a3=4λ∞−λ0T3,a4=λ0−λ∞T4.Define the tracking error variablee1=x1−xd, and set e1 satisfies the following prescribed performance boundary (PPB):(10)pt,T,a0,a∞<e1<pt,T,b0,b∞,where pt,T,a0,a∞ and pt,T,b0,b∞ are FTPFs and a0,b0,a∞,b∞ satisfy a∞<a0<b0 and a∞<b∞<b0.The transformation variablez1, which corresponding to PPB (10), is defined as follows:(11)z1=lnρt1−ρt,where ρt=e1−pt,T,a0,a∞/pt,T,b0,b∞−pt,T,a0,a∞.Lemma 4 (see [17]).
Ifz1 is bounded, e1 can be limited within PPB (10).The time derivative ofz1 becomes(12)z˙1=r1φf1Tx¯1θ^f1+φf1Tx¯1θ˜f1+x2+εf1x¯1−x˙d+Π,where(13)r1=1ρt1−ρtpt,T,b0,b∞−pt,T,a0,a∞>0,Π=pt,T,a0,a∞p˙t,T,b0,b∞−p˙t,T,a0,a∞pt,T,b0,b∞−ep˙t,T,b0,b∞−p˙t,T,a0,a∞pt,T,b0,b∞−pt,T,a0,a∞.Remark 4.
Compared with the traditional predetermined performance function [24], the proposed prescribed performance function (8) in this paper can guarantee that e can be limited within PPB (10) after t≥T. Meanwhile, we can obtain the following results: (i) r1=1/ρt1−ρtpt,T,b0,b∞−pt,T,a0,a∞>0 and (ii) r˙1/r12 tends to a small neighborhood of zero. Therefore, z12/2r1 will be used in Lyapunov function to explore the boundedness of z1.


## 2.3. Control Aim

The aim of this paper is (i) to design the composite prescribed performance control method so that the tracking errore1 satisfies PPB (10), (ii) to construct prediction errors by using historical data for neural weights update, and (iii) to compare with an adaptive neural network PPC method to show the effectiveness of the proposed control method in this paper.


## 3. Control Design

### 3.1. Adaptive Neural Network PPC Method

In order to guarantee thatz1 is bounded, backstepping technique will be employed, and the whole control process is divided into n steps. Step 1:
from (12), the virtual control α2 is designed as(14)α2=−φf1Tx¯1θ^f1+x˙d−Π−c1z1,where c1 is a positive design constant.Definez2=x2−ζ1, where ζ1 is used to estimate α2 through the following first-order filter:(15)τ1ζ˙1=−ζ1−α2,ζ10=α20,where τ1 is design time constant. By Lemma 1, we know that there exists a positive constant κ1 such as ζ1−α2≤κ1. Define ε1t=ζ1−α2; then, x2=z2+ε1t+α2 and (12) can be rewritten as(16)z˙1=r1φf1Tx¯1θ˜f1+εf1x¯1+z2+ε1t−c1z1.The adaptive law ofθ^f1 is chosen as(17)θ^.f1=λf1z1φf1x¯1−δ1θ^f1,where λf1 and δ1 are positive design parameters.Stepj:j=2,…,n−1. According to the definition of zj=xj−ζj−1, we have(18)z˙j=φfjTx¯jθ˜fj+εfjx¯j+θ^fjTφfjx¯j+xj+1−ζ˙j−1,Choose virtual controlαj+1 as(19)αj+1=−φfjTx¯jθ^fj+ζ˙j−1−cjzj−zj−1,where cj is a positive design constant.Definezj+1=xj+1−ζj and ζj satisfies the following first-order filter:(20)τjζ˙j=−ζj−αj+1,ζj0=αj+10,where τj is design time constant. Define εjt=ζj−αj+1, and there exists a positive constant κj such as εjt≤κj by Lemma 1. Similar to step 1, we can rewrite (18) as follows:(21)z.j=φfjTx¯jθ˜fj+εfjx¯j+zj+1+εjt−cjzj−zj−1.And, the parameter adaptive law ofθ^fj is designed as(22)θ^.fj=λfjziφfjx¯j−δjθ^fj,where λfj and δj are positive design parameters.Stepn: the time derivative of zn produces(23)z˙n=φfnTx¯nθ˜fn+εfnx¯n+θ^fnTφfnx¯n+u−ζ˙n−1.The controlleru is designed as(24)u=−φfnTx¯nθ^fn+ζ˙n−1−cnzn−zn−1,where cn is a positive constant. Submitting (24) into (23), we obtain(25)z.n=φfnTx¯nθ˜fn+εfnx¯n−cnzn−zn−1.The adaptive law forθ^fn is chosen as(26)θ^.fn=λfnznφfnx¯n−δnθ^fn,where λfn and δn are positive design parameters.Theorem 1.
Consider uncertainnth-order strict-feedback nonlinear system (1) with assumptions 1-2. If virtual controllers (14) and (19), real controller 24), and RBFNN updating laws (17), (22), and (26) are designed, then all signals in the closed-loop system (27) are uniformly ultimately bounded, and then, the boundedness of z1 ensures that e1 satisfies PPB (10).Proof.
Consider the following Lyapunov function:(27)V1=12r1z12+12λf1θ˜f1Tθ˜f1+∑i=2n12zi2+12λfiθ˜fiTθ˜fi.
According to (16), (21), and (25) and RBFNN updating laws (17), (22), and (26), we have(28)V˙1=−c1+r˙12r12z12−∑i=2ncizi2+∑k=1nzkεfkx¯k+zkφfkTx¯kθ˜fk−∑k=1nθ˜fkTzkφfkx¯k−δkθ^fk+∑j=1n−1zjεjt,=−c1+r˙12r12z12−∑i=2ncizi2+∑j=1n−1zjεjt+∑k=1nzkεfkx¯k+δkθ˜kkTθ^fk.
Letcj=cj1+2cj2,cn=cn1+cn2, and cj1,cj2>0, j=1,2,…,n−1. By using Young’s inequality, we obtain(29)δkθ˜fkTθ^fk≤−δk2θ˜fk2+δk2θ˜fk∗2,−ck2zk2+zkεfkx¯k≤εfk2x¯k4ck2≤εfk∗24ck2,k=1,2,…,n,−cj2zj2+zjεjt≤εj2t4cj2≤κj24cj2,j=1,2,…,n−1.
Substituting (29) into (28), we obtain(30)V˙1≤−c11+r˙12r12z12−∑i=2nci1zi2−∑k=1nδk2θ˜fk2︸θ˜fkTθ˜fk+R1,where R1=∑k=1nδk/2θfk∗2+εfk∗2/4ck2+∑j=1n−1κj2/4cj2. According to 5, we choose c11>−r˙1/2r12 and define the following compact sets:(31)Ωz1=z1|z1≤R1c11+r˙1/2r12,Ωzi=zi|zi≤R1ci1,Ωθfk=θfk|θ˜fk≤2R1δk.
Obviously,z1∉Ωz1 or zi∉Ωzi or θ˜fk∉Ωθ˜fk, we have V˙1<0. Thus, z1,z˜i,zn and θ˜fk are all semiglobally uniformly bounded. According to Lemma 4, the boundedness of z1 ensures that e1 meets PPB (10). This concludes the proof.Remark 5.
In fact, Theorem1 also implies that all system states x1,x2,…,xn are recurrent for t≥T. Because e1 is limited within PPB (10), i.e., a∞<x1−xd<b∞, t≥T, so, from Lemma 3, we obtain that x1 is recurrent. And, because fix¯i is smooth, this leads to x2, x3,…,xn are recurrent in turn. It also implies that there exists a regression subvector φf¯ix¯i such that φf¯ix¯i satisfies partial PE condition by Lemma 2.Remark 6.
It should be pointed out that RBFNN updating laws (17), (22), and (26) only use the instantaneous data, and it may not estimate the unknown function fx¯i accurately. Therefore, in the next section, we will use the composite learning control method to ensure the boundedness of z1 and the accurate estimation of fx¯i.


### 3.2. Adaptive Neural Network PPC Method with Learning

In this section, a prediction errorϖit is constructed by using online data and instantaneous data. According to Remark 5 in Section 3.1, system states x1,x2,…,xn in system (1) will be recurrent after T. So, for x¯i, there is a regression subvector φf¯ix¯i, which satisfies partial PE condition by Lemma 2, i.e., ∫t−τdtφf¯ix¯iφf¯iTx¯idτ≥σiI, where σi>0 and t≥T, i=1,2,…,n. And, fix¯i can also be estimated by φf¯iTx¯iθ^f¯i instead of φfiTx¯iθ^fi after T. The specific modification is as follows.Firstly, substitutingfix¯i=φf¯ix¯iTθf¯i∗+εf¯ix¯it≥T into system (1), one obtains(32)x˙i=φf¯ix¯iTθf¯i∗+εf¯ix¯i+xi+1,i=1,2,…,n−1,x˙n=φf¯nx¯nTθf¯n∗+εf¯nx¯n+u,where εf¯ix¯i≤εf¯i∗, εf¯i∗ is a positive constant. Let Qit:=∫t−τdtφf¯ix¯iφf¯iTx¯idτ and ε¯f¯it:=∫t−τdtφf¯ix¯iεf¯ix¯idτ. Multiplying both sides of above equalities in (32) by φf¯ix¯i and integrating equalities over t,t−τd, one obtains(33)Qitθf¯i∗+ε¯f¯it=∫t−τdtφf¯ix¯ix˙i−xi+1dτ,Qntθf¯n∗+ε¯f¯nt=∫t−τdtφf¯nx¯nx˙n−udτ.So, the prediction errorϖit is defined as(34)ϖit=Qitθf¯i∗+ε¯f¯it−Qitθ^f¯i=Qitθ˜f¯i+ε¯f¯it.Obviously,ε¯f¯it≤τdχiεf¯i∗; let χ¯=maxχ1,χ2,…,χn and ε¯=maxεf¯1∗,εf¯2∗,…,εf¯n∗; then, for ∀i, ε¯f¯it≤τdψ¯ε¯ holds.Then, we replaceφfiTx¯iθ^fi with φf¯iTx¯iθ^f¯i in (12), (18), and (23) after T, and one obtains(35)z˙1=r1φf¯1Tx¯1θ^f¯1+φf¯1Tx¯1θ˜f¯1+x2+εf¯1x¯1−x˙d+Π,z˙j=φf¯jTx¯jθ^f¯j+φf¯jTx¯jθ˜f¯j+xj+1+εf¯jx¯j−ζ˙j−1,z˙n=φf¯nTx¯nθ^f¯n+φf¯nTx¯nθ˜f¯j+u+εf¯nx¯n−ζ˙n−1.where zj=xj−ζj−1, ζj−1 is the estimation of αj, which is the same as the one in Section 3.1, j=2,3,…,n. And, the modified virtual controller αj is designed as follows.According to (35), the virtual control α2 in (14) is modified as(36)α2=−φf1Tx¯1θ^f1+x˙d−Π−c1z1,t<T,−φf¯1Tx¯1θ^f¯1+x˙d−Π−c1z1,t≥T.And, composite learning laws ofθ^f1 and θ^f¯1 are given as(37)θ^.f1=λf1z1φf1x¯1−δ1θ^f1,t<T,θ^.f¯1=λf¯1z1φf¯1x¯1+δ1′ϖ1t,t≥T.where λf¯1 and δ1′ are positive design parameters.The virtual controlαj+1 in (19) is modified as(38)αj+1=−θ^fjTφfjx¯j+ζ˙j−1−cjzj−zj−1,t<T,−θ^f¯jTφf¯jx¯j+ζ˙j−1−cjzj−zj−1,t≥T.Similarly, composite learning laws ofθ^fj and θ^f¯j are designed as(39)θ^.fj=λfjziφfjx¯j−δjθ^fj,t<T,θ^.f¯j=λf¯jziφf¯jx¯j+δj′ϖjt,t≤T,where λf¯1 and δj′ are positive design parameters.The controlleru in (24) is modified as(40)u=−θ^fnTφfnx¯n+ζ˙n−1−cnzn−zn−1,t<T,−θ^f¯nTφf¯nx¯n+ζ˙n−1−cnzn−zn−1,t≥T.and composite learning laws for θ^fn and θ^f¯n are designed as(41)θ^.fn=λfnznφfnx¯n−δnθ^fn,t<T,θ^.f¯n=λf¯nznφf¯nx¯n+δn′ϖnt,t≥T,where λf¯1 and δn′ are positive design parameter. Other parameters ci,λfi, and δi are the same as those in Section 3.1.Next, a main theorem of this paper is given as follows.Theorem 2.
Consider uncertainnth-order strict-feedback nonlinear system (1) with Assumptions 1-2. The virtual controllers (36) and (38) and real controller (40) combine with RBFNN updating composite learning laws (37), (39), and (41) guarantee both the convergence of zi and θ^f¯i during t∈T,+∞, and then, tracking error e1 meets PPB (10).Proof.
Theorem1 and Remark 5 have been proved that x1,x2,…,xn are recurrent after time T. So, it is necessary to prove whether the composite learning law of θ^f¯i can ensure that zi and θ˜f¯i enter a small neighborhood of zero after time T. Consider the following Lyapunov function at t∈T,+∞:(42)V2=12r1z12+12λf¯1θ˜f¯1Tθ˜f¯1+∑i=2n12zi2+12λf¯iθ˜f¯iTθ˜f¯i.
According to (36)–(41), one obtains(43)V˙2=−c1+r˙12r12z12−∑i=2ncizi2+∑k=1nzkεf¯kx¯k+zkφf¯kTx¯kθ˜f¯k−∑k=1nθ˜f¯kTzkφf¯kx¯k+δk′ϖkt+∑j=1n−1zjεjt,=−c1+r˙12r12z12−∑i=2ncizi2+∑j=1n−1zjεjt+∑k=1nzkεf¯kx¯k−δk′θ˜f¯kTQktθ˜f¯k−δk′θ˜f¯kTε¯f¯kt.
Letcj=cj1+2cj2,cn=cn1+cn2, and cj1,cj2>0. By using Young’s inequality and Lemma 2, one obtains(44)−cj2zj2+zjεjt≤εj2t4cj2≤κj24cj2,−ck2zk2+zkεf¯kx¯k≤εf¯k2x¯k4ck2≤εf¯k∗24ck2,−δk′θ˜f¯kTQktθ˜f¯k≤−σkδk′θ˜f¯kTθ˜f¯k,−δk′σk2θ˜f¯kTθ˜f¯k+2σkθ˜f¯kTε¯f¯kt≤δk′σk2ε¯f¯kt2σk2≤δk′τdχ¯ε¯22σk.
Substituting (44) into (43), one has(45)V˙2≤−c11+r˙12r12z12−∑i=2nci1zi2−∑k=1nσkδk′2θ˜f¯kTθ˜f¯k+R2,where R2=∑k=1nδk′τdχ¯ε¯2/2σk+εf¯k∗2/4ck2+∑j=1n−1κj2/4cj2. And, we choose a positive constant γ2 such that γ2≤min2r1c11+r˙1/r1,2c21,…,2c2n,λf¯1σ1δ1′,…,λf¯nσnδn′, which yields(46)V˙2≤γ2V2+R2,∀t∈T,+∞.
Obviously, solving the above inequality (46) leads to(47)V2≤V2Texp−γ2t+R2γ2.
Inequality (47) implies that zi and θ^f¯i exponentially converge to R2/γ2-neighborhood of zero. And, then the boundedness of z1 derives that e1 satisfies PPB (10) by Lemma 4. This concludes the proof.Remark 7.
The difference between Theorem1 and Theorem 2 is that Theorem 2 introduces prediction errors ϖit, and online recorded data and instantaneous data is used for neural weights update. The comparative simulation results show the control effect of the two methods directly and highlight the proposed composite learning PPC method proposed in this paper is better.


## 3.1. Adaptive Neural Network PPC Method

In order to guarantee thatz1 is bounded, backstepping technique will be employed, and the whole control process is divided into n steps. Step 1:
from (12), the virtual control α2 is designed as(14)α2=−φf1Tx¯1θ^f1+x˙d−Π−c1z1,where c1 is a positive design constant.Definez2=x2−ζ1, where ζ1 is used to estimate α2 through the following first-order filter:(15)τ1ζ˙1=−ζ1−α2,ζ10=α20,where τ1 is design time constant. By Lemma 1, we know that there exists a positive constant κ1 such as ζ1−α2≤κ1. Define ε1t=ζ1−α2; then, x2=z2+ε1t+α2 and (12) can be rewritten as(16)z˙1=r1φf1Tx¯1θ˜f1+εf1x¯1+z2+ε1t−c1z1.The adaptive law ofθ^f1 is chosen as(17)θ^.f1=λf1z1φf1x¯1−δ1θ^f1,where λf1 and δ1 are positive design parameters.Stepj:j=2,…,n−1. According to the definition of zj=xj−ζj−1, we have(18)z˙j=φfjTx¯jθ˜fj+εfjx¯j+θ^fjTφfjx¯j+xj+1−ζ˙j−1,Choose virtual controlαj+1 as(19)αj+1=−φfjTx¯jθ^fj+ζ˙j−1−cjzj−zj−1,where cj is a positive design constant.Definezj+1=xj+1−ζj and ζj satisfies the following first-order filter:(20)τjζ˙j=−ζj−αj+1,ζj0=αj+10,where τj is design time constant. Define εjt=ζj−αj+1, and there exists a positive constant κj such as εjt≤κj by Lemma 1. Similar to step 1, we can rewrite (18) as follows:(21)z.j=φfjTx¯jθ˜fj+εfjx¯j+zj+1+εjt−cjzj−zj−1.And, the parameter adaptive law ofθ^fj is designed as(22)θ^.fj=λfjziφfjx¯j−δjθ^fj,where λfj and δj are positive design parameters.Stepn: the time derivative of zn produces(23)z˙n=φfnTx¯nθ˜fn+εfnx¯n+θ^fnTφfnx¯n+u−ζ˙n−1.The controlleru is designed as(24)u=−φfnTx¯nθ^fn+ζ˙n−1−cnzn−zn−1,where cn is a positive constant. Submitting (24) into (23), we obtain(25)z.n=φfnTx¯nθ˜fn+εfnx¯n−cnzn−zn−1.The adaptive law forθ^fn is chosen as(26)θ^.fn=λfnznφfnx¯n−δnθ^fn,where λfn and δn are positive design parameters.Theorem 1.
Consider uncertainnth-order strict-feedback nonlinear system (1) with assumptions 1-2. If virtual controllers (14) and (19), real controller 24), and RBFNN updating laws (17), (22), and (26) are designed, then all signals in the closed-loop system (27) are uniformly ultimately bounded, and then, the boundedness of z1 ensures that e1 satisfies PPB (10).Proof.
Consider the following Lyapunov function:(27)V1=12r1z12+12λf1θ˜f1Tθ˜f1+∑i=2n12zi2+12λfiθ˜fiTθ˜fi.
According to (16), (21), and (25) and RBFNN updating laws (17), (22), and (26), we have(28)V˙1=−c1+r˙12r12z12−∑i=2ncizi2+∑k=1nzkεfkx¯k+zkφfkTx¯kθ˜fk−∑k=1nθ˜fkTzkφfkx¯k−δkθ^fk+∑j=1n−1zjεjt,=−c1+r˙12r12z12−∑i=2ncizi2+∑j=1n−1zjεjt+∑k=1nzkεfkx¯k+δkθ˜kkTθ^fk.
Letcj=cj1+2cj2,cn=cn1+cn2, and cj1,cj2>0, j=1,2,…,n−1. By using Young’s inequality, we obtain(29)δkθ˜fkTθ^fk≤−δk2θ˜fk2+δk2θ˜fk∗2,−ck2zk2+zkεfkx¯k≤εfk2x¯k4ck2≤εfk∗24ck2,k=1,2,…,n,−cj2zj2+zjεjt≤εj2t4cj2≤κj24cj2,j=1,2,…,n−1.
Substituting (29) into (28), we obtain(30)V˙1≤−c11+r˙12r12z12−∑i=2nci1zi2−∑k=1nδk2θ˜fk2︸θ˜fkTθ˜fk+R1,where R1=∑k=1nδk/2θfk∗2+εfk∗2/4ck2+∑j=1n−1κj2/4cj2. According to 5, we choose c11>−r˙1/2r12 and define the following compact sets:(31)Ωz1=z1|z1≤R1c11+r˙1/2r12,Ωzi=zi|zi≤R1ci1,Ωθfk=θfk|θ˜fk≤2R1δk.
Obviously,z1∉Ωz1 or zi∉Ωzi or θ˜fk∉Ωθ˜fk, we have V˙1<0. Thus, z1,z˜i,zn and θ˜fk are all semiglobally uniformly bounded. According to Lemma 4, the boundedness of z1 ensures that e1 meets PPB (10). This concludes the proof.Remark 5.
In fact, Theorem1 also implies that all system states x1,x2,…,xn are recurrent for t≥T. Because e1 is limited within PPB (10), i.e., a∞<x1−xd<b∞, t≥T, so, from Lemma 3, we obtain that x1 is recurrent. And, because fix¯i is smooth, this leads to x2, x3,…,xn are recurrent in turn. It also implies that there exists a regression subvector φf¯ix¯i such that φf¯ix¯i satisfies partial PE condition by Lemma 2.Remark 6.
It should be pointed out that RBFNN updating laws (17), (22), and (26) only use the instantaneous data, and it may not estimate the unknown function fx¯i accurately. Therefore, in the next section, we will use the composite learning control method to ensure the boundedness of z1 and the accurate estimation of fx¯i.


## 3.2. Adaptive Neural Network PPC Method with Learning

In this section, a prediction errorϖit is constructed by using online data and instantaneous data. According to Remark 5 in Section 3.1, system states x1,x2,…,xn in system (1) will be recurrent after T. So, for x¯i, there is a regression subvector φf¯ix¯i, which satisfies partial PE condition by Lemma 2, i.e., ∫t−τdtφf¯ix¯iφf¯iTx¯idτ≥σiI, where σi>0 and t≥T, i=1,2,…,n. And, fix¯i can also be estimated by φf¯iTx¯iθ^f¯i instead of φfiTx¯iθ^fi after T. The specific modification is as follows.Firstly, substitutingfix¯i=φf¯ix¯iTθf¯i∗+εf¯ix¯it≥T into system (1), one obtains(32)x˙i=φf¯ix¯iTθf¯i∗+εf¯ix¯i+xi+1,i=1,2,…,n−1,x˙n=φf¯nx¯nTθf¯n∗+εf¯nx¯n+u,where εf¯ix¯i≤εf¯i∗, εf¯i∗ is a positive constant. Let Qit:=∫t−τdtφf¯ix¯iφf¯iTx¯idτ and ε¯f¯it:=∫t−τdtφf¯ix¯iεf¯ix¯idτ. Multiplying both sides of above equalities in (32) by φf¯ix¯i and integrating equalities over t,t−τd, one obtains(33)Qitθf¯i∗+ε¯f¯it=∫t−τdtφf¯ix¯ix˙i−xi+1dτ,Qntθf¯n∗+ε¯f¯nt=∫t−τdtφf¯nx¯nx˙n−udτ.So, the prediction errorϖit is defined as(34)ϖit=Qitθf¯i∗+ε¯f¯it−Qitθ^f¯i=Qitθ˜f¯i+ε¯f¯it.Obviously,ε¯f¯it≤τdχiεf¯i∗; let χ¯=maxχ1,χ2,…,χn and ε¯=maxεf¯1∗,εf¯2∗,…,εf¯n∗; then, for ∀i, ε¯f¯it≤τdψ¯ε¯ holds.Then, we replaceφfiTx¯iθ^fi with φf¯iTx¯iθ^f¯i in (12), (18), and (23) after T, and one obtains(35)z˙1=r1φf¯1Tx¯1θ^f¯1+φf¯1Tx¯1θ˜f¯1+x2+εf¯1x¯1−x˙d+Π,z˙j=φf¯jTx¯jθ^f¯j+φf¯jTx¯jθ˜f¯j+xj+1+εf¯jx¯j−ζ˙j−1,z˙n=φf¯nTx¯nθ^f¯n+φf¯nTx¯nθ˜f¯j+u+εf¯nx¯n−ζ˙n−1.where zj=xj−ζj−1, ζj−1 is the estimation of αj, which is the same as the one in Section 3.1, j=2,3,…,n. And, the modified virtual controller αj is designed as follows.According to (35), the virtual control α2 in (14) is modified as(36)α2=−φf1Tx¯1θ^f1+x˙d−Π−c1z1,t<T,−φf¯1Tx¯1θ^f¯1+x˙d−Π−c1z1,t≥T.And, composite learning laws ofθ^f1 and θ^f¯1 are given as(37)θ^.f1=λf1z1φf1x¯1−δ1θ^f1,t<T,θ^.f¯1=λf¯1z1φf¯1x¯1+δ1′ϖ1t,t≥T.where λf¯1 and δ1′ are positive design parameters.The virtual controlαj+1 in (19) is modified as(38)αj+1=−θ^fjTφfjx¯j+ζ˙j−1−cjzj−zj−1,t<T,−θ^f¯jTφf¯jx¯j+ζ˙j−1−cjzj−zj−1,t≥T.Similarly, composite learning laws ofθ^fj and θ^f¯j are designed as(39)θ^.fj=λfjziφfjx¯j−δjθ^fj,t<T,θ^.f¯j=λf¯jziφf¯jx¯j+δj′ϖjt,t≤T,where λf¯1 and δj′ are positive design parameters.The controlleru in (24) is modified as(40)u=−θ^fnTφfnx¯n+ζ˙n−1−cnzn−zn−1,t<T,−θ^f¯nTφf¯nx¯n+ζ˙n−1−cnzn−zn−1,t≥T.and composite learning laws for θ^fn and θ^f¯n are designed as(41)θ^.fn=λfnznφfnx¯n−δnθ^fn,t<T,θ^.f¯n=λf¯nznφf¯nx¯n+δn′ϖnt,t≥T,where λf¯1 and δn′ are positive design parameter. Other parameters ci,λfi, and δi are the same as those in Section 3.1.Next, a main theorem of this paper is given as follows.Theorem 2.
Consider uncertainnth-order strict-feedback nonlinear system (1) with Assumptions 1-2. The virtual controllers (36) and (38) and real controller (40) combine with RBFNN updating composite learning laws (37), (39), and (41) guarantee both the convergence of zi and θ^f¯i during t∈T,+∞, and then, tracking error e1 meets PPB (10).Proof.
Theorem1 and Remark 5 have been proved that x1,x2,…,xn are recurrent after time T. So, it is necessary to prove whether the composite learning law of θ^f¯i can ensure that zi and θ˜f¯i enter a small neighborhood of zero after time T. Consider the following Lyapunov function at t∈T,+∞:(42)V2=12r1z12+12λf¯1θ˜f¯1Tθ˜f¯1+∑i=2n12zi2+12λf¯iθ˜f¯iTθ˜f¯i.
According to (36)–(41), one obtains(43)V˙2=−c1+r˙12r12z12−∑i=2ncizi2+∑k=1nzkεf¯kx¯k+zkφf¯kTx¯kθ˜f¯k−∑k=1nθ˜f¯kTzkφf¯kx¯k+δk′ϖkt+∑j=1n−1zjεjt,=−c1+r˙12r12z12−∑i=2ncizi2+∑j=1n−1zjεjt+∑k=1nzkεf¯kx¯k−δk′θ˜f¯kTQktθ˜f¯k−δk′θ˜f¯kTε¯f¯kt.
Letcj=cj1+2cj2,cn=cn1+cn2, and cj1,cj2>0. By using Young’s inequality and Lemma 2, one obtains(44)−cj2zj2+zjεjt≤εj2t4cj2≤κj24cj2,−ck2zk2+zkεf¯kx¯k≤εf¯k2x¯k4ck2≤εf¯k∗24ck2,−δk′θ˜f¯kTQktθ˜f¯k≤−σkδk′θ˜f¯kTθ˜f¯k,−δk′σk2θ˜f¯kTθ˜f¯k+2σkθ˜f¯kTε¯f¯kt≤δk′σk2ε¯f¯kt2σk2≤δk′τdχ¯ε¯22σk.
Substituting (44) into (43), one has(45)V˙2≤−c11+r˙12r12z12−∑i=2nci1zi2−∑k=1nσkδk′2θ˜f¯kTθ˜f¯k+R2,where R2=∑k=1nδk′τdχ¯ε¯2/2σk+εf¯k∗2/4ck2+∑j=1n−1κj2/4cj2. And, we choose a positive constant γ2 such that γ2≤min2r1c11+r˙1/r1,2c21,…,2c2n,λf¯1σ1δ1′,…,λf¯nσnδn′, which yields(46)V˙2≤γ2V2+R2,∀t∈T,+∞.
Obviously, solving the above inequality (46) leads to(47)V2≤V2Texp−γ2t+R2γ2.
Inequality (47) implies that zi and θ^f¯i exponentially converge to R2/γ2-neighborhood of zero. And, then the boundedness of z1 derives that e1 satisfies PPB (10) by Lemma 4. This concludes the proof.Remark 7.
The difference between Theorem1 and Theorem 2 is that Theorem 2 introduces prediction errors ϖit, and online recorded data and instantaneous data is used for neural weights update. The comparative simulation results show the control effect of the two methods directly and highlight the proposed composite learning PPC method proposed in this paper is better.


## 4. Simulation Studies

In this section of the simulation, the Chua system [25] is considered as an example:(48)x˙1=βx2,x˙2=f2x¯2+x3,x˙2=f3x¯3+u,y=x1.where f2x¯2=x1−x2 and f3x¯3=γx2−x3−m1x3−1/2m0−m1x3+1−x3−1. Denote the method in Section 3.1 as the ANPPC method and the method in Section 3.2 as the CLPPC method. The reference signal xd=cost. For the NN design of the ANPPC method, the number of hidden nodes is chosen as 3×3 and 3×3×3, respectively, and their centers x¯2 and x¯3 are being evenly spaced on −2,2×−0.5,0.5 and −2,2×−0.5,0.5×−1.5,1.5. Initial weights are θ^f20=0 and θ^f30=0 and the initial states are x10,x20,x30T=−0.8,0.6,0.1T. Parameters of system (48) are selected as β=−25, γ=15.6,m0=−1.028,andm1=−0.714, and design parameters of ANPPC-method are chosen as λf2=λf3=10, c1=c2=c3=10, τ1=τ2=0.002, and δ2=δ3=0.005. FTPFs are selected as pt,T,−4,−0.05 and pt,T,−0.5,0.05 with T=5s. For the parameters in the CLPPC method, the selection of parameters is the same as that in the ANPPC method when t<T=5s and the parameters λf¯2=λf¯3=10,τd=5,andδ2′=δ3′=6 and other parameters are the same as those in ANPPC-method when t≥T=5s. Notice that the system states x1, x2, and x3 are recurrent after T=5s seconds, so Lemma 2 can be applied to redesign the NN after 5 seconds: the number of hidden nodes is chosen as 3×3 and 3×3×3, respectively, and their centers x¯2 and x¯3 are being evenly spaced on −1,1×−0.05,0.05 and −1,1×−0.05,0.05×−1,1. And, initial weights are θ^f¯20=0 and θ^f¯30=0. The simulation results are shown in Figures 1-6.Figure 1 shows that the tracking error e1 can be limited within PPB (10) under the ANPPC method and the CLPPC method, but Figure 1(b) shows that, after 10s, the control effect of e1 is greatly improved by using the CLPPC method compared with the ANPPC method. The ANPPC method and CLPPC method are used to estimate unknown functions f2x¯2 and f2x¯3. The estimation effect is shown in Figures 2 and 3, respectively. Obviously, the estimation effect of the CLPPC method is better than that of the ANPPC method after 10 s. Figure 4 shows composite learning laws of θ^f¯2 and θ^f¯3 after T=5s. Figure 5 shows the control inputs u under above methods. In the whole process of control input, the energy consume is not significant difference, except that the CLPPC method has two jumps at 5 seconds and 10 seconds. In addition, compared with the prescribed performance control method used in [11–16], the proposed CLPPC method in this paper can realize the finite time adjustment of tracking error e1, which is shown in Figure 6. In general, the control effect of tracking error and estimation effect of unknown functions by using the CLPPC method are better than that by using the ANPPC method.Figure 1
Trajectories ofe1 in (a)0,30; (b)5,20.
(a)(b)Figure 2
Estimation of (a)f2x¯2 and (b) f2x¯3 by using the ANPPC method.
(a)(b)Figure 3
Estimation of (a)f2x¯2 and (b)f2x¯3 by using the CLPPC method.
(a)(b)Figure 4
Trajectories of (a)θ^f¯2 and (b)θ^f¯3 by using the CLPPC method.
(a)(b)Figure 5
Control trajectories ofu.Figure 6
Tracking errore1 by the CLPPC method with (a) T=3s, (b) T=10s, and (c) T=20s.
(a)(b)(c)


## 5. Conclusion

In this paper, the composite learning prescribed performance control method is investigated for uncertain strict-feedback system. Firstly, a new finite-time performance was proposed and predictive errors were introduced by applying online recorded date and instantaneous date to update NN weights. The boundedness of the closed-loop system was ensured by the proposed CLPPC method. Simulation results show the effectiveness of the proposed method. Because the reference signal may not be recurrent, it will lead to partial PE condition is not tenable. So, the accurate estimation of unknown function under an interval excitation (IE) condition becomes the next research direction.



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*Source: 9988328-2021-05-10.xml*