# Prescribed Performance Neural Control of Strict-Feedback Systems via Disturbance Observers

Authors: Wei Xiang; Guangkui Xu; Fang Zhu; Chunzhi Yang
Journal: Complexity
 (2020)

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## Abstract

This paper provides a disturbance observer-based prescribed performance control method for uncertain strict-feedback systems. To guarantee that the tracking error meets a design prescribed performance boundary (PPB) condition, an improved prescribed performance function is introduced. And radial basis function neural networks (RBFNNs) are used to approximate nonlinear functions, while second-order filters are employed to eliminate the “explosion-complexity” problem inherent in the existing method. Meanwhile, disturbance observers are constructed to estimate the compounded disturbance which includes time-varying disturbances and network construction errors. The stability of the whole closed-loop system is guaranteed via Lyapunov theory. Finally, comparative simulation results confirm that the proposed control method can achieve better tracking performance.


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## Body

## 1. Introduction

Backstepping technology is widely used for nonlinear systems with the strict-feedback structure [1–3]. This method requires repeated differentiation of the virtual controller, which leads to the problem of “explosion-complexity.” In order to solve this problem and estimate the virtual controller, dynamic surface control (DSC) technique [4–7] is applied. In [4], DSC technology was applied to the state-feedback adaptive control law, which eliminated the problem of “explosion-complexity” and ensured that all signals in the closed-loop system were uniformly ultimately bounded. By using DCS technology and fuzzy logic systems, Aung et al. [7] proposed a fuzzy adaptive control scheme that can guarantee tracking error to a small residual set. To further estimate the derivative of the virtual controller αt, a famous first-order Levant differentiator was constructed in [8]. By applying the same differentiator, Yu et al. [9] proposed a backstepping approach, which enables the tracking error to converge to a small neighborhood in finite time. It can be seen from the above literature that the nonlinear system contains unknown nonlinear function terms and external disturbances. For the former, there are two approximate methods: one is using the composite learning control method [10–13] which uses online recorded data to produce a prediction error to construct a composite learning signal, and the other is using neural networks (NNs) or fuzzy logic systems (FLSs) [14–19]. However, adaptive laws of estimating parameters in [14–19] only consider whether the Lyapunov stability condition is satisfied but does not consider whether the parameter law can accurately estimate the unknown nonlinear function itself. For the latter, we cannot just assume that the external disturbance is bounded and the upper bound is unknown. And this assumption will lead to a problem where the disturbance cannot be accurately estimated. Currently, the effective method is to build a disturbance observer. The basic principle is that all disturbances are regarded as system states. In [20], a neural network disturbance observer (NNDOB) was developed for a 3-DOF model helicopter system. In [21], disturbance observer-based composite control was proposed for the flexible-link manipulator. In order to estimate unknown system nonlinearity and unknown disturbance, composite learning control strategy for uncertain systems in strict-feedback form was proposed in [22, 23]. Some other interesting results about disturbance observer-based control can be seen in [24–27].On the contrary, especially in practical control, it is better to specify a prescribed performance boundary (PPB) of tracking error rather than the approach tracking error to a small neighborhood. Therefore, how to combine the prescribed performance control (PPC) [28, 29] and the disturbance observer to make the error meet the PPB condition has become a research focus. Usually, the traditional PPB [30] of tracking error e is described as(1)−λμt<e<μtife0>0,−μt<e<λμtife0>0,where λ∈0,1 and μt=μ0−μ∞e−λ1t+μ∞; μ0,μ∞,andλ1 are design parameters. However, the traditional PPC strategy has two shortcomings. One is that the design of the controller needs to consider the sign of the initial value e0, and the other is that the small convergence overshoot of e cannot be guaranteed. To tackle the above defects, an improved PPC methodology was proposed in [31]. And Bu et al. [31] only discussed the situation that external disturbances were unknown. However, some known information of nonlinear functions may not be able to provide in the practical control system. Therefore, it is necessary to discuss that how to accurately estimate unknown nonlinear functions and external disturbances under the condition that the tracking error meets the PPB.Inspired by the above discussion, a class of SISO uncertain nonlinear system in strict-feedback form with unknown time-varying disturbance will be studied in this paper. Different from previous designs, an improved prescribed performance control scheme is proposed. The overall framework of this paper is as follows: firstly, a specific performance function and PPB condition are given. Then, the original system is transformed into an equivalent transformation system. Afterwards, unknown nonlinear functions are approximated by using RBFNNs, and a disturbance observer is designed to estimate the compounded disturbance which includes time-varying disturbances and network construction errors. The proposed composite prescribed performance control scheme can not only guarantee the prescribed performance but also make accurate estimates of system uncertainties. The main contributions of this paper are summarized as follows: (1) the proposed PPC method guarantees a small convergence overshoot of the tracking error; (2) system uncertainties (include unknown nonlinear functions and external disturbances) can be effectively estimated by the disturbance observer and RBFNNs; and (3) compared with the traditional PPC method, the proposed method has better control performance. The remainder of this paper is organized as follows: Section2 lists some preliminaries. In Section 3, a novel PPC scheme is designed, and the stability of the closed-loop system is considered. Simulation studies are included in Section 4. Finally, brief conclusion is presented in Section 5.


## 2. Preliminaries

### 2.1. System Description

Consider the following SISO strict-feedback nonlinear system:(2)x˙1=f1x¯1+g1x¯1x2+d1t,x˙2=f2x¯2+g1x¯2x3+d2t,⋯x˙n=fnx¯n+gnx¯nu+dnt,y=x1,where x¯i=x1,x2,…,xiT∈Ri, x1=x¯1∈R, x=x¯n=x1,x2,…,xnT∈Rn, u∈R is the control input, y∈R is the output, fix¯i is an unknown smooth function, gix¯i is a known nonzero function, and dit is the external time-varying disturbance, i=1,2,…,n.Assumption 1.
The reference signalxd and its derivative x˙d are available and bounded.Assumption 2.
There exist positive constantsd¯i,di∗,andg¯i such that |dit|≤d¯i, |d˙it|≤di∗, and |gix¯i|≤g¯i, i=1,2,⋅⋅⋅,n.Remark 1.
Assumptions1 and 2 are common assumptions for strict-feedback systems [4, 21, 22]. Here, the boundedness of the external disturbance and its derivative is to construct the disturbance observer, and positive constants d¯i,di∗,andg¯i are only used for stability analysis.
For SISO strict-feedback nonlinear system (2), backstepping technique will be used in the design process of controller u, and the tedious derivative calculation of the virtual control law will cause the problem of “explosion of terms.” To avoid such problems, the following second-order filters are used in this paper.Lemma 1 (see [28]).
Consider the following system:(3)χ˙1=χ2,χ˙2=ϖ12−ϖ2tanhχ1−ξ−ϖ3tanhχ2ϖ1,where ϖ1,ϖ2, and ϖ3 are design parameters and ξ is the input signal. From [28], it can be concluded that there exists a positive constant ϱ such that χi−ξi−1=O1/ϖ1ϱω−i+1, i=1,2, where ω=1−ω¯/ω¯ and ω¯∈minϱ/1+ϱ,1/2. Therefore, there exist sufficiently small positive constants κ1andκ2 such that ϵ1≤κ1 and ϵ2≤κ2, where ϵ1=χ1−ξ and ϵ2=χ2−ξ˙.In order to estimate the functionfix¯i, we define Fix¯i=lifix¯i; li is the positive design constant. According to the approximation property of RBFNNs, the approximation form of Fix¯i can be expressed as(4)F^ix¯i=θ^FiTφFix¯i,where θ^Fi=θ^Fi1,θ^Fi2,…,θ^FisT∈Rs is the weight vector, s is the node number, and φFix¯i=φFi1x¯i,φFi2x¯i,…,φFisx¯iT∈Rs is the basis function vector. Here, φFijx¯i,j=1,2,…,s, is(5)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 [2–4], choosing appropriate cjandbj and some sufficiently large integer s, there exists an ideal weight vector θFi∗∈Rs such that(6)Fix¯i=θFi∗TφFix¯i+εFi,where εFi is the approximate error. So, we can rewrite Fix¯i in (6) as(7)Fix¯i=θ˜FiTφFix¯i+θ^FiTφFix¯i+εFi,where θ˜Fi=θFi∗−θ^Fi, and θ^Fi is the estimation of θFi∗.Notice thatFix¯i=lifix¯i, so we have(8)fix¯i=li−1Fix¯i=li−1θ˜FiTφFix¯i+li−1θ^FiTφFix¯i+li−1εFi.Substituting (8) into (2), we get(9)x˙1=l1−1θ^F1TφF1x¯1+l1−1θ˜F1TφF1x¯1+g1x¯1x2+D1,x˙2=l2−1θ^F2TφF2x¯2+l2−1θ˜F2TφF2x¯2+g2x¯2x3+D2,⋯x˙n=ln−1θ^FnTφFnx¯n+ln−1θ˜FnTφFnx¯n+gnx¯nu+Dn,where Di=dit+li−1εFi,i=1,2,…,n.Remark 2.
From (9), the design parameters l1,l2,…,ln need to meet certain conditions, which will be given in the subsequent stability analysis. In [31], Bu et al. did not consider the case that nonlinear functions fix¯i,i=1,2,…,n, are unknown and how to estimate them accurately. Therefore, the results of this paper can be regarded as a further study of [31]. For parameters l1,l2,…,ln in (9), the stability conditions of them will be given in the later analytical proof.


### 2.2. Prescribed Performance

Define the tracking error variablee1=x1−xd, and set e satisfies the following prescribed performance boundary (PPB) [31]:(10)p1t<e1<p2t,where p1t and p2t are performance functions and defined as(11)p1t=signe10−λ1μt−μ∞signe10,p2t=signe10+λ2μt−μ∞signe10,μt=μ0−μ∞e−λ3t+μ∞,where 0≤λ1,λ2≤1,μ0>μ∞>0, and λ3>0.In order to ensure that the tracking errore is limited within PPB (10), we introduce the following transformation variable z1:(12)z1=lnht1−ht,where ht=e1t−p1t/p2t−p1t.Lemma 2.
Ifz1 is bounded, e1 can be limited within PPB (10).Proof.
Sincez1 is bounded, there exists a constant M>0 such that z1≤M. Notice z1=lnht/1−ht is equivalent to ht=ez1/1+ez1. So,(13)0<e−M11+e−M1<ht=e1t−p1tp2t−p1t<eM11+eM1<1.
Clearly, we havep1t<e1t<p2t. The proof is completed.Remark 3.
Compared with [30], prescribed performance function (11) in this paper can guarantee the convergence of e1 in a small overshoot, which is depicted in Figure 1.
When the tracking errore1 is limited within PPB (10) by choosing appropriate parameters μ0,μ∞,λ1,λ2, and λ3, we can conclude that (i) r1=1/ht1−htp2t−p1t>0; (ii) r1 tends to a small neighborhood of a certain constant, and then r˙1 also tends to a small neighborhood of zero. Therefore, based on the above two properties of r1 and Lemma 2, we use 1/2r1z12 instead of 1/2z12 to explore the boundedness of z1.Figure 1
Graphical illustration of prescribed performance function (11) with (a) e10>0; (b) e10<0.
(a)(b)


### 2.3. Control Aim

The aim of this paper is (i) to design the composite prescribed performance control scheme so that the tracking errore1 satisfies PPB (10) and (ii) to construct the disturbance observer D^i and neural network rules (NNs) so that fix¯i+dit can be estimated accurately.Remark 4.
It is necessary to point out that the disturbance observerD^i in this paper is not to estimate the external disturbance dit but to estimate accurately the external disturbance dit and unknown function fix¯i,i=1,2,…,n.


## 2.1. System Description

Consider the following SISO strict-feedback nonlinear system:(2)x˙1=f1x¯1+g1x¯1x2+d1t,x˙2=f2x¯2+g1x¯2x3+d2t,⋯x˙n=fnx¯n+gnx¯nu+dnt,y=x1,where x¯i=x1,x2,…,xiT∈Ri, x1=x¯1∈R, x=x¯n=x1,x2,…,xnT∈Rn, u∈R is the control input, y∈R is the output, fix¯i is an unknown smooth function, gix¯i is a known nonzero function, and dit is the external time-varying disturbance, i=1,2,…,n.Assumption 1.
The reference signalxd and its derivative x˙d are available and bounded.Assumption 2.
There exist positive constantsd¯i,di∗,andg¯i such that |dit|≤d¯i, |d˙it|≤di∗, and |gix¯i|≤g¯i, i=1,2,⋅⋅⋅,n.Remark 1.
Assumptions1 and 2 are common assumptions for strict-feedback systems [4, 21, 22]. Here, the boundedness of the external disturbance and its derivative is to construct the disturbance observer, and positive constants d¯i,di∗,andg¯i are only used for stability analysis.
For SISO strict-feedback nonlinear system (2), backstepping technique will be used in the design process of controller u, and the tedious derivative calculation of the virtual control law will cause the problem of “explosion of terms.” To avoid such problems, the following second-order filters are used in this paper.Lemma 1 (see [28]).
Consider the following system:(3)χ˙1=χ2,χ˙2=ϖ12−ϖ2tanhχ1−ξ−ϖ3tanhχ2ϖ1,where ϖ1,ϖ2, and ϖ3 are design parameters and ξ is the input signal. From [28], it can be concluded that there exists a positive constant ϱ such that χi−ξi−1=O1/ϖ1ϱω−i+1, i=1,2, where ω=1−ω¯/ω¯ and ω¯∈minϱ/1+ϱ,1/2. Therefore, there exist sufficiently small positive constants κ1andκ2 such that ϵ1≤κ1 and ϵ2≤κ2, where ϵ1=χ1−ξ and ϵ2=χ2−ξ˙.In order to estimate the functionfix¯i, we define Fix¯i=lifix¯i; li is the positive design constant. According to the approximation property of RBFNNs, the approximation form of Fix¯i can be expressed as(4)F^ix¯i=θ^FiTφFix¯i,where θ^Fi=θ^Fi1,θ^Fi2,…,θ^FisT∈Rs is the weight vector, s is the node number, and φFix¯i=φFi1x¯i,φFi2x¯i,…,φFisx¯iT∈Rs is the basis function vector. Here, φFijx¯i,j=1,2,…,s, is(5)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 [2–4], choosing appropriate cjandbj and some sufficiently large integer s, there exists an ideal weight vector θFi∗∈Rs such that(6)Fix¯i=θFi∗TφFix¯i+εFi,where εFi is the approximate error. So, we can rewrite Fix¯i in (6) as(7)Fix¯i=θ˜FiTφFix¯i+θ^FiTφFix¯i+εFi,where θ˜Fi=θFi∗−θ^Fi, and θ^Fi is the estimation of θFi∗.Notice thatFix¯i=lifix¯i, so we have(8)fix¯i=li−1Fix¯i=li−1θ˜FiTφFix¯i+li−1θ^FiTφFix¯i+li−1εFi.Substituting (8) into (2), we get(9)x˙1=l1−1θ^F1TφF1x¯1+l1−1θ˜F1TφF1x¯1+g1x¯1x2+D1,x˙2=l2−1θ^F2TφF2x¯2+l2−1θ˜F2TφF2x¯2+g2x¯2x3+D2,⋯x˙n=ln−1θ^FnTφFnx¯n+ln−1θ˜FnTφFnx¯n+gnx¯nu+Dn,where Di=dit+li−1εFi,i=1,2,…,n.Remark 2.
From (9), the design parameters l1,l2,…,ln need to meet certain conditions, which will be given in the subsequent stability analysis. In [31], Bu et al. did not consider the case that nonlinear functions fix¯i,i=1,2,…,n, are unknown and how to estimate them accurately. Therefore, the results of this paper can be regarded as a further study of [31]. For parameters l1,l2,…,ln in (9), the stability conditions of them will be given in the later analytical proof.


## 2.2. Prescribed Performance

Define the tracking error variablee1=x1−xd, and set e satisfies the following prescribed performance boundary (PPB) [31]:(10)p1t<e1<p2t,where p1t and p2t are performance functions and defined as(11)p1t=signe10−λ1μt−μ∞signe10,p2t=signe10+λ2μt−μ∞signe10,μt=μ0−μ∞e−λ3t+μ∞,where 0≤λ1,λ2≤1,μ0>μ∞>0, and λ3>0.In order to ensure that the tracking errore is limited within PPB (10), we introduce the following transformation variable z1:(12)z1=lnht1−ht,where ht=e1t−p1t/p2t−p1t.Lemma 2.
Ifz1 is bounded, e1 can be limited within PPB (10).Proof.
Sincez1 is bounded, there exists a constant M>0 such that z1≤M. Notice z1=lnht/1−ht is equivalent to ht=ez1/1+ez1. So,(13)0<e−M11+e−M1<ht=e1t−p1tp2t−p1t<eM11+eM1<1.
Clearly, we havep1t<e1t<p2t. The proof is completed.Remark 3.
Compared with [30], prescribed performance function (11) in this paper can guarantee the convergence of e1 in a small overshoot, which is depicted in Figure 1.
When the tracking errore1 is limited within PPB (10) by choosing appropriate parameters μ0,μ∞,λ1,λ2, and λ3, we can conclude that (i) r1=1/ht1−htp2t−p1t>0; (ii) r1 tends to a small neighborhood of a certain constant, and then r˙1 also tends to a small neighborhood of zero. Therefore, based on the above two properties of r1 and Lemma 2, we use 1/2r1z12 instead of 1/2z12 to explore the boundedness of z1.Figure 1
Graphical illustration of prescribed performance function (11) with (a) e10>0; (b) e10<0.
(a)(b)


## 2.3. Control Aim

The aim of this paper is (i) to design the composite prescribed performance control scheme so that the tracking errore1 satisfies PPB (10) and (ii) to construct the disturbance observer D^i and neural network rules (NNs) so that fix¯i+dit can be estimated accurately.Remark 4.
It is necessary to point out that the disturbance observerD^i in this paper is not to estimate the external disturbance dit but to estimate accurately the external disturbance dit and unknown function fix¯i,i=1,2,…,n.


## 3. Control Design

In order to prove thatz1 is bounded, the prescribed performance control and backstepping technique will be employed in this section. And the controller u will be designed at step n.Step 1.
The time derivative ofz1 becomes(14)z˙1=r1l1−1θ˜F1TφF1x¯1+l1−1θ^F1TφF1x¯1+g1x¯1x2+D1−x˙d+Π,where(15)r1=1ht1−htp2t−p1t>0,Π=p1tp˙2t−p˙1tp2t−e1p˙2t−p1tp2t−p1t.Remark 5.
According to (10) and (12), we know that p2t−p1t>0, 1>ht>0, and 1−ht>0, so r1=1/ht1−htp2t−p1t>0.
Choose virtual controlα1 as(16)α1=−l1−1θ^F1TφF1x¯1−D^1+x˙d−Π−c1z1g1x¯1,where D^1 is the estimation of D1andc1 is a positive design constant.
Definez2=x2−ξ21, where ξ21 is used to estimate α1 through the following second-order filter:(17)ξ˙21=ξ22,ξ˙22=λ212−λ22tanhξ21−α1−λ23tanhξ22λ21,where λ21,λ22, and λ23 are design parameters. By Lemma 1, we know that there exists a positive constant M1 such that ξ21−α1≤M1. Define e2=ξ21−α1 and a prediction error z˜1 as z˜1=z^1−z1, where z^1 satisfies(18)z^˙1=r1l1−1θ^F1TφF1x¯1+g1x¯1x2+D^1−x˙d+Π−k1z˜1,where k1 is a positive design constant. The disturbance observer is designed as(19)D^1=l01z1−ω1,ω˙1=r1l1−1θ^F1TφF1x¯1+g1x¯1x2+D^1−x˙d+Π+l01−1γ1z˜1−z1,where l01 and γ1 are positive constants. It should be pointed out that the purpose of introducing the prediction error here is to provide the information of the error state so that the following parameter adaptive law can ensure that the NN effectively estimates the unknown function.
The composite parameter adaptive law forθ^F1 is designed as(20)θ^.F1=λF1l1−1z1φF1x¯1−γ1l1−1z˜1φF1x¯1−δ1θ^F1,where λF1 and δ1 are positive design parameters.
Define the following Lyapunov function:(21)V1=12r1z12+γ1z˜12+D˜12+12λF1θ˜F1Tθ˜F1,where D˜1=D1−D^1. According to (14)–(20) and x2=z2+e2+α1, the time derivative of V1 is(22)V˙1=−c1+r˙12r12z12−γ1k1+r˙12r12z˜12+z1g1x¯1z2+z1g1x¯1e2,−l01+r˙12r12D˜12+D˜1D˙1r1−l01l1−1D˜1θ˜F1TφF1x¯1+δ1θ˜F1Tθ^F1T.
Ifz2 is bounded, i.e., there exists a positive constant K2 such that z2≤K2, and according to Young’s inequality, we have(23)D˜1D˙1r1≤D˜12r12+D˙122≤D˜12r12+D1∗22,−l01l1−1D˜1θ˜F1TφF1x¯1≤l01l1−1D˜122+l01l1−1χF122θ˜F12,δ1θ˜F1Tθ^F1≤−δ12θ˜F12+δ12θ˜F1∗2,z1e2g1x¯1≤z122+g¯1M122,z1g1x¯1z2≤z122+g¯12K222,where φF1x¯1≤χF1. Here, the boundedness of φF1x¯1 can be referred to in [12, 13]. Substituting (23) into (22), we obtain(24)V˙1≤−c1−1+r˙12r12z12−γ1k1+r˙12r12z˜12−l01−l01l1−12+r˙1−12r12D˜12−δ12−l01l1−1χF122θ˜F12︸θ˜F1Tθ˜F1+R1,where R1=D1∗2/2+δ1/2θF1∗2+g¯12K22/2+g¯1M12/2. Let c1>1−r˙1/2r12, k1>−r˙1/2r12, l01>l01l1−1/2−r˙1−1/2r12, and δ1/2>l01l1−1χF12/2, and define the following compact sets:(25)Ωz1=z1|z1≤R1c1−1+r˙1/2r12,Ωz˜1=z˜1|z1≤R1γ1k1+r˙1/2r12,ΩD˜1=D˜1|D˜1≤R1l01−l01l1−1/2+r˙1−1/2r12,Ωθ˜F1=θ˜F1|θ˜F1≤R1δ1/2−l01l1−1χF12/2.
It can be seen thatz1∉Ωz1 or z˜1∉Ωz˜1 or D˜1∉ΩD˜1 or θ˜F1∉Ωθ˜F1, and we have V˙1<0. Thus, z1,z˜1,D˜1, and θ˜F1 are all semiglobally uniformly bounded. The above conclusion is based on the fact that the variable z2 is bounded. Now, we need to prove that z2 is bounded.Step i.
i=2,…,n−1.
According to the definition ofzi=xi−ξi1, we have(26)z˙i=li−1θ˜FiTφFix¯i+li−1θ^FiTφFix¯i+gix¯ixi+1+Di−ξi2.
Choose virtual controlαi as(27)αi=−li−1θ^FiTφFix¯i−D^i+ξi2−cizigix¯i,where D^i is the estimation of Diandci is a positive design constant.
Definezi+1=xi+1−ξi+1,1, and ξi+1,1 satisfies the following second-order filter:(28)ξ˙i+1,1=ξi+1,2,ξ˙i+1,2=λi+1,12−λi+1,2tanhξi+i,1−αi−λi+1,1tanhξi+1,3λi+1,1,where λi+1,1,λi+1,2, and λi+1,3 are design parameters. Define ei+1=ξi+i,1−αi, and there exists a positive constant Mi such that ei+1=ξi+i,1−αi≤Mi by Lemma 1. Similar to Step 1, define the prediction variable z^i as(29)z^˙i=li−1θ^FiTφFix¯i+gix¯ixi+1+D^i−ξi2−kiz˜i,where z˜i=z^i−zi and ki>0.
The disturbance observer is designed as(30)D^i=l0izi−wi,w˙i=li−1θ^FiTφFix¯i+gix¯ixi+1+D^i−ξi2+l0i−1γiz˜i−zi,where l0i is a positive design parameter. And the composite parameter adaptive law for θ^Fi is designed as(31)θ^.Fi=λFili−1z1φFix¯i−γili−1z˜iφFix¯i−δiθ^Fi,where λFi and δi are positive design parameters.
Define the following Lyapunov function:(32)Vi=12zi2+γiz˜i2+D˜i2+1λFiθ˜FiTθ˜Fi,where D˜i=Di−D^i.
According to (27)–(31) and xi+1=zi+1+ei+1+αi, the time derivative of Vi is(33)V˙i=−c1z12−γikiz˜i2+zigix¯izi+1+zigix¯iei+1−l0iD˜12+D˜iD˙i−l0ili−1D˜iθ˜FiTφFix¯i+δiθ˜FiTθ^FiT.
Ifzi+1 is bounded, i.e., zi+1≤Ki+1, Ki+1 is a positive constant, and according to Young’s inequality, we have(34)D˜iD˙i≤D˜i2+D˙i22≤D˜i2+Di∗22,−l0ili−1D˜iθ˜FiTφFix¯i≤l0ili−1D˜i22+l0ili−1χFi22θ˜Fi2,δiθ˜FiTθ^Fi≤−δi2θ˜Fi2+δi2θ˜Fi∗2,ziei+1gix¯i≤zi22+g¯iMi22,zigix¯izi+1≤zi22+g¯i2Ki+122,where φFix¯i≤χFi. Substituting (34) into (33), we obtain(35)V˙i≤−ci−1zi2−γikiz˜i2−l0i−12−l0ili−12D˜i2−δi2−l0ili−1χFi22θ˜Fi2︸θ˜FiTθ˜Fi+Ri,where Ri=γi2/2+δi/2θFi∗2+g¯i2Ki+12/2+g¯iMi2/2. Let ci>1, ki>0, l0i>l0ili−1/2+1/2, and δi/2>l0ili−1χFi2/2, and define the following compact sets:(36)Ωzi=zi|zi≤Rici−1,Ωz˜i=z˜i|z˜i≤Riγiki,ΩD˜i=D˜i|D˜i≤Ril0i−l0ili−1/2−1/2,Ωθ˜Fi=θ˜Fi|θ˜Fi≤Riδi/2−l0ili−1χFi2/2.
We know that ifzi∉Ωzi or z˜i∉Ωz˜i or D˜i∉ΩD˜i or θ˜Fi∉Ωθ˜Fi, we have V˙i<0. Thus, zi,z˜i,D˜i, and θ˜Fi are semiglobally uniformly bounded. So, to prove that zi is bounded, we have to prove that zi+1 is bounded.Step n.
The time derivative ofzn produces(37)z˙n=ln−1θ^FnTφFnx¯n+ln−1θ˜FnTφFnx¯n+gnx¯nu+Dn−ξn2.
The controlu is designed as(38)u=−ln−1θ^FnTφFnx¯n−D^n+ξn2−cnzngnx¯n,where D^n is the estimation of Dn and cn is a positive constant. The prediction error z˜n=z^n−zn, and z^n satisfies(39)z^˙n=ln−1θ^FnTφFnx¯n+gnx¯nu+D^n−ξn2−knz˜n.
The disturbance observerD^n is designed as(40)D^n=l0nzn−wn,w˙n=ln−1θ^FnTφFnx¯n+gnx¯nu+D^n−ξn2+l0n−1γnz˜n−zn,where l0n is a positive design parameter. And the composite parameter adaptive law for θ^Fn is designed as(41)θ^.Fn=λFnln−1znφFnx¯n−γnln−1z˜nφFnx¯n−δnθ^Fn,where λFn and δn are positive design parameters.
Consider the following Lyapunov function:(42)Vn=12zn2+γnz˜n2+D˜n2+1λFnθ˜FnTθ˜Fn,where D˜n=Dn−D^n.
According to (38)–(41), the time derivative of Vn is(43)V˙n=−cnzn2−γnknz˜n2−l0nD˜n2+D˜nD˙n−l0nln−1D˜nθ˜FnTφFnx¯n+δnθ˜FnTθ^FnT.
Consider the following facts:(44)D˜nD˙n≤D˜n2+Dn∗22,−l0nln−1D˜nθ˜FnTφFnx¯n+δnθ˜FnTθ^FnT,δnθ˜FnTθ^Fn≤−δn2θ˜Fn2+δn2θ˜Fn∗2,where ‖φFnx¯n‖≤χFn. Substituting (44) into (43), we obtain(45)V˙n≤−cnzn2−γnknz˜n2−l0n−12−l0nln−12D˜n2−δn2−l0nln−1χFn22θ˜Fn2+Rn,where Rn=Dn∗2/2+δn/2θFn∗2. Let cn>0, kn>0, l0n>l0nln−1/2+1/2, and δn/2>l0nln−1χFn2/2, and define the following compact sets:(46)Ωzn=zi|zn≤Rncn,Ωz˜n=z˜n|zn≤Rnγnkn,ΩD˜n=D˜n|D˜n≤Ril0n−l0nln−1/2−1/2,Ωθ˜Fn=θ˜Fn|θ˜Fn≤Rnδn/2−l0nln−1χFn2/2.
Obviously,zn∉Ωzn or z˜n∉Ωz˜n or D˜n∉ΩD˜n or θ˜Fn∉Ωθ˜Fn, and we have V˙n<0. Thus, zn,z˜n,D˜n, and θ˜Fn are semiglobally uniformly bounded. It can be seen that the controller u can guarantee that zn is bounded, the boundedness of zn guarantees that zn−1 is bounded, and the like, and z2 guarantees that z1 is bounded. So, we get z1 is bounded in turn. According to Lemma 2, the tracking error e1 satisfies PPB (10). We obtain the following main result.Theorem 1.
Consider strict-feedback nonlinear system (2) with uncertainties and unknown time-varying disturbances. If virtual controllers (16), (27), disturbance observers (19), (30), (40), the RBFNN updating laws (20), (31), (41), and real controller (37) are designed, it is guaranteed that the tracking error e1 satisfies PPB (10).Remark 6.
In this paper, neural network technology is adopted to transform system (2) into the equivalent of system (9). If the external disturbance Di=0 in (9), it can be found from the construction of the disturbance observer that li=0 and l0i=0, i=1,2,⋅⋅⋅,n. So, the control method in this paper can be applied to a wider range.Remark 7.
The traditional PPC control method can also be used in this paper, but the control effect is less than the proposed control method in this paper, which will be given intuitively through the figures in the simulations.


## 4. Simulation Studies

In the simulation, the following second-order strict-feedback nonlinear system is considered:(47)x˙1=−sinx12+sinx1︸f1x¯1+12+0.1sinx1︸g1x¯1x2+sin0.15t︸d1t,x˙2=−x1x2+sinx1−0.25x2︸f2x¯2+3−0.2sinx2︸g2x¯2u+2sin0.15t︸d2t,y=x1.Denote the method in this paper as DOB-PPC. In order to compare with DOB-PPC, defineF¯1x¯1=f1x¯1+d1t and F¯2x¯2=f2x¯2+d2t, and the traditional PPC control method (denoted as T-PPC) is designed as(48)u=−ϑ^F¯2TφF¯2x¯2+ξ22−c2z¯2g2x¯2,ξ˙21=ξ22,ξ˙22=λ212−λ22tanhξ21−α1−λ23tanhξ22λ21,α1=ϑ^F¯1TφF¯1x¯1+x˙d−Π¯−c1z¯1g1x¯1,ϑ^.F¯1=λF¯1z¯1φF¯1x¯1−δ1ϑ^F¯1,ϑ^.F¯2=λF¯2z¯2φF¯2x¯2−δ2ϑ^F¯2,z¯1=tanheμ,μ=2.5−0.3exp−3.0t+0.3.The stability proof of T-PPC (48) is given in Appendix 1. It can be found that only the RBFNN is used to estimate system uncertainty in T-PPC (48). The parameters of T-PPC are chosen as λF1=λF2=40, c1=c2=5, λ21=150, λ22=λ23=5, and δ1=δ2=0.5. N1=17andN2=45 (N1andN2 are the dimensions of ϑ^F1 and ϑ^F2, respectively). The centers of x¯1 and x¯2 are evenly spaced in 0,8 and 0,8×−1,3, respectively.For method of DOB-PPC, the above parameters are the same. In addition, selectl1=l2=10, l01=l02=20, γ1=γ2=3,k1=k2=5, and λ1=λ2=1/2. The dimension of θ^F1 is N1=17, and the dimension of θ^F2 is N2=81. The centers of x¯1 and x¯2 are the same as ones in the method of T-PPC. The initial value x0=x10,x20T=2,2T, and xd=sint. Firstly, the tracking error effect of T-PPC and DOB-PPC is compared, and the result in Figure 2 confirms that the error effect of DOB-PPC is better than that of T-PPC.Figure 2
Tacking errore of system (47) for two methods in time period (a) [0, 15 s]; (b) [10 s, 15 s].
(a)(b)The estimation off1x1+d1t and f2x2+d2t for T-PPC and DOB-PPC is shown in Figures 3 and 4. It can be found that the estimated effect of f1x1+d1t and f2x2+d2t is not satisfactory by using T-PPC, while this estimated effect is greatly improved by using DOB-PPC.Figure 3
Estimation off1x¯1+d1t by using T-PPC in time period (a) [0, 15 s]; (b) [10 s, 15 s]; estimation of f2x¯2+d2t by using T-PPC in time period (c) [0, 15 s]; (d) [10 s, 15 s].
(a)(b)(c)(d)Figure 4
Estimation off1x¯1+d1t by using DOB-PPC in time period (a) [0, 15 s]; (b) [10 s, 15 s]; estimation of f2x¯2+d2t by using DOB-PPC in time period (c) [0, 15 s]; (d) [10 s, 15 s].
(a)(b)(c)(d)From (25), r˙1 appears in compact sets Ωz1,Ωz˜1, and ΩD˜1. If r˙1 is infinite, this will cause the above compact sets to not exist. By observing Figure 5, we find that r1 tends to a certain positive value, and r˙1 tends to zero, and this simulation result also confirms the rationality of z12/2r1 in Remark 3. Finally, the control input of the two methods is shown in Figure 6. It can be seen from Figure 6 that the energy of DOB-PPC is greater than that of T-PPC at the beginning of the control process, and then the energy of the two control methods quickly approaches the same level. The results of the two methods are that the proposed DOB-PPC can significantly improve the tracking performance and the estimation accuracy of system uncertainties. So, it can be concluded that the proposed DOB-PPC is more effective than T-PPC.Figure 5
Time response of (a)r1 and (b) r˙1.
(a)(b)Figure 6
Time response of controlleru for two methods in time period (a) [0, 15 s]; (b) [0, 1.5 s].
(a)(b)


## 5. Conclusion

In this paper, the composite prescribed performance control scheme is investigated for uncertain strict-feedback systems with unknown external disturbances. To estimate the unknown nonlinear functionsfix¯i and unknown disturbances dit, RBFNNs and disturbance observers are employed. The observer D^i constructed here does not estimate the external disturbance dit but estimates the external disturbance dit and approximates error εFi. With the composite NN updating laws and disturbance observers, fix¯i+dit can be accurately estimated. Meanwhile, the whole closed-loop system can ensure stability by using PPC and backstepping technique. Moreover, the tracking error e1 satisfies PPB (10). Finally, an example shows the effectiveness of the proposed method. Through the research in this paper, DOB-PPC with nonlinear control inputs will become the next research direction.



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*Source: 8835512-2020-10-29.xml*