# Neural Network Command Filtered Control of Fractional-Order Chaotic Systems

Authors: Hua Zhang
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

An adaptive neural network (NN) backstepping control method based on command filtering is proposed for a class of fractional-order chaotic systems (FOCSs) in this paper. In order to solve the problem of the item explosion in the classical backstepping method, a command filter method is adopted and the error compensation mechanism is introduced to overcome the shortcomings of the dynamic surface method. Moreover, an adaptive neural network method for unknown FOCSs is proposed. Compared with the existing control methods, the advantage of the proposed control method is that the design of the compensation signals eliminates the filtering errors, which makes the control effect of the actual system improve well. Finally, two examples are given to prove the effectiveness and potential of the proposed method.


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## Body

## 1. Introduction

The fractional calculus equations are increasingly used to describe problems in optical and thermal systems, material and mechanical systems, signal processing, system identification, control, robotics, and other applications. The theory of fractional calculus has also been paid more and more attention by scholars at home and abroad, and especially, the fractional differential equations abstracted from practical problems have become the research focus of many mathematicians [1–3]. Bao and Cao [4] proved that fractional-order differential equations can well simulate the dynamics of various chemical materials and special materials in practical applications. The traditional integer-order differential equation can be regarded as a special model of fractional-order differential equation. More information on fractional-order differential integration can be found in [5, 6]. In recent years, many achievements have been made in the study of fractional-order calculus equations, which are used to study the problems of stability analysis and control of fractional-order strictly feedback nonlinear systems. For example, Li et al. [7] proposed a Lyapunov direct method for fractional-order nonlinear systems. Boroujeni and Momeni [8] proposed a fractional-order state observer for fractional-order nonlinear systems. Yang and Chen [9] presented a finite-time stabilization method for fractional-order switching systems. Adaptive control methods had been used in the control of fractional-order nonlinear systems [5, 6, 10]. Huang et al. [11] proposed a fuzzy feedback control method for uncertain fractional-order chaotic systems (FOCSs). Therefore, the research on the stability and control of FOCSs is becoming more and more popular.The backstepping control was proposed for the first time to obtain asymptotic tracking and global stability of strictly feedback nonlinear systems [12]. The backstepping technique has always been a powerful tool for the design of controllers for nonlinear dynamic systems [13, 14]. However, the classical backstepping control has two obvious disadvantages: one is the explosion of terms caused by the repeated derivation of virtual control signals [15–17], and the other is certain functions in the systems with uncertain parameters must be linear [18–20]. Li et al. [20] presented a fuzzy adaptive output feedback control method to improve the tracking effect of the systems. In addition, Shao-Cheng Tong et al. [21] proposed a backstepping method based on command filters to solve the problem of item explosion caused by repeated derivation of the virtual controller. Then, the command filtered backstepping control was extended to the adaptive case in [22]. In each step of the control design, the output of the command filter was used to approximate the differential coefficient of the virtual control signal, which eliminates the problem of item explosion. Note that the error compensation mechanism was designed to eliminate errors generated by the command filters [23]. However, the above works considered only the cases of systems with unknown parameters which are constant and systems without unknown parameters. In addition, the above research results only consider the case of integer-order systems, but the research results of the backstepping control method based on command filtering are relatively few for fractional-order strict feedback systems. In this paper, a backstepping control method based on command filtering is presented for a class of fractional-order strictly feedback nonlinear systems.Based on the radial-basis-function neural network (NN), an adaptive control method based on approximation theory was proposed to deal with nonlinear systems with uncertain parameters [24–28] or NN [29–31] approximation. The neural network control based on the backstepping control method of command filtering is used to solve nonlinear problems in uncertain systems. It can solve the nonlinear system which does not meet the matching condition and the certain functions are nonlinear [32–35]. For a class of FOCSs, an adaptive controller via the backstepping technology similar to [36, 37] was designed in [38]. However, because it is difficult to solve the fractional-order derivative of quadratic function, the adaptive backstepping control technology of integer-order nonlinear system cannot be directly applied to the fractional nonlinear system. For FOCSs, an adaptive backstepping control method was proposed in [39], and the stability of the controller was analyzed by the integer-order Lyapunov method. Therefore, for fractional-order nonlinear systems, how to design an adaptive backstepping controller via Lyapunov stability theory is an urgent problem to be solved.According to the above discussion, an NN adaptive backstepping control method based on command filtering is proposed for FOCSs in this paper. The NN is used to deal with unknown functions in the system. The command filters are proposed to solve the problem of item explosion caused by repeated derivation of virtual controllers, and compensation signals are designed to eliminate the errors caused by the command filter. Based on NN approximation theory, an adaptive backstepping method based on command filtering is presented. The results show that this control method can adjust the tracking error to the small neighborhood of origin and ensure that all the closed-loop system signals have the limit boundedness of half blade uniformity.The main advantages of this approach over current results can be summarized as follows:(i)
The command filtered adaptive NN backstepping control can solve two problems of classical backstepping of a class of fractional-order nonlinear systems and reduce the calculation burden.(ii)
An error compensation mechanism is introduced to overcome the shortcomings of the dynamic surface method, and the tracking error is controlled in the small neighborhood of zero.The rest of this article is organized as follows: Section2 is about the fuzzy logic system, fractional calculus description, and the fractional-order nonlinear systems. In Section 3, detailed controller design and stability analysis are given. In Section 4, simulation results are given to verify the effectiveness of the backstepping adaptive control method based on command filtering. Finally, Section 5 is the conclusion of this paper.


## 2. Preliminaries and Problem Description

### 2.1. Preliminaries of Fractional-Order Calculus

In recent several decades, the relatively common two kinds of fractional-order calculus are Riemann–Liouville (RL) fractional-order calculus and Caputo fractional-order calculus. The nice property of the Caputo fractional-order differential equation is that its integral value at zero makes sense. Therefore, this paper will adopt the definition of Caputo fractional-order calculus.The fractional-order calculus operators can be regarded as a broader concept of integral calculus operators.Thus, the definition of the fractional-order integral can be expressed as [1](1)I0Ctβxt=1Γβ∫0txζt−ζ1−βdζ,where Γ⋅ is the gamma function and Γs=∫0+∞ts−1e−tdt. Consequently, the fractional-order derivative operator is represented by [1](2)D0Ctβxt=1Γn−β∫0txnζt−ζβ+1−ndζ.The Laplace transform of (2) can be expressed as(3)LD0Ctβxt=sβXs−∑k=0n−1sβ−k−1xk0,where Xs is the Laplace transform of Xt.In the following sections, we examine only the case ofβ∈0,1.Definition 1 (see [1]).
The Mittag–Leffler function can be expressed as(4)Eβ,γz=∑k=0+∞zkΓkβ+γ,with β,γ>0, and z∈C. The Laplace transform of (5) can be described as(5)Ltγ−1Eβ,γ−atβ=sβ−γsβ+a.Lemma 1 (see [1]).
If there is a complex numberγ and two real numbers β (0<β<1) and ϖ, where ϖ satisfies the following condition:(6)πβ2<ϖ<minπ,πβ,then, the following formula is true for all integers n≥1:(7)Eβ,γz=∑k=0n1Γγ−βjzj+o1zn+1,when z⟶∞ and ϖ≤argz≤n.Lemma 2 (see [1]).
Ifβ (0<β<2) and γ are two arbitrary real numbers, and there exists a positive constant ϱ satisfying the following condition:(8)vπ2<ϱ≤minπ,βπ,then the following inequality holds:(9)Eβ,γz≤C1+z,where C>0, ϱ≤argz≤π, and z≥0.Definition 2 (see [7]).
Supposew:0,c⟶ℐ is a continuous function. If the continuous function is strictly increasing and w0=0, then it is said to be class K.Lemma 3 (see [7]).
If an equilibrium point of fractional-order nonlinear systemDtβyt=gt,tx is origin, which g:ℐ×Ω⟶ℛ is Lipschitz continuous. There exist a Lyapunov function Vt,yt and class K functions wii=1,2,3 to satisfy(10)w1yt≤Vt,yt≤w2yt,DtβVt,yt≤w3yt.Then, Dtβyt=gt,tx is asymptotically stable; i.e., limt⟶∞yt=0.Lemma 4 (see [2]).
Assume thatyt∈C1t0,∞,ℛn, then(11)12DtβyTtyt≤yTtDtβyt,∀t∈ℐ.


### 2.2. Preliminaries of Radial-Basis-Function NN

Lethχt be a continuous unknown nonlinear function defined over a compact set Ω. Then, there exists a radial-basis-function NN to appropriate the unknown nonlinear function hχt as follows:(12)h^χt=θTψχt,where h^:ℛn⟶ℛ (n∈N) is a continuous mapping, χt=χ1t,χ2t,…,χntT is the NN input vector, t≥0, χit∈C1ℐ,Ω (i=1,2,…,n), θ=θ1,θ2,…,θNT∈ℛN (N>1) is the weight vector, ψχt=ψ1χt,ψ2χt,…,ψNχtT∈ℛN is a regressor, and N is the number of NN nodes. The regression variable ψıχt selected in this paper is the Gaussian radial-basis-function(13)ψıχt=exp−x−δi22ιi2,where ı=1,2,…,N, δı=δ1ı,δ2ı,…,δnıT∈Rn, δJı= (J=1,2,…,n) represent the centers of the Gaussian function, and ιı∈R+ represent the widths of the Gaussian function.Therefore, according to the above notation, the optimal estimate ofhχt can be expressed as(14)hχt=θ∗Tψχt+ϵχt,in which ϵχt=ϵ1χ1t,ϵ2χ2t,…,ϵNχNt and θ∗ is the optimal constant weight vector and satisfies the following condition:(15)θ∗t=argminθsupχthχt−h^χt.Let(16)θ˜=θ−θ∗,in which θ˜ is the parameter estimation error. According to the properties of the radial-basis-function NN, it is assumed that the optimal approximation error remains bounded. For any ϵ¯, there exists a sufficient number of NN nodes such that(17)ϵıχıt≤ϵ¯ı,where ϵ¯ı is a known positive constant and ı=1,2,…,N.Then, one can obtain(18)h^ıχıt,θı−hıχıt=h^ıχıt,θı−h^ıχıt,θı∗+h^ıχıt,θı∗−hıχıt=θıTψıχıt−θı∗Tψıχıt+ϵıχıt=θ˜ıTψıχıt+ϵıχıt,in which ı=1,2,…,N.


### 2.3. FOCSs

Consider the following FOCSs:(19)Dtβx1t=x2t+f1x1¯,Dtβxit=xi+1t+fixi¯,Dtβxnt=fnxt+dt+ut,yt=x1t,where β0<β<1 is the system commensurate order, xt=1t,x2t,…,xntT∈ℛn is the state vector of the system, x¯t=x1t,x2t,…,xitT∈ℛi, fix¯i∈ℛi=1,2,…,n are unknown smooth nonlinear functions, dt∈ℛ is an unknown external disturbance, and ut∈ℛ and yt∈ℛ are control input and control output of the fractional-order strictly feedback nonlinear systems.Assumption 1.
For the FOCSs, ifdt is bounded and unknown, then there exists an unknown constant d¯ (d¯>0) such that dt≤d¯ for all t≥0.


## 2.1. Preliminaries of Fractional-Order Calculus

In recent several decades, the relatively common two kinds of fractional-order calculus are Riemann–Liouville (RL) fractional-order calculus and Caputo fractional-order calculus. The nice property of the Caputo fractional-order differential equation is that its integral value at zero makes sense. Therefore, this paper will adopt the definition of Caputo fractional-order calculus.The fractional-order calculus operators can be regarded as a broader concept of integral calculus operators.Thus, the definition of the fractional-order integral can be expressed as [1](1)I0Ctβxt=1Γβ∫0txζt−ζ1−βdζ,where Γ⋅ is the gamma function and Γs=∫0+∞ts−1e−tdt. Consequently, the fractional-order derivative operator is represented by [1](2)D0Ctβxt=1Γn−β∫0txnζt−ζβ+1−ndζ.The Laplace transform of (2) can be expressed as(3)LD0Ctβxt=sβXs−∑k=0n−1sβ−k−1xk0,where Xs is the Laplace transform of Xt.In the following sections, we examine only the case ofβ∈0,1.Definition 1 (see [1]).
The Mittag–Leffler function can be expressed as(4)Eβ,γz=∑k=0+∞zkΓkβ+γ,with β,γ>0, and z∈C. The Laplace transform of (5) can be described as(5)Ltγ−1Eβ,γ−atβ=sβ−γsβ+a.Lemma 1 (see [1]).
If there is a complex numberγ and two real numbers β (0<β<1) and ϖ, where ϖ satisfies the following condition:(6)πβ2<ϖ<minπ,πβ,then, the following formula is true for all integers n≥1:(7)Eβ,γz=∑k=0n1Γγ−βjzj+o1zn+1,when z⟶∞ and ϖ≤argz≤n.Lemma 2 (see [1]).
Ifβ (0<β<2) and γ are two arbitrary real numbers, and there exists a positive constant ϱ satisfying the following condition:(8)vπ2<ϱ≤minπ,βπ,then the following inequality holds:(9)Eβ,γz≤C1+z,where C>0, ϱ≤argz≤π, and z≥0.Definition 2 (see [7]).
Supposew:0,c⟶ℐ is a continuous function. If the continuous function is strictly increasing and w0=0, then it is said to be class K.Lemma 3 (see [7]).
If an equilibrium point of fractional-order nonlinear systemDtβyt=gt,tx is origin, which g:ℐ×Ω⟶ℛ is Lipschitz continuous. There exist a Lyapunov function Vt,yt and class K functions wii=1,2,3 to satisfy(10)w1yt≤Vt,yt≤w2yt,DtβVt,yt≤w3yt.Then, Dtβyt=gt,tx is asymptotically stable; i.e., limt⟶∞yt=0.Lemma 4 (see [2]).
Assume thatyt∈C1t0,∞,ℛn, then(11)12DtβyTtyt≤yTtDtβyt,∀t∈ℐ.


## 2.2. Preliminaries of Radial-Basis-Function NN

Lethχt be a continuous unknown nonlinear function defined over a compact set Ω. Then, there exists a radial-basis-function NN to appropriate the unknown nonlinear function hχt as follows:(12)h^χt=θTψχt,where h^:ℛn⟶ℛ (n∈N) is a continuous mapping, χt=χ1t,χ2t,…,χntT is the NN input vector, t≥0, χit∈C1ℐ,Ω (i=1,2,…,n), θ=θ1,θ2,…,θNT∈ℛN (N>1) is the weight vector, ψχt=ψ1χt,ψ2χt,…,ψNχtT∈ℛN is a regressor, and N is the number of NN nodes. The regression variable ψıχt selected in this paper is the Gaussian radial-basis-function(13)ψıχt=exp−x−δi22ιi2,where ı=1,2,…,N, δı=δ1ı,δ2ı,…,δnıT∈Rn, δJı= (J=1,2,…,n) represent the centers of the Gaussian function, and ιı∈R+ represent the widths of the Gaussian function.Therefore, according to the above notation, the optimal estimate ofhχt can be expressed as(14)hχt=θ∗Tψχt+ϵχt,in which ϵχt=ϵ1χ1t,ϵ2χ2t,…,ϵNχNt and θ∗ is the optimal constant weight vector and satisfies the following condition:(15)θ∗t=argminθsupχthχt−h^χt.Let(16)θ˜=θ−θ∗,in which θ˜ is the parameter estimation error. According to the properties of the radial-basis-function NN, it is assumed that the optimal approximation error remains bounded. For any ϵ¯, there exists a sufficient number of NN nodes such that(17)ϵıχıt≤ϵ¯ı,where ϵ¯ı is a known positive constant and ı=1,2,…,N.Then, one can obtain(18)h^ıχıt,θı−hıχıt=h^ıχıt,θı−h^ıχıt,θı∗+h^ıχıt,θı∗−hıχıt=θıTψıχıt−θı∗Tψıχıt+ϵıχıt=θ˜ıTψıχıt+ϵıχıt,in which ı=1,2,…,N.


## 2.3. FOCSs

Consider the following FOCSs:(19)Dtβx1t=x2t+f1x1¯,Dtβxit=xi+1t+fixi¯,Dtβxnt=fnxt+dt+ut,yt=x1t,where β0<β<1 is the system commensurate order, xt=1t,x2t,…,xntT∈ℛn is the state vector of the system, x¯t=x1t,x2t,…,xitT∈ℛi, fix¯i∈ℛi=1,2,…,n are unknown smooth nonlinear functions, dt∈ℛ is an unknown external disturbance, and ut∈ℛ and yt∈ℛ are control input and control output of the fractional-order strictly feedback nonlinear systems.Assumption 1.
For the FOCSs, ifdt is bounded and unknown, then there exists an unknown constant d¯ (d¯>0) such that dt≤d¯ for all t≥0.


## 3. Adaptive NN Backstepping Control Design

Let(20)s1t=x1t−xd,(21)sit=xit−xi,c,in which xd is a known reference signal and xi,c (i=2,3,…,n) are the command filters.The command filtering mechanism is defined as follows:(22)Dtβxi+1,ct=−κixi+1,ct−αit,where κi is a positive constant and αit and xi+1,ct (i=1,2,…,n−1) are the input signal and the output signal in the command filtering mechanism, respectively.It is worth noting that command filtering produces errors that add to the burden of getting better tracking results. To solve this problem, an error compensation mechanism is presented to overcome the errorsxi+1,c−αi generated during the filtering process.Then, the error compensation mechanism is defined as(23)Dtβλ1t=−k11λ1t+λ2+x2,c−α1,Dtβλit=−k1iλit−λi−1+λi+1+xi+1,c−αi,Dtβλnt=−k1nλnt−λn−1,where k1i>0 are design parameters and λi (i=1,2,…,n−1) are the error compensating signals.The compensated tracking error signalseit are designed as(24)eit=sit−λit,in which i=1,2,…,n.Our goal is to design the control inputu of the nonlinear systems such that the control output yt of the system to track xd and the compensated tracking error of the nonlinear systems e1t eventually converge to a sufficiently small region of origin.Then, the virtual control functionsαi (i=1,2,…,n) are constructed as(25)α1=−θ1Tψ1x¯1t−k11s1t−k21signe1t+Dtβxdt,(26)α2=−θ2Tψ2x¯2t−k12s2t−k22signe2t−s1t+Dtβx2,ct,(27)αi=−θiTψix¯it−k1isit−k2isigneit−si−1t+Dtβxi,ct,where k2i>ϵ¯i (i=1,2,…,n−1) are design parameters and ϵ¯i will be determined later.The controllerut is designed as follows:(28)ut=−θnTψnxnt−k1nsnt−sn−1t−k2n+d¯^tsignent+Dtβxn,ct,in which d¯^t is the estimation of d¯, k2n>ϵ¯n is a design parameter, and ϵ¯n will be determined later.Then, the backstepping control algorithms of the fractional-order nonlinear system are shown as follows.Step 1.
Consider the first Lyapunov functionv1x as follows:(29)v1t=12e12t.
According to Lemma4, differentiating v1x can gain(30)Dtβv1t≤e1tDtβe1t=e1tDtβx1t−Dtβxdt−Dtβλ1t=e1tx2t+f1x¯1t−Dtβxdt−Dtβλ1t=e1ts2t+x2,ct+f1x¯1t−f^1x¯1t,θ1∗+f^1x¯1t,θ1∗−Dtβxdt+k11λ1t−λ2t−x2,ct+β1t=e1te2t−θ˜1Tψ1x¯1t−ϵ1x¯1t+θ1Tψ1x¯1t−Dtβxdt+k11λ1t+α1t,in which ϵ1x¯1t=f^1x¯1t,θ∗−f1x¯1t and θ˜1=θ1−θ1∗.
Substituting (25) into (30), we can obtain(31)Dtβv1t≤e1te2t−k11e1t−θ˜1Tψ1x¯1t−ϵ1x¯1t−k21signe1t≤e1te2t−k11e12t−e1tθ˜1Tψ1x¯1t+ϵ¯1e1t−k21e1t≤e1te2t−k11e12t−e1tθ˜1Tψ1x¯1t,where k11>0 and k21>ϵ¯1 are design parameters.
Choose the Lyapunov functionV1x as follows:(32)V1t=v1t+12ρ1θ˜1Ttθ˜1t.
The design of adaptive law is as follows:(33)Dtβθ1=ρ1e1tψ1x¯1t−γ1θ1,where ρ1 and γ1 are positive design parameters.
Becauseθ1∗ is a constant, then we have(34)Dtβθ˜1=Dtβθ1.
Then, according to Lemma4, we can have(35)DtβV1t≤Dtβv1t+1ρ1θ˜1TDtβθ˜1.
Substituting formulas (31) and (33) into the above inequality, the following can be obtained:(36)DtβV1t≤−k11e12t+e1te2t−1ρ1θ˜1Tθ1≤−k11e12t+e1te2t−1ρ1θ˜1Tθ˜1−1ρ1θ˜1Tθ1∗.
According to Young’s inequality, then we can obtain(37)−1ρ1θ˜1Tθ1∗≤12ρ1θ1∗Tθ1∗+12ρ1θ˜1Tθ˜1.
According to formulas (36) and (37), it can be obtained(38)DtβV1t≤−k11e12t+e1te2t−12ρ1θ˜1Tθ˜1+12ρ1θ1∗Tθ1∗≤−k1V1t+e1te2t+Θ1,in which k1=min2k11,γ1 and Θ1=γ1/2ρ1θ1∗Tθ1∗ are two positive constants.Step 2.
Consider the second Lyapunov functionv2x as follows:(39)v2t=12e22t.
According to Lemma4, differentiating v2x can gain(40)Dtβv2t≤e2tDtβe2t=e2tDtβx2t−Dtβx2,ct−Dtβλ2t=e2tx3t+f2x¯2t−Dtβx2,ct−Dtβλ2t=e2ts3t+x3,ct+f2x¯2t−f^2x¯2t,θ2∗+f^2x¯2t,θ2∗−Dtβx2,ct+k12λ2+λ1t−λ3t−x3,ct+α2t=e2te3t−θ˜2Tψ2x¯2t−ϵ2x¯2t+θ2Tψ2x¯2t−Dtβx2,ct+k12λ2+λ1t+α2t,in which ϵ2x¯2t=f^2x¯2t,θ∗−f2x¯2t andθ˜2=θ2−θ2∗.
Substituting (27) into (41), we can obtain(41)Dtβv2t≤e2te3t−k12e2t−θ˜2Tψ2x¯2t−ϵ2x¯2t−k22signe2t−et≤e2te3t−k12e22t−e2tθ˜2Tψ2x¯2t+ϵ¯2e2t−k22e2t−e1te2t≤−k12e22+e2te3t−e2tθ˜2Tψ2x¯2t−e1te2t,where k12>0 and k22>ε¯2 are design parameters.
Choose the Lyapunov functionV2x as follows:(42)V2t=V1t+v2t+12ρ2θ˜2Ttθ˜2t.
The design of adaptive law is as follows:(43)Dtβθ2=ρ2e2tψ2x¯2t−γ2θ2,where ρ2 and γ2 are positive design parameters.
Becauseθ2∗ is a constant, then we have(44)Dtβθ˜2=Dtβθ2.
Then, according to Lemma4 and formulas (42) and (44), we can have(45)DtβV2t≤−kV1t+e1te2t+Θ1+Dtβv2t+1ρ2θ˜2TDtβθ˜2.
According to formulas (45) and (46), the above equation can be reduced to(46)DtβV2t≤−k1V1t+Θ1−k12e22t+e2te3t−12ρ2θ˜2Tθ˜2+12ρ2θ2∗Tθ2∗≤−k2V2t+e2te3t+Θ2,in which k2=mink1,2k12,γ2 and Θ2=Θ1+γ2/2ρ2θ2∗Tθ2∗ are two positive constants.Step 3.
ii=3,4,…,n−1.
Consider theith Lyapunov function vix as follows:(47)vit=12ei2t.
According to Lemma4, differentiating vix can gain(48)Dtβvit≤eitDtβeit=eitDtβxit−Dtβxi,ct−Dtβλit=eitxi+1t+fix¯it−Dtβxi,ct−Dtβλit=eitsi+1t+xi+1,ct+fix¯it−f^ix¯it,θi∗+f^ix¯it,θi∗−Dtβxi,ct+k1iλi+λi−1t−λi+1t−xi+1,ct+αit=eitei+1t−θ˜iTψix¯it−ϵix¯it+θiTψix¯it−Dtβxi,ct+k1iλi+λi−1t+αit,in which ϵix¯it=f^ix¯it,θ∗−fix¯it and θ˜i=θi−θi∗.
Substituting (27) into (49), we can obtain(49)Dtβvit≤eitei+1t−k1ieit−θ˜iTψix¯it−ϵix¯it−k2isigneit−ei−1t≤eitei+1t−k1iei2t−eitθ˜iTψix¯it+ϵ¯ieit−k2ieit−ei−1teit≤eitei+1t−k1iei2−eitθ˜iTψix¯it−ei−1teit,where k1i>0 and k2i>ϵ¯i are design parameters.
Choose the Lyapunov functionVix as follows:(50)Vit=Vi−1t+vit+12ρiθ˜iTtθ˜it.
The design of adaptive law is as follows:(51)Dtβθi=ρieitψix¯it−γiθi,where ρi and γi are positive design parameters.
Becauseθi∗ is a constant, then we have(52)Dtβθ˜i=Dtβθi.
Then, according to Lemma4, we can have(53)DtβVit≤−ki−1Vi−1t+ei−1teit+Θi−1+Dtβvit+1ρiθ˜iTDtβθ˜i.
From formulas (50), (52), and (54), we have(54)DtβVit≤−ki−1Vi−1t+Θi−1+eitei+1t−k1iei2−1ρiθ˜iTθi≤−ki−1Vi−1t+Θi−1−k1iei2t+eitei+1t−1ρiθ˜iTθ˜i−1ρiθ˜iTθi∗.
According to Young’s inequality, then we can obtain(55)−1ρiθ˜iTθi∗≤12ρiθi∗Tθi∗+12ρiθ˜iTθ˜i.
Substitute the above formula into (55), the following can be obtained(56)DtβVit≤−ki−1Vi−1t+Θi−1−k1iei2t+eitei+1t−12ρiθ˜iTθ˜i+12ρiθi∗Tθi∗≤−kiVit+ei+1teit+Θi,in which ki=minki−1,2k1i,γi and Θi=Θi−1+γi/2ρiθi∗Tθi∗ are two positive constants.Step 4.
n.
Consider thenth Lyapunov function vnx as follows:(57)vnt=12en2t.
According to Lemma4, differentiating vnx can gain(58)Dtβvnt≤entDtβent=entDtβxnt−Dtβxn,ct−Dtβλnt=entfnx¯nt+dt+ut−Dtβxn,ct−Dtβλnt=entfnx¯nt−f^nx¯nt,θn∗+f^nx¯nt,θn∗+dt+ut−Dtβxn,ct+k1nλn+λn−1=ent−θ˜nTψnx¯nt−ϵnx¯nt+θnTψnx¯nt+dt+ut−Dtβxn,ct+k1nλn+λn−1,in which ϵnx¯nt=f^nx¯nt,θ∗−fnx¯nt and θ˜n=θn−θn∗.
Substituting (28) into (59), one obtains(59)Dtβvnt≤ent−k1nent−θ˜nTψnx¯nt−ϵnx¯nt−k2n+d¯^tsignent−en−1t+dt≤−k1nen2t−entθ˜nTψnx¯nt+ϵ¯nent−k2n+d¯^tent−en−1tent+d¯ent≤−k1nen2−entθ˜nTψnx¯nt−en−1tent−d¯˜tent,where k1n>0 and k2n>ϵ¯n are design parameters and d¯˜t=d¯^t−d¯ is the estimation error.
Choose the Lyapunov functionVnx as follows:(60)Vnt=Vn−1t+vnt+12ρnθ˜nTtθ˜nt+12ξ1d¯˜2t.
The design of adaptive law is as follows:(61)Dtβθn=ρnentψnx¯nt−γnθn,(62)Dtβd¯^t=ξ1en−ξ2d¯^t,where ρn, γn, ξ1, and ξ2 are positive design parameters.
Becauseθn∗ is a constant, then we have(63)Dtβθ˜n=Dtβθn.
Then, according to Lemma4, we can have(64)DtβVnt≤−kn−1Vn−1t+en−1tent+Θn−1+Dtβvnt+1ρnθ˜nTDtβθ˜n+1ξ1d¯˜tDtβd¯˜t.
From formulas (60) and (65), we have(65)DtβVnt≤−kn−1Vn−1t+Θn−1−k1nen2−entθ˜nTψnx¯nt−d¯˜tent+1ρnθ˜nTDtβθ˜n+1ξ1d¯˜tDtβd¯˜t.
By substituting (62) and (63) into the above inequality, it can be obtained(66)DtβVnt≤−kn−1Vn−1t+Θn−1−k1nen2−γnρnθ˜nTθn−ξ2ξ1d¯˜td¯^t≤−kn−1Vn−1t+Θn−1−k1nen2−γnρnθ˜nTθ˜n−γnρnθ˜nTθn∗−ξ2ξ1d¯˜2t−ξ2ξ1d¯˜td¯.
According to Young’s inequality, we can obtain(67)−1ρiθ˜nTθn∗≤12ρnθn∗Tθn∗+12ρnθ˜nTθ˜n,(68)−ξ2ξ1d¯˜td¯t≤ξ22ξ1d¯˜2t+ξ22ξ1d¯2.
Substitute the above formulas into (67), one obtains(69)DtβVit≤−kn−1Vn−1t+Θn−1−k1nen2−γn2ρnθ˜nTθ˜n+γn2ρnθn∗Tθn∗−ξ22ξ1d¯˜2t+ξ22ξ1d¯2≤−knVnt+Θn,in which kn=minkn−1,2k1n,γn,ξ2 and Θn=Θn−1+γn/2ρnθn∗Tθn∗+ξ2/2ξ1d¯2 are two positive constants.
The stability analysis of fractional-order nonlinear systems is shown below:Theorem 1.
Consider system (20) under Assumption 1. The control input is constructed as (29) with (26)–(28), and the adaptation laws are designed as (34), (44), (52), (62), and (63); then, there exist appropriate design parameters, such that the tracking error e1t tends to an arbitrarily small region of the origin.Proof.
According to (70), there exists a function qt≥0 such that the following equation holds on:(70)DtβVit+qt=−knVnt+Θn.
The Laplace transform of equation (71) is applied to obtain(71)Vns=sβ−1sβ+knVn0+Θnssβ+kn−Qssβ+kn=sβ−1sβ+knVn0+sβ−1+βΘnsβ+kn−Qssβ+kn,in which Vns and Qs are Laplace transforms of Vnt and qt, respectively, and Vn0is the initial condition. According to (6) and (71), it can be represented as(72)Vnt=Vn0Eβ,1−kntβ+ΘntβEβ,1+β−kntβ−qt∗t−1Eβ,0−kntβ,where ∗ is the convolution operator. If sine qt and t−1Eβ,0−kntβ are both nonnegative functions, then qt∗t−1Eβ,0−kntβ≥0 can be obtained. Hence, the following inequality holds on:(73)Vnt≤Vn0Eβ,1−kntβ+ΘntβEβ,1+β−kntβ.
Becausearg−kntβ=−π, −kntβ≥0, for all t≥0 and β∈0,2 and according to Lemma 2, then there exists a positive constant A such that(74)Eβ,1−kntβ≤A1+kntβ.
According to (75), one can obtain(75)limt⟶∞Vn0Eβ,1−kntβ=0.
Consequently, for everyη>0, there exists a constant tr>0, such that t>tr holds on, then one can obtain(76)Vn0Eβ,1−kntβ<η3.
In addition, according to Lemma1, one can obtain(77)Eβ,β+1−kntβ=1Γ1kntβ+o1−kntβ1+1,in which the integer n is chosen as 1. According to (78), for every η>0, there exists a constant tb>0, such that t>tb holds on, one obtains(78)ΘntηEη,η+1−kntη≤Θnkn+η3.
Then, we can select appropriate design parameters such thatΘn/n≤η/3. Therefore, from (74), (77), and (79), the following inequality can be obtained:(79)Vnt<η.
According to (80) and the definition of Vnt, all signals in the closed-loop system are bounded and the tracking error e1t tends to an arbitrary small region of the origin, i.e., for all t>maxtr,tb, such that 1/2e12t≤η.


## 4. Simulation Results

In this section, two examples will be given to demonstrate the effectiveness of the adaptive neural network backstepping control based on the command filtering.Example 1.
The fractional-order Duffing’s Oscillator chaos system is as follows:(80)Dtβx1t=x2t,Dtβx2t=x1t−x13t−0.5x2t+1.3cost+dt+ut,in which f1x1t=0 and f2x1t,x2t=x1t−x13t−0.5x2t+1.3cost are unknown functions. Let the fractional-order β=0.95 and the initial conditions x1=0.21andx2=0.13. In addition, it can be seen from Figures 1 and 2 that the nonlinear system is unstable when both the controller ut and the disturbance signal dt of the system are zero.
The known smooth reference signalxdt and the unknown external disturbance signal dt are chosen as sint and sint+cost, respectively. The parameters to be designed are selected as follows: k11=k12=1, k21=k22=1, ρ1=ρ2=5, γ1=γ2=0.5, ξ1=ξ2=1, and κ1=50. In the process of designing the controller, sign⋅ is used to represent arctan10 to avoid the chattering phenomenon.
Next, let us design the radial-basis-functions. According to the property of the radial-basis-function NN, the input variable isx1t in the first radial-basis-function and six Gaussian functions evenly distributed within the interval −5,5 are designed. The second radial-basis-function uses x1t and x2t as input variables. Same as the first radial-basis-function, each input variable corresponds to six Gaussian functions evenly distributed on −5,5. The initial conditions are presented to be θ10=1,1,1,1,1,1T∈ℛ6 and θ20=1,1,…,1T∈ℛ36.
When the above initial conditions are met, the simulation of the nonlinear system (81) is shown in Figures 3–7.
As can be seen from Figures3 and 4, the state variables x1t and x2t of the system are tracked by signals xdt and x2ct and the tracking effect is relatively well. As shown in Figure 5, the tracking error e1t rapidly converges to a relatively small neighborhood of the origin. From Figure 6, the control input is large at first, then decreases rapidly, and when the system reaches stability, it is controlled in a relatively small neighborhood. As shown in Figure 7, the adaptive laws of the system are bounded and converge rapidly to the neighborhood of the origin. Consequently, the effectiveness of the adaptive neural network backstepping control method based on command filtering for a class of classical fractional-order nonlinear strictly feedback systems.Figure 1
The uncontrolled fractional-order Duffing’s oscillator chaos system (81).Figure 2
The time response of the state variable of (81) when ut=0 and dt=0.Figure 3
The trace ofx1 and xd of system (81).Figure 4
The trace ofx2 and x2c of system (81).Figure 5
The tracking error of the outputyt of system (81).Figure 6
The control input of system (81).Figure 7
The adaptive laws of system (81).Example 2.
The fractional-order Arneodo’s chaos system is as follows:(81)Dtβx1t=x2t,Dtβx2t=x3t,Dtβx2t=5.5x1t−3.5x2t−0.8x3t+x13t+dt+ut,where β=0.97,x10=−0.2,x20=0.5,andx30=0.2. If the input and disturbance of the system satisfy ut=0 and dt=0, the state variables of the system are shown in Figures 8 and 9.
The known smooth reference signalxdt and the unknown external disturbance signal dt are sint and 2sintcost, respectively. The parameters to be designed are selected as follows:k11=40, k12=25, k13=30, k21=k22=k23=15, ρ1=ρ2=ρ2=15, γ1=γ2=γ3=0.5, ξ1=ξ2=1, κ1=1, and κ2=20.
In Example2, there are three radial-basis-function neural networks. In the first radial-basis-function NN, the input variable is x1t and four Gaussian functions evenly distributed within the interval −3,3 are designed. The second radial-basis-function uses x1t and x2t as input variables. Same as the first radial-basis-function, each input variable corresponds to four Gaussian functions evenly distributed on −3,3. The third radial-basis-function uses x1t, x2t, and x3t as input variables. Same as the before radial-basis-functions, four Gaussian functions in every input variable are evenly distributed on −3,3. The initial conditions are presented to be θ10=1,1,1,1T∈ℛ4, θ20=1,1,…,1T∈ℛ16, and θ30=1,1,…,1T∈ℛ64. When the above initial conditions are met, the simulation of the nonlinear system (81) is shown in Figures 10–15.
The simulation results of Example2 are shown above, which is similar to the result analysis of Example 1, and the final results are the same as that of Example 1.Figure 8
The uncontrolled fractional-order Arneodo’s chaos system (81).Figure 9
The time response of the state variable of (81) when ut=0 and dt=0.Figure 10
The trace ofx1 and xd of system (81).Figure 11
The trace ofx2 and x2c of system (81).Figure 12
The trace ofx3 and x3c of system (81).Figure 13
The tracking error of the outputyt of system (81).Figure 14
The control input of system (81).Figure 15
The adaptive laws of system (81).


## 5. Conclusion

In this paper, an adaptive neural network backstepping control method based on command filtering is proposed for a class of fractional-order strictly feedback nonlinear systems. The command filtering technology is used to deal with the explosive of terms in the traditional backstepping technology, and the error compensation mechanism is introduced to overcome the shortcomings of the dynamic surface method. The simulation results of the fractional-order nonlinear strictly feedback system show the effectiveness of the proposed method. The future research will be adaptive neural network control of fractional-order nonlinear strictly feedback system with disturbance observer based on command filtering.



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