# Adaptive Fuzzy Fast Finite-Time Tracking Control for Nonlinear Systems in Pure-Feedback Form with Unknown Disturbance

Authors: Ke Xu; Huanqing Wang; Xiaoping Liu; Ming Chen
Journal: Complexity
 (2020)

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## Abstract

In this paper, based on the fast finite-time stability theorem, an adaptive fuzzy control problem is considered for a class of nonlinear systems in pure-feedback form with unknown disturbance. In the controller design process, the mean value theorem is applied to address the nonaffine structure of the pure-feedback plant, the universal approximation capability of the fuzzy logic system (FLS) is utilized to compensate the unknown uncertainties, and the adaptive backstepping technique is used to design the controller model. Combined with the selection of the appropriate Lyapunov function at each step, a fuzzy-based adaptive tracking control scheme is proposed, which ensures that all signals in the closed-loop system are bounded and tracking error converges to a small neighborhood of the origin in fast finite-time. Finally, simulation results illustrate the validity of the proposed approach.


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## Body

## 1. Introduction

During the recent years, the topics related to the field of nonlinear control have attracted a lot of attention [1–3]. Many approaches for controller design have been investigated, such as backstepping control, dynamic surface control, adaptive control, and so on. Among them, the adaptive control combined with the backstepping technique has provided a systematic framework model-based for control design. The adaptive backstepping control method solves the control design problem for nonlinear systems with unmatched conditions and uncertain parameters and ensures the stability of the closed-loop system successfully. Besides, it supplies an approach which can achieve the transient performance of the systems better by tuning design parameters. Until now, it has already become one of the popular control methods for nonlinear systems in [4–7]. In [8], an adaptive tracking control scheme was proposed for nonlinear strict-feedback systems with additive disturbances. In [9], the authors focused on the position control for a gear transmission servo system by using the backstepping technique. In [10], an adaptive controller was designed via backstepping for nonlinear systems with quantized states. In [11], the adaptive backstepping control approach was developed for a class of stochastic cascade nonlinear time-delay systems.Despite the adaptive backstepping control method having a few merits, there is a need for large-enough gains to suppress uncertainties, which will degrade control performances. It is not feasible for a controlled system with unknown nonlinear functions to use this method alone. Then, the fuzzy logic system (FLS) proposed by Wang and Mendel [12] and the neural network (NN) proposed by Polycarpou [13] have solved this problem well. Because of their universal approximation capabilities, they have become a set of powerful tools to compensate the unknown nonlinear functions of closed-loop systems. During the past decades, many scholars have obtained a lot of meaningful results by using FLSs and NNs combined with the adaptive backstepping technique in [14–26]. An adaptive NN-based fault-tolerant controller for the nonlinear system was investigated in [27]. Combining the adaptive backstepping technique with FLSs and NNs, several interesting control strategies were designed for uncertain stochastic nonlinear systems in [28, 29]. In [30], the authors presented a NNs-based robust adaptive tracking control scheme for hexacopter UAVs. In [31], an adaptive fuzzy-based event-triggered controller was considered for strict-feedback nonlinear systems.Although the fuzzy-based or NNs-based adaptive backstepping control approaches have made great progress, the existing literatures are restricted to the strict-feedback nonlinear systems, and only few results are available about the control of nonlinear systems in pure-feedback form. Different from the strict-feedback nonlinear systems, pure-feedback nonlinear systems possess nonaffine property. It means that there are not state variables to be used as virtual control signals and the actual control input in the pure-feedback form. Therefore, it is more hard and challenging to address the problems of controller design and stability analysis in the pure-feedback systems. In [32, 33], the authors presented the adaptive neural control for nonlinear pure-feedback systems. A fuzzy-based adaptive controller was designed for pure-feedback nonlinear systems in [34]. The control schemes were proposed for pure-feedback nonlinear systems with unknown uncertainties in [35, 36], which were designed by combining dynamic surface control with adaptive backstepping algorithm.It is worth pointing out that the abovementioned literatures are developed on the basis of the Lyapunov asymptotically stability theorem. However, in practical applications, the finite-time control method has lots of advantages such as higher tracking precision, better robustness, and the ability to achieve systems transient performance faster. Recently, plenty of meaningful research results have been produced about the finite-time control problem in [37–42]. Based on the finite-time fault-tolerant control, a new control approach was introduced for robot manipulators by utilizing time-delay estimation in [43]. The adaptive finite-time control problem was addressed for a class of nonlinear systems with the actuator faults in [44]. In [45], the authors designed an adaptive decentralized controller for time-varying output-constrained nonlinear large-scale systems in finite-time. The Lyapunov theorem of finite-time stability was proposed for the first time in [46]. Then, in order to obtain the faster convergence rate, the fast finite-time stability was introduced in [47]. However, compared with the asymptotic control design process, the procedure of adaptive finite-time or fast finite-time controller design is more complex for the nonlinear strict-feedback systems. Furthermore, it is a difficult but meaningful unsolved issue to develop an adaptive fuzzy fast finite-time tracking control scheme for nonlinear systems in pure-feedback form. It is the main motivation of this paper.Inspired by the aforementioned observations, in this paper, the problem of adaptive fuzzy fast finite-time tracking control is considered for a class of nonlinear systems in pure-feedback form with unknown disturbance. During the controller design, the mean value theorem is applied to deal with the nonaffine problem of the pure-feedback systems, FLSs are adopted to approximate packaged unknown nonlinearities, and an improved adaptive fuzzy fast finite-time controller is designed via the backstepping technique. The stability of the closed-loop systems is guaranteed in fast finite-time. To sum up, the main contributions in this paper are listed below:(1)
The fast finite-time theorem is extended to pure-feedback systems for the first time. Also, a fuzzy-based adaptive fast finite-time tracking control scheme for nonlinear systems in pure-feedback form with unknown disturbance is proposed for the first time, too.(2)
Combined the traditional adaptive backstepping technique with the characteristics of the radial basis function of fuzzy logic systems, by applying the mean value theorem and the fast finite-time theory, the system structure is simplified so that reduces complexity of the controller design.The remaining parts are organized as follows. Section2 introduces problem formulation and preliminaries. Section 3 presents the controller design procedure in detail and stability analysis. Section 4 provides simulation results. Section 5 concludes this research.


## 2. Problem Formulation and Preliminaries

### 2.1. System Descriptions and Control Problem

Consider a class of nonlinear pure-feedback systems described as follows:(1)x˙it=fix¯i,xi+1,1≤i≤n−1,x˙nt=fnx¯n,u+dt,yt=x1t,in which x¯it=x1t,…,xitT∈Ri,i=1,…,n−1 is the vector of the states and x¯nt=xnt,…,xntT∈Rn; yt∈R represents the system output; u∈R denotes input signal; dt is a bounded disturbance; and fi· and fn· represent unknown smooth functions.Using the mean value theorem [48], we can express fi· and fn· in (1) as follows:(2)fix¯i,xi+1=fix¯i,xi+10+∂fix¯i,xi+1∂xi+1xi+1=xi+1δi×xi+1−xi+10,1≤i≤n−1,(3)fnx¯n,u=fnx¯n,u0+∂fnx¯n,u∂uu=uδn×u−u0,where xi+1δi=δixi+1+1−δixi+10, with 0<δi<1, and uδn=δnu+1−δnu0, with 0<δn<1.For convenience of writing, we definehix¯i,xi+1δi=∂fix¯i,xi+1/∂xi+1xi+1=xi+1δi and hnx¯n,uδn=∂fnx¯n,u/∂uu=uδn, which are unknown nonlinear functions. Then, substituting (2) with (3) into (4) and choosing xi+10=0, u0=0, we get(4)x˙it=fix¯i,0+hix¯i,xi+1δixi+1,x˙nt=fnx¯n,0+hnx¯n,uδnu+dt,yt=x1t.The objective of this paper is to design a fuzzy-based fast finite-time tracking controller such that all signals in the closed-loop system are bounded and the outputy can follow the specified desired trajectory yd.Throughout this paper, the following assumptions and lemmas are imposed on system (4).Assumption 1.
(see [49]). The desired trajectory signal yd and that up to the nth derivative are smooth and bounded.Assumption 2.
(see [33]). There exists an unknown bounded constant d∗ such that dt≤d∗.Assumption 3.
(see [50]). The function hix¯i,xi+1δi satisfies 0<h¯i≤hi·≤h¯i<∞, for i=1,…,n, where h¯i and h¯i are unknown constants.Remark 1.
According to Assumption3, it is reasonable that the unknown smooth function hi is strictly either positive or negative. Without losing generality, we assume that hi>0. For facilitating the actual controller design, h¯n is known.Lemma 1 (see [51]).
Consider the systemx˙=fx, if there exists continuous function Vx, μ1>0,μ2>0,0<γ<1, and 0<η<∞, so that V˙x≤−μ1Vx−μ2Vγx+η, then the trajectory of system x˙=fx is practical finite-time stable, and the residual set of the solution of system x˙=fx is given by(5)limt⟶TrVx≤minη1−θ0μ1,η1−θ0μ21/γ,where θ0 satisfies 0<θ0<1. The settling time is bounded as(6)Tr≤maxt0+1θ0μ11−γlnθ0μ1V1−γt0+μ2μ2,t0+1μ11−γlnμ1V1−γt0+θ0μ2θ0μ2.Remark 2.
For the convenience of derivation and the proof of the process in this paper, the parameter in the aforementioned inequality is chosen asγ=3/4.Lemma 2 (see [52]).
For any constantε>0 and variable z∈R, we have(7)0≤z−z2z2+ε2<ε.Lemma 3 (see [53]).
Forxi∈R,i=1,…,n and ϕ∈0,1, we have(8)∑k=1nxkϕ≤∑k=1nxkϕ.Lemma 4 (see [54]).
Foro1>0,o2>0,o3>0,γ1≥0,γ2≥0,γ3≥0, the following inequality holds:(9)γ1o1γ2o2γ3≤o3γ1o1+o2+o2o1+o2×o1o3o1+o2o1/o2γ2o1+o2γ3o1+o2/o2.


### 2.2. Fuzzy Logic Systems

An FLS is composed of the knowledge base, the fuzzifier, the fuzzy inference engine, and the defuzzifier. The knowledge base comprises a collection of fuzzy If-Then rules of the following form:Rl: If Z1 is F1l,…, and Zn is Fnl, then ϖ is Gl,l=1,…,NWhereZ=Z1,…,ZnT∈R and ϖ represent the FLS input and output, Fil and Gl are fuzzy sets, their fuzzy membership functions are μFilZi and μGlϖ, and N is rule number of If-Then. ϖ can be expressed as(10)ϖZ=∑l=1NW¯j∏l=1nμFilZi∑l=1N∏l=1nμFilZi,where W¯j=maxϖ∈RμGlϖ.Let(11)SjZ=∏l=1nμFilZi∑l=1N∏l=1nμFilZi.DenotingW=W¯1,…,W¯NT and SZ=S1Z,…,SNZT, the FLS can be reformulated as(12)ϖZ=WTSZ.Lemma 5 (see [12]).
For a continuous functionfZ defined on a compact set Ω and ∀ε>0, there exists an FLS (12) satisfying(13)supZ∈ΩfZ−WTSZ≤ε.


## 2.1. System Descriptions and Control Problem

Consider a class of nonlinear pure-feedback systems described as follows:(1)x˙it=fix¯i,xi+1,1≤i≤n−1,x˙nt=fnx¯n,u+dt,yt=x1t,in which x¯it=x1t,…,xitT∈Ri,i=1,…,n−1 is the vector of the states and x¯nt=xnt,…,xntT∈Rn; yt∈R represents the system output; u∈R denotes input signal; dt is a bounded disturbance; and fi· and fn· represent unknown smooth functions.Using the mean value theorem [48], we can express fi· and fn· in (1) as follows:(2)fix¯i,xi+1=fix¯i,xi+10+∂fix¯i,xi+1∂xi+1xi+1=xi+1δi×xi+1−xi+10,1≤i≤n−1,(3)fnx¯n,u=fnx¯n,u0+∂fnx¯n,u∂uu=uδn×u−u0,where xi+1δi=δixi+1+1−δixi+10, with 0<δi<1, and uδn=δnu+1−δnu0, with 0<δn<1.For convenience of writing, we definehix¯i,xi+1δi=∂fix¯i,xi+1/∂xi+1xi+1=xi+1δi and hnx¯n,uδn=∂fnx¯n,u/∂uu=uδn, which are unknown nonlinear functions. Then, substituting (2) with (3) into (4) and choosing xi+10=0, u0=0, we get(4)x˙it=fix¯i,0+hix¯i,xi+1δixi+1,x˙nt=fnx¯n,0+hnx¯n,uδnu+dt,yt=x1t.The objective of this paper is to design a fuzzy-based fast finite-time tracking controller such that all signals in the closed-loop system are bounded and the outputy can follow the specified desired trajectory yd.Throughout this paper, the following assumptions and lemmas are imposed on system (4).Assumption 1.
(see [49]). The desired trajectory signal yd and that up to the nth derivative are smooth and bounded.Assumption 2.
(see [33]). There exists an unknown bounded constant d∗ such that dt≤d∗.Assumption 3.
(see [50]). The function hix¯i,xi+1δi satisfies 0<h¯i≤hi·≤h¯i<∞, for i=1,…,n, where h¯i and h¯i are unknown constants.Remark 1.
According to Assumption3, it is reasonable that the unknown smooth function hi is strictly either positive or negative. Without losing generality, we assume that hi>0. For facilitating the actual controller design, h¯n is known.Lemma 1 (see [51]).
Consider the systemx˙=fx, if there exists continuous function Vx, μ1>0,μ2>0,0<γ<1, and 0<η<∞, so that V˙x≤−μ1Vx−μ2Vγx+η, then the trajectory of system x˙=fx is practical finite-time stable, and the residual set of the solution of system x˙=fx is given by(5)limt⟶TrVx≤minη1−θ0μ1,η1−θ0μ21/γ,where θ0 satisfies 0<θ0<1. The settling time is bounded as(6)Tr≤maxt0+1θ0μ11−γlnθ0μ1V1−γt0+μ2μ2,t0+1μ11−γlnμ1V1−γt0+θ0μ2θ0μ2.Remark 2.
For the convenience of derivation and the proof of the process in this paper, the parameter in the aforementioned inequality is chosen asγ=3/4.Lemma 2 (see [52]).
For any constantε>0 and variable z∈R, we have(7)0≤z−z2z2+ε2<ε.Lemma 3 (see [53]).
Forxi∈R,i=1,…,n and ϕ∈0,1, we have(8)∑k=1nxkϕ≤∑k=1nxkϕ.Lemma 4 (see [54]).
Foro1>0,o2>0,o3>0,γ1≥0,γ2≥0,γ3≥0, the following inequality holds:(9)γ1o1γ2o2γ3≤o3γ1o1+o2+o2o1+o2×o1o3o1+o2o1/o2γ2o1+o2γ3o1+o2/o2.


## 2.2. Fuzzy Logic Systems

An FLS is composed of the knowledge base, the fuzzifier, the fuzzy inference engine, and the defuzzifier. The knowledge base comprises a collection of fuzzy If-Then rules of the following form:Rl: If Z1 is F1l,…, and Zn is Fnl, then ϖ is Gl,l=1,…,NWhereZ=Z1,…,ZnT∈R and ϖ represent the FLS input and output, Fil and Gl are fuzzy sets, their fuzzy membership functions are μFilZi and μGlϖ, and N is rule number of If-Then. ϖ can be expressed as(10)ϖZ=∑l=1NW¯j∏l=1nμFilZi∑l=1N∏l=1nμFilZi,where W¯j=maxϖ∈RμGlϖ.Let(11)SjZ=∏l=1nμFilZi∑l=1N∏l=1nμFilZi.DenotingW=W¯1,…,W¯NT and SZ=S1Z,…,SNZT, the FLS can be reformulated as(12)ϖZ=WTSZ.Lemma 5 (see [12]).
For a continuous functionfZ defined on a compact set Ω and ∀ε>0, there exists an FLS (12) satisfying(13)supZ∈ΩfZ−WTSZ≤ε.


## 3. Adaptive Fuzzy Controller Design

In this section, a fuzzy-based adaptive control scheme is proposed by using the backstepping technique and FLSs for system (4). The design process of the controller contains n steps based on the following change of coordinates:(14)z1=x1−yd,zi=xi−αi−1,i=2,…,n,where αi−1 is an intermediate control which will be developed for the corresponding i-subsystem combined with choosing the proper Lyapunov functions. The actual control law u will be constructed at Step n to cope with the stability problem of the closed-loop system and the unknown disturbance.In each step, we apply an FLSWiTSiZi to approximate the unknown nonlinearities and define an uncertain parameter θi, θ^i is the estimate of θi, and θ˜i=θi−θ^i. For the sake of simplicity, hix¯i,xi+1δi and hnx¯n,uδn will be abbreviated to hi for i=1,…,n−1 and hn.Step 1: according toz1=x1−yd, and x2=z2+α1, it follows from (14) that(15)z˙1=x˙1−y˙d,=f1+h1x2−y˙d,=f1+h1z2+α1−y˙d,=f1+h1z2+h1α1−y˙d.Consider Lyapunov functionV1 as(16)V1=12z12+h1′θ˜122r1,whereh1′=h¯1 and the design parameter r1>0.Then, we get(17)V˙1=z1z˙1−h1′r1θ˜1θ^.1,=z1h1z2+h1α1+Λ1−z122−h1′r1θ˜1θ^.1,whereΛ1=f1−y˙d+z1/2 is an unknown smooth function, and it can be approximated by FLS such that(18)Λ1Z1=W1TS1Z1+δ1Z1,δ1Z1≤ε1,whereZ1=x1,yd,y˙dT, δ1Z1 is the estimate error.By using Lemma5 and the completion of squares, one has(19)z1Λ1Z1=z1W1TS1Z1+δ1Z1≤|z1|W1S1Z1+ε1≤h1′2a12z12θ1S1TS1+a122+z122+ε122,whereθ1=W12/h1′ and the constant a1>0.Substituting (19) into (17) yields(20)V1.≤z1h1z2+z1h1α1+a122+ε122+h1′2a12z12θ1S1TS1−h1′r1θ˜1θ^.1.The virtual control lawα1is designed as follows:(21)α1=−z1αˇ12z12αˇ12+ε12.Combining with Lemma2, one has(22)z1h1α1≤−h¯1z12αˇ12z12αˇ12+ε12≤ε1−z1αˇ1h1′.Then, chooseαˇ1 and adaption law θ^.1 as(23)αˇ1=12K11z1+K12123/4z121/4+12a12z1θ^1S1TS1,(24)θ^.1=r12a12z12S1TS1−ε1θ^1,where the design parametersa1,K11,K12>0.Substituting (21)–(24) into (20) yields(25)V˙1≤−K¯11z122−K¯12z1223/4+z1h1z2+h1′ε1θ˜1θ^1r1+σ1,whereK¯11=K11h1′, K¯12=K12h1′, and σ1=a12/2+ε12/2+ε1>0.Stepi2≤i≤n−1: utilizing the coordinate transformation in (14), the derivative of zi is(26)z˙i=x˙i−α˙i−1,=fi+hixi+1−α˙i−1,=fi+hizi+1+αi−α˙i−1,=fi+hizi+1+hiαi−α˙i−1,whereα˙i−1=∑j=1i−1∂αi−1/∂xjfj+hjxj+1+∑j=1i−1∂αi−1/∂θ^jθ^.j+∑j=0i−1∂αi−1/∂ydjydj+1.Select a Lyapunov function as(27)Vi=Vi−1+12zi2+hi′θ˜i22ri,wherehi′=h¯i and the parameter ri>0.Next, the derivative ofVi is(28)V˙i=V˙i−1+ziz˙i−hi′θ˜iθ^.iri,=−∑j=1i−1K¯j1zj22−∑j=1i−1K¯j2zj223/4+∑j=1i−1hj′εjθ˜jθ^jrj+zihizi+1+hiαi+Λi−zi22−hi′θ˜iθ^.iri+σi−1,whereΛi=fi−α˙i−1+zi−1hi−1+zi/2 is an unknown smooth function, and we adopt an FLS to approximate it such that(29)ΛiZi=WiTSiZi+δiZi,δiZi≤εi.Combing Lemma5 with the completion of squares, the following result holds:(30)ziΛiZi=ziWiTSiZi+δiZi≤ziWiSiZi+εi≤hi′2ai2zi2θiSiTSi+ai22+zi22+εi22,whereθi=Wi2/hi′ and ai>0 is a constant.Furthermore, substituting (30) into (28) yields(31)V˙i≤−∑j=1i−1K¯j1zj22−∑j=1i−1K¯j2zj223/4+∑j=1i′1hj′εjθ˜jθ^jrj−hi′θ˜iθ^.iri+zihizi+1+zihiαi+σi−1+hi′2ai2zi2θiSiTSi+ai22+εi22.The virtual control lawαi is chosen as follows:(32)αi=−ziαˇi2zi2αˇi2+εi2.By using Lemma2, we have(33)zihiαi≤−h¯izi2αˇi2zi2αˇi2+εi2≤εi−ziαˇihi′.Next,αˇi and θ^.i are chosen as(34)αˇi=12Ki1zi+Ki2123/4zi21/4+12ai2ziθ^iSiTSi,(35)θ^.i=ri2ai2zi2SiTSi−εiθ^i,whereai,Ki1,Ki2 are positive design parameters.By substituting (33)–(35) into (31), the following result holds:(36)V˙i≤−∑j=1iK¯j1zj22−∑j=1iK¯j2zj223/4+∑j=1ihj′εjθ˜jθ^jrj+zihizi+1+σi,where∑j=1iK¯j1=∑j=1iKj1hj′, ∑j=1iK¯j2=∑j=1iKj2hj′, and σi=σi−1+ai2/2+εi2/2+εi>0.Stepn: the control input u will be designed in this step. Since zn=xn−αn−1 in (14), the derivative of zn is(37)z˙n=x˙n−α˙n−1,=fn+hnu+dt−α˙n−1.Consider the following Lyapunov function:(38)Vn=Vn−1+12zn2+hn′θ˜n22rn+12rdd˜2,where hn′=h¯n, d˜=d∗−d^, d^ is the estimate of the disturbance d∗, and rn and rd are positive design parameters.Then, the time derivative ofVn is expressed as(39)V˙n=V˙n−1+znz˙n−hn′θ˜nθ^.nrn−1rdd˜d^⋅,=−∑j=1n−1K¯j1zj22−∑j=1n−1K¯j2zj223/4+∑j=1n−1hj′εjθ˜jθ^jrj+znhnu+Λn−zn22+zndt−hn′θ˜nθ^.nrn+1rdd˜d˜⋅+σn−1,where Λn=fn−α˙n−1+zn−1hn−1+zn/2 is an unknown smooth function, and it can be approximated by FLS such that(40)ΛnZn=WnTSnZn+δnZn,δnZn≤εn.With the help of the completion of squares and Lemma5, one has(41)znΛnZn=znWnTSnZn+δnZn≤znWnSnZn+εn≤hn′2an2zn2θnSnTSn+an22+zn22+εn22,where θn=Wn2/hn′ and an>0 is a constant.Furthermore, based on Assumption2, (39) becomes(42)V˙n≤−∑j=1n−1K¯j1zj22−∑j=1n−1K¯j2zj223/4+∑j=1n−1hj′εjθ˜jθ^jrj−hn′θ˜nθ^.nrn+znhnu+hn′2an2zn2θnSnTSn+an22+εn22+σn−1+|zn|d∗−1rdd˜d^⋅.Design the actual control signalu(43)u=−znαˇn2zn2αˇn2+εn2−tanhzn/ωd^h¯n,where ω is a positive constant.Combing (43) with Lemma 2, one has(44)znhnu≤−h¯nzn2αˇn2zn2αˇn2+εn2−zntanhznωd^≤εn−znαˇnhn′−zntanhznωd^.Then, chooseαˇn and θ^.n as follows:(45)αˇn=12Kn1zn+Kn2123/4zn21/4+12an2znθ^nSnTSn,(46)θ^.n=rn2an2zn2SnTSn−εnθ^n,where an,Kn1,Kn2 are positive constants.Combing (43)–(46), Vn can be rewritten as(47)V˙n≤−∑j=1n−1K¯j1zj22−∑j=1n−1K¯j2zj223/4+∑j=1n−1hj′εjθ˜jθ^jrj−12Kn1hn′zn2−Kn2hn′zn223/4−zntanhznωd∗−d˜+znd∗+an22+εn22+σn−1+εn−d˜d^⋅rd.Utilizing the following property of the hyperbolic tangent function(48)0≤zn−zntanhznω≤0.2785ω,we have(49)V˙n≤−∑j=1n−1K¯j1zj22−∑j=1n−1K¯j2zj223/4+∑j=1n−1hj′εjθ˜jθ^jrj−12Kn1hn′zn2−Kn2hn′zn223/4+d˜zntanhznω−d^⋅rd+0.2785ωd∗+an22+εn22+σn−1+εn,and d^⋅ can be constructed as(50)d^⋅=rdzntanhznω−σdd^,where σd is a positive constant.By Young’s inequality, the following inequality holds:(51)σdd˜d^=σdd˜d∗−d˜≤−σdd˜22+σdd∗22.Substituting (50) and (51) into (49), it yields(52)V˙n≤−∑j=1nK¯j1zj22−∑j=1nK¯j2zj223/4+∑j=1nhj′εjθ˜jθ^jrj+0.2785ωd∗+σdd∗22−σdd˜22+σn,where ∑j=1nK¯j1=∑j=1nKj1hj′, ∑j=1nK¯j2=∑j=1nKj2hj′, and σn=σn−1+an2/2+εn2/2+εn>0.Forμ¯1=minK¯11,K¯21,…,K¯n1,μ¯2=minK¯12,K¯22,…,K¯n2, and Lemma 3, we have(53)−∑j=1nK¯j1zj22≤−μ¯1∑j=1nzj22≤−μ¯1∑j=1nzj22,(54)−∑j=1nK¯j2zj223/4≤−μ¯2∑j=1nzj223/4≤−μ¯2∑j=1nzj223/4.Furthermore, by applying Young’s inequality, we obtain(55)∑j=1nhj′εjθ˜jθ^jrj≤−∑j=1nhj′εjθ˜j22rj+∑j=1nhj′εjθj22rj.Substituting (53)–(55) into (52), one gets(56)V˙n≤−μ¯1∑j=1nzj22−μ¯2∑j=1nzj223/4−∑j=1nhj′εjθ˜j22rj+∑j=1nhj′εjθj22rj−∑j=1nθ˜j22rj3/4+∑j=1nθ˜j22rj3/4−d˜22rd3/4+d˜22rd3/4−σdd˜22+0.2785ωd∗+σdd∗22+σn.Using Lemma4, for o1=1−o2,o2=3/4,o3=o1o23,γ1=1,γ2=∑j=1nθ˜j2/2rj,γ3=1, the following inequality holds:(57)∑j=1nθ˜j22rj3/4≤o3+∑j=1nθ˜j22rj.Similarly, foro1=1−o2,o2=3/4,o3=o1o23,γ1=1,γ2=d˜2/2rd,γ3=1, one has(58)d˜22rd3/4≤o3+d˜22rd.Substituting (57) and (58) into (56), we get(59)V˙n≤−μ¯1∑j=1nzj22−μ¯2∑j=1nzj223/4−∑j=1nθ˜j22rj3/4−hj′εj−1∑j=1nθ˜j22rj−d˜22rd3/4−rdσd−1d˜22rd+Ξ,where Ξ=∑j=1nhj′εjθj2/2rj+0.2785ωd∗+σdd∗2/2+2o3+σn.Finally, forμ1=minμ¯1,hj′εj−1,rdσd−1,μ2=minμ¯2,1, according to Lemma 3, we obtain(60)V˙n≤−μ1∑j=1nzj22+∑j=1nθ˜j22rj+d˜22rd−μ2∑j=1nzj223/4+∑j=1nθ˜j22rj3/4+d˜22rd3/4+Ξ≤−μ1Vn−μ2Vn3/4+Ξ.In the present stage, the controller design has been finished and the following theorem concludes the main result of this research:Theorem 1.
consider the closed-loop system consisting of the plant (1) and the control input (43) with the adaptive laws (24), (35), and (46). Under Assumptions 1–3 and the bounded initial conditions, all the signals defined in the closed-loop system are fast finite-time bounded and the tracking error z1 satisfies(61)z1≤2Ξθ0μ1,with assured settling time Tr as(62)Tr≤maxt0+4θ0μ1lnθ0μ1V1/4t0+μ2μ2,t0+4μ1lnμ1V1/4t0+θ0μ2θ0μ2.Proof.
According to (60), it can be concluded that Vn is bounded, since when Vn≥Ξ/μ1,V˙n≤−μ2Vn3/4≤0. Also, zj,θ˜j,d˜ are bounded from the boundedness of Vn. The boundedness of θ˜j and θj guarantee the boundedness of θ^j. Similarly, d^ is also bounded since d˜=d^−d∗. From (24), αˇ1 is bounded which is the result from the boundedness of z1 and θ^1. The boundedness of α1 in (21) can be also inferred. As z2=x2−α1, x2 is bounded because of the boundedness of α1 and z2. In an inductive manner, the boundedness of θ^2,…,θ^n,α2,…,αn, as well as x3,…,xn, can be guaranteed. The boundedness of actual control signal u in (43) can also be inferred. Thus, the boundedness of all closed-loop signals can be explained.
Furthermore, we transform (60) V˙n≤−μ1Vn−μ2Vn3/4+Ξ into the following form V˙n≤−1−θ0μ1Vn−θ0μ1Vn−μ2Vn3/4+Ξ, where 0<θ0<1. When Vn≥Ξ/θ0μ1, namely, Ξ≤θ0μ1Vn, one has V˙n≤−1−θ0μ1Vn−μ2Vn3/4.
Based on (60) and Lemma 1, we obtain that Vn converges to the set ΩV=Vn:Vn<Ξ/θ0μ1 in fast finite-time with the settling-time estimation(63)Tr≤maxt0+4θ0μ1lnθ0μ1V1/4t0+μ2μ2,t0+4μ1lnμ1V1/4t0+θ0μ2θ0μ2.
Combing with the definition ofVn in (38), it can be observed that 1/2z12≤Vn<Ξ/θ0μ1. Therefore, we have z1≤2Ξ/θ0μ1 in fast finite-time with guaranteed convergence time estimated as Tr in (63).
Through the abovementioned analysis, we have completed the proof.


## 4. Simulation

This part gives an example to prove that the proposed control scheme is valid. Consider the second-order pure-feedback nonlinear system with disturbance as follows:(64)x˙1=0.5cosx1+x2+0.05sinx2,x˙2=cosx22+2u+0.5sinu+0.1sint,y=x1,where x1 and x2 represent the state variables and y and u denote the output and input signal, respectively. f1x¯1,x2=0.5cosx1+x2+0.05sinx2, f2x¯2,u=cosx22+2u+0.5sinu, and dt=0.1sint. The purpose is to establish a controller which can guarantee that all signals remain bounded in the closed-loop system, the system output y=x1 tracks the reference signal yd=0.4sint+sin0.5t, and the tracking error z1=x1−yd is convergent.By using Theorem1, the virtual control signals and actual controller are expressed as follows:(65)α1=−z1αˇ12z12αˇ12+ε12,αˇ1=12K11z1+K12123/4z121/4+12a12z1θ^1S1TS1,u=−z2αˇ22z22αˇ22+ε22−tanhz2/ωd^h¯2,αˇ2=12K21z2+K22123/4z221/4+12a22z2θ^2S2TS2,and the adaptive laws are constructed in the following forms:(66)θ^.1=r12a12z12S1TS1−ε1θ^1,θ^.2=r22a22z22S2TS2−ε2θ^2,d^⋅=rdz2tanhz2ω−σdd^.In the simulation, the initial conditions are set asx10x20d^0θ^10θ^20T=0.10.100.10T. Also, the simulation is run by taking the design parameters as K11=K12=40,K21=K22=20,a1=a2=ε1=ε2=ε2=1,r1=r2=ε1=0.01,ω=rd=0.1,σd=10,h¯2=1.Figures1–5 illustrate the simulation results. Figure 1 shows the desired signal yd and the output y=x1. Figure 2 depicts the trajectory of the tracking error z1. Figure 3 shows the curve of the state x2. From the abovementioned three figures, we can observe that the states x1t and x2t of (64) are bounded and x1 can tracks yd under our design controller in Figures 1 and 3. Moreover, the tracking error z1−x1−yd in Figure 2 is very small and converges into a small neighborhood of zero. Figure 4 displays that the adaptive laws θ^1,θ^2, and d^ are bounded. The trajectory of ut is depicted in Figure 5, from which it can be seen that ut is bounded. Through the numerical simulation in Figures 1–5, the proposed control scheme achieves that y tracks yd in a quick and precise way and desired convergence with the control performance.Figure 1
Reference signalyd and system output x1.Figure 2
The tracking errorz1.Figure 3
State variablex2.Figure 4
The adaptive parametersθ^1,θ^2,d^.Figure 5
The input signalu.


## 5. Conclusions

The adaptive fuzzy-based fast finite-time tracking control via the backstepping technique has been developed for nonlinear systems in pure-feedback form with unknown disturbance. The mean value theorem was used to address the nonaffine problem of the pure-feedback systems, and FLSs were applied to approximate packaged unknown nonlinearities. All signals in the closed-loop system are bounded, and the reference signal can be tracked in fast finite-time. Simulation results have been given to prove the effectiveness of the suggested scheme. In this paper, the approximation error of the FLS is not taken into account. The future work will be concentrated on extending the results to more general nonlinear systems.



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