# Finite-Time Adaptive Tracking Control for a Class of Pure-Feedback Nonlinear Systems with Disturbances via Decoupling Technique

Authors: Qiangqiang Zhu; Ben Niu; Shengtao Li; Peiyong Duan; Dong Yang
Journal: Complexity
 (2020)

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## Abstract

This paper addresses the finite-time adaptive tracking control problem for a class of pure feedback nonlinear systems whose nonaffine functions may not be differentiable. By properly modeling the nonaffine function, the design difficulty of the pure feedback structure is overcome without using the median value theorem. In our design procedure, an finite-time adaptive controller is elaborately developed using the decoupling technology, which eliminates the limitation assumption on the partial derivatives of nonaffine functions. Furthermore, the constructed controller can stabilize the system within a finite-time so that all signals in the closed-loop system are semiglobally uniformly finite-time bounded (SGUFB), while ensuring the tracking performance. Finally, the simulation results prove the effectiveness of the proposed method.


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## Body

## 1. Introduction

In the past few decades, there have been various research results on the nonfinite time stability of nonlinear systems [1–10], and these results are widely applied to practical systems. However, in actual engineering, the control goal is always expected to be achieved within a finite time. Nonfinite time stable schemes cannot accomplish such control objective because nonfinite time stable control often requires a long transient response. Therefore, the definition of finite-time stability was first proposed in [11, 12] and has received great attention. The finite-time stability can ensure that the system state variables quickly converge to equilibrium within a limited time. At present, the finite-time control of nonlinear systems has become a new research hot spot [13–15]. At the same time, there are many challenging problems which needs hard work to overcome.On the contrary, the rapid development of computer technology has made great progress in the research of adaptive control [16–23]. It is worth mentioning that when there exist completely unknown nonlinear functions in a nonlinear system with a strict feedback structure, radial basis function neural networks (RBF NNs) and fuzzy logic systems play an important role in its adaptive control [24–35]. Using the approximation ability of RBF NNs or fuzzy logic systems, there have been many meaningful research results on adaptive intelligent control for strict feedback nonlinear systems [36–39]. Despite great success in the research on adaptive intelligent control for strict feedback nonlinear systems have been achieved, the research on finite-time control for the nonlinear system is not fully considered [40–42].In recent years, the finite-time adaptive control schemes for strict-feedback nonlinear systems have been developed in [43–47]. However, the finite-time control strategies in [43–47] are only applicable to the strict feedback nonlinear systems, but not applicable to the pure feedback nonlinear systems. It is worth noting that the study on the finite-time tracking control of pure feedback systems has achieved some results [48, 49], but almost all the results are obtained based on the use of differential median theorem, which can convert the pure feedback structure into the strict feedback structure. This requires us to make restrictive assumptions for the partial derivatives of nonaffine functions. However, it is well known that nonsmooth nonlinearities, such as dead zones and hysteresis, exist in a wide range of practical control systems. Thus, not all system functions are differentiable in actual control, which requires exploring a new design technique to deal with the pure feedback structure.In order to meet the actual requirements better, we consider the finite-time adaptive control problem for a class of nonlinear systems with the pure-feedback structure as well as external disturbances. A finite-time adaptive control method based on the decoupling technology is proposed to make the system have better transient response. By selecting the design parameters appropriately, the generated tracking error can converge to a smaller neighborhood of the origin so that the system output follows the desired trajectory within a limited time. The main contributions of this article are as follows. First, we consider a more general class of pure-feedback systems with nonaffine nonlinear functions that may not be differentiable. In order to make our method more practical in industrial control systems than the existing methods [50–59], the limitation of using the differential median theorem in the study of pure feedback systems is eliminated. Second, we construct a suitable controller to stabilize the system in a finite time, which not only ensures that the system state variables quickly converge to equilibrium within a limited time but also improves the robustness of the system and reduces the effects of approximation errors. Third, the appropriate scaling technique is applied to reduce the number of adaptive parameters in the process of designing the controller such that the developed result is more suitable for the actual operation process, which also reduces the complexity of the design procedure. In the end, even if the control direction of the system is unknown, our method can still make all signals in the closed-loop system which are SGUFB.


## 2. Mathematical Preliminaries

Consider a class of pure-feedback nonlinear systems given by(1)τ˙i=giτ¯i,τi+1+rit,1≤i≤n−1,τ˙n=gnτ¯n,u+rnt,y=τ1,where τ¯i=τ1,τ2,…,τiT∈Ri, y∈R, and u∈R are the system state, output, and control input, respectively, giτ¯i,τi+1,1≤i≤n are unknown nonaffine nonlinear functions, and rit are the unknown external disturbances.Definition 1.
(see [60]). The equilibrium ζ=0 of nonlinear system ζ˙=gζ is semiglobal practical finite-time stable (SGPFS) if for all ζt0=ζ0, there exists ε>0 and a settling time Tε,ζ0<∞ to make ζt<ε, for all t≥t0+T.Lemma 1.
(see [60]). Consider the system ζ˙=fζ. If there is a smooth positive definite function Vζ and scalars κ>0, 0<η<1, and ρ>0 such that(2)V˙ζ≤−κVηζ+ρ,t≥0,then this nonlinear system ζ˙=fζ is SGPFS.Proof.
For∀0<θ≤1, from (2), one has(3)V˙ζ≤−θκVηζ−1−θκVηζ+ρ.
LetΩζ=ζVηζ≤ρ/1−θκ and Ω¯ζ=ζVηζ>ρ/1−θκ, if ζt∈Ω¯ζ, one yields V˙ζ≤−θκVηζ. By solving differential equations, we can obtain that(4)Treach=11−ηθκV1−ηζ0−ρ1−θκ1−η/η.
So,ζt∈Ωζ is held for ∀T≥Treach, otherwise, the trajectory of ζt does not exceed the set Ωζ. This means that the time to reach the set Ωζ is bounded as Treach. In other words, the solution of ζ˙=fζ is bounded in a finite time.Lemma 2.
(see [61]). For real variables z and ς and any positive constants β,μ, and ι, the following relation holds:(5)zμςβ≤μμ+βιzμ+β+βμ+βι−μ/βςμ+β.Lemma 3.
(see [62]). For zi∈R,i=1,…,n,0≤p≤1, the following inequality is true:(6)∑i=1nzip≤∑i=1nzip≤n1−p∑i=1nzip.Remark 1.
It is worth noting that system functiongiτ¯i,τi+1 is always assumed to satisfy 0≤p¯i≤∂giτ¯i,τi+1/∂τi+1≤p¯i,p¯i,p¯i∈R in existing articles [52, 54, 55]. However, it is well known that not all system functions are differentiable in actual control, which requires exploring a new design technique to deal with the pure feedback structures. Next, we will introduce a decoupling technique to deal with the unknown nonaffine nonlinear functions of the pure feedback system (1) rather than the median theorem.Lemma 4.
(see [63]). In order to effectively design the control input of the system, the decoupling technology is utilized to deal with the nonaffine terms. After a series of processing, the following formula can be obtained:(7)giτ¯i,τi+1=giτ¯i,0+Piτ¯i+1τi+1+Γiτ¯i+1,where Piτ¯i+1 and Γiτ¯i+1 are defined immediately below.Proof.
DefineGiτ¯i,τi+1=giτ¯i,τi+1−giτ¯i,0,i=1,2,…,n, where τn+1=u and τ¯n+1=τ¯nT,uT. We assume that the function Giτ¯i,τi+1 satisfies(8)G¯iτi+1+Δ1i≤Giτ¯i,τi+1≤G¯iτi+1+Δ2i,τi+1≥0,G¯i′τi+1+Δ3i≤Giτ¯i,τi+1≤G¯i′τi+1+Δ4i,τi+1≤0,where G¯i,G¯i,G¯i′,andG¯i′ are unknown positive constants and Δ1i,Δ2i,Δ3i,andΔ4i,i=1,…,n, are unknown constants.
It can be shown that there exist functionsϑi1τ¯i+1 and ϑi2τ¯i+1, taking values in the closed interval [0, 1] and satisfying(9)Giτ¯i,τi+1=1−ϑi1τ¯i+1G¯iτi+1+Δ1i+ϑi1τ¯i+1G¯iτi+1+Δ2i,τi+1≥0,Giτ¯i,τi+1=1−ϑi2τ¯i+1G¯i′τi+1+Δ3i+ϑi2τ¯i+1G¯i′τi+1+Δ4i,τi+1≤0.
To facilitate the controller design, we have defined the following simplified symbolsPiτ¯i+1 and Γiτ¯i+1 as(10)Piτ¯i+1=1−ϑi1τ¯i+1G¯i+ϑi1τ¯i+1G¯i,τi+1≥0,ϑi2τ¯i+1G¯i′+1−ϑi2τ¯i+1G¯i′,τi+1≤0,(11)Γiτ¯i+1=1−ϑi1τ¯i+1Δ1i+ϑi1τ¯i+1Δ2i,τi+1≥0,ϑi2τ¯i+1Δ4i+1−ϑi2τ¯i+1Δ3i,τi+1≤0.
We can infer from the above definition thatPiτ¯i+1 and Γiτ¯i+1 are bounded. Then, we can model the nonaffine terms Giτ¯i,τi+1 as(12)Giτ¯i,τi+1=Piτ¯i+1τi+1+Γiτ¯i+1.
Hence, (7) was established, and we can rewrite (1) as(13)τ˙i=giτ¯i,0+Piτ¯i+1τi+1+Γiτ¯i+1+rit,1≤i≤n−1,τ˙n=gnτ¯n,0+Pnτ¯n+1u+Γnτ¯n+1+rnt,y=τ1.
In backstepping design, the variableτi+1 is usually taken as the virtual control input for the ith subsystem. So, the virtual control coefficient function Piτ¯i+1 should not pass though the zero point. Therefore, the following assumption is pressed on the system (13).Assumption 1.
The desired trajectoryyd and its derivatives y˙d and y¨d are continuous and bounded.Assumption 2.
Due to realistic considerations, fori=1,…,n, there exist unknown positive constants ri∗ such that rit≤ri∗.Remark 2.
DefinePm=mini=1,2,…,ρG¯i,G¯i,G¯i′,G¯i′, PM=maxi=1,2,…,ρG¯i,G¯i,G¯i′,G¯i′, and Ai∗=maxi=1,2,…,ρΔ1i+Δ2i,Δ3i+Δ4i. It can be inferred from definitions (10) and (11) that the functions Piτ¯i+1 and Γiτ¯i+1 satisfy(14)0≤Pm≤Piτ¯i+1≤PM,(15)0≤Γiτ¯i+1≤Ai∗.
RBFNNs: in the design of this article, the following radial basis function neural networks (RBF NNs) is used to approximate the continuous function hZ:Rn⟶R:(16)hnnZ=WTξZ,where W=W1,W2,…,Wl∈Rl is the weight vector and the neural network node number l>1. ξZ=ξ1Z,…,ξlZT is the basic vector being chosen as the commonly used Gaussian functions, which has the form:(17)ξiZ=exp−Z−μiTZ−μiκ2,where Z∈ΩZ⊂Rn is the input vector, μi=μi1,…,μinT is the center of the respective field, and κ is the width of the Gaussian function.
As shown in [64], the neural network can approximate any continuous function on the compact set ΩZ⊂Rq to any desired accuracy ε∗ as follows:(18)hnnZ=W∗TξZ+εZ,Z∈ΩZ⊂Rq,where W∗ is the ideal constant weight vector and εZ is the approximation error satisfying εZ≤ε∗, ε∗>0 is a very small constant.


## 3. Adaptive State-Feedback Controller Design

In this section, the finite-time adaptive controller is proposed for the backstepping control of system (13). To start, consider the following change of coordinates:(19)Ξi=τi−αi−1,i=1,2,3,…,n,where αi−1 is the virtual control signal constructed in step i−1 and α0=yd.Step 1: differentiatingΞ1 through the first system of (13), we have(20)Ξ˙1=τ˙1−y˙d=g1τ¯1,0+P1τ¯2τ2+Γ1τ¯2+r1t−y˙d.Choose Lyapunov function candidate to construct the virtual control signal of this system as(21)V1=12Ξ12+Pm2γ1ψ˜12.By substituting (20), we can get the time derivative of V1 as(22)V˙1=Ξ1g1τ¯1,0+P1τ¯2Ξ2+α1+Γ1τ¯2+r1t−y˙d−Pmγ1ψ˜1ψ∧˙1,whereΞ2=τ2−α1. Now, we define a new function as g1′=g1τ¯1,0+1/2PMΞ1−y˙d+k1Ξ12η−1+3/2Ξ1 with η=2n−1/2n+1, where n is a natural number and k1>0 is a constant. Then, (22) can be rewritten as(23)V˙1=Ξ1g1′−k1Ξ12η−1−12PMΞ1−32Ξ1+P1τ¯2Ξ2+α1+Γ1τ¯2+r1t−Pmγ1ψ˜1ψ∧˙1=Ξ1g1′+P1τ¯2Ξ2+α1+Ξ1Γ1τ¯2+r1t−12PMΞ12−k1Ξ12η−32Ξ12−Pmγ1ψ˜1ψ∧˙1.Next, based on (15) and Assumption 2, we can obtain(24)V˙1≤−PM2Ξ12−k1Ξ12η+Ξ1g1′+P1τ¯2Ξ2+α1+Ξ1A1∗+r1∗−32Ξ12−Pmγ1ψ˜1ψ∧˙1.Because the unknown functiong1′ cannot be used for the controller design, we can infer from (18) that(25)g1′=W1∗Tξ1τ1′+ε1τ1′,ε1τ1′≤ε1∗,whereτ1′=τ1,yd,y˙dT. For simplicity, we use ξ1 and ε1 instead of ξ1τ1′ and ε1τ1′, respectively. Define ψ1=W1∗2/Pm, and ψ˜1=ψ1−ψ∧1 is the parameter estimation error; then, using Yang’s inequality and Remark 2, one yields(26)Ξ1P1τ¯2Ξ2≤PM2Ξ12+PM2Ξ22,(27)Ξ1A1∗+r1∗≤Ξ12+12A1∗2+12r1∗2,(28)Ξ1g1′=Ξ1W1∗Tξ1+Ξ1ε1≤Pm2a12Ξ12ψ1ξ1Tξ1+12a12+12Ξ12+12ε1∗2,wherea1>0 is the design positive constant. Substituting (26) and (28) into (24) produces(29)V˙1≤−c11+PmΞ12η+Ξ1Pm2a12Ξ1ψ1ξ1Tξ1+P1τ¯2α1+12δ1∗2−Pmγ1ψ˜1ψ∧˙1+PM2Ξ22,wherec1=k11+Pm and δ1∗2=a12+ε1∗2+A1∗2+r1∗2. Next, we construct a virtual signal as(30)α1=−k1Ξ12η−1−ψ∧12a12Ξ1ξ1Tξ1.Substituting (30) into (29) yields(31)V˙1≤−c1Ξ12η+Pmγ1ψ˜1γ12a12Ξ12ξ1Tξ1−ψ∧˙1+12δ1∗2+PM2Ξ22.Next, we construct the adaptive rateψ∧˙1 as(32)ψ∧˙1=γ12a12Ξ12ξ1Tξ1−ϒ1ψ∧1,ψ∧10≥0,whereγ1>0 and Υ1>0 are two design constants. As a result, one can obtain the following formula:(33)V˙1≤−c1Ξ12η+PmΥ1γ1ψ˜1ψ∧1+12δ1∗2+PM2Ξ22.Stepii=2,…,n−1: from (19), we can know that Ξi=τi−αi−1. Next, we use Pi and Γi instead of Piτ¯i+1 and Γiτ¯i+1 for simplicity. Then, the dynamic equation of Ξi is constructed as follows:(34)Ξ˙i=giτ¯i,0+Piτi+1+Γi+rit−α˙i−1,where(35)α˙i−1=∑k=1i−1∂αi−1∂τkgkτ¯k,0+Pkτk+1+Γk+∑k=1i−1∂αi−1∂ψ∧kψ∧˙k+∑k=0i−1∂αi−1∂ydkydk+1.Choose a Lyapunov function candidate as(36)Vi=Vi−1+12Ξi2+Pm2γiψ˜i2.DifferentiatingVi results in(37)V˙i=V˙i−1+Ξigiτ¯i,0+Piτi+1+Γi+rit−α˙i−1−Pmγiψ˜iψ∧˙i,whereψi=Wi∗2/Pm, ψ˜i=ψi−ψ∧i, represents the parameter estimation error. Similar to the processing in the first step, we need to define a new function as gi′=giτ¯i,0+PMΞi−α˙i−1+kiΞi2η−1+3/2Ξi, with ki>0 is a design constant. Then, (37) can be rewritten as(38)V˙i=V˙i−1+Ξigi′+PiΞi+1+αi+ΞiΓi+rit−PMΞi2−kiΞi2η−32Ξi2−Pmγiψ˜iψ∧˙i.Now, based on Assumptions2 and Remark 2, one can obtain(39)V˙i≤V˙i−1+Ξigi′+PiΞi+1+αi+ΞiAi∗+ri∗−PMΞi2−kiΞi2η−32Ξi2−Pmγiψ˜iψ∧˙i.According to (18), we can choose the following neural network system:(40)gi′=Wi∗∧∧∗Tξiτi′+εiτi′,εiτi′≤εi∗,whereτ≤i′=τ¯iT,ψ∧¯i−1T,y¯diTT∈Ωτi⊂R3i and ψ∧¯i−1=ψ∧1,ψ∧2,…,ψ∧i−1T. For simplicity, we use ξi and εi to represent ξiτi′ and εiτi′, respectively. Thus, we can obtain(41)ΞiPiΞi+1≤PM2Ξi2+PM2Ξi+12,(42)ΞiAi∗+ri∗≤Ξi2+12Ai∗2+12ri∗2,(43)Ξigi′=ΞiWi∗Tξi+Ξiεi≤Pm2ai2Ξi2ψiξiTξi+12ai2+12Ξi2+12εi∗2,whereai>0 is the design positive constant. Like the first step, substituting (41)–(43) into (39), the following inequality holds:(44)V˙i≤−ci1+PmΞi2η+ΞiPm2ai2ΞiψiξiTξi+Piαi+12δi∗2−Pmγiψ˜iψ∧˙i+PM2Ξi+12−PM2Ξi2+V˙i−1,whereci=ki1+Pm and δi∗2=ai2+εi∗2+Ai∗2+ri∗2. Next, we construct the virtual signal αi as well as the adaptive rate ψ∧˙i as follows:(45)αi=−kiΞi2η−1−ψ∧i2ai2ΞiξiTξi,(46)ψ∧˙i=γi2ai2Ξi2ξiTξi−Υiψ∧i,ψ∧i0≥0,whereγi>0 and ϒi>0 are two design constants. As a result, substituting (45) and (46) into (44), one can get the following formula:(47)V˙i≤V˙i−1−PM2Ξi2−ciΞi2η+PmΥiγiψ˜iψ∧i+12δi∗2+PM2Ξi+12.Comparing (33) and (47), we can get the following formula by mathematical induction:(48)V˙i≤−∑k=1ickΞk2η+∑k=1iPmΥkγkψ˜kψ∧k+∑k=1i12δk∗2+PM2Ξi+12.Remark 3.
As can be seen from formulas (28) and (43), we used Yang’s inequality to obtain ψi=Wi∗2/Pm in advance such that only one adaptive parameter should be estimated in each step of the controller design. However, multidimensional vectors (weight vectors) are directly estimated in some literatures such as W∧i=Γi1−mZiziSiZi−σiW∧i in [65], which makes the design of the adaptive rate more difficult. Therefore, the method we adopt can reduce the number of adaptive parameters compared to the previous method in [65].Stepn: define ψn=Wn∗2/Pm, where Wn∗ is the ideal weight vector, and ψ˜n=ψn−ψ∧n is the parameter estimation error. Choose the Lyapunov function candidate for system (13) as follows:(49)Vn=Vn−1+12Ξn2+Pm2γnψ˜n2,whereγn>0 is a design positive constant. It can be seen from the previous n−1 step that the virtual control signal αn−1 can be constructed such that the following inequality can be obtained:(50)V˙n−1≤−∑k=1n−1ckΞk2η+∑k=1n−1PmΥkγkψ˜kψ∧k+∑k=1n−112δk∗2+PM2Ξn2.As we all know, the dynamic equation ofΞn is as follows:(51)Ξ˙n=gnτ¯n,0+Pnu+Γn+rnt−α˙n−1,with(52)α˙n−1=∑k=1n−1∂αn−1∂τkgkτ¯k,0+Pkτk+1+Γk+∑k=1n−1∂αn−1∂ψ∧kψ∧˙k+∑k=0n−1∂αn−1∂ydkydk+1.From (49), one can get the time derivative of Vn along (51) as(53)V˙n=V˙n−1+Ξngnτ¯n,0+Pnu+Γn+rnt−α˙n−1−Pmγnψ˜nψ∧˙n.Similar to the processing in the above steps, we define a new function asgn′=gn+1/2PMΞn−α˙n−1+knΞn2η−1+3/2Ξn, where kn>0 is a design constant. Then, (53) can be rewritten as(54)V˙n=V˙n−1+Ξngn′+Pnu+ΞnΓn+rnt−12PMΞn2−knΞn2η−32Ξn2−Pmγnψ˜nψ∧˙n.We designed the actual controller and the adaptation law as follows:(55)u=−knΞn2η−1−ψ∧n2an2ΞnξnTξn,(56)ψ∧˙n=γn2an2Ξn2ξnTξn−Υnψ∧n,ψ∧n0≥0,whereγn>0 and Υn>0 are two design constants. Like (38)–(48), it is easy to obtain(57)V˙n≤−∑j=1ncjΞj2η+∑j=1nPmΥjγjψ˜jψ∧j+∑j=1nδj∗22,wherecj=kj1+Pm and δj∗2=aj2+εj∗2+Aj∗2+rj∗2.


## 4. Stability Analysis

In this section, the main result will be summarized in Theorem1.Theorem 1.
Consider system (1) satisfying Assumptions 1–2, and suppose that the finite-time adaptive controller (55) and the adaptive law (46) as well as (56) are constructed based on the decoupling technology. As long as the design parameters η,k,a,γ,andΥ are properly selected, it can be ensured that the system output y follows the desired trajectory yd, and at the same time all the signals of the pure-feedback nonlinear systems (1) are SGUFB.Proof.
For the Lyapunov function candidateV=Vn, define c=mincj,Υj,j=1,2,...,n. Then, it follows from (57) that(58)V˙≤−c∑j=1nΞj2η+∑j=1nPmΥjγjψ˜jψ∧j+∑k=1nδj∗22.
From the definition ofψ˜, we can get ψ˜jψ∧j≤1/2ψj2−1/2ψ˜j2; further rewrite (58) as(59)V˙≤−2ηc∑j=1nΞj22η−c∑j=1nPm2γjψ˜j2+∑k=1nδj∗22+∑j=1nPmΥj2γjψj2.
For Lemma2, we choose the appropriate parameters for z=1,ς=c1/η∑j=1nPm/2γjψ˜j2, and μ=1−η,β=η,ι=cη/η−η/1−η. Then, one can obtain(60)c∑j=1nPm2γjψ˜j2η≤1−ηι+c∑j=1nPm2γjψ˜j2.
Substituting (60) into (59) and using the zoom method of (6), the following inequality holds:(61)V˙≤−2ηc∑j=1nΞj22η−c∑j=1nPm2γjψ˜j2η+c∑j=1nPm2γjψ˜j2η−c∑j=1nPm2γjψ˜j2+∑k=1nδj∗22+∑j=1nPmΥj2γjψj2,≤−2ηc∑j=1nΞj22η−c∑j=1nPm2γjψ˜j2η+1−ηι+∑k=1nδj∗22+∑j=1nPmΥj2γjψj2.
Applying Lemma3, we can further simplify the time derivative of V as(62)V˙≤−κVη+ρ,where(63)κ=min2ηc,c,ρ=1−ηι+∑k=1nδj∗22+∑j=1nPmΥj2γjψj2.
Now, defineT∗=1/1−ηθκV1−ηΞ0,Φ0−ρ/1−θκ1−η/η with Ξ0=Ξ10, Ξ20,…,Ξn0T, and Φ0=ψ10,ψ20,…,ψn0T. Based on Lemma 1, we can get Vηζ≤ρ/1−θκ for ∀T≥T∗. So, the solution of ζ˙=fζ is bounded in a finite time and all the signals in the nonlinear system (1) are SGUFB. To be more precise, the finite-time controller proposed by us can converge the tracking error to a small neighborhood of zero and remains there after the finite time T∗. In order to be more intuitive, we will confirm the research results through a simulation example.


## 5. A Simulation Example

In this section, we will demonstrate the effectiveness of the proposed scheme through the following simulation example.Let us consider a two-dimensional nonaffine pure-feedback nonlinear system with disturbance as follows:(64)τ˙1=τ1+τ2+τ235+0.2sin2τ1τ2,τ˙2=τ1τ2+u+0.1sinτ1τ2,y=τ1,where τ¯i=τ1,τ2,…,τiT∈Ri,i=1,…,n, y∈R are the system state and output and u∈R are the system control input, respectively. The reference signal of the system output is given as yd=0.1sint. According Lemma 3, we construct virtual control signal and actual controller as well as the adaptation law for system (64) as follows:(65)α1=−k1Ξ12η−1−ψ∧12a12Ξ1ξ1Tξ1,ψ∧˙1=γ12a12Ξ12ξ1Tξ1−Υ1ψ∧1,ψ∧10≥0,u=−k2Ξ22η−1−ψ∧22a22Ξ2ξ2Tξ2,ψ∧˙2=γ22a22Ξ22ξ2Tξ2−Υ2ψ∧2,ψ∧n0≥0.Then, the initial conditions are given asτ10=0,τ20=0, and ψ∧0=0. We choose the design parameter in the simulation as follows: η=99/101,k1=15,k2=8,a1=1,a2=1,γ1=10,γ2=10,Υ1=0.5,andΥ2=0.5. Finally, we get Figures 1–6. Figure 1 denotes the responses of the system out yt and reference signal yd of the example. The tracking error Ξ1 of the example between the output of the system and the reference signal converges to a small neighborhood, which can be observed intuitively in Figure 2. Figure 3 shows the response of the state τ2 variable. From the trends in Figures 4 and 5, it is clear to see that the boundedness of the adaptive rate ψ∧˙1 and ψ∧˙2. It can be seen from these results that even though the nonaffine function of our simulation system is not differentiable, it has achieved excellent control performance. Finally, the response of the control law uf is shown in Figure 6.Figure 1
The responses of the system outyt and reference signal yd of the example.Figure 2
The response of the tracking errorΞ1 of the example.Figure 3
The response of the state variableτ2 of the example.Figure 4
The response of the adaptive rateψ^1 of the example.Figure 5
The response of the adaptive rateψ^2 of the example.Figure 6
The response of the control law inputu.


## 6. Conclusions

A novel finite-time adaptive controller has been presented for the considered pure-feedback nonlinear system in this paper. The first design difficulty in this paper is to decouple the pure feedback system without using the median value theorem. The second design difficulty is the extremely complicated formula derivation when designing the finite-time controller, in order to make the system variables converge to the equilibrium quickly in a limited time. Compared with the existing results, the developed method addressed the finite-time adaptive tracking control problem for the pure feedback system whose nonaffine functions may not be differentiable. Furthermore, the decoupling technology has been used in our design frame to eliminate the restrictive assumption of partial derivatives of nonaffine functions, which makes the method more widely used. It is worth noting that the finite-time controller constructed by us can not only ensure that the system state variables quickly converge to equilibrium within a limited time but also improve the robustness of the closed-loop system. In the future, the finite-time adaptive control of various types of complex switched nonlinear systems can be further discussed, such as multiple input multiple output stochastic switched nonlower triangular systems and stochastic switched nonlower triangular pure feedback nonlinear systems.



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*Source: 1354340-2020-08-17.xml*