# Constrained Adaptive Neural Control of Nonlinear Strict-Feedback Systems with Input Dead-Zone

Authors: Jingping Shi; Zhonghua Wu; Jingchao Lu
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper focuses on a single neural network tracking control for a class of nonlinear strict-feedback systems with input dead-zone and time-varying output constraint via prescribed performance method. To release the limit condition on previous performance function that the initial tracking error needs to be known, a new modified performance function is first constructed. Further, to reduce the computational burden of traditional neural back-stepping control approaches which require all the virtual controllers to be necessarily carried out in each step, the nonlinear items are transmitted to the last step such that only one neural network is required in this design. By regarding the input-coefficients of the dead-zone slopes as a system uncertainty and introducing a new concise system transformation technique, a composite adaptive neural state-feedback control approach is developed. The most prominent feature of this scheme is that it not only owes low-computational property but also releases the previous limitations on performance function and is capable of guaranteeing the output confined within the new form of prescribed bound. Moreover, the closed-loop stability is proved using Lyapunov function. Comparative simulation is induced to verify the effectiveness.


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## Body

## 1. Introduction

In recent years, actuated by practical requirements and theoretical developments, numerous adaptive back-stepping control schemes have been proposed for uncertain nonlinear systems in lower-triangular form including strict-feedback and pure-feedback systems [1]. Specially, fuzzy logic systems (FLS) or neural networks (NNs) based control schemes have attracted great concern due to their inherent approximation capabilities as well as relaxing linearly in parameters assumption [2, 3]. The benefit of applying FLS or NNs is that the problem of spending much effort on system modeling can be elegantly overcome. Although there has been significant progress in aforementioned literatures, the problem of complexity (POC) is the main drawback of traditional adaptive back-stepping schemes caused by two reasons [4, 5]: the first one is the reduplicative derivations of virtual controllers, and the other one is that there exist numerous NNs/FLS. In the existing literature, several so-called dynamic surface control [6], sliding mode differentiator (or tracking differentiator) [7], and command filter techniques [8] have been incorporated into adaptive back-stepping control design to avoid the repeated derivations problem. Meanwhile, those techniques have been extensively employed to multifarious practical systems such as flexible-joint robot system [9], flight control [10], and static Var compensator [11]. By using those techniques, the repeated differentiations of virtual control laws can be avoided. However, numerous NNs employed to construct the virtual and actual control laws are still necessary for high-order systems. To thoroughly refrain from this problem, single neural (SN) approximation-based adaptive back-stepping control has been proposed for strict-feedback nonlinear systems where the control gains in each subsystem are equal to one [4]. After that, Pan and Sun have extended SN control structure to a general class of uncertain strict-feedback [5] and pure-feedback nonlinear systems [12], respectively. Meanwhile, some researchers have also utilized the single neural control technique (SNC) to discrete-time nonlinear systems [13, 14]. However, the works described in [4, 5, 12–14] only focus on the stabilization of interested systems, and the output-constraint problems are rarely considered. Meanwhile, it must be pointed out that violation of the output constraint may lower the system performance or even lead to systems instability [15, 16].To manage the problem of output constraint, a number of attempts regarding model predictive control [17], barrier Lyapunov function (BLF) [18–24], and prescribed performance control (PPC) [25–28] techniques have been investigated from both academia and industries. Using the property of BLF, a constrained back-stepping control approach is proposed for nonlinear strict-feedback systems to ensure that the static constraint is not transgressed [20]. More specifically, some constrained adaptive neural/fuzzy back-stepping schemes are investigated for nonlinear systems subject to external disturbance and unknown functions [19, 22, 29]. Apart from the technique about BLF, Bechiloulis et al. have also proposed an alternative approach named PPC to conquer the problem of output constraint [25].The dead-zone characteristic on behalf of the most important input nonlinearities widely exists in actuators and sensors among numerous industrial processes, which also severely deteriorate the system performance [30–34]. Thus, many researchers have devoted themselves to improving the performance of the control systems in the presence of dead-zones. Some researches apply an inverse dead-zone model to minimize the influence of the dead-zone; see reference [30]. On the other hand, by exploring the bounds of the dead-zone slopes and regarding the coefficients of the dead-zone as a system uncertainty, some robust adaptive neural/fuzzy control schemes are proposed of nonlinear systems with nonsymmetric dead-zone inputs [31, 35–37]. Na [38] and Chen [39] both proposed an adaptive neural prescribed performance control for nonlinear system with dead-zone input. Unfortunately, aforementioned PPC approaches still suffer from the POC and use numerous NNs, which result in a complex control structure. Moreover, there also exists restriction that the initial tracking errors must be known in advance. However, actually, the initial tracking error is hard to obtain in the presence of uncertainties and additional disturbances. Thus, how to concurrently tackle such restriction and aforementioned problems needs further investigations.To simultaneously solve the aforementioned problems, a systematic design procedure is developed to derive a composite single neural network control scheme of nonlinear strict-feedback systems subject to dead-zone input, system uncertainties, and time-varying output constraint. In the controller design, to release the limit condition on previous performance function that the initial tracking error needs to be known, a modified performance function is first constructed. Meanwhile, depending on a new system transformation technique, the prescribed performance of the constrained system can be equivalent to ensure the stability of the transformed system. Regarding the coefficients of the dead-zone as a system uncertainty, an adaptive NN approach is developed via single NN control structure to stabilize the transformed system. Finally, stability analysis is conducted by using Lyapunov theory. The main contributions of this paper are summarized as follows:(1)
A new modified prescribed performance function is constructed to stipulate the predefined region of the tracking error. Thus, the previous restriction on initial tracking error is removed.(2)
In construct to previous output-constraint neural/fuzzy back-stepping control approach [15, 25, 38, 40, 41], the proposed control scheme does not need numerous NNs/FLS to construct virtual and practical control law in each step, only one neural network including one adaptive laws is required to approximate the lumped unknown function, thus deriving a low-computational control scheme.(3)
By regarding time-varying input-coefficients of input dead-zone as a system uncertainty and using a new system transformation technique in the control design, an integrated single NN adaptive controller, which is capable of arbitrarily prescribing the system performance and dealing with dead-zone input nonlinearity, simultaneously, is first presented for a class of nonlinear strict-feedback systems.


## 2. Problem Formulation and Preliminaries

### 2.1. Problem Formulation

Consider uncertain nonlinear strict-feedback systems in the following form:(1)x˙1=f1x1+g1x1x2x˙i=fix-i+gix-ixi+1,2≤i≤n-1x˙n=fnx-n+gnx-nuu=Dvy=x1,where x-i=[x1,x2,…,xi]T∈Ri, i=2,…,n, u∈R, and y∈R are state variables, system input, and output, respectively. fi(x-i) are the unknown smooth functions; gi(x-i) denote unknown smooth virtual control gain functions. v is the output from the controller. According to [30], u is a nonsymmetric dead-zone input nonlinearity which is defined as follows:(2)u=Dv=mlv+bl,v≤-bl0,-bl<v<brmrv-br,v≥br,where bl and br denote breakpoints of the input nonlinearity.To facilitate control design, we need the following assumptions.Assumption 1.
The desired trajectoryyd(t) and its derivatives yd(1),yd(2),…,yd(n) are bounded and known.Assumption 2.
There exist constantsgi0,gi1∈R+ such that 0<gi0≤gi(x-i)≤gi1 with i=1,2,…,n. Without loss of generality, it is assumed that gi(x-i)>0.Assumption 3 (see [36]).
The dead-zone parametersml, mr, br, and bl, are unknown bounded positive constants.The dead-zone nonlinearity (2) can be rewritten as a slowly time-varying function:(3)u=Dv=Hv+dv,where (4)H=ml,v≤0mr,v≥0,dv=mlbl,v≤-bl-Hv,bl<v<br-mrbr,v≥br.Here, we can easily conclude that dv≤d∗, d∗=max⁡{mlbl,mrbr}, and there exist positive constants H- and H_ such that H_≤H≤H-.Control Objective. The control goal is to design a controller such that (1) the output y tracks the desired trajectory yd well; (2) meanwhile, the output y can be always confined within the predefined time-varying constraint. That is to say, the output y is needed to satisfy(5)y_t<yt<y-t,where y_(t) and y-(t) are the bounds of the output.Assumption 4 (see [15]).
The output boundsy_(t) and y-(t) are smooth functions and their derivatives from 1 to nth are all available.


### 2.2. Some Preliminaries

Lemma 5 (see [42]).
Assume thatyd(t) and its n-1 derivatives are bounded. Consider system (6)λς˙1=ς2λς˙2=ς3⋮λς˙n=-d1ςn-d2ςn-1-⋯-dn-1ς2-ς1+ydt,where λ is a constant and d1,…,dn-1 are selected such that sn+d1sn-1+⋯+dn-1s+1 is Hurwitz. Thus, there exist positive constants θ and t∗ such that ∀t>t∗ one has (7)Yd-Y^d≤λθ,where Yd=yd,yd(1),…,yd(n-1) and Y^d=yd,ς2/λ,…,ςn/λn-1 are the estimation of Yd.Neural Network. Radial basis function (RBF) NN is often employed as an effective tool to approximate nonlinear functions. The following NN is applied to approximate f(Z):Rq→R(8)f^Z=W^TΦZ,where Z∈ΩZ⊂Rq is the input vector; W^=[w1,…,wl]T∈Rl represents weight vector; l>1 denotes the node number; and Φ(Z)=[Φ1(Z),…,Φl(Z)]T has the form(9)ΦiZ=exp⁡-Z-μiTZ-μi2ηi2,i=1,…,l,where μi=[μi1,…,μiq]T and ηi denote the center and width of the Gaussian function.RBF network (8) has the ability to approximate any continuous function to arbitrary accuracy over the set ΩZ⊂Rq(10)fZ=W∗TΦZ+ε,∀Z∈ΩZ,where W∗ is ideal constant weights and ε is the approximation error.Assumption 6.
There are ideal constant weights such thatε≤ε∗ for all Z∈ΩZ.Lemma 7 (see [43]).
The Gaussian function described in (9) with Z^=Z-rψ being the input vector, where ψ is a bounded vector and r is a positive, is given as (11)ΦiZ^=exp⁡-Z^-μiTZ^-μi2ηi2,i=1,…,l(12)ΦZ^=ΦZ+rΦt,where Φt is a bounded vector.Remark 8.
Assumption1 imposes a controllability condition for system (1) and can be found in most existing adaptive back-stepping neural network control approaches.


## 2.1. Problem Formulation

Consider uncertain nonlinear strict-feedback systems in the following form:(1)x˙1=f1x1+g1x1x2x˙i=fix-i+gix-ixi+1,2≤i≤n-1x˙n=fnx-n+gnx-nuu=Dvy=x1,where x-i=[x1,x2,…,xi]T∈Ri, i=2,…,n, u∈R, and y∈R are state variables, system input, and output, respectively. fi(x-i) are the unknown smooth functions; gi(x-i) denote unknown smooth virtual control gain functions. v is the output from the controller. According to [30], u is a nonsymmetric dead-zone input nonlinearity which is defined as follows:(2)u=Dv=mlv+bl,v≤-bl0,-bl<v<brmrv-br,v≥br,where bl and br denote breakpoints of the input nonlinearity.To facilitate control design, we need the following assumptions.Assumption 1.
The desired trajectoryyd(t) and its derivatives yd(1),yd(2),…,yd(n) are bounded and known.Assumption 2.
There exist constantsgi0,gi1∈R+ such that 0<gi0≤gi(x-i)≤gi1 with i=1,2,…,n. Without loss of generality, it is assumed that gi(x-i)>0.Assumption 3 (see [36]).
The dead-zone parametersml, mr, br, and bl, are unknown bounded positive constants.The dead-zone nonlinearity (2) can be rewritten as a slowly time-varying function:(3)u=Dv=Hv+dv,where (4)H=ml,v≤0mr,v≥0,dv=mlbl,v≤-bl-Hv,bl<v<br-mrbr,v≥br.Here, we can easily conclude that dv≤d∗, d∗=max⁡{mlbl,mrbr}, and there exist positive constants H- and H_ such that H_≤H≤H-.Control Objective. The control goal is to design a controller such that (1) the output y tracks the desired trajectory yd well; (2) meanwhile, the output y can be always confined within the predefined time-varying constraint. That is to say, the output y is needed to satisfy(5)y_t<yt<y-t,where y_(t) and y-(t) are the bounds of the output.Assumption 4 (see [15]).
The output boundsy_(t) and y-(t) are smooth functions and their derivatives from 1 to nth are all available.


## 2.2. Some Preliminaries

Lemma 5 (see [42]).
Assume thatyd(t) and its n-1 derivatives are bounded. Consider system (6)λς˙1=ς2λς˙2=ς3⋮λς˙n=-d1ςn-d2ςn-1-⋯-dn-1ς2-ς1+ydt,where λ is a constant and d1,…,dn-1 are selected such that sn+d1sn-1+⋯+dn-1s+1 is Hurwitz. Thus, there exist positive constants θ and t∗ such that ∀t>t∗ one has (7)Yd-Y^d≤λθ,where Yd=yd,yd(1),…,yd(n-1) and Y^d=yd,ς2/λ,…,ςn/λn-1 are the estimation of Yd.Neural Network. Radial basis function (RBF) NN is often employed as an effective tool to approximate nonlinear functions. The following NN is applied to approximate f(Z):Rq→R(8)f^Z=W^TΦZ,where Z∈ΩZ⊂Rq is the input vector; W^=[w1,…,wl]T∈Rl represents weight vector; l>1 denotes the node number; and Φ(Z)=[Φ1(Z),…,Φl(Z)]T has the form(9)ΦiZ=exp⁡-Z-μiTZ-μi2ηi2,i=1,…,l,where μi=[μi1,…,μiq]T and ηi denote the center and width of the Gaussian function.RBF network (8) has the ability to approximate any continuous function to arbitrary accuracy over the set ΩZ⊂Rq(10)fZ=W∗TΦZ+ε,∀Z∈ΩZ,where W∗ is ideal constant weights and ε is the approximation error.Assumption 6.
There are ideal constant weights such thatε≤ε∗ for all Z∈ΩZ.Lemma 7 (see [43]).
The Gaussian function described in (9) with Z^=Z-rψ being the input vector, where ψ is a bounded vector and r is a positive, is given as (11)ΦiZ^=exp⁡-Z^-μiTZ^-μi2ηi2,i=1,…,l(12)ΦZ^=ΦZ+rΦt,where Φt is a bounded vector.Remark 8.
Assumption1 imposes a controllability condition for system (1) and can be found in most existing adaptive back-stepping neural network control approaches.


## 3. Controller Design and Stability Analysis

### 3.1. System Transformation

Inspired by [15], the tracking error is defined as (13)z1=yt-ydt.By subtracting yd from both sides of (5), we have(14)y_t-ydt<z1t<y-t-ydt.For clarity, we define φ-1(t)=y-(t)-yd(t) and φ_1(t)=y_(t)-yd(t), where φ-1(t)>0 and φ_1(t)<0.Then, (14) becomes (15)φ_1t<z1t<φ-1t.Therefore, the output constraints given by (5) are now transformed into tracking error constraints shown as (15). To establish the relationship between z1(t) and φ-1(t), φ_1(t), we first define φ1(t)=z1(t)-φ_1(t)/φ-1(t)-φ_1(t). Then, the following novel transformed error is established as (16)s1t=ln⁡φ1t1-φ1t,where s1(t) is an error variable.The inverse transformation of (16) with respect to φ1(t) is shown as(17)φ1t=es1t1+es1t.Therefore, if s1(t) is bounded, we have(18)0<φ1t<1,0<z1t-φ_1tφ-1t-φ_1t<1,φ_1t<z1t<φ-1t.Adding yd to every side of (18), condition (5) can be easily got. Therefore, the goal is presently converted to designing a controller to ensure s1 to be bounded.The derivative of (16) is shown as (19)s˙1=f1x1+g1x1x2-y˙d+φ_1tφ-˙1t-φ_˙1tφ-1t-z1tφ-˙1t-φ_˙1t/φ-1t-φ_1tφ1t1-φ1tφ-1t-φ_1t.Therefore, (19) can be rewritten as(20)s˙1=P+R-f1x1+g1x1x2,where (21)R-=1φ1t1-φ1tφ-1t-φ_1t,P=R--y˙d+φ_1tφ-˙1t-φ_˙1tφ-1t-z1tφ-˙1t-φ_˙1tφ-1t-φ_1t.Apparently, we have the transformed systems shown as(22)s˙1=P+R-f1x1+g1x1x2x˙n=fnx-n+gnx-nu,n≥2.Remark 9.
It should be noted thatz1, M1-1, φ-˙1, φ1, φ˙1, R-, and P are definitely known and can be used in the controller design.Remark 10.
From (20) and (18), we know that R-=1/φ11-φ1φ-1-φ_1, φ_1<z1<φ-1. The expression of R- can be rewritten as (23)R-=φ-1-φ_1z1-φ_1φ-1-z1.Define a new function fz1=1/(z1-φ_1)(φ-1-z1) with φ_1<z1<φ-1; the derivative of fz1 with respect to z1 can be described as(24)∂fz1∂z1=2z1-φ-1-φ_1z1-φ_1φ-1-z12.It is easy for us to obtain that function fz1 will get minimum in the set φ_1<z1<φ-1 as long as z1=φ-1+φ_1/2. By considering Assumption 4, therefore, we can conclude that R-≥2/φ-1-φ_1, which also means that there exists a time instant t′ such that R-≥R-min with R-min=2/φ-1(t′)-φ_1(t′).


### 3.2. Constrained Functions Design

To study the constrained character of the tracking errorz1(t), the following positive functions are chosen as (25)ρt=coth⁡l0t+μ0-1+ρ∞σ1t=σ10-σ1∞e-l1t+σ1∞σ2t=σ20-σ2∞e-l2t+σ2∞,where l0, μ0, ρ∞, σ10, σ1∞, l1, σ20, σ2∞, and l2 are positive design constants and σ10>σ1∞, σ20>σ2∞; coth⁡(·) denotes the hyperbolic cotangent function. Then, the control objective can be achieved if the following conditions hold:(26)-ρtσ1t<z1t<ρtσ2t,where φ_1(t)=-ρ(t)σ1(t),φ-1(t)=ρ(t)σ2(t).Remark 11.
From the definition ofρ(t), apparently, ρ(t) satisfies ρ(0)=eμ0+e-μ0/eμ0-e-μ0-1+ρ∞>ρ∞ and limμ0→0⁡ρ(0)→∞ and limt→∞⁡ρ(t)=ρ∞. Thus, the initial and ultimate tracking accuracy of (26) are confined within the bounds -∞<limt→0⁡z1(t)<+∞ and -ρ∞σ1∞<limt→∞⁡z1(t)<ρ∞σ2∞, respectively. It is worth mentioning that traditional prescribed performance function [25, 44, 45] requires the initial tracking error z1(0) to be precisely known in advance and needs the initial value of performance function satisfying strict condition such as -ρ0σ10<z1(0)<ρ0σ20. However, in practical, the precise initial tracking error is uneasy to obtain in real systems with consideration of the uncertainty. Fortunately, the proposed prescribed performance can naturally release this limit condition.Remark 12.
In contrast to previous prescribed performance functions with constant parametersσ1 and σ2, where the designed σ1(t) and σ2(t) are time-varying functions. It implies that the proposed constrained function owes more degrees of freedom to adjust the prescribed performance bounds of the tracking error. This is quite beneficial for the controller design.


### 3.3. Controller Design

Before the controller design, we defineκi0≔1, κij≔kiki-1⋯ki-j-1,j≤i, and κij=kiκi-1j-1 for i>1, where km∈R+ with m=1,2,…,i are design control gains, i is a positive integer, and j is a nonnegative integer. Moreover, let Ydi(t)≔[yd1(t),…,ydi(t)]∈Ri, Y-i(t)≔[y-1(t),…,y-i(t)]∈Ri, Y_i(t)≔[y_1(t),…,y_i(t)]∈Ri, Yd(n+1)(t)≔[Ydn(t),yd(n)(t)]T∈Rn+1, Y-n+1(t)≔[Y-n(t),y-(n)(t)]T∈Rn+1, and Y_n+1(t)≔[Y_n(t),y_(n)(t)]T∈Rn+1, where yd1(t)=yd(t), ydi(t)=yd(i-1)(t), y-1(t)=y-(t), y-i(t)=y-(i-1)(t), y_1(t)=y_(t), and y_i(t)=y_(i-1)(t) with i=2,…,n. Here, ydi(t), y-i(t), and y_i(t) denote the ith derivative of yd(t), y-1(t), and y_1(t), respectively. Meanwhile, we define the virtual tracking errors zi=xi-αi with i=2,…,n, where αi are virtual control inputs which will be defined later. Let Xei=[x-iT,Yd(i+1)T,Y-(i+1)(t),Y_(i+1)(t)]T with i=1,2,…,n. Xei denotes the neural input vector which will be used in the following derivation.Step 1.
Consider the transformed system (22), s˙1=P+R-[f1(x1)+g1(x1)x2]; the first virtual control input is chosen as (27)α2=-k1s1-h1Xe1,where h1(Xe1)=P+R-f1(x1)/R-g1(x1).
Applyingx2=z2+α2 and (27), we have(28)s˙1=g1x1R--k1s1+z2.Applying (27) to z2, one obtains (29)z2=x2+k1s1+h1Xe1=k1s1+∑j=22κ12-jxj-ydj+h1∗Xe1,where h1∗(Xe1)=h1(Xe1)+yd2.Step 2.
The derivative ofz2 is shown as follows:(30)z˙2=f2x-2+g2x-2x3-α˙2.From (27), (28), and (13), one get α2=α2(Xe1); thus we have (31)α˙2=∂α2∂x1Tx˙1+∂α2∂Yd2TY˙d2+∂α2∂Y-3TY-3+∂α2∂Y_3TY_3Defining α2∗(Xe2)=α˙2 and using (31), (30) can be rewritten as (32)z˙2=f2x-2+g2x-2x3-α2∗Xe2.To remove the term R-g1(x1)z2 in (28), a related term R-g1(x1)s1 is added and subtracted in equality (32), then we have (33)z˙2=f2x-2+R-g1x1s1+g2x-2x3-α2∗Xe2-R-g1x1s1.Consider the virtual control input(34)α3=-k2z2-h2Xe2,where h2(Xe2)=f2(x-2)+R-g1(x1)s1-α2∗(Xe2)/g2(x-2).
Applyingx3=z3+α3 and (34) to (33) yields (35)z˙2=g2x-2-k2z2+z3-R-g1x1s1.Utilizing (34) and (29), z3 can be changed to(36)z3=x3+k2z2+h2Xe2=x3-yd3+κ22s1+∑j=22κ23-jxj-ydj+k2h1∗Xe1+h2∗Xe2=κ22s1+∑j=23κ23-jxj-ydj+∑j=12κ22-jhj∗Xej,where h2∗(Xe2)=h2(Xe2)+yd3.Step i (3≤i≤n-1). The derivative of zi is as follows:(37)z˙i=fix-i+gix-ixi+1-α˙i,where α˙i=(∂αi/∂x-iT)x-˙i+(∂αi/∂Yi+1T)Y˙i+1+(∂αi/∂Y-i+1T)Y-˙i+1+(∂αi/∂Y_i+1T)Y_˙i+1.Defineα˙i=αi∗(Xei), and (37) can be rewritten as (38)z˙i=fix-i+gix-ixi+1-αi∗Xei.As done previously, we add and subtractgi-1(x-i-1)zi in equality (38) to remove this interconnected term gi-1(x-i-1)zi, then (38) can be changed to(39)z˙i=fix-i+gi-1x-i-1zi+gix-ixi+1-αi∗Xei-gi-1x-i-1zi.The virtual control is designed as(40)αi+1=-kiz1-hiXei,where hi(Xei)=fi(x-i)+gi-1(x-i-1)zi-αi∗(Xei)/gi(x-i).Usingxi=zi+αi and (40), (39) can be written as (41)z˙i=gix-i-kizi+zi+1-gi-1x-i-1zi-1.Consider (41) and (36), we can obtain that (42)zi=κi-1i-1s1+∑j=2iκi-1i-jxj-ydj+∑j=1i-1κi-1i-1-jhj∗Xej,where hj∗(Xej)=hi(Xei)+yd(i+1).Step n. The derivative of zn is shown as (43)z˙n=fnx-n+gnx-nu-α˙n,where α˙n can be written as(44)α˙n=∂αn∂x-n-1Tx-˙n-1+∂αn∂Yn+1TY˙n+1+∂αn∂Y-n+1TY-˙n+1+∂αn∂Y_n+1TY_˙n+1.Define αn∗(Xen)=α˙n, then, we can get(45)z˙n=fnx-n+gnx-nu-αn∗Xen.As done previously, we add and subtractgn-1(x-n-1)zn-1 in equality (45) to remove this interconnected term and consider the dead-zone input (3), then (45) can be changed to(46)z˙n=fnx-n+gn-1x-n-1zn-1-αn∗Xen+gnx-nHv+gnx-ndv-gn-1x-n-1zn-1.The desired control law v∗ is chosen as (47)v∗=-knzn-hnXen,where hn(Xen)=fn(x-n)+gn-1(x-n-1)zn-1-αn∗(Xen)/gn(x-n)H.Applying (42) and (40), zn can be rewritten as (48)zn=κn-1n-1s1+∑j=2nκn-1n-jxj-ydj+∑j=1n-1κn-1n-1-jhj∗Xej.It must be noted that v∗ is unreliable due to the unknown functions fi, gi, and H with i=1,2,…,n. The expression of zn in (48) is substituted into (47), then we have(49)v∗=-κnns1-∑j=2nκnn+1-jxj-ydj-FXen,where F(Xen)=∑j=1nκnn-jhj∗(Xej) and hn∗(Xen)=hn(Xen).Hence, we use NN to approximate the lumped unknown uncertaintyF(Xen).(50)F^Xen,W^=W^ΦXen,where Φ(·) satisfies Φ(·)≤Φ-; Φ- is a certain positive constant.Therefore, the actual control law and adaptive law are shown as(51)v=-κnns1-∑j=2nκnn+1-jxj-ydj-W^ΦXen(52)W^˙=γs1ΦXen-σs1W^,where γ and σ are positive design constants.Consider (47), (49), and (51); (46) can be converted to(53)z˙n=gnx-nHfnx-n+gn-1x-n-1zn-1-αn∗Xengnx-nH+v+gnx-ndv-gn-1x-n-1zn-1=gnx-nH-knzn+W~ΦXen+ε+gnx-ndv-gn-1x-n-1zn-1,where W^=W∗-W^.


### 3.4. Stability Analysis

Lemma 13.
For updated law (52), there exists a compact set (54)ΩW=W^∣W^≤Φ-σ,where Φ(Xen)≤Φ-, such that W^(t)∈ΩW, ∀t≥0 in case that W^(0)∈ΩW.Proof.
Consider the following Lyapunov candidateVW=1/2γW^TW^. The time derivative of function VW along with (52) is derived as(55)V˙W=1γW^TW^˙=W^Ts1ΦXen-σs1W^2≤-s1W^σW^-Φ-.It follows that V˙W<0 as long as W^>Φ-/σ. Therefore, W^(t)∈ΩW if W^(0)∈ΩW∀t≥0.Theorem 14.
Consider system (1) satisfying Assumptions 1 to 4, controller (51), and the updated law (52). Then, we know that the output time-varying output constraint can be achieved and the signals s1, zi(i=2,…,n) in the closed-loop are bounded.Proof.
Consider the following Lyapunov candidate:(56)V=12s12+∑i=2n12zi2.By using (28), (35), (41), and (53), the derivative of (56) is shown(57)V˙=s1s˙1+∑i=2nziz˙i=-k1g1R-s12-∑i=2n-1kigizi2-kngnHzn2+zngnHW~Φ+zngnHε+zngndv.
SinceW^1 is bounded as presented in Lemma 13, it follows that W~1≤εW, where εW denote the upper bound of the approximation error. We note that Φ(Xen)≤Φ-. Consider the following facts, we have (58)zngnHW~Φ≤gn1H-zn24+εW2Φ-2,zngnHε≤H-gn1zn24+ε∗2,zngndv≤gn1H-zn22+d∗22H-.we have(59)V˙≤-k1R-ming10s12-∑i=2n-1kigi0zi2-kngn0H_-gn1H-zn2+εW2H-gn1Φ-2+H-gn1ε∗2+gn1d∗22H-.By choosing ki and σ such that ki>0,i=1,2,…,n-1, kn>gn1H-/gn0H_, and σ>0 and defining C1=min⁡{2R-ming10,2ki,2kngn0H_-gn1H-}, i=2,…,n, C2=εW2H-gn1Φ-2+H-gn1ε∗2+gn1d∗2/2H-. The derivative of V can be reformed as (60)V˙≤-k1R-ming10s12-∑i=2n-1kigi0zi2-kngn0H_-gn1H-gn0zn2+C2≤-C1V+C2.If s1∉Ωs1=s1∣s1≤C2/k1R-ming10, zi∉Ωzi=zi∣zi≤C2/kigi0, i=2,…,n-1, and zn∉Ωzn=zn∣zn≤C2/kngn0H_-gn1H-, V˙ will become negative. Thus, we can conclude that the transformed error s1, tracking errors zi, and W~ are uniformly ultimately boundedness. Then, the boundedness of s1 can ensure the time-varying output constraints.


### 3.5. Further Design

It must be noted that the control design described in (51) needs the high-order derivatives of the reference signal yd. In practice, high-order derivatives are hard to obtain in the implantation of the controller. Motivated by [46, 47], we can employ a high-gain observer (HGO) or high-order sliding mode observer (HOSM) to estimate y(i), i=2,…,n. As indicated in [46], the prominent features of those observers lie in the fact that they can achieve finite-time observer error convergence and thus can be utilized in almost any feedback with separation principle being trivially fulfilled. If we use HGO to estimate the high-order derivatives of yd, the actual control law (51) and updated law (52) can be modified as (61)v=-κnns1-∑j=2nκnn+1-jxj-y^dj-W^ΦX^enW^˙=γs1ΦX^en-σs1W^,where y^dj and X^en are the estimation of ydj and Xen, respectively.By using HGO to estimate the high-order derivative ofyd, the actual controller and updated law have been changed to (61). The stability of the closed-loop system can be demonstrated the same as Section 3.4. From Lemma 7, it has Φ(X^en)≤Φ-- with Φ--=Φ(Xen)+rΦt. Employing the same procedure of Lemma 13, we can easily prove that there also exists a compact set (62)ΩW=W^∣W^≤Φ--σ,where Φ(X^en)≤Φ--, such that W^(t)∈ΩW, ∀t≥0 in case that W^(0)∈ΩW.Using the property of Gaussian radial basis function (12), (53) can be rewritten as(63)z˙n=gnx-nH-knzn+WΦXen+ε-W^ΦX^en+M1+gnx-ndv-gn-1x-n-1zn-1=gnx-nH-knzn+W~ΦX^en+ε-rWΦt+M1+gnx-ndv-gn-1x-n-1zn-1,where M1=∑j=2nκnn+1-j(ydj-y^dj).Substituting (63) into the derivative of Lyapunov function, we have(64)V˙=s1s˙1+∑i=2nziz˙i=-k1g1R-s12-∑i=2n-1kigizi2-kngnHzn2+zngnHM+zngndv,where M=ε-rWΦt+M1+W~Φ(X^en).SinceW^1 is bounded, it follows that W~1≤εW where εW denote the upper bound of the approximation error. We note that Φ(X^en)≤Φ--. Using Lemma 5, we can also get that there exist a constant M-1 such that M1≤M-1. Until now, it can be easily concluded that there is a constant M- with M-=ε+rWΦt+M-1+W~Φ(X^en) such that M≤M-.Consider the following fact:(65)zngnHM≤gn1H-zn22+gn1H-M-22.we have (66)V˙≤-k1R-ming10s12-∑i=2n-1kigi0zi2-kngn0H_-gn1H-zn2+H-gn1M-22+gn1d∗22H-.The same analysis with Theorem 14, the signals s1, and zi(i=2,…,n) in the closed-loop are bounded.Remark 15.
Compared with the previously proposed output-constraint [40] or prescribed performance control [41, 44, 45] adaptive back-stepping approach, the proposed structure is extremely simple and the computational burden is really low since there is only actual controller required to be implemented and there exists single neural network to approximate the lump uncertainty. It also be noted that the issue of explosion of the complexity inherent in the traditional back-stepping approach is completely removed without employing DSC, command filter, or differentiator technique.Remark 16.
In contrast to the results [4, 5, 12] using cascade low-pass filter to approximate the high-order derivatives of the reference without considering the influence of estimated error in the closed-loop stability analysis, the HGO is applied to approach the derivatives of the reference and the estimated error effect is also taken into account in the control design and stability analysis.


## 3.1. System Transformation

Inspired by [15], the tracking error is defined as (13)z1=yt-ydt.By subtracting yd from both sides of (5), we have(14)y_t-ydt<z1t<y-t-ydt.For clarity, we define φ-1(t)=y-(t)-yd(t) and φ_1(t)=y_(t)-yd(t), where φ-1(t)>0 and φ_1(t)<0.Then, (14) becomes (15)φ_1t<z1t<φ-1t.Therefore, the output constraints given by (5) are now transformed into tracking error constraints shown as (15). To establish the relationship between z1(t) and φ-1(t), φ_1(t), we first define φ1(t)=z1(t)-φ_1(t)/φ-1(t)-φ_1(t). Then, the following novel transformed error is established as (16)s1t=ln⁡φ1t1-φ1t,where s1(t) is an error variable.The inverse transformation of (16) with respect to φ1(t) is shown as(17)φ1t=es1t1+es1t.Therefore, if s1(t) is bounded, we have(18)0<φ1t<1,0<z1t-φ_1tφ-1t-φ_1t<1,φ_1t<z1t<φ-1t.Adding yd to every side of (18), condition (5) can be easily got. Therefore, the goal is presently converted to designing a controller to ensure s1 to be bounded.The derivative of (16) is shown as (19)s˙1=f1x1+g1x1x2-y˙d+φ_1tφ-˙1t-φ_˙1tφ-1t-z1tφ-˙1t-φ_˙1t/φ-1t-φ_1tφ1t1-φ1tφ-1t-φ_1t.Therefore, (19) can be rewritten as(20)s˙1=P+R-f1x1+g1x1x2,where (21)R-=1φ1t1-φ1tφ-1t-φ_1t,P=R--y˙d+φ_1tφ-˙1t-φ_˙1tφ-1t-z1tφ-˙1t-φ_˙1tφ-1t-φ_1t.Apparently, we have the transformed systems shown as(22)s˙1=P+R-f1x1+g1x1x2x˙n=fnx-n+gnx-nu,n≥2.Remark 9.
It should be noted thatz1, M1-1, φ-˙1, φ1, φ˙1, R-, and P are definitely known and can be used in the controller design.Remark 10.
From (20) and (18), we know that R-=1/φ11-φ1φ-1-φ_1, φ_1<z1<φ-1. The expression of R- can be rewritten as (23)R-=φ-1-φ_1z1-φ_1φ-1-z1.Define a new function fz1=1/(z1-φ_1)(φ-1-z1) with φ_1<z1<φ-1; the derivative of fz1 with respect to z1 can be described as(24)∂fz1∂z1=2z1-φ-1-φ_1z1-φ_1φ-1-z12.It is easy for us to obtain that function fz1 will get minimum in the set φ_1<z1<φ-1 as long as z1=φ-1+φ_1/2. By considering Assumption 4, therefore, we can conclude that R-≥2/φ-1-φ_1, which also means that there exists a time instant t′ such that R-≥R-min with R-min=2/φ-1(t′)-φ_1(t′).


## 3.2. Constrained Functions Design

To study the constrained character of the tracking errorz1(t), the following positive functions are chosen as (25)ρt=coth⁡l0t+μ0-1+ρ∞σ1t=σ10-σ1∞e-l1t+σ1∞σ2t=σ20-σ2∞e-l2t+σ2∞,where l0, μ0, ρ∞, σ10, σ1∞, l1, σ20, σ2∞, and l2 are positive design constants and σ10>σ1∞, σ20>σ2∞; coth⁡(·) denotes the hyperbolic cotangent function. Then, the control objective can be achieved if the following conditions hold:(26)-ρtσ1t<z1t<ρtσ2t,where φ_1(t)=-ρ(t)σ1(t),φ-1(t)=ρ(t)σ2(t).Remark 11.
From the definition ofρ(t), apparently, ρ(t) satisfies ρ(0)=eμ0+e-μ0/eμ0-e-μ0-1+ρ∞>ρ∞ and limμ0→0⁡ρ(0)→∞ and limt→∞⁡ρ(t)=ρ∞. Thus, the initial and ultimate tracking accuracy of (26) are confined within the bounds -∞<limt→0⁡z1(t)<+∞ and -ρ∞σ1∞<limt→∞⁡z1(t)<ρ∞σ2∞, respectively. It is worth mentioning that traditional prescribed performance function [25, 44, 45] requires the initial tracking error z1(0) to be precisely known in advance and needs the initial value of performance function satisfying strict condition such as -ρ0σ10<z1(0)<ρ0σ20. However, in practical, the precise initial tracking error is uneasy to obtain in real systems with consideration of the uncertainty. Fortunately, the proposed prescribed performance can naturally release this limit condition.Remark 12.
In contrast to previous prescribed performance functions with constant parametersσ1 and σ2, where the designed σ1(t) and σ2(t) are time-varying functions. It implies that the proposed constrained function owes more degrees of freedom to adjust the prescribed performance bounds of the tracking error. This is quite beneficial for the controller design.


## 3.3. Controller Design

Before the controller design, we defineκi0≔1, κij≔kiki-1⋯ki-j-1,j≤i, and κij=kiκi-1j-1 for i>1, where km∈R+ with m=1,2,…,i are design control gains, i is a positive integer, and j is a nonnegative integer. Moreover, let Ydi(t)≔[yd1(t),…,ydi(t)]∈Ri, Y-i(t)≔[y-1(t),…,y-i(t)]∈Ri, Y_i(t)≔[y_1(t),…,y_i(t)]∈Ri, Yd(n+1)(t)≔[Ydn(t),yd(n)(t)]T∈Rn+1, Y-n+1(t)≔[Y-n(t),y-(n)(t)]T∈Rn+1, and Y_n+1(t)≔[Y_n(t),y_(n)(t)]T∈Rn+1, where yd1(t)=yd(t), ydi(t)=yd(i-1)(t), y-1(t)=y-(t), y-i(t)=y-(i-1)(t), y_1(t)=y_(t), and y_i(t)=y_(i-1)(t) with i=2,…,n. Here, ydi(t), y-i(t), and y_i(t) denote the ith derivative of yd(t), y-1(t), and y_1(t), respectively. Meanwhile, we define the virtual tracking errors zi=xi-αi with i=2,…,n, where αi are virtual control inputs which will be defined later. Let Xei=[x-iT,Yd(i+1)T,Y-(i+1)(t),Y_(i+1)(t)]T with i=1,2,…,n. Xei denotes the neural input vector which will be used in the following derivation.Step 1.
Consider the transformed system (22), s˙1=P+R-[f1(x1)+g1(x1)x2]; the first virtual control input is chosen as (27)α2=-k1s1-h1Xe1,where h1(Xe1)=P+R-f1(x1)/R-g1(x1).
Applyingx2=z2+α2 and (27), we have(28)s˙1=g1x1R--k1s1+z2.Applying (27) to z2, one obtains (29)z2=x2+k1s1+h1Xe1=k1s1+∑j=22κ12-jxj-ydj+h1∗Xe1,where h1∗(Xe1)=h1(Xe1)+yd2.Step 2.
The derivative ofz2 is shown as follows:(30)z˙2=f2x-2+g2x-2x3-α˙2.From (27), (28), and (13), one get α2=α2(Xe1); thus we have (31)α˙2=∂α2∂x1Tx˙1+∂α2∂Yd2TY˙d2+∂α2∂Y-3TY-3+∂α2∂Y_3TY_3Defining α2∗(Xe2)=α˙2 and using (31), (30) can be rewritten as (32)z˙2=f2x-2+g2x-2x3-α2∗Xe2.To remove the term R-g1(x1)z2 in (28), a related term R-g1(x1)s1 is added and subtracted in equality (32), then we have (33)z˙2=f2x-2+R-g1x1s1+g2x-2x3-α2∗Xe2-R-g1x1s1.Consider the virtual control input(34)α3=-k2z2-h2Xe2,where h2(Xe2)=f2(x-2)+R-g1(x1)s1-α2∗(Xe2)/g2(x-2).
Applyingx3=z3+α3 and (34) to (33) yields (35)z˙2=g2x-2-k2z2+z3-R-g1x1s1.Utilizing (34) and (29), z3 can be changed to(36)z3=x3+k2z2+h2Xe2=x3-yd3+κ22s1+∑j=22κ23-jxj-ydj+k2h1∗Xe1+h2∗Xe2=κ22s1+∑j=23κ23-jxj-ydj+∑j=12κ22-jhj∗Xej,where h2∗(Xe2)=h2(Xe2)+yd3.Step i (3≤i≤n-1). The derivative of zi is as follows:(37)z˙i=fix-i+gix-ixi+1-α˙i,where α˙i=(∂αi/∂x-iT)x-˙i+(∂αi/∂Yi+1T)Y˙i+1+(∂αi/∂Y-i+1T)Y-˙i+1+(∂αi/∂Y_i+1T)Y_˙i+1.Defineα˙i=αi∗(Xei), and (37) can be rewritten as (38)z˙i=fix-i+gix-ixi+1-αi∗Xei.As done previously, we add and subtractgi-1(x-i-1)zi in equality (38) to remove this interconnected term gi-1(x-i-1)zi, then (38) can be changed to(39)z˙i=fix-i+gi-1x-i-1zi+gix-ixi+1-αi∗Xei-gi-1x-i-1zi.The virtual control is designed as(40)αi+1=-kiz1-hiXei,where hi(Xei)=fi(x-i)+gi-1(x-i-1)zi-αi∗(Xei)/gi(x-i).Usingxi=zi+αi and (40), (39) can be written as (41)z˙i=gix-i-kizi+zi+1-gi-1x-i-1zi-1.Consider (41) and (36), we can obtain that (42)zi=κi-1i-1s1+∑j=2iκi-1i-jxj-ydj+∑j=1i-1κi-1i-1-jhj∗Xej,where hj∗(Xej)=hi(Xei)+yd(i+1).Step n. The derivative of zn is shown as (43)z˙n=fnx-n+gnx-nu-α˙n,where α˙n can be written as(44)α˙n=∂αn∂x-n-1Tx-˙n-1+∂αn∂Yn+1TY˙n+1+∂αn∂Y-n+1TY-˙n+1+∂αn∂Y_n+1TY_˙n+1.Define αn∗(Xen)=α˙n, then, we can get(45)z˙n=fnx-n+gnx-nu-αn∗Xen.As done previously, we add and subtractgn-1(x-n-1)zn-1 in equality (45) to remove this interconnected term and consider the dead-zone input (3), then (45) can be changed to(46)z˙n=fnx-n+gn-1x-n-1zn-1-αn∗Xen+gnx-nHv+gnx-ndv-gn-1x-n-1zn-1.The desired control law v∗ is chosen as (47)v∗=-knzn-hnXen,where hn(Xen)=fn(x-n)+gn-1(x-n-1)zn-1-αn∗(Xen)/gn(x-n)H.Applying (42) and (40), zn can be rewritten as (48)zn=κn-1n-1s1+∑j=2nκn-1n-jxj-ydj+∑j=1n-1κn-1n-1-jhj∗Xej.It must be noted that v∗ is unreliable due to the unknown functions fi, gi, and H with i=1,2,…,n. The expression of zn in (48) is substituted into (47), then we have(49)v∗=-κnns1-∑j=2nκnn+1-jxj-ydj-FXen,where F(Xen)=∑j=1nκnn-jhj∗(Xej) and hn∗(Xen)=hn(Xen).Hence, we use NN to approximate the lumped unknown uncertaintyF(Xen).(50)F^Xen,W^=W^ΦXen,where Φ(·) satisfies Φ(·)≤Φ-; Φ- is a certain positive constant.Therefore, the actual control law and adaptive law are shown as(51)v=-κnns1-∑j=2nκnn+1-jxj-ydj-W^ΦXen(52)W^˙=γs1ΦXen-σs1W^,where γ and σ are positive design constants.Consider (47), (49), and (51); (46) can be converted to(53)z˙n=gnx-nHfnx-n+gn-1x-n-1zn-1-αn∗Xengnx-nH+v+gnx-ndv-gn-1x-n-1zn-1=gnx-nH-knzn+W~ΦXen+ε+gnx-ndv-gn-1x-n-1zn-1,where W^=W∗-W^.


## 3.4. Stability Analysis

Lemma 13.
For updated law (52), there exists a compact set (54)ΩW=W^∣W^≤Φ-σ,where Φ(Xen)≤Φ-, such that W^(t)∈ΩW, ∀t≥0 in case that W^(0)∈ΩW.Proof.
Consider the following Lyapunov candidateVW=1/2γW^TW^. The time derivative of function VW along with (52) is derived as(55)V˙W=1γW^TW^˙=W^Ts1ΦXen-σs1W^2≤-s1W^σW^-Φ-.It follows that V˙W<0 as long as W^>Φ-/σ. Therefore, W^(t)∈ΩW if W^(0)∈ΩW∀t≥0.Theorem 14.
Consider system (1) satisfying Assumptions 1 to 4, controller (51), and the updated law (52). Then, we know that the output time-varying output constraint can be achieved and the signals s1, zi(i=2,…,n) in the closed-loop are bounded.Proof.
Consider the following Lyapunov candidate:(56)V=12s12+∑i=2n12zi2.By using (28), (35), (41), and (53), the derivative of (56) is shown(57)V˙=s1s˙1+∑i=2nziz˙i=-k1g1R-s12-∑i=2n-1kigizi2-kngnHzn2+zngnHW~Φ+zngnHε+zngndv.
SinceW^1 is bounded as presented in Lemma 13, it follows that W~1≤εW, where εW denote the upper bound of the approximation error. We note that Φ(Xen)≤Φ-. Consider the following facts, we have (58)zngnHW~Φ≤gn1H-zn24+εW2Φ-2,zngnHε≤H-gn1zn24+ε∗2,zngndv≤gn1H-zn22+d∗22H-.we have(59)V˙≤-k1R-ming10s12-∑i=2n-1kigi0zi2-kngn0H_-gn1H-zn2+εW2H-gn1Φ-2+H-gn1ε∗2+gn1d∗22H-.By choosing ki and σ such that ki>0,i=1,2,…,n-1, kn>gn1H-/gn0H_, and σ>0 and defining C1=min⁡{2R-ming10,2ki,2kngn0H_-gn1H-}, i=2,…,n, C2=εW2H-gn1Φ-2+H-gn1ε∗2+gn1d∗2/2H-. The derivative of V can be reformed as (60)V˙≤-k1R-ming10s12-∑i=2n-1kigi0zi2-kngn0H_-gn1H-gn0zn2+C2≤-C1V+C2.If s1∉Ωs1=s1∣s1≤C2/k1R-ming10, zi∉Ωzi=zi∣zi≤C2/kigi0, i=2,…,n-1, and zn∉Ωzn=zn∣zn≤C2/kngn0H_-gn1H-, V˙ will become negative. Thus, we can conclude that the transformed error s1, tracking errors zi, and W~ are uniformly ultimately boundedness. Then, the boundedness of s1 can ensure the time-varying output constraints.


## 3.5. Further Design

It must be noted that the control design described in (51) needs the high-order derivatives of the reference signal yd. In practice, high-order derivatives are hard to obtain in the implantation of the controller. Motivated by [46, 47], we can employ a high-gain observer (HGO) or high-order sliding mode observer (HOSM) to estimate y(i), i=2,…,n. As indicated in [46], the prominent features of those observers lie in the fact that they can achieve finite-time observer error convergence and thus can be utilized in almost any feedback with separation principle being trivially fulfilled. If we use HGO to estimate the high-order derivatives of yd, the actual control law (51) and updated law (52) can be modified as (61)v=-κnns1-∑j=2nκnn+1-jxj-y^dj-W^ΦX^enW^˙=γs1ΦX^en-σs1W^,where y^dj and X^en are the estimation of ydj and Xen, respectively.By using HGO to estimate the high-order derivative ofyd, the actual controller and updated law have been changed to (61). The stability of the closed-loop system can be demonstrated the same as Section 3.4. From Lemma 7, it has Φ(X^en)≤Φ-- with Φ--=Φ(Xen)+rΦt. Employing the same procedure of Lemma 13, we can easily prove that there also exists a compact set (62)ΩW=W^∣W^≤Φ--σ,where Φ(X^en)≤Φ--, such that W^(t)∈ΩW, ∀t≥0 in case that W^(0)∈ΩW.Using the property of Gaussian radial basis function (12), (53) can be rewritten as(63)z˙n=gnx-nH-knzn+WΦXen+ε-W^ΦX^en+M1+gnx-ndv-gn-1x-n-1zn-1=gnx-nH-knzn+W~ΦX^en+ε-rWΦt+M1+gnx-ndv-gn-1x-n-1zn-1,where M1=∑j=2nκnn+1-j(ydj-y^dj).Substituting (63) into the derivative of Lyapunov function, we have(64)V˙=s1s˙1+∑i=2nziz˙i=-k1g1R-s12-∑i=2n-1kigizi2-kngnHzn2+zngnHM+zngndv,where M=ε-rWΦt+M1+W~Φ(X^en).SinceW^1 is bounded, it follows that W~1≤εW where εW denote the upper bound of the approximation error. We note that Φ(X^en)≤Φ--. Using Lemma 5, we can also get that there exist a constant M-1 such that M1≤M-1. Until now, it can be easily concluded that there is a constant M- with M-=ε+rWΦt+M-1+W~Φ(X^en) such that M≤M-.Consider the following fact:(65)zngnHM≤gn1H-zn22+gn1H-M-22.we have (66)V˙≤-k1R-ming10s12-∑i=2n-1kigi0zi2-kngn0H_-gn1H-zn2+H-gn1M-22+gn1d∗22H-.The same analysis with Theorem 14, the signals s1, and zi(i=2,…,n) in the closed-loop are bounded.Remark 15.
Compared with the previously proposed output-constraint [40] or prescribed performance control [41, 44, 45] adaptive back-stepping approach, the proposed structure is extremely simple and the computational burden is really low since there is only actual controller required to be implemented and there exists single neural network to approximate the lump uncertainty. It also be noted that the issue of explosion of the complexity inherent in the traditional back-stepping approach is completely removed without employing DSC, command filter, or differentiator technique.Remark 16.
In contrast to the results [4, 5, 12] using cascade low-pass filter to approximate the high-order derivatives of the reference without considering the influence of estimated error in the closed-loop stability analysis, the HGO is applied to approach the derivatives of the reference and the estimated error effect is also taken into account in the control design and stability analysis.


## 4. Simulation Studies

In this section, the effectiveness of the proposed method is illustrated by one-link manipulator with a brush DC (BDC) motor. One-link manipulator actuated by a BDC motor can be expressed as [48](67)Dq¨+Bq˙+Nsin⁡q=IMI˙=-HI-Kmq˙+V,where q, q˙, and q¨ represent the link angular position, velocity, and acceleration, respectively. I denotes the motor current. V represents the input control voltage. The parameter values with appropriate units are given by D=1, B=1, M=0.05, H=0.5, N=10, and Km=10.Settingx1=q, x2=q˙, and x3=I, (67) can be written as (68)x˙1=x2x˙2=f2x1,x2+g2x3x˙3=f3x1,x2,x3+g3uy=x1,where f2(x1,x2)=(-Nsin⁡(x1)-Bx2)/D, g2=1/D, f2(x1,x2,x3)=(-Kmx2-HDx3)/M, and g3=1/M.The dead-zone model is assumed as(69)u=Dv=1.3v+4,v≤-40,-4<v<1.51.4v-1.5,v≥1.5.The control goal is to derive the output to follow the desired trajectoryyd=0.5πsin⁡(1-e-0.1t2) with dead-zone input nonlinearity shown in (69) while ensuring predefined time-varying output constraints (5) with y_(t)=yd+φ_1(t) and y-(t)=yd+φ-1(t). Here, we define ρ(t)=coth⁡(t+0.01)-1+0.5, σ1(t)=(1-0.2)e-10t+0.2, and σ2(t)=(1-0.2)e-10t+0.2.System (68) satisfies Assumptions 1, 2, and 3. Neural network W^Φ(Xen) contains 128 nodes (i.e., l1=128), Xen=Yd4,x-3,Y-4,Y_4, with centers μl evenly spaced in their corresponding scopes and widths ηl=5. The design parameters of the controller are chosen as k1=2, k2=10, k3=2, γ=10, and σ=0.02. The parameters of HGO are selected as λ=0.05, d1=10, d2=8, and d3=6. In the simulation, the initial conditions are set as [x10,x20,x30]=[0,0,0], W^(0)=0, ς10,ς20,ς30,ς40T=[0,0,0,0]T.To verify the proposed controller, a comparative simulation study [49] has been conducted in this section. For fair comparison, the control gains and initial conditions of control scheme in [49] are the same with this approach. The comparative simulation results are depicted in Figures 1, 2, 3, and 4, where yN, zN, uN, and W^N denote the simulation results of comparative study [49]. In Figures 1 and 2, we plot the controller tracking performance and tracking errors trajectory along with their bounds. Figures 1 and 2 imply that the proposed control method can ensure the output and its tracking error both confined within their bounds. However, the tracking performance and tracking error of comparative study are given an unsatisfactory effect. The control input and norm of NN weights are depicted in Figures 3 and 4, respectively. It is easy for us to find that the control input and norm of NN weights are bounded. Apparently, as opposed to the output tracking performance achieved by control scheme [49], the proposed control approach has a satisfactory result which can ensure the output and tracking error without overstepping their bounds.Figure 1
System output along with its bounds.Figure 2
Trajectory of errorz1 and its bounds.Figure 3
Control input.Figure 4
Trajectory ofW^.


## 5. Conclusion

In this paper, a new concise adaptive neural tracking control scheme has been developed for a class of strict-feedback system subject to input dead-zone and time-varying output constraint. Firstly, to release the limit condition on previous performance function that the initial tracking error needs to be known, a new modified performance function is constructed. By utilizing a system transformation technique, after we transform the original constrained nonlinear system into an unconstrained one, a composite adaptive neural state-feedback control approach is investigated. All unknown functions from1 to n-1 steps are integrated to the n step such that only one neural network is required to approximate the lump uncertainty, thus deriving a low-computational scheme. Meanwhile, the output constraint and input dead-zone are both considered in this scheme. Finally, the closed-loop stability is proved using Lyapunov technique. Comparative simulation is used to demonstrate the effectiveness of this scheme.



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*Source: 2981518-2017-11-23.xml*