# Finite-Time Tracking Control for Nonstrict-Feedback State-Delayed Nonlinear Systems with Full-State Constraints and Unmodeled Dynamics

Authors: Yangang Yao; Jieqing Tan; Jian Wu
Journal: Complexity
 (2020)

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## Abstract

The problem of finite-time tracking control is discussed for a class of uncertain nonstrict-feedback time-varying state delay nonlinear systems with full-state constraints and unmodeled dynamics. Different from traditional finite-control methods, aC1 smooth finite-time adaptive control framework is introduced by employing a smooth switch between the fractional and cubic form state feedback, so that the desired fast finite-time control performance can be guaranteed. By constructing appropriate Lyapunov-Krasovskii functionals, the uncertain terms produced by time-varying state delays are compensated for and unmodeled dynamics is coped with by introducing a dynamical signal. In order to avoid the inherent problem of “complexity of explosion” in the backstepping-design process, the DSC technology with a novel nonlinear filter is introduced to simplify the structure of the controller. Furthermore, the results show that all the internal error signals are driven to converge into small regions in a finite time, and the full-state constraints are not violated. Simulation results verify the effectiveness of the proposed method.


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## Body

## 1. Introduction

During the past few decades, great achievements have been proposed for uncertain nonlinear systems based on adaptive control technique, especially for pure-feedback systems (e.g., see [1–5]) and strict-feedback systems (e.g., see [6–9]) with the lower-triangular structure. Lately, the authors in [10] introduced a more general nonlinear system named nonstrict-feedback nonlinear systems. By employing the variable separation method, the tracking control problem has been well solved. Since then, many control techniques for nonstrict-feedback systems and extensions to other fields were achieved (e.g., see [11–17]).It is known to all that many practical systems encounter the effect of the constraints, such as the temperature of chemical reactor and physical stoppages. Thus, the research about the systems with state constraints is very meaningful and necessary on account of the existence of state constraints which may undermine the stability of the system. In order to tackle the problem of state constraints, some effective control techniques (e.g., model predictive control (MPC) [18, 19], reference governors (RGs) [20], one-to-one nonlinear mapping (NM) [21–23], and barrier Lyapunov functions (BLFs) [24–28]) have been presented. Due to the fact that MPC and RGs require strong online computing capability to guarantee constraints, this requirement restricts their applications in engineering design. Therefore, one-to-one NM and the BLFs-based methods become the main methods to deal with the constrained nonlinear systems. There exist many significant results which focus on lower-triangular structure nonlinear systems with different constraints (e.g., input constraints [3], output constraints [24], partial-state constraints [25], and full-state constraints [21–23, 26, 27]). In addition, the rate of convergence is also an essential consideration for most practical systems. The works mentioned above only obtain asymptotic or exponential stability with infinite time, which cannot meet the requirement of finite-time control in most practical control systems. As a consequence, a considerable number of meaningful researches (e.g., see [28–33]) have been proposed on finite-time control for nonlinear systems. However, most of the works are to present C0 finite-time controller by using a backstepping technique together with a nonsmooth fractional feedback design method. In order to achieve a faster convergence rate, the authors in [34] originally proposed a C1 smooth finite-time adaptive NN controller by using a smooth switch between the fractional and cubic form state feedback. Moreover, there are other significant results presented in [35–41], such that two globally stable adaptive controllers were proposed in [35, 36]. To obtain the tracking accuracy, a practical adaptive fuzzy tracking controller for a class of perturbed nonlinear systems with backlash nonlinearity has been designed in [37]. An adaptive fuzzy output-feedback tracking control technique for switched stochastic pure-feedback nonlinear systems has been presented in [38]. The authors in [39] proposed an observed-based adaptive finite-time tracking control technique for a class of nonstrict-feedback nonlinear systems with input saturation. An adaptive finite-time output-feedback controller for switched pure-feedback nonlinear systems with average dwell time has been given in [40]. A decentralized event-triggered controller for interconnected systems with unknown disturbances has been proposed in [41].Furthermore, due to the fact that unmodeled dynamics can severely degrade the closed-loop system performance, dealing with the effects of unmodeled dynamics is essential for practical nonlinear control systems. Therefore, several results were proposed by employing backstepping or DSC in [4, 21–23, 42–47]. Generally, unmodeled dynamics was disposed by introducing a dynamic signal in [4, 21–23, 42–46] or a Lyapunov function description in [47].In addition, time delays frequently occur in some practical engineering systems. As stated in [48], their existence can deteriorate the transient performance and even can destroy the stability of the control systems. Thus, the research on nonlinear time-delay systems has become one of the hot topics and some meaningful results have been achieved during the past decades [49–53]. For uncertain nonlinear time-delay systems, the effective controller was developed originally in [50] by combining the backstepping technique with Lyapunov-Krasovskii functionals. Soon afterward, this method was extended to nonlinear strict-feedback time-delay system with unknown control gain functions [51] and uncertain multi-input/multi-output nonlinear systems with time delays [52]. Later, some improved control schemes based on [50] were proposed (e.g., see [35, 53, 54]).Although many significant research results on adaptive neural network control for uncertain nonstrict-feedback systems have been obtained in [11–17], their considered systems did not include unmodeled dynamics or full-state constraints. In [21–28], the effective controllers have been designed for the lower-triangular structure nonlinear systems with state constraints and unmodeled dynamics, but their considered systems did not include state delay and their control methods may be invalid to nonstrict-feedback systems on account of subsystem function which contains the whole state variables. Furthermore, the above-mentioned control methods only obtain asymptotic or exponential stability with infinite time. To the best knowledge of the authors, finite-time tracking control for a class of uncertain nonstrict-feedback time-varying state-delayed nonlinear systems with full-state constraints and unmodeled dynamics has not been fully discussed in the literature, which is still open and remains unsolved. In this paper, we are committed to solving the problem mentioned above. The main contributions of the paper are summarized as follows:(i)
In contrast to the existing results reported in [21–28, 47] where the control methods have been proposed for nonlinear strict-feedback or pure-feedback systems with state or output constraints and unmodeled dynamics, a generalization of the results is proposed for a class of nonstrict-feedback state delay systems with state constraints and unmodeled dynamics of which the subsystem function contains the whole state variables. To the best of authors’ knowledge, it is the first time to develop an adaptive DSC method for uncertain nonstrict-feedback state delay systems with state constraints and unmodeled dynamics.(ii)
Different from the finite-control methods in [31–33], a C1 smooth finite-time adaptive control framework is introduced by employing a smooth switch between the fractional and cubic form state feedback reported in [34], so that the desired fast finite-time control performance can be guaranteed. Moreover, unmodeled dynamics is coped with by introducing a dynamical signal and the uncertain terms produced by time-varying state delays are compensated for by constructing appropriate Lyapunov-Krasovskii functionals. The results show that all the error signals are driven to converge into small regions in a finite time, and the full-state constraints are never violated.The remainder of this paper is organized as follows. In Section2, the problem formulation and preliminaries are presented. Adaptive DSC design and stability analysis are given in Section 3. Simulation results verify the effectiveness of the proposed control approach in Section 4, followed by Section 5, which concludes this paper.Notation. In this paper, R denotes a set of real numbers, R+ denotes a set of nonnegative real numbers, Rm×n denotes a set of m×n real matrices, Rn denotes a set of n-dimensional real vectors, sup⋅ denotes the least upper bound, ⋅ denotes 2-norm of a vector or matrix, ⋅ denotes an absolute value of a real number ·, exp⋅ denotes an exponential function of ·, and log⋅ denotes the natural logarithm of ·.


## 2. Problem Formulation and Preliminaries

### 2.1. Problem Statement

Consider a class of uncertain nonstrict-feedback state-delayed nonlinear systems with unmodeled dynamics fori=1,2,…,n−1 in the following form:(1)ξ˙=qξ,x,t,x˙i=fix+gix¯ixi+1+δiξ,x,t+dix¯it−Tit,x˙n=fnx+gnx¯nu+δnξ,x,t+dnx¯nt−Tnt,y=x1,where x=x1,x2,…,xnT∈Rn is the state vector, ξ∈Rn0 is the unmodeled dynamics, and u,y,Tit denote the system input, the system output, and the unknown time-varying delays, respectively. fix,gix¯i, and dix¯it−Tit are the unknown smooth functions. Let x¯i=x1,x2,…,xiT and δiξ,x,t be the unknown uncertain disturbances. All the states xi are required to remain in the sets Ωxi=xi:xi<kci, where kci are positive constants.Remark 1.
System (1) is called a nonstrict-feedback form in which the system function fi⋅ and its bounding function contain all the state variables [10]. Apparently, strict-feedback and pure-feedback structures are the special cases of system (1). The methods proposed in [21–28, 31–33, 47] cannot be directly applied to system (1) on account of its nonstrict-feedback structure.
The control objective of this paper is to construct an adaptive NN controllerut to make sure that the output y follows the desired trajectory yr in a finite time, while every state xi∈Ωxi is never violated.


### 2.2. RBFNN Approximation

In this paper, fori=1,…,n, the unknown smooth nonlinear functions F¯iZi:Rm⟶R will be approximated on a compact set Ωi⊂Rm by the following RBFNN:(2)F¯iZi=WiTSiZi+εiZi,where Zi,Wi,l denote input vectors, weight vectors, and NN node number, respectively. εiZi are the NN inherent approximation errors which are bounded over the compact sets; that is, εiZi≤εi, where ϵi are unknown constants and SiZi=s1Zi,…,slZiT:Ωi⟶Rl are known smooth vector functions with sqZi being chosen as the commonly used Gaussian functions, which have the form(3)sqZi=exp−Zi−μqTZi−μqηq2,q=1,…,l,where μq=μq1,…,μqmT is the center vector and ηq is the spreads of the Gaussian function. The optimal weight vector Wi is defined as(4)Wi=argminW^i∈RlsupZi∈ΩiFZi−W^iTSZ,where W^i is the estimate of Wi.


### 2.3. Key Definition and Lemmas

Definition 1 (see [21]).
The unmodeled dynamicsξ is said to be exponentially input-state-practically stable (exp-ISpS), that is, for system ξ˙=qξ,x,t, if there exist functions α¯1,α¯2 of class K∞ and a Lyapunov function Vξ, such that(5)α¯1ξ≤Vξ≤α¯2ξ,and there exist two constants c>0,d≥0 and a class K∞ function γ, such that(6)∂Vξ∂ξqξ,x,t≤−cVξ+γx1+d,∀t≥0,where c and d are known positive constants and γ⋅ is a known function of class K∞.Lemma 1 (see [21]).
IfV is an exp-ISpS Lyapunov function for a system ξ˙=qξ,x,t, that is, (5) and (6) hold, then, for any constant c¯∈0,c, any initial instant t0>0, any initial condition ξ0=ξt0,r0>0, and any continuous function γ¯, such that γ¯x1≥γx1, there exist a finite T0=max0,logVξ0/r0/c−c¯≥0, a nonnegative function Dt0,t defined for all t≥t0, and a signal described by(7)r˙=−c¯r+γ¯x1+d,rt0=r0,such that Dt0,t=0 for t≥t0+T0 and Vξ≤rt+Dt0,t with Dt0,t=max0,e−ct−t0Vz0−e−c¯t−t0r0.Lemma 2 (see [11]).
LetSZ be the basis function vector of an RBFNN and Z be the input vector, where SZ=s1Z,…,slZT and Z=z1,…znT. For any positive integer m≤n, let Zm=z1,…,zmT, and the following inequality holds:(8)SZ2≤SZm2.Lemma 3 (see [55]).
For any real numbersζ1>0,ζ2>0 and 0<h<1, an extended Lyapunov condition of finite-time stability can be given in the form of fast terminal sliding mode as V˙x+ζ1Vx+ζ2Vhx≤0; then, Vx is in fast finite-time convergent with a finite settling time T∗≤1/ζ11−hlogζ1V1−hx0+ζ2/ζ2.Lemma 4 (see [56]).
Forx,y∈R, if 0<h=h2/h1<1, where h1,h2>0 are odd integers, then xyh≤−ς1x1+h+ς2x+y1+h, where ς1=1/1+h2h−1−2h−1h+1 and ς2=1/1+h1+2h/1+h+2−h−12h+1/1+h2h−1.Lemma 5 (see [34]).
Consider the dynamic system(9)ϕ˙t=−l1ϕt−l2ϕht+ϱt,where ϕt∈R,0<h=h2/h1<1 (h1 and h2 are positive odd integers), l1 and l2 are positive constants, and ϱt is a positive function. Then, for any given bounded initial condition ϕ0≥0, one has that ϕt≥0, ∀t≥0.Lemma 6.
(see [57]). For xi∈R,i=1,2,…,n, and 0<h≤1, then ∑i=1nxih≤∑i=1nxih≤n1−h∑i=1nxih.To obtain the control objective, the following assumptions are needed.Assumption 1.
The unmodeled dynamicsξ is exp-ISpS.Assumption 2.
There exist unknown nonnegative continuous functionsφi1 and nondecreasing continuous functions φi2 such that(10)δiξ,x,t≤φi1x¯i+φi2ξ,∀ξ,x,t∈Rn0×Rn×R+,where φi20=0,i=1,…,n.Remark 2.
From Definition1 and Assumption 1, we have ξ≤α¯1−1Vξ. According to Lemma 1, there exists a positive constant D0 such that ξ≤α¯1−1r+D0,∀t≥0. This inequality will be used to cope with the uncertain terms in the following controller design.Assumption 3.
The sign ofgix¯i is known, and there exist some unknown positive constants ai and bi such that 0<bi≤gix¯i≤ai. Without loss of generality, this paper assumes that gix¯i>0.Assumption 4.
The reference trajectoryyrt and its derivatives about time y˙r and y¨r are in a bounded region Ωd, and there exists a known constant A0, such that yr≤A0<kc1.Assumption 5.
The unknown continuous functionsdix¯it−Tit satisfy the following inequality:(11)dix¯it−Tit≤∑j=1iρijxjt−Tjt,and the time-varying state delays Tit satisfy the inequalities 0≤Tit≤Tmax and T˙it≤T¯max<1, where ρijxjt−Tjt are unknown positive smooth functions and Tmax and T¯max are unknown constants.


## 2.1. Problem Statement

Consider a class of uncertain nonstrict-feedback state-delayed nonlinear systems with unmodeled dynamics fori=1,2,…,n−1 in the following form:(1)ξ˙=qξ,x,t,x˙i=fix+gix¯ixi+1+δiξ,x,t+dix¯it−Tit,x˙n=fnx+gnx¯nu+δnξ,x,t+dnx¯nt−Tnt,y=x1,where x=x1,x2,…,xnT∈Rn is the state vector, ξ∈Rn0 is the unmodeled dynamics, and u,y,Tit denote the system input, the system output, and the unknown time-varying delays, respectively. fix,gix¯i, and dix¯it−Tit are the unknown smooth functions. Let x¯i=x1,x2,…,xiT and δiξ,x,t be the unknown uncertain disturbances. All the states xi are required to remain in the sets Ωxi=xi:xi<kci, where kci are positive constants.Remark 1.
System (1) is called a nonstrict-feedback form in which the system function fi⋅ and its bounding function contain all the state variables [10]. Apparently, strict-feedback and pure-feedback structures are the special cases of system (1). The methods proposed in [21–28, 31–33, 47] cannot be directly applied to system (1) on account of its nonstrict-feedback structure.
The control objective of this paper is to construct an adaptive NN controllerut to make sure that the output y follows the desired trajectory yr in a finite time, while every state xi∈Ωxi is never violated.


## 2.2. RBFNN Approximation

In this paper, fori=1,…,n, the unknown smooth nonlinear functions F¯iZi:Rm⟶R will be approximated on a compact set Ωi⊂Rm by the following RBFNN:(2)F¯iZi=WiTSiZi+εiZi,where Zi,Wi,l denote input vectors, weight vectors, and NN node number, respectively. εiZi are the NN inherent approximation errors which are bounded over the compact sets; that is, εiZi≤εi, where ϵi are unknown constants and SiZi=s1Zi,…,slZiT:Ωi⟶Rl are known smooth vector functions with sqZi being chosen as the commonly used Gaussian functions, which have the form(3)sqZi=exp−Zi−μqTZi−μqηq2,q=1,…,l,where μq=μq1,…,μqmT is the center vector and ηq is the spreads of the Gaussian function. The optimal weight vector Wi is defined as(4)Wi=argminW^i∈RlsupZi∈ΩiFZi−W^iTSZ,where W^i is the estimate of Wi.


## 2.3. Key Definition and Lemmas

Definition 1 (see [21]).
The unmodeled dynamicsξ is said to be exponentially input-state-practically stable (exp-ISpS), that is, for system ξ˙=qξ,x,t, if there exist functions α¯1,α¯2 of class K∞ and a Lyapunov function Vξ, such that(5)α¯1ξ≤Vξ≤α¯2ξ,and there exist two constants c>0,d≥0 and a class K∞ function γ, such that(6)∂Vξ∂ξqξ,x,t≤−cVξ+γx1+d,∀t≥0,where c and d are known positive constants and γ⋅ is a known function of class K∞.Lemma 1 (see [21]).
IfV is an exp-ISpS Lyapunov function for a system ξ˙=qξ,x,t, that is, (5) and (6) hold, then, for any constant c¯∈0,c, any initial instant t0>0, any initial condition ξ0=ξt0,r0>0, and any continuous function γ¯, such that γ¯x1≥γx1, there exist a finite T0=max0,logVξ0/r0/c−c¯≥0, a nonnegative function Dt0,t defined for all t≥t0, and a signal described by(7)r˙=−c¯r+γ¯x1+d,rt0=r0,such that Dt0,t=0 for t≥t0+T0 and Vξ≤rt+Dt0,t with Dt0,t=max0,e−ct−t0Vz0−e−c¯t−t0r0.Lemma 2 (see [11]).
LetSZ be the basis function vector of an RBFNN and Z be the input vector, where SZ=s1Z,…,slZT and Z=z1,…znT. For any positive integer m≤n, let Zm=z1,…,zmT, and the following inequality holds:(8)SZ2≤SZm2.Lemma 3 (see [55]).
For any real numbersζ1>0,ζ2>0 and 0<h<1, an extended Lyapunov condition of finite-time stability can be given in the form of fast terminal sliding mode as V˙x+ζ1Vx+ζ2Vhx≤0; then, Vx is in fast finite-time convergent with a finite settling time T∗≤1/ζ11−hlogζ1V1−hx0+ζ2/ζ2.Lemma 4 (see [56]).
Forx,y∈R, if 0<h=h2/h1<1, where h1,h2>0 are odd integers, then xyh≤−ς1x1+h+ς2x+y1+h, where ς1=1/1+h2h−1−2h−1h+1 and ς2=1/1+h1+2h/1+h+2−h−12h+1/1+h2h−1.Lemma 5 (see [34]).
Consider the dynamic system(9)ϕ˙t=−l1ϕt−l2ϕht+ϱt,where ϕt∈R,0<h=h2/h1<1 (h1 and h2 are positive odd integers), l1 and l2 are positive constants, and ϱt is a positive function. Then, for any given bounded initial condition ϕ0≥0, one has that ϕt≥0, ∀t≥0.Lemma 6.
(see [57]). For xi∈R,i=1,2,…,n, and 0<h≤1, then ∑i=1nxih≤∑i=1nxih≤n1−h∑i=1nxih.To obtain the control objective, the following assumptions are needed.Assumption 1.
The unmodeled dynamicsξ is exp-ISpS.Assumption 2.
There exist unknown nonnegative continuous functionsφi1 and nondecreasing continuous functions φi2 such that(10)δiξ,x,t≤φi1x¯i+φi2ξ,∀ξ,x,t∈Rn0×Rn×R+,where φi20=0,i=1,…,n.Remark 2.
From Definition1 and Assumption 1, we have ξ≤α¯1−1Vξ. According to Lemma 1, there exists a positive constant D0 such that ξ≤α¯1−1r+D0,∀t≥0. This inequality will be used to cope with the uncertain terms in the following controller design.Assumption 3.
The sign ofgix¯i is known, and there exist some unknown positive constants ai and bi such that 0<bi≤gix¯i≤ai. Without loss of generality, this paper assumes that gix¯i>0.Assumption 4.
The reference trajectoryyrt and its derivatives about time y˙r and y¨r are in a bounded region Ωd, and there exists a known constant A0, such that yr≤A0<kc1.Assumption 5.
The unknown continuous functionsdix¯it−Tit satisfy the following inequality:(11)dix¯it−Tit≤∑j=1iρijxjt−Tjt,and the time-varying state delays Tit satisfy the inequalities 0≤Tit≤Tmax and T˙it≤T¯max<1, where ρijxjt−Tjt are unknown positive smooth functions and Tmax and T¯max are unknown constants.


## 3. Adaptive DSC Design and Stability Analysis

### 3.1. Adaptive DSC Design

Similar to traditional backstepping, the backstepping-design procedure withn steps is developed to construct the adaptive neural controller in this part.By using the backstepping technique, the proposed adaptive DSC scheme containsn steps as follows.Step 1.
Define the first surface errorz1=x1−yr; the time derivative of z1 is defined as(12)z˙1=f1x+g1x¯1x2+δ1ξ,x,t+d1x¯1t−T1t−y˙r.The virtual control lawα1 and the update law for ϖ^˙1 are designed as(13)α1=−c1z1−μ1z1kb12−z12−z1ϖ^1S1TΞ1S1Ξ12l1kb12−z12−κ1β1z1,(14)ϖ^˙1=ρ1−σ11ϖ^1−σ12ϖ^1h+z12S1TΞ1S1Ξ12l1kb12−z122,where c1,μ1,kb1,κ1,ρ1,σ11,σ12,l1 are positive design parameters, ϖ^1 is an estimate of ϖ1, ϖ˜1=ϖ1−b1ϖ^1, ϖ1=W12,b1 is defined in Assumption 3. β1z1 is defined as(15)β1z1=z1hkb12−z121−h/2,ifz1≥τ1,ι11z1+ι12z13,ifz1<τ1,where 0<h=h1/h2<1,h1 and h2 are the positive odd integers, ι11=τ1h−1kb12−τ121−h/2−ι12τ12,ι12=1/2τ13h−1τ1hkb12−τ121−h/2+τ12kb12−τ12−1+h/2, and τ1<kb1 is a small positive constant.Consider the BLF candidateVz1 as(16)Vz1=12logkb12kb12−z12+12b1ρ1ϖ˜12.Obviously,Vz1 is positive definite and continuously differentiable. Based on Assumptions 2 and 5 and Young’s inequality, we obtain the time derivative of Vz1 as follows:(17)V˙z1=z1kb12−z12f1x+g1x¯1x2+δ1ξ,x,t+d1x¯1t−T1t−y˙r−1ρ1ϖ˜1ϖ^˙1,≤z1kb12−z12f1x+g1x¯1x2−y˙r+z1kb12−z12φ11x¯1+φ12α¯1−1r+D0+ρ11x1t−T1t−1ρ1ϖ˜1ϖ^˙1,≤z1kb12−z12f1x+g1x¯1x2−y˙r+z12kb12−z122φ11x¯1+φ12α¯1−1r+D02+14,+z122kb12−z122+12ρ112x1t−T1t−1ρ1ϖ˜1ϖ^˙1,≤z1kb12−z12f1x+g1x¯1x2+z1kb12−z12φ11x¯1+φ12α¯1−1r+D02+z12kb12−z12−y˙r,+14+12ρ112x1t−T1t−1ρ1ϖ˜1ϖ^˙1,≤z1kb12−z12F¯1Z1+g1x¯1x2+14+12ρ112x1t−T1t−1ρ1ϖ˜1ϖ^˙1,=z1kb12−z12F¯1Z1+g1x¯1z2+y2+α1+14+12ρ112x1t−T1t−1ρ1ϖ˜1ϖ^˙1,where(18)F¯1Z1=f1x+z1kb12−z12φ11x¯1+φ12α¯1−1r+D02+z12kb12−z12−y˙r.Note thatF¯1Z1 is an unknown continuous function and RBFNN can be used to approximate it. Hence, from (2), the following equation holds:(19)F¯1Z1=W1TS1Z1+ε1Z1,where W1TS1Z1 is an NN, ε1Z1≤ε1,Z1=x¯n,z1,r,y˙rT, and ε1>0 is any given.By using Young’s inequality and Lemma2, one has(20)z1kb12−z12F¯1Z1=z1kb12−z12W1TS1Z1+ε1Z1,≤z1kb12−z12W1S1Z1+ε1Z1z1kb12−z12,≤z1kb12−z12W1S1Ξ1+ε1Z1z1kb12−z12,≤12l1z12kb12−z122W1|2S1Ξ12+l12+ε1z1kb12−z12,=12l1z12kb12−z122ϖ1S1Ξ12+l12+ε1z1kb12−z12,where Ξ1=x1,z1,r,y˙rT.Substituting (13), (14), and (20) into (17), we can obtain(21)V˙z1≤g1x¯1z1z2kb12−z12+g1x¯1z1y2kb12−z12−c1g1x¯1z12kb12−z12−μ1g1x¯1z12kb12−z122−g1x¯1z12ϖ^1S1Ξ122l1kb12−z122+l12,−κ1g1x¯1z1β1z1kb12−z12+z12ϖ1S1Ξ122l1kb12−z122+14+ε1z1kb12−z12+12ρ112x1t−T1t,−1ρ1ϖ˜1ρ1−σ11ϖ^1−σ12ϖ^1h+z12S1TΞ1S1Ξ12l1kb12−z122,≤g1x¯1z1z2kb12−z12+g1x¯1z1y2kb12−z12−b1c1z12kb12−z12−μ1b1z12kb12−z122−z12ϖ1S1Ξ12−z12ϖ˜1S1Ξ122l1kb12−z122,+l12−κ1g1x¯1z1β1z1kb12−z12+z12ϖ1S1Ξ122l1kb12−z122+14+ε1z1kb12−z12+12ρ112x1t−T1t,+σ11ϖ˜1ϖ^1+σ12ϖ˜1ϖ^1h−z12ϖ˜1S1TΞ1S1Ξ12l1kb12−z122,≤g1x¯1z1z2kb12−z12+g1x¯1z1y2kb12−z12−b1c1z12kb12−z12−μ1b1z12kb12−z122+l12−κ1g1x¯1z1β1z1kb12−z12+14,+ε1z1kb12−z12+12ρ112x1t−T1t+σ11ϖ˜1ϖ^1+σ12ϖ˜1ϖ^1h.By utilizing Young’s inequality, the following inequalities can be obtained:(22)g1x¯1z1y2kb12−z12−μ1b1z122kb12−z122≤g12x¯1y222μ1b1≤a12y222μ1b1,ε1z1kb12−z12−μ1b1z122kb12−z122≤ε122μ1b1.Therefore, we have(23)V˙z1≤g1x¯1z1z2kb12−z12+a12y222μ1b1+ε122μ1b1−b1c1z12kb12−z12+l12−κ1g1x¯1z1β1z1kb12−z12+14,+12ρ112x1t−T1t+σ11ϖ˜1ϖ^1+σ12ϖ˜1ϖ^1h.According to the inequality2b1ϖ˜1ϖ^1≤ϖ12−ϖ˜12 and Lemma 4, one as(24)V˙z1≤g1x¯1z1z2kb12−z12+a12y222μ1b1+ε122μ1b1−b1c1z12kb12−z12+l12−κ1g1x¯1z1β1z1kb12−z12+14,+12ρ112x1t−T1t+σ112b1ϖ12−σ112b1ϖ˜12−σ12ζ1b1hϖ˜11+h+σ12ζ2b1hϖ11+h,where ζ1 and ζ2 are defined in Lemma 4.To deal with the time delay in equation (24), define the Lyapunov-Krasovskii functional as follows:(25)VU1=e−γt−Tmax21−T¯max∫t−T1tteγsρ112x1sds,where γ>0 is a positive constant. Using Assumption 5, we obtain that the derivative of VU1 is(26)V˙U1=e−γt−Tmax21−T¯maxeγtρ112x1t−eγt−T1tρ112x1t−T1t1−T˙1t−γVU1,≤eγTmax21−T¯maxρ112x1t−12ρ112x1t−T1t−γVU1.From equations (24) and (26), we have(27)V˙z1+V˙U1≤g1x¯1z1z2kb12−z12+a12y222μ1b1+ε122μ1b1−b1c1z12kb12−z12+l12−κ1g1x¯1z1β1z1kb12−z12+14,+σ112b1ϖ12−σ112b1ϖ˜12−σ12ζ1b1hϖ˜11+h+σ12ζ2b1hϖ11+h+Φ1−γVU1,where Φ1=eγTmax/21−T¯maxρ112x1t.To move on, introduce the coordinate transformation(28)zi=xi−wi,yi=wi−αi−1,where zi,αi−1, and yi denote the tracking error, the virtual control input, and the boundary layer error for i=2,3,…,n, respectively. wi is the output of the following first-order filter:(29)w˙i=−τi1yi−τi2yih,where τi1 and τi2 are the positive design constants and h is defined in (15).Remark 3.
From (29), it can be seen that the proposed filter involves both the linear and fractional terms. In particular, when τi1=0 or τi2=0, filter (29) degrades into the fractional filter used in [58] and the linear filter as widely used in the literature [21–23], respectively. It is the key to ensure the fast finite-time stability of the closed-loop system, which will be detailed in the following analysis.Remark 4.
As mentioned in [34], by designing ι11 and ι12 properly, both the virtual control input α1 and its derivative α˙1 are ensured to be inherently continuous in the set Ωxi. It means that the virtual control input α1 defined in (13) is C1 continuous in the set Ωxi. From (13), it is not hard to see that α1 and its derivative α˙1 are the functions of the variables z1,ϖ^1,yr. and z1,z2,ϖ^˙1,y2,y˙r,y¨r, respectively. Combining the continuity of α˙1 and (28) and (29), it can be seen that there exists a continuous function λ2z1,z2,ϖ^˙1,y2,y˙r,y¨r which satisfies(30)y˙2≤−τ21y2−τ22y2h+λ2z1,z2,ϖ^˙1,y2,y˙r,y¨r.Step 2.
i=2,3,…,n−1 Define the ith surface error zi=xi−wi; the time derivative of zi is defined as(31)z˙i=fix+gix¯ixi+1+δiξ,x,t+dix¯it−Tit−w˙i,=fix+gix¯izi+1+yi+1+αi+δiξ,x,t+dix¯it−Tit−w˙i.
The virtual control lawαi and the update law ϖ^˙i are designed as(32)αi=−cizi−μizikbi2−zi2−ziϖ^iSiTΞiSiΞi2likbi2−zi2−κiβizi,(33)ϖ^˙i=ρi−σi1ϖ^i−σi2ϖ^ih+zi2SiTΞiSiΞi2likbi2−zi22,where ci,μi,kbi,κi,ρi,σi1,σi2,li are positive design parameters, ϖ^i is an estimate of ϖi,ϖ˜i=ϖi−biϖ^i, ϖi=Wi2,βizi is defined as(34)βizi=zihkbi2−zi21−h/2,ifzi≥τi,ιi1zi+ιi2zi3,ifzi<τi,where h is defined in (15), ιi1=τih−1kbi2−τi21−h/2−ιi2τi2,ιi2=1/2τi3h−1τihkbi2−τi21−h/2+τi2kbi2−τi2−1+h/2, and τi<kbi is a small positive constant.
Consider the BLF candidateVzi as(35)Vzi=12logkbi2kbi2−zi2+12biρiϖ˜i2,where Vzi is also positive definite and continuously differentiable in the set zi<kbi. Similar to (17), the time derivative of Vzi is(36)V˙zi≤zikbi2−zi2F¯iZi+gix¯izi+1+yi+1+αi+14−1ρiϖ˜iϖ^˙i,+12∑j=1iρij2xjt−Tjt−gi−1x¯i−1zi−1zikbi−12−zi−12,where(37)F¯iZi=fix+zikbi2−zi2φi1x¯i+φi2α¯i−1r+D02−w˙i+gi−1x¯i−1kbi2−zi2zi−1kbi−12−zi−12+izi2kbi2−zi2.
Note thatF¯iZi is an unknown continuous function and RBFNN can be used to approximate it. Hence, from (2), the following equation holds:(38)F¯iZi=WiTSiZi+εiZi,where WiTSiZi is an NN, εiZi≤εi,Zi=x¯n,zi−1,zi,r,wi−1,w˙iT, and εi>0 is any given.
By using Young’s inequality and Lemma2, one has(39)zikbi2−zi2F¯iZi=zikbi2−zi2WiTSiZi+εiZi,≤zikbi2−zi2WiSiZi+εiZizikbi2−zi2,≤zikbi2−zi2WiSiΞi+εiZizikbi2−zi2,≤12lizi2kbi2−zi22Wi2SiΞi2+li2+εiZizikbi2−zi2.
Substituting (32), (33), and (39) into (36), we can obtain(40)V˙zi≤gix¯izizi+1kbi2−zi2+gix¯iziyi+1kbi2−zi2−cigix¯izi2kbi2−zi2−μigix¯izi2kbi2−zi22−gix¯izi2ϖ^iSiΞi22likbi2−zi22+li2+14,−κigix¯iziβizikbi2−zi2+zi2ϖiSiΞi22likbi2−zi22+εiZizikbi2−zi2+12∑j=1iρij2xjt−Tjt,−gi−1x¯i−1zi−1zikbi−12−zi−12−1ρiϖ˜iρi−σi1ϖ^i−σi2ϖ^ih+zi2SiTΞiSiΞi2likbi2−zi22,≤gix¯izizi+1kbi2−zi2+gix¯iziyi+1kbi2−zi2−bicizi2kbi2−zi2−μibizi2kbi2−zi22−zi2ϖiSiΞi2−zi2ϖ˜iSiΞi22likbi2−zi22,+li2+14−κigix¯iziβizikbi2−zi2+zi2ϖiSiΞi22likbi2−zi22+εiZizikbi2−zi2+12∑j=1iρij2xjt−Tjt,+σi1ϖ˜iϖ^i+σi2ϖ˜iϖ^ih−zi2ϖ˜iSiTΞiSiΞi2likbi2−zi22−gi−1x¯i−1zi−1zikbi−12−zi−12,≤gix¯izizi+1kbi2−zi2+gix¯iziyi+1kbi2−zi2−bicizi2kbi2−zi2−μibizi2kbi2−zi22+li2+14−κigix¯iziβizikbi2−zi2+εiZizikbi2−zi2,+12∑j=1iρij2xjt−Tjt+σi1ϖ˜iϖ^i+σi2ϖ˜iϖ^ih−gi−1x¯i−1zi−1zikbi−12−zi−12.
By utilizing Young’s inequality, the following inequalities can be obtained:(41)gix¯iziyi+1kbi2−zi2−μibizi22kbi2−zi22≤gi2x¯iyi+122μibi≤ai2yi+122μibi,εiZizikbi2−zi2−μibizi22kbi2−zi22≤εi2Zi2μibi≤εi22μibi.
Therefore, we have(42)V˙zi≤gix¯izizi+1kbi2−zi2+ai2yi+122μibi+εi22μibi−bicizi2kbi2−zi2−κigix¯iziβizikbi2−zi2,+12∑j=1iρij2xjt−Tjt+li2+14+σi1ϖ˜iϖ^i+σi2ϖ˜iϖ^ih−gi−1x¯i−1zi−1zikbi−12−zi−12.
According to the inequality2biϖ˜iϖ^i≤ϖi2−ϖ˜i2 and Lemma 4, one has(43)V˙zi≤gix¯izizi+1kbi2−zi2+ai2yi+122μibi+εi22μibi−bicizi2kbi2−zi2−κigix¯iziβzikbi2−zi2,+12∑j=1iρij2xjt−Tjt+li2+14+σi12biϖi2−σi12biϖ˜i2−σi2ζ1bihϖ˜i1+h+σi2ζ2bihϖi1+h−gi−1x¯i−1zi−1zikbi−12−zi−12,where ζ1 and ζ2 are defined in Lemma 4.
To handle the time delay, define the Lyapunov-Krasovskii functional as follows:(44)VUi=e−γt−Tmax21−T¯max∑j=1i∫t−Tjtteγsρij2xjsds,where γ>0 is a positive constant. By using Assumption 5, we obtain that the derivative of VUi is(45)V˙Ui=e−γt−Tmax21−T¯max∑j=1ieγtρij2xjt−eγt−Tjtρij2xjt−Tjt1−T˙jt−γVUi,≤∑j=1ieγTmax21−T¯maxρij2xjt−12∑j=1iρij2xjt−Tjt−γVUi.
From in equations (43) and (45), we have(46)V˙zi+V˙Ui≤gix¯izizi+1kbi2−zi2+ai2yi+122μibi+εi22μibi−bicizi2kbi2−zi2−κigix¯iziβizikbi2−zi2+li2+14+σi12biϖi2,−σi12biϖ˜i2−σi2ζ1bihϖ˜i1+h+σi2ζ2bihϖi1+h−gi−1x¯i−1zi−1zikbi−12−zi−12+Φi−γVUi,where Φi=∑j=1ieγTmax/21−T¯maxρij2xjt.
Similar to the analysis in Remark4, there exists a continuous function λi+1z¯i+1,ϖ^¯i,y2,y3,\\…,yi+1,y˙r,y¨r which satisfies(47)y˙i+1≤−τi+11yi+1−τi+12yi+1h+λi+1z¯i+1,ϖ^¯i,y2,y3,…,yi+1,y˙r,y¨r.Step 3.
Define thenth surface error zn=xn−wn; the time derivative of zn is defined as(48)z˙n=fnx+gnx¯nu+δnξ,x,t+dnx¯nt−Tnt−w˙n.The actual control lawu and the update law ϖ^˙n are designed as(49)u=−cnzn−μnzn2kbn2−zn2−znϖ^nSnTΞnSnΞn2lnkbn2−zn2−κnβnzn,(50)ϖ^˙n=ρn−σn1ϖ^n−σn2ϖ^nh+zn2SnTΞnSnΞn2lnkbn2−zn22,where cn,μn,kbn,κn,ρn,σn1,σn2,ln are positive design parameters, ϖ^n is an estimate of ϖn, ϖ˜n=ϖn−bnϖ^n, ϖn=Wn2,βnzn is defined as(51)βnzn=znhkbn2−zn21−h/2,ifzn≥τn,ιn1zn+ιn2zn3,ifzn<τn,where h is defined in (15), ιn1=τnh−1kbn2−τn21−h/2−ιn2τn2,ιn2=1/2τn3h−1τnhkbn2−τn21−h/2+τn2kbn2−τn2−1+h/2, and τn<kbn is a small positive constant.Consider the BLF candidateVzn as(52)Vzn=12logkbn2kbn2−zn2+12bnρnϖ˜n2.Similar to (17) and (36), we can obtain the time derivative of Vzn as follows:(53)V˙zn≤znkbn2−zn2F¯nZn+gnx¯nu+14+12∑j=1nρnj2xjt−Tjt−1ρnϖ˜nϖ^˙n−gn−1x¯n−1zn−1znkbn−12−zn−12,where(54)F¯nZn=fnx+znkbn2−zn2φn1x¯n+φn2α¯n−1r+D02−w˙n,+gn−1x¯n−1kbn2−zn2zn−1kbn−12−zn−12+nzn2kbn2−zn2.Note thatF¯nZn is an unknown continuous function and RBFNN can be used to approximate it. Hence, from (2), the following equation holds:(55)F¯nZn=WnTSnZn+εnZn,where WnTSnZn is an NN, εnZn≤εn,Zn=x¯n,zn−1,zn,r,wn−1,w˙nT, and εn>0 is any given.By using Young’s inequality and Lemma2, one has(56)znkbn2−zn2F¯nZn=znkbn2−zn2WnTSnZn+εnZn,≤znkbn2−zn2WnSnZn+εnZnznkbn2−zn2,≤znkbn2−zn2WnSnΞn+εnZnznkbn2−zn2,≤12lnzn2kbn2−zn22Wn2SnΞn2+ln2+εnZnznkbn2−zn2,=12lnzn2kbn2−zn22ϖnSnΞn2+ln2+εnZnznkbn2−zn2,where Ξn=Zn=x¯n,zn−1,zn,r,wn−1,w˙nT.Substituting (49), (50), and (56) into (53), we can obtain(57)V˙zn≤−bncnzn2kbn2−zn2−μnbnzn22kbn2−zn22−κngnx¯nznβnznkbn2−zn2+ln2+14+εnZnznkbn2−zn2,+12∑j=1nρnj2xjt−Tjt+σn1ϖ˜nϖ^n+σn2ϖ˜nϖ^nh−gn−1x¯n−1zn−1znkbn−12−zn−12.By utilizing Young’s inequality, the following inequality can be obtained:(58)εnZnznkbn2−zn2−μnbnzn22kbn2−zn22≤εn2Zn2μnbn≤εn22μnbn.Therefore, we have(59)V˙zn≤−bncnzn2kbn2−zn2+εn22μnbn−κngnx¯nznβnznkbn2−zn2+ln2+14+12∑j=1nρnj2xjt−Tjt,+σn1ϖ˜nϖ^n+σn2ϖ˜nϖ^nh−gn−1x¯n−1zn−1znkbn−12−zn−12.According to the inequality2bnϖ˜nϖ^n≤ϖn2−ϖ˜n2 and Lemma 4, one has(60)V˙zn≤−bncnzn2kbn2−zn2+εn22μnbn−κngnx¯nznβnznkbn2−zn2+ln2+14+12∑j=1nρnj2xjt−Tjt,+σn12bnϖn2−σn12bnϖ˜n2−σn2ζ1bnhϖ˜n1+h+σn2ζ2bnhϖn1+h−gn−1x¯n−1zn−1znkbn−12−zn−12,where ζ1 and ζ2 are defined in Lemma 4.To handle the time delay, define the Lyapunov-Krasovskii functional as follows:(61)VUn=e−γt−Tmax21−T¯max∑j=1n∫t−Tjtteγsρnj2xjsds,where γ>0 is a positive constant. By using Assumption 5, we obtain that the derivative of VUn is(62)V˙Un=e−γt−Tmax21−T¯max∑j=1neγtρnj2xjt−eγt−Tjtρnj2xjt−Tjt1−T˙jt−γVUn,≤∑j=1neγTmax21−T¯maxρnj2xjt−12∑j=1nρnj2xjt−Tjt−γVUn.From equations (60) and (62), we have(63)V˙zn+V˙Un≤−bncnzn2kbn2−zn2+εn22μnbn−κngnx¯nznβnznkbn2−zn2+ln2+14+σn12bnϖn2−σn12bnϖ˜n2,−σn2ζ1bnhϖ˜n1+h+σn2ζ2bnhϖn1+h−gn−1x¯n−1zn−1znkbn−12−zn−12+Φn−γVUn,where Φn=∑j=1neγTmax/21−T¯maxρnj2xjt.


### 3.2. Stability Analysis

In this subsection, we present the stability analysis of the resulting closed-loop system. The main results are presented by the following theorem.Theorem 1.
Consider the nonlinear system (1) with Assumptions 1–5. Let the actual control input and the NN adaptive law be designed as (49) and (50), respectively. If the initial conditions satisfy V0≤Δ,zi0≤kbi, in which Δ>kbi is any positive constant for i=1,2,…,n and kbi are properly chosen, such that kc1>kb1+A0 and kci>w¯i+kbi with w¯i=supwi for i=2,3,…,n, one has that all internal signals zi,ϖ˜i and yi+1 in the closed-loop system are semiglobally uniformly ultimately bounded and the tracking error will converge into the arbitrarily small regions in a finite time. Meanwhile, each state xi will remain in the set Ωxi; that is, the full-state constraints are never violated.Proof.
Construct the overall Lyapunov function candidate(64)V=∑i=1nVzi+∑i=1nVUi+∑i=1n−1Vyi,where Vyi=ai2/2biyi+12 and Vzi,VUi are defined in (35) and (44), respectively.
From (28), (29), and (47), the derivative of ∑i=1n−1Vyi is(65)∑i=1n−1V˙yi≤−∑i=1n−1ai2biτi+11yi+12+τi+12yi+11+h+∑i=1n−1ai2biyi+1λi+1z¯i+1,ϖ^¯i,y2,y3,…,yi+1,y˙r,y¨r.
Define a compact set asΩn=z¯n,ϖ^¯n,y2,y3,…,yn:V≤Δ with Δ being a positive constant. If V≤Δ, together with Assumption 4 and (65), it can be obtained that there exists a positive constant Λi+1i=1,2,…,n−1, such that λi+1⋅≤Λi+1 on the compact set Ωn×Ωd. Then, applying Young’s inequality to (65) yields(66)∑i=1n−1V˙yi≤−∑i=1n−1ai2biτi+11−12χi+1yi+12−∑i=1n−1ai2biτi+12yi+11+h+∑i=1n−1ai22biΛi+12χi+1,where χi+1 are positive constants.
According to the above analysis, we can obtain the derivative of the overall Lyapunov function candidateV as(67)V˙≤−∑i=1nbicizi2kbi2−zi2−∑i=1nκigix¯iziβizikbi2−zi2−∑i=1nσi12biϖ˜i2−∑i=1nσi2ζ1bihϖ˜i1+h−∑i=1n−1τ^i+11yi+12,−∑i=1n−1ai2biτi+12yi+11+h−γ∑i=1nVUi+d0,where(68)τ^i+11=ai2biτi+11−12μi−12χi+1,d0=∑i=1nεi22μibi+li2+14+σi12biϖi2+σi2ζ2bihϖi1+h+Φi+∑i=1n−1ai22biΛi+12χi+1.Here, we choose τi+11>1/2μi+1/2χi+1, such that τ^i+11>0.
From the definition ofβizii=1,2,…,n in (15), (34), and (51), the following two cases should be considered.Case 1: Whenzi<τi,i=1,2,…,n, substituting βizi=ιi1zi+ιi2zi3 into (67) gives(69)V˙≤−∑i=1nci+κiιi1bizi2kbi2−zi2−∑i=1nκibiιi2zi4kbi2−zi2−∑i=1nσi12biϖ˜i2−∑i=1nσi2ζ1bihϖ˜i1+h,−∑i=1n−1τ^i+11yi+12−∑i=1n−1ai2biτi+12yi+11+h−γ∑i=1nVUi+d0,≤−∑i=1nci+κiιi1bizi2kbi2−zi2−∑i=1nσi12biϖ˜i2−∑i=1n−1τ^i+11yi+12−γ∑i=1nVUi+d0.Noting (69), we can have(70)V˙≤−υV+d0,withυ=min2b1c1+κ1ι11,…,2bncn+κnιn1,ρ1σ11,…,ρnσn1,γ,2b1τ^2,1/a12,…,2bn−1τ^n,1/an−12, which further implies that all the internal signals are uniformly ultimately bounded.Case 2: Whenzi≥τi,i=1,2,…,n, substituting βizi=zihkbi2−zi21−h/2 into (67) gives(71)V˙≤−∑i=1nbicizi2kbi2−zi2−∑i=1nκigix¯izi1+hkbi2−zi21+h/2−∑i=1nσi12biϖ˜i2−∑i=1nσi2ζ1bihϖ˜i1+h−∑i=1n−1τ^i+11yi+12,−∑i=1n−1ai2biτi+12yi+11+h−γ2∑i=1nVUi−γ2∑i=1nVUi1+h/2+γ21−h21+h21+h/1−h+d0,≤−∑i=1nbicizi2kbi2−zi2−∑i=1nκigix¯izi1+hkbi2−zi21+h/2−∑i=1nσi12biϖ˜i2−∑i=1nσi2ζ1bihϖ˜i1+h−∑i=1n−1τ^i+11yi+12,−∑i=1n−1ai2biτi+12yi+11+h−γ2∑i=1nVUi−γ2n1−h/2∑i=1nVUi1+h/2+γ21−h21+h21+h/1−h+d0,≤−υ1V−υ2V1+h/2+d1,where(72)υ1=min2b1c1,…,2bncn,σ11ρ1,…,σn1ρn,γ2,2b1τ^21a12,…,2bn−1τ^n1an−12,υ2=minb1c121+h/2,…,bncn21+h/2,σ12ζ1b1h2b1ρ1σ111+h/2,…,σn2ζ1bnh2bnρnσn11+h/2,γ2n1−h/2,a12τ22b12b1a121+h/2,…,an−12τn2bn−12bn−1an−121+h/2,d1=d0+γ21−h21+h21+h/1−h.
By virtue of [[59], Th.5.2], there always exists a finite-time t∗, such that V≥2d1/υ21+h/2 for all t∈0,t∗. Thus, for all t∈0,t∗, one has V˙≤−υ1V−υ2/2V1+h/2, and it then comes from Lemma 3 that the fast finite-time stability of the closed-loop system can be ensured with a finite settling time T∗≤2/υ11−hlog2υ1V1−h/20+υ2/υ2. Furthermore, it is readily seen that t∗≤T∗. Therefore, ∀t>T∗,V≤2d1/υ22/1+h. Then, the internal error signals zi,ϖ˜i, and yi+1 will converge into the following compact sets:(73)zi≤kbi1−e−22d1/υ22/1+h1/2,i=1,…,n,ϖ˜i≤2ρibi1/22d1υ21/1+h,i=1,…,n,yi+1≤2biai21/22d1υ21/1+h,i=1,…,n−1,in a finite-time T∗ with T∗≤2/υ11−hlog2υ1V1−h/20+υ2/υ2. It is readily seen that the regions (73) can be made as small as possible by adjusting 2d/υ2 with proper control parameters.
Then, we will prove that the full-state constraints are never violated. According to [[60], Lemma 1], we can conclude from (70) and (71) that zi≤kbi,i=1,…,n, for all t≥0. Noting that yr≤A0 from Assumption 4 and z1=x1−yr, we have that x1≤kb1+A0. To get x2≤kc2, we need to show the boundedness of w2. From (73), one has that y2 is bounded and b1ϖ^1=ϖ1−ϖ˜1 is also bounded. With the proper choices of ι1 and ι2, α1 is a continuous function of ϖ^1,x1, and y˙r. Then, there exists an upper bound w¯2 , such that w2=y2+α1≤w¯2. From z2=x2−w2 and z2<kb2, we get that x2≤z2+w2≤kc2. Similarly and iteratively, we have that αi−1and yi for i=3,…,n are bounded, which together with zi<kbi ensures that xi≤kci,i=3,…,n. Therefore, each state xi,i=3,…,n will remain in the set Ωxi. The proof is completed.


## 3.1. Adaptive DSC Design

Similar to traditional backstepping, the backstepping-design procedure withn steps is developed to construct the adaptive neural controller in this part.By using the backstepping technique, the proposed adaptive DSC scheme containsn steps as follows.Step 1.
Define the first surface errorz1=x1−yr; the time derivative of z1 is defined as(12)z˙1=f1x+g1x¯1x2+δ1ξ,x,t+d1x¯1t−T1t−y˙r.The virtual control lawα1 and the update law for ϖ^˙1 are designed as(13)α1=−c1z1−μ1z1kb12−z12−z1ϖ^1S1TΞ1S1Ξ12l1kb12−z12−κ1β1z1,(14)ϖ^˙1=ρ1−σ11ϖ^1−σ12ϖ^1h+z12S1TΞ1S1Ξ12l1kb12−z122,where c1,μ1,kb1,κ1,ρ1,σ11,σ12,l1 are positive design parameters, ϖ^1 is an estimate of ϖ1, ϖ˜1=ϖ1−b1ϖ^1, ϖ1=W12,b1 is defined in Assumption 3. β1z1 is defined as(15)β1z1=z1hkb12−z121−h/2,ifz1≥τ1,ι11z1+ι12z13,ifz1<τ1,where 0<h=h1/h2<1,h1 and h2 are the positive odd integers, ι11=τ1h−1kb12−τ121−h/2−ι12τ12,ι12=1/2τ13h−1τ1hkb12−τ121−h/2+τ12kb12−τ12−1+h/2, and τ1<kb1 is a small positive constant.Consider the BLF candidateVz1 as(16)Vz1=12logkb12kb12−z12+12b1ρ1ϖ˜12.Obviously,Vz1 is positive definite and continuously differentiable. Based on Assumptions 2 and 5 and Young’s inequality, we obtain the time derivative of Vz1 as follows:(17)V˙z1=z1kb12−z12f1x+g1x¯1x2+δ1ξ,x,t+d1x¯1t−T1t−y˙r−1ρ1ϖ˜1ϖ^˙1,≤z1kb12−z12f1x+g1x¯1x2−y˙r+z1kb12−z12φ11x¯1+φ12α¯1−1r+D0+ρ11x1t−T1t−1ρ1ϖ˜1ϖ^˙1,≤z1kb12−z12f1x+g1x¯1x2−y˙r+z12kb12−z122φ11x¯1+φ12α¯1−1r+D02+14,+z122kb12−z122+12ρ112x1t−T1t−1ρ1ϖ˜1ϖ^˙1,≤z1kb12−z12f1x+g1x¯1x2+z1kb12−z12φ11x¯1+φ12α¯1−1r+D02+z12kb12−z12−y˙r,+14+12ρ112x1t−T1t−1ρ1ϖ˜1ϖ^˙1,≤z1kb12−z12F¯1Z1+g1x¯1x2+14+12ρ112x1t−T1t−1ρ1ϖ˜1ϖ^˙1,=z1kb12−z12F¯1Z1+g1x¯1z2+y2+α1+14+12ρ112x1t−T1t−1ρ1ϖ˜1ϖ^˙1,where(18)F¯1Z1=f1x+z1kb12−z12φ11x¯1+φ12α¯1−1r+D02+z12kb12−z12−y˙r.Note thatF¯1Z1 is an unknown continuous function and RBFNN can be used to approximate it. Hence, from (2), the following equation holds:(19)F¯1Z1=W1TS1Z1+ε1Z1,where W1TS1Z1 is an NN, ε1Z1≤ε1,Z1=x¯n,z1,r,y˙rT, and ε1>0 is any given.By using Young’s inequality and Lemma2, one has(20)z1kb12−z12F¯1Z1=z1kb12−z12W1TS1Z1+ε1Z1,≤z1kb12−z12W1S1Z1+ε1Z1z1kb12−z12,≤z1kb12−z12W1S1Ξ1+ε1Z1z1kb12−z12,≤12l1z12kb12−z122W1|2S1Ξ12+l12+ε1z1kb12−z12,=12l1z12kb12−z122ϖ1S1Ξ12+l12+ε1z1kb12−z12,where Ξ1=x1,z1,r,y˙rT.Substituting (13), (14), and (20) into (17), we can obtain(21)V˙z1≤g1x¯1z1z2kb12−z12+g1x¯1z1y2kb12−z12−c1g1x¯1z12kb12−z12−μ1g1x¯1z12kb12−z122−g1x¯1z12ϖ^1S1Ξ122l1kb12−z122+l12,−κ1g1x¯1z1β1z1kb12−z12+z12ϖ1S1Ξ122l1kb12−z122+14+ε1z1kb12−z12+12ρ112x1t−T1t,−1ρ1ϖ˜1ρ1−σ11ϖ^1−σ12ϖ^1h+z12S1TΞ1S1Ξ12l1kb12−z122,≤g1x¯1z1z2kb12−z12+g1x¯1z1y2kb12−z12−b1c1z12kb12−z12−μ1b1z12kb12−z122−z12ϖ1S1Ξ12−z12ϖ˜1S1Ξ122l1kb12−z122,+l12−κ1g1x¯1z1β1z1kb12−z12+z12ϖ1S1Ξ122l1kb12−z122+14+ε1z1kb12−z12+12ρ112x1t−T1t,+σ11ϖ˜1ϖ^1+σ12ϖ˜1ϖ^1h−z12ϖ˜1S1TΞ1S1Ξ12l1kb12−z122,≤g1x¯1z1z2kb12−z12+g1x¯1z1y2kb12−z12−b1c1z12kb12−z12−μ1b1z12kb12−z122+l12−κ1g1x¯1z1β1z1kb12−z12+14,+ε1z1kb12−z12+12ρ112x1t−T1t+σ11ϖ˜1ϖ^1+σ12ϖ˜1ϖ^1h.By utilizing Young’s inequality, the following inequalities can be obtained:(22)g1x¯1z1y2kb12−z12−μ1b1z122kb12−z122≤g12x¯1y222μ1b1≤a12y222μ1b1,ε1z1kb12−z12−μ1b1z122kb12−z122≤ε122μ1b1.Therefore, we have(23)V˙z1≤g1x¯1z1z2kb12−z12+a12y222μ1b1+ε122μ1b1−b1c1z12kb12−z12+l12−κ1g1x¯1z1β1z1kb12−z12+14,+12ρ112x1t−T1t+σ11ϖ˜1ϖ^1+σ12ϖ˜1ϖ^1h.According to the inequality2b1ϖ˜1ϖ^1≤ϖ12−ϖ˜12 and Lemma 4, one as(24)V˙z1≤g1x¯1z1z2kb12−z12+a12y222μ1b1+ε122μ1b1−b1c1z12kb12−z12+l12−κ1g1x¯1z1β1z1kb12−z12+14,+12ρ112x1t−T1t+σ112b1ϖ12−σ112b1ϖ˜12−σ12ζ1b1hϖ˜11+h+σ12ζ2b1hϖ11+h,where ζ1 and ζ2 are defined in Lemma 4.To deal with the time delay in equation (24), define the Lyapunov-Krasovskii functional as follows:(25)VU1=e−γt−Tmax21−T¯max∫t−T1tteγsρ112x1sds,where γ>0 is a positive constant. Using Assumption 5, we obtain that the derivative of VU1 is(26)V˙U1=e−γt−Tmax21−T¯maxeγtρ112x1t−eγt−T1tρ112x1t−T1t1−T˙1t−γVU1,≤eγTmax21−T¯maxρ112x1t−12ρ112x1t−T1t−γVU1.From equations (24) and (26), we have(27)V˙z1+V˙U1≤g1x¯1z1z2kb12−z12+a12y222μ1b1+ε122μ1b1−b1c1z12kb12−z12+l12−κ1g1x¯1z1β1z1kb12−z12+14,+σ112b1ϖ12−σ112b1ϖ˜12−σ12ζ1b1hϖ˜11+h+σ12ζ2b1hϖ11+h+Φ1−γVU1,where Φ1=eγTmax/21−T¯maxρ112x1t.To move on, introduce the coordinate transformation(28)zi=xi−wi,yi=wi−αi−1,where zi,αi−1, and yi denote the tracking error, the virtual control input, and the boundary layer error for i=2,3,…,n, respectively. wi is the output of the following first-order filter:(29)w˙i=−τi1yi−τi2yih,where τi1 and τi2 are the positive design constants and h is defined in (15).Remark 3.
From (29), it can be seen that the proposed filter involves both the linear and fractional terms. In particular, when τi1=0 or τi2=0, filter (29) degrades into the fractional filter used in [58] and the linear filter as widely used in the literature [21–23], respectively. It is the key to ensure the fast finite-time stability of the closed-loop system, which will be detailed in the following analysis.Remark 4.
As mentioned in [34], by designing ι11 and ι12 properly, both the virtual control input α1 and its derivative α˙1 are ensured to be inherently continuous in the set Ωxi. It means that the virtual control input α1 defined in (13) is C1 continuous in the set Ωxi. From (13), it is not hard to see that α1 and its derivative α˙1 are the functions of the variables z1,ϖ^1,yr. and z1,z2,ϖ^˙1,y2,y˙r,y¨r, respectively. Combining the continuity of α˙1 and (28) and (29), it can be seen that there exists a continuous function λ2z1,z2,ϖ^˙1,y2,y˙r,y¨r which satisfies(30)y˙2≤−τ21y2−τ22y2h+λ2z1,z2,ϖ^˙1,y2,y˙r,y¨r.Step 2.
i=2,3,…,n−1 Define the ith surface error zi=xi−wi; the time derivative of zi is defined as(31)z˙i=fix+gix¯ixi+1+δiξ,x,t+dix¯it−Tit−w˙i,=fix+gix¯izi+1+yi+1+αi+δiξ,x,t+dix¯it−Tit−w˙i.
The virtual control lawαi and the update law ϖ^˙i are designed as(32)αi=−cizi−μizikbi2−zi2−ziϖ^iSiTΞiSiΞi2likbi2−zi2−κiβizi,(33)ϖ^˙i=ρi−σi1ϖ^i−σi2ϖ^ih+zi2SiTΞiSiΞi2likbi2−zi22,where ci,μi,kbi,κi,ρi,σi1,σi2,li are positive design parameters, ϖ^i is an estimate of ϖi,ϖ˜i=ϖi−biϖ^i, ϖi=Wi2,βizi is defined as(34)βizi=zihkbi2−zi21−h/2,ifzi≥τi,ιi1zi+ιi2zi3,ifzi<τi,where h is defined in (15), ιi1=τih−1kbi2−τi21−h/2−ιi2τi2,ιi2=1/2τi3h−1τihkbi2−τi21−h/2+τi2kbi2−τi2−1+h/2, and τi<kbi is a small positive constant.
Consider the BLF candidateVzi as(35)Vzi=12logkbi2kbi2−zi2+12biρiϖ˜i2,where Vzi is also positive definite and continuously differentiable in the set zi<kbi. Similar to (17), the time derivative of Vzi is(36)V˙zi≤zikbi2−zi2F¯iZi+gix¯izi+1+yi+1+αi+14−1ρiϖ˜iϖ^˙i,+12∑j=1iρij2xjt−Tjt−gi−1x¯i−1zi−1zikbi−12−zi−12,where(37)F¯iZi=fix+zikbi2−zi2φi1x¯i+φi2α¯i−1r+D02−w˙i+gi−1x¯i−1kbi2−zi2zi−1kbi−12−zi−12+izi2kbi2−zi2.
Note thatF¯iZi is an unknown continuous function and RBFNN can be used to approximate it. Hence, from (2), the following equation holds:(38)F¯iZi=WiTSiZi+εiZi,where WiTSiZi is an NN, εiZi≤εi,Zi=x¯n,zi−1,zi,r,wi−1,w˙iT, and εi>0 is any given.
By using Young’s inequality and Lemma2, one has(39)zikbi2−zi2F¯iZi=zikbi2−zi2WiTSiZi+εiZi,≤zikbi2−zi2WiSiZi+εiZizikbi2−zi2,≤zikbi2−zi2WiSiΞi+εiZizikbi2−zi2,≤12lizi2kbi2−zi22Wi2SiΞi2+li2+εiZizikbi2−zi2.
Substituting (32), (33), and (39) into (36), we can obtain(40)V˙zi≤gix¯izizi+1kbi2−zi2+gix¯iziyi+1kbi2−zi2−cigix¯izi2kbi2−zi2−μigix¯izi2kbi2−zi22−gix¯izi2ϖ^iSiΞi22likbi2−zi22+li2+14,−κigix¯iziβizikbi2−zi2+zi2ϖiSiΞi22likbi2−zi22+εiZizikbi2−zi2+12∑j=1iρij2xjt−Tjt,−gi−1x¯i−1zi−1zikbi−12−zi−12−1ρiϖ˜iρi−σi1ϖ^i−σi2ϖ^ih+zi2SiTΞiSiΞi2likbi2−zi22,≤gix¯izizi+1kbi2−zi2+gix¯iziyi+1kbi2−zi2−bicizi2kbi2−zi2−μibizi2kbi2−zi22−zi2ϖiSiΞi2−zi2ϖ˜iSiΞi22likbi2−zi22,+li2+14−κigix¯iziβizikbi2−zi2+zi2ϖiSiΞi22likbi2−zi22+εiZizikbi2−zi2+12∑j=1iρij2xjt−Tjt,+σi1ϖ˜iϖ^i+σi2ϖ˜iϖ^ih−zi2ϖ˜iSiTΞiSiΞi2likbi2−zi22−gi−1x¯i−1zi−1zikbi−12−zi−12,≤gix¯izizi+1kbi2−zi2+gix¯iziyi+1kbi2−zi2−bicizi2kbi2−zi2−μibizi2kbi2−zi22+li2+14−κigix¯iziβizikbi2−zi2+εiZizikbi2−zi2,+12∑j=1iρij2xjt−Tjt+σi1ϖ˜iϖ^i+σi2ϖ˜iϖ^ih−gi−1x¯i−1zi−1zikbi−12−zi−12.
By utilizing Young’s inequality, the following inequalities can be obtained:(41)gix¯iziyi+1kbi2−zi2−μibizi22kbi2−zi22≤gi2x¯iyi+122μibi≤ai2yi+122μibi,εiZizikbi2−zi2−μibizi22kbi2−zi22≤εi2Zi2μibi≤εi22μibi.
Therefore, we have(42)V˙zi≤gix¯izizi+1kbi2−zi2+ai2yi+122μibi+εi22μibi−bicizi2kbi2−zi2−κigix¯iziβizikbi2−zi2,+12∑j=1iρij2xjt−Tjt+li2+14+σi1ϖ˜iϖ^i+σi2ϖ˜iϖ^ih−gi−1x¯i−1zi−1zikbi−12−zi−12.
According to the inequality2biϖ˜iϖ^i≤ϖi2−ϖ˜i2 and Lemma 4, one has(43)V˙zi≤gix¯izizi+1kbi2−zi2+ai2yi+122μibi+εi22μibi−bicizi2kbi2−zi2−κigix¯iziβzikbi2−zi2,+12∑j=1iρij2xjt−Tjt+li2+14+σi12biϖi2−σi12biϖ˜i2−σi2ζ1bihϖ˜i1+h+σi2ζ2bihϖi1+h−gi−1x¯i−1zi−1zikbi−12−zi−12,where ζ1 and ζ2 are defined in Lemma 4.
To handle the time delay, define the Lyapunov-Krasovskii functional as follows:(44)VUi=e−γt−Tmax21−T¯max∑j=1i∫t−Tjtteγsρij2xjsds,where γ>0 is a positive constant. By using Assumption 5, we obtain that the derivative of VUi is(45)V˙Ui=e−γt−Tmax21−T¯max∑j=1ieγtρij2xjt−eγt−Tjtρij2xjt−Tjt1−T˙jt−γVUi,≤∑j=1ieγTmax21−T¯maxρij2xjt−12∑j=1iρij2xjt−Tjt−γVUi.
From in equations (43) and (45), we have(46)V˙zi+V˙Ui≤gix¯izizi+1kbi2−zi2+ai2yi+122μibi+εi22μibi−bicizi2kbi2−zi2−κigix¯iziβizikbi2−zi2+li2+14+σi12biϖi2,−σi12biϖ˜i2−σi2ζ1bihϖ˜i1+h+σi2ζ2bihϖi1+h−gi−1x¯i−1zi−1zikbi−12−zi−12+Φi−γVUi,where Φi=∑j=1ieγTmax/21−T¯maxρij2xjt.
Similar to the analysis in Remark4, there exists a continuous function λi+1z¯i+1,ϖ^¯i,y2,y3,\\…,yi+1,y˙r,y¨r which satisfies(47)y˙i+1≤−τi+11yi+1−τi+12yi+1h+λi+1z¯i+1,ϖ^¯i,y2,y3,…,yi+1,y˙r,y¨r.Step 3.
Define thenth surface error zn=xn−wn; the time derivative of zn is defined as(48)z˙n=fnx+gnx¯nu+δnξ,x,t+dnx¯nt−Tnt−w˙n.The actual control lawu and the update law ϖ^˙n are designed as(49)u=−cnzn−μnzn2kbn2−zn2−znϖ^nSnTΞnSnΞn2lnkbn2−zn2−κnβnzn,(50)ϖ^˙n=ρn−σn1ϖ^n−σn2ϖ^nh+zn2SnTΞnSnΞn2lnkbn2−zn22,where cn,μn,kbn,κn,ρn,σn1,σn2,ln are positive design parameters, ϖ^n is an estimate of ϖn, ϖ˜n=ϖn−bnϖ^n, ϖn=Wn2,βnzn is defined as(51)βnzn=znhkbn2−zn21−h/2,ifzn≥τn,ιn1zn+ιn2zn3,ifzn<τn,where h is defined in (15), ιn1=τnh−1kbn2−τn21−h/2−ιn2τn2,ιn2=1/2τn3h−1τnhkbn2−τn21−h/2+τn2kbn2−τn2−1+h/2, and τn<kbn is a small positive constant.Consider the BLF candidateVzn as(52)Vzn=12logkbn2kbn2−zn2+12bnρnϖ˜n2.Similar to (17) and (36), we can obtain the time derivative of Vzn as follows:(53)V˙zn≤znkbn2−zn2F¯nZn+gnx¯nu+14+12∑j=1nρnj2xjt−Tjt−1ρnϖ˜nϖ^˙n−gn−1x¯n−1zn−1znkbn−12−zn−12,where(54)F¯nZn=fnx+znkbn2−zn2φn1x¯n+φn2α¯n−1r+D02−w˙n,+gn−1x¯n−1kbn2−zn2zn−1kbn−12−zn−12+nzn2kbn2−zn2.Note thatF¯nZn is an unknown continuous function and RBFNN can be used to approximate it. Hence, from (2), the following equation holds:(55)F¯nZn=WnTSnZn+εnZn,where WnTSnZn is an NN, εnZn≤εn,Zn=x¯n,zn−1,zn,r,wn−1,w˙nT, and εn>0 is any given.By using Young’s inequality and Lemma2, one has(56)znkbn2−zn2F¯nZn=znkbn2−zn2WnTSnZn+εnZn,≤znkbn2−zn2WnSnZn+εnZnznkbn2−zn2,≤znkbn2−zn2WnSnΞn+εnZnznkbn2−zn2,≤12lnzn2kbn2−zn22Wn2SnΞn2+ln2+εnZnznkbn2−zn2,=12lnzn2kbn2−zn22ϖnSnΞn2+ln2+εnZnznkbn2−zn2,where Ξn=Zn=x¯n,zn−1,zn,r,wn−1,w˙nT.Substituting (49), (50), and (56) into (53), we can obtain(57)V˙zn≤−bncnzn2kbn2−zn2−μnbnzn22kbn2−zn22−κngnx¯nznβnznkbn2−zn2+ln2+14+εnZnznkbn2−zn2,+12∑j=1nρnj2xjt−Tjt+σn1ϖ˜nϖ^n+σn2ϖ˜nϖ^nh−gn−1x¯n−1zn−1znkbn−12−zn−12.By utilizing Young’s inequality, the following inequality can be obtained:(58)εnZnznkbn2−zn2−μnbnzn22kbn2−zn22≤εn2Zn2μnbn≤εn22μnbn.Therefore, we have(59)V˙zn≤−bncnzn2kbn2−zn2+εn22μnbn−κngnx¯nznβnznkbn2−zn2+ln2+14+12∑j=1nρnj2xjt−Tjt,+σn1ϖ˜nϖ^n+σn2ϖ˜nϖ^nh−gn−1x¯n−1zn−1znkbn−12−zn−12.According to the inequality2bnϖ˜nϖ^n≤ϖn2−ϖ˜n2 and Lemma 4, one has(60)V˙zn≤−bncnzn2kbn2−zn2+εn22μnbn−κngnx¯nznβnznkbn2−zn2+ln2+14+12∑j=1nρnj2xjt−Tjt,+σn12bnϖn2−σn12bnϖ˜n2−σn2ζ1bnhϖ˜n1+h+σn2ζ2bnhϖn1+h−gn−1x¯n−1zn−1znkbn−12−zn−12,where ζ1 and ζ2 are defined in Lemma 4.To handle the time delay, define the Lyapunov-Krasovskii functional as follows:(61)VUn=e−γt−Tmax21−T¯max∑j=1n∫t−Tjtteγsρnj2xjsds,where γ>0 is a positive constant. By using Assumption 5, we obtain that the derivative of VUn is(62)V˙Un=e−γt−Tmax21−T¯max∑j=1neγtρnj2xjt−eγt−Tjtρnj2xjt−Tjt1−T˙jt−γVUn,≤∑j=1neγTmax21−T¯maxρnj2xjt−12∑j=1nρnj2xjt−Tjt−γVUn.From equations (60) and (62), we have(63)V˙zn+V˙Un≤−bncnzn2kbn2−zn2+εn22μnbn−κngnx¯nznβnznkbn2−zn2+ln2+14+σn12bnϖn2−σn12bnϖ˜n2,−σn2ζ1bnhϖ˜n1+h+σn2ζ2bnhϖn1+h−gn−1x¯n−1zn−1znkbn−12−zn−12+Φn−γVUn,where Φn=∑j=1neγTmax/21−T¯maxρnj2xjt.


## 3.2. Stability Analysis

In this subsection, we present the stability analysis of the resulting closed-loop system. The main results are presented by the following theorem.Theorem 1.
Consider the nonlinear system (1) with Assumptions 1–5. Let the actual control input and the NN adaptive law be designed as (49) and (50), respectively. If the initial conditions satisfy V0≤Δ,zi0≤kbi, in which Δ>kbi is any positive constant for i=1,2,…,n and kbi are properly chosen, such that kc1>kb1+A0 and kci>w¯i+kbi with w¯i=supwi for i=2,3,…,n, one has that all internal signals zi,ϖ˜i and yi+1 in the closed-loop system are semiglobally uniformly ultimately bounded and the tracking error will converge into the arbitrarily small regions in a finite time. Meanwhile, each state xi will remain in the set Ωxi; that is, the full-state constraints are never violated.Proof.
Construct the overall Lyapunov function candidate(64)V=∑i=1nVzi+∑i=1nVUi+∑i=1n−1Vyi,where Vyi=ai2/2biyi+12 and Vzi,VUi are defined in (35) and (44), respectively.
From (28), (29), and (47), the derivative of ∑i=1n−1Vyi is(65)∑i=1n−1V˙yi≤−∑i=1n−1ai2biτi+11yi+12+τi+12yi+11+h+∑i=1n−1ai2biyi+1λi+1z¯i+1,ϖ^¯i,y2,y3,…,yi+1,y˙r,y¨r.
Define a compact set asΩn=z¯n,ϖ^¯n,y2,y3,…,yn:V≤Δ with Δ being a positive constant. If V≤Δ, together with Assumption 4 and (65), it can be obtained that there exists a positive constant Λi+1i=1,2,…,n−1, such that λi+1⋅≤Λi+1 on the compact set Ωn×Ωd. Then, applying Young’s inequality to (65) yields(66)∑i=1n−1V˙yi≤−∑i=1n−1ai2biτi+11−12χi+1yi+12−∑i=1n−1ai2biτi+12yi+11+h+∑i=1n−1ai22biΛi+12χi+1,where χi+1 are positive constants.
According to the above analysis, we can obtain the derivative of the overall Lyapunov function candidateV as(67)V˙≤−∑i=1nbicizi2kbi2−zi2−∑i=1nκigix¯iziβizikbi2−zi2−∑i=1nσi12biϖ˜i2−∑i=1nσi2ζ1bihϖ˜i1+h−∑i=1n−1τ^i+11yi+12,−∑i=1n−1ai2biτi+12yi+11+h−γ∑i=1nVUi+d0,where(68)τ^i+11=ai2biτi+11−12μi−12χi+1,d0=∑i=1nεi22μibi+li2+14+σi12biϖi2+σi2ζ2bihϖi1+h+Φi+∑i=1n−1ai22biΛi+12χi+1.Here, we choose τi+11>1/2μi+1/2χi+1, such that τ^i+11>0.
From the definition ofβizii=1,2,…,n in (15), (34), and (51), the following two cases should be considered.Case 1: Whenzi<τi,i=1,2,…,n, substituting βizi=ιi1zi+ιi2zi3 into (67) gives(69)V˙≤−∑i=1nci+κiιi1bizi2kbi2−zi2−∑i=1nκibiιi2zi4kbi2−zi2−∑i=1nσi12biϖ˜i2−∑i=1nσi2ζ1bihϖ˜i1+h,−∑i=1n−1τ^i+11yi+12−∑i=1n−1ai2biτi+12yi+11+h−γ∑i=1nVUi+d0,≤−∑i=1nci+κiιi1bizi2kbi2−zi2−∑i=1nσi12biϖ˜i2−∑i=1n−1τ^i+11yi+12−γ∑i=1nVUi+d0.Noting (69), we can have(70)V˙≤−υV+d0,withυ=min2b1c1+κ1ι11,…,2bncn+κnιn1,ρ1σ11,…,ρnσn1,γ,2b1τ^2,1/a12,…,2bn−1τ^n,1/an−12, which further implies that all the internal signals are uniformly ultimately bounded.Case 2: Whenzi≥τi,i=1,2,…,n, substituting βizi=zihkbi2−zi21−h/2 into (67) gives(71)V˙≤−∑i=1nbicizi2kbi2−zi2−∑i=1nκigix¯izi1+hkbi2−zi21+h/2−∑i=1nσi12biϖ˜i2−∑i=1nσi2ζ1bihϖ˜i1+h−∑i=1n−1τ^i+11yi+12,−∑i=1n−1ai2biτi+12yi+11+h−γ2∑i=1nVUi−γ2∑i=1nVUi1+h/2+γ21−h21+h21+h/1−h+d0,≤−∑i=1nbicizi2kbi2−zi2−∑i=1nκigix¯izi1+hkbi2−zi21+h/2−∑i=1nσi12biϖ˜i2−∑i=1nσi2ζ1bihϖ˜i1+h−∑i=1n−1τ^i+11yi+12,−∑i=1n−1ai2biτi+12yi+11+h−γ2∑i=1nVUi−γ2n1−h/2∑i=1nVUi1+h/2+γ21−h21+h21+h/1−h+d0,≤−υ1V−υ2V1+h/2+d1,where(72)υ1=min2b1c1,…,2bncn,σ11ρ1,…,σn1ρn,γ2,2b1τ^21a12,…,2bn−1τ^n1an−12,υ2=minb1c121+h/2,…,bncn21+h/2,σ12ζ1b1h2b1ρ1σ111+h/2,…,σn2ζ1bnh2bnρnσn11+h/2,γ2n1−h/2,a12τ22b12b1a121+h/2,…,an−12τn2bn−12bn−1an−121+h/2,d1=d0+γ21−h21+h21+h/1−h.
By virtue of [[59], Th.5.2], there always exists a finite-time t∗, such that V≥2d1/υ21+h/2 for all t∈0,t∗. Thus, for all t∈0,t∗, one has V˙≤−υ1V−υ2/2V1+h/2, and it then comes from Lemma 3 that the fast finite-time stability of the closed-loop system can be ensured with a finite settling time T∗≤2/υ11−hlog2υ1V1−h/20+υ2/υ2. Furthermore, it is readily seen that t∗≤T∗. Therefore, ∀t>T∗,V≤2d1/υ22/1+h. Then, the internal error signals zi,ϖ˜i, and yi+1 will converge into the following compact sets:(73)zi≤kbi1−e−22d1/υ22/1+h1/2,i=1,…,n,ϖ˜i≤2ρibi1/22d1υ21/1+h,i=1,…,n,yi+1≤2biai21/22d1υ21/1+h,i=1,…,n−1,in a finite-time T∗ with T∗≤2/υ11−hlog2υ1V1−h/20+υ2/υ2. It is readily seen that the regions (73) can be made as small as possible by adjusting 2d/υ2 with proper control parameters.
Then, we will prove that the full-state constraints are never violated. According to [[60], Lemma 1], we can conclude from (70) and (71) that zi≤kbi,i=1,…,n, for all t≥0. Noting that yr≤A0 from Assumption 4 and z1=x1−yr, we have that x1≤kb1+A0. To get x2≤kc2, we need to show the boundedness of w2. From (73), one has that y2 is bounded and b1ϖ^1=ϖ1−ϖ˜1 is also bounded. With the proper choices of ι1 and ι2, α1 is a continuous function of ϖ^1,x1, and y˙r. Then, there exists an upper bound w¯2 , such that w2=y2+α1≤w¯2. From z2=x2−w2 and z2<kb2, we get that x2≤z2+w2≤kc2. Similarly and iteratively, we have that αi−1and yi for i=3,…,n are bounded, which together with zi<kbi ensures that xi≤kci,i=3,…,n. Therefore, each state xi,i=3,…,n will remain in the set Ωxi. The proof is completed.


## 4. Simulation Results

Example 1.
Consider the following nonlinear system:(74)ξ˙=−ξ+0.5x12sinx1t,x˙1=x2e−0.5x1+1+x12x2+δ1ξ,x1,x2,t+2x12t−T1t,x˙2=x1x22+2.5ut+δ2ξ,x1,x2,t+0.2x22t−T2t,y=x1,where δ1ξ,x1,x2,t=0.2ξx1sinx2t,δ2ξ,x1,x2,t=0.1ξcos0.5x2t,T1t=0.24+sint,T2=4+0.5sint, and the dynamic signal r˙=−r+2.5x14+0.625. The desired tracking trajectory yr=sin0.5t. u is the control input. The design parameters of the controller are taken as kb1=0.4,kb2=2,c1=10,c2=15,τ1=τ2=0.01,σ11=σ12=0.01,ρ1=ρ2=50,l1=l2=1,T=0.001,μ1=μ2=3,κ1=κ2=0.5,h=0.6. There are 68 nodes with the center placed on −2,2×−2,2×−2,2×−2,2 and the width of Gaussian functions is η1=1 in the first RBF vector. There are 85 nodes with the center placed on −2,2×−2,2×−2,2×−2,2×−2,2 and the width of Gaussian functions is η2=1 in the second RBF vector. With the initial conditions, x10=0.2,x20=0.1,w10=0.1,ϖ^1=2,ϖ^2=0.5,r0=0.1. Simulation results are shown in Figures 1–6. The profiles of the system output y and the desired signal yr are shown in Figure 1, which indicates that the output y follows the specified desire trajectory yr. From Figure 2, we know that all state constraints are not violated.Figure 1
Outputy and desired trajectory yr.Figure 2
Phase portrait of statesx1 and x2.Figure 3
Profiles of control inputsu and α1.
(a)(b)Figure 4
Profiles of the tracking errorsz1 and z2.
(a)(b)Figure 5
Estimated parametersϖ^1 and ϖ^2.Figure 6
Profiles of the boundary layer errory2.Example 2.
A Spring-Mass-Damper system is provided in this part. The system model is as follows:(75)p˙=V,MV˙=−KP−CV+F,where P,VandF are the position, the velocity, and the force applied to the object, respectively. Let x1=P,x2=V,u=F. Assuming that the controlled system (75) gives unmodeled dynamics and time delay, let δ1ξ,x1,x2,t=0.2ξx1sinx2t,δ2ξ,x1,x2,t=0.1ξcos0.5x2t,T1t=0.24+sint,T2=4+0.5sint, and the dynamic signal r˙=−r+2.5x14+0.625. Then, system (75) can be rewritten as(76)ξ˙=−ξ+0.5x12sinx1t,x˙1=x2+δ1ξ,x1,x2,t+2x12t−T1t,x˙2=−KMx1−CMx2+1Mut+δ2ξ,x1,x2,t+0.2x22t−T2t,y=x1,
The desired tracking trajectoryyr=sin0.5t+0.5sint. The design parameters of the controller are taken as kb1=0.4,kb2=2,c1=10,c2=15,τ1=τ2=0.01,σ11=σ12=0.01,ρ1=ρ2=50,l1=l2=1,T=0.001,μ1=μ2=3,κ1=κ2=0.5,h=0.6. There are 68 nodes with the center placed on −2,2×−2,2×−2,2×−2,2 and the width of Gaussian functions is η1=1 in the first RBF vector. There are 85 nodes with the center placed on −2,2×−2,2×−2,2×−2,2×−2,2 and the width of Gaussian functions is η2=1 in the second RBF vector. With the initial conditions, x10=0.2,x20=0.1,w10=0.1,ϖ^1=2,ϖ^2=0.5,r0=0.1. Simulation results are shown in Figures 7–12.Figure 7
Outputy and desired trajectory yr.Figure 8
Phase portrait of statesx1 and x2.Figure 9
Profiles of control inputsu and α1.
(a)(b)Figure 10
Profiles of the tracking errorsz1 and z2.Figure 11
Estimated parametersϖ^1 and ϖ^2.Figure 12
Profiles of the boundary layer errory2.


## 5. Conclusions

The problem of finite-time tracking control for a class of uncertain nonstrict-feedback state-delayed nonlinear systems with full-state constraints and unmodeled dynamics has been proposed in this paper. Unmodeled dynamics is dealt with by introducing a dynamical signal and the uncertain terms produced by time-varying state delays are compensated for by constructing appropriate Lyapunov-Krasovskii functionals. By utilizing a smooth switch between the fractional and cubic form state feedback, novelC1 smooth finite-time NN control laws have been provided for nonlinear systems with full-state constraints. Based on a modified DSC method and adaptive NN control, together with the BLFs, the fast finite-time control performance of the closed-loop nonlinear systems can be ensured, while the full-state constraints are never violated. Theoretical proofs and experimental simulation show that all the internal signals in the closed-loop system are uniformly bounded, and the tracking error signals can converge into compact sets in a finite time with sufficient accuracy, respectively. To extend this control scheme to solve the finite-time tracking control problem for some more complicated systems, such as MIMO nonlinear systems, switched nonlinear systems are also the direction of our future efforts.



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