# Approximation-Based Robust Adaptive Backstepping Prescribed Performance Control for a Huger Class of Nonlinear Systems

Authors: Jihui Xu; Xiaolin Wang; Lei Zhang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper proposes an innovative adaptive neural prescribed performance control (PPC) scheme for large classes of nonlinear, nonstrict-feedback systems under input saturation constraint. A restrictive hypothesis under which the upper and lower bounds of control gain functions exist a priori is first relieved by constructing appropriate compact sets within which all state trajectories are held. A novel asymmetry error transformed variable is then introduced to cope with the nondifferentiable obstacle and complex deductions corresponding to traditional PPC schemes. To efficiently manage the input saturation constraint, a new auxiliary dynamic system with a bounded compensation tangent function term is established as the strictly bounded assumption of the dynamic system is canceled. It is rigorously proven that all signals in the closed-loop systems are semiglobally uniformly ultimately bounded under both Lyapunov and invariant set theories. The tracking errors converge to a small tunable residual set with prescribed performance under the effect of the input saturation constraint. The effectiveness of the proposed control scheme is thoroughly verified by two simulation examples.


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## Body

## 1. Introduction

The approximation-based adaptive control of uncertain nonlinear systems is a significant theoretical challenge that has garnered a great deal of research interest in recent years [1–3]. Many researchers have utilized fuzzy logic systems or neural networks (NNs) for this purpose [4–10]. When combined with the backstepping methodology, approximation-based adaptive approaches can achieve global stability for many classes of nonlinear systems [11–15]. However, as the traditional backstepping controller repeatedly differentiates the virtual controllers at each step, its complexity drastically increases as the order of systems increases. The dynamic surface control (DSC) technique is designed to mitigate this problem by introducing a first-order low-pass filter at each step. The DSC technique has allowed scholars to construct approximation-based adaptive control schemes for many nonlinear strict-feedback and nonstrict-feedback systems [16–21]; however, these control schemes are all based on the assumption that the control gain functions must be bounded. To relax this restrictive hypothesis, two adaptive NNs control schemes were developed for strict-feedback and nonstrict-feedback systems by assuming that control gain functions are continuous and are bounded on a compact set [22, 23]. However, these schemes do not consider the simultaneous occurrence of prescribed performance and input saturation constraints due to inherent difficulties in the design.The prescribed performance constraint is unavoidable in many industrial control systems, such as precise microinstruments and robotics, and may result in degradation, hazards, or system failure [24, 25]. Thus, it represents an issue worth of careful attention during the control system design process [26]. The prescribed performance control (PPC) concept has been extensively employed in controller design for systems of various forms [27–35]. For example, in [27], an adaptive control scheme for SISO strict-feedback nonlinear systems capable of guaranteeing prescribed performance bounds is considered. In [30], an improved prescribed performance control scheme is proposed for a strict-feedback nonlinear dynamic system based on the backstepping technique. Recently, an adaptive NN-based decentralized control scheme under a prescribed performance constraint was recently presented for uncertain switched nonstrict-feedback interconnected nonlinear systems [35]. These control schemes share the common assumption that bounded control gain functions are required, which is very restrictive [27–35].The input saturation constraint also appears in many actual systems, such as industrial robots and numerical control machines [36–38]. If it is ignored during control system design, it can cause inaccuracy, instability, or even deteriorate the performance of the entire closed-loop system. The input saturation problem has received much scholarly interest in recent years [39–43]. For example, an adaptive neural control method is proposed for a class of strict-feedback stochastic nonlinear system in the presence of input saturation constraint in [40]. Moreover, in [42], an adaptive fuzzy prescribed performance control scheme is presented for a class of nonstrict-feedback systems subject to input saturation constraint. Recently, an efficient fuzzy controller for a larger class of stochastic nonlinear systems is constructed to manage the input saturation constraint in [43]. However, previously published techniques [39–43] depend upon the assumption that the control gain functions are always bounded, which is overly restrictive; the upper and lower constants bounds of the control gain functions may be difficult to acquire in some practical systems or may even be nonexistent [41–43].To the best of our knowledge, there are extremely few extant schemes applicable to the control of large classes of nonstrict-feedback systems under both prescribed performance and input saturation constraints where the control gain functions are possibly unbounded. This is yet an open problem with theoretical and practical significance. In this study, we developed a novel adaptive neural PPC scheme for a large class of nonlinear nonstrict-feedback systems with both prescribed performance and input saturation constraints. The main contributions of our work can be summarized as follows.(1) Unlike other strategies [27–43], this is the first instance in which an adaptive control problem of a large class of nonstrict-feedback systems with continuous, possibly unbounded control gain functions fall under both prescribed performance and input saturation constraints.(2) As discussed in this paper, we constructed a novel error constraint transformation to overcome the nondifferentiable obstacle and complex deductions existing in traditional PPC schemes. We also obtained a new asymmetry error constraint variable.(3) By contrast to traditional backstepping techniques, no additional first-order filter or repeated differentiations of intermediate control signals are needed to operate the proposed technique. NNs are utilized to approximate the unknown continuous functions, which contain all states of the whole system.The remainder of this paper is organized as follows. Section2 provides the problem formulation and preliminaries. The adaptive neural prescribed performance controller is discussed in Section 3. Section 4 is devoted to stability analysis. In Section 5, two simulation examples are presented to demonstrate the effectiveness of the proposed control scheme. Section 6 concludes the paper.


## 2. Problem Statement and Preliminaries

### 2.1. Problem Formulation

Consider the following large class of uncertain nonstrict-feedback nonlinear systems [41]:(1)ξ˙j,ij=hj,ijξ-j,ρj+θj,ijξ-j,ijξj,ij+1+Dj,ijξ,t,1≤ij≤ρj-1ξ˙j,ρj=hj,ρjξ-j,ρj+θj,ρjξ-j,ρjujιj+Dj,ρjξ,tyj=ξj,1,j=1,2,…,mwhere ξj,ij∈R is the state of jth subsystem and ξ=ξ-1,ρ1T,…,ξ-j,ρjT,…,ξ-m,ρmTT∈RN represents the state vector of the whole system N=ρ1+⋯+ρm, where ξ-j,ρjT=ξj,1,⋯,ξj,ρjT∈Rρj and ρj is the order of the jth subsystem. ξ-j,ijT=ξj,1,⋯,ξj,ijT∈Rij, uj∈R, and yj∈R denote the system input and output of the jth subsystem, respectively. hj,ij· are unknown continuous functions with hj,ij0=0; θj,ij· are unknown continuous control gain functions and Dj,ijξ,t are uncertainties consisting of dynamical coupling terms and external disturbances. The system states are all assumed to be available in the control design process. Input saturation ujιj can be written as follows:(2)ujιj=satιj=signιjuj,M,ιj≥uj,Mιj,ιj<uj,Mwhere uj,M means the bound of ujιj, ιj∈R is the input to the saturator, and uj=ujιj.Control Objective: the purpose of this study is to establish a novel adaptive neural prescribed performance controller uj for System (1) to guarantee two performance indicators. (1) All closed-loop signals in the systems are semiglobally uniformly ultimately bounded (SGUUB) and (2) the prescribed output tracking error bound for the tracking error, ej,1t=yjt-rjt, is always satisfied. In this paper, the given reference signal rj satisfies Assumption 1.Assumption 1.
The reference signalrj is a sufficiently smooth function and there exists a constant Rj0>0 such that ∏j0≔rj,r˙j,r¨jT∣rj2+r˙j2+r¨j2≤Rj0.Definition 2 (see [18]).
The solution of System (1) is SGUUB if, for any Π, a compact subset of Rn, and all ξ-nk0∈Π there exist an ζ>0 and a number Nζ,ξ-nk0 such that ξ-nk∈ζ for all k≥k0+N.
Unlike other methods [27–43], we aim here to achieve this objective in the presence of the following assumptions and lemmas.Assumption 3.
The continuous control gain functions satisfyθj,ij(ξ-j,ij)>0 for ij=1,2,...,ρj,j=1,2,…,m.Remark 4.
Unlike other scholars [27–43], we assume only the signs of the control gain functions to be known. This effectively relaxes thea priori boundedness assumption. Despite some other techniques available to relax this assumption [22, 23], the case has limited application in practice if the prescribed performance constraint and input saturation constraint are not taken into account. This complicates the control design of nonlinear systems due to the couplings between the output and input constraints.Remark 5.
To the best of our knowledge, the present study marks the first time that the adaptive control problem of a general class of nonstrict-feedback nonlinear systems with both prescribed performance and input saturation constraints has been investigated.Assumption 6.
For∀t>0, there exist unknown constants Dj,ij∗>0 satisfying Dj,ijξ,t≤Dj,ij∗(ij=1,2,…,ρj,j=1,2,…,m.Lemma 7 (see [12]).
Consider the following dynamic system:(3)ς˙t=-aςt+pυtwhere a and p are positive constants and υt is a positive function. For ς0≥0, we have ςt≥0, ∀t≥0.Lemma 8 (see [15]).
The hyperbolic tangent functiontanh(·) is uninterrupted and differentiable. For ∀χ∈R,∀ε>0, we have(4)0≤χ-ξtanh⁡χε≤0.2785ε,0≤χtanh⁡χεHere, RBF NNs [4, 19] are employed to approximate any continuous functions hZ:Rn→R as follows:(5)hZ=Θ∗TΞ-Z+oZ,∀Z∈ΠZ⊂Rnwhere Z is the input vector and n is the NN input dimension. Θ∗=Θ1,Θ2,…,ΘmT∈Rm is the weight vector, m>1 is the NN node number, and oZ is the approximation error satisfying oZ≤o∗with o∗>0 as an unknown constant. Ξ-Z=Ξ1Z,Ξ2Z,…,ΞmZT∈Rm is the basic function vector and ΞjZ is a commonly used Gaussian function, i.e., ΞjZ=exp⁡-Z-ωjTZ-ωj/dj2,j=1,2,…,m, with center vector of ωj=ωj1,ωj2,…,ωjnTand width of dj∈R.


## 2.1. Problem Formulation

Consider the following large class of uncertain nonstrict-feedback nonlinear systems [41]:(1)ξ˙j,ij=hj,ijξ-j,ρj+θj,ijξ-j,ijξj,ij+1+Dj,ijξ,t,1≤ij≤ρj-1ξ˙j,ρj=hj,ρjξ-j,ρj+θj,ρjξ-j,ρjujιj+Dj,ρjξ,tyj=ξj,1,j=1,2,…,mwhere ξj,ij∈R is the state of jth subsystem and ξ=ξ-1,ρ1T,…,ξ-j,ρjT,…,ξ-m,ρmTT∈RN represents the state vector of the whole system N=ρ1+⋯+ρm, where ξ-j,ρjT=ξj,1,⋯,ξj,ρjT∈Rρj and ρj is the order of the jth subsystem. ξ-j,ijT=ξj,1,⋯,ξj,ijT∈Rij, uj∈R, and yj∈R denote the system input and output of the jth subsystem, respectively. hj,ij· are unknown continuous functions with hj,ij0=0; θj,ij· are unknown continuous control gain functions and Dj,ijξ,t are uncertainties consisting of dynamical coupling terms and external disturbances. The system states are all assumed to be available in the control design process. Input saturation ujιj can be written as follows:(2)ujιj=satιj=signιjuj,M,ιj≥uj,Mιj,ιj<uj,Mwhere uj,M means the bound of ujιj, ιj∈R is the input to the saturator, and uj=ujιj.Control Objective: the purpose of this study is to establish a novel adaptive neural prescribed performance controller uj for System (1) to guarantee two performance indicators. (1) All closed-loop signals in the systems are semiglobally uniformly ultimately bounded (SGUUB) and (2) the prescribed output tracking error bound for the tracking error, ej,1t=yjt-rjt, is always satisfied. In this paper, the given reference signal rj satisfies Assumption 1.Assumption 1.
The reference signalrj is a sufficiently smooth function and there exists a constant Rj0>0 such that ∏j0≔rj,r˙j,r¨jT∣rj2+r˙j2+r¨j2≤Rj0.Definition 2 (see [18]).
The solution of System (1) is SGUUB if, for any Π, a compact subset of Rn, and all ξ-nk0∈Π there exist an ζ>0 and a number Nζ,ξ-nk0 such that ξ-nk∈ζ for all k≥k0+N.
Unlike other methods [27–43], we aim here to achieve this objective in the presence of the following assumptions and lemmas.Assumption 3.
The continuous control gain functions satisfyθj,ij(ξ-j,ij)>0 for ij=1,2,...,ρj,j=1,2,…,m.Remark 4.
Unlike other scholars [27–43], we assume only the signs of the control gain functions to be known. This effectively relaxes thea priori boundedness assumption. Despite some other techniques available to relax this assumption [22, 23], the case has limited application in practice if the prescribed performance constraint and input saturation constraint are not taken into account. This complicates the control design of nonlinear systems due to the couplings between the output and input constraints.Remark 5.
To the best of our knowledge, the present study marks the first time that the adaptive control problem of a general class of nonstrict-feedback nonlinear systems with both prescribed performance and input saturation constraints has been investigated.Assumption 6.
For∀t>0, there exist unknown constants Dj,ij∗>0 satisfying Dj,ijξ,t≤Dj,ij∗(ij=1,2,…,ρj,j=1,2,…,m.Lemma 7 (see [12]).
Consider the following dynamic system:(3)ς˙t=-aςt+pυtwhere a and p are positive constants and υt is a positive function. For ς0≥0, we have ςt≥0, ∀t≥0.Lemma 8 (see [15]).
The hyperbolic tangent functiontanh(·) is uninterrupted and differentiable. For ∀χ∈R,∀ε>0, we have(4)0≤χ-ξtanh⁡χε≤0.2785ε,0≤χtanh⁡χεHere, RBF NNs [4, 19] are employed to approximate any continuous functions hZ:Rn→R as follows:(5)hZ=Θ∗TΞ-Z+oZ,∀Z∈ΠZ⊂Rnwhere Z is the input vector and n is the NN input dimension. Θ∗=Θ1,Θ2,…,ΘmT∈Rm is the weight vector, m>1 is the NN node number, and oZ is the approximation error satisfying oZ≤o∗with o∗>0 as an unknown constant. Ξ-Z=Ξ1Z,Ξ2Z,…,ΞmZT∈Rm is the basic function vector and ΞjZ is a commonly used Gaussian function, i.e., ΞjZ=exp⁡-Z-ωjTZ-ωj/dj2,j=1,2,…,m, with center vector of ωj=ωj1,ωj2,…,ωjnTand width of dj∈R.


## 3. Adaptive Neural PPC Controller Methodology

In this section, a novel adaptive neural prescribed performance controller is constructed for a larger class of nonstrict-feedback nonlinear system (1) under the framework of backstepping technology. Before designing the virtual control functions, we need to first introduce the prescribed performance concept. Traditional PPC schemes [25–27, 30, 35] have certain shortcomings such as complex deductions and nondifferentiable obstacles. We attempted to resolve those shortcomings in this study via a novel asymmetry prescribed performance form:(6)Ψjt<ej,1t<Ωjtwhere Ωjt=τΩj-τj,∞e-ℵΩjt+τj,∞ and Ψjt=τΨj+τj,∞e-ℵΨjt-τj,∞ with τΩj and τΨj being the initial values of boundary functions Ωjt and Ψjt, respectively. The design parameters ℵΩj>0 and ℵΨj>0 are convergence rates of exponential functions. Ωj∞=-Ψj∞=τj,∞>0. τΩj, τΨj, and τj,∞ are properly chosen constants satisfying Ψj0<ej,10<Ωj0 and Ψjt<Ωjt. To this effect, τΩj and τΨj limit the maximum overshoot of tracking error ej,1t at the transient.The termej,1t in (6) cannot be utilized directly to construct controllers, so we introduce the novel transformed error νj,1t:(7)νj,1t=ƛj,1t1-ƛj,12twith the normalized error(8)ƛj,1t=2ej,1t-Ωjt+ΨjtΩjt-Ψjtwhere ej,1t=yjt-rjt. The time derivative of νj,1t can be given by(9)ν˙j,1t=ϖj,1e˙j,1t+ιj,1where ϖj,1=21+ƛj,12t/1-ƛj,12t2Ωjt-Ψjt and ιj,1=ƛj,1tΩ˙jt-Ψ˙jt-Ω˙jt+Ψ˙jt/2.Lemma 9.
For∀t≥0, if Ψj0<ej,10<Ωj0 and there exists a constant ν-j,1>0 satisfying νj,1t≤ν-j,1, the tracking error satisfies Ψjt<ej,1t<Ωjt.Proof.
See the Appendix.Then, the design procedure is given in a step-by-step way as follows.Step 1.
Considering the standard backstepping control design method, we define(10)zj,1=νj,1,zj,2=ξj,2-sj,1where sj,1 is a virtual control law to be designed later.
Differentiatingzj,1 and considering (1) and (9) yield(11)z˙j,1=ϖj,1Qj,1+ϖj,1θj,1ξj,1ξj,2+Dj,1ξ,twhere Qj,1=hj,1ξ-j,ρj-r˙j+ιj,1.
SinceQj,1 is an unknown continuous function, we use a RBF NNs Θj,1∗TΞj,1ξ-j,ρj to approximate the function:(12)Qj,1=Θj,1∗TΞ-j,1ξ-j,ρj+oj,1,∀ξ-j,ρj∈Πξ-j,ρjwhere Ξ-j,1ξ-j,ρj=Ξj,1,1ξ-j,ρj,Ξj,1,2ξ-j,ρj,…,Ξj,1,lj,1ξ-j,ρjT with Ξj,1,nξ-j,ρj being Gaussian functions for n=1,2,…,lj,1, and oj,1 is the approximation error, satisfying oj,1≤oj,1∗ with oj,1∗>0 being an unknown constant.
Consider the following quadratic Lyapunov function candidate:(13)Vzj,1=12zj,12
According to (11) and (12), the time derivative of (13) can be given by(14)V˙zj,1=zj,1ϖj,1Θj,1∗TΞ-j,1ξ-j,ρj+oj,1+zj,1ϖj,1Dj,1ξ,t+zj,1ϖj,1θj,1ξj,1ξj,2Define a compact setΠj,1≔zj,1∣zj,12≤2ζ, with ζ>0 being a design constant. For the compact set Πj,1×Πj0, the following lemma holds.Lemma 10.
The functionθj,1(ξj,1) has a maximum and a minimum in Πj,1×Πj0; namely, there exist constants θ_j,1>0 and θ-j,1>0 such that θ_j,1=minΠj,1×Πj0⁡θj,1(ξj,1) and θ-j,1=maxΠj,1×Πj0⁡θj,1(ξj,1).Proof.
See the Appendix.Design the virtual controller lawsj,1 as(15)sj,1=-cj,1zj,1ϖj,1-ϑ^j,1zj,1ϖj,12bj,12-ψ^j,1tanh⁡zj,1ϖj,1γj,1where cj,1>0 and γj,1>0 are design parameters.The corresponding parameters adaptation lawsϑ^j,1 and ψ^j,1 are given by(16)ϑ^˙j,1=μj,1zj,1ϖj,122bj,12-σj,1ϑ^j,1(17)ψ^˙j,1=ρj,1zj,1ϖj,1tanh⁡zj,1ϖj,1γj,1-σj,1ψ^j,1where μj,1>0, ρj,1>0, and σj,1>0 are design parameters; ϑ^j,1 and ψ^j,1 are estimates of the unknown constants ϑj,1=θ_j,1-1Θj,1∗2lj,1 and ψj,1∗=θ_j,1-1oj,1∗+Dj,1∗, respectively, with lj,1 being the dimension of Ξ-j,1χ-j,ρj. According to Lemma 7, we have ϑ^j,1t≥0, ψ^j,1t≥0 for ∀t≥0 after selecting ϑ^j,10≥0 and ψ^j,10≥0.In view of Young’s inequality, we have(18)zj,1ϖj,1Θj,1∗TΞ-j,1ξ-j,ρj≤bj,122+zj,1ϖj,12Θj,1∗22bj,12Ξ-j,1Tξ-j,ρjΞ-j,1ξ-j,ρjwhere bj,1 is any positive constant. From [22], we have Ξ-j,1Tξ-j,ρjΞ-j,1ξ-j,ρj≤lj,1 since Ξ-j,1ξ-j,ρj=Ξj,1,1ξ-j,ρj,Ξj,1,2ξ-j,ρj,…,Ξj,1,lj,1ξ-j,ρjT and Ξj,1,nξ-j,ρj≤1,n=1,2,…,lj,1. Thus, we further have(19)zj,1ϖj,1Θj,1∗TΞ-j,1ξ-j,ρj≤zj,1ϖj,12Θj,1∗2lj,12bj,12+bj,122Considering (14) and (19), we obtain the time derivative of Vsj,1 as(20)V˙zj,1≤zj,1ϖj,12Θj,1∗2lj,12bj,12+bj,122+zj,1ϖj,1θj,1ξj,1χj,2+zj,1ϖj,1θ_j,1ψj,1∗where ψj,1∗=θ_j,1-1oj,1∗+Dj,1∗.Let the Lyapunov function candidate be(21)Vj,1=Vzj,1+θ_j,1ϑ~j,122μj,1+θ_j,1ψ~j,122ρj,1where ϑ~j,1=ϑj,1-ϑ^j,1 and ψ~j,1=ψj,1∗-ψ^j,1 are the estimation errors of ϑj,1 and ψj,1, respectively.From (15), (20) and χj,2=zj,2+sj,1, the time derivative of (21) is(22)V˙j,1≤zj,1zj,2ϖj,1θj,1ξj,1+zj,1ϖj,1θ_j,1ψj,1∗-cj,1θ_j,1zj,1ϖj,12-ψ^j,1θ_j,1zj,1ϖj,1tanh⁡zj,1ϖj,1γj,1-θ_j,1ϑ^j,1zj,1ϖj,122bj,12+bj,122-θ_j,1ψ~j,1ψ^˙j,1ρj,1-θ_j,1ϑ~j,1ϑ^˙j,1μj,1+zj,1ϖj,12Θj,1∗2lj,12bj,12Substituting (16) and (17) into (22) yields(23)V˙j,1≤zj,1zj,2ϖj,1θj,1ξj,1+θ_j,1σj,1ϑ~j,1ϑ^j,1+θ_j,1σj,1ψ~j,1ψ^j,1+0.2785θ_j,1ψj,1∗γj,1-cj,1θ_j,1zj,1ϖj,12+bj,122Step (j,ij2≤ij≤ρj-1,j=1,…,m). In the j,ijth step, we define(24)zj,ij=ξj,ij-sj,ij-1,zj,ij+1=ξj,ij+1-sj,ij,where sj,ij-1 and sj,ij are the virtual control laws of the j,ij-1th step and the j,ijth step, respectively.Considering (1) and differentiating zj,ij, one has(25)z˙j,ij=Qj,ij+θj,ijξ-j,ijξj,ij+1+Dj,ijξ,twhere Qj,ij=hj,ijξ-j,ρj-s˙j,ij-1.Similarly, a RBF NNsΘj,ij∗TΞ-j,ijξ-j,ρj is employed to approximate the functions Qj,ij:(26)Qj,ij=Θj,ij∗TΞ-j,ijξ-j,ρj+oj,ij,ξ-j,ρj∈Πξ-j,ρjwith Ξ-j,ijξ-j,ρj=[Ξj,ij,1ξ-j,ρj,Ξj,ij,2ξ-j,ρj,…, Ξj,ij,lj,ijξ-j,ρj]T and oj,ij being the approximation error, satisfying oj,ij≤oj,ij∗.Consider the following quadratic Lyapunov function candidate:(27)Vzj,ij=12zj,ij2It follows from (25) and (26) that the time derivative of Vzj,ij is(28)V˙zj,ij=zj,ijθj,ijξ-j,ijξj,ij+1+oj,ij+zj,ijDj,ijξ,t+zj,ijΘj,ij∗TΞ-j,ijξ-j,ρjDesign the virtual control lawssj,ij as(29)sj,ij=-cj,ijzj,ij-ϑ^j,ijzj,ij2bj,ij2-ψ^j,ijtanh⁡zj,ijγj,ijThe design process of parameters is similar to Step1, and the parameters adaptation laws ϑ^j,ij and ψ^j,ij are given by(30)ϑ^˙j,ij=μj,ijzj,ij22bj,ij2-σj,ijϑ^j,ij(31)ψ^˙j,ij=ρj,ijzj,ijtanh⁡zj,ijγj,ij-σj,ijψ^j,ijThe design process of parameters is also similar to Step1. The functions θj,ijξ-j,ij can be rewritten as(32)θj,ijξ-j,ij=Φj,ijz-j,ij,ϑ^-j,ij-1,ψ^-j,ij-1,rjwhere z-j,ij=zj,1,zj,2,…,zj,ijT, ϑ^-j,ij-1=ϑ^j,1,ϑ^j,2,…,ϑ^j,ij-1T, ψ^-j,ij-1=ψ^j,1,ψ^j,2,…ψ^j,ij-1T, and Φj,ij· is a continuous function.Define the compact setsΠj,ij as(33)Πj,ij≔z-j,ijT,ϑ^-j,ij-1T,ψ^-j,ij-1TT∣∑c=1ijzj,c2+∑c=1ij-1θ_j,cϑ~j,c2μj,c+θ_j,cψ~j,c2ρj,c≤2ζwhere ζ>0. Similarly, we can know that the functions θj,ij(ξ-j,ij) have a maximum and a minimum in the compact set Πj,ij×Πj0; namely, there exist constants θ-j,ij>0 and θ_j,ij>0 satisfying(34)0<θ_j,ij≤θj,ijξ-j,ij≤θ-j,ij,ξ-j,ij∈Πj,ij×Πj0Let the Lyapunov function candidate be(35)Vj,ij=Vzj,ij+θ_j,ijϑ~j,ij22μj,ij+θ_j,ijψ~j,ij22ρj,ijwhere ϑ~j,ij=ϑj,ij-ϑ^j,ij and ψ~j,ij=ψj,ij∗-ψ^j,ij.According to Young’s inequality, one has(36)zj,ijΘj,ij∗TΞ-j,ijξ-j,ρj≤zj,ij2Θj,ij∗2lj,ij2bj,ij2+bj,ij22where bj,ij is any positive constant. lj,ij is the dimension of Ξ-j,ijξ-j,ρj=Ξj,ij,1ξ-j,ρj,Ξj,ij,2ξ-j,ρj,…,Ξj,ij,lj,ijξ-j,ρjT with Ξj,ij,nξ-j,ρj≤1, for n=1,2,…,lj,ij.From (36), we can rewrite (28) as(37)V˙zj,ij≤zj,ijθj,ijξ-j,ijξj,ij+1+zj,ij2Θj,ij∗2lj,ij2bj,ij2+bj,ij22+zj,ijθ_j,ijψj,ij∗where ψj,ij∗=θ_j,ij-1οj,ij∗+Dj,ij∗.According to (29), (30), (31), and (37), we obtain the time derivative of Vj,ij as(38)V˙j,ij≤zj,ijzj,ij+1θj,ijξ-j,ij+θ_j,ijσj,ijϑ~j,ijϑ^j,ij+θ_j,ijσj,ijψ~j,ijψ^j,ij-cj,ijθ_j,ijzj,ij2+0.2785θ_j,ijψj,ij∗γj,ij+bj,ij22Step (j,ρjj=1,…,m). Define zj,ρj=ξj,ρj-sj,ρj-1-ħjtanh⁡βj, with ħj>0 being a design parameter, and βj is defined as(39)β˙j=cosh2βjħj-αjtanh⁡βj+sat⁡ιj-ιj,βj0=0where αj>0 is a design parameter.Considering (1) and differentiating zj,ρj with respect to time yields(40)z˙j,ρj=Qj,ρj+θj,ρjξ-j,ρjuj+Dj,ρjξ,t+αjtanh⁡βj-satιj+ιjwhere Qj,ρj=hj,ρjξ-j,ρj-sj,ρj-1. For uj=satιj≤uj,M, there exists a continuous function Gj,ρjξ-j,ρj such that(41)θj,ρjξ-j,ρjuj=θj,ρjξ-j,ρjsatιj≤Gj,ρjξ-j,ρjDefine the following compact set(42)Πj,ρj≔z-j,ρjT,ϑ^-j,ρj-1T,ψ^-j,ρj-1TT∣∑c=1ρjzj,c2+∑c=1ρj-1θ_j,cϑ~j,c2μj,c+θ_j,cψ~j,c2ρj,c≤2ζThe functionsGj,ρjξ-j,ρj=Gj,ρjξ-j,ρj-1,zj,ρj+sj,ρj-1 have maximums on Πj,ρj×Πj0. Thus, we have Gj,ρjξ-j,ρj≤Gj,ρj∗ with Gj,ρj∗>0 being the unknown constants.Choose the actual control lawιj as(43)ιj=-cj,ρjzj,ρj-ϑ^j,ρjzj,ρj2bj,ρj2-ψ^j,ρjtanh⁡zj,ρjγj,ρjThe design process of parameters is similar to stepj,ij and Step 1, and the parameters adaptation laws ϑ^j,ρj and ψ^j,ρj are given by(44)ϑ^˙j,ρj=μj,ρjzj,ρj22bj,ρj2-σj,ρjϑ^j,ρj(45)ψ^˙j,ρj=ρj,ρjzj,ρjtanh⁡zj,ρjγj,ρj-σj,ρjψ^j,ρjwhere ψj,ρj∗=oj,ρj∗+Dj,ρj∗+Gj,ρj∗+αj+uj,M, and the design process of other parameters is also similar to step j,ij and Step 1.Similar to the former steps, the functionθj,ρjξ-j,ρj can be expressed as(46)θj,ρjξ-j,ρj=Φj,ρjz-j,ρj,ϑ^-j,ρj-1,ψ^-j,ρj-1,rjwith Φj,ρj· being a continuous function.In line with Lemma10, the function Φj,ρj· has maximum θ-j,ρj and minimum θ_j,ρj such that (47)0<θ_j,ρj≤θj,ρjξ-j,ρj≤θ-j,ρjSimilarly to the previous steps, defineVzj,ρj=zj,ρj2/2, and a RBF NNs Θj,ρj∗TΞ-j,ρjχ-j,ρj is utilized to approximate the functions Qj,ρj. According to (40), (41), and (47), the time derivative of Vzj,ρj is(48)V˙zj,ρj≤zj,ρj2Θj,ρj∗2lj,ρj2bj,ρj2+bj,ρj22+zj,ρjψj,ρj∗+ιjsj,ρjwhere bj,ρj>0 and ψj,ρj∗=oj,ρj∗+Dj,ρj∗+Gj,ρj∗+αj+uj,M.Consider the Lyapunov function candidate as(49)Vj,ρj=Vzj,ρj+ϑ~j,ρj22μj,ρj+ψ~j,ρj22ρj,ρjwhere ϑ~j,ρj=ϑj,ρj-ϑ^j,ρj and ψ~j,ρj=ψj,ρj∗-ψ^j,ρj.Substituting (43), (44), (45), and (48) into (49), the time derivative of Vj,ρj is(50)V˙j,ρj≤σj,ρjϑ~j,ρjϑ^j,ρj+σj,ρjψ~j,ρjψ^j,ρj-cj,ρjzj,ρj2+0.2785ψj,ρj∗γj,ρj+bj,ρj22


## 4. Stability Analysis

Choose the Lyapunov function as follows:(51)V=∑j=1mVjwhere Vj is the Lyapunov function for the jth subsystem(52)Vj=12∑ij=1ρj-1+θ_j,ijϑ~j,ij2μj,ij+θ_j,ijψ~j,ij2ρj,ij+12∑ij=1ρjzj,ij2+ϑ~j,ρj22μj,ρj+ψ~j,ρj22ρj,ρjThe main stability result of the proposed scheme is summarized in the following Theorem11.Theorem 11.
Consider Assumptions1–6 and the intermediate virtual control laws (15), (29), the actual control law (43), and the adaptive laws (16), (17), (30), (31), (44), and (45). For Ψj0<ej,10<Ωj0,∀ζ>0, ϑ^j,ijt≥0, ψ^j,ijt≥0, and Vj0≤ζ, there exist design parameters σj,ij, γj,ij, bj,ij, cj,ij, ρj,ij, and μj,ij such that
(i) all the closed-loop system signals are SGUUB;
(ii) the whole system output tracking errore1 remains in a neighborhood of the origin within the preselected transient and steady bounds; and
(iii) the closed-loop system variableβj is bounded.Proof.
In view of (23), (38), and (50), the time derivative of Vj is(53)V˙j≤∑ij=1ρj-1θ_j,ijσj,ijϑ~j,ijϑ^j,ij+θ_j,ijσj,ijψ~j,ijψ^j,ij+∑ij=1ρjbj,ij22+σj,ρjϑ~j,ρjϑ^j,ρj+σj,ρjψ~j,ρjψ^j,ρj-cj,1θ_j,1zj,1ϖj,12+∑ij=1ρj-10.2785θ_j,ijψj,ij∗γj,ij-∑ij=2ρj-1cj,ijθ_j,ijzj,ij2+0.2785ψj,ρj∗γj,ρj+zj,1zj,2ϖj,1θj,1ξj,1+∑ij=2ρj-1zj,ijzj,ij+1θj,ijξ-j,ij-cj,ρjzj,ρj2
By completion of squares, it holds that(54)ϑ~j,ijϑ^j,ij=ϑ~j,ijϑj,ij-ϑ~j,ij≤ϑj,ij22-ϑ~j,ij22ψ~j,ijψ^j,ij=ψ~j,ijψj,ij∗-ψ~j,ij≤ψj,ij∗22-ψ~j,ij22zj,1zj,2ϖj,1θj,1ξj,1≤zj,1ϖj,12θ-j,1kj,12+θ-j,1zj,222kj,1zj,ijzj,ij+1θj,ijξ-j,ij≤zj,ij2θ-j,ijkj,ij2+θ-j,ijzj,ij+122kj,ijwhere kj,1 and kj,ij are unknown positive constants.
Then, (53) can be rewritten as(55)V˙j≤-∑ij=2ρj-1cj,ijθ_j,ij-θ-j,ij-12kj,ij-1-θ-j,ijkj,ij2zj,ij2+Σj-cj,1θ_j,1-θ-j,1kj,12ϖj,12zj,12-σj,ρjψ~j,ρj22-∑ij=1ρj-1θ_j,ijσj,ijϑ~j,ij22+θ_j,ijσj,ijψ~j,ij22-cj,ρj-θ-j,ρj-12kj,ρj-1zj,ρj2-σj,ρjϑ~j,ρj22where Σj=∑ij=1ρj-1θ_j,ijσj,ijϑj,ij2/2+θ_j,ijσj,ijψj,ij∗2/2+0.2785θ_j,ijψj,ij∗γj,ij+σj,ρjϑ~j,ρjϑ^j,ρj+σj,ρjψ~j,ρjψ^j,ρj+∑ij=1ρjbj,ij2/2+0.2785ψj,ρj∗γj,ρj.
Fromϖj,1=21+ƛj,12t/1-ƛj,12t2Ωjt-Ψjt, we can further have 2/Ωj0-Ψj0≤ϖj,1 on compact set Πj,ρj×Πj0. Considering ϖj,12≥4/Ωj0-Ψj02, choose the control parameters cj,1≥κj,1Ωj0-Ψj02/4+θ-j,1kj,1/2/θ_j,1,cj,ij≥θ-j,ijkj,ij/2+θ-j,ij/2kj,ij+κj,ij/θ_j,ij,cj,ρj≥θ-j,ρj-1/2kj,ρj-1+κj,ρj/θ_j,ρj, σj,ij≥κj,ij/θ_j,ijij=1,2,…,ρj-1,j=1,2,…,m, σj,ρj≥κj,ρj with κj,ij being a positive constant.
Invoking (55), we can obtain(56)V˙j≤-κjVj+Σjwhere κj=min2κj,ij,ρj,ijκj,ij,μj,ijκj,ijij=1,2,…,ρj.
Note thatΣj/κj can be made arbitrarily small by decreasing σj,ij, γj,ij, and bj,ij, increasing cj,ij, ρj,ij, and μj,ij in the meanwhile. Thus, we can have Σj/κj≤ζ by selecting proper design parameters. It follows from Σj/κj≤ζ and (56) that V˙j≤0 on the level set Vj=ζ. Subsequently, the compact set Πj,ρj×Πj0 is an invariant set and the signals zj,ij, ϑ~j,ij and ψ~j,ijij=1,2,…,ρj of the closed-loop system are SGUUB. The property (i) of Theorem 11 is proved.
Multiplying (56) by eκjt and integrating over 0,t yields(57)0≤Vjt≤Vj0e-κjt+Τj,∀t≥0where Τj=Σj/κj.
According to (21) and (57), we have(58)12zj,12=12νj,12≤Vj0e-κjt+Τj,∀t≥0
Fromzj,1≤2Vj0+Τj,∀t≥0 and Lemma 9, we have Ψjt<ej,1t<Ωjt, for ∀t≥0. Now let us consider the Lyapunov function candidate for the whole systems as V=∑j=1mVj. From (57), it can be derived that(59)V˙=∑j=1mV˙j≤∑j=1m-κjVj+Σj≤-ΛV+Χwhere Λ=minκ1,…,κm and Χ=∑j=1mΣj. Then, we further have(60)Vt≤V0e-Λt+Υ,∀t≥0where Υ=Χ/Λ is a positive constant.
Similarly, from (21) and (60), we can obtain(61)12z12=12ν12≤V0e-Λt+Υ,∀t≥0
Fromz1≤2V0+Υ,∀t≥0 and Lemma 9, we have Ψt<e1t<Ωt, for ∀t≥0. Consequently, by appropriately choosing the design parameters, the tracking control error can be shown to converge to a small neighborhood of the origin and its prescribed performance is satisfied. Thus, property (ii) of Theorem 11 is proved.
Furthermore, there exists a scalarωj≥0 to satisfy Γj≤ωj with Γj=satιj-ιj and αj>ωj. Define Vβj=ħjβj2/2, we can have(62)V˙βj≤cosh2βj0.2785αj-αj-ωjβj
Ifβj>0.2785αj/αj-ωj, we can obtain V˙βj<0. Thus βj will lie in the compact set βj∣βj≤0.2785αj/αj-ωj, for ∀t≥0. This completes the proof.


## 5. Simulation Analysis

In this section, to illustrate the effectiveness of the proposed control scheme, two simulation examples are considered.Example 12.
Consider the following uncertain nonlinear systems(63)ξ˙1,1=h1,1ξ-1,2+θ1,1ξ-1,1ξ1,2+D1,1ξ,tξ˙1,2=h1,2ξ-1,2+θ1,2ξ-1,2u1ι1+D1,2ξ,tξ˙2,1=h2,1ξ-2,2+θ2,1ξ-2,1ξ2,2+D2,1ξ,tξ˙2,2=h2,2ξ-2,2+θ2,2ξ-2,2u2ι2+D2,2ξ,ty1=ξ1,1y2=ξ2,1where h1,1ξ-1,2=ξ1,1ξ1,2e-1.5ξ1,22, h1,2ξ-1,2=cosξ1,1ξ1,2ξ1,22, h2,1ξ-2,2=0.8+sinξ1,2ξ2,12, h2,2ξ-2,2=ξ1,1ξ1,22+ξ2,12ξ2,2 and θ1,1ξ-1,1=1+eξ1,12, θ2,1ξ-2,1=eξ1,1ξ2,1, θ1,2ξ-1,2=1.2+eξ1,1ξ1,23, θ2,2ξ-2,2=2.5+eξ1,2ξ2,1ξ2,2. D1,1=0.5cos0.5tsinξ1,1ξ1,22ξ2,1, D2,1=0.5cosξ1,1ξ1,22ξ2,22, D1,2=2sinξ1,22+ξ1,1ξ2,1, and D2,2=cosξ2,22+ξ2,12sint2. The input saturations u1ι1 and u2ι2 are described by(64)u1ι1=satι1=signι13.5,ι1≥3.5ι1,ι1<3.5u2ι2=satι2=signι24.5,ι2≥4.5ι2,ι2<4.5
Choose the reference signal as the van der pol oscillator system described as follows:(65)ξ˙r1=ξr2ξ˙r2=-ξr1+Τ-1-ξr22ξr2yr=ξr1where Τ-=0.3 and ξr1,ξr2T=1.2,0.7T. Based on Theorem 11, the virtual stabilizing control laws and actual control laws are chosen as(66)s1,1=-0.8z1,1ϖ1,1-ϑ^1,1z1,1ϖ1,12×0.252-ψ^1,1tanh⁡z1,1ϖ1,10.5s2,1=-0.6z2,1ϖ2,1-ϑ^2,1z2,1ϖ2,12×0.12-ψ^2,1tanh⁡z2,1ϖ2,10.3ι1=-1.2z1,2-ϑ^1,2z1,22×0.52-ψ^1,2tanh⁡z1,20.75ι2=-1.5z2,2-ϑ^2,2z2,22×0.32-ψ^2,2tanh⁡z2,20.55with adaptation laws(67)ϑ^˙1,2=z1,222×0.52-0.3ϑ^1,2,ψ^˙1,2=1.2z1,2tanh⁡z1,20.75-0.3ψ^1,2ϑ^˙1,1=z1,1ϖ1,122×0.252-0.3ϑ^1,1,ψ^˙1,1=z1,1ϖ1,1tanh⁡z1,1ϖ1,10.5-0.3ψ^1,1ϑ^˙2,2=1.5z2,222×0.32-0.25ϑ^2,2,ψ^˙2,2=2z2,2tanh⁡z2,20.55-0.25ψ^2,2ϑ^˙2,1=1.5z2,1ϖ2,122×0.12-0.25ϑ^2,1,ψ^˙2,1=1.2z2,1ϖ2,1tanh⁡z2,1ϖ2,10.3-0.25ψ^2,1and performance functions(68)Ω1t=-Ψ1t=1.1-0.07e-t+0.07Ω2t=-Ψ2t=0.6-0.06e-t+0.06
Choose the initial conditions asξ1,10,ξ1,20T=0.7,-0.7T, ξ2,10,ξ2,20T=0.3,-0.3T, ϑ^1,10=ϑ^1,20=ϑ^2,10=ϑ^2,20=0 and ψ^1,10=ψ^1,20=ψ^2,10=ψ^2,20=0. The simulation results of Example 12 are shown in Figures 1–5.
Figure1 shows that the system outputs y1 and y2 follow the desired trajectories r1 and r2 as closely as possible, and the control inputs u1ι1 and u2ι2 are shown in Figure 2. The prescribed performance bounds Ω1, Ψ1, Ω2, Ψ2 and the tracking errors e1,1, e2,1 are illustrated in Figures 3-4, from which we can see the output tracking errors achieve preselected transient and steady bounds. Finally, Figure 5 is given to explain the boundedness of adaptive parameters ϑ^1,1, ψ^1,1, ϑ^1,2, ψ^1,2, ϑ^2,1, ψ^2,1, ϑ^2,2, and ψ^2,2.Figure 1
System outputs and reference signals.Figure 2
Curves of control inputs.Figure 3
Curves ofe1,1 and performance bounds.Figure 4
Curves ofe2,1 and performance bounds.Figure 5
Curves of adaptive parameters.Example 13.
Consider the following uncertain nonlinear systems(69)ξ˙1,1=h1,1ξ-1,2+θ1,1ξ-1,1ξ1,2+D1,1ξ,tξ˙1,2=h1,2ξ-1,2+θ1,2ξ-1,2u1ι1+D1,2ξ,tξ˙2,1=h2,1ξ-2,2+θ2,1ξ-2,1ξ2,2+D2,1ξ,tξ˙2,2=h2,2ξ-2,2+θ2,2ξ-2,2u2ι2+D2,2ξ,ty1=ξ1,1y2=ξ2,1where h1,1ξ-1,2=ξ1,1+ξ1,2, h1,2ξ-1,2=cosξ1,12ξ1,22ξ1,2+ξ1,1ξ1,22, h2,1ξ-2,2=ξ1,1+ξ1,2ξ2,1eξ2,1, h2,2ξ-2,2=ξ2,1+ξ1,2ξ2,22 and θ2,1ξ-2,1=eξ2,1, θ1,1ξ-1,1=1.3+eξ1,12, θ1,2ξ-1,2=0.5+eξ1,1ξ1,2, θ2,2ξ-2,2=1.2+eξ1,1ξ2,1ξ2,2. D1,1=0.5costsin0.5t+ξ1,1ξ2,1ξ2,22, D1,2=ξ1,2ξ2,1+0.3sint2cost+ξ1,12, D2,1=0.5sint+ξ1,2ξ2,12ξ2,2 and D2,2=cos0.2tsint2+ξ1,1ξ1,2+ξ2,22. u1ι1 and u2ι2 are described by(70)u1ι1=satι1=signι14,ι1≥4ι1,ι1<4u2ι2=satι2=signι25.5,ι2≥5.5ι2,ι2<5.5
Then, the virtual stabilizing control laws and actual control laws are constructed as(71)s1,1=-2z1,1ϖ1,1-ϑ^1,1z1,1ϖ1,12×0.52-ψ^1,1tanhz1,1ϖ1,10.75s2,1=-2.5z2,1ϖ2,1-ϑ^2,1z2,1ϖ2,12×0.12-ψ^2,1tanhz2,1ϖ2,10.25ι1=-3z1,2-ϑ^1,2z1,22×0.252-ψ^1,2tanhz1,20.75ι2=-4z2,2-ϑ^2,2z2,22×0.22-ψ^2,2tanhz2,20.45with adaptive laws(72)ψ^˙1,2=1.4z1,2tanhz1,20.75-0.4ψ^1,2,ϑ^˙1,2=z1,222×0.252-0.4ϑ^1,2ψ^˙1,1=z1,1ϖ1,1tanhz1,1ϖ1,10.75-0.4ψ^1,1,ϑ^˙1,1=1.3z1,1ϖ1,122×0.52-0.4ϑ^1,1ψ^˙2,2=1.5z2,2tanhz2,20.45-0.35ψ^2,2,ϑ^˙2,2=1.5z2,222×0.22-0.35ϑ^2,2ψ^˙2,1=1.1z2,1ϖ2,1tanhz2,1ϖ2,10.25-0.35ψ^2,1,ϑ^˙2,1=2z2,1ϖ2,122×0.12-0.35ϑ^2,1and performance functions(73)Ω1t=0.5-0.08e-t+0.08ħ1t=-2+0.08e-t-0.08Ω2t=1-0.09e-2t+0.09ħ2t=-3+0.09e-2t-0.09where the initial conditions ξ1,10,ξ1,20T=-0.9,0.9T, ξ2,10,ξ2,20T=-0.2,0.2T, ϑ^1,10=ϑ^1,20=ϑ^2,10=ϑ^2,20=0, and ψ^1,10=ψ^1,20=ψ^2,10=ψ^2,20=0. The simulation results of Example 13 are shown in Figures 6–9.
Figure6 shows that the system outputs track the desired trajectories, perfectly, and Figure 7 reveals that the system inputs are bounded. From Figures 8-9, the tracking errors e1,1, e2,1satisfy the asymmetry prescribed bounds and the transformed errors z1,1, z2,1 are bounded, from which we can see that a fairly good tracking performance is achieved.Figure 6
System outputs and reference signals.Figure 7
Curves of control inputs.Figure 8
Tracking errors and performance bounds.Figure 9
Curves of transformed errorsz1,1 and z2,1.


## 6. Conclusion

This paper proposed a novel adaptive neural PPC scheme for large classes of nonlinear nonstrict-feedback systems with prescribed performance under the effect of input saturation. Compared with previously published methods [27–43], the restrictive assumption that the upper and lower bounds of control gain functions must be positive constants or coefficients is relieved under the proposed method. The innovative error transformation proposed in this paper also overcomes the nondifferentiable obstacle and complex deductions corresponding to traditional PPC schemes. A novel auxiliary system with a bounded compensation term was also proposed to relax the strictly bounded assumption of the dynamic system. The stability of the closed-loop system was proven under Lyapunov and invariant set theories. Tracking errors were made arbitrarily small by appropriately selecting the design parameters. Finally, the simulation results show that the proposed control laws do achieve the desired performance. In the future, we will address control problems for uncertain nonlinear systems with an unbounded or nondifferentiable time derivative of disturbance; this will allow us to utilize the PPC approach combined with set-invariance mechanisms.



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