# Adaptive Fuzzy Output Feedback Control for Partial State Constrained Nonlinear Pure Feedback Systems

Authors: Liping Wang; Weiwei Sun; You Wu
Journal: Mathematical Problems in Engineering
 (2018)

---

## Abstract

The adaptive fuzzy output feedback control problem for a class of pure feedback systems with partial state constraints is addressed in this paper. The fuzzy state observers are designed to estimate the unmeasured state while the fuzzy logic systems are used to approximate the unknown nonlinear functions. The proposed adaptive fuzzy output feedback controller can guarantee that the partial state constraints are not violated, and all closed-loop signals remain bounded by use of Barrier Lyapunov Functions (BLFs). A numerical example is presented to illustrate the effectiveness of the results in this paper.


---

## Body

## 1. Introduction

During the last decades, control design of nonlinear systems has attracted increasing interests. All kinds of control techniques have been proposed for both theoretical analysis and practical applications [1–6]. Many practical systems are inherently nonlinear and subject to many forms of constraints such as saturation and physical stoppages. Violation of the constraints may degrade the control performance and even make the system unstable. Therefore, the constraints handling in control design has attracted considerable attention [6–15]. There exist various techniques to tackle the constraints for nonlinear systems like nonlinear reference governor [7], invariance control [9], nonlinear model predictive control [11], etc.Backstepping methodology is more effective in synthesis of robust and adaptive nonlinear controllers for various systems with parametric or dynamic nonlinearities and uncertainties [16–22]. The work in [20] constructs an adaptive tracking controller by introducing an auxiliary integrator subsystem and using the improved backstepping method such that the closed-loop system has a unique solution that is globally bounded in probability. The concept of Barrier Lyapunov Function (BLF), which is developed via Control Lyapunov Function (CLF) [23] in backstepping design method, was first proposed in [24]. The characteristic of BLF is that it will approach infinity whenever its arguments approach some limits. Transgression of constraints can be prevented through keeping BLF bounded in the closed-loop system. BLF-based backstepping control has been applied to many constrained nonlinear systems control synthesis [25–29]. Therein, [24, 30, 31] have solved the BLF-based control problem of strict feedback nonlinear systems. The work [27] investigates the output tracking control problem of constrained nonlinear switch systems. The work [29] deals with the problem of adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form. And the problem with respect to full state constraints is solved in [32, 33].As we all know, the adaptive fuzzy control has an automatic learning capability which can adjust the adaptive parameters to deal with the uncertainty. Using the approximation property, fuzzy logic systems (FLSs) have been employed to tackle unknown nonlinear systems [26, 28, 34–36]. A control for nonlinear sampled systems with the guaranteed suboptimal performance achieved robust tracking by using fuzzy disturbance observer approach [26]. The work [34] studied an adaptive fuzzy dynamic surface control for nonlinear systems with fuzzy dead zone, unmodeled dynamics, dynamical disturbances, and unknown control gain functions. And the unknown system functions are approximated by the Takagi-Sugeno-type fuzzy logic systems. The work [36] addressed the adaptive control problem for nonlinear pure feedback systems based on fuzzy backstepping approach. Moreover, [28] constructed the approximation-based adaptive fuzzy tracking controller for non-strict-feedback stochastic nonlinear time-delay systems. As for the adaptive fuzzy observer design, to the best of the authors’ knowledge, the existing results consider only the influence of full state constraints, and there is no further discussion about partial state constraints. There is seldom adaptive control method subject to partial state constraints and unmeasured state.In this paper, we present an adaptive fuzzy backstepping tracking controller for a class of pure feedback nonlinear systems subject to unmeasured states and unknown nonlinear function. Firstly, the fuzzy state observers are designed to estimate the unmeasured state while the fuzzy logic systems are used to approximate the unknown nonlinear functions. Secondly, the proposed adaptive fuzzy output feedback controller can guarantee that the partial state constraints are not exceeded, and all closed-loop signals remain bounded with the using of BLF while the adaptive law for the estimations on uncertain parameters is constructed. A new coordinate transform is introduced during the process. Finally, a numerical example is given to validate our results presented in this paper.The paper is organized as follows. Section2 presents the problem formulation and some preliminaries. The main results are proposed in Section 3. Section 4 illustrates the effectiveness of the results by a numerical example.


## 2. Problem Statement and Preliminaries

### 2.1. Systems Description

Consider the following nonlinear pure feedback systems:(1)x˙i=xi+1+fix¯i,xi+1,x˙n=u+fnx¯n,u,y=x1,where x¯i=[x1,x2,…,xi]T∈Ri,i=1,2,…,n, are the state vectors of the systems, u and y are the input and output, respectively. The partition of the full states is constrained, i.e., constrained states xs=[x1,x2,…,xns]T and free states xr=[xns+1,xns+2,…,xn]T. And the number sequences, {1,2,…,ns} and {ns+1,ns+2,…,n}, are both ascending. The states xi(t),i=1,…,ns are required to remain in the set |xi|<kci with kci being positive constant, ∀t≥0. The nonlinear functions fi(x¯i,xi+1),i=1,2,…,n-1 and fn(x¯n,u) are unknown nonlinear smooth functions, supposing that only the output signal is measured and other states are unmeasurable.This paper is concerned with the problem of adaptive fuzzy output feedback control for system (1). Because of the existing unknown nonlinear functions, the fuzzy logic systems are employed to approximate the unknown nonlinear functions, and the fuzzy state observers are designed to handle the unmeasurable states. The following lemmas and assumptions related to backstepping design are given which will be used in the further analysis in the sequel.Lemma 1 (see [24]).
For any positive constantskbi, let Z:={z∈Rl:zi<kbi,i=1,2,…,n}⊂Rn and N:=Rl×Z⊂Rn+l be open sets. Consider the system(2)η˙=ht,η,where η:=[w,z]T∈N is the state, and the function h:R+×N→Rn+l is piecewise continuous in t and locally Lipschitz in η, uniformly in t, on R+×N. Let Zi:={zi∈R:|zi|<kbi}⊂R. Suppose that there exist positive definite functions U:Rl→R+ and Vi:Zi→R+(i=1,2,…,n), both of which are also continuously differentiable on Rl and Zi, respectively, such that(3)Vizi→∞aszi→±kbi.Let Vη:=∑i=1nVi(zi)+U(w) and z(0)∈Z. If the inequality(4)V˙=∂V∂ηh≤-ρV+c,with constants ρ>0, c>0, holds in the set z∈Z, then z(t)∈Z,∀t∈[0,∞).Lemma 2 (see [30]).
For any positive constantskb, positive integer p, and any z∈R satisfying |z|<kb, one has(5)log⁡kb2pkb2p-z2p≤z2pkb2p-z2p.Proof.
We define(6)q=z2pkb2p-z2p-log⁡kb2pkb2p-z2p=z2pkb2p-z2p-log⁡1-z2pkb2p-z2p.As |z|<kb, we have |z|2p<kb2p. Then, we can get the inequalities 0≤z2p/(kb2p-z2p)<kb2p/(kb2p-z2p)=1-z2p/(kb2p-z2p) and 0≤z2p/(kb2p-z2p)<1/2. Let ς=z2p/(kb2p-z2p) and then q=-log⁡(1-ς)+ς. The derivative of q is given as q˙=(2-ς)/(1-ς)>0. It shows that q is continuously increasing and the minimum of q is qmin=0. Thus, we get log⁡(kb2p/(kb2p-z2p))≤z2p/(kb2p-z2p), and log⁡(kb2p/(kb2p-z2p))=z2p/(kb2p-z2p) if and only if z=0.Lemma 3 (see [37]).
Letf(x) be a continuous function defined on a compact set Ω, and, for any constant ε>0, there exists fuzzy logic system (12) such as (7)fx=f^x∣θ+εx,εx≤ε.Lemma 4 (see [31]).
For anyβ∈Rn and ϑ>0, the following inequality holds: (8)0<β-βtanh⁡βϑ≤μϑ,μ=0.2785.Assumption 5 (see [24]).
For anykc1>0, there exist positive constants A0,Y1,Y2,…,Yn such that the desired trajectory yd(t) and its time derivatives satisfy |yd(t)|≤A0<kc1 and ydit<Yi,i=1,2,…,n, for all t≥0.Assumption 6.
There exist known constantsmi>0 such that (9)fix1-fix2≤mix1-x2,i=1,2,…,n.Rewrite (1) as(10)x˙i=xi+1+fix¯^i,x^i+1,f+∆fi,x˙n=u+fnx¯^n,uf+∆fn,y=x1,where ∆fi=fi(x¯i,xi+1)-fi(x¯^i,x^i+1,f),i=1,2,…,n-1,∆fn=fn(x¯n,u)-fn(x¯^n,uf).  x¯^i are the estimates of x¯i, which will be obtained by the state observer designed later; x^i+1,f and uf are the filtered signals for x^i+1 and u, respectively. There exist known constants τi,i=1,2,…,n, such that |x^i+1-x^i+1,f|≤τi+1(τ1=0). The filtered signals are defined as follows:(11)x^i,f=HLsx^i,uf=HLsu,where HL(s) is a Butterworth low-pass filter (LPF) [26, 36, 38] with the cutoff frequency ωc=1rad/s for different values of n.Remark 7.
The filtered signalsx^i+1,f and uf are employed to avoid the so-called algebraic loop problem existing in [29, 39] and to design the state observer and controller for nonlinear pure feedback systems.Remark 8.
Based on the statements in [26, 36, 38], most actuators have low-pass property and the replacements x^i,f≈x^i and uf≈u are reasonable in the controller design. Therefore, assume that |x^i-x^i,f|≤τi with τi being known constants.Then (10) can be further rewritten into the following state space form:(12)x˙=Ax+Ky+∑i=1n-1Bifix¯^i,x¯^i+1,f+∆fi+Bnfnx¯^n,uf+∆fn+u,where x=x1⋮xn,A=-k1⋮I-kn0⋯0,K=k1⋮kn,Bi=0⋮1⋮0,Bn=0⋮1. The vector K is chosen to make matrix A to be a strict Hurwitz matrix; i.e., for given a matrix Q=QT>0 there exists a matrix P=PT>0 satisfying(13)ATP+PA=-2Q.


### 2.2. Design of FLSs and State Observer

To tackle the unknown nonlinear functions, the fuzzy logic systems are introduced as follows.Rulej: if x1 is Nj1, and x2 is Nj2 and ⋯ and xn is Njn, then (14)yxisMj,j=1,2,…,m, where y is the output of the system, Nj1 and Mj denote fuzzy sets, and m represents the number of fuzzy rules.By using the singleton fuzzifier, center average defuzzification and product inference [25, 28], the final output can be expressed as follows:(15)yx=∑j=1my~j∏l=1nμNj1xl∑j=1m∏l=1nμNj1xl,where y~j=maxy∈RμMj(y); μNj1(xl) and μMj(y) stand for membership functions with respect to fuzzy sets Nj1 and Mj, respectively.Define(16)φx=∏l=1nμNj1xl∑j=1m∏l=1nμNj1xl as the basis function vector. The ideal constant weight vector is =[y~1,y~2,…,y~m]T=[θ1,θ2,…,θm]T, φ(x)=[φ1(x),φ2(x),…,φm(x)]T, and φi(x) is chosen as Gaussian function; i.e., for i=1,2,…,m, (17)φix=exp⁡-x-μiTx-μiηi2, where μi=[μi1,μi2,…,μin]T is the center vector, and ηi is the width of the Gaussian function.Then, the fuzzy outputy in (15) is described as (18)yx=θTφx.Accordingly, assume that the unknown nonlinear functions in (1) are approximated by the following fuzzy logic systems:(19)f^ix¯^i,x^i+1,f∣θ^i=θ^iTφix¯^i,x^i+1,f,1≤i≤n,where x^n+1,f=uf. The optimal weight vector θi∗ is defined as (20)θi∗=arg⁡minθi∈Ωisupx¯^i,x^i+1,f∈Ui1×Ui2f^ix¯^i,x^i+1,f∣θ^i-f^ix¯^i,x^i+1,f∣θi∗, where Ωi and Ui1×Ui2 are compact regions for θi and (x¯^i,x^i+1,f), respectively.Define(21)fix=θi∗Tφix+εix=θ^iTφix+δix,where εi∗ and δi∗(i=1,2,…,n) are known constants satisfying |εi|≤εi∗ and |δi|≤δi∗. Letting wi=εi-δi,i=1,2,…,n, it is clear that there is an unknown constant wi∗>0 such that |wi|≤wi∗=εi∗+δi∗.Based on (12), design a fuzzy state observer as(22)x^˙i=x^i+1+f^ix¯^i,x^i+1,f+kiy-x^1,x^˙n=u+f^nx¯^n,uf+kny-x^1,y=x1.Let ei=xi-x^i be an observer error vector. Then from (12) and (22), we have(23)e˙=Ae+δ+∆F,where δ=[δ1,δ2,…,δn]T and ∆F=[∆f1,∆f2,…,∆fn]T. Substituting (23) into (22), we have(24)e˙i=Ae+δ+∆F,x^˙i=x^i+1+f^ix¯^i,x^i+1,f+kie1,x^˙n=u+f^nx¯^n,uf+kne1,y=x1.Consider the following Lyapunov function candidate:(25)V0=12eTPe.Computing the time derivative of V0, one has(26)V˙0=eTPe˙=eTPAe+δ+∆F.According to Assumption 6, we get(27)∆fi=fix-i,xi+1-fix-^i,x^i+1,f≤mie2+mix^i+1-x^i+1,f≤mie2+miτi+1=mie2+τi+1′,where τi+1′=miτi+1. It is clear that(28)∆F2≤nm2e2+∑i=2n+1τi′2,eTP∆F≤ηP2e2+14ηnm2e2+ηP2e2+14η∑i=2n+1τi′2≤2ηP2e2+14ηnm2e2+14η∑i=2n+1τi′2.Using Young’s inequality and (28), we get(29)V˙0≤-λminQe2+ξP2e2+14ξδ∗2+2ηP2e2+14ηnm2e2+14η∑i=2n+1τi′2.To simplify the notation, let V˙0≜-p0e2+E0, where -p0=-λmin(Q)+ξP2+2ηP2+(1/4η)nm2,E0=(1/4ξ)δ∗2+(1/4η)(∑i=2n+1τi′)2.


## 2.1. Systems Description

Consider the following nonlinear pure feedback systems:(1)x˙i=xi+1+fix¯i,xi+1,x˙n=u+fnx¯n,u,y=x1,where x¯i=[x1,x2,…,xi]T∈Ri,i=1,2,…,n, are the state vectors of the systems, u and y are the input and output, respectively. The partition of the full states is constrained, i.e., constrained states xs=[x1,x2,…,xns]T and free states xr=[xns+1,xns+2,…,xn]T. And the number sequences, {1,2,…,ns} and {ns+1,ns+2,…,n}, are both ascending. The states xi(t),i=1,…,ns are required to remain in the set |xi|<kci with kci being positive constant, ∀t≥0. The nonlinear functions fi(x¯i,xi+1),i=1,2,…,n-1 and fn(x¯n,u) are unknown nonlinear smooth functions, supposing that only the output signal is measured and other states are unmeasurable.This paper is concerned with the problem of adaptive fuzzy output feedback control for system (1). Because of the existing unknown nonlinear functions, the fuzzy logic systems are employed to approximate the unknown nonlinear functions, and the fuzzy state observers are designed to handle the unmeasurable states. The following lemmas and assumptions related to backstepping design are given which will be used in the further analysis in the sequel.Lemma 1 (see [24]).
For any positive constantskbi, let Z:={z∈Rl:zi<kbi,i=1,2,…,n}⊂Rn and N:=Rl×Z⊂Rn+l be open sets. Consider the system(2)η˙=ht,η,where η:=[w,z]T∈N is the state, and the function h:R+×N→Rn+l is piecewise continuous in t and locally Lipschitz in η, uniformly in t, on R+×N. Let Zi:={zi∈R:|zi|<kbi}⊂R. Suppose that there exist positive definite functions U:Rl→R+ and Vi:Zi→R+(i=1,2,…,n), both of which are also continuously differentiable on Rl and Zi, respectively, such that(3)Vizi→∞aszi→±kbi.Let Vη:=∑i=1nVi(zi)+U(w) and z(0)∈Z. If the inequality(4)V˙=∂V∂ηh≤-ρV+c,with constants ρ>0, c>0, holds in the set z∈Z, then z(t)∈Z,∀t∈[0,∞).Lemma 2 (see [30]).
For any positive constantskb, positive integer p, and any z∈R satisfying |z|<kb, one has(5)log⁡kb2pkb2p-z2p≤z2pkb2p-z2p.Proof.
We define(6)q=z2pkb2p-z2p-log⁡kb2pkb2p-z2p=z2pkb2p-z2p-log⁡1-z2pkb2p-z2p.As |z|<kb, we have |z|2p<kb2p. Then, we can get the inequalities 0≤z2p/(kb2p-z2p)<kb2p/(kb2p-z2p)=1-z2p/(kb2p-z2p) and 0≤z2p/(kb2p-z2p)<1/2. Let ς=z2p/(kb2p-z2p) and then q=-log⁡(1-ς)+ς. The derivative of q is given as q˙=(2-ς)/(1-ς)>0. It shows that q is continuously increasing and the minimum of q is qmin=0. Thus, we get log⁡(kb2p/(kb2p-z2p))≤z2p/(kb2p-z2p), and log⁡(kb2p/(kb2p-z2p))=z2p/(kb2p-z2p) if and only if z=0.Lemma 3 (see [37]).
Letf(x) be a continuous function defined on a compact set Ω, and, for any constant ε>0, there exists fuzzy logic system (12) such as (7)fx=f^x∣θ+εx,εx≤ε.Lemma 4 (see [31]).
For anyβ∈Rn and ϑ>0, the following inequality holds: (8)0<β-βtanh⁡βϑ≤μϑ,μ=0.2785.Assumption 5 (see [24]).
For anykc1>0, there exist positive constants A0,Y1,Y2,…,Yn such that the desired trajectory yd(t) and its time derivatives satisfy |yd(t)|≤A0<kc1 and ydit<Yi,i=1,2,…,n, for all t≥0.Assumption 6.
There exist known constantsmi>0 such that (9)fix1-fix2≤mix1-x2,i=1,2,…,n.Rewrite (1) as(10)x˙i=xi+1+fix¯^i,x^i+1,f+∆fi,x˙n=u+fnx¯^n,uf+∆fn,y=x1,where ∆fi=fi(x¯i,xi+1)-fi(x¯^i,x^i+1,f),i=1,2,…,n-1,∆fn=fn(x¯n,u)-fn(x¯^n,uf).  x¯^i are the estimates of x¯i, which will be obtained by the state observer designed later; x^i+1,f and uf are the filtered signals for x^i+1 and u, respectively. There exist known constants τi,i=1,2,…,n, such that |x^i+1-x^i+1,f|≤τi+1(τ1=0). The filtered signals are defined as follows:(11)x^i,f=HLsx^i,uf=HLsu,where HL(s) is a Butterworth low-pass filter (LPF) [26, 36, 38] with the cutoff frequency ωc=1rad/s for different values of n.Remark 7.
The filtered signalsx^i+1,f and uf are employed to avoid the so-called algebraic loop problem existing in [29, 39] and to design the state observer and controller for nonlinear pure feedback systems.Remark 8.
Based on the statements in [26, 36, 38], most actuators have low-pass property and the replacements x^i,f≈x^i and uf≈u are reasonable in the controller design. Therefore, assume that |x^i-x^i,f|≤τi with τi being known constants.Then (10) can be further rewritten into the following state space form:(12)x˙=Ax+Ky+∑i=1n-1Bifix¯^i,x¯^i+1,f+∆fi+Bnfnx¯^n,uf+∆fn+u,where x=x1⋮xn,A=-k1⋮I-kn0⋯0,K=k1⋮kn,Bi=0⋮1⋮0,Bn=0⋮1. The vector K is chosen to make matrix A to be a strict Hurwitz matrix; i.e., for given a matrix Q=QT>0 there exists a matrix P=PT>0 satisfying(13)ATP+PA=-2Q.


## 2.2. Design of FLSs and State Observer

To tackle the unknown nonlinear functions, the fuzzy logic systems are introduced as follows.Rulej: if x1 is Nj1, and x2 is Nj2 and ⋯ and xn is Njn, then (14)yxisMj,j=1,2,…,m, where y is the output of the system, Nj1 and Mj denote fuzzy sets, and m represents the number of fuzzy rules.By using the singleton fuzzifier, center average defuzzification and product inference [25, 28], the final output can be expressed as follows:(15)yx=∑j=1my~j∏l=1nμNj1xl∑j=1m∏l=1nμNj1xl,where y~j=maxy∈RμMj(y); μNj1(xl) and μMj(y) stand for membership functions with respect to fuzzy sets Nj1 and Mj, respectively.Define(16)φx=∏l=1nμNj1xl∑j=1m∏l=1nμNj1xl as the basis function vector. The ideal constant weight vector is =[y~1,y~2,…,y~m]T=[θ1,θ2,…,θm]T, φ(x)=[φ1(x),φ2(x),…,φm(x)]T, and φi(x) is chosen as Gaussian function; i.e., for i=1,2,…,m, (17)φix=exp⁡-x-μiTx-μiηi2, where μi=[μi1,μi2,…,μin]T is the center vector, and ηi is the width of the Gaussian function.Then, the fuzzy outputy in (15) is described as (18)yx=θTφx.Accordingly, assume that the unknown nonlinear functions in (1) are approximated by the following fuzzy logic systems:(19)f^ix¯^i,x^i+1,f∣θ^i=θ^iTφix¯^i,x^i+1,f,1≤i≤n,where x^n+1,f=uf. The optimal weight vector θi∗ is defined as (20)θi∗=arg⁡minθi∈Ωisupx¯^i,x^i+1,f∈Ui1×Ui2f^ix¯^i,x^i+1,f∣θ^i-f^ix¯^i,x^i+1,f∣θi∗, where Ωi and Ui1×Ui2 are compact regions for θi and (x¯^i,x^i+1,f), respectively.Define(21)fix=θi∗Tφix+εix=θ^iTφix+δix,where εi∗ and δi∗(i=1,2,…,n) are known constants satisfying |εi|≤εi∗ and |δi|≤δi∗. Letting wi=εi-δi,i=1,2,…,n, it is clear that there is an unknown constant wi∗>0 such that |wi|≤wi∗=εi∗+δi∗.Based on (12), design a fuzzy state observer as(22)x^˙i=x^i+1+f^ix¯^i,x^i+1,f+kiy-x^1,x^˙n=u+f^nx¯^n,uf+kny-x^1,y=x1.Let ei=xi-x^i be an observer error vector. Then from (12) and (22), we have(23)e˙=Ae+δ+∆F,where δ=[δ1,δ2,…,δn]T and ∆F=[∆f1,∆f2,…,∆fn]T. Substituting (23) into (22), we have(24)e˙i=Ae+δ+∆F,x^˙i=x^i+1+f^ix¯^i,x^i+1,f+kie1,x^˙n=u+f^nx¯^n,uf+kne1,y=x1.Consider the following Lyapunov function candidate:(25)V0=12eTPe.Computing the time derivative of V0, one has(26)V˙0=eTPe˙=eTPAe+δ+∆F.According to Assumption 6, we get(27)∆fi=fix-i,xi+1-fix-^i,x^i+1,f≤mie2+mix^i+1-x^i+1,f≤mie2+miτi+1=mie2+τi+1′,where τi+1′=miτi+1. It is clear that(28)∆F2≤nm2e2+∑i=2n+1τi′2,eTP∆F≤ηP2e2+14ηnm2e2+ηP2e2+14η∑i=2n+1τi′2≤2ηP2e2+14ηnm2e2+14η∑i=2n+1τi′2.Using Young’s inequality and (28), we get(29)V˙0≤-λminQe2+ξP2e2+14ξδ∗2+2ηP2e2+14ηnm2e2+14η∑i=2n+1τi′2.To simplify the notation, let V˙0≜-p0e2+E0, where -p0=-λmin(Q)+ξP2+2ηP2+(1/4η)nm2,E0=(1/4ξ)δ∗2+(1/4η)(∑i=2n+1τi′)2.


## 3. Main Results

In this section, we propose a generalised design to deal with partial state constraint and the adaptive fuzzy output feedback controller which based on the backstepping technique and fuzzy state observer will be proposed. To guarantee the system performance, the virtual control signals and adaptive laws are designed. A new design procedure is presented which may cover some results related to the full state constraint [37]. To ensure that xi remains in the constrained region, we give the feasibility conditions with respect to the design parameters and an initial state region, i.e., x(0)∈Ωx(0), where Ωx(0):={x∈Rn:-ai≤xi≤bi,i=1,2,…,n} with ai<kci and bi<kci.Let the tracking error(30)z1=y-ydand the variables(31)zi=x^i-αi-1,i=2,…,n,where αi-1 is a virtual controller to be designed in Step i.DefineKzi=zi/(kbi2-zi2),kb1=kc1-Y0 and Ωzi={|zi|<kbi,i=1,2,…,n}. Consider the BLF candidate combined with quadratic Lyapunov function as(32)Vn=V0+∑i=1ns12log⁡kbi2kbi2-zi2+∑i=1n12γiθ~iTθ~i+12γ1¯ε~12+∑i=2n12γi¯w~i2+U,where U=∑j=ns+1n(1/2)zj2,j=ns+1,…,n, and log(∗) stands for the natural logarithm of ∗. γ1,γi,γ¯1, and γ¯i are positive constants. It can be proved that Vn is continuously differentiable and positive definite on Ωzi.The detailed design procedures are given below.Step 1. According to (19), (21), (24), and (30), the derivative of z1 is calculated as follows:(33)z˙1=y˙-y˙d=z2+α1+θ1∗φ1+ε1x+∆F1-y˙d.The Lyapunov function is defined as(34)V1=V0+12log⁡kb12kb12-z12+12γ1θ~1Tθ~1+12γ¯1ε~12.Then, substituting (33) into (34), one can have(35)V˙1=V˙0+z1z˙1kb12-z12-1γ1θ~1Tθ^˙1-1γ¯1ε~1ε^˙1=V˙0+Kz1z2+α1+θ1∗Tφ1+ε1x+∆F1-y˙d-1γ1θ~1Tθ^˙1-1γ¯1ε~1ε^˙1.Design a virtual controller α1, adaptive law θ^˙1, and ε^˙1 as(36)α1=-λ1z1-θ^1Tφ1+y˙d-ε^1tanh⁡Kz1k-12Kz1,θ^˙1=-σ1θ^1+γ1Kz1φ1,ε^˙1=-σ¯1ε^1+γ¯1Kz1tanh⁡Kz1k,where θ^1 is the estimation of θ1∗ and θ~j=θj∗-θ^j, ε~1=ε1∗-ε^1, σ1,σ¯1, and k are the positive constants to be designed, respectively. Substituting (36) into (35), it yields(37)V˙1=V˙0+Kz1z2+α1+θ1∗Tφ1+ε1x+∆F1-y˙d-1γ1θ~1T-σ1θ^1+γ1Kz1φ1-1γ¯1ε~1-σ¯1ε^1+γ¯1Kz1tanh⁡Kz1k=V˙0+Kz1z2-λ1z1-12Kz1+∆F1+ε1xKz1-ε1∗Kz1tanh⁡Kz1k+σ1γ1θ~1Tθ^1+σ¯1γ¯1ε~1ε^1≤V˙0+Kz1z2-λ1z1-12Kz1+∆F1+0.2785kε1∗+σ1γ1θ~1Tθ^1+σ¯1γ¯1ε~1ε^1.S t e p  i  i = 2,3 , … , n s . The derivative of zi is calculated as follows:(38)z˙i=x^˙i-α˙i-1=zi+1+αi+θi∗φi+wix+kie1-α˙i-1.Choose the following Lyapunov function candidates:(39)Vi=Vi-1+12log⁡kbi2kbi2-zi2+12γiθ~iTθ~i+12γ¯iw~i2,where w~i=wi∗-w^i. Similar to (35), one can have(40)V˙i=V˙i-1+ziz˙ikbi2-zi2-1γiθ~iTθ^˙i-1γ¯iw~iw^˙i=V˙i-1-1γiθ~iTθ^˙i-1γ¯iw~iw^˙i+Kzizi+1+αi+θi∗φi+wix+kie1-α˙i-1.The virtual controller αi, adaptive law θ^˙i, and ε^˙i are designed as(41)αi=-λizi-θ^iTφi-kie1+α˙i-1-w^itanh⁡Kzik-Kzi-1ziKzi,θ^˙i=-σiθ^i+γiKziφi,w^˙i=-σ¯iw^i+γ¯iKzitanh⁡Kzik, where σi and σ¯i are positive constants. Substituting (41) into (40), the following inequality can be obtained:(42)V˙i=V˙i-1-1γ¯iw~i-σ¯iw^i+γ¯iKzitanhKzik+Kzizi+1+αi+θi∗φi+wix+kie1-α˙i-1-1γiθ~iT-σiθ^i+γiKziφi≤V˙0+Kzizi+1-∑r=1iλrKzrzr+12∆F12+0.2785kε1∗+0.2785k∑r=2iwr∗+∑r=1iσrγrθ~rTθ^r+σ¯1γ¯1ε~1ε^1+∑r=2iσ¯rγ¯rw~rw^r.Step n s + 1. The derivative of zns+1 is calculated as follows:(43)z˙ns+1=x^˙ns+1-α˙ns=zns+2+αns+1+θns+1∗φns+1+wns+1x+kns+1e1-α˙ns.The Lyapunov function is chosen as(44)Vns+1=Vns+12zns+12+12γns+1θ~ns+1Tθ~ns+1+12γ¯ns+1w~ns+12.Then, we have(45)V˙ns+1=V˙ns+zns+1z˙ns+1-1γns+1θ~ns+1Tθ^˙ns+1-1γ¯ns+1w~ns+1w^˙ns+1=V˙ns+zns+1zns+2+αns+1+θns+1∗φns+1+wns+1x+kns+1e1-α˙ns-1γns+1θ~ns+1Tθ^˙ns+1-1γ¯ns+1w~ns+1w^˙ns+1.Design a virtual controller αns+1, adaptive law θ^˙ns+1, and ε^˙ns+1 as(46)αns+1=-λns+1zns+1-θ^ns+1Tφns+1-kns+1e1+α˙ns-w^ns+1tanh⁡zns+1k-Kzns,θ^˙ns+1=-σns+1θ^ns+1+γns+1zns+1φns+1,w^˙ns+1=-σ¯ns+1w^ns+1+γ¯ns+1zns+1tanhzns+1k.Substituting (46) into (45), the following inequality is obtained:(47)V˙ns+1=V˙ns+zns+1zns+2+αns+1+θns+1∗φns+1+wns+1x+kns+1e1-α˙ns-1γ¯ns+1w~ns+1-σ¯ns+1w^ns+1+γ¯ns+1zns+1tanhzns+1k-1γns+1θ~ns+1T-σns+1θ^ns+1+γns+1zns+1φns+1≤V˙0+zns+1zns+2-∑r=1nsλrKzrzr-λns+1zns+12+12∆F12+0.2785kε1∗+0.2785k∑r=2ns+1wr∗+∑r=1ns+1σrγrθ~rTθ^r+σ¯1γ¯1ε~1ε^1+∑r=2ns+1σ¯rγ¯rw~rw^r.S t e p j j = n s + 2 , … , n - 1 . The derivative of zj is calculated as follows:(48)z˙j=x^˙j-α˙j-1=zj+1+αj+θj∗φj+wjx+kje1-α˙j-1.The Lyapunov function is defined as(49)Vj=Vj-1+12zj2+12γjθ~jTθ~j+12γ¯jw~j2.Then, one can have(50)V˙j=V˙j-1+zjz˙j-1γjθ~jTθ^˙j-1γ¯jw~jw^˙j=V˙j-1-1γjθ~jTθ^˙j-1γ¯jw~jw^˙j+zjzj+1+αj+θj∗φj+wjx+kje1-α˙j-1.Design a virtual controller αs+1, adaptive law θ^˙s+1, and ε^˙s+1 as(51)αj=-λjzj-θ^jTφj-kje1+α˙j-1-w^jtanh⁡zjk-zj-1,θ^˙j=-σjθ^j+γjzjφj,w^˙j=-σ¯jw^j+γ¯jzjtanhznjk. Substituting (51) into (50), the following inequality can be obtained:(52)V˙j=V˙j-1-1γjθ~jT-σjθ^j+γjzjφj+zjzj+1+αj+θj∗φj+wjx+kje1-α˙j-1-1γ¯jw~j-σ¯jw^j+γ¯jzjtanhzjk≤V˙0+zjzj+1-∑r=1nsλrKzrzr-∑r=ns+1jλrzr2+12∆F12+0.2785kε1∗+0.2785k∑r=2jwr∗+∑r=1jσrγrθ~rTθ^r+σ¯1γ¯1ε~1ε^1+∑r=2jσ¯rγ¯rw~rw^r.Step n . The derivative of zn is calculated as follows:(53)z˙n=x^˙n-α˙n-1=u+θn∗φn+wnx+kne1-α˙n-1.Define the Lyapunov function as(54)Vn=Vn-1+12zn2+12γnθ~nTθ~n+12γ¯nw~n2.Then, one can have(55)V˙n=V˙n-1+znz˙n-1γnθ~nTθ^˙n-1γ¯nw~nw^˙n=V˙n-1+znu+θn∗φn+wnx+kne1-α˙n-1-1γnθ~nTθ^˙n-1γ¯nw~nw^˙n.We choose the control law and the adaptive laws as(56)u=-λnzn-θ^nTφn-kne1+α˙n-1-w^ntanh⁡znk-zn-1,θ^˙n=-σnθ^n+γnznφn,w^˙n=-σ¯nw^n+γ¯nzntanh⁡znk.Substituting (56) into (55), the following equality can be obtained:(57)V˙n=V˙n-1+znu+θn∗φn+wnx+kne1-α˙n-1-1γ¯nw~n-σ¯nw^n+γ¯nzntanh⁡znk-1γnθ~nT-σnθ^n+γnznφn≤V˙0-∑r=1nsλrKzrzr-∑r=ns+1nλrzr2+12∆F12+0.2785kε1∗+0.2785k∑r=2nwr∗+∑r=1nσrγrθ~rTθ^r+σ¯1γ¯1ε~1ε^1+∑r=2nσ¯rγ¯rw~rw^r.According to Young’s inequality, we have (58)σiγiθ~iTθ^i=σiγiθ~iTθi∗-θ~i≤σi2γiθi∗Tθi∗-θ~iTθ~i. Then, the derivative of Vn is given by(59)V˙n≤V˙0-∑r=1nsλrKzrzr-∑r=ns+1nλrzr2+12∆F12+0.2785kε1∗+0.2785k∑r=2nwr∗+∑r=1nσr2γrθr∗Tθr∗-θ~rTθ~r+σ¯12γ¯1ε1∗Tε1∗-ε~1Tε~1+∑r=2nσ¯r2γ¯rwr∗Twr∗-w~rTw~r.Combined with the above analysis, we have come to the following conclusions.Theorem 9.
Consider system (1). Assumptions 5 and 6 hold on the sets Ωzi. For the virtual controller αi,i=1,2,…,n-1, in (36), (41), and (51) and the actual controller u in (56) and the adaptive laws in (36), (41), (51), and (56), the following properties hold:
(i) The proposed adaptive control scheme can guarantee that the tracking error converges to a bounded compact setΩD={|z1|<D}.
(ii) All the signals in the closed-loop systems are bounded.
(iii) The partial state constraints are not violated.Proof.
(i) From Lemma1, (59) can be rearranged into the form(60)V˙≤-ρV+c,where ρ=min(4η+(1/η)P2)/8λminQ-ξ+2ηP2-(1/4η)nm2-1/2m12,1/2λi,1/σj,1/σ¯1,1/σ¯j, c=En+∑j=1n(σj/γj)θj∗Tθj∗+(σ¯1/γ¯1)ε1∗2+∑j=2nσ¯j/γ¯jwj∗2,En=E0+(1/2)τ2′2+k′ε1∗+∑j=1nk′wj∗2, and k′=0.2785k. Adding eρt to both sides of the above inequality and integrating it over [0,t], it has(61)Vt≤V0-cρe-ρt+cρ≤V0+cρ≜V0+C.
From the preceding inequality andV(t), we can conclude that log⁡(kbj2/(kbj2-zj2)), ei, θ~j, ε~1, w~j and zj are bounded, and Kzi is bounded. Due to the boundedness of θj∗, ε1∗, wj∗ and θ~j=θj∗-θ^j, ε~1=ε1∗-ε^1, w~j=wj∗-w^j, then θ^j, ε^1, and w^j are bounded. It is easy to see that the tracking error converges to a bounded compact set.
(ii) According to (61), we define B=2(V(0)+C)/λmin(P). From the form of V(t), we have ei≤e≤B. Since x1=z1+yd(t) and |yd(t)|≤Y0, it can be shown that |x1|≤|z1|+|yd(t)|<kb1+Y0. Suppose kc1=kb1+Y0, and then |x1|<kc1. To show |x2|<kc2, it needs to confirm that there exists a positive constant α¯1 such that |α1|≤α¯1. The boundedness of x1, yd, y˙d, θ^1, ε^1, Kz1 can be guaranteed because α1=α1(x1,yd,y˙d,θ^1,ε^1,Kz1). It is easy to know from the definition of α1 that the supremum of α1 exists. In view of |z2|<kb2 and x^2=z2+α1, it has |x^2|<kb2+α¯1. Due to x2=x^2+e2, the inequality |x2|≤|x^2|+|e2|<kb2+α¯1+B holds. Let kc2=kb2+α¯1+B, and then |x2|<kc2. Similarly, it can show that |xi+1|<kci+1,i=2,3,…,ns-1, after verifying |αi|≤α¯i. Since (41), (51), and |αi-1|≤α¯i-1, then the controller u is bounded. From the above analysis, we conclude that all the signals of the closed-loop system x,αi,u,z,θ^j,ε^1,w^j are bounded.
(iii) From the construction of (61), we get (62)log⁡kb12kb12-z12≤2V0-Ce-ρt+2C.Taking exponentials on both sides, one has (63)kb12kb12-z12≤e2V0-Ce-ρt+2C. Define (64)D=kb11-e-2V0-Ce-ρt-2C. Because kb12-z12>0, it is easy to get |z1|≤D. If V(0)=C then |z1|≤kb11-e-2C=D holds. If V(0)≠C, it can be concluded that, given any D>kb11-e-2C, there exists T such that, for any t>T, it has |z1|≤D. This implies that |z1|≤kb11-e-2C as t→∞. That means z1 can be made arbitrarily small. From above analysis, we can get that |xi+1|<kci+1(i=2,3,…,ns-1), |αi-1|≤α¯i-1, and the controller u is bounded. The variables zi=x^i-αi-1, so that zi(i=1,2,…,n) is bound, and the systems states are not violated. This completes the proof.


## 4. Illustrative Example

In this section, we give an example to show how to apply the results proposed in this paper to investigate the stabilization of nonlinear pure feedback systems subject to partial state constraints.Let us consider the following nonlinear systems:(65)x˙1=x1cos⁡x1+2x12+0.6x2,x˙2=x1x2+u+0.4sin⁡u,y=x1,where the state constraints are |x1|<0.5; the reference signal is given as yd=0.1sin(πt/3)+0.1cos⁡(t/3) for the tracking problem. It is unnecessary to give precise knowledge of the initial state x(0). In this simulation, the figures are specific cases. Let x1=0.05,x2=0.2,x^1=0,x^2=0. Fuzzy membership functions for the variables x^1,x^2,x^2f, and uf are given as follows:(66)μF1′x^1=exp⁡-x^1-5+2l22,μF2′x^2=exp⁡-x^2-3+l25,μF3′x^2f=exp⁡-x^2f-3+l25,μF4′uf=exp⁡-uf-5+3l27,l=1,2,…,5.From [35], the fuzzy logic systems can be represented as(67)f^1x^1,x^2f∣θ1=θ1Tφx^1,x^2f,f^2x^1,x^2,uf∣θ2=θ2Tφx^1,x^2,uf.Letting k1=5 and k2=5, then the state observer can be constructed as(68)x^˙1=x^2+f^1x^1,x^2f∣θ1+5y-x^1,x^˙2=u+f^2x^1,x^2,uf∣θ2+5y-x^1.The virtual control function α1 and controller u as well as the adaptive laws θ^1,θ^2,ε^1 and w^2 are as follows:(69)α1=-λ1z1-θ^1Tφ1+y˙d-ε^1tanh⁡Kz1k-12Kz1,u=-λ2z2-θ^2Tφ2-k2e1+α˙1-w^2tanh⁡z2k-z1,θ^˙1=-σ1θ^1+γ1Kz1φ1,ε^˙1=-σ¯1ε^1+γ¯1Kz1tanhKz1k,θ^˙2=-σ2θ^2+γ2z2φ2,w^˙2=-σ¯2w^2+γ¯2z2tanh⁡z2k.Set the design parameters in the above control scheme as (70)λ1=1,λ2=2,γ1=1,γ2=1,γ¯1=1,γ¯2=1;σ1=1,σ2=1,σ¯1=1.5,σ¯2=0.5,k=0.2.Let kb1=0.2, and from calculation we know it is valid. Choose the initial conditions of adaptive parameters as θ^1=(1,1,1,1,1),ε^1=1,θ^2=(1,1,1,1,1),w^2=1.The simulation is given in Figures1–8. Figure 1 shows the trajectory of the state x. Figure 2 is the swing curve of the control signal u. Figure 3 is the state x1 which remains in the constraint region; it shows the trajectory of the state x. Figure 4 stands for the variables of z1 and z2, and these error variables can not violate their bounds. Figures 5 and 6 are used to illustrate the trajectories of the system states x1,x2 and the observer states x^1,x^2. Figures 7 and 8 show that all the signals in the closed-loop system are bounded. It is clear that the nonlinear pure feedback system subject to partial constraints under the output feedback law is bounded. From the simulation, we can conclude that the results proposed in Theorem 9 are very practicable in stability of nonlinear pure feedback systems with partial constraints. Meanwhile, it is a good tool in analyzing the stability problems of some classes of nonlinear pure feedback systems in the presence of constraint.Figure 1
The tracking result.Figure 2
The controlu.Figure 3
Statesx1 and x2.Figure 4
Variablesz1 and z2.Figure 5
Statesx1 and x^1.Figure 6
Statesx2 and x^2.Figure 7
Parameter estimatesθ^1, θ^2.Figure 8
Parameter estimatesε^1, w^2.


## 5. Conclusions

In this paper, the tracking control problem of a class of nonlinear pure feedback systems subject to partial state constraints and adaptive fuzzy output feedback controls has been investigated by use of BLFs and backstepping. The output feedback control law, by which the stability of the closed-loop system is guaranteed, is determined by constraints. Simulations show that the results obtained in this paper are very practicable in analyzing the stability of some classes of nonlinear pure feedback systems.



---

*Source: 9624938-2018-07-24.xml*