# Adaptive Fuzzy Control for Stochastic Pure-Feedback Nonlinear Systems with Unknown Hysteresis and External Disturbance

Authors: Xikui Liu; Yingying Ge; Yan Li
Journal: Complexity
 (2018)

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## Abstract

This paper solves the tracking control problem of a class of stochastic pure-feedback nonlinear systems with external disturbances and unknown hysteresis. By using the mean-value theorem, the problem of pure-feedback nonlinear function is solved. The direction-unknown hysteresis problem is solved with the aid of the Nussbaum function. The external disturbance problems can be solved by defining new Lyapunov functions. Using the backstepping technique, a new adaptive fuzzy control scheme is proposed. The results show that the proposed control scheme ensures that all signals of the closed-loop system are semiglobally uniformly bounded and the tracking error converges to the small neighborhood of origin in the sense of mean quartic value. Simulation results illustrate the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

Hysteresis is widely found in mechanical equipment, which severely limits performance of the system and even leads to system instability. Therefore, the control problem of the system with hysteresis has been paid more and more attention. For the adaptive control system in [1–7], scholars study problems in different directions, such as hysteresis input in [1], dead zone input in [2], and time-delay input in [3]. There are several common hysteresis phenomena. The author solves the backlash-like hysteresis problem in [4]. The authors solve a class of traditional P-I hysteresis problems and propose an adaptive backstepping scheme in [5]. Scholars have studied a class of nonlinear systems with generalized P-I hysteresis inputs in [6, 7]. In addition, scholars have studied unmodeled dynamics deterministic systems in [8] and uncertain nonsmooth deterministic systems in [9]. The finite-time control problem of nonlinear deterministic systems is studied in [10]. However, the system studied above is a deterministic system and ignores the effects of stochastic disturbance.Stochastic disturbance often occur in many systems, and the adaptive control problems of stochastic nonlinear systems are more difficult than those of deterministic nonlinear systems. Stochastic disturbance is added to the system, and differential operations on Lyapunov functions are more complicated. The research of stochastic nonlinear systems has been increasingly discussed in [11–21]. A finite time control method of switched stochastic systems is proposed in [11]. The control problem of nonlinear stochastic systems is discussed in [12–14]. Using the mean value theorem to solve the pure-feedback nonlinear function, the complexity of the system is increased. Further, more researchers have studied other types of stochastic nonlinear systems, such as from stochastic systems with unknown backlash-like hysteresis in [15] to pure-feedback stochastic nonlinear systems with unknown dead-zone input in [16], from SISO systems in [17] to pure-feedback MIMO systems in [18]. The stochastic systems with time-varying delays are proposed in [19, 20] and the stochastic systems with unknown direction hysteresis are proposed in [21]. The above studies have considered the effect of stochastic disturbance, without considering the external disturbances.However, external disturbance often exists in practice. External disturbance cannot be ignored; it is also a source of system instability in practice. Scholars have extended the system without external disturbances in [22–25] to systems with external disturbances in [26–30], such as a determination system with external disturbances in [29] and a stochastic system with external disturbances in [30]. The system with external disturbances makes the design of the controller more difficult.In this paper, the control problem of pure-feedback stochastic nonlinear system with external disturbance and unknown hysteresis is studied. For the determination system with external disturbance in [26], stochastic terms are not considered. The adaptive fuzzy control problem for stochastic nonlinear systems is studied in [14], without considering external disturbances. Therefore, a more general nonlinear system is processed in this paper. Furthermore, the difficulty is to deal with unknown direction hysteresis in [21]. The difficulty of this paper is how to solve the influence of external disturbance on the unknown direction hysteresis and ensure the stability of the nonlinear system. This problem can be solved by designing appropriate Lyapunov functions. The major contributions of this paper are described below:(1)
The tracking control problem of the stochastic pure-feedback nonlinear systems with stochastic disturbances, direction-unknown hysteresis, and external disturbances is solved in this paper.(2)
In thenth step of the backstepping design, the Lyapunov function with the external disturbance term Δ(t) is defined, and the external disturbance problem is solved. A new adaptive control scheme is proposed.The remainder of this article is as follows. The second part puts forward the preparation work and the problem formulation. The third part is the design process of the adaptive control method. The fourth part gives the simulation example. The fifth part summarizes the full text.


## 2. Preparation and Problem Formulation

### 2.1. Preliminary Knowledge

The stochastic nonlinear system is expressed as follows:(1)dx=fx,tdt+hx,tdw,where x∈Rn is the state variable, f:Rn×R+→Rn, h:Rn×R+→Rn×r are continuous functions. w indicates that an independent r-dimension standard Brownian motion, which is defined on the complete probability space.Definition 1 (see [31]).
For a quadratic continuous differentiable functionV(x,t), define a derivative operator L expressed as follows:(2)LV=∂V∂t+∂V∂xf+12TrhT∂2V∂x2h,where Tr is a trace of matrix.Remark 2.
The∂2V/∂x2 in Ito^ correction term 1/2Tr{hT∂2V/∂x2h} makes the design of the control scheme in the stochastic system more complicated than the design of the control scheme in the determined system.Lemma 3 (see [32]).
For the stochastic system (1), let V(t) and ζ(t) are smooth functions defined on [0,tf), V(t)≥0, ∀t∈[0,tf); the function ξ satisfes 0<|ξ(t)|≤l<∞ (l is a constant); N(.) is a Nussbaum-type function. The following inequality is satisfied: (3)Vt≤e-ct∫0tξtNζζ˙+ζ˙ecτdτ+η+Mt,∀t∈0,tf,where η is a nonnegative variable, M(t) is a real valued continuous local martingale. Then the functions V(t), ζ(t) and ∫0t‍(ξ(t)N(ζ)ζ˙+ζ˙)dτ are bounded on [0,tf).Lemma 4 (see [33]).
For∀(x,y)∈R2, the following inequality is established:(4)xy≤εppxp+1qεqyq,where ε>0, p>1, and (p-1)(q-1)=1.Lemma 5 (see [3]).
Consider the following dynamic system:(5)θ^˙t=-γθ^t+kρt,where γ and k are positive constants and ρ(t) is a positive function. Then for ∀t≥t0 and any bounded initial condition θ^(t0)≥0, we have θ^(t)≥0.


### 2.2. Problem Formulation

This paper considers the following stochastic pure-feedback nonlinear system:(6)dxi=fix¯i,xi+1dt+ψiTx¯idw,1≤i≤n-1,dxn=fnx¯n,u+Δtdt+ψnTx¯ndw,y=x1,where x=[x1,x2,…,xn]T∈Rn and x¯i=[x1,x2,…,xi]T∈Ri are the state vectors, y∈R is system output, and w is defined as (1). fi(.):Ri→R and ψi(.):Ri→Rr(i=1,2,…,n) are unknown nonlinear functions. Δ(t) is a bounded external disturbance. u∈R is the system input and the output of an unknown Bouc-Wen hysteresis is defined as follows [1]:(7)u=Hv=μ1v+μ2ζ,where μ1 and μ2 are unknown constants and have the same sign. v(t)∈R is the input of the hysteresis. ζ is the auxiliary variable, ζ(t0)=0. In [1] we know that ζ is bounded and can be expressed as: (8)ζ≤1β+χn,where β, χ, n are the unknown hysteresis parameters, and β>|χ|, n>1.Remark 6.
This article has stochastic termψiT(x¯i)dw, and the hysteresis output u is different from [29]. If we ignore the external disturbance Δ(t), the results of this paper are the same as [21]. Therefore, this paper considers a more general nonlinear system.
For the system (6), define (9)gix¯i,xi+1=∂fix¯i,xi+1∂xi+1,i=1,2,…,n,with xi+1=u.Assumption 7.
For1≤i≤n, there is an unknown constant bM such that(10)1≤gix¯i,xi+1≤bM<∞,∀x¯i,xi+1∈Ri×R.By using the mean-value theorem, the pure-feedback nonlinear functions in (6) can be expressed as(11)fix¯i,xi+1-fix¯i,xi+10=gix¯i,ηixi+1-xi+10,where xn+10=u0=H(v0), xn+1=u=H(v), and ηi is point between xi+1 and xi+10.
Substituting (11) into (6), the control system can be rewritten as(12)dxi=fix¯i,xi+10+gix¯i,ηixi+1-xi+10dt+ψiTx¯idw,1≤i≤n-1,dxn=fnx¯n,u0+gnx¯n,ηnu-u0+Δtdt+ψnTx¯ndw,y=x1.


### 2.3. Fuzzy Logic Systems

In order to approximate a continuous functionf(x) with a fuzzy logic system, consider the following fuzzy rules:Rl: If x1 is F1l and … and xn is Fnl.Theny is Gl, l=1,2,…,N,wherex=[x1,x2,…,xn]T∈Rn is the input of system, y∈R is the output of the system, Fil and Gl are fuzzy sets in R, and N is the number of rules. The output form of the fuzzy logic system is as follows:(13)yx=∑l=1NΦl∏i=1nμFilxi∑l=1N∏i=1nμFilxi,where(14)Φl=maxy∈RμGly,Φ=Φ1,Φ2,…,ΦNT.Letting(15)ξlx=∏i=1nμFilxi∑l=1N∏i=1nμFilxi,ξX=ξ1x,ξ2x,…,ξNxT,the fuzzy system is written as(16)yx=ΦTξx.Lemma 8 (see [27]).
Letf(x) be a continuous function defined on a compact set Ω. Then, for ∀ε>0, there exists a fuzzy logic system ΦTξ(x) such that(17)supx∈Ωfx-ΦTξx≤ε.
The goal of this paper is to design an adaptive controller, so that the system outputy converges to the reference signal yd and all signals of the closed-loop system are bounded.
Define a vector function asy¯d(i)=[yd,yd(1),…,yd(i)]T, i=1,2,…,n, where yd(i) denotes the ith order derivative of yd.Assumption 9 (see [21]).
The reference signalyd(t) and its order derivatives up to the nth time are continuous and bounded.


## 2.1. Preliminary Knowledge

The stochastic nonlinear system is expressed as follows:(1)dx=fx,tdt+hx,tdw,where x∈Rn is the state variable, f:Rn×R+→Rn, h:Rn×R+→Rn×r are continuous functions. w indicates that an independent r-dimension standard Brownian motion, which is defined on the complete probability space.Definition 1 (see [31]).
For a quadratic continuous differentiable functionV(x,t), define a derivative operator L expressed as follows:(2)LV=∂V∂t+∂V∂xf+12TrhT∂2V∂x2h,where Tr is a trace of matrix.Remark 2.
The∂2V/∂x2 in Ito^ correction term 1/2Tr{hT∂2V/∂x2h} makes the design of the control scheme in the stochastic system more complicated than the design of the control scheme in the determined system.Lemma 3 (see [32]).
For the stochastic system (1), let V(t) and ζ(t) are smooth functions defined on [0,tf), V(t)≥0, ∀t∈[0,tf); the function ξ satisfes 0<|ξ(t)|≤l<∞ (l is a constant); N(.) is a Nussbaum-type function. The following inequality is satisfied: (3)Vt≤e-ct∫0tξtNζζ˙+ζ˙ecτdτ+η+Mt,∀t∈0,tf,where η is a nonnegative variable, M(t) is a real valued continuous local martingale. Then the functions V(t), ζ(t) and ∫0t‍(ξ(t)N(ζ)ζ˙+ζ˙)dτ are bounded on [0,tf).Lemma 4 (see [33]).
For∀(x,y)∈R2, the following inequality is established:(4)xy≤εppxp+1qεqyq,where ε>0, p>1, and (p-1)(q-1)=1.Lemma 5 (see [3]).
Consider the following dynamic system:(5)θ^˙t=-γθ^t+kρt,where γ and k are positive constants and ρ(t) is a positive function. Then for ∀t≥t0 and any bounded initial condition θ^(t0)≥0, we have θ^(t)≥0.


## 2.2. Problem Formulation

This paper considers the following stochastic pure-feedback nonlinear system:(6)dxi=fix¯i,xi+1dt+ψiTx¯idw,1≤i≤n-1,dxn=fnx¯n,u+Δtdt+ψnTx¯ndw,y=x1,where x=[x1,x2,…,xn]T∈Rn and x¯i=[x1,x2,…,xi]T∈Ri are the state vectors, y∈R is system output, and w is defined as (1). fi(.):Ri→R and ψi(.):Ri→Rr(i=1,2,…,n) are unknown nonlinear functions. Δ(t) is a bounded external disturbance. u∈R is the system input and the output of an unknown Bouc-Wen hysteresis is defined as follows [1]:(7)u=Hv=μ1v+μ2ζ,where μ1 and μ2 are unknown constants and have the same sign. v(t)∈R is the input of the hysteresis. ζ is the auxiliary variable, ζ(t0)=0. In [1] we know that ζ is bounded and can be expressed as: (8)ζ≤1β+χn,where β, χ, n are the unknown hysteresis parameters, and β>|χ|, n>1.Remark 6.
This article has stochastic termψiT(x¯i)dw, and the hysteresis output u is different from [29]. If we ignore the external disturbance Δ(t), the results of this paper are the same as [21]. Therefore, this paper considers a more general nonlinear system.
For the system (6), define (9)gix¯i,xi+1=∂fix¯i,xi+1∂xi+1,i=1,2,…,n,with xi+1=u.Assumption 7.
For1≤i≤n, there is an unknown constant bM such that(10)1≤gix¯i,xi+1≤bM<∞,∀x¯i,xi+1∈Ri×R.By using the mean-value theorem, the pure-feedback nonlinear functions in (6) can be expressed as(11)fix¯i,xi+1-fix¯i,xi+10=gix¯i,ηixi+1-xi+10,where xn+10=u0=H(v0), xn+1=u=H(v), and ηi is point between xi+1 and xi+10.
Substituting (11) into (6), the control system can be rewritten as(12)dxi=fix¯i,xi+10+gix¯i,ηixi+1-xi+10dt+ψiTx¯idw,1≤i≤n-1,dxn=fnx¯n,u0+gnx¯n,ηnu-u0+Δtdt+ψnTx¯ndw,y=x1.


## 2.3. Fuzzy Logic Systems

In order to approximate a continuous functionf(x) with a fuzzy logic system, consider the following fuzzy rules:Rl: If x1 is F1l and … and xn is Fnl.Theny is Gl, l=1,2,…,N,wherex=[x1,x2,…,xn]T∈Rn is the input of system, y∈R is the output of the system, Fil and Gl are fuzzy sets in R, and N is the number of rules. The output form of the fuzzy logic system is as follows:(13)yx=∑l=1NΦl∏i=1nμFilxi∑l=1N∏i=1nμFilxi,where(14)Φl=maxy∈RμGly,Φ=Φ1,Φ2,…,ΦNT.Letting(15)ξlx=∏i=1nμFilxi∑l=1N∏i=1nμFilxi,ξX=ξ1x,ξ2x,…,ξNxT,the fuzzy system is written as(16)yx=ΦTξx.Lemma 8 (see [27]).
Letf(x) be a continuous function defined on a compact set Ω. Then, for ∀ε>0, there exists a fuzzy logic system ΦTξ(x) such that(17)supx∈Ωfx-ΦTξx≤ε.
The goal of this paper is to design an adaptive controller, so that the system outputy converges to the reference signal yd and all signals of the closed-loop system are bounded.
Define a vector function asy¯d(i)=[yd,yd(1),…,yd(i)]T, i=1,2,…,n, where yd(i) denotes the ith order derivative of yd.Assumption 9 (see [21]).
The reference signalyd(t) and its order derivatives up to the nth time are continuous and bounded.


## 3. Adaptive Control Design

In this part, the adaptive fuzzy control is proposed by the backstepping technique, and the following coordinate transformation is defined to develop the backstepping technique:(18)z1=y-yd,zi=xi-αi-1,i=2,…,n,where αi-1 is an intermediate function to be determined next.In each step of the backstepping method, a fuzzy logic systemΦiTξi(Xi) will be used to approximate an unknown function f¯i. We define a constant θi=Φi2, the estimation error is θ~i=θi-θ^i, θ^i as the estimation of θi, i=1,2,…,n.Step 1. For stochastic pure-feedback systems (12), according to z1=x1-yd, we know that dynamic error is satisfied (19)dz1=f1x¯1,x20+g1x¯1,η1x2-x20-y˙ddt+ψ1Tdw.We choose Lyapunov function as follows:(20)V1=14z14+12λ1θ~12,where λ1 is a positive constant. By (2), (18), and (19), one has (21)LV1=z13f1x¯1,x20+g1x¯1,η1z2+α1-x20-y˙d+32z12ψ1Tψ1-1λ1θ~1θ^˙1.Applying Lemma 4 and Assumption 7, the following inequalities hold:(22)32z12ψ1Tψ1≤34l1-2z14ψ14+34l12,(23)g1x¯1,η1z13z2≤34bMz14+bM4z24,where l1 is a positive constant. Substituting (22) and (23) into (21), we can get(24)LV1≤z13f1x¯1,x20+g1x¯1,η1α1-x20-y˙d+34bMz1+34l1-2z1ψ14+34l12+bM4z24-1λ1θ~1θ^˙1.Defining a new function f¯1=f1+3/4bMz1-y˙d+3/4l1-2z1ψ14, then the above inequality can be rewritten as(25)LV1≤z13f¯1+z13g1α1-x20+34l12+bM4z24-1λ1θ~1θ^˙1.Because f¯1 contains the unknown function f1 and ψ1, it cannot be directly controlled in practice. Therefore, according to Lemma 8, for any given ε1>0, there exists a fuzzy logic system Φ1Tξ1(X1) such that(26)f¯1=Φ1Tξ1X1+δ1X1,δ1X1≤ε1,where X1=(x1,yd,y˙d). According to Lemma 4, it follows that(27)z13f¯1=z13Φ1Tξ1X1+z13δ1X1≤z162a12Φ12ξ1Tξ1+12a12+34z14+14ε14=z162a12θ1ξ1Tξ1+12a12+34z14+14ε14,where a1 is a positive parameter. We choose the following virtual control signal and adaptive law: (28)α1=-k1+34z1-12a12θ^1z13ξ1Tξ1+x20,(29)θ^˙1=λ12a12z16ξ1Tξ1-γ1θ^1,θ^10≥0,where k1 and γ1 are positive constants. Based on (28), Assumption 7, one has (30)z13g1α1-x20≤-k1+34z14-12a12z16θ^1ξ1Tξ1.Substituting (27), (29), and (30) into (25), we have(31)LV1≤-k1z14+bM4z24+34l12+14ε14+γ1λ1θ~1θ^1+12a12.It is noted that(32)γ1λ1θ~1θ^1≤-γ12λ1θ~12+γ12λ1θ12.Substituting (32) into (31), we have(33)LV1≤-k1z14-γ12λ1θ~12+ϱ1+bM4z24,where ϱ1=3/4l12+1/2a12+1/4ε14+γ1/2λ1θ12.Step 2. Since z2=x2-α1 and Ito^ formula, one has(34)dz2=f2x¯2,x30+g2x¯2,η2x3-x30-Lα1dt+ψ2-∂α1∂x1ψ1Tdw,with (35)Lα1=∂α1∂x1f1x¯1,x20+g1x¯1,η1x2-x20+∂α1∂θ^1θ^˙1+∑i=01∂α1∂ydiydi+1+12∂2α1∂x12ψ1Tψ1.Choose stochastic Lyapunov function as (36)V2=V1+14z24+12λ2θ~22,where λ2 is a positive design constant. Using the similar procedure as (21), it follows that (37)LV2=LV1+z23f2x¯2,x30+g2x¯2,η2z3+α2-x30-Lα1+32z22ψ2-∂α1∂x1ψ1Tψ2-∂α1∂x1ψ1-1λ2θ~2θ^˙2.It is noticed that(38)32z22ψ2-∂α1∂x1ψ12≤34l2-2z24ψ2-∂α1∂x1ψ14+34l22,(39)g2x¯2,η2z23z3≤34bMz24+bM4z34,where l2 is a positive parameter. Substituting (33), (38), and (39) into (37), we have(40)LV2≤-k1z14-γ12λ1θ~12+ϱ1+z23f2x¯2,x30+g2x¯2,η2α2-x30-Lα1+bMz2+34l2-2z2ψ2-∂α1∂x1ψ14+34l22+bM4z34-1λ2θ~2θ^˙2.Define a function as f¯2=f2-Lα1+bMz2+3/4l2-2z2ψ2-∂α1/∂x1ψ14. Furthermore, (40) can be rewritten as (41)LV2≤-k1z14-γ12λ1θ~12+ϱ1+z23f¯2+z23g2α2-x30+34l22+bM4z34-1λ2θ~2θ^˙2.Since f¯2 contains the unknown function f1, ψ1, and ψ2, it is not possible in practice. Thus, the fuzzy logic system Φ2Tξ2(X2) is used to approximate f¯2, where X2=[x¯2T,θ^1,y¯d(2)T]T∈ΩZ2. Due to Lemma 8, f¯2 can be written as(42)f¯2=Φ2Tξ2X2+δ2X2,δ2X2≤ε2,where ε2 is any given positive constant. Repeating the method of (27), we have(43)z23f¯2≤z262a22θ2ξ2Tξ2+12a22+34z24+14ε24,where a2 is a positive parameter. We choose the following virtual control signal and adaptive law:(44)α2=-k2+34z2-12a22θ^2z23ξ2Tξ2+x30,(45)θ^˙2=λ22a22z26ξ2Tξ2-γ2θ^2,θ^20≥0,where k2, γ2 are design constants. Similar to (30), the following inequality is obtained:(46)z23g2α2-x30≤-k2+34z24-12a22z26θ^2ξ2Tξ2.Substituting (43), (45) and (46) into (41), we have(47)LV2≤-k1z14-γ12λ1θ~12+ϱ1-k2z24+bM4z34+34l22+14ε24+γ2λ2θ~2θ^2+12a22.It is noted that(48)γ2λ2θ~2θ^2≤-γ22λ2θ~22+γ22λ2θ22,(47), can be rewritten in the form(49)LV2≤-∑j=12kjzj4+γj2λjθ~j2+∑j=12ϱj+bM4z34,where ϱj=γj/2λjθj2+3/4lj2+1/4εj4+1/2aj2, j=1,2.Step i   ( 3 ≤ i ≤ n - 1 ). According to zi=xi-αi-1 and Ito^ formula, one has (50)dzi=fix¯i,xi+10+gix¯i,ηixi+1-xi+10-Lαi-1dt+ψi-∑j=1i-1∂αi-1∂xjψjTdw,with (51)Lαi-1=∑j=1i-1∂αi-1∂xjfj+gjxj+1-xj+10+∑j=1i-1∂αi-1∂θ^jθ^˙j+∑j=0i-1∂αi-1∂ydjydj+1+12∑p,q=1i-1∂2αi-1∂xp∂xqψpTψq.We consider the following Lyapunov function:(52)Vi=Vi-1+14zi4+12λiθ~i2,where λi is a positive constant. Using the similar procedure as (21), it follows that(53)LVi=LVi-1+zi3fi+gizi+1+αi-xi+10-Lαi-1+32zi2ψi-∑j=1i-1∂αi-1∂xjψjTψi-∑j=1i-1∂αi-1∂xjψj-1λiθ~iθ^˙i.It is noticed that(54)32zi2ψi-∑j=1i-1∂αi-1∂xjψj2≤34li2+34li-2zi4ψi-∑j=1i-1∂αi-1∂xjψj4,(55)gix¯i,ηizi3zi+1≤34bMzi4+bM4zi+14,where li is a positive constant. Using the similar procedure as (24), we have(56)LVi≤-∑j=1i-1kjzj4+γj2λjθ~j2+∑j=1i-1ϱj+zi3fi+giαi-xi+10-Lαi-1+bMzi+34li-2ziψi-∑j=1i-1∂αi-1∂xjψj4+34li2+bM4zi+14-1λiθ~iθ^˙i.Define a function as f¯i=fi-Lαi-1+bMzi+3/4li-2ziψi-∑j=1i-1‍∂αi-1/∂xjψj4. Furthermore, (56) can be rewritten as(57)LVi≤-∑j=1i-1kjzj4+γj2λjθ~j2+∑j=1i-1ϱj+zi3f¯i+zi3giαi-xi+10+34li2+bM4zi+14-1λiθ~iθ^˙i.Similarly, The fuzzy logic system ΦiTξi(Xi) is used to approximate f¯i, where Xi=[x¯iT,θ^¯i-1T,y¯d(i)T]T∈ΩZi, with θ^¯i-1=[θ^1,θ^2,…,θ^i-1]T. According to Lemma 8, f¯i can be expressed as (58)f¯i=ΦiTξiXi+δiXi,δiXi≤εi,where εi is any given positive constant. According to the method of (27), we can get (59)zi3f¯i≤zi62ai2θiξiTξi+12ai2+34zi4+14εi4,where ai is a design constant. We choose the following virtual control signal and adaptive law:(60)αi=-ki+34zi-12ai2θ^izi3ξiTξi+xi+10,(61)θ^˙i=λi2ai2zi6ξiTξi-γiθ^i,θ^i0≥0,where ki and γi are positive parameters. Similar to (30), we have (62)zi3giαi-xi+10≤-ki+34zi4-12ai2zi6θ^iξiTξi.Similar to (31), we have (63)LVi≤-∑j=1i-1kjzj4+γj2λjθ~j2+∑j=1i-1ϱj-kizi4+bM4zi+14+34li2+14εi4+γiλiθ~iθ^i+12ai2.It is noted that (64)γiλiθ~iθ^i≤-γi2λiθ~i2+γi2λiθi2.The above inequality can be rewritten as (65)LVi≤-∑j=1ikjzj4+γj2λjθ~j2+∑j=1iϱj+bM4zi+14,where ϱj=γj/2λjθj2+3/4lj2+1/4εj4+1/2aj2, j=1,2,…,i.Step n. Based on the coordinate transformation zn=xn-αn-1 and Ito^ formula, we can get (66)dzn=fnx¯n,u0+gnx¯n,ηnu-u0+Δt-Lαn-1dt+ψn-∑j=1n-1∂αn-1∂xjψjTdw,with (67)Lαn-1=∑j=1n-1∂αn-1∂xjfj+gjxj+1-xj+10+∑j=1n-1∂αn-1∂θ^jθ^˙j+∑j=0n-1∂αn-1∂ydjydj+1+12∑p,q=1n-1∂2αn-1∂xp∂xqψpTψq.Consider stochastic Lyapunov function as follows: (68)Vn=Vn-1+14zn4+12λnθ~n2+12σΔ~2,where λn and σ are positive constants. Denote Δ^ as the estimation of Δ, and the estimation error is Δ~=Δ-Δ^. Similar to procedure (21), it follows that (69)LVn=LVn-1+zn3fn+gnHv-u0+Δ-Lαn-1+32zn2ψn-∑j=1n-1∂αn-1∂xjψjTψn-∑j=1n-1∂αn-1∂xjψj-1λnθ~nθ^˙n-1σΔ~Δ^˙.It is noticed that (70)32zn2ψn-∑j=1n-1∂αn-1∂xjψj2≤34ln2+34ln-2zn4ψn-∑j=1n-1∂αn-1∂xjψj4,where ln is a positive constant. Based on (7), the the following inequality holds (71)gnzn3Hv-u0=gnzn3μ1v-v0+gnzn3μ2ς-ς0.As [21], an even Nussbaum-type function N(ζ)=ζ2cosζ is defined, and the following equality holds: (72)v=-Nζv¯+v0,(73)ζ˙=-γv¯zn3,where v¯ is the auxiliary virtual controller and γ is a positive parameter. Then, the following equality can be obtained: (74)gnzn3μ1v-v0=zn3gnμ1-Nζv¯+v0-v0=-zn3gnμ1Nζv¯-zn3v¯+zn3v¯=-zn3gnμ1Nζ+1v¯+zn3v¯.According to Lemma 4 and (8), it follows that(75)gnzn3μ2ς-ς0≤34bMμ24/3zn4+12bMβ+χ-4/n.Substituting (73)-(75) into (71), we have (76)gnzn3Hv-u0≤1γgnμ1Nζ+1ζ˙+zn3v¯+34bMμ24/3zn4+12bMβ+χ-4/n.According to (65) with (i=n-1), (70), and (76), (69) can be rewritten as follows: (77)LVn≤-∑j=1n-1kjzj4+γj2λjθ~j2+∑j=1n-1ϱj+zn3fn+14bMzn+v¯+34bMμ24/3zn-Lαn-1+Δ+34ln-2znψn-∑j=1n-1∂αn-1∂xjψj4+1γgnμ1Nζ+1ζ˙+12bMβ+χ-4/n+34ln2-1λnθ~nθ^˙n-1σΔ~nΔ^˙.Define a function as f¯n=1/4bMzn+fn+3/4bMμ24/3zn+Δ-Lαn-1+3/4ln-2znψn-∑j=1n-1‍∂αn-1/∂xjψj4. Furthermore, (77) can be rewritten as (78)LVn≤-∑j=1n-1kjzj4+γj2λjθ~j2+∑j=1n-1ϱj+zn3f¯n+zn3v¯+1γgnμ1Nζ+1ζ˙+12bMβ+χ-4/n+34ln2-1λnθ~nθ^˙n-1σΔ~nΔ^˙.For any positive constant εn>0, the fuzzy logic system ΦnTξn(Xn) existed, such that(79)f¯n=ΦnTξnXn+δnXn,δnXn≤εn,Similarly, we can obtain (80)zn3f¯n≤zn62an2θnξnTξn+12an2+34zn4+14εn4,where an is a positive parameter. We choose the following virtual control signal and adaptive law: (81)v¯=-kn+34zn-12an2θ^nzn3ξnTξn,(82)θ^˙n=λn2an2zn6ξnTξn-γnθ^n,(83)Δ^˙=-σdΔ^,where kn, γn, and σd are positive constants. Substituting (80)-(83) into (78), we have (84)LVn≤-∑j=1n-1kjzj4+γj2λjθ~j2+∑j=1n-1ϱj+1γgnμ1Nζ+1ζ˙+12bMβ+χ-4/n+34ln2+12an2+14εn4-knzn4+γnλnθ~θ^+σdσΔ~Δ^.Furthermore, (85)γnλnθ~nθ^n≤-γn2λnθ~n2+γn2λnθn2,σdσΔ~Δ^≤-σd2σΔ~2+σd2σΔ2.The above inequality can be rewritten as (86)LVn≤-∑j=1nkjzj4+γj2λjθ~j2+∑j=1nϱj-σd2σΔ~2+1γgnμ1Nζ+1ζ˙+12bMβ+χ-4/n,where ϱj=γj/2λjθj2+3/4lj2+1/4εj4+1/2aj2, j=1,2,…,n-1, and ϱn=γn/2λnθn2+3/4ln2+1/4εn4+1/2an2+σd/2σΔ2.The control design of the adaptive fuzzy logic system has been completed by using the backstepping technique. The main theorem is described below.Theorem 10.
Consider the stochastic pure-feedback nonlinear system (6) with Assumptions 7–9. For bounded initial conditions, combine with the virtual control signal and the adaptation law (60)-(61) and (81)-(83) guarantee that (i)
all the signals of the closed-loop system are semi-globally uniformly bounded on[0,tf), ∀tf>0;(ii)
the steady-state tracking errorszj converge to ΩZ in the sense of mean quartic value, which is defined as (87)ΩZ=zj∣∑j=1nEzj4≤4ρ,where ρ is defined in (97).Proof.
(i) Let Lyapunov function asV=Vn, defining c=min⁡{σd,4kj,γj,j=1,2,…,n} and d=∑j=1n‍ϱj+1/2bM(β+χ)-4/n. According to (86), it follows that(88)LV≤-cV+1γgnμ1Nζ+1ζ˙+d,t≥0.Multiplying V by ect and based on Ito^ formula, one has (89)dectV=ectcV+LVdt+ectMtdw,where(90)Mt=∂V∂z1ψ1Tx1+∑i=2n∂V∂ziψix¯i-∑j=1i-1∂αi-1∂xjψix¯iT.According to (88)-(89), the following inequality can be obtained: (91)dectV≤ect1γgnμ1Nζ+1ζ˙+ddt+ectMtdw.Integrating (91) on [0,t], one has (92)ectVt-V0≤∫0tecτ1γgnμ1Nζ+1ζ˙dτ+dcect-1+∫0tecτMτdw.Furthermore,(93)Vt≤e-ct1γ∫0tecτgnμ1Nζ+1ζ˙dτ+e-ctV0+dc1-e-ct+e-ct∫0tecτMτdw,∀t∈0,tf.
Letξ(t)=gnμ1; according to Assumption 7 and the definition of μ1, we have 0<|ξ(t)|≤bM|μ1|<+∞. Thus, according to the boundedness of ξ(t) and Lemma 3, we can get that V(t) and ζ and ∫0t‍(gnμ1N(ζ)+1)ζ˙dτ are bounded on [0,tf).
Next according to the definition ofV(t), we can get that zj and θ~j and Δ~ are bounded on [0,tf). Thereby, θ^j and Δ^ are also bounded on [0,tf). Due to z1=x1-yd and yd being bounded, we can get that x1 is bounded. Based on α1 is the function of z1 and θ^1, thus x2=z2+α1 is also bounded. Furthermore, we know that xj,j=3,4,…,n is bounded. Thus, all the signals of the closed-loop system are semiglobally uniformly bounded on [0,tf), ∀tf>0.
(ii) Defineβ=sup1/γ∫0t‍|(gnμ1N(ζ)+1)ζ˙|dτ.(94)e-ct1γ∫0tecτgnμ1Nζ+1ζ˙dτ=1γ∫0tecτ-tgnμ1Nζ+1ζ˙dτ≤1γ∫0tgnμ1Nζ+1ζ˙dτ≤β,t≥0.By taking expectation on (93) and applying (94), E[w]=0, we can get (95)EVt≤β+e-ctEV0+dc1-e-ct,t≥0,and therefore,(96)EVt≤β+dc,t⟶+∞.Let (97)ρ=β+dc,and further, due to V=Vn in the definition of (68), we have (98)E∑j=1nzj4≤4EVt≤4ρ,t⟶+∞.Thus, the steady-state tracking errors zj converge to ΩZ in the sense of mean quartic value.


## 4. Simulation Example

In this section, a simulation example is given to prove the effectiveness of the proposed adaptive control scheme.Example. Consider a second-order pure-feedback stochastic system with unknown hysteresis and external disturbance (99)dx1=x12+5x2+0.5sin⁡x12dt+sin⁡x1dw,dxn=x22sin⁡x22+3u+sin⁡t2dt+cos⁡x1x2dw,y=x1,where x1 and x2 are the state variable, t is the time variable, y is the system output, and u is the unknown hysteresis output. These hysteresis parameters are chosen as μ1=0.5, μ2=1.2, β=1.5, χ=0.8, and n=2. It is obvious that (99) satisfies the assumptions of 7-9. The system control target is that the system output y is tracked to the reference signal yd=0.6sin⁡(0.5x)+0.5sin⁡(x).For all state variables, the fuzzy sets are defined on the interval[-1.5,1.5], the fuzzy membership functions are as follows: (100)μFi1x=exp⁡-0.5x+1.52,μFi2x=exp⁡-0.5x+12,μFi3x=exp⁡-0.5x+0.52,μFi4x=exp⁡-0.5x2,μFi5x=exp⁡-0.5x-0.52,μFi6x=exp⁡-0.5x-12,μFi7x=exp-0.5x-1.52.The virtual control signal and the adaptive law are selected as follows:(101)α1=-k1+34z1-12a12θ^1z13ξ1Tξ1+x20,θ^˙1=λ12a12z16ξ1Tξ1-γ1θ^1,θ^10≥0,v¯=-k2+34z2-12a22θ^2z23ξ2Tξ2,θ^˙2=λ22a22z26ξ2Tξ2-γ2θ^2,Δ^˙=-σdΔ^,where z1=x1-yd, z2=x2-α1. The Nussbaum function is chosen as N(ζ)=ζ2cos⁡ζ; its control law is v=-N(ζ)v¯+v0, ζ˙=-γv¯z23. The control parameters are designed as k1=k2=10, a1=a2=2, λ1=λ2=8, γ1=γ2=0.5, γ=8, σd=0.25, and x20=v0=0. The initial condition is selected as [x1(0),x2(0)]T=[0.1,0.2]T, Δ^(0)=0, and [θ^1(0)=θ^2(0)]T=[0,0]T. The simulation results are shown in Figures 1–5. Figure 1 shows the tracking error z1=y-yd. Figure 2 shows the state variable x2. Figure 3 shows the true control input v¯. Figure 4 presents the adaptive parameter θ^2. Figure 5 presents estimation of disturbance bound Δ^.Figure 1
Tracking errorz1=y-yd.Figure 2
State variablex2.Figure 3
The true control inputv¯.Figure 4
The adaptive parameterθ^2.Figure 5
Estimation of disturbance boundΔ^.


## 5. Conclusion

In this paper, we study a class of stochastic pure-feedback nonlinear systems with bounded external disturbance and unknown hysteresis. This paper holds that stochastic disturbance and external disturbance exist simultaneously. By using the characteristics of the Nussbaum function, the unknown hysteresis problem is solved. Based on the approximation ability of fuzzy logic system, a new adaptive fuzzy control scheme is proposed. The scheme ensures that all signals of the closed-loop system are bounded and the tracking error converges to small domain of the origin. Finally, the simulation results show the effectiveness of the proposed scheme.



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*Source: 1487134-2018-11-08.xml*