# The Adaptive Neural Control for a Class of High-Order Uncertain Stochastic Nonlinear Systems

Authors: Xiaoyan Qin
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper studies the problem of the adaptive neural control for a class of high-order uncertain stochastic nonlinear systems. By using some techniques such as the backstepping recursive technique, Young’s inequality, and approximation capability, a novel adaptive neural control scheme is constructed. The proposed control method can guarantee that the signals of the closed-loop system are bounded in probability, and only one parameter needs to be updated online. One example is given to show the effectiveness of the proposed control method.


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## Body

## 1. Introduction

Ever since the stochastic stability theory was established by [1–3], the design and analysis of backstepping controller for stochastic nonlinear systems have achieved remarkable development in recent years; see [4–20] and the references therein. Based on the backstepping technique, Pan and Basar [8] firstly studied a class of stochastic nonlinear systems under a risk-sensitive cost criterion. Then, by combining backstepping technique with different nonlinear control methods, [9–13] obtained the state-feedback stabilization results of stochastic nonlinear systems in various structures. In the case of system states being unmeasurable, [14–19] further studied the problem of the output-feedback stabilization for stochastic nonlinear systems with the help of observer design. In addition, by applying the backstepping design and Lyapunov stability analysis, the finite-time control with fast convergence rate has been achieved for stochastic nonlinear systems in [10, 20, 21].Note that when stochastic nonlinear system is of high-order, it may be nonsmooth and in general not stabilizable. How to deal with this problem is difficult. To handle this case, [9–13, 17, 20] have done remarkable work on stochastic high-order nonlinear systems and obtained different control results. Particularly, the homogeneous domination approach was extended to stochastic nonlinear system in [17], which provides an effective design methods for high-order stochastic nonlinear systems. However, the above-mentioned results were subjected to the nonlinear dynamics models which are known exactly or unknown parameters existing linearly. Thus, these results cannot be used for the stochastic systems with structured uncertainties. Naturally, one raises the problem of how to design the controller for the high-order stochastic nonlinear systems with structured uncertainties.To handle the structured uncertainties for the stochastic nonlinear systems, the radial basis function neural network (RBF NN) or the fuzzy logic is used to approximate the uncertain functions, which ensures the growth assumptions can be weakened or removed. Based on these methods and some useful adaptive backstepping control approaches, fruitful results have been introduced and obtained in [22–30] and the references therein. Reference [22] studied the problem of fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems. Reference [23] considered the adaptive neural network output-feedback control for nonlinear systems with dynamical uncertainties. Reference [25] considered NN output-feedback control for stochastic nonlinear systems with unknown control coefficients. Reference [24] studied NN output-feedback control for stochastic time-delay nonlinear systems with unknown control coefficients. Furthermore, more problems of stochastic nonlinear systems with unmodeled dynamics were studied in [26–30].Motivated by the aforementioned literatures, one raises the following meaningful problem:how to relax or remove the matching conditions on drift and diffusion terms by RBF NN? And how to design the adaptive NN state-feedback controller for a class of high-order stochastic nonlinear systems with unknown control directions?In this paper, we will discuss the problem of adaptive neural control for a class of high-order stochastic nonlinear systems with structured uncertainties. By using backstepping recursive approach, Young’s inequality, etc., the restrictions on systems nonlinearities are removed and the procedure of the design is simpler. A novel adaptive neural controller is constructed, which assures that the closed-loop system is bounded in probability. In addition, in the design progress, there is only one parameter that needs to be updated online.The rest of this paper is organized as follows. The notations and some preliminaries are provided in Section2. In Section 3, we present the main results. In Section 4, we give the simulation example to illustrate the effectiveness of the proposed results, and the conclusion is drawn in Section 5.


## 2. Preliminaries and Problem Formulation

Notations. R denotes the set of all real numbers, R+ denotes the set of all nonnegative real numbers, and Rn denotes the real n-dimensional space. Ci stands for the family of the functions with ith continuous partial derivations. For a given vector or matrix X, XT denotes its transpose, Tr{X} denotes its trance when X is square, and X is the Euclidean norm of a vector X. For simplicity, the smooth function ϱ(·) is sometimes denoted by ϱ.Consider a class of high-order stochastic nonlinear systems as follows:(1)dxi=hitxi+1Pidt+φit,x¯idt+gix¯i∑tdω,i=1,…,n-1,dxn=hntuPndt+φnt,x¯ndt+gnx¯n∑tdωwhere x=[x1,…,xn]T∈Rn and u∈R are the system states and the control input, respectively. x¯i=[x1,…,xi]T∈Ri,i=1,2,…,n and x¯n=x. ω∈Rr is an r-dimensional standard Wiener process defined on the complete probability space (Ω,F,P) with Ω being a sample space, F being a filtration, and P being measure. φi(·) and gi(·) are unknown smooth functions, with φi(0,0)=0, gi(0)=0 for i=1,2,…,n. The disturbed virtual control coefficients hi(t):R→R (i=1,2,…,n) are unknown and continuous functions, respectively; ∑‍(t):R+→Rr×r is the Borel bounded measurable functions.


### 2.1. Preliminary Results

Next we introduce several technical lemmas which will play an important role in our later control design.Consider the following stochastic nonlinear system:(2)dx=ft,xdt+ht,xdωwhere f and g are the Borel measurable functions. f:R+×Rn+1→Rn and g:R+×Rn+1→Rn×r are assumed to be C1 in their arguments.Definition 1 (see [2]).
GivenV(x,t)∈C1,2 for stochastic nonlinear system (2), the differential operator L is defined as follows:(3)LVx,t=∂V∂t+∂V∂xf+12TrhT∂2V∂x2hwhere C1,2(Rn×R+;R+) denotes all nonnegative functions V(x,t) on Rn×R+; i.e., V(x,t) satisfies C1 in t and C2 in x. Simply, the smooth function f(·) is denoted by f.Definition 2 (see [15]).
The solution of stochastic system (2) is said to be bounded in probability if it satisfies (4)limc→∞⁡sup0≤t≤∞⁡Pxt>c=0.Definition 3 (see [15]).
Consider system (2) with f(t,0)=0 and h(t,0)=0; the equilibrium x=0 is globally stable in probability if, for any ε>0, there exists a class K∞ function γ(·) such that (5)Pxt<γx0≥1-ε,∀t≥0,x0∈Rn∖0.Lemma 4 (Young’s inequality [4]).
For any(x,y)∈R2, the following inequality holds:(6)xy≤εppxp+1qεqyq.Lemma 5 (see [15]).
Consider the stochastic system (2). Assume that f(x,t) and h(x,t) are C1 in their arguments, and f(0,t) and h(0,t) are bounded uniformly in t if there exist functions V(x,t)∈C1,2(Rn×R+,R+), u1(·),u2(·)∈K∞ and constants a0>0,b0>0 such that (7)u1x≤Vx,t≤u2x,LV≤-a0Vx,t+b0.Then the solution of (2) is bounded in probability.The purpose of this paper is to construct a smooth adaptive neural state-feedback controller such that the solution process of system (1) is bounded in probability.To design the controller for system (1), the following assumptions are needed:(A1) p1=p2=⋯pn=p≥1 are odd integers.(A2) For any Y∈ΩY, there exists an ideal constant weight vector W∗ such that 0<W∗∞≤Wmax and 0<δ≤δmax.(A3) For i=1,…,n, there are positive constants γi1 and γi2, such that γi1≤hit≤γi2.(A4) There exists constant M>0 such that ∑Tt∑tF2≤M.Remark 6.
Ifp1=p2=⋯pn=1, system (1) becomes the strict-feedback form. The problem of the feedback control has been studied in [22–24, 26, 27]. However, they did not consider p1=p2=⋯pn>1. In this paper, we will consider the problem of the feedback control under the case p1=p2=⋯pn>1.The following radial basis function neural network (RBFNN) will be considered and used to approximate unknown continuous functions:(8)ΨY:Rn→R,ΨnnY=WTSYwhere Y∈ΩY⊂Rq is the input vector with q being the neural networks input dimension. W=[w1,w2,…,wl]T∈Rl denotes the weight vector. l>1 is the neural network mode number. S(Y)=[s1(Y),s2(Y),…,sl(Y)]T and si(Y)=exp⁡[-Y-ui2/δi],i=1,2,…,l are the basis function vectors. Here ui=[ui1,ui2,…,uiq]T is the center of the receptive field, and δi is the width of the Gaussian function. Equation (8) can approximate any unknown continuous function over the compact set ΩY⊂Rq with arbitrary accuracy. Namely,(9)ΨY=W∗TSY+δY,∀Y∈ΩY.The ideal constant weight vector W∗ is defined as W∗≔arg⁡minW∈Rl⁡{supY∈ΩY⁡ΨY-WTSY}, and δ(Y) is the approximation error.From (9), we can easily get(10)W∗TSY+δ≤W∗TSY+δ≤∑i=1lSiYWmax+δmax≤ΘπYwhere π(Y)=(l+1)(∑i=1lSi2(Y)+1), Θ=max(Wmax,δmax).To design a state-feedback controller, we first introduce the following transformation:(11)ηi=xi-αix¯i,Θ^,αi=-ηi-1βi-1·.i=1,2,…,nwhere α1=0, αi(·) is the virtual control law and βi(·)>0 can be designed in the following form.Using (11), we have(12)dη1=dx1=h1x2pdt+φ¯1dt+G1∑tdωdηi=dxi-αi=hixi+1pdt+φ¯idt-∂αi∂Θ^Θ^˙dt+Gi∑tdωi=2,…,n,where xn+1=u, φ¯i=φi-∑l=1i-1∂αi/∂xlφl-∑l=1i-1∂αi/∂xlxl+1p-1/2∑k,m=1i-1∂2αi/∂xk∂xmgk∑(t)∑T(t)gmT, and Gi=gi-∑l=1i-1∂αi/∂xlgl.


## 2.1. Preliminary Results

Next we introduce several technical lemmas which will play an important role in our later control design.Consider the following stochastic nonlinear system:(2)dx=ft,xdt+ht,xdωwhere f and g are the Borel measurable functions. f:R+×Rn+1→Rn and g:R+×Rn+1→Rn×r are assumed to be C1 in their arguments.Definition 1 (see [2]).
GivenV(x,t)∈C1,2 for stochastic nonlinear system (2), the differential operator L is defined as follows:(3)LVx,t=∂V∂t+∂V∂xf+12TrhT∂2V∂x2hwhere C1,2(Rn×R+;R+) denotes all nonnegative functions V(x,t) on Rn×R+; i.e., V(x,t) satisfies C1 in t and C2 in x. Simply, the smooth function f(·) is denoted by f.Definition 2 (see [15]).
The solution of stochastic system (2) is said to be bounded in probability if it satisfies (4)limc→∞⁡sup0≤t≤∞⁡Pxt>c=0.Definition 3 (see [15]).
Consider system (2) with f(t,0)=0 and h(t,0)=0; the equilibrium x=0 is globally stable in probability if, for any ε>0, there exists a class K∞ function γ(·) such that (5)Pxt<γx0≥1-ε,∀t≥0,x0∈Rn∖0.Lemma 4 (Young’s inequality [4]).
For any(x,y)∈R2, the following inequality holds:(6)xy≤εppxp+1qεqyq.Lemma 5 (see [15]).
Consider the stochastic system (2). Assume that f(x,t) and h(x,t) are C1 in their arguments, and f(0,t) and h(0,t) are bounded uniformly in t if there exist functions V(x,t)∈C1,2(Rn×R+,R+), u1(·),u2(·)∈K∞ and constants a0>0,b0>0 such that (7)u1x≤Vx,t≤u2x,LV≤-a0Vx,t+b0.Then the solution of (2) is bounded in probability.The purpose of this paper is to construct a smooth adaptive neural state-feedback controller such that the solution process of system (1) is bounded in probability.To design the controller for system (1), the following assumptions are needed:(A1) p1=p2=⋯pn=p≥1 are odd integers.(A2) For any Y∈ΩY, there exists an ideal constant weight vector W∗ such that 0<W∗∞≤Wmax and 0<δ≤δmax.(A3) For i=1,…,n, there are positive constants γi1 and γi2, such that γi1≤hit≤γi2.(A4) There exists constant M>0 such that ∑Tt∑tF2≤M.Remark 6.
Ifp1=p2=⋯pn=1, system (1) becomes the strict-feedback form. The problem of the feedback control has been studied in [22–24, 26, 27]. However, they did not consider p1=p2=⋯pn>1. In this paper, we will consider the problem of the feedback control under the case p1=p2=⋯pn>1.The following radial basis function neural network (RBFNN) will be considered and used to approximate unknown continuous functions:(8)ΨY:Rn→R,ΨnnY=WTSYwhere Y∈ΩY⊂Rq is the input vector with q being the neural networks input dimension. W=[w1,w2,…,wl]T∈Rl denotes the weight vector. l>1 is the neural network mode number. S(Y)=[s1(Y),s2(Y),…,sl(Y)]T and si(Y)=exp⁡[-Y-ui2/δi],i=1,2,…,l are the basis function vectors. Here ui=[ui1,ui2,…,uiq]T is the center of the receptive field, and δi is the width of the Gaussian function. Equation (8) can approximate any unknown continuous function over the compact set ΩY⊂Rq with arbitrary accuracy. Namely,(9)ΨY=W∗TSY+δY,∀Y∈ΩY.The ideal constant weight vector W∗ is defined as W∗≔arg⁡minW∈Rl⁡{supY∈ΩY⁡ΨY-WTSY}, and δ(Y) is the approximation error.From (9), we can easily get(10)W∗TSY+δ≤W∗TSY+δ≤∑i=1lSiYWmax+δmax≤ΘπYwhere π(Y)=(l+1)(∑i=1lSi2(Y)+1), Θ=max(Wmax,δmax).To design a state-feedback controller, we first introduce the following transformation:(11)ηi=xi-αix¯i,Θ^,αi=-ηi-1βi-1·.i=1,2,…,nwhere α1=0, αi(·) is the virtual control law and βi(·)>0 can be designed in the following form.Using (11), we have(12)dη1=dx1=h1x2pdt+φ¯1dt+G1∑tdωdηi=dxi-αi=hixi+1pdt+φ¯idt-∂αi∂Θ^Θ^˙dt+Gi∑tdωi=2,…,n,where xn+1=u, φ¯i=φi-∑l=1i-1∂αi/∂xlφl-∑l=1i-1∂αi/∂xlxl+1p-1/2∑k,m=1i-1∂2αi/∂xk∂xmgk∑(t)∑T(t)gmT, and Gi=gi-∑l=1i-1∂αi/∂xlgl.


## 3. Controller Design and Stability Analysis

### 3.1. Controller Design

In this section, by using the backstepping method and the BRFNN, we construct the adaptive neural controller and approximate the unknown nonlinear functions, respectively.Step 1.
Consider the Lyapunov functionV1=1/4η14+1/2Θ~2, where Θ~=Θ-Θ^ is the parameter error. By (3) and (12), we have(13)LV1=η13h1x2p+φ-1+32η12TrG1T∑Tt∑tG1-Θ~Θ^˙.From Lemma 4 and assumptions (A2)-(A4), there exists constant a1j>0, j=1,2, such that(14)η13φ¯1≤η1p+33p+3a113/p+3φ¯1p+3/3+1p+3a11p+3and(15)32η12TrG1T∑Tt∑tG1≤32η12G12∑Tt∑tF2≤2p+3a122/p+3η1p+3G1p+3M+1p+3a12p+3M.Substituting (14) and (15) into (13), we obtain(16)LV1≤η13h1x2p+a1-Θ~Θ^˙+η1p+3Ψ1where Ψ1=3/p+3a113/p+3φ¯1p+3/3+2/p+3a122/p+3G1p+3M, a1=1/p+3a11p+3+2/p+3a12p+3M. Obviously, Ψ1 is an unknown function since it has unknown function φ1 and g1. In practice, it cannot be used directly. Moreover, there exists a neural network W1∗TS1(Y1), Y1=x1∈ΩY1⊂R1, such that(17)Ψ1=W1∗TS1Y1+δ1Y1≤Θπ1·where π1(·)=(l+1)∑k=1lS1k2+1. In the view of (16) and (17), we can get(18)LV1≤-Θ~Θ^˙+η13h1x2p-α2p+η1p+3Θπ1+η13h1α2p+a1.Now we choose the virtual control laws(19)α2=-c1-Θ^π1γ111/pη1=-β1η1,β1=c1+Θ^π1γ111/p>0where c1>0 is a constant to be chosen.
Substitutingα2 into (18), it can be rewritten as(20)LV1≤Θ~η1p+3π1-Θ^˙+η13h1x2p-α2p-c1η1p+3+a1.By (11), Lemma 4, assumptions (A2), (A3), and (a+b)n=∑i=0nCnkakbn-k, we obtain(21)η13h1x2p-α2p=η13h1η2+α2p-α2p=η13h1∑k=0p-1Cpkα2kη2p-k=η13h1∑k=0p-1Cpk-η1β1kη2p-k≤γ12∑k=0p-1/2Cp2kη13+2kη2p-2kβ12k≤∑k=0p-1/23+2kp+3ϵ1kp+3/3+2kη13+p+∑k=0p-1/2p-2kp+3ϵ1kp-2k/p+3Cp2kβ12kγ12p+3/p-2kη23+p≤ϵ1η1p+3+γ20η2p+3where ϵ1≥∑k=0p-1/23+2k/p+3ϵ1kp+3/3+2k, γ20≥∑k=0p-1/2p-2k/p+3ϵ1kp-2k/p+3(Cp2kβ12kγ12)p+3/p-2k, and ϵ1k>0(k=0,1,⋯p-1/2) are constants.
Substituting (21) into (20), we can get(22)LV1≤Θ~η1p+3π1-Θ^˙-c1-ϵ1η1p+3+γ20η2p+3+a1.Step 2.
We choose the Lyapunov functionV2=V1+1/4η24 to design the control law α3. From (11), (12), and (22), we have(23)LV2=LV1+η23h2x3p+φ¯2-∂α2∂Θ^Θ^˙+32η22TrG2T∑Tt∑tG2=η23h2x3p+φ¯2-∂α2∂Θ^Θ^˙+32η22TrG2T∑Tt∑tG2+Θ~η1p+3π1-Θ^˙-c1-ϵ1η1p+3+γ20η2p+3+a1.From the definition of φ¯2 and Lemma 4, there exist constants a2j>0, j=1,2, such that(24)η23φ¯2≤η2p+33p+3a213/p+3φ¯2p+3/3+1p+3a21p+3and(25)32η22TrG2T∑Tt∑tG2=32η22G22∑Tt∑tF2≤η2p+32p+3a222/p+3G2p+3M+1p+3a22p+3M.By (23), (24), and (25), we can obtain(26)LV2≤η23h2x3p-∂α2∂Θ^Θ^˙+Θ~η1p+3π1-Θ^˙-c1-ϵ1η1p+3+γ20η2p+3+a1+η2p+32p+3a222/p+3G2p+3M+3p+3a213/p+3φ¯2p+3/3+1p+3a21p+3+1p+3a22p+3M.Further, adding and subtracting η23f2 in the first bracket in (26) and using Lemma 4, there exists a constant a23>0, such that(27)-η23f2≤η2p+33p+3a233/p+3f2p+3/3+1p+3a23p+3.Then, substituting (27) into (26), we get(28)LV2≤η23h2x3p-∂α2∂Θ^Θ^˙+f2+Θ~η1p+3π1-Θ^˙-c1-ϵ1η1p+3η2p+33p+3a233/p+3f2p+3/3+1p+3a23p+3+γ20η2p+3+η2p+3Ψ2+∑i=12aiwhere a2=1/p+3a21p+3+1/p+3a23p+3+1/p+3a22p+3M, Ψ2=2/p+3a222/p+3G2p+3M+3/p+3a21-3/p+3φ¯2p+3/3+3/p+3a233/p+3f2p+3/3, and f2=-∂α2/∂Θ^c0Θ^+∂α2/∂Θ^η1p+3π1-∂α2/∂Θ^η2p+3π2.
Similar to Step1, obviously Ψ2 is an unknown function. Hence there exist the neural networks W2∗TS2(Y2) and Y2=[x2¯,Θ^]∈ΩY2⊂R3 such that(29)Ψ2=W2∗TS2Y2+δ2Y2≤Θπ2·where π2(·)=(l+1)∑k=1lS2k2+1.
We add and subtracth2η23α3p in the formula (26), respectively, and, substituting (29) into (26), we can get(30)LV2≤η23h2x3p-α3p+η23f2-∂α2∂Θ^Θ^˙+Θ~η1p+3π1-Θ^˙+h2η2pα3p-c1-ϵ1η1p+3+γ20η2p+3+∑i=12ai+η2p+3Θπ2.Now we choose the virtual control law α3=(-c2-Θ^π2-γ20/γ11)1/pη2=-β2η2 and β2=(c2+Θ^π2+γ20/γ11)1/p>0, where c2>0 is a constant to be chosen. Substitution α3 into (30), (30) can be calculated as(31)LV2≤η23h2x3p-α3p+η23f2-∂α2∂Θ^Θ^˙+Θ~∑k=12ηkp+3πk-Θ^˙-c1-ϵ1η1p+3+∑i=12ai-c2η2p+3.By (11), Lemma 4, assumptions (A2)-(A4), and (a+b)n=∑i=0nCnkakbn-k, we have(32)η23h2x3p-α3p=η23h2η3+α3p-α3p=η23h2∑k=0p-1Cpkα3kη3p-k=η23h2∑k=0p-1Cpk-η2β2kη3p-k≤∑k=0p-1/2Cp2kη23+2kη3p-2kβ22k≤∑k=0p-1/23+2kp+3ϵ2kp+3/3+2kη23+p+∑k=0p-1/2p-2kp+3ϵ2kp-2k/p+3Cp2kβ22kγ22p+3/p-2kη33+p≤ϵ2η2p+3+γ30η2p+3where ϵ2≥∑k=0p-1/23+2k/p+3ϵ2kp+3/3+2k, γ30≥∑k=0p-1/2p-2k/p+3ϵ2kp-2k/p+3(Cp2kβ22kγ22)p+3/p-2k, ϵ2k>0, k=0,1,…,p-1/2.
Substituting (32) into (31), we get(33)LV2≤Θ~∑i=12ηip+3πi-Θ^˙-∑i=12ci-ϵiηip+3+γ30η3p+3+η23f2-∂α2∂Θ^Θ^˙+∑i=12ai.Remark 7.
SinceΘ^˙ contains η1, η2, the term ∂α2/∂ΘΘ^˙ cannot be used directly to design the virtual control law α2. And the function f2 will be used to compensate for ∂α2/∂ΘΘ^˙. As a result, the term η23(f2-∂α2/∂ΘΘ^˙) will be considered in the later section.Step i (3≤i≤n).
We choose the following Lyapunov functionVi=Vi-1+1/4ηi4. From (3) and (12), we have(34)LVi=LVi-1+ηi3hixi+1p+ηi3φ¯i+32ηi2TrGiT∑Tt∑tGi+ηi3∂αi∂Θ^Θ^˙where Vi-1=1/4∑j=1i-1ηj4+1/2Θ~2, and(35)LVi-1≤-∑j=1i-1cj-ϵjηjp+3+Θ~∑j=1i-1ηjp+3πj-Θ^˙+∑j=2i-1ηj3fj-∂αj∂Θ^Θ^˙+γioηip+3+∑j=1i-1ajwhere (36)fj=-∂αj∂Θ^c0Θ^+∂αj∂Θ^∑k=1i-1ηkp+3πk-ηjpπj∑l=2j∂αl∂Θ^ηl3.
Similar to the Steps1 and 2, from the definition of φ¯i, Lemma 4, and assumptions (A2)-(A4), there exist constants aij>0, j=1,2, such that(37)ηi3φ¯i≤ηip+33p+3ai13/p+3φ¯ip+3/3+1p+3ai1p+3and(38)32ηi2TrGiT∑Tt∑tGi=32ηi2Gi2∑Tt∑tF2≤ηip+32p+3ai22/p+3Gip+3M+1p+3ai2p+3M.Substituting LVi-1, (37), and (38) into (34), we can get(39)LVi≤ηi3hixi+1p-∂αi∂Θ^Θ^˙+Θ~∑j=2i-1ηjp+3πj-Θ^˙-∑j=1i-1cj-ϵjηjp+3+γi0ηip+3+∑j=1i-1aj+ηip+32p+3ai22/p+3Gip+3M+3p+3ai13/p+3φ¯ip+3/3+ai+∑j=2i-1ηj3fj-∂αj∂Θ^Θ^˙+1p+3ai1p+3+1p+3ai2p+3M.Further, add and subtract ηi3fi in the first bracket in (39), and, using the Lemma 4, there exists a constant ai3>0, such that(40)-ηi3fi≤ηip+33p+3ai33/p+3fip+3/3+1p+3ai3p+3Then, substituting (40) into (39), we get(41)LVi≤ηi3hixi+1p-∂α2∂Θ^Θ^˙+fi+Θ~∑j=2i-1ηjp+3πj-Θ^˙-∑j=1i-1cj-ϵjηjp+3+γi0ηip+3+∑j=1i-1aj+ηip+32p+3ai22/p+3Gip+3M+3p+3ai13/p+3φ¯ip+3/3+ai+∑j=2i-1ηj3fj-∂αj∂Θ^Θ^˙+1p+3ai1p+3+1p+3ai2p+3M+ηip+33p+3ai33/p+3fip+3/3+1p+3ai3p+3=ηi3hixi+1p+Θ~∑j=2i-1ηjp+3πj-Θ^˙-∑j=1i-1cj-ϵjηjp+3+γi0ηip+3+∑j=1iaj+ηip+3Ψi+∑j=2iηj3fj-∂αj∂Θ^Θ^˙+γi0ηip+3.
where(42)ai=1p+3ai1p+3+1p+3ai2p+3M+1p+3ai3p+3,Ψi=2p+3ai22/p+3Gip+3M+3p+3ai13/p+3φ¯ip+3/3+3p+3ai33/p+3fip+3/3and(43)fi=-∂αi∂Θ^c0Θ^+∂αi∂Θ^∑j=1i-1ηjp+3πj-ηipπi∑j=2i∂αj∂Θ^ηj3.
Since the unknown functionsφi and gi can be derived by Ψi, it cannot be used to design the control law directly. Thus there exist the neural networks Wi∗TSi(Yi) and Yi=[xi¯,Θ^]∈ΩYi⊂Ri+1 such that(44)Ψi=Wi∗TSiYi+δiYi≤Θπi·where πi(·)=(l+1)∑k=1lSik2+1, fi=-∂αi/∂Θ^c0Θ^+∂αi/∂Θ^∑j=1i-1ηjp+3πj-ηipπi∑j=2i∂αj/∂Θ^ηj3.
We add and subtracthiηi3αi+1p in the formula (39), respectively, and, substituting (44) into (39), it has(45)LVi≤ηi3hixi+1p-αi+1p+ηi3hiαi+1p+Θ~∑j=2i-1ηjp+3πj-Θ^˙-∑j=1i-1cj-ϵjηjp+3+γi0ηip+3+∑i=1iaj+ηip+3πiΘ+∑j=2iηj3fj-∂αj-1∂Θ^Θ^˙.Choose the virtual control law (46)αi+1=-ci-Θ^πi-γi0γi11/pηi=-βiηi,βi=ci+Θ^πi+γi0γi11/p>0,where ci>0 is a constant to be chosen. Then, by substituting αi+1 into (45) and using (11), Lemma 5, and (a+b)n=∑i=0nCnkakbn-k, one yields (47)ηi3hixi+1p-αi+1p=ηi3hiηi+1+αi+1p-αi+1p=ηi3hi∑k=0p-1Cpkαi+1kηi+1p-k=ηi3hi∑k=0p-1Cpk-ηiβikηi+1p-k≤∑k=0p-1/2Cp2kηi3+2kηi+1p-2kβi2k≤∑k=0p-1/23+2kp+3ϵikp+3/3+2kηi3+p+∑k=0p-1/2p-2kp+3ϵikp-2k/p+3Cp2kβi2kγi2p+3/p-2kηi+13+p≤ϵiηip+3+γi+10ηi+1p+3where ϵi≥∑k=0p-1/23+2k/p+3ϵikp+3/3+2k, γi+10≥∑k=0p-1/2p-2k/p+3ϵikp-2k/p+3(Cp2kβi2kγi2)p+3/p-2k, ϵik>0, k=0,1,…,p-1/2.
In the view of (47), (45) can be calculated as(48)LVi≤Θ~∑j=2iηjp+3πj-Θ^˙-∑j=1icj-ϵjηjp+3+γi+10ηi+1p+3+∑j=1iaj+∑j=2iηj3fj-∂αj∂Θ^Θ^˙.Remark 8.
SinceΘ^˙ contains ηj,j=1,2,…,i, the term ∂αj/∂ΘΘ^˙ cannot be used directly to design the virtual control law αi. And the function fj will be used to compensate for ∂αj/∂ΘΘ^˙. As a result, the term ηj3(fj-∂αj/∂ΘΘ^˙) will be considered in the later section.Finally, wheni=n, ηn+1=u is the actual control. Choose the actual controller(49)u=-ηnβn,βn=cn+γn0+πnΘ^γn11/p>0where cn>0 is a design parameter to be chosen. It can be deduced that(50)LVn≤-∑j=1ncj-εjηjp+3+∑j=1naj+Θ~∑j=2nηjp+3πj-Θ^˙+∑j=2nηj3fj-∂αj∂Θ^Θ^˙where Vn(η,Θ^)=∑k=1n1/4ηk4+1/2(Θ^-Θ)2,η=(η1,⋯ηn). We finish the controller design procedure.


### 3.2. Analysis of Stability

We have obtained the main result in the following theorem.Theorem 9.
Suppose that assumptions(A1)-(A4) hold for the stochastic nonlinear system (1). Furthermore, suppose that the unknown functions Ψi(1≤i≤n) can be approximated by the RBF neural networks. Given a control law with the virtual control signals αi, it is constructed in (49), and the adaptive law satisfies(51)Θ^˙=∑k=1nηkp+3πk-c0Θ^where the design parameter c0>0. Then the signals of the closed-loop system are bounded in probability.Proof.
Choose the Lyapunov functionV=Vn such that(52)LV≤-∑j=1ncj-εjηjp+3+∑j=1naj+Θ~∑j=2nηjp+3πj-Θ^˙+∑j=2nηj3fj-∂αj∂Θ^Θ^˙.Substituting the adaptive law (51) into the penultimate term in (52) results in(53)LV≤-∑j=1ncj-εjηjp+3+∑j=1naj+c0Θ~Θ^+∑j=2nηj3fj-∂αj∂Θ^Θ^˙.
In the following, we will prove that the last term∑j=2nηj3(fj-∂αj/∂Θ^Θ^˙) in (53) is negative. It is clear that(54)-∑j=2nηj3∂αj∂Θ^Θ^˙=∑j=2nηj3∂αj∂Θ^c0Θ^-∑j=2nηj3∂αj∂Θ^∑i=1nηip+3πi=∑j=2nηj3∂αj∂Θ^c0Θ^-∑j=2nηj3∂αj∂Θ^∑i=1nηip+3πiand(55)-∑j=2nηj3∂αj∂Θ^∑i=1nηip+3πi=-∑j=2nηj3∂αj∂Θ^∑i=1j-1ηip+3πi-∑j=2nηj3∂αj∂Θ^∑i=jnηip+3πi≤-∑j=2nηj3∂αj∂Θ^∑i=1j-1ηip+3πi+∑j=2nηjp+3πj∑i=2jηi3∂αi∂Θ^.Substituting (55) into (54), one arrives at(56)-∑j=2nηj3∂αj∂Θ^Θ^˙=∑j=2nηj3∂αj∂Θ^c0Θ^-∑i=1nηip+3πi≤∑j=2nηj3∂αj∂Θ^c0Θ^-∑j=2nηj3∂αj∂Θ^∑i=1j-1ηip+3πi+∑j=2nηjp+3πj∑i=2jηi3∂αi∂Θ^=-∑j=2nηj3-∂αj∂Θ^c0Θ^+∂αj∂Θ^∑i=1j-1ηip+3πi-ηjpπj∑i=2jηi3∂αi∂Θ^=-∑j=2nηj3fj.By the definition of fi, which implies that(57)∑j=2nηj3fj-∂αj∂Θ^Θ^˙≤0and(58)c0Θ~Θ^=c0Θ~Θ-Θ~=c0Θ~Θ-c0Θ~2≤c02Θ2-c02Θ~2substituting (57) and (58) into (53) yields(59)LV≤-∑j=1ncj-εjηjp+3+∑j=1naj+c02Θ2-c02Θ~2.Furthermore, let a0=min⁡{4(cj-ϵj),c0}, j=1,2,⋯n, and b0=∑j=1naj+c0/2Θ2, and it follows that(60)LV≤-a0V+b0,t≥0.Therefore, according to Lemma 5, ηi,i=1,2,…,n, and Θ~ are bounded in probability. Since Θ is a constant, Θ^ is bounded in probability. It can be obtained that the control law αj,j=1,2,…,n, is also bounded in probability because αj is the function of ηj and Θ^. So far we get that all the states of the closed-loop system (1) are bounded in probability.


## 3.1. Controller Design

In this section, by using the backstepping method and the BRFNN, we construct the adaptive neural controller and approximate the unknown nonlinear functions, respectively.Step 1.
Consider the Lyapunov functionV1=1/4η14+1/2Θ~2, where Θ~=Θ-Θ^ is the parameter error. By (3) and (12), we have(13)LV1=η13h1x2p+φ-1+32η12TrG1T∑Tt∑tG1-Θ~Θ^˙.From Lemma 4 and assumptions (A2)-(A4), there exists constant a1j>0, j=1,2, such that(14)η13φ¯1≤η1p+33p+3a113/p+3φ¯1p+3/3+1p+3a11p+3and(15)32η12TrG1T∑Tt∑tG1≤32η12G12∑Tt∑tF2≤2p+3a122/p+3η1p+3G1p+3M+1p+3a12p+3M.Substituting (14) and (15) into (13), we obtain(16)LV1≤η13h1x2p+a1-Θ~Θ^˙+η1p+3Ψ1where Ψ1=3/p+3a113/p+3φ¯1p+3/3+2/p+3a122/p+3G1p+3M, a1=1/p+3a11p+3+2/p+3a12p+3M. Obviously, Ψ1 is an unknown function since it has unknown function φ1 and g1. In practice, it cannot be used directly. Moreover, there exists a neural network W1∗TS1(Y1), Y1=x1∈ΩY1⊂R1, such that(17)Ψ1=W1∗TS1Y1+δ1Y1≤Θπ1·where π1(·)=(l+1)∑k=1lS1k2+1. In the view of (16) and (17), we can get(18)LV1≤-Θ~Θ^˙+η13h1x2p-α2p+η1p+3Θπ1+η13h1α2p+a1.Now we choose the virtual control laws(19)α2=-c1-Θ^π1γ111/pη1=-β1η1,β1=c1+Θ^π1γ111/p>0where c1>0 is a constant to be chosen.
Substitutingα2 into (18), it can be rewritten as(20)LV1≤Θ~η1p+3π1-Θ^˙+η13h1x2p-α2p-c1η1p+3+a1.By (11), Lemma 4, assumptions (A2), (A3), and (a+b)n=∑i=0nCnkakbn-k, we obtain(21)η13h1x2p-α2p=η13h1η2+α2p-α2p=η13h1∑k=0p-1Cpkα2kη2p-k=η13h1∑k=0p-1Cpk-η1β1kη2p-k≤γ12∑k=0p-1/2Cp2kη13+2kη2p-2kβ12k≤∑k=0p-1/23+2kp+3ϵ1kp+3/3+2kη13+p+∑k=0p-1/2p-2kp+3ϵ1kp-2k/p+3Cp2kβ12kγ12p+3/p-2kη23+p≤ϵ1η1p+3+γ20η2p+3where ϵ1≥∑k=0p-1/23+2k/p+3ϵ1kp+3/3+2k, γ20≥∑k=0p-1/2p-2k/p+3ϵ1kp-2k/p+3(Cp2kβ12kγ12)p+3/p-2k, and ϵ1k>0(k=0,1,⋯p-1/2) are constants.
Substituting (21) into (20), we can get(22)LV1≤Θ~η1p+3π1-Θ^˙-c1-ϵ1η1p+3+γ20η2p+3+a1.Step 2.
We choose the Lyapunov functionV2=V1+1/4η24 to design the control law α3. From (11), (12), and (22), we have(23)LV2=LV1+η23h2x3p+φ¯2-∂α2∂Θ^Θ^˙+32η22TrG2T∑Tt∑tG2=η23h2x3p+φ¯2-∂α2∂Θ^Θ^˙+32η22TrG2T∑Tt∑tG2+Θ~η1p+3π1-Θ^˙-c1-ϵ1η1p+3+γ20η2p+3+a1.From the definition of φ¯2 and Lemma 4, there exist constants a2j>0, j=1,2, such that(24)η23φ¯2≤η2p+33p+3a213/p+3φ¯2p+3/3+1p+3a21p+3and(25)32η22TrG2T∑Tt∑tG2=32η22G22∑Tt∑tF2≤η2p+32p+3a222/p+3G2p+3M+1p+3a22p+3M.By (23), (24), and (25), we can obtain(26)LV2≤η23h2x3p-∂α2∂Θ^Θ^˙+Θ~η1p+3π1-Θ^˙-c1-ϵ1η1p+3+γ20η2p+3+a1+η2p+32p+3a222/p+3G2p+3M+3p+3a213/p+3φ¯2p+3/3+1p+3a21p+3+1p+3a22p+3M.Further, adding and subtracting η23f2 in the first bracket in (26) and using Lemma 4, there exists a constant a23>0, such that(27)-η23f2≤η2p+33p+3a233/p+3f2p+3/3+1p+3a23p+3.Then, substituting (27) into (26), we get(28)LV2≤η23h2x3p-∂α2∂Θ^Θ^˙+f2+Θ~η1p+3π1-Θ^˙-c1-ϵ1η1p+3η2p+33p+3a233/p+3f2p+3/3+1p+3a23p+3+γ20η2p+3+η2p+3Ψ2+∑i=12aiwhere a2=1/p+3a21p+3+1/p+3a23p+3+1/p+3a22p+3M, Ψ2=2/p+3a222/p+3G2p+3M+3/p+3a21-3/p+3φ¯2p+3/3+3/p+3a233/p+3f2p+3/3, and f2=-∂α2/∂Θ^c0Θ^+∂α2/∂Θ^η1p+3π1-∂α2/∂Θ^η2p+3π2.
Similar to Step1, obviously Ψ2 is an unknown function. Hence there exist the neural networks W2∗TS2(Y2) and Y2=[x2¯,Θ^]∈ΩY2⊂R3 such that(29)Ψ2=W2∗TS2Y2+δ2Y2≤Θπ2·where π2(·)=(l+1)∑k=1lS2k2+1.
We add and subtracth2η23α3p in the formula (26), respectively, and, substituting (29) into (26), we can get(30)LV2≤η23h2x3p-α3p+η23f2-∂α2∂Θ^Θ^˙+Θ~η1p+3π1-Θ^˙+h2η2pα3p-c1-ϵ1η1p+3+γ20η2p+3+∑i=12ai+η2p+3Θπ2.Now we choose the virtual control law α3=(-c2-Θ^π2-γ20/γ11)1/pη2=-β2η2 and β2=(c2+Θ^π2+γ20/γ11)1/p>0, where c2>0 is a constant to be chosen. Substitution α3 into (30), (30) can be calculated as(31)LV2≤η23h2x3p-α3p+η23f2-∂α2∂Θ^Θ^˙+Θ~∑k=12ηkp+3πk-Θ^˙-c1-ϵ1η1p+3+∑i=12ai-c2η2p+3.By (11), Lemma 4, assumptions (A2)-(A4), and (a+b)n=∑i=0nCnkakbn-k, we have(32)η23h2x3p-α3p=η23h2η3+α3p-α3p=η23h2∑k=0p-1Cpkα3kη3p-k=η23h2∑k=0p-1Cpk-η2β2kη3p-k≤∑k=0p-1/2Cp2kη23+2kη3p-2kβ22k≤∑k=0p-1/23+2kp+3ϵ2kp+3/3+2kη23+p+∑k=0p-1/2p-2kp+3ϵ2kp-2k/p+3Cp2kβ22kγ22p+3/p-2kη33+p≤ϵ2η2p+3+γ30η2p+3where ϵ2≥∑k=0p-1/23+2k/p+3ϵ2kp+3/3+2k, γ30≥∑k=0p-1/2p-2k/p+3ϵ2kp-2k/p+3(Cp2kβ22kγ22)p+3/p-2k, ϵ2k>0, k=0,1,…,p-1/2.
Substituting (32) into (31), we get(33)LV2≤Θ~∑i=12ηip+3πi-Θ^˙-∑i=12ci-ϵiηip+3+γ30η3p+3+η23f2-∂α2∂Θ^Θ^˙+∑i=12ai.Remark 7.
SinceΘ^˙ contains η1, η2, the term ∂α2/∂ΘΘ^˙ cannot be used directly to design the virtual control law α2. And the function f2 will be used to compensate for ∂α2/∂ΘΘ^˙. As a result, the term η23(f2-∂α2/∂ΘΘ^˙) will be considered in the later section.Step i (3≤i≤n).
We choose the following Lyapunov functionVi=Vi-1+1/4ηi4. From (3) and (12), we have(34)LVi=LVi-1+ηi3hixi+1p+ηi3φ¯i+32ηi2TrGiT∑Tt∑tGi+ηi3∂αi∂Θ^Θ^˙where Vi-1=1/4∑j=1i-1ηj4+1/2Θ~2, and(35)LVi-1≤-∑j=1i-1cj-ϵjηjp+3+Θ~∑j=1i-1ηjp+3πj-Θ^˙+∑j=2i-1ηj3fj-∂αj∂Θ^Θ^˙+γioηip+3+∑j=1i-1ajwhere (36)fj=-∂αj∂Θ^c0Θ^+∂αj∂Θ^∑k=1i-1ηkp+3πk-ηjpπj∑l=2j∂αl∂Θ^ηl3.
Similar to the Steps1 and 2, from the definition of φ¯i, Lemma 4, and assumptions (A2)-(A4), there exist constants aij>0, j=1,2, such that(37)ηi3φ¯i≤ηip+33p+3ai13/p+3φ¯ip+3/3+1p+3ai1p+3and(38)32ηi2TrGiT∑Tt∑tGi=32ηi2Gi2∑Tt∑tF2≤ηip+32p+3ai22/p+3Gip+3M+1p+3ai2p+3M.Substituting LVi-1, (37), and (38) into (34), we can get(39)LVi≤ηi3hixi+1p-∂αi∂Θ^Θ^˙+Θ~∑j=2i-1ηjp+3πj-Θ^˙-∑j=1i-1cj-ϵjηjp+3+γi0ηip+3+∑j=1i-1aj+ηip+32p+3ai22/p+3Gip+3M+3p+3ai13/p+3φ¯ip+3/3+ai+∑j=2i-1ηj3fj-∂αj∂Θ^Θ^˙+1p+3ai1p+3+1p+3ai2p+3M.Further, add and subtract ηi3fi in the first bracket in (39), and, using the Lemma 4, there exists a constant ai3>0, such that(40)-ηi3fi≤ηip+33p+3ai33/p+3fip+3/3+1p+3ai3p+3Then, substituting (40) into (39), we get(41)LVi≤ηi3hixi+1p-∂α2∂Θ^Θ^˙+fi+Θ~∑j=2i-1ηjp+3πj-Θ^˙-∑j=1i-1cj-ϵjηjp+3+γi0ηip+3+∑j=1i-1aj+ηip+32p+3ai22/p+3Gip+3M+3p+3ai13/p+3φ¯ip+3/3+ai+∑j=2i-1ηj3fj-∂αj∂Θ^Θ^˙+1p+3ai1p+3+1p+3ai2p+3M+ηip+33p+3ai33/p+3fip+3/3+1p+3ai3p+3=ηi3hixi+1p+Θ~∑j=2i-1ηjp+3πj-Θ^˙-∑j=1i-1cj-ϵjηjp+3+γi0ηip+3+∑j=1iaj+ηip+3Ψi+∑j=2iηj3fj-∂αj∂Θ^Θ^˙+γi0ηip+3.
where(42)ai=1p+3ai1p+3+1p+3ai2p+3M+1p+3ai3p+3,Ψi=2p+3ai22/p+3Gip+3M+3p+3ai13/p+3φ¯ip+3/3+3p+3ai33/p+3fip+3/3and(43)fi=-∂αi∂Θ^c0Θ^+∂αi∂Θ^∑j=1i-1ηjp+3πj-ηipπi∑j=2i∂αj∂Θ^ηj3.
Since the unknown functionsφi and gi can be derived by Ψi, it cannot be used to design the control law directly. Thus there exist the neural networks Wi∗TSi(Yi) and Yi=[xi¯,Θ^]∈ΩYi⊂Ri+1 such that(44)Ψi=Wi∗TSiYi+δiYi≤Θπi·where πi(·)=(l+1)∑k=1lSik2+1, fi=-∂αi/∂Θ^c0Θ^+∂αi/∂Θ^∑j=1i-1ηjp+3πj-ηipπi∑j=2i∂αj/∂Θ^ηj3.
We add and subtracthiηi3αi+1p in the formula (39), respectively, and, substituting (44) into (39), it has(45)LVi≤ηi3hixi+1p-αi+1p+ηi3hiαi+1p+Θ~∑j=2i-1ηjp+3πj-Θ^˙-∑j=1i-1cj-ϵjηjp+3+γi0ηip+3+∑i=1iaj+ηip+3πiΘ+∑j=2iηj3fj-∂αj-1∂Θ^Θ^˙.Choose the virtual control law (46)αi+1=-ci-Θ^πi-γi0γi11/pηi=-βiηi,βi=ci+Θ^πi+γi0γi11/p>0,where ci>0 is a constant to be chosen. Then, by substituting αi+1 into (45) and using (11), Lemma 5, and (a+b)n=∑i=0nCnkakbn-k, one yields (47)ηi3hixi+1p-αi+1p=ηi3hiηi+1+αi+1p-αi+1p=ηi3hi∑k=0p-1Cpkαi+1kηi+1p-k=ηi3hi∑k=0p-1Cpk-ηiβikηi+1p-k≤∑k=0p-1/2Cp2kηi3+2kηi+1p-2kβi2k≤∑k=0p-1/23+2kp+3ϵikp+3/3+2kηi3+p+∑k=0p-1/2p-2kp+3ϵikp-2k/p+3Cp2kβi2kγi2p+3/p-2kηi+13+p≤ϵiηip+3+γi+10ηi+1p+3where ϵi≥∑k=0p-1/23+2k/p+3ϵikp+3/3+2k, γi+10≥∑k=0p-1/2p-2k/p+3ϵikp-2k/p+3(Cp2kβi2kγi2)p+3/p-2k, ϵik>0, k=0,1,…,p-1/2.
In the view of (47), (45) can be calculated as(48)LVi≤Θ~∑j=2iηjp+3πj-Θ^˙-∑j=1icj-ϵjηjp+3+γi+10ηi+1p+3+∑j=1iaj+∑j=2iηj3fj-∂αj∂Θ^Θ^˙.Remark 8.
SinceΘ^˙ contains ηj,j=1,2,…,i, the term ∂αj/∂ΘΘ^˙ cannot be used directly to design the virtual control law αi. And the function fj will be used to compensate for ∂αj/∂ΘΘ^˙. As a result, the term ηj3(fj-∂αj/∂ΘΘ^˙) will be considered in the later section.Finally, wheni=n, ηn+1=u is the actual control. Choose the actual controller(49)u=-ηnβn,βn=cn+γn0+πnΘ^γn11/p>0where cn>0 is a design parameter to be chosen. It can be deduced that(50)LVn≤-∑j=1ncj-εjηjp+3+∑j=1naj+Θ~∑j=2nηjp+3πj-Θ^˙+∑j=2nηj3fj-∂αj∂Θ^Θ^˙where Vn(η,Θ^)=∑k=1n1/4ηk4+1/2(Θ^-Θ)2,η=(η1,⋯ηn). We finish the controller design procedure.


## 3.2. Analysis of Stability

We have obtained the main result in the following theorem.Theorem 9.
Suppose that assumptions(A1)-(A4) hold for the stochastic nonlinear system (1). Furthermore, suppose that the unknown functions Ψi(1≤i≤n) can be approximated by the RBF neural networks. Given a control law with the virtual control signals αi, it is constructed in (49), and the adaptive law satisfies(51)Θ^˙=∑k=1nηkp+3πk-c0Θ^where the design parameter c0>0. Then the signals of the closed-loop system are bounded in probability.Proof.
Choose the Lyapunov functionV=Vn such that(52)LV≤-∑j=1ncj-εjηjp+3+∑j=1naj+Θ~∑j=2nηjp+3πj-Θ^˙+∑j=2nηj3fj-∂αj∂Θ^Θ^˙.Substituting the adaptive law (51) into the penultimate term in (52) results in(53)LV≤-∑j=1ncj-εjηjp+3+∑j=1naj+c0Θ~Θ^+∑j=2nηj3fj-∂αj∂Θ^Θ^˙.
In the following, we will prove that the last term∑j=2nηj3(fj-∂αj/∂Θ^Θ^˙) in (53) is negative. It is clear that(54)-∑j=2nηj3∂αj∂Θ^Θ^˙=∑j=2nηj3∂αj∂Θ^c0Θ^-∑j=2nηj3∂αj∂Θ^∑i=1nηip+3πi=∑j=2nηj3∂αj∂Θ^c0Θ^-∑j=2nηj3∂αj∂Θ^∑i=1nηip+3πiand(55)-∑j=2nηj3∂αj∂Θ^∑i=1nηip+3πi=-∑j=2nηj3∂αj∂Θ^∑i=1j-1ηip+3πi-∑j=2nηj3∂αj∂Θ^∑i=jnηip+3πi≤-∑j=2nηj3∂αj∂Θ^∑i=1j-1ηip+3πi+∑j=2nηjp+3πj∑i=2jηi3∂αi∂Θ^.Substituting (55) into (54), one arrives at(56)-∑j=2nηj3∂αj∂Θ^Θ^˙=∑j=2nηj3∂αj∂Θ^c0Θ^-∑i=1nηip+3πi≤∑j=2nηj3∂αj∂Θ^c0Θ^-∑j=2nηj3∂αj∂Θ^∑i=1j-1ηip+3πi+∑j=2nηjp+3πj∑i=2jηi3∂αi∂Θ^=-∑j=2nηj3-∂αj∂Θ^c0Θ^+∂αj∂Θ^∑i=1j-1ηip+3πi-ηjpπj∑i=2jηi3∂αi∂Θ^=-∑j=2nηj3fj.By the definition of fi, which implies that(57)∑j=2nηj3fj-∂αj∂Θ^Θ^˙≤0and(58)c0Θ~Θ^=c0Θ~Θ-Θ~=c0Θ~Θ-c0Θ~2≤c02Θ2-c02Θ~2substituting (57) and (58) into (53) yields(59)LV≤-∑j=1ncj-εjηjp+3+∑j=1naj+c02Θ2-c02Θ~2.Furthermore, let a0=min⁡{4(cj-ϵj),c0}, j=1,2,⋯n, and b0=∑j=1naj+c0/2Θ2, and it follows that(60)LV≤-a0V+b0,t≥0.Therefore, according to Lemma 5, ηi,i=1,2,…,n, and Θ~ are bounded in probability. Since Θ is a constant, Θ^ is bounded in probability. It can be obtained that the control law αj,j=1,2,…,n, is also bounded in probability because αj is the function of ηj and Θ^. So far we get that all the states of the closed-loop system (1) are bounded in probability.


## 4. Simulation Example

In this section, we will give an example to show the effectiveness of the proposed control method in this paper.Example 1.
Consider the following stochastic nonlinear system:(61)dx1=1+12sin⁡tx23dt+x13sin⁡x1tdt+3x1sin⁡x11+14sin⁡tdωdx2=1-14sin⁡tu3dt.Obviously, the system satisfies(A1)-(A4) and p1=p2=3>1. Now according to Theorem 9, the virtual control function α2, the control law u, and the adaptive law Θ^˙ are chosen, respectively, as(62)α2=-η1β1=-η1c1+Θ^π11/3,u=-η2β2=-η2c2+Θ^π2+γ201/3,Θ^˙=∑i=12ηi6πi-c0Θ^,where η1=x1, η2=x2-α2, Y1=η1, Y2=[η1,η2,Θ^]T. In the simulation, neural network W1∗TS1(Y1) includes 7 nodes with centers spaced evenly in [-3,-3], neural network W2∗TS2(Y2) includes 81 nodes with centers spaced evenly in [-3,-3]×[-3,-3]×[0,3], and all the widths are chosen as 2. The design parameters are chosen as c1=2, c2=0.5, c0=0.1. γ20=∑k=013-2k/4ϵ1k3-2k/4(c32kβ12k)4/3-2k, and ϵ10=ϵ11=1/4. The initial condition [x1(0),x2(0),Θ^(0)]=[0.05,-0.13,-0.25]. Figures 1 and 2 show the simulation results. From the figures, we can see that the proposed adaptive control method can guarantee that all the variables for the closed-loop system are bounded.Figure 1
The responses of statesx1 and x2 relative to time.Figure 2
The responses of the adaptive lawΘ^ and the controller u relative to time.


## 5. Conclusions

This paper has investigated the adaptive neural control for a class of high-order uncertain stochastic nonlinear systems. With the help of backstepping technique and separation technique, a smooth adaptive controller is constructed, and it ensures the closed-loop system is the global bounded in probability. Only one adaptive learning parameter needed to be updated online. One example has been given to show the effectiveness of the proposed analytical results. A further work is how to design the output-feedback tracking control for more high-order stochastic systems.



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