# High-Order Stochastic Adaptive Controller Design with Application to Mechanical System

Authors: Jie Tian; Wei Feng; Yuzhen Wang
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The main purpose of this paper is to apply stochastic adaptive controller design to mechanical system. Firstly, by a series of coordinate transformations, the mechanical system can be transformed to a class of special high-order stochastic nonlinear system, based on which, a more general mathematical model is considered, and the smooth state-feedback controller is designed. At last, the simulation for the mechanical system is given to show the effectiveness of the design scheme.


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## Body

## 1. Introduction

In recent years, the study for deterministic high-order nonlinear systems has achieved remarkable development, see, for example, [1–3] and references herein. Inspired by these interesting and important results, it is natural to generalize their results to the following stochastic high-order nonlinear systems which are neither necessarily feedback linearizable nor affine in the control input:(1.1)dz=f0(z,x1)dt+g0T(z,x1)dω,dxi=(di(x̅i,t)xi+1pi+fi(z,x̅i))dt+giT(z,x̅i)dω,i=1,…,n-1,dxn=(dn(x̅n,t)upn+fn(z,x̅n))dt+gnT(z,x̅n)dω,
where (zT,x1,…,xn)T∈ℝm+n, and u∈ℝ are the measurable state and the input of system, respectively, x̅i=(x1,…,xi)T, i=1,…,n, z=(z1,…,zm)T∈ℝm is referred to as the state of the stochastic inverse dynamics, ω is an r-dimensional standard Wiener process defined on a probability space (Ω,ℱ,P) with Ω being a sample space, ℱ being a σ-algebra, and P being a probability measure, pi≥1, i=1,…,n are odd integers, and the functions fi(·) and gi(·), i=0,1,…,n are assumed to be smooth, vanishing at the origin (zT,x̅nT)=(01×m,01×n).For (1.1) with di(·)=1, Xie and Tian in [4] considered the state-feedback stabilization problem for the first time. After considering the stabilization of high-order stochastic nonlinear systems, [5] further addressed the problem of state-feedback inverse optimal stabilization in probability, that is, the designed stabilizing backstepping controller is also optimal with respect to meaningful cost functionals. When di(·)≠1, [6] designed an adaptive state-feedback controller for a class of stochastic nonlinear uncertain systems with 0<λi≤di(·)≤μi≤μ, and [7] designed a smooth adaptive state-feedback controller for high-order stochastic systems with λi(x̅i)≤di(·)≤μ̅i(x̅i,θ) by using the parameter separation lemma and some flexible algebraic techniques. Recently, more excellent results [8–28] were achieved by Xie and his group.However, all these theoretical results mentioned above are demonstrated only by some numerical simulation examples. Since many practical application systems in aerospace industry, industrial process control, and so forth, can be described by (or transformed to) stochastic high-order nonlinear systems, so it is very necessary to apply the control schemes to these systems. Based on this reason, we consider a practical example of mechanical movement in this paper. By a series of coordinate transformations, the mechanical system can be transformed to a high-order stochastic nonlinear system, based on which, we consider a more general mathematical model and design a smooth state-feedback control law. At last, the simulation for the mechanical system is given to show the effectiveness of the design scheme.This paper is organized as follows. Section2 gives a practical example. Section 3 provides preliminary knowledge and presents problem statement. Controller design and stability analysis are given in Section 4. The simulation for the practical example is provided to demonstrate the control scheme in Section 5. Section 6 gives some concluding remarks.


## 2. A Practical Example

Let us consider the following mechanical system which consists of two massesm1 and m2 on a horizontal smooth surface as shown in Figure 1. The mass m1 is interconnected to the wall by a linear spring and to the mass m2 by a nonlinear spring which has cubic force-deformation relation. Let x be the displacement of mass m1 and y the displacement of mass m2 such that at x=0 and y=0, that is, the springs are unstretched. A control force u acts on m1.Figure 1
A mechanical system.Where the units ofm1, x, and u are “kg”, “m”, and “N”, respectively, and y1=x-y. The equations of motion for the system are described by(2.1)ÿ=k1m2(x-y)3,ẍ=-km1x-k1m1(x-y)3+um1,
where k and k1 are the spring coefficients, and their units are “N/m” and “N/m3”, respectively.Introducing the smooth change of coordinates(2.2)x1=y,x2=ẋ1=ẏ,x3=(x-y)k1m13,x4=ẋ3,
one gets(2.3)y=x1,ẏ=x2,x=x3k1/m13+y,x4=x4k1/m13+x2.The linear spring constantk has a specific nominal value k0=1.5 which is considered uncertain, and k∈[0.75,2.25]. Let Δ(t)=k(t)-k0. For all t≥0, Δ(t) is the Gaussian white noise process with EΔ(t)=0 and EΔ2(t)=σ2. We can choose the value of parameter σ such that k(t) obeys the bound 0.75≤k≤2.25 with a sufficiently high probability. This model of spring rate variations leads to an uncertain stochastic system. By (2.2), one chooses the smooth state-feedback control(2.4)u=m1vk1/m13+m1+m2m2m1x33,
which together with the property of Δ(t) leads to(2.5)dx1=x2dt,dx2=m1m2x33dt,dx3=x4dt,dx4=vdt+k0f(x)dt+σf(x)dω,y=x1,
where f(x)=-x3/m1-(k1/m1)3(x1/m1), and ω is standard Wiener process.This stochastic high-order nonlinear systems can be generalized to a more general system which will be given in the following section.


## 3. Preliminary Knowledge and Problem Statement

### 3.1. Preliminary Knowledge

In this section, we will introduce the concept of input-to-state practical stability (ISpS) in probability.Consider the following stochastic nonlinear system(3.1)dx=f(x,u)dt+gT(x,u)dω,x(0)=x0∈Rn,
where x∈ℝn, u∈ℝm are the state and the input of system, respectively. The Borel measurable functions f:ℝn+m→ℝn and g:ℝn+m→ℝn×r are locally Lipschitz in x, and ω∈ℝr is an r-dimensional independent standard Wiener process defined on the complete probability space (Ω,ℱ,P).The following definitions and lemmas will be used throughout the paper.Definition 3.1 (see [29]).
For any givenV(x)∈𝒞2,associated with stochastic system (3.1), the differential operator ℒ is defined as follows:(3.2)LV(x)=∂V(x)∂xf(x,u)+12Tr⁡{g(x,u)∂2V(x)∂x2gT(x,u)}.Definition 3.2 (see [30]).
The stochastic system (3.1) is input-to-state practically stable (ISpS) in probability if for any ε>0, there exist a class 𝒦ℒ-function β(·),a class 𝒦∞-function γ(·),and a constant d0 such that
(3.3)P{|x(t)|<β(|x0|,t)+γ(|ut|)+d0}≥1-ε,x0∈Rn∖{0}.Lemma 3.3 (see [30]).
For system (3.1), if there exist a  𝒞2 function  V(x), class  𝒦∞ functions  α1,  α2,  χ, a class  𝒦 function  α, and a constant  d̅ such that
(3.4)α1(|x|)≤V(x)≤α2(|x|),(3.5)LV(x)≤-α(|x|)+χ(|u|)+d̅,
then(1)
There exists an almost surely unique solution on[0,∞);(2)
The system (3.1) is ISpS in probability.Lemma 3.4 (see [6]).
Letx and  y be real variables. Then, for any positive integers  m,  n and any nonnegative smooth function  b(·), the following inequality holds:(3.6)|xmyn|≤mm+nb(⋅)|x|m+n+nm+nb(⋅)-m/n|y|m+n.Lemma 3.5 (see [2]).
For real variablesx≥0,  y>0, and real number  m≥1, the following inequality holds:
(3.7)x≤y+(xm)m(m-1y)m-1.


### 3.2. Problem Statement

From (2.5), we introduce a more general class of stochastic nonlinear systems as follows:(3.8)dxi=di(x)xi+1pidt+fi(x̅i+1)dt+gi(x̅i)Tdω,i=1,…,n-1,dxn=dn(x)upndt+fn(x̅n)dt+gn(x̅n)Tdω,y=x1,
where x=(x1,…,xn)T∈ℝn, u,y∈ℝ are the state, the input, and the measurable output of system, respectively, x̅i=(x1,…,xi)T, pi, i=1,…,n, are positive odd integers, fi(·):ℝi+1→ℝ and gi(·):ℝi→ℝ×ℝr are smooth functions with fi(0)=0 and gi(0)=0,di(x) is unknown control coefficient with known sign, and ω is an r-dimensional standard Wiener process defined on the complete probability space (Ω,ℱ,P).The following assumptions are made on system (3.8).A1: for eachdi(x), there exist unknown constant θ′>0 and known nonnegative smooth functions bi(x̅i) and b̅i(x̅i+1) such that(3.9)0≤bi(x̅i)≤di(x)≤θ′b̅i(x̅i+1).A2: for functionsfi(·), gi(·), i=1,2,…,n, there exist known nonnegative smooth functions φij(x̅i) and ψ̅i(x̅i) such that(3.10)|fi(x̅i+1)|≤∑j=0pi-1|xi+1|jφij(x̅i),|gi(x̅i)|≤(|x1|(pi+1)/2+⋯+|xi|(pi+1)/2)ψ̅i(x̅i).A3: the reference signalyr and its derivative ẏr are bounded.The objective of this paper is to design an adaptive controller such that the closed-loop system is ISpS in probability and the tracking errorξ1=y-yr can be regulated to a neighborhood of the origin with radius as small as possible.


## 3.1. Preliminary Knowledge

In this section, we will introduce the concept of input-to-state practical stability (ISpS) in probability.Consider the following stochastic nonlinear system(3.1)dx=f(x,u)dt+gT(x,u)dω,x(0)=x0∈Rn,
where x∈ℝn, u∈ℝm are the state and the input of system, respectively. The Borel measurable functions f:ℝn+m→ℝn and g:ℝn+m→ℝn×r are locally Lipschitz in x, and ω∈ℝr is an r-dimensional independent standard Wiener process defined on the complete probability space (Ω,ℱ,P).The following definitions and lemmas will be used throughout the paper.Definition 3.1 (see [29]).
For any givenV(x)∈𝒞2,associated with stochastic system (3.1), the differential operator ℒ is defined as follows:(3.2)LV(x)=∂V(x)∂xf(x,u)+12Tr⁡{g(x,u)∂2V(x)∂x2gT(x,u)}.Definition 3.2 (see [30]).
The stochastic system (3.1) is input-to-state practically stable (ISpS) in probability if for any ε>0, there exist a class 𝒦ℒ-function β(·),a class 𝒦∞-function γ(·),and a constant d0 such that
(3.3)P{|x(t)|<β(|x0|,t)+γ(|ut|)+d0}≥1-ε,x0∈Rn∖{0}.Lemma 3.3 (see [30]).
For system (3.1), if there exist a  𝒞2 function  V(x), class  𝒦∞ functions  α1,  α2,  χ, a class  𝒦 function  α, and a constant  d̅ such that
(3.4)α1(|x|)≤V(x)≤α2(|x|),(3.5)LV(x)≤-α(|x|)+χ(|u|)+d̅,
then(1)
There exists an almost surely unique solution on[0,∞);(2)
The system (3.1) is ISpS in probability.Lemma 3.4 (see [6]).
Letx and  y be real variables. Then, for any positive integers  m,  n and any nonnegative smooth function  b(·), the following inequality holds:(3.6)|xmyn|≤mm+nb(⋅)|x|m+n+nm+nb(⋅)-m/n|y|m+n.Lemma 3.5 (see [2]).
For real variablesx≥0,  y>0, and real number  m≥1, the following inequality holds:
(3.7)x≤y+(xm)m(m-1y)m-1.


## 3.2. Problem Statement

From (2.5), we introduce a more general class of stochastic nonlinear systems as follows:(3.8)dxi=di(x)xi+1pidt+fi(x̅i+1)dt+gi(x̅i)Tdω,i=1,…,n-1,dxn=dn(x)upndt+fn(x̅n)dt+gn(x̅n)Tdω,y=x1,
where x=(x1,…,xn)T∈ℝn, u,y∈ℝ are the state, the input, and the measurable output of system, respectively, x̅i=(x1,…,xi)T, pi, i=1,…,n, are positive odd integers, fi(·):ℝi+1→ℝ and gi(·):ℝi→ℝ×ℝr are smooth functions with fi(0)=0 and gi(0)=0,di(x) is unknown control coefficient with known sign, and ω is an r-dimensional standard Wiener process defined on the complete probability space (Ω,ℱ,P).The following assumptions are made on system (3.8).A1: for eachdi(x), there exist unknown constant θ′>0 and known nonnegative smooth functions bi(x̅i) and b̅i(x̅i+1) such that(3.9)0≤bi(x̅i)≤di(x)≤θ′b̅i(x̅i+1).A2: for functionsfi(·), gi(·), i=1,2,…,n, there exist known nonnegative smooth functions φij(x̅i) and ψ̅i(x̅i) such that(3.10)|fi(x̅i+1)|≤∑j=0pi-1|xi+1|jφij(x̅i),|gi(x̅i)|≤(|x1|(pi+1)/2+⋯+|xi|(pi+1)/2)ψ̅i(x̅i).A3: the reference signalyr and its derivative ẏr are bounded.The objective of this paper is to design an adaptive controller such that the closed-loop system is ISpS in probability and the tracking errorξ1=y-yr can be regulated to a neighborhood of the origin with radius as small as possible.


## 4. Controller Design and Stability Analysis

With the aid of Lemmas3.3–3.5, we are ready to present the main results of this paper. In this section, we show that under A1–A3, it is possible to construct a globally stabilizing, state-feedback smooth controller for system (3.8). Introduce the odd positive integer p=maxi=1,…,n{pi}, and the following coordinate change(4.1)ξ1=x1-yr,ξi=xi-xi*(x̅i-1,yr,θ̂),i=2,…,n,
where xi*(x̅i-1,yr,θ̂), i=2,…,n, are virtual smooth controllers to be designed later, θ:=max{θ′,θ′(p+3)/(p-pi+3)}, and θ̂ denotes the estimate of θ. Then, according to Itô differentiation rule, one has(4.2)dξ1=d1x2p1dt+f1dt+g1Tdω-ẏrdt,dξi=dixi+1pidt+fidt-∑k=1i-1∂xi*∂xk(xk+1pk+fk)dt-12∑j,k=1i-1∂2xi*∂xj∂xkgjTgkdt-∂xi*∂θ̂θ̂̇dt-∂xi*∂yrẏrdt+(giT-∑k=1i-1∂xi*∂xkgkT)dω,i=2,…,n-1,dξn=dnupndt+fndt-∑k=1n-1∂xn*∂xk(xk+1pk+fk)dt-12∑j,k=1n-1∂2xn*∂xj∂xkgjTgkdt-∂xn*∂θ̂θ̂̇dt-∂xn*∂yrẏrdt+(gnT-∑k=1n-1∂xn*∂xkgkT)dω.
Let GiT=giT-∑k=1i-1(∂xi*/∂xk)gkT, i=2,…,n. Next, we design the controller step by step by backstepping.Step 1.
Consider the 1st Lyapunov candidate function(4.3)V1(ξ1,θ̃)=1p-p1+4ξ1p-p1+4+12θ̃2,
where θ̃=θ-θ̂ is the parameter estimation error. In view of (3.2), (4.1), and (4.2), one has
(4.4)LV1(ξ1,θ̃)=ξ1p-p1+3(d1(x)x2p1+f1(x̅2)-ẏr)+12Tr⁡{g1(x1)(p-p1+3)ξ1p-p1+2g1T(x1)}-θ̃θ̂̇≤d1(x)ξ1p-p1+3x2p1+|ξ1|p-p1+3|f1(x̅2)-ẏr|+12(p-p1+3)ξ1p-p1+2|g1(x1)|2-θ̃θ̂̇.
By Lemma 3.4 and A2, there exist nonnegative smooth functions φ̅1(x1) and ψ1(x1) such that
(4.5)|f1(x̅2)|≤∑j=0p1-1|x2|jφ1j(x1)=∑j=0p1-1|x2|j(φ1j1/(p1-j)(x1))p1-j≤∑j=0p1-1(jp1(12jb1(x1))|x2|p1+p1-jp1(2jb1(x1))j/(p1-j)φ1jp1/(p1-j)(x1))≤b1(x1)2|x2|p1+φ̅1(x1),|g1(x1)|≤|ξ1|(p1+1)/2ψ1(x1),
which together with the boundedness of ẏr imply that
(4.6)|f1-ẏr|≤b1(x1)2|x2|p1+φ1′(x1,yr),
where φ1′(x1,yr) is a nonnegative smooth function, ψ1(x1)=ψ̅1(x1). Then, for any real number δ1>0, choosing a=|ξ1p-p1+3|φ1′(x1,yr), b=δ1, m=(p+3)/(p-p1+3), by Lemma 3.5, there is a smooth function ϕ11(x1,yr) such that
(4.7)|ξ1|p-p1+3|f1-ẏr|≤|ξ1|p-p1+3(b1(x1)2|x2|p1+φ1′(x1,yr))≤|ξ1|p-p1+3b1(x1)2|x2|p1+δ1+((p-p1+3)|ξ1|p-p1+3φ1′(x1,yr)p+3)(p+3)/(p-p1+3)×(p1δ1(p-p1+3))p1/(p-p1+3)=|ξ1|p-p1+3b1(x1)2|x2|p1+ξ1p+3ϕ11(x1,yr)+δ1,
where ϕ11(x1,yr)=((p-p1+3)φ1′(x1,yr)/(p+3))(p+3)/(p-p1+3)(p1/δ1(p-p1+3))p1/(p-p1+3). Substituting (4.5) and (4.7) into (4.4), and adding and subtracting (b1(x1)/2)ξ1p-p1+3x2*p1 on the right-hand side of (4.4), we have
(4.8)LV1≤d1(x)ξ1p-p1+3x2p1+b1(x1)2|ξ1p-p1+3x2p1|+ξ1p+3ϕ11(x1,yr)+δ1+p-p1+32ξ1p-p1+2ξ1p1+1ψ12(x1)-θ̃θ̂̇=d1(x)ξ1p-p1+3x2p1+b1(x1)2|ξ1|p-p1+3|x2|p1+b1(x1)2ξ1p-p1+3x2*p1-b1(x1)2ξ1p-p1+3x2*p1+ξ1p+3ϕ11(x1,yr)+ξ1p+3ϕ12(x1)+δ1-θ̃θ̂̇,
where ϕ12(x1)=((p-p1+3)/2)ξ1p-p1ψ12(x1). Suppose the virtual smooth controller x2*=-ξ1β1(x1,yr,θ̂) with β1(x1,yr,θ̂)>0, which together with A1 lead to
(4.9)0≤-b1(x1)ξ1p-p1+3x2*p1≤-d1(x)ξ1p-p1+3x2*p1,-b1(x1)2ξ1p-p1+3x2*p1+b1(x1)2|ξ1p-p1+3x2*p1|≤-d1(x)ξ1p-p1+3x2*p1,-b1(x1)2ξ1p-p1+3x2*p1≤-d1(x)ξ1p-p1+3x2*p1-b1(x1)2|ξ1p-p1+3x2*p1|.
Substituting (4.9) into (4.8), one can obtain
(4.10)LV1≤d1(x)ξ1p-p1+3x2p1+b1(x1)2|ξ1p-p1+3x2p1|+b1(x1)2ξ1p-p1+3x2*p1-d1(x)ξ1p-p1+3x2*p1-b1(x1)2|ξ1p-p1+3x2*p1|+δ1-θ̃θ̂̇-c̅1θξ1p+3+c̅1θξ1p+3+ξ1p+3ϕ11(x1,yr)+ξ1p+3ϕ12(x1)≤-c̅1θξ1p+3+(θb̅1(x̅2)+b1(x1)2)|ξ1|p-p1+3|x2p1-x2*p1|+b1(x1)2ξ1p-p1+3x2*p1+c̅1θ̂ξ1p+3+ξ1p+3ϕ11(x1,yr)+ξ1p+3ϕ12(x1)+δ1+θ̃(τ1-θ̂̇),
where τ1=c̅1ξ1p+3 is a nonnegative smooth function. Choose x2* as follows:
(4.11)x2*(x1,yr,θ̂)=-ξ1β1(x1,yr,θ̂),β1(x1,yr,θ̂)=(2b1(x1)(c1+ϕ11(x1,yr)+ϕ12(x1)+c̅11+θ̂2))1/p1,
where β1(x1,yr,θ̂)≥0 is a smooth function. Then,
(4.12)LV1≤-c1ξ1p+3-c̅1θξ1p+3+(θb̅1(x̅2)+b1(x1)2)|ξ1|p-p1+3|x2p1-x2*p1|+δ1+θ̃(τ1-θ̂̇).Step i.
2≤i≤n: Assume that at Step i-1, there exists a smooth state-feedback virtual control
(4.13)xi*(x̅i-1,yr,θ̂)=-βi-1(x̅i-1,yr,θ̂)ξi-1,
such that
(4.14)LVi-1≤-∑j=1i-1(cj-∑k=j+1i-1ckj)ξjp+3-∑j=1i-1(c̅j-∑k=j+1i-1c̅kj)θξjp+3+(θb̅i-1+bi-12)|ξi-1|p-pi-1+3|xipi-1-xi*pi-1|+∑j=1i-1δj+(θ̃+∑k=2i-1ξkp-pk+3∂xk*∂θ̂)(τi-1-θ̂̇),
where βi-1>0 is a smooth function, and Vi-1=(1/4)∑k=1i-1ξkp-pk+4+(1/2)θ̃2. We will prove that (4.14) still holds for Step i.
Define theith Lyapunov candidate function
(4.15)Vi=Vi-1+14ξip-pi+4.
From (4.2) and (4.14), it follows that
(4.16)LVi≤-∑j=1i-1(cj-∑k=j+1i-1ckj)ξjp+3-∑j=1i-1(c̅j-∑k=j+1i-1c̅kj)θξjp+3+(θb̅i-1+bi-12)|ξi-1|p-pi-1+3|xipi-1-xi*pi-1|+∑j=1i-1δj+(θ̃+∑k=2i-1ξkp-pk+3∂xk*∂θ̂)(τi-1-θ̂̇)+ξip-pi+3di(x)xi+1pi+ξip-pi+3×(fi-∑k=1i-1∂xi*∂xk(dk(x)xkpk+fk)-12∑j,k=1i-1∂2xi*∂xj∂xkgjTgk-∂xi*∂yrẏr-∂xi*∂θ̂θ̂̇)+12Tr⁡{Gi(p-pi+3)ξip-pi+2GiT}.
By A2 and Lemma 3.4, there is a smooth nonnegative function φ̅i(x̅i) such that
(4.17)|fi(x̅i+1)|≤∑j=0pi-1|xi+1|jφij(x̅i)≤bi(x̅i)|xi+1pi2+φ̅i(x̅i),
then,
(4.18)|fi-∑k=1i-1∂xi*∂xkfk-12∑j,k=1i-1∂2xi*∂xj∂xkgjTgk-∂xi*∂yrẏr|≤bi(x̅i)|xi+1pi2+φi(x̅i,yr,θ̂),
where φi(x̅i,yr,θ̂) is a smooth function. By A2, (4.1) and (4.13), there exists a nonnegative smooth function ψi′(x̅i,yr,θ̂) such that
(4.19)|Gi(x̅i)|≤(|ξ1|(pi+1)/2+⋯+|ξi|(pi+1)/2)ψi′(x̅i,yr,θ̂).
By (4.13), we have
(4.20)(θb̅i-1(x̅i)+bi-1(xi-1)2)|ξi-1|p-pi-1+3|xipi-1-xi*pi-1|=(θb̅i-1(x̅i)+bi-1(xi-1)2)∑k=1pi-1Cpi-1k|ξi|k|ξ1|p-k+3β1pi-1-k≤∑k=1i-1cik1ξkp+3+∑k=1i-1c̅ikθξkp+3+φi1(x̅i,yr,θ̂)ξip+3+θφi2ξip+3,
where φi1(x̅i,yr,θ̂) and φi2(x̅i,yr,θ̂) are two smooth functions. From A1, (4.1), and (4.13), it follows that
(4.21)|-ξip-pi+3∑k=1i-1∂xi*∂xkdk(x)xkpk|≤θ′|ξi|p-pi+3∑k=1i-1bk-1(x̅k)|∂xi*∂xk||ξk+xk*|pk≤θ′(p+3)/(p-pi+3)ξip+3φi3(x̅i,yr,θ̂)+δi1≤θξip+3φi3(x̅i,yr,θ̂)+δi1,(4.22)|ξi|p-pi+3|fi-∑k=1i-1∂xi*∂xkfk-12∑j,k=1i-1∂2xi*∂xj∂xkgjTgk-∂xi*∂yrẏr|≤|ξi|p-pi+3(bi(x̅i)|xi+1|pi2+φi′(x̅i,yr,θ̂))≤bi(x̅i)2|ξi|p-pi+3|xi+1|pi+φi4(x̅i,yr,θ̂)ξip+3+δi2,
where φi3(x̅i,yr,θ̂) and φi4(x̅i,yr,θ̂) are two smooth functions. From (4.19), one can obtain
(4.23)12Tr⁡{Gi(p-pi+3)ξip-pi+2GiT}≤p-pi+32ξ2p-pi+2(|ξ1|(pi+1)/2+⋯+|ξi|(pi+1)/2)2ψi′2(x̅i,yr,θ̂)≤∑k=1i-1cik2ξkp+3+φi5(x̅i,yr,θ̂)ξip+3,
where φi5(x̅i,yr,θ̂) is a smooth nonnegative function. Substituting (4.20)–(4.23) into (4.16), one gets
(4.24)LVi≤-∑j=1i-1(cj-∑k=j+1i-1ckj)ξjp+3-∑j=1i(c̅j-∑k=j+1i-1c̅kj)θξjp+3+∑j=1i-1cij′ξjp+3+∑j=1i-1δj+θ∑j=1i-1c̅ijξjp+3+hi1′ξip+3+θhi2ξip+3-c̅iθξip+3+c̅iθξip+3+ξip-pi+3di(x)xi+1pi+bi(x̅i)2|ξi|p-pi+3|xi+1|pi+bi(x̅i)2ξip-pi+3x3*pi-bi(x̅i)2ξip-pi+3x3*pi+(θ̃+∑k=2i-1ξkp-pk+3∂xk*∂θ̂)(τi-1-θ̂̇)-ξip-pi+3∂xi*∂θ̂θ̂̇,
where
(4.25)cij′=cij1+cij2,j=1,…,i-1,hi1′=φi1+φi4+φi5,hi2=φi2+φi3.
Suppose the virtual smooth controller xi+1*=-ξiβi(x̅i,yr,θ̂) with βi(x̅i,yr,θ̂)>0, which together with A2 render
(4.26)-bi(x̅i)2ξip-pi+3xi+1*pi≤-di(x)ξip-pi+3xi+1*pi-bi(x̅i)2|ξip-pi+3xi+1*pi|.
Substituting (4.26) into (4.24) leads to
(4.27)LVi≤-∑j=1i-1(cj-∑k=j+1i-1ckj)ξjp+3-∑j=1i(c̅j-∑k=j+1ic̅kj)θξjp+3+∑j=1i-1cij′ξjp+3+hi1′ξip+3+(θ̂+θ̃)hi2ξip+3+c̅i(θ̂+θ̃)ξip+3+ξip-pi+3di(x)xi+1pi+bi(x̅i)2|ξi|p-pi+3|xi+1|pi+bi(x̅i)2ξip-pi+3x3*pi-di(x)ξip-pi+3xi+1*p2-bi(x̅i)2|ξip-pi+3xi+1*pi|+(θ̃+∑k=2i-1ξkp-pk+3∂xk*∂θ̂)(τi-1-θ̂̇)-ξip-pi+3∂xi*∂θ̂θ̂̇+∑k=2i-1ξkp-pk+3∂xk*∂θ̂(hi2+c̅i)ξip+3-∑k=2i-1ξkp-pk+3∂xk*∂θ̂(hi2+c̅i)ξip+3+ξip-pi+3∂xi*∂θ̂τi-ξip-pi+3∂xi*∂θ̂τi,
where τi=τi-1+(hi2+c̅i)ξip+3. For (4.27), we have
(4.28)|-∑k=2i-1ξkp-pk+3∂xk*∂θ̂(hi2+c̅i)ξip+3|≤φi6(x̅i,yr,θ̂)ξip+3,|-ξip-pi+3∂xi*∂θ̂τi|≤∑k=1i-1cik3ξkp+3+φi7(x̅i,yr,θ̂)ξip+3,
where cik3 is a design parameter, φi6(x̅i,yr,θ̂) and φi7(x̅i,yr,θ̂) are the smooth functions. Let cij=cij′+cij3, hi1=hi1′+φi6+φi7. (4.27) becomes
(4.29)LVi≤-∑j=1i-1(cj-∑k=j+1i-1ckj)ξjp+3-∑j=1i(c̅j-∑k=j+1ic̅kj)θξjp+3+∑j=1i-1cijξjp+3+hi1′ξip+3+θ̂hi2ξip+3+(θb̅i(x̅i+1)+bi(xi)2)|ξi|p-pi+3|xi+1pi-xi+1*pi|+bi(x̅i)2ξip-pi+3xi+1*pi+(θ̃+∑k=2iξkp-pk+3∂xk*∂θ̂)(τi-θ̂̇)+∑j=1iδj≤-∑j=1i(cj-∑k=j+1ickj)ξjp+3-∑j=1i(c̅j-∑k=j+1ic̅kj)θξjp+3+(θb̅i(x̅i+1)+bi(xi)2)|ξi|p-pi+3|xi+1pi-xi+1*pi|+∑j=1iδj+(θ̃+∑k=2iξkp-pk+3∂xk*∂θ̂)(τi-θ̂̇),
by choosing
(4.30)xi+1*(x̅i,yr,θ̂)=-ξiβi(x̅i,yr,θ̂),βi(x̅i,yr,θ̂)=(2bi(x̅i)(ci+hi1+(hi2+c̅i)1+θ̂2))1/pi,
where βi(x̅i,yr,θ̂)≥0 is a smooth function.
Finally, wheni=n,xn+1=xn+1*=u is the actual control. By choosing the actual control law and the adaptive law:
(4.31)u(x̅n,yr,θ̂)=-βn(x̅n,yr,θ̂)ξn,θ̂̇=τn=∑k=1nHk2ξkp+3,
where βn≥0 and H12,…,Hn2 are smooth functions, one gets
(4.32)LVn≤-∑j=1n(cj-∑k=j+1nckj)ξjp+3-∑j=1n(c̅j-∑k=j+1nc̅kj)θξjp+3+∑j=1nδj.Theorem 4.1.
If A1–A3 hold for the high-order stochastic nonlinear system (3.8), under the smooth adaptive state-feedback controller (4.32), the closed-loop system is ISpS in probability, and the tracking error ξ1=y-yr can be regulated to a neighborhood of the origin in probability with radius as small as possible (Figure 2).Gives the response of the closed-loop system, from which, the effectiveness of the controller is demonstrated.
(a)(b)Proof.
ForVn=∑i=1n(1/4)ξip-pi+4+(1/2)θ̃2, it is obvious that Vn satisfies (3.4). Choosing all the design parameters cj and c̅j to satisfy
(4.33)cj>∑k=j+1nckj,c̅j>∑k=j+1nc̅kj,j=1,…,n,
such that (3.5) holds, and then using Lemma 3.3, one can prove Theorem 4.1.


## 5. Simulation

Now, we apply the control scheme to the mechanical system (2.5). Let ξ1=x1-yr be the tracking error, where yr=sint is a bounded smooth reference signal. For (2.5), di(·)=1, and p=max{1,3}=3.ChooseV1(ξ1)=(1/(p-p1+4))ξ1p-p1+4=ξ16/6. Then,(5.1)LV1(ξ1)=ξ15(x2-ẏr).
The smooth virtual controller can be chosen as x2*(x1,yr)=-c1ξ1+ẏr, which renders(5.2)LV1(ξ1)=-c1ξ16+ξ15(x2-x2*).Next, definingV2(ξ1,ξ2)=V1+(1/(p-p2+4))ξ2p-p2+4=ξ16/6+ξ24/4, a direct calculation gives(5.3)LV2=-c1ξ16+ξ15ξ2+ξ23(x33-∂x2*∂x1x2-∂x2*∂yrẏr)=-c1ξ16+ξ15ξ2+ξ23(x33-h2),
where ξ2=x2-x2*. By Lemma 3.5, choosing m=3/2, one can obtain that for any constant δ2>0,(5.4)|ξ24h2|≤δ2+(2ξ24h23)3/2(12δ2)1/2≤δ2+ξ26φ2(x̅2).
Then, by (5.4) and (5.5), it is easy to see that(5.5)LV(ξ1,ξ2)=-(c1-c21)ξ16-c2ξ26+ξ23(x33-x3*3)+δ2,
by choosing x3*(x1,x2,yr)=-ξ2(c2+d2+φ2)1/3.Definingξ3=x3-x3* and the Lyapunov function V3(ξ1,ξ2,ξ3)=V2(ξ1,ξ2)+(1/6)ξ36, one gets(5.6)LV3≤-(c1-c21)ξ16-c2ξ26+ξ23(x33-x3*3)+δ2+ξ35(x4-∂x3*∂x1x2-∂x3*∂x2x33-∂x3*∂yrẏr)≤-(c1-c21)ξ16-c2ξ26+δ31+ξ36φ31+δ32+ξ36φ32+ξ35(x4-x4*)+ξ35x4*=-(c1-c21)ξ16-c2ξ26-c3ξ36+ξ35(x4-x4*)+δ2+δ3,
by choosing x4*(x1,x2,x3,yr)=-ξ3(c3+φ31+φ32). At last, choosing ξ4=x4-x4*,V4(ξ1,ξ2,ξ3,ξ4)=V3(ξ1,ξ2,ξ3)+(1/6)ξ46, a direct calculation gives(5.7)V4≤-(c1-c21)ξ16-c2ξ26-c3ξ36+ξ35(x4-x4*)+δ2+δ3+ξ45(v+k0f-∂x4*∂x1x2-∂x4*∂x2x33-∂x4*∂x3x4-∂x4*∂yrẏr)+5ξ44σ2f2≤-(c1-c21)ξ16-c2ξ26-c3ξ36-c4ξ46+δ2+δ3+δ41+ξ6φ41+δ42+ξ6φ42+ξ45v=-(c1-c21)ξ16-c2ξ26-c3ξ36-c4ξ46+δ2+δ3+δ4,
by choosing(5.8)v=-ξ4(c4+φ41+φ42).Choose the design parametersσ=0.125,δ2=0.01,δ3=0.01, and δ4=0.01. Moreover, to satisfy (5.3), we choose c1=1>c21=5/6,c2=1.5,c3=0.5 and c4=0.5. Choose the initial values x1(0)=0.45,x2(0)=0.5,x3(0)=0.5,andx4(0)=0.5.


## 6. Concluding Remarks

In this paper, a mechanical system is firstly introduced. Then, by a series of coordinate transformations, the mechanical system can be transformed to a class of high-order stochastic nonlinear system, based on which, a more general mathematical model is considered and the smooth state-feedback controller is designed which guarantees that the tracking errorξ1=y-yr can be regulated to a neighborhood of the origin in probability with radius as small as possible. At last, the simulation is given to show the effectiveness of the design scheme.



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*Source: 718913-2012-02-28.xml*