# Adaptive Exponential Stabilization for a Class of Stochastic Nonholonomic Systems

Authors: Xiaoyan Qin
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper investigates the adaptive stabilization problem for a class of stochastic nonholonomic systems with strong drifts. By using input-state-scaling technique, backstepping recursive approach, and a parameter separation technique, we design an adaptive state feedback controller. Based on the switching strategy to eliminate the phenomenon of uncontrollability, the proposed controller can guarantee that the states of closed-loop system are global bounded in probability.


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## Body

## 1. Introduction

The nonholonomic systems cannot be stabilized by stationary continuous state feedback, although it is controllable, due to Brockett’s theorem [1]. So the well-developed smooth nonlinear control theory and the method cannot be directly used in these systems. Many researchers have studied the control and stabilization of nonholonomic systems in the nonlinear control field and obtained some success [2–6]. It should be mentioned that many literatures consider the asymptotic stabilization of nonholonomic systems; the exponential convergence is also an important topic theme, which is demanded in many practical applications. However, the exponential regulation problem, particularly the systems with parameterization, has received less attention. Recently, [3] firstly introduced a class of nonholonomic systems with strong nonlinear uncertainties and obtained global exponential regulation. References [4, 5] studied a class of nonholonomic systems with output feedback control. Reference [6] combined the idea of combined input-state-scaling and backstepping technology, achieving the asymptotic stabilization for nonholonomic systems with nonlinear parameterization.It is well known that when the backstepping designs were firstly introduced, the stochastic nonlinear control had obtained a breakthrough [7]. Based on quartic Lyapunov functions, the asymptotical stabilization control in the large of the open-loop system was discussed in [8]. Further research was developed by the recent work [9–16]. [17–19] studied a class of nonholonomic systems with stochastic unknown covariance disturbance. Since stochastic signals are very prevalent in practical engineering, the study of nonholonomic systems with stochastic disturbances is very significant. So, there exists a natural problem that is how to design an adaptive exponential stabilization for a class of nonholonomic systems with stochastic drift and diffusion terms. Inspired by these papers, we will study the exponential regulation problem with nonlinear parameterization for a class of stochastic nonholonomic systems. We use the input-state-scaling, the backstepping technique, and the switching scheme to design a dynamic state-feedback controller with∑T∑≠I; the closed-loop system is globally exponentially regulated to zero in probability.This paper is organized as follows. In Section2, we give the mathematical preliminaries. In Section 3, we construct the new controller and offer the main result. In the last section, we present the conclusions.


## 2. Problem Statement and Preliminaries

In this paper, we consider a class of stochastic nonholonomic systems as follows:(1)dx0=d0(t)u0dt+f0(t,x0)dtdxi=di(t)xi+1u0dt+fi(t,x0,x-i)dt+φi(x-i)∑(t)dω,=di(t)xi+1u0dt+fi(t,x0,x-i)dt+gli=1,…,n-1,dxn=dn(t)u1dt+fn(t,x0,x)dt+φn(x-)∑(t)dω,
wherex0∈Randx=[x1,…,xn]T∈Rnare the system states andu0∈Randu1∈Rare the control inputs, respectively.x-i=[x1,x2,…,xi]T∈Ri,(i=1,2,…,n), andx-n=x;ω∈Rris anr-dimensional standard Wiener process defined on the complete probability space(Ω,F,P)withΩbeing a sample space,Fbeing a filtration, andPbeing measure. The drift and diffusion termsfi(·),φi(·)are assumed to be smooth, vanishing at the origin(x1,x2,…,xi)=(0,0,…,0);∑(t):R+→Rr×ris the Borel bounded measurable functions and is nonnegative definite for eacht≥0.di(t)are disturbed virtual control coefficients, wherei=0,1…n.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=f(x,t)dt+g(x,t)dω,x(0)=x0∈Rn,
wherex∈Rnis the state of system (2), the Borel measurable functions:f:Rn+1→Rnandg:Rn+1→Rn×rare assumed to beC1in their arguments, andω∈Rris anr-dimensional standard Wiener process defined on the complete probablity space(Ω,F,P).Definition 1 (see [8]).
Given anyV(x,t)∈C1,2, for stochastic nonlinear system (2), the differential operatorLis defined as follows:
(3)LV(x,t)=∂V∂t+∂V∂xf+12tr(gT∂2V∂x2g),
whereC1,2(Rn×R+;R+)denotes all nonnegative functionsV(x,t)onRn×R+, which areC1intandC2inx, and for simplicity, the smooth functionf(·)is denoted byf.Lemma 2 (see [8]).
Letxandybe real variables. Then, for any positive integersm,n, and any real numberε>0, the following inequality holds:
(4)α(·)xmyn≤ε|x|m+n+nm+n(m+nm)-m/n×α(·)(m+n)/nε-m/n|y|m+n.Lemma 3 (see [7]).
Considering the stochastic nonlinear system (2), if there exist aC1,2functionV(x,t),K∞class functionsα_andα-, constantc-, and a nonnegative functionsW(x,t)such that
(5)α_|(x)|≤V(x)≤α-|(x)|,LV(x)≤-W(x,t)+c-,
then for eachx0∈Rn. (1) For (2), there exists an almost surely unique solution on[0,∞]. (2) Whenc-=0,f(0,t)=0,g(0,t)=0, andW(x,t)=W(x)is continuous, the equilibriumx=0is globally stable in probability, and the solutionx(t)satisfiesP{limt→∞W(x(t)=0}=1.(3) For any givenε>0, there exist a classKLfunctionβc(·,·)andKfunctionγ(·)such thatP{|(x(t))|<βc(|x0|,t)+γ(c)}≥1-εfor anyt≥0,x0∈Rn∖{0}.Lemma 4 (see [20]).
For any real-valued continuous functionf(x,y),x∈Rm,y∈Rn, there exist smooth scalar-value funcionsa(x)≥0,b(y)≥0,c(x)>1,andd(y)≥1, such that|f(x,y)|≤a(x)+b(y), and|f(x,y)|≤c(x)d(y).


## 3. Controller Design and Analysis

The purpose of this paper is to construct a smooth state-feedback control law such that the solution process of system (1) is bounded in probability. For clarity, the case thatx0(t0)≠0is firstly considered. Then, the case where the initialx0(t0)=0is dealt with later. The triangular structure of system (1) suggests that we should design the control inputsu0andu1in two separate stages.To design the controller for system (1), the following assumptions are needed.Assumption 5.
For0≤i≤n, there are some positive constantsλi1andλi2that satisfy the inequalityλi1≤di(t)≤λi2.Assumption 6.
Forf0(t,x0), there exists a nonnegative smooth functionγ0(t,x0), such that|f0(t,x0)|≤|x0|γ0(t,x0).
For eachfi(t,x0,x-i),φi(x-i), there exist nonnegative smooth functionsγi(t,x0,x-i)andρi(x-i), such that|fi(t,x0,x-i)|≤(∑k=1i|xk|)γi(t,x0,x-i),|φi(x-i)|≤(∑k=1i|xk|)ρi(x-i).


### 3.1. Designingu0forx0-Subsystem

Forx0-subsystem, the controlu0can be chosen as
(6)u0=-λ0x0,
whereλ0=(k0+γ0)/λ01andk0is a positive design parameter.Consider the Lyapunov function candidateV0=x02/2. From (6) and Assumptions 5 and 6, we have
(7)LV0=x0(d0u0+f0(t,x0))≤d0u0x0+x02γ0≤-k0x02=-2k0V0.
So, we obtain the first result of this paper.Theorem 7.
Thex0-subsystem, under the control law (6) with an appropriate choice of the parametersk0,λ01,λ02, is globally exponentially stable.Proof.
Clearly, from (7),LV0≤0, which implies that|x0(t)|≤|x0(t0)|e-k0(t-t0). Therefore,x0is globally exponentially convergent. Consequently,x0can be zero only att=t0, whenx(t0)=0ort=∞. It is concluded thatx0does not cross zero for allt∈(t0,∞)provided thatx(t0)≠0.Remark 8.
Ifx(t0)≠0,u0exists and does not cross zero for allt∈(t0,∞)independent of thex-subsystem from (6).


### 3.2. Backstepping Design foru1

From the above analysis, thex0-state in (1) can be globally exponentially regulated to zero ast→∞, obviously. In this subsection, we consider the control lawu1for thex-subsystem by using backstepping technique. To design a state-feedback controller, one first introduces the following discontinuous input-state-scaling transformation:
(8)ηi=eαtxiu0n-i,i=1…,n,u=eαtu1.
Under the newx-coordinates,x-subsystems is transformed into
(9)dηi=diηi+1dt+f-idt+ϕiT∑T(t)dω,i=1,…,n-1,dηn=dnudt+f-ndt+ϕnT∑T(t)dω,
where
(10)f-i=αηi+eαtfiu0n-i-(n-i)ηiu0∂u0∂x0(d0u0+f0),ϕi=eαtφiu0n-i.In order to obtain the estimations for the nonlinear functionsf-iandϕi, the following Lemma can be derived by Assumption 6.Lemma 9.
Fori=1,2…n,there exist nonnegative smooth functionsγ-i(·),ρ-i(·), such that
(11)|f-i|≤(∑k=1i‍|ηk|)γ-i(x0,x-i),(12)|ϕi|≤(∑k=1i‍|ηk|)ρ-i(x-i).Proof.
We only prove (11). The proof of (12) is similar to that of (11). In view of (6), (8), (10) and Assumption 6, one obtains
(13)|f-i|=|αηi+eαtfiu0n-i-(n-i)ηiu0∂u0∂x0(d0x0+f0)|≤|αηi|+(∑k=1i‍eαt|xk|u0n-k|u0i-k|)γi+(n-i)(λ0λ02+γ0)|ηi|≤|α||ηi|+(∑k=1i‍|ηk|λ0i-k|x0i-k|)γi+(n-i)(λ0λ02+γ0)|ηi|≤(∑k=1i‍|ηk|)(|α|+|λ0i-k||x0i-k|γi+(n-i)(λ0λ02+γ0))≤(∑k=1i‍|ηk|)γ-i(x0,x-i),
whereγ-i(x0,x-i)≥|α|+|λ0i-k|x0i-k|γi+(n-i)(λ0λ02+γ0).
To design a state-feedback controller, one introduces the coordinate transformation(14)z1=η1,zi=ηi-αi(z-i-1),i=1,2…,n,
whereα2,…,αnare smooth virtual control laws and will be designed later andα1=0.θ^denotes the estimate ofθ, where
(15)θ=supt≥0{max{∥∑(t)∑T(t)∥2,∥∑(t)∑T(t)∥4/3,∥∑(t)∑T(t)∥∥∑(t)∑T(t)∥2,∥∑(t)∑T(t)∥4/3}}.
Then using (9), (10), (14) and Ito^differentiation rule, one has
(16)dzi=d(ηi-αi)=(diηi+1+Fi(z-i,x0)-∂αi∂θ^θ^˙)dt+GiT(z-i)∑T(t)dω-12∑k,m=1i-1‍∂2αi∂zk∂zmϕkT(z-k)∑T(t)∑(t)ϕm(z-m)dt,=(diηi+1+Fi(z-i,x0)-∂αi∂θ^θ^˙)dt+GiTi=1,2…n,
whereηn+1=u,Fi(z-i,x0)=f-i+∑k=1i-1‍(∂αi/∂zk)(dkηk+1+f-k), andGi(z-i,x0)=ϕi+∑k=1i-1‍(∂αi/∂zk)ϕk,wherei=1,2…n.Using Lemmas 2, 4, and 9 and (14), we easily obtain the following lemma.Lemma 10.
For1≤i≤n,there exist nonnegative smooth functionsγi1(z-i,x0),pi1(z-i), andp-i(z-i), such that
(17)|Fi|≤(∑k=1i‍|zk|)γi1(z-i,x0),|Gi|≤(∑k=1i‍|zk|)pi1(z-i),|Φi|≤(∑k=1i‍|zk|)p-i(z-i).The proof of Lemma10 is similar to that of Lemma 9, so we omitted it.We now give the design process of the controller.Step 1.
Consider the first Lyapunov functionV1(z1,θ^)=(1/4)z14+(1/2)(θ^-θ)2. By (14), (15), and (16), we have
(18)LV1=z13(d1η2+F1)+32z12Tr(G1T∑T(t)∑(t)G1)+(θ^-θ)θ^˙.
Using Lemma10 and Lemma 4, we have
(19)|z13F1|≤z14γ11(z1,x0)|32z12Tr(G1T∑T(t)∑(t)G1)|≤z14p112(z1,x0)|∑T(t)∑(t)|≤z14p112(z1,x0)θ.
Substituting (19) into (18) and using (14), we have
(20)LV1≤d1z13(η2-α2)+d1z13α2+z14p112(z1,x0)θ+z14γ11(z1,x0)+(θ^-θ)θ^˙≤d1z13z2+d1z13α2+z14p112(z1,x0)θ+z14γ11(z1,x0)+(θ^-θ)θ^˙,
whereα2=-z1β1=-z1((c1+γ11+p112θ^)/λ11). Substitutingα2into (20), we have
(21)LV1≤d1z13z2-c1z14+(θ^-θ)(θ^˙-τ1),
whereτ1=z14p112.Step i.
(2≤i≤n). Assume that at stepi-1, there exists a smooth state-feedback virtual controlαi=-zi-1βi-1(z-i-1,θ^)=-zi-1((ci-1+θ^1+(ψi-12+ψi-13)2+bi-1+ψi-11+ψi-14)/λi-11), such that
(22)LVi-1≤-∑j=1i-2‍(cj-εj-ej)zj4-ci-1zi-14+di-1zi-13zi+(θ^-θ-∑k=2i-1zk3∂αk∂θ^)(θ^˙-τi-1),
whereVi-1=∑j=1i-1‍(1/4)zj4+(1/2)(θ^-θ)2,τi-1=τ1+∑k=2i-1‍zk4(ψi-12+ψi-13), andεj=∑k=1j‍(εk1+εk2+εk3+εk4),wherej=1,…,n.
Then, define theith Lyapunov candidate functionVi(z-i,θ^)=Vi-1+(1/4)zi4. From (16) and (22), it follows that
(23)LVi≤-∑j=1i-2‍(cj-εj-ej)zj4-ci-1zi-14+di-1zi-13zi+zi3(∑k,m=1i-1‍diηi+1+Fi(z-i,x0)-∂αi∂θ^θ^˙+zi3(-12∑k,m=1i-1‍∂2αi∂zk∂zmϕkT(z-k)∑T(t)∑(t)ϕm(z-m))+32zi2Tr(GiT(z-i)∑T(t)∑(t)Gi(z-i))+(θ^-θ-∑k=2i-1‍zk3∂αk∂θ^)(θ^˙-τi-1).
Using Lemmas 9 and 4, there are always known nonnegative smooth functionsψi1(z-i),ψi2(z-i),ψi3(z-i),ψi4(z-i)and constantεi>0,  εij>0, where  i=1,…,nand  j=1,2,3,4.
Consider(24)zi3Fi≤|zi3|(∑k=1i-1‍|zk|)γi1(z-i,x0)≤γi1zi4+∑k=1i-1‍(εk1zk4+34(4εk1)-1/3γi14/3zi4)≤∑k=1i-1‍εk1zk4+ψi1zi4,
whereψi1≥γi1+∑k=1i-1‍(3/4)(4εk1)-1/3γi14/3.(25)-12zi3∑k,m=1i-1‍∂2αi∂zk∂zmϕkT∑T(t)∑(t)ϕm≤12zi3∑k,m=1i-1‍|∂2αi∂zk∂zm|(∑j=1k‍|zj|)p-k(z-k)×(∑j=1m‍|zj|)p-m(z-m)|∑T(t)∑(t)|≤zi3(∑k=1i-1‍zk2)p--i(z-i)|∑T(t)∑(t)|≤zi4ψi2(z-i)θ+∑k=1i-1‍εk2zk4,
whereψi2≥∑k=1i-1‍(3/4)(4εk2)-1/3(p--i(z-i))3/4.
(26)32zi2Tr(GiT∑T(t)∑(t)Gi)≤32zi2pi22(z-i)(∑k=1i‍|zk|)2|∑T(t)∑(t)|≤32zi2ipi22(z-i)(∑k=1i‍zk2)|∑T(t)∑(t)|≤32zi4ipi22(z-i)θ+∑k=1i-1‍εk3zk4+∑k=1i-1‍14εk3(32ipi22)2zi4θ≤∑k=1i-1‍εk3zk4+ψi3zi4θ,
whereψi3≥(3/2)ipi22+∑k=1i-1‍(1/4εk3)((3/2)ipi22)2. 									(27)di-1zi-13zi≤λi-12|zi-13zi|≤ei-1zi-14+14(43ei-1)-3zi4λi24≤ei-1zi-14+bizi4,
wherebi≥(1/4)((4/3)ei-1)-3λi24,τi-1=z14p112+∑k=2i-1‍zk4(ψk2+ψk3), andτi=τi-1+(ψi2+ψi3)zi4.
(28)-zi3∂αi∂θ^τi≤zi3|∂αi∂θ^|(τi-1+zi4(ψi2+ψi3))≤zi41+(zi3∂αi∂θ^)2(ψi2+ψi3)+zi3|∂αi∂θ^|(z14p112+∑k=2i-1‍zk4(ψk2+ψk3))+34(4)-1/3(|∂α2∂θ^|z13p112)4/3εi4-1/3zi4≤εi4z14+34(4)-1/31+(∂αi∂θ^z13P112)24/3εi4-1/3zi4+∑k=2i-1‍εk4zk4+∑k=2i-1‍34(4)-1/31+(∂αi∂θ^zk3(ψk2+ψk3))24/3εk4-1/3zi4≤∑k=1i-1‍εk4zk4+ψi4zi4,
whereψi4≥(3/4)(4εi4)-1/31+((∂αi/∂θ^)z13P112)24/3+∑k=2i-1‍(3/4)(4εk4)-1/31+((∂αi/∂θ^)zk3(ψk2+ψk3))24/3.
(29)αi+1(z-i,θ^)=-ziβi(z-i,θ^),βi(z-i,θ^)=ci+ψi1+ψi4+bi+1+(ψi2+ψi3)2θ^λi1,
whereci>0is a design parameter to be chosen.
With the aid of (24)–(29) and (14), (23) can be simplified as
(30)LVi≤-∑j=1i-1‍(cj-εj-ej)zj4-cizi4+dizi3zi+1+(θ^-θ-∑k=2i‍∂αk∂θ^zk3)(θ^˙-τi).
Finally, wheni=n,  zn+1=uis the actual control. By choosing the actual control law and the adaptive law,
(31)u(z-n,θ^)=-znβn(z-n,θ^),θ^˙=τn=z14p112+∑k=2n‍zk4(ψk2+ψk3),βn(z-n,θ^)=cn+bn+ψn1+ψn4+1+(ψn2+ψn3)2θ^λn1,u1=e-αtu,
wherecn>0is a design parameter to be chosen andψni,i=1,…4are smooth functions; we get
(32)LVn≤-∑j=1n‍(cj-εj-ej)zj4,
whereVn(z,θ^)=∑k=1n‍(1/4)zk4+(1/2)(θ^-θ)2,z=(z1,…zn). We have finished the controller design procedure forx0(t0)≠0and the parameter identification. Without loss of generality, we can assume thatt0≠0.


### 3.3. Switching Control and Main Result

In the preceding subsection, we have given controller design forx0≠0.Now, we discuss how to choose the control lawsu0andu1whenx0=0.We chooseu0asu0=-λ0x0+u0*,u0*>0. And choose the Lyapunov functionV0=(1/2)x02. Its time derivative is given byLV0=-λ0x02+u0*, which leads to the bounds ofx0. During the time period[0,ts), usingu0=-λ0x0+u0*, new control lawucan be obtained by the control procedure described above to the originalx-subsystem in (1). Then, we can conclude that thex-state of (1) cannot be blown up during the time period[0,ts). Since atx(ts)≠0,we can switch the control inputsu0anduto (6) and (31), respectively.Now, we state the main results as follows.Theorem 11.
Under Assumption5, if the proposed adaptive controller (31) together with the above switching control strategy is used in (1), then for any initial contidion(x0,x,θ^)∈Rn, the closed-loop system has an almost surely unique solution on[0,∞), the solution process is bounded in probability, andP{limt→∞θ^(t)existsandisfinite}=1.Proof.
According to the above analysis, it suffices to prove in the casex0(0)≠0.Since we have already proven thatx0can be globally exponentially convergent to zero in probability in Section 3.1, we only need prove thatx(t)is convergent to zero in probability also. In this case, we choose the Lyapunov functionV=Vn, andci>εi+ei; from (32) and Lemma 3, we know that the closed-loop system has an almost surely unique solution on[0,∞), and the solution process is bounded in probability.


## 3.1. Designingu0forx0-Subsystem

Forx0-subsystem, the controlu0can be chosen as
(6)u0=-λ0x0,
whereλ0=(k0+γ0)/λ01andk0is a positive design parameter.Consider the Lyapunov function candidateV0=x02/2. From (6) and Assumptions 5 and 6, we have
(7)LV0=x0(d0u0+f0(t,x0))≤d0u0x0+x02γ0≤-k0x02=-2k0V0.
So, we obtain the first result of this paper.Theorem 7.
Thex0-subsystem, under the control law (6) with an appropriate choice of the parametersk0,λ01,λ02, is globally exponentially stable.Proof.
Clearly, from (7),LV0≤0, which implies that|x0(t)|≤|x0(t0)|e-k0(t-t0). Therefore,x0is globally exponentially convergent. Consequently,x0can be zero only att=t0, whenx(t0)=0ort=∞. It is concluded thatx0does not cross zero for allt∈(t0,∞)provided thatx(t0)≠0.Remark 8.
Ifx(t0)≠0,u0exists and does not cross zero for allt∈(t0,∞)independent of thex-subsystem from (6).


## 3.2. Backstepping Design foru1

From the above analysis, thex0-state in (1) can be globally exponentially regulated to zero ast→∞, obviously. In this subsection, we consider the control lawu1for thex-subsystem by using backstepping technique. To design a state-feedback controller, one first introduces the following discontinuous input-state-scaling transformation:
(8)ηi=eαtxiu0n-i,i=1…,n,u=eαtu1.
Under the newx-coordinates,x-subsystems is transformed into
(9)dηi=diηi+1dt+f-idt+ϕiT∑T(t)dω,i=1,…,n-1,dηn=dnudt+f-ndt+ϕnT∑T(t)dω,
where
(10)f-i=αηi+eαtfiu0n-i-(n-i)ηiu0∂u0∂x0(d0u0+f0),ϕi=eαtφiu0n-i.In order to obtain the estimations for the nonlinear functionsf-iandϕi, the following Lemma can be derived by Assumption 6.Lemma 9.
Fori=1,2…n,there exist nonnegative smooth functionsγ-i(·),ρ-i(·), such that
(11)|f-i|≤(∑k=1i‍|ηk|)γ-i(x0,x-i),(12)|ϕi|≤(∑k=1i‍|ηk|)ρ-i(x-i).Proof.
We only prove (11). The proof of (12) is similar to that of (11). In view of (6), (8), (10) and Assumption 6, one obtains
(13)|f-i|=|αηi+eαtfiu0n-i-(n-i)ηiu0∂u0∂x0(d0x0+f0)|≤|αηi|+(∑k=1i‍eαt|xk|u0n-k|u0i-k|)γi+(n-i)(λ0λ02+γ0)|ηi|≤|α||ηi|+(∑k=1i‍|ηk|λ0i-k|x0i-k|)γi+(n-i)(λ0λ02+γ0)|ηi|≤(∑k=1i‍|ηk|)(|α|+|λ0i-k||x0i-k|γi+(n-i)(λ0λ02+γ0))≤(∑k=1i‍|ηk|)γ-i(x0,x-i),
whereγ-i(x0,x-i)≥|α|+|λ0i-k|x0i-k|γi+(n-i)(λ0λ02+γ0).
To design a state-feedback controller, one introduces the coordinate transformation(14)z1=η1,zi=ηi-αi(z-i-1),i=1,2…,n,
whereα2,…,αnare smooth virtual control laws and will be designed later andα1=0.θ^denotes the estimate ofθ, where
(15)θ=supt≥0{max{∥∑(t)∑T(t)∥2,∥∑(t)∑T(t)∥4/3,∥∑(t)∑T(t)∥∥∑(t)∑T(t)∥2,∥∑(t)∑T(t)∥4/3}}.
Then using (9), (10), (14) and Ito^differentiation rule, one has
(16)dzi=d(ηi-αi)=(diηi+1+Fi(z-i,x0)-∂αi∂θ^θ^˙)dt+GiT(z-i)∑T(t)dω-12∑k,m=1i-1‍∂2αi∂zk∂zmϕkT(z-k)∑T(t)∑(t)ϕm(z-m)dt,=(diηi+1+Fi(z-i,x0)-∂αi∂θ^θ^˙)dt+GiTi=1,2…n,
whereηn+1=u,Fi(z-i,x0)=f-i+∑k=1i-1‍(∂αi/∂zk)(dkηk+1+f-k), andGi(z-i,x0)=ϕi+∑k=1i-1‍(∂αi/∂zk)ϕk,wherei=1,2…n.Using Lemmas 2, 4, and 9 and (14), we easily obtain the following lemma.Lemma 10.
For1≤i≤n,there exist nonnegative smooth functionsγi1(z-i,x0),pi1(z-i), andp-i(z-i), such that
(17)|Fi|≤(∑k=1i‍|zk|)γi1(z-i,x0),|Gi|≤(∑k=1i‍|zk|)pi1(z-i),|Φi|≤(∑k=1i‍|zk|)p-i(z-i).The proof of Lemma10 is similar to that of Lemma 9, so we omitted it.We now give the design process of the controller.Step 1.
Consider the first Lyapunov functionV1(z1,θ^)=(1/4)z14+(1/2)(θ^-θ)2. By (14), (15), and (16), we have
(18)LV1=z13(d1η2+F1)+32z12Tr(G1T∑T(t)∑(t)G1)+(θ^-θ)θ^˙.
Using Lemma10 and Lemma 4, we have
(19)|z13F1|≤z14γ11(z1,x0)|32z12Tr(G1T∑T(t)∑(t)G1)|≤z14p112(z1,x0)|∑T(t)∑(t)|≤z14p112(z1,x0)θ.
Substituting (19) into (18) and using (14), we have
(20)LV1≤d1z13(η2-α2)+d1z13α2+z14p112(z1,x0)θ+z14γ11(z1,x0)+(θ^-θ)θ^˙≤d1z13z2+d1z13α2+z14p112(z1,x0)θ+z14γ11(z1,x0)+(θ^-θ)θ^˙,
whereα2=-z1β1=-z1((c1+γ11+p112θ^)/λ11). Substitutingα2into (20), we have
(21)LV1≤d1z13z2-c1z14+(θ^-θ)(θ^˙-τ1),
whereτ1=z14p112.Step i.
(2≤i≤n). Assume that at stepi-1, there exists a smooth state-feedback virtual controlαi=-zi-1βi-1(z-i-1,θ^)=-zi-1((ci-1+θ^1+(ψi-12+ψi-13)2+bi-1+ψi-11+ψi-14)/λi-11), such that
(22)LVi-1≤-∑j=1i-2‍(cj-εj-ej)zj4-ci-1zi-14+di-1zi-13zi+(θ^-θ-∑k=2i-1zk3∂αk∂θ^)(θ^˙-τi-1),
whereVi-1=∑j=1i-1‍(1/4)zj4+(1/2)(θ^-θ)2,τi-1=τ1+∑k=2i-1‍zk4(ψi-12+ψi-13), andεj=∑k=1j‍(εk1+εk2+εk3+εk4),wherej=1,…,n.
Then, define theith Lyapunov candidate functionVi(z-i,θ^)=Vi-1+(1/4)zi4. From (16) and (22), it follows that
(23)LVi≤-∑j=1i-2‍(cj-εj-ej)zj4-ci-1zi-14+di-1zi-13zi+zi3(∑k,m=1i-1‍diηi+1+Fi(z-i,x0)-∂αi∂θ^θ^˙+zi3(-12∑k,m=1i-1‍∂2αi∂zk∂zmϕkT(z-k)∑T(t)∑(t)ϕm(z-m))+32zi2Tr(GiT(z-i)∑T(t)∑(t)Gi(z-i))+(θ^-θ-∑k=2i-1‍zk3∂αk∂θ^)(θ^˙-τi-1).
Using Lemmas 9 and 4, there are always known nonnegative smooth functionsψi1(z-i),ψi2(z-i),ψi3(z-i),ψi4(z-i)and constantεi>0,  εij>0, where  i=1,…,nand  j=1,2,3,4.
Consider(24)zi3Fi≤|zi3|(∑k=1i-1‍|zk|)γi1(z-i,x0)≤γi1zi4+∑k=1i-1‍(εk1zk4+34(4εk1)-1/3γi14/3zi4)≤∑k=1i-1‍εk1zk4+ψi1zi4,
whereψi1≥γi1+∑k=1i-1‍(3/4)(4εk1)-1/3γi14/3.(25)-12zi3∑k,m=1i-1‍∂2αi∂zk∂zmϕkT∑T(t)∑(t)ϕm≤12zi3∑k,m=1i-1‍|∂2αi∂zk∂zm|(∑j=1k‍|zj|)p-k(z-k)×(∑j=1m‍|zj|)p-m(z-m)|∑T(t)∑(t)|≤zi3(∑k=1i-1‍zk2)p--i(z-i)|∑T(t)∑(t)|≤zi4ψi2(z-i)θ+∑k=1i-1‍εk2zk4,
whereψi2≥∑k=1i-1‍(3/4)(4εk2)-1/3(p--i(z-i))3/4.
(26)32zi2Tr(GiT∑T(t)∑(t)Gi)≤32zi2pi22(z-i)(∑k=1i‍|zk|)2|∑T(t)∑(t)|≤32zi2ipi22(z-i)(∑k=1i‍zk2)|∑T(t)∑(t)|≤32zi4ipi22(z-i)θ+∑k=1i-1‍εk3zk4+∑k=1i-1‍14εk3(32ipi22)2zi4θ≤∑k=1i-1‍εk3zk4+ψi3zi4θ,
whereψi3≥(3/2)ipi22+∑k=1i-1‍(1/4εk3)((3/2)ipi22)2. 									(27)di-1zi-13zi≤λi-12|zi-13zi|≤ei-1zi-14+14(43ei-1)-3zi4λi24≤ei-1zi-14+bizi4,
wherebi≥(1/4)((4/3)ei-1)-3λi24,τi-1=z14p112+∑k=2i-1‍zk4(ψk2+ψk3), andτi=τi-1+(ψi2+ψi3)zi4.
(28)-zi3∂αi∂θ^τi≤zi3|∂αi∂θ^|(τi-1+zi4(ψi2+ψi3))≤zi41+(zi3∂αi∂θ^)2(ψi2+ψi3)+zi3|∂αi∂θ^|(z14p112+∑k=2i-1‍zk4(ψk2+ψk3))+34(4)-1/3(|∂α2∂θ^|z13p112)4/3εi4-1/3zi4≤εi4z14+34(4)-1/31+(∂αi∂θ^z13P112)24/3εi4-1/3zi4+∑k=2i-1‍εk4zk4+∑k=2i-1‍34(4)-1/31+(∂αi∂θ^zk3(ψk2+ψk3))24/3εk4-1/3zi4≤∑k=1i-1‍εk4zk4+ψi4zi4,
whereψi4≥(3/4)(4εi4)-1/31+((∂αi/∂θ^)z13P112)24/3+∑k=2i-1‍(3/4)(4εk4)-1/31+((∂αi/∂θ^)zk3(ψk2+ψk3))24/3.
(29)αi+1(z-i,θ^)=-ziβi(z-i,θ^),βi(z-i,θ^)=ci+ψi1+ψi4+bi+1+(ψi2+ψi3)2θ^λi1,
whereci>0is a design parameter to be chosen.
With the aid of (24)–(29) and (14), (23) can be simplified as
(30)LVi≤-∑j=1i-1‍(cj-εj-ej)zj4-cizi4+dizi3zi+1+(θ^-θ-∑k=2i‍∂αk∂θ^zk3)(θ^˙-τi).
Finally, wheni=n,  zn+1=uis the actual control. By choosing the actual control law and the adaptive law,
(31)u(z-n,θ^)=-znβn(z-n,θ^),θ^˙=τn=z14p112+∑k=2n‍zk4(ψk2+ψk3),βn(z-n,θ^)=cn+bn+ψn1+ψn4+1+(ψn2+ψn3)2θ^λn1,u1=e-αtu,
wherecn>0is a design parameter to be chosen andψni,i=1,…4are smooth functions; we get
(32)LVn≤-∑j=1n‍(cj-εj-ej)zj4,
whereVn(z,θ^)=∑k=1n‍(1/4)zk4+(1/2)(θ^-θ)2,z=(z1,…zn). We have finished the controller design procedure forx0(t0)≠0and the parameter identification. Without loss of generality, we can assume thatt0≠0.


## 3.3. Switching Control and Main Result

In the preceding subsection, we have given controller design forx0≠0.Now, we discuss how to choose the control lawsu0andu1whenx0=0.We chooseu0asu0=-λ0x0+u0*,u0*>0. And choose the Lyapunov functionV0=(1/2)x02. Its time derivative is given byLV0=-λ0x02+u0*, which leads to the bounds ofx0. During the time period[0,ts), usingu0=-λ0x0+u0*, new control lawucan be obtained by the control procedure described above to the originalx-subsystem in (1). Then, we can conclude that thex-state of (1) cannot be blown up during the time period[0,ts). Since atx(ts)≠0,we can switch the control inputsu0anduto (6) and (31), respectively.Now, we state the main results as follows.Theorem 11.
Under Assumption5, if the proposed adaptive controller (31) together with the above switching control strategy is used in (1), then for any initial contidion(x0,x,θ^)∈Rn, the closed-loop system has an almost surely unique solution on[0,∞), the solution process is bounded in probability, andP{limt→∞θ^(t)existsandisfinite}=1.Proof.
According to the above analysis, it suffices to prove in the casex0(0)≠0.Since we have already proven thatx0can be globally exponentially convergent to zero in probability in Section 3.1, we only need prove thatx(t)is convergent to zero in probability also. In this case, we choose the Lyapunov functionV=Vn, andci>εi+ei; from (32) and Lemma 3, we know that the closed-loop system has an almost surely unique solution on[0,∞), and the solution process is bounded in probability.


## 4. Conclusions

This paper investigates the globally exponential stabilization problem for a class of stochastic nonholonomic systems in chained form. To deal with the nonlinear parametrization problem, a parameter separation technique is introduced. With the help of backstepping technique, a smooth adaptive controller is constructed which ensures that the closed-loop system is globally asymptotically stable in probability. A further work is how to design the output-feedback tracking control for more high-order stochastic nonholonomic systems.



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*Source: 658050-2013-11-26.xml*