# Adaptive State-Feedback Stabilization for Stochastic Nonholonomic Mobile Robots with Unknown Parameters

Authors: Wenli Feng; Qingli Sun; Zhijun Cao; Dongkai Zhang; Hua Chen
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

The stabilizing problem of  stochastic nonholonomic mobile robots
with uncertain parameters is addressed in this paper. The
nonholonomic mobile robots with kinematic unknown parameters are
extended to the stochastic case. Based on backstepping technique,
adaptive state-feedback stabilizing controllers are
designed for nonholonomic mobile robots with kinematic unknown parameters
whose linear velocity and angular velocity are subject to some stochastic disturbances simultaneously.
A switching control strategy for the original system is
presented. The proposed controllers that guarantee the states of
closed-loop system are asymptotically stabilized at the zero
equilibrium point in probability.


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## Body

## 1. Introduction

In the past decades, the control of nonholonomic systems has been widely pursued. By the results of Brockett [1], the nonholonomic system cannot be stabilized at a single equilibrium point by any static smooth pure state-feedback controller. To solve this problem, lots of novel approaches have been considered: discontinuous feedback control [2–4], smooth time-varying feedback controller [5], and the method of LMI [6]. The control of nonholonomic mobile robots plays an important role in that of nonholonomic systems because they are a benchmark for these systems. There is much attention devoted to the control of nonholonomic mobile robots. The nonholonomic mobile robots were classified into four types, which were characterized by generic structures of the model equations [7]. Based on the backstepping technique, the control for nonholonomic mobile robots was discussed: tracking problems [8] and stabilizing problems [9, 10]. Hespanha et al. introduced the mobile robot with parametric uncertainties [11], which were further discussed [12, 13]. But all the above articles discussed the nonholonomic systems in the deterministic case, which was not considered a stochastic disturbance.In recent years, stochastic nonlinear systems have received much attention [14, 15], especially for stochastic control when backstepping designs were firstly introduced [16, 17]. For stochastic nonholonomic systems, there were a few papers. The almost global adaptive asymptotical controllers of stochastic nonholonomic chained form systems were discussed by using discontinuous control [18]. The adaptive stabilization problem of stochastic nonholonomic systems with nonlinear drifts was considered [19–21]. By using state-scaling method, backstepping controllers were proposed to deal with exponential stabilization for nonholonomic mobile robots with stochastic disturbance [22, 23]. But the above two papers did not consider unknown parameters. To our knowledge, the problem of adaptive state-feedback stabilization for nonholonomic mobile robots with kinematic unknown parameters, whose linear velocity and angular velocity are subject to some stochastic disturbances simultaneously, has not been reported. So, there exists a natural problem which is how to extend the models in [11–13] to the stochastic case and design an adaptive state-feedback stabilizing controller for stochastic nonholonomic mobile robots with uncertain parameters.The purpose of this paper is to design adaptive state-feedback stabilizing controllers for stochastic nonholonomic mobile robots with unknown parameters. The main idea of this paper is highlighted as follows.(i)
We extend the models of nonholonomic mobile robots with unknown parameters in [11–13] to the stochastic case. The stabilizing controllers are designed for stochastic nonholonomic mobile robots with unknown parameters by adaptive state-feedback backstepping technique.(ii)
A switching control strategy for the original system is presented. It guarantees that the states of closed-loop system are asymptotically stabilized at the zero equilibrium point in probability.The paper is organized as follows. Section1 begins with the mathematical preliminaries. In Section 2, the adaptive state-feedback backstepping controller is designed. In Section 3, a switching control strategy for the original system is discussed. Finally, a simulation example is given to show the effectiveness of the controller in Section 4.


## 2. Preliminaries and Problem Formulation

### 2.1. Preliminaries

Consider the following stochastic nonlinear system:(1)dx=f(x)dt+g(x)dB,x(t0)∈ℝn,
where x∈ℝn is the state, the Borel measurable functions f:ℝn→ℝn and g:ℝn→ℝn×r are locally Lipschitz in x, and B∈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 in the paper.Definition 1 (see [16]).
For any givenV(x)∈𝒞2, associated with stochastic system (1), the differential operator ℒ is defined as follows:
(2)ℒV(x)=∂V∂xf(x)+12Tr{gT(x)∂2V∂x2g(x)}.Definition 2 (see [24]).
The equilibriumx=0 of system (1) is(i)
globally stable in probability if for∀ε>0, there exists a class 𝒦 function γ(·) such that
(3)P{|x(t)|<γ(|x(t0)|)}≥1-ε,∀t≥0,x(t0)∈ℝn∖{0};(ii)
globally asymptotically stable in probability if it is globally stable in probability and(4)P{limt→∞|x(t)|=0}=1,∀x(t0)∈ℝn.Definition 3 (see [25]).
A stochastic processx(t) is said to be bounded in probability if the random variable |x(t)| is bounded in probability uniformly in t; that is,
(5)limR→∞supt>t0P{|x(t)|>R}=0.Lemma 4 (see [24]).
Considering the stochastic system (1), if there exist a 𝒞2 function V(x), class 𝒦∞ functions α1(·) and α2(·), constants c1>0, c2≥0, and a nonnegative function W(x) such that
(6)α1(|x|)≤V(x)≤α2(|x|),ℒV(x)=∂V∂xf+12Tr{gT∂2V∂x2g}≤-c1W(x)+c2,
then(i)
for (1), there exists an almost surely unique solution on [t0,∞) for each x(t0)∈ℝn;(ii)
whenc2=0, f(0)=0, g(0)=0, and W(x) is continuous, then the equilibrium x=0 is globally stable in probability and
(7)P{limt→∞W(x(t))=0}=1,for∀x(t0)∈ℝn.Lemma 5 (see [26]).
Letx and y be real variables. Then, for any positive integers m, n and any real number ε>0, the following inequality holds:
(8)|x|m|y|n≤mm+nε|x|m+n+nm+nε-m/n|y|m+n.


### 2.2. Problem Formulation

Hespanha et al. introduced the mobile robot with parametric uncertainties [11], which were further discussed in [12, 13] as follows:
(9)θ˙=p1*ω,x˙c=p2*vcosθ,y˙c=p2*vsinθ,
where v and ω are two control inputs to denote the forward velocity and angular velocity, respectively.Here we assume that the forward velocityv and the angular velocity ω are subject to some stochastic disturbances. Based on the similar methods in [27, Page 1-2], velocity v and the angular velocity ω with stochastic disturbances can be expressed as follows:
(10)ω(θ)=ω1(θ)+ω2(θ)B˙(t),v(xc,yc,θ)=v1(xc,yc,θ)+v2(xc,yc,θ)B˙(t),
where B˙(t) is the derivative of a Brownian motion B(t).Remark 6.
The second equality of (10) is the same as that of Remark 2 in [19]. Moreover, (10) means that ω(θ) can be divided into two parts, with the second parts being stochastic disturbances and the same for  v(xc,yc,θ).Substituting (10) into (9), the system (9) can be transformed into
(11)dθ=p1*ω1dt+p1*ω2dB,dxc=p2*v1cosθdt+p2*v2cosθdB,dyc=p2*v1sinθdt+p2*v2sinθdB,
where p1* is unknown parameter taking values in a known interval [pmin,pmax] with 0<pmin<pmax<∞; p2* is unknown positive parameter.For system (11), we introduce the following state and input transformation:
(12)x0=θ,u0=ω1,u=v1,x1=xcsinθ-yccosθ,x2=xccosθ+ycsinθ,
and it is easy to see that(13a)dx0=p1*u0dt+p1*ω2(x0)dB,(13b)dx1=p2*x2u0dt-12(p1*)2x1ω22dt+p1*p2*v2ω2dt+p1*x2ω2dB,dx2=p2*udt-(p1*x1u0+12(p1*)2x2ω22)dt+(p2*v2-p1*x1ω2)dB.Remark 7.
The main difference between this paper and [22] is that the unknown parameter exists in this paper. The controller design of systems (13a) and (13b) will be more difficult.Remark 8.
For system (13a) and (13b), the variable x2 appears in the term p1*x2ω2dB in the first equation of (13b); this is different from the traditional stochastic backstepping technique in [16, 17, 24].


## 2.1. Preliminaries

Consider the following stochastic nonlinear system:(1)dx=f(x)dt+g(x)dB,x(t0)∈ℝn,
where x∈ℝn is the state, the Borel measurable functions f:ℝn→ℝn and g:ℝn→ℝn×r are locally Lipschitz in x, and B∈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 in the paper.Definition 1 (see [16]).
For any givenV(x)∈𝒞2, associated with stochastic system (1), the differential operator ℒ is defined as follows:
(2)ℒV(x)=∂V∂xf(x)+12Tr{gT(x)∂2V∂x2g(x)}.Definition 2 (see [24]).
The equilibriumx=0 of system (1) is(i)
globally stable in probability if for∀ε>0, there exists a class 𝒦 function γ(·) such that
(3)P{|x(t)|<γ(|x(t0)|)}≥1-ε,∀t≥0,x(t0)∈ℝn∖{0};(ii)
globally asymptotically stable in probability if it is globally stable in probability and(4)P{limt→∞|x(t)|=0}=1,∀x(t0)∈ℝn.Definition 3 (see [25]).
A stochastic processx(t) is said to be bounded in probability if the random variable |x(t)| is bounded in probability uniformly in t; that is,
(5)limR→∞supt>t0P{|x(t)|>R}=0.Lemma 4 (see [24]).
Considering the stochastic system (1), if there exist a 𝒞2 function V(x), class 𝒦∞ functions α1(·) and α2(·), constants c1>0, c2≥0, and a nonnegative function W(x) such that
(6)α1(|x|)≤V(x)≤α2(|x|),ℒV(x)=∂V∂xf+12Tr{gT∂2V∂x2g}≤-c1W(x)+c2,
then(i)
for (1), there exists an almost surely unique solution on [t0,∞) for each x(t0)∈ℝn;(ii)
whenc2=0, f(0)=0, g(0)=0, and W(x) is continuous, then the equilibrium x=0 is globally stable in probability and
(7)P{limt→∞W(x(t))=0}=1,for∀x(t0)∈ℝn.Lemma 5 (see [26]).
Letx and y be real variables. Then, for any positive integers m, n and any real number ε>0, the following inequality holds:
(8)|x|m|y|n≤mm+nε|x|m+n+nm+nε-m/n|y|m+n.


## 2.2. Problem Formulation

Hespanha et al. introduced the mobile robot with parametric uncertainties [11], which were further discussed in [12, 13] as follows:
(9)θ˙=p1*ω,x˙c=p2*vcosθ,y˙c=p2*vsinθ,
where v and ω are two control inputs to denote the forward velocity and angular velocity, respectively.Here we assume that the forward velocityv and the angular velocity ω are subject to some stochastic disturbances. Based on the similar methods in [27, Page 1-2], velocity v and the angular velocity ω with stochastic disturbances can be expressed as follows:
(10)ω(θ)=ω1(θ)+ω2(θ)B˙(t),v(xc,yc,θ)=v1(xc,yc,θ)+v2(xc,yc,θ)B˙(t),
where B˙(t) is the derivative of a Brownian motion B(t).Remark 6.
The second equality of (10) is the same as that of Remark 2 in [19]. Moreover, (10) means that ω(θ) can be divided into two parts, with the second parts being stochastic disturbances and the same for  v(xc,yc,θ).Substituting (10) into (9), the system (9) can be transformed into
(11)dθ=p1*ω1dt+p1*ω2dB,dxc=p2*v1cosθdt+p2*v2cosθdB,dyc=p2*v1sinθdt+p2*v2sinθdB,
where p1* is unknown parameter taking values in a known interval [pmin,pmax] with 0<pmin<pmax<∞; p2* is unknown positive parameter.For system (11), we introduce the following state and input transformation:
(12)x0=θ,u0=ω1,u=v1,x1=xcsinθ-yccosθ,x2=xccosθ+ycsinθ,
and it is easy to see that(13a)dx0=p1*u0dt+p1*ω2(x0)dB,(13b)dx1=p2*x2u0dt-12(p1*)2x1ω22dt+p1*p2*v2ω2dt+p1*x2ω2dB,dx2=p2*udt-(p1*x1u0+12(p1*)2x2ω22)dt+(p2*v2-p1*x1ω2)dB.Remark 7.
The main difference between this paper and [22] is that the unknown parameter exists in this paper. The controller design of systems (13a) and (13b) will be more difficult.Remark 8.
For system (13a) and (13b), the variable x2 appears in the term p1*x2ω2dB in the first equation of (13b); this is different from the traditional stochastic backstepping technique in [16, 17, 24].


## 3. Adaptive Controller Design

In this section, we will design state-feedback controllers such that all the signals in closed-loop system are regulated to the origin in probability. The following assumptions are needed.Assumption 9.
For the smooth functionω2(θ), there exists a known positive constant m1, such that
(14)ω2(θ)=m1θ.Assumption 10.
For smooth functionv2(xc,yc,θ) and any positive constant C, there exists a known nonnegative constant m2, such that
(15)|v2(xc,yc,θ)|≤m2|x2-Cx1u0|.Remark 11.
For the adaptive controllers’ design in the following, if we letC=c1, this assumption will change to |v2(xc,yc,θ)|≤m2|ε2|, where ε2 is defined in (25) and c1 is the same as that in (28) in the following Section 3.2.Firstly, we will consider the problem of stabilization for systems (13a) and (13b) under the condition of x0(t0)≠0. The case of x0(t0)=0 will be discussed in Section 3.


### 3.1. The First State Stabilization

Let us consider the subsystem (13a) of stochastic nonholonomic nonlinear systems (13a) and (13b):
(16)dx0=p1*u0dt+p1*ω2(x0)dB.In order to guarantee thatx0 converges to zero, one can take u0 as follows:
(17)u0=-η0x0,η0=λ+32m12pmax,
where λ>0 is a design parameter.If we employ a Lyapunov function of the form:(18)V0(x0)=14p1*x04.
From (13a), (17), (18), and Assumption 9, one can obtain
(19)ℒV0≤x03u0+32p1*x02(p1*)2ω22≤-λx04.Theorem 12.
If Assumption9 holds, one can choose positive constants m1, λ, and pmax  and the controller u0 as (17), respectively, then(i)
the closed-loop subsystem composed by (13a) and (17) has an almost surely unique solution on [t0,∞) for ∀x0(t0);(ii)
the equilibriumx0=0 of the closed-loop subsystem composed by (13a) and (17) is globally asymptotically stable in probability.Proof.
Choosing Lyapunov function as (18), by (19), λ>0, and Lemma 4, (i) holds and the equilibrium x0=0 of the closed-loop subsystem which contained (13a) and (17) is globally stable in probability and for ∀x0(t0)≠0, P{limt→∞(|x0(t)|=0)}=1. From Definition 2, (ii) holds.Remark 13.
From Theorem12, one has the state x0 bounded in probability; that is, there exists a positive constant m3, such that
(20)limm3→∞supt>t0P{|x0(t)|>m3}=0.
Substituting (17) into the subsystem (13a), one gets
(21)dx0=-p1*η0x0dt+p1*m1x0dB.Proposition 14.
For initial statex0(t0)≠0, the solution of (21) will never reach the zero, which avoids the uncontrollability of the subsystem (13b).Proof.
From Lemma 2.3 ([27, Page 93]), the following equality will be the solution of (21):
(22)x0(t)=x0(t0)exp{∫t0t‍(-p1*η0-12(p1*m1)2)dsx0(t)=x0x0(t0)exp+∫t0t‍p1*m1dω}.
From the above expression of x0(t) and x0(t0)≠0, it is easy to see that x0(t) will never cross the origin at the time interval t∈(t0,+∞).In the following Section3.2, the other states will be regulated to the origin in probability by the design of the control input u.


### 3.2. Other States Stabilization

In order to design a smooth adaptive state-feedback controller, the following state-input scaling discontinuous transformation is needed:(23)z1=x1u0,z2=x2.Remark 15.
For the initial statex0(t0)≠0, from Proposition 14, one can obtain that transformation (23) is meaningful.Under the newz-coordinate, the subsystem (13b) is transformed into
(24)dz1=p1*(-η0z2-0.5p1*z1ω22-p1*z2ω22x02+η0z1dz1+p2*v2ω2x0+p1*z1ω22x02)dt+p1*(z2ω2x0-z1ω2x0)dB,dz2=p2*udt+(p1*η0z1x02-0.5(p1*)2z2ω22)dt+(p2*v2-p1*z1x0ω2)dB.To invoke the backstepping method, the error variablesε1 and ε2 are given by
(25)ε1=z1,ε2=z2-α1(ε1).Step 1.
Define the first Lyapunov candidate function:(26)V1=14p1*ε14.
By (24)–(26) and Definition 1, one has
(27)ℒV1≤ε13{-η0z2-12p1*z1ω22-p1*z2ω22x02+η0z1+p2*v2ω2x0+p1*z1ω22x02}+3ε12{(p1*)2z22ω22x02+(p1*)2z12ω22x02}.
The virtual control can be chosen as(28)α1(ε1)=c1ε1,
where c1 is a positive constant, which will be chosen later. From (27), Lemma 5, and simple operation, we have the following inequalities:
(29)-η0z2ε13≤η0{3d4ε14+34d3ε24}-η0c1ε14,-12ε13p1*z1ω22≤12pmaxm12m32ε14,-p1*ε13z2ω22x02≤pmaxm12{34+c1}ε14+14pmaxm12ε24,p2*v2ω2x0≤14ε14+34(m1m2)4/3(p2*)4/3ε24,p1*z1ω22x02≤pmaxm12ε14,3(p1*)2ε13z22ω22x02≤3pmax2m12{2c12+1}ε14+3pmax2m12ε24,3(p1*)2ε13z12ω22x02≤3pmax2m12ε14,
where d>0 is a design parameter. Substituting these above inequalities into (27), it is easy to see that(30)ℒV1≤{pmaxm12(36pmax+7)+14-η0c1e+(1+3d4)η0-c1η0(1-e)+pmaxm12(2m32+4c1+24pmaxc12)4+pmaxm12(36pmax+7)+14}ε14+{η04d3+pmaxm124+3pmax2m12}ε24+34(m1m2)4/3(p2*)4/3ε24,
where e is a design parameter and 0<e<1. If we select parameters η0 and c1 to satisfy
(31)c1≥4+3d4e,η0≥pmaxm12(2m32+4c1+24pmaxc12+36pmax+7)+12c1(1-e),
one has
(32)ℒV1≤-c1η0(1-e)2ε14+34(m1m2)4/3(p2*)4/3ε24+{η04d3+pmaxm124+3pmax2m12}ε24.Step 2.
By (24), (25), (32), and Ito^ formula (Theorem 6.2, [27, Page 32]), one gets
(33)dε2=p2*udt+{p1*p2*c1v2ω2x0-(p1*)2c1z1ω22x02p1*η0z1x02-12(p1*)2z2ω22p2*udt+p2*u+p1*c1η0z2+12(p1*)2c1z1ω22p2*udt+p2*u-p1*c1η0z1+(p1*)2c1z2ω22x02p2*udt+p2*u-p1*p2*c1v2ω2x0-(p1*)2c1z1ω22x02}dt+(p2*v2-p1*z1x0ω2-p1*c1z2ω2x0+p1*c1z1ω2x0)dB.
To deal with the uncertain parameterp2*, define parameter
(34)Θ=max{p2*,(p2*)4/3,1p2*,(1p2*)4/3,(1p2*)2},
and Θ~=Θ-Θ^ being the parameter estimation error, Θ^ being the estimate of Θ. Define the second Lyapunov candidate function:
(35)V2=V1+14p2*ε24+12Θ~2.
From (33), (35), and Definition 1, one can obtain
(36)ℒV2≤-c1η0(1-e)2ε14+34(m1m2)4/3(p2*)4/3ε24+{η04d3+pmaxm124+3pmax2m12}ε24+ε23{-p1*c1v2ω2x0-(p1*)2p2*c1z1ω22x02u+p1*p2*η0z1x02-12(p1*)2p2*z2ω22+ε23+ε23ε23+p1*p2*c1η0z2+12(p1*)2p2*c1z1ω22+ε23+ε23ε23-p1*p2*c1η0z1+(p1*)2p2*c1z2ω22x02+ε23+ε23ε23-p1*c1v2ω2x0-(p1*)2p2*c1z1ω22x02}dt+6ε22{p2*v22+(p1*)2p2*z12x02ω22+6ε22+6ε22+(p1*)2p2*c12z22ω22x02+(p1*)2p2*c12z12ω22x02}-Θ~Θ^.
By (34), (36), and Lemma 5, we have the following inequalities:
(37)p1*p2*η0z1x02ε23≤Θ34(pmaxη0m32)4/3ε24+14ε14,-12(p1*)2p2*z2ω22ε24≤Θ{12pmax2m32m12+38(c1pmax2m32m12)4/3}ε24+18ε14,p1*p2*c1η0z2ε23≤Θ{c1pmaxη0+34c18/3pmax4/3η04/3}ε24+14ε14,12(p1*)2p2*c1z1ω22ε23≤Θ38(c1pmax2m32m12)4/3ε24+18ε14,-p1*p2*c1η0z1ε23≤Θ34(c1η0pmax)4/3ε24+14ε14,(p1*)2p2*c1z2ω22x02ε23≤Θc1pmax2m12ε24+Θ34(c12m12pmax2)4/3ε24+14ε14,-p1*c1v2ω2x0ε23≤pmaxm1m2c1ε24,-(p1*)2p2*c1z1ω22x02ε23≤14ε14+Θ34(pmax2c1m12)4/3ε24,6ε22{p2*v22z12x02+(p1*)2p2*ω22+(p1*)2p2*c12z22ω22x026ε226ε22+(p1*)2p2*c12z12ω22x02}≤Θ{6m22+3pmax4m38m14+12pmax2m12≤Θ≤Θ+6pmax4m14c14+3pmax4m14c14}ε24+12ε14.
Substituting the above inequalities into (36) and adding and subtracting the term c2ε24 on the right-hand side of (36), we have
(38)ℒV2≤-{c1η0(1-e)2-13.5}ε14-c2ε24+H21ε24+ε23u+ΘH22ε24-Θ~Θ^˙≤-c¯1ε14-c2ε24+ε23u+Θ~{H22ε24-Θ^˙}+{H21+1+Θ^2H22}ε24,
where
(39)H21=η04d3+pmaxm124+3pmax2m12+pmaxm1m2c1,c¯1=c1η0(1-e)2-13.5,H22=c2+34(pmaxη0m32)4/3+12pmax2m32m12+c1pmaxη0+38(c1pmax2m32m12)4/3+34c18/3pmax4/3η04/3+38(c1pmax2m32m12)4/3+34(c1η0pmax)4/3+c1pmax2m12+34(c12m12pmax2)4/3+6m22+12pmax2m12+3pmax4m38m14+6pmax4m14c14+3pmax4m14c14+34(pmax3c1m12)4/3+34(m1m2)4/3.
One can choose the actual control lawu and the adaptive laws Θ^ as follows:
(40)u=-{H21+1+Θ^2H22}ε2,Θ^˙=H22ε24.
Substituting (40) into (38), one gets
(41)ℒV2≤-c¯1ε14-c2ε24.
Choosing the Lyapunov function as(42)V=V0+V2,
together with (19) and (41), we have
(43)ℒV≤-λx02-c¯1ε14-c2ε24.Theorem 16.
If Assumptions9 and 10 hold, one can choose positive constants λ, m1, m2, m3, and  pmax, with  d>0 and 0<e<1 satisfying c¯1>0 and (31); for positive constant c2, one has the following.(i)
The closed-loop system composed by (13a), (17), (24), and (40) has an almost surely unique solution on [t0,∞) for ∀x0(t0),z(t0) and Θ^(t0).(ii)
The equilibrium(x0,z,Θ~)=(0,0,0) of the closed-loop system is globally stable in probability.(iii)
For initial condition∀x0(t0), z(t0), and Θ^(t0), P{limt→∞(|x0(t)|+|z(t)|)=0}=1, P{limt→∞Θ^(t)existsandisfinite}=1, where Θ~=Θ-Θ^ and z(t)=(z1(t),z2(t)).Proof.
From conditions in Theorem16, it is easy to see that constants λ>0, c¯1>0, and c2>0. So, ℒV in (43) becomes the same form as (3.19) in [21]. Using (43) and Lemma 4, Theorem 16 can be proved.


## 3.1. The First State Stabilization

Let us consider the subsystem (13a) of stochastic nonholonomic nonlinear systems (13a) and (13b):
(16)dx0=p1*u0dt+p1*ω2(x0)dB.In order to guarantee thatx0 converges to zero, one can take u0 as follows:
(17)u0=-η0x0,η0=λ+32m12pmax,
where λ>0 is a design parameter.If we employ a Lyapunov function of the form:(18)V0(x0)=14p1*x04.
From (13a), (17), (18), and Assumption 9, one can obtain
(19)ℒV0≤x03u0+32p1*x02(p1*)2ω22≤-λx04.Theorem 12.
If Assumption9 holds, one can choose positive constants m1, λ, and pmax  and the controller u0 as (17), respectively, then(i)
the closed-loop subsystem composed by (13a) and (17) has an almost surely unique solution on [t0,∞) for ∀x0(t0);(ii)
the equilibriumx0=0 of the closed-loop subsystem composed by (13a) and (17) is globally asymptotically stable in probability.Proof.
Choosing Lyapunov function as (18), by (19), λ>0, and Lemma 4, (i) holds and the equilibrium x0=0 of the closed-loop subsystem which contained (13a) and (17) is globally stable in probability and for ∀x0(t0)≠0, P{limt→∞(|x0(t)|=0)}=1. From Definition 2, (ii) holds.Remark 13.
From Theorem12, one has the state x0 bounded in probability; that is, there exists a positive constant m3, such that
(20)limm3→∞supt>t0P{|x0(t)|>m3}=0.
Substituting (17) into the subsystem (13a), one gets
(21)dx0=-p1*η0x0dt+p1*m1x0dB.Proposition 14.
For initial statex0(t0)≠0, the solution of (21) will never reach the zero, which avoids the uncontrollability of the subsystem (13b).Proof.
From Lemma 2.3 ([27, Page 93]), the following equality will be the solution of (21):
(22)x0(t)=x0(t0)exp{∫t0t‍(-p1*η0-12(p1*m1)2)dsx0(t)=x0x0(t0)exp+∫t0t‍p1*m1dω}.
From the above expression of x0(t) and x0(t0)≠0, it is easy to see that x0(t) will never cross the origin at the time interval t∈(t0,+∞).In the following Section3.2, the other states will be regulated to the origin in probability by the design of the control input u.


## 3.2. Other States Stabilization

In order to design a smooth adaptive state-feedback controller, the following state-input scaling discontinuous transformation is needed:(23)z1=x1u0,z2=x2.Remark 15.
For the initial statex0(t0)≠0, from Proposition 14, one can obtain that transformation (23) is meaningful.Under the newz-coordinate, the subsystem (13b) is transformed into
(24)dz1=p1*(-η0z2-0.5p1*z1ω22-p1*z2ω22x02+η0z1dz1+p2*v2ω2x0+p1*z1ω22x02)dt+p1*(z2ω2x0-z1ω2x0)dB,dz2=p2*udt+(p1*η0z1x02-0.5(p1*)2z2ω22)dt+(p2*v2-p1*z1x0ω2)dB.To invoke the backstepping method, the error variablesε1 and ε2 are given by
(25)ε1=z1,ε2=z2-α1(ε1).Step 1.
Define the first Lyapunov candidate function:(26)V1=14p1*ε14.
By (24)–(26) and Definition 1, one has
(27)ℒV1≤ε13{-η0z2-12p1*z1ω22-p1*z2ω22x02+η0z1+p2*v2ω2x0+p1*z1ω22x02}+3ε12{(p1*)2z22ω22x02+(p1*)2z12ω22x02}.
The virtual control can be chosen as(28)α1(ε1)=c1ε1,
where c1 is a positive constant, which will be chosen later. From (27), Lemma 5, and simple operation, we have the following inequalities:
(29)-η0z2ε13≤η0{3d4ε14+34d3ε24}-η0c1ε14,-12ε13p1*z1ω22≤12pmaxm12m32ε14,-p1*ε13z2ω22x02≤pmaxm12{34+c1}ε14+14pmaxm12ε24,p2*v2ω2x0≤14ε14+34(m1m2)4/3(p2*)4/3ε24,p1*z1ω22x02≤pmaxm12ε14,3(p1*)2ε13z22ω22x02≤3pmax2m12{2c12+1}ε14+3pmax2m12ε24,3(p1*)2ε13z12ω22x02≤3pmax2m12ε14,
where d>0 is a design parameter. Substituting these above inequalities into (27), it is easy to see that(30)ℒV1≤{pmaxm12(36pmax+7)+14-η0c1e+(1+3d4)η0-c1η0(1-e)+pmaxm12(2m32+4c1+24pmaxc12)4+pmaxm12(36pmax+7)+14}ε14+{η04d3+pmaxm124+3pmax2m12}ε24+34(m1m2)4/3(p2*)4/3ε24,
where e is a design parameter and 0<e<1. If we select parameters η0 and c1 to satisfy
(31)c1≥4+3d4e,η0≥pmaxm12(2m32+4c1+24pmaxc12+36pmax+7)+12c1(1-e),
one has
(32)ℒV1≤-c1η0(1-e)2ε14+34(m1m2)4/3(p2*)4/3ε24+{η04d3+pmaxm124+3pmax2m12}ε24.Step 2.
By (24), (25), (32), and Ito^ formula (Theorem 6.2, [27, Page 32]), one gets
(33)dε2=p2*udt+{p1*p2*c1v2ω2x0-(p1*)2c1z1ω22x02p1*η0z1x02-12(p1*)2z2ω22p2*udt+p2*u+p1*c1η0z2+12(p1*)2c1z1ω22p2*udt+p2*u-p1*c1η0z1+(p1*)2c1z2ω22x02p2*udt+p2*u-p1*p2*c1v2ω2x0-(p1*)2c1z1ω22x02}dt+(p2*v2-p1*z1x0ω2-p1*c1z2ω2x0+p1*c1z1ω2x0)dB.
To deal with the uncertain parameterp2*, define parameter
(34)Θ=max{p2*,(p2*)4/3,1p2*,(1p2*)4/3,(1p2*)2},
and Θ~=Θ-Θ^ being the parameter estimation error, Θ^ being the estimate of Θ. Define the second Lyapunov candidate function:
(35)V2=V1+14p2*ε24+12Θ~2.
From (33), (35), and Definition 1, one can obtain
(36)ℒV2≤-c1η0(1-e)2ε14+34(m1m2)4/3(p2*)4/3ε24+{η04d3+pmaxm124+3pmax2m12}ε24+ε23{-p1*c1v2ω2x0-(p1*)2p2*c1z1ω22x02u+p1*p2*η0z1x02-12(p1*)2p2*z2ω22+ε23+ε23ε23+p1*p2*c1η0z2+12(p1*)2p2*c1z1ω22+ε23+ε23ε23-p1*p2*c1η0z1+(p1*)2p2*c1z2ω22x02+ε23+ε23ε23-p1*c1v2ω2x0-(p1*)2p2*c1z1ω22x02}dt+6ε22{p2*v22+(p1*)2p2*z12x02ω22+6ε22+6ε22+(p1*)2p2*c12z22ω22x02+(p1*)2p2*c12z12ω22x02}-Θ~Θ^.
By (34), (36), and Lemma 5, we have the following inequalities:
(37)p1*p2*η0z1x02ε23≤Θ34(pmaxη0m32)4/3ε24+14ε14,-12(p1*)2p2*z2ω22ε24≤Θ{12pmax2m32m12+38(c1pmax2m32m12)4/3}ε24+18ε14,p1*p2*c1η0z2ε23≤Θ{c1pmaxη0+34c18/3pmax4/3η04/3}ε24+14ε14,12(p1*)2p2*c1z1ω22ε23≤Θ38(c1pmax2m32m12)4/3ε24+18ε14,-p1*p2*c1η0z1ε23≤Θ34(c1η0pmax)4/3ε24+14ε14,(p1*)2p2*c1z2ω22x02ε23≤Θc1pmax2m12ε24+Θ34(c12m12pmax2)4/3ε24+14ε14,-p1*c1v2ω2x0ε23≤pmaxm1m2c1ε24,-(p1*)2p2*c1z1ω22x02ε23≤14ε14+Θ34(pmax2c1m12)4/3ε24,6ε22{p2*v22z12x02+(p1*)2p2*ω22+(p1*)2p2*c12z22ω22x026ε226ε22+(p1*)2p2*c12z12ω22x02}≤Θ{6m22+3pmax4m38m14+12pmax2m12≤Θ≤Θ+6pmax4m14c14+3pmax4m14c14}ε24+12ε14.
Substituting the above inequalities into (36) and adding and subtracting the term c2ε24 on the right-hand side of (36), we have
(38)ℒV2≤-{c1η0(1-e)2-13.5}ε14-c2ε24+H21ε24+ε23u+ΘH22ε24-Θ~Θ^˙≤-c¯1ε14-c2ε24+ε23u+Θ~{H22ε24-Θ^˙}+{H21+1+Θ^2H22}ε24,
where
(39)H21=η04d3+pmaxm124+3pmax2m12+pmaxm1m2c1,c¯1=c1η0(1-e)2-13.5,H22=c2+34(pmaxη0m32)4/3+12pmax2m32m12+c1pmaxη0+38(c1pmax2m32m12)4/3+34c18/3pmax4/3η04/3+38(c1pmax2m32m12)4/3+34(c1η0pmax)4/3+c1pmax2m12+34(c12m12pmax2)4/3+6m22+12pmax2m12+3pmax4m38m14+6pmax4m14c14+3pmax4m14c14+34(pmax3c1m12)4/3+34(m1m2)4/3.
One can choose the actual control lawu and the adaptive laws Θ^ as follows:
(40)u=-{H21+1+Θ^2H22}ε2,Θ^˙=H22ε24.
Substituting (40) into (38), one gets
(41)ℒV2≤-c¯1ε14-c2ε24.
Choosing the Lyapunov function as(42)V=V0+V2,
together with (19) and (41), we have
(43)ℒV≤-λx02-c¯1ε14-c2ε24.Theorem 16.
If Assumptions9 and 10 hold, one can choose positive constants λ, m1, m2, m3, and  pmax, with  d>0 and 0<e<1 satisfying c¯1>0 and (31); for positive constant c2, one has the following.(i)
The closed-loop system composed by (13a), (17), (24), and (40) has an almost surely unique solution on [t0,∞) for ∀x0(t0),z(t0) and Θ^(t0).(ii)
The equilibrium(x0,z,Θ~)=(0,0,0) of the closed-loop system is globally stable in probability.(iii)
For initial condition∀x0(t0), z(t0), and Θ^(t0), P{limt→∞(|x0(t)|+|z(t)|)=0}=1, P{limt→∞Θ^(t)existsandisfinite}=1, where Θ~=Θ-Θ^ and z(t)=(z1(t),z2(t)).Proof.
From conditions in Theorem16, it is easy to see that constants λ>0, c¯1>0, and c2>0. So, ℒV in (43) becomes the same form as (3.19) in [21]. Using (43) and Lemma 4, Theorem 16 can be proved.


## 4. Switching Control Stability

In Section2, the case of x0(t0)≠0 is discussed. We design controllers u0 and u for systems (13a) and (13b) as in (17) and (40), respectively. Now we turn to the case of x0(t0)=0. When the initial x0(t0)=0, one can choose an open loop control u0=-u0*≠0 to drive the state x0 away from zero in a limited time.In fact, when we choose an open loop controlu0=-u0*≠0, system (13a) will be in the following form:
(44)dx0=-u0*dt+p1*ω2(x0)dB.
For a given constant l>0, define a stopping time τl=inf{t:t≥t0,|x0(t)|≥l}. With the similar analysis in Section V in [22], we have P(τl-t0≥T)≤l/Tu0*, which means that P(τl=∞)=0 for any l>0. Letting ts*=τl, it is easy to see that
(45)|x0(ts*)|=|x0(τl)|=l≠0.
So, there exists ts*>0, such that x0(ts*)≠0. After that, at the time t=ts*, we switch the control inputs u0 and u to (17) and (40) in t∈[ts*,+∞), respectively.Theorem 17.
If Assumptions9 and 10 hold, one can apply the following switching control procedure to system (11): (i)
when the initial state belongs to(46){(θ(t0),xc(t0),yc(t0))∈ℝ3∣θ(t0)≠0},(ii)
when the initial state belongs to(47){(θ(t0),xc(t0),yc(t0))∈ℝ3∣θ(t0)=0}.One designs control inputs u0 and u in form (17) and (40), respectively; for t∈[t0,ts*), one can choose the control law u0=-u0*≠0 and u=u*; for t∈[ts*,+∞), at the time t=ts*, one switches the control inputs u0 and u to (17) and (40), respectively.
Then, for any initial condition in the state space, the states of system (11) are asymptotically regulated to zero in probability.Proof.
Firstly, we consider the case that the initial state belongs to(48){(θ(t0),xc(t0),yc(t0))∈ℝ3∣θ(t0)≠0}.
From Theorems 12 and 16, for the closed-loop system composed by (13a), (17), (24), and (40), states x0 and z(t) are regulated to zero in probability, Θ^(t) is bounded in probability, and P{limt→∞(|x0(t)|+|z(t)|)=0}=1. This implies that states x0 and z(t) are globally asymptotically regulated to zero in probability and bounded in probability. As a result of (23), one gets that the states x0, x1, and x2 of closed-loop system composed by (11), (17), and (40) asymptotically converge to zero in probability and all bounded in probability. By orthogonal transformation (12), one can obtain that the states θ, xc, and yc of closed-loop system composed by (11), (17), and (40) are asymptotically stabilized in probability.
Secondly, when the initial state belongs to(49){(θ(t0),xc(t0),yc(t0))∈ℝ3∣θ(t0)=0},
we use the constant control u0=-u0*≠0 in order to drive x0 far away from the origin, which guarantees that all the signals are bounded in probability during [t0,ts*). Then, in view of x0(ts*)≠0, the switching control strategy is applied to system (11) at the time instant ts*>0. This completes the proof.


## 5. A Simulation Example

Consider the system (11) with ω2=0.5θ and v2=x2-(3.5x1/u0). In simulation, one can choose p1*=0.2, p2*=0.1, e=0.5, pmax=0.25, d=1, c1=5.5, c2=0.2, η0=10.5, m1=0.5, m2=1, and m3=3 and the initial values θ(0)=2.8, xc(0)=-0.678, yc(0)=0.528, and Θ^(0)=0.38. Figures 1, 2, and 3 give the responses of the closed-loop system consisting of (11), (17), and (40).Figure 1
The responses of statesθ, xc, and yc with respect to time.Figure 2
The responses of controllersu0 and u with respect to time.Figure 3
The response of estimate parameterΘ^ with respect to time.From Figure1, it is easy to see that the states θ, xc, and yc are asymptotically regulated to zero in probability in spite of the stochastic disturbances. As shown in Figure 2, the control inputs u0 and u are convergent to a small neighborhood of zero asymptotically. Figure 3 indicates that the estimated parameter Θ^ is bounded.


## 6. Conclusions

In this paper, we extend the nonholonomic mobile robots with unknown parameters to the stochastic case. Based on backstepping technique, adaptive state-feedback stabilizing controllers are designed for stochastic nonholonomic mobile robots with unknown parameters. A switching control strategy for the original system is given, which guarantees that the states of closed-loop system are asymptotically stabilized at the zero equilibrium point in probability.There exist some problems to be discussed, for example, how to design the controller for the dynamic stochastic nonholonomic systems with unknown parameters.



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*Source: 808249-2013-11-27.xml*