# A New Approach to Adaptive Stabilization of Stochastic High-Order Nonholonomic Systems

Authors: Guangju Li; Kemei Zhang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper studies the problem of adaptive stabilization for a class of stochastic high-order nonholonomic systems. Under the weaker assumptions, by constructing the appropriate Lyapunov function and combining sign function technique, an adaptive state feedback controller is designed to guarantee global asymptotic stability in probability of the closed-loop system. The effectiveness of the controller is demonstrated by a mechanical system.


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## Body

## 1. Introduction

Ever since the stochastic stability theory was established by [1, 2], the design and analysis of backstepping controller for stochastic nonlinear systems has achieved remarkable development in recent years; see [1, 3–12] and the references therein. But these papers do not consider stochastic nonholonomic systems.In this paper, we consider stochastic high-order nonholonomic systems(1)dx0=d0t,θu0p0dt+f0x0,θdt+g0⊤x0,θdω,dxi=dit,θu0qixi+1pidt+fix0,x,θdt+gi⊤x0,x,θdω,dxn=dnt,θu0qnupndt+fnx0,x,θdt+gn⊤x0,x,θdω,where x0∈R and x=(x1,…,xn)⊤∈Rn are system states, u0 and u are control inputs, pi∈Rodd≥1≜{p/q|pandqarepositiveoddintegers,andp≥q} are odd integers, and qi≥0 are constants, i=1,…,n. θ∈Rm is an unknown constant vector. ω is a r-dimensional standard Wiener process defined on a probability space (Ω,F,P) with Ω being a sample space, F being a filtration, and P being a probability measure. f0(x0,θ):R×Rm→R, g0(x0,θ):R×Rm→Rr, fi(x0,x,θ):R×Rn×Rr→R, gi(x0,x,θ):R×Rn×Rr→Rr, i=1,…,n, are locally Lipschitz functions. di(t,θ), i=0,…,n, are nonlinear control coefficients.Since many mechanical systems can be modeled by system (1), a series of theoretical results have been obtained. To mention a few, [13–16] investigated the deterministic case of gi=0. Specially when di=pi=qi=1 and fi=0, the authors designed a continuous controller by proposing a sliding mode control approach in [13]. When systems contain nonlinear drifts and unknown time-varying coefficients, [15] presented an adaptive control approach to achieve sate-feedback stabilization. High-order nonholonomic systems, that is, pi≥1, were introduced in [16]. The design procedure in [16] combined the idea of a discontinuous change of coordinate and adding a power integrator. However, it did not consider nonlinear parameterizations. Because many practical control systems such as biochemical processes and machines with friction often contain unknown parameters, [14] studied the problem of adaptive stabilization control design for a class of high-order nonholonomic systems with strong nonlinear drifts. Since stochastic noise frequently arises and is inevitable in practical control systems, how to extend these approaches to stochastic high-order nonholonomic systems is a very interesting problem. When gi≠0, [17] studied the problem of state feedback stabilization for a class of high-order stochastic nonholonomic systems with disturbed control directions and more general nonlinear drifts. However, nonlinear parameterization and nonlinear drift term in x0-subsystem were not discussed in [17]. Unfortunately, all these mentioned results require that the nonlinearities fi(·) are dependent on (x1,…,xi). Naturally, an interesting problem is put forward:For system (1), under weaker assumptions, can a stabilizable state feedback controller be designed?In this paper, we will provide a satisfactory answer to this problem. By constructing the appropriate Lyapunov function, skillfully combining parameter separation, sign function, and backstepping design approach, an adaptive state feedback controller is designed to guarantee global asymptotic stability in probability of the closed-loop system. Finally, a simulation example is used to demonstrate the effectiveness of this approach.The contributions and difficulties of this paper are highlighted from three aspects.(i) The system under consideration is more general than those investigated in [13–17]. System (1) has unknown control coefficients and permits unknown parameters to enter nonlinear equations. Nonlinear functions fi(·) are dependent on (x1,…,xi+1), which makes discontinuous change of coordinates be inapplicable to the adaptive state feedback control of systems (1). In this paper, we propose a novel design approach to solve this obstacle.(ii) The unknown growth rates of the upper bounds offi(·) and gi(·) are extended; see Remark 13. Therefore, some new mathematical tools, such as sign function and transformation technique, are introduced to simplify the construction of Lyapunov function.(iii) An practical example for mobile robot with small angle measurement error is modeled and solved by the proposed approach.This paper is organized as follows. Section2 gives some preliminaries. Section 3 presents the design and analysis of the adaptive controller, following a practical example in Section 4. Section 5 concludes this paper. The proofs of Propositions 16–19 are given in Appendix.


## 2. Mathematical Preliminaries

Some notations and definitions will be used throughout this paper.R+ stands for the set of all nonnegative real numbers and Rn denotes real n-dimensional space. For vector χ=(χ1,…,χn)⊤, χ¯i represents (χ1,…,χi)⊤. For a given vector or matrix x, x⊤ denotes its transpose, |x| is the Euclidean norm of a vector x, and Tr{x} denotes its trace when x is square. Ci denotes the set of all functions with continuous ith partial derivatives. K denotes the set of all functions: R+→R+ that are continuous, strictly increasing, and vanishing at zero; K∞ denotes the set of all functions that are of class K and unbounded. For any y∈R, sign function sgn(y) is defined as sgn(y)=1 if y>0; sgn(y)=0 if y=0 and sgn(y)=-1 if y<0. For simplicity, we denote yp=|y|psgn(y). The arguments of functions are sometimes omitted or simplified, for example, a function f(x(t)) is denoted by f(x) or f.In what follows, we present some definitions and lemmas which will be frequently used in the design and analysis of controller. Consider stochastic nonlinear system(2)dx=fx,θdt+gx,θdω,xt0=x0∈Rn,where ω is a r-dimensional standard Wiener process defined on a probability space (Ω,F,P), x∈Rn is system state, θ∈Rm is an unknown constant vector and the functions f:Rn×Rm→Rn, and g:Rn×Rm→Rn×r are locally bounded and locally Lipschitz with respect to x.Definition 1 (see [1]).
The equilibriumx(t)=0 of system (2) with f(0,θ)=0, g(0,θ)=0 is
(i) globally stable in probability if for anyε>0, there exists a class K function r(·) such that P{|x(t)|<r(x0)}≥1-ε, for any t≥0, x0∈Rn∖{0}
(ii) globally asymptotically stable in probability if it is globally stable in probability andP{limt→∞|x(t)=0|}=1, for any x0∈RnLemma 2 (see [1]).
For system (2), if there exists a C2 function V(x):Rn→R+ and class K∞ functions α1(·) and α2(·), such that(3)α1x≤Vx≤α2x,LVx=∂Vx∂xfx,θ+12TrgTx,θ∂2Vx∂x2gx,θ≤-Wx,where W(x) is a nonnegative continuous function, then, for any x(t0)∈Rn,
(i) there exists an almost surely unique solution on[t0,∞)
(ii) the equilibriumx=0 is globally stable in probability, and the solution x(t) satisfies P{limt→∞W(x(t))=0}=1Lemma 3 (see [18]).
For any real-valued continuous functionh(x,y), where x∈Rm, y∈Rn, there are smooth scalar functions c(x)≥1 and d(y)≥1, such that |h(x,y)|≤c(x)d(y).Lemma 4 (see [19]).
Letx, y be real variables, then for any positive real numbers b, c, d and continuous function a(·)≥0, one has a(·)|x|c|y|d≤b|x|c+d+(d/c+d)(c+d/c)-c/da(·)c+d/db-c/d|y|c+d.Lemma 5 (see [20]).
f ( x ) = x a is continuously differentiable and f˙(x)=a|x|a-1, where a≥1, x∈R. Moreover, df(x(t))/dt=a|x(t)|a-1x˙(t).Lemma 6 (see [20]).
Ifp=a/b∈Rodd≥1, b≥1, then |xp-yp|≤21-1/b|xa-ya|1/b for any x,y∈R.Lemma 7 (see [20]).
For givenr≥0 and every x∈R, y∈R, |x+y|r≤Cr(|x|r+|y|r) holds, where Cr=2r-1 if r≥1 and Cr=1 if 0≤r<1.Lemma 8 (see [20]).
Letf:[a,b]→R be a continuous monotone function with f(a)=0; then |∫ab‍f(x)dx|≤f(b)b-a.


## 3. Design and Analysis of Adaptive Controller

### 3.1. Problem Formulation and Assumptions

The aim of this paper is to design an adaptive state feedback controller for system (1) to guarantee global asymptotic stability in probability of the closed-loop system. We need the following assumptions to achieve this aim.Assumption 9.
The sign ofdi(t,θ), i=0,…,n, is assumed to be positive, and there exist smooth functions 0<λi,1(t)≤λi,2(t,θ), such that(4)0<λi,1t≤dit,θ≤λi,2t,θ.Assumption 10.
For smooth functionsf0(x0,θ) and g0(x0,θ), there exist bounded smooth functions α0(x0,θ), β0(x0,θ) such that(5)f0x0,θ=x0α0x0,θ,g0x0,θ=x0β0x0,θ.Assumption 11.
Fori=1,…,n, there exist nonnegative continuous functions αi(x0,x¯i,θ), βi(x0,x¯i,θ) and a constant τ satisfying -rn<τ<0 such that(6)fix0,x,θ≤x1ri+τ/r1+⋯+xiri+τ/riαix0,x¯i,θ+ciu0qidit,θxi+1ri+τ/ri+1,gix0,x,θ≤x12ri+τ/2r1+⋯+xi2ri+τ/2riβix0,x¯i,θ,where rn+1 is a given positive constant and r1,⋯, rn are recursively defined by ri=piri+1-τ and nonnegative constants ci<1/2pi-1.Remark 12.
Assumptions10-11 imply that f0(0,θ)=0, g0(0,θ)=0, fi(0,0,θ)=0, gi(0,0,θ)=0, i=1,…,n; that is, origin is the equilibrium of system (1).Remark 13.
Assumption11 enlarges the scope of nonholonomic systems. Specifically, if state xi+1 does not appear in the nonlinear function fi(·), Assumption 11 is degenerated into the following form:(7)fix0,x,θ≤x1ri+τ/r1+⋯+xiri+τ/riαix0,x¯i,θ,gix0,x,θ≤x12ri+τ/2r1+⋯+xi2ri+τ/2riβix0,x¯i,θ,where r1=1, ri=1+(i-1)τ with τ being a constant for each i=1,…,n. Particularly, when τ=0, (7) becomes the following growth condition:(8)fix0,x,θ≤x1+⋯+xiαix0,x¯i,θ,gix0,x,θ≤x1+⋯+xiβix0,x¯i,θ.If αi(x0,x¯i,θ) and βi(x0,x¯i,θ) are constants, (8) becomes the linear growth condition in [21]. If αi(x0,x¯i,θ) and βi(x0,x¯i,θ) are smooth nonnegative functions, (8) becomes the linear growth condition in [22, 23]. When τ=-p/q∈(-2/4n+1,0) with p being an even integer and q being an odd integer, condition (7) becomes the low-order growth condition in [24, 25]. Assumption 11 extends the value of τ to an explicit interval rather than a ratio of an even integer over an odd integer.Remark 14.
By Lemma3, there are smooth scalar functions λ~i,2(t)≥1, α¯i(x0,x¯i)≥1, β¯i(x0,x¯i)≥1, i=0,…,n, and Θ(θ)≥1 such that(9)λi,1t≤dit,θ≤λ~i,2tΘ,f0x0,θ≤x0α¯0x0Θ,g0x0,θ≤x0β¯0x0Θ,fix0,x,θ≤x1ri+τ/r1+⋯+xiri+τ/riα¯ix0,x¯iΘ+ciu0qidit,θxi+1ri+τ/ri+1,gix0,x,θ≤x12ri+τ/2r1+⋯+xi2ri+τ/2riβ¯ix0,x¯iΘ.


### 3.2. Stability and Convergence Analysis

We state the main result in this paper.Theorem 15.
If Assumptions9–11 hold for system (1), under an appropriate controller, the origin of the closed-loop system is globally asymptotically stable in probability for any initial condition.Proof.The proof is based on inductive argument. Firstly, we design an adaptive controller by considering two cases: x0(t0)≠0 and x0(t0)=0.Case I (x0(t0)≠0). The structure of system (1) means that the design procedure is divided into two separate parts.Part I (design of controller u0). Let us consider x0-subsystem in system (1), define Θ0=λ0,1-1max{λ0,1-1,Θ,Θ2}, and choose V0=x04/4+λ0,1/2Θ~02, where Θ~0=Θ0-Θ^0. From Remark 14, it follows that(10)LV0=x03d0u0p0+f0+32Trg0x02g0⊤-λ0,1Θ~0Θ^˙0≤-aλ0,1x04+aλ0,1x04+x03d0u0p0+Θα¯0x0x04+32Θ2β¯02x0x04-λ0,1Θ~0Θ^˙0≤-aλ0,1x04+λ0,1x03d0λ0,1u0p0+hx0Θ^0x0-λ0,1Θ~0Θ^˙0-x04hx0,where a is a positive constant and h(x0)=a+α¯0(x0)+3/2β¯02(x0) is a positive smooth function. Substituting the adaptive controller(11)u0=-x0hx0Θ^02+11/p0,Θ^˙0=x04hx0into (10) leads to(12)LV0≤-aλ0,1x04.The succeeding proposition characterizes the features of x0-subsystem.Proposition 16.
If Assumptions9-10 hold for x0-subsystem, controller (11) guarantees that
(i)x0-subsystem has an almost surely unique solution on [t0,∞) for any x0(t0)≠0
(ii) the equilibriumx0=0 of x0-subsystem is globally asymptotically stable in probability
(iii) the solution ofx0-subsystem and u0 does not cross zero for any x0(t0)≠0Proof.
See Appendix.Part II (design of controller u). In this part, we need to consider xi-subsystem in system (1). With the help of Proposition 16, controller u will be recursively constructed by applying the adding a power integrator approach. Before the beginning of recursive design, we define Θ1=max{1,Θ,Θ2} and state transformation(13)ξ1=x1σ/r1-x1∗σ/r1,x1∗=0,ξ2=x2σ/r2-x2∗σ/r2,x2∗=-ϕ1x0,x1,Θ^1ξ1r2/σ,⋮ξk=xkσ/rk-xk∗σ/rk,xk∗=-ϕk-1x0,x¯k-1,Θ^1ξk-1rk/σ,where σ is a positive constant satisfying the relationship σ≥max{p1r2,…,pnrn+1}, Θ^1 is the estimate of Θ1, and functions ϕk(·) satisfying sgn(ϕk)=sgn(u0qk) will be determined later. To solve the problem caused by sign function, the definition of Wk(x0,x¯k,Θ^1) is given by(14)Wkx0,x¯k,Θ^1=∫xk∗xksσ/rk-xk∗σ/rk4lσ-τ-rk/σds,where k=1,…,n and 4l is an even number and satisfies (4l-2)σ≥τ+rk.Proposition 17.
W k is C2 function and satisfies Ck1|xk-xk∗|4lσ-τ/rk≤Wk≤Ck2|ξk|4lσ-τ/σ, where Ck1=2(σ-rk)(4lσ-τ-rk)/σrkrk/4lσ-τ, Ck2=21-rk/σ, k=1,…,n.Proof.
​See Appendix.To obtain the expression ofu, we determine ϕ1,⋯, ϕn by induction, the design procedure is implemented as follows.Step  1. Let V1=W1+1/2Θ~12, where Θ~1=Θ1-Θ^1. By using (13), (A.3), (A.7), and Lemma 7, we get(15)LV1≤ξ14lσ-τ-r1/σd1u0q1x2p1+f1+4lσ-τ-r12r1ξ14l-1σ-τ-r1/σx1σ-r1/r1g12-Θ~1Θ^˙1≤-nξ14l+nξ14l+ξ14lσ-τ-r1/σd1u0q1x2p1-x2∗p1+ξ14lσ-τ-r1/σd1u0q1x2∗p1-Θ~1Θ^˙1+4lσ-τ-r12r1ξ14l-1σ-τ-r1/σx1σ+r1+τ/r1β¯12Θ2+ξ14lσ-τ-r1/σx1r1+τ/r1α¯1Θ+c1d1u0q1x2p1.Clearly, the application of the fact ξ14lσ-τ-r1/σd1u0q1x2∗p1≤0 implies(16)ξ14lσ-τ-r1/σu0q1d1x2∗p1+ξ14lσ-τ-r1/σc1d1u0q1x2p1≤ξ14lσ-τ-r1/σu0q1d1x2∗p1+ξ14lσ-τ-r1/σc¯1d1u0q1x2∗p1+ξ14lσ-τ-r1/σc¯1d1u0q1ξ2r1+τ/σ≤ξ14lσ-τ-r1/σd1u0q11-c¯1x2∗p1+ξ14lσ-τ-r1/σc¯1d1u0q1ξ2r1+τ/σ,where c¯1=2p1-1c1 is a positive constant. Substituting (16) into (15) and choosing(17)x2∗=-1λ1,1u0q11-c¯1Θ^1α¯1+n+4lσ-τ-r12r1β¯121/p1ξ1r2/σ≜-ϕ1x0,x1,Θ^1ξ1r2/σ,one has(18)LV1≤-nξ14l+ξ14lσ-τ-r1/σd1u0q1x2p1-x2∗p1+Ψ1-Θ^˙1Θ~1+η1+c¯1d1u0q1ξ14lσ-τ-r1/σξ2r1+τ/σ,where Ψ1=ξ14l(α¯1+n+4lσ-τ-r1/2r1β¯12)≥0 and η1=0.Step k(k=2,…,n). At Step k-1, assume that there exists a C2 function Vk-1 and virtual controllers x2∗,⋯, xk∗ such that(19)LVk-1≤-n-k+2∑i=1k-1ξi4l+ξk-14lσ-τ-rk-1/σdk-1u0qk-1xkpk-1-xk∗pk-1+Ψk-1-Θ^˙1Θ~1+ηk-1+c¯k-1dk-1u0qk-1ξk-14lσ-τ-rk-1/σξkrk-1+τ/σ,where c¯k-1=2pk-1-1ck-1 is a nonnegative constant and(20)0≤Ψk-1≤ξ14l+⋯+ξk-14lφk-1,for a C∞ function φk-1≥0. In what follows, we prove that (19) also holds at Step k. To prove this point, considering Vk=Vk-1+Wk, we deduce from (19) and (A.3)-(A.13) that(21)LVk≤-n-k+2∑i=1k-1ξi4l+ξk-14lσ-τ-rk-1/σdk-1u0qk-1xkpk-1-xk∗pk-1+∂Wk∂Θ^1Θ^˙1+Ψk-1-Θ^˙1Θ~1+ηk-1+c¯k-1dk-1u0qk-1ξk-14lσ-τ-rk-1/σξkrk-1+τ/σ+∂Wk∂x0d0u0p0+f0+∂Wk∂xkdku0qkxk+1pk+fk+12∂2Wk∂x02g02+∑i=1k-1∂Wk∂xidiu0qixi+1pi+fi+12∂2Wk∂xk2gk2+12∑i=1k-1∂2Wk∂xi2gi2+12∂2Wk∂xk∂x0gkg0⊤+12∑i=1k-1∂2Wk∂xi∂x0gig0⊤+12∑i=1k-1∂2Wk∂xi∂xkgigk⊤+12∑i,j=1,i≠jk-1∂2Wk∂xi∂xjgigj⊤.We give the following proposition, whose proof is placed in the Appendix.Proposition 18.
There is a smooth functionσk such that(22)ξk-14lσ-τ-rk-1/σdk-1u0qk-1xkpk-1-xk∗pk-1+∂Wk∂x0d0u0q0+f0+∑i=1k-1∂Wk∂xidiu0qixi+1pi+fi+∂Wk∂xkfk+12∂2Wk∂x02g02+12∂2Wk∂xk2gk2+12∑i=1k-1∂2Wk∂xi∂xkgigk⊤+12∑i=1k-1∂2Wk∂xi2gi2+12∂2Wk∂xk∂x0gkg0⊤+12∑i=1k-1∂2Wk∂xi∂x0gig0⊤+12∑i,j=1,i≠jk-1∂2Wk∂xi∂xjgigj⊤+c¯k-1dk-1u0qk-1ξk-14lσ-τ-rk-1/σξkrk-1+τ/σ≤1213∑i=1k-1ξi4l+σkΘ1ξk4l+c¯kdku0qkξk4lσ-τ-rk/σξk+1rk+τ/σ+ξk4lσ-τ-rk/σdkc¯ku0qkxk+1∗pk,where c¯k=2pk-1ck is a nonnegative constant. Substituting (22) into (21), we arrive at(23)LVk≤-n-k+1413∑i=1k-1ξi4l+Ψk-1-Θ^˙1Θ~1+ηk-1+∂Wk∂xkdku0qkxk+1pk+σkξk4lΘ1+∂Wk∂Θ^1Θ^˙1+c¯kdku0qkξk4lσ-τ-rk/σξk+1rk+τ/σ+ξk4lσ-τ-rk/σdkc¯ku0qkxk+1∗pk.Obviously, defining Ψk=Ψk-1+ξk4l(σk+n-k+14/13) and ηk=ηk-1-∂Wk/∂Θ^1, it follows from (20) that 0≤Ψk≤(ξ14l+⋯+ξk4l)φk, where φk=max{φk-1,σk+n-k+14/13} is a nonnegative smooth function. With this in mind, (23) can be rewritten as(24)LVk≤-n-k+1413∑i=1kξi4l+Ψk-Θ^˙1Θ~1+ηk+∂Wk∂xkdku0qkxk+1pk+c¯kdku0qkξk4lσ-τ-rk/σξk+1rk+τ/σ+ξk4lσ-τ-rk/σdkc¯ku0qkxk+1∗pk+ξk4lΘ^1σk+n-k+1413+ηk-1Ψk-1-ηkΨk.With the help of Lemma 4, one has(25)ηk-1Ψk-1-ηkΨk≤ηk-1ξk4lΘ^1σk+n-k+1413+∂Wk∂Θ^1Ψk≤113∑i=1kξi4l+φ¯kΘ^1ξk4l,which, together with (24), implies that(26)LVk≤-n-k+1∑i=1kξi4l+Ψk-Θ^˙1Θ~1+ηk+ξk4lσ-τ-rk/σdku0qkxk+1pk-xk+1∗pk+ξk4lσ-τ-rk/σdku0qk1-c¯kxk+1∗pk+ξkrk+τ/σΘ^1φ¯k+σk+n-k+1413+c¯kdku0qkξk4lσ-τ-rk/σξk+1rk+τ/σ,where φ¯k≥0 is a smooth function. It is easy to see that the virtual controller(27)xk+1∗=-1λk,1u0qk1-c¯kΘ^1φ¯k+σk+n-k+14131/pkξkrk+1/σ≜-ϕkx0,x¯k,Θ^1ξkrk+1/σ,renders(28)LVk≤-n-k+1∑i=1kξi4l+ξk4lσ-τ-rk/σdku0qkxk+1pk-xk+1∗pk+Ψk-Θ^˙1Θ~1+ηk+c¯kdku0qkξk4lσ-τ-rk/σξk+1rk+τ/σ,which still holds for k=n. Hence at last step, we can explicitly construct a positive-definite and proper Lyapunov function Vn=Vn-1+Wn and a smooth controller xn+1∗ with form (27) such that(29)LVn≤-∑i=1nξi4l+Ψn-Θ^˙1Θ~1+ηn+ξn4lσ-τ-rn/σdnu0qnupn-xn+1∗pn+c¯ndnu0qnξn4lσ-τ-rn/σξn+1rn+τ/σ.Noting ξn+1=0, by choosing the smooth actual controller u and the adaptive law for Θ^1,(30)u=xn+1∗=-ϕnx0,x¯n,Θ^1ξnrn+1/σ,Θ^˙1=Ψn,we get(31)LVn≤-∑i=1nξi4l.The succeeding proposition characterizes the features of xi-subsystem.Proposition 19.
If Assumptions9–11 hold for system (1), controllers (11) and (30) guarantee that
(i) the closed-loop system has an almost surely unique solution on[t0,∞) for each (x1(t0),…, xn(t0),Θ^1(t0))
(ii) the equilibrium of the closed-loop system is globally asymptotically stable in probability,P{limt→∞|x(t)|=0}=1, P{limt→∞Θ^1(t) exist and is finite}=1 for each (x1(t0),…,xn(t0), Θ^1(t0))Proof.
See Appendix.Case II (x0(t0)=0). When x0(t0)=0, how to select the controllers u0 and u is an interesting topic. If the initial state is zero, one chooses an open loop controller u0=u0∗≠0 to drive the state x0 away from zero; a novel controller u=u∗ can be obtained by the above procedure of original xi-subsystem in (1). So there exists ts>0 such that x0(ts)≠0; after that, controllers u0 in (11) and u in (30) can be used.In view of the argument above, there exists an adaptive controller, such that the equilibrium of the closed-loop system is globally asymptotically stable in probability for any initial condition. The proof of Theorem15 is completed.


## 3.1. Problem Formulation and Assumptions

The aim of this paper is to design an adaptive state feedback controller for system (1) to guarantee global asymptotic stability in probability of the closed-loop system. We need the following assumptions to achieve this aim.Assumption 9.
The sign ofdi(t,θ), i=0,…,n, is assumed to be positive, and there exist smooth functions 0<λi,1(t)≤λi,2(t,θ), such that(4)0<λi,1t≤dit,θ≤λi,2t,θ.Assumption 10.
For smooth functionsf0(x0,θ) and g0(x0,θ), there exist bounded smooth functions α0(x0,θ), β0(x0,θ) such that(5)f0x0,θ=x0α0x0,θ,g0x0,θ=x0β0x0,θ.Assumption 11.
Fori=1,…,n, there exist nonnegative continuous functions αi(x0,x¯i,θ), βi(x0,x¯i,θ) and a constant τ satisfying -rn<τ<0 such that(6)fix0,x,θ≤x1ri+τ/r1+⋯+xiri+τ/riαix0,x¯i,θ+ciu0qidit,θxi+1ri+τ/ri+1,gix0,x,θ≤x12ri+τ/2r1+⋯+xi2ri+τ/2riβix0,x¯i,θ,where rn+1 is a given positive constant and r1,⋯, rn are recursively defined by ri=piri+1-τ and nonnegative constants ci<1/2pi-1.Remark 12.
Assumptions10-11 imply that f0(0,θ)=0, g0(0,θ)=0, fi(0,0,θ)=0, gi(0,0,θ)=0, i=1,…,n; that is, origin is the equilibrium of system (1).Remark 13.
Assumption11 enlarges the scope of nonholonomic systems. Specifically, if state xi+1 does not appear in the nonlinear function fi(·), Assumption 11 is degenerated into the following form:(7)fix0,x,θ≤x1ri+τ/r1+⋯+xiri+τ/riαix0,x¯i,θ,gix0,x,θ≤x12ri+τ/2r1+⋯+xi2ri+τ/2riβix0,x¯i,θ,where r1=1, ri=1+(i-1)τ with τ being a constant for each i=1,…,n. Particularly, when τ=0, (7) becomes the following growth condition:(8)fix0,x,θ≤x1+⋯+xiαix0,x¯i,θ,gix0,x,θ≤x1+⋯+xiβix0,x¯i,θ.If αi(x0,x¯i,θ) and βi(x0,x¯i,θ) are constants, (8) becomes the linear growth condition in [21]. If αi(x0,x¯i,θ) and βi(x0,x¯i,θ) are smooth nonnegative functions, (8) becomes the linear growth condition in [22, 23]. When τ=-p/q∈(-2/4n+1,0) with p being an even integer and q being an odd integer, condition (7) becomes the low-order growth condition in [24, 25]. Assumption 11 extends the value of τ to an explicit interval rather than a ratio of an even integer over an odd integer.Remark 14.
By Lemma3, there are smooth scalar functions λ~i,2(t)≥1, α¯i(x0,x¯i)≥1, β¯i(x0,x¯i)≥1, i=0,…,n, and Θ(θ)≥1 such that(9)λi,1t≤dit,θ≤λ~i,2tΘ,f0x0,θ≤x0α¯0x0Θ,g0x0,θ≤x0β¯0x0Θ,fix0,x,θ≤x1ri+τ/r1+⋯+xiri+τ/riα¯ix0,x¯iΘ+ciu0qidit,θxi+1ri+τ/ri+1,gix0,x,θ≤x12ri+τ/2r1+⋯+xi2ri+τ/2riβ¯ix0,x¯iΘ.


## 3.2. Stability and Convergence Analysis

We state the main result in this paper.Theorem 15.
If Assumptions9–11 hold for system (1), under an appropriate controller, the origin of the closed-loop system is globally asymptotically stable in probability for any initial condition.Proof.The proof is based on inductive argument. Firstly, we design an adaptive controller by considering two cases: x0(t0)≠0 and x0(t0)=0.Case I (x0(t0)≠0). The structure of system (1) means that the design procedure is divided into two separate parts.Part I (design of controller u0). Let us consider x0-subsystem in system (1), define Θ0=λ0,1-1max{λ0,1-1,Θ,Θ2}, and choose V0=x04/4+λ0,1/2Θ~02, where Θ~0=Θ0-Θ^0. From Remark 14, it follows that(10)LV0=x03d0u0p0+f0+32Trg0x02g0⊤-λ0,1Θ~0Θ^˙0≤-aλ0,1x04+aλ0,1x04+x03d0u0p0+Θα¯0x0x04+32Θ2β¯02x0x04-λ0,1Θ~0Θ^˙0≤-aλ0,1x04+λ0,1x03d0λ0,1u0p0+hx0Θ^0x0-λ0,1Θ~0Θ^˙0-x04hx0,where a is a positive constant and h(x0)=a+α¯0(x0)+3/2β¯02(x0) is a positive smooth function. Substituting the adaptive controller(11)u0=-x0hx0Θ^02+11/p0,Θ^˙0=x04hx0into (10) leads to(12)LV0≤-aλ0,1x04.The succeeding proposition characterizes the features of x0-subsystem.Proposition 16.
If Assumptions9-10 hold for x0-subsystem, controller (11) guarantees that
(i)x0-subsystem has an almost surely unique solution on [t0,∞) for any x0(t0)≠0
(ii) the equilibriumx0=0 of x0-subsystem is globally asymptotically stable in probability
(iii) the solution ofx0-subsystem and u0 does not cross zero for any x0(t0)≠0Proof.
See Appendix.Part II (design of controller u). In this part, we need to consider xi-subsystem in system (1). With the help of Proposition 16, controller u will be recursively constructed by applying the adding a power integrator approach. Before the beginning of recursive design, we define Θ1=max{1,Θ,Θ2} and state transformation(13)ξ1=x1σ/r1-x1∗σ/r1,x1∗=0,ξ2=x2σ/r2-x2∗σ/r2,x2∗=-ϕ1x0,x1,Θ^1ξ1r2/σ,⋮ξk=xkσ/rk-xk∗σ/rk,xk∗=-ϕk-1x0,x¯k-1,Θ^1ξk-1rk/σ,where σ is a positive constant satisfying the relationship σ≥max{p1r2,…,pnrn+1}, Θ^1 is the estimate of Θ1, and functions ϕk(·) satisfying sgn(ϕk)=sgn(u0qk) will be determined later. To solve the problem caused by sign function, the definition of Wk(x0,x¯k,Θ^1) is given by(14)Wkx0,x¯k,Θ^1=∫xk∗xksσ/rk-xk∗σ/rk4lσ-τ-rk/σds,where k=1,…,n and 4l is an even number and satisfies (4l-2)σ≥τ+rk.Proposition 17.
W k is C2 function and satisfies Ck1|xk-xk∗|4lσ-τ/rk≤Wk≤Ck2|ξk|4lσ-τ/σ, where Ck1=2(σ-rk)(4lσ-τ-rk)/σrkrk/4lσ-τ, Ck2=21-rk/σ, k=1,…,n.Proof.
​See Appendix.To obtain the expression ofu, we determine ϕ1,⋯, ϕn by induction, the design procedure is implemented as follows.Step  1. Let V1=W1+1/2Θ~12, where Θ~1=Θ1-Θ^1. By using (13), (A.3), (A.7), and Lemma 7, we get(15)LV1≤ξ14lσ-τ-r1/σd1u0q1x2p1+f1+4lσ-τ-r12r1ξ14l-1σ-τ-r1/σx1σ-r1/r1g12-Θ~1Θ^˙1≤-nξ14l+nξ14l+ξ14lσ-τ-r1/σd1u0q1x2p1-x2∗p1+ξ14lσ-τ-r1/σd1u0q1x2∗p1-Θ~1Θ^˙1+4lσ-τ-r12r1ξ14l-1σ-τ-r1/σx1σ+r1+τ/r1β¯12Θ2+ξ14lσ-τ-r1/σx1r1+τ/r1α¯1Θ+c1d1u0q1x2p1.Clearly, the application of the fact ξ14lσ-τ-r1/σd1u0q1x2∗p1≤0 implies(16)ξ14lσ-τ-r1/σu0q1d1x2∗p1+ξ14lσ-τ-r1/σc1d1u0q1x2p1≤ξ14lσ-τ-r1/σu0q1d1x2∗p1+ξ14lσ-τ-r1/σc¯1d1u0q1x2∗p1+ξ14lσ-τ-r1/σc¯1d1u0q1ξ2r1+τ/σ≤ξ14lσ-τ-r1/σd1u0q11-c¯1x2∗p1+ξ14lσ-τ-r1/σc¯1d1u0q1ξ2r1+τ/σ,where c¯1=2p1-1c1 is a positive constant. Substituting (16) into (15) and choosing(17)x2∗=-1λ1,1u0q11-c¯1Θ^1α¯1+n+4lσ-τ-r12r1β¯121/p1ξ1r2/σ≜-ϕ1x0,x1,Θ^1ξ1r2/σ,one has(18)LV1≤-nξ14l+ξ14lσ-τ-r1/σd1u0q1x2p1-x2∗p1+Ψ1-Θ^˙1Θ~1+η1+c¯1d1u0q1ξ14lσ-τ-r1/σξ2r1+τ/σ,where Ψ1=ξ14l(α¯1+n+4lσ-τ-r1/2r1β¯12)≥0 and η1=0.Step k(k=2,…,n). At Step k-1, assume that there exists a C2 function Vk-1 and virtual controllers x2∗,⋯, xk∗ such that(19)LVk-1≤-n-k+2∑i=1k-1ξi4l+ξk-14lσ-τ-rk-1/σdk-1u0qk-1xkpk-1-xk∗pk-1+Ψk-1-Θ^˙1Θ~1+ηk-1+c¯k-1dk-1u0qk-1ξk-14lσ-τ-rk-1/σξkrk-1+τ/σ,where c¯k-1=2pk-1-1ck-1 is a nonnegative constant and(20)0≤Ψk-1≤ξ14l+⋯+ξk-14lφk-1,for a C∞ function φk-1≥0. In what follows, we prove that (19) also holds at Step k. To prove this point, considering Vk=Vk-1+Wk, we deduce from (19) and (A.3)-(A.13) that(21)LVk≤-n-k+2∑i=1k-1ξi4l+ξk-14lσ-τ-rk-1/σdk-1u0qk-1xkpk-1-xk∗pk-1+∂Wk∂Θ^1Θ^˙1+Ψk-1-Θ^˙1Θ~1+ηk-1+c¯k-1dk-1u0qk-1ξk-14lσ-τ-rk-1/σξkrk-1+τ/σ+∂Wk∂x0d0u0p0+f0+∂Wk∂xkdku0qkxk+1pk+fk+12∂2Wk∂x02g02+∑i=1k-1∂Wk∂xidiu0qixi+1pi+fi+12∂2Wk∂xk2gk2+12∑i=1k-1∂2Wk∂xi2gi2+12∂2Wk∂xk∂x0gkg0⊤+12∑i=1k-1∂2Wk∂xi∂x0gig0⊤+12∑i=1k-1∂2Wk∂xi∂xkgigk⊤+12∑i,j=1,i≠jk-1∂2Wk∂xi∂xjgigj⊤.We give the following proposition, whose proof is placed in the Appendix.Proposition 18.
There is a smooth functionσk such that(22)ξk-14lσ-τ-rk-1/σdk-1u0qk-1xkpk-1-xk∗pk-1+∂Wk∂x0d0u0q0+f0+∑i=1k-1∂Wk∂xidiu0qixi+1pi+fi+∂Wk∂xkfk+12∂2Wk∂x02g02+12∂2Wk∂xk2gk2+12∑i=1k-1∂2Wk∂xi∂xkgigk⊤+12∑i=1k-1∂2Wk∂xi2gi2+12∂2Wk∂xk∂x0gkg0⊤+12∑i=1k-1∂2Wk∂xi∂x0gig0⊤+12∑i,j=1,i≠jk-1∂2Wk∂xi∂xjgigj⊤+c¯k-1dk-1u0qk-1ξk-14lσ-τ-rk-1/σξkrk-1+τ/σ≤1213∑i=1k-1ξi4l+σkΘ1ξk4l+c¯kdku0qkξk4lσ-τ-rk/σξk+1rk+τ/σ+ξk4lσ-τ-rk/σdkc¯ku0qkxk+1∗pk,where c¯k=2pk-1ck is a nonnegative constant. Substituting (22) into (21), we arrive at(23)LVk≤-n-k+1413∑i=1k-1ξi4l+Ψk-1-Θ^˙1Θ~1+ηk-1+∂Wk∂xkdku0qkxk+1pk+σkξk4lΘ1+∂Wk∂Θ^1Θ^˙1+c¯kdku0qkξk4lσ-τ-rk/σξk+1rk+τ/σ+ξk4lσ-τ-rk/σdkc¯ku0qkxk+1∗pk.Obviously, defining Ψk=Ψk-1+ξk4l(σk+n-k+14/13) and ηk=ηk-1-∂Wk/∂Θ^1, it follows from (20) that 0≤Ψk≤(ξ14l+⋯+ξk4l)φk, where φk=max{φk-1,σk+n-k+14/13} is a nonnegative smooth function. With this in mind, (23) can be rewritten as(24)LVk≤-n-k+1413∑i=1kξi4l+Ψk-Θ^˙1Θ~1+ηk+∂Wk∂xkdku0qkxk+1pk+c¯kdku0qkξk4lσ-τ-rk/σξk+1rk+τ/σ+ξk4lσ-τ-rk/σdkc¯ku0qkxk+1∗pk+ξk4lΘ^1σk+n-k+1413+ηk-1Ψk-1-ηkΨk.With the help of Lemma 4, one has(25)ηk-1Ψk-1-ηkΨk≤ηk-1ξk4lΘ^1σk+n-k+1413+∂Wk∂Θ^1Ψk≤113∑i=1kξi4l+φ¯kΘ^1ξk4l,which, together with (24), implies that(26)LVk≤-n-k+1∑i=1kξi4l+Ψk-Θ^˙1Θ~1+ηk+ξk4lσ-τ-rk/σdku0qkxk+1pk-xk+1∗pk+ξk4lσ-τ-rk/σdku0qk1-c¯kxk+1∗pk+ξkrk+τ/σΘ^1φ¯k+σk+n-k+1413+c¯kdku0qkξk4lσ-τ-rk/σξk+1rk+τ/σ,where φ¯k≥0 is a smooth function. It is easy to see that the virtual controller(27)xk+1∗=-1λk,1u0qk1-c¯kΘ^1φ¯k+σk+n-k+14131/pkξkrk+1/σ≜-ϕkx0,x¯k,Θ^1ξkrk+1/σ,renders(28)LVk≤-n-k+1∑i=1kξi4l+ξk4lσ-τ-rk/σdku0qkxk+1pk-xk+1∗pk+Ψk-Θ^˙1Θ~1+ηk+c¯kdku0qkξk4lσ-τ-rk/σξk+1rk+τ/σ,which still holds for k=n. Hence at last step, we can explicitly construct a positive-definite and proper Lyapunov function Vn=Vn-1+Wn and a smooth controller xn+1∗ with form (27) such that(29)LVn≤-∑i=1nξi4l+Ψn-Θ^˙1Θ~1+ηn+ξn4lσ-τ-rn/σdnu0qnupn-xn+1∗pn+c¯ndnu0qnξn4lσ-τ-rn/σξn+1rn+τ/σ.Noting ξn+1=0, by choosing the smooth actual controller u and the adaptive law for Θ^1,(30)u=xn+1∗=-ϕnx0,x¯n,Θ^1ξnrn+1/σ,Θ^˙1=Ψn,we get(31)LVn≤-∑i=1nξi4l.The succeeding proposition characterizes the features of xi-subsystem.Proposition 19.
If Assumptions9–11 hold for system (1), controllers (11) and (30) guarantee that
(i) the closed-loop system has an almost surely unique solution on[t0,∞) for each (x1(t0),…, xn(t0),Θ^1(t0))
(ii) the equilibrium of the closed-loop system is globally asymptotically stable in probability,P{limt→∞|x(t)|=0}=1, P{limt→∞Θ^1(t) exist and is finite}=1 for each (x1(t0),…,xn(t0), Θ^1(t0))Proof.
See Appendix.Case II (x0(t0)=0). When x0(t0)=0, how to select the controllers u0 and u is an interesting topic. If the initial state is zero, one chooses an open loop controller u0=u0∗≠0 to drive the state x0 away from zero; a novel controller u=u∗ can be obtained by the above procedure of original xi-subsystem in (1). So there exists ts>0 such that x0(ts)≠0; after that, controllers u0 in (11) and u in (30) can be used.In view of the argument above, there exists an adaptive controller, such that the equilibrium of the closed-loop system is globally asymptotically stable in probability for any initial condition. The proof of Theorem15 is completed.


## 4. Simulation Example

Consider the bilinear model of a mobile robot with small angle measurement error [26], which is described by(32)xc˙=1-ε22ν,yc˙=θν+εν,θ˙=ω,where ε is a small bias in orientation. ν and ω are two control inputs to denote the linear velocity and angular velocity, respectively. Since stochastic disturbance frequently arises in practical control systems, when the angular velocity ω is subject to some stochastic disturbances, ω can be expressed as ω(xc,yc,θ)=ω1(xc,yc,θ)+ω2(xc,yc,θ)B˙(t), where B(t) is the so-called white noise [2]. Then system (32) is transformed into(33)dxc=1-ε22νdt,dyc=θν+ενdt,dθ=ω1dt+ω2dB.For system (33), by taking the state and input transformation x0=xc, x1=yc, x2=θ+ε, u0=ν, u1=ω1, one has(34)dx0=1-ε22u0dt,dx1=x2u0dt,dx2=u1dt+ω2dB.System (34) is a special case of system (1). For simplicity, we assume 1-ε2/2>0, ω2=x2sin⁡x1, and x0(0)≠0. Following the above design process, an adaptive controller can be explicitly given(35)u0=-x0,u1=-Θ^1φ¯2+σ+1413ξ1,Θ^˙1=2ξ14+ξ24σ+1413,where ξ1=⌈x1⌉, ξ2=⌈x2⌉-2Θ^1u0-1⌈ξ1⌉, φ¯2=12|u0|-4/3(σ+14/13)-4/3(ξ14+ξ24), σ=231.7u04Θ^13+6Θ^1+12Θ^14/3+231.7. By choosing ε=1 and initial values (x0(0),x1(0),x2(0),Θ^1(0))=(2,-0.25,2,1), Figure 1 demonstrates the effectiveness of the control scheme.Figure 1
Trajectories ofx0, x1, x2, Θ^1, u0u1.


## 5. Conclusions

This paper investigates adaptive state feedback stabilization for more general stochastic high-order nonholonomic systems. There still exist some problems to be investigated; for instance, we have the following.(1) The result in this paper can be applied to the case of pi∈Rodd≥1. However, if pi is an even number or a ratio of odd integer and even integer, it is unclear whether the control strategy can be applied or not. (2) Recently, some results on stochastic nonlinear systems with SiISS dynamic uncertainty have been obtained [27–35]. An important problem is how to solve adaptive feedback control for stochastic nonholonomic nonlinear systems with SiISS dynamic and parametric uncertainties.



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