# Further Result on Finite-Time Stabilization of Stochastic Nonholonomic Systems

Authors: Fangzheng Gao; Fushun Yuan; Jian Zhang; Yuqiang Wu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper further investigates the problem of finite-time state feedback stabilization for a class of stochastic nonholonomic systems in chained form.
Compared with the existing literature, the stochastic nonholonomic systems under
investigation have more uncertainties, such as thex0-subsystem contains stochastic
disturbance. This renders the existing finite-time control methods highly difficult
to the control problem of the systems or even inapplicable. In this paper, by extending adding a power integrator design method to a stochastic system and by
skillfully  constructing C2 Lyapunov function, a novel switching control strategy
is proposed, which renders that the states of closed-loop system are almost surely
regulated to zero in a finite time. A simulation example is provided to demonstrate
the effectiveness of the theoretical results.


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## Body

## 1. Introduction

The nonholonomic systems, which can be used to model many frequently met mechanical systems, such as wheeled mobile robot, knife edge and  rolling disk, have been an active research field over the past decades. From Brockett’s necessary condition [1], it is well known that the nonholonomic systems cannot be stabilized to the origin by any static continuous state feedback, so the classical smooth control theory cannot be applied directly. In order to overcome this obstruction, several novel approaches have been developed for the problem, such as discontinuous time-invariant stabilization [2, 3], smooth time-varying stabilization [4–6],  and hybrid stabilization [7]. Using these valid approaches, many fruitful results have been developed [8–15]. Particularly, the stochastic nonholonomic systems, which can be viewed as the extension of the classical nonholonomic systems, have been recently achieved investigations [16–18]. However, it should be noted that those aforementioned papers consider the feedback stabilizer that makes the trajectories of the systems converge to the equilibrium as the time goes to infinity.Compared to asymptotic stabilization, the closed-loop system with finite-time convergence usually demonstrates faster convergence rates, higher accuracies, better disturbance rejection properties, and robustness against uncertainties [19]. Hence, it is more meaningful to investigate the finite-time stabilization problem than the classical asymptotical stability. For the deterministic case, in [20], a novel switching finite-time control strategy was proposed to nonholonomic systems in a chained form with uncertain parameters and perturbed terms by the use of time rescaling and Lyapunov based method. Later this result was essentially extended under weaker constraints on drift terms in [21]. As a natural extension, the finite-time stabilization for stochastic nonholonomic systems is an interesting and challenging subject of intensive study. However, it cannot be solved by simply extending the methods for deterministic systems because of the presence of stochastic disturbance. As pointed out by Yin et al. [22], “the existence of a unique solution and the non-satisfaction of local Lipschitz condition are two prerequisites of discussing the finite-time stability for a stochastic nonlinear system. That is, as a common assumption condition to guarantee the existence of a unique solution for stochastic nonlinear system, local Lipschitz condition cannot be applied to study stochastic finite-time stability.” Therefore, the finite-time controller design for stochastic nonholonomic systems should solve the following questions: under what conditions, the stochastic nonholonomic systems exist possibly finite-time stabilizer? Under these conditions, how can one design a finite-time state feedback stabilizing controller?To the best of the authors’ knowledge, only the authors in [23] considered the aforementioned problems when thex0-subsystem is ordinary differential equation. This paper continues the investigation in [23] and addresses the finite-time stabilizing control design for stochastic nonholonomic systems with more uncertainties than those in [16–18, 23], which cannot be handled by general existing methods. A constructive method in designing finite-time stabilizing controller for such uncertain system is proposed. The contribution of this paper is highlighted as follows. First, inspired by the deterministic works in [24, 25], we generalize adding a power integrator design method [26] to a stochastic system, and a novel C2 Lyapunov function is constructed to deal with stochastic disturbances. Second, based on a new switching method, a systematic control design procedure is proposed to solve the finite-time stabilization problem for all plants in the considered class.The remainder of this paper is organized as follows. Section2 presents some necessary notations, definitions, and preliminary results. Section 3 describes the systems to be studied and formulates the control problem. Section 4 presents the design scheme to the controller and the main result. Section 5 gives a simulation example to illustrate the theoretical finding of this paper. Finally, concluding remarks are proposed in Section 6.


## 2. Notations and Preliminary Results

The following notations, definitions, and lemmas are to be used throughout the paper.R+ denotes the set of all nonnegative real numbers, and Rn denotes the real n-dimensional space. (a,b)odddenotes the set{x∣x=p/qandx∈(a,b), wherep and q are odd integers}. (a,b)even denotes the set {x∣x=p/q andx∈(a,b), where p is an even integer and q is an odd integer}. For a given vector or matrix X, XT denotes its transpose, Tr{X} denotes its trace when X is square, and |X| is the Euclidean norm of a vector X. Ci denotes the set of all functions with continuous ith partial derivatives. K denotes the set of all functions:R+→R+,  which are continuous, strictly increasing, and vanishing at zero; K∞ denotes the set of all functions which are of class K and unbounded. The arguments of the functions (or the functionals) will be omitted or simplified, whenever no confusion can arise from the context. For instance, we sometimes denote a function  f(x(t)) by simply f(x) or f.We begin with some basic concepts and terminologies related to the notion of stochastic finite-time stability. The reader is referred to [22, 27, 28] as well as the references therein for additional details.Consider the stochastic nonlinear system(1)dx=f(t,x)dt+g(t,x)dω,
where x∈Rn is the system state with the initial condition x(0)=x0; ω is an m-dimensional independent standard Wiener process defined on a complete probability space (Ω,ℱ,{ℱt}t≥0,P) withΩbeing a sample space, ℱ being a σ-field,{ℱt}t≥0 being a filtration, and P being a probability measure. The functions: f:R+×Rn→Rn and g:R+×Rn→Rn×m are piecewise continuous and continuous with respect to the first and second arguments, respectively, and satisfy f(t,0)≡0 and g(t,0)≡0.  Moreover, system (1) is assumed to have a pathwise unique strong solution, denoted by x(t,x0).Definition 1.
For system (1), define T0(x0,w)=inf{T≥0:x(t,x0)=0, for all  t≥T},  which is called the stochastic settling time function. Especially,T0(x0,w):=+∞ifx(t,x0)≠0, for all t≥0.Definition 2.
The equilibriumx≡0 of the system (1) is said to be a stochastic finite-time stable equilibrium if(i)
it is stable in probability: for every pair ofε∈(0,1) andr>0, there existsδ>0such that P{|x(t,x0)|<r,  for all t≥0}≥1-ε, whenever|x|<δ;(ii)
its stochastic settling-time functionT0(x0,w)exists finitely with probability andE[T0(x0,w)]<∞.Next, we give two lemmas where the first one provides sufficient conditions to ensure the existence of pathwise unique strong solution to system (1), and the other one has been used to determine the finite-time stability of stochastic nonlinear systems.Lemma 3 (see [23, 29]).
Assume thatf(t,x) and g(t,x) are continuous in x. Further, for any 0<δ<1, each N=1,2,…, and each 0≤T<∞, if the following conditions hold:
(2)|f(t,x)|≤c(t)(1+|x|),|g(t,x)|2≤c(t)(1+|x|2),|f(t,x1)-f(t,x2)|≤cTN(t)|x1-x2|,|g(t,x1)-g(t,x2)|≤cTN(t)|x1-x2|,
as  0<δ≤|xi|≤N,i=1,2,t∈[0,T],  where c(t) andcTN(t)are nonnegative functions such that ∫0T‍c(t)dt<∞ and ∫0T‍cTN(t)dt<∞.  Then for any givenx0∈Rn, system (1) has a pathwise unique strong solution.Lemma 4 (see [30]).
Consider the stochastic nonlinear system described in (1). Suppose that there exist a C2 function V(x),  class  K∞   functions μ1 and μ2, real numbersc>0and0<α<1, such that
(3)μ1(|x|)≤V(x)≤μ2(|x|),ℒV(x)=∂V∂xf+12Tr{gT∂2V∂x2g}≤-cVα(x).
Then it is globally finite-time stable in probability, and the stochastic settling time function T0(x0,w) satisfies
(4)E[T0(x0,w)]≤V1-α(x0)c(1-α).Then we list two lemmas that serve as the basis of the key tools for the adding a power integrator technique.Lemma 5 (see [31]).
Forx∈R, y∈R,  and p≥1 being a constant, the following inequalities hold:
(5)|x+y|p≤2p-1|xp+yp|,(|x|+|y|)1/p≤|x|1/p+|y|1/p≤2(p-1)/p(|x|+|y|)1/p.
Ifp≥1is odd, then
(6)|x-y|p≤2p-1|xp-yp|,|x1/p-y1/p|≤2(p-1)/p(|x-y|)1/p.Lemma 6 (see [32]).
Letx,ybe real variables; then for any positive real numbersa,m, andn, one has
(7)a|x|m|y|n≤b|x|m+n+nm+n(m+nm)-m/n×a(m+n)/nb-m/n|y|m+n,
whereb>0is any real number.


## 3. Problem Formulation

Motivated by the statement in [18, 23] that many nonholonomic mechanical systems, such as wheeled mobile robot, subject to stochastic disturbances, can be transformed to a kind of stochastic nonholonomic systems in the so-called chained form, this paper considers the following class of stochastic nonholonomic systems in chained form:
(8)dx0=d0(t)u0dt+f0(t,x0)dt+g0T(t,x0)dw,dxi=di(t)xi+1u0dt+fi(t,x0,x)dt+giT(t,x0,x)dw,i=1,…,n-1,dxn=dn(t)u1dt+fn(t,x0,x)dt+gnT(t,x0,x)dw,
wherex0∈Rand x=(x1,…,xn)T∈Rn are system states and u0∈R and u1∈R are control inputs, respectively; di:R+→R, i=0,…,n are uncertain and continuous, called the control coefficients. The nonlinear functions f0:R+×R→R and g0:R+×R→Rm, fi:R+×Ri+1→R and gi:R+×Ri+1→Rm, i=1,…,n are assumed to be C1 with their arguments with f0(t,0)=0, g0(t,0)=0, fi(t,x0,0)=0 and gi(t,x0,0)=0.  and ω is an m-dimensional independent standard Wiener process defined on a complete probability space (Ω,F,P) with Ω being a sample space, F being a filtration, and P being a probability measure.Remark 7.
Obviously, the stochastic nonholonomic system (8) contains more uncertainties in drift and diffusion terms than those in the closely related papers [16–18, 23]. This, especially the existence of stochastic disturbance in thex0-subsystem, renders the procedure of control design in [23] inapplicable to the finite-time control problem of the system. Up to now, how to design a stabilizing controller to achieve the finite-time stabilization of the system (8) is unsolved. It is precisely our intention of this paper.
The following assumptions regarding system (8) are imposed throughout the paper.Assumption 8.
Fori=0,…,n,  there are known positive constants ci1 and ci2 such that
(9)ci1≤di(t)≤ci2.Assumption 9.
Forf0 and g0, there are constantsa>0and r∈(0,1)odd such that
(10)|f0(t,x0)-f0(t,x^0)|≤a|x0-x^0|r,|g0(t,x0)-g0(t,x^0)|≤a|x0-x^0|(r+1)/2.Assumption 10.
Fori=1,…,n, there is a constantb>0such that for anyτ∈(-2/(4n+1),0),
(11)|fi(t,x0,x)-fi(t,x0,x^)|≤b(|x1-x^1|(mi+τ)/m1+⋯+|xi-x^i|(mi+τ)/mi),|gi(t,x0,x)-gi(t,x0,x^)|≤b(|x1-x^1|(2mi+τ)/2m1+⋯+|xi-x^i|(2mi+τ)/2mi),
where  mi=1+(i-1)τ.
For simplicity, in this paper we assume thatτ=-p/q with p being any even integer and q being any odd integer, under which and the definition of mi in Assumption 10, we know that mi is an odd number.Remark 11.
Noting thatf0(t,0)=0, g0(t,0)=0, fi(t,x0,0)=0 and gi(t,x0,0)=0 are assumed, Assumptions 9 and 10 imply that
(12)|f0(t,x0)|≤a|x0|r,|g0(t,x0)|≤a|x0|(r+1)/2,|fi(t,x0,x)|≤b(|x1|(mi+τ)/m1+⋯+|xi|(mi+τ)/mi),|gi(t,x0,x)|≤b(|x1|(2mi+τ)/2m1+⋯+|xi|(2mi+τ)/2mi).
It is noteworthy that Assumption 10 is a generalization of the homogeneous growth condition introduced in [33] where x^=0 and τ≥0. Assumptions 9 and 10 are necessary, which play an essential role in ensuring the existence of finite-time stabilizer for stochastic nonholonomic system (8).


## 4. Finite-Time Control Design

In this section, we give a constructive procedure for the finite-time stabilizer of system (8). The design of finite-time state feedback controller is divided into two steps.Step
A
. We first stabilize the x-subsystem in a finite time almost surely.For thex0-subsystem, we choose the control u0 as
(13)u0≡u0*,
where  u0*  is a positive constant. In this case, the  x0-subsystem becomes
(14)dx0=(d0u0*+f0)dt+g0Tdw.With the help of Assumptions8 and 9 and Lemma 3, it easy to obtain that system (14) has a unique solution on [0,T], for any given finite time T>0. Hence, x0 is well defined on [0,T]. Next, we stabilize the x-subsystem
(15)dxi=di(t)u0*xi+1dt+fi(t,x0,x)dt+giT(t,x0,x)dw,i=1,…,n-1,dxn=dn(t)u1dt+fn(t,x0,x)dt+gnT(t,x0,x)dw,
within a finite time. The control law u1 can be recursively constructed by applying the method of adding a power integrator.Step   1. Let ξ1=x1σ, where σ∈(2,∞)odd is a constant. Choosing the Lyapunov function V1=x1(4lσ-τ)/(4lσ-τ), where l is a constant satisfying 4l∈(0,∞)even  and (4l-2)σ≥1. With the help of Remark 11 and Lemma 4, it can be verified that
(16)ℒV1≤d1u0*x14lσ-τ-1(x2-x2*)+d1u0*x14lσ-τ-1x2*+bx14lσ+12(4σ-τ-1)b2x14lσ.Obviously, the first virtual controller(17)x2*=-1c11u0*[M+n-1+b+12(4σ-τ-1)b2]x11+τ:=-β1m2/σξ1m2/σ,
with design constant M>0, results in
(18)ℒV1≤-(M+n-1)ξ14l+d1u0*ξ1(4lσ-τ-1)/σ(x2-x2*).Inductive Step. Suppose at step k-1, there are a C2, proper and positive definite Lyapunov function Vk-1, and a set of virtual controllers x1*,…,xk* defined by
(19)x1*=0,ξ1=x1σ/m1-x1*σ/m1,x2*=-β1m2/σξ1m2/σ,ξ2=x2σ/m2-x2*σ/m2,⋮⋮xk*=-βk-1mk/σξk-1mk/σ,ξk=xkσ/mk-xk*σ/mk,
with constants β1>0,…,βk-1>0, such that
(20)ℒVk-1≤-(M+n-k+1)(ξ14l+⋯+ξk-14l)+dk-1u0*ξk-1(4lσ-τ-mk-1)/σ(xk-xk*).
To complete the induction, at the kth step, we choose the following Lyapunov function:
(21)Vk(x1,…,xk)=Vk-1(x1,…,xk-1)+Wk(x1,…,xk),
where
(22)Wk(x1,…,xk)=∫xk*xk‍(sσ/mk-xk*σ/mk)(4lσ-τ-mk)/σds.Noting that(23)xk*σ/mk=-βk-1ξk-1,ξk=xkσ/mk-βk-1xk-1σ/mk-1+⋯+(-1)k-1×βk-1⋯β1x1σ/m1,
with the help of the fact σ/mi>2, the function Vk can be shown to be C2, proper and positive definite using a similar method as in [34]. Moreover, we can obtain
(24)∂Wk∂xk=ξk(4lσ-τ-mk)/σ,∂2Wk∂xk2=4lσ-τ-mkmkξk((4l-1)σ-τ-mk)/σxk(σ-mk)/mk,∂2Wk∂xk∂xi=∂2Wk∂xi∂xk=4lσ-τ-mkσ×ξk((4l-1)σ-τ-mk)/σ∂xk*σ/mk∂xi,∂Wk∂xi=-4lσ-τ-mkσ∂xk*σ/mk∂xi×∫xk*xk‍(sσ/mk-xk*σ/mk)((4l-1)σ-τ-mk)/σds,∂2Wk∂xi2=4lσ-τ-mkσ(4l-1)σ-τ-mkσ(∂xk*σ/mk∂xi)2×∫xk*xk‍(sσ/mk-xk*σ/mk)((4l-2)σ-τ-mk)/σds-4σ-τ-mkσ∂2xk*σ/mk∂xi2×∫xk*xk‍(sσ/mk-xk*σ/mk)((4l-1)σ-τ-mk)/σds,∂2Wk∂xi∂xj=4lσ-τ-mkσ·(4l-1)σ-τ-mkσ∂xk*σ/mk∂xi∂xk*σ/mk∂xj×∫xk*xk‍(sσ/mk-xk*σ/mk)((4l-2)σ-τ-mk)/σds,
where i,j=1,…,k-1,   i≠j.Using (20)–(22) and (24), it follows that
(25)ℒVk≤-(M+n-k+1)(ξ14l+⋯+ξk-14l)+dku0*ξk(4lσ-τ-mk)/σxk+1+dk-1u0*ξk-1(4lσ-τ-mk-1)/σ(xk-xk*)+ξk(4lσ-τ-mk)/σfk+∑i=1k-1‍∂Wk∂xi(diu0*xi+1+fi)+12∑i,j=1,i≠jk-1‍|∂2Wk∂xi∂xj||giT||gjT|+12∑i=1k-1‍|∂2Wk∂xi2||giT|2+12∑i=1k-1‍|∂2Wk∂xk∂xi||gkT||giT|+12|∂2Wk∂xk2||gkT|2.Based on Lemmas5 and 6, the following proposition is given to estimate the last seven terms on the right-hand side of (25).Proposition 12.
There exists a positive constantlk such that
(26)dk-1u0*ξk-1(4lσ-τ-mk)/σ(xk-xk*)+ξk(4lσ-τ-mk)/σfk+∑i=1k-1‍∂Wk∂xi(diu0*xi+1+fi)+12∑i,j=1,i≠jk-1‍|∂2Wk∂xi∂xj||giT||gjT|+12∑i=1k-1‍|∂2Wk∂xi2||giT|2+12∑i=1k-1‍|∂2Wk∂xk∂xi||gkT||giT|+12|∂2Wk∂xk2||gkT|2≤ξ14l+⋯+ξk-14l+lkξk4l.Proof.
The similar proof can be found in [23] and thus it is omitted here.
Substituting (26) into (25), we obtain
(27)ℒVk≤-(M+n-k)(ξ14l+⋯+ξk-14l)+dku0*ξk(4lσ-τ-mk)/σxk+1+lkξk4l.
Clearly, theC0 virtual controller
(28)xk+1*=-1ck1u0*ξk(mk+τ)/σ(M+n-k+lk):=-βkmk+1/σξkmk+1/σ,
with βk>0  being constant, results in
(29)ℒVk≤-(M+n-k)(ξ14l+⋯+ξk4l)+dku0*ξk(4lσ-τ-mk)/σ(xk+1-xk+1*).
This completes the proof of the inductive step.Using the previous inductive argument, one concludes that at thenth step, there exists a non-Lipschitz continuous state feedback control law of the form
(30)u=xn+1*=-βnmn+1/σξnmn+1/σ,
with βn>0  being constant and a C2 positive definite and proper Lyapunov function Vn of the form (21)-(22), such that
(31)ℒVn≤-M(ξ14l+⋯+ξn4l).So the following result is obtained.Lemma 13.
If Assumptions8 and 10 hold for the stochastic nonlinear system (15), then the solution of the closed-loop system consisting of (15) and (30) is finite-time stable in probability.Proof.
From (23) and (30), it is not hard to verify that all conditions in Lemma 3 are satisfied, which means that the closed-loop system admits a unique solution. Next we prove that the closed-loop system is globally finite-time stable in probability.
First of all, by using Lemma5, it is easy to see that
(32)Wk=∫xk*xk‍(sσ/mk-xk*σ/mk)(4lσ-τ-mk)/σds≤|ξk|(4lσ-τ-mk)/σ|xk-xk*|≤2(σ-mk)/σ|ξk|(4lσ-τ)/σ.
So we have the following estimate(33)Vn=1(4lσ-τ)x14lσ-τ+∑k=2n‍Wk≤1(4lσ-τ)ξ1(4lσ-τ)/σ+∑k=2n‍2(σ-mk)/σ|ξk|(4lσ-τ)/σ≤2∑k=1n‍ξk(4lσ-τ)/σ.
Letα=4lσ/(4lσ-τ). With (33) and (31) in mind, by Lemma 5, it is not difficult to obtain that
(34)ℒVn≤-MVnα2α.
Therefore, by Lemma4, the uncertain system (15) with the controller (30) is globally finite-time stable in probability and its stochastic settling time T1(x(0),w) satisfies
(35)E[T1(x(0),w)]≤2αVn1-α(x(0))M(1-α).Step
B
. Then we design a state feedback controller such that the x0-subsystem is finite-time stable in probability.From StepA, we know that x(t)≡0 when t≥T1(x(0),w). Therefore, we just need to stabilize the x0-subsystem in a finite time almost surely. When t≥T1(x(0),w), for the x0-subsystem, we can take the following control law:
(36)u0(x0)=-1c01(k0+a+32a2)x0r,
where k0 is a positive constant.Taking the Lyapunov functionV0=x04/4, a simple computation gives
(37)ℒV0≤-k0x03+r.Thus by Lemma4, x0 stochastically tends to 0 within a settling time denoted by T2(x0(0),w)  and
(38)E[T2(x0(0),w)]≤2V0(1-r)/4(x0(0))k0(1-r).Up to now, we have finished the finite-time stabilizing controller design of the system (8). Consequently, the following theorem can be obtained to summarize the main result of the paper.Theorem 14.
Under Assumptions8–10, if the proposed control design procedure together with the previous switching control strategy is applied to system (8), then, for any initial conditions in the state space (x0,x)∈Rn+1, the closed-loop system is globally finite time regulated at origin in probability.Remark 15.
The settling time of the closed-loop system depends on design parametersk0, M and initial conditions, that is, when the initial conditions are known, the settling-time can be arbitrarily adjusted by the choice of appropriate design parameters k0 and M.


## 5. Simulation Example

To verify our proposed controller, we consider the following low-dimensional system:(39)dx0=(1.5+0.5cost)u0dt+14x04/5dw,dx1=x2dt+14x19/11dt+18sinx1dw,dx2=udt+14x17/11dt+18cos2x2dw.We chooseτ=-2/11, which together with m1=1 implies that m2=9/11.  By Lemma 5, it is verified that Assumptions 9-10 are satisfied with a=b=1/2. Assumption 8 holds with 1≤1.5+0.5cost≤2; hence the controller proposed in this paper is applicable.When(x0(0),x1(0),x2(0))=(1,-1,1), by choosing σ=3 and l=2/3, according to the design procedure proposed in Section 4, a state feedback stabilizing controller can be explicitly given. Using MATLAB, Figures 1 and 2 are obtained to exhibit the trajectories of the closed-loop system. From these figures, we can see that the states of the closed-loop system are finite time regulated to zero almost surely.States of the closed-loop system.
(a)
x
0
(b)
x
1
(c)
x
2Control inputs of the closed-loop system.
(a)
u
0
(b)
u
1


## 6. Conclusion

In this paper, the finite-time state feedback stabilization problem has been investigated for a class of nonholonomic systems with more general stochastic disturbances. With the help of adding a power integrator technique, a systematic control design procedure is developed in the stochastic setting. It should be noted that the proposed controller can only work well when the whole state vector is measurable. Therefore, a natural and more interesting problem is how to design output feedback stabilizing controller for the system studied in the paper if only partial state vector is measurable, which is now under our further investigation.



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