# Finite-Time Stabilization of Stochastic Nonholonomic Systems and Its Application to Mobile Robot

Authors: Fangzheng Gao; Fushun Yuan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper investigates the problem of finite-time stabilization for a class of stochastic nonholonomic systems in chained form. By using stochastic finite-time stability theorem and the method of adding a power integrator, a recursive controller design procedure in the stochastic setting is developed. Based on switching strategy to overcome the uncontrollability problem associated withx0(0)=0, global stochastic finite-time regulation of the closed-loop system states is achieved. The proposed scheme can be applied to the finite-time control of nonholonomic mobile robot subject to stochastic disturbances. The simulation results demonstrate the validity of the presented algorithm.


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## Body

## 1. Introduction

The nonholonomic systems, which can model many classes of mechanical systems such as mobile robots and wheeled vehicles, have attracted intensive attention 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 although it is controllable. As a consequence, the classical smooth control theory cannot be applied directly used to such systems. In order to overcome this obstruction, several approaches have been proposed 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, considering the unavoidability of stochastic disturbance, the asymptotic stabilization for stochastic nonholonomic systems was achieved in [16–18]. However, it should be mentioned 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 the asymptotic stabilization, the finite-time stabilization, which renders the trajectories of the closed-loop systems convergent to the origin in a finite time, has many advantages such as fast response, high tracking precision, and disturbance-rejection properties. In many practical situations, the finite-time stabilization problem is more meaningful than the classical asymptotical stability. For the deterministic case, a sufficient and necessary condition for finite-time stability has been proposed in [19]. Its improvements and extensions have been given in [20, 21], for continuous systems satisfying uniqueness of solutions in forward time and for nonautonomous continuous systems, respectively. Reference [22] defined finite-time input-to-state stability for continuous systems with locally essentially bounded input. Accordingly, the problem of finite-time stabilization for nonlinear systems has been studied and numerous theoretical control design methods were presented and developed for various types of nonlinear systems over the last years [23–27]. Especially with help of time-rescaling and Lyapunov based method [28] proposed a novel switching finite time control strategy to nonholonomic chained systems in the deterministic setting.However, the finite-time stabilization for stochastic nonholonomic systems 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. [29], the existence of a unique solution and the nonsatisfaction of local Lipschitz condition are the preconditions of discussing the finite-time stability for a stochastic nonlinear system. Therefore, the finite-time controller design for stochastic nonholonomic systems in this paper 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? Inspired by the works [25, 28], we generalize adding a power integrator design method to a stochastic system and based on stochastic finite-time stability theorem, by skillfully constructing C2 Lyapunov functions, a state feedback controller is successfully achieved to guarantee that the closed-loop system states are globally regulated to zero within a given settling time almost surely.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 gives the main contributions of this paper and presents the design scheme to the controller. Section 5 gives a practical example, the model of which falls into our class of uncertain nonlinear system (3.1) via some technical transformations, to demonstrate the effectiveness of the theoretical results. 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. 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.Consider the stochastic nonlinear system(2.1)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) with Ω being 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.The following Lemma is a corollary of Theorem 170 in [30], which provides a sufficient condition to ensure the existence and uniqueness of solution for the system (2.1).Lemma 2.1.
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:
(2.2)|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) and cTN(t) are nonnegative functions such that ∫0T‍c(t)dt<∞ and ∫0T‍cTN(t)dt<∞. Then for any given x0∈Rn, system (2.1) has a pathwise unique strong solution.Definition 2.2 (see [31]).
For system (2.1), define τ(0,x0)=inf{T≥0:x(t,x0)=0, for all t≥T}, which is called the stochastic settling time function of system (2.1), where x0∈Rn.Definition 2.3 (see [31]).
The equilibriumx≡0 of the system (2.1) is said to be a stochastic finite-time stable equilibrium if(i)
it is stable in probability: for every pair ofε∈(0,1) and r>0, there exists δ>0 such that P{|x(t,x0)|<r,forallt≥0}≥1-ε, whenever |x|<δ.(ii)
its stochastic settling-time functionτ(t0,x0) exists finitely with probability and E[τ(0,x0)]<∞.Lemma 2.4 (see [32]).
Consider the stochastic nonlinear system described in (2.1). Suppose there exists a C2 function V(x), class K∞ functions μ1 and μ2, real numbers c>0 and 0<α<1, such that
(2.3)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 τ(0,x0) satisfies
(2.4)E[τ(0,x0)]≤V1-α(x0)c(1-α).Lemma 2.5 (see [25]).
For any real numbersxi, i=1,…,n and 0<b<1, the following inequality holds:
(2.5)(|x1|+⋯+|xn|)b≤|x1|b+⋯+|xn|b,
when b=p/q<1, where p>0 and q>0 are odd integers,
(2.6)|xb-yb|≤21-b|x-y|b.Lemma 2.6 (see [25]).
Letc,d be positive real numbers and π(x,y)>0 be a real-valued function. Then,
(2.7)|x|c|y|d≤cπ(x,y)|x|c+dc+d+dπ-c/d(x,y)|y|c+dc+d.


## 3. Problem Formulation

In this paper, we focus our attention on the following class of stochastic nonholonomic systems:(3.1)dx0=d0(t)u0dt,dxi=di(t)xi+1u0dt+giT(x0,x[i])dw,i=1,…,n-1,dxn=dn(t)u1dt+gnT(x0,x[n])dw,
where x0∈R and x=(x1,…,xn)T∈Rn are system states, u0∈R and u1∈R are control inputs, respectively; x[i]=(x1,…,xi)T, x[n]=x; di,i=1,…,n represent the possible modeling error, refered to as disturbed virtual control coefficients; gi:R×Ri→Rm,i=1,…,n, are uncertain continuous functions satisfying gi(0,0)=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 3.1.
It should be mentioned that the system investigated in this paper, which emphasizes the effect of stochastic disturbance on thex-subsystem, is a special one; however it can be found in many real systems, such as the angular velocity of mobile robot subject to stochastic disturbances (see Section 5).
The objective of this paper is to find a robust state feedback controller of the form(3.2)u0=u0(x0),u1=u1(x0,x),
such that the stochastic finite-time regulation of closed-loop system states is achieved, that is, P{limt→T(|x0(t)|+|x(t)|)=0}=1 and P{(x0(t),x(t))=(0,0)}=1 for any t≥T, where T is a given settling time.
To achieve the above control objective, we need the following assumptions.Assumption 3.2.
Fori=0,1,…,n, there are known positive constants ci1 and ci2 such that
(3.3)ci1≤di(t)≤ci2.Assumption 3.3.
Fori=1,…,n, there are constants b and τ∈(-2/(4n+1),0) such that
(3.4)|gi(x0,x[i])-gi(x0,x^[i])|≤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 assumeτ=-p/q with p being any even integer and q being any odd integer, under which and the definition of mi in Assumption 3.3, we know that mi is an odd number.Remark 3.4.
Noting thatgi(0,0)=0 is assumed, Assumption 3.3 implies that
(3.5)|gi(t,x,u)|≤b(|x1|(2mi+τ)/2m1+⋯+|xi|(2mi+τ)/2mi).
In fact, Assumption 3.3 is a generalization of the homogeneous growth condition introduced in [33] where x^[i]=0 and τ≥0. The assumption is necessary, which plays an essential role in ensuring the existence of finite-time stabilizer for stochastic nonholonomic system (3.1). Furthermore, it is worthwhile to point out that there exist some nonlinearities such as sinx that can be bounded by a function |x|m for any constant m∈(0,1) and satisfies Assumption 3.3.


## 4. Finite-Time Stabilization

In this section, we give a constructive procedure for the finite-time stabilizing control of system (3.1) within any given settling time T. For clarity, the case that x0(0)≠0 is considered first. Then, the case where the initial x0(0)=0 is dealt later. The inherently triangular structure of system (3.1) suggests that we should design the control inputs u0 and u1 in two separate stages.


### 4.1. Control forx0(0)≠0

Forx0-subsystem, we take the following control law:
(4.1)u0=-k0x0α0,0<α0=pq<1,
where k0 is a positive design parameter, and p, q are positive odd numbers.Taking the Lyapunov functionV0=x02/2, a simple computation gives
(4.2)-c02k0x01+α0≤V˙0≤-c01k0x01+α0≤0,
which implies |x0(t)|≤|x0(0)|.Furthermore, we have(4.3)-c02k0(2V0)(1+α0)/2≤V˙0≤-c01k0(2V0)(1+α0)/2.
Thus, x0 tends to 0 within a settling time denoted by T0. Moreover,
(4.4)|x0(0)|1-α0c02k0(1-α0)≤T0≤|x0(0)|1-α0c01k0(1-α0).
To secure finite-time convergence within T for any x0(0)≠0, we need to keep T0≤|x0(0)|1-α0/c01k0(1-α0)≤T by taking k0>|x0(0)|1-α0/c01T(1-α0). If we take T*=(c01T0)/(2c02), then we obtain x0(t)∈R does not change its sign when t<T*, x0(0)≠0 and moreover
(4.5)|x0(0)|≥|x0(t)|≥|x0(0)|2,∀t∈[0,T*].Therefore,u0 is bounded and does not change sign during [0,T*]. Furthermore, from this and Assumption 3.2, the following result can be obtained.Lemma 4.1.
Fori=1,…,n, there are positive constants λi and μi such that
(4.6)λi≤di(t)αiu0≤μi,λn≤dn(t)≤μn,
where αi=-sgn(x0(0)). Besides, for the simplicity of expression in later use, let αn=1.Since we have already proven thatx0 can be globally finite-time regulated to zero as t→T0. Next, we only need to stabilize the time-varying x-subsystem
(4.7)dxi=di(t)xi+1u0dt+giT(x0,x[i])dw,i=1,…,n-1,dxn=dn(t)u1dt+gnT(x0,x[n])dw,
within the given settling time T*. The control law u1 can be recursively constructed by applying the method of adding a power integrator.Step 1.
Letξ1=x1σ, where σ>2 is a odd number and choose V1=x14σ-τ/(4σ-τ) to be the candidate Lyapunov function for this step. Then, along the trajectories of system (4.7), we have
(4.8)ℒV1≤d1u0x14σ-τ-1(x2-x2*)+d1u0x14σ-τ-1x2*+12(4σ-τ-1)b2x14σ.
Obviously, the first virtual controller(4.9)x2*=-1λ1α1x11+τ[M+n-1+12(4σ-τ-1)b2]≜-α1β1ξ1m2/σ,
with design constant M>0, results in
(4.10)ℒV1≤-(M+n-1)ξ14+d1u0x14σ-τ-1(x2-x2*).Inductive Step (2 ≤ k ≤ n)
Suppose at stepk-1, there is a C2 Lyapunov function Vk-1, which is positive definite and proper, satisfying
(4.11)Vk-1≤2(ξ1(4σ-τ)/σ+⋯+ξk-1(4σ-τ)/σ),
and a set of virtual controllers x1*,…,xk-1* defined by
(4.12)x1*=0,ξ1=x1σ/m1-x1*σ/m1,x2*=-α1β1ξ1m2/σ,ξ2=x2σ/m2-x2*σ/m2,⋮⋮xk*=-αk-1βk-1ξk-1mk/σ,ξk=xkσ/mk-xk*σ/mk,
with constants β1>0,…,βk-1>0, such that
(4.13)ℒVk-1≤-(M+n-k+1)(ξ14+⋯+ξk-14)+dk-1u0ξk-1(4σ-τ-mk-1)/σ(xk-xk*).
We claim that (4.11) and (4.13) also hold at step k. To prove this claim, consider
(4.14)Vk(x[k])=Vk-1(x[k-1])+Wk(x[k]),
where
(4.15)Wk(x[k])=∫xk*xk‍(sσ/mk-xk*σ/mk)(4σ-τ-mk)/σds.
Noting that(4.16)xk*σ/mk=-αk-1σ/mkβk-1σ/mkξk-1,ξk=ckxkσ/mk+ck-1xk-1σ/mk-1+⋯+c1x1σ/m1,
where
(4.17)ci={αk-1σ/mkβk-1σ/mk⋯αiσ/mi+1βiσ/mi+1,i=1,…,k-1,1,i=k.
Similar to the corresponding proof in [25], it is easy to verify that the Lyapunov function Vk(x[k]) thus defined has several useful properties collected in the following propositions.Proposition 4.2.
W
k
(
x
[
k
]
) is C2. Moreover
(4.18)∂Wk∂xi=-ci4σ-τ-mkσxi(σ-mi)/mi×∫xk*xk‍(sσ/mk-xk*σ/mk)(3σ-τ-mk)/σds,∂Wk∂xk=ξk(4σ-τ-mk)/σ,∂2Wk∂xi∂xj=cicj4σ-τ-mkσ·3σ-τ-mkσxi(σ-mi)/mixj(σ-mj)/mj×∫xk*xk‍(sσ/mk-xk*σ/mk)(2σ-τ-mk)/σds,∂2Wk∂xi2=ci24σ-τ-mkσ·3σ-τ-mkσxi2(σ-mi)/mi×∫xk*xk‍(s1/mk-xk*1/mk)(2σ-τ-mk)/σds-ci24σ-τ-mkσxi(σ-2mi)/mi×∫xk*xk‍(sσ/mk-xk*σ/mk)(3σ-τ-mk)/σds,∂2Wk∂xk2=ck4σ-τ-mkσξk(3σ-τ-mk)/σxk(σ-mk)/mk,∂2Wk∂xk∂xi=∂2Wk∂xi∂xk=ci4σ-τ-mkσξk(3σ-τ-mk)/σxi(σ-mi)/mi,
where i,j=1,…,k-1, i≠j.Proposition 4.3.
V
k
(
x
[
k
]
) is C2, positive definite and proper, satisfying
(4.19)Vk≤2(ξ1(4σ-τ)/σ+⋯+ξk(4σ-τ)/σ).Using Proposition4.2, it is deduced from (4.13) that
(4.20)ℒVk≤-(M+n-k+1)(ξ14+⋯+ξk-14)+dk-1u0ξk-1(4σ-τ-mk-1)/σ(xk-xk*)+dku0ξk(4σ-τ-mk)/σxk+1+∑i=1k-1‍∂Wk∂xidiu0xi+1+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.
To estimate the second term in (4.20), by Lemma 2.5, we have
(4.21)|xk-xk*|=|(xkσ/mk)mk/σ-(xk*σ/mk)mk/σ|≤2(σ-mk)/σ|ξk|mk/σ.
Noting that mk=mk-1+τ, by applying (4.6) and Lemma 2.6, we have
(4.22)dk-1u0ξk-1(4σ-τ-mk-1)/σ(xk-xk*)≤16ξk-14+lk1ξk4,
where lk1 is a positive constant.Based on Proposition4.2 and Lemma 2.6, the following propositions are given to estimate the other terms on the right hand side of inequality (4.20), whose proofs are included in the appendix.Proposition 4.4.
There exists a positive constantlk2 such that
(4.23)∑i=1k-1‍∂Wk∂xidiu0xi+1≤16(ξ14+⋯+ξk-14)+lk2ξk4.Proposition 4.5.
There exists a positive constantlk3 such that
(4.24)12∑i,j=1,i≠jk-1‍|∂2Wk∂xi∂xj||giT||gjT|≤16(ξ14+⋯+ξk-14)+lk3ξk4.Proposition 4.6.
There exists a positive constantlk4 such that
(4.25)12∑i=1k-1‍|∂2Wk∂xi2||giT|2≤16(ξ14+⋯+ξk-14)+lk4ξk4.Proposition 4.7.
There exists a positive constantlk5 such that
(4.26)12∑i=1k-1‍|∂2Wk∂xk∂xi||gkT||giT|≤16(ξ14+⋯+ξk-14)+lk5ξk4.Proposition 4.8.
There exists a positive constantlk6 such that
(4.27)12|∂2Wk∂xk2||gkT|2≤16(ξ14+⋯+ξk-14)+lk6ξk4.Substituting (4.22)–(4.27) into (4.20) yields
(4.28)ℒVk≤-(M+n-k)(ξ14+⋯+ξk-14)+dku0ξk(4σ-τ-mk)/σxk+1+ξk4(lk1+⋯+lk6).
Clearly, the C0 virtual controller
(4.29)xk+1*=-1λkαkξk(mk+τ)/σ(M+n-k+lk1+⋯+lk6)≜-αkβkξkmk+1,
with βk>0 being constant, results in
(4.30)ℒVk≤-(M+n-k)(ξ14+⋯+ξk4)+dku0ξk(4σ-τ-mk)/σ(xk+1-xk+1*).This completes the proof of the inductive step.The inductive argument shows that (4.30) holds for k=n. Hence, we conclude that at the last step the actual controller
(4.31)u1=xn+1*=-βnξnmn+1/σ,
with βn>0 being constant and a C2 positive definite and proper Lyapunov function Vn(x[n]) of the form (4.14) and (4.15), such that
(4.32)ℒVn≤-M(ξ14+⋯+ξn4).We have thus far completed the controller design procedure forx0(0)≠0.


### 4.2. Control forx0(0)=0

In the last subsection, we gave the controller expressions (4.1) and (4.31) for u0 and u1 of system (3.1) in the case of x0(0)≠0. Now, we consider finite-time control laws for the case of x0(0)=0. In the absence of the disturbances, most of the commonly used control strategies use constant control u0=u0*≠0 in time interval [0,ts). In this paper, we also use this method when x0(0)=0, with u0 chosen as follows:
(4.33)u0=u0*,u0*>0.Sincex0(0)=0, from the x0-subsystem we know that
(4.34)x˙(0)=u0*(0)=u0*≠0.
We have x0 does not escape and x(ts)≠0, for given any finite ts>0.During the time period[0,ts), using u0 defined in (4.33), new control law u1=u1*(x0,x) can be obtained by the control procedure described above to the original x-subsystem in (3.1). Then we can conclude that the x-state of (3.1) cannot blow up during the time period [0,ts). Since x0(ts)≠0 at t=ts, we can switch the control inputs u0 and u1 to (4.1) and (4.31), respectively.The following theorem summarizes the main result of this paper.Theorem 4.9.
Under Assumptions3.2 and 3.3, if the proposed control design procedure together with the above switching control strategy is applied to system (3.1), 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.Proof.
According to the above analysis, it suffices to prove the statement in the case wherex0(0)≠0.
Since we have already proven thatx0 can be globally finite-time regulated to zero in Section 4.1, we just need to show that x(t) is globally stochastically convergence to zero in a finite time. For the system (4.7)+(4.31), it is not hard to verify that all conditions in Lemma 2.1 are satisfied, which means that the closed-loop system admits a unique solution. In this case, choose the Lyapunov function V=Vn, from (4.32), its time derivative is given by
(4.35)ℒV≤-M(ξ14+⋯+ξn4).
Let α=4σ/(4σ-τ), by Proposition 4.3 and Lemma 2.5, we have
(4.36)Vα≤2(ξ14+⋯+ξn4).
Then, putting (4.36) back to (4.35) gives
(4.37)ℒV≤-MVα/2.
By Lemma2.4, system (4.7) under control law (4.31) is finite-time stable in probability with its settling time Tx satisfying
(4.38)Tx≤2V(1-α)(0)M(1-α).
Hence with the choice ofM satisfying M>2V(1-α)(0)/[T*(1-α)], Tx<T* is guaranteed. Thus, the conclusion follows.Remark 4.10.
As seen from (4.31) and (4.1), the control law u1 may exhibit extremely large value when x0(0)≠0 is sufficiently small. This is unacceptable from a practical point of view. It is therefore recommended to apply (4.33) in order to enlarge the initial value of x0 before we appeal to the finite-time converging controllers (4.1) and (4.31).


## 4.1. Control forx0(0)≠0

Forx0-subsystem, we take the following control law:
(4.1)u0=-k0x0α0,0<α0=pq<1,
where k0 is a positive design parameter, and p, q are positive odd numbers.Taking the Lyapunov functionV0=x02/2, a simple computation gives
(4.2)-c02k0x01+α0≤V˙0≤-c01k0x01+α0≤0,
which implies |x0(t)|≤|x0(0)|.Furthermore, we have(4.3)-c02k0(2V0)(1+α0)/2≤V˙0≤-c01k0(2V0)(1+α0)/2.
Thus, x0 tends to 0 within a settling time denoted by T0. Moreover,
(4.4)|x0(0)|1-α0c02k0(1-α0)≤T0≤|x0(0)|1-α0c01k0(1-α0).
To secure finite-time convergence within T for any x0(0)≠0, we need to keep T0≤|x0(0)|1-α0/c01k0(1-α0)≤T by taking k0>|x0(0)|1-α0/c01T(1-α0). If we take T*=(c01T0)/(2c02), then we obtain x0(t)∈R does not change its sign when t<T*, x0(0)≠0 and moreover
(4.5)|x0(0)|≥|x0(t)|≥|x0(0)|2,∀t∈[0,T*].Therefore,u0 is bounded and does not change sign during [0,T*]. Furthermore, from this and Assumption 3.2, the following result can be obtained.Lemma 4.1.
Fori=1,…,n, there are positive constants λi and μi such that
(4.6)λi≤di(t)αiu0≤μi,λn≤dn(t)≤μn,
where αi=-sgn(x0(0)). Besides, for the simplicity of expression in later use, let αn=1.Since we have already proven thatx0 can be globally finite-time regulated to zero as t→T0. Next, we only need to stabilize the time-varying x-subsystem
(4.7)dxi=di(t)xi+1u0dt+giT(x0,x[i])dw,i=1,…,n-1,dxn=dn(t)u1dt+gnT(x0,x[n])dw,
within the given settling time T*. The control law u1 can be recursively constructed by applying the method of adding a power integrator.Step 1.
Letξ1=x1σ, where σ>2 is a odd number and choose V1=x14σ-τ/(4σ-τ) to be the candidate Lyapunov function for this step. Then, along the trajectories of system (4.7), we have
(4.8)ℒV1≤d1u0x14σ-τ-1(x2-x2*)+d1u0x14σ-τ-1x2*+12(4σ-τ-1)b2x14σ.
Obviously, the first virtual controller(4.9)x2*=-1λ1α1x11+τ[M+n-1+12(4σ-τ-1)b2]≜-α1β1ξ1m2/σ,
with design constant M>0, results in
(4.10)ℒV1≤-(M+n-1)ξ14+d1u0x14σ-τ-1(x2-x2*).Inductive Step (2 ≤ k ≤ n)
Suppose at stepk-1, there is a C2 Lyapunov function Vk-1, which is positive definite and proper, satisfying
(4.11)Vk-1≤2(ξ1(4σ-τ)/σ+⋯+ξk-1(4σ-τ)/σ),
and a set of virtual controllers x1*,…,xk-1* defined by
(4.12)x1*=0,ξ1=x1σ/m1-x1*σ/m1,x2*=-α1β1ξ1m2/σ,ξ2=x2σ/m2-x2*σ/m2,⋮⋮xk*=-αk-1βk-1ξk-1mk/σ,ξk=xkσ/mk-xk*σ/mk,
with constants β1>0,…,βk-1>0, such that
(4.13)ℒVk-1≤-(M+n-k+1)(ξ14+⋯+ξk-14)+dk-1u0ξk-1(4σ-τ-mk-1)/σ(xk-xk*).
We claim that (4.11) and (4.13) also hold at step k. To prove this claim, consider
(4.14)Vk(x[k])=Vk-1(x[k-1])+Wk(x[k]),
where
(4.15)Wk(x[k])=∫xk*xk‍(sσ/mk-xk*σ/mk)(4σ-τ-mk)/σds.
Noting that(4.16)xk*σ/mk=-αk-1σ/mkβk-1σ/mkξk-1,ξk=ckxkσ/mk+ck-1xk-1σ/mk-1+⋯+c1x1σ/m1,
where
(4.17)ci={αk-1σ/mkβk-1σ/mk⋯αiσ/mi+1βiσ/mi+1,i=1,…,k-1,1,i=k.
Similar to the corresponding proof in [25], it is easy to verify that the Lyapunov function Vk(x[k]) thus defined has several useful properties collected in the following propositions.Proposition 4.2.
W
k
(
x
[
k
]
) is C2. Moreover
(4.18)∂Wk∂xi=-ci4σ-τ-mkσxi(σ-mi)/mi×∫xk*xk‍(sσ/mk-xk*σ/mk)(3σ-τ-mk)/σds,∂Wk∂xk=ξk(4σ-τ-mk)/σ,∂2Wk∂xi∂xj=cicj4σ-τ-mkσ·3σ-τ-mkσxi(σ-mi)/mixj(σ-mj)/mj×∫xk*xk‍(sσ/mk-xk*σ/mk)(2σ-τ-mk)/σds,∂2Wk∂xi2=ci24σ-τ-mkσ·3σ-τ-mkσxi2(σ-mi)/mi×∫xk*xk‍(s1/mk-xk*1/mk)(2σ-τ-mk)/σds-ci24σ-τ-mkσxi(σ-2mi)/mi×∫xk*xk‍(sσ/mk-xk*σ/mk)(3σ-τ-mk)/σds,∂2Wk∂xk2=ck4σ-τ-mkσξk(3σ-τ-mk)/σxk(σ-mk)/mk,∂2Wk∂xk∂xi=∂2Wk∂xi∂xk=ci4σ-τ-mkσξk(3σ-τ-mk)/σxi(σ-mi)/mi,
where i,j=1,…,k-1, i≠j.Proposition 4.3.
V
k
(
x
[
k
]
) is C2, positive definite and proper, satisfying
(4.19)Vk≤2(ξ1(4σ-τ)/σ+⋯+ξk(4σ-τ)/σ).Using Proposition4.2, it is deduced from (4.13) that
(4.20)ℒVk≤-(M+n-k+1)(ξ14+⋯+ξk-14)+dk-1u0ξk-1(4σ-τ-mk-1)/σ(xk-xk*)+dku0ξk(4σ-τ-mk)/σxk+1+∑i=1k-1‍∂Wk∂xidiu0xi+1+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.
To estimate the second term in (4.20), by Lemma 2.5, we have
(4.21)|xk-xk*|=|(xkσ/mk)mk/σ-(xk*σ/mk)mk/σ|≤2(σ-mk)/σ|ξk|mk/σ.
Noting that mk=mk-1+τ, by applying (4.6) and Lemma 2.6, we have
(4.22)dk-1u0ξk-1(4σ-τ-mk-1)/σ(xk-xk*)≤16ξk-14+lk1ξk4,
where lk1 is a positive constant.Based on Proposition4.2 and Lemma 2.6, the following propositions are given to estimate the other terms on the right hand side of inequality (4.20), whose proofs are included in the appendix.Proposition 4.4.
There exists a positive constantlk2 such that
(4.23)∑i=1k-1‍∂Wk∂xidiu0xi+1≤16(ξ14+⋯+ξk-14)+lk2ξk4.Proposition 4.5.
There exists a positive constantlk3 such that
(4.24)12∑i,j=1,i≠jk-1‍|∂2Wk∂xi∂xj||giT||gjT|≤16(ξ14+⋯+ξk-14)+lk3ξk4.Proposition 4.6.
There exists a positive constantlk4 such that
(4.25)12∑i=1k-1‍|∂2Wk∂xi2||giT|2≤16(ξ14+⋯+ξk-14)+lk4ξk4.Proposition 4.7.
There exists a positive constantlk5 such that
(4.26)12∑i=1k-1‍|∂2Wk∂xk∂xi||gkT||giT|≤16(ξ14+⋯+ξk-14)+lk5ξk4.Proposition 4.8.
There exists a positive constantlk6 such that
(4.27)12|∂2Wk∂xk2||gkT|2≤16(ξ14+⋯+ξk-14)+lk6ξk4.Substituting (4.22)–(4.27) into (4.20) yields
(4.28)ℒVk≤-(M+n-k)(ξ14+⋯+ξk-14)+dku0ξk(4σ-τ-mk)/σxk+1+ξk4(lk1+⋯+lk6).
Clearly, the C0 virtual controller
(4.29)xk+1*=-1λkαkξk(mk+τ)/σ(M+n-k+lk1+⋯+lk6)≜-αkβkξkmk+1,
with βk>0 being constant, results in
(4.30)ℒVk≤-(M+n-k)(ξ14+⋯+ξk4)+dku0ξk(4σ-τ-mk)/σ(xk+1-xk+1*).This completes the proof of the inductive step.The inductive argument shows that (4.30) holds for k=n. Hence, we conclude that at the last step the actual controller
(4.31)u1=xn+1*=-βnξnmn+1/σ,
with βn>0 being constant and a C2 positive definite and proper Lyapunov function Vn(x[n]) of the form (4.14) and (4.15), such that
(4.32)ℒVn≤-M(ξ14+⋯+ξn4).We have thus far completed the controller design procedure forx0(0)≠0.


## 4.2. Control forx0(0)=0

In the last subsection, we gave the controller expressions (4.1) and (4.31) for u0 and u1 of system (3.1) in the case of x0(0)≠0. Now, we consider finite-time control laws for the case of x0(0)=0. In the absence of the disturbances, most of the commonly used control strategies use constant control u0=u0*≠0 in time interval [0,ts). In this paper, we also use this method when x0(0)=0, with u0 chosen as follows:
(4.33)u0=u0*,u0*>0.Sincex0(0)=0, from the x0-subsystem we know that
(4.34)x˙(0)=u0*(0)=u0*≠0.
We have x0 does not escape and x(ts)≠0, for given any finite ts>0.During the time period[0,ts), using u0 defined in (4.33), new control law u1=u1*(x0,x) can be obtained by the control procedure described above to the original x-subsystem in (3.1). Then we can conclude that the x-state of (3.1) cannot blow up during the time period [0,ts). Since x0(ts)≠0 at t=ts, we can switch the control inputs u0 and u1 to (4.1) and (4.31), respectively.The following theorem summarizes the main result of this paper.Theorem 4.9.
Under Assumptions3.2 and 3.3, if the proposed control design procedure together with the above switching control strategy is applied to system (3.1), 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.Proof.
According to the above analysis, it suffices to prove the statement in the case wherex0(0)≠0.
Since we have already proven thatx0 can be globally finite-time regulated to zero in Section 4.1, we just need to show that x(t) is globally stochastically convergence to zero in a finite time. For the system (4.7)+(4.31), it is not hard to verify that all conditions in Lemma 2.1 are satisfied, which means that the closed-loop system admits a unique solution. In this case, choose the Lyapunov function V=Vn, from (4.32), its time derivative is given by
(4.35)ℒV≤-M(ξ14+⋯+ξn4).
Let α=4σ/(4σ-τ), by Proposition 4.3 and Lemma 2.5, we have
(4.36)Vα≤2(ξ14+⋯+ξn4).
Then, putting (4.36) back to (4.35) gives
(4.37)ℒV≤-MVα/2.
By Lemma2.4, system (4.7) under control law (4.31) is finite-time stable in probability with its settling time Tx satisfying
(4.38)Tx≤2V(1-α)(0)M(1-α).
Hence with the choice ofM satisfying M>2V(1-α)(0)/[T*(1-α)], Tx<T* is guaranteed. Thus, the conclusion follows.Remark 4.10.
As seen from (4.31) and (4.1), the control law u1 may exhibit extremely large value when x0(0)≠0 is sufficiently small. This is unacceptable from a practical point of view. It is therefore recommended to apply (4.33) in order to enlarge the initial value of x0 before we appeal to the finite-time converging controllers (4.1) and (4.31).


## 5. Application to Mobile Robot

In this section, we illustrate systematic controller design method proposed above by means of the example of mobile robot.Consider the tricycle-type mobile robot with parametric uncertainty [34], which is described by
(5.1)x˙c=p1*vcosθ,y˙c=p1*vsinθ,θ˙=p2*ω,
where p1* and p2* are unknown parameters taking values in a known interval [pmin,pmax] with 0<pmin<pmax<∞, v and ω are two control inputs to denote the linear velocity and angular velocity, respectively.When the angular velocityω is subject to some stochastic disturbances, that is,
(5.2)ω(xc,yc,θ)=ω1(xc,yc,θ)+ω2(xc,yc,θ)B˙(t),
where B(t) is the so-called white noise. Then system (5.1) is transformed into
(5.3)dxc=p1*vcosθdt,dyc=p1*vsinθdt,dθ=p2*ω1dt+p2*ω2dB.For system (5.3), by taking the following state and input transformation:
(5.4)x0=xc,x1=yc,x2=tanθ,u0=vcosθ,u1=ω1sec2θ,
we obtain
(5.5)dx0=p1*u0dt,dx1=p1*x2u0dt,dx2=p2*u1dt+p2*(1+x22)ω2dB.Clearly, system (5.5) is a special case of system (3.1). As discussed in Remark 3.4, there always exist some nonlinearities satisfy Assumption 3.3. For simplicity, it is assumed the ω2=(x2sinx1)/(1+x22)2. And as in [8], pmin=1, pmax=2, which are known for us in contructing control laws to make (5.5) finite-time stable, and parameters p1*=p1*=1.5 are unknown. When (x0(0),x1(0),x2(0))=(1,1,1), by choosing τ=-2/11 and σ=3, according to the design procedure proposed in Section 4, the following controllers can be obtained for a given settling time T=6:
(5.6)u0=-12x01/3,u1=-[x211/3-(2M+2)11/3x13]7/33(M+l21+l22+l24),
where l21, l22 and l24 are known positive constants. Choosing design parameter as M=1, the simulation results in Figures 1 and 2 show that the effectiveness of the controller.Figure 1
Time history of system states.Figure 2
Time history of control inputs.


## 6. Conclusion

In this paper, the finite-time state feedback stabilization problem has been investigated for a class of nonholonomic systems with stochastic disturbances. With the help of adding a power integrator technique, a systematic control design procedure is developed in the stochastic setting. To get around the stabilization burden associated with nonholonomic systems, a switching control strategy is proposed. It is shown that the designed control laws can guarantee that the closed-loop system states are globally finite-time regulated to zero in probability. In addition, the proposed approach can be applied to mobile robot with stochastic disturbances.There are some related problems to investigate, for example, how to design a finite-time state-feedback stabilizing controller for stochastic nonholonomic systems when thex0-subsystem contains stochastic disturbances. Furthermore, if only partial state vector being measurable, how to design a finite-time output feedback stabilizing controller for stochastic nonholonomic systems.



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*Source: 361269-2012-10-30.xml*