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

and Applied Analysis 3 Consider the stochastic nonlinear system dx f t, x dt g t, x dω, 2.1 where x ∈ R is the system state with the initial condition x 0 x0, ω is an mdimensional 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 → R 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 that f 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: ∣ ∣f t, x ∣ ∣ ≤ c t 1 |x| , ∣∣g t, x ∣2 ≤ c t ( 1 |x| ) , ∣ ∣f t, x1 − f t, x2 ∣ ∣ ≤ c T t |x1 − x2|, ∣ ∣g t, x1 − g t, x2 ∣ ∣ ≤ c T t |x1 − x2|, 2.2 as 0 < δ ≤ |xi| ≤ N, i 1, 2, t ∈ 0, T , where c t and c T t are nonnegative functions such that ∫T 0 c t dt < ∞ and ∫T 0 c N T t dt < ∞. Then for any given x0 ∈ R, 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 ∈ R. Definition 2.3 see 31 . The equilibrium x ≡ 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 of ε ∈ 0, 1 and r > 0, there exists δ > 0 such that P{|x t, x0 | < r, for all t ≥ 0} ≥ 1 − ε, whenever |x| < δ. ii its stochastic settling-time function τ 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 , classK∞ functions μ1 and μ2, real numbers c > 0 and 0 < α < 1, such that μ1 |x| ≤ V x ≤ μ2 |x| , LV x ∂V ∂x f 1 2 Tr { g ∂2V ∂x2 g } ≤ −cV α x . 2.3 4 Abstract and Applied Analysis Then it is globally finite-time stable in probability and the stochastic settling time function τ 0, x0 satisfies E τ 0, x0 ≤ V 1−α x0 c 1 − α . 2.4 Lemma 2.5 see 25 . For any real numbers xi, i 1, . . . , n and 0 < b < 1, the following inequality holds: |x1| · · · |xn| b ≤ |x1| · · · |xn|, 2.5 when b p/q < 1, where p > 0 and q > 0 are odd integers, ∣ ∣ ∣x − y ∣ ∣ ∣ ≤ 21−b ∣ ∣x − y∣∣b. 2.6 Lemma 2.6 see 25 . Let c, d be positive real numbers and π x, y > 0 be a real-valued function. Then, |x|c∣∣y∣∣d ≤ cπ ( x, y )|x|c d c d dπ−c/d ( x, y )∣ ∣y ∣ ∣ c d


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][9][10][11][12][13][14][15] . Particularly, considering the unavoidability of stochastic disturbance, the asymptotic stabilization for stochastic nonholonomic systems was achieved in [16][17][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 disturbancerejection 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 finitetime 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 C 2 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. Section 2 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.

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 R n denotes the real n-dimensional space. For a given vector or matrix X, X T denotes its transpose, Tr{X} denotes its trace when X is square, and |X| is the Euclidean norm of a vector X. C i 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 where x ∈ R n is the system state with the initial condition x 0 x 0 , ω is an mdimensional independent standard Wiener process defined on a complete probability space Ω, F, {F t } t≥0 , P with Ω being a sample space, F being a σ-field, {F t } t≥0 being a filtration, and P being a probability measure. The functions: f : R × R n → R n and g : R × R n → R n×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 that f 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:  The equilibrium x ≡ 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 of ε ∈ 0, 1 and r > 0, there exists δ > 0 such that ii its stochastic settling-time function τ t 0 , x 0 exists finitely with probability and E τ 0, x 0 < ∞.

Lemma 2.4 see 32 .
Consider the stochastic nonlinear system described in 2.1 . Suppose there exists a C 2 function V x , class K ∞ functions μ 1 and μ 2 , real numbers c > 0 and 0 < α < 1, such that

4 Abstract and Applied Analysis
Then it is globally finite-time stable in probability and the stochastic settling time function τ 0, x 0 satisfies Lemma 2.5 see 25 . For any real numbers x i , i 1, . . . , n and 0 < b < 1, the following inequality holds: where p > 0 and q > 0 are odd integers, Lemma 2.6 see 25 . Let c, d be positive real numbers and π x, y > 0 be a real-valued function. Then,

Problem Formulation
In this paper, we focus our attention on the following class of stochastic nonholonomic systems: where x 0 ∈ R and x x 1 , . . . , x n T ∈ R n are system states, u 0 ∈ R and u 1 ∈ R are control inputs, respectively; . . , n represent the possible modeling error, refered to as disturbed virtual control coefficients; g i : R × R i → R m , i 1, . . . , n, are uncertain continuous functions satisfying g i 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 the x-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 Abstract and Applied Analysis 5 such that the stochastic finite-time regulation of closed-loop system states is achieved, that is, where T is a given settling time.
To achieve the above control objective, we need the following assumptions.
Assumption 3.3. For i 1, . . . , n, there are constants b and τ ∈ −2/ 4n 1 , 0 such that For simplicity, in this paper we assume τ −p/q with p being any even integer and q being any odd integer, under which and the definition of m i in Assumption 3.3, we know that m i is an odd number.
Remark 3.4. Noting that g i 0, 0 0 is assumed, Assumption 3.3 implies that 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 sin x that can be bounded by a function |x| m for any constant m ∈ 0, 1 and satisfies Assumption 3.3.

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 x 0 0 / 0 is considered first. Then, the case where the initial x 0 0 0 is dealt later. The inherently triangular structure of system 3.1 suggests that we should design the control inputs u 0 and u 1 in two separate stages.

4.1.
Control for x 0 0 / 0 For x 0 -subsystem, we take the following control law: where k 0 is a positive design parameter, and p, q are positive odd numbers.

Abstract and Applied Analysis
Taking the Lyapunov function V 0 x 2 0 /2, a simple computation gives Thus, x 0 tends to 0 within a settling time denoted by T 0 . Moreover, To secure finite-time convergence within T for any If we take T * c 01 T 0 / 2c 02 , then we obtain x 0 t ∈ R does not change its sign when t < T * , x 0 0 / 0 and moreover Therefore, u 0 is bounded and does not change sign during 0, T * . Furthermore, from this and Assumption 3.2, the following result can be obtained.
where α i −sgn x 0 0 . Besides, for the simplicity of expression in later use, let α n 1.
Since we have already proven that x 0 can be globally finite-time regulated to zero as t → T 0 . Next, we only need to stabilize the time-varying x-subsystem within the given settling time T * . The control law u 1 can be recursively constructed by applying the method of adding a power integrator.
Step 1. Let ξ 1 x σ 1 , where σ > 2 is a odd number and choose V 1 x 4σ−τ 1 / 4σ − τ to be the candidate Lyapunov function for this step. Then, along the trajectories of system 4.7 , we have

4.8
Abstract and Applied Analysis 7 Obviously, the first virtual controller with design constant M > 0, results in

4.10
Inductive Step (2 ≤ k ≤ n) Suppose at step k − 1, there is a C 2 Lyapunov function V k−1 , which is positive definite and proper, satisfying

4.11
and a set of virtual controllers x * 1 , . . . , x * k−1 defined by

4.12
with constants β 1 > 0, . . . , β k−1 > 0, such that We claim that 4.11 and 4.13 also hold at step k. To prove this claim, consider Noting that Similar to the corresponding proof in 25 , it is easy to verify that the Lyapunov function V k x k thus defined has several useful properties collected in the following propositions.

4.19
Using Proposition 4.2, it is deduced from 4.13 that Abstract and Applied Analysis

4.20
To estimate the second term in 4.20 , by Lemma 2.5, we have Noting that m k m k−1 τ, by applying 4.6 and Lemma 2.6, we have where l k1 is a positive constant. Based on Proposition 4.2 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.

4.24
Proposition 4.6. There exists a positive constant l k4 such that

4.25
Proposition 4.7. There exists a positive constant l k5 such that

4.26
Proposition 4.8. There exists a positive constant l k6 such that Substituting 4.22 -4.27 into 4.20 yields Clearly, the C 0 virtual controller

4.29
with β k > 0 being constant, results in 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 · · · ξ 4 n .

4.32
We have thus far completed the controller design procedure for x 0 0 / 0.

Control for x 0 0 0
In the last subsection, we gave the controller expressions 4.1 and 4.31 for u 0 and u 1 of system 3.1 in the case of x 0 0 / 0. Now, we consider finite-time control laws for the case of x 0 0 0. In the absence of the disturbances, most of the commonly used control strategies use constant control u 0 u * 0 / 0 in time interval 0, t s . In this paper, we also use this method when x 0 0 0, with u 0 chosen as follows: Since x 0 0 0, from the x 0 -subsystem we know thaṫ We have x 0 does not escape and x t s / 0, for given any finite t s > 0. During the time period 0, t s , using u 0 defined in 4.33 , new control law u 1 u * 1 x 0 , 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, t s . Since x 0 t s / 0 at t t s , we can switch the control inputs u 0 and u 1 to 4.1 and 4.31 , respectively.
The following theorem summarizes the main result of this paper. Proof. According to the above analysis, it suffices to prove the statement in the case where x 0 0 / 0. Since we have already proven that x 0 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 V n , from 4.32 , its time derivative is given by Hence with the choice of M satisfying M > 2V 1−α 0 / T * 1 − α , T x < T * is guaranteed. Thus, the conclusion follows. Remark 4.10. As seen from 4.31 and 4.1 , the control law u 1 may exhibit extremely large value when x 0 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 x 0 before we appeal to the finite-time converging controllers 4.1 and 4.31 .

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 byẋ where p * 1 and p * 2 are unknown parameters taking values in a known interval p min , p max with 0 < p min < p max < ∞, v and ω are two control inputs to denote the linear velocity and angular velocity, respectively.
When the angular velocity ω is subject to some stochastic disturbances, that is, where B t is the so-called white noise. Then system 5.1 is transformed into dx c p * 1 v cos θdt, dy c p * 1 v sin θdt, dθ p * 2 ω 1 dt p * 2 ω 2 dB.

5.3
For system 5.3 , by taking the following state and input transformation: we obtain 5.5 Clearly, system 5.5 is a special case of system 3.  where l 21 , l 22 and l 24 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.

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 finitetime state-feedback stabilizing controller for stochastic nonholonomic systems when the x 0subsystem 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.