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Abstract and Applied Analysis

Volume 2012 (2012), Article ID 361269, 18 pages

http://dx.doi.org/10.1155/2012/361269

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

School of Mathematics and Statistics, Anyang Normal University, Anyang 455002, Henan Province, China

Received 19 July 2012; Accepted 27 September 2012

Academic Editor: Ahmed El-Sayed

Copyright © 2012 Fangzheng Gao and Fushun Yuan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 with , 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.

#### 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 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.

#### 2. Notations and Preliminary Results

The following notations, definitions, and lemmas are to be used throughout the paper. denotes the set of all nonnegative real numbers and denotes the real -dimensional space. For a given vector or matrix , denotes its transpose, denotes its trace when is square, and is the Euclidean norm of a vector . denotes the set of all functions with continuous ith partial derivatives. denotes the set of all functions: , which are continuous, strictly increasing, and vanishing at zero; denotes the set of all functions which are of class and unbounded.

Consider the stochastic nonlinear system where is the system state with the initial condition is an -dimensional independent standard Wiener process defined on a complete probability space with being a sample space, being a -field, being a filtration, and being a probability measure. The functions: and are piecewise continuous and continuous with respect to the first and second arguments, respectively, and satisfy and .

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 and are continuous in . Further, for any , each , and each , if the following conditions hold:
**
as , where and are nonnegative functions such that and . Then for any given , system (2.1) has a pathwise unique strong solution.*

*Definition 2.2 (see [31]). * For system (2.1), define , for all , which is called the stochastic settling time function of system (2.1), where .

*Definition 2.3 (see [31]). *The equilibrium 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 and , there exists such that , whenever .(ii)its stochastic settling-time function exists finitely with probability and .

Lemma 2.4 (see [32]). *Consider the stochastic nonlinear system described in (2.1). Suppose there exists a function , class functions and , real numbers and , such that
**
Then it is globally finite-time stable in probability and the stochastic settling time function satisfies
*

Lemma 2.5 (see [25]). *For any real numbers , and , the following inequality holds:
**
when , where and are odd integers,
*

Lemma 2.6 (see [25]). *Let be positive real numbers and be a real-valued function. Then,
*

#### 3. Problem Formulation

In this paper, we focus our attention on the following class of stochastic nonholonomic systems:
where and are system states, and are control inputs, respectively; , ; represent the possible modeling error, refered to as disturbed virtual control coefficients; , are uncertain continuous functions satisfying ; and is an *m*-dimensional independent standard Wiener process defined on a complete probability space with being a sample space, being a filtration, and 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
such that the stochastic finite-time regulation of closed-loop system states is achieved, that is, and for any , where is a given settling time.

To achieve the above control objective, we need the following assumptions.

*Assumption 3.2. *For , there are known positive constants and such that

*Assumption 3.3. *For , there are constants and such that
where .

For simplicity, in this paper we assume with being any even integer and being any odd integer, under which and the definition of in Assumption 3.3, we know that is an odd number.

*Remark 3.4. *Noting that is assumed, Assumption 3.3 implies that
In fact, Assumption 3.3 is a generalization of the homogeneous growth condition introduced in [33] where and . 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 that can be bounded by a function for any constant 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 . For clarity, the case that is considered first. Then, the case where the initial is dealt later. The inherently triangular structure of system (3.1) suggests that we should design the control inputs and in two separate stages.

##### 4.1. Control for

For -subsystem, we take the following control law: where is a positive design parameter, and , are positive odd numbers.

Taking the Lyapunov function , a simple computation gives which implies .

Furthermore, we have Thus, tends to within a settling time denoted by . Moreover, To secure finite-time convergence within for any , we need to keep by taking . If we take , then we obtain does not change its sign when , and moreover

Therefore, is bounded and does not change sign during . Furthermore, from this and Assumption 3.2, the following result can be obtained.

Lemma 4.1. *For , there are positive constants and such that
**
where . Besides, for the simplicity of expression in later use, let .*

Since we have already proven that can be globally finite-time regulated to zero as . Next, we only need to stabilize the time-varying -subsystem within the given settling time . The control law can be recursively constructed by applying the method of adding a power integrator.

*Step 1. *Let , where is a odd number and choose to be the candidate Lyapunov function for this step. Then, along the trajectories of system (4.7), we have

Obviously, the first virtual controller
with design constant , results in

*Inductive Step ( 2 ≤ k ≤ n)*

Suppose at step , there is a Lyapunov function , which is positive definite and proper, satisfying and a set of virtual controllers defined by with constants , such that

We claim that (4.11) and (4.13) also hold at step . To prove this claim, consider where

Noting that where

Similar to the corresponding proof in [25], it is easy to verify that the Lyapunov function thus defined has several useful properties collected in the following propositions.

Proposition 4.2. * is . Moreover
**
where , .*

Proposition 4.3. * is , positive definite and proper, satisfying
*

Using Proposition 4.2, it is deduced from (4.13) that To estimate the second term in (4.20), by Lemma 2.5, we have Noting that , by applying (4.6) and Lemma 2.6, we have where 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.

Proposition 4.4. *There exists a positive constant such that
*

Proposition 4.5. *There exists a positive constant such that
*

Proposition 4.6. *There exists a positive constant such that
*

Proposition 4.7. *There exists a positive constant such that
*

Proposition 4.8. *There exists a positive constant such that
*

Substituting (4.22)–(4.27) into (4.20) yields Clearly, the virtual controller with being constant, results in

This completes the proof of the inductive step.

The inductive argument shows that (4.30) holds for . Hence, we conclude that at the last step the actual controller with being constant and a positive definite and proper Lyapunov function of the form (4.14) and (4.15), such that

We have thus far completed the controller design procedure for .

##### 4.2. Control for

In the last subsection, we gave the controller expressions (4.1) and (4.31) for and of system (3.1) in the case of . Now, we consider finite-time control laws for the case of . In the absence of the disturbances, most of the commonly used control strategies use constant control in time interval . In this paper, we also use this method when , with chosen as follows:

Since , from the -subsystem we know that We have does not escape and , for given any finite .

During the time period , using defined in (4.33), new control law can be obtained by the control procedure described above to the original -subsystem in (3.1). Then we can conclude that the -state of (3.1) cannot blow up during the time period . Since at , we can switch the control inputs and to (4.1) and (4.31), respectively.

The following theorem summarizes the main result of this paper.

Theorem 4.9. *Under Assumptions 3.2 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 , 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 where .

Since we have already proven that can be globally finite-time regulated to zero in Section 4.1, we just need to show that 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 , from (4.32), its time derivative is given by
Let , by Proposition 4.3 and Lemma 2.5, we have
Then, putting (4.36) back to (4.35) gives

By Lemma 2.4, system (4.7) under control law (4.31) is finite-time stable in probability with its settling time satisfying

Hence with the choice of satisfying , is guaranteed. Thus, the conclusion follows.

*Remark 4.10. *As seen from (4.31) and (4.1), the control law may exhibit extremely large value when 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 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 where and are unknown parameters taking values in a known interval with , and are two control inputs to denote the linear velocity and angular velocity, respectively.

When the angular velocity is subject to some stochastic disturbances, that is, where is the so-called white noise. Then system (5.1) is transformed into

For system (5.3), by taking the following state and input transformation: we obtain

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 . And as in [8], , , which are known for us in contructing control laws to make (5.5) finite-time stable, and parameters are unknown. When , by choosing and , according to the design procedure proposed in Section 4, the following controllers can be obtained for a given settling time : where , and are known positive constants. Choosing design parameter as , the simulation results in Figures 1 and 2 show that the effectiveness of the controller.

#### 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 the -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.

#### Appendix

*Proof of Proposition 4.4. *Recall that and . By Lemma 2.5, for ,
where .

With (4.17), (4.18), and , we get
where and are positive constants.

Applying Lemma 2.6 to , we can find a positive constant such that

*Proof of Proposition 4.5. *According to (4.17), (4.18), , and Remark 3.4, we have
where , and are positive constants.

Applying Lemma 2.6 to the above inequality, we have
where is a positive constant.

*Proof of Proposition 4.6. *From (4.17), (4.18), , and Remark 3.4, with Lemma 2.6, it is deduced that
where , , and are positive constants.

*Proof of Proposition 4.7. *From (4.17), (4.18), , and Remark 3.4, with Lemma 2.6, it follows that
where , , and are positive constants.

*Proof of Proposition 4.8. *By (4.17), (4.18), , and Remark 3.4, with Lemma 2.6, we can obtain
where , , and are positive constants.

#### Acknowledgments

This work has been supported in part by National Nature Science Foundation of China under Grant 61073065 and Nature Science Foundation of Henan Province under Grant 092300410145. The authors would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions for improving the quality of the paper.

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