Abstract and Applied Analysis

Volume 2013, Article ID 658050, 6 pages

http://dx.doi.org/10.1155/2013/658050

## Adaptive Exponential Stabilization for a Class of Stochastic Nonholonomic Systems

College of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, China

Received 21 July 2013; Revised 22 October 2013; Accepted 22 October 2013

Academic Editor: Mark McKibben

Copyright © 2013 Xiaoyan Qin. 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 adaptive stabilization problem for a class of stochastic nonholonomic systems with strong drifts. By using input-state-scaling technique, backstepping recursive approach, and a parameter separation technique, we design an adaptive state feedback controller. Based on the switching strategy to eliminate the phenomenon of uncontrollability, the proposed controller can guarantee that the states of closed-loop system are global bounded in probability.

#### 1. Introduction

The nonholonomic systems cannot be stabilized by stationary continuous state feedback, although it is controllable, due to Brockett’s theorem [1]. So the well-developed smooth nonlinear control theory and the method cannot be directly used in these systems. Many researchers have studied the control and stabilization of nonholonomic systems in the nonlinear control field and obtained some success [2–6]. It should be mentioned that many literatures consider the asymptotic stabilization of nonholonomic systems; the exponential convergence is also an important topic theme, which is demanded in many practical applications. However, the exponential regulation problem, particularly the systems with parameterization, has received less attention. Recently, [3] firstly introduced a class of nonholonomic systems with strong nonlinear uncertainties and obtained global exponential regulation. References [4, 5] studied a class of nonholonomic systems with output feedback control. Reference [6] combined the idea of combined input-state-scaling and backstepping technology, achieving the asymptotic stabilization for nonholonomic systems with nonlinear parameterization.

It is well known that when the backstepping designs were firstly introduced, the stochastic nonlinear control had obtained a breakthrough [7]. Based on quartic Lyapunov functions, the asymptotical stabilization control in the large of the open-loop system was discussed in [8]. Further research was developed by the recent work [9–16]. [17–19] studied a class of nonholonomic systems with stochastic unknown covariance disturbance. Since stochastic signals are very prevalent in practical engineering, the study of nonholonomic systems with stochastic disturbances is very significant. So, there exists a natural problem that is how to design an adaptive exponential stabilization for a class of nonholonomic systems with stochastic drift and diffusion terms. Inspired by these papers, we will study the exponential regulation problem with nonlinear parameterization for a class of stochastic nonholonomic systems. We use the input-state-scaling, the backstepping technique, and the switching scheme to design a dynamic state-feedback controller with; the closed-loop system is globally exponentially regulated to zero in probability.

This paper is organized as follows. In Section 2, we give the mathematical preliminaries. In Section 3, we construct the new controller and offer the main result. In the last section, we present the conclusions.

#### 2. Problem Statement and Preliminaries

In this paper, we consider a class of stochastic nonholonomic systems as follows: whereandare the system states andandare the control inputs, respectively.,, and;is an-dimensional standard Wiener process defined on the complete probability spacewithbeing a sample space,being a filtration, andbeing measure. The drift and diffusion termsare assumed to be smooth, vanishing at the origin;is the Borel bounded measurable functions and is nonnegative definite for each.are disturbed virtual control coefficients, where.

Next we introduce several technical lemmas which will play an important role in our later control design.

Consider the following stochastic nonlinear system: whereis the state of system (2), the Borel measurable functions:andare assumed to bein their arguments, andis an-dimensional standard Wiener process defined on the complete probablity space.

*Definition 1 (see [8]). *Given any, for stochastic nonlinear system (2), the differential operatoris defined as follows:
wheredenotes all nonnegative functionson, which areinandin, and for simplicity, the smooth functionis denoted by.

Lemma 2 (see [8]). *Letandbe real variables. Then, for any positive integers,, and any real number, the following inequality holds:
*

Lemma 3 (see [7]). *Considering the stochastic nonlinear system (2), if there exist afunction,class functionsand, constant, and a nonnegative functionssuch that
**
then for each. (1) For (2), there exists an almost surely unique solution on. (2) When,,, andis continuous, the equilibriumis globally stable in probability, and the solutionsatisfies(3) For any given, there exist a classfunctionandfunctionsuch thatfor any,.*

Lemma 4 (see [20]). *For any real-valued continuous function, there exist smooth scalar-value funcionsand, such that, and.*

#### 3. Controller Design and Analysis

The purpose of this paper is to construct a smooth state-feedback control law such that the solution process of system (1) is bounded in probability. For clarity, the case thatis firstly considered. Then, the case where the initialis dealt with later. The triangular structure of system (1) suggests that we should design the control inputsandin two separate stages.

To design the controller for system (1), the following assumptions are needed.

*Assumption 5. *For, there are some positive constantsandthat satisfy the inequality.

*Assumption 6. *For, there exists a nonnegative smooth function, such that.

For each, there exist nonnegative smooth functionsand, such that,.

##### 3.1. Designingfor-Subsystem

For-subsystem, the controlcan be chosen as whereandis a positive design parameter.

Consider the Lyapunov function candidate. From (6) and Assumptions 5 and 6, we have So, we obtain the first result of this paper.

Theorem 7. *The-subsystem, under the control law (6) with an appropriate choice of the parameters, is globally exponentially stable.*

*Proof. *Clearly, from (7),, which implies that. Therefore,is globally exponentially convergent. Consequently,can be zero only at, whenor. It is concluded thatdoes not cross zero for allprovided that.

*Remark 8. *If,exists and does not cross zero for allindependent of the-subsystem from (6).

##### 3.2. Backstepping Design for

From the above analysis, the-state in (1) can be globally exponentially regulated to zero as, obviously. In this subsection, we consider the control lawfor the-subsystem by using backstepping technique. To design a state-feedback controller, one first introduces the following discontinuous input-state-scaling transformation: Under the new-coordinates,-subsystems is transformed into where

In order to obtain the estimations for the nonlinear functionsand, the following Lemma can be derived by Assumption 6.

Lemma 9. *Forthere exist nonnegative smooth functions,, such that
*

*Proof. *We only prove (11). The proof of (12) is similar to that of (11). In view of (6), (8), (10) and Assumption 6, one obtains
where.

To design a state-feedback controller, one introduces the coordinate transformation
whereare smooth virtual control laws and will be designed later and.denotes the estimate of, where
Then using (9), (10), (14) and Itdifferentiation rule, one has
where,, andwhereUsing Lemmas 2, 4, and 9 and (14), we easily obtain the following lemma.

Lemma 10. *Forthere exist nonnegative smooth functions,, and, such that
*

The proof of Lemma 10 is similar to that of Lemma 9, so we omitted it.

We now give the design process of the controller.

*Step 1. *Consider the first Lyapunov function. By (14), (15), and (16), we have

Using Lemma 10 and Lemma 4, we have
Substituting (19) into (18) and using (14), we have
where. Substitutinginto (20), we have
where.

*Step i. *(). Assume that at step, there exists a smooth state-feedback virtual control, such that
where,, andwhere.

Then, define theth Lyapunov candidate function. From (16) and (22), it follows that
Using Lemmas 9 and 4, there are always known nonnegative smooth functions,,and constant, , where and .

Consider
where
where.
where.
where,, and.
where.
whereis a design parameter to be chosen.

With the aid of (24)–(29) and (14), (23) can be simplified as

Finally, when, is the actual control. By choosing the actual control law and the adaptive law,
whereis a design parameter to be chosen andare smooth functions; we get
where. We have finished the controller design procedure forand the parameter identification. Without loss of generality, we can assume that

##### 3.3. Switching Control and Main Result

In the preceding subsection, we have given controller design forNow, we discuss how to choose the control lawsandwhenWe chooseas. And choose the Lyapunov function. Its time derivative is given by, which leads to the bounds of. During the time period, using, new control lawcan be obtained by the control procedure described above to the original-subsystem in (1). Then, we can conclude that the-state of (1) cannot be blown up during the time period. Since atwe can switch the control inputsandto (6) and (31), respectively.

Now, we state the main results as follows.

Theorem 11. *Under Assumption 5, if the proposed adaptive controller (31) together with the above switching control strategy is used in (1), then for any initial contidion, the closed-loop system has an almost surely unique solution on, the solution process is bounded in probability, and.*

*Proof. *According to the above analysis, it suffices to prove in the caseSince we have already proven thatcan be globally exponentially convergent to zero in probability in Section 3.1, we only need prove thatis convergent to zero in probability also. In this case, we choose the Lyapunov function, and; from (32) and Lemma 3, we know that the closed-loop system has an almost surely unique solution on, and the solution process is bounded in probability.

#### 4. Conclusions

This paper investigates the globally exponential stabilization problem for a class of stochastic nonholonomic systems in chained form. To deal with the nonlinear parametrization problem, a parameter separation technique is introduced. With the help of backstepping technique, a smooth adaptive controller is constructed which ensures that the closed-loop system is globally asymptotically stable in probability. A further work is how to design the output-feedback tracking control for more high-order stochastic nonholonomic systems.

#### Acknowledgments

This work was supported by the university research projects of Department of Education in Shandong Province, China (J13LI03). The author would like to thank the reviewers for their helpful comments.

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