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.

The nonholonomic systems cannot be stabilized by stationary continuous state feedback, although it is controllable, due to Brockett’s theorem [

It is well known that when the backstepping designs were firstly introduced, the stochastic nonlinear control had obtained a breakthrough [

This paper is organized as follows. In Section

In this paper, we consider a class of stochastic nonholonomic systems as follows:

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

Consider the following stochastic nonlinear system:

Given any

Let

Considering the stochastic nonlinear system (

For any real-valued continuous function

The purpose of this paper is to construct a smooth state-feedback control law such that the solution process of system (

To design the controller for system (

For

For

For each

For

Consider the Lyapunov function candidate

The

Clearly, from (

If

From the above analysis, the

In order to obtain the estimations for the nonlinear functions

For

We only prove (

To design a state-feedback controller, one introduces the coordinate transformation

For

The proof of Lemma

We now give the design process of the controller.

Consider the first Lyapunov function

Using Lemma

(

Then, define the

Consider

With the aid of (

Finally, when

In the preceding subsection, we have given controller design for

Now, we state the main results as follows.

Under Assumption

According to the above analysis, it suffices to prove in the case

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.

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.