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This paper investigates the problem of adaptive output feedback stabilization for a class of nonholonomic systems with nonlinear parameterization and strong nonlinear drifts. A parameter separation technique is introduced to transform nonlinearly parameterized system into a linear-like parameterized system. Then, by using the integrator backstepping approach based on observer and parameter estimator, a constructive design procedure for output feedback adaptive control is given. And a switching strategy is developed to eliminate the phenomenon of uncontrollability. It is shown that, under some conditions, the proposed controller can guarantee that all the system states globally converge to the origin, while other signals remain bounded. An illustrative example is also provided to demonstrate the effectiveness of the proposed scheme.

Control of nonholonomic systems has received a great deal of attention over the last few years. It has been shown in [

However, it should be noticed that most of these papers were concerned with the systems with linear parameterization. Since nonlinear parameterization is exceptionally difficult to estimate, there are very few reports in literature for adaptive control of nonlinearly parameterized nonholonomic systems. Exceptions include [

In this paper, the case of only partial state vector being measurable and adaptive control for chained form systems with nonlinear parameterization by output feedback is considered. The contributions of this paper are listed as follows: (i) a new observer design method is proposed, and, based on the observer, the unmeasurable states of the system involved are reconstructed; (ii) using parameter separation technique [

The rest of this paper is organized as follows. In Section

In this paper, we present an output feedback adaptive control design procedure for a class of uncertain chained form systems with nonlinear parameterization:

The objective of this paper is to design an output feedback adaptive stabilization control in the form

A full characterization of the class of system (

For

For

Assumptions

For any real-valued continuous function

By Assumption

For

In this section, we focus on designing the control input

Let

Note the control

Consider the Lyapunov function candidate

From (

Choose the adaptation law

Therefore,

Under the control law (

Consequently,

From the above analysis, we can see the

We design the following observer for the system (

The estimation error

The eigenvalues of the matrix A defined by (

The proof can be found in [

By Assumption

We now turn to the constructive design procedure of the control.

This step can be regarded as the initial assignation of the entire design procedure. At this step, we introduce a Lyapunov function for the estimation error

Then, taking time-derivation of

Substituting (

Take

Suppose at step

We intend to establish a similar property for

Since

After lengthy but simple calculations based on the completion of squares, there is a smooth nonnegative function

Using (

By Lemma

Based on the completion of squares, it is deduced that there is a smooth function

Substituting (

Moreover, (

Putting (

Now, it easy to see that the smooth virtual controller

As

Therefore, by choosing the smooth actual control

In Section

In the preceding section, we have given controller design for

Choosing the same Lyapunov function (

During the time period

We are now ready to state the main theorem of our paper.

Under Assumption

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

Since we have already proven that

As seen from (

To verify our proposed controller, we consider the following low-dimensional system with parametric uncertainty:

If

For simplicity, in the first subsystem, we can choose the control law

Solving the matrix equation

Using the method given in Section

The unknown parameter

System states.

Observer states.

Parameter estimate.

Control inputs.

In this paper, a constructive adaptive output feedback control strategy is presented for a class of nonlinearly parameterized nonholonomic systems with strong nonlinear drifts. To deal with the nonlinear parameterization problem, a parameter separation technique is introduced to transform the nonlinear parameterized nonholonomic system into a linear-like parameterized nonholonomic system. We estimate only

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the editor and the anonymous reviewers for their constructive comments and suggestions for improving the quality of the paper. This work has been supported in part by National Nature Science Foundation of China under Grant no. 61073065 and the Key Program of Science Technology Research of Education Department of Henan Province under Grants nos. 13A120016, 14A520003.