This paper investigates the problem of global finite-time stabilization by output feedback for a class of nonholonomic systems in chained form with uncertainties. By using backstepping recursive technique and the homogeneous domination approach, a constructive design procedure for output feedback control is given. Together with a novel switching control strategy, the designed controller renders that the states of closed-loop system are regulated to zero in a finite time. A simulation example is provided to illustrate the effectiveness of the proposed approach.

Over the past decade, nonholonomic systems have attracted much attention because they can be used to model many real systems, such as mobile robots, car-like vehicle, and under-actuated satellites. An important feature of a nonholonomic system is that the number of its inputs is less than the number of its degree of freedom, which makes the control problems of a nonholonomic system challenging. As pointed out by Brockett in [

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 [

To illustrate the difficulties in finite-time control of nonholonomic systems via output feedback, let us consider a problem of finite time stabilizing the following simple system at the origin:

In discontinuous approach, as seen, for example, in [

Next, we need to stabilize the

However, the system (

Motivated by the aforementioned discussion, in this paper we aim to tackle this challenging question and provide a solution to the problem of global finite-time output feedback stabilization for nonholonomic systems with uncertainties by applying the homogeneous domination approach. The main contribution of this paper is twofold. (i) Compared to the existing output feedback stabilization results for nonholonomic systems, the finite-time stabilizer proposed in this paper leads to faster convergence rate. (ii) As the common assumption to guarantee the existence of global finite-time output feedback stabilizer for a nonlinear system, the low-order growth (the order less than one) of system nonlinearities renders the discontinuous change of coordinates (i.e., the

The rest of this paper is organized as follows. Section

In this paper, we consider the following uncertain nonholonomic systems:

The objective of this paper is to design an output feedback controller in the form

To this end, the following assumptions regarding system (

For

For

For

For simplicity, in this paper we assume

Assumptions

The following definitions and lemmas will serve as the basis of the coming control design and performance analysis.

Consider a system

Consider the nonlinear system described in (

Then, the origin of system (

Weighted homogeneity: for fixed coordinates

The dilation

A function

A vector field

A homogeneous

Suppose that

There is a constant

For

If

Let

For

In this section, we give a constructive procedure for the finite-time stabilizer of system (

We first stabilize the

Then we design a controller such that the

For the

Noting that

Next we consider the finite-time output feedback stabilizer for system (

It is worth pointing out that, in terms of the transformation (

To construct a global output feedback controller for system (

In this subsection, we will construct an output feedback stabilizer for the following nominal system:

The design of output feedback controller is divided into two steps. In Step A, we suppose that all the states are measurable and develop a recursive design method to explicitly construct a state feedback control law for system (

Choose the Lyapunov function

In this step, we can obtain the following property.

For the

The detailed proof can be found in [

From the inductive steps, we can design

Considering

Each term on the right-hand side of (

There exists a positive constant

For

For the controller

For

Choosing

By (

There exists a positive constant

With the help of Proposition

Since

It can be shown that (

Combining (

It should be pointed out that the output feedback controller (

Together with the homogeneous controller and observer established previously, in this subsection we are ready to use the homogeneous domination approach to globally stabilize (

Now we construct an observer with a gain

In addition, we design

Now, the closed-loop system (

Hence, it can be concluded from (

From (

Noting that, for

With (

Substituting (

Furthermore, it can be deduced from (

By Lemma

From Section

Taking the Lyapunov function

Thus, by Lemma

Up to now, we have finished the finite-time output feedback stabilizing controller design of the system (

Under Assumptions

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

It should be mentioned that, when

For simplicity, it is assumed that

If we pick

Then, when

In the simulation, we assume

The responses of the closed-loop system (

Although system (

This paper has solved the problem of global finite-time output feedback stabilization for a class of nonholonomic systems in chained form with uncertainties. With the help of backstepping recursive technique and the homogeneous domination approach, a constructive design procedure for output feedback control is given. It is shown that the designed control laws can guarantee that the closed-loop system states are globally finite-time regulated to zero. In this direction, there are still remaining problems to be investigated. For example, an interesting research problem is how to design a finite-time output feedback stabilizing controller for nonholonomic systems in stochastic setting.

By Lemma

Using

By (

Similar to (

From

According to

By (

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. This work has been supported in part by the National Natural Science Foundation of China under Grant 61073065 and the Key Program of Science Technology Research of Education Department of Henan Province under Grant 13A120016.