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This paper considers the state feedback stabilization problem for a class of stochastic feedforward nonlinear systems. By using the homogeneous domination approach, a state feedback controller is constructed to render the closed-loop system globally asymptotically stable in probability. A simulation example is provided to show the effectiveness of the designed controller.

Consider the following stochastic feedforward nonlinear systems described by

Since the stochastic stability theory was established, the stabilization problems for stochastic lower-triangular nonlinear systems have made a great number of achievements in recent years; see, for example, [

Feedforward system is an another important class of nonlinear systems. From the theoretical viewpoint, they are not feedback linearizable and cannot be stabilized by the conventional backstepping method; to some extent, the control problem of these systems is more difficult than feedback systems. On the other hand, some simple physical models, for example, the cart-pendulum system in [

The purpose of this paper is to solve the state feedback stabilization problem of system (

The paper is organized as follows. Section

The following notations, definitions, and lemmas are to be used throughout the paper.

Consider the following stochastic nonlinear system:

For any given

For system (

For fixed coordinates

The dilation

A function

A vector field

A homogeneous

Consider system (

Given a dilation weight

Suppose that

there is a constant

Let

For system (

For

Obviously, system (

Due to the special form of stochastic feedforward system, almost all the existing methods fail to be applicable to solve the stabilization problem of system (

To achieve this objective, we first introduce the following coordinate transformation:

We construct a state feedback controller for the following nominal nonlinear system of (

Introducing

Suppose that at Step

From (

Using (

Choosing

At Step

We state the main result in this paper.

If Assumption

the closed-loop system has an almost surely unique solution on

the equilibrium at the origin of the closed-loop system is GAS in probability.

We prove Theorem

By Steps 1-2 and Lemma

This paper extends the homogeneous domination idea from deterministic systems to stochastic system (

Consider the following stochastic nonlinear system:

Now, we give the controller design of system (

In simulation, we choose the initial values

The responses of the closed-loop system (

In this paper, the homogeneous domination approach is introduced to solve the state feedback stabilization problem for the stochastic feedforward nonlinear system (

The authors would like to express their sincere gratitude to the editor and reviewers for their helpful suggestions in improving the quality of this paper. This work was partially supported by the National Natural Science Foundation of China (nos. 61304002, 61304003, 61203123, and 61304054), the Fundamental Research Funds for the Central Universities of China (no. 11CX04044A), the Shandong Provincial Natural Science Foundation of China (no. ZR2012FQ019), and Doctoral Start-up Fund of Bohai University.