Mathematical Problems in Engineering

Volume 2013, Article ID 908459, 6 pages

http://dx.doi.org/10.1155/2013/908459

## State Feedback Control for Stochastic Feedforward Nonlinear Systems

^{1}College of Engineering, Bohai University, Liaoning 121013, China^{2}College of Mathematics and Physics, Bohai University, Liaoning 121013, China

Received 13 October 2013; Accepted 13 December 2013

Academic Editor: Xudong Zhao

Copyright © 2013 Liang Liu and Ming Gao. 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 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.

#### 1. Introduction

Consider the following stochastic feedforward nonlinear systems described by where and are the system state and input, respectively. , . is an -dimensional standard Wiener process defined on the complete probability space with being a sample space, being a -field, being a filtration, and being a probability measure. and are assumed to be locally Lipschitz with and , .

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, [1–16] and the other references.

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 [17] and the ball-beam with a friction term in [18], can be described by equations with the feedforward form. In recent papers on feedforward systems, [19] studied delay-adaptive feedback for linear systems. The input delay compensation for forward complete and strict-feedforward nonlinear systems was solved by [20]. Reference [21] considered the adaptive stabilization problem for feedforward nonlinear systems with time delays by taking a nested saturation feedback. The global output feedback stabilization problem for system (1) without stochastic noise was addressed by [22]. Reference [23] investigated the state and output feedback control for a class of feedforward nonlinear time-delay systems. For high-order nonlinear feedforward systems, [24] considered global stabilization problem by using the generalized adding a power integrator method and a series of nested saturation functions, [25, 26] respectively dealt with the state feedback control for this kind of systems with time delay, but all these results are limited to deterministic systems. Due to the special form of this system, there are few results on stochastic feedforward systems at present.

The purpose of this paper is to solve the state feedback stabilization problem of system (1) by using the homogeneous domination approach in [22]. The underlying idea of this approach is that the homogeneous controller is first developed without considering the drift and diffusion terms, and then a low gain is introduced to the state feedback controller to dominate the drift and diffusion terms. By adopting this method, a state feedback controller is explicitly constructed to render the closed-loop system globally asymptotically stable in probability.

The paper is organized as follows. Section 2 provides some preliminary results. The design and analysis of state feedback controller is given in Sections 3 and 4, following a simulation example in Section 5. Section 6 concludes this paper.

#### 2. Preliminary Results

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

denotes the set of all nonnegative real numbers and denotes the real -dimensional space. For a given vector or matrix , denotes its transpose, denotes its trace when is square, and is the Euclidean norm of a vector . denotes the set of all functions with continuous th partial derivatives. denotes the set of all functions: , which are continuous, strictly increasing, and vanishing at zero; denotes the set of all functions which are of class and unbounded; denotes the set of all functions : , which are of for each fixed and decrease to zero as for each fixed .

Consider the following stochastic nonlinear system: where is the system state and is an -dimensional standard Wiener process defined on the complete probability space . The Borel measurable functions and are locally Lipschitz with and .

*Definition 1 (see [1]). *For any given associated with stochastic system (2), the differential operator is defined as

*Definition 2 (see [1]). *For system (2), the equilibrium is globally asymptotically stable (GAS) in probability if for any , there exists a class function such that for any and .

*Definition 3 (see [22]). *For fixed coordinates and real numbers , .(i)The dilation is defined by for any and are called the weights of the coordinates. For simplicity, we define dilation weight as .(ii)A function is said to be homogeneous of degree if there is a real number such that for any , .(iii)A vector field is said to be homogeneous of degree if there is a real number such that for any , , .(iv)A homogeneous -norm is defined as for any , where is a constant. For simplicity, in this paper, we choose and write for .

Lemma 4 (see [1]). *Consider system (2) and suppose that there exist a function , class functions and , and a class function such that
**
Then there exists an almost surely unique solution on , the equilibrium is GAS in probability, and .*

Lemma 5 (see [22]). *Given a dilation weight , suppose that and are homogeneous functions of degrees and , respectively. Then is also homogeneous with respect to the same dilation weight . Moreover, the homogeneous degree of is .*

Lemma 6 (see [22]). *Suppose that is a homogeneous function of degree with respect to the dilation weight ; then*(i)* is homogeneous of degree with being the homogeneous weight of ;*(ii)*there is a constant such that . Moreover, if is positive definite, then , where is a positive constant.*

Lemma 7 (see [4]). *Let and be positive constants. For any positive number , then .*

#### 3. Design of State Feedback Controller

##### 3.1. Assumption

For system (1), we need the following Assumption.

*Assumption 8. *For , there exist positive constants and such that
where .

*Remark 9. *Obviously, system (1) satisfying Assumption 8 is a stochastic feedforward nonlinear system. As discussed in the deterministic feedforward references such as [21–26], that is, for , and stochastic feedforward reference [27], Assumption 8 is a general and frequently used condition.

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 (1). Based on this reason, the objective of this paper is to design a state feedback controller for system (1) such that the equilibrium of the closed-loop system is globally asymptotically stable in probability.

To achieve this objective, we first introduce the following coordinate transformation:
where is a designed constant. With the help of (6), (1) can be rewritten as
where and , .

##### 3.2. State Feedback Control of Nominal Nonlinear System

We construct a state feedback controller for the following nominal nonlinear system of (7):

*Step 1. *Introducing and choosing , by (3) and (8), it can be verified that . The first virtual controller
leads to .*Step * *.* In this step, one can obtain the similar property for the th subsystem, which is presented by the following lemma.

Lemma 10. *Suppose that at Step there are a set of virtual controllers defined by
**
such that the th Lyapunov function satisfies
**
where , , , are positive constants. Then there exists a virtual control law such that
**
where .*

* Proof. *From (3), (8), (10), and (11), it follows that
We concentrate on the last two terms on the right-hand side of (13).

Using (10) and Lemma 7, one obtains
where , , , and are positive constants, .

Choosing
and substituting (14)-(15) into (13), one gets the desired result.

At Step , choosing and by (3), (12), and (16), one gets where , , , and , are positive constants. The system (7) and (16) can be written as where , , , and . Introducing the dilation weight , by (10) and , one obtains from which and Definition 3, we know that is homogeneous of degree 4.

#### 4. Stability Analysis

We state the main result in this paper.

Theorem 11. *If Assumption 8 holds for the stochastic feedforward nonlinear system (1), under the state feedback controller and (16), then*(i)*the closed-loop system has an almost surely unique solution on ;*(ii)*the equilibrium at the origin of the closed-loop system is GAS in probability.*

* Proof. *We prove Theorem 11 by three steps.*Step 1.* Since and are assumed to be locally Lipschitz, so the system consisting of (7) and (16) satisfies the locally Lipschitz condition. *Step 2.* By Lemma 6 and (17), there exists a positive constant such that
By Assumption 8, (6), (16), and , one has
where is a positive constant. According to Lemmas 5–6 and (21), one obtains
where is a positive constant. Similar to (21), there is a positive constant such that
from which and Lemmas 5-6, one leads to
where is a positive constant. By (3), (18), (20), (22), and (24), one has
Since is a constant independent of and , by choosing
(25) becomes , where is a positive constant.

By Steps 1-2 and Lemma 4, the system consisting of (7) and (16) has an almost surely unique solution on , is GAS in probability, and .*Step 3.* Since (6) is an equivalent transformation, the closed-loop system consisting of (1), , and (16) has the same properties as the system (7) and (16). Theorem 11 holds.

*Remark 12. *This paper extends the homogeneous domination idea from deterministic systems to stochastic system (1) and explicitly constructs a state feedback controller. It should be emphasized that the rigorous proof of Theorem 11 is not an easy work.

#### 5. A Simulation Example

Consider the following stochastic nonlinear system: It is easy to verify that Assumption 8 is satisfied with and .

Now, we give the controller design of system (27). Introducing the coordinate transformation system (27) becomes Choosing and , we obtain , where . By and , a direct calculation leads to By Lemma 7, one has Choosing and substituting (31) into (30), it leads to By (28) and (32), one obtains the actual controller Defining , one gets . From (22) and (24), it follows that By (29), one has , from which one obtains .

In simulation, we choose the initial values , , and . Figure 1 demonstrates the effectiveness of the state feedback controller.

#### 6. A Concluding Remark

In this paper, the homogeneous domination approach is introduced to solve the state feedback stabilization problem for the stochastic feedforward nonlinear system (1). There still exist some problems to be investigated. One is to consider the more general switched stochastic feedforward nonlinear systems by adopting average dwell time method in [28]. Another is to consider stochastic feedforward networked or fuzzy systems (similar to [29–33]).

#### Acknowledgments

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.

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