State Feedback Control for Stochastic Feedforward Nonlinear Systems

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.

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.

Mathematical Problems in Engineering
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.

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, Tr{} denotes its trace when  is square, and || is the Euclidean norm of a vector .C  denotes the set of all functions with continuous th partial derivatives.K denotes the set of all functions:  + →  + , which are continuous, strictly increasing, and vanishing at zero; K ∞ denotes the set of all functions which are of class K and unbounded; KL denotes the set of all functions (, ):  + ×  + →  + , which are of K 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 (Ω, F, {F  } ≥0 , ).The Borel measurable functions :   →   and :   →  × are locally Lipschitz with (0) = 0 and (0) = 0.
(iv) A homogeneous -norm is defined as for any  ∈   , where  ≥ 1 is a constant.For simplicity, in this paper, we choose  = 2 and write ‖‖ Δ for ‖‖ Δ,2 .

Design of State Feedback Controller
where  +1 = 0.
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 closedloop system is globally asymptotically stable in probability.

State Feedback Control of Nominal Nonlinear
System.We construct a state feedback controller for the following nominal nonlinear system of (7): Step 1. Introducing  1 =  1 and choosing  1 ( 1 ) = (1/4) 4  1 , by ( 3) and ( 8), it can be verified that L 1 =  3  1  2 .The first virtual controller Step  ( = 2, . . ., ).In this step, one can obtain the similar property for the th subsystem, which is presented by the following lemma.

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  =   V and (16), then (i) the closed-loop system has an almost surely unique solution on [0, ∞); (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.
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.

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][30][31][32][33]).