# State-Feedback Stabilization for a Class of Stochastic Feedforward Nonlinear Time-Delay Systems

Authors: Liang Liu; Zhandong Yu; Qi Zhou; Hamid Reza Karimi
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate the state-feedback stabilization problem for a class of stochastic
feedforward nonlinear time-delay systems. By using the homogeneous domination approach and choosing
an appropriate Lyapunov-Krasovskii functional, the delay-independent state-feedback controller is
explicitly constructed such that the closed-loop system is globally asymptotically stable in probability.
A simulation example is provided to demonstrate the effectiveness of the proposed design method.


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## Body

## 1. Introduction

In recent years, the study on stochastic lower-triangular nonlinear systems has received considerable attention from both theoretical and practical point of views see, for instance, [1–19] and the references therein. This paper will further consider the following stochastic feedforward nonlinear time-delay systems described by
(1)dx1=x2dt+f1(x~3,x~3(t-d(t)))dt+g1T(x~2,x~2(t-d(t)))dω,⋮dxn-2=xn-1dt+fn-2(x~n,x~n(t-d(t)))dt+gn-2T(x~n-1,x~n-1(t-d(t)))dω,dxn-1=xndt+gn-1T(x~n,x~n(t-d(t)))dω,dxn=udt,
where x=(x1,…,xn)T∈Rn and u∈R are the system state and input signal, respectively, x~i=(xi,…,xn)T, x~i(t-d(t))=(xi(t-d(t)),…,xn(t-d(t)))T is the time-delayed state vector, and d(t):R+→[0,d] is the time-varying delay. ω is an m-dimensional standard Wiener process defined on the complete probability space (Ω,ℱ,{ℱt}t≥0,P) with Ω being a sample space, ℱ being a σ-field, {ℱt}t≥0 being a filtration, and P being a probability measure. fi:Rn-i-1×Rn-i-1→R and gj:Rn-j×Rn-j→Rm are assumed to be locally Lipschitz with fi(0,0)=0 and gj(0,0)=0, i=1,…,n-2, j=1,…,n-1.Feedforward (also called upper-triangular) system is another important class of nonlinear systems. Firstly, from a theoretical point of view, since they are not feedback linearizable and maybe not stabilized by applying the conventional backstepping method, the stabilization problem of these systems is more difficult than that of lower-triangular systems. Secondly, many physical devices, such as the cart-pendulum system in [20] and the ball-beam system with a friction term in [21], can be described by equations with the feedforward structure. In the recent papers, the stabilization problems for feedforward nonlinear (or time-delay) systems have achieved remarkable development; see, for example, [22–29] and the references therein.However, all these above-mentioned results are limited to deterministic systems. There are fewer results on stochastic feedforward nonlinear systems until now, due to the special characteristics of this system. To the best of the authors’ knowledge, [30] is the only paper to consider this kind of stochastic feedforward nonlinear systems, but the assumptions on the nonlinearities are restrictive.The purpose of this paper is to further weaken the assumptions on the drift and diffusion terms of system (1) and solve the state-feedback stabilization problem. By using the homogeneous domination approach in [26] and choosing an appropriate Lyapunov-Krasovskii functional, a delay-independent state-feedback controller is explicitly constructed such that the closed-loop system is globally asymptotically stable in probability.The paper is organized as follows. Section2 provides some preliminary results. The design and analysis of state-feedback controller are given in Sections 3 and 4, respectively, 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.R
+ denotes the set of all nonnegative real numbers, and Rn denotes the real n-dimensional space. For a given vector or matrix X, XT denotes its transpose, Tr{X} denotes its trace when X is square, and |X| is the Euclidean norm of a vector X. 𝒞([-d,0];Rn) denotes the space of continuous Rn-value functions on [-d,0] endowed with the norm ∥·∥ defined by ∥f∥=supx∈[-d,0]|f(x)| for f∈𝒞([-d,0];Rn); 𝒞ℱ0b([-d,0];Rn) denotes the family of all ℱ0-measurable bounded 𝒞([-d,0];Rn)-valued random variables ξ={ξ(θ):-d≤θ≤0}. 𝒞i denotes the set of all functions with continuous ith partial derivatives; 𝒞2,1(Rn×[-d,∞);R+) denotes the family of all nonnegative functions V(x,t) on Rn×[-d,∞) which are 𝒞2 in x and 𝒞1 in t; 𝒞2,1 denotes the family of all functions which are 𝒞2 in the first argument and 𝒞1 in the second argument. 𝒦 denotes the set of all functions R+→R+, which are continuous, strictly increasing, and vanishing at zero; 𝒦∞ denotes the set of all functions which are of class 𝒦 and unbounded; 𝒦ℒ is the set of all functions β(s,t): R+×R+→R+, which are of 𝒦 for each fixed t and decrease to zero as t→∞ for each fixed s.Consider the following stochastic time-delay system:(2)dx(t)=f(x(t),x(t-d(t)),t)dt+g(x(t),x(t-d(t)),t)dω,hhhhhhhhihhhhhhhhh∀t≥0,
with initial data {x(θ):-d≤θ≤0}=ξ∈𝒞ℱ0b([-d,0];Rn), where d(t):R+→[0,d] is a Borel measurable function, ω is an m-dimensional standard Wiener process defined on the complete probability space (Ω,ℱ,{ℱt}t≥0,P), and f:Rn×Rn×R+→Rn and g:Rn×Rn×R+→Rn×m are locally Lipschitz in (x(t), x(t-d(t))) uniformly in t with f(0,0,t)≡0 and g(0,0,t)≡0.Definition 1 (see [6]).
For any givenV(x(t),t)∈𝒞2,1 associated with system (2), the differential operator ℒ is defined as
(3)ℒV=∂V∂t+∂V∂xf+12Tr{gT∂2V∂x2g}.Definition 2 (see [6]).
The equilibriumx(t)=0 of system (2) is said to be globally asymptotically stable (GAS) in probability if for any ϵ>0 there exists a function β(·,·)∈𝒦ℒ such that P{|x(t)|≤β(∥ξ∥,t)}≥1-ϵ for any t≥0, ξ∈𝒞ℱ0b([-d,0];Rn)∖{0}, where ∥ξ∥=supθ∈[-d,0]|x(θ)|.Definition 3 (see [26]).
For fixed coordinates(x1,…,xn)T∈Rn and real numbers ri>0, i=1,…,n, one has the following.(i)
The dilationΔɛ(x) is defined by Δɛ(x)=(ɛr1x1,…,ɛrnxn) for any ɛ>0; r1,…,rn are called as the weights of the coordinates. For simplicity, we define dilation weight Δ=(r1,…,rn).(ii)
A functionV∈𝒞(Rn,R) is said to be homogeneous of degree τ if there is a real number τ∈R such that V(Δɛ(x))=ɛτV(x1,…,xn) for any x∈Rn∖{0}, ɛ>0.(iii)
A vector fieldh∈𝒞(Rn,Rn) is said to be homogeneous of degree τ if there is a real number τ∈R such that hi(Δɛ(x))=ɛτ+rihi(x) for any x∈Rn∖{0}, ɛ>0, i=1,…,n.(iv)
A homogeneousp-norm is defined as ∥x∥Δ,p=(∑i=1n‍|xi|p/ri)1/p for any x∈Rn, where p≥1 is a constant. For simplicity, in this paper, one chooses p=2 and writes ∥x∥Δ for ∥x∥Δ,2.Lemma 4 (see [6]).
For system (2), if there exist a function V(x(t),t)∈𝒞2,1(Rn×[-d,∞);R+), two class 𝒦∞ functions α1, α2, and a class 𝒦 function α3 such that
(4)α1(|x(t)|)≤V(x(t),t)≤α2(sup-d≤s≤0|x(t+s)|),ℒV(x(t),t)≤-α3(|x(t)|),
then there exists a unique solution on [-d,∞) for (2), the equilibrium x(t)=0 is GAS in probability, and P{limt→∞|x(t)|=0}=1.Lemma 5 (see [26]).
Given a dilation weightΔ=(r1,…,rn), suppose that V1(x) and V2(x) are homogeneous functions of degrees τ1 and τ2, respectively. Then V1(x)V2(x) is also homogeneous with respect to the same dilation weight Δ. Moreover, the homogeneous degree of V1·V2 is τ1+τ2.Lemma 6 (see [26]).
Suppose thatV:Rn→R is a homogeneous function of degree τ with respect to the dilation weight Δ; then (i) ∂V/∂xi is homogeneous of degree τ-ri with ri being the homogeneous weight of xi; (ii) there is a constant c such that V(x)≤c∥x∥Δτ. Moreover, if V(x) is positive definite, then V(x)≥c_∥x∥Δτ, where c_ is a positive constant.Lemma 7 (see [5]).
Letc and d be positive constants. For any positive number γ¯, then |x|c|y|d≤(c/(c+d))γ¯|x|c+d+(d/(c+d))γ¯-c/d|y|c+d.


## 3.  Design of State-Feedback Controller

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.


### 3.1. Assumptions

For system (1), we need the following assumptions.Assumption 8.
Fori=1,…,n-1, there exist positive constants a1 and a2 such that
(5)|fi(x~i+2,x~i+2(t-d(t)))|≤a1(∑j=i+2n‍|xj|+∑j=i+2n‍|xj(t-d(t))|),|gi(x~i+1,x~i+1(t-d(t)))|≤a2(∑j=i+1n‍|xj|+∑j=i+1n‍|xj(t-d(t))|),
where xn+1=xn+1(t-d(t))=0.Assumption 9.
The time-varying delayd(t) satisfies d˙(t)≤γ<1 for a constant γ.Remark 10.
Whenxi+1=xi+1(t-d(t))=0 in diffusion term gi(i=1,…,n-1), Assumption 8 reduces to the same form as in [30], from which one can see that system (1) is more general than [30]. The significance and reasonability of Assumption 8 are illustrated in that paper.
Firstly, we introduce the following coordinate transformation:(6)ηi=xiκi-1,v=uκn,i=1,…,n,
where 0<κ<1 is a scalar to be designed. By (6), (1) can be expressed as
(7)dη1=κη2dt+f¯1(η~3,η~3(t-d(t)))dt+g¯1T(η~2,η~2(t-d(t)))dω,⋮dηn-2=κηn-1dt+f¯n-2(η~n,η~n(t-d(t)))dt+g¯n-2T(η~n-1,η~n-1(t-d(t)))dω,dηn-1=κηndt+g¯n-1T(η~n,η~n(t-d(t)))dω,dηn=κvdt,
where f¯i=fi/κi-1, g¯i=gi/κi-1, i=1,…,n-1, f¯n-1=0.


### 3.2. State-Feedback Controller Design

We construct a state-feedback controller for system (7).Step 1.
Introducingξ1=η1 and choosing V1(η1)=(1/4)ξ14, from (3) and (7), it follows that
(8)ℒV1=κξ13η2+∂V1∂η1f¯1+12Tr{g¯1∂2V1∂η12g¯1T}.
The first virtual controller(9)η2*=-c11ξ1=:-α1ξ1,c11>0,
leads to ℒV1≤-κc11ξ14+κξ13(η2-η2*)+(∂V1/∂η1)f¯1+(1/2)Tr{g¯1(∂2V1/∂η12)g¯1T}.Step i (i=2,…,n).
In this step, we can get the following lemma.Lemma 11.
Suppose that at stepi-1, there is a set of virtual controllers η1*,…,ηi* defined by
(10)η1*=0,ξ1=η1-η1*=η1,ηk*=-αk-1ξk-1,ξk=ηk-ηk*,k=2,…,i,
such that the (i-1)th Lyapunov function Vi-1(η¯i-1)=(1/4)∑j=1i-1‍ξj4 satisfies
(11)ℒVi-1≤-κ∑j=1i-1‍ci-1,jξj4+κξi-13(ηi-ηi*)+∑j=1i-1‍∂Vi-1∂ηjf¯j+12∑p,q=1i-1‍Tr{g¯p∂2Vi-1∂ηp∂ηqg¯qT},
where αj, ci-1,j, j=1,…,i-1, are positive constants. Then there exists a virtual control law ηi+1*=-αiξi such that
(12)ℒVi≤-κ∑j=1i‍cijξj4+κξi3(ηi+1-ηi+1*)+∑j=1i‍∂Vi∂ηjf¯j+12∑p,q=1i‍Tr{g¯p∂2Vi∂ηp∂ηqg¯qT},
where Vi(η¯i)=(1/4)∑j=1i‍ξj4=:Vi-1(η¯i-1)+Wi(η¯i).Proof.
See the Appendix.At stepn, choosing Vn(η¯n)=(1/4)∑i=1n‍ξi4 and
(13)v=ηn+1*=-αnξn=-(α¯nηn+α¯n-1ηn-1+⋯+α¯1η1),
with the help of (3), (12), and (13), one obtains
(14)ℒVn≤-κ∑i=1n‍cniξi4+κξn3(v-ηn+1*)+∑i=1n‍∂Vn∂ηif¯i+12∑p,q=1n‍Tr{g¯p∂2Vn∂ηp∂ηqg¯qT}=-κ∑i=1n‍cniξi4+∂Vn∂ηF+12Tr{G∂2Vn∂η2GT},
where F=(f¯1,…,f¯n-2,0,0)T, G=(g¯1,…,g¯n-1,0), ξn=ηn-ηn*, α¯i=αn⋯αi, cni, i=1,…,n, are positive constants. The system (7) and (13) can be written as
(15)dη=κE(η)dt+F(η,η(t-d(t)))dt+GT(η,η(t-d(t)))dω,
where η=η¯n=(η1,…,ηn)T, E(η)=(η2,…,ηn,v)T, and F and G are defined as in (14). Introducing the dilation weight Δ=(1,1,…,1︸forη1,…,ηn), by (10) and Vn(η)=(1/4)∑i=1n‍ξi4, one has
(16)Vn(Δɛ(η))=14∑i=1n‍(ɛηi+αi-1ɛηi-1+⋯+αi-1⋯α1ɛη1)4=ɛ4Vn(η),
from which and Definition 3, we know that Vn(η) is homogeneous of degree 4.


## 3.1. Assumptions

For system (1), we need the following assumptions.Assumption 8.
Fori=1,…,n-1, there exist positive constants a1 and a2 such that
(5)|fi(x~i+2,x~i+2(t-d(t)))|≤a1(∑j=i+2n‍|xj|+∑j=i+2n‍|xj(t-d(t))|),|gi(x~i+1,x~i+1(t-d(t)))|≤a2(∑j=i+1n‍|xj|+∑j=i+1n‍|xj(t-d(t))|),
where xn+1=xn+1(t-d(t))=0.Assumption 9.
The time-varying delayd(t) satisfies d˙(t)≤γ<1 for a constant γ.Remark 10.
Whenxi+1=xi+1(t-d(t))=0 in diffusion term gi(i=1,…,n-1), Assumption 8 reduces to the same form as in [30], from which one can see that system (1) is more general than [30]. The significance and reasonability of Assumption 8 are illustrated in that paper.
Firstly, we introduce the following coordinate transformation:(6)ηi=xiκi-1,v=uκn,i=1,…,n,
where 0<κ<1 is a scalar to be designed. By (6), (1) can be expressed as
(7)dη1=κη2dt+f¯1(η~3,η~3(t-d(t)))dt+g¯1T(η~2,η~2(t-d(t)))dω,⋮dηn-2=κηn-1dt+f¯n-2(η~n,η~n(t-d(t)))dt+g¯n-2T(η~n-1,η~n-1(t-d(t)))dω,dηn-1=κηndt+g¯n-1T(η~n,η~n(t-d(t)))dω,dηn=κvdt,
where f¯i=fi/κi-1, g¯i=gi/κi-1, i=1,…,n-1, f¯n-1=0.


## 3.2. State-Feedback Controller Design

We construct a state-feedback controller for system (7).Step 1.
Introducingξ1=η1 and choosing V1(η1)=(1/4)ξ14, from (3) and (7), it follows that
(8)ℒV1=κξ13η2+∂V1∂η1f¯1+12Tr{g¯1∂2V1∂η12g¯1T}.
The first virtual controller(9)η2*=-c11ξ1=:-α1ξ1,c11>0,
leads to ℒV1≤-κc11ξ14+κξ13(η2-η2*)+(∂V1/∂η1)f¯1+(1/2)Tr{g¯1(∂2V1/∂η12)g¯1T}.Step i (i=2,…,n).
In this step, we can get the following lemma.Lemma 11.
Suppose that at stepi-1, there is a set of virtual controllers η1*,…,ηi* defined by
(10)η1*=0,ξ1=η1-η1*=η1,ηk*=-αk-1ξk-1,ξk=ηk-ηk*,k=2,…,i,
such that the (i-1)th Lyapunov function Vi-1(η¯i-1)=(1/4)∑j=1i-1‍ξj4 satisfies
(11)ℒVi-1≤-κ∑j=1i-1‍ci-1,jξj4+κξi-13(ηi-ηi*)+∑j=1i-1‍∂Vi-1∂ηjf¯j+12∑p,q=1i-1‍Tr{g¯p∂2Vi-1∂ηp∂ηqg¯qT},
where αj, ci-1,j, j=1,…,i-1, are positive constants. Then there exists a virtual control law ηi+1*=-αiξi such that
(12)ℒVi≤-κ∑j=1i‍cijξj4+κξi3(ηi+1-ηi+1*)+∑j=1i‍∂Vi∂ηjf¯j+12∑p,q=1i‍Tr{g¯p∂2Vi∂ηp∂ηqg¯qT},
where Vi(η¯i)=(1/4)∑j=1i‍ξj4=:Vi-1(η¯i-1)+Wi(η¯i).Proof.
See the Appendix.At stepn, choosing Vn(η¯n)=(1/4)∑i=1n‍ξi4 and
(13)v=ηn+1*=-αnξn=-(α¯nηn+α¯n-1ηn-1+⋯+α¯1η1),
with the help of (3), (12), and (13), one obtains
(14)ℒVn≤-κ∑i=1n‍cniξi4+κξn3(v-ηn+1*)+∑i=1n‍∂Vn∂ηif¯i+12∑p,q=1n‍Tr{g¯p∂2Vn∂ηp∂ηqg¯qT}=-κ∑i=1n‍cniξi4+∂Vn∂ηF+12Tr{G∂2Vn∂η2GT},
where F=(f¯1,…,f¯n-2,0,0)T, G=(g¯1,…,g¯n-1,0), ξn=ηn-ηn*, α¯i=αn⋯αi, cni, i=1,…,n, are positive constants. The system (7) and (13) can be written as
(15)dη=κE(η)dt+F(η,η(t-d(t)))dt+GT(η,η(t-d(t)))dω,
where η=η¯n=(η1,…,ηn)T, E(η)=(η2,…,ηn,v)T, and F and G are defined as in (14). Introducing the dilation weight Δ=(1,1,…,1︸forη1,…,ηn), by (10) and Vn(η)=(1/4)∑i=1n‍ξi4, one has
(16)Vn(Δɛ(η))=14∑i=1n‍(ɛηi+αi-1ɛηi-1+⋯+αi-1⋯α1ɛη1)4=ɛ4Vn(η),
from which and Definition 3, we know that Vn(η) is homogeneous of degree 4.


## 4. Stability Analysis

We state the main result in this paper.Theorem 12.
If Assumptions8 and 9 hold for the stochastic feedforward nonlinear time-delay system (1), under the state-feedback controller u=κnv and (13), then(i)
the closed-loop system has a unique solution on[-d,∞);(ii)
the equilibrium at the origin of the closed-loop system is GAS in probability.Proof.
We prove Theorem12 by four steps.
Step  1. Since fi, gi, i=1,…,n, are assumed to be locally Lipschitz, so the system consisting of (13) and (15) satisfies the locally Lipschitz condition.
Step  2. We consider the following entire Lyapunov function for system (15):
(17)V(η)=Vn(η)+(c¯02+c¯03)κ21-γ∫t-d(t)t‍∥η(σ)∥Δ4dσ,
where c¯02 and c¯03 are positive parameters to be determined. It is easy to verify that V(η) is 𝒞2 on η. Since Vn(η) is continuous, positive definite, and radially unbounded, by Lemma 4.3 in [31], there exist two class 𝒦∞ functions β1 and α21 such that
(18)β1(|η|)≤Vn(η)≤α21(|η|).
By Lemma 4.3 in [31] and Lemma 6, there exist positive constants c_ and c¯, class 𝒦∞ functions α_22 and α¯22, and a positive definite function U(η) whose homogeneous degree is 4 such that
(19)c_∥η∥Δ4≤U(η)≤c¯∥η∥Δ4,α_22(|η|)≤U(η)≤α¯22(|η|).
Fromd(t):R+→[0,d] and (19), it follows that
(20)(c¯02+c¯03)κ21-γ∫t-d(t)t‍∥η(σ)∥Δ4dσ≤c~∫t-d(t)t‍α¯22(|η(σ)|)dσ=σ=s+tc~∫-d(t)0‍α¯22(|η(s+t)|)d(s+t)≤c~∫-d0‍α¯22(|η(s+t)|)d(s+t)≤csup-d≤s≤0α¯22(|η(s+t)|)≤α22(sup-d≤s≤0|η(s+t)|),
where c~, c are positive constants and α22 is a class 𝒦∞ function. Since |η|≤sup-d≤s≤0|η(s+t)|, α21(|η|)≤α21(sup-d≤s≤0|η(s+t)|). Defining β2=α21+α22, by (17), (18), and (20), one gets
(21)β1(|η|)≤V(η)≤β2(sup-d≤s≤0|η(s+t)|).
Step  3. By Lemma 6 and (14), there exists a positive constant c01 such that
(22)∂Vn∂ηκE(η)≤-c01κ∥η∥Δ4.
By Assumption8, (6), and 0<κ<1, one has
(23)|f¯i(η~i+2,η~i+2(t-d(t)))|≤a1(∑j=i+2n|κj-1ηj|+∑j=i+2n|κj-1ηj(t-d(t))|)κi-1≤a1κ2(∑j=i+2n‍|ηj|+∑j=i+2n‍|ηj(t-d(t))|)≤δ1κ2(∥η∥Δ+∥η(t-d(t))∥Δ),
where δ1 is a positive constant. Using Lemmas 5–7 and (23), one gets
(24)∂Vn∂ηF(η,η(t-d(t)))=∑i=1n-2‍∂Vn∂ηif¯i(η~i+2,η~i+2(t-d(t)))≤c~02κ2∑i=1n-2‍∥η∥Δ3(∥η∥Δ+∥η(t-d(t))∥Δ)≤κ2(c02∥η∥Δ4+c¯02∥η(t-d(t))∥Δ4),
where c02, c¯02, and c~02 are positive constants. Similar to (23), there is a positive constant δ2 such that
(25)|g¯i(η~i+1,η~i+1(t-d(t)))|≤δ2κ(∥η∥Δ+∥η(t-d(t))∥Δ),
from which and Lemmas 5–7, one gets
(26)12Tr{G(η,η(t-d(t)))∂2Vn∂η2GT(η,η(t-d(t)))}≤12mm∑i,j=1n-1‍|∂2Vn∂ηi∂ηj||g¯i(η~i+1,η~i+1(t-d(t)))|≤12mm∑i,j=1n-1‍×|g¯j(η~j+1,η~j+1(t-d(t)))|≤c~03κ2∑i,j=1n-1‍∥η∥Δ2(∥η∥Δ+∥η(t-d(t))∥Δ)2≤κ2(c03∥η∥Δ4+c¯03∥η(t-d(t))∥Δ4),
where c03, c¯03, and c~03 are positive constants. With the help of (3), (15), (17), (22), (24), (26), and Assumption 9, one has
(27)ℒV≤∂Vn∂ηκE(η)+∂Vn∂ηF(η,η(t-d(t)))+12Tr{G(η,η(t-d(t)))∂2Vn∂η2GT(η,η(t-d(t)))}+(c¯02+c¯03)κ2(11-γ∥η∥Δ4-∥η(t-d(t))∥Δ4)≤-c01κ∥η∥Δ4+(c02+c03+c¯02+c¯031-γ)κ2∥η∥Δ4=-κ(c01-(c02+c03+c¯02+c¯031-γ)κ)∥η∥Δ4.
Sincec01 is a constant independent of c02, c03, c¯02, c¯03, and γ, by choosing
(28)0<κ<κ*=:min{1,c01c02+c03+((c¯02+c¯03)/(1-γ))}
Equation (27) becomes ℒV≤-c0∥η∥Δ4, where c0 is a positive constant. By (19), one obtains
(29)ℒV≤-c0c¯α_22(|η|).
By Steps 1–3 and Lemma4, the system consisting of (13) and (15) has a unique solution on [-d,∞), η=0 is GAS in probability, and P{limt→∞|η|=0}=1.
Step  4. Since (6) is an equivalent transformation, so the closed-loop system consisting of (1), u=κnv, and (13) has the same properties as the system (13) and (15). Theorem 12 holds.Remark 13.
In this paper, the homogeneous domination idea is generalized to stochastic feedforward nonlinear time-delay systems (1). The underlying philosophy of this approach is that the state-feedback controller is first constructed for system (7) without considering the drift and diffusion terms, and then a low gain κ in (6) (whose the value range is (28)) is introduced to state-feedback controller to dominate the drift and diffusion terms.Remark 14.
Due to the special upper-triangular structure and the appearance of time-varying delay, there is no efficient method to solve the stabilization problem of system (1). By combining the homogeneous domination approach with stochastic nonlinear time-delay system criterion, the state-feedback stabilization of system (1) was perfectly solved in this paper.Remark 15.
One of the main obstacles in the stability analysis is how to deal with the effect of time-varying delay. In this paper, by constructing an appropriate Lyapunov-Krasovskii functional (17), this problem was effectively solved.Remark 16.
It is worth pointing out that the rigorous proof of Theorem12 is not an easy job.


## 5.  A Simulation Example

Consider the following stochastic nonlinear system:(30)dx1=x2dt+110(x2+x2(t-d(t))cosx2)dω,dx2=udt,
where d(t)=1+(1/2)sint. It is easy to verify that Assumptions 8 and 9 are satisfied with a1=0, a2=1/10, and d˙(t)=(1/2)cost<1.Design of Controller. Introducing the following coordinate transformation:
(31)η1=x1,η2=x2κ,v=uκ2,
system (30) becomes
(32)dη1=κη2dt+g¯1dω,dη2=κvdt,
where g¯1=(1/10)(κη2+κη2(t-d(t))cosκη2). Choosing ξ1=η1 and V1(η1)=(1/4)ξ14, we obtain ℒV1≤-2κξ14+κξ13(η2-η2*)+(1/2)(∂2V1/∂η12)g¯12, where η2*=-2η1=:-α1ξ1. By ξ2=η2-η2* and V2(η¯2)=V1(η1)+(1/4)ξ24, a direct calculation leads to
(33)ℒV2≤-2κξ14+κξ13ξ2+κξ23v+κα1ξ23η2+12∂2V2∂η12g¯12.By Lemma7, one has
(34)ξ13ξ2≤0.5ξ14+0.8438ξ24,α1ξ23η2≤0.5ξ14+5.7797ξ24.Choosing(35)v=-7.6235ξ2=:-α2ξ2
and substituting (34) into (33), one gets
(36)ℒV2≤-κ(ξ14+ξ24)+12∂2V2∂η12g¯12.By (31) and (35), one obtains the actual controller
(37)u=-α2(κx2+α1κ2x1).The Choice of  κ*. Defining ∥η∥Δ=(η12+η22)1/2 and choosing
(38)V(η)=V2(η)+κ2∫t-d(t)t‍∥η(σ)∥Δ4dσ,
by (3), (36), and d(t)=1+(1/2)sint, one obtains
(39)ℒV≤-κ∥η∥Δ4+κ2(1.5∥η∥Δ4+0.5∥η(t-d(t))∥Δ4)+κ2(∥η∥Δ4-0.5∥η(t-d(t))∥Δ4)=-κ(1-2.5κ)∥η∥Δ4,
from which we get the critical value κ*=0.4; that is, κ∈(0,0.4).In simulation, we choose the initial valuesx1(0)=-0.8, x2(0)=1, and κ=0.3. Figure 1 demonstrates the effectiveness of the state-feedback controller.(a) The response of the closed-loop system (30) and (b) the response of the controller (37).
(a)
(b)


## 6.  A Concluding Remark

By using the homogeneous domination approach, this paper further studied the state-feedback stabilization problem for a class of stochastic feedforward nonlinear time-delay systems (1). The delay-independent state-feedback controller is explicitly constructed such that the closed-loop system is globally asymptotically stable in probability.There still exist some problems to be investigated. One is to consider the output-feedback control of switched stochastic system (1) by using average dwell time method in [32]. Another is to find a practical example (similar to [33–35]) for system (1). The last is to generalize the networked control systems (such as [36–41]) to stochastic feedforward networked systems.



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*Source: 151374-2013-12-09.xml*