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

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.

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 𝑑𝑥₁=𝑥₂𝑑𝑡+𝑓₁𝑥₃,𝑥₃(𝑡−𝑑(𝑡))𝑑𝑡+𝑔^𝑇₁𝑥₂,𝑥₂((𝑡⋮𝑡−𝑑))𝑑𝜔,𝑑𝑥_𝑛−2=𝑥_𝑛−1𝑑𝑡+𝑓_𝑛−2𝑥_𝑛,𝑥_𝑛(𝑡−𝑑(𝑡))𝑑𝑡+𝑔^𝑇_𝑛−2𝑥_𝑛−1,𝑥_𝑛−1(𝑡−𝑑(𝑡))𝑑𝜔,𝑑𝑥_𝑛−1=𝑥_𝑛𝑑𝑡+𝑔^𝑇_𝑛−1𝑥_𝑛,𝑥_𝑛(𝑡−𝑑(𝑡))𝑑𝜔,𝑑𝑥_𝑛=𝑢𝑑𝑡, (1) where 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈𝑅^𝑛 and 𝑢∈𝑅 are the system state and input signal, respectively, 𝑥_𝑖=(𝑥_𝑖,…,𝑥_𝑛)^𝑇, 𝑥_𝑖(𝑡−𝑑(𝑡))=(𝑥_𝑖(𝑡−𝑑(𝑡)),…,𝑥_𝑛(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,{ℱ_𝑡}_𝑡≥0,𝑃) with Ω being a sample space, ℱ being a 𝜎-field, {ℱ_𝑡}_𝑡≥0 being a filtration, and 𝑃 being a probability measure. 𝑓_𝑖∶𝑅^{𝑛−𝑖−1}×𝑅^{𝑛−𝑖−1}→𝑅 and 𝑔_𝑗∶𝑅^𝑛−𝑗×𝑅^𝑛−𝑗→𝑅^𝑚 are assumed to be locally Lipschitz with 𝑓_𝑖(0,0)=0 and 𝑔_𝑗(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. Section 2 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 𝑅^𝑛 denotes the real 𝑛-dimensional space. For a given vector or matrix 𝑋, 𝑋^𝑇 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];𝑅^𝑛) denotes the space of continuous 𝑅^𝑛-value functions on [−𝑑,0] endowed with the norm ‖⋅‖ defined by ‖𝑓‖=sup_{𝑥∈[−𝑑,0]}|𝑓(𝑥)| for 𝑓∈𝒞([−𝑑,0];𝑅^𝑛); 𝒞^𝑏_ℱ0([−𝑑,0];𝑅^𝑛) denotes the family of all ℱ₀-measurable bounded 𝒞([−𝑑,0];𝑅^𝑛)-valued random variables 𝜉={𝜉(𝜃)∶−𝑑≤𝜃≤0}. 𝒞^𝑖 denotes the set of all functions with continuous 𝑖th partial derivatives; 𝒞^2,1(𝑅^𝑛×[−𝑑,∞);𝑅₊) denotes the family of all nonnegative functions 𝑉(𝑥,𝑡) on 𝑅^𝑛×[−𝑑,∞) which are 𝒞² in 𝑥 and 𝒞¹ in 𝑡; 𝒞^2,1 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: (𝑡)(𝑥(𝑡)((𝑡)(𝑥(𝑡)((𝑡)𝑑𝑥=𝑓,𝑥𝑡−𝑑)),𝑡𝑑𝑡+𝑔,𝑥𝑡−𝑑)),𝑡𝑑𝜔,∀𝑡≥0, (2) with initial data {𝑥(𝜃)∶−𝑑≤𝜃≤0}=𝜉∈𝒞^𝑏_ℱ0([−𝑑,0];𝑅^𝑛), 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,{ℱ_𝑡}_𝑡≥0,𝑃), and 𝑓∶𝑅^𝑛×𝑅^𝑛×𝑅₊→𝑅^𝑛 and 𝑔∶𝑅^𝑛×𝑅^𝑛×𝑅₊→𝑅^𝑛×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 given 𝑉(𝑥(𝑡),𝑡)∈𝒞^2,1 associated with system (2), the differential operator ℒ is defined as ℒ𝑉=𝜕𝑉+𝜕𝑡𝜕𝑉1𝜕𝑥𝑓+2Tr𝑔^𝑇𝜕²𝑉𝜕𝑥²𝑔. (3)

Definition 2 (see [6]). The equilibrium 𝑥(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∈𝒞^𝑏_ℱ0([−𝑑,0];𝑅^𝑛)⧵{0}, where ‖𝜉‖=sup_{𝜃∈[−𝑑,0]}|𝑥(𝜃)|.

Definition 3 (see [26]). For fixed coordinates (𝑥₁,…,𝑥_𝑛)^𝑇∈𝑅^𝑛 and real numbers 𝑟_𝑖>0, 𝑖=1,…,𝑛, one has the following.(i)The dilation Δ_𝜀(𝑥) is defined by Δ_𝜀(𝑥)=(𝜀^𝑟1𝑥₁,…,𝜀^𝑟𝑛𝑥_𝑛) for any 𝜀>0; 𝑟₁,…,𝑟_𝑛 are called as the weights of the coordinates. For simplicity, we define dilation weight Δ=(𝑟₁,…,𝑟_𝑛).(ii)A function 𝑉∈𝒞(𝑅^𝑛,𝑅) is said to be homogeneous of degree 𝜏 if there is a real number 𝜏∈𝑅 such that 𝑉(Δ_𝜀(𝑥))=𝜀^𝜏𝑉(𝑥₁,…,𝑥_𝑛) for any 𝑥∈𝑅^𝑛⧵{0}, 𝜀>0.(iii)A vector field ℎ∈𝒞(𝑅^𝑛,𝑅^𝑛) is said to be homogeneous of degree 𝜏 if there is a real number 𝜏∈𝑅 such that ℎ_𝑖(Δ_𝜀(𝑥))=𝜀^{𝜏+𝑟𝑖}ℎ_𝑖(𝑥) for any 𝑥∈𝑅^𝑛⧵{0}, 𝜀>0, 𝑖=1,…,𝑛.(iv)A homogeneous 𝑝-norm is defined as ‖𝑥‖_Δ,𝑝=(∑^𝑛_𝑖=1|𝑥_𝑖|^{𝑝/𝑟𝑖})^1/𝑝 for any 𝑥∈𝑅^𝑛, 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)∈𝒞^2,1(𝑅^𝑛×[−𝑑,∞);𝑅₊), two class 𝒦_∞ functions 𝛼₁, 𝛼₂, and a class 𝒦 function 𝛼₃ such that 𝛼₁(|𝑥(𝑡)|)≤𝑉(𝑥(𝑡),𝑡)≤𝛼₂sup_{−𝑑≤𝑠≤0}|𝑥(,(𝑥(𝑡))𝑡+𝑠)|ℒ𝑉,𝑡≤−𝛼₃(|𝑥(𝑡)|), (4) then there exists a unique solution on [−𝑑,∞) for (2), the equilibrium 𝑥(𝑡)=0 is GAS in probability, and 𝑃{lim_𝑡→∞|𝑥(𝑡)|=0}=1.

Lemma 5 (see [26]). Given a dilation weight Δ=(𝑟₁,…,𝑟_𝑛), 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 that 𝑉∶𝑅^𝑛→𝑅 is a homogeneous function of degree 𝜏 with respect to the dilation weight Δ; then (i) 𝜕𝑉/𝜕𝑥_𝑖 is homogeneous of degree 𝜏−𝑟_𝑖 with 𝑟_𝑖 being the homogeneous weight of 𝑥_𝑖; (ii) there is a constant 𝑐 such that 𝑉(𝑥)≤𝑐‖𝑥‖^𝜏_Δ. Moreover, if 𝑉(𝑥) is positive definite, then 𝑉(𝑥)≥𝑐‖𝑥‖^𝜏_Δ, where 𝑐 is a positive constant.

Lemma 7 (see [5]). Let 𝑐 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. For 𝑖=1,…,𝑛−1, there exist positive constants 𝑎₁ and 𝑎₂ such that 𝑓_𝑖𝑥_𝑖+2,𝑥_𝑖+2(𝑡−𝑑(𝑡))≤𝑎₁^𝑛_𝑗=𝑖+2𝑥_𝑗+^𝑛_𝑗=𝑖+2𝑥_𝑗((𝑡,𝑔𝑡−𝑑))_𝑖𝑥_𝑖+1,𝑥_𝑖+1(𝑡−𝑑(𝑡))≤𝑎₂^𝑛_𝑗=𝑖+1𝑥_𝑗+^𝑛_𝑗=𝑖+1𝑥_𝑗(𝑡−𝑑(𝑡)), (5) where 𝑥_𝑛+1=𝑥_𝑛+1(𝑡−𝑑(𝑡))=0.

Assumption 9. The time-varying delay 𝑑(𝑡) satisfies 𝑑(̇𝑡)≤𝛾<1 for a constant 𝛾.

Remark 10. When 𝑥_𝑖+1=𝑥_𝑖+1(𝑡−𝑑(𝑡))=0 in diffusion term 𝑔_𝑖(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: 𝜂_𝑖=𝑥_𝑖𝜅^𝑖−1𝑢,𝑣=𝜅^𝑛,𝑖=1,…,𝑛, (6) where 0<𝜅<1 is a scalar to be designed. By (6), (1) can be expressed as 𝑑𝜂₁=𝜅𝜂₂𝑑𝑡+𝑓₁𝜂₃,𝜂₃((𝑡+𝑡−𝑑))𝑑𝑡𝑔^𝑇₁𝜂₂,𝜂₂((𝑡⋮𝑡−𝑑))𝑑𝜔,𝑑𝜂_𝑛−2=𝜅𝜂_𝑛−1𝑑𝑡+𝑓_𝑛−2𝜂_𝑛,𝜂_𝑛((𝑡+𝑡−𝑑))𝑑𝑡𝑔^𝑇_𝑛−2𝜂_𝑛−1,𝜂_𝑛−1(𝑡−𝑑(𝑡))𝑑𝜔,𝑑𝜂_𝑛−1=𝜅𝜂_𝑛𝑑𝑡+𝑔^𝑇_𝑛−1𝜂_𝑛,𝜂_𝑛(𝑡−𝑑(𝑡))𝑑𝜔,𝑑𝜂_𝑛=𝜅𝑣𝑑𝑡, (7) where 𝑓_𝑖=𝑓_𝑖/𝜅^𝑖−1, 𝑔_𝑖=𝑔_𝑖/𝜅^𝑖−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. Introducing 𝜉₁=𝜂₁ and choosing 𝑉₁(𝜂₁)=(1/4)𝜉⁴₁, from (3) and (7), it follows that ℒ𝑉₁=𝜅𝜉³₁𝜂₂+𝜕𝑉₁𝜕𝜂₁𝑓₁+12Tr𝑔₁𝜕²𝑉₁𝜕𝜂²₁𝑔^𝑇₁. (8)
The first virtual controller 𝜂^∗₂=−𝑐₁₁𝜉₁=∶−𝛼₁𝜉₁,𝑐₁₁>0, (9) leads to ℒ𝑉₁≤−𝜅𝑐₁₁𝜉⁴₁+𝜅𝜉³₁(𝜂₂−𝜂^∗₂)+(𝜕𝑉₁/𝜕𝜂₁)𝑓₁+(1/2)Tr{𝑔₁(𝜕²𝑉₁/𝜕𝜂²₁)𝑔^𝑇₁}.

Step i (𝑖=2,…,𝑛). In this step, we can get the following lemma.

Lemma 11. Suppose that at step 𝑖−1, there is a set of virtual controllers 𝜂^∗₁,…,𝜂^∗_𝑖 defined by 𝜂^∗₁=0,𝜉₁=𝜂₁−𝜂^∗₁=𝜂₁,𝜂^∗_𝑘=−𝛼_𝑘−1𝜉_𝑘−1,𝜉_𝑘=𝜂_𝑘−𝜂^∗_𝑘,𝑘=2,…,𝑖, (10) such that the (𝑖−1)th Lyapunov function 𝑉_𝑖−1(𝜂_𝑖−1)=(1/4)∑^𝑖−1_𝑗=1𝜉⁴_𝑗 satisfies ℒ𝑉_𝑖−1≤−𝜅^𝑖−1_𝑗=1𝑐_{𝑖−1,𝑗}𝜉⁴_𝑗+𝜅𝜉³_𝑖−1𝜂_𝑖−𝜂^∗_𝑖+^𝑖−1_𝑗=1𝜕𝑉_𝑖−1𝜕𝜂_𝑗𝑓_𝑗+12^𝑖−1_𝑝,𝑞=1Tr𝑔_𝑝𝜕²𝑉_𝑖−1𝜕𝜂_𝑝𝜕𝜂_𝑞𝑔^𝑇_𝑞, (11) where 𝛼_𝑗, 𝑐_{𝑖−1,𝑗}, 𝑗=1,…,𝑖−1, are positive constants. Then there exists a virtual control law 𝜂^∗_𝑖+1=−𝛼_𝑖𝜉_𝑖 such that ℒ𝑉_𝑖≤−𝜅^𝑖_𝑗=1𝑐_𝑖𝑗𝜉⁴_𝑗+𝜅𝜉³_𝑖𝜂_𝑖+1−𝜂^∗_𝑖+1+^𝑖_𝑗=1𝜕𝑉_𝑖𝜕𝜂_𝑗𝑓_𝑗+12^𝑖_𝑝,𝑞=1Tr𝑔_𝑝𝜕²𝑉_𝑖𝜕𝜂_𝑝𝜕𝜂_𝑞𝑔^𝑇_𝑞, (12) where 𝑉_𝑖(𝜂_𝑖)=(1/4)∑^𝑖_𝑗=1𝜉⁴_𝑗=∶𝑉_𝑖−1(𝜂_𝑖−1)+𝑊_𝑖(𝜂_𝑖).

Proof. See the Appendix.

At step 𝑛, choosing 𝑉_𝑛(𝜂_𝑛)=(1/4)∑^𝑛_𝑖=1𝜉⁴_𝑖 and 𝑣=𝜂^∗_𝑛+1=−𝛼_𝑛𝜉_𝑛=−𝛼_𝑛𝜂_𝑛+𝛼_𝑛−1𝜂_𝑛−1+⋯+𝛼₁𝜂₁, (13) with the help of (3), (12), and (13), one obtains ℒ𝑉_𝑛≤−𝜅^𝑛_𝑖=1𝑐_𝑛𝑖𝜉⁴_𝑖+𝜅𝜉³_𝑛𝑣−𝜂^∗_𝑛+1+^𝑛_𝑖=1𝜕𝑉_𝑛𝜕𝜂_𝑖𝑓_𝑖+12^𝑛_𝑝,𝑞=1Tr𝑔_𝑝𝜕²𝑉_𝑛𝜕𝜂_𝑝𝜕𝜂_𝑞𝑔^𝑇_𝑞=−𝜅^𝑛_𝑖=1𝑐_𝑛𝑖𝜉⁴_𝑖+𝜕𝑉_𝑛1𝜕𝜂𝐹+2𝐺𝜕Tr²𝑉_𝑛𝜕𝜂²𝐺^𝑇, (14) where 𝐹=(𝑓₁,…,𝑓_𝑛−2,0,0)^𝑇, 𝐺=(𝑔₁,…,𝑔_𝑛−1,0), 𝜉_𝑛=𝜂_𝑛−𝜂^∗_𝑛, 𝛼_𝑖=𝛼_𝑛⋯𝛼_𝑖, 𝑐_𝑛𝑖, 𝑖=1,…,𝑛, are positive constants. The system (7) and (13) can be written as 𝑑𝜂=𝜅𝐸𝜂𝑑𝑡+𝐹𝜂,𝜂(𝑡−𝑑(𝑡))𝑑𝑡+𝐺^𝑇𝜂,𝜂(𝑡−𝑑(𝑡))𝑑𝜔, (15) where 𝜂=𝜂_𝑛=(𝜂₁,…,𝜂_𝑛)^𝑇, 𝐸(𝜂)=(𝜂₂,…,𝜂_𝑛,𝑣)^𝑇, and 𝐹 and 𝐺 are defined as in (14). Introducing the dilation weight Δ=(1,1,…,1_{for𝜂1,…,𝜂𝑛}), by (10) and 𝑉_𝑛(𝜂)=(1/4)∑^𝑛_𝑖=1𝜉⁴_𝑖, one has 𝑉_𝑛Δ_𝜀𝜂=14^𝑛_𝑖=1𝜀𝜂_𝑖+𝛼_𝑖−1𝜀𝜂_𝑖−1+⋯+𝛼_𝑖−1⋯𝛼₁𝜀𝜂₁⁴=𝜀⁴𝑉_𝑛𝜂, (16) from which and Definition 3, we know that 𝑉_𝑛(𝜂) is homogeneous of degree 4.

4. Stability Analysis

We state the main result in this paper.

Theorem 12. If Assumptions 8 and 9 hold for the stochastic feedforward nonlinear time-delay system (1), under the state-feedback controller 𝑢=𝜅^𝑛𝑣 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 Theorem 12 by four steps.
Step  1. Since 𝑓_𝑖, 𝑔_𝑖, 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): 𝑉𝜂=𝑉_𝑛𝜂+𝑐₀₂+𝑐₀₃𝜅²1−𝛾^𝑡_{(𝑡)𝑡−𝑑}𝜂(𝜎)⁴_Δ𝑑𝜎, (17) where 𝑐₀₂ and 𝑐₀₃ are positive parameters to be determined. It is easy to verify that 𝑉(𝜂) is 𝒞² on 𝜂. Since 𝑉_𝑛(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 𝛽₁𝜂≤𝑉_𝑛𝜂≤𝛼₂₁𝜂. (18)
By Lemma 4.3 in [31] and Lemma 6, there exist positive constants 𝑐 and 𝑐, class 𝒦_∞ functions 𝛼₂₂ and 𝛼₂₂, and a positive definite function 𝑈(𝜂) whose homogeneous degree is 4 such that 𝑐𝜂⁴_Δ≤𝑈𝜂≤𝑐𝜂⁴_Δ,𝛼₂₂𝜂≤𝑈𝜂≤𝛼₂₂𝜂. (19)
From 𝑑(𝑡)∶𝑅₊→[0,𝑑] and (19), it follows that 𝑐₀₂+𝑐₀₃𝜅²1−𝛾^𝑡_{(𝑡)𝑡−𝑑}𝜂(𝜎)⁴_Δ≤𝑐̃𝑑𝜎_{𝑡(𝑡)𝑡−𝑑}𝛼₂₂𝜂(𝜎)𝑑𝜎_{𝜎=𝑠+𝑡}=𝑐̃⁰_{(𝑡)−𝑑}𝛼₂₂𝜂(𝑑()≤𝑐̃𝑠+𝑡)𝑠+𝑡_0−𝑑𝛼₂₂𝜂(𝑠+𝑡)𝑑(𝑠+𝑡)≤𝑐sup_{−𝑑≤𝑠≤0}𝛼₂₂𝜂(𝑠+𝑡)≤𝛼₂₂sup_{−𝑑≤𝑠≤0}𝜂(𝑠+𝑡), (20) where 𝑐̃, 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 𝛽₁𝜂≤𝑉𝜂≤𝛽₂sup_{−𝑑≤𝑠≤0}𝜂(𝑠+𝑡). (21)
Step  3. By Lemma 6 and (14), there exists a positive constant 𝑐₀₁ such that 𝜕𝑉_𝑛𝜂𝜕𝜂𝜅𝐸≤−𝑐₀₁𝜅𝜂⁴_Δ. (22)
By Assumption 8, (6), and 0<𝜅<1, one has 𝑓_𝑖𝜂_𝑖+2,𝜂_𝑖+2((𝑡≤𝑎𝑡−𝑑))₁∑^𝑛_𝑗=𝑖+2𝜅^𝑗−1𝜂_𝑗+∑^𝑛_𝑗=𝑖+2𝜅^𝑗−1𝜂_𝑗(𝑡−𝑑(𝑡))𝜅^𝑖−1≤𝑎₁𝜅^2𝑛_𝑗=𝑖+2𝜂_𝑗+^𝑛_𝑗=𝑖+2𝜂_𝑗(𝑡−𝑑(𝑡))≤𝛿₁𝜅²𝜂_Δ+𝜂(𝑡−𝑑(𝑡))_Δ, (23) where 𝛿₁ is a positive constant. Using Lemmas 5–7 and (23), one gets 𝜕𝑉_𝑛𝐹((𝑡=𝜕𝜂𝜂,𝜂𝑡−𝑑))_{𝑛−2𝑖=1}𝜕𝑉_𝑛𝜕𝜂_𝑖𝑓_𝑖𝜂_𝑖+2,𝜂_𝑖+2((𝑡≤𝑐𝑡−𝑑))₀₂𝜅^2𝑛−2_𝑖=1𝜂³_Δ𝜂_Δ+𝜂(𝑡−𝑑(𝑡))_Δ≤𝜅²𝑐₀₂𝜂⁴_Δ+𝑐₀₂𝜂(𝑡−𝑑(𝑡))⁴_Δ, (24) where 𝑐₀₂, 𝑐₀₂, and 𝑐₀₂ are positive constants. Similar to (23), there is a positive constant 𝛿₂ such that 𝑔_𝑖𝜂_𝑖+1,𝜂_𝑖+1(𝑡−𝑑(𝑡))≤𝛿₂𝜅𝜂_Δ+𝜂(𝑡−𝑑(𝑡))_Δ, (25) from which and Lemmas 5–7, one gets 12𝐺((𝑡𝜕Tr𝜂,𝜂𝑡−𝑑))²𝑉_𝑛𝜕𝜂²𝐺^𝑇((𝑡≤1𝜂,𝜂𝑡−𝑑))2𝑚√𝑚^𝑛−1_𝑖,𝑗=1𝜕²𝑉_𝑛𝜕𝜂_𝑖𝜕𝜂_𝑗𝑔_𝑖𝜂_𝑖+1,𝜂_𝑖+1((𝑡×𝑡−𝑑))𝑔_𝑗𝜂_𝑗+1,𝜂_𝑗+1((𝑡≤𝑐𝑡−𝑑))₀₃𝜅^2𝑛−1_𝑖,𝑗=1𝜂²_Δ𝜂_Δ+𝜂(𝑡−𝑑(𝑡))_Δ²≤𝜅²𝑐₀₃𝜂⁴_Δ+𝑐₀₃𝜂(𝑡−𝑑(𝑡))⁴_Δ, (26) where 𝑐₀₃, 𝑐₀₃, and 𝑐₀₃ are positive constants. With the help of (3), (15), (17), (22), (24), (26), and Assumption 9, one has ℒ𝑉≤𝜕𝑉_𝑛𝜂+𝜕𝜂𝜅𝐸𝜕𝑉_𝑛𝐹((𝑡+1𝜕𝜂𝜂,𝜂𝑡−𝑑))2𝐺((𝑡𝜕Tr𝜂,𝜂𝑡−𝑑))²𝑉_𝑛𝜕𝜂²𝐺^𝑇((𝑡+𝜂,𝜂𝑡−𝑑))𝑐₀₂+𝑐₀₃𝜅²1𝜂1−𝛾^4Δ−𝜂(𝑡−𝑑(𝑡))⁴_Δ≤−𝑐₀₁𝜅𝜂⁴_Δ+𝑐₀₂+𝑐₀₃+𝑐₀₂+𝑐₀₃𝜅1−𝛾²𝜂⁴_Δ=−𝜅𝑐₀₁−𝑐₀₂+𝑐₀₃+𝑐₀₂+𝑐₀₃𝜅𝜂1−𝛾^4Δ. (27)
Since 𝑐₀₁ is a constant independent of 𝑐₀₂, 𝑐₀₃, 𝑐₀₂, 𝑐₀₃, and 𝛾, by choosing 0<𝜅<𝜅^∗𝑐=∶min1,⁰¹𝑐₀₂+𝑐₀₃+𝑐₀₂+𝑐₀₃/1−𝛾 (28)
Equation (27) becomes ℒ𝑉≤−𝑐₀‖𝜂‖⁴_Δ, where 𝑐₀ is a positive constant. By (19), one obtains 𝑐ℒ𝑉≤−⁰𝑐𝛼₂₂𝜂. (29)
By Steps 1–3 and Lemma 4, the system consisting of (13) and (15) has a unique solution on [−𝑑,∞), 𝜂=0 is GAS in probability, and 𝑃{lim_𝑡→∞|𝜂|=0}=1.
Step  4. Since (6) is an equivalent transformation, so the closed-loop system consisting of (1), 𝑢=𝜅^𝑛𝑣, 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 Theorem 12 is not an easy job.

5. A Simulation Example

Consider the following stochastic nonlinear system: 𝑑𝑥₁=𝑥₂1𝑑𝑡+𝑥10²+𝑥₂(𝑡−𝑑(𝑡))cos𝑥₂𝑑𝜔,𝑑𝑥₂=𝑢𝑑𝑡, (30) 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: 𝜂₁=𝑥₁,𝜂₂=𝑥₂𝜅𝑢,𝑣=𝜅², (31) system (30) becomes 𝑑𝜂₁=𝜅𝜂₂𝑑𝑡+𝑔₁𝑑𝜔,𝑑𝜂₂=𝜅𝑣𝑑𝑡, (32) where 𝑔₁=(1/10)(𝜅𝜂₂+𝜅𝜂₂(𝑡−𝑑(𝑡))cos𝜅𝜂₂). Choosing 𝜉₁=𝜂₁ and 𝑉₁(𝜂₁)=(1/4)𝜉⁴₁, we obtain ℒ𝑉₁≤−2𝜅𝜉⁴₁+𝜅𝜉³₁(𝜂₂−𝜂^∗₂)+(1/2)(𝜕²𝑉₁/𝜕𝜂²₁)𝑔²₁, where 𝜂^∗₂=−2𝜂₁=∶−𝛼₁𝜉₁. By 𝜉₂=𝜂₂−𝜂^∗₂ and 𝑉₂(𝜂₂)=𝑉₁(𝜂₁)+(1/4)𝜉⁴₂, a direct calculation leads to ℒ𝑉₂≤−2𝜅𝜉⁴₁+𝜅𝜉³₁𝜉₂+𝜅𝜉³₂𝑣+𝜅𝛼₁𝜉³₂𝜂₂+12𝜕²𝑉₂𝜕𝜂²₁𝑔²₁. (33)

By Lemma 7, one has 𝜉³₁𝜉₂≤0.5𝜉⁴₁+0.8438𝜉⁴₂,𝛼₁𝜉³₂𝜂₂≤0.5𝜉⁴₁+5.7797𝜉⁴₂. (34)

Choosing 𝑣=−7.6235𝜉₂=∶−𝛼₂𝜉₂ (35) and substituting (34) into (33), one gets ℒ𝑉₂≤−𝜅𝜉⁴₁+𝜉⁴₂+12𝜕²𝑉₂𝜕𝜂²₁𝑔²₁. (36)

By (31) and (35), one obtains the actual controller 𝑢=−𝛼₂𝜅𝑥₂+𝛼₁𝜅²𝑥₁. (37)

The Choice of  𝜅^∗. Defining ‖𝜂‖_Δ=(𝜂²₁+𝜂²₂)^1/2 and choosing 𝑉𝜂=𝑉₂𝜂+𝜅^2𝑡_{𝑡−𝑑(𝑡)}𝜂(𝜎)⁴_Δ𝑑𝜎, (38) by (3), (36), and 𝑑(𝑡)=1+(1/2)sin𝑡, one obtains ℒ𝑉≤−𝜅𝜂⁴_Δ+𝜅²1.5𝜂⁴_Δ+0.5𝜂(𝑡−𝑑(𝑡))⁴_Δ+𝜅²𝜂⁴_Δ−0.5𝜂(𝑡−𝑑(𝑡))⁴_Δ=−𝜅(1−2.5𝜅)𝜂⁴_Δ, (39) from which we get the critical value 𝜅^∗=0.4; that is, 𝜅∈(0,0.4).

In simulation, we choose the initial values 𝑥₁(0)=−0.8, 𝑥₂(0)=1, and 𝜅=0.3. Figure 1 demonstrates the effectiveness of the state-feedback controller.

(a)

(b)

Figure 1 

(a) The response of the closed-loop system (30) and (b) the response of the controller (37).

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.

Appendix

Proof of Lemma 11. According to (3), (7), (10), and (11), one has ℒ𝑉_𝑖≤−𝜅^𝑖−1_𝑗=1𝑐_{𝑖−1,𝑗}𝜉⁴_𝑗+𝜅𝜉³_𝑖−1𝜉_𝑖+^𝑖−1_𝑗=1𝜕𝑉_𝑖−1𝜕𝜂_𝑗𝑓_𝑗+12^𝑖−1_𝑝,𝑞=1Tr𝑔_𝑝𝜕²𝑉_𝑖−1𝜕𝜂_𝑝𝜕𝜂_𝑞𝑔^𝑇_𝑞+^𝑖_𝑘=1𝜕𝑊_𝑖𝜕𝜂_𝑘𝜅𝜂_𝑘+1+𝑓_𝑘+12Tr𝑔_𝑖𝜕²𝑊_𝑖𝜕𝜂²_𝑖𝑔^𝑇_𝑖+^𝑖−1_𝑗=1Tr𝑔_𝑖𝜕²𝑊_𝑖𝜕𝜂_𝑖𝜕𝜂_𝑗𝑔^𝑇_𝑗+12^𝑖−1_𝑝,𝑞=1Tr𝑔_𝑝𝜕²𝑊_𝑖𝜕𝜂_𝑝𝜕𝜂_𝑞𝑔^𝑇_𝑞=−𝜅^𝑖−1_𝑗=1𝑐_{𝑖−1,𝑗}𝜉⁴_𝑗+𝜅𝜉³_𝑖−1𝜉_𝑖+𝜕𝑊_𝑖𝜕𝜂_𝑖𝜅𝜂_𝑖+1+^𝑖−1_𝑗=1𝜕𝑉_𝑖−1𝜕𝜂_𝑗𝑓_𝑗+^𝑖_𝑘=1𝜕𝑊_𝑖𝜕𝜂_𝑘𝑓_𝑘+^𝑖−1_𝑘=1𝜕𝑊_𝑖𝜕𝜂_𝑘𝜅𝜂_𝑘+1+12^𝑖−1_𝑝,𝑞=1Tr𝑔_𝑝𝜕²𝑉_𝑖−1𝜕𝜂_𝑝𝜕𝜂_𝑞𝑔^𝑇_𝑞+12Tr𝑔_𝑖𝜕²𝑉_𝑖𝜕𝜂²_𝑖𝑔^𝑇_𝑖+^𝑖−1_𝑗=1Tr𝑔_𝑖𝜕²𝑉_𝑖𝜕𝜂_𝑖𝜕𝜂_𝑗𝑔^𝑇_𝑗+12^𝑖−1_𝑝,𝑞=1Tr𝑔_𝑝𝜕²𝑊_𝑖𝜕𝜂_𝑝𝜕𝜂_𝑞𝑔^𝑇_𝑞=−𝜅^𝑖−1_𝑗=1𝑐_{𝑖−1,𝑗}𝜉⁴_𝑗+𝜅𝜉³_𝑖𝜂_𝑖+1+^𝑖_𝑗=1𝜕𝑉_𝑖𝜕𝜂_𝑗𝑓_𝑗+12^𝑖_𝑝,𝑞=1Tr𝑔_𝑝𝜕²𝑉_𝑖𝜕𝜂_𝑝𝜕𝜂_𝑞𝑔^𝑇_𝑞+𝜅𝜉³_𝑖−1𝜉_𝑖−𝜅𝜉³_𝑖^𝑖−1_𝑘=1𝜕𝜂^∗_𝑖𝜕𝜂_𝑘𝜂_𝑘+1. (A.1) We concentrate on the last two terms on the right-hand side of (A.1).
Using (10) and Lemma 7, one obtains 𝜉³_𝑖−1𝜉_𝑖≤𝑙_{𝑖,𝑖−1,1}𝜉⁴_𝑖−1+𝜌_𝑖1𝜉⁴_𝑖,−𝜉³_𝑖^𝑖−1_𝑘=1𝜕𝜂^∗_𝑖𝜕𝜂_𝑘𝜂_𝑘+1≤𝜉_𝑖^3𝑖−1_𝑘=1𝛼_𝑖−1⋯𝛼_𝑘𝜉_𝑘+1−𝛼_𝑘𝜉_𝑘≤𝜉_𝑖^3𝑖−1_𝑘=1𝛼_𝑖−1⋯𝛼_𝑘−1+𝛼_𝑖−1⋯𝛼_𝑘+1𝛼²_𝑘𝜉_𝑘+𝛼_𝑖−1𝜉_𝑖≤^𝑖−1_𝑘=1𝑙_𝑖𝑘2𝜉⁴_𝑘+𝜌_𝑖2𝜉⁴_𝑖, (A.2) where 𝑙_{𝑖,𝑖−1,1}, 𝑙_𝑖𝑘2(𝑘=1,…,𝑖−1), 𝜌_𝑖1, and 𝜌_𝑖2 are positive constants, 𝛼₀=0.
Choosing 𝑐_𝑖𝑗=𝑐_{𝑖−1,𝑗}−𝑙_𝑖𝑗2𝑐>0,𝑗=1,…,𝑖−2,_{𝑖−1,𝑖−1}−𝑙_{𝑖,𝑖−1,1}−𝑙_{𝑖,𝑖−1,2}𝜂>0,𝑗=𝑖−1,_∗𝑖+1=−𝑐_𝑖𝑖+𝜌_𝑖1+𝜌_𝑖2𝜉_𝑖=∶−𝛼_𝑖𝜉_𝑖,𝑐_𝑖𝑖>0, (A.3) and substituting (A.2)-(A.3) into (A.1), one gets the desired result.

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgments

The authors would like to express sincere gratitude to 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 the Polish-Norwegian Research Programme operated by the National Center for Research and 24 Development under the Norwegian Financial Mechanism 2009–2014 in the frame of Project Contract (no. Pol-Nor/200957/47/2013).