# State-Feedback Stabilization for Stochastic High-Order Nonlinear Systems with Time-Varying Delays

Authors: Fangzheng Gao; Zheng Yuan; Fushun Yuan
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper investigates the problem of state-feedback stabilization for a class of stochastic high-order nonlinear systems with time-varying delays. Under the weaker conditions on the power order and the nonlinear growth, by using the method of adding a power integrator, a state-feedback controller is successfully designed, and the global asymptotic stability in the probability of the resulting closed-loop system is proven with the help of an appropriate Lyapunov-Krasovskii functional. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.


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## Body

## 1. Introduction

It is well known that stochastic modeling has come to play an important role in the field of engineering where stochastic differential equations have been applied for the analysis and control of stochastic systems. As a consequence, the study of stochastic nonlinear systems has drawn an increasing attention in the past decades [1–9]. Especially, by applying the so-called method of adding a power integrator [10], which can be viewed as the latest achievement of the traditional backstepping method, a series of research results for stochastic high-order nonlinear systems have been achieved [11–18].On the other hand, time delay extensively exists in various engineering systems, such as electrical networks, microwave oscillator, and hydraulic systems. It is well known that the existence of time delay often deteriorates the control performance of systems and even causes the instability of closed-loop systems [19]. Therefore, the control design and stability analysis of time-delay systems has been an active research area within the automation and control community. In recent years, by employing the Lyapunov-Krasovskii method or Lyapunov-Razumikhin method to deal with the time delay, control theory and techniques for time-delay linear systems were developed and many advanced methods such as time-delay partitioning method and input-output method have been established; see, for instance, [20–25] and reference therein. However, for the time-delay nonlinear systems, there exist many open problems which are so important and interesting at least from the theoretical point of view and have been paid careful attention; see, for example, the lastly published papers [26–29].In this paper, we consider a class of stochastic high-order systems with time-varying delays described by(1)dxi(t)=hi(t)xi+1pi(t)+fi(t,x¯i(t),x¯i(t-d(t)))dt+giT(t,x¯i(t),x¯i(t-d(t)))dw,i=1,…,n-1,dxn(t)=hn(t)upn(t)+fn(t,x¯n(t),x¯n(t-d(t)))dt+gnT(t,x¯n(t),x¯n(t-d(t)))dw,
where x=(x1,…,xn)T∈Rn is system state vector and u∈R is control input, respectively; x¯i=(x1,…,xi)T, x¯n=x; d(t):R+→[0,d] is the time-varying delay satisfying d˙(t)≤η<1 for a known constant η; pi∈Rodd≥1:={p/q∣p and q are positive odd integers, and p≥q} are said to be the high orders of the system; hi(t), i=1,…,n, are disturbed virtual control coefficients; w is an m-dimensional independent standard Wiener process defined on a 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. The functions fi:R+×Ri×Ri→R and gi:R+×Ri×Ri→Rm are assumed to be locally Lipschitz with fi(t,0,0)=0 and gi(t,0,0)=0.It is worth pointing out that, whenpi=1 and hi(t)=1, system (1) degenerates to the stochastic time-delay system in strict-feedback form, whose study on feedback control problem has achieved great development in recent years; see, for example, [30–33] and reference therein. However, when pi>1, the control design for system (1) is challenging because on one hand, the presence of the fractional power makes system (1) not only nonsmooth but also having the uncontrollable unstable linearization, and on the other hand, the existence of time-delay effect makes the common assumption on the high-order systems nonlinearities infeasible and which conditions should be placed to the nonlinearities remains unanswered. To the best of the authors’ knowledge, up to now only [34, 35] two papers considered the state-feedback stabilization problem of system (1) under the strict power order restriction (p1=p2=⋯=pn=p) and some rigorous growth condition. However, from both practical and theoretical points of view, it is somewhat restrictive to require system (1) to satisfy such restrictions. Immediately, the following interesting questions are proposed. Is it possible to relax the power order restriction and the nonlinear growth condition? Under these weaker conditions, how to design a state-feedback controller such that the closed-loop system is globally asymptotically stable in probability?Inspired by the recent works [15, 29], we will solve the aforementioned problems here. The contribution of this paper is highlighted as follows.(i)
By comparison with the existing results in [34, 35], the power order restriction is completely removed, the nonlinear growth condition is largely relaxed, and a much weaker sufficient condition is given.(ii)
Different from the result in the deterministic case [29], in which the controller need not satisfy the locally Lipschitz condition, for stochastic system, to guarantee the existence and uniqueness of the solution, the designed controller must guarantee that the closed-loop system satisfies the locally Lipschitz condition. Hence, how to construct the controller constitutes one of the main contributions.(iii)
Due to the appearance of high-order, time-varying delay and nonlinear assumption, how to construct an appropriate Lyapunov-Krasovskii functional for system (1), especially under the assumption that fi(t,x¯i(t),x¯i(t-d(t))) and gi(t,x¯i(t),x¯i(t-d(t))) are not bounded linearly, is a nontrivial work.The remainder of this paper is organized as follows. Section2 offers notations and some preliminary results. Section 3 presents the control design procedure and the main result, while Section 4 gives a simulation example to illustrate the theoretical finding of this paper. Finally, concluding remarks are proposed in Section 5.


## 2. Notations and Preliminary Results

Throughout this paper, the following notations are adopted.R+ denotes the set of all nonnegative real numbers and Rn denotes the real n-dimensional space. Rodd+:={p/q∣p and q positive integers}, Rodd≥1:={p/q∣p and q are positive odd integers, and p≥q}. 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≤s≤0}. Ci denotes the set of all functions with continuous ith partial derivatives; C1,2(R+;Rn×[-d,∞)) denotes the family of all nonnegative functions V(t,x) on [-d,∞)×Rn which are C1 in t and C2 in x; C1,2 denotes the family of all functions which are C1 in the first argument and C2 in the second argument.Consider the following stochastic time-delay system:(2)dx(t)=f(t,x(t),x(t-d(t)))dt+g(t,x(t),x(t-d(t)))dw,
with initial data {x(θ):-d≤s≤0}=ξ∈Cℱ0b, where d(t):R+→[0,d] is a Borel measurable function; w is an m-dimensional independent standard Wiener process defined on a complete probability space (Ω,ℱ,{ℱt}t≥0,P). The functions f:R+×Rn×Rn→Rn and g:R+×Rn×Rn→Rn×m are locally Lipschitz with f(t,0,0)=0 and g(t,0,0)=0.Definition 1 (see [32]).
For any givenV(t,x(t))∈C1,2 associated with system (2), the differential operator ℒ is defined as
(3)ℒV=∂V∂t+∂V∂xf+12Tr{gT∂2V∂x2g}.Lemma 2 (see [32]).
For system (2), if there is a function V(t,x(t))∈C1,2(R+;Rn×[-d,∞)) and two class K∞  functions β1(·) and β2(·), such that
(4)β1(|x(t)|)≤V(t,x(t))≤β2(sup-d≤s≤0|x(t+s)|),ℒV≤-W(x(t)),
where W(x(t)) is a nonnegative continuous function, then(i)
there exists a unique solution on[-d,∞);(ii)
the equilibriumx=0 is globally stable in probability and the solution x(t) satisfies
(5)P{limt→∞W(x(t))=0}=1.
Especially, ifW(x(t)) is a positive definite function, then the equilibrium x=0 is globally asymptotically stable (GAS) in probability.In the remainder of this section, we list several lemmas that serve as the basis for the development of a state-feedback controller for system (1). They are the key tools for adding a power integrator technique.Lemma 3 (see [36]).
Forx∈R, y∈R, and p≥1 being a constant, the following inequalities hold:
(6)|x+y|p≤2p-1|xp+yp|,(|x|+|y|)1/p≤|x|1/p+|y|1/p≤2(p-1)/p(|x|+|y|)1/p.
If p≥1 is odd, then
(7)|x-y|p≤2p-1|xp-yp|,|x1/p-y1/p|≤2(p-1)/p(|x-y|)1/p.Lemma 4 (see [37]).
Forx,y∈R and positive real number p, the following inequality holds:
(8)|xp-yp|≤p|x-y||xp-1+yp-1|≤c|x-y||(x-y)p-1+yp-1|,
where c=p for 1<p≤2 and c=p2p-1 for p>2.Lemma 5 (see [38]).
Letx and y be real variables; then for any positive real numbers a, m, and n, one has
(9)a|x|m|y|n≤b|x|m+n+nm+n(m+nm)-m/na(m+n)/nb-m/n|y|m+n,
where b>0 is any real number.


## 3. Controller Design and Main Result

The objective of this paper is to design a state-feedback controller of the form(10)u=u(x),
such that system (1) is GAS in probability at the origin.To this end, the following assumptions regarding system (1) are imposed.Assumption 6.
Fori=1,…,n, there are positive constants ci1 and ci2 such that
(11)ci1≤hi(t)≤ci2.Assumption 7.
Fori=1,…,n, there are nonnegative smooth functions ai1(x¯i), ai2(x¯i) and constants τ1≥τ2≥⋯≥τn≥0 such that
(12)|fi(t,x¯i(t),x¯i(t-d(t)))|≤ai1(x¯i)∑j=1i‍(|xj(t)|(ri+τi)/rj0000.0000000000+|xj(t-d(t))|(ri+τi)/rj),|gi(t,x¯i(t),x¯i(t-d(t)))|≤ai2(x¯i)∑j=1i‍(|xj(t)|(2ri+τi)/2rj00000000000000+|xj(t-d(t))|(2ri+τi)/2rj),
where r1=1 and ri+1=(ri+τi)/pi>0. Let r0=max{ri} and αi=r0/ri, i=1,…,n. Meanwhile, one of the following conditions should be satisfied:(i)
rn+τn≥ri is required if αi=1 or αi≥2, i=1,…,n;(ii)
rn+τn≥2ri is required if condition (i) does not hold.Remark 8.
Assumption7 is a generalization of the homogeneous growth condition introduced in [15], where d(t)=0. The assumption plays an essential role in ensuring the locally Lipschitz condition of the closed-loop system, which guarantees the existence and uniqueness of the solution. Moreover, it is worth pointing out that Assumption 7 encompasses the assumption in existing results [34, 35]. Specifically, when pi=p∈Rodd>2, τi=τ, and ai1(x¯i)=ai2(x¯i)=a, Assumption 7 can be seen to reduce to the condition used in [35]. Moreover, when pi=p and ai1(x¯i)=ai2(x¯i)=a by choosing τi=p-1, Assumption 7 reduces to the following condition:
(13)|fi(t,x¯i(t),x¯i(t-d(t)))|≤a∑j=1i‍(|xj(t)|p+|xj(t-d(t))|p),|gi(t,x¯i(t),x¯i(t-d(t)))|≤a∑j=1i‍(|xj(t)|(p+1)/2+|xj(t-d(t))|(p+1)/2),
which is equivalent to that in [34].Remark 9.
It should be emphasized that the available state-feedback results for high-order stochastic time-delay system (1) in [34, 35] are all based on the strict power order restriction (p1=p2=⋯=pn=p). In this paper, for pi, we only need a much more general condition: pi∈Rodd≥1, i=1,…,n, which means that the power order restriction is completely removed in this paper.Without loss of generality, we assume thatτi=pi/di with pi being any even integer and di being any odd integer, under which and with the definition of ri in Assumption 7, we know that ri∈Rodd+. For general ri, a similar technique in [39] can be used.Choosel≥1 to satisfy rn+τn≥(ri+τi)/l and r0≥(ri+τi)/l, i=1,…,n. With Assumption 7, σ is chosen in the following manner.(a)
Chooseσ=r0 if condition (i) of Assumption 7 is satisfied.(b)
σ can be chosen as any σ∈Rodd+ satisfying rn+τn≥σ≥max1≤i≤n{(ri+τi)/l,2ri} if condition (ii) of Assumption 7 holds.Now, we are in the position to give the controller design procedure by applying the method of adding a power integrator. To simplify the deduction procedure, we sometimes denoteχ(t) by χ.Step 1.
Defineξ1=x1σ and choose the Lyapunov functional V1=W1+(n/(1-η))∫t-d(t)t‍x14lσ(s)ds, where W1=(1/(4lσ-τ1))x14lσ-τ1. With the help of Assumptions 6 and 7 and Lemma 5, we have
(14)ℒV1≤h1x14lσ-τ1-1x2p1+a11x14lσ-τ1-1(|x1|1+τ1+|x1(t-d(t))|1+τ1)+(4σ-τ1-1)a122x14lσ-τ1-2×(|x1|2+τ1+|x1(t-d(t))|2+τ1)+n1-ηx14lσ-n(1-d˙(t))1-ηx14lσ(t-d(t))≤h1x14lσ-τ1-1(x2p1-x2*p1)-(n-1)x14lσ(t-d(t))+h1x14lσ-τ1-1x2*p1+x14lσ(n1-η+a11+(4σ-τ1-1)a122+l11+l12),
where l11(x1)≥((4lσ-τ1-1)/4lσ) × ((1+τ1)/2lσ)(1+τ1)/(4lσ-τ1-1) × a114lσ/(4lσ-τ1-1) and l12(x1)≥((4lσ-τ1-2)/4lσ) × ((2+τ1)/2lσ)(2+τ1)/(4lσ-τ1-2) × [(4σ-τ1-1)a122]4lσ/(4lσ-τ1-2) are nonnegative smooth functions.Obviously, the first virtual controller(15)x2*=-1c11(n+n1-η+a11+(4σ-τ1-1)a12200000000+l11+l12n1-η)x1(1+τ1)/p1:=-β1r2/σξ1r2/σ,
with β1>0 being smooth results in
(16)ℒV1≤-nξ14l-(n-1)ξ14l(t-d(t))+h1ξ1(4lσ-τ1-1)/σ(x2p1-x2*p1).Remark 10.
It is necessary to mention that in the first design step, the functionsl11(x1) and l12(x1) have been provided with explicit expression in order to deduce the completely explicit virtual controller. However, in the later design steps, sometimes for the sake of briefness, we will not explicitly write out the functions which are easily defined.Inductive Step. Suppose at step k-1 that there are a C2, proper and positive definite Lyapunov function Vk-1, and a set of virtual controllers x1*,…,xk-1* defined by
(17)x1*=0,ξ1=x1σ/r1-x1*σ/r1,x2*=-β1r2/σξ1r2/σ,ξ2=x2σ/r2-x2*σ/r2,⋮⋮xk*=-βk-1rk/σξk-1rk/σ,ξk=xkσ/rk-xk*σ/rk,
with β1>0,…,βk-1>0 being smooth, such that
(18)ℒVk-1≤-(n-k+2)∑i=1k-1‍ξi4l-(n-k+1)∑i=1k-1‍ξi4l(t-d(t))+hk-1ξk-1(4lσ-τk-1-rk-1)/σ(xkpk-1-xk*pk-1).To complete the induction, at thekth step, we choose the following Lyapunov functional candidate
(19)Vk(x¯k)=Vk-1(x¯k-1)+Wk(x¯k)+n-k+11-η∫t-d(t)t‍ξk4l(s)ds,
where
(20)Wk(x¯k)=∫xk*xk‍(sσ/rk-xk*σ/rk)(4lσ-τk-rk)/σds.Noting thatσ/ri≥2, using a similar method as in [40], Vk can be shown to be C2, proper and positive definite. Moreover, we can obtain
(21)∂Wk∂xk=ξk(4lσ-τk-rk)/σ,∂2Wk∂xk2=4lσ-τk-rkrkξk((4l-1)σ-τk-rk)/σxk(σ-rk)/rk,∂2Wk∂xk∂xi=∂2Wk∂xi∂xk=4lσ-τk-rkσξk((4l-1)σ-τk-rk)/σ∂xk*σ/rk∂xi,∂Wk∂xi=-4lσ-τk-rkσ∂xk*σ/rk∂xi×∫xk*xk‍(sσ/rk-xk*σ/rk)((4l-1)σ-τk-rk)/σds,∂2Wk∂xi2=4lσ-τk-rkσ·(4l-1)σ-τk-rkσ(∂xk*σ/rk∂xi)2×∫xk*xk‍(sσ/rk-xk*σ/rk)((4l-2)σ-τk-rk)/σds-4σ-τk-rkσ∂2xk*σ/rk∂xi2×∫xk*xk‍(sσ/rk-xk*σ/rk)((4l-1)σ-τk-rk)/σds,∂2Wk∂xi∂xj=4lσ-τk-rkσ·(4l-1)σ-τk-rkσ×∂xk*σ/rk∂xi∂xk*σ/rk∂xj×∫xk*xk‍(sσ/rk-xk*σ/rk)((4l-2)σ-τk-rk)/σds,
where i,j=1,…,k-1, i≠j.Using (19), (20), and (21), it follows that
(22)ℒVk≤-(n-k+2)∑i=1k-1‍ξi4l-(n-k+1)∑i=1k-1‍ξi4l(t-d(t))+hk-1ξk-1(4lσ-τk-1-rk-1)/σ(xkpk-1-xk*pk-1)+hkξk(4lσ-τk-rk)/σxk+1pk+ξk(4lσ-τk-rk)/σfk+∑i=1k-1‍∂Wk∂xi(hixi+1+fi)+12∑i,j=1,i≠jk-1‍|∂2Wk∂xi∂xj||giT||gjT|+12∑i=1k-1‍|∂2Wk∂xi2||giT|2+12∑i=1k-1‍|∂2Wk∂xk∂xi||gkT||giT|+12|∂2Wk∂xk2||gkT|2+n-k+11-ηξk4l-(n-k+1)(1-d˙(t))1-ηξk4l(t-d(t)).In order to proceed further, an appropriate bounding estimate should be given for each term on the right-hand side of (22). This is accomplished in the following propositions whose technical proofs are given in the Appendix.Proposition 11.
There exists a positive constantlk1 such that
(23)hk-1ξk-1(4lσ-τk-1-rk-1)/σ(xkpk-1-xk*pk-1)≤14ξk-14l+ξk4llk1.Proposition 12.
There exists a nonnegative smooth functionlk2 such that
(24)ξk(4lσ-τk-rk)/σfk≤18∑i=1k-1‍ξi4l+18∑i=1k‍ξi4l(t-d(t))+ξk4llk2.Proposition 13.
There exists a nonnegative smooth functionlk3 such that
(25)∑i=1k-1‍∂Wk∂xi(hixi+1+fi)≤18∑i=1k-1‍ξi4l+18∑i=1k‍ξi4l(t-d(t))+ξk4llk3.Proposition 14.
There exists a nonnegative smooth functionlk4 such that
(26)12∑i,j=1,i≠jk-1‍|∂2Wk∂xi∂xj||giT||gjT|≤18∑i=1k-1‍ξi4l+18∑i=1k‍ξi4l(t-d(t))+ξk4llk4.Proposition 15.
There exists a nonnegative smooth functionlk5 such that
(27)12∑i=1k-1‍|∂2Wk∂xi2||giT|2≤18∑i=1k-1‍ξi4l+18∑i=1k‍ξi4l(t-d(t))+ξk4llk5.Proposition 16.
There exists a nonnegative smooth functionlk6 such that
(28)12∑i=1k-1‍|∂2Wk∂xk∂xi||gkT||giT|≤18∑i=1k-1‍ξi4l+18∑i=1k‍ξi4l(t-d(t))+ξk4llk6.Proposition 17.
There exists a nonnegative smooth functionlk7 such that
(29)12|∂2Wk∂xk2||gkT|2≤18∑i=1k-1‍ξi4l+18∑i=1k‍ξi4l(t-d(t))+ξk4llk7.Substituting (23)–(29) into (22) yields
(30)ℒVk≤-(n-k+1)∑i=1k-1‍ξi4l-(n-k)∑i=1k‍ξi4l(t-d(t))+hkξk(4lσ-τk-rk)/σ(xk+1pk-xk+1*pk)+hkξk(4lσ-τk-rk)/σxk+1*pk+ξk4l(∑j=17‍lkj+n-k+11-η).
Now, it easy to see that the virtual controller
(31)xk+1*=-1ck1(n-k+1+∑j=17‍lkj+n-k+11-η)ξk(rk+τk)/σpk:=-βkrk+1/σξkrk+1/σ
renders
(32)ℒVk≤-(n-k+1)∑i=1k‍ξi4l-(n-k)∑i=1k‍ξi4l(t-d(t))+hkξk(4lσ-τk-rk)/σ(xk+1pk-xk+1*pk).Finally, whenk=n, xn+1=xn+1*=u is the actual control. By choosing the actual control law,
(33)u=-βnrn+1/σξnrn+1/σ,
we get
(34)ℒVn≤-∑i=1n‍ξi4l,
where
(35)Vn=∑i=1n‍Wi+∑i=1n‍n-i+11-η∫t-d(t)t‍ξi4l(s)ds.We are now ready to state the main theorem of this paper.Theorem 18.
If Assumptions6 and 7 hold for stochastic high-order nonlinear time-delay system (1), under the state-feedback controller (33), then the closed-loop system has a unique solution on [-d,∞) and the equilibrium at the origin of the closed-loop system is GAS in probability.Proof .
We prove Theorem18 in two steps.
Step
1. We first prove that upn in (1) satisfies the locally Lipschitz condition.
From (17), (33), and rn+1pn=rn+τn, one has
(36)upn=-(β¯1x1σ/r1+⋯+β¯nxnσ/rn)(rn+τn)/σ,
where β¯i=βn,…,βi. Then, for i=1,…,n,
(37)∂upn∂xi=-Ai(β¯1x1σ/r1+⋯+β¯nxnσ/rn)(rn+τn-σ)/σ,
where
(38)Ai=rn+τnriβ¯ixi(σ-ri)/ri+∑j=1n-1‍rj+τjσ∂β¯j∂xixjσ/rj.
By the definition ofσ, we know that rn+τ≥σ≥max1≤i≤n{ri}>0, which implies that
(39)σ-riri≥0,rn+τn-σσ≥0,
from which, (37), (38), and βi’s being smooth, we obtain that ∂upn/∂xi is continuous. Therefore, upn is C1.
Sincefi and gi are assumed to be locally Lipschitz, so the system consisting of (1), (17), and (33) satisfies the locally Lipschitz condition.
Step
2. Then we prove that the origin of closed-loop system is GAS in probability.
From the definitions ofWi’s, we easily see that ∑i=1n‍Wi, as the function of x, is positive, definite, and radially unbounded. Then, by (34), (35), and Lemma 4.3 in [41], there exist K∞ functions β1, β2, and β3 such that
(40)β1(|ξ(t)|)≤Vn(t,ξt(θ))≤β2(sup-d≤θ≤0|ξt(t+θ)|),ℒVn≤-β3(|ξ(t)|).
With the help of Lemma 2, we conclude that the closed-loop system has a unique solution on [-d,∞), and the equilibrium ξ=0 is GAS in probability. This together with the definitions of xi*’s directly concludes that the origin x=0 of system (1) is also GAS in probability. Thus, the proof is completed.Remark 19.
In the deterministic work [29], strong stability theory [42] guarantees the existence of a solution; hence, the locally Lipschitz condition need not to be satisfied. While for stochastic system, according to the existing stochastic stability theory, to guarantee the existence and uniqueness of the solution, the designed controller must guarantee that the closed-loop system satisfies the locally Lipschitz condition. What should be emphasized is that how to guarantee the constructed Lyapunov functional being C2 and the closed-loop systems satisfying the locally Lipschitz condition simultaneously is one of the difficulties in this paper.Remark 20.
Since Itô stochastic differentiation involves not only the gradient but also the higher order Hessian term in the Lyapunov design procedure, many tedious nonlinear terms will arise in the design process, especially in the case of the appearance of high order, time-varying delay and nonlinear assumption. How to deal with these terms is another difficulty of this paper.Remark 21.
From the above design procedure, we can see that the upper bound of the change rate of time delays has important impact on the control effort. To keep the control effort within the certain range, the upper bound of the change rate of time delays cannot be arbitrarily close to 1, which should be considered in practical engineering design.


## 4. Simulation Example

To illustrate the effectiveness of the proposed controller, we consider the following low-dimensional system(41)dx1=(1.5+0.5cost)x25/3dt+14x15/3(t-d(t))dt,dx2=u3+14x1sinx1dt+18x24/3(t-d(t))dw,
where d(t)=(1/3)(1+sin(t)).It is evident that, even though system (41) is simple, it could not be globally asymptotically stabilized using the design method presented in [34, 35] because of the unsatisfiability of the power order restriction. However, if we choose τ1=τ2=2/3 which together with r1=1 and p1=5/3implies that r2=1, it is easy to get |f1|≤(1/4)|x1(t-d(t))|5/3, |f2|≤(1/4)|x1|5/3, and |g2|≤(1/8)|x2(t-d(t))|4/3. Clearly, Assumption 7 is satisfied. Moreover, noting that 1≤1.5+0.5cost≤2 and d˙(t)=(1/3)cost≤(1/3)<1, the controller proposed in this paper is applicable.By choosingl=1 and σ=1, following the design procedure given in Section 3, we can get
(42)x2*=-(5+l11)x1,u=-(52+l21+l22+l23)5/27ξ25/27,
where β1=5+l11, ξ2=x2+β1x1, l11=(7/12)×(5/6)5/7×(1/4)12/7, l21=54×(10/3)4+(5/12) × (3/7)7/5 × (10/3)12/5, l22=(5/12)×(3/7)7/5 × (1/4)12/5, and l23=2+2β14/3+(3/4)×(1+β14/3)3.In the simulation, we choose the initial valuesx1(0)=1 and x2(0)=-1. Figure 1 gives the responses of (41) and (42). From the figure, we can see that under the constructed controller, the solution process of the closed-loop system asymptotically converges to zero almost surely. We can also see that a little larger control effort is needed at the beginning. As mentioned in [32], when there exist stochastic disturbances and time delays, the effort of a controller designed based on the backstepping method is bigger than the common case, to which attention should be paid in practical use.The responses of closed-loop system (41) and (42).
(a)
System states
(b)
Control input


## 5. Conclusion

This paper deals with the state-feedback stabilization problem for a class of stochastic high-order nonlinear time-delay systems under weaker conditions. The designed state-feedback controller ensures that the origin of the closed-loop system is GAS in probability. It should be noted that the proposed controller can only work well when the whole state vector is measurable. Therefore, a natural and more interesting problem is how to design output feedback stabilization controller for the systems studied in the paper if only partial state vector is being measurable, which is now under our further investigation.



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*Source: 673869-2014-01-30.xml*