# Output-Feedback Control for a Class of Stochastic High-Order Feedforward Nonlinear Systems with Delay

Authors: Qian Ma; Ticao Jiao; Guozeng Cui
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

The problem of global output-feedback stabilization for a class of stochastic high-order time-delay feedforward nonlinear systems with different power orders is investigated. By combining the adding one power integrator technique with the homogeneous domination approach, an output-feedback controller design is proposed, which ensures the global asymptotical stability in probability of the closed-loop system.


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## Body

## 1. Introduction

In the last decades, stochastic systems have received much more attention since stochastic modeling has come to play an important role in many branches of science and engineering applications. For this type of systems, the authors in [1–10] presented the basic stability theory of the stochastic control systems.Global stabilization of triangular structural stochastic nonlinear systems has been a major issue in control theory over the last decades. For lower-triangular systems (namely, feedback systems), the stochastic asymptotic stabilization has been studied by using backstepping approach; see [11–13] and the references therein. Upper-triangular systems, which are also called feedforward systems, have been fully used to model many physical devices, such as the ball-beam with a friction term [14] and the cart-pendulum system [15]. By using a homogeneous domination approach, the problem of global output-feedback stabilization was addressed in [16] for a class of upper-triangular systems with higher-order nonlinearities. Then, the case of lower-order nonlinearities was considered in [17]. With the help of a generalized definition of homogeneity, the problem of using small controls to globally stabilize a class of upper-triangular systems was investigated in [18]. Inspired by the works [19, 20], the problem of using a sampled data controller to globally stabilize a class of uncertain upper-triangular systems was considered in [21]. When parameter uncertainties appeared in a system model, an adaptive stabilizer for feedforward nonlinear systems with general dynamic order in [22] was extended to [23].On the other hand, time delays may arise naturally, which are usually the key factors that influence the stability of nonlinear systems. Many researchers have paid more attention to studying the stochastic nonlinear time-delay systems over the last decades and various results concerning stochastic lower-triangular nonlinear systems with time delays have been reported in [24–27]. For the case where the delays are of unknown length, an adaptive control design was presented in [28] for a system in feedforward form. An approach for compensating input delay of arbitrary length was presented [29] for forward complete and strict-feedforward nonlinear systems. When the nonlinear functions were higher order in states, the problem of global stabilization by state feedback was addressed in [30, 31]. It should be pointed out that these results were obtained for the case of state measurement. Naturally, one may ask a challenging and interesting question: if not all the states are measurable, how do we design an output-feedback controller for stochastic high-order feedforward systems with time delay? To the best of our knowledge, no output-feedback stabilization control scheme has so far been proposed for stochastic high-order feedforward systems with time delay.Inspired by the aforementioned discussion, we deal with the problem of global output-feedback stabilization for a class of stochastic high-order feedforward nonlinear systems with time-varying delay and different power orders. Firstly, we design a state-feedback controller for the nominal system using the adding one power integrator technique. Then, by designing a homogeneous observer for the nominal system, we explicitly construct a homogeneous output-feedback stabilizer under the certainty equivalence principle. At last, we propose a scaled controller which guarantees global asymptotic stability in probability of the closed-loop system.The outline of this paper is as follows. Sections2 and 3 offer some preliminary results and problem formulation, respectively. The output-feedback controller is designed and analyzed in Section 4. This paper is concluded in Section 5.Notations. The following standard notations are used throughout this paper. Rn denotes the real n-dimensional space. |x|=(∑i=1n‍xi2)1/2 for any x∈Rn. For any matrix Q∈Rn×Rm, |Q| denotes the Frobenius norm |Q|=(∑i,j‍qij2)1/2 and |Q|∞=max⁡1≤i≤n(∑j=1m‍qij2)1/2. For the sake of simplicity, sometimes function x(t) is denoted by x; xi represents the ith element of vector x and x¯i=(x1,…,xi)T. Ci denotes the set of all functions with continuous ith partial derivative. 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.


## 2. Preliminary Results

Consider the following stochastic time-delay system(1)dx(t)=f(x(t),x(t-σ(t)))dt+g(x(t),x(t-σ(t)))Tdω
with an initial condition {x(s):-σ0≤s≤0}=ξ∈CF0b×([-σ0,0],Rn), where σ(t):R+→[0,σ0] is a Borel measurable function; x(t)∈Rn denotes the state vector and x(t-σ(t)) is the state vectors with time-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; f(·)∈Rn, g(·)∈Rn×m are locally Lipschitz functions and satisfy f(0,0)=0,  g(0,0)=0.In the following, we borrow some definitions and lemmas which play an important role in stabilizing and analyzing the stochastic time-delay systems for later development in this paper.Definition 1 (see [26]).
Define the infinitesimal generatorℒ of function V(x)∈C2 along system (1) as follows:
(2)ℒV(x)=∂V∂zf(x,x(t-σ(t)))+12Tr⁡{g(x,x(t-σ(t)))∂2V∂x2g×(x,x(t-σ(t)))T(x,x(t-σ(t)))∂2V∂x2},
in which (1/2)Tr⁡{g(x,x(t-σ(t)))(∂2V/∂x2)g(x,x(t-σ(t)))T} is called the Hessian term of ℒ.Definition 2 (see [26]).
The equilibriumx=0 of system (1) is globally asymptotically stable (GAS) in probability if for any ϵ>0, there exists a function γ(·,·)∈𝒦ℒ such that P{|x|≤γ(∥ζ∥,t)}≥1-ϵ for any t≥0, ζ∈CF0b×([-τ,0],Rn)∖{0}, where ∥ζ∥=sup⁡θ∈[-τ,0]|ζ(θ)|.Definition 3 (see [16]).
For fixed coordinates(x1,…,xn) and real numbers ri>0,i=1,…,n, one has the following.(i)
The dilationΔɛ(x) is defined by Δɛ(x)=(ɛr1x1,…,ɛrnxn),  for all ɛ>0, with ri being called as the weights of the coordinates. For simplicity of notation, 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 ∀x∈Rn∖{0},  ɛ>0.(iii)
A vector fieldf∈C(Rn,Rn) is said to be homogeneous of degree τ if there is a real number τ∈R such that fi(Δɛ(x))=ɛτ+rifi(x1,…,xn), for ∀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 all x∈Rn, for a constant p≥1. For the simplicity, we choose p=2 and write ∥x∥Δ for ∥x∥Δ,2 in this paper.Lemma 4 (see [26]).
Consider the stochastic time-delay system (1); if there exist a function V(x)∈C2([-σ0,∞]×Rn) and class 𝒦∞ functions α1,α2 satisfying the following inequalities:
(3)α1(|x(t)|)≤V(x)≤α2(sup⁡(-σ0≤s≤0)⁡|x(t+s)|),ℒV(x)≤-W(x(t)),
where W(x(t)) is continuous and positive definite, then(1)
there exists a unique solution on[-σ0,∞),(2)
the equilibriumx=0 of the system (1) is GAS in probability and P{lim⁡t→∞|x(t)|=0}=1.Lemma 5 (see [16]).
Given a dilation weightΔ=(r1,…,rn), suppose V1(x) and V2(x) are homogeneous functions of degree 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(x)·V2(x) is τ1+τ2.Lemma 6 (see [16]).
SupposeV:Rn→R is a homogeneous function of degree τ with respect to the dilation weight Δ. Then, the following holds.(i)
∂V/∂xi is homogeneous of degree τ-ri with ri being the homogeneous weight of xi.(ii)
There is a constantc such that V(x)≤c∥x∥Δτ. Moreover, if V(x) is positive definite, then
(4)c_∥x∥Δτ≤V(x),foraconstantc_>0.Lemma 7 (see [16]).
For anyx∈R,  y∈R, and p≥1, the following inequalities hold:
(5)|x+y|p≤2p-1|xp+yp|,(|x|+|y|)1/p≤|x|1/p+|y|1/p.
Moreover, when p∈Rodd≥1,
(6)|x-y|p≤2p-1|xp-yp|,|xp-yp|≤p|x-y||xp-1+yp-1|≤c|x-y||(x-y)p-1+yp-1|,
where c>0 is a constant.Lemma 8 (see [27]).
Letx and y be any real numbers and p∈Rodd≥1; then
(7)-(x-y)(xp-yp)≤-12p-1(x-y)p+1.Lemma 9 (see [27]).
Letc and d be two positive real numbers. For any positive number γ>0, the following inequality holds
(8)|x|c|y|d≤cc+dγ|x|c+d+dc+dγ-c/d|y|c+d.


## 3.  Problem Formulation

In this paper, we will consider the following stochastic high-order feedforward nonlinear system with time-varying delay in the form(9)dz1=z2p1dt+f1(z,z(t-σ(t)))+g1T(z,z(t-σ(t)))dω,dz2=z3p2dt+f2(z,z(t-σ(t)))+g2T(z,z(t-σ(t)))dω,⋮dzn=υpn+fn(z,z(t-σ(t)))+gnT(z,z(t-σ(t)))dω,y=z1,
where z=[z1,…,zn]T∈Rn, y∈R, υ∈R denote the system state, output, and input, respectively; z(t-σ(t))=[z1(t-σ(t)),…,zn(t-σ(t))] is the state vector with time-delay; σ(t):R+→[0,σ0] is a time-varying delay. ω is an m-dimensional standard Wiener process. For i=1,…,n, pi∈Rodd≥3≜{p∈R:p≥3 and p is a ratio of odd integers}. The drift functions fi:Rn×Rn→R and the diffusion functions gi:Rn×Rn→Rm are locally Lipschitz in (z,z(t-σ(t))) and vanish at the origin.In order to obtain the main results of this paper, the following assumptions are needed.Assumption 10.
Fori=1,…,n-1, there are constants ai>0,bi>0, such that
(10)|fi(z,z(t-σ(t)))|≤ai∑j=i+2n+1‍(|zj(t)|(ri+τ)/rj+|zj(t-σ(t))|(ri+τ)/rj),|gi(z,z(t-σ(t)))|≤bi∑j=i+2n+1‍(|zj(t)|(2ri+τ)/2rj+|zj(t-σ(t))|(2ri+τ)/2rj),
where pi∈Rodd≥3, τ∈[d0,∞), r1=1,  ri+1=(ri+τ)/pi,  d1=(2-1/(p1⋯pn-1))/(1+∑s=1n-1‍(1/(ps⋯pn-1))), di+1=(2/(p1⋯pi)-1/(p1⋯pn-1))/(1+∑s=1n-1‍(1/(ps⋯pn-1))-∑s=1i‍(2/(ps⋯pi))), and d0=max⁡j=1,…,n{dj}.Assumption 11.
The time-varying delayσ(t) satisfies σ˙(t)≤σ<1 for a constant σ.In this paper, we aim to constructively design a homogeneous output-feedback controller(11)χ˙(t)=ν(χ(t),y(t)),u(t)=ω(χ(t),y(t)),
for system (9) under Assumptions 10-11, such that the closed-loop system is GAS in probability.


## 4. Controller Design and Stability Analysis

In this section, an output-feedback controller will be explicitly constructed for the nonlinear system (9). We will combine the adding one power integrator technique with the homogeneous domination approach for output-feedback stabilization. The design procedure can be divided into three steps: (i) we first design a state-feedback controller for the nominal system without the drift and diffusion functions using the adding one power integrator technique; (ii) then, by designing a homogeneous observer for the nominal system, we explicitly construct a homogeneous output-feedback stabilizer under the certainty equivalence principle; (iii) in the end, we propose a scaled controller which guarantees global asymptotic stability in probability of the closed-loop system.For simplicity, we assumeτ=m/n in this paper with m being an even integer and n being an odd integer. Based on this assumption, it is obvious that ri is a ratio of odd integers.


### 4.1.  Homogeneous State-Feedback Control of Nominal Nonlinear System

In this subsection, we firstly introduce a key lemma, which avoids the zero-division problem and serves as a basis in the following design procedure. The proof is similar to that in [32].Lemma 12.
Ifpi∈Rodd≥3, τ∈[d0,+∞) are guaranteed, then we can find l0,μ0∈Rodd+, such that
(12)max⁡1≤i≤n{2ri,ri+τl0}≤μ0≤rn+τ
holds.Now, we design a homogeneous state-feedback controller for the nominal system(13)dx1=x2p1dt,dx2=x3p2dt,⋮dxn=upndt.Lemma 13.
For system (13), there are positive definite, proper, and C2 Lyapunov function Vn, a state-feedback controller u*, and two positive constants, such that
(14)ℒVn≤-∑j=1n‍πn,jηj4l0+ηnqn/μ0(upn-u*pn),
where Vn,  u* are defined in the following form:
(15)Vn(x¯n)=∑i=1n‍∫xi*xi‍(sμ0/ri-xi*μ0/ri)qi/μ0ds,x1*=0,η1=x1μ0/r1,x2*=-β1r2/μ0η1r2/μ0,η2=x2μ0/r2-x2*μ0/r2,⋮⋮xn*=-βn-1rn/μ0ηn-1rn/μ0,ηn=xnμ0/rn-xn*μ0/rn,u*=-(πnn+bn1+bn2)1/pnηnrn+1/μ0≜-βnrn+1/μ0ηnrn+1/μ0=-(β¯nxnμ0/rn+β¯n-1xn-1μ0/rn-1+⋯+β¯1x1μ0/r1)rn+1/μ0,β¯j=βn⋯βj,j=1,…,n.Remark 14.
For the deterministic work, it is well known that there only needs a positive definite, proper, andC1 Lyapunov function for the controller design and analysis. However, for the stochastic system, the Lyapunov function must be positive definite, proper, and at least C2 because of the appearance of the Hessian term.


### 4.2. Construction of a Homogeneous Observer

Sincex2,…,xn are unmeasurable, we design a reduced-order homogeneous observer
(16)η^˙2=-k2x^2p1,x^2=[η^2+k2x^1]r2/r1,η^˙3=-k3x^3p2,x^3=[η^3+k3x^2]r3/r2,⋮⋮η^˙n*=-knx^npn-1,x^n=[η^n+knx^n-1]rn/rn-1,
where k2>0,…,kn>0 are constant gains to be determined later and x^1=x1.According to the certainty equivalence principle, we use the estimated statesx^2,…,x^n to construct the implementable controller
(17)u(x^)=-(β¯nx^nμ0/rn+β¯n-1x^n-1μ0/rn-1+⋯+β¯1x^1μ0/r1)rn+1/μ0,
where x^=(x^1,x^2,…,x^n).In order to prove the global stability for the closed-loop system (13)–(16)-(17), we define
(18)Uj=∫λjqj-1/rj-1xjqj-1/rj‍(srj-1/qj-1-λj)ds,λj=η^j+kjx^j-1,
and let
(19)ej=(xjpj-1-x^jpj-1)μ0/rjpj-1,
for j=2,…,n.Therefore, the differential operatorℒ of Uj is
(20)ℒ(Uj)=qj-1rjxjqj-1/rj-1(xjrj-1/rj-λj)xj+1pj-kj(xjqj-1/rj-λjqj-1/rj-1)xjpj-1+kj(xjqj-1/rj-λjqj-1/rj-1)x^jpj-1=qj-1rjxjqj-1/rj-1(xjrj-1/rj-λj)xj+1pj-kj[(xjqj-1/rj-x^jqj-1/rj)+(x^jqj-1/rj-λjqj-1/rj-1)]ejrjpj-1/μ0,
where xn+1=u(x^).Just as in [16], each term of the right-hand side of (20) can be estimated by the following propositions, whose proofs are omitted here.Proposition 15.
Forj=2,…,n-1,
(21)qj-1rjxjqj-1/rj-1(xjrj-1/rj-λj)xj+1pj≤∑k=j-1j+1‍δj,k,3ηk4l0+gj1(kj)ej-14l0+bj3ej4l0,
where δj,k,3>0,  bj3>0, gj1(kj) is a continuous function with g21(k2)=0.Proposition 16.
For the implementable controlleru(x^) in (17), one obtains
(22)qj-1rjxjqj-1/rj-1(xjrj-1/rj-λj)u(x^)pn≤∑j=1n‍δn,j,4ηj4l0+gn1(kn)en-14l0+∑j=2n‍bj4ej4l0
with positive constants δn,j,4,bj4 and with  gn1(kn)  being a continuous function.Proposition 17.
Forj=2,…,n, there exists a constant ϖj=21-qj-1/rjpj-1>0 such that
(23)-kj(xjqj-1/rj-x^jqj-1/rj)ejrjpj-1/μ0≤-kjϖjej4l0.Proposition 18.
Forj=3,…,n,
(24)-kj(x^jqj-1/rj-λjqj-1/rj-1)ejrjpj-1/μ0≤∑k=j-1j‍δj,k,5ηk4l0+gj2(kj)ej-14l0+bj5ej4l0
with positive constants δj,k,5,  bj5 and with gj2(kj) being a continuous function.Considering the Lyapunov functionU=∑j=2n‍Uj and substituting (21)–(24) into (20), one concludes
(25)ℒU≤∑j=1n‍ϵj1ηj4l0+e24l0×[-k2ϖ2+b23+b24+g31(k3)+g32(k3)]+∑j=3n-1‍ej4l0[-kjϖj+bj3+bj4+bj5+gj+1,1(kj+1)+gj+1,2(kj+1)]+en4l0[-knϖn+bn4+bn5].To determinate the observer gainkj in (16), we have the following proposition to deal with the redundant term ηnqn/μ0(upn(x^)-u*pn) in (14).Proposition 19.
There are constantsϵj2>0,bj6>0 such that
(26)ηnqn/μ0(upn(x^)-u*pn)≤∑j=1n‍ϵj2ηj4l0+∑j=2n‍bj6ej4l0.Defining the Lyapunov functionV=Vn+U and combining (14), (25), and (26) together yield
(27)ℒV≤-∑j=1n‍πn,jηj4l0+∑j=1n‍ϵj2ηj4l0+∑j=1n‍ϵj1ηj4l0+∑j=2n‍bj6ej4l0+e24l0×[-k2ϖ2+b23+b24+g31(k3)+g32(k3)]+∑j=3n-1‍ej4l0[-kjϖj+bj3+bj4+bj5+gj+1,1(kj+1)+gj+1,2(kj+1)]+en4l0[-knϖn+bn4+bn5].It is clear that by choosing(28)lj≜πn,j-ϵj1-ϵj2>0,(j=1,…,n)kn≥1ϖn(φn+bn4+bn5+bn6),kn-1≥1ϖn-1((kn)φn-1+bn-1,3+bn-1,4+bn-1,5+bn-1,6+gn,1(kn)+gn,2(kn)),⋮k2≥1ϖ2(φ2+b2,3+b2,4+b2,6+g3,1(k3)+g3,2(k3)),
one leads to
(29)ℒV≤-∑j=1n‍ljηj4l0-∑j=2n‍φjej4l0
with positive constants φ2,…,φn.By denotingΦ=(x1,…,xn,η^2,…,η^n)T, the closed-loop system (13), (16), and (17) can be rewritten as
(30)dΦ=E(Φ)dt=[x2p1,…,xn-1pn-2,upn(x^),η^˙2,…,η^˙n]dt,
which is homogeneous of degree τ with respect to Δ=(r1,…,rn︸forx1,…,xn,r1,…,rn-1︸forη^2,…,η^n). Following a similar way in [16], it can be shown that
(31)V(Φ)=∑j=1n‍∫xj*xj‍(sμ0/rj-xi*μ0/rj)qj/μ0ds+∑j=2n‍∫λjqj-1/rj-1xjqj-1/rj‍(srj-1/qj-1-λj)ds
is homogeneous of degree 4l0μ0-τ about Δ.


### 4.3. Stability Analysis

To state the main result in this paper, we first introduce the following coordinate transformation:(32)x1=z1,xi=ziLsi,upn=vpnLsn+1,
where s1=0,  si=(si-1+1)/pi-1,  i=2,…,n,  0<L<1 is a constant to be determined later.It follows from (32) that the system (9) can be rewritten as
(33)dx1=Lx2p1dt+f1(x,x(t-σ(t)))+g1T(x,x(t-σ(t)))dω,dx2=Lx3p2dt+f2(x,x(t-σ(t)))Ls2+g2T(x,x(t-σ(t)))Ls2dω,⋮dxn=Lupn+fn(x,x(t-σ(t)))Lsn+gnT(x,x(t-σ(t)))Lsndω,y=z1.Next, we construct a scaled homogeneous observer(34)η^˙2=-Lk2x^2p1,x^2=[η^2+k2x^1]r2/r1,η^˙3=-Lk3x^3p2,x^3=[η^3+k3x^2]r3/r2,⋮⋮η^˙n*=-Lknx^npn-1,x^n=[η^n+knx^n-1]rn/rn-1,
where k2>0,…,kn>0 are the gains selected in (28) and the controller u is designed with the same construction of (17). The closed-loop system (33), (34), and (17) can be expressed as
(35)dΦ=LE(Φ)dt+Fdt+GTdω,
where F=(f1,f2/Ls2,…,fn/Lsn,0,…,0), G=(g1,g2/Ls2,…,gn/Lsn,0,…,0). Based on the above discussion, we are ready to prove the main result of this paper.Theorem 20.
Under Assumptions10 and 11, there is an output-feedback controller vpn=upnLsn+1 such that the closed-loop system consisting of (9), (16), and (17) has a global unique solution and the equilibrium x=0 is GAS in probability.Proof.
We prove Theorem20 as follows.
Step  1. From (17), for j=1,…,n, one obtains
(36)∂upn(x^)∂x^j=-rn+τrjβ¯jx^jμ0/rj-1×(β¯nx^nμ0/rn+β¯n-1x^n-1μ0/rn-1+⋯+β¯1x^1μ0/r1)(rn+τ)/μ0-1.
By Lemma12, we know μ0/rj-1≥0, (rn+τ)/μ0-1≥0, which implies that upn(x^) is C1. Therefore, the closed-loop system satisfies the locally Lipschitz condition.
Step  2. Construct a Lyapunov-Krasovskii functional
(37)W(Φ)=V(Φ)+L1+ɛ0(a0+b0)1-σ∫t-σ(t)t‍∥Φ(s)∥Δ4l0μ0ds,
which is positive definite, proper, and C2 on Φ, where ɛ0,a0,b0 are positive parameters to be determined later. According to Lemma 4.3 in [33], there are two 𝒦∞ functions α1 and α21 such that
(38)α1(|Φ(t)|)≤V(Φ)≤α21(|Φ(t)|).
Then, with the notation in [26], it is easy to verify that
(39)L1+ɛ0(a0+b0)1-σ∫t-σ(t)t‍∥Φ(s)∥Δ4l0μ0ds≤α22(sup⁡-σ0≤s≤0|Φ(t+s)|),
where α22 is a 𝒦∞ function. Because α21(|Φ(t)|)≤α21(sup⁡-σ0≤s≤0|Φ(t+s)|), then
(40)α1(|Φ(t)|)≤W(Φ)≤α2(sup⁡-σ0≤s≤0|Φ(t+s)|)
with α2=α21+α22.
Step  3. Since V(Φ) and E(Φ) are homogenous of degree 4l0μ0-τ and τ, respectively, by Lemmas 5 and 6, one has
(41)∂W(Φ)∂XLE(Φ)=∂V(Φ)∂XLE(X)≤-Lγ1∥Φ(s)∥Δ4l0μ0,
where γ1>0 is a constant.
Using Assumption10 and the new coordinates in (32), we have
(42)|fiLsi|≤ai∑j=i+2n+1‍(|Lsjxj(t)|(ri+τ)/rj+|Lsjxj(t-σ(t))|(ri+τ)/rjLsi)
and the power of L in (42) is
(43)sjri+τrj-si=1+sj(ri+τ)-(si+1)rjrj=1+pirj(sjri+1-si+1rj).
Owing tos1=0,  si=(si-1+1)/pi-1, one has
(44)si=∑s=1i-1‍1ps⋯pi-1,i=2,…,n.
Forj≥i+2, (44) gives
(45)sjri+1-si+1rj=sj(1p1⋯pi+si+1τ)-si+1(1p1⋯pj-1+sjτ)=sjp1⋯pi-si+1p1⋯pj-1>0.
In summary, it can be concluded that there exists a positive constant(46)ɛ1=min⁡{pirj(sjri+1-si+1rj):1≤i≤n-1,i+2≤j≤n}
such that
(47)|fiLsi|≤aiL1+ɛ1∑j=i+2n+1‍(|xj(t)|(ri+τ)/rj+|xj(t-σ(t))|(ri+τ)/rj)≤a^iL1+ɛ1(∥Φ(t)∥Δri+τ+∥Φ(t-σ(t))∥Δri+τ)
with a^i>0. According to Lemma 6, we deduce that
(48)|∂V(Φ)∂ΦF|≤∑i=1n-1‍|∂V(Φ)∂Φi||fiLsi|≤∑i=1n-1‍a^iL1+ɛ1∥Φ(t)∥Δ4l0μ0-ri-τ×(∥Φ(t)∥Δri+τ+∥Φ(t-σ(t))∥Δri+τ)≤a0L1+ɛ1(∥Φ(t)∥Δ4l0μ0+∥Φ(t-σ(t))∥Δ4l0μ0),
where a0 is a positive constant.
Similarly, there is a positive constant(49)ɛ2=min⁡{sj(2ri+τ)-(2si+1)rj2rj:1≤i≤n-1,aaaaaaaaaaaaaaaaaaaaaaaaaai+2≤j≤nsj(2ri+τ)-(2si+1)rj2rj},
such that
(50)|giLsi|≤biL1/2+ɛ2×∑j=i+2n+1‍(|Φj(t)|(2ri+τ)/2rj+|Φj(t-σ(t))|(2ri+τ)/2rj)≤b^iL1/2+ɛ2(∥Φ(t)∥Δri+τ/2+∥Φ(t-σ(t))∥Δri+τ/2)
with b^i>0. Then,
(51)12Tr⁡{G∂2V(Φ)∂Φ2GT}≤12mm∑k,l=1n-1‍|∂2V(Φ)∂Φk∂Φl||gkLsk||glLsl|≤b~L1+2ɛ2×∑k,l=1n-1‍∥Φ(t)∥4l0μ0-τ-rk-rl×(∥Φ(t)∥Δrk+τ/2+∥Φ(t-σ(t))∥Δrk+τ/2)×(∥Φ(t)∥Δrl+τ/2+∥Φ(t-σ(t))∥Δrl+τ/2)≤b0L1+2ɛ2(∥Φ(t)∥Δ4l0μ0+∥Φ(t-σ(t))∥Δ4l0μ0),
where b~,b0 are positive constants.
By choosingɛ0=min⁡{ɛ1,2ɛ2} and Definition 1, Assumption 11, (41), (48), and (51), one can obtain
(52)ℒW(Φ)≤∂W(Φ)∂XLE(Φ)+∂V(Φ)∂XF+12Tr⁡{G∂2V(Φ)∂Φ2GT}+L1+ɛ0(a0+b0)×(11-σ∥Φ(t)∥Δ4l0μ0-∥Φ(t-σ(t))∥Δ4l0μ0)≤-L(γ1-Lɛ0(a0+b0+a0+b01-σ))∥Φ(t)∥Δ4l0μ0,
where 0<L<1. Apparently, by choosing 0<L<min⁡{(γ1/(a0+b0+(a0+b0)/(1-σ)))1/ɛ0,1}, there is a constant δ, such that ℒW(Φ)≤-δ∥Φ(t)∥Δ4l0μ0.
From Steps  1 to 3 and Lemma4, it is obtained that the closed-loop system (33), (34), and (17) has a global unique solution and the equilibrium Φ=0 is GAS in probability.
Step  4. Because of the equivalence in (32), then there is a global unique solution for the closed-loop system consisting of (9), (16), and (17) and vpn=upnLsn+1 and the equilibrium z=0 is GAS in probability.Remark 21.
Using the adding one power integrator method and the homogeneous domination technique, [34] has designed a state-feedback stabilizer for a class of stochastic high-order feedforward nonlinear systems. However, to deal with the unmeasurable states and time-delay terms in this paper, we choose a reduced-order homogeneous observer and an appropriate Lyapunov-Krasovskii functional, which is not an easy work.


## 4.1.  Homogeneous State-Feedback Control of Nominal Nonlinear System

In this subsection, we firstly introduce a key lemma, which avoids the zero-division problem and serves as a basis in the following design procedure. The proof is similar to that in [32].Lemma 12.
Ifpi∈Rodd≥3, τ∈[d0,+∞) are guaranteed, then we can find l0,μ0∈Rodd+, such that
(12)max⁡1≤i≤n{2ri,ri+τl0}≤μ0≤rn+τ
holds.Now, we design a homogeneous state-feedback controller for the nominal system(13)dx1=x2p1dt,dx2=x3p2dt,⋮dxn=upndt.Lemma 13.
For system (13), there are positive definite, proper, and C2 Lyapunov function Vn, a state-feedback controller u*, and two positive constants, such that
(14)ℒVn≤-∑j=1n‍πn,jηj4l0+ηnqn/μ0(upn-u*pn),
where Vn,  u* are defined in the following form:
(15)Vn(x¯n)=∑i=1n‍∫xi*xi‍(sμ0/ri-xi*μ0/ri)qi/μ0ds,x1*=0,η1=x1μ0/r1,x2*=-β1r2/μ0η1r2/μ0,η2=x2μ0/r2-x2*μ0/r2,⋮⋮xn*=-βn-1rn/μ0ηn-1rn/μ0,ηn=xnμ0/rn-xn*μ0/rn,u*=-(πnn+bn1+bn2)1/pnηnrn+1/μ0≜-βnrn+1/μ0ηnrn+1/μ0=-(β¯nxnμ0/rn+β¯n-1xn-1μ0/rn-1+⋯+β¯1x1μ0/r1)rn+1/μ0,β¯j=βn⋯βj,j=1,…,n.Remark 14.
For the deterministic work, it is well known that there only needs a positive definite, proper, andC1 Lyapunov function for the controller design and analysis. However, for the stochastic system, the Lyapunov function must be positive definite, proper, and at least C2 because of the appearance of the Hessian term.


## 4.2. Construction of a Homogeneous Observer

Sincex2,…,xn are unmeasurable, we design a reduced-order homogeneous observer
(16)η^˙2=-k2x^2p1,x^2=[η^2+k2x^1]r2/r1,η^˙3=-k3x^3p2,x^3=[η^3+k3x^2]r3/r2,⋮⋮η^˙n*=-knx^npn-1,x^n=[η^n+knx^n-1]rn/rn-1,
where k2>0,…,kn>0 are constant gains to be determined later and x^1=x1.According to the certainty equivalence principle, we use the estimated statesx^2,…,x^n to construct the implementable controller
(17)u(x^)=-(β¯nx^nμ0/rn+β¯n-1x^n-1μ0/rn-1+⋯+β¯1x^1μ0/r1)rn+1/μ0,
where x^=(x^1,x^2,…,x^n).In order to prove the global stability for the closed-loop system (13)–(16)-(17), we define
(18)Uj=∫λjqj-1/rj-1xjqj-1/rj‍(srj-1/qj-1-λj)ds,λj=η^j+kjx^j-1,
and let
(19)ej=(xjpj-1-x^jpj-1)μ0/rjpj-1,
for j=2,…,n.Therefore, the differential operatorℒ of Uj is
(20)ℒ(Uj)=qj-1rjxjqj-1/rj-1(xjrj-1/rj-λj)xj+1pj-kj(xjqj-1/rj-λjqj-1/rj-1)xjpj-1+kj(xjqj-1/rj-λjqj-1/rj-1)x^jpj-1=qj-1rjxjqj-1/rj-1(xjrj-1/rj-λj)xj+1pj-kj[(xjqj-1/rj-x^jqj-1/rj)+(x^jqj-1/rj-λjqj-1/rj-1)]ejrjpj-1/μ0,
where xn+1=u(x^).Just as in [16], each term of the right-hand side of (20) can be estimated by the following propositions, whose proofs are omitted here.Proposition 15.
Forj=2,…,n-1,
(21)qj-1rjxjqj-1/rj-1(xjrj-1/rj-λj)xj+1pj≤∑k=j-1j+1‍δj,k,3ηk4l0+gj1(kj)ej-14l0+bj3ej4l0,
where δj,k,3>0,  bj3>0, gj1(kj) is a continuous function with g21(k2)=0.Proposition 16.
For the implementable controlleru(x^) in (17), one obtains
(22)qj-1rjxjqj-1/rj-1(xjrj-1/rj-λj)u(x^)pn≤∑j=1n‍δn,j,4ηj4l0+gn1(kn)en-14l0+∑j=2n‍bj4ej4l0
with positive constants δn,j,4,bj4 and with  gn1(kn)  being a continuous function.Proposition 17.
Forj=2,…,n, there exists a constant ϖj=21-qj-1/rjpj-1>0 such that
(23)-kj(xjqj-1/rj-x^jqj-1/rj)ejrjpj-1/μ0≤-kjϖjej4l0.Proposition 18.
Forj=3,…,n,
(24)-kj(x^jqj-1/rj-λjqj-1/rj-1)ejrjpj-1/μ0≤∑k=j-1j‍δj,k,5ηk4l0+gj2(kj)ej-14l0+bj5ej4l0
with positive constants δj,k,5,  bj5 and with gj2(kj) being a continuous function.Considering the Lyapunov functionU=∑j=2n‍Uj and substituting (21)–(24) into (20), one concludes
(25)ℒU≤∑j=1n‍ϵj1ηj4l0+e24l0×[-k2ϖ2+b23+b24+g31(k3)+g32(k3)]+∑j=3n-1‍ej4l0[-kjϖj+bj3+bj4+bj5+gj+1,1(kj+1)+gj+1,2(kj+1)]+en4l0[-knϖn+bn4+bn5].To determinate the observer gainkj in (16), we have the following proposition to deal with the redundant term ηnqn/μ0(upn(x^)-u*pn) in (14).Proposition 19.
There are constantsϵj2>0,bj6>0 such that
(26)ηnqn/μ0(upn(x^)-u*pn)≤∑j=1n‍ϵj2ηj4l0+∑j=2n‍bj6ej4l0.Defining the Lyapunov functionV=Vn+U and combining (14), (25), and (26) together yield
(27)ℒV≤-∑j=1n‍πn,jηj4l0+∑j=1n‍ϵj2ηj4l0+∑j=1n‍ϵj1ηj4l0+∑j=2n‍bj6ej4l0+e24l0×[-k2ϖ2+b23+b24+g31(k3)+g32(k3)]+∑j=3n-1‍ej4l0[-kjϖj+bj3+bj4+bj5+gj+1,1(kj+1)+gj+1,2(kj+1)]+en4l0[-knϖn+bn4+bn5].It is clear that by choosing(28)lj≜πn,j-ϵj1-ϵj2>0,(j=1,…,n)kn≥1ϖn(φn+bn4+bn5+bn6),kn-1≥1ϖn-1((kn)φn-1+bn-1,3+bn-1,4+bn-1,5+bn-1,6+gn,1(kn)+gn,2(kn)),⋮k2≥1ϖ2(φ2+b2,3+b2,4+b2,6+g3,1(k3)+g3,2(k3)),
one leads to
(29)ℒV≤-∑j=1n‍ljηj4l0-∑j=2n‍φjej4l0
with positive constants φ2,…,φn.By denotingΦ=(x1,…,xn,η^2,…,η^n)T, the closed-loop system (13), (16), and (17) can be rewritten as
(30)dΦ=E(Φ)dt=[x2p1,…,xn-1pn-2,upn(x^),η^˙2,…,η^˙n]dt,
which is homogeneous of degree τ with respect to Δ=(r1,…,rn︸forx1,…,xn,r1,…,rn-1︸forη^2,…,η^n). Following a similar way in [16], it can be shown that
(31)V(Φ)=∑j=1n‍∫xj*xj‍(sμ0/rj-xi*μ0/rj)qj/μ0ds+∑j=2n‍∫λjqj-1/rj-1xjqj-1/rj‍(srj-1/qj-1-λj)ds
is homogeneous of degree 4l0μ0-τ about Δ.


## 4.3. Stability Analysis

To state the main result in this paper, we first introduce the following coordinate transformation:(32)x1=z1,xi=ziLsi,upn=vpnLsn+1,
where s1=0,  si=(si-1+1)/pi-1,  i=2,…,n,  0<L<1 is a constant to be determined later.It follows from (32) that the system (9) can be rewritten as
(33)dx1=Lx2p1dt+f1(x,x(t-σ(t)))+g1T(x,x(t-σ(t)))dω,dx2=Lx3p2dt+f2(x,x(t-σ(t)))Ls2+g2T(x,x(t-σ(t)))Ls2dω,⋮dxn=Lupn+fn(x,x(t-σ(t)))Lsn+gnT(x,x(t-σ(t)))Lsndω,y=z1.Next, we construct a scaled homogeneous observer(34)η^˙2=-Lk2x^2p1,x^2=[η^2+k2x^1]r2/r1,η^˙3=-Lk3x^3p2,x^3=[η^3+k3x^2]r3/r2,⋮⋮η^˙n*=-Lknx^npn-1,x^n=[η^n+knx^n-1]rn/rn-1,
where k2>0,…,kn>0 are the gains selected in (28) and the controller u is designed with the same construction of (17). The closed-loop system (33), (34), and (17) can be expressed as
(35)dΦ=LE(Φ)dt+Fdt+GTdω,
where F=(f1,f2/Ls2,…,fn/Lsn,0,…,0), G=(g1,g2/Ls2,…,gn/Lsn,0,…,0). Based on the above discussion, we are ready to prove the main result of this paper.Theorem 20.
Under Assumptions10 and 11, there is an output-feedback controller vpn=upnLsn+1 such that the closed-loop system consisting of (9), (16), and (17) has a global unique solution and the equilibrium x=0 is GAS in probability.Proof.
We prove Theorem20 as follows.
Step  1. From (17), for j=1,…,n, one obtains
(36)∂upn(x^)∂x^j=-rn+τrjβ¯jx^jμ0/rj-1×(β¯nx^nμ0/rn+β¯n-1x^n-1μ0/rn-1+⋯+β¯1x^1μ0/r1)(rn+τ)/μ0-1.
By Lemma12, we know μ0/rj-1≥0, (rn+τ)/μ0-1≥0, which implies that upn(x^) is C1. Therefore, the closed-loop system satisfies the locally Lipschitz condition.
Step  2. Construct a Lyapunov-Krasovskii functional
(37)W(Φ)=V(Φ)+L1+ɛ0(a0+b0)1-σ∫t-σ(t)t‍∥Φ(s)∥Δ4l0μ0ds,
which is positive definite, proper, and C2 on Φ, where ɛ0,a0,b0 are positive parameters to be determined later. According to Lemma 4.3 in [33], there are two 𝒦∞ functions α1 and α21 such that
(38)α1(|Φ(t)|)≤V(Φ)≤α21(|Φ(t)|).
Then, with the notation in [26], it is easy to verify that
(39)L1+ɛ0(a0+b0)1-σ∫t-σ(t)t‍∥Φ(s)∥Δ4l0μ0ds≤α22(sup⁡-σ0≤s≤0|Φ(t+s)|),
where α22 is a 𝒦∞ function. Because α21(|Φ(t)|)≤α21(sup⁡-σ0≤s≤0|Φ(t+s)|), then
(40)α1(|Φ(t)|)≤W(Φ)≤α2(sup⁡-σ0≤s≤0|Φ(t+s)|)
with α2=α21+α22.
Step  3. Since V(Φ) and E(Φ) are homogenous of degree 4l0μ0-τ and τ, respectively, by Lemmas 5 and 6, one has
(41)∂W(Φ)∂XLE(Φ)=∂V(Φ)∂XLE(X)≤-Lγ1∥Φ(s)∥Δ4l0μ0,
where γ1>0 is a constant.
Using Assumption10 and the new coordinates in (32), we have
(42)|fiLsi|≤ai∑j=i+2n+1‍(|Lsjxj(t)|(ri+τ)/rj+|Lsjxj(t-σ(t))|(ri+τ)/rjLsi)
and the power of L in (42) is
(43)sjri+τrj-si=1+sj(ri+τ)-(si+1)rjrj=1+pirj(sjri+1-si+1rj).
Owing tos1=0,  si=(si-1+1)/pi-1, one has
(44)si=∑s=1i-1‍1ps⋯pi-1,i=2,…,n.
Forj≥i+2, (44) gives
(45)sjri+1-si+1rj=sj(1p1⋯pi+si+1τ)-si+1(1p1⋯pj-1+sjτ)=sjp1⋯pi-si+1p1⋯pj-1>0.
In summary, it can be concluded that there exists a positive constant(46)ɛ1=min⁡{pirj(sjri+1-si+1rj):1≤i≤n-1,i+2≤j≤n}
such that
(47)|fiLsi|≤aiL1+ɛ1∑j=i+2n+1‍(|xj(t)|(ri+τ)/rj+|xj(t-σ(t))|(ri+τ)/rj)≤a^iL1+ɛ1(∥Φ(t)∥Δri+τ+∥Φ(t-σ(t))∥Δri+τ)
with a^i>0. According to Lemma 6, we deduce that
(48)|∂V(Φ)∂ΦF|≤∑i=1n-1‍|∂V(Φ)∂Φi||fiLsi|≤∑i=1n-1‍a^iL1+ɛ1∥Φ(t)∥Δ4l0μ0-ri-τ×(∥Φ(t)∥Δri+τ+∥Φ(t-σ(t))∥Δri+τ)≤a0L1+ɛ1(∥Φ(t)∥Δ4l0μ0+∥Φ(t-σ(t))∥Δ4l0μ0),
where a0 is a positive constant.
Similarly, there is a positive constant(49)ɛ2=min⁡{sj(2ri+τ)-(2si+1)rj2rj:1≤i≤n-1,aaaaaaaaaaaaaaaaaaaaaaaaaai+2≤j≤nsj(2ri+τ)-(2si+1)rj2rj},
such that
(50)|giLsi|≤biL1/2+ɛ2×∑j=i+2n+1‍(|Φj(t)|(2ri+τ)/2rj+|Φj(t-σ(t))|(2ri+τ)/2rj)≤b^iL1/2+ɛ2(∥Φ(t)∥Δri+τ/2+∥Φ(t-σ(t))∥Δri+τ/2)
with b^i>0. Then,
(51)12Tr⁡{G∂2V(Φ)∂Φ2GT}≤12mm∑k,l=1n-1‍|∂2V(Φ)∂Φk∂Φl||gkLsk||glLsl|≤b~L1+2ɛ2×∑k,l=1n-1‍∥Φ(t)∥4l0μ0-τ-rk-rl×(∥Φ(t)∥Δrk+τ/2+∥Φ(t-σ(t))∥Δrk+τ/2)×(∥Φ(t)∥Δrl+τ/2+∥Φ(t-σ(t))∥Δrl+τ/2)≤b0L1+2ɛ2(∥Φ(t)∥Δ4l0μ0+∥Φ(t-σ(t))∥Δ4l0μ0),
where b~,b0 are positive constants.
By choosingɛ0=min⁡{ɛ1,2ɛ2} and Definition 1, Assumption 11, (41), (48), and (51), one can obtain
(52)ℒW(Φ)≤∂W(Φ)∂XLE(Φ)+∂V(Φ)∂XF+12Tr⁡{G∂2V(Φ)∂Φ2GT}+L1+ɛ0(a0+b0)×(11-σ∥Φ(t)∥Δ4l0μ0-∥Φ(t-σ(t))∥Δ4l0μ0)≤-L(γ1-Lɛ0(a0+b0+a0+b01-σ))∥Φ(t)∥Δ4l0μ0,
where 0<L<1. Apparently, by choosing 0<L<min⁡{(γ1/(a0+b0+(a0+b0)/(1-σ)))1/ɛ0,1}, there is a constant δ, such that ℒW(Φ)≤-δ∥Φ(t)∥Δ4l0μ0.
From Steps  1 to 3 and Lemma4, it is obtained that the closed-loop system (33), (34), and (17) has a global unique solution and the equilibrium Φ=0 is GAS in probability.
Step  4. Because of the equivalence in (32), then there is a global unique solution for the closed-loop system consisting of (9), (16), and (17) and vpn=upnLsn+1 and the equilibrium z=0 is GAS in probability.Remark 21.
Using the adding one power integrator method and the homogeneous domination technique, [34] has designed a state-feedback stabilizer for a class of stochastic high-order feedforward nonlinear systems. However, to deal with the unmeasurable states and time-delay terms in this paper, we choose a reduced-order homogeneous observer and an appropriate Lyapunov-Krasovskii functional, which is not an easy work.


## 5. Conclusions

For a class of stochastic high-order feedforward nonlinear systems with different power orders and time-varying delay, an output-feedback stabilizer has been designed by virtue of the adding one power integrator technique and homogeneous domination approach. It globally stabilizes the origin of the closed-loop system. The proof of stability we have adopted depends on the construction of a Lyapunov-Krasovskii functional.



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