# Adaptive NN State-Feedback Control for Stochastic High-Order Nonlinear Systems with Time-Varying Control Direction and Delays

Authors: Huifang Min; Na Duan
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Nussbaum-type gain function and neural network (NN) approximation approaches are extended to investigate the adaptive state-feedback stabilization problem for a class of stochastic high-order nonlinear time-delay systems. The distinct features of this paper are listed as follows. Firstly, the power order condition is completely removed; the restrictions on system nonlinearities and time-varying control direction are greatly weakened. Then, based on Lyapunov-Krasovskii function
and dynamic surface control technique, an adaptive NN controller is constructed to render the closed-loop system semiglobally uniformly ultimately bounded (SGUUB). Finally, a simulation example is shown to demonstrate the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

During the past decades, the control of stochastic nonlinear systems has been an interesting field and received fruitful development based on the stochastic stability theory in [1–3] and other references. However, the existences of nonlinearities and time delays cause the instability of system performance and give rise to much greater design difficulty in the stabilization procedure. To handle nonlinearities, neural network (NN), especially radial basis function neural network (RBF NN), has been successfully extended to stochastic nonlinear systems due to its inherent approximation capacity; see [4–8] and the references therein. To deal with time delays, there are often two ways, namely, Lyapunov-Krasovskii function and Lyapunov-Razumikhin technique. Recently, together with NN and backstepping, Lyapunov-Krasovskii theory was used in [9–13] and Lyapunov-Razumikhin approach was utilized in [14–17] to stabilize stochastic nonlinear systems with time delays, respectively. However, to the best of our knowledge, for stochastic high-order nonlinear time-delay systems, there are still a few results.In this paper, we further consider the following stochastic high-order time-delay system:(1)dxi=xi+1pidt+fit,xt,xt-ditdt+git,xt,xt-ditdω,i=1,…,n-1,dxn=ηtupndt+fnt,xt,xt-dntdt+gnt,xt,xt-dntdω,y=x1,where x=(x1,…,xn)T∈Rn and x(t-di(t))=(x1(t-di(t)),…,xn(t-di(t)))T are system state vectors; x1,…,xn, u∈R and y are system measurable states, control input and output, respectively. For 1≤i≤n, di(t):R+→[0,di] are time-varying delays, which are Borel measurable functions. Define d=max1≤i≤n{di} with initial value {x(θ):-d≤θ≤0}=ξ∈CF0b([-d,0];Rn). p1,…,pn≥1 are odd integers called the high-order of the system. η(t)≠0 is unknown bounded time-varying continuous function called control direction. For 1≤i≤n, the drift terms fi:R+×Rn×Rn→R and the diffusion terms gi:R+×Rn×Rn→Rr are assumed to be locally Lipschitz in (x(t),x(t-di(t))) and piecewise continuous in t with fi(t,0,0)=0 and gi(t,0,0)=0. ω is an r-dimensional standard Wiener process defined on a probability space {Ω,F,{F}t≥0,P}, where Ω is a sample space, F is a σ-field, {F}t≥0 is filtration, and P is the probability measure.Whendi(t)=0, [18–27] investigated different feedback control problems for system (1) with different structures. When di(t)≠0, [28] studied the output-feedback control with pi=p and di(t)=d(t) under restrictive growth conditions on fi and gi. To relax the growth conditions, [29, 30] firstly generalized the homogeneous domination approach proposed by [31] to stochastic high-order time-delay systems. Subsequently, [32–34] further solved the stabilization problems for more general systems based on homogeneous domination approach. However, in [29, 30, 32–34], the nonlinear functions were still assumed to satisfy somewhat restrictive growth conditions. In addition, [29, 30, 32–34] were all achieved with η(t) being bounded by positive constants or η(t)=1. For η(t)≠1, the Nussbaum-type gain function approach proposed by [35] has been proven to be a useful tool and is frequently used in [11, 12, 36–38] for stochastic nonlinear systems with pi=1. Immediately, one raises the following problem:How does one generalize RBF NN and Nussbaum-type gain function approaches to system (1) to further relax the restrictions on nonlinear functions fi and gi and time-varying control direction η(t)?This paper focuses on solving this problem. The main contributions are listed as follows. (i) Compared with the existing results [29, 30, 32–34], we largely weaken the restrictions on the drift and diffusion terms by extending NN approximation approach to system (1). Furthermore, to handle unknown control direction, we utilize Nussbaum-type gain function approach to allow the sign and the bounds of η(t) to be unknown. (ii) By introducing dynamic surface control (DSC), the problem of “explosion of complexity” generalized by the repeated differentiation of virtual controllers is avoided, which greatly simplifies the control algorithm. (iii) An adaptive state-feedback controller is constructed to ensure the closed-loop system to be semiglobally uniformly ultimately bounded (SGUUB) by using backstepping technique and choosing Lyapunov-Krasovskii function skillfully.The remainder of this paper is organized as follows. Section2 begins with mathematical preliminaries. The design process and analysis of adaptive state-feedback controller are given in Sections 3 and 4, respectively. In Section 5, a simulation example is shown. Section 6 summarizes the paper. Finally, the necessary proof is provided in the Appendix.


## 2. Mathematical Preliminaries

The following notations, definitions, and lemmas are to be used.Notations. R+ denotes the set of all the nonnegative real numbers; Rn denotes the n-dimensional Euclidean space. Ci denotes the family of all the functions with continuous ith partial derivations; 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. C([-d,0];Rn) denotes the space of continuous Rn-valued functions on [-d,0] endowed with the norm · defined by f=supx∈[-d,0]fx for f∈C([-d,0],Rn); · denotes the Euclidean norm of a vector; 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}. XT denotes the transpose of a given vector or matrix X; Tr⁡{X} denotes its trace when X is square. 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. To simplify the procedure, we sometimes denote X(t) by X for any variable X(t).Consider the following stochastic time-delay system:(2)dxt=ft,xt,xt-dtdt+gt,xt,xt-dtdω,∀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 r-dimensional standard wiener process defined as in (1). f:R+×Rn×Rn→Rn and g:R+×Rn×Rn→Rn×r are locally Lipschitz with f(0,0,t)=0 and g(0,0,t)=0. For any given V(x(t),t)∈C2,1, together with stochastic system (2), the differential operator L is defined as(3)LV=∂V∂t+∂V∂xf+12TrgT∂2V∂x2g.Definition 1 (see [35]).
A functionN(ζ) is called a Nussbaum-type function if it has the following properties:(4)limsups→∞∫s0sNζdζ=+∞,liminfs→∞∫s0sNζdζ=-∞.
Many functions satisfy these properties, for example,eζ2cos⁡(πζ/2), ln⁡(ζ+1)cos⁡(ln⁡(ζ+1)), and ζ2cos⁡(πζ/2). Throughout this paper, the Nussbaum function N(ζ)=eζ2cos⁡(πζ/2) is exploited.Definition 2 (see [14]).
Letp≥1; considering stochastic nonlinear time-delay system (2), the solution {x(t),t≥0} with initial condition ξ∈S0 (S0 is some compact set containing the origin) is said to be p-moment semiglobally uniformly ultimately bounded (SGUUB) if there exists a constant d such that E{x(t,ξ)p}≤d, ∀t≥T, holds for some T≥0.To facilitate the control design, the following lemmas are applied.Lemma 3 (Young’s inequality).
For∀(x,y)∈R2, xy≤(εp/p)|x|p+(1/qεq)|y|q holds, where ɛ>0, p,q>1, and 1/p+1/q=1.Lemma 4 (see [26]).
For real variablesx≥0 and y>0; then, x≤y+(x/m)m((m-1)/y)m-1, where m≥1 is a real number.Lemma 5 (see [31]).
Letx and y be real variables; then, for any real numbers m,n,b>0 and continuous function a(·)≥0, one has a(·)xmyn≤b|x|m+n+(n/m+n)((m+n)/m)-m/na(·)(m+n)/nb-m/n|y|m+n.Lemma 6 (see [31]).
Forx,y∈R, where p≥1 is a constant, the following inequalities hold: |x+y|p≤2p-1|xp+yp| and (|x|+|y|)1/p≤|x|1/p+|y|1/p.In the sequel, radial basis function neural network (RBF NN) is to be applied to estimate the unknown nonlinear functions. By choosing sufficiently large node number, for any continuous functionf(x) over a compact set Sx⊂Rq, there is a RBF NN W∗TS(x) such that, for an ideal level of accuracy ɛ (0<ɛ<1),(5)fx=W∗TSx+δx,δx≤ɛ,where δ(x) is the approximation error and S(x)=[s1(x),…,sN(x)]T is the known function vector with N>1 being the RBF NN node number. The basis functions si(x) (1≤i≤N) are chosen as si(x)=exp⁡[-(x-bi)T(x-bi)/ς2], where ς is the width of the function and bi=[bi1,…,bin]T is the center of the receptive field. W∗ is the ideal constant weight vector with the form W∗=argminW∈RN⁡supx∈Sx⁡fx-WTSx, where argmin is the value of variable W when the objective function supx∈Sx⁡fx-WTSx is minimum with W=[w1,…,wN]T being the weight vector.


## 3. Design of State-Feedback Controller

The objective of this paper is to design an adaptive NN state-feedback controller for system (1) under weaker conditions such that the closed-loop system is SGUUB. To achieve the above objective, we need the following assumptions.Assumption 7.
The time-varying delaysdi(t), i=1,…,n, in system (1) satisfy 0≤di(t)≤hi and d˙i(t)≤γi<1 for positive constants hi and γi.Assumption 8.
η
(
t
)
≠
0 is unknown sign and takes value in the unknown closed interval H≔[l-,l+] with 0∉H.Assumption 9.
Nonlinear functionsfi and gi satisfy(6)fit,xt,xt-dit≤fi1y+fi2yt-dit,git,xt,xt-dit≤gi1y+gi2yt-dit,for i=1,…,n, where fi1, fi2, gi1, and gi2 are positive functions with fij(s)=sf¯ij(s) and gij(s)=sg¯ij(s) for j=1,2.Remark 10.
We emphasize two points. (i) To the best of our knowledge, only [33] consider the unknown control directions for stochastic high-order time-delay systems. However, in [33], the unknown control directions are of known signs and are bounded by positive constants. We allow the sign of η(t) to be unknown and remove the bounds limitations in Assumption 8. (ii) Motivated by [14, 15] for stochastic time-delay systems, the restrictions on fi and gi are greatly relaxed in Assumption 9 compared with the existing results in [29, 30, 32–34].To simplify the design process, define(7)θ=max⁡NiWi∗2,i=1,…,n,where N1,…,Nn are the number of RBF NN nodes and W1∗,…,Wn∗ are the ideal constant weight vectors. Before the design procedure, introduce the following coordinate transformation:(8)z1=x1,zi=xi-αif,i=2,…,n,where α2f,…,αnf are the outputs of the first-order filter with virtual control laws α2,…,αn being inputs. Now, we give the backstepping design procedure by utilizing the technique of DSC and RBF NN approximation approach.Step 1.
Choosing the 1st Lyapunov function candidate asV1(z1,θ^)=(k1/p-p1+4)z1p-p1+4+(1/2Γ)θ~ and using (1), (3), and (8), one has(9)LV1≤k1z1p-p1+3x2p1+f1·+p-p1+32k1z1p-p1+2g1·g1T·-1Γθ~θ^˙,where p=max{pi,i=1,…,n}, k1, Γ>0 are design constants, θ^ is the estimate of θ, and θ~=θ-θ^ is the estimation error of θ.In the sequel, we estimate the terms of (9) by using Lemmas 4-5 and Assumption 9 as(10)k1z1p-p1+3f1·≤k1z1p-p1+3f11y+f12yt-d1t≤k1z1p-p1+4f¯11y+l11z1p-p1+4+l¯11f12p-p1+4yt-d1t≤k1z1p-p1+4f¯11y+l11z1p-p1+4+ξ11+ξ¯11f12p+3yt-d1t,p-p1+32k1z1p-p1+2g1·g1T·≤k1p-p1+3z1p-p1+2g112y+g122yt-d1t≤k1p-p1+3z1p-p1+4g¯112y+l12z1p-p1+4+l¯12g12p-p1+4yt-d1t≤k1p-p1+3z1p-p1+4g¯112y+l12z1p-p1+4+ξ12+ξ¯12g12p+3yt-d1t,where l11, l12, ξ11, and ξ12 are positive parameters, l¯11=(1/p-p1+4)((p-p1+4/p-p1+3)l11)-(p-p1+3)k1p-p1+4, ξ¯11=(((p-p1+4)/(p+3))l¯11)(p+3)/(p-p1+4)((p1-1)/(p-p1+4)ξ11)(p1-1)/(p-p1+4), l¯12=(2/p-p1+4)((p-p1+4/p-p1+2)l12)-(p-p1+2)/2(k1(p-p1+3))(p-p1+4)/2, and ξ¯12=(((p-p1+4)/(p+3))l¯12)(p+3)/(p-p1+4)((p1-1)/(p-p1+4)ξ12)(p1-1)/(p-p1+4) are constants that can be designed. Substituting (10) into (9) and choosing the intermediate variable α^2f as(11)α^2fp1X1=-z1k1f¯11+k1p-p1+3g¯112+l11+l12+ν1·+ϕ12θ^z1p1k1,one yields(12)LV1≤k1z1p-p1+3x2p1-α^2fp1-ν1·z1p+3-ϕ12θ^z1p+3-1Γθ~θ^˙+ξ11+ξ12+yp+3t-d1tS1yt-d1t,where X1=(z1,θ^)∈SX1={X1∣x∈Sx}, Sx is a compact set through which the state trajectories may travel, ν1(·) and ϕ12(θ^) are continuous functions to be designed later, and S1(y(t-d1(t)))=ξ¯11f¯12p+3(y(t-d1(t)))+ξ¯12g¯12p+3(y(t-d1(t))).To proceed further, applying Lemmas3-4 and RBF NN (5) to estimate α^2fp1 as α^2fp1=W1∗TS1+δ1 and δ1<ε1, this leads to(13)-k1z1p-p1+3α^2fp1≤z1p-p1+3k12+θ2+ε122≤z1p-p1+3φ11θ^+12z1p-p1+3θ~≤ξ1+z1p+3ϕ11θ^+12z1p-p1+3θ~,where ξ1>0 is a design constant, W1∗T2S12≤|W1∗T|2N1≤θ, φ11(θ^)=k12+1+θ^2/2+ε12/2 and ϕ11≥((p-p1+3/p+3)φ11(θ^))(p+3)/(p-p1+3)(p1/(p-p1+3)ξ1)p1/(p-p1+3). Choose the first control law as(14)α2z1,θ^=-z1c1+ϕ11θ^k11/p1=-z1β1θ^,which together with (13) changes (12) into(15)LV1≤-c1z1p+3+k1z1p-p1+3x2p1-α2p1-ν1·z1p+3-ϕ12θ^z1p+3-1Γθ~θ^˙-τ1+Δ1+yp+3t-d1tS1yt-d1t,where c1>0, τ1=(Γ/2)z1p-p1+3, and Δ1=ξ1+ξ11+ξ12.In order to avoid the repeated differentiation of virtual control law, we introduce dynamic surface control technique. Letα2f be obtained by a first-order filter with time constant q2; then, it has q2α˙2f+α2f=α2 and α2f(0)=α2(0). Define Y2=α2f-α2 as the output error of the filter; thus, α˙2f=-Y2/q2 and Y˙2=-Y2/q2+D2(X1), where D2(X1)=z˙1β1(θ^)+z1β˙1(θ^)θ^˙. Then, by using Lemmas 5-6 and (a+b)n=∑i=0nCnian-ibi, it holds that(16)k1z1p-p1+3x2p1-α2p1=k1z1p-p1+3z2+Y2+α2p1-α2p1≤k1z1p-p1+3∑j=0p1-1Cp1jz2+Y2p1-j-z1β1θ^j≤k1∑j=0p1-1/2Cp12j2p1-1-2jz1p-p1+3+2jz2p1-2j+Y2p1-2jβ12j≤γ11z1p+3+ϕ20θ^z2p+3+γ12Y2p+3+ϕ12θ^z1p+3,where γ11=∑j=0(p1-1)/2γ11j, γ12=∑j=0(p1-1)/2γ12j with γ11j and γ12j being positive design parameters, ϕ20(θ^) ≥ (p1-2j/p+3)((p+3/p-p1+3+2j)γ11j)-p-p1+3+2j/p1-2j(k12p1-1-2jCp12j1+β14j)(p+3)/(p1-2j), and ϕ12(θ^)≥p-p1+3+2j/p+3·(((p+3)/(p1-2j))γ12j)-(p1-2j)/(p-p1+3+2j)(k12p1-1-2jCp12j1+β14j)(p+3)/(p-p1+3+2j). Substituting (16) into (15), one gets(17)LV1≤-c1-γ11z1p+3-ν1·z1p+3+ϕ20θ^z2p+3+γ12Y2p+3-1Γθ~θ^˙-τ1+Δ1+yp+3t-d1tS1yt-d1t.Step i  (
2
≤
i
≤
n
-
1
). A similar result to Step 1 is obtained at this step. We state it in the following proposition.Proposition 11.
For theith Lyapunov function(18)Vi·=∑j=1ikjp-pj+4zjp-pj+4+∑j=2i1p+3Yjp+3+12Γθ~,there exists a virtual control law in the following form:(19)αi+1z¯i,θ^=-zici+ϕi0θ^+ϕi1θ^ki1/pi=-ziβiθ^,such that(20)LVi≤-∑j=1icj-γj1zjp+3-ν1·z1p+3+ϕi+1,0θ^zi+1p+3+∑j=1iγj2Yj+1p+3-1Γθ~θ^˙-τi+Δi+∑j=1iyp+3t-djtSjyt-djt+∑j=1i-1Yj+1p+2-Yj+1qj+1+Dj+1,where ci, ki, and γj1 are positive design constants, φi(θ^)=ki2+1+θ^2/2+εi2/2, τi=∑j=1i(Γ/2)zjp-pj+3, ϕi1(θ^)≥((p-pi+3/p+3)φi(θ^))(p+3)/(p-pi+3)(pi/(p-pi+3)ξi)pi/(p-pi+3), Δi=∑j=1i(ξj0+ξj1+ξj2+ξj), ξ10=0, ϕi0(θ^)≥(pi-1-2j/p+3)((p+3)γi-1,1j/(p-pi-1+3+2j))-(p-pi-1+3+2j)/(pi-1-2j)(ki-12pi-1-1-2jCpi-12j1+βi-14j)(p+3)/(pi-1-2j), and Sj(·)=ξ¯j1f¯j2p+3(·)+ξ¯j2g¯j2p+3(·).Proof.
See the Appendix.Hence, at stepn, we choose(21)Vn·=∑i=1nkip-pi+4zip-pi+4+∑i=2n1p+3Yip+3+12Γθ~.By exactly following the design procedure in Proposition 11 and introducing Nussbaum function, one can construct the adaptive control laws as(22)u=znNζβnθ^,ζ˙=znp+3βnpnθ^,βnθ^=cn+ϕn0θ^+ϕn1θ^1/pn,θ^˙=∑i=1nΓ2zip-pi+3-k0θ^;thus, this leads to(23)LVn≤-∑i=1nci-γi1zip+3-ν1·z1p+3+∑i=1n-1γi2-1qi+1Yi+1p+3+∑i=1n-1Yi+1p+2Di+1+Δn+k0Γθ~θ^+∑i=1nyp+3t-ditSiyt-dit+knηtNpnζζ˙+ζ˙,where cn, k0>0, γn1=0 are design constants, φn(θ^)=kn2+1+θ^2/2+εn2/2, Δn=∑i=1n(ξi0+ξi1+ξi2+ξi), ϕn0(θ^)≥(pn-1-2j/p+3)((p+3)γn-1,1j/(p-pn-1+3+2j))-(p-pn-1+3+2j)/(pn-1-2j)(kn-12pn-1-1-2jCpn-12j1+βn-14j)(p+3)/(pn-1-2j), ϕn1(θ^)≥((p-pn+3)φn(θ^)/(p+3))(p+3)/(p-pn+3)(pn/(p-pn+3)ξn)pn/(p-pn+3), ξn0=0, and Si(y(t-di(t)))=ξ¯i1f¯i2p+3(y(t-di(t)))+ξ¯i2g¯i2p+3(y(t-di(t))).Remark 12.
During the design procedure, the drift and diffusion terms are technically handled for more general system (1) by RBF NN approximation approach compared with [29, 30, 32–34]. Furthermore, the repeated differentiation of virtual control laws is avoided and the unknown time-varying control direction is successfully handled by introducing DSC and Nussbaum-type gain function.


## 4. Stability Analysis

The main result of this paper is stated in the following theorem.Theorem 13.
For stochastic high-order time-delay system (1) satisfying Assumptions 7–9, when the adaptive control laws are chosen as (14), (19), and (22) and the initial condition satisfies V(0)≤σ, the closed-loop system consisting of (1), (8), (14), (19), and (22) is semiglobally uniformly ultimately bounded (SGUUB), where V is the closed-loop Lyapunov-Krasovskii functional and σ is some constant.Proof.
Choosing Lyapunov-Krasovskii functionalV=Vn+W as(24)V=Vn+∑i=1n11-γi∫t-dittyp+3Siysds,in terms of Assumption 7 and (23)-(24), one obtains(25)LV≤-∑i=1nci-γi1zip+3-ν1·z1p+3+∑i=1n11-γiyp+3Siy+∑i=1n-1γi2-1qi+1Yi+1p+3+∑i=1n-1Yi+1p+2Di+1+Δn+k0Γθ~θ^+knηtNpnζζ˙+ζ˙.Let ν1(·)-∑i=1n(1/1-γi)Si(y)≥0 and use Lemma 3 to estimate the terms in (25) as(26)Yi+1p+2Di+1≤p+2p+3ε0Di+1p+3/p+2Yi+1p+3+1p+3ε0p+3,θ~θ^≤-12θ~2+12θ2;one can get(27)LV≤-∑i=1nci-γi1zip+3-∑i=1n-11qi+1-γi2-p+2p+3ε0Di+1p+3/p+2Yi+1p+3-k02Γθ~2+Δn+1+knηtNpnζζ˙+ζ˙,where ε0>0 is a parameter and Δn+1=Δn+1/(p+3)ε0p+3+(k0/2Γ)θ2.
By estimatingVn as V-W≤∑i=1nρizip+3+Δ0+∑i=1n-11/p+3Yi+1p+3+(1/2Γ)θ~2 and choosing(28)λ¯=min⁡ci-γi1ρi,p+31qi+1-γi2-p+2p+3ε0Di+1p+3/p+2,k0,1,μ=Δn+1+λ¯Δ0,this leads to(29)LV≤-λ¯V-W+μ+knηtNpnζζ˙+ζ˙≤-λV+μ+knηtNpnζζ˙+ζ˙,where Δ0 is a design constant, ρi=((p-pi+4)ki/(p+3))(p+3)/(p-pi+4)((pi-1)/(p-pi+4)Δ0)(pi-1)/(p-pi+4), and λ is a positive parameter to be chosen.
Multiplying (29) by eλt, taking expectations on both sides and integrating it over [0,t], one has(30)EeλtV≤EV0+μλeλt-1+∫0tknηsNpnζ+1ζ˙eλsds;then,(31)EV≤EV0e-λt+μλ1-e-λt+e-λt∫0tknηsNpnζ+1ζ˙eλsds≤EV0e-λt+μλ+∫0tknηsNpnζ+1ζ˙e-λt-sds.It is easy to see that, in (31), 0<e-λ(t-s)≤1 for s∈[0,t]. Let e-λ(t-s)Npn(ζ) be a new Nussbaum function; according to Definition 1, one has limζ→∞sup⁡∫e-λ(t-s)Npn(ζ)ζ˙→+∞ and limζ→∞inf⁡∫e-λ(t-s)Npn(ζ)ζ˙→-∞. Then, it can be obtained that E(V)<0, but this counteracts the fact that E(V)≥0; therefore, ζ and e-λ(t-s)Npn(ζ)ζ˙ must be bounded by some constant, which shows that E(V) must be bounded to some constant.
From the above discussion and Definition2, one gets that V is SGUUB. Thus, from the forms of V and Vi in the design procedure, one concludes that all the variables in the closed-system are SGUUB; that is to say the closed-loop system consisting of (1), (8), (14), (19), and (22) is SGUUB. The proof is completed.


## 5. A Simulation Example

Consider the following stochastic high-order nonlinear system:(32)dx1=x23dt+f1·dt+g1·dω,dx2=ηtx25dt+f2·dt+g2·dω,y=x1,where p=max⁡{3,5}=5, f1=x13/(1+x12)+x1t-d1tsin⁡(x2), g1=x12x1(t-d1(t)), f2=x1cos⁡(x2)+(1/5)x1x12(t-d1(t)), g2=x12(t-d1(t))/(1+x22)+x12/(1+x12), η(t)=1+0.1sin⁡(t), d1(t)=(1/4)(1+sin⁡(t)), and d2(t)=(1/5)(1+sin⁡(t)). It can be verified that Assumptions 7–9 hold true.By exactly following the design procedure in Section3, we construct the adaptive control laws (33)z1=x1,z2=x2-α2f,α2=-z1c1+ϕ11θ^k11/3=-z1β1θ^,u=z2Nζβ2θ^,ζ˙=z28β25θ^,β2=c2+ϕ20θ^+ϕ21θ^1/5,θ^˙=Γ2z15+z23-k0θ^,where c1, c2, k0, k1, k2, and Γ are positive design constants, q2α˙2f+α2f=α2 with q2>0 being a time constant of a filter, φ1(θ^)=k12+1+θ^2/2+ε12/2, ϕ11=((5/8)φ1(θ^))8/5(3/5ξ1)3/5, φ2(θ^)=k22+1+θ^2/2+ε22/2, ϕ20=(3/8)((8/5)γ110)-5/3((2/4)k1)8/3+(1/8)((8/7)γ111)-7(3k11+β14)8, ϕ21=((3/8)φ2(θ^))8/3(3/5ξ2)3/5, and ε1, ε2, ξ1, ξ2, γ110, and γ111 are positive parameters.In simulation, choose the Nussbaum function asN(ζ)=eζ2cos⁡((π/2)ζ) and let the design constants and parameters be c1=12.5, c2=5, k0=1, k1=2, k2=0.5, Γ=50, ε1=0.01, ε2=0.01, ξ1=1, ξ2=1, γ110=2, γ111=2, and q2=0.1. The initial conditions are given by x1(0)=1.35, x2(0)=-0.85, θ^(0)=0, and ζ(0)=0. From Figure 1, one can see that system states, control input, adaptive law, and variable α2f converge to the origin uniformly, which means that all the signals of the closed-loop (32)-(33) are SGUUB. Figure 1 demonstrates the effectiveness of the control scheme.Figure 1
The responses of the closed-loop system (32)-(33).


## 6. Conclusions

This paper considers the unknown time-varying control coefficient by generalizing Nussbaum function to a class of stochastic high-order time-delay systems. Under weaker conditions on the drift and diffusion terms, the adaptive state-feedback control is solved by using RBF NN approximation approach. The control scheme ensures the closed-loop system to be SGUUB. An issue to be investigated is how to further handle the output-feedback problem for system (1).



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*Source: 723425-2015-10-25.xml*