# Adaptive Output Feedback Stabilization of Random Nonlinear Systems with Unmodeled Dynamics Driven by Colored Noise

Authors: Mingyue Cui
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper focuses on the problem of adaptive output feedback stabilization for random nonlinear systems with unmodeled dynamics and uncertain nonlinear functions driven by colored noise. Under the assumption of unmodeled dynamics having enough stability margin, an adaptive output feedback stabilization controller is designed based on a reduced-order observer such that the state of the closed-loop system has an asymptotic gain in the 2-th moment (AG-2-M) and the mean square of the output can be made arbitrarily small by tuning parameters. A simulation example is used to illustrate the effectiveness of the control scheme.


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## Body

## 1. Introduction

During the past decades, the control problem for systems with unmodeled dynamics mainly caused by simplifications in modeling process has received much attention. To deal with the unmodeled dynamics, an available dynamic signal is constructed to bound the unmodeled dynamics by exploiting prior information on the system in [1]. By the aid of small-gain technique in [2, 3], the unmodeled dynamics were treated by a worst-case design on the basis of small-gain arguments in [4]. By introducing input-to-state stability for time-varying control systems, sufficient conditions for global stabilization of triangular systems are given in [5]. For time-varying control systems, various equivalent characterizations of the nonuniform in time input-to-state stability (ISS) property are established in [6]. By using a suitable extension of the small-gain theorem, a uniform input-to-state stabilization controller is given in [7]. For MIMO nonlinear systems, a robust adaptive observer was given in [8]. By introducing K-filters and dynamic signal, [9] designed an adaptive output feedback controller such that the closed-loop control system is semiglobally uniformly ultimately bounded. By combining small-gain theorem and changing supply function techniques, [10] proposed a robust adaptive output feedback controller. However, most of the existing references mainly focus on the deterministic case.With the development of stochastic theory [11–15], many researchers pay great attention to the control problem of stochastic differential equations (SDEs) in the presence of unmodeled dynamics. Under the assumption of unmodeled dynamics having enough stability margin, [16] presented an output feedback controller based on minimal-order observer. In [17], a unifying solution to stochastic adaptive output feedback stabilization was presented by introducing dynamic signal and changing supply function technique. Also, some other control strategies such as adaptive neural networks control [18–20], adaptive fuzzy control [21, 22], and adaptive tracking control [23] were proposed.Since the stochastic disturbance is more reasonably described as colored noise than white noise in practical engineering, the models described by SDEs may not suit very well. In [24], a theoretical framework on stability of random nonlinear systems where stochastic disturbance is colored noise and applications to controller design are given. For a class of random nonlinear systems with unmodeled dynamics, under the assumption of unmodeled dynamics having enough stability margin, [25] designed a feedback stabilization controller by backstepping method. Until now, to our knowledge, there are no many results on adaptive output feedback control for random nonlinear systems with unmodeled dynamics and uncertain nonlinear functions in the literature. In this paper, motivated by [16, 17, 24, 25], the purpose of this paper is to solve this problem under some milder assumptions. The main work consists of the following aspects.(i) Because it is difficult to deal with Hessian terms caused by Wiener process, the stochastic disturbance is regarded as colored noise whose second-order moment is bounded in this paper. Distinguished from the existing stochastic analysis method, the method of ordinary differential equations is used to analyze the stability of the closed-loop system.(ii) For easier-to-implement and more reliable practical purposes, a reduced-order observer and1-dimension adaptive law are introduced, which lead to a simple controller. The state of the closed-loop system has an asymptotic gain in the 2-th moment (AG-2-M) and the mean square of the output can be regulated to an arbitrarily small neighborhood of the origin.The paper is organized as follows: in Section2, the mathematical preparation is given and the problem is formulated. Section 3 gives the observer-based backstepping adaptive controller design procedure. The main result is presented in Section 4. A simulation example is included in Section 5 to illustrate effectiveness of the proposed design method. Section 6 concludes the paper.Notations: the following notations are used throughout the paper. For a vectorx, xT denotes its transpose; for a matrix X, λmin(X) and λmax(X) denote its smallest and largest eigenvalue, respectively; · denotes the usual Euclidean norm of “·”; E denotes the mathematical expectation; R+ denotes the set of all nonnegative real numbers; Rn denotes the real n-dimensional space; Rn×m denotes the real n×m matrix space; Ci denotes the set of all functions with continuous i-th partial derivative. For simplicity, sometimes the arguments of functions are dropped.


## 2. Mathematical Preliminaries and Problem Formulation

### 2.1. Mathematical Preliminaries

Consider the following random nonlinear affine system:(1)x˙=fx,t+gx,tξt,xt0=x0,where x∈Rn is the state of system, ξ(t)∈Rm is a stochastic process, and the underlying complete probability space is taken to be the quartet (Ω,F,Ft,P) with a filtration Ft satisfying the usual condition (i.e., it is increasing and right continuous while F0 contains all P-null sets). Both functions f:Rn×R+→Rn and g:Rn×R+→Rn×m are locally Lipschitz in x piecewise continuous in t; i.e., for each R>0, there exists a constant LR such that(2)fx1,t-fx2,t+gx1,t-gx2,t≤LRx2-x1for any t∈R+ and x1,x2∈UR={x:x≤R},x1≠x2. Moreover, f(0,t) and g(0,t) are bounded a.s. Process ξ is a Ft-adapted process and piecewise continuous, and there exists a positive constant K such that(3)supt0≤s≤t⁡Eξs2≤K,∀t≥t0.The following definition, criterion, and inequality are represented now for the stability analysis.Definition 1 ([24]).
The state of system (1) has an asymptotic gain in the m-th moment (AG-m-M) if there exists a function γ(·) of class K such that, for any x0∈Rn,(4)limt→∞⁡Extm≤γδK,where δK=δ1K+δ2 with δ1>0,δ2≥0.Lemma 2.
For system (1) with conditions (3), if there exist parameters a1>0, a2>0, c>0, δ1>0, δ2≥0 and a function V∈C1 such that(5)a1xm≤Vx≤a2xm,(6)V˙xt≤-cVxt+δ1ξt2+δ2,then system (1) has a unique global solution, and the state of system has an AG-m-M.Proof.
Following the same lines as the proof of [24, Theorem 3], the result can be obtained.Lemma 3 ([26]).
Consider the continuous functionsk(t), h(t), and they are integrable over every finite interval. If a continuous function y(t) satisfies the inequality(7)y˙t≤ktyt+ht,∀t≥t0,then(8)yt≤yt0e∫t0tksds+∫t0te∫stkuduhsds,∀t≥t0.


### 2.2. Problem Formulation

Consider the following system(9)χ˙=qχ,x1+pχ,x1ξ0,x˙1=x2+ϕ1x,χ+Ω1Tx1ξ,⋮x˙i=xi+1+ϕix,χ+ΩiTx1ξ,⋮x˙n=u+ϕnx,χ+ΩnTx1ξ,y=x1,where x=[x1,x2,…,xn]T∈Rn, χ∈Rn0, u∈R and y∈R are the state, the unmodeled dynamics, the input and the output of system, respectively. ξ0∈Rm0 and ξ∈Rm are Ft-adapted stochastic processes. In this paper, it is assumed that only the output y can be measured and uncertain functions q, p, ϕi, and Ωi are smooth.The objective in this paper is to design an adaptive output feedback controller such that the state of the closed-loop system has an AG-m-M, and the mean square of the output can be regulated to an arbitrarily small neighborhood of the origin.Throughout the paper, the following assumptions are made on system (9).Assumption 4.
ξ 0 ∈ R m 0 and ξ∈Rm are Ft-adapted processes and piecewise continuous, and there exists a positive constant K such that(10)supt0≤s≤t⁡Eξ0s2≤K,supt0≤s≤t⁡Eξs2≤K,supt0≤s≤t⁡Eξs4≤K,∀t≥t0.Assumption 5.
For theχ-system, there exists a function V0∈C1, a K∞-function δ(·), a smooth function δ0(·):δ0(0)=0, and constants a_>0, a¯>0, c0>0, γ0>0, d0>0 such that (11)a_χ2≤V0χ≤a¯χ2,∂V0∂χqχ,x1+14d0∂V0∂χpχ,x12≤-c0V0χ-γ0δχ+yδ0y.Assumption 6.
For each1≤i≤n, there exits an unknown positive constant θi such that(12)ϕix,χ≤θiδiy+γiπχ,Ωix1≤θiψiy,where δi, ψi are known nonnegative smooth functions and δi(0)=ψi(0)=0 are assumed.Assumption 7.
There exists a constantγ>0 such that(13)δμ≥γπ2μ,∀μ∈0,∞.Remark 8.
Assumption5 describes the dynamical behavior of the unobservable state χ. It has some stability margin with respect to the unmodeled dynamics. The term yδ0(y) imposes restrictions on the influence of state x on the stability of the unobservable state χ.Remark 9.
In Assumption6, using the identity f(x)-f(x0)=∫01df/dλλ=αx-x0+x0dαx-x0=f¯x,x0x-x0 to δiy, there exist smooth functions δ¯i such that(14)δiy=yδ¯iy,which will be frequently used in the subsequent sections.Remark 10.
Assumption7 depicts the connection between the stability margin of the unobservable state χ and the unmodeled dynamics.


## 2.1. Mathematical Preliminaries

Consider the following random nonlinear affine system:(1)x˙=fx,t+gx,tξt,xt0=x0,where x∈Rn is the state of system, ξ(t)∈Rm is a stochastic process, and the underlying complete probability space is taken to be the quartet (Ω,F,Ft,P) with a filtration Ft satisfying the usual condition (i.e., it is increasing and right continuous while F0 contains all P-null sets). Both functions f:Rn×R+→Rn and g:Rn×R+→Rn×m are locally Lipschitz in x piecewise continuous in t; i.e., for each R>0, there exists a constant LR such that(2)fx1,t-fx2,t+gx1,t-gx2,t≤LRx2-x1for any t∈R+ and x1,x2∈UR={x:x≤R},x1≠x2. Moreover, f(0,t) and g(0,t) are bounded a.s. Process ξ is a Ft-adapted process and piecewise continuous, and there exists a positive constant K such that(3)supt0≤s≤t⁡Eξs2≤K,∀t≥t0.The following definition, criterion, and inequality are represented now for the stability analysis.Definition 1 ([24]).
The state of system (1) has an asymptotic gain in the m-th moment (AG-m-M) if there exists a function γ(·) of class K such that, for any x0∈Rn,(4)limt→∞⁡Extm≤γδK,where δK=δ1K+δ2 with δ1>0,δ2≥0.Lemma 2.
For system (1) with conditions (3), if there exist parameters a1>0, a2>0, c>0, δ1>0, δ2≥0 and a function V∈C1 such that(5)a1xm≤Vx≤a2xm,(6)V˙xt≤-cVxt+δ1ξt2+δ2,then system (1) has a unique global solution, and the state of system has an AG-m-M.Proof.
Following the same lines as the proof of [24, Theorem 3], the result can be obtained.Lemma 3 ([26]).
Consider the continuous functionsk(t), h(t), and they are integrable over every finite interval. If a continuous function y(t) satisfies the inequality(7)y˙t≤ktyt+ht,∀t≥t0,then(8)yt≤yt0e∫t0tksds+∫t0te∫stkuduhsds,∀t≥t0.


## 2.2. Problem Formulation

Consider the following system(9)χ˙=qχ,x1+pχ,x1ξ0,x˙1=x2+ϕ1x,χ+Ω1Tx1ξ,⋮x˙i=xi+1+ϕix,χ+ΩiTx1ξ,⋮x˙n=u+ϕnx,χ+ΩnTx1ξ,y=x1,where x=[x1,x2,…,xn]T∈Rn, χ∈Rn0, u∈R and y∈R are the state, the unmodeled dynamics, the input and the output of system, respectively. ξ0∈Rm0 and ξ∈Rm are Ft-adapted stochastic processes. In this paper, it is assumed that only the output y can be measured and uncertain functions q, p, ϕi, and Ωi are smooth.The objective in this paper is to design an adaptive output feedback controller such that the state of the closed-loop system has an AG-m-M, and the mean square of the output can be regulated to an arbitrarily small neighborhood of the origin.Throughout the paper, the following assumptions are made on system (9).Assumption 4.
ξ 0 ∈ R m 0 and ξ∈Rm are Ft-adapted processes and piecewise continuous, and there exists a positive constant K such that(10)supt0≤s≤t⁡Eξ0s2≤K,supt0≤s≤t⁡Eξs2≤K,supt0≤s≤t⁡Eξs4≤K,∀t≥t0.Assumption 5.
For theχ-system, there exists a function V0∈C1, a K∞-function δ(·), a smooth function δ0(·):δ0(0)=0, and constants a_>0, a¯>0, c0>0, γ0>0, d0>0 such that (11)a_χ2≤V0χ≤a¯χ2,∂V0∂χqχ,x1+14d0∂V0∂χpχ,x12≤-c0V0χ-γ0δχ+yδ0y.Assumption 6.
For each1≤i≤n, there exits an unknown positive constant θi such that(12)ϕix,χ≤θiδiy+γiπχ,Ωix1≤θiψiy,where δi, ψi are known nonnegative smooth functions and δi(0)=ψi(0)=0 are assumed.Assumption 7.
There exists a constantγ>0 such that(13)δμ≥γπ2μ,∀μ∈0,∞.Remark 8.
Assumption5 describes the dynamical behavior of the unobservable state χ. It has some stability margin with respect to the unmodeled dynamics. The term yδ0(y) imposes restrictions on the influence of state x on the stability of the unobservable state χ.Remark 9.
In Assumption6, using the identity f(x)-f(x0)=∫01df/dλλ=αx-x0+x0dαx-x0=f¯x,x0x-x0 to δiy, there exist smooth functions δ¯i such that(14)δiy=yδ¯iy,which will be frequently used in the subsequent sections.Remark 10.
Assumption7 depicts the connection between the stability margin of the unobservable state χ and the unmodeled dynamics.


## 3. Controller Design

### 3.1. Reduced-Order Observer Design

To counteract the unavailable statexi(i=1,2,…,n), a reduced-order partial-state observer is introduced as follows:(15)x^˙i=x^i+1+ki+1y-kix^1+k1y,i=1,…,n-2,x^˙n-1=u-kn-1x^1+k1y,where k=(k1,k2,…,kn-1)T is chosen such that(16)A0=-kIn-20⋯0is asymptotically stable. For each i=1,2,…,n-1, denote the observer error as(17)εi=1θ∗xi+1-x^i-kiy,where θ∗=max⁡{1,θ1,…,θn}. From (9), (15), and (17), one has(18)ε˙i=1θ∗xi+2+ϕi+1+Ωi+1Tξ-x^i+1+ki+1y-kix^1+k1y-kix2+ϕ1+Ω1Tξ=εi+1-kiε1+1θ∗ϕi+1-kiϕ1+1θ∗Ωi+1-kiΩ1Tξwith xn+1=u, x^n=u, and kn=0, whose compact from is(19)ε˙=A0ε+Δy,χ+ΩTyξ,where Δ(y,χ)=1/θ∗(ϕ2-k1ϕ1,…,ϕn-kn-1ϕ1)T and Ω(y)=1/θ∗(Ω2-k1Ω1,…,Ωn-kn-1Ω1).From Assumption4, we have(20)Δy,χ2=∑i=1n-11θ∗ϕi+1-kiϕ12≤∑i=1n-1δi+1y+kiδ1y+γi+1πχ+kiγ1πχ2≤∑i=1n-12δi+1y+kiδ1y2+∑i=1n-12γi+1+kiγ12π2χ≤y2Φ1y+aδχ,where Φ1(y)=∑i=1n-12(δ¯i+1(y)+kiδ¯1(y))2 and a=∑i=1n-12/γ(γi+1+kiγ1)2.(21)Ωy2=∑i=1n-11θ∗Ωi+1-kiΩ12≤∑i=1n-1ψi+1y+kiψ1y2=y2Ψ1y,where Ψ1(y)=∑i=1n-1(ψ¯i+1(y)+kiψ¯1(y))2. Since A0 is stable, there exists a symmetric positive definite matrix P such that PA0+A0TP=-2In-1. Along the solution of (19), differentiating the quadratic function Vε=1/2εTPε yields(22)V˙ε=-εTε+εTPΔy,χ+ΩTyξ≤-εTε+12εTε+12P2Δy,χ+Ωyξ2≤-12εTε+P2Φ1yy2+aP2δχ+P2Ψ1yy2ξ2≤-12εTε+P2Φ1yy2+aP2δχ+14dεP4Ψ12yy4+dεξ4,where dε is a design parameter.


### 3.2. Adaptive Backstepping Controller Design

From (9) and (17), the derivative of output is represented as(23)y˙=θ∗ε1+x^1+k1y+ϕ1x,χ+Ω1Tyξ,which, together with (9), (15), and (19), consists of the following system(24)χ˙=qχ,y+pχ,yξ0,ε˙=A0ε+Δy,χ+ΩTξ,y˙=x^1+θ∗ε1+k1y+ϕ1x,χ+Ω1Tyξ,x^˙i=x^i+1+ki+1y-kix^1+k1y,i=1,…,n-2,x^˙n-1=u-kn-1x^1+k1y.An adaptive backstepping controller based on the (y,x^1,…,x^n-1)-system of (24) will be developed.Step 1.
Introduce the first two error variables(25)z1=y,z2=x^1-α1,where α1 will be given in later. Consider the Lyapunov function candidate(26)V1=12z12+12rθθ~2,where θ~=θ-θ^, θ^ is the estimate of θ=max⁡{θ∗,θ∗2}. In view of (24) and (25), the derivative of V1 satisfies(27)V˙1≤yz2+α1+θ∗ε1+k1y+ϕ1x,χ+Ω1Tyξ-1rθθ~θ^˙.Applying Young’s inequality, one has(28)yθ∗ε1≤r1y2θ+14r1ε12,where r1>0 is a design parameter. By Assumption 4, one has(29)yϕ1x,χ≤yθ1δ1y+γ1πχ≤12ϵ1y2θ12+ϵ12δ1y+γ1π1χ2≤12ϵ1y2θ+ϵ1y2δ¯12y+ϵ1γ12γδχ,(30)yΩ1Tyξ≤yθ1ψ1yξ≤14d1y2θ1ψ12y+d1ξ2≤14d1y2ψ12yθ+d1ξ2,where ϵ1>0 and d1 are design parameters. Substituting (28)-(30) into (27), one has(31)V˙1≤yz2+yα1+k1y+ϵ1yδ¯12y+ω1θ^+ϵ1γ12γδχ+d1ξ2+θ~ω1y-1rθθ^˙+14r1ε12,where ω1=r1y+1/2ϵ1y+1/4d1yψ12(y). The stabilizing function α1 and the tuning function τ1 are designed as(32)α1=-c1z1-k1y-ω1θ^-βy,τ1=-σθ^+ω1y,where c1>0 is a design parameter and β(y) will be chosen in later. Then(33)V˙1≤-c1z12+yϵ1yδ¯12y-βy+z1z2+ϵ1γ12γδχ+d1ξ2+σθ~θ^+θ~τ1-1rθθ^˙+14r1ε12.Step i = 2 , … , n. Introduce the coordinate change(34)zi+1=x^i-αiy,x^1,…,x^i-1,θ^with zn+1=0 and x^n=u. The differential of zi is given as follows:(35)z˙i=x^i+kiy-ki-1x^1+k1y-∂αi-1∂yx^1+θ∗ε1+k1y+ϕ1x,χ+Ω1Tyξ-∑j=1i-2∂αi-1∂x^jx^j+1+kj+1y-kjx^1+k1y-∂αi-1∂θ^θ^˙=zi+1+αi+ηi-∂αi-1∂yθ∗ε1+ϕ1x,χ+Ω1Tyξ-∂αi-1∂θ^θ^˙,where ηi=kiy-ki-1(x^1+k1y)-∂αi-1/∂y(x^1+k1y)-∑j=1i-2∂αi-1/∂x^j(x^j+1+kj+1y-kj(x^1+k1y)). Assume that one has designed smooth function αj, τj(j=2,…,i-1) such that the following inequality holds for Vi-1=Vi-2+1/4zi-12,(36)V˙i-1≤-∑j=1i-1cjzj2+y∑j=1i-1ϵjyδ¯12y-βy+zi-1zi+∑j=1i-1ϵjγ12γδχ+∑j=1i-1djξ2+∑j=1i-2zj+1∂αj∂θ^rθτi-1-θ^˙+σθ~θ^+θ~τi-1-1rθθ^˙+∑j=1i-114rjε12,where ϵj>0, dj>0, rj>0 (2≤j≤i-1) are design parameter. In the following, we will prove that (36) holds for the i-th Lyapunov function candidate(37)Vi=Vi-1+14zi2.The derivative of Vi satisfies(38)V˙i≤-∑j=1i-1cjzj2+y∑j=1i-1ϵjyδ¯12y-βy+zi-1zi+∑j=1i-1ϵjγ12γδχ+∑j=1i-1djξ2+∑j=1i-2zj+1∂αj∂θ^rθτi-1-θ^˙+σθ~θ^+θ~τi-1-1rθθ^˙+∑j=1i-114rjε12+zizi+1+αi+ηi-∂αi-1αyθ∗ε1+ϕ1x,χ+Ω1Tyξ-∂αi-1∂θ^θ^˙.Applying Young’s inequality, one has(39)zi∂αi-1αyθ∗ε1≤rizi2∂αi-1αy2θ+14riε12,where ri>0 is a design parameter. By Assumption 4, one has(40)zi∂αi-1αyϕ1x,χ≤zi∂αi-1αyθ1δ1y+γ1πχ≤12ϵizi2∂αi-1αy2θ12+ϵi2yδ¯1y+γ1πχ2≤12ϵizi2∂αi-1αy2θ+ϵiy2δ¯12y+ϵiγ12γδχ,(41)zi∂αi-1αyΩ1Tyξ≤zi∂αi-1αyθ1ψ1yξ≤14dizi2∂αi-1αy2θ12ψ12y+diξ2≤14dizi2∂αi-1αy2θψ12y+diξ2,where ϵi>0 and di>0 are design parameters. With (39)-(41), it is obtained that(42)V˙i≤-∑j=1i-1cjzj2+y∑j=1i-1ϵjyδ¯12y-βy+zi-1zi+∑j=1i-1ϵjγ12γδχ+∑j=1i-1djξ2+∑j=1i-2zj+1∂αj∂θ^rθτi-1-θ^˙+σθ~θ^+θ~τi-1-1rθθ^˙+∑j=1i-114rjε12+zizi+1+αi+ηi+rizi2∂αi-1αy2θ+14riε12+12ϵizi2∂αi-1αy2θ+ϵiy2δ¯12y+ϵiγ12γδχ+14dizi2∂αi-1αy2θψ12y+diξ2-zi∂αi-1∂θ^θ^˙.By choosing the i-th tuning function as τi=τi-1+ωizi with ωi=zi∂αi-1/∂y2(ri+1/2ϵi+1/4diψ12(y)), one has(43)V˙i≤-∑j=1i-1cjzj2+y∑j=1iϵjyδ¯12y-βy+zizi-1+∑j=1iϵjγ12γδχ+∑j=1idjξ2+∑j=1i-2zj+1∂αj∂θ^rθτi-θ^˙+σθ~θ^+θ~τi-1rθθ^˙+∑j=1i14rjε12+zizi-1+αi+ηi+ωiθ^-∂αi-1∂θ^rθτi-∑j=1i-2zj+1∂αj∂θ^rθωi.Choose(44)αi=-zi-1-cizi-ηi-ωiθ^+∂αi-1∂θ^rθτi+∑j=1i-2zj+1∂αj∂θ^rθωi,where ci>0 is a design parameter. Then(45)V˙i≤-∑j=1icjzj2+y∑j=1iϵjyδ¯12y-βy+zizi+1+∑j=1iϵjγ12γδχ+∑j=1idjξ2+∑j=1i-1zj+1∂αj∂θ^rθτi-θ^˙+σθ~θ^+θ~τi-1rθθ^˙+∑j=1i14rjε12.At the end of the recursive procedure, the control law and adaptive law are chosen as(46)u=αny,x^1,…,x^n-1,θ^,θ^˙=rθτn.By (45) and (46), one gets(47)V˙n≤-∑j=1ncjzj2+y∑j=1nϵjyδ¯12y-βy+∑j=1nϵjγ12γδχ+∑j=1ndjξ2+σθ~θ^+∑j=1n14rjε12.The Lyapunov function for the whole system is(48)V=ε0V0+rεVε+Vn.Then, by Assumption 6, (22), and (47), we obtain(49)V˙≤-c0ε0V0χ-∑j=1ncjzj2+yrεP2Φ1yy+rε4dεP4Ψ12yy3+ε0δ0y+∑j=1nϵjyδ¯12y-βy+rεaP2+∑j=1nϵjγ12γ-ε0γ0δχ+d0ξ02+rεdεξ4+∑j=1ndjξ2+σθ~θ^-rε2ε2+∑j=1n14rjε12.By choosing(50)rε≥∑j=1n1rj,ε0≥1γ0rεaP2+∑j=1nϵjγ12γ,βy=rεP2Φ1yy+rε4dεP4Ψ12yy3+ε0δ0y+∑j=1nϵjyδ¯12y,it is obtained that(51)V˙≤-c0ε0V0χ-∑j=1ncjzj2-rε4ε2-12σθ~2+d0ξ02+rεdεξ4+∑j=1ndjξ2+12σθ2≤-cVe+d0ξ02+rεdεξ4+∑j=1ndjξ2+12σθ2,where c=min⁡c0,2c1,…,2cn,1/2λmax(P),rθσ.Remark 11.
Because the system of this paper contains the colored noise and uncertain nonlinear functions (see (9)), the small-gain technique in [1–7] which is applied to deterministic nonlinear system is not applicable any more. Because the system of this paper is not an Itô type stochastic differential equation and the changing supply function technique in [17] cannot deal with the colored noise, the unmodeled dynamics is assumed to have enough stability margin motivated by [16] in this paperRemark 12.
Different from the deterministic nonlinear systems in [1–10], a class of random nonlinear systems driven by colored noise is considered in this paper. The stochastic disturbance is regarded as colored noise other than the white noise in [16, 17] which is more reasonable. Different from [25] only considering the problem of the state feedback stabilization, an adaptive output feedback stabilization controller is designed in this paper.


## 3.1. Reduced-Order Observer Design

To counteract the unavailable statexi(i=1,2,…,n), a reduced-order partial-state observer is introduced as follows:(15)x^˙i=x^i+1+ki+1y-kix^1+k1y,i=1,…,n-2,x^˙n-1=u-kn-1x^1+k1y,where k=(k1,k2,…,kn-1)T is chosen such that(16)A0=-kIn-20⋯0is asymptotically stable. For each i=1,2,…,n-1, denote the observer error as(17)εi=1θ∗xi+1-x^i-kiy,where θ∗=max⁡{1,θ1,…,θn}. From (9), (15), and (17), one has(18)ε˙i=1θ∗xi+2+ϕi+1+Ωi+1Tξ-x^i+1+ki+1y-kix^1+k1y-kix2+ϕ1+Ω1Tξ=εi+1-kiε1+1θ∗ϕi+1-kiϕ1+1θ∗Ωi+1-kiΩ1Tξwith xn+1=u, x^n=u, and kn=0, whose compact from is(19)ε˙=A0ε+Δy,χ+ΩTyξ,where Δ(y,χ)=1/θ∗(ϕ2-k1ϕ1,…,ϕn-kn-1ϕ1)T and Ω(y)=1/θ∗(Ω2-k1Ω1,…,Ωn-kn-1Ω1).From Assumption4, we have(20)Δy,χ2=∑i=1n-11θ∗ϕi+1-kiϕ12≤∑i=1n-1δi+1y+kiδ1y+γi+1πχ+kiγ1πχ2≤∑i=1n-12δi+1y+kiδ1y2+∑i=1n-12γi+1+kiγ12π2χ≤y2Φ1y+aδχ,where Φ1(y)=∑i=1n-12(δ¯i+1(y)+kiδ¯1(y))2 and a=∑i=1n-12/γ(γi+1+kiγ1)2.(21)Ωy2=∑i=1n-11θ∗Ωi+1-kiΩ12≤∑i=1n-1ψi+1y+kiψ1y2=y2Ψ1y,where Ψ1(y)=∑i=1n-1(ψ¯i+1(y)+kiψ¯1(y))2. Since A0 is stable, there exists a symmetric positive definite matrix P such that PA0+A0TP=-2In-1. Along the solution of (19), differentiating the quadratic function Vε=1/2εTPε yields(22)V˙ε=-εTε+εTPΔy,χ+ΩTyξ≤-εTε+12εTε+12P2Δy,χ+Ωyξ2≤-12εTε+P2Φ1yy2+aP2δχ+P2Ψ1yy2ξ2≤-12εTε+P2Φ1yy2+aP2δχ+14dεP4Ψ12yy4+dεξ4,where dε is a design parameter.


## 3.2. Adaptive Backstepping Controller Design

From (9) and (17), the derivative of output is represented as(23)y˙=θ∗ε1+x^1+k1y+ϕ1x,χ+Ω1Tyξ,which, together with (9), (15), and (19), consists of the following system(24)χ˙=qχ,y+pχ,yξ0,ε˙=A0ε+Δy,χ+ΩTξ,y˙=x^1+θ∗ε1+k1y+ϕ1x,χ+Ω1Tyξ,x^˙i=x^i+1+ki+1y-kix^1+k1y,i=1,…,n-2,x^˙n-1=u-kn-1x^1+k1y.An adaptive backstepping controller based on the (y,x^1,…,x^n-1)-system of (24) will be developed.Step 1.
Introduce the first two error variables(25)z1=y,z2=x^1-α1,where α1 will be given in later. Consider the Lyapunov function candidate(26)V1=12z12+12rθθ~2,where θ~=θ-θ^, θ^ is the estimate of θ=max⁡{θ∗,θ∗2}. In view of (24) and (25), the derivative of V1 satisfies(27)V˙1≤yz2+α1+θ∗ε1+k1y+ϕ1x,χ+Ω1Tyξ-1rθθ~θ^˙.Applying Young’s inequality, one has(28)yθ∗ε1≤r1y2θ+14r1ε12,where r1>0 is a design parameter. By Assumption 4, one has(29)yϕ1x,χ≤yθ1δ1y+γ1πχ≤12ϵ1y2θ12+ϵ12δ1y+γ1π1χ2≤12ϵ1y2θ+ϵ1y2δ¯12y+ϵ1γ12γδχ,(30)yΩ1Tyξ≤yθ1ψ1yξ≤14d1y2θ1ψ12y+d1ξ2≤14d1y2ψ12yθ+d1ξ2,where ϵ1>0 and d1 are design parameters. Substituting (28)-(30) into (27), one has(31)V˙1≤yz2+yα1+k1y+ϵ1yδ¯12y+ω1θ^+ϵ1γ12γδχ+d1ξ2+θ~ω1y-1rθθ^˙+14r1ε12,where ω1=r1y+1/2ϵ1y+1/4d1yψ12(y). The stabilizing function α1 and the tuning function τ1 are designed as(32)α1=-c1z1-k1y-ω1θ^-βy,τ1=-σθ^+ω1y,where c1>0 is a design parameter and β(y) will be chosen in later. Then(33)V˙1≤-c1z12+yϵ1yδ¯12y-βy+z1z2+ϵ1γ12γδχ+d1ξ2+σθ~θ^+θ~τ1-1rθθ^˙+14r1ε12.Step i = 2 , … , n. Introduce the coordinate change(34)zi+1=x^i-αiy,x^1,…,x^i-1,θ^with zn+1=0 and x^n=u. The differential of zi is given as follows:(35)z˙i=x^i+kiy-ki-1x^1+k1y-∂αi-1∂yx^1+θ∗ε1+k1y+ϕ1x,χ+Ω1Tyξ-∑j=1i-2∂αi-1∂x^jx^j+1+kj+1y-kjx^1+k1y-∂αi-1∂θ^θ^˙=zi+1+αi+ηi-∂αi-1∂yθ∗ε1+ϕ1x,χ+Ω1Tyξ-∂αi-1∂θ^θ^˙,where ηi=kiy-ki-1(x^1+k1y)-∂αi-1/∂y(x^1+k1y)-∑j=1i-2∂αi-1/∂x^j(x^j+1+kj+1y-kj(x^1+k1y)). Assume that one has designed smooth function αj, τj(j=2,…,i-1) such that the following inequality holds for Vi-1=Vi-2+1/4zi-12,(36)V˙i-1≤-∑j=1i-1cjzj2+y∑j=1i-1ϵjyδ¯12y-βy+zi-1zi+∑j=1i-1ϵjγ12γδχ+∑j=1i-1djξ2+∑j=1i-2zj+1∂αj∂θ^rθτi-1-θ^˙+σθ~θ^+θ~τi-1-1rθθ^˙+∑j=1i-114rjε12,where ϵj>0, dj>0, rj>0 (2≤j≤i-1) are design parameter. In the following, we will prove that (36) holds for the i-th Lyapunov function candidate(37)Vi=Vi-1+14zi2.The derivative of Vi satisfies(38)V˙i≤-∑j=1i-1cjzj2+y∑j=1i-1ϵjyδ¯12y-βy+zi-1zi+∑j=1i-1ϵjγ12γδχ+∑j=1i-1djξ2+∑j=1i-2zj+1∂αj∂θ^rθτi-1-θ^˙+σθ~θ^+θ~τi-1-1rθθ^˙+∑j=1i-114rjε12+zizi+1+αi+ηi-∂αi-1αyθ∗ε1+ϕ1x,χ+Ω1Tyξ-∂αi-1∂θ^θ^˙.Applying Young’s inequality, one has(39)zi∂αi-1αyθ∗ε1≤rizi2∂αi-1αy2θ+14riε12,where ri>0 is a design parameter. By Assumption 4, one has(40)zi∂αi-1αyϕ1x,χ≤zi∂αi-1αyθ1δ1y+γ1πχ≤12ϵizi2∂αi-1αy2θ12+ϵi2yδ¯1y+γ1πχ2≤12ϵizi2∂αi-1αy2θ+ϵiy2δ¯12y+ϵiγ12γδχ,(41)zi∂αi-1αyΩ1Tyξ≤zi∂αi-1αyθ1ψ1yξ≤14dizi2∂αi-1αy2θ12ψ12y+diξ2≤14dizi2∂αi-1αy2θψ12y+diξ2,where ϵi>0 and di>0 are design parameters. With (39)-(41), it is obtained that(42)V˙i≤-∑j=1i-1cjzj2+y∑j=1i-1ϵjyδ¯12y-βy+zi-1zi+∑j=1i-1ϵjγ12γδχ+∑j=1i-1djξ2+∑j=1i-2zj+1∂αj∂θ^rθτi-1-θ^˙+σθ~θ^+θ~τi-1-1rθθ^˙+∑j=1i-114rjε12+zizi+1+αi+ηi+rizi2∂αi-1αy2θ+14riε12+12ϵizi2∂αi-1αy2θ+ϵiy2δ¯12y+ϵiγ12γδχ+14dizi2∂αi-1αy2θψ12y+diξ2-zi∂αi-1∂θ^θ^˙.By choosing the i-th tuning function as τi=τi-1+ωizi with ωi=zi∂αi-1/∂y2(ri+1/2ϵi+1/4diψ12(y)), one has(43)V˙i≤-∑j=1i-1cjzj2+y∑j=1iϵjyδ¯12y-βy+zizi-1+∑j=1iϵjγ12γδχ+∑j=1idjξ2+∑j=1i-2zj+1∂αj∂θ^rθτi-θ^˙+σθ~θ^+θ~τi-1rθθ^˙+∑j=1i14rjε12+zizi-1+αi+ηi+ωiθ^-∂αi-1∂θ^rθτi-∑j=1i-2zj+1∂αj∂θ^rθωi.Choose(44)αi=-zi-1-cizi-ηi-ωiθ^+∂αi-1∂θ^rθτi+∑j=1i-2zj+1∂αj∂θ^rθωi,where ci>0 is a design parameter. Then(45)V˙i≤-∑j=1icjzj2+y∑j=1iϵjyδ¯12y-βy+zizi+1+∑j=1iϵjγ12γδχ+∑j=1idjξ2+∑j=1i-1zj+1∂αj∂θ^rθτi-θ^˙+σθ~θ^+θ~τi-1rθθ^˙+∑j=1i14rjε12.At the end of the recursive procedure, the control law and adaptive law are chosen as(46)u=αny,x^1,…,x^n-1,θ^,θ^˙=rθτn.By (45) and (46), one gets(47)V˙n≤-∑j=1ncjzj2+y∑j=1nϵjyδ¯12y-βy+∑j=1nϵjγ12γδχ+∑j=1ndjξ2+σθ~θ^+∑j=1n14rjε12.The Lyapunov function for the whole system is(48)V=ε0V0+rεVε+Vn.Then, by Assumption 6, (22), and (47), we obtain(49)V˙≤-c0ε0V0χ-∑j=1ncjzj2+yrεP2Φ1yy+rε4dεP4Ψ12yy3+ε0δ0y+∑j=1nϵjyδ¯12y-βy+rεaP2+∑j=1nϵjγ12γ-ε0γ0δχ+d0ξ02+rεdεξ4+∑j=1ndjξ2+σθ~θ^-rε2ε2+∑j=1n14rjε12.By choosing(50)rε≥∑j=1n1rj,ε0≥1γ0rεaP2+∑j=1nϵjγ12γ,βy=rεP2Φ1yy+rε4dεP4Ψ12yy3+ε0δ0y+∑j=1nϵjyδ¯12y,it is obtained that(51)V˙≤-c0ε0V0χ-∑j=1ncjzj2-rε4ε2-12σθ~2+d0ξ02+rεdεξ4+∑j=1ndjξ2+12σθ2≤-cVe+d0ξ02+rεdεξ4+∑j=1ndjξ2+12σθ2,where c=min⁡c0,2c1,…,2cn,1/2λmax(P),rθσ.Remark 11.
Because the system of this paper contains the colored noise and uncertain nonlinear functions (see (9)), the small-gain technique in [1–7] which is applied to deterministic nonlinear system is not applicable any more. Because the system of this paper is not an Itô type stochastic differential equation and the changing supply function technique in [17] cannot deal with the colored noise, the unmodeled dynamics is assumed to have enough stability margin motivated by [16] in this paperRemark 12.
Different from the deterministic nonlinear systems in [1–10], a class of random nonlinear systems driven by colored noise is considered in this paper. The stochastic disturbance is regarded as colored noise other than the white noise in [16, 17] which is more reasonable. Different from [25] only considering the problem of the state feedback stabilization, an adaptive output feedback stabilization controller is designed in this paper.


## 4. Stability Analysis

Theorem 13.
For the random system (9), under Assumptions 4–7, the control law and adaptive law (46), the closed-loop system has a unique solution on [t0,∞), and the state of the closed-loop system has an AG-2-M. Furthermore, the output y(t) satisfies(52)limt→∞⁡Ey2≤2δKc,where δK=(d0+rεdε+∑j=1ndj)K+1/2σθ2 and the right-hand can be made small enough by tuning parameters.Proof.
DefineΞ=(χ,εT,z1,…,zn,θ~)T. From the definition of V, V satisfies(53)a1Ξ2≤V≤a2Ξ2,with a1=1/2min⁡{2ε0a_,rελmin(P),1,1/rθ} and a2=1/2max⁡{2ε0a¯,rελmax(P),1,1/rθ}. Since the functions of the closed-loop system satisfy the local Lipschitz condition, from Lemma 2, (53), and (51), then the closed-loop system has a unique solution on [t0,∞), and the state of the closed-loop system has an AG-2-M.
Furthermore, by definingv(t)=EV(t), from (51), one has(54)v˙t=EV˙t≤-cvt+d0Eξ02+rεdεEξ4+∑j=1ndjEξ2+12σθ2.By Lemma 3 and (3), it is obtained that(55)vt≤v0e-ct-t0+δK,which together with (48) implies(56)Ey2≤2EVet≤2Ve0e-ct-t0+2δKc,which leads to (52).
Notingc=min⁡c0,2c1,…,2cn,1/2λmax(P),rθσ and δK=(d0+rεdε+∑j=1ndj)K+1/2σθ2, it is clear that the right-hand sides of (52) can be made small enough by choosing c1,…,cn large enough and d1,…,dn small enough.Remark 14.
By Chebyshev’s inequality, for anyε>0 and ε0>0, there exists a moment T>0 such that when t>T, Pyt>ε≤1/ε2(ε0+2δK/c)≤ε′, where ε′ can be regulated to small enough, which implies the asymptotically stabilization in probability in some sense.


## 5. A Simulation Example

Consider the following nonlinear system(57)χ˙=qχ,x1+pχ,x1ξ0,x˙1=x2+ϕ1x,χ+Ω1Tx1ξ,x˙2=u+ϕ2x,χ+Ω2Tx1ξ,y=x1,where q(χ,x1)=-6χ+2x1+tanh⁡x1, p(χ,x1)=cos⁡χ+sin⁡x1, ϕ1(x,χ)=θ1x1+χsin⁡x2, ϕ2(x,χ)=θ2χcos⁡x2, Ω1T(x1)=[θ1x1,0], Ω2T(x1)=[0,θ2x1] with θ1=0.3, θ2=0.2.Forχ-subsystem of (57), by choosing the Lyapunov function V0(χ)=a_χ2=a¯χ2=1/2χ2, one can verify(58)∂V0∂χqχ,x1+14d0∂V0∂χpχ,x12≤-V0-92-1d0χ2+x12,which implies that Assumption 5 holds for d0=1/2, c0=1, γ0=5/2, δ(χ)=χ2, δ0(y)=y. By verifying Assumptions 6 and 7, it is easy to obtain that δ1=y, δ2=0, π=χ, γ1=1, γ2=1, γ=1, ψ1=y and ψ2=y.The following observer is needed(59)x^1=u-k1x^1+k1y,where k1>0 is a design parameter. By defining two error variables (25), the adaptive output feedback control law is given by the recursive design procedure in Section 3, i.e.,(60)α1=-c1z1-k1y-rεP2Φ1yy-rε4dεP4Ψ12yy3-r1yθ^-12ϵ1yθ^-14d1yψ12yθ^-ϵ1yδ¯12-ϵ2yδ¯12-ε0δ0,τ1=σθ^+r1y2+12ϵ1y2+14d1y2ψ12y,θ^˙=rθτ2=rθτ1+z22∂α1αy2r2+12ϵ2+14d2ψ12y,u=α2=-z1-c2z2-η2-z2∂α1∂y2r2+12ϵ2+14d2ψ12yθ^+∂α1∂θ^rθτ2,η2=-k1x^1+k1y-∂α1∂yx^1+k1y,where P=1/k1, Φ1=2k12, Ψ1=(1+k1)2 and δ¯1=sign(y).In the simulation, the disturbanceξi(i=0,1,2) is produced by(61)ξ˙t=-ξt+biwt,ξi0=0,where b1=0.5, b1=-0.8, b1=1 and w(t)∈R is a zero-mean white noise whose spectral function equals 1. Therefore, ξi is a zero-mean widely stationary process and Eξi2=bi/2; thus, K=0.5. Choose the initial values χ(0)=-0.3, x1(0)=0.15, x2(0)=0.2 and the design parameters k1=3.7, dε=0.5, r1=1, ϵ1=0.3, d1=0.5, r2=1, ϵ2=0.5, d2=0.2, c1=0.5, c2=0.4, rθ=0.0075, σ=2.5, x^1(0)=0, θ^(0)=0, rε=1/r1+1/r2, ε0=1/γ0(rε2/γ(γ2+k1γ1)2P2+∑j=12ϵjγ12/γ)=2/5rε(1+k1)21/k12+ϵ1+ϵ2.Figure1 shows the system responses of closed-loop system. From the results of simulation, the response of closed-loop system is all bounded and the output y can be practically regulated to zero.Figure 1
The response of closed-loop system.


## 6. Conclusion

In this paper, a class of random nonlinear systems with unmodeled dynamics and uncertain nonlinear functions is investigated. Based on a reduced-order observer, an adaptive output feedback stabilization controller is designed such that the mean square of the output can be made small enough by choosing the appropriate parameters. The method of ordinary differential equations is used to analyze the stability of the closed-loop system. A simulation example demonstrates the effectiveness of the proposed scheme.There are other problems under current investigation such as the tracking control, the control based on stochastic small-gain technique, the control based on changing supply function technique, and their practical application. In addition, one can try to apply these methods to generalized triangular form systems driven by colored noise.



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