# Adaptive Output Feedback Control for a Class of Stochastic Nonlinear Systems with SiISS Inverse Dynamics

Authors: Na Duan; Hai-Kuan Liu
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The adaptive stabilization scheme based on tuning function for stochastic nonlinear systems with stochastic integral input-to-state stability (SiISS) inverse dynamics is investigated. By combining the stochastic LaSalle theorem and small-gain type conditions on SiISS, an adaptive output feedback controller is constructively designed. It is shown that all the closed-loop signals are bounded almost surely and the stochastic closed-loop system is globally stable in probability.


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## Body

## 1. Introduction

Global stabilization control design of stochastic nonlinear systems is one of the most important topics in nonlinear control theory, which has received and is increasingly receiving a great deal of attention; see, for example, [1–37] and the references therein. For a class of stochastic nonlinear systems with stochastic inverse dynamics, much progress has been made in the design of the global stabilization controller [12, 13, 15, 16, 24, 25, 31]. However, all these controllers are only robust against stochastic inverse dynamics with stringent stability margin. To weaken the stringent condition on stochastic inverse dynamics, a natural idea is to benefit from input-to-state stability (ISS) in [38] and integral input-to-state stability (iISS) in [39] which are now recognized as the central unifying concepts in feedback design and stability analysis of deterministic nonlinear systems. Tsinias in [21], Tang and Basar in [19] first proposed the concept of stochastic input-to-state stability (SISS) independently. Further in-depth study on SISS and its applications are presented in [9–11, 18, 22]. Motivated by these aforementioned important results, [34] showed that SISS condition can be weakened to stochastic integral input-to-state stability (SiISS) and developed a unifying output feedback framework for global regulation.Nonlinear small-gain theorem plays an important role in the controller design and stability analysis of deterministic nonlinear systems in [40, 41]. While for stochastic nonlinear systems, there are fewer results on the small-gain theorem. [25] firstly established a gain-function-based stochastic nonlinear small-gain theorem for ISpS in probability. In some succeeding research work, [9, 11, 35] presented some Lyapunov-based small-gain type conditions on SISS and SiISS, respectively. [4] further discusses the relationship of small-gain type conditions on SiISS and studies the problem of GAS in probability via output feedback.In this paper, inspired by [4], a more general class of stochastic nonlinear systems with uncertain parameters and stochastic integral input-to-state stability (SiISS) inverse dynamics is investigated. By combining the stochastic LaSalle theorem and small-gain type conditions on SiISS, an adaptive output feedback controller is proposed to guarantee that all the closed-loop signals are bounded almost surely and the stochastic closed-loop system is globally stable in probability.The paper is organized as follows. Section2 begins with the mathematical preliminaries. Section 3 presents statement of the problem. The design of adaptive output feedback controller is given in Section 4. The corresponding analysis is given in Section 5. Section 6 concludes the paper.


## 2. Mathematical Preliminaries

The following notations are used throughout the paper.ℝ+ stands for the set of all nonnegative real numbers, ℝn is the n-dimensional Euclidean space, and ℝn×m is the space of real n×m-matrices. For x=(x1,…,xn), one defines x̅i=(x1,…,xi), i=1,…,n-1. Ci denotes the family of all the functions with continuous ith partial derivatives. L1(ℝ+;ℝ+) is the family of all functions l:ℝ+→ℝ+ such that ∫0∞l(t)dt<∞. For a given vector or matrix X, XT denotes its transpose, Tr{X} denotes its trace when X is square. |X| denotes the Euclidean norm of a vector X, and ∥X∥=(Tr{XXT})1/2 for a matrix X. 𝒦 denotes the set of all functions: ℝ+→ℝ+, which are continuous, strictly increasing, and vanishing at zero; 𝒦∞ is the set of all functions which are of class 𝒦 and unbounded; 𝒦ℒ denotes the set of all functions β(s,t): ℝ+×ℝ+→ℝ+, which are of 𝒦 for each fixed t and decrease to zero as t→∞ for each fixed s.Consider the stochastic nonlinear delay-free system(2.1)dx=f(x,t)dt+g(x,t)dw,∀x(0)=x0∈Rn,
where x∈ℝn, w is an m-dimensional standard Wiener process defined in 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 the probability measure. Borel measurable functions f: ℝn×ℝ+→ℝn and g: ℝn×ℝ+→ℝn×m are piecewise continuous in t and locally bounded and locally Lipschitz continuous in x uniformly in t∈ℝ+. Let ℒV(x) denote infinitesimal generator of function V∈C2 along stochastic system (2.1) with the definition of(2.2)LV(x)=∂V(x)∂xf(x,t)+12Tr⁡{gT(x,t)∂2V(x)∂x2g(x,t)}.Definition 2.1 (see [34]).
Stochastic process{ξ(t)}t≥t0 is said to be bounded almost surely if supt≥t0|ξ(t)|<∞ a.s.Lemma 2.2 (Stochastic LaSalle Theorem [14]).
For system (2.1), if there exist functions V∈C2, α1,α2∈𝒦∞, l∈L1(ℝ+;ℝ+), and a continuous nonnegative function W:ℝn→ℝ such that for all x∈ℝn, t≥0,
(2.3)α1(|x|)≤V(x)≤α2(|x|),LV(x)≤-W(x)+l(t),
then for each x0∈ℝn,(i)
system (2.1) has a unique strong solution on [0,∞), and solution x(t) is bounded almost surely;(ii)
whenf(0,t)≡0, g(0,t)≡0, l(t)≡0, the equilibrium x=0 is globally stable in probability.In the following, we cite two small-gain type conditions on SiISS in [35].Consider the following stochastic nonlinear system(2.4)dx=f(x,v,t)dt+g(x,v,t)dw,∀x(0)=x0∈Rn,
where x∈ℝn is the state, v∈ℝm is the input, and w is an r-dimensional standard Wiener process defined as in (2.1). Borel measurable functions f: ℝn×ℝm×ℝ+→ℝn and g: ℝn×ℝm×ℝ+→ℝn×r are locally bounded and locally Lipschitz continuous with respect to (x,v) uniformly in t∈ℝ+.Definition 2.3 (see [34]).
System (2.4) is said to be stochastic integral input-to-state stable (SiISS) using Lyapunov function if there exist functions V∈C2(ℝn;ℝ), α1,α2,γ∈𝒦∞, and a merely positive definite continuous function α such that
(2.5)α1(|x|)≤V(x)≤α2(|x|),LV(x)≤-α(|x|)+γ(|v|).
The function V satisfying (2.5) is said to be a SiISS-Lyapunov function, and (α,γ) in (2.5) is called the SiISS supply rate of system (2.4).Lemma 2.4 (see [35]).
For system (2.4) satisfying (2.5), if there exists a positive definite continuous function α̃ such that limsups→0+α̃(s)/α(s)<∞, limsups→∞α̃(s)/α(s)<∞, then there exists a function γ̃∈𝒦∞ such that (α̃,γ̃) is a new SiISS supply rate of system (2.4). Moreover, if limsups→0+γ(s)/sm<∞, then limsups→0+γ̃(s)/sm<∞, where m is any positive integer.The following lemma shows that the condition at infinity can be removed if more prior information on stochastic system is known.Assumption H.
For functionsg, V, α given in (2.4), (2.5) with liminfs→∞α(s)=∞, there exist known smooth positive definite functions ϕ1, ϕ2 such that ∥g(x,v,t)∥≤ϕ1(|x|), |∂V(x)/∂x|≤ϕ2(|x|) and limsups→0+ϕ12(s)ϕ22(s)/α(s)<∞.Lemma 2.5 (see [35]).
For system (2.4) satisfying (2.5) and Assumption H, if there exists a positive definite function α̃ such that
(2.6)limsup⁡s→0+α̃(s)α(s)<∞,∫0∞e-∫0s(1/ζ(α1-1(τ)))dτ[ξ(α1-1(s))]′ds<∞,
where ξ(·)≥0, ζ(·)>0 are smooth increasing functions with ξ(s)α(s)≥2α̃(s), ζ(s)α(s)≥ϕ12(s)ϕ22(s) for any s≥0, then there exists a function γ̃∈𝒦∞ such that (α̃,γ̃) is a new SiISS supply rate of system (2.4). Moreover, if limsups→0+γ(s)/sm<∞, then limsups→0+γ̃(s)/sm<∞, where m is any positive integer.Lemma 2.6.
Letx,y be real variables, then for any positive integers m, n and continuous function a(·)≥0, a(·)xmyn≤b|x|m+n+(n/(m+n))((m+n)/m)-m/na(m+n)/n(·)bm/n|y|m+n, where b>0 is any real number.


## 3. Problem Statement

In this paper, we consider a class of stochastic nonlinear systems described by(3.1)dη=φ0(η,x1)dt+ψ0(η,x1)dw,dx1=x2dt+φ1(η,x)dt+ψ1(η,x)dw,⋮dxn-1=xndt+φn-1(η,x)dt+ψn-1(η,x)dw,dxn=udt+φn(η,x)dt+ψn(η,x)dw,y=x1,
where (x2,…,xn)∈ℝn-1, u,y∈ℝ represent the unmeasurable state, the control input, and the measurable output, respectively. η∈ℝn0 is referred to as the stochastic inverse dynamics. The initial value (ηT(0),x1(0),…,xn(0)) can be chosen arbitrarily. w is an m-dimensional standard Wiener process defined as in (2.1). Uncertain functions φi:ℝn0×ℝn→ℝ, ψi:ℝn0×ℝn→ℝm, 1≤i≤n, are smooth functions. It is assumed that φ0 and ψ0 are locally Lipschitz continuous functions.The research purpose of this paper is to design an adaptive output feedback controller for system (3.1) by using stochastic LaSalle theorem and small-gain type conditions on SiISS, in such a way that, for all initial conditions, the solutions of the closed-loop system are bounded almost surely and the closed-loop systems are globally stable in probability. To achieve the control purpose, we need the following assumptions.Assumption 3.1.
For each1≤i≤n, there exist the unknown constant θi>0, the known nonnegative smooth functions φi0, φi1, ψi0 and ψi1 with φij(0)=0, ψij(0)=0, j=0,1, such that |φi(η,x)|≤θi(φi0(|η|)+φi1(x1)), |ψi(η,x)|≤θi(ψi0(|η|)+ψi1(x1)).For Assumption3.1, there exist smooth functions φ̅i1 and ψ̅i1 satisfying(3.2)φi1(x1)=x1φ̅i1(x1),ψi1(x1)=x1ψ̅i1(x1),
which will be frequently used in the subsequent sections.Assumption 3.2.
For theη-subsystem, there exists an SiISS-Lyapunov function V0(η). Namely, V0 satisfies α̲(|η|)≤V0(η)≤α¯(|η|), ℒV0(η)≤-α(|η|)+γ(|x1|), where α̲,α¯,γ are class 𝒦∞ functions, and α is merely a continuous positive definite function.


## 4. Design of an Adaptive Output Feedback Controller

### 4.1. Reduced-Order Observer Design

Introduce the following reduced-order observer:(4.1)x̂̇i=x̂i+1+ki+1y-ki(x̂1+k1y),1≤i≤n-2,x̂̇n-1=u-kn-1(x̂1+k1y),
where k=(k1,…,kn-1)T is chosen such that A0=[-kIn-20…0] is asymptotically stable. Define the error variable(4.2)ei=1θ*(xi+1-x̂i-kix1),1≤i≤n-1,θ*=max⁡{1,θ1,…,θn}.
By (3.1), (4.1), and (4.2), one gets
(4.3)dei=(ei+1-kie1)dt+1θ*(φi+1(η,x)-kiφ1(η,x))dt+1θ*(ψi+1(η,x)-kiψ1(η,x))dw,
whose compact form is
(4.4)de=(A0e+Φ(η,x))dt+Ψ(η,x)dw,
where(4.5)e=(e1,…,en-1)T,Φ(η,x)=1θ*(φ2(η,x)-k1φ1(η,x),φ3(η,x)-k2φ1(η,x),…,φn(η,x)-kn-1φ1(η,x))T,Ψ(η,x)=1θ*(ψ2(η,x)-k1ψ1(η,x),ψ3(η,x)-k2ψ1(η,x),…,ψn(η,x)-kn-1ψ1(η,x))T.


### 4.2. The Design of Adaptive Backstepping Controller

From (3.1), (4.1), (4.2), and (4.4), the interconnected system is represented as(4.6)dη=φ0(η,y)dt+ψ0(η,y)dw,de=(A0e+Φ(η,x))dt+Ψ(η,x)dw,dy=(x̂1+k1y+θ*e1+φ1(η,x))dt+ψ1(η,x)dw,dx̂1=(x̂2+k2y-k1(x̂1+k1y))dt,dx̂2=(x̂3+k3y-k2(x̂1+k1y))dt,⋮dx̂n-1=(u-kn-1(x̂1+k1y))dt.
Next, we will develop an adaptive backstepping controller by using the backstepping method. Firstly, a coordinate transformation is introduced(4.7)z1=y,zi+1=x̂i-αi(y,x̂1,…,x̂i-1,θ̂),i=1,…,n-1.Step 1.
By (4.6) and (4.7), one has(4.8)dz1=(α1+z2+k1y+θ*e1+φ1(η,x))dt+ψ1(η,x)dw.
Since A0 is asymptotically stable, there exists a positive definite matrix P such that PA0+A0TP=-In-1. Choose(4.9)V1(e,z1,θ̂)=δ2(eTPe)2+12Γθ̃2+14z14,δ>0,Γ>0,
where θ̃=θ̂-θ, θ̂ is the estimate of θ=max{θ*,θ*(4/3),θ*4}. In view of (2.2), (4.4), (4.8), and (4.9), then(4.10)LV1=2δeTPeeTP(A0e+Φ(η,x))+12Tr⁡{ΨT(η,x)(4δ(eTP)TeTP+2δeTPeP)Ψ(η,x)}+1Γθ̃θ̂̇+z13(α1+z2+k1y+θ*e1+φ1(η,x))+32z12Tr⁡{ψ1T(η,x)ψ1(η,x)}=δeTPeeT(PA0+A0TP)e+2δeTPeeTPΦ(η,x)+Tr⁡{ΨT(η,x)(2δ(eTP)TeTP+δeTPeP)Ψ(η,x)}+1Γθ̃θ̂̇+z13(α1+z2+k1y+θ*e1+φ1(η,x))+32z12Tr⁡{ψ1T(η,x)ψ1(η,x)}≤-δλmin⁡(P)|e|4+2δ‖P‖2|e|3|Φ(η,x)|+3δ‖P‖2|e|2‖Ψ(η,x)‖2+1Γθ̃θ̂̇+z13(α1+z2+k1y+θ*e1+φ1(η,x))+32z12Tr⁡{ψ1T(η,x)ψ1(η,x)}.
Applying Assumption 3.1, (3.2), (4.3), (4.7), and Lemma 2.6, it follows that(4.11)2δ‖P‖2|e|3Φ|(η,x)|≤a01|e|4+a̅01|Φ(η,x)|4,3δ‖P‖2|e|2‖Ψ(η,x)‖2≤a02|e|4+a̅02‖Ψ(η,x)‖4,z13z2≤a10z14+a̅10z24,θ*z13e1≤a11e14+θ*(4/3)γ11(z1)z14≤a11e14+θγ11(z1)z14,z13φ1(η,x)≤|z1|3θ*(φ10(|η|)+φ11(z1))≤a12φ104(|η|)+θγ12(z1)z14,32z12Tr⁡{ψ1T(η,x)ψ1(η,x)}≤3z12θ*2(ψ102(|η|)+ψ112(z1))≤a13ψ104(|η|)+θγ13(z1)z14,where a̅01, a̅02, γ11, γ12, and γ13 are smooth functions, a01,a02,a10,a̅10,a11,a12,a13>0 are constants.Using(a1+⋯+an)2≤(x+a)n=n∑i=1nai2, (a+b)4≤8a4+8b4, y=x1, Assumption 3.1, (3.2), and (4.5), one gets(4.12)|Φ(η,x)|4≤64(n-1)((k14+⋯+kn-14)(φ104+y4φ̅114)+φ204+⋯+φn04+y4(φ̅214+⋯+φ̅n14)),‖Ψ(η,x)‖4≤64(n-1)((k14+⋯+kn-14)(ψ104+y4ψ̅114)+ψ204+⋯+ψn04+y4(ψ̅214+⋯+ψ̅n14)).
Substituting (4.11)-(4.12) into (4.10) and using z1=y lead to(4.13)LV1≤-a00|e|4+a11e14+Δ1(|η|)+a̅10z24+z13(α1+k1z1+Δ00(z1)z1+a10z1)+θ(γ11(z1)+γ12(z1)+γ13(z1))z14+1Γθ̃θ̂̇,where(4.14)a00=δλmin⁡(P)-a01-a02,Δ1(|η|)=64(n-1)a̅01((k14+⋯+kn-14)φ104(|η|)+φ204(|η|)+⋯+φn04(|η|))+64(n-1)a̅02((k14+⋯+kn-14)ψ104(|η|)+ψ204(|η|)+⋯+ψn04(|η|))+a12φ104(|η|)+a13ψ104(|η|),Δ00(z1)=64(n-1)a̅01((k14+⋯+kn-14)φ̅114(z1)+φ̅214(z1)+⋯+φ̅n14(z1))+64(n-1)a̅02((k14+⋯+kn-14)ψ̅114(z1)+ψ̅214(z1)+⋯+ψ̅n14(z1)).
Choosing the virtual control α1(·) and the tuning function τ1(·)(4.15)α1(y,θ̂)=-z1(c1+ν1(z1)+k1+Δ00(z1)+a10+θ̂(γ11(z1)+γ12(z1)+γ13(z1))),τ1(y)=Γ(γ11(z1)+γ12(z1)+γ13(z1))z14,
one gets(4.16)LV1≤-c1z14-ν1(z1)z14-a00|e|4+a11e14+Δ1(|η|)+a̅10z24+1Γθ̃(θ̂̇-τ1),
where c1>0 is design parameter and ν1(z1)>0 is a smooth function to be chosen later.Stepi [i=2,…,n]
For notation coherence, denoteu=αn, zn+1=0. At this step, we can obtain a property similar to (4.16), which is presented by the following lemma.Lemma 4.1.
For theith Lyapunov function Vi(e,z̅i,θ̂)=(δ/2)(eTPe)2+(1/2Γ)θ̃2+∑j=1i(zj4/4), there are the virtual control law αi(y,x̂1,…,x̂i-1,θ̂) and the tuning function τi with the form
(4.17)αi(y,x̂1,…,x̂i-1,θ̂)=-Ωi-zi(∑j=1i-1ci+a̅i-1,0+ai0+θ̂(γi1+γi2+γi3+γi4)+Γ(γi1+γi2+γi3+γi4)∑j=1i-1∂αj∂θ̂zj+13)+∂αi-1∂θ̂τi,τi(y,x̂1,…,x̂i-1,θ̂)=τi-1+Γ(γi1+γi2+γi3+γi4)zi4,
such that
(4.18)LVi≤-∑j=1icjzj4+∑j=2ibjz14-ν1(z1)z14-a00|e|4+∑j=1iaj1e14+Δi(|η|)+a̅i0zi+14+(1Γθ̃-∑j=1i-1∂αj∂θ̂zj+13)(θ̂̇-τi),
where ci>0 is the designed parameters, ai0, a̅i-1,0, ai1, bi are some positive constants, γi1, γi2, γi3, γi4 are smooth nonnegative functions, whose choices are given in the proof.Proof.
See the appendix.Therefore, at the end of the recursive procedure, the controller can be constructed as(4.19)u=αn(y,x̂1,…,x̂n-1,θ̂),θ̂̇=τn(y,x̂1,…,x̂n-1,θ̂).
Choosing parameters δ, a01, a02, a11, …, an1, c2,…,cn to satisfy(4.20)a0=δλmin⁡(P)-a01-a02-∑i=1nai1>0,c2,…,cn>0,
by (4.16) and (4.18), one has(4.21)LVn≤-∑i=1ncizi4+∑j=2nbjz14-ν1(z1)z14-a0|e|4+Δn(|η|),
where(4.22)Vn(e,z,θ̂)=δ2(eTPe)2+12Γθ̃2+14∑i=1nzi4,Δn(|η|)=64(n-1)a̅01((k14+…+kn-14)φ104(|η|)+φ204(|η|)+…+φn04(|η|))+64(n-1)a̅02((k14+…+kn-14)ψ104(|η|)+ψ204(|η|)+…+ψn04(|η|))+∑i=1n(ai2φ104(|η|)+ai3ψ104(|η|))+∑j=2naj4ψ104(|η|).


## 4.1. Reduced-Order Observer Design

Introduce the following reduced-order observer:(4.1)x̂̇i=x̂i+1+ki+1y-ki(x̂1+k1y),1≤i≤n-2,x̂̇n-1=u-kn-1(x̂1+k1y),
where k=(k1,…,kn-1)T is chosen such that A0=[-kIn-20…0] is asymptotically stable. Define the error variable(4.2)ei=1θ*(xi+1-x̂i-kix1),1≤i≤n-1,θ*=max⁡{1,θ1,…,θn}.
By (3.1), (4.1), and (4.2), one gets
(4.3)dei=(ei+1-kie1)dt+1θ*(φi+1(η,x)-kiφ1(η,x))dt+1θ*(ψi+1(η,x)-kiψ1(η,x))dw,
whose compact form is
(4.4)de=(A0e+Φ(η,x))dt+Ψ(η,x)dw,
where(4.5)e=(e1,…,en-1)T,Φ(η,x)=1θ*(φ2(η,x)-k1φ1(η,x),φ3(η,x)-k2φ1(η,x),…,φn(η,x)-kn-1φ1(η,x))T,Ψ(η,x)=1θ*(ψ2(η,x)-k1ψ1(η,x),ψ3(η,x)-k2ψ1(η,x),…,ψn(η,x)-kn-1ψ1(η,x))T.


## 4.2. The Design of Adaptive Backstepping Controller

From (3.1), (4.1), (4.2), and (4.4), the interconnected system is represented as(4.6)dη=φ0(η,y)dt+ψ0(η,y)dw,de=(A0e+Φ(η,x))dt+Ψ(η,x)dw,dy=(x̂1+k1y+θ*e1+φ1(η,x))dt+ψ1(η,x)dw,dx̂1=(x̂2+k2y-k1(x̂1+k1y))dt,dx̂2=(x̂3+k3y-k2(x̂1+k1y))dt,⋮dx̂n-1=(u-kn-1(x̂1+k1y))dt.
Next, we will develop an adaptive backstepping controller by using the backstepping method. Firstly, a coordinate transformation is introduced(4.7)z1=y,zi+1=x̂i-αi(y,x̂1,…,x̂i-1,θ̂),i=1,…,n-1.Step 1.
By (4.6) and (4.7), one has(4.8)dz1=(α1+z2+k1y+θ*e1+φ1(η,x))dt+ψ1(η,x)dw.
Since A0 is asymptotically stable, there exists a positive definite matrix P such that PA0+A0TP=-In-1. Choose(4.9)V1(e,z1,θ̂)=δ2(eTPe)2+12Γθ̃2+14z14,δ>0,Γ>0,
where θ̃=θ̂-θ, θ̂ is the estimate of θ=max{θ*,θ*(4/3),θ*4}. In view of (2.2), (4.4), (4.8), and (4.9), then(4.10)LV1=2δeTPeeTP(A0e+Φ(η,x))+12Tr⁡{ΨT(η,x)(4δ(eTP)TeTP+2δeTPeP)Ψ(η,x)}+1Γθ̃θ̂̇+z13(α1+z2+k1y+θ*e1+φ1(η,x))+32z12Tr⁡{ψ1T(η,x)ψ1(η,x)}=δeTPeeT(PA0+A0TP)e+2δeTPeeTPΦ(η,x)+Tr⁡{ΨT(η,x)(2δ(eTP)TeTP+δeTPeP)Ψ(η,x)}+1Γθ̃θ̂̇+z13(α1+z2+k1y+θ*e1+φ1(η,x))+32z12Tr⁡{ψ1T(η,x)ψ1(η,x)}≤-δλmin⁡(P)|e|4+2δ‖P‖2|e|3|Φ(η,x)|+3δ‖P‖2|e|2‖Ψ(η,x)‖2+1Γθ̃θ̂̇+z13(α1+z2+k1y+θ*e1+φ1(η,x))+32z12Tr⁡{ψ1T(η,x)ψ1(η,x)}.
Applying Assumption 3.1, (3.2), (4.3), (4.7), and Lemma 2.6, it follows that(4.11)2δ‖P‖2|e|3Φ|(η,x)|≤a01|e|4+a̅01|Φ(η,x)|4,3δ‖P‖2|e|2‖Ψ(η,x)‖2≤a02|e|4+a̅02‖Ψ(η,x)‖4,z13z2≤a10z14+a̅10z24,θ*z13e1≤a11e14+θ*(4/3)γ11(z1)z14≤a11e14+θγ11(z1)z14,z13φ1(η,x)≤|z1|3θ*(φ10(|η|)+φ11(z1))≤a12φ104(|η|)+θγ12(z1)z14,32z12Tr⁡{ψ1T(η,x)ψ1(η,x)}≤3z12θ*2(ψ102(|η|)+ψ112(z1))≤a13ψ104(|η|)+θγ13(z1)z14,where a̅01, a̅02, γ11, γ12, and γ13 are smooth functions, a01,a02,a10,a̅10,a11,a12,a13>0 are constants.Using(a1+⋯+an)2≤(x+a)n=n∑i=1nai2, (a+b)4≤8a4+8b4, y=x1, Assumption 3.1, (3.2), and (4.5), one gets(4.12)|Φ(η,x)|4≤64(n-1)((k14+⋯+kn-14)(φ104+y4φ̅114)+φ204+⋯+φn04+y4(φ̅214+⋯+φ̅n14)),‖Ψ(η,x)‖4≤64(n-1)((k14+⋯+kn-14)(ψ104+y4ψ̅114)+ψ204+⋯+ψn04+y4(ψ̅214+⋯+ψ̅n14)).
Substituting (4.11)-(4.12) into (4.10) and using z1=y lead to(4.13)LV1≤-a00|e|4+a11e14+Δ1(|η|)+a̅10z24+z13(α1+k1z1+Δ00(z1)z1+a10z1)+θ(γ11(z1)+γ12(z1)+γ13(z1))z14+1Γθ̃θ̂̇,where(4.14)a00=δλmin⁡(P)-a01-a02,Δ1(|η|)=64(n-1)a̅01((k14+⋯+kn-14)φ104(|η|)+φ204(|η|)+⋯+φn04(|η|))+64(n-1)a̅02((k14+⋯+kn-14)ψ104(|η|)+ψ204(|η|)+⋯+ψn04(|η|))+a12φ104(|η|)+a13ψ104(|η|),Δ00(z1)=64(n-1)a̅01((k14+⋯+kn-14)φ̅114(z1)+φ̅214(z1)+⋯+φ̅n14(z1))+64(n-1)a̅02((k14+⋯+kn-14)ψ̅114(z1)+ψ̅214(z1)+⋯+ψ̅n14(z1)).
Choosing the virtual control α1(·) and the tuning function τ1(·)(4.15)α1(y,θ̂)=-z1(c1+ν1(z1)+k1+Δ00(z1)+a10+θ̂(γ11(z1)+γ12(z1)+γ13(z1))),τ1(y)=Γ(γ11(z1)+γ12(z1)+γ13(z1))z14,
one gets(4.16)LV1≤-c1z14-ν1(z1)z14-a00|e|4+a11e14+Δ1(|η|)+a̅10z24+1Γθ̃(θ̂̇-τ1),
where c1>0 is design parameter and ν1(z1)>0 is a smooth function to be chosen later.Stepi [i=2,…,n]
For notation coherence, denoteu=αn, zn+1=0. At this step, we can obtain a property similar to (4.16), which is presented by the following lemma.Lemma 4.1.
For theith Lyapunov function Vi(e,z̅i,θ̂)=(δ/2)(eTPe)2+(1/2Γ)θ̃2+∑j=1i(zj4/4), there are the virtual control law αi(y,x̂1,…,x̂i-1,θ̂) and the tuning function τi with the form
(4.17)αi(y,x̂1,…,x̂i-1,θ̂)=-Ωi-zi(∑j=1i-1ci+a̅i-1,0+ai0+θ̂(γi1+γi2+γi3+γi4)+Γ(γi1+γi2+γi3+γi4)∑j=1i-1∂αj∂θ̂zj+13)+∂αi-1∂θ̂τi,τi(y,x̂1,…,x̂i-1,θ̂)=τi-1+Γ(γi1+γi2+γi3+γi4)zi4,
such that
(4.18)LVi≤-∑j=1icjzj4+∑j=2ibjz14-ν1(z1)z14-a00|e|4+∑j=1iaj1e14+Δi(|η|)+a̅i0zi+14+(1Γθ̃-∑j=1i-1∂αj∂θ̂zj+13)(θ̂̇-τi),
where ci>0 is the designed parameters, ai0, a̅i-1,0, ai1, bi are some positive constants, γi1, γi2, γi3, γi4 are smooth nonnegative functions, whose choices are given in the proof.Proof.
See the appendix.Therefore, at the end of the recursive procedure, the controller can be constructed as(4.19)u=αn(y,x̂1,…,x̂n-1,θ̂),θ̂̇=τn(y,x̂1,…,x̂n-1,θ̂).
Choosing parameters δ, a01, a02, a11, …, an1, c2,…,cn to satisfy(4.20)a0=δλmin⁡(P)-a01-a02-∑i=1nai1>0,c2,…,cn>0,
by (4.16) and (4.18), one has(4.21)LVn≤-∑i=1ncizi4+∑j=2nbjz14-ν1(z1)z14-a0|e|4+Δn(|η|),
where(4.22)Vn(e,z,θ̂)=δ2(eTPe)2+12Γθ̃2+14∑i=1nzi4,Δn(|η|)=64(n-1)a̅01((k14+…+kn-14)φ104(|η|)+φ204(|η|)+…+φn04(|η|))+64(n-1)a̅02((k14+…+kn-14)ψ104(|η|)+ψ204(|η|)+…+ψn04(|η|))+∑i=1n(ai2φ104(|η|)+ai3ψ104(|η|))+∑j=2naj4ψ104(|η|).


## 5. Stability Analysis

We state the main theorems in this paper. This section is divided into two parts.


### 5.1. Case I

Theorem 5.1.
Assume that Assumptions3.1 and 3.2 hold with the following properties:
(5.1)limsup⁡s→0+Δn(|η|)α(s)<∞,limsup⁡s→∞Δn(|η|)α(s)<∞.
If limsups→0+γ(s)/s4<∞ in Assumption 3.2, by appropriately choosing the positive smooth function ν1(·) in (4.15) and the parameters δ, a01, a02, a11, …, an1, c2, …, cn to satisfy (4.20), then(i)
the closed-loop system consisting of (3.1), (4.1), (4.2), (4.7), (4.15), (4.17), and (4.19) has a unique and almost surely bounded strong solution on [0,∞);(ii)
for each initial value(η(0),x(0),x̂(0),θ̂(0)), the equilibrium (η,x)=(0,0) is globally stable in probability, where x̂=(x̂1,…,x̂n-1).Proof.
For any constantϵ>0, by (5.1), one has
(5.2)limsup⁡s→0+(1+ϵ)Δn(|η|)α(s)<∞,limsup⁡s→∞(1+ϵ)Δn(|η|)α(s)<∞.
For the η-subsystem, by Lemma 2.4, there exists a positive and radially unbounded Lyapunov function Ṽ0(η)∈C2 and γ̃=ργ such that
(5.3)LṼ0(η)≤-(1+ϵ)Δn(|η|)+γ̃(|z1|),(5.4)limsup⁡s→0+γ̃(s)s4<∞,
where ρ is a positive constant satisfying (1+ϵ)Δn(|η|)≤ρα(s) for all s≥0. Choosing the following Lyapunov function for the entire closed-loop system
(5.5)V(η,e,z,θ̂)=Ṽ0(η)+Vn(e,z,θ̂)
and combining (4.21) and (5.3), one obtains
(5.6)LV(η,e,z,θ̂)≤-∑i=1ncizi4-(ν1(z1)-∑j=2nbj)z14-a0|e|4-ϵΔn(|η|)+γ̃(|z1|).
By (5.4), there always exists a smooth function ν1(·) to satisfy the following two inequalities:
(5.7)z14(ν1(z1)-∑j=2nbj)≥γ̃(|z1|),ν1(z1)≥∑j=2nbj.
Substituting (5.7) into (5.6) leads to
(5.8)LV(η,e,z,θ̂)≤-∑i=1ncizi4-a0|e|4-ϵΔn(|η|)≜-W(η,e,z).
Noting that φi0(·) and ψi0(·) are nonnegative smooth functions and using (4.22), it follows that Δn(·) and W(·) are continuous nonnegative functions. By (5.5), (5.8), and Lemma 2.2, one concludes that all the solutions of the closed-loop system are bounded almost surely, the equilibrium (η,e,z)=(0,0,0) is globally stable in probability. By (3.1), (4.1), (4.2), (4.7), (4.15), (4.17), (4.19), and the almost sure boundedness of all the signals, it is not difficult to recursively prove that the equilibrium (η,x)=(0,0) is globally stable in probability.


### 5.2. Case II

If more information aboutα in Assumption 3.2 is known, that is, liminfs→∞α(s)=∞, further results under the weaker conditions is given as follows.Assumption 5.2.
For functionsψ0 and V0 given by (4.22) and Assumption 3.2, there exist known smooth nonnegative functions ψ̃1 and ψ̃2 with ψ̃1(0)=ψ̃2(0)=0, such that ∥ψ0(η,x1)∥≤ψ̃1(|η|) and |∂V0(η)/∂η|≤ψ̃2(|η|).Lemma 5.3.
ForΔn, α̲, α, ψ̃1 and ψ̃2 given by (4.22), Assumptions 3.2 and 5.2, if
(5.9)limsup⁡s→0+Δn(s)+ψ̃12(s)ψ̃22(s)α(s)<∞,∫0∞e-∫0s(1/ζ1(α̲-1(τ)))dτ[ξ1(α̲-1(s))]′ds<∞,
where ξ1(·)≥0 and ζ1(·)>0 are smooth increasing functions satisfying
(5.10)ξ1(s)α(s)≥2(1+ϵ)Δn(s),ζ1(s)α(s)≥ψ̃12(s)ψ̃22(s),∀s≥0,ϵ is any positive constant. Then there exists a function γ̃∈𝒦∞ such that ((1+ϵ)Δn,γ̃) is a new SiISS supply rate of the η-subsystem in (3.1). Moreover, if γ in Assumptions 3.2 satisfies limsups→0+γ(s)/s4<∞, then limsups→0+γ̃(s)/s4<∞.Since the condition (5.9) is weaker than (5.1), by using Lemma 5.3, we give further results under the weaker condition (5.9).Theorem 5.4.
Suppose that Assumptions3.1–5.2 and the conditions of Lemma 5.3 hold. If limsups→0+γ(s)/s4<∞ in Assumption 3.2, by appropriately choosing the positive smooth function ν1(·) in (4.15) and the parameters δ, a01, a02, a11, …, an1, c2, …, cn to satisfy (4.20), then(1)
the closed-loop system consisting of (3.1), (4.1), (4.2), (4.7), (4.15), (4.17), and (4.19) has a unique and almost surely bounded strong solution on [0,∞);(2)
for each initial value(η(0),x(0),x̂(0),θ̂(0)), the equilibrium (η,x)=(0,0) is globally stable in probability.


## 5.1. Case I

Theorem 5.1.
Assume that Assumptions3.1 and 3.2 hold with the following properties:
(5.1)limsup⁡s→0+Δn(|η|)α(s)<∞,limsup⁡s→∞Δn(|η|)α(s)<∞.
If limsups→0+γ(s)/s4<∞ in Assumption 3.2, by appropriately choosing the positive smooth function ν1(·) in (4.15) and the parameters δ, a01, a02, a11, …, an1, c2, …, cn to satisfy (4.20), then(i)
the closed-loop system consisting of (3.1), (4.1), (4.2), (4.7), (4.15), (4.17), and (4.19) has a unique and almost surely bounded strong solution on [0,∞);(ii)
for each initial value(η(0),x(0),x̂(0),θ̂(0)), the equilibrium (η,x)=(0,0) is globally stable in probability, where x̂=(x̂1,…,x̂n-1).Proof.
For any constantϵ>0, by (5.1), one has
(5.2)limsup⁡s→0+(1+ϵ)Δn(|η|)α(s)<∞,limsup⁡s→∞(1+ϵ)Δn(|η|)α(s)<∞.
For the η-subsystem, by Lemma 2.4, there exists a positive and radially unbounded Lyapunov function Ṽ0(η)∈C2 and γ̃=ργ such that
(5.3)LṼ0(η)≤-(1+ϵ)Δn(|η|)+γ̃(|z1|),(5.4)limsup⁡s→0+γ̃(s)s4<∞,
where ρ is a positive constant satisfying (1+ϵ)Δn(|η|)≤ρα(s) for all s≥0. Choosing the following Lyapunov function for the entire closed-loop system
(5.5)V(η,e,z,θ̂)=Ṽ0(η)+Vn(e,z,θ̂)
and combining (4.21) and (5.3), one obtains
(5.6)LV(η,e,z,θ̂)≤-∑i=1ncizi4-(ν1(z1)-∑j=2nbj)z14-a0|e|4-ϵΔn(|η|)+γ̃(|z1|).
By (5.4), there always exists a smooth function ν1(·) to satisfy the following two inequalities:
(5.7)z14(ν1(z1)-∑j=2nbj)≥γ̃(|z1|),ν1(z1)≥∑j=2nbj.
Substituting (5.7) into (5.6) leads to
(5.8)LV(η,e,z,θ̂)≤-∑i=1ncizi4-a0|e|4-ϵΔn(|η|)≜-W(η,e,z).
Noting that φi0(·) and ψi0(·) are nonnegative smooth functions and using (4.22), it follows that Δn(·) and W(·) are continuous nonnegative functions. By (5.5), (5.8), and Lemma 2.2, one concludes that all the solutions of the closed-loop system are bounded almost surely, the equilibrium (η,e,z)=(0,0,0) is globally stable in probability. By (3.1), (4.1), (4.2), (4.7), (4.15), (4.17), (4.19), and the almost sure boundedness of all the signals, it is not difficult to recursively prove that the equilibrium (η,x)=(0,0) is globally stable in probability.


## 5.2. Case II

If more information aboutα in Assumption 3.2 is known, that is, liminfs→∞α(s)=∞, further results under the weaker conditions is given as follows.Assumption 5.2.
For functionsψ0 and V0 given by (4.22) and Assumption 3.2, there exist known smooth nonnegative functions ψ̃1 and ψ̃2 with ψ̃1(0)=ψ̃2(0)=0, such that ∥ψ0(η,x1)∥≤ψ̃1(|η|) and |∂V0(η)/∂η|≤ψ̃2(|η|).Lemma 5.3.
ForΔn, α̲, α, ψ̃1 and ψ̃2 given by (4.22), Assumptions 3.2 and 5.2, if
(5.9)limsup⁡s→0+Δn(s)+ψ̃12(s)ψ̃22(s)α(s)<∞,∫0∞e-∫0s(1/ζ1(α̲-1(τ)))dτ[ξ1(α̲-1(s))]′ds<∞,
where ξ1(·)≥0 and ζ1(·)>0 are smooth increasing functions satisfying
(5.10)ξ1(s)α(s)≥2(1+ϵ)Δn(s),ζ1(s)α(s)≥ψ̃12(s)ψ̃22(s),∀s≥0,ϵ is any positive constant. Then there exists a function γ̃∈𝒦∞ such that ((1+ϵ)Δn,γ̃) is a new SiISS supply rate of the η-subsystem in (3.1). Moreover, if γ in Assumptions 3.2 satisfies limsups→0+γ(s)/s4<∞, then limsups→0+γ̃(s)/s4<∞.Since the condition (5.9) is weaker than (5.1), by using Lemma 5.3, we give further results under the weaker condition (5.9).Theorem 5.4.
Suppose that Assumptions3.1–5.2 and the conditions of Lemma 5.3 hold. If limsups→0+γ(s)/s4<∞ in Assumption 3.2, by appropriately choosing the positive smooth function ν1(·) in (4.15) and the parameters δ, a01, a02, a11, …, an1, c2, …, cn to satisfy (4.20), then(1)
the closed-loop system consisting of (3.1), (4.1), (4.2), (4.7), (4.15), (4.17), and (4.19) has a unique and almost surely bounded strong solution on [0,∞);(2)
for each initial value(η(0),x(0),x̂(0),θ̂(0)), the equilibrium (η,x)=(0,0) is globally stable in probability.


## 6. Conclusions

This paper further considers a more general class of stochastic nonlinear systems with uncertain parameters and SiISS inverse dynamics. By combining the stochastic LaSalle theorem and small-gain type conditions on SiISS, an adaptive output feedback controller is designed to guarantee that all the closed-loop signals are bounded almost surely and the stochastic closed-loop system is globally stable in probability.There are two remaining problems to be investigated: (1) an essential problem is to find a practical example with explicit physical meaning for system (3.1). A preliminary attempt on high-order stochastic nonlinear system can be found in [28]. (2) How to design an output feedback controller by using this method in this paper for system (3.1) in which the drift and diffusion vector fields depend on the unmeasurable states besides the measurable output?



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*Source: 673878-2012-02-27.xml*