This paper further considers more general high-order stochastic nonlinear system driven by noise of unknown covariance and its adaptive state-feedback stabilization problem. A smooth state-feedback controller is designed to guarantee that the origin of the closed-loop system is globally stable in probability.

1. Introduction

In this paper, we consider the following high-order stochastic nonlinear system:dx1=x2pdt+f1(x1)dt+g1(x1)Σdω,dx2=x3pdt+f2(x¯2)dt+g2(x¯2)Σdω,⋮dxn=updt+fn(x¯n)dt+gn(x¯n)Σdω,
where x=(x1,…,xn)∈Rn, u∈R, are the state and control input, respectively. x¯i=(x1,…,xi), i=1,…,n. p≥1 is odd integer. w is an r-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. Σ:R+→Rr×r is an unknown bounded nonnegative definite Borel measurable matrix function and ΣΣT denotes the infinitesimal covariance function of the driving noise Σdω. fi:Ri→R and gi:Ri→Rr, i=1,…,n, are assumed to be smooth with fi(0)=0 and gi(0)=0.

When p=1, system (1.1) reduced to the well-known normal form whose study on feedback control problem has achieved great development in recent years. According to the difference of selected Lyapunov functions, the existing literature on controller design can be mainly divided into two types. One type is based on quadratic Lyapunov functions which are multiplied by different weighting functions, see, for example, [1–5] and the references therein. Another essential improvement belongs to Krstić and Deng. By introducing the quartic Lyapunov function, [6, 7] presented asymptotical stabilization control in the large under the assumption that the nonlinearities equal to zero at the equilibrium point of the open-loop system. Subsequently, for several classes of stochastic nonlinear systems with unmodeled dynamics and uncertain nonlinear functions, by combining Krstić and Deng’s method with stochastic small-gain theorem [8], and with dynamic signal and changing supply function [9, 10], different adaptive output-feedback control schemes are studied.

When p>1, some intrinsic features of (1.1), such as that its Jacobian linearization is neither controllable nor feedback linearizable, lead to the existing design tools hardly applicable to this kind of systems. Motivated by the fruitful deterministic results in [11, 12] and the related papers and based on stochastic stability theory in [13–15], and so forth, [16] firstly considered, this class of systems with stochastic inverse dynamics. Subsequently, [17–21] considered respectively, the state-feedback stabilization problem for more general systems with different structures. [22, 23] solved the output-feedback stabilization, and [24] addressed the inverse optimal stabilization.

All the papers mentioned above, however, only consider the case of ΣΣT=I. In this paper, we will further consider more general high-order stochastic nonlinear system driven by noise of unknown covariance and its adaptive state-feedback stabilization problem. A smooth state-feedback controller is designed to guarantee that the origin of the closed-loop system is globally stable in probability. A simulation example verifies the effectiveness of the control scheme.

The paper is organized as follows. Section 2 provides some preliminary results. Section 3 gives the state-feedback controller design and stability analysis, following a simulation example in Section 4. In Section 5, we conclude the paper.

2. Preliminary Results

The following notations definitions and lemmas are to be used throughout the paper.

R+ stands for the set of all nonnegative real numbers, Rn is the n-dimensional Euclidean space, Rn×m is the space of real n×m-matrixes. 𝒞2 denotes the family of all the functions with continuous second partial derivatives. |x| is the usual Euclidean norm of a vector x. ∥X∥=(Tr{XXT})1/2, where Tr{X} is its trace when X is a square matrix and XT denotes the transpose of 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 class 𝒦 for each fixed t and decrease to zero as t→∞ for each fixed s. To simplify the procedure, we sometimes denote χ(t) by χ for any variable χ(t).

Consider the nonlinear stochastic system
dx=f(x)dt+g(x)dω,
where x∈Rn is the state, w is an r-dimensional independent Wiener process with incremental covariance ΣΣTdt, that is, E{dωdωT}=ΣΣTdt, where Σ is a bounded function taking values in the set of nonnegative definite matrices, f:Rn→Rn and g:Rn→Rn×r are locally Lipschitz functions.

Definition 2.1 (see [<xref ref-type="bibr" rid="B13">13</xref>]).

For any given V(x)∈𝒞2 associated with stochastic system (2.1), the differential operator ℒ is defined as
LV(x)≜∂V(x)∂xf(x)+12Tr{gT(x)∂2V(x)∂x2g(x)}.

Definition 2.2 (see [<xref ref-type="bibr" rid="B25">25</xref>]).

For the stochastic system (2.1) with f(0)=0, g(0)=0, the equilibrium x(t)=0 is globally asymptotically stable (GAS) in probability if for any ξ>0, there exists a class 𝒦ℒ function β(·,·) such that
P{|x(t)|<β(|x0|,t)}≥1-ξ,t≥0,∀x0∈Rn∖{0}.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

Consider the stochastic system (2.1). If there exist a 𝒞2 function V(x), class 𝒦∞ function α1 and α2, constants c1>0 and c2≥0, and a nonnegative function W(x) such that for all x∈Rn, t≥0α1(|x|)≤V(x)≤α2(|x|),LV≤-c1W(x)+c2,
then,

there exists an almost surely unique solution on [0,∞) for each x0∈Rn,

when c2=0, f(0)=0, g(0)=0, and W(x) is continuous, then the equilibrium x=0 is globally stable in probability and the solution x(t) satisfies P{limt→∞W(x(t))=0}=1.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let x,y be real variables, for any positive integers m,n, positive real number b and nonnegative continuous function a(·), then
a(⋅)xmyn≤b|x|m+n+nm+n(m+nm)-m/n(a(⋅))(m+n)/nb-m/n|y|m+n,
when a(·)=1,b=(m/(m+n))d, d is a positive constant, then the above inequality is
xmyn≤mm+nd|x|m+n+nm+nd-m/n|y|m+n.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let x,y and zi, i=1,…,p, be real variables and let l1(·) and l2(·) be smooth mappings. Then for any positive integers m, n and real number N>0, there exist two nonnegative smooth functions h1(·) and h2(·) such that the following inequalities hold:

Lemma 2.6 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let x1,…,xn, p, be positive real variables, then
(x1+⋯+xn)p≤max{np-1,1}(x1p+⋯+xnp).

3. Controller Design and Stability Analysis

The objective of this paper is to design a smooth state-feedback controller for system (1.1), such that the solution of the closed-loop system is GAS in probability. To achieve the control objective, we need the following assumption.

Assumption 3.1.

There are nonnegative smooth functions fi1, gi1, i=1,…,n, such that
|fi(x¯i)|≤(∑j=1i|xj|p)fi1(x¯i),|gi(x¯i)|≤(∑j=1i|xj|p)gi1(x¯i).

3.1. Controller Design

Now, we give the controller design procedure by using the backstepping method.

First, we introduce the following coordinate change:
z1=x1,zi=xi-αi-1(x¯i-1,θ̂),i=2,…,n,
where αi-1(x¯i-1,θ̂), i=2,…,n, are smooth virtual controllers which will be designed later, θ̂ is the estimation of θ, and
θ≜maxt≥0{‖ΣΣT‖(p+3)/2,‖ΣΣT‖(p+3)/3,‖ΣΣT‖}.
Then, by Itô’s differentiation rule, one hasdzi=d(xi-αi-1(x¯i-1,θ̂))=(xi+1p+Fi(x¯i)-∑l=1i-1∂αi-1∂xlxl+1p-12∑k,m=1i-1∂2αi-1∂xk∂xmgk(x¯k)ΣΣTgmT(x¯m)-∂αi-1∂θ̂θ̂̇)dt+Gi(x¯i)Σdω,
where
Fi(x¯i)=fi(x¯i)-∑l=1i-1∂αi-1∂xlfl(x¯l),Gi(x¯i)=gi(x¯i)-∑l=1i-1∂αi-1∂xlgl(x¯l).

Step 1.

Define the first Lyapunov function
V1(z1,θ̂)=14z14+12Γθ̃2,
where Γ is a positive constant, θ̃=θ-θ̂ is the parameter estimation error. By (3.2)–(3.4) and Assumption 3.1, there exist nonnegative smooth functions μ11 and μ15 such that
LV1=z13x2p+z13f1(x1)+32z12g1(x1)ΣΣTg1T(x1)-θ̃Γθ̂̇≤z13(x2p-α1p)+z13α1p+z1p+3μ11(z1)+z1p+3μ15(z1)θ-θ̃Γθ̂̇≤z13(x2p-α1p)+z13α1p+z1p+3μ11(z1)+z1p+3μ15(z1)1+θ̂2-θ̃Γ(θ̂̇-Γz1p+3μ15(z1)).
Choosing the first smooth virtual controller
α1(x1,θ̂)=-z1β1(z1,θ̂),β1(z1,θ̂)=(c11+μ11(z1)+μ15(z1)1+θ̂2)1/p,
and the tuning function
τ1(z1)=Γz1p+3μ15(z1),
one has
LV1≤-c11z1p+3+z13(x2p-α1p)-θ̃Γ(θ̂̇-τ1),
where c11>0 is a design parameter.

Step i (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M137"><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>i</mml:mi><mml:mo>≤</mml:mo><mml:mi>n</mml:mi></mml:math></inline-formula>).

For notational coherence, denote u=xn+1. Assuming that at step i-1, one has
LVi-1≤-∑j=1i-1cj,i-1zjp+3-(θ̃Γ+∑j=2i-1zj3∂αj-1∂θ̂)(θ̂̇-τi-1)+zi-13(xip-αi-1p),
where τi-1=τi-2+Γzi-1p+3(μi-1,4+μi-1,5). In the sequel, we will prove that (3.11) still holds for the ith Lyapunov function Vi(z¯i,θ̂)=Vi-1(z¯i-1,θ̂)+(1/4)zi4. By (3.4) and (3.11), one has
LVi≤LVi-1+zi3(xi+1p+Fi(x¯i)-∑l=1i-1∂αi-1∂xlxl+1p-12∑k,m=1i-1∂2αi-1∂xk∂xmgk(x¯k)ΣΣTgmT(x¯m))-zi3∂αi-1∂θ̂θ̂̇+32zi2Tr{ΣTGiT(x¯i)Gi(x¯i)Σ}≤-∑j=1i-1cj,i-1zjp+3-(θ̃Γ+∑j=2i-1zj3∂αj-1∂θ̂)(θ̂̇-τi-1)+zi-13(xip-αi-1p)+zi3(xi+1p+Fi(x¯i)-∑l=1i-1∂αi-1∂xlxl+1p-12∑k,m=1i-1∂2αi-1∂xk∂xmgk(x¯k)ΣΣTgmT(x¯m))-zi3∂αi-1∂θ̂θ̂̇+32zi2Tr{ΣTGiT(x¯i)Gi(x¯i)Σ}+zi3∂αi-1∂θ̂τi-1-zi3∂αi-1∂θ̂τi-1.
To proceed further, an estimate for each term in the right-hand side of (3.12) is needed. Using Itô’s differentiation rule, Lemmas 2.4–2.6, (3.2), and (3.3), it follows that
zi-13(xip-αi-1p)=zi-13((zi+αi-1)p-αi-1p)≤ξi1zi-1p+3+μi1(z¯i,θ̂)zip+3,zi3Fi(x¯i)≤|zi|3∑j=1i|zj|pρi2j(z¯i,θ̂)≤∑j=1i-1ξi2jzjp+3+μi2(z¯i,θ̂)zip+3,-zi3∑l=1i-1∂αi-1∂xlxl+1p≤|zi|3∑j=1i-1|zj|pρi3j(z¯i,θ̂)≤∑j=1i-1ξi3jzjp+3+μi3(z¯i,θ̂)zip+3,-12zi3∑k,m=1i-1∂2αi-1∂xk∂xmgk(x¯k)ΣΣTgmT(x¯m)≤|zi|3∑j=1i-1|zj|2pρi4j(z¯i,θ̂)‖ΣΣT‖≤∑j=1i-1ξi4jzjp+3+μi4(z¯i,θ̂)zip+3θ,32zi2Tr{ΣTGiT(x¯i)Gi(x¯i)Σ}≤zi2∑j=1izjp+1ρi5j(z¯i,θ̂)‖ΣΣT‖≤∑j=1i-1ξi5jzjp+3+μi5(z¯i,θ̂)zip+3θ,-zi3∂αi-1∂θ̂τi-1≤|zi|3∑j=1i-1|zj|pρi6j(z¯i,θ̂)≤∑j=1i-1ξi6jzjp+3+μi6(z¯i,θ̂)zip+3,
where ξi1, ξi2j, ξi3j, ξi4j, ξi5j, ξi6j, j=1,…,i-1, are positive constants and μi1, μi2, μi3, μi4, μi5, μi6, ρi2j, ρi3j, ρi4j, ρi5j, ρi6j, j=1,…,i, are nonnegative smooth functions. Substituting (3.13) into (3.12), one has
LVi≤-∑j=1i-1cj,i-1zjp+3+ξi1zi-1p+3-(θ̃Γ+∑j=2i-1zj3∂αj-1∂θ̂)(θ̂̇-τi-1)+zi3αip+zip+3(μi1+μi2+μi3+μi6+(μi4+μi5)1+θ̂2)+θ̃Γ(μi4+μi5)Γzip+3+∑j=1i-1(ξi2j+ξi3j+ξi4j+ξi5j+ξi6j)zjp+3+zi3(xi+1p-αip).
Choosing the ith smooth virtual controller αiαi(x¯i,θ̂)=-ziβi(z¯i,θ̂),βi(z¯i,θ̂)=(cii+μi1+μi2+μi3+μi6+(μi4+μi5)(1+θ̂2+∑j=2izj3∂αj-1∂θ̂Γ))1/p,
and tuning function τiτi(z¯i)=τi-1(z¯i-1)+Γzip+3(μi4+μi5),
and substituting (3.15) and (3.16) into (3.14), it follows that
LVi(z¯i,θ̂)≤-∑j=1icjizjp+3-(θ̃Γ+∑j=2izj3∂αj-1∂θ̂)(θ̂̇-τi)+zi3(xi+1p-αip),
where cji=cjj-ξj+1,1-∑k=26ξikj, j=1,…,i-1.

Hence at step n, the smooth adaptive state-feedback controlleru=αn(x¯n,θ̂)=-znβn(z¯n,θ̂),θ̂̇=τn(z¯n),βn(z¯n,θ̂)=(cnn+μn1+μn2+μn3+μn6+(μn4+μn5)(1+θ̂2+∑j=2nzj3∂αj-1∂θ̂Γ))1/p,τn(z¯n)=Γz1p+3μ15(z1)+∑j=2nΓzjp+3(μj4+μj5),
such that the nth Lyapunov functionVn(z¯n,θ̂)=14∑j=1nzj4+12Γθ̃2
satisfiesLVn≤-∑j=1ncjnzjp+3,
where μnl, l=1,…,6, are nonnegative smooth functions, cjn, j=1,…,n, are constants, and
cjn=cjj-ξj+1,1-∑k=26ξnkj,j=1,…,n-1.

3.2. Stability AnalysisTheorem 3.2.

If Assumption 3.1 holds for the high-order stochastic nonlinear system (1.1), under the state-feedback controller (3.18), then

the closed-loop system consisting of (1.1), (3.2), (3.8), (3.9), (3.15), (3.16), and (3.18) has an almost surely unique solution on [0,∞) for each (x0,θ̃(0))∈Rn+1,

the origin of the closed-loop system is globally stable in probability,

P{limt→∞|x(t)|=0}=1 and P{limt→∞θ̂(t) exists and is finite}=1.

Proof.

It is easy to verify that Vn(z¯n,θ̂) is 𝒞2 on z¯n and θ̂. For j=1,…,n-1, choose the design parameter cjj>ξj+1,1+∑k=26ξnkj, cnn>0, then by (3.21), cjn>0, j=1,…,n-1. Since Vn(z¯n,θ̂) is continuous, positive, and radially bounded, by (3.20), (3.21), and Lemma 4.3 in [25], there exist two class 𝒦∞ functions α1 and α2 such that α1(|x|,|θ̃|)≤Vn(z¯n,θ̂)≤α2(|x|,|θ̃|). Hence, the condition of Lemma 2.3 holds.

By Lemma 2.3, it follows that conclusion (i), (ii) hold, and P{limt→∞|z(t)|=0}=1. In view of αi(0,θ̂)=0 and xi=zi+αi-1(x¯i-1,θ̂), one has P{limt→∞|x(t)|=0}=1. By (3.20) and the definition of Vn(z¯n,θ̂) in (3.19), it holds that θ̃(t) converges a.s. to a finite limit θ̃∞ as t→∞, therefore θ̂(t) converges a.s. to a finite limit as t→∞.

4. A Simulation Example

Consider a two-order nonlinear stochastic system
dx1=x23dt+f1(x1)dt+x13Σdω,dx2=u3dt+f2(x¯2)dt+x23Σdω,
where f1(x1)=x13, f2(x¯2)=x1x22. By Lemma 2.4, one gets |f1(x1)|≤|x1|3, |g1(x1)|≤|x1|3, |f2(x¯2)|≤(1/3)|x1|3+(2/3)|x2|3, g2(x¯2)≤|x2|3. We choose f11(x1)=1, g11(x1)=1, f21(x¯2)=2/3, g21(x¯2)=1, Assumption 3.1 is satisfied.

We now give the design of state-feedback controller for system (4.1).

Step 1.

Define z1=x1, V1(z1,θ̂)=(1/4)z14+(1/2Γ)θ̃2. A smooth virtual controller
α1(x1,θ̂)=-z1β1(z1,θ̂),β1(z1,θ̂)=(c11+1+μ15(z1)1+θ̂2)1/3,
and the tuning function
τ1(z1)=Γz16μ15(z1)
yield ℒV1(z1,θ̂)≤-c11z16+z13(x23-α13)+(θ̃/Γ)(θ̂̇-τ1), where
μ15(z1)=32z12,θ(t)=maxt≥0{‖Σ(t)Σ(t)T‖3,‖Σ(t)Σ(t)T‖2,‖Σ(t)Σ(t)T‖}.

Step 2.

Defining z2=x2-α1(x1,θ̂), V2(z¯2,θ̂)=V1(z1,θ̂)+(1/4)z24, by (3.12), one has
LV2(z1,θ̂)≤-c11z16+z13(x23-α13)+θ̃Γ(θ̂̇-τ1)+z23(u3+F2(x¯2)-∂α1∂x1x23-12∂2α1∂x12g1(x¯1)ΣΣTg1T(x¯1))-z23∂α1∂θ̂θ̂̇+32z22Tr{ΣTG2T(x¯2)G2(x¯2)Σ},
where F2(x¯2)=f2(x¯2)-(∂α1/∂x1)f1(x1), G2(x¯2)=g2(x¯2)-(∂α1/∂x1)g1(x1). By Lemma 2.4, the definition of z2, and (4.2), one can obtain
z13(x23-α13)≤12d11z16+12d11-1z26+3(23d12z16+13d12-1β13z26+56d13z16+16d13-1β112z26)=ξ21z16+μ21(z1,θ̂)z26,z23F2(x¯2)≤2|z2|3(13|z1|3+23|z2|3+|z1|3β12-∂α1∂x1z13)=ξ221z16+μ22(z¯2,θ̂),-z23∂α1∂x1x23≤|z2|3|∂α1∂x1|(z23-3z22z1β1+3z2z1β1-z13β13)≤ξ231z16+μ23(z¯2,θ̂)z26,-12z23∂2α1∂x12g13ΣΣTg13≤|z2|3∂2α1∂x12z16‖ΣΣT‖≤ξ241z16+μ24(z¯2,θ̂)z26θ;
by (4.3), Lemmas 2.4, 2.6, and the definitions of z2 and G2(x¯2), one has
-z23∂α1∂θ̂τ1≤|z2|3|z1|3(32∂α1∂θ̂Γ|z1|5)≤ξ261z16+μ26(z¯2,θ̂)z26,32z22Tr{ΣTG2T(x¯2)G2(x¯2)Σ}≤32z22((z2-z1β1)3-∂α1∂x1z13)2‖ΣΣT‖≤z22((3⋅25β16+3⋅(∂α1∂x1)2)z16+25z26)‖ΣΣT‖≤ξ251z16+μ25(z¯2,θ̂)z26θ,
where ξ21=(1/2)d11+2d12+(5/2)d13, ξ221=(1/3)d21+d22+d23, ξ231=(1/2)d31+(1/2)d32, ξ241=(1/2)d41, ξ251=(2/3)d51, ξ261=(1/2)d61, μ21(z1,θ̂)=(1/2)d11-1+d12-1β13(z1,θ̂)+(1/2)d13-5β112(z1,θ̂), μ22(z¯2,θ̂)=(1/3)d21-1+(4/3)+d22-1β14+d23-1(∂α1/∂x1)2, μ23(z¯2,θ̂)=(∂α1/∂x1)(1-2β13/2+β16)+(∂α1/∂x1)2((1/2)d31-1+(1/2)d32-1), μ24(z¯2,θ̂)=(1/2)d41-1(∂2α1/∂x12)2z16, μ25(z¯2,θ̂)=32z22+d51-1(128β118+4(∂α1/∂x1)6)z16, μ26(z¯2,θ̂)=(9/8)d61-1(∂α1/∂θ̂)2Γ2z110, d11, d12, d13, d21, d22, d23, d31, d32, d41, d51, d61 are positive constants.

Choosing the smooth adaptive controller
u=-z2β2(z¯2,θ̂),θ̂̇=τ2(z¯2),τ2(z¯2)=Γz16μ15(z1)+Γz26(μ24(z¯2,θ̂)+μ25(z¯2,θ̂)),β2(z¯2,θ̂)=(c22+μ21+μ22+μ23+μ26+(μ24+μ25)(1+θ̂2+Γz23∂α1∂θ̂))1/3,
and substituting (4.6)–(4.8) into (4.5), one has
LV2≤-c12z16-c22z26,
where c12=c11-ξ21-ξ221-ξ231-ξ241-ξ251-ξ261>0.

In simulation, we choose Σ(t)≡1, the parameters Γ=1, c11=11, c22=1, d11=1, d12=1, d13=1, d21=1, d22=1, d23=1, d31=0.01, d32=1, d41=1, d51=1, d61=1, the initial values θ(0)=0, x1(0)=0, x2(0)=-0.5, the sampling period =0.01. Figure 1 verifies the effectiveness of the control scheme.

The responses of closed-loop system (4.1)–(4.3), (4.8).

5. Conclusion

In this paper, we further consider more general high-order stochastic nonlinear system driven by noise of unknown covariance and its adaptive state-feedback stabilization problem.

There is a still remaining problem to be investigated: under current investigation, how to design an output feedback controller for system (1.1) with Assumption 3.1?

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 10971256 and 61104222), the Specialized Research Fund for the Doctoral Program of Higher Education of China (no. 20103705110002), and Natural Science Foundation of Jiangsu Province (BK2011205), Natural Science Foundation of Xuzhou Normal University.

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