# Output-Feedback and Inverse Optimal Control of a Class of Stochastic Nonlinear Systems with More General Growth Conditions

Authors: Liu Jianwei; Guo Longchuan; Zuo Xin; Liang Huaqing
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper investigates the problem of output-feedback stabilization for a class of stochastic nonlinear systems in which the nonlinear terms depend on unmeasurable states besides measurable output. We extend linear growth conditions to power growth conditions and reduce the control effort. By using backstepping technique, choosing a high-gain parameter, an output-feedback controller is designed to ensure the closed-loop system to be globally asymptotically stable in probability, and the inverse optimal stabilization in probability is achieved. The efficiency of the output-feedback controller is demonstrated by a simulation example.


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## Body

## 1. Introduction

The design of output-feedback controller for stochastic nonlinear systems has achieved remarkable research development, because output feedback control is more suitable for practical engineering systems; for example, see [1–12] and references therein. In recent years such research hotspot has mainly focused on a class of special nonlinear stochastic systems in which the nonlinear vector terms depend on the unmeasurable states besides the measurable output; for example, see [13–17] and references therein. The work of [13] discussed the output-feedback controller design by introducing a stability concept named globally asymptotically stable in probability. Based on the purpose of reducing the amount of control, [14] considered the output-feedback stabilization problem.However, in [13–17], the nonlinear vector terms satisfy the linear growth conditions strictly, which greatly narrows the scope of application of the research results. Naturally, one may ask about an interesting and challenging problem: can we further relax the linear growth conditions? To our knowledge, the existing research results on this problem are as in [18–20]. In [18], authors discussed the output-feedback stabilization problem by introducing a rescaling transformation under more relaxed growth conditions. On the basis of [18], the work of [19] and [20] further considered the output-feedback controller design problem for high-order stochastic nonlinear systems. However, for [18–20], the observer gain K is usually larger than 1, and the choice can lead to a controller design which needs larger control effort. So another challenging problem is proposed that is whether the assumption K>1 can be removed.In this paper, we investigate the output-feedback stabilization problem for a class of stochastic nonlinear systems satisfying power growth conditions. Inspired by [13, 14], we find the maximum value interval of observer gain for the desired controller by using backstepping technique. For this interval, the designed output-feedback controller ensures that the equilibrium at the origin is globally asymptotically stable in probability and the inverse optimal stabilization in probability is achieved. The main contributions of this paper are characterized as follows. (i) We extend the linear growth conditions to the power growth conditions. (ii) The assumption of K>1 in [18–20] is removed so that we can get less control effort.The paper is organized as follows. Section2 provides some preliminary results. In Section 3, the problem to be investigated is presented. In Sections 4 and 5, an output-feedback controller is designed and analysed. In Section 6, the inverse optimal stabilization in probability is achieved. Section 7 provides a simulation example. Section 8 concludes this paper.


## 2. Notations and Preliminary Results

Throughout this paper, the following notations will be used.R denotes the set of all real numbers; R+ denotes the set of all nonnegative real numbers; Rn denotes the real n-dimensional space; Rn×r denotes the real n×r matrix space; Tr(·) denotes the trace for square matrix X; |X| denotes the Euclidean norm of a vector X, and ∥X∥ is the Frobenius norm of matrix X defined by ∥X∥=(∑i=1n∑j=1rxij2)1/2; for a given vector x=(x1,…,xn)T, x[i] denotes (x1,…,xi)T; Ci denotes the set of all function with continuous ith partial derivatives; C2,1(Rn×R+,R+) is the family of all nonnegative functions V(x,t) on Rn×R+, which are C2 in x and C1 in t; K denotes the set of all functions: R+→R+, which are continuous, strictly increasing, and vanish at zero; K∞ denotes 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.Lemma 1.
The inequality(|x|+|y|)p≤2p-1(|x|p+|y|p) is established for any x∈R, y∈R, p≥1, p∈Z.Proof.
For an assumption ofα=(a1,a2)=(|x|,|y|), β=(b1,b2)=(1,1), inspired by Holder inequality [21], we can get
(1)|x|+|y|≤(|x|p+|y|p)1/p21-1/p
and further get
(2)(|x|+|y|)p≤2p-1(|x|p+|y|p).
Then the proof is completed.Consider the following stochastic system:(3)dx=f(x,u)dt+g(x)dω,∀x0∈Rn,
where x=(x1,…,xn)∈Rn is the state of the system. u∈Rm is the control input of the system. ω is an r-dimensional standard Wiener process defined on a probability space {Ω,F,P}. The nonlinear functions f:Rm+n→Rn and g:Rn→Rn×r are locally Lipschitz with f(0)=0,  g(0)=0. For any given V∈C2(Rn;R) associated with stochastic system (3), the differential operator L is defined as follows:
(4)LV(x)=∂V∂xf(x)+12Tr(gT(x)∂2V∂x2g(x)).
In order to discuss the stability of stochastic nonlinear systems, we introduce the following stability notion.Definition 2 (see [22]).
For the stochastic nonlinear system (3) with f(0,u)=0, g(0)=0, the equilibrium x(t)=0 of (3) is said to be globally asymptotically stable (GAS) in probability if, for any ε>0, there exists a class KL function β(·,·) such that P{|x(t)|<β(|x0|,t)}≥1-ε, ∀t≥0, x0∈Rn∖{0}.The following lemmas give some sufficient conditions ensuring global asymptotical stability in probability.Lemma 3 (see [23]).
For system (3), if there exist V(x)∈C2, class K∞ functions α1, α2, and a class K function α3 such that α1(|x|)≤V(x)≤α2(|x|), LV(x)≤-α3(|x|), then there exists an almost surely unique solution to system (3) on [0,∞), and the equilibrium x(t)=0 is globally asymptotically stable in probability.Lemma 4 (see [24]).
Consider the following control law:(5)u=α(x)=-R2-1(Lφ2V)Tℓγ(|Lφ2VR2-1/2|)|Lφ2VR2-1/2|2,
where γ(·) is a class K∞ function, ℓγ(s)=s(γ˙)-1(s)-γ((γ˙)-1(s)), and R2(x) is a matrix valued function such that R2(x)=R2T(x)>0. If the control law (5) to be ensures system (3) globally asymptotically stable in probability, then the control law
(6)u*=β*(x)=-θ2R2-1(Lφ2V)Tℓγ(|Lφ2VR2-1/2|)|Lφ2VR2-1/2|2,θ≥2,
solves the problem of inverse optimal stabilization in probability for system (3) by minimizing the cost functional
(7)J(u)=E{∫0∞[l(x)+θ2γ(2θ|R2(x)1/2u|)]dτ},
where l(x) is a positive definite radially unbounded function satisfying
(8)l(x)=2θ[ℓγ(|Lφ2VR2-1/2|)-12Tr{φ1T∂2V∂x2φ1}]+θ(θ-2)ℓγ(|Lφ2VR2-1/2|).


## 3. Problem Formulation

Consider the following stochastic nonlinear systems:(9)dxi=xi+1dt+φi(x)dω,i=1,…,n-1,dxn=udt+φn(x)dω,y=x1,
where x=(x1,…,xn)∈Rn, u∈R, and y∈R are the states, the control input, and the measurable output of the system, ω∈Rr is defined as in (3), and x2,…,xn are the unmeasurable states. φi:Rn→Rr, i=1,…,n, are locally Lipschitz with φi(0)=0 and satisfy the following power growth conditions.Assumption 5.
For each1≤i≤n, there exists the known positive constant d≥0 such that |φi(x)|≤d(|x1|p+|x2|p+⋯+|xi|p), where p is any positive integer.Remark 6.
Assumption5 can be simplified into linear growth conditions when p=1. Therefore, linear growth conditions as a special case are included in Assumption 5. This paper extends previous work and gets a new result.The objective of this paper is to design a smooth output-feedback controller for system (9), such that the closed-loop system is globally asymptotically stable in probability at the origin and achieves the design of the inverse optimal stabilization in probability.


## 4. Output-Feedback Controller Design

Sincex2,…,xn are unmeasured, the following observer is introduced:
(10)x^˙i=x^i+1+Kihi(x1-x^1),i=1,…,n-1,x^˙n=u+Knhn(x1-x^1),
where x^i is the estimated value of xi, K∈R+ is the observer gain to be determined, and hi>0,   i=1,…,n, are chosen such that matrix A=(-h1⋮In-1-hn0⋯0) is asymptotically stable; thus there exists a positive definite matrix P satisfying ATP+PA=-I. Let x~=(x~1,…,x~n)T, where x~i=(xi-x^i)/Ki-1 for each i=1,…,n. By (9) and (10), we can get error system
(11)dx~=KAx~dt+Φ(x)dω,
where Φ(x)=(φ1(x),φ2(x)/K,…,φn(x)/Kn-1)T.Now we give the backstepping controller design procedure.Step  0. Choosing the zeroth Lyapunov function V0(x~)=(n+1)x~TPx~, applying 2ab≤2(a2+b2), (a+b)2≤2(a2+b2), ∑i=1n|ai|2≤(∑i=1n|ai|)2, (∑i=1nai)2≤n∑i=1nai2, Lemma 1, Assumption 5, and (4),  we can get
(12)LV0=-(n+1)K|x~|2+(n+1)Tr(ΦT(x)PΦ(x))≤-(n+1)K|x~|2+(n+1)∥P∥(∑i=1n|φi(x)Ki-1|2)≤-(n+1)K|x~|2+(n+1)∥P∥(∑i=1n|φi(x)Ki-1|)2≤-(n+1)K|x~|2+(n+1)∥P∥d2(∑i=1n1Ki-1)2×(|x1|p+|x2|pK+⋯+|xn|pKn-1)≤-(n+1)K|x~|2+22p-1nd*×(∑i=1n(x^ipKi-1)2+∑i=1n(K(i-1)p-(i-1)x~ip)2)≤-((n+1)nK-22p-1nd*∑i=1n(K(i-1)p-(i-1))2)∥x~∥∞2p+22p-1nd*(x^12p+x^22pK2+⋯+x^n2pK2n-2),
where
(13)d*=(n+1)∥P∥d2(∑i=1n1Ki-1)2=(n+1)∥P∥d2(∑i=0n-1(i+1)Ki+∑i=n2n-2(2n-i-1)Ki)K2n-2
and ∥x~∥∞=maxi|xi|.We introduce a series of coordinate changes as follows:(14)w1=x^1,wi=x^i-βi-1(x^[i-1]),
where βi-1(x^[i-1])  (i=2,…,n) is the virtual control law to be designed.Step  1. Constructing the 1st Lyapunov function
(15)V1(x~,w1)=V0(x~)+1p+1w1p+1,
using (3), (10), (12)–(15), and Young’s inequality [22], we can obtain
(16)LV1≤-((n+1)nK-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)∥x~∥∞2p+22p-1nd*(x^32pK4+⋯+x^n2pK2n-2)+22p-1nd*w12p+22p-1nd*x^22pK2+w1pw2+w1pβ1+Kx~12+K4h12w12p.
Applying (14) and Lemma 1, choosing K≥22pnd*, we can get
(17)22p-1nd*w12≤K2w12,22p-1nd*x^22pK2≤42p-1nd*1K2w22p+42p-1nd*1K2β12p,
which one substitutes in (16) to obtain
(18)LV1≤-((n+1)nK-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)×∥x~∥∞2p+22p-1nd*(x^32pK4+⋯+x^n2pK2n-2)+22p-1nd*w12+42P-1nd*1K2w22p+42p-1nd*1K2β12p+w1pw2+w1pβ1+Kx~12+K4h12w12p≤-(((n+1)n-1)K-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)×∥x~∥∞2p+22p-1nd*(x^32pK4+⋯+x^n2pK2n-2)+42p-1nd*1K2w22p+42p-1nd*1K2w12p+w1pw2+w1pβ1+K(12+h124)w12p≤-(((n+1)n-1)K-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)×∥x~∥∞2p+22p-1nd*(x^32pK4+⋯+x^n2pK2n-2)-nKw12p+42p-1nd*v12pw12p⋯w12p︸2p+42p-1nd*1K2w22p+w1pw2≤-(((n+1)n-1)K-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)×∥x~∥∞2p+22p-1nd*(x^32pK4+⋯+x^n2pK2n-2)-nKw12p+42p-1nd*v12p(2pw12p)2+42p-1nd*1K2w22p+w1pw2=-(((n+1)n-1)K-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)×∥x~∥∞2p+22p-1nd*(x^32pK4+⋯+x^n2pK2n-2)-(nK-24p-1pnd*v12p)w12p+42p-1nd*1K2w22p+w1pw2,
by choosing the 1st virtual control law
(19)β1=-Kv1w1p,v1=12+h124+n.Step  2. Using (10), (14), and (18), we can get
(20)dw2=(x^3+K2h2x~1+Kv1pw1p-1(x^2+Kh1x~1))dt=(x^3+K2h2x~1+K2h1v1pw1p-1x~1+Kv1pw1p-1(w2+β1))dt.
Constructing the 2nd Lyapunov function
(21)V2(x~,w[2])=V1(x~,w1)+1K2·1p+1w2p+1,
applying (14), (18)–(21), K≥22pnd*, Lemma 1, and Young’s inequality [20], we obtain
(22)LV2≤-(((n+1)n-1)K-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)×∥x~∥∞2p+22p-1nd*(x^42pK6+⋯+x^n2pK2n-2)-(nK-24p-1pnd*v12p)w12p+42p-1nd*1K2w22p+w1pw2+22p-1nd*x^32pK4+1K2w2p(x^3+K2h2x~1+K2h1v1pw1p-1x~1+Kv1pw1p-1w2-K2v12pw12p-1)≤-(((n+1)n-1)K-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)×∥x~∥∞2p+22p-1nd*(x^42pK6+⋯+x^n2pK2n-2)-(nK-24p-1pnd*v12p)w12p+42p-1nd*1K2w22p+w1pw2+22p-1nd*x^32pK4+1K2w2px^3+w2ph2x~1+w2ph1v1pw1p-1x~1+1Kw2p+1v1pw1p-1-v12pw12p-1w2p≤-(((n+1)n-1)K-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)×∥x~∥∞2p+22p-1nd*(x^42pK6+⋯+x^n2pK2n-2)-(nK-24p-1pnd*v12p)w12p+42p-1nd*1K4w32p+42p-1nd*1K4β22p+1K2w2pw3+1K2w2pβ2+(w2p(h2+h1v1pw1p-1))x~1+42p-1nd*1K2w22p+w1pw2+1Kw2p+1v1pw1p-1-v12pw12p-1w2p≤-(((n+1)n-1)K-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)×∥x~∥∞2p+22p-1nd*(x^42pK6+⋯+x^n2pK2n-2)-(nK-24p-1pnd*v12p)w12p+42p-1nd*1K4w32p+42p-1nd*1K4β22p+1K2w2pw3+1K2w2pβ2+(w2p(h2+h1v1pw1p-1))x~1+(14nd*-1)w22p+w12p+14w22+1Kw22p+2;
then we can get the 2nd virtual control law
(23)β2(x^[2])=-K2v2w2p,v2=h224+v12h124+v144+v1+1+n-1,
which satisfies
(24)LV2≤-(((n+1)n-2)K-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)×∥x~∥∞2p+22p-1nd*(x^42pK6+⋯+x^n2pK2n-2)-((n-1)K-24p-1pnd*v12p)w12p-1K2((n-4p-1)K-24p-1pnd*v22p)w22p+42p-1nd*1K4w32p+1K2w2pw3.Step
i  (i=3,…,n-1). Suppose at (i-1)th, that there are a set of virtual control laws β1(x^1),…,βi-1(x^[i-1]): as follows
(25)β1(x^1)=-Kv1w1p,β2(x^[2])=-K2v2w2p,⋮βi-1(x^[i-1])=-Ki-1vi-1wi-1p,v1=12+h124+n,v2=h224+v12h124+v144+v1+1+n-1,⋮vi-1=14(hi-1+vi-2hi-2+vi-2vi-3hi-3+⋯+vi-2vi-3⋯v1h1)2+⋯+14(vi-3vi-2-vi-22)2+vi-2+1+n-(i-2),
with vj>0  (j=1,…,i-1) being independent of K such that the ith Lyapunov function
(26)Vi-1(x~,w[i-1])=V0(x~)+1p+1∑j=1i-11Kjwjp+1
satisfies
(27)LVi-1≤-((∑i=1nK(i-1)p-(i-1))2((n+1)n-(i-1))K-22p-1nd*×(∑i=1nK(i-1)p-(i-1))2)∥x~∥∞2p+22p-1nd*(x^i+12pK2i+⋯+x^n2pK2n-2)-∑j=1i-11K2j-2((n-4(p-1)(j-1))K-24p-1pnd*vj2p)wj2p+42p-1nd*1K2(i-1)wi2p+1K2(i-2)wi-1pwi.
In the sequel, we will prove that (27) still holds for
(28)Vi(x~,w[i])=Vi-1(x~,w[i-1])+1p+1·1Kiwip+1.
Using (14) and (25), a direct calculation leads to
(29)dzi=(x^i+1+Kihix~i+Khi-1pwi-1p-1(x^i+Ki-1hi-1x~1)+K2vi-1vi-2pwi-2p-1(x^i-1+Ki-2hi-2x~1)+⋯+Ki-1vi-1vi-2⋯v1pw1p-1(x^2+Kh1x~1))dt.
Using K≥22pnd*, Lemma 1, Young’s inequality [22], (14), (27), (28), and (29), we obtain
(30)LVi≤-((∑i=1nK(i-1)p-(i-1))2((n+1)n-(i-1))K-22p-1nd*×(∑i=1nK(i-1)p-(i-1))2)∥x~∥∞2p+22p-1nd*(x^i+22pK2(i+1)+⋯+x^n2pK2n-2)-∑j=1i-11K2j-2((n-4(p-1)(j-1))K-24p-1pnd*vj2p)wj2p+22p-1nd*1K2ix^i+12p+1K2(i-1)wi2px^i+1+1Kiwip×(x^i+1+Kihix~i+Kvi-1pwi-1p-1(x^i+Ki-1hi-1x~1)+⋯+Ki-1vi-1vi-2⋯v1pw1p-1(x^2+Kh1x~1))≤-(((n+1)n-i)K-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)×∥x~∥∞2p+22p-1nd*(x^i+22pK2(i+1)+⋯+x^n2pK2n-2)-∑j=1i1K2j-2((n-4(p-1)j)K-24p-1pnd*vj2p)wj2p+42p-1nd*1K2iwi+12p+42p-1nd*1K2iβi2p+1K2(i-1)wipwi+1+1K2(i-1)wipβi+1K2i-3wi2p(14(hi+vi-1hi-1+vi-1vi-2hi-2+⋯+vi-1vi-2⋯v1h1)2+⋯+14(vi-3vi-2-vi-22)2+vi-1+1),
then we can choose the ith smooth virtual control law
(31)βi(x^[i])=-Kiviwip,vi=14(hi+vi-1hi-1+vi-1vi-2hi-2+⋯+vi-1vi-2⋯v1h1)2+⋯+14(vi-3vi-2-vi-22)2+vi-1+1+n-(i-1)
and get
(32)LVi≤-(((n+1)n-i)K-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)×∥x~∥∞2p+22p-1nd*(x^i+22pK2(i+1)+⋯+x^n2pK2n-2)-∑j=1i1K2j-2((n-4(p-1)j)K-24p-1pnd*vj2p)wj2p+42p-1nd*1K2iwi+12p+1K2(i-1)wipwi+1.Step
n. Using repeatedly the previous arguments, at the n-1th step, we can get
(33)LVn-1≤-((∑i=1nK(i-1)p-(i-1))2((n+1)n-(n-1))K-22p-1nd*×(∑i=1nK(i-1)p-(i-1))2)∥x~∥∞2p-∑j=1n-11K2j-2×((n-4(p-1)(j-1))K-24p-1pnd*vj2p)wj2p+42p-1nd*1K2(i-1)wn2p+1K2(i-2)wn-1pwn,
where
(34)Vn-1(x~,w[n-1])=Vn-2(x~,w[n-2])+1Kn-1·1p+1wn2p.
At the end of the recursive procedure, choosing the controller
(35)u(x^[n])=-Knvnwnp,
where vn>0 satisfies (25) and is independent of K, we can get
(36)LVn≤-(n2K-22p-1nd*(∑i=1nK(i-1)p-(i-1))2)∥x~∥∞2p-∑j=1n-11K2j-2×((n-4(p-1)(j-1))K-24p-1pnd*vj2p)wj2p-1K2n-5wn2p,
where
(37)Vn(x~,w[n])=(n+1)x~TPx~+1p+1∑j=1n1Kjwjp+1.Remark 7.
The item(x^12p+x^22p/K2+⋯+x^n2p/K2n-2) is canceled at step n-1. By the following analysis, we obtain the maximum value interval of K to ensure the system to be globally asymptotically stable in probability at the origin.


## 5. Performance Analysis

Next, we give the main result in this paper.Theorem 8.
If Assumption5 holds for stochastic nonlinear system (9) under the controllers (10) and (35), then there always exists a constant K*≥0, such that for any K>K*,(1)
the closed-loop system has an almost surely unique solution on[0,∞) for any x0;(2)
the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability.Proof.
UsingK≥0, (18), (23), and (31), obviously, if
(38)K>max{|24p-1pnd*v12pn|,|24p-1pnd*v22pn-4p-1|,…,|24p-1pnd*vn2pn-4(p-1)(n-1)|}=|24p-1pnd*vn2pn-4(p-1)(n-1)|
holds, then conclusions (1) and (2) follow from (36), (37), and Lemma 3. In the following, we analyze (38). From (13), (25), and (31), it is easy to find that d* depends on K. Substituting (13) into (38) leads to
(39)K>24p-1pnvn2pn-4(p-1)(n-1)·(n+1)∥P∥d2K2n-2·(∑i=0n-1(i+1)Ki+∑i=n2n-2(2n-i-1)Ki);
equivalently,
(40)K2n-1>nn-4(p-1)(n-1)·24p-1·p·vn2p·(n+1)∥P∥d2(∑i=0n-1(i+1)Ki+∑i=n2n-2(2n-i-1)Ki),
which is equivalent to
(41)K2n-1+∑i=02n-2aiKi>0
with the real numbers
(42)a0=-Δ,a1=-2Δ,⋮an-1=-nΔ,an=-(n-1)Δ,an+1=-(n-2)Δ,⋮a2n-2=-Δ-(n+1)∥P∥2,Δ=24p-1pn(n+1)d2vn-12p.
According to the factorization theorem of real coefficient polynomial, (41) can be further expressed as
(43)(K-K1)m1⋯(K-Kr)mr(K2+p1K+q1)n1⋯(K2+psK+qs)ns>0,
where mi,  nj are positive integers with ∑i=1rmi+2∑j=1snj=2n-1, Ki, i≤r, are different real numbers, and (pj,qj), j≥s, satisfy pj2-4qj<0 for all j=1,…,s. Obviously, K2+pjK+Kj>0 for all j=1,…,s. Now, we divide into two cases to discuss the choice of Ki. (1) If there is at least one positive number for K1,…,Kr under the condition of appropriate value of p, one chooses K*=max1≤i≤r{Ki}. (2) Otherwise, K*=0. Thus there always exists K*≥0, such that for any K>K*, (38) holds.


## 6. Inverse Optimal Controller Design

In this section we will design the inverse optimal controller on the basis of Theorem8 to meet specific performance indicators besides achieving control objectives.Theorem 9.
The control law(44)u*=-θKnvnwnp,θ≥2,
solves the problem of inverse optimal stabilization in probability for (9) by minimizing the cost function
(45)J(u)=E{∫0∞[l(x~,x^)+1K2(n-1)vn-1(x^)Knu2]dr},
where φ-1(x~,x^)=(ΦT(x),0,…,0)T,  φ-2(x~,x^)=(0,…,0,1)T, V=Vn.Proof.
Equations (10) and (11) can be represented as
(46)(dx~dx^)=φ-1(x~,x^)dω+φ-2(x~,x^)udt,
where φ-1,  φ-2 are identified in Theorem 9. Choosing γ(r)=(1/2K2(n-1))r2, we can get (γ′)-1(r)=K2(n-1)r and ℓγ(r)=(1/2)K2(n-1)r2. Applying Lemma 4, we get
(47)u=β(x^)=-R2-1(x^)1K2(n-1)wnp12K2(n-1)=-12R2-1(x^)wnp.
According to Theorem 8 and Lemma 4, the inverse optimal controller can be designed as follows:
(48)u*=β*(x^)=-θ2R2-1(x^)1K2(n-1)wnpK2(n-1)=-θ2R2-1(x^)wnp=θβ(x^)=θu,θ≥2,
where R2(x^)=1/2Knvn.


## 7. Simulation Examples

In this section, for a numerical example, we design the output-feedback controller by using two methods, where one method is introduced in this paper and the other is introducted in [19, 20].Consider the following stochastic system:(49)dx1=x2dt+110x13sinx2dω,dx2=udt+110(x13+x23)dω,y=x1,
where
(50)|φ1(x)|=|110x13sinx2|≤110|x1|3,|φ2(x)|=|110(x13+x23)|≤110(|x1|3+|x2|3).
With the notation of Assumption 5, one can take p=3, c=1/10.In line with design method discussed in Section4, we can design the observer states as follows:
(51)x^˙1=x^2+K(y-x^1),x^˙2=u+K2(y-x^1).
According to the design procedure in Section 4, we construct the controller as follows:
(52)β1(x^1)=-Kv1w13,v1=2.75,u=-K2v2w23,v2=8.27,
where K will be chosen later, h1=h2=1, and d*=(1/12)×10-4∥P∥(1+1/K)2, ∥P∥=((15+55)/8)1/2. With Theorem 8, one gets K>42.35. According to the design procedure in Section 6, we choose θ=2 and construct the inverse optimal controller as follows:
(53)u*=-2K2v2w23.In simulation, we choose the initial valuesx1(0)=0.02, x2(0)=0.01, x^1(0)=0.01, x^2(0)=-0.01, and K=50. Figure 1 shows the responses of the closed-loop system (49)~(53), which demonstrate the effectiveness of the control scheme.The responses of the closed-loop systems (49)~(53).
(a)
(b)
(c)If the method in [19, 20] is adopted for the same systems, Figure 2 gives the corresponding responses of the systems (here the controller design theory of [19, 20] is not tackled details, and the interested readers can consult the relevant literature).The responses of the closed-loop systems (49)~(53) when adopting the method in [19, 20].
(a)
(b)
(c)Remark 10.
By comparing the two figures, we can observe that the value of the control of Figure1 is far less than Figure 2. In other words, our method requires less control effort to ensure the closed-loop system to be globally asymptotically stable in probability, and it demonstrates the advantage of this method clearly.


## 8. Concluding and Outlook

In this paper, we have studied the output-feedback stabilization for a class of stochastic nonlinear systems. We have given a design of the output-feedback controller so as to make the equilibrium at the origin of the closed-loop system globally asymptotically stable in probability by using the backstepping design technique and choosing a high-gain parameter, and the inverse optimal stabilization in probability is achieved. Our main contribution is extending the linear growth conditions to the more general power growth conditions so as to enable the result to be more general and to have a broader field of use.There are two problems to be investigated.(1)
By extending the value ofp in Assumption 5 from positive integers to the rationales, it can further weaken the conditions of the system (5). For this system, output-feedback problem deserves further research.(2)
Another is to extend stochastic nonlinear systems in this paper to delay systems and study the design of controller.



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*Source: 251340-2013-08-05.xml*