Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 246579, 13 pages
doi:10.1155/2012/246579
Research Article
Adaptive State-Feedback Stabilization for
High-Order Stochastic Nonlinear Systems Driven
by Noise of Unknown Covariance
Cong-Ran Zhao,1, 2 Xue-Jun Xie,1 and Na Duan1
1
2
School of Electrical Engineering and Automation, Xuzhou Normal University, Xuzhou 221116, China
Institute of Automation, Qufu Normal University, Qufu 273165, China
Correspondence should be addressed to Xue-Jun Xie, xuejunxie@126.com
Received 20 July 2011; Accepted 10 September 2011
Academic Editor: Weihai Zhang
Copyright q 2012 Cong-Ran Zhao et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
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 statefeedback 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:
p
dx1 x2 dt f1 x1 dt g1 x1 Σdω,
p
dx2 x3 dt f2 x2 dt g2 x2 Σdω,
..
.
1.1
dxn up dt fn xn dt gn xn Σdω,
where x x1 , . . . , xn ∈ Rn , u ∈ R, are the state and control input, respectively. xi 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 σfield, {Ft }t≥0 being a filtration, and P being the probability measure. Σ : R → Rr×r is an
2
Mathematical Problems in Engineering
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{XX T }1/2 , where Tr{X} is its trace when X is a square matrix and
X T 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 βs, 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 χt by χ for any variable χt.
Consider the nonlinear stochastic system
dx fxdt gxdω,
2.1
Mathematical Problems in Engineering
3
where x ∈ Rn is the state, w is an r-dimensional independent Wiener process with incremental covariance ΣΣT dt, that is, E{dωdωT } ΣΣT dt, 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 13
. For any given V x ∈ C2 associated with stochastic system 2.1, the
differential operator L is defined as
∂2 V x
1
∂V x
T
fx Tr g x
LV x gx .
∂x
2
∂x2
2.2
Definition 2.2 see 25
. For the stochastic system 2.1 with f0 0, g0 0, the equilibrium xt 0 is globally asymptotically stable GAS in probability if for any ξ > 0, there exists
a class KL function β·, · such that
P |xt| < β|x0 |, t ≥ 1 − ξ,
t ≥ 0, ∀x0 ∈ Rn \ {0}.
2.3
Lemma 2.3 see 14
. 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 Wx such that for
all x ∈ Rn , t ≥ 0
α1 |x| ≤ V x ≤ α2 |x|,
LV ≤ −c1 Wx c2 ,
2.4
then,
a there exists an almost surely unique solution on 0, ∞ for each x0 ∈ Rn ,
b when c2 0, f0 0, g0 0, and Wx is continuous, then the equilibrium x 0 is
globally stable in probability and the solution xt satisfies P {limt → ∞ Wxt 0} 1.
Lemma 2.4 see 12
. Let x, y be real variables, for any positive integers m, n, positive real number
b and nonnegative continuous function a·, then
a·xm yn ≤ b|x|mn mn
m n −m/n
n
a·mn/n b−m/n y ,
mn
m
2.5
when a· 1, b m/m nd, d is a positive constant, then the above inequality is
xm y n ≤
mn
m
n
d|x|mn d−m/n y .
mn
mn
2.6
Lemma 2.5 see 12
. 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:
i |xm y xl1 xn − xl1 xn | ≤ |x|mn /N |y|mn h1 x, y,
p
ii |ym zn1 · · · znp yn l2 z1 , . . . , zp , y| ≤ 1/N k1 |zk |mn |y|mn h2 z1 , . . . , zp , y.
4
Mathematical Problems in Engineering
Lemma 2.6 see 12
. Let x1 , . . . , xn , p, be positive real variables, then
p
p
x1 · · · xn p ≤ max np−1 , 1 x1 · · · xn .
2.7
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
⎛
⎞
i p
fi xi ≤ ⎝ xj ⎠fi1 xi ,
⎛
⎞
i p
gi xi ≤ ⎝ xj ⎠gi1 xi .
j1
3.1
j1
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 xi−1 , θ ,
i 2, . . . , n,
3.2
i 2, . . . , n, are smooth virtual controllers which will be designed later, θ
where αi−1 xi−1 , θ,
is the estimation of θ, and
p3/2 p3/3 , ΣΣT , ΣΣT .
θ max ΣΣT 3.3
t≥0
Then, by Itô’s differentiation rule, one has
dzi d xi − αi−1 xi−1 , θ
p
xi1
Fi xi −
i−1
∂αi−1
l1
∂xl
p
xl1
i−1
∂2 αi−1
1 ∂αi−1 ˙
T T
−
gk xk ΣΣ gm xm −
θ dt
2 k,m1 ∂xk ∂xm
∂θ
3.4
Gi xi Σdω,
where
Fi xi fi xi −
i−1
∂αi−1
l1
∂xl
fl xl ,
Gi xi gi xi −
i−1
∂αi−1
l1
∂xl
gl xl .
3.5
Mathematical Problems in Engineering
5
Step 1. Define the first Lyapunov function
1
1 2
V1 z1 , θ z41 θ ,
4
2Γ
3.6
where Γ is a positive constant, θ θ − θ is the parameter estimation error. By 3.2–3.4 and
Assumption 3.1, there exist nonnegative smooth functions μ11 and μ15 such that
3
θ
p
LV1 z31 x2 z31 f1 x1 z21 g1 x1 ΣΣT g1T x1 − θ˙
2
Γ
p
θ
p
p
p3
p3
≤ z31 x2 − α1 z31 α1 z1 μ11 z1 z1 μ15 z1 θ − θ˙
Γ
p
θ ˙
p
p
p3
p3
p3
≤ z31 x2 − α1 z31 α1 z1 μ11 z1 z1 μ15 z1 1 θ2 −
θ − Γz1 μ15 z1 .
Γ
3.7
Choosing the first smooth virtual controller
1/p
β1 z1 , θ c11 μ11 z1 μ15 z1 1 θ2
,
α1 x1 , θ −z1 β1 z1 , θ ,
3.8
and the tuning function
p3
τ1 z1 Γz1 μ15 z1 ,
3.9
one has
p3
LV1 ≤ −c11 z1
p
θ p
z31 x2 − α1 −
θ˙ − τ1 ,
Γ
3.10
where c11 > 0 is a design parameter.
Step i 2 ≤ i ≤ n. For notational coherence, denote u xn1 . Assuming that at step i − 1, one
has
LVi−1
⎛
⎞
i−1
∂α
θ
j−1
p3
⎠ θ˙ − τi−1 z3 xp − αp ,
≤ − cj,i−1 zj − ⎝ z3j
i−1
i
i−1
Γ j2
∂θ
j1
i−1
3.11
6
Mathematical Problems in Engineering
p3
where τi−1 τi−2 Γzi−1 μi−1,4 μi−1,5 . In the sequel, we will prove that 3.11 still holds for
Vi−1 zi−1 , θ
1/4z4 . By 3.4 and 3.11, one has
the ith Lyapunov function Vi zi , θ
i
LVi ≤ LVi−1 z3i
p
xi1
Fi xi −
i−1
∂αi−1
l1
∂xl
p
xl1
i−1
∂2 αi−1
1 T T
−
gk xk ΣΣ gm xm 2 k,m1 ∂xk ∂xm
∂αi−1 ˙ 3 2 T T
θ zi Tr Σ Gi xi Gi xi Σ
2
∂θ
⎛
⎞
i−1
i−1
∂α
θ
j−1
p3
⎠ θ˙ − τi−1 z3 xp − αp
≤ − cj,i−1 zj − ⎝ z3j
i−1
i
i−1
Γ j2
∂θ
j1
− z3i
z3i
p
xi1
Fi xi −
i−1
∂αi−1
l1
− z3i
∂xl
p
xl1
i−1
∂2 αi−1
1 T
−
gk xk ΣΣT gm
xm 2 k,m1 ∂xk ∂xm
3.12
∂αi−1 ˙ 3 2 T T
∂αi−1
∂αi−1
θ zi Tr Σ Gi xi Gi xi Σ z3i
τi−1 − z3i
τi−1 .
2
∂θ
∂θ
∂θ
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
p
p
p
z3i−1 xi − αi−1 z3i−1 zi αi−1 p − αi−1
p3
p3
≤ ξi1 zi−1 μi1 zi , θ zi ,
z3i Fi xi ≤ |zi |3
i zj p ρi2j zi , θ
j1
≤
i−1
p3
ξi2j zj
p3
μi2 zi , θ zi ,
j1
−z3i
i−1
∂αi−1
l1
∂xl
p
xl1 ≤ |zi |3
≤
i−1
i−1 zj p ρi3j zi , θ
j1
p3
ξi3j zj
p3
μi3 zi , θ zi ,
j1
i−1 i−1
1 ∂2 αi−1
2p
T
− z3i
gk xk ΣΣT gm
xm ≤ |zi |3 zj ρi4j zi , θ ΣΣT 2 k,m1 ∂xk ∂xm
j1
≤
i−1
p3
ξi4j zj
p3
μi4 zi , θ zi θ,
j1
i
3 2 T T
p1
zi Tr Σ Gi xi Gi xi Σ ≤ z2i zj ρi5j zi , θ ΣΣT 2
j1
Mathematical Problems in Engineering
7
i−1
≤
p3
ξi5j zj
p3
μi5 zi , θ zi θ,
j1
−z3i
i−1 ∂αi−1
p
τi−1 ≤ |zi |3 zj ρi6j zi , θ
∂θ
j1
i−1
≤
p3
ξi6j zj
p3
μi6 zi , θ zi ,
j1
3.13
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 ≤ −
i−1
⎛
p3
cj,i−1 zj
p3
ξi1 zi−1
j1
⎞
i−1
∂α
θ
j−1
⎠ θ˙ − τi−1 z3 αp
− ⎝ z3j
i i
Γ j2
∂θ
p3
θ
μi1 μi2 μi3 μi6 μi4 μi5 1 θ2 μi4 μi5 Γzi
Γ
p3
zi
3.14
p
p3
p
ξi2j ξi3j ξi4j ξi5j ξi6j zj z3i xi1 − αi .
i−1
j1
Choosing the ith smooth virtual controller αi
αi xi , θ −zi βi zi , θ ,
⎛
⎛
⎞⎞1/p
i
∂αj−1
3
βi zi , θ ⎝cii μi1 μi2 μi3 μi6 μi4 μi5 ⎝ 1 θ2 zj
Γ⎠⎠ ,
∂θ
j2
3.15
and tuning function τi
p3 τi zi τi−1 zi−1 Γzi
μi4 μi5 ,
3.16
and substituting 3.15 and 3.16 into 3.14, it follows that
⎛
⎞
i
i
∂α
θ
j−1
p3
⎠ θ˙ − τi z3 xp − αp ,
LVi zi , θ ≤ − cji zj − ⎝ z3j
i
i
i1
Γ j2
∂θ
j1
where cji cjj − ξj1,1 −
6
k2 ξikj ,
j 1, . . . , i − 1.
3.17
8
Mathematical Problems in Engineering
Hence at step n, the smooth adaptive state-feedback controller
u αn xn , θ −zn βn zn , θ ,
βn
θ˙ τn zn ,
⎛
⎛
⎞⎞1/p
n
∂αj−1
3
zn , θ ⎝cnn μn1 μn2 μn3 μn6 μn4 μn5 ⎝ 1 θ2 zj
Γ⎠⎠ ,
∂θ
j2
p3
τn zn Γz1 μ15 z1 n
p3 Γzj μj4 μj5 ,
j2
3.18
such that the nth Lyapunov function
n
1
1 2
Vn zn , θ z4 θ
4 j1 j 2Γ
3.19
satisfies
LVn ≤ −
n
p3
cjn zj ,
3.20
j1
where μnl , l 1, . . . , 6, are nonnegative smooth functions, cjn , j 1, . . . , n, are constants, and
cjn cjj − ξj1,1 −
6
ξnkj ,
j 1, . . . , n − 1.
3.21
k2
3.2. Stability Analysis
Theorem 3.2. If Assumption 3.1 holds for the high-order stochastic nonlinear system 1.1, under
the state-feedback controller 3.18, then
i 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 , θ0
∈ Rn1 ,
ii the origin of the closed-loop system is globally stable in probability,
exists and is finite} 1.
iii P {limt → ∞ |xt| 0} 1 and P {limt → ∞ θt
is C2 on zn and θ.
For j 1, . . . , n − 1, choose the design
Proof. It is easy to verify that Vn zn , θ
6
parameter cjj > ξj1,1 k2 ξnkj , cnn > 0, then by 3.21, cjn > 0, j 1, . . . , n − 1. Since Vn zn , θ
is continuous, positive, and radially bounded, by 3.20, 3.21, and Lemma 4.3 in 25
, there
≤ Vn zn , θ
≤ α2 |x|, |θ|.
Hence,
exist two class K∞ functions α1 and α2 such that α1 |x|, |θ|
the condition of Lemma 2.3 holds.
By Lemma 2.3, it follows that conclusion i, ii hold, and P {limt → ∞ |zt| 0} 1.
0 and xi zi αi−1 xi−1 , θ,
one has P {limt → ∞ |xt| 0} 1. By 3.20
In view of αi 0, θ
Mathematical Problems in Engineering
9
in 3.19, it holds that θt
converges a.s. to a finite limit θ∞ as
and the definition of Vn zn , θ
converges a.s. to a finite limit as t → ∞.
t → ∞, therefore θt
4. A Simulation Example
Consider a two-order nonlinear stochastic system
dx1 x23 dt f1 x1 dt x13 Σdω,
dx2 u3 dt f2 x2 dt x23 Σdω,
4.1
where f1 x1 x13 , f2 x2 x1 x22 . By Lemma 2.4, one gets |f1 x1 | ≤ |x1 |3 , |g1 x1 | ≤ |x1 |3 ,
|f2 x2 | ≤ 1/3|x1 |3 2/3|x2 |3 , g2 x2 ≤ |x2 |3 . We choose f11 x1 1, g11 x1 1, f21 x2 2/3, g21 x2 1, Assumption 3.1 is satisfied.
We now give the design of state-feedback controller for system 4.1.
1/4z4 1/2Γθ2 . A smooth virtual controller
Step 1. Define z1 x1 , V1 z1 , θ
1
α1 x1 , θ −z1 β1 z1 , θ ,
1/3
β1 z1 , θ c11 1 μ15 z1 1 θ2
,
4.2
τ1 z1 Γz61 μ15 z1 4.3
and the tuning function
≤ −c11 z6 z3 x3 − α3 θ/Γ
yield LV1 z1 , θ
θ˙ − τ1 , where
1
1 2
1
μ15 z1 3 2
z,
2 1
3 2 θt max ΣtΣtT , ΣtΣtT , ΣtΣtT .
t≥0
4.4
V2 z2 , θ
V1 z1 , θ
1/4z4 , by 3.12, one has
Step 2. Defining z2 x2 − α1 x1 , θ,
2
θ LV2 z1 , θ ≤ −c11 z61 z31 x23 − α31 θ˙ − τ1
Γ
∂α1 3 1 ∂2 α1
3
3
T T
z2 u F2 x2 −
x −
g1 x1 ΣΣ g1 x1 ∂x1 2 2 ∂x12
− z32
∂α1 ˙ 3 2 T T
θ z2 Tr Σ G2 x2 G2 x2 Σ ,
2
∂θ
4.5
10
Mathematical Problems in Engineering
where F2 x2 f2 x2 − ∂α1 /∂x1 f1 x1 , G2 x2 g2 x2 − ∂α1 /∂x1 g1 x1 . By Lemma 2.4,
the definition of z2 , and 4.2, one can obtain
1
1 −1 6
2
1 −1 3 6 5
1 −1 12 6
z2 3 d12 z61 d12
β1 z2 d13 z61 d13
β1 z2
z31 x23 − α31 ≤ d11 z61 d11
2
2
3
3
6
6
ξ21 z61 μ21 z1 , θ z62 ,
z32 F2 x2 −z32
2
∂α1 3
1
z
≤ 2|z2 |
|z1 |3 |z2 |3 |z1 |3 β12 −
3
3
∂x1 1
ξ221 z61 μ22 z2 , θ ,
3
∂α1 3
∂α1 3
z − 3z2 z1 β1 3z2 z1 β1 − z3 β3
x2 ≤ |z2 |3 2
2
1 1
∂x1
∂x1
≤ ξ231 z61 μ23 z2 , θ z62 ,
2
1 ∂2 α1 3 T 3
3 ∂ α1 6 T
− z32
g
ΣΣ
g
≤
z
|z
|
ΣΣ
2
1
2 ∂x12 1
∂x12 1
≤ ξ241 z61 μ24 z2 , θ z62 θ;
4.6
by 4.3, Lemmas 2.4, 2.6, and the definitions of z2 and G2 x2 , one has
−z32
∂α1
3 ∂α1
τ1 ≤ |z2 |3 |z1 |3
Γ|z1 |5
2 ∂θ
∂θ
≤ ξ261 z61 μ26 z2 , θ z62 ,
3 3 ∂α1 3 2 3 2 T T
z2 Tr Σ G2 x2 G2 x2 Σ ≤ z22 z2 − z1 β1 −
z1 ΣΣT 2
2
∂x1
∂α1 2
2
5 6
6
5 6 ≤ z2
3 · 2 β1 3 ·
z1 2 z2 ΣΣT ∂x1
≤ ξ251 z61 μ25 z2 , θ z62 θ,
4.7
where ξ21 1/2d11 2d12 5/2d13 , ξ221 1/3d21 d22 d23 , ξ231 1/2d31 1/2d32 ,
1/2d−1 d−1 β3 z1 , θ
ξ241 1/2d41 , ξ251 2/3d51 , ξ261 1/2d61 , μ21 z1 , θ
11
12 1
2
−5 12
−1
−1 4
−1
1/2d13 β1 z1 , θ, μ22 z2 , θ 1/3d21 4/3 d22 β1 d23 ∂α1 /∂x1 , μ23 z2 , θ ∂α1 /
1/2d−1 ∂2 α1 /∂x2 2 z6 ,
∂x1 1 − 2β3/2 β6 ∂α1 /∂x1 2 1/2d−1 1/2d−1 , μ24 z2 , θ
1
1
31
32
41
1
1
32z2 d−1 128β18 4∂α1 /∂x1 6 z6 , μ26 z2 , θ
9/8d−1 ∂α1 /∂θ
2 Γ2 z10 , d11 , d12 ,
μ25 z2 , θ
2
1
1
1
51
61
d13 , d21 , d22 , d23 , d31 , d32 , d41 , d51 , d61 are positive constants.
11
×10−5
−2.5
5
−3
4
−3.5
3
Control
State
Mathematical Problems in Engineering
−4
−4.5
2
1
−5
0.01
0.1
1
0
10
0.01
Time (s)
0.1
1
10
Time (s)
x1
u
a
b
0
×10
8.04
−0.05
8.02
−0.1
8
Parameter
State
−6
−0.15
−0.2
7.98
7.96
7.94
−0.25
0.01
0.1
1
10
0.01
0.1
Time (s)
1
10
Time (s)
x2
c
d
Figure 1: The responses of closed-loop system 4.1–4.3, 4.8.
Choosing the smooth adaptive controller
u −z2 β2 z2 , θ ,
θ˙ τ2 z2 ,
τ2 z2 Γz61 μ15 z1 Γz62 μ24 z2 , θ μ25 z2 , θ ,
β2 z2 , θ 4.8
1/3
3 ∂α1
2
c22 μ21 μ22 μ23 μ26 μ24 μ25 1 θ Γz2
,
∂θ
and substituting 4.6–4.8 into 4.5, one has
LV2 ≤ −c12 z61 − c22 z62 ,
where c12 c11 − ξ21 − ξ221 − ξ231 − ξ241 − ξ251 − ξ261 > 0.
4.9
12
Mathematical Problems in Engineering
In simulation, we choose Σt ≡ 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 θ0 0, x1 0 0, x2 0 −0.5, the sampling period 0.01. Figure 1 verifies
the effectiveness of the control scheme.
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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