Finite-time Stabilization of A Class of High-order Stochastic
Nonlinear Systems
Suiyang Khoo
1,2,3
, Juliang Yin
1,2
, Zhihong Man
3
1. School of Engineering, Deakin University, VIC Australia
E-mail: sui.khoo@deakin.edu.au
2. Department of Statistic and Mathematics, Jinan University, China
E-mail: yin
juliang@hotmail.com
3. School of Robotics and Mechatronics, Swinburne University of Technology, VIC Australia
E-mail: zman@swin.edu.au
Abstract: This paper studies the problem of almost surely finite-time stabilization for stochastic nonlinear systems. We prove that
global almost surely finite-time stabilization of stochastic nonlinear systems in triangular form can be reached by the continuous
feedback control law. The proof is based on the recently proposed almost surely finite-time stability theorem in [17]. A recursive
design algorithm is developed for the construction of the control. The adding an integrator technique is employed to construct
the global almost surely finite-time stabilizer. Simulation example is given to illustrate the theoritical analysis.
Key Words: Stochastic differentia equations, Finite-time stability, Nonlinear systems.
1 Introduction
Consider the stochastic nonlinear system described by:
dx
i
= x
i+1
dt + g
T
i
(¯x
i
)dw, i =1,...,n− 1,
dx
n
=(f(x)+u)dt + g
T
n
(¯x
n
)dw, (1)
where (Ω, F ,P) is a probability space and F
t
isafil-
tration of sub-σ-fields of F , x := (x
1
,...,x
n
)
T
=
{x
t
, F
t
;0≤ t<∞} is a continuous, adapted R
n
-valued
measurable process, w = {w
t
, F
t
;0≤ t<∞} is an m-
dimensional Brownian motion, u ∈Rrepresents the control
input of the system. Let ¯x
i
=(x
1
,...,x
i
)
T
, the functions
f(x):R
n
→Rand g
i
(¯x
i
):R
i
→R
m
, i =1,...,n, also
called the coefficients of the equation, are Borel measurable,
continuous, and satisfy f(0) = 0,g(0) = 0 for all t ≥ 0.
Here, we define ¯g
T
i
(¯x
i
)=[g
T
1
(¯x
1
), ···,g
T
i
(¯x
i
)].
Since ordinary differential equations are special case of
stochastic differential equations, it is very realistic to model
engineering systems with stochastic differential equations.
Stochastic stability describes the most important character-
istic for those systems modelled by stochastic differential
equations. Deng & Krstic [2], Has’minskii [7] and Mao
[13] present the basic Lyapunov stability theory for stochas-
tic nonlinear systems which has been widely used for the de-
sign and analysis of the stochastic nonlinear control systems
[3–6, 10, 11, 14, 18].
Motivated by the asymptotic stability theorem for stochas-
tic differential equation systems in [7, 13], complete finite-
time stability and instability theorems were proposed and
analysed by the authors in [17]. For deterministic case, the
design of finite-time global stabilizer has been discussed in
[1, 8, 9, 12, 15, 16]. Naturally, a fundamental and unsolved
problem is: Can the finite-time control design for determin-
istic nonlinear systems be extended to stochastic nonlinear
systems?
This work is partly supported by the Natural Science Foundation of
China Under Grant 61174202, the Natural Science Foundation of Guang-
dong Under Grant 10151063201000042, and the Australian Research
Council under the Discovery Scheme (project number DP0986376).
In this paper, we will concentrate on this problem. The
main contribution are as follows.
1) New definition on finite-time stable in probability is
given. It is seen that the proof of almost surely finite-
time stability theorem in [17] can be applied directly to
proof the proposed finite-time stability theorem in this
paper.
2) By using the adding a power integrator technique and
skillfully choosing the design parameters and construct
the C
2
Lyapunov function, a smooth state-feedback
controller is explicitly constructed.
3) It is proven that the solution process can reach zero in
finite time with probability one.
2 Background and preliminaries
In this section we review some terminologies related to
the notion of finite-time stability and the corresponding Lya-
punov stability theory for one-dimensional stochastic non-
linear system of the form:
dx = f(x)dt + g
T
(x)dw(t),t≥ 0,
x
0
= ξ. (2)
Definition 1. The trivial solution of (1) is said to be finite-
time stable in probability if the solution exist for any initial
data x
0
∈R, denoted by x(t; x
0
), moreover the following
statement hold:
(i) Finite-time attractiveness in probability: For every ini-
tial value x
0
∈R
n
\{0}, the first hitting time τ
x
0
=
inf{t; x(t; x
0
)=0}, which is called the stochastic
settling time, is finite almost surely, that is, P {τ
x
0
<
∞} =1;
(ii) Stability in probability: For every pair of ε ∈ (0, 1)
and r>0, there exists a δ = δ(ε, r) > 0 such that
P {|x(t; x
0
)| < r, for all t ≥ 0}≥1 − ε, whenever
|x
0
| <δ;
(iii) The solution x((t + τ
x
0
); x
0
) is unique for t ≥ 0.
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