Globally State-feedback Control for Stochastic
High-order Nonlinear Systems in the Presence of
Time Delays
Huifang Min
School of Automation, Nanjing University
of Science and Technology, Nanjing 210094,
P. R. China
Email: jiejie1043640772@126.com
Na Duan, Shuliang Wang, Xingxing Chen
School of Electrical Engineering
& Automation, Jiangsu Normal University,
Xuzhou 221116, P. R. China
Email: duanna08@163.com
Abstract
—This paper is concerned with the state-feedback
control for a class of stochastic high-order nonlinear systems with
multiple time delays. A distinctive novelty lies in the removal
of the traditional growth assumptions imposed on the delay-
dependent nonlinear terms. Then, without any extra assumptions,
a state-feedback controller is explicitly constructed based on
the Lyapunov-Krasovskii functionals and recursive backstepping
design, which guarantees the closed-loop system to be globally
uniformly ultimately bounded (GUUB). Finally, a simulation
example is shown to demonstrate the effectiveness of the proposed
scheme.
Keywords: Stochastic Nonlinear Systems, Time Delays, State-
feedback, GUUB.
I. I
NTRODUCTION
In nonlinear control fields, the existences of stochastic
noises including parameter perturbance, stochastic errors and
environment variations, have made stochastic nonlinear system
control be gradually into the researchers’ version. During
the past decades, with the help of stochastic stability theory
established and improved in [1]-[5], great developments have
been obtained for the control and analysis of stochastic non-
linear systems in various structures. Particularly, considering
time delays extensively occur in practical systems, which may
destroy system stability, the control and design for stochastic
nonlinear time-delay systems has also attracted an unfading
attraction; see, for example [6]-[10] and the references therein.
In this paper, we further consider stochastic high-order non-
linear time-delay systems with the form
dx
i
= x
p
i+1
dt + f
i
(¯x
i
(t − d
i
))dt + g
i
(x
1
)dω,
i = 1, · · · , n − 1,
dx
n
= u
p
dt + f
n
(x(t − d
n
))dt + g
n
(x
1
)dω,
(1)
where
u ∈ R
and
x
i
∈ R
are system control input and measur-
able states, respectively;
¯x
i
(t − d
i
) = (x
1
(t − d
i
), · · · , x
i
(t −
d
i
))
T
and
x(t − d
n
) = (x
1
(t − d
n
), · · · , x
n
(t − d
n
))
T
;
d
1
, · · · , d
n
> 0
are constant time delays;
p ≥ 1
is an odd inte-
ger called as high-order;
ω
is an
r
-dimensional standard wiener
process defined on a complete probability space
{Ω, F, P }
with
Ω
being a sample space,
F
being a
σ
-field, and
P
being
the probability measure; for
i = 1, · · · , n
, the drift terms
f
i
(¯x
i
(t − d
i
)) : R
i
→ R
are
C
1
functions with
f
i
(0) = 0
and
the diffusion terms
g
i
(x
1
) : R → R
1×r
are smooth functions
with
g
i
(0) = 0
.
When
p > 1
and
d
i
= 0
, the main difficulty in the design of
system (1) is how to relax or further remove the common used
assumptions on high-order and nonlinear terms. In [11]-[12],
the state-feedback and output-feedback control were solved for
system (1) under restrictive growth conditions, respectively.
Then, by generalizing the homogeneous domination approach
in [13] to stochastic nonlinear systems and using the adding
a power integrator technique, the stabilization problems were
further considered in [14]-[16] under somewhat weaker growth
assumptions.
However, how to further weaken even remove
the growth assumptions is still an interesting and impending
problem.
The above discussions motivate the main work and con-
tributions of this paper as follows: (i). Compared with [11]-
[12], [14]-[16] and the existing references, the traditional
growth conditions assumed on
f
i
are removed; (ii). Without
imposing any growth assumptions, a state-feedback controller
is designed by applying Lyapunov-Krasovskii functionals and
backstepping technique. It is proven that the constructed con-
troller can render the closed-loop system be globally uniformly
ultimately bounded (GUUB).
The rest of this paper is organized as follows. Section
2 begins with the preliminaries; Section 3 and Section 4
present the controller design procedure and stability analysis,
respectively; the simulation results are given in Section 5 and
Section 6 concludes this paper.
II. P
RELIMINARIES
The following definition and lemmas will be used through-
out the paper. Consider a normal stochastic nonlinear time-
delay system
dx = f(x, x(t − d))dt + g(x, x(t − d))dω, ∀t ≥ 0,
(2)
where
x ∈ R
n
is system state with the initial data
x(θ) = ξ
for
−d ≤ θ ≤ 0
;
d > 0
is a constant delay;
ω
is defined as in