Adaptive Control of Inherently Time-Delay Nonlinear Systems
with Nonlinear Parameterization
Li-Na Lv, Zong-Yao Sun
1
, Xue-Jun Xie
1. Institute of Automation, Qufu Normal University, Qufu 273165, P. R. China
E-mail: sunzongyao@sohu.com
Abstract: This paper is concerned with global adaptive state-feedback stabilization for a class of inherently time-delay nonlinear
systems with nonlinear parameterization. Constraint condition on system growth is relaxed. With the help of two dynamic gains,
the uncertainty and nonlinear growth rate of the system are successfully managed. Without precise information about time-delay
being needed and only by like Lyapunov function, a new control tactic is presented based on homogeneous domination idea and
two necessary transformations. Finally, a numerical example is provided to illustrate the effectiveness of the theoretical result.
Key Words: High-order uncertain nonlinear system; multiple delays; adaptive stabilization; homogeneous domination
1 Introduction
As is well-known, time-delay phenomena are usually en-
countered in engineering system such as chemical process,
long transmission lines in pneumatic system, and ignoring
their effect will result in catastrophic consequences, so the
study of time-delay system has been a topic of great impor-
tance and received increasing attention, see [1–9] and the
references therein.
This paper considers high-order uncertain time-delay non-
linear system of the form
˙x
i
= x
p
i
i+1
+ f
i
(¯x
i
,x
1
(t − τ
1
),...,x
i
(t − τ
i
),d(t)),
˙x
n
= u
p
n
+ f
n
(x, x
1
(t − τ
1
),...,x
n
(t − τ
n
),d(t)),
(1)
where i =1,...,n− 1, ¯x
j
[x
1
,...,x
j
]
, j =
1,...,n,x(t)=[x
1
(t),..., x
n
(t)]
∈ IR
n
is system state,
x
n+1
(t) u(t) ∈ IR is control input, and d :IR→ IR
r
is an
unknown vector denoting uncertainty/unknown. τ
1
, ..., τ
n
are unknown constant time-delays. Initial condition is
x(θ)=ζ
0
(θ), ∀θ ∈ [−τ, 0] with τ ≥ max{τ
1
,...,τ
n
}
and ζ
0
(·) being a specified continuous function. For i =
1,...,n, f
i
is continuous with f
i
(0, 0,d(t)) = 0, and p
i
∈
IR
≥1
odd
{
p
q
|p and q are positive odd integers, and p ≥ q} is
system high-order.
In the existing literature, for system (1) with τ
i
=0,
fruitful results have been achieved by the adding a power
integrator method and adaptive technique, see [10–18] and
the references therein. When τ
i
=0, intractable difficulties
arise in control design, such as trade-off of time-delay ef-
fect, identification of time-delay restriction. Delightedly, to
a certain extent the stabilization problem has been resolved
for the special case of d(t)=0in [19–21], by imposing
∗
This work is supported by the National Natural Science Foundation of
China under Grant 61004013, 61273125 and 61203013, the Program for the
Scientific Research Innovation Team in Colleges and Universities of Shan-
dong Province, the Specialized Research Fund for the Doctoral Program of
Higher Education under Grant 22220103705110002 and 20113705120003,
the Shandong Provincial Natural Science Foundation of China under Grant
ZR2010FQ003 and ZR2012FM018, the Project of Taishan Scholar of Shan-
dong Province and the Doctoral Scientific Research Start-Up Foundation of
Qufu Normal University.
some reasonable conditions on nonlinearity f
i
(·). Specif-
ically, [19] proposed an interesting problem, that is, how
to achieve global stabilization for system (1), under the as-
sumption |f
i
(·)|≤C
i
j=1
|x
j
(t)|
δ
i
+C
i
j=1
|x
j
(t−τ)|
δ
i
with C being an unknown positive constant or δ
i
taking val-
ues on certain interval. If C is known in prior, in virtue of
the extended adding a power integrator method and homoge-
neous domination idea, [20, 21] discussed the existing range
of δ
i
, [22, 23] further studied it in stochastic setting. How-
ever, to our knowledge, no result has ever been reported
on the case that C is unknown. In this paper, with the aid
of the homogeneous domination idea whose novelty is that
no precise information about the nonlinearity is needed, and
its application brings about many breakthrough results on
control design of high-order nonlinear system without time-
delay [13, 15, 17], we will make an attempt to explore this
possibility for the first time.
On the other hand, although Lyapunov-Krasovskii (L-K)
functional method is prevailing in control design and analy-
sis of time-delay system, most of time it fails to satisfy the
desired assumption in real space. Therefore, a good question
is spontaneously proposed: Whether like Lyapunov function
method can be conveniently used to solve global adaptive
stabilization for high-order time-delay nonlinear system?
This paper is to achieve some progress towards this ques-
tion. In particular, for the use of homogeneous domination
idea, a suitable state transformation is introduced to convert
system (1) into a new one with a dynamic gain. Next, a con-
tinuous adaptive controller is constructed in a step-by-step
manner based on adaptive technique and a like Lyapunov
function. Then, by the delicate selection of dynamic gain
and updating law, the constructed controller guarantees the
boundedness of the closed-loop system state and asymptotic
convergence of the original system state.
2 Preliminary Results
The following notations will be used throughout the pa-
per. IR denotes the set of all real numbers. IR
n
denotes the
real space with dimension n. C
τ
C([−τ,0]; IR
n
) denotes
the set of all the continuous functions mapping [−τ, 0] into
Proceedings of the 33rd Chinese Control Conference
Jul
28-30, 2014, Nan
in
, China
2011