非线性参数化时滞非线性系统自适应控制研究

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"具有非线性参数化的固有时滞非线性系统的自适应控制" 这篇研究论文探讨了全球自适应状态反馈稳定化对于一类具有非线性参数化和固有时滞的非线性系统的方法。作者Li-Na Lv、Zong-Yao Sun和Xue-Jun Xie来自曲阜师范大学自动化研究所。论文主要关注的是如何处理那些由于时间延迟和非线性参数化带来的控制挑战。在传统的控制系统理论中，时滞现象通常被视为干扰或不稳定因素，尤其是在化学工程、远程传输线路和其他各种工程系统中普遍存在。这篇论文的一个关键贡献是放宽了对系统增长的约束条件。通过引入两个动态增益，作者们成功地管理了系统的不确定性以及非线性增长速率。这种方法的独特之处在于，它不需要准确的时间延迟信息，只需要类似Lyapunov函数的工具，就能基于同质性主导思想和两次必要的变换提出新的控制策略。同质性主导思想是一种处理非线性系统的方法，它允许通过对系统进行适当的变换，将非线性项的影响转化为可控制的形式。这样的方法对于处理具有复杂动态特性的时滞系统尤其有效，因为它可以简化稳定性分析并提供设计自适应控制器的基础。论文中提出的新控制策略旨在实现全局稳定，这意味着无论初始条件如何，系统都能够收敛到一个稳定的平衡点。这在实际应用中非常重要，因为无法预知所有可能的初始状态。此外，通过自适应控制，系统能够在线调整控制器参数以适应不确定性，这对于处理时滞和非线性问题非常关键。在结论部分，作者提供了一个数值例子来验证理论结果的有效性。这个例子展示了所提出的控制策略如何在具体情境中工作，并证明了其对于解决具有非线性参数化和固有时滞的非线性系统的稳定控制问题的能力。关键词包括：高阶不确定非线性系统、多重延迟、自适应稳定化、同质性主导。这些关键词突出了论文研究的核心问题和采用的技术手段。这篇论文为处理具有复杂动态特性的时滞非线性系统提供了新的理论框架和控制策略，为未来在相关领域的研究和工程实践提供了重要的参考。

Adaptive Control of Inherently Time-Delay Nonlinear Systems

with Nonlinear Parameterization

Li-Na Lv, Zong-Yao Sun

, 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

= x

i+1

+ f

(¯x

(t − τ

),...,x

(t − τ

),d(t)),

˙x

= u

+ f

(x, x

(t − τ

),...,x

(t − τ

),d(t)),

(1)

where i =1,...,n− 1, ¯x

 [x

,...,x

]



, j =

1,...,n,x(t)=[x

(t),..., x

(t)]



∈ IR

is system state,

n+1

(t)  u(t) ∈ IR is control input, and d :IR→ IR

is an

unknown vector denoting uncertainty/unknown. τ

, ..., τ

are unknown constant time-delays. Initial condition is

x(θ)=ζ

(θ), ∀θ ∈ [−τ, 0] with τ ≥ max{τ

,...,τ

}

and ζ

(·) being a speciﬁed continuous function. For i =

1,...,n, f

is continuous with f

(0, 0,d(t)) = 0, and p

∈

≥1

odd

 {

|p and q are positive odd integers, and p ≥ q} is

system high-order.

In the existing literature, for system (1) with τ

=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

=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

(·). Specif-

ically, [19] proposed an interesting problem, that is, how

to achieve global stabilization for system (1), under the as-

sumption |f

(·)|≤C



j=1

(t)|



j=1

(t−τ)|

with C being an unknown positive constant or δ

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

, [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

denotes the

real space with dimension n. C

 C([−τ,0]; IR

) denotes

the set of all the continuous functions mapping [−τ, 0] into

Proceedings of the 33rd Chinese Control Conference

Jul

28-30, 2014, Nan

, China

2011

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