不确定时滞非线性高阶系统自适应控制方法

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"一类不确定时滞非线性高阶系统的自适应控制设计" 本文是一篇研究论文，主要关注的是不确定时滞非线性高阶系统的自适应控制设计问题。在作者的研究范围内，尚未发现类似的结果。因此，这项工作首次提出了处理延迟效应和控制设计框架的方法。在研究中，针对系统的非线性特性，论文采用添加功率积分器的方法以及相关的自适应技术，设计出了一种连续自适应状态反馈控制器，而且避免了过度参数化的问题。这种控制器设计旨在确保系统的稳定性，即使系统存在不确定性和时间延迟。论文通过一些必要的系统非线性限制条件，利用Lyapunov-Krasovskii函数，严格证明了由此产生的闭环系统的均匀稳定性和收敛性。Lyapunov-Krasovskii函数在系统稳定性分析中起着关键作用，它能够帮助验证系统的渐近稳定性。最后，作者提供了一个数值例子，以实际案例展示理论结果的有效性和实用性。这个例子展示了在给定条件下，所设计的自适应控制器如何成功地应对非线性、时滞和不确定性，实现系统的稳定控制。关键词包括：高阶时滞非线性系统、添加功率积分器、自适应稳定化、Lyapunov-Krasovskii函数。这篇论文深入探讨了如何在具有不确定性和时间延迟的复杂非线性系统中，通过自适应控制策略来实现系统的稳定运行。这一研究对于理解和解决实际工程中的控制问题，尤其是在自动化、航空航天、电力系统等领域具有重要的理论与实践意义。

ADAPTIVE CONTROL DESIGN FOR A CLASS OF UNCERTAIN

HIGH-ORDER NONLINEAR SYSTEMS WITH TIME DELAY

Zong-Yao Sun and Yun-Gang Liu

ABSTRACT

This paper is concerned with adaptive stabilization for a class of uncertain high-order nonlinear systems with time delays.

To the authors’ knowledge, there has been no analogous result. Hence during investigation, the conditions on delay effect and the

control design framework should be established for the ﬁrst time. In this paper, under somewhat necessary restrictions on the

system nonlinearities, by the method of adding a power integrator and the related adaptive technique, a procedure is developed

to design the continuous adaptive state-feedback controller without overparametrization. Moreover, the uniform stability and

convergence of the resulting closed-loop system are rigorously proven, with the aid of a suitable Lyapunov-Krasovskii functional.

Finally, a numerical example is provided to illustrate the effectiveness of the theoretical result.

Key Words: High-order time-delay nonlinear systems, adding a power integrator, adaptive stabilization, Lyapunov-Krasovskii

functional.

I. INTRODUCTION

Time delay systems have received considerable atten-

tion continuously over the past decades, since time delay

phenomena exist in many physical systems, such as mechani-

cal, biological and economical systems, and are often the

primary cause of instability and poor performance. So far,

great progress has been made for time delay linear systems

[1–3]. Nevertheless for time delay nonlinear systems, there

are many rather challenging control problems that remain

unsolved and need further investigation.

This paper considers adaptive stabilization for the high-

order time delay uncertain nonlinear system



x t x t f t xt xt dt

xt u t ft

() () (, (), ( ), ()),

() () (

=+ −

,, ( ), ( ) ( )),xt xt dt−

⎧

⎨

⎩

(1)

where i = 1,...,n −1,x(t) = [x

(t),...,x

(t)]

∈ R

is the

system state vector and u(t) ∈ R is the control input; d:R→R

is an unknown vector denoting the uncertainties or

unknowns; τ ≥ 0 denotes the time delay of the system state;

the system initial time is t

and the system initial condition is

x(θ) = ξ

(θ), θ ∈ [t

− τ, t

], with ξ

being a speciﬁed

continuous function; for i = 1,...,n ,

∈

≥



odd

is called the

powers of the system where



odd

≥1

denotes the set of all of the

numbers in the form of

with p and q being positive odd

integers and p ≥ q; and f

, named by system nonlinearity, is an

unknown continuous function and satisﬁes f

(t,0,0,d(t)) = 0.

When p

= 1, System (1) is known as the strict-feedback

nonlinear system, which has been extensively studied since

the early 1990s [4–6]. When p

> 1, the most remarkable

feature of the system is that its Jacobian linearization at the

origin is neither controllable nor observable, and this makes

the traditional design tools, such as feedback linearization [6]

or the backstepping method [5], difﬁcult to apply to System

(1). Mainly due to the addition of a power integrator method,

a series of research results have been achieved over the past

decade for the case of p

> 1, see [7–12] and [13–14] for

stabilization and tracking, respectively.

When τ > 0, for System (1), which is actually retarded

functional differential equations (FDEs), we have solved two

special cases with weak uncertainties [15,16]. Speciﬁcally,

[15] attempted to construct a continuous controller by impos-

ing a strong condition on nonlinearities f

’s, and [16] relaxed

the restriction on f

’s in [15] and studied the case of multiple

state time delays. To the best of our knowledge, no adaptive

control results on such a class of systems with serious uncer-

tainties have ever been reported. Nevertheless there are plen-

tiful research results for the special case of p

= 1, for example

[17–22], where backstepping is the main method to control

design. Speciﬁcally, merged with a wavelet neural network,

[17–19] constructed neural network controllers for serval

Manuscript received March 15, 2013; revised July 18, 2013; accepted September

19, 2013.

Zong-Yao Sun (corresponding author, e-mail: sunzongyao@sohu.com) is with

institute of Automation, Qufu Normal University, Qufu, Shandong, 273165, China.

Yun-Gang Liu is with School of Control Science and Engineering, Shandong

University, Jinan, Shandong, 250061, China (e-mail: lygfr@sdu.edu.cn).

This work was supported by National Natural Science Foundation of China

(61004013, 61203013, 61273084, 61233014), the Shandong Provincial Natural

Science Foundation of China (ZR2010FQ003), the Independent Innovation Founda-

tion of Shandong University (2012JC014), the Specialized Research Fund for the

Doctoral Program of Higher Education (2011370 5120003), the Project of Taishan

Scholar of Shandong Province and the Doctoral Scientiﬁc Research Start-Up Foun-

dation of Qufu Normal University.

Asian Journal of Control, Vol. 17, No. 2, pp. –

, March 2015

Published online 8 May 2014 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.895

535

543

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