2007年：不确定时变脉冲切换系统状态延迟下的鲁棒指数稳定性研究

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本文档探讨了2007年发表在《控制理论与应用》期刊上的"Robust Exponential Stability of Uncertain Discrete Time Impulsive Switching Systems with State Delay"一文。作者Guangdeng ZONG、Yuqiang WU、Baoyong ZHANG和Yangyang KONG，分别来自南京科技大学自动化学院和曲阜师范大学自动化研究所，共同研究了一种新型的混合脉冲和切换系统模型及其稳定性分析和控制器设计。研究的核心内容聚焦于一种混合系统，它由稳定的子系统和不稳定的子系统组成。这个系统的特点在于包含了状态延迟和参数不确定性，同时存在子系统之间的脉冲切换效应。这些特性使得系统的动态行为更为复杂，稳定性分析面临挑战。论文针对这一问题，提出了系统维数的扩展方法，并引入了平均稳定期的概念，从而设计出一种实用的切换规则。这种规则确保了在存在不确定性和状态延迟的情况下，系统的稳健指数增长稳定性得到保持。文章的主要贡献在于提出了一个基于切换时间的控制策略，通过这种策略，即使在面对不完全信息和动态变化的条件时，也能有效地保证系统的稳定性。此外，论文还给出了一个切换状态反馈控制器的设计，这是一种关键的控制手段，能够帮助系统在切换过程中实现稳定的过渡，并对抗外部干扰。关键词包括"指数稳定性"、"切换系统"、"状态延迟"、"参数不确定性"和"脉冲控制"，这些都是理解和评估该论文核心内容的关键术语。这篇论文为处理带有状态延迟和不确定性的一类复杂系统提供了重要的理论支持和方法论，对于控制工程领域的研究者和实践者具有较高的参考价值。

Journal of Control Theory and Applications 2007 5 (4) 351–356 DOI 10.1007/s11768-006-6051-x

Robust exponential stability of uncertain discrete

time impulsive switching systems with state delay

Guangdeng ZONG

1,2

, Yuqiang WU

, Baoyong ZHANG

, Yangyang KONG

(1. School of Automation, Nanjing University of Science and Technology, Nanjing Jiangsu 210094, China;

2. Institute of Automation, Qufu Normal University, Qufu Shandong 273165, China )

Abstract: A new class of hybrid impulsive and switching models are introduced and their robust exponential stability

and control synthesis are addressed. The proposed switched system is composed of stable subsystems and unstable subsys-

tems, which not only involves state delay and norm-bounded time-varying parameter uncertainties, but also contains the

impulsive switching effects between the subsystems. Based on the extension of the system dimension and the concept of

average dwell time, a kind of practically useful switching rule is presented which guarantees the desired robust exponential

stability. A switched state feedback controller is also given.

Keywords: Exponential stability; Switched systems; Switching control; State delay; Average dwell time

1 Introduction

In recent years, there has been growing interest in the sta-

bility analysis and controller design of switched systems

(see e.g., [1∼4] for some details). Switched systems are

an important class of hybrid dynamic systems consisting

of a family of continuous time or discrete time subsystems

and a switching rule specifying the switching among them.

The motivation for studying switched systems or switching

control mainly comes from three aspects. 1) Many systems

encountered in practice exhibit switching phenomena be-

tween several subsystems due to the inherent multi-model

or various environmental factors [1, 2]. 2) The methods of

intelligent control design are based on the idea of switch-

ing between different controllers [1, 5]. 3) Switching con-

trollers can achieve a better performance than the traditional

feedback controllers. For example, Narendra and Balakrish-

nan improved system performance using a set of switching

adaptive controllers [6].

Stability analysis is a hot issue in system studies. The sta-

bility of switched systems has been studied using the multi-

ple Lyapunov function method. Although the multiple Lya-

punov function method can also be used to study the robust

stability and exponential stability of switched linear sys-

tems, it is very difﬁcult to ﬁnd the Lyapunov functions for

unstable subsystems. To ensure the exponential stability of

switched systems, a piecewise Lyapunov function together

with dwell time is utilized. For systems without impulsive

effect, it is shown in [9] that if all the subsystems are sta-

ble and the dwell time (i.e. the time between consecutive

switchings) is sufﬁciently long, the entire system is expo-

nentially stable for any switching between the subsystems.

In [10], Hespanha extends the dwell time concept to the av-

erage case in the sense that a switching signal has ﬁnite av-

erage dwell time if the number of discontinuities it exhibits

in any given interval grows no longer than linearly with the

length of the interval. It is proved that if the average dwell

time is sufﬁciently large, the given switched system is expo-

nentially stable. However, the considered switched systems

in [9, 10] are only composed of stable subsystems. Moti-

vated by the previous work, Hu et al. [11] and Zhai et al.

[12] considered a more general class of switched systems,

in which both stable subsystems and unstable subsystems

exist. Based on the average dwell time concept and by us-

ing the idea of specifying the total activation time period ra-

tio between the stable subsystems and unstable subsystems,

the exponential stability is guaranteed. Similar to [12], [13]

considered the exponential stability of a class of discrete

time switched systems.

As is well known, time delays commonly appear in many

real control systems, which give rise to instability and poor

performance. During the past decades, the stability of con-

trol systems with time delay has been widely studied. How-

ever, few results for switched systems have considered the

factor of time delay. In [3], based on the switched Lyapunov

function technique, the authors present two different linear

matrix inequality conditions, which guaranteed the asymp-

Received 27 March 2006; revised 3 April 2007.

This work was supported by the National Natural Science Foundation of China (No. 60674027), China Postdoctoral Science Foundation (No. 200704

10336), and the Postdoctor Foundation of Jiangsu Province (No. 0602042B).

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