时变交换系统收敛域的研究与多个李雅普诺夫函数

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"这篇研究论文关注的是时变切换系统的收敛域问题。作者Jinghui Suo和Jitao Sun来自 Tongji University 的数学系。文章在2015年发表于《系统与控制信函》(Systems & Control Letters)期刊，探讨了时变切换系统（Time-varying Switched Systems）的收敛域，并通过构建多个时间变权重的李亚普诺夫函数（Lyapunov functions）来分析系统在外球体之外的递减行为。论文提供了这种一般性时变切换系统的相对精确的收敛域定义，并通过一个具体的数值例子——质量-弹簧-阻尼系统（mass-spring-damper system）来展示理论结果的有效性。" 在本文中，作者首先强调稳定性对于动态系统的重要性。时变切换系统是一种包含多个子系统且在运行过程中会根据预定规则或随机事件在这些子系统间切换的复杂系统。这些系统广泛应用于工程、自动化、生物学等领域，但其稳定性分析往往更具挑战性，因为系统的行为会随时间和切换策略的变化而变化。核心贡献在于提出了多Lyapunov函数方法，这是一种常用于证明系统稳定性的工具。每个Lyapunov函数在半径随时间变化的球体外部都是递减的，这种方法允许作者更细致地描述系统的动态行为。通过这种方式，他们能够为一般的时变切换系统提供一个相对精确的收敛域，即系统能够保持稳定并最终达到平衡状态的区域。最后，通过一个质量-弹簧-阻尼系统的数值示例，作者展示了所提出的理论在实际问题中的应用和有效性。这个例子通常被用来模拟物理系统的动力学行为，如车辆悬挂系统或者机械结构的振动控制，它帮助读者直观理解理论分析如何与实际系统行为相吻合。这篇论文为理解和控制时变切换系统的动态行为提供了一种新的分析框架，对于设计更稳定的控制系统以及优化切换策略具有重要的理论价值和实践意义。

Systems & Control Letters 78 (2015) 19–24

Contents lists available at ScienceDirect

Systems & Control Letters

journal homepage: www.elsevier.com/locate/sysconle

Convergence domain for time-varying switched systems

Jinghui Suo, Jitao Sun

∗

Department of Mathematics, Tongji University, Shanghai 200092, PR China

a r t i c l e i n f o

Article history:

Received 13 December 2013

Received in revised form

19 September 2014

Accepted 10 January 2015

Available online 18 February 2015

Keywords:

Switched systems

Convergence domain

Multiple Lyapunov functions

a b s t r a c t

In this paper, we investigate the convergence domain for time-varying switched systems. We construct

multiple Lyapunov functions and each Lyapunov function is decreasing outside a ball whose radius is time-

varying. We give a relatively accurate convergence domain for a general time-varying switched system.

Finally, a numerical example, a mass–spring–damper system, is provided to show the effectiveness of the

theoretical results.

1. Introduction

Stability is an important issue for dynamical systems. However

sometimes there is no need to achieve stability in practice. In cer-

tain engineering applications, it is more natural to identify whether

the solutions of dynamical systems are bounded, see [1–9] and ref-

erences therein.

Lyapunov analysis is very useful to show boundedness of the

solutions. When we observe the derivative of a Lyapunov function

along the solution of a differential dynamical system, it is common

that this derivative is not always negative but is indeed negative

when the state trajectory satisfies some conditions. [2,3,5,10]

considered the case that the derivative of a Lyapunov function was

negative when the norm of the state was greater than or equal to

a constant and provided a theorem about ultimate boundedness of

the solution. Recently, Zhao et al. [4] investigated a more general

case that the derivative of a Lyapunov function was negative when

the norm of the state was greater than or equal to a certain function

of time and gave a convergence domain for a general class of time-

varying nonlinear systems.

A switched system is a dynamical system consisting of a family

of subsystems and a switching signal that determines the switch-

ing between them. In recent years, switched systems have gained

considerable attention in science and engineering and a number

of criteria have been derived, see [7–19] and references therein.

In view of the interaction between continuous dynamics and dis-

crete dynamics, switched systems may have very complicated be-

haviors. For instance, switching may destabilize a switched system

∗

Corresponding author. Tel.: +86 21 65983241x1307; fax: +86 21 65981985.

E-mail address: sunjt@sh163.net (J. Sun).

even if all individual subsystems are stable, whereas suitably con-

strained switching between unstable subsystems may give rise to

stability. Motivated by this, some scientists have focused on the

boundedness of the solutions of switched systems [7–10]. How-

ever, these researches do not provide the relatively accurate con-

vergence domain for a general switched system. In this paper, we

study the convergence domain for time-varying switched systems.

Since a common Lyapunov function may not exist for a switched

system, we use multiple Lyapunov function method and assume

that for each Lyapunov function, its derivative is negative when

the norm of the state is no less than a function of time. To the best

of our knowledge, there has been no result of such research. There-

fore the study in this aspect is meaningful and challenging.

This paper is organized as follows. In Section 2, we give the main

results about convergence domain for a time-varying switched

system. An application example, a mass–spring–damper system, is

provided to illustrate the theoretical results in Section 3, following

by conclusions in Section 4.

Notation: ∥ · ∥ refers to the Euclidean norm for vectors. I

represents the identity matrix. B(

x, r) = {x| ∥x −

x∥ ≤ r} denotes

a closed ball with radius r > 0 centered at a point

x. A function β is

said to be of class K if it is a continuous, strictly increasing function

satisfying β(0) = 0.

2. Main results

We consider the following switched system

x = f

σ (t)

(t, x), (1)

where x ∈ R

, σ : [0, ∞) → P = {1, 2, . . . , m} is a switching

signal, m is the number of subsystems. σ is a piecewise constant

http://dx.doi.org/10.1016/j.sysconle.2015.01.007

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