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A proof of global attractivity of a class of switching systems using a non-Lyapunov approach

Shorten, Robert and Ó Cairbre , Fiacre (2001) A proof of global attractivity of a class of switching systems using a non-Lyapunov approach. IMA Journal of Mathematical Control and Information, 18 (3). pp. 341-353. ISSN 1471-6887

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Official URL: http://imamci.oxfordjournals.org/cgi/reprint/18/3/...

Abstract

A sufficient condition for the existence of a Lyapunov function of the form V(x)= xTpx, P=PT > 0, P ∈ Rnxn, for the stable linear time invariant systems x = Aix, Ai ∈ Rnxn, Ai ∈ A =∆ {A1,...,Am}, is that the matrices Ai are Hurwitz, and that a non-singular matrix T exists, such that TAiT-1, i ∈ {1,...,m}, is upper triangular (Mori, Mori & Kuroe 1996, Mori, Mori & Kuroe 1997, Liberzon, Hespanha & Morse 1998, Shorten & Narendra 1998b). The existence of such a function referred to as a common quadratic Lyapunov function (CQLF) is sufficient to guarantee the exponential stability of the switching system x = A(t)x, A(t)∈ A. In this paper we investigate the stability properties of a related class of switching systems. We consider sets of matrices A, where no single matrix T exists that simultaneously transforms each Ai ∈ A to upper triangular form, but where a set of non-singular matrices Tij exist such that the matrices TijAiTij-1,TijAjTij-1} i, j ∈ are upper triangular. We show that for a special class of such systems the origin of the switching system x = A(t)x, A(t) ∈ A, is globally attractive. A novel technique is developed to derive this result and the applicability of this technique to more general systems is discussed towards the end of the paper.

Item Type: Article
Additional Information: This is an electronic version of an article published in IMA Journal of Mathematical Control and Information (2001) 18(3) 341-353 http://imamci.oxfordjournals.org/
Keywords: Stability; Switching-systems; Hybrid-systems; Lyapunov; Hamilton Institute.
Subjects: Science & Engineering > Computer Science
Science & Engineering > Hamilton Institute
Science & Engineering > Mathematics
Item ID: 1862
Depositing User: Hamilton Editor
Date Deposited: 23 Feb 2010 16:14
Journal or Publication Title: IMA Journal of Mathematical Control and Information
Publisher: Oxford University Press
Refereed: Yes
URI:

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