Shorten, Robert N. and Ó Cairbre, Fiacre (2001) A proof of global attractivity of a class of switching systems using a nonLyapunov approach. IMA Journal of Mathematical Control and Information, 18 (3). pp. 341353. ISSN 14716887
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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 nonsingular matrix T exists, such that TAiT1, 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 nonsingular matrices Tij exist such that the matrices TijAiTij1,TijAjTij1} 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) 341353 http://imamci.oxfordjournals.org/ 
Keywords:  Stability; Switchingsystems; Hybridsystems; Lyapunov; Hamilton Institute. 
Academic Unit:  Faculty of Science and Engineering > Computer Science Faculty of Science and Engineering > Mathematics and Statistics Faculty of Science and Engineering > Research Institutes > Hamilton Institute 
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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