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.