Abstract

A sufficient condition for the existence of a Lyapunov function of the form V(x)=xTPx, P=PT>0, P ∈ IRnxn, for the stable linear time invariant systems x=Aix, Ai ∈ IRnxn, 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 1998). 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 related classes 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 {fTijAiT-1ij, TijAjT-1ij}, i,j ∈ {1,...,m}are upper triangular. We show that in general, this condition does not imply the existence of a common quadratic Lyapunov function CQLF Further we also show by means of a simple example that the condition of pairwise triangularisability is not sucient to guarantee stability of an associated switching system However we show that for special classes of related systems the origin of the switching system. x = A(t)x, A(t)∈ A is globally attractive A novel technique referred to in this paper as state-space-embedding is developed to derive this result. State-space-embedding is based upon the observation that the stability properties of an n-dimensional switching system may, on occasion, be analysed by embedding the n-dimensional system in a higher dimensional system The efficacy of this technique is demonstrated by showing the stability of two distinct classes of switching systems and by utilising these results to design a control system for a real industrial application namely the design of a stable automobile speed control system.