© 2001 by Institute of Mathematics and its Applications
A proof of global attractivity for a class of switching systems using a non-quadratic Lyapunov approach
1 Department of Computer Science, National University of Ireland, Maynooth, Co. Kildare, Republic of Ireland 2 Department of Mathematics, National University of Ireland, Maynooth, Co. Kildare, Republic of Ireland
A sufficient condition for the existence of a Lyapunov function of the form V(x) = xT xP, P = PT > 0, P 
Rnxn, for the stable linear time invariant systems 
= iA x, iA Rnxn, iA A 
{A1, ..., mA}, is that the matrices iA are Hurwitz, and that a non-singular matrix T exists, such that iT A T–1, i {1, ..., m}, is upper triangular [Mori, Y., Mori, T., & Kuroe, Y., Proceedings of Electronic Information and Systems Conference (1996); Proceedings of 36th Conference on Decision and Control, (1997); Liberzon, D., Hespanha, J.P., & Morse, S., Technical Report, Laboratory for Control Science and Engineering, Yale University, (1998); Shorten, R. & Narendra, K., Proceedings of Conference on Decision and Control, (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 = 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 iA A to upper triangular form, but where a set of non-singular matrices i jT exist such that the matrices {i jiT A i jT–1, i jjT A i jT–1}, i, j {1, ..., m}, are upper triangular. We show that, for a special class of such systems, the origin of the switching system = 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.
Keywords: Stability; switching-; systems; hybrid-; systems; Lyapunov.