Journal Article

A proof of global attractivity for a class of switching systems using a non‐quadratic Lyapunov approach

Robert Shorten and Fiacre Ó Cairbre

in IMA Journal of Mathematical Control and Information

Published on behalf of Institute of Mathematics and its Applications

Volume 18, issue 3, pages 341-353
Published in print September 2001 | ISSN: 0265-0754
Published online September 2001 | e-ISSN: 1471-6887 | DOI: http://dx.doi.org/10.1093/imamci/18.3.341
A proof of global attractivity for a class of switching systems using a non‐quadratic Lyapunov approach

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A sufficient condition for the existence of a Lyapunov function of the form V(x) = xT xP, P = PT > 0, P ∈ Rn×n, for the stable linear time invariant systems = iA x, iA ∈ Rn×n, iAA ≜ {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 iAA 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

Journal Article.  0 words. 

Subjects: Mathematics

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