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Preservation of Common Quadratic Lyapunov Functions and Padé Approximations

Sajja, Surya and Solmaz, Selim and Shorten, Robert and Corless, Martin (2010) Preservation of Common Quadratic Lyapunov Functions and Padé Approximations. 49th IEEE Conference on Decision and Control (CDC), 2010. 7334 -7338 . ISSN 0743-1546

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Abstract

It is well known that the bilinear transform, or first order diagonal Padé approximation to the matrix exponential, preserves quadratic Lyapunov functions between continuous-time and corresponding discrete-time linear time invariant (LTI) systems, regardless of the sampling time. It is also well known that this mapping preserves common quadratic Lyapunov functions between continuous-time and discrete-time switched systems. In this note we show that while diagonal Padé approximations do not in general preserve other types of Lyapunov functions (or even stability), it is true that diagonal Padé approximations of the matrix exponential, of any order and sampling time, preserve quadratic stability. A consequence of this result is that the quadratic stability of switched systems is robust with respect to certain discretization methods.

Item Type: Article
Additional Information: The definitive version of this paper is published at 49th IEEE Conference on Decision and Control (CDC), 2010. ISBN 978-1-4244-7745-6 ©2010 IEEE. DOI: 10.1109/CDC.2010.5717670
Keywords: Lyapunov methods; approximation theory; bilinear systems; continuous time systems; discrete time systems; linear systems; matrix algebra; stability; time-varying systems; transforms;
Subjects: Science & Engineering > Hamilton Institute
Item ID: 3826
Depositing User: Hamilton Editor
Date Deposited: 22 Aug 2012 15:41
Journal or Publication Title: 49th IEEE Conference on Decision and Control (CDC)
Publisher: IEEE
Refereed: No
URI:

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