Relations between Common Lyapunov Functions of Quadratic and Infinity-Norm Forms for a Set of Discrete-Time LTI Systems

Thang Viet NGUYEN  Takehiro MORI  Yoshihiro MORI  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E89-A   No.6   pp.1794-1798
Publication Date: 2006/06/01
Online ISSN: 1745-1337
DOI: 10.1093/ietfec/e89-a.6.1794
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Systems and Control
common quadratic Lyapunov function,  common infinity-norm Lyapunov function,  robust stability,  set relations,  existence conditions,  discrete-time systems,  

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This paper studies the problem of the relations between existence conditions of common quadratic and those of common infinity-norm Lyapunov functions for sets of discrete-time linear time-invariant (LTI) systems. Based on the equivalence between the robust stability of a class of time-varying systems and the existence of a common infinity-norm Lyapunov function for the corresponding set of LTI systems, the relations are determined. It turns out that although the relation is an equivalent one for single stable systems, the existence condition of common infinity-norm type is strictly implied by that of common quadratic type for the set of systems. Several existence conditions of a common infinity-norm Lyapunov functions are also presented for the purpose of easy checking.