On infinity norms as Lyapunov functions: Alternative necessary and sufficient conditions

Abstract

Citation for published version (APA): Lazar, M. (2010). On infinity norms as Lyapunov functions : alternative necessary and sufficient conditions. In Proceedings of the 49th IEEE Conference on Decision and Control (CDC), 15-17 December 2010, Atlanta, Georgia (pp. 5936-5942 Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at:openaccess@tue.nl providing details and we will investigate your claim. Abstract-This paper considers the synthesis of infinity norm Lyapunov functions for discrete-time linear systems. A proper conic partition of the state-space is employed to construct a finite set of linear inequalities in the elements of the Lyapunov weight matrix. Under typical assumptions, it is proven that the feasibility of the derived set of linear inequalities is equivalent with the existence of an infinity norm Lyapunov function. Furthermore, it is shown that the developed solution extends naturally to several relevant classes of discrete-time nonlinear systems.

Construction of invariant polytopic sets with specified complexity

“…These methods can be grouped in two categories, according to the approach used. The first category exploits the algebraic necessary and sufficient conditions of positive invariance and existence of Lyapunov functions, developed initially by (Molchanov and Pyatnitskii 1986a,b,c) for bounded polyhedral sets, by (Bitsoris 1988(Bitsoris , 1991 for both bounded and unbounded polyhedral sets, by (Blanchini 1990) for polytopic sets under vertex representation, and extended later by (Hennet 1995, Polański 1995, Lazar 2010 and others. These conditions can be used directly to verify invariance of a given set.…”

Finite Bisimulations for Switched Linear Systems

“…In this case, the dynamics corresponding to (2) becomes a difference inclusion, i.e., x ∈ Φ(x), Φ(x) := {Ax | A ∈ A} for some set A ⊆ R n×n . It has been shown in [24], [25], that all the above definitions (invariant set, Lyapunov function) and results apply directly to difference inclusions in the absolute sense, i.e., given x, the corresponding conditions must hold for all x ∈ Φ(x).…”

“…The matrices, {L σ } σ∈Σ , of the Lyapunov function (25) are omitted due to space reasons, but can be found in [28]. We set Γ X = 19.75 (P Γ X = {x ∈ R n | V (x) ≤ Γ X } as before) Γ D = 10, and X σ = X ∩ Ω σ for all σ ∈ Σ.…”

Finite bisimulations for switched linear systems

“…We assume that a common polyhedral Lyapunov function (LF) of the form (5) with contraction rate ρ ∈ (0, 1) is known for system (6). The algorithm proposed in [25] is employed to construct such a function with a desired contraction rate.…”