Further comments on "Vector norms as Lyapunov functions for linear systems"

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell time requirements and state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.

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“…A generalization of this result for norm-based Lyapunov functions of the form V (x) = ||W x|| p can be found in [79,93].…”

Section: Common Piecewise Linear Lyapunov Functionsmentioning

confidence: 99%

Stability Criteria for Switched and Hybrid Systems

Shorten¹,

Wirth²,

Mason³

et al. 2007

SIAM Rev.

914

558

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“…A classical problem related to polyhedral LFs is the existence and synthesis of a Lyapunov function defined using a weighted infinity norm. For stable systems described by a linear polytopic differential or difference inclusion it is known [3]- [5] that existence of an infinity norm LF is a necessary condition. Moreover, it is also known [6], [7] that existence of an infinity norm LF is equivalent with existence of a 0-symmetric polyhedral contractive (invariant) set.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, it is also known [6], [7] that existence of an infinity norm LF is equivalent with existence of a 0-symmetric polyhedral contractive (invariant) set. As such, available methods for constructing an infinity norm LF for a linear equation or polytopic inclusion either search (i) for a matrix that satisfies the corresponding standard Lyapunov conditions, see, e.g., [3]- [5], [8]- [11] or, (ii) for a 0-symmetric polyhedral contractive (invariant) set, see, e.g., [7], [12]- [14]. For an overview the interested reader is referred to the excellent monograph [15].…”

Section: Introductionmentioning

confidence: 99%

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

Lazăr

2010

49th IEEE Conference on Decision and Control (CDC)

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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.

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“…Notice that trading quadratic forms for infinity norms as Lyapunov functions does not necessarily make any of the above-mentioned problems easier, on the contrary. The application of most Lyapunov criteria expressed using infinity norms, see, for example, the seminal papers [14][15][16][17][18], is limited to stability analysis for linear systems or linear polytopic inclusions [19]. An extension of stability analysis via piecewise linear Lyapunov functions to certain classes of smooth nonlinear systems was proposed in [20].…”

Section: Introductionmentioning

confidence: 99%

“…Perhaps the main reason for the limited (in terms of synthesis in particular) applicability of infinity norms as Lyapunov functions lies in the corresponding necessary stabilization conditions [14,15,18] that require the solution of a bilinear matrix equation subject to a full-column rank constraint. Starting from the standard Lyapunov sufficient conditions, in this work we propose a novel, geometric approach to infinity norms as Lyapunov functions that leads to a new set of sufficient conditions that can be expressed via a finite number of linear inequalities.…”

Section: Introductionmentioning

confidence: 99%

On infinity norms as Lyapunov functions for piecewise affine systems

Lazăr

Jokic

2010

Proceedings of the 13th ACM International Conference on Hybrid Systems: Computation and Control

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This paper considers off-line synthesis of stabilizing static feedback control laws for discrete-time piecewise affine (PWA) systems. Two of the problems of interest within this framework are: (i) incorporation of the S -procedure in synthesis of a stabilizing state feedback control law and (ii) synthesis of a stabilizing output feedback control law. Tackling these problems via (piecewise) quadratic Lyapunov function candidates yields a bilinear matrix inequality at best. A new solution to these problems is proposed in this work, which uses infinity norms as Lyapunov function candidates and, under certain conditions, requires solving a single linear program. This solution also facilitates the computation of piecewise polyhedral positively invariant (or contractive) sets for discrete-time PWA systems.

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Further comments on "Vector norms as Lyapunov functions for linear systems"

Cited by 23 publications

References 9 publications

Stability Criteria for Switched and Hybrid Systems

Stability Criteria for Switched and Hybrid Systems

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

On infinity norms as Lyapunov functions for piecewise affine systems

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