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Stability Analysis of Linear Delay Systems via Internally Positive Representations: An Overview

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Accounting for Constraints in Delay Systems

Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 12))

Abstract

The stability analysis of time-delay systems is a tough task even in the linear case, especially when time-varying delays are considered. In recent years, for the class of positive delay systems, a number of insightful works appeared in the literature, highlighting their peculiar stability properties, which lead to simple and easy to check necessary and sufficient conditions for the delay-independent stability. This fact poses the question whether some results that hold for positive systems can be exported to arbitrary systems. In some recent papers we gave an answer to this question by extending to the time-delay setting the Internally Positive Representation technique, a method that allows to systematically construct positive representations of arbitrary systems. In this work, after a survey on recent stability results for positive delay systems, an overview of the Internally Positive Representation technique and of its implications on the stability analysis of arbitrary systems is reported. Three classes of linear systems are considered: delay differential, delay difference, and coupled delay differential-difference systems.

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Authors and Affiliations

  1. Dipartimento di Ingegneria e Scienze dell’Informazione, e Matematica, Università degli Studi dell’Aquila, Via Vetoio, 67100, Coppito, AQ, Italy

    Vittorio De Iuliis, Alfredo Germani & Costanzo Manes

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  1. Vittorio De Iuliis
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  2. Alfredo Germani
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  3. Costanzo Manes
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Correspondence to Costanzo Manes .

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Editors and Affiliations

  1. Laboratoire des Signaux et Systèmes, CentraleSupélec, Gif-sur-Yvette, France

    Giorgio Valmorbida

  2. Department of Computer Science, KU Leuven, Heverlee, Belgium

    Wim Michiels

  3. Department of Information Engineering, Computer Science, and Mathematics, University of L’Aquila, L'Aquila, Italy

    Pierdomenico Pepe

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De Iuliis, V., Germani, A., Manes, C. (2022). Stability Analysis of Linear Delay Systems via Internally Positive Representations: An Overview. In: Valmorbida, G., Michiels, W., Pepe, P. (eds) Accounting for Constraints in Delay Systems. Advances in Delays and Dynamics, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-030-89014-8_2

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