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Delays and Interconnections: Methodology, Algorithms and Applications pp 153–167Cite as

Scanning the Space of Parameters for Stability Regions of a Class of Time-Delay Systems; A Lyapunov Matrix Approach

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Part of the Advances in Delays and Dynamics book series (ADVSDD,volume 10)

Abstract

A computationally efficient approach for carrying out the second step in the production of stability charts for a class of time-delay systems is presented. The methodology consists of detecting stable regions in the delay-parameter space based on a stability test in terms of the delay Lyapunov matrix, which plays a central role in the framework of complete type Lyapunov–Krasovskii functionals. Our approach is tested on a number of challenging academic examples found in the literature, and it is also employed on the analysis of the effects of non-intentional input delays in delay-based control schemes.

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Notes

  1. 1.

    Local stability analysis of non-linear systems follows from linearization, meaning that the technique presented here may be used as a first approximation to the stability analysis of non-linear dynamics.

  2. 2.

    It is worthy of mention that the delay Lyapunov matrix has exact solution only in the commensurate delays case, in the rest of the chapter, we take advantage of this characteristic to optimize the scanning process such that efficiency is improved.

  3. 3.

    That is, the characteristic equation of (1), namely \(\det (sI-\sum _{j=0}^mA_je^{-h_js})=0\), has no symmetric eigenvalues with respect to the imaginary axis in the complex plane.

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Acknowledgements

The presented research has been supported by Conacyt, grant 180725

Author information

Authors and Affiliations

  1. Cuerpo Académico de Electrónica y Control, CITIS, Universidad Autónoma del Estado de Hidalgo, Carretera Pachuca-Tulancingo km. 4.5, C. U., 42084, Pachuca, Hidalgo, Mexico

    Carlos Cuvas

  2. Departamento de Control Automático, CINVESTAV-IPN, Av. IPN 2508, 07360, Ciudad de México, Mexico

    Adrián Ramírez, Luis Juárez & Sabine Mondié

Authors
  1. Carlos Cuvas
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  2. Adrián Ramírez
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  3. Luis Juárez
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  4. Sabine Mondié
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Corresponding author

Correspondence to Sabine Mondié .

Editor information

Editors and Affiliations

  1. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupèlec-Université Paris-Saclay, Gif-sur-Yvette, France

    Prof. Giorgio Valmorbida

  2. Laboratoire d’Analyse et d’Architecture de Systèmes-CNRS, Toulouse, France

    Dr. Alexandre Seuret

  3. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupèlec-Université Paris-Saclay, Gif-sur-Yvette, France

    Islam Boussaada

  4. Mechanical and Industrial Engineering, Northeastern University, Boston, MA, USA

    Prof. Rifat Sipahi

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Cuvas, C., Ramírez, A., Juárez, L., Mondié, S. (2019). Scanning the Space of Parameters for Stability Regions of a Class of Time-Delay Systems; A Lyapunov Matrix Approach. In: Valmorbida, G., Seuret, A., Boussaada, I., Sipahi, R. (eds) Delays and Interconnections: Methodology, Algorithms and Applications. Advances in Delays and Dynamics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-030-11554-8_10

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