Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay

Fazia Bedouhene; Islam Boussaada; Silviu-Iulian Niculescu

doi:10.5802/crmath.112

Théorie du contrôle

Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay

Fazia Bedouhene ¹ ; Islam Boussaada ^2,³ ; Silviu-Iulian Niculescu ³

¹ Laboratoire de Mathématiques Pures et Appliquées (LMPA), Mouloud Mammeri University of Tizi‐Ouzou, Tizi-Ouzou, BP No 17, RP 15000, Algeria
² IPSA, Ivry sur Seine, France
³ Université Paris Saclay, L2S, CNRS-CentraleSupélec, Inria Saclay-Île-de-France, Equipe DISCO 91192 Gif-sur-Yvette cedex, France

Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 1011-1032.

Résumé
Abstract

Ce travail exploite les propriétés structurelles d’une classe de matrices de Vandermonde fonctionnelles, pour mettre en évidence certaines propriétés qualitatives d’une classe d’équation différentielle d’ordre $n$ , autonome linéaire avec un terme source dépendant de la variable retardée. Plus précisément, il traite de l’effet stabilisateur du paramètre de retard couplé à la coexistence du nombre maximal de valeurs spectrales réelles. Les conditions dérivées sont nécessaires et suffisantes et, à la connaissance des auteurs, représentent une nouveauté dans la littérature. Sous des conditions appropriées, une telle configuration caractérise l’abscisse spectrale correspondant à l’équation étudiée. Un nouveau critère de stabilité est proposé. Ce critère étend les résultats récents sur la factorisation de fonctions quasi-polynomiales. Le potentiel applicatif du procédé proposé est illustré par la stabilisation d’oscillateurs couplés.

This work exploits structural properties of a class of functional Vandermonde matrices to emphasize some qualitative properties of a class of linear autonomous $n^{th}$ order differential equation with forcing term consisting in the delayed dependent-variable. More precisely, it deals with the stabilizing effect of delay parameter coupled with the coexistence of the maximal number of real spectral values. The derived conditions are necessary and sufficient and, to the best of the authors’ knowledge, represent a novelty in the literature. Under appropriate conditions, such a configuration characterizes the spectral abscissa corresponding to the studied equation. A new stability criterion is proposed. This criterion extends recent results in factorizing quasipolynomial functions. The applicative potential of the proposed method is illustrated through the stabilization of coupled oscillators.

Reçu le : 2019-08-20
Révisé le : 2020-09-02
Accepté le : 2020-09-04
Publié le : 2021-01-05

DOI : 10.5802/crmath.112

Classification : 34K20, 39B82, 70Q05, 47N70

Affiliations des auteurs :

Fazia Bedouhene ¹ ; Islam Boussaada ^{2, 3} ; Silviu-Iulian Niculescu ³

¹ Laboratoire de Mathématiques Pures et Appliquées (LMPA), Mouloud Mammeri University of Tizi‐Ouzou, Tizi-Ouzou, BP No 17, RP 15000, Algeria
² IPSA, Ivry sur Seine, France
³ Université Paris Saclay, L2S, CNRS-CentraleSupélec, Inria Saclay-Île-de-France, Equipe DISCO 91192 Gif-sur-Yvette cedex, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Fazia Bedouhene and Islam Boussaada and Silviu-Iulian Niculescu},
     title = {Real spectral values coexistence and their effect on the stability of time-delay systems: {Vandermonde} matrices and exponential decay},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1011--1032},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {9-10},
     year = {2020},
     doi = {10.5802/crmath.112},
     language = {en},
}

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Fazia Bedouhene; Islam Boussaada; Silviu-Iulian Niculescu. Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay. Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 1011-1032. doi : 10.5802/crmath.112. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.112/

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Ahlem Sassi; Islam Boussaada; Silviu-Iulian Niculescu Some Insights into Partial Pole Placement Method in Observers Design for Time-Delay Systems, Control Theory and Inverse Problems (2025), p. 207 | DOI:10.1007/978-3-031-68046-5_10
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Karim L. Trabelsi; Islam Boussaada; Amina Benarab; Csenge Molnar; Silviu-Iulian Niculescu; Tamas Insperger Central nervous system action on rolling balance board robust stabilization: computer algebra and MID-based feedback design, Advances in partial differential equations and control. The 2023 conference in Seville, Spain, March 27–29, 2023. Based on the conference on control related fields, Cham: Birkhäuser, 2024, pp. 215-247 | DOI:10.1007/978-3-031-62265-6_11 | Zbl:7951305
Ahlem Sassi; Islam Boussaada; Silviu-Iulian Niculescu; Michel Zasadzinski Observer-based Controller Design for LTI SISO Systems with Feedback Delay. A Partial Pole Placement Approach, IFAC-PapersOnLine, Volume 58 (2024) no. 27, p. 90 | DOI:10.1016/j.ifacol.2024.10.305
Timothée Schmoderer; Islam Boussaada; Silviu-Iulian Niculescu; Fazia Bedouhene Insights on equidistributed real spectral values in second-order delay systems: perspectives in partial pole placement, Systems Control Letters, Volume 185 (2024), p. 9 (Id/No 105728) | DOI:10.1016/j.sysconle.2024.105728 | Zbl:1537.93280
Amira Remadna; Islam Boussaada; Fazia Bedouhene; Silviu-Iulian Niculescu; Azzedine Benchettah, 2023 27th International Conference on System Theory, Control and Computing (ICSTCC) (2023), p. 192 | DOI:10.1109/icstcc59206.2023.10308493
Ahlem Sassi; Islam Boussaada; Silviu-Iulian Niculescu, 2023 IEEE 11th International Conference on Systems and Control (ICSC) (2023), p. 919 | DOI:10.1109/icsc58660.2023.10449817
Amina Benarab; Islam Boussaada; Silviu-Iulian Niculescu; Karim Trabelsi On the spectrum distribution of parametric second-order delay differential equations: perspectives in partial pole placement, Control and inverse problems. The 2022 spring workshop in Monastir, Tunisia, May 9–11, 2022, Cham: Birkhäuser, 2023, pp. 49-75 | DOI:10.1007/978-3-031-35675-9_4 | Zbl:7916370
Timothée Schmoderer; Islam Boussaada; Silviu-Iulian Niculescu On Boundary Control of the Transport Equation. Assigning Real Spectra and Exponential Decay, IEEE Control Systems Letters, Volume 7 (2023), p. 3591 | DOI:10.1109/lcsys.2023.3337591
Islam Boussaada; Guilherme Mazanti; Silviu-Iulian Niculescu The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: when delay-systems characteristics meet the zeros of Kummer functions, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 360 (2022), pp. 349-369 | DOI:10.5802/crmath.293 | Zbl:1500.34066
Ahlem Sassi; Islam Boussaada; Silviu-Iulian Niculescu Observer Design in LTI Time-Delay Systems using Partial Pole Placement with Applications, IFAC-PapersOnLine, Volume 55 (2022) no. 36, p. 157 | DOI:10.1016/j.ifacol.2022.11.350
Islam Boussaada; Guilherme Mazanti; Silviu-Iulian Niculescu; Ayrton Hammoumou; Titouan Millet; Jayvir Raj; Julien Huynh New Features of P3δ Software. Insights and Demos, IFAC-PapersOnLine, Volume 55 (2022) no. 36, p. 246 | DOI:10.1016/j.ifacol.2022.11.365
Hitay Özbay; A. Nazli Gündes PID and low-order controller design for guaranteed delay margin and pole placement, International Journal of Robust and Nonlinear Control, Volume 32 (2022) no. 18, pp. 9438-9451 | DOI:10.1002/rnc.5521 | Zbl:1529.93034
Tamas Balogh; Islam Boussaada; Tamas Insperger; Silviu-Iulian Niculescu Conditions for stabilizability of time-delay systems with real-rooted plant, International Journal of Robust and Nonlinear Control, Volume 32 (2022) no. 6, pp. 3206-3224 | DOI:10.1002/rnc.5698 | Zbl:1527.93324
Islam Boussaada; Guilherme Mazanti; Silviu-Iulian Niculescu; Adrien Leclerc; Jayvir Raj; Max Perraudin New Features of P3δ software: Partial Pole Placement via Delay Action, IFAC-PapersOnLine, Volume 54 (2021) no. 18, p. 215 | DOI:10.1016/j.ifacol.2021.11.142
Silviu-Iulian Niculescu; Islam Boussaada; Xu-Guang Li; Guilherme Mazanti; César-Fernando Méndez-Barrios Stability, Delays and Multiple Characteristic Roots in Dynamical Systems: A Guided Tour, IFAC-PapersOnLine, Volume 54 (2021) no. 18, p. 222 | DOI:10.1016/j.ifacol.2021.11.143

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