Linear systems subject to input saturation and time delay: Global asymptotic and $ {L}^{p}$-stabilization

Karim Yakoubi; Yacine Chitour

doi:10.1016/j.crma.2005.03.010

Automation (theoretical)

Linear systems subject to input saturation and time delay: Global asymptotic and

L^{p}

-stabilization
[Des systèmes linéaires saturés et retardés : stabilisation asymptotique globale et stabilisation-

L^{p}

]

Présenté par : Alain Bensoussan

Karim Yakoubi ¹ ; Yacine Chitour ¹

¹ Laboratoire des signaux et systèmes, université Paris-sud, CNRS, Supélec, 91192 Gif-sur-Yvette cedex, France

Comptes Rendus. Mathématique, Volume 340 (2005) no. 9, pp. 703-708.

Résumé
Abstract

Dans cette Note on traite deux problèmes de stabilisation de systèmes linéaires par des feedbacks statiques retardés et bornés : la stabilisation asymptotique globale et la stabilisation- $L^{p}$ avec gain fini. Pour le premier problème, sous les conditions nécessaires standard, on fournit deux solutions, avec une borne d'amplitude arbitrairement petite sur la commande et pour tout retard. La première solution utilise l'existance d'un feedback stabilisant pour le système sans retard. La seconde est de type saturation emboîtée, ce qui généralise les resultats de Mazenc et al. [IEEE Trans. Automat. Contr. 48 (1) (2003) 57–63]. Pour la stabilisation- $L^{p}$ , le système est supposé stable. On donne un feedback linéaire qui assure la stabilité- $L^{p}$ en respectant toute $L^{p}$ -norme, $p \in [1, \infty]$ , pour tout retard et toute borne d'amplitude assez petite de la commande. le $L^{p}$ -gain correspendant est indépendant du retard $h > 0$ .

This Note deals with two problems on stabilization of linear systems by static feedbacks which are bounded and time-delayed, namely global asymptotic stabilization and finite gain $L^{p}$ -stabilization, $p \in [1, \infty]$ . Regarding the first issue, we provide, under standard necessary conditions, two types of solutions for arbitrary small bound on the control and large (constant) delay. The first solution is based on the knowledge of a static stabilizing feedback in the zero-delay case and the second solution is of nested saturation type, which extends results of Mazenc et al. [IEEE Trans. Automat. Contr. 48 (1) (2003) 57–63]. For the finite-gain $L^{p}$ -stabilization issue, we assume that the system is neutrally stable. We show the existence of a linear feedback such that, for arbitrary small bound on the control and large (constant) delay, finite gain $L^{p}$ -stability holds with respect to every $L^{p}$ -norm, $p \in [1, \infty]$ . Moreover, the corresponding $L^{p}$ -gain is delay-independent.

Reçu le : 2005-01-27
Accepté le : 2005-02-22
Publié le : 2005-05-01

DOI : 10.1016/j.crma.2005.03.010

Affiliations des auteurs :

Karim Yakoubi ¹ ; Yacine Chitour ¹

¹ Laboratoire des signaux et systèmes, université Paris-sud, CNRS, Supélec, 91192 Gif-sur-Yvette cedex, France

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     author = {Karim Yakoubi and Yacine Chitour},
     title = {Linear systems subject to input saturation and time delay: {Global} asymptotic and $ {L}^{p}$-stabilization},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {703--708},
     publisher = {Elsevier},
     volume = {340},
     number = {9},
     year = {2005},
     doi = {10.1016/j.crma.2005.03.010},
     language = {en},
}

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Karim Yakoubi; Yacine Chitour. Linear systems subject to input saturation and time delay: Global asymptotic and $ {L}^{p}$-stabilization. Comptes Rendus. Mathématique, Volume 340 (2005) no. 9, pp. 703-708. doi : 10.1016/j.crma.2005.03.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.03.010/

Bibliographie
Cité par

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[6] H.J. Sussmann; E.D. Sontag; Y. Yang A general result on the stabilization of linear systems using bounded controls, IEEE Trans. Automat. Contr., Volume 39 (1994), pp. 2411-2425

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[8] A.J. Van Der Schaft $L^{2}$ -gain and Passivity Techniques in Nonlinear Control, Lecture Notes in Control and Inform., Springer, London, 1996

[9] K. Yakoubi, Y. Chitour, Linear systems subject to input saturation and time delay: global asymptotic stabilization, submitted for publication

[10] K. Yakoubi, Y. Chitour, Linear systems subject to input saturation and time delay: finite-gain $L^{p}$ -stabilization, submitted for publication

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