Séminaire d'Automatique du Plateau de Saclay :On Control Lyapunov-Krasovskii Functionals and Stabilization in the Sample-and-Hold Sense of Nonlinear Time-Delay Systems

Séminaire le 8 Juin 2016, 10h00 à CentraleSupelec (Gif-sur-Yvette) Salle du conseil du L2S - B4.40
Pierdomenico Pepe (Università degli Studi dell'Aquila)

This talk deals with the stabilization in the sample-and-hold sense of nonlinear systems described by retarded functional differential equations. The notion of stabilization in the sample-and-hold sense has been introduced in 1997 by Clarke, Ledyaev, Sontag and Subbotin, for nonlinear delay-free systems. Roughly speaking, a state feedback (continuous or not) is said to be a stabilizer in the sample-and-hold sense if, for any given large ball and small ball of the origin, there exists a suitable small sampling period such that the feedback control law obtained by sampling and holding the above state feedback, with the given sampling period, keeps uniformly bounded all the trajectories starting in any point of the large ball and, moreover, drives all such trajectories into the small ball, uniformly in a maximum finite time, keeping them in, thereafter. In this talk suitable control Lyapunov-Krasovski functionals will be introduced and suitable induced state feedbacks (continuous or not), and it will be shown that these state feedbacks are stabilizers in the sample-and- hold sense, for fully nonlinear time-delay systems. Moreover, in the case of time-delay systems, implementation by means of digital devices often requires some further approximation due to non availability in the buffer of the value of the system variables at some past times, as it can be frequently required by the proposed state feedback. In order to cope with this problem, well known approximation schemes based on first order splines are used. It is shown, for fully nonlinear retarded systems, that, by sampling at suitable high frequency the system (finite dimensional) variable, stabilization in the sample-and-hold sense is still guaranteed, when the holden input is obtained as a feedback of the (first order) spline approximation of the (infinite dimensional) system state, whose entries are available at sampling times, and the state feedback is Lipschitz on any bounded subset of the Banach state space