On the L^p-stabilization of the double integrator subject to input saturation

Université Paris XI, Département de Mathématiques, 91405 Orsay, France; Yacine.Chitour@math.u-psud.fr.

Received: 22 November 1999
Received: January 2001

Abstract

We consider a finite-dimensional control system , such that there exists a feedback stabilizer k that renders globally asymptotically stable. Moreover, for (H,p,q) with H an output map and , we assume that there exists a -function α such that , where x_u is the maximal solution of , corresponding to u and to the initial condition x(0)=0. Then, the gain function of (H,p,q) given by 14.5cm is well-defined. We call profile of k for (H,p,q) any -function which is of the same order of magnitude as . For the double integrator subject to input saturation and stabilized by , we determine the profiles corresponding to the main output maps. In particular, if is used to denote the standard saturation function, we show that the L₂-gain from the output of the saturation nonlinearity to u of the system with , is finite. We also provide a class of feedback stabilizers k_F that have a linear profile for (x,p,p), . For instance, we show that the L₂-gains from x and to u of the system with , are finite.