Robustness of linear time‐varying systems with relaxed excitation

Abstract

SummaryIt is a well‐known fact that linear time‐varying systems with a persistently excited state matrix are exponentially converging and input‐to‐state stable with respect to additive perturbations. Recently, several relaxed conditions of persistent excitation have been presented, which ensure an asymptotic convergence rate in the system. In the present work, it is shown that these conditions are similar and that, under such a relaxed excitation, only nonuniform in time input‐to‐state stability and integral input‐to‐state stability properties can be obtained. The results are illustrated by simulations for a problem of estimation in the linear regression model.