Constructive Robust Stabilization by Using Square Wave Dithers: A Time-Delay Approach

This article studies robust stabilization of the second- and third-order (with relative degree 3) linear uncertain systems by a fast-varying square wave dither with high frequency
${1\over \varepsilon }$
and high gain, where
$\varepsilon >0$
is small. In contrast to the existing methods for control by fast oscillations that are all qualitative, we present constructive quantitative results for finding an upper bound on
$\varepsilon$
that ensures the exponential stability. Our method consists of two steps: 1) we construct appropriate coordinate transformations that cancel the high gains and lead to a stable averaged system; 2) we apply the time-delay approach to periodic averaging of the system in new coordinates and derive linear matrix inequalities for finding an upper bound on
$\varepsilon$
. Three numerical examples illustrate the efficiency of the method.