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/ϵ and high gain, where ϵ >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 ϵ 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 ϵ. Three numerical examples illustrate the efficiency of the method.