Constructive robust stabilization by using square wave dithers: A time-delay approach

This paper 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 <inline-formula><tex-math notation="LaTeX">${1\over \varepsilon }$</tex-math></inline-formula> and high gain, where <inline-formula><tex-math notation="LaTeX">$\varepsilon &gt;0$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula>. Three numerical examples illustrate the efficiency of the method.