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

Jin Zhang, Emilia Fridman

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)1-8
Number of pages8
JournalIEEE Transactions on Automatic Control
StateAccepted/In press - 2022


  • Linear matrix inequalities
  • Stability criteria
  • Stabilization by fast oscillations
  • Symmetric matrices
  • Time-varying systems
  • Uncertain systems
  • Uncertainty
  • Upper bound
  • averaging
  • time-delay systems


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