## Abstract

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">$\varepsilon >0$</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 language | English |
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Pages (from-to) | 1-8 |

Number of pages | 8 |

Journal | IEEE Transactions on Automatic Control |

DOIs | |

State | Accepted/In press - 2022 |

## Keywords

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