Stability analysis of an extremum seeking controller for mode-matching in vibrating microgyros

The aim of this paper is to provide a (local) stability analysis of an extremum seeking controller that we have designed in order to achieve the mode-matching condition in vibrating microgyroscopes. It is shown that the proposed adaptive add-on control scheme reaches the mode-matching condition at least locally about the driving mode, i.e. when the initial mismatch of the vibrating modes in the microgyro is sufficiently small (local convergence). To prove that, we make use of several interwound steps in which the averaging and the singular perturbation methods are applied.