Necessary and sufficient conditions for convergence of DREM-based estimators with applications in adaptive control

Abstract

In this paper, a necessary and sufficient condition for finite-time identification of a regression model, obtained using the dynamic regressor extension and mixing (DREM) method, is established. Estimators designed to satisfy transient and robust specifications via a time-varying gain are then proposed to have this condition as necessary and sufficient for their convergence to the true values when continuous functions are involved. These estimators are then used as a part of an adaptive control scheme, following a modular approach, to solve a tracking control problem for a nonlinear system in the strict feedback form with parametric and non-parametric uncertainty. It is shown that the necessary and sufficient condition can be expressed without using closed-loop signals, which allows attaining finite-time identification, exponential convergence to the tracking aim, and local/global robustness to non-parametric perturbations with minimal excitation conditions on the tracked trajectory. An example is developed to illustrate the main results.